Landau levels from the Bethe Ansatz equations
Hoshi, K.; Hatsugai, Y.
2000-01-01
The Bethe ansatz (BA) equations for the two-dimensional Bloch electrons in a uniform magnetic field are treated in the weak-field limit. We have calculated energies near the lower boundary of the energy spectrum up to the first nontrivial order. It corresponds to calculating a finite size correction for the excitation energies of the BA solvable lattice models and gives the Landau levels in the present problem.
Landau Levels from the Bethe Ansatz Equations
Hoshi, K.; Hatsugai, Y.
1999-01-01
The Bethe ansatz (BA) equations for the two-dimensional Bloch electrons in a uniform magnetic field are treated in the weak field limit. We have calculated energies near the lower boundary of the energy spectrum up to the first nontrivial order. It corresponds to calculating a finite size correction for the excitation energies of the BA solvable lattice models and gives the Landau levels in the present problem.
Levkovich-Maslyuk, Fedor
2016-08-01
We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S-matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU(2) chiral Gross–Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU(3) chiral Gross–Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU(3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completely different setting, namely for the 1D oscillator in quantum mechanics.
Explicit Solutions of the Bethe Ansatz Equations for Bloch Electrons in a Magnetic Field
Hatsugai, Yasuhiro; Kohmoto, Mahito; Wu, Yong-Shi
1994-01-01
For Bloch electrons in a magnetic field, explicit solutions are obtained at the center of the spectrum for the Bethe ansatz equations of Wiegmann and Zabrodin. When the magnetic flux per plaquette is 1 / Q with Q an odd integer, distribution of the roots of the Bethe ansatz equation is uniform except at two points on the unit circle in the complex plane. For the semiclassical limit Q→∞, the wave function is
On the algebraic Bethe ansatz for the XXX spin chain: creation operators 'beyond the equator'
Considering the XXX spin-1/2 chain in the framework of the algebraic Bethe ansatz, we make the following short comment: the product of the creation operators corresponding to the recently found solution of the Bethe equations 'on the wrong side of the equator' is just zero (not only its action on the pseudovacuum). (author). Letter-to-the-editor
Levkovich-Maslyuk, Fedor
2016-01-01
We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S-matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU(2) chiral Gross-Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU(3) chiral Gross-Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU(3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completley different setting, namely for the 1d oscillator in quantum mechani...
ODE/IM correspondence and Bethe ansatz for affine Toda field equations
Ito, Katsushi
2015-01-01
We study the linear problem associated with modified affine Toda field equation for the Langlands dual $\\hat{\\mathfrak{g}}^\\vee$, where $\\hat{\\mathfrak{g}}$ is an untwisted affine Lie algebra. The connection coefficients for the asymptotic solutions of the linear problem correspond to the Q-functions for $\\mathfrak{g}$-type quantum integrable models. The $\\psi$-system for the solutions associated with the fundamental representations of $\\mathfrak{g}$ leads to Bethe ansatz equations associated with the affine Lie algebra $\\hat{\\mathfrak{g}}$. We also study the $A^{(2)}_{2r}$ affine Toda field equation in massless limit in detail and find its Bethe ansatz equations as well as T-Q relations.
ODE/IM correspondence and Bethe ansatz for affine Toda field equations
Katsushi Ito
2015-07-01
Full Text Available We study the linear problem associated with modified affine Toda field equation for the Langlands dual gˆ∨, where gˆ is an untwisted affine Lie algebra. The connection coefficients for the asymptotic solutions of the linear problem are found to correspond to the Q-functions for g-type quantum integrable models. The ψ-system for the solutions associated with the fundamental representations of g leads to Bethe ansatz equations associated with the affine Lie algebra gˆ. We also study the A2r(2 affine Toda field equation in massless limit in detail and find its Bethe ansatz equations as well as T–Q relations.
Analysis of the Bethe-ansatz equations of the chiral-invariant Gross-Neveu model
The Bethe-ansatz equations of the chiral-invariant Gross-Neveu model are reduced to a simple form in which the parameters of the vacuum solution have been eliminated. The resulting system of equations involves only the rapidities of physical particles and a minimal set of complex parameters needed to distinguish the various internal symmetry states of these particles. The analysis is performed without invoking the time-honored assumption that the solutions of the Bethe-ansatz equations, in the infinite-volume limit, are comprised entirely of strings ('bound states'). Surprisingly, it is found that the correct description of the n-particle states involves no strings of length greater than two (except for special values of the momenta). (orig.)
Bethe-ansatz equations for quantum Heisenberg chains with elliptic exchange
Inozemtsev, V. I.
1999-01-01
The eigenvectors of the Hamiltonian ${\\cal H}_{N}$ of $N$-sites quantum spin chains with elliptic exchange are connected with the double Bloch meromorphic solutions of the quantum continuous elliptic Calogero-Moser problem. This fact allows one to find the eigenvectors via the solutions to the system of highly transcendental equations of Bethe-ansatz type which is presented in explicit form.
osp (1 vertical bar 2) off-shell Bethe ansatz equations
The semiclassical limit of the algebraic quantum inverse scattering method is used to solve the theory of the Gaudin model. Via Off-shell Bethe ansatz equations of an integrable representation of the graded osp(1 vertical bar 2) vertex model we find the spectrum of the N - 1 independents Hamiltonians of Gaudin. Integral representations of the N-point correlators are presented as solutions of the Knizhnik-Zamolodchikov equation. These results are extended for highest representations of the osp (1 vertical bar 2) Gaudin algebra
Explicit Solutions of the Bethe Ansatz Equations for Bloch Electrons in a Magnetic Field
Hatsugai, Yasuhiro; Kohmoto, Mahito; Wu, Yong-Shi
1994-01-01
For Bloch electrons in a magnetic field, explicit solutions are obtained at the center of the spectrum for the Bethe ansatz equations recently proposed by Wiegmann and Zabrodin. When the magnetic flux per plaquette is $1/Q$ where $Q$ is an odd integer, distribution of the roots is uniform on the unit circle in the complex plane. For the semi-classical limit, $ Q\\rightarrow\\infty$, the wavefunction obeys the power low and is given by $|\\psi(x)|^2=(2/ \\sin \\pi x)$ which is critical and unnormal...
Crossover from droplet to flat initial conditions in the KPZ equation from the replica Bethe ansatz
Le Doussal, Pierre
2014-04-01
We show how our previous result based on the replica Bethe ansatz for the Kardar-Parisi-Zhang (KPZ) equation with the ‘half-flat’ initial condition leads to the Airy2 to Airy1 (i.e. GUE (Gaussian unitary ensemble) to GOE (Gaussian orthogonal ensemble)) universal crossover one-point height distribution in the limit of large time. It involves a ‘decoupling assumption’ in that limit, validated by the result. Equivalently, we obtain the distribution of the free energy of a long directed polymer (DP) in a random potential with one fixed endpoint and the other one on a half-line. We generalize to a DP when each endpoint is free on its own half-line. This yields, in the large time limit, a conjecture for the distribution of the maximum of the transition process Airy2→1 (minus a half-parabola) on a half-line.
Crossover from droplet to flat initial conditions in the KPZ equation from the replica Bethe ansatz
We show how our previous result based on the replica Bethe ansatz for the Kardar–Parisi–Zhang (KPZ) equation with the ‘half-flat’ initial condition leads to the Airy2 to Airy1 (i.e. GUE (Gaussian unitary ensemble) to GOE (Gaussian orthogonal ensemble)) universal crossover one-point height distribution in the limit of large time. It involves a ‘decoupling assumption’ in that limit, validated by the result. Equivalently, we obtain the distribution of the free energy of a long directed polymer (DP) in a random potential with one fixed endpoint and the other one on a half-line. We generalize to a DP when each endpoint is free on its own half-line. This yields, in the large time limit, a conjecture for the distribution of the maximum of the transition process Airy2→1 (minus a half-parabola) on a half-line. (paper)
Bracken, Anthony J.; Ge Xiangyu; Gould, Mark D.; Links, Jon; Zhou Huanqiang [Centre for Mathematical Physics, University of Queensland, Brisbane, QLD (Australia)
2001-06-01
Integrable extended Hubbard models arising from symmetric group solutions are examined in the framework of the graded quantum inverse scattering method. The Bethe ansatz equations for all these models are derived by using the algebraic Bethe ansatz method. (author)
Kitanine, N; Niccoli, G
2014-01-01
We solve the longstanding problem to define a functional characterization of the spectrum of the transfer matrix associated to the most general spin-1/2 representations of the 6-vertex reflection algebra for general inhomogeneous chains. The corresponding homogeneous limit reproduces the spectrum of the Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most general integrable boundaries. The spectrum is characterized by a second order finite difference functional equation of Baxter type with an inhomogeneous term which vanishes only for some special but yet interesting non-diagonal boundary conditions. This functional equation is shown to be equivalent to the known separation of variable (SOV) representation hence proving that it defines a complete characterization of the transfer matrix spectrum. The polynomial character of the Q-function allows us then to show that a finite system of equations of generalized Bethe type can be similarly used to describe the complete transfer matrix spectru...
Off-diagonal Bethe ansatz for exactly solvable models
This book serves as an introduction of the off-diagonal Bethe Ansatz method, an analytic theory for the eigenvalue problem of quantum integrable models. It also presents some fundamental knowledge about quantum integrability and the algebraic Bethe Ansatz method. Based on the intrinsic properties of R-matrix and K-matrices, the book introduces a systematic method to construct operator identities of transfer matrix. These identities allow one to establish the inhomogeneous T-Q relation formalism to obtain Bethe Ansatz equations and to retrieve corresponding eigenstates. Several longstanding models can thus be solved via this method since the lack of obvious reference states is made up. Both the exact results and the off-diagonal Bethe Ansatz method itself may have important applications in the fields of quantum field theory, low-dimensional condensed matter physics, statistical physics and cold atom systems.
Introduction to the thermodynamic Bethe ansatz
van Tongeren, Stijn J.
2016-08-01
We give a pedagogical introduction to the thermodynamic Bethe ansatz, a method that allows us to describe the thermodynamics of integrable models whose spectrum is found via the (asymptotic) Bethe ansatz. We set the stage by deriving the Fermi–Dirac distribution and associated free energy of free electrons, and then in a similar though technically more complicated fashion treat the thermodynamics of integrable models, focusing first on the one-dimensional Bose gas with delta function interaction as a clean pedagogical example, secondly the XXX spin chain as an elementary (lattice) model with prototypical complicating features in the form of bound states, and finally the {SU}(2) chiral Gross–Neveu model as a field theory example. Throughout this discussion we emphasize the central role of particle and hole densities, whose relations determine the model under consideration. We then discuss tricks that allow us to use the same methods to describe the exact spectra of integrable field theories on a circle, in particular the chiral Gross–Neveu model. We moreover discuss the simplification of TBA equations to Y systems, including the transition back to integral equations given sufficient analyticity data, in simple examples.
Algebraic Bethe Ansatz for O(2N) sigma models with integrable diagonal boundaries
Gombor, Tamas
2015-01-01
The finite volume problem of O(2N) sigma models with integrable diagonal boundaries on a finite interval is investigated. The double row transfer matrix is diagonalized by Algebraic Bethe Ansatz. The boundary Bethe Yang equations for the particle rapidities and the accompanying Bethe Ansatz equations are derived.
The exactly integrable isotropic Heisenberg chain of N spins s is studied, and numerical solutions to the Bethe ansatz equations corresponding to the antiferromagnetic vacuum (for sN ≤ 128) and the simplest excitations have been obtained. For s = 1, a complete set of states for N = 6 is given, and the vacuum solution for finite N is estimated analytically. The deviations from the string picture at large N are discussed
Matrix coordinate Bethe Ansatz: applications to XXZ and ASEP models
We present the construction of the full set of eigenvectors of the open asymmetric simple exclusion process (ASEP) and XXZ models with special constraints on the boundaries. The method combines both recent constructions of coordinate Bethe Ansatz and the old method of matrix Ansatz specific to the ASEP. This 'matrix coordinate Bethe Ansatz' can be viewed as a non-commutative coordinate Bethe Ansatz, the non-commutative part being related to the algebra appearing in the matrix Ansatz. (paper)
Bethe ansatz for the Temperley–Lieb spin chain with integrable open boundaries
In this paper we study the spectrum of the spin-1 Temperley–Lieb spin chain with integrable open boundary conditions. We obtain the eigenvalue expressions as well as its associated Bethe ansatz equations by means of the coordinate Bethe ansatz. These equations provide the complete description of the spectrum of the model. (paper)
Introduction to the thermodynamic Bethe ansatz
van Tongeren, Stijn J
2016-01-01
We give a pedagogical introduction to the thermodynamic Bethe ansatz, a method that allows us to describe the thermodynamics of integrable models whose spectrum is found via the (asymptotic) Bethe ansatz. We set the stage by deriving the Fermi-Dirac distribution and associated free energy of free electrons, and then in a similar though technically more complicated fashion treat the thermodynamics of integrable models, focusing on the one dimensional Bose gas with delta function interaction as a clean pedagogical example, secondly the XXX spin chain as an elementary (lattice) model with prototypical complicating features in the form of bound states, and finally the SU(2) chiral Gross-Neveu model as a field theory example. Throughout this discussion we emphasize the central role of particle and hole densities, whose relations determine the model under consideration. We then discuss tricks that allow us to use the same methods to describe the exact spectra of integrable field theories on a circle, in particular ...
Bethe ansatz solution of the open XX spin chain with non-diagonal boundary terms
We consider the integrable open XX quantum spin chain with non-diagonal boundary terms. We derive an exact inversion identity, by which we obtain the eigenvalues of the transfer matrix and the Bethe ansatz equations. For generic values of the boundary parameters, the Bethe ansatz solution is formulated in terms of the Jacobian elliptic functions. (author)
Off-diagonal Bethe ansatz solution of the XXX spin-chain with arbitrary boundary conditions
Cao, Junpeng; Shi, Kangjie; Wang, Yupeng
2013-01-01
With the off-diagonal Bethe ansatz method proposed recently by the present authors, we exactly diagonalize the $XXX$ spin chain with arbitrary boundary fields. By constructing a functional relation between the eigenvalues of the transfer matrix and the quantum determinant, the associated $T-Q$ relation and the Bethe ansatz equations are derived.
Off-diagonal Bethe ansatz solution of the XXX spin chain with arbitrary boundary conditions
Employing the off-diagonal Bethe ansatz method proposed recently by the present authors, we exactly diagonalize the XXX spin chain with arbitrary boundary fields. By constructing a functional relation between the eigenvalues of the transfer matrix and the quantum determinant, the associated T–Q relation and the Bethe ansatz equations are derived
Off-diagonal Bethe ansatz solution of the XXX spin chain with arbitrary boundary conditions
Cao, Junpeng; Yang, Wen-Li; Shi, Kangjie; Wang, Yupeng
2013-10-01
Employing the off-diagonal Bethe ansatz method proposed recently by the present authors, we exactly diagonalize the XXX spin chain with arbitrary boundary fields. By constructing a functional relation between the eigenvalues of the transfer matrix and the quantum determinant, the associated T-Q relation and the Bethe ansatz equations are derived.
Algebraic Bethe ansatz for 19-vertex models with upper triangular K-matrices
By means of an algebraic Bethe ansatz approach, we study the Zamolodchikov–Fateev and Izergin–Korepin vertex models with non-diagonal boundaries, characterized by reflection matrices with an upper triangular form. Generalized Bethe vectors are used to diagonalize the associated transfer matrix. The eigenvalues as well as the Bethe equations are presented. (paper)
Algebraic Bethe Ansatz for Open XXX Model with Triangular Boundary Matrices
Belliard, Samuel; Crampé, Nicolas; Ragoucy, Eric
2013-05-01
We consider an open XXX spin chain with two general boundary matrices whose entries obey a relation, which is equivalent to the possibility to put simultaneously the two matrices in a upper-triangular form. We construct Bethe vectors by means of a generalized algebraic Bethe ansatz. As usual, the method uses Bethe equations and provides transfer matrix eigenvalues.
Algebraic Bethe ansatz for the gl(1|2) generalized model: II. the three gradings
The algebraic Bethe ansatz can be performed rather abstractly for whole classes of models sharing the same R-matrix, the only prerequisite being the existence of an appropriate pseudo vacuum state. Here we perform the algebraic Bethe ansatz for all models with 9 x 9, rational, gl(1|2) invariant R-matrix and all three possibilities of choosing the grading. Our Bethe ansatz solution applies, for instance, to the supersymmetric t-J model, the supersymmetric U model and a number of interesting impurity models. It may be extended to obtain the quantum transfer matrix spectrum for this class of models. The properties of a specific model enter the Bethe ansatz solution (i.e. the expression for the transfer matrix eigenvalue and the Bethe ansatz equations) through the three pseudo vacuum eigenvalues of the diagonal elements of the monodromy matrix which in this context are called the parameters of the model
Matrix coordinate Bethe Ansatz: applications to XXZ and ASEP models
Crampe, N [Laboratoire Charles Coulomb, UMR 5221, Universite Montpellier 2, F-34095 Montpellier (France); Ragoucy, E [Laboratoire de Physique Theorique LAPTH, CNRS and Universite de Savoie, 9 chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux Cedex (France); Simon, D, E-mail: nicolas.crampe@univ-montp2.fr, E-mail: ragoucy@lapp.in2p3.fr, E-mail: damien.simon@upmc.fr [LPMA, Universite Pierre et Marie Curie, Case Courrier 188, 4 place Jussieu, 75252 Paris Cedex 05 (France)
2011-10-07
We present the construction of the full set of eigenvectors of the open asymmetric simple exclusion process (ASEP) and XXZ models with special constraints on the boundaries. The method combines both recent constructions of coordinate Bethe Ansatz and the old method of matrix Ansatz specific to the ASEP. This 'matrix coordinate Bethe Ansatz' can be viewed as a non-commutative coordinate Bethe Ansatz, the non-commutative part being related to the algebra appearing in the matrix Ansatz. (paper)
Cyclotomic Gaudin Models: Construction and Bethe Ansatz
Vicedo, Benoît; Young, Charles
2016-05-01
To any finite-dimensional simple Lie algebra g and automorphism {σ: gto g we associate a cyclotomic Gaudin algebra. This is a large commutative subalgebra of {U(g)^{⊗ N}} generated by a hierarchy of cyclotomic Gaudin Hamiltonians. It reduces to the Gaudin algebra in the special case {σ =id}. We go on to construct joint eigenvectors and their eigenvalues for this hierarchy of cyclotomic Gaudin Hamiltonians, in the case of a spin chain consisting of a tensor product of Verma modules. To do so we generalize an approach to the Bethe ansatz due to Feigin, Frenkel and Reshetikhin involving vertex algebras and the Wakimoto construction. As part of this construction, we make use of a theorem concerning cyclotomic coinvariants, which we prove in a companion paper. As a byproduct, we obtain a cyclotomic generalization of the Schechtman-Varchenko formula for the weight function.
Integrability in three dimensions: Algebraic Bethe ansatz for anyonic models
Khachatryan, Sh.; Ferraz, A.; Klümper, A.; Sedrakyan, A.
2015-10-01
We extend basic properties of two dimensional integrable models within the Algebraic Bethe Ansatz approach to 2 + 1 dimensions and formulate the sufficient conditions for the commutativity of transfer matrices of different spectral parameters, in analogy with Yang-Baxter or tetrahedron equations. The basic ingredient of our models is the R-matrix, which describes the scattering of a pair of particles over another pair of particles, the quark-anti-quark (meson) scattering on another quark-anti-quark state. We show that the Kitaev model belongs to this class of models and its R-matrix fulfills well-defined equations for integrability.
Integrability in three dimensions: Algebraic Bethe ansatz for anyonic models
Sh. Khachatryan
2015-10-01
Full Text Available We extend basic properties of two dimensional integrable models within the Algebraic Bethe Ansatz approach to 2+1 dimensions and formulate the sufficient conditions for the commutativity of transfer matrices of different spectral parameters, in analogy with Yang–Baxter or tetrahedron equations. The basic ingredient of our models is the R-matrix, which describes the scattering of a pair of particles over another pair of particles, the quark-anti-quark (meson scattering on another quark-anti-quark state. We show that the Kitaev model belongs to this class of models and its R-matrix fulfills well-defined equations for integrability.
Algebraic Bethe ansatz for Q-operators: the Heisenberg spin chain
Frassek, Rouven
2015-07-01
We diagonalize Q-operators for rational homogeneous {sl}(2)-invariant Heisenberg spin chains using the algebraic Bethe ansatz. After deriving the fundamental commutation relations relevant for this case from the Yang-Baxter equation we demonstrate that the Q-operators act diagonally on the Bethe vectors if the Bethe equations are satisfied. In this way we provide a direct proof that the eigenvalues of the Q-operators studied here are given by Baxter's Q-functions.
Algebraic Bethe ansatz for the XXX chain with triangular boundaries and Gaudin model
Cirilo António, N.; Manojlović, N.; Salom, I.
2014-12-01
We implement fully the algebraic Bethe ansatz for the XXX Heisenberg spin chain in the case when both boundary matrices can be brought to the upper-triangular form. We define the Bethe vectors which yield the strikingly simple expression for the off shell action of the transfer matrix, deriving the spectrum and the relevant Bethe equations. We explore further these results by obtaining the off shell action of the generating function of the Gaudin Hamiltonians on the corresponding Bethe vectors through the so-called quasi-classical limit. Moreover, this action is as simple as it could possibly be, yielding the spectrum and the Bethe equations of the Gaudin model.
Algebraic Bethe ansatz for the XXX chain with triangular boundaries and Gaudin model
Cirilo António, N., E-mail: nantonio@math.ist.utl.pt [Centro de Análise Funcional e Aplicações, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa (Portugal); Manojlović, N., E-mail: nmanoj@ualg.pt [Grupo de Física Matemática da Universidade de Lisboa, Av. Prof. Gama Pinto 2, PT-1649-003 Lisboa (Portugal); Departamento de Matemática, F.C.T., Universidade do Algarve, Campus de Gambelas, PT-8005-139 Faro (Portugal); Salom, I., E-mail: isalom@ipb.ac.rs [Institute of Physics, University of Belgrade, P.O. Box 57, 11080 Belgrade (Serbia)
2014-12-15
We implement fully the algebraic Bethe ansatz for the XXX Heisenberg spin chain in the case when both boundary matrices can be brought to the upper-triangular form. We define the Bethe vectors which yield the strikingly simple expression for the off shell action of the transfer matrix, deriving the spectrum and the relevant Bethe equations. We explore further these results by obtaining the off shell action of the generating function of the Gaudin Hamiltonians on the corresponding Bethe vectors through the so-called quasi-classical limit. Moreover, this action is as simple as it could possibly be, yielding the spectrum and the Bethe equations of the Gaudin model.
Algebraic Bethe ansatz for the XXX chain with triangular boundaries and Gaudin model
We implement fully the algebraic Bethe ansatz for the XXX Heisenberg spin chain in the case when both boundary matrices can be brought to the upper-triangular form. We define the Bethe vectors which yield the strikingly simple expression for the off shell action of the transfer matrix, deriving the spectrum and the relevant Bethe equations. We explore further these results by obtaining the off shell action of the generating function of the Gaudin Hamiltonians on the corresponding Bethe vectors through the so-called quasi-classical limit. Moreover, this action is as simple as it could possibly be, yielding the spectrum and the Bethe equations of the Gaudin model
Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz
Samuel Belliard
2013-11-01
Full Text Available We propose a generalization of the algebraic Bethe ansatz to obtain the eigenvectors of the Heisenberg spin chain with general boundaries associated to the eigenvalues and the Bethe equations found recently by Cao et al. The ansatz takes the usual form of a product of operators acting on a particular vector except that the number of operators is equal to the length of the chain. We prove this result for the chains with small length. We obtain also an off-shell equation (i.e. satisfied without the Bethe equations formally similar to the ones obtained in the periodic case or with diagonal boundaries.
Heisenberg XXX Model with General Boundaries: Eigenvectors from Algebraic Bethe Ansatz
Belliard, Samuel; Crampé, Nicolas
2013-11-01
We propose a generalization of the algebraic Bethe ansatz to obtain the eigenvectors of the Heisenberg spin chain with general boundaries associated to the eigenvalues and the Bethe equations found recently by Cao et al. The ansatz takes the usual form of a product of operators acting on a particular vector except that the number of operators is equal to the length of the chain. We prove this result for the chains with small length. We obtain also an off-shell equation (i.e. satisfied without the Bethe equations) formally similar to the ones obtained in the periodic case or with diagonal boundaries.
Heisenberg XXX model with general boundaries: Eigenvectors from Algebraic Bethe ansatz
Belliard, S
2013-01-01
We propose a generalization of the algebraic Bethe ansatz to obtain the eigenvectors of the Heisenberg spin chain with general boundaries associated to the eigenvalues and the Bethe equations found recently by Cao et al. The ansatz takes the usual form of a product of operators acting on a particular vector except that the number of operators is equal to the length of the chain. We prove this result for the chains with small length. We obtain also an off-shell equation (i.e. satisfied without the Bethe equations) formally similar to the ones obtained in the periodic case or with diagonal boundaries.
Bethe ansatz solution of the $\\tau_2$-model with arbitrary boundary fields
Xu, Xiaotian; Yang, Tao; Cao, Junpeng; Yang, Wen-Li; Shi, Kangjie
2016-01-01
The quantum $\\tau_2$-model with generic site-dependent inhomogeneity and arbitrary boundary fields is studied via the off-diagonal Bethe Ansatz method. The eigenvalues of the corresponding transfer matrix are given in terms of an inhomogeneous T-Q relation, which is based on the operator product identities among the fused transfer matrices and the asymptotic behavior of the transfer matrices. Moreover, the associated Bethe Ansatz equations are also obtained.
Spin-1/2 XYZ model revisit: General solutions via off-diagonal Bethe ansatz
Cao, Junpeng [Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190 (China); Collaborative Innovation Center of Quantum Matter, Beijing (China); Cui, Shuai [Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190 (China); Yang, Wen-Li, E-mail: wlyang@nwu.edu.cn [Institute of Modern Physics, Northwest University, Xian 710069 (China); Beijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing 100048 (China); Shi, Kangjie [Institute of Modern Physics, Northwest University, Xian 710069 (China); Wang, Yupeng, E-mail: yupeng@iphy.ac.cn [Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190 (China); Collaborative Innovation Center of Quantum Matter, Beijing (China)
2014-09-15
The spin-1/2 XYZ model with both periodic and anti-periodic boundary conditions is studied via the off-diagonal Bethe ansatz method. The exact spectra of the Hamiltonians and the Bethe ansatz equations are derived by constructing the inhomogeneous T–Q relations, which allow us to treat both the even N (the number of lattice sites) and odd N cases simultaneously in a unified approach.
Algebraic Bethe ansatz for the Temperley-Lieb spin-1 chain
Nepomechie, Rafael I
2016-01-01
We use the algebraic Bethe ansatz to obtain the eigenvalues and eigenvectors of the spin-1 Temperley-Lieb open quantum chain with "free" boundary conditions. We exploit the associated reflection algebra in order to prove the off-shell equation satisfied by the Bethe vectors.
Modified algebraic Bethe ansatz for XXZ chain on the segment - II - general cases
Belliard, Samuel
2015-01-01
The spectral problem of the Heisenberg XXZ spin-$\\frac{1}{2}$ chain on the segment is investigated within a modified algebraic Bethe ansatz framework. We consider in this work the most general boundaries allowed by integrability. The eigenvalues and the eigenvectors are obtained. They are characterised by a set of Bethe roots with cardinality equal to $N$, the length of the chain, and which satisfies a set of Bethe equations with an additional term.
Bethe Ansatz for the Ruijsenaars Model of BC_1-Type
Oleg Chalykh
2007-02-01
Full Text Available We consider one-dimensional elliptic Ruijsenaars model of type $BC_1$. It is given by a three-term difference Schrödinger operator $L$ containing 8 coupling constants. We show that when all coupling constants are integers, $L$ has meromorphic eigenfunctions expressed by a variant of Bethe ansatz. This result generalizes the Bethe ansatz formulas known in the $A_1$-case.
Algebraic Bethe ansatz for the XXX chain with triangular boundaries and Gaudin model
António, N Cirilo; Salom, I
2014-01-01
We implement fully the algebraic Bethe ansatz for the XXX Heisenberg spin chain in the case when both boundary matrices can be brought to the upper-triangular form. We define the Bethe vectors which yield the strikingly simple expression for the off shell action of the transfer matrix, deriving the spectrum and the corresponding Bethe equations. We explore further these results by obtaining the off shell action of the generating function of the Gaudin Hamiltonians on the Bethe vectors through the so-called quasi-classical limit.
Large and small density approximations to the thermodynamic Bethe ansatz
We provide analytical solutions to the thermodynamic Bethe ansatz equations in the large and small density approximations. We extend results previously obtained for leading order behaviour of the scaling function of affine Toda field theories related to simply laced Lie algebras to the non-simply laced case. The comparison with semi-classical methods shows perfect agreement for the simply laced case. We derive the Y-systems for affine Toda field theories with real coupling constant and employ them to improve the large density approximations. We test the quality of our analysis explicitly for the sinh-Gordon model and the (G2(1),D4(3)) -affine Toda field theory
Hofstadter problem on the honeycomb and triangular lattices: Bethe ansatz solution
Kohmoto, M.; Sedrakyan, A.
2006-06-01
We consider Bloch electrons on the honeycomb lattice under a uniform magnetic field with 2πp/q flux per cell. It is shown that the problem factorizes to two triangular lattices. Treating magnetic translations as a Heisenberg-Weyl group and by the use of its irreducible representation on the space of theta functions, we find a nested set of Bethe equations, which determine the eigenstates and energy spectrum. The Bethe equations have simple form which allows us to consider them further in the limit p,q→∞ by the technique of thermodynamic Bethe ansatz and analyze the Hofstadter problem for the irrational flux.
Bethe ansatz for the XXX-S chain with non-diagonal open boundaries
We consider the algebraic Bethe ansatz solution of the integrable and isotropic XXX-S Heisenberg chain with non-diagonal open boundaries. We show that the corresponding K-matrices are similar to diagonal matrices with the help of suitable transformations independent of the spectral parameter. When the boundary parameters satisfy certain constraints we are able to formulate the diagonalization of the associated double-row transfer matrix by means of the quantum inverse scattering method. This allows us to derive explicit expressions for the eigenvalues and the corresponding Bethe ansatz equations. We also present evidences that the eigenvectors can be build up in terms of multiparticle states for arbitrary S
On the algebraic Bethe ansatz: Periodic boundary conditions
Lima-Santos, A.
2006-01-01
In this paper, the algebraic Bethe ansatz with periodic boundary conditions is used to investigate trigonometric vertex models associated with the fundamental representations of the non-exceptional Lie algebras. This formulation allow us to present explicit expressions for the eigenvectors and eigenvalues of the respective transfer matrices.
Coordinate Bethe Ansatz for Spin s XXX Model
Nicolas Crampé; Eric Ragoucy; Ludovic Alonzi
2010-01-01
We compute the eigenfunctions and eigenvalues of the periodic integrable spin s XXX model using the coordinate Bethe ansatz. To do so, we compute explicitly the Hamiltonian of the model. These results generalize what has been obtained for spin 1/2 and spin 1 chains.
Characters in Conformal Field Theories from Thermodynamic Bethe Ansatz
Kuniba, A.; Nakanishi, T; Suzuki, J.
1993-01-01
We propose a new $q$-series formula for a character of parafermion conformal field theories associated to arbitrary non-twisted affine Lie algebra $\\widehat{g}$. We show its natural origin from a thermodynamic Bethe ansatz analysis including chemical potentials.
The Yangians, Bethe ansatz and combinatorics
An axiomatic definition of a quantum monodromy matrix and the representations of its corresponding Hopf algebra are discussed. The connection between the quantum inverse transform method and the representation theory of a symmetric group is considered. A new approach to the completeness problem of Bethe vectors is also given. (orig.)
Wiegmann, P. B.; Zabrodin, A. V.
1993-01-01
We present a new approach to the problem of Bloch electrons in magnetic field,\\\\ by making explicit a natural relation between magnetic translations and the\\\\quantum group $U_{q}(sl_2)$. The approach allows to express the spectrum and\\\\\\ the Bloch function as solutions of the Bethe-Ansatz equations typical for com\\\\pletely integrable quantum systems
Algebraic Bethe ansatz for the sl(2) Gaudin model with boundary
António, N Cirilo; Ragoucy, E; Salom, I
2015-01-01
Following Sklyanin's proposal in the periodic case, we derive the generating function of the Gaudin Hamiltonians with boundary terms. Our derivation is based on the quasi-classical expansion of the linear combination of the transfer matrix of the XXX Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function. By defining the appropriate Bethe vectors which yield strikingly simple off shell action of the generating function, we fully implement the algebraic Bethe ansatz, obtaining the spectrum of the generating function and the corresponding Bethe equations.
Universal Bethe ansatz solution for the Temperley-Lieb spin chain
Nepomechie, Rafael I
2016-01-01
We consider the Temperley-Lieb (TL) open quantum spin chain with "free" boundary conditions associated with the spin-$s$ representation of quantum-deformed $sl(2)$. We construct the transfer matrix, and determine its eigenvalues and the corresponding Bethe equations using analytical Bethe ansatz. We show that the transfer matrix has quantum group symmetry, and we propose explicit formulas for the number of solutions of the Bethe equations and the degeneracies of the transfer-matrix eigenvalues. We propose an algebraic Bethe ansatz construction of the off-shell Bethe states, and we conjecture that the on-shell Bethe states are highest-weight states of the quantum group. We also propose a determinant formula for the scalar product between an off-shell Bethe state and its on-shell dual, as well as for the square of the norm. We find that all of these results, except for the degeneracies and a constant factor in the scalar product, are universal in the sense that they do not depend on the value of the spin. In an...
Nested Bethe ansatz for "all" closed spin chains
Belliard, S.; Ragoucy, E.
2008-01-01
We present in an unified and detailed way the Nested Bethe Ansatz for closed spin chains based on Y(gl(n)), Y(gl(m|n)), U_q(gl(n)) or U_q(gl(m|n)) (super)algebras, with arbitrary representations (i.e. `spins') on each site of the chain. In particular, the case of indecomposable representations of superalgebras is studied. The construction extends and unifies the results already obtained for spin chains based on Y(gl(n)) or U_q(gl(n)) and for some particular super-spin chains. We give the Beth...
The asymmetric simple exclusion process with open boundaries, which is a very simple model of out-of-equilibrium statistical physics, is known to be integrable. In particular, its spectrum can be described in terms of Bethe roots. The large deviation function of the current can be obtained as well by diagonalizing a modified transition matrix, which is still integrable: the spectrum of this new matrix can also be described in terms of Bethe roots for special values of the parameters. However, due to the algebraic framework used to write the Bethe equations in previous works, the nature of the excitations and the full structure of the eigenvectors remained unknown. This paper explains why the eigenvectors of the modified transition matrix are physically relevant, gives an explicit expression for the eigenvectors and applies it to the study of atypical currents. It also shows how the coordinate Bethe ansatz developed for the excitations leads to a simple derivation of the Bethe equations and of the validity conditions of this ansatz. All the results obtained by de Gier and Essler are recovered and the approach gives a physical interpretation of the exceptional points. The overlap of this approach with other tools such as the matrix ansatz is also discussed. The method that is presented here may be not specific to the asymmetric exclusion process and may be applied to other models with open boundaries to find similar exceptional points
How algebraic Bethe ansatz works for integrable model
Fadeev, L
1996-01-01
I study the technique of Algebraic Bethe Ansatz for solving integrable models and show how it works in detail on the simplest example of spin 1/2 XXX magnetic chain. Several other models are treated more superficially, only the specific details are given. Several parameters, appearing in these generalizations: spin s, anisotropy parameter \\ga, shift \\om in the alternating chain, allow to include in our treatment most known examples of soliton theory, including relativistic model of Quantum Field Theory.
Bethe ansatz for higher spin eight vertex models
Takebe, T
1995-01-01
A generalization of the eight vertex model by means of higher spin representations of the Sklyanin algebra is investigated by the quantum inverse scattering method and the algebraic Bethe Ansatz. Under the well-known string hypothesis low-lying excited states are considered and scattering phase shifts of two physical particles are calculated. The S matrix of two particle states is shown to be proportional to the Baxter's elliptic R matrix with a different elliptic modulus from the original one.
Bethe ansatz solvable multi-chain quantum systems
In this article we review recent developments in the one-dimensional Bethe ansatz solvable multi-chain quantum models. The algebraic version of the Bethe ansatz (the quantum inverse scattering method) permits us to construct new families of integrable Hamiltonians using simple generalizations of the well known constructions of the single-chain model. First we consider the easiest example ('basic' model) of this class of models: the antiferromagnetic two-chain spin-1/2 model with the nearest-neighbour and next-nearest-neighbour spin-frustrating interactions (zigzag chain). We show how the algebra of the quantum inverse scattering method works for this model, and what are the important features of the Hamiltonian (which reveal the topological properties of two dimensions together with the one-dimensional properties). We consider the solution of the Bethe ansatz for the ground state (in particular, commensurate-incommensurate quantum phase transitions present due to competing spin-frustrating interactions are discussed) and construct the thermal Bethe ansatz (in the form of the 'quantum transfer matrix') for this model. Then possible generalizations of the basic model are considered: an inclusion of a magnetic anisotropy, higher-spin representations (including the important case of a quantum ferrimagnet), the multi-chain case, internal degrees of freedom of particles at each site, etc. We observe the similarities and differences between this class of models and related exactly solvable models: other groups of multi-chain lattice models, quantum field theory models and magnetic impurity (Kondo-like) models. Finally, the behaviour of non-integrable (less constrained) multi-chain quantum models is discussed. (author)
Hierarchical Structure of Azbel-Hofstader Problem: Strings and loose ends of Bethe Ansatz
Abanov, A. G.; Talstra, J. C.; Wiegmann, P. B.
1997-01-01
We present numerical evidence that solutions of the Bethe Ansatz equations for a Bloch particle in an incommensurate magnetic field (Azbel-Hofstadter or AH model), consist of complexes-"strings". String solutions are well-known from integrable field theories. They become asymptotically exact in the thermodynamic limit. The string solutions for the AH model are exact in the incommensurate limit, where the flux through the unit cell is an irrational number in units of the elementary flux quantu...
Bethe Ansatz Solutions of the Bose-Hubbard Dimer
Jon Links
2006-12-01
Full Text Available The Bose-Hubbard dimer Hamiltonian is a simple yet effective model for describing tunneling phenomena of Bose-Einstein condensates. One of the significant mathematical properties of the model is that it can be exactly solved by Bethe ansatz methods. Here we review the known exact solutions, highlighting the contributions of V.B. Kuznetsov to this field. Two of the exact solutions arise in the context of the Quantum Inverse Scattering Method, while the third solution uses a differential operator realisation of the su(2 Lie algebra.
G/G gauged WZW model and Bethe Ansatz for the phase model
Okuda, Satoshi
2012-01-01
We investigate the G/G gauged Wess-Zumino-Witten model on a Riemann surface from the point of view of the algebraic Bethe Ansatz for the phase model. After localization procedure is applied to the G/G gauged Wess-Zumino-Witten model, the diagonal components for group elements satisfy Bethe Ansatz equations for the phase model. We show that the partition function of the G/G gauged Wess-Zumino-Witten model is identified as the summation of norms with respect to all the eigenstates of the Hamiltonian with the fixed number of particles in the phase model. We also consider relations between the Chern-Simons theory on $S^1\\times\\Sigma_h$ and the phase model.
Spectral Theory for Interacting Particle Systems Solvable by Coordinate Bethe Ansatz
Borodin, Alexei; Corwin, Ivan; Petrov, Leonid; Sasamoto, Tomohiro
2015-11-01
We develop spectral theory for the q-Hahn stochastic particle system introduced recently by Povolotsky. That is, we establish a Plancherel type isomorphism result that implies completeness and biorthogonality statements for the Bethe ansatz eigenfunctions of the system. Owing to a Markov duality with the q-Hahn TASEP (a discrete-time generalization of TASEP with particles' jump distribution being the orthogonality weight for the classical q-Hahn orthogonal polynomials), we write down moment formulas that characterize the fixed time distribution of the q-Hahn TASEP with general initial data. The Bethe ansatz eigenfunctions of the q-Hahn system degenerate into eigenfunctions of other (not necessarily stochastic) interacting particle systems solvable by the coordinate Bethe ansatz. This includes the ASEP, the (asymmetric) six-vertex model, and the Heisenberg XXZ spin chain (all models are on the infinite lattice). In this way, each of the latter systems possesses a spectral theory, too. In particular, biorthogonality of the ASEP eigenfunctions, which follows from the corresponding q-Hahn statement, implies symmetrization identities of Tracy and Widom (for ASEP with either step or step Bernoulli initial configuration) as corollaries. Another degeneration takes the q-Hahn system to the q-Boson particle system (dual to q-TASEP) studied in detail in our previous paper (2013). Thus, at the spectral theory level we unify two discrete-space regularizations of the Kardar-Parisi-Zhang equation/stochastic heat equation, namely, q-TASEP and ASEP.
Bethe ansatz matrix elements as non-relativistic limits of form factors of quantum field theory
M. Kormos; G. Mussardo; B. Pozsgay
2010-01-01
We show that the matrix elements of integrable models computed by the algebraic Bethe ansatz (BA) can be put in direct correspondence with the form factors of integrable relativistic field theories. This happens when the S-matrix of a Bethe ansatz model can be regarded as a suitable non-relativistic
Twist-three at five loops, Bethe ansatz and wrapping
Beccaria, Matteo; Forini, Valentina; Łukowski, Tomasz; Zieme, Stefan
2009-03-01
We present a formula for the five-loop anomalous dimension of Script N = 4 SYM twist-three operators in the fraktur sfraktur l(2) sector. We obtain its asymptotic part from the Bethe Ansatz and finite volume corrections from the generalized Lüscher formalism, considering scattering processes of spin chain magnons with virtual particles that travel along the cylinder. The complete result respects the expected large spin scaling properties and passes non-trivial tests including reciprocity constraints. We analyze the pole structure and find agreement with a conjectured resummation formula. In analogy with the twist-two anomalous dimension at four-loops wrapping effects are of order (log2M/M2) for large values of the spin.
Correlation functions of the spin chains. Algebraic Bethe Ansatz approach
Spin chains are the basic elements of integrable quantum models. These models have direct applications in condense matter theory, in statistical physics, in quantum optics, in field theory and even in string theory but they are also important because they enable us to solve, in an exact manner, non-perturbative phenomena that otherwise would stay unresolved. The method described in this work is based on the algebraic Bethe Ansatz. It is shown how this method can be used for the computation of null temperature correlation functions of the Heisenberg 1/2 spin chain. The important point of this approach is the solution of the inverse quantum problem given by the XXZ spin chain. This solution as well as a simple formulae for the scalar product of the Bethe states, have enabled us to get the most basic correlation functions under the form of multiple integrals. The formalism of multiple integrals open the way for asymptotic analysis for a few physical quantities like the probability of vacuum formation. It is worth noticing that this formalism can give exact results for two-point functions that are the most important correlation functions for applications. A relationship has been discovered between these multiple integrals and the sum of the form factors. The results have been extended to dynamical correlation functions. (A.C.)
Quantum Group and Magnetic Translations. Bethe-Ansatz Solution for Azbel-Hofstadter Problem
Wiegmann, P. B.; Zabrodin, A. V.
1993-01-01
We present a new approach to the problem of Bloch electrons in magnetic ( sometimes called Azbel-Hofstadter problem) field, by making explicit a natural relation between the group of magnetic translations and the quantum group $U_{q}(sl_2)$. The approach allows us to express the "mid" band spectrum of the model and the Bloch wave function as solutions of the Bethe-Ansatz equations typical for completely integrable quantum systems. The zero mode wave functions are found explicitly in terms of ...
Quantum Group, Bethe Ansatz and Bloch Electrons in a Magnetic Field
Hatsugai, Y.; Kohmoto, M.; Wu, Y.-S.
1995-01-01
The wave functions for two dimensional Bloch electrons in a uniform magnetic field at the mid-band points are studied with the help of the algebraic structure of the quantum group $U_q(sl_2)$. A linear combination of its generators gives the Hamiltonian. We obtain analytical and numerical solutions for the wave functions by solving the Bethe Ansatz equations, proposed by Wiegmann and Zabrodin on the basis of above observation. The semi-classical case with the flux per plaquette $\\phi=1/Q$ is ...
Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity
Gainutdinov, Azat M
2016-01-01
For generic values of q, all the eigenvectors of the transfer matrix of the U_q sl(2)-invariant open spin-1/2 XXZ chain with finite length N can be constructed using the algebraic Bethe ansatz (ABA) formalism of Sklyanin. However, when q is a root of unity (q=exp(i pi/p) with integer p>1), the Bethe equations acquire continuous solutions, and the transfer matrix develops Jordan cells. Hence, there appear eigenvectors of two new types: eigenvectors corresponding to continuous solutions (exact complete p-strings), and generalized eigenvectors. We propose general ABA constructions for these two new types of eigenvectors. We present many explicit examples, and we construct complete sets of (generalized) eigenvectors for various values of p and N.
Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity
Azat M. Gainutdinov
2016-08-01
Full Text Available For generic values of q, all the eigenvectors of the transfer matrix of the Uqsl(2-invariant open spin-1/2 XXZ chain with finite length N can be constructed using the algebraic Bethe ansatz (ABA formalism of Sklyanin. However, when q is a root of unity (q=eiπ/p with integer p≥2, the Bethe equations acquire continuous solutions, and the transfer matrix develops Jordan cells. Hence, there appear eigenvectors of two new types: eigenvectors corresponding to continuous solutions (exact complete p-strings, and generalized eigenvectors. We propose general ABA constructions for these two new types of eigenvectors. We present many explicit examples, and we construct complete sets of (generalized eigenvectors for various values of p and N.
Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity
Gainutdinov, Azat M.; Nepomechie, Rafael I.
2016-08-01
For generic values of q, all the eigenvectors of the transfer matrix of the Uq sl (2)-invariant open spin-1/2 XXZ chain with finite length N can be constructed using the algebraic Bethe ansatz (ABA) formalism of Sklyanin. However, when q is a root of unity (q =e iπ / p with integer p ≥ 2), the Bethe equations acquire continuous solutions, and the transfer matrix develops Jordan cells. Hence, there appear eigenvectors of two new types: eigenvectors corresponding to continuous solutions (exact complete p-strings), and generalized eigenvectors. We propose general ABA constructions for these two new types of eigenvectors. We present many explicit examples, and we construct complete sets of (generalized) eigenvectors for various values of p and N.
Functional Bethe ansatz methods for the open XXX chain
We study the spectrum of the integrable open XXX Heisenberg spin chain subject to non-diagonal boundary magnetic fields. The spectral problem for this model can be formulated in terms of functional equations obtained by separation of variables or, equivalently, from the fusion of transfer matrices. For generic boundary conditions the eigenvalues cannot be obtained from the solution of finitely many algebraic Bethe equations. Based on careful finite size studies of the analytic properties of the underlying hierarchy of transfer matrices we devise two approaches to analyze the functional equations. First we introduce a truncation method leading to Bethe-type equations determining the energy spectrum of the spin chain. In a second approach, the hierarchy of functional equations is mapped to an infinite system of nonlinear integral equations of TBA type. The two schemes have complementary ranges of applicability and facilitate an efficient numerical analysis for a wide range of boundary parameters. Some data are presented on the finite-size corrections to the energy of the state which evolves into the antiferromagnetic ground state in the limit of parallel boundary fields.
Quantum integrability and Bethe ansatz solution for interacting matter-radiation systems
A unified integrable system, generating a new series of interacting matter-radiation models with interatomic coupling and different atomic frequencies, is constructed and exactly solved through an algebraic Bethe ansatz. Novel features in Rabi oscillation and vacuum Rabi splitting are shown on the example of an integrable two-atom Buck-Sukumar model with resolution of some important controversies in the Bethe ansatz solution including its possible degeneracy for such models. (letter to the editor)
Algebraic Bethe ansatz for scalar products in SU(3)-invariant integrable models
Belliard, S; Ragoucy, E; Slavnov, N A
2012-01-01
We study SU(3)-invariant integrable models solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for particular case of scalar products of Bethe vectors. This representation can be used for the calculation of form factors and correlation functions of XXX SU(3)-invariant Heisenberg chain.
Masoero, Davide; Raimondo, Andrea; Valeri, Daniele
2016-06-01
We study the ODE/IM correspondence for ODE associated to {widehat{mathfrak{g}}}-valued connections, for a simply-laced Lie algebra {mathfrak{g}}. We prove that subdominant solutions to the ODE defined in different fundamental representations satisfy a set of quadratic equations called {Ψ}-system. This allows us to show that the generalized spectral determinants satisfy the Bethe Ansatz equations.
Twisted Bethe equations from a twisted S-matrix
Ahn, Changrim; Bombardelli, Diego; Nepomechie, Rafael I
2010-01-01
All-loop asymptotic Bethe equations for a 3-parameter deformation of AdS5/CFT4 have been proposed by Beisert and Roiban. We propose a Drinfeld twist of the AdS5/CFT4 S-matrix, together with c-number diagonal twists of the boundary conditions, from which we derive these Bethe equations. Although the undeformed S-matrix factorizes into a product of two su(2|2) factors, the deformed S-matrix cannot be so factored. Diagonalization of the corresponding transfer matrix requires a generalization of the conventional algebraic Bethe ansatz approach, which we first illustrate for the simpler case of the twisted su(2) principal chiral model. We also demonstrate that the same twisted Bethe equations can alternatively be derived using instead untwisted S-matrices and boundary conditions with operatorial twists.
Masoero, Davide; Valeri, Daniele
2015-01-01
We assess the ODE/IM correspondence for the quantum $\\mathfrak{g}$-KdV model, for a non-simply laced Lie algebra $\\mathfrak{g}$. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra ${\\mathfrak{g}}^{(1)}$, and constructing the relevant $\\Psi$-system among subdominant solutions. We then use the $\\Psi$-system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum $\\mathfrak{g}$-KdV model. We also consider generalized Airy functions for twisted Kac--Moody algebras and we construct new explicit solutions to the Bethe Ansatz equations. The paper is a continuation of our previous work on the ODE/IM correspondence for simply-laced Lie algebras.
Instantaneous Bethe-Salpeter equation
We present a systematic algebraic and numerical investigation of the instantaneous Beth-Salpeter equation. Emphasis is placed on confining interaction kernels of the Lorentz scalar, time component vector, and full vector-types. We explore the stability of the solutions and Regge behavior for each of these interactions, and conclude that only time component vector confinement leads to normal Regge structure and stable solutions for all quark masses
Particle-hole symmetry in algebraic Bethe Ansatz for the XXX model
It is well known that the space of all quantum states of the XXX model for a magnetic ring of N nodes, each with the spin 1/2, decomposes into sectors with r spin deviations, r = 0,1, 2,..., N [1, 2, 3, 4]. The sectors r and N - r are related mutually by the particle-hole transformation which exchanges the signs + and - on each node. We discuss here effects of this transformation on the formalism of algebraic Bethe Ansatz, in particular on the form of the monodromy matrix, the main tool of this formalism. We derive explicitly appropriate transformation rules for CN- orbits of magnetic configurations and the corresponding Fourier transformations within the bases of wavelets. In particular, we point out some important phase relations between orbits on both sides of the equator.
Bethe Ansatz and exact form factors of the O(N) Gross Neveu-model
Babujian, Hrachya M; Karowski, Michael
2015-01-01
We apply the algebraic nested O(N) Bethe Ansatz to construct a general form factor formula for the O(N) Gross-Neveu model. We examine this formula for several operators, such as the energy momentum, the spin-field and the current. We also compare these results with the 1/N expansion of this model and obtain full agreement. We discuss bound state form factors, in particular for the three particle form factor of the field. In addition for the two particle case we prove a recursion relation for the K-functions of the higher level Bethe Ansatz.
Bethe Ansatz and exact form factors of the O ( N) Gross Neveu-model
Babujian, Hrachya M.; Foerster, Angela; Karowski, Michael
2016-02-01
We apply previous results on the O ( N) Bethe Ansatz [1-3] to construct a general form factor formula for the O ( N) Gross-Neveu model. We examine this formula for several operators, such as the energy momentum, the spin-field and the current. We also compare these results with the 1 /N expansion of this model and obtain full agreement. We discuss bound state form factors, in particular for the three particle form factor of the field. In addition for the two particle case we prove a recursion relation for the K-functions of the higher level Bethe Ansatz.
Bethe Ansatz Matrix Elements as Non-Relativistic Limits of Form Factors of Quantum Field Theory
Kormos, M.; Mussardo, G.; Pozsgay, B.
2010-01-01
We show that the matrix elements of integrable models computed by the Algebraic Bethe Ansatz can be put in direct correspondence with the Form Factors of integrable relativistic field theories. This happens when the S-matrix of a Bethe Ansatz model can be regarded as a suitable non-relativistic limit of the S-matrix of a field theory, and when there is a well-defined mapping between the Hilbert spaces and operators of the two theories. This correspondence provides an efficient method to compu...
Integrable achiral D5-brane reflections and asymptotic Bethe equations
Correa, Diego H; Young, Charles A S
2011-01-01
We study the reflection of magnons from a D5-brane in the framework of the AdS/CFT correspondence. We consider two possible orientations of the D5-brane with respect to the reference vacuum state, namely vacuum states aligned along "vertical" and "horizontal" directions. We show that the reflections are of the achiral type. We also show that the reflection matrices satisfy the boundary Yang-Baxter equations for both orientations. In the horizontal case the reflection matrix can be interpreted in terms of a bulk S-matrix, S(p, -p), and factorizability of boundary scattering therefore follows from that of bulk scattering. Finally, we solve the nested coordinate Bethe ansatz for the system in the vertical case to find the Bethe equations. In the horizontal case, the Bethe equations are of the same form as those for the closed string.
Milewski, J., E-mail: jsmilew@wp.pl [Institute of Mathematics, Poznań University of Technology, Piotrowo 3A, 60-965 Poznań (Poland); Lulek, B., E-mail: barlulek@amu.edu.pl [East European State Higher School, ul. Tymona Terleckiego 6, 37-700 Przemyśl (Poland); Lulek, T., E-mail: tadlulek@prz.edu.pl [Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznań (Poland); East European State Higher School, ul. Tymona Terleckiego 6, 37-700 Przemyśl (Poland); Łabuz, M., E-mail: labuz@univ.rzeszow.pl [University of Rzeszow, Institute of Physics, Rejtana 16a, 35-959 Rzeszów (Poland); Stagraczyński, R., E-mail: rstag@prz.edu.pl [Rzeszow University of Technology, The Faculty of Mathematics and Applied Physics, Powstańców Warszawy 6, 35-959 Rzeszów (Poland)
2014-02-01
The exact Bethe eigenfunctions for the heptagonal ring within the isotropic XXX model exhibit a doubly degenerated energy level in the three-deviation sector at the centre of the Brillouin zone. We demonstrate an explicit construction of these eigenfunctions by use of algebraic Bethe Ansatz, and point out a relation of degeneracy to parity conservation, applied to the configuration of strings for these eigenfunctions. Namely, the internal structure of the eigenfunctions (the 2-string and the 1-string, with opposite quasimomenta) admits generation of two mutually orthogonal eigenfunctions due to the fact that the strings which differ by their length are distinguishable objects.
Massless Lüscher terms and the limitations of the AdS3 asymptotic Bethe ansatz
Abbott, Michael C.; Aniceto, Inês
2016-05-01
In AdS5/CFT4 integrability the Bethe ansatz gives the spectrum of long strings, accurate up to exponentially small corrections. This is no longer true in three-dimensional anti-de Sitter (AdS3 ) space, as we demonstrate here by studying Lüscher F-terms with a massless particle running in the loop. We apply this to the classic test of Hernández and López, in which the s u (2 ) sector Bethe equations (including the one-loop dressing phase) should match the semiclassical string theory result for a circular spinning string. These calculations do not agree in AdS3×S3×T4 , and we show that the sum of all massless Lüscher F-terms can reproduce the difference.
Generalized Coordinate Bethe Ansatz for open spin chains with non-diagonal boundaries
We introduce a generalization of the original Coordinate Bethe Ansatz that allows to treat the case of open spin chains with non-diagonal boundary matrices. We illustrate it on two cases: the XXX and XXZ chains. Short review on a joint work with N. Crampe (L2C) and D. Simon (LPMA), see arXiv:1009.4119, arXiv:1105.4119 and arXiv:1106.3264.
Yang-Baxter algebras, integrable theories and Bethe Ansatz
This paper presents the Yang-Baxter algebras (YBA) in a general framework stressing their power to exactly solve the lattice models associated to them. The algebraic Behe Ansatz is developed as an eigenvector construction based on the YBA. The six-vertex model solution is given explicitly. The generalization of YB algebras to face language is considered. The algebraic BA for the SOS model of Andrews, Baxter and Forrester is described using these face YB algebras. It is explained how these lattice models yield both solvable massive QFT and conformal models in appropriated scaling (continuous) limits within the lattice light-cone approach. This approach permit to define and solve rigorously massive QFT as an appropriate continuum limit of gapless vertex models. The deep links between the YBA and Lie algebras are analyzed including the quantum groups that underlay the trigonometric/hyperbolic YBA. Braid and quantum groups are derived from trigonometric/hyperbolic YBA in the limit of infinite spectral parameter. To conclude, some recent developments in the domain of integrable theories are summarized
Algebraic and geometric properties of Bethe Ansatz eigenfunctions on a pentagonal magnetic ring
The exact solution of the eigenproblem of the Heisenberg Hamiltonian for the XXX model in the case of a magnetic ring with N=5 nodes is presented in a closed algebraic form. It is demonstrated that the eigenproblem itself is expressible within the extension of the prime field Q of rationals by the primitive fifth root of unity, whereas the associated Bethe parameters, i.e. pseudomomenta, phases of scattering, and spectral parameters, require some finite field extensions, such that the nonlinearity remains algebraic rather than transcendental. Classification of exact Bethe Ansatz eigenstates in terms of rigged string configurations is presented. -- Research Highlights: → The eigenproblem is expressed in a finite extension of the field Q. → The Galois symmetry gives rise to operators which reproduce the energy band structure. → Original Bethe parameters can be derived by the inverse BA problem. → String hypothesis, expected to work as N goes to infinity, is almost satisfied for N=5.
Kitanine, N
2007-09-15
Spin chains are the basic elements of integrable quantum models. These models have direct applications in condense matter theory, in statistical physics, in quantum optics, in field theory and even in string theory but they are also important because they enable us to solve, in an exact manner, non-perturbative phenomena that otherwise would stay unresolved. The method described in this work is based on the algebraic Bethe Ansatz. It is shown how this method can be used for the computation of null temperature correlation functions of the Heisenberg 1/2 spin chain. The important point of this approach is the solution of the inverse quantum problem given by the XXZ spin chain. This solution as well as a simple formulae for the scalar product of the Bethe states, have enabled us to get the most basic correlation functions under the form of multiple integrals. The formalism of multiple integrals open the way for asymptotic analysis for a few physical quantities like the probability of vacuum formation. It is worth noticing that this formalism can give exact results for two-point functions that are the most important correlation functions for applications. A relationship has been discovered between these multiple integrals and the sum of the form factors. The results have been extended to dynamical correlation functions. (A.C.)
Bethe ansatz solvability and supersymmetry of the M2 model of single fermions and pairs
A detailed study of a model for strongly-interacting fermions with exclusion rules and lattice N=2 supersymmetry is presented. A submanifold in the space of parameters of the model where it is Bethe-ansatz solvable is identified. The relation between this manifold and the existence of additional, so-called dynamic, supersymmetries is discussed. The ground states are analysed with the help of cohomology techniques, and their exact finite-size Bethe roots are found. Moreover, through analytical and numerical studies it is argued that the model provides a lattice version of the N=1 super-sine-Gordon model at a particular coupling where an additional N=(2,2) supersymmetry is present. The dynamic supersymmetry is shown to allow an exact determination of the gap scaling function of the model. (paper)
Algebraic Bethe ansatz for the six vertex model with upper triangular K-matrices
We consider a formulation of the algebraic Bethe ansatz for the six vertex model with non-diagonal open boundaries. Specifically, we study the case where both left and right K-matrices have an upper triangular form. We show that the main difficulty entailed by those forms of the K-matrices is the construction of the excited states. However, it is possible to treat this problem with the aid of an auxiliary transfer matrix and by means of a generalized creation operator. (paper)
A Bethe ansatz solvable model for superpositions of Cooper pairs and condensed molecular bosons
Hibberd, K. E.; Dunning, C.; Links, J.
2006-08-01
We introduce a general Hamiltonian describing coherent superpositions of Cooper pairs and condensed molecular bosons. For particular choices of the coupling parameters, the model is integrable. One integrable manifold, as well as the Bethe ansatz solution, was found by Dukelsky et al. [J. Dukelsky, G.G. Dussel, C. Esebbag, S. Pittel, Phys. Rev. Lett. 93 (2004) 050403]. Here we show that there is a second integrable manifold, established using the boundary quantum inverse scattering method. In this manner we obtain the exact solution by means of the algebraic Bethe ansatz. In the case where the Cooper pair energies are degenerate we examine the relationship between the spectrum of these integrable Hamiltonians and the quasi-exactly solvable spectrum of particular Schrödinger operators. For the solution we derive here the potential of the Schrödinger operator is given in terms of hyperbolic functions. For the solution derived by Dukelsky et al., loc. cit. the potential is sextic and the wavefunctions obey PT-symmetric boundary conditions. This latter case provides a novel example of an integrable Hermitian Hamiltonian acting on a Fock space whose states map into a Hilbert space of PT-symmetric wavefunctions defined on a contour in the complex plane.
A Bethe ansatz solvable model for superpositions of Cooper pairs and condensed molecular bosons
Hibberd, K.E. [Centre for Mathematical Physics, University of Queensland, 4072 (Australia); Dunning, C. [Institute of Mathematics, Statistics and Actuarial Science, University of Kent (United Kingdom); Links, J. [Centre for Mathematical Physics, University of Queensland, 4072 (Australia)]. E-mail: jrl@maths.uq.edu.au
2006-08-07
We introduce a general Hamiltonian describing coherent superpositions of Cooper pairs and condensed molecular bosons. For particular choices of the coupling parameters, the model is integrable. One integrable manifold, as well as the Bethe ansatz solution, was found by Dukelsky et al. [J. Dukelsky, G.G. Dussel, C. Esebbag, S. Pittel, Phys. Rev. Lett. 93 (2004) 050403]. Here we show that there is a second integrable manifold, established using the boundary quantum inverse scattering method. In this manner we obtain the exact solution by means of the algebraic Bethe ansatz. In the case where the Cooper pair energies are degenerate we examine the relationship between the spectrum of these integrable Hamiltonians and the quasi-exactly solvable spectrum of particular Schrodinger operators. For the solution we derive here the potential of the Schrodinger operator is given in terms of hyperbolic functions. For the solution derived by Dukelsky et al., loc. cit. the potential is sextic and the wavefunctions obey PT-symmetric boundary conditions. This latter case provides a novel example of an integrable Hermitian Hamiltonian acting on a Fock space whose states map into a Hilbert space of PT-symmetric wavefunctions defined on a contour in the complex plane.
Kundu, Anjan
2016-01-01
Integrable quantum field models are known to exist mostly in one space-dimension. Exploiting the concept of multi-time in integrable systems and a Lax matrix of higher scaling order, we construct a novel quantum field model in quasi-two dimensions involving interacting fields. The Yang-Baxter integrability is proved for the model by finding a new kind of commutation rule for its basic fields, representing nonstandard scalar fields along the transverse direction. In spite of a close link with the quantum Landau-Lifshitz equation, the present model differs widely from it, in its content and the result obtained. Using further the algebraic Bethe ansatz we solve exactly the eigenvalue problem of this quantum field model for all its higher conserved operators. The idea presented here should instigate the construction of a novel class of integrable field and lattice models and exploration of a new type of underlying algebras.
Log-gamma directed polymer with fixed endpoints via the replica Bethe Ansatz
We study the model of a discrete directed polymer (DP) on a square lattice with homogeneous inverse gamma distribution of site random Boltzmann weights, introduced by Seppalainen (2012 Ann. Probab. 40 19–73). The integer moments of the partition sum, Zn-bar , are studied using a transfer matrix formulation, which appears as a generalization of the Lieb–Liniger quantum mechanics of bosons to discrete time and space. In the present case of the inverse gamma distribution the model is integrable in terms of a coordinate Bethe Ansatz, as discovered by Brunet. Using the Brunet-Bethe eigenstates we obtain an exact expression for the integer moments of Zn-bar for polymers of arbitrary lengths and fixed endpoint positions. Although these moments do not exist for all integer n, we are nevertheless able to construct a generating function which reproduces all existing integer moments and which takes the form of a Fredholm determinant (FD). This suggests an analytic continuation via a Mellin–Barnes transform and we thereby propose a FD ansatz representation for the probability distribution function (PDF) of Z and its Laplace transform. In the limit of a very long DP, this ansatz yields that the distribution of the free energy converges to the Gaussian unitary ensemble (GUE) Tracy-Widom distribution up to a non-trivial average and variance that we calculate. Our asymptotic predictions coincide with a result by Borodin et al (2013 Commun. Math. Phys. 324 215–32) based on a formula obtained by Corwin et al (2011 arXiv:1110.3489) using the geometric Robinson–Schensted–Knuth (gRSK) correspondence. In addition we obtain the dependence on the endpoint position and the exact elastic coefficient at a large time. We argue the equivalence between our formula and that of Borodin et al. As we will discuss, this provides a connection between quantum integrability and tropical combinatorics. (paper)
Log-gamma directed polymer with fixed endpoints via the replica Bethe Ansatz
Thiery, Thimothée; Le Doussal, Pierre
2014-10-01
We study the model of a discrete directed polymer (DP) on a square lattice with homogeneous inverse gamma distribution of site random Boltzmann weights, introduced by Seppalainen (2012 Ann. Probab. 40 19-73). The integer moments of the partition sum, \\overline{Z^n} , are studied using a transfer matrix formulation, which appears as a generalization of the Lieb-Liniger quantum mechanics of bosons to discrete time and space. In the present case of the inverse gamma distribution the model is integrable in terms of a coordinate Bethe Ansatz, as discovered by Brunet. Using the Brunet-Bethe eigenstates we obtain an exact expression for the integer moments of \\overline{Z^n} for polymers of arbitrary lengths and fixed endpoint positions. Although these moments do not exist for all integer n, we are nevertheless able to construct a generating function which reproduces all existing integer moments and which takes the form of a Fredholm determinant (FD). This suggests an analytic continuation via a Mellin-Barnes transform and we thereby propose a FD ansatz representation for the probability distribution function (PDF) of Z and its Laplace transform. In the limit of a very long DP, this ansatz yields that the distribution of the free energy converges to the Gaussian unitary ensemble (GUE) Tracy-Widom distribution up to a non-trivial average and variance that we calculate. Our asymptotic predictions coincide with a result by Borodin et al (2013 Commun. Math. Phys. 324 215-32) based on a formula obtained by Corwin et al (2011 arXiv:1110.3489) using the geometric Robinson-Schensted-Knuth (gRSK) correspondence. In addition we obtain the dependence on the endpoint position and the exact elastic coefficient at a large time. We argue the equivalence between our formula and that of Borodin et al. As we will discuss, this provides a connection between quantum integrability and tropical combinatorics.
Twisting singular solutions of Bethe's equations
Nepomechie, Rafael I
2014-01-01
The Bethe equations for the periodic XXX and XXZ spin chains admit singular solutions, for which the corresponding eigenvalues and eigenvectors are ill-defined. We use a twist regularization to derive conditions for such singular solutions to be physical, in which case they correspond to genuine eigenvalues and eigenvectors of the Hamiltonian.
Hofstadter Problem on the Honeycomb and Triangular Lattices: Bethe Ansatz Solution
Kohmoto, M.; Sedrakyan, A.
2006-01-01
We consider Bloch electrons on the honeycomb lattice under a uniform magnetic field with $2 \\pi p/q$ flux per cell. It is shown that the problem factorizes to two triangular lattices. Treating magnetic translations as Heisenberg-Weyl group and by the use of its irreducible representation on the space of theta functions, we find a nested set of Bethe equations, which determine the eigenstates and energy spectrum. The Bethe equations have simple form which allows to consider them further in the...
Bethe Ansatz for Supersymmetric Model Constructed from Uq[osp(2|2)(2)] R-Matrix
YANG Wen-Li; ZHEN Yi
2001-01-01
Using the algebraic Bethe ansatz method, we obtain the eigenvalues of transfer matrix of the supersymmetric model constructed from the R-matrix of the twisted affine superalgebra Uq[osp(2|2)(2)] in periodic boundary condition and twisted boundary condition.``
Gaudin-type models, non-skew-symmetric classical r-matrices and nested Bethe ansatz
T. Skrypnyk
2015-02-01
Full Text Available We consider quantum integrable systems associated with the Lie algebra gl(n and Cartan-invariant non-dynamical non-skew-symmetric classical r-matrices. We describe the sub-class of Cartan-invariant non-skew-symmetric r-matrices for which exists the standard procedure of the nested Bethe ansatz associated with the chain of embeddings gl(n⊃gl(n−1⊃gl(n−2⊃⋯⊃gl(1. We diagonalize the corresponding quantum integrable systems by its means. We illustrate the obtained results by the examples of the generalized Gaudin systems with and without external magnetic field associated with three classes of non-dynamical non-skew-symmetric classical r-matrices.
Bethe-salpeter equation from many-body perturbation theory
Sander, Tobias; Starke, Ronald; Kresse, Georg [Computational Materials Physics, University of Vienna, Sensengasse 8/12, 1090 Vienna (Austria)
2013-07-01
The Green function formalism is a powerful tool to calculate not only electronic structure within the quasi-particle (QP) picture, but it also gives access to optical absorption spectra. Starting from QP energies within the GW method, the polarizability, as central quantity, is calculated from the solution of a Bethe-Salpeter-like equation (BSE). It is usually solved within the Tamm-Dancoff Approximation (TDA) which neglects the coupling of resonant (positive frequency branch) and anti-resonant (negative frequency branch) excitations. In this work we solve the full BSE (beyond TDA) based on self-consistently calculated QP orbitals and energies for typical systems. The dielectric function is averaged over many low dimensional shifted k-meshes to obtain k-point converged results. We compare the results to recently introduced approximation to the BSE kernel. Additionally, the time-evolution ansatz is employed to calculate the polarizability, which avoids the direct solution of the BSE.
Bethe-salpeter equation from many-body perturbation theory
The Green function formalism is a powerful tool to calculate not only electronic structure within the quasi-particle (QP) picture, but it also gives access to optical absorption spectra. Starting from QP energies within the GW method, the polarizability, as central quantity, is calculated from the solution of a Bethe-Salpeter-like equation (BSE). It is usually solved within the Tamm-Dancoff Approximation (TDA) which neglects the coupling of resonant (positive frequency branch) and anti-resonant (negative frequency branch) excitations. In this work we solve the full BSE (beyond TDA) based on self-consistently calculated QP orbitals and energies for typical systems. The dielectric function is averaged over many low dimensional shifted k-meshes to obtain k-point converged results. We compare the results to recently introduced approximation to the BSE kernel. Additionally, the time-evolution ansatz is employed to calculate the polarizability, which avoids the direct solution of the BSE.
RPA equations and the instantaneous Bethe-Salpeter equation
Resag, J
1993-01-01
We give a derivation of the particle-hole RPA equations for an interacting multi-fermion system by applying the instantaneous approximation to the amputated two-fermion propagator of the system. In relativistic field theory the same approximation leads from the fermion-antifermion Bethe-Salpeter equation to the Salpeter equation. We show that RPA equations and Salpeter equation are indeed equivalent.
We extend the exact periodic Bethe ansatz solution for one-dimensional bosons and fermions with δ-interaction and arbitrary internal degrees of freedom to the case of hard wall boundary conditions. We give an analysis of the ground-state properties of fermionic systems with two internal degrees of freedom, including expansions of the ground-state energy in the weak and strong coupling limits
Choudhury, A.G.; Chowdhury, A.R. [Jadavpur Univ., Calcutta (India)
1996-08-01
Intertwining relations for the quantum R-matrix of the SU{sub p,q}(2) invariant spin chain are obtained and the corresponding face model is deduced. An important difference is seen to arise due to the asymmetry generated by the parameters p and q, which leads to a asymmetric face model. An algebraic Bethe ansatz is set up and solved with the help of these intertwining vectors.
Oelkers, N; Batchelor, M T; Bortz, M; Guan, X-W [Department of Theoretical Physics, Research School of Physical Sciences and Engineering and Mathematical Sciences Institute, The Australian National University, Canberra, ACT 0200 (Australia)
2006-02-03
We extend the exact periodic Bethe ansatz solution for one-dimensional bosons and fermions with {delta}-interaction and arbitrary internal degrees of freedom to the case of hard wall boundary conditions. We give an analysis of the ground-state properties of fermionic systems with two internal degrees of freedom, including expansions of the ground-state energy in the weak and strong coupling limits.
We consider the problem of consistence between the Bethe ansatz (BA) wave function and the multiparticle (more than two) scattering in one-dimensional δ-function interacting SU(4) fermions, which the approach of BA does not explicitly take into account. We find the scattering conditions of three and four particles located at the same position and show that the conditions can be fulfilled by the two-particle connection conditions of the BA wave function. So the definition of the BA wave function can be exactly extended to those cases with multiple occupancies. The inconsistence between the BA and multiparticle interacting on a same site in the degenerate Hubbard model, which makes the BA fail for the model, is shown to vanish in the limit of small site spacing. A correspondence relation of the BA equation and SU(4) symmetry of the system is also indicated for the fermions. The degeneracy of state with BA eigenenergy is given. Singlet lies in the case when there are equal numbers of particles in each inner component
Gainutdinov, A M; Nepomechie, Rafael I; Sommese, Andrew J
2015-01-01
We consider the sl(2)_q-invariant open spin-1/2 XXZ quantum spin chain of finite length N. For the case that q is a root of unity, we propose a formula for the number of admissible solutions of the Bethe ansatz equations in terms of dimensions of irreducible representations of the Temperley-Lieb algebra; and a formula for the degeneracies of the transfer matrix eigenvalues in terms of dimensions of tilting sl(2)_q-modules. These formulas include corrections that appear if two or more tilting modules are spectrum-degenerate. For the XX case (q=exp(i pi/2)), we give explicit formulas for the number of admissible solutions and degeneracies. We also consider the cases of generic q and the isotropic (q->1) limit. Numerical solutions of the Bethe equations up to N=8 are presented. Our results are consistent with the Bethe ansatz solution being complete.
Two-body bound states & the Bethe-Salpeter equation
Pichowsky, M. [Argonne National Lab., IL (United States); Kennedy, M. [Univ. of New Hampshire, Durham, NH (United States). Physics Dept.; Strickland, M. [Duke Univ., Durham, NC (United States)
1995-01-18
The Bethe-Salpeter formalism is used to study two-body bound states within a scalar theory: two scalar fields interacting via the exchange of a third massless scalar field. The Schwinger-Dyson equation is derived using functional and diagrammatic techniques, and the Bethe-Salpeter equation is obtained in an analogous way, showing it to be a two-particle generalization of the Schwinger-Dyson equation. The authors also present a numerical method for solving the Bethe-Salpeter equation without three-dimensional reduction. The ground and first excited state masses and wavefunctions are computed within the ladder approximation and space-like form factors are calculated.
Colored Quantum Algebra and Its Bethe State
We investigate the colored Yang—Baxter equation. Based on a trigonometric solution of colored Yang—Baxter equation, we construct a colored quantum algebra. Moreover we discuss its algebraic Bethe ansatz state and highest wight representation. (general)
The same-position scattering (SPS) of more than two electrons in a one-dimensional model of two-band electrons with spin-exchange interaction is discussed. The boundary conditions of three- and four-particle SPS are given. It is shown that the conditions can be fulfilled by the two-particle boundary conditions for the Bethe ansatz (BA) wavefunction. Consequently, the definition of the BA wavefunction can be extended to those cases of more than two particles occupying the same position. Therefore, unlike the case in lattice models in which configurations with more than two particles at one site are excluded in applying the approach, the BA is valid without the exclusion of multi-particle SPS in the spin-exchange model. A relation between the SU(2)xSU(2) symmetry and the BA equation is also indicated. (author)
Bethe states of the integrable spin-s chain with generic open boundaries
Based on the inhomogeneous T –Q relation and the associated Bethe ansatz equations obtained via the off-diagonal Bethe ansatz, we construct the Bethe-type eigenstates of the SU(2)-invariant spin-s chain with generic non-diagonal boundaries by employing certain orthogonal basis of the Hilbert space. (paper)
Glueball properties from the Bethe-Salpeter equation
For over thirty years bound states of gluons are an outstanding problem of both theoretical and experimental physics. Being predicted by Quantum-Chromodynamics their experimental confirmation is one of the foremost goals of large experimental facilities currently under construction like FAIR in Darmstadt. This thesis presents a novel approach to the theoretical determination of physical properties of bound states of two gluons, called glueballs. It uses the consistent combination of Schwinger-Dyson equations for gluons and ghosts and appropriate Bethe-Salpeter equations describing their corresponding bound-states. A rigorous derivation of both sets of equations, starting from an 2PI effective action is given as well as a general determination of appropriate decompositions of Bethe-Salpeter amplitudes to a given set of quantum numbers of a glueball. As an application example bound state masses of glueballs in a simple truncation scheme are calculated. (orig.)
Castro-Alvaredo, Olalla; Doyon, Benjamin; Hoogeveen, Marianne
2013-01-01
We evaluate the exact energy current and scaled cumulant generating function (related to the large-deviation function) in non-equilibrium steady states with energy flow, in any integrable model of relativistic quantum field theory (IQFT) with diagonal scattering. Our derivations are based on various recent results of D. Bernard and B. Doyon. The steady states are built by connecting homogeneously two infinite halves of the system thermalized at different temperatures $T_l$, $T_r$, and waiting for a long time. We evaluate the current $J(T_l,T_r)$ using the exact QFT density matrix describing these non-equilibrium steady states and using Al.B. Zamolodchikov's method of the thermodynamic Bethe ansatz (TBA). The scaled cumulant generating function is obtained from the extended fluctuation relations which hold in integrable models. We verify our formula in particular by showing that the conformal field theory (CFT) result is obtained in the high-temperature limit. We analyze numerically our non-equilibrium steady-...
Excited charmonium states from Bethe-Salpeter Equation
Šauli, Vladimír; Bicudo, P.
2012-01-01
Roč. 7, 043 (2012), s. 1-10. ISSN 1824-8039. [International Workshop on QCD Green’s Functions. Tranto, 05.09.2011-09.09.2011] R&D Projects: GA MŠk(CZ) LG11005 Institutional research plan: CEZ:AV0Z10480505 Keywords : charmonium * Bethe-Salpeter Equation Subject RIV: BE - Theoretical Physics http://pos.sissa.it/archive/conferences/136/043/QCD- TNT -II_043.pdf
Fast analytic solver of rational Bethe equations
Marboe, Christian
2016-01-01
In this note we propose an approach for a fast analytic determination of all possible eigenstates of rational GL(N|M) integrable spin chains of given not too large length, in terms of Baxter Q-functions. We observe that all exceptional solutions, if any, are automatically correctly accounted. The key intuition behind the approach is that the equations on the Q-functions are determined solely by the Young diagram, and not by the choice of the rank of the GL symmetry. Hence we can choose arbitrary N and M that accommodate the desired representation. Then we consider all distinguished Q-functions at once, not only those following a certain Kac-Dynkin path.
Bethe vectors in GL(3)-based quantum integrable models
Pakuliak, S; Slavnov, N A
2015-01-01
We consider a composite generalized quantum integrable model solvable by the nested algebraic Bethe ansatz. Using explicit formulas of the action of the monodromy matrix elements onto Bethe vectors in the GL(3)-based quantum integrable models we prove a formula for the Bethe vectors of composite model. We show that this representation is a particular case of general coproduct property of the weight functions (Bethe vectors) found in the theory of the deformed Knizhnik--Zamolodchokov equation.
Bethe-Salpeter equation for elastic nucleon-nucleon scattering
The Bethe-Salpeter equation for NN scattering with one-boson exchange is investigated for the case in which the pion-nucleon coupling is described by axial-vector theory. In contrast to the results with pseudoscalar coupling, good agreement with the experimental data can be obtained for all partial waves. Also, the deviations from the Blankenbecler-Sugar equation are not as large as they are for pseudoscalar coupling. In addition, cancellations between the direct and the crossed box graph with pseudoscalar πN coupling are investigated for the 3S1 phase shift in the framework of the variational operator Pade approximation
Hutsalyuk, A; Pakuliak, S Z; Ragoucy, E; Slavnov, N A
2016-01-01
We study integrable models with $\\mathfrak{gl}(2|1)$ symmetry and solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for scalar products of Bethe vectors, when the Bethe parameters obey some relations weaker than the Bethe equations. This representation allows us to find the norms of on-shell Bethe vectors and obtain determinant formulas for form factors of the diagonal entries of the monodromy matrix.
Bethe states of the XXZ spin-12 chain with arbitrary boundary fields
Xin Zhang
2015-04-01
Full Text Available Based on the inhomogeneous T−Q relation constructed via the off-diagonal Bethe Ansatz, the Bethe-type eigenstates of the XXZ spin-12 chain with arbitrary boundary fields are constructed. It is found that by employing two sets of gauge transformations, proper generators and reference state for constructing Bethe vectors can be obtained respectively. Given an inhomogeneous T−Q relation for an eigenvalue, it is proven that the resulting Bethe state is an eigenstate of the transfer matrix, provided that the parameters of the generators satisfy the associated Bethe Ansatz equations.
Excited charmonium states from Bethe-Salpeter equation
Sauli, Vladimir
2011-01-01
We solve the Bethe-Salpeter equation for a system of a heavy quark-antiquark pair interacting with a screened linear confining potential. First we show the spinless QFT model is inadequate and fail to describe even gross feature of the quarkonia spectrum. In order to get reliable description the spine degrees of freedom has to be considered. Within the approximation employed we reasonably reproduce known radial excitation of vector charmonium. The BSE favors relatively large string breaking scale $\\mu\\simeq 350MeV$ . Using free charm quark propagators we observe that $J/\\Psi$ is the only charmonium left bellow naive quark-antiquark threshold $2m_c$, while the all excited states are situated above this threshold. Within the numerical method we overcome obstacles related with threshold singularity and discuss the consequences of the use of free propagators for calculation of excited states above the threshold.
A systematic approach to sketch Bethe-Salpeter equation
Qin, Si-xue
2016-01-01
To study meson properties, one needs to solve the gap equation for the quark propagator and the Bethe-Salpeter (BS) equation for the meson wavefunction, self-consistently. The gluon propagator, the quark-gluon vertex, and the quark--anti-quark scattering kernel are key pieces to solve those equations. Predicted by lattice-QCD and Dyson-Schwinger analyses of QCD's gauge sector, gluons are non-perturbatively massive. In the matter sector, the modeled gluon propagator which can produce a veracious description of meson properties needs to possess a mass scale, accordingly. Solving the well-known longitudinal Ward-Green-Takahashi identities (WGTIs) and the less-known transverse counterparts together, one obtains a nontrivial solution which can shed light on the structure of the quark-gluon vertex. It is highlighted that the phenomenologically proposed anomalous chromomagnetic moment (ACM) vertex originates from the QCD Lagrangian symmetries and its strength is proportional to the magnitude of dynamical chiral symm...
Two-body bound states ampersand the Bethe-Salpeter equation
The Bethe-Salpeter formalism is used to study two-body bound states within a scalar theory: two scalar fields interacting via the exchange of a third massless scalar field. The Schwinger-Dyson equation is derived using functional and diagrammatic techniques, and the Bethe-Salpeter equation is obtained in an analogous way, showing it to be a two-particle generalization of the Schwinger-Dyson equation. The authors also present a numerical method for solving the Bethe-Salpeter equation without three-dimensional reduction. The ground and first excited state masses and wavefunctions are computed within the ladder approximation and space-like form factors are calculated
Skrypnyk, T.
2016-09-01
We consider quantum integrable models based on the Lie algebra gl(n) and non-skew-symmetric classical r-matrices associated with Z 2-gradings of gl(n) of the following type: {gl}(n)={gl}{(n)}\\bar{0}+{gl}{(n)}\\bar{1}, where {gl}{(n)}\\bar{0}={gl}({n}1)\\oplus {gl}(n-{n}1). Among the considered models are Gaudin-type models with an external magnetic field, used in nuclear physics to produce proton–neutron Bardeen–Cooper–Schrieer-type models, n-level many-mode Jaynes–Cummings–Dicke-type models of quantum optics, matrix generalization of Bose–Hubbard dimers, etc. We diagonalize the constructed models by means of the ‘generalized’ nested Bethe ansatz.
Bethe's quantum numbers and rigged configurations
Anatol N. Kirillov
2016-04-01
Full Text Available We propose a method to determine the quantum numbers, which we call the rigged configurations, for the solutions to the Bethe ansatz equations for the spin-1/2 isotropic Heisenberg model under the periodic boundary condition. Our method is based on the observation that the sums of Bethe's quantum numbers within each string behave particularly nicely. We confirm our procedure for all solutions for length 12 chain (totally 923 solutions.
Bethe's quantum numbers and rigged configurations
Kirillov, Anatol N.; Sakamoto, Reiho
2016-01-01
We propose a method to determine the quantum numbers, which we call the rigged configurations, for the solutions to the Bethe ansatz equations for the spin-1/2 isotropic Heisenberg model under the periodic boundary condition. Our method is based on the observation that the sums of Bethe's quantum numbers within each string behave particularly nicely. We confirm our procedure for all solutions for length 12 chain (totally 923 solutions).
Efficient implementation of core-excitation Bethe Salpeter equation calculations
Gilmore, K; Shirley, E L; Prendergast, D; Pemmaraju, C D; Kas, J J; Vila, F D; Rehr, J J
2016-01-01
We present an efficient implementation of the Bethe-Salpeter equation (BSE) method for obtaining core-level spectra including x-ray absorption (XAS), x-ray emission (XES), and both resonant and non-resonant inelastic x-ray scattering spectra (N/RIXS). Calculations are based on density functional theory (DFT) electronic structures generated either by abinit or Quantumespresso, both plane-wave basis, pseudopotential codes. This electronic structure is improved through the inclusion of a GW self energy. The projector augmented wave technique is used to evaluate transition matrix elements between core-level and band states. Final two-particle scattering states are obtained with the NIST core-level BSE solver (NBSE). We have previously reported this implementation, which we refer to as ocean (Obtaining Core Excitations from Ab initio electronic structure and NBSE) [Phys. Rev. B 83, 115106 (2011)]. Here, we present additional efficiencies that enable us to evaluate spectra for systems ten times larger than previous...
Scattering solutions of Bethe-Salpeter equation in Minkowski and Euclidean spaces
Carbonell, J
2016-01-01
We shortly review different methods to obtain the scattering solutions of the Bethe-Salpeter equation in Minkowski space. We emphasize the possibility to obtain the zero energy observables in terms of the Euclidean scattering amplitude.
Light Pseudoscalar Mesons in Bethe-Salpeter Equation with Instantaneous Interaction
Lucha, Wolfgang
2015-01-01
The light pseudoscalar mesons play a twofold role: they may or have to be regarded both as low-lying bound states of the fundamental degrees of freedom of quantum chromodynamics as well as the (pseudo-) Goldstone bosons of the spontaneously broken chiral symmetries of quantum chromodynamics. We interrelate these aspects in a single quantum-field-theoretic approach relying on the Bethe-Salpeter formalism in instantaneous approximation by very simple means: the shape of the pseudoscalar-meson Bethe-Salpeter wave function dictated by chiral symmetry is used in Bethe-Salpeter equations for bound states of vanishing mass, in order to deduce analytically the interactions which govern the bound states under study. In this way, we obtain exact Bethe-Salpeter solutions for pseudoscalar mesons, in the sense of establishing the rigorous relationship between, on the one hand, the relevant interactions and, on the other hand, the Bethe-Salpeter amplitudes that characterize the bound states.
Use of the Bethe equation for inner-shell ionization by electron impact
Powell, Cedric J.; Llovet, Xavier; Salvat, Francesc
2016-05-01
We analyzed calculated cross sections for K-, L-, and M-shell ionization by electron impact to determine the energy ranges over which these cross sections are consistent with the Bethe equation for inner-shell ionization. Our analysis was performed with K-shell ionization cross sections for 26 elements, with L-shell ionization cross sections for seven elements, L3-subshell ionization cross sections for Xe, and M-shell ionization cross sections for three elements. The validity (or otherwise) of the Bethe equation could be checked with Fano plots based on a linearized form of the Bethe equation. Our Fano plots, which display theoretical cross sections and available measured cross sections, reveal two linear regions as predicted by de Heer and Inokuti [in Electron Impact Ionization, edited by T. D. Märk and G. H. Dunn, (Springer-Verlag, Vienna, 1985), Chap. 7, pp. 232-276]. For each region, we made linear fits and determined values of the two element-specific Bethe parameters. We found systematic variations of these parameters with atomic number for both the low- and the high-energy linear regions of the Fano plots. We also determined the energy ranges over which the Bethe equation can be used.
Bethe states for the two-site Bose-Hubbard model: a binomial approach
Santos, Gilberto; Foerster, Angela; Roditi, Itzhak
2015-01-01
We calculate explicitly the Bethe vectors states by the algebraic Bethe ansatz method with the $gl(2)$-invariant $R$-matrix for the two-site Bose-Hubbard model. Using a binomial expansion of the n-th power of a sum of two operators we get and solve a recursion equation. We calculate the scalar product and the norm of the Bethe vectors states. The form factors of the imbalance current operator are also computed.
Solution of the Bethe-Goldstone Equation without Partial Wave Decomposition
White, L; Sammarruca, F.
2013-01-01
We present a method for solving the nucleon-nucleon scattering equation without the use of a partial wave expansion of the scattering amplitude. After verifying the accuracy of the numerical solutions, we proceed to apply the method to the in-medium scattering equation (the Bethe-Goldstone equation) in three dimensions. A focal point is a study of Pauli blocking effects calculated in the (angle-dependent) three-dimensional formalism as compared to the usual spherical approximation. We discuss...
Normalization and perturbation theory for tightly bound states of the spinor Bethe-Salpeter equation
L.G. Suttorp
1976-01-01
The normalisation integrals for the tightly-bound-state solutions of the spinor Bethe-Salpeter equation that have been derived recently are evaluated. Ghost states are found to appear when the continuous parameters characterising the type of fermion-boson interaction reach a critical value. Perturba
Exact solutions of the spinor Bethe-Salpeter equation for tightly bound states
L.G. Suttorp
1975-01-01
Exact solutions are obtained for the spinor Bethe-Salpeter equation that describes tightly bound states of spin-/sup 1///sub 2/ fermions with massless-boson exchange. The corresponding coupling constants form a discrete spectrum that depends continuously on the parameters characterizing the type of
Calculation of Spin Observables for Proton-Neutron Elastic Scattering in the Bethe-Salpeter Equation
Kinpara, Susumu
2016-01-01
Bethe-Salpeter equation is applied to $p$-$n$ elastic scattering. The spin observables are calculated by the M matrix similar to $p$-$p$ case. The parameters of the meson-exchange model are used with the cut-off for the pion exchange interaction. Change of the M matrix indicates breaking of the charge independence in the nucleon-nucleon system.
Stochastic integration of the Bethe-Salpeter equation for two bound fermions
A non-perturbative method using a Monte Carlo algorithm is used to integrate the Bethe-Salpeter equation in momentum space. Solutions for two scalars and two fermions with an arbitrary coupling constant are calculated for bound states in the ladder approximation. The results are compared with other numerical methods. (Author) (13 refs., 2 figs.)
The connection of two-particle relativistic quantum mechanics with the Bethe-Salpeter equation
We show the formal equivalence between the wave equations of two-particle relativistic quantum mechanics, based on the manifestly covariant hamiltonian formalism with constraints, and the Bethe-Salpeter equation. This is achieved by algebraically transforming the latter so as to separate it into two independent equations which match the equations of hamiltonian relativistic quantum mechanics. The first equation determines the relative time evolution of the system, while the second one yields a three-dimensional eigenvalue equation. A connection is thus established between the Bethe-Salpeter wave function and its kernel on the one hand and the quantum mechanical wave function and interaction potential on the other. For the sector of solutions of the Bethe-Salpeter equation having non-relativistic limits, this relationship can be evaluated in perturbation theory. We also device a generalized form of the instantaneous approximation which simplifies the various expressions involved in the above relations. It also permits the evaluation of the normalization condition of the quantum mechanical wave function as a three-dimensional integral
Solving the inhomogeneous Bethe-Salpeter Equation in Minkowski space: the zero-energy limit
Frederico, T; Viviani, M
2015-01-01
For the first time, the inhomogeneous Bethe-Salpeter Equation for an interacting system, composed by two massive scalars exchanging a massive scalar, is numerically investigated in ladder approximation, directly in Minkowski space, by using an approach based on the Nakanishi integral representation. In this paper, the limiting case of zero-energy states is considered, extending the approach successfully applied to bound states presented in Phys. Rev. D 89, (2014) 016010, where the Nakanishi integral representation has been exploited for solving the homogeneous Bethe-Salpeter Equation. The numerical values of scattering lengths, evaluated by using two different integral equations that stem within the Nakanishi framework, are compared with the analogous quantities recently obtained, within a totally different framework. Moreover, relevant functions, like the Nakanishi weight functions and the distorted part of the zero-energy Light-front wave functions are also presented. Interestingly, a highly non trivial iss...
Light composite Higgs boson from the normalized Bethe-Salpeter equation
Doff, A.(Universidade Tecnológica Federal do Paraná – UTFPR – DAFIS, Av. Monteiro Lobato Km 04, 84016-210 Ponta Grossa, PR, Brazil); Natale, A. A.; da Silva, P. S. Rodrigues
2009-01-01
Scalar composite boson masses have been computed in QCD and Technicolor theories with the help of the homogeneous Bethe-Salpeter equation (BSE), resulting in a scalar mass that is twice the dynamically generated fermion or technifermion mass ($m_{dyn}$). We show that in the case of walking (or quasi-conformal) technicolor theories, where the $m_{dyn}$ behavior with the momenta may be quite different from the one predicted by the standard operator product expansion, this result is incomplete a...
Gluon bound state and asymptotic freedom derived from the Bethe--Salpeter equation
Fukamachi, Hitoshi; Nishino, Shogo; Shinohara, Toru
2016-01-01
In this paper we study the two-body bound states for gluons and ghosts in a massive Yang-Mills theory which is obtained by generalizing the ordinary massless Yang-Mills theory in a manifestly Lorentz covariant gauge. First, we give a systematic derivation of the coupled Bethe-Salpeter equations for gluons and ghosts by using the Cornwall-Jackiw-Tomboulis effective action of the composite operators within the framework of the path integral quantization. Then, we obtain the numerical solutions for the Bethe-Salpeter amplitude representing the simultaneous bound states of gluons and ghosts by solving the homogeneous Bethe-Salpeter equation in the ladder approximation. We study how the inclusion of ghosts affects the two-gluon bound states in the cases of the standing and running gauge coupling constant. Moreover, we show explicitly that the approximate solutions obtained for the gluon-gluon amplitude are consistent with the ultraviolet asymptotic freedom signaled by the negative $\\beta$ function.
Symmetry preserving truncations of the gap and Bethe-Salpeter equations
Binosi, Daniele; Chang, Lei; Papavassiliou, Joannis; Qin, Si-Xue; Roberts, Craig D.
2016-05-01
Ward-Green-Takahashi (WGT) identities play a crucial role in hadron physics, e.g. imposing stringent relationships between the kernels of the one- and two-body problems, which must be preserved in any veracious treatment of mesons as bound states. In this connection, one may view the dressed gluon-quark vertex, Γμa , as fundamental. We use a novel representation of Γμa , in terms of the gluon-quark scattering matrix, to develop a method capable of elucidating the unique quark-antiquark Bethe-Salpeter kernel, K , that is symmetry consistent with a given quark gap equation. A strength of the scheme is its ability to expose and capitalize on graphic symmetries within the kernels. This is displayed in an analysis that reveals the origin of H -diagrams in K , which are two-particle-irreducible contributions, generated as two-loop diagrams involving the three-gluon vertex, that cannot be absorbed as a dressing of Γμa in a Bethe-Salpeter kernel nor expressed as a member of the class of crossed-box diagrams. Thus, there are no general circumstances under which the WGT identities essential for a valid description of mesons can be preserved by a Bethe-Salpeter kernel obtained simply by dressing both gluon-quark vertices in a ladderlike truncation; and, moreover, adding any number of similarly dressed crossed-box diagrams cannot improve the situation.
Symmetry preserving truncations of the gap and Bethe-Salpeter equations
Binosi, Daniele; Chang, Lei; Papavassiliou, Joannis; Qin, Si-Xue; Roberts, Craig D.
2016-05-01
Ward-Green-Takahashi (WGT) identities play a crucial role in hadron physics, e.g. imposing stringent relationships between the kernels of the one-and two-body problems, which must be preserved in any veracious treatment of mesons as bound states. In this connection, one may view the dressed gluon-quark vertex, Gamma(alpha)(mu), as fundamental. We use a novel representation of Gamma(alpha)(mu), in terms of the gluon-quark scattering matrix, to develop a method capable of elucidating the unique quark-antiquark Bethe-Salpeter kernel, K, that is symmetry consistent with a given quark gap equation. A strength of the scheme is its ability to expose and capitalize on graphic symmetries within the kernels. This is displayed in an analysis that reveals the origin of H-diagrams in K, which are two-particle-irreducible contributions, generated as two-loop diagrams involving the three-gluon vertex, that cannot be absorbed as a dressing of Gamma(alpha)(mu) in a Bethe-Salpeter kernel nor expressed as a member of the class of crossed-box diagrams. Thus, there are no general circumstances under which the WGT identities essential for a valid description of mesons can be preserved by a Bethe-Salpeter kernel obtained simply by dressing both gluon-quark vertices in a ladderlike truncation; and, moreover, adding any number of similarly dressed crossed-box diagrams cannot improve the situation.
Advances in solving the two-fermion homogeneous Bethe-Salpeter equation in Minkowski space
de Paula, W; Salmè, G; Viviani, M
2016-01-01
Actual solutions of the Bethe-Salpeter equation for a two-fermion bound system are becoming available directly in Minkowski space, by virtue of a novel technique, based on the so-called Nakanishi integral representation of the Bethe-Salpeter amplitude and improved by expressing the relevant momenta through light-front components, i.e. $k^\\pm=k^0 \\pm k^3$. We solve a crucial problem that widens the applicability of the method to real situations by providing an analytically exact treatment of the singularities plaguing the two-fermion problem in Minkowski space, irrespective of the complexity of the irreducible Bethe-Salpeter kernel. This paves the way for feasible numerical investigations of relativistic composite systems, with any spin degrees of freedom. We present a thorough comparison with existing numerical results, evaluated in both Minkowski and Euclidean space, fully corroborating our analytical treatment, as well as fresh light-front amplitudes illustrating the potentiality of non perturbative calcula...
On string solutions of Bethe equations in N=4 supersymmetric Yang-Mills theory
The Bethe equations, arising in description of the spectrum of the dilatation operator for the su(2) sector of the N=4 supersymmetric Yang-Mills theory, are considered in the anti-ferromagnetic regime. These equations are deformation of those for the Heisenberg XXX magnet. It is proven that in the thermodynamic limit roots of the deformed equations group into strings. It is proven that the corresponding Yang's action is convex, which implies uniqueness of solution for centers of the strings. The state formed of strings of length (2n+1) is considered and the density of their distribution is found. It is shown that the energy of such a state decreases as n grows. It is observed that non-analyticity of the left hand side of the Bethe equations leads to an additional contribution to the density and energy of strings of even length. Whence it is concluded that the structure of the anti-ferromagnetic vacuum is determined by the behaviour of exponential corrections to string solutions in the thermodynamic limit and possibly involves strings of length 2. (orig.)
Norm of Bethe Wave Function as a Determinant
Korepin, Vladimir E
2009-01-01
This is a historical note. Bethe Ansatz solvable models are considered, for example XXZ Heisenberg anti-ferromagnet and Bose gas with delta interaction. Periodic boundary conditions lead to Bethe equation. The square of the norm of Bethe wave function is equal to a determinant of linearized system of Bethe equations (determinant of matrix of second derivatives of Yang action). The proof was first published in Communications in Mathematical Physics, vol 86, page 391 in l982. Also domain wall boundary conditions for 6 vertex model were discovered in the same paper [see Appendix D]. These play an important role for algebraic combinatorics: alternating sign matrices, domino tiling and plane partition. Many publications are devoted to six vertex model with domain wall boundary conditions.
Exact treatment of the Pauli exclusion operator in the nuclear matter Bethe-Goldstone equation
We solve the Bethe-Goldstone equation in nuclear matter for the scattering of a nucleon of a few hundred MeV. The angular momentum coupling caused by the nonspherical part of the Pauli blocking operator is treated exactly. It is found that standard approximation of angle averaging the Pauli operator is quite accurate for bulk properties of the reaction matrix even at energies as high as 300 MeV. Our result justifies one of the common approximations previously untested in the microscopic calculation of the optical model potential
Abe, Yuya
2016-01-01
We investigate a simple model using the numerical simulation in the complex Langevin equation (CLE) and the analytical approximation with the Gaussian Ansatz. We find that the Gaussian Ansatz captures the essential and even quantitative features of the CLE results quite well including unwanted behavior in the unstable region where the CLE converges to a wrong answer. The Gaussian Ansatz is therefore useful for looking into this convergence problem and we find that the exact answer in the unstable region is nicely reproduced by another solution that is naively excluded from the stability condition. We consider the Gaussian probability distributions corresponding to multiple solutions along the Lefschetz thimble to discuss the stability and the locality. Our results suggest a prescription to improve the convergence of the CLE simulation to the exact answer.
Solution to Bethe-Salpeter equation via Mellin-Barnes transform
Allendes, Pedro [Concepcion Univ. (Chile). Dept. de Fisica; Kniehl, Bernd [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik; Kondrashuk, Igor; Rojas Medar, Marko [Univ. del Bio-Bio, Chillan (Chile). Dept. de Ciencias Basicas; Notte Cuello, Eduardo A. [Univ. de La Serena (Chile). Facultad de Ciencias
2012-06-15
We consider Mellin-Barnes transform of triangle ladder-like scalar diagram in d=4 dimensions. It is shown how multi-fold MB transform of the momentum integral corresponding to any number of rungs is reduced to two-fold MB transform. For this purpose we use Belokurov-Usyukina reduction method for four-dimensional scalar integrals in the position space. The result is represented in terms of Euler {psi}-function and its derivatives. We derive new formulas for MB two-fold integration in the complex planes of two complex variables. We demonstrate that these formulas solve Bethe-Salpeter equation. We comment on further applications of solution to Bethe-Salpeter equation for vertices in N=4 supersymmetric Yang-Mills theory. We show that the recursive property of MB transforms observed in the present work for that kind of diagrams has nothing to do with quantum field theory, theory of integral transforms, or with theory of polylogarithms in general, but has an origin in a simple recursive property for smooth functions which can be shown by using basic methods of mathematical analysis.
Ground State Mass Spectrum for Scalar Diquarks with Bethe-Salpeter Equation
WANG Zhi-Gang; WAN Shao-Long; YANG Wei-Min
2007-01-01
In this article,we study the structures of the pseudoscalar mesons π,K and the scalar diquarks Ua,Da,Sa in the framework of the coupled rainbow Schwinger-Dyson equation and ladder Bethe-Salpeter equation with the confining effective potential.The u,d,s quarks have small current masses,and the renormalization is very large,the mass poles in the timelike region are absent which implements confinement naturally.The Bethe-Salpeter wavefunctions of the pseudoscalar mesons π,K,and the scalar diquarks Ua,Da,Sa have the same type (Gaussian type) momentum dependence,center around zero momentum and extend to the energy scale about q2 = 1 GeV2,which happens to be the energy scale for the chiral symmetry breaking,the strong interactions in the infrared region result in bound (or quasi-bound) states.The numerical results for the masses and decay constants of the π and K mesons can reproduce the experimental values,and the ground state masses of the scalar diquarks Ua,Da,Sa are consistent with the existing theoretical calculations.We suggest a new Lagrangian which may explain the uncertainty of the masses of the scalar diquarks.
Many-body-QED perturbation theory: Connection to the two-electron Bethe-Salpeter equation
Lindgren, I.; Salomonson, S.; Hedendahl, D.
2005-03-01
The connection between many-body perturbation theory (MBPT) and quantum electrodynamics (QED) is reviewed for systems of two fermions in an external field. The treatment is mainly based on the recently developed covariant-evolution-operator method for QED calculations (I. Lindgren, S. Salomonson, and B. Asen. Phys. Rep. 389, 161 (2004)), which is quite similar in structure to MBPT. At the same time, this procedure is closely related to the S-matrix and Green's-function formalisms and can therefore serve as a bridge connecting various approaches. It is demonstrated that the MBPT-QED scheme, when carried to all orders, leads to a Schrodinger-like equation, equivalent to the Bethe-Salpeter (BS) equation. A Bloch equation in commutator form that can be used for an "extended" or quasi-degenerate model space is derived. This is a multi-state equation that has the same relation to the single-state BS equation as the standard Bloch equation has to the ordinary Schrodinger equation. It can be used to generate a perturbation expansion compatible with the BS equation even in the case of a quasi-degenerate model space.
Intriguin solutions of Bethe-Salpeter equation for radially excited pseudoscalar charmonia
Sauli, Vladimir
2012-01-01
When generalizing recent various quantum mechanical models of heavy quarkonia to Quantum Filed theoretical approach based on Bethe-Salpeter equation one is faced to the solutions that do not exist in nonrelativistic limit. Mainly, there is unexpected doubling of the spectrum when comparing to the experimentally known spectrum as well as the ones obtained from the solution of the Schroedinger equation. These additional states are not apriory unphysical as both of them have the same symmetry. Our study strongly suggests that these solutions appear due to the sensitivity of BSE to the details of the analytical form of the constituents quark and antiquark propagators, more specifically they are consequence of using unconfining free propagators. To show this explicitly we develop and describe the efficient method of the numerical solution for quarkonium BSE and numerically solve it for the case of pseudoscalar charmonia. For the bare propagators of constituents we are able to find BSE solution for arbitrarily high...
Symmetry preserving truncations of the gap and Bethe-Salpeter equations
Binosi, Daniele; Papavassiliou, Joannis; Qin, Si-Xue; Roberts, Craig D
2016-01-01
Ward-Green-Takahashi (WGT) identities play a crucial role in hadron physics, e.g. imposing stringent relationships between the kernels of the one- and two-body problems, which must be preserved in any veracious treatment of mesons as bound-states. In this connection, one may view the dressed gluon-quark vertex, $\\Gamma_\\mu^a$, as fundamental. We use a novel representation of $\\Gamma_\\mu^a$, in terms of the gluon-quark scattering matrix, to develop a method capable of elucidating the unique quark-antiquark Bethe-Salpeter kernel, $K$, that is symmetry-consistent with a given quark gap equation. A strength of the scheme is its ability to expose and capitalise on graphic symmetries within the kernels. This is displayed in an analysis that reveals the origin of $H$-diagrams in $K$, which are two-particle-irreducible contributions, generated as two-loop diagrams involving the three-gluon vertex, that cannot be absorbed as a dressing of $\\Gamma_\\mu^a$ in a Bethe-Salpeter kernel nor expressed as a member of the class...
Bethe-Salpeter equation for non-self conjugate mesons in a power-law potential
We develop an approach to the solution of the spinless Bethe-Salpeter equation for the different-mass case. Although the calculations are developed for spin-zero particles in any arbitrary spherically symmetric potential, the non-Coulombic effective power-law potential is used as a kernel to produce the spin-averaged bound states of the non-self-conjugate mesons. The analytical formulae are also applicable to the self-conjugate mesons in the equal-mass case. The flavor-independent case is investigated in this work. The calculations are carried out to the third-order correction of the energy series. Results are consistent with those obtained before. (author). 14 refs, 1 tab
The Bethe-Salpeter equation is solved in closed form with the help of a four dimensional separable 'potential'. For possible applications to three-nucleon investigations the authors have fitted all nucleon-nucleon S-wave phase shifts in a sufficient way by this method; in addition they also present an example for a P-wave. (Auth.)
Gadjiev, S A
2001-01-01
Scattering amplitude of fermions and bosons in the ladder approximation at high energies is investigated. For the imaginary part of the scattering amplitude the set of Bethe-Salpeter type integral equations is constructed. Solutions of this set in the Regge asymptotic form are found. The impact of mass parameters on the behavior of the amplitude at high energies is studied.
Solving the inhomogeneous Bethe-Salpeter equation in Minkowski space: the zero-energy limit
Frederico, Tobias; Salmè, Giovanni; Viviani, Michele
2015-08-01
The inhomogeneous Bethe-Salpeter equation for an interacting system, composed of two massive scalars exchanging a massive scalar, is numerically investigated in the ladder approximation directly in Minkowski space, by using for the first time in the continuum an approach based on the Nakanishi integral representation. In this paper, the limiting case of zero-energy states is considered, thus extending an approach that has already been successfully applied to bound states. The numerical values of scattering lengths, are calculated for several values of the Yukawa coupling constant, by using two different integral equations that stem from the Nakanishi framework. Those low-energy observables are compared with (1) the analogous quantities recently obtained in literature, within a totally different framework, and (2) the non-relativistic evaluations, to illustrate the relevance of a nonperturbative, genuine field theoretical treatment in Minkowski space, even in the low-energy regime. Moreover, dynamical functions, like the Nakanishi weight functions and the distorted part of the zero-energy light-front wave functions are also presented. Interestingly, a highly non-trivial issue related to the abrupt change in the width of the support of the Nakanishi weight function, when the zero-energy limit is approached, is elucidated, ensuring a sound basis to the forthcoming evaluation of phase shifts.
Solving the inhomogeneous Bethe-Salpeter equation in Minkowski space: the zero-energy limit
Frederico, Tobias [Instituto Tecnologico de Aeronautica, DCTA, Dept. de Fisica, Sao Paulo (Brazil); Salme, Giovanni [Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Roma (Italy); Viviani, Michele [Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Pisa (Italy)
2015-08-15
The inhomogeneous Bethe-Salpeter equation for an interacting system, composed of two massive scalars exchanging a massive scalar, is numerically investigated in the ladder approximation directly in Minkowski space, by using for the first time in the continuum an approach based on the Nakanishi integral representation. In this paper, the limiting case of zero-energy states is considered, thus extending an approach that has already been successfully applied to bound states. The numerical values of scattering lengths, are calculated for several values of the Yukawa coupling constant, by using two different integral equations that stem from the Nakanishi framework. Those low-energy observables are compared with (1) the analogous quantities recently obtained in literature, within a totally different framework, and (2) the non-relativistic evaluations, to illustrate the relevance of a nonperturbative, genuine field theoretical treatment in Minkowski space, even in the low-energy regime. Moreover, dynamical functions, like the Nakanishi weight functions and the distorted part of the zero-energy light-front wave functions are also presented. Interestingly, a highly non-trivial issue related to the abrupt change in the width of the support of the Nakanishi weight function, when the zero-energy limit is approached, is elucidated, ensuring a sound basis to the forthcoming evaluation of phase shifts. (orig.)
Radiative decay widths are calculated for the radiative decay processes observed experimentally in the charmonium system. The model uses a Bethe-Salpeter equation with a static kernel and harmonic oscillator potentials to model the c-anti c system. Each decay width is calculated for 21 different choices of the c-quark mass. The potential used was a linear combination of a vector coupled and a scalar coupled harmonic oscillator potential. The quark mass and the scalar to vector coupling ratio were determined by trying to fit simultaneously the psi'(3685) - psi(3095) mass difference, the psi(3095) → e+ + e-decay width and the 3P/sub J/ mass splittings. A single choice of the quark mass and scalar to vector coupling ratio could not simultaneously fit all these constraints. The best fit to these constraints occurred when the quark mass was 5.5 and the scalar to vector coupling ratio parameter was -0.16. The decay width calculations are shown graphically for values of the quark mass from 0.00 to 16 GeV. The decay widths were calculated two different ways: (1) using the matrix elements of the quark momentum; (2) using the matrix elements of the quark position. Most of the published calculations use method (2). The widths computed by methods (1) and (2) are quite different for all masses and all transitions implying that the usual method (2) give incorrect results, and the fits with experimental data are fortuitous
Non-regular eigenstate of the XXX model as some limit of the Bethe state
For the one-dimensional XXX model under the periodic boundary conditions, we discuss two types of eigenvectors, regular eigenvectors which have finite-valued rapidities satisfying the Bethe ansatz equations and non-regular eigenvectors which are descendants of some regular eigenvectors under the action of the SU(2) spin-lowering operator. It has been pointed out by many authors that the non-regular eigenvectors should correspond to the Bethe ansatz wavefunctions which have multiple infinite rapidities. However, it has not been explicitly shown whether such a delicate limiting procedure is possible. In this paper, we discuss it explicitly at the level of wavefunctions: we prove that any non-regular eigenvector of the XXX model is derived from the Bethe ansatz wavefunctions through some limit of infinite rapidities. We formulate the regularization also in terms of the algebraic Bethe ansatz method. As an application of infinite rapidity, we discuss the period of the spectral flow under the twisted periodic boundary conditions. (author)
无
2005-01-01
The interaction kernel in the Bethe-Salpeter equation for quark-antiquark bound states is derived newly from QCD in the case where the quark and the antiquark are of different flavors. The technique of the derivation is the usage of the irreducible decomposition of the Green's functions involved in the Bethe-Salpeter equation satisfied by the quark-antiquark four-point Green's function. The interaction kernel derived is given a closed and explicit expression which shows a specific structure of the kernel since the kernel is represented in terms of the quark, antiquark and gluon propagators and some kinds of quark, antiquark and/or gluon three, four, five and six-point vertices. Therefore,the expression of the kernel is not only convenient for perturbative calculations, but also suitable for nonperturbative investigations.
Multi-Regge limit of the n-gluon bubble ansatz
Bartels, J. [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik; Schomerus, V.; Sprenger, M. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
2012-07-15
We investigate n-gluon scattering amplitudes in the multi-Regge region of N=4 supersymmetric Yang-Mills theory at strong coupling. Through a careful analysis of the thermodynamic bubble ansatz (TBA) for surfaces in AdS{sub 5} with n-g(lu)on boundary conditions we demonstrate that the multi-Regge limit probes the large volume regime of the TBA. In reaching the multi-Regge regime we encounter wall-crossing in the TBA for all n>6. Our results imply that there exists an auxiliary system of algebraic Bethe ansatz equations which encode valuable information on the analytical structure of amplitudes at strong coupling.
Osano, Bob
2016-01-01
We present an ansatz for the relationship between magnetic flux density and fluid vorticity evolution equations. We also suggest that the magnetic flux density evolution equations be compared to the evolution equation for an effective vorticity ($\\omega_{eff}$), which bears a power law relation to the ordinary vorticity.
Bethe vectors for XXX-spin chain
The paper deals with algebraic Bethe ansatz for XXX-spin chain. Generators of Yang-Baxter algebra are expressed in basis of free fermions and used to calculate explicit form of Bethe vectors. Their relation to N-component models is used to prove conjecture about their form in general. Some remarks on inhomogeneous XXX-spin chain are included
Bethe vectors for XXX-spin chain
Burdík, Čestmír; Fuksa, Jan; Isaev, Alexei
2014-11-01
The paper deals with algebraic Bethe ansatz for XXX-spin chain. Generators of Yang-Baxter algebra are expressed in basis of free fermions and used to calculate explicit form of Bethe vectors. Their relation to N-component models is used to prove conjecture about their form in general. Some remarks on inhomogeneous XXX-spin chain are included.
Bethe vectors of GL(3)-invariant integrable models
We study GL(3)-invariant integrable models solvable by the nested algebraic Bethe ansatz. Different formulas are given for the Bethe vectors and the actions of the generators of the Yangian Y(gl3) on the Bethe vectors are considered. These actions are relevant for the calculation of correlation functions and form factors of local operators of the underlying quantum models. (paper)
Hadronic Observables from Dyson-Schwinger and Bethe-Salpeter equations
Sanchis-Alepuz, Helios
2015-01-01
In these proceedings we present a mini-review on the topic of the Dyson-Schwinger/Bethe-Salpeter approach to the study of relativistic bound-states in physics. In particular, we present a self-contained discussion of their derivation, as well as their truncation such that important symmetries are maintained.
On the history of a stochastic ansatz for solving the transport equation
A very useful approximate tool for understanding the role of random material properties on solutions of the transport equation is described and its historical derivation given. The development of this stochastic tool, from its introduction by Randall, to its use in describing current problems involving dichotomic or pseudo-dichotomic Markov processes is discussed.
Gaudin, Michel
2014-01-01
Michel Gaudin's book La fonction d'onde de Bethe is a uniquely influential masterpiece on exactly solvable models of quantum mechanics and statistical physics. Available in English for the first time, this translation brings his classic work to a new generation of graduate students and researchers in physics. It presents a mixture of mathematics interspersed with powerful physical intuition, retaining the author's unmistakably honest tone. The book begins with the Heisenberg spin chain, starting from the coordinate Bethe Ansatz and culminating in a discussion of its thermodynamic properties. Delta-interacting bosons (the Lieb-Liniger model) are then explored, and extended to exactly solvable models associated to a reflection group. After discussing the continuum limit of spin chains, the book covers six- and eight-vertex models in extensive detail, from their lattice definition to their thermodynamics. Later chapters examine advanced topics such as multi-component delta-interacting systems, Gaudin magnets and...
Mass of Y(3940) in Bethe-Salpeter equation for quarks
Chen, Xiaozhao [Shandong University of Science and Technology, Department of Foundational Courses, Taian (China); Lue, Xiaofu [Sichuan University, Department of Physics, Chengdu (China); The Chinese Academy of Sciences, Institute of Theoretical Physics, Beijing (China); CCAST (World Laboratory), Beijing (China)
2015-03-01
The general form of the Bethe-Salpeter wave functions for the bound states composed of two vector fields of arbitrary spin and definite parity is corrected. Using the revised general formalism, we investigate the observed Y(3940) state, which is considered as a molecule state consisting of D{sup *0} anti D{sup *0}. Though the attractive potential between D{sup *0} and anti D{sup *0} including one light meson (σ, π, ω, ρ) exchange is considered, we find that in our approach the contribution from one-π exchange is equal to zero and consider SU(3) symmetry breaking. The obtained mass of Y(3940) is consistent with the experimental value. (orig.)
In this article, we investigate the structures of the pseudoscalar mesons (π, K, D, Ds, B and Bs) in the framework of the coupled rainbow Schwinger-Dyson equation and ladder Bethe-Salpeter equation with the confining effective potential (infrared modified flat bottom potential). The Schwinger-Dyson functions for the u, d and s quarks are greatly renormalized at small momentum region and the curves are steep at about q2=1 GeV2 which indicates an explicitly dynamical symmetry breaking. The Euclidean time Fourier transformed quark propagators have no mass poles in the time-like region which naturally implements confinement. As for the c and b quarks, the current masses are very large, the renormalization are more tender, however, mass poles in the time-like region are also absent. The Bethe-Salpeter wavefunctions for those mesons have the same type (Gaussian type) momentum dependence and center around small momentum which indicate that the bound states exist in the infrared region. The decay constants for those pseudoscalar mesons are compatible with the values of experimental extractions and theoretical calculations, such as lattice simulations and QCD sum rules
Gamma Matrix Expansion of the Bethe-Salpeter Equation for Nucleon-Nucleon System
Kinpara, Susumu
2016-01-01
For the coefficients of the amplitude a set of simultaneous equations is derived in momentum space. By the auxiliary conditions they are equivalent to nonrelativistic equations and suitable for the investigation of two-nucleon system.
In this paper the general structure of leptonic decay constants of vector mesons is evaluated in the framework of Bethe-Salpeter Equation under Covariant Instantaneous Ansatz (CIA) with a modified structure of the Hqq-bar vertex function Γ which is generalized to include Dirac covariants other than the leading Dirac covariant γμ within its structure. The numerical values of fv in this CIA framework are calculated. (author)
Gaind, Vaibhav
Fluorescence resonance energy transfer (FRET) has found many applications in in vitro imaging as an indicator of molecular activity. However, till now, in vivo FRET imaging has been restricted to near-surface multiphoton microscopy. Optical diffusion tomography (ODT) is an emerging tool for deep tissue imaging. In this work, FRET was incorporated in an ODT framework, thereby allowing FRET to be applied in deep tissue imaging. Using simulations and tissue phantom and small animal imaging experiments, the possibility of imaging molecular activity on the nanometer scale using macroscopic measurements is demonstrated. The diffusion equation model is limited to regions of high scatter and low absorption. The Bethe-Salpeter equation has been used extensively to explain various scattering phenomena and is more fundamental than the Boltzmann transport equation. In this work, the Bethe-Salpeter equation has been investigated for modeling photon transport in the non-diffusive regime.
We realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schrödinger equation with delta-potential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schrödinger equation in terms of generalized Bethe wave functions. It is shown that the generalized Bethe wave functions satisfy affine difference Knizhnik-Zamolodchikov equations as functions of the momenta. The relation to the vector valued root system analogs of the quantum Bose gas on the circle with delta-function interactions is indicated.
Meson states from the Bethe-Salpeter equation: successes and challenges
Full text: Dyson-Schwinger equations provide a nonperturbative continuum approach to QCD. The infinite coupled system of integral equations is truncated in a symmetry preserving manner to allow for both proof of exact results as well as sophisticated model calculations to illustrate these results and to perform qualitative as well as quantitative studies of hadronic observables. Over the past years a lot of investigations have used the so-called rainbow-ladder truncation. I will report the successes and ongoing progress within this truncation and demonstrate the need for a 'step beyond' with the help of examples taken from meson physics. (author)
Bruneval, Fabien; Hamed, Samia M.; Neaton, Jeffrey B.
2015-01-01
The predictive power of the ab initio Bethe-Salpeter equation (BSE) approach, rigorously based on many-body Green's function theory but incorporating information from density functional theory, has already been demonstrated for the optical gaps and spectra of solid-state systems. Interest in photoactive hybrid organic/inorganic systems has recently increased, and so has the use of the BSE for computing neutral excitations of organic molecules. However, no systematic benchmarks of the BSE for ...
On the Dressing Factors, Bethe Equations and Yangian Symmetry of Strings on AdS3 x S3 x T4
Borsato, Riccardo; Sfondrini, Alessandro; Stefanski, Bogdan; Torrielli, Alessandro
2016-01-01
Integrability is believed to underlie the AdS3/CFT2 correspondence with sixteen supercharges. We elucidate the role of massless modes within this integrable framework. Firstly, we find the dressing factors that enter the massless and mixed-mass worldsheet S matrix. Secondly, we derive a set of all-loop Bethe Equations for the closed strings, determine their symmetries and weak-coupling limit. Thirdly, we investigate the underlying Yangian symmetry in the massless sector and show that it fits into the general framework of Yangian integrability. In addition, we compare our S matrix in the near-relativistic limit with recent perturbative worldsheet calculations of Sundin and Wulff.
The method of spherical harmonics as an ansatz for the solution of non-linear Boltzmann equations
A new coordinate-free representation of the differential scattering probability function of the binary self-collision leads to a scattering kernel which is particularly appropriate for the expansion in Legendre polynomials. Thus, the non-linear transport equation can be treated using the spherical harmonics method. Assuming the scattering in the centre-of-mass system to be isotropic, the non-linear moment equations of the particle distribution function are derived. (orig.)
A series of simple stopping power (SP) formulas, modified from the relativistic Bethe equation, is presented that is based on the concepts of target effective atomic number and mean excitation energy (MEE). The analytical model function is constructed to approximate experimental or calculated SPs at low electron energies and tend asymptotically to the relativistic Bethe function at high energies. The energy dependencies of our effective values, in contrast with theoretical approaches, are defined empirically by parametrization with tuning parameters. A least-squares fitting routine based on the Levenberg–Marquardt algorithm was developed. We utilize the material parameters and numerical calculations of SPs from optical data using the full Penn-algorithm. Our formula is thought to be applicable for energies above 60 eV. Our simulations of SPs for 41 elemental solids are found to be in good agreement with published numerical results. The flexibility of a general empirical formula is shown. Shortened formulas were developed that are applicable for particular energy ranges, and effective MEEs are proposed that differ from previously recommended values. The presented formulas may be used for analytical calculation of SPs over a broad projectile energy region
Dirac Equation and Some Quasi-Exact Solvable Potentials in the Turbiner's Classification
Aghaei, S.; Chenaghlou, A.
2013-09-01
In this paper quasi-exact solvability (QES) of Dirac equation with some scalar potentials based on sl(2) Lie algebra is studied. According to the quasi-exact solvability theory, we construct the configuration of the classes II, IV, V, and X potentials in the Turbiner's classification such that the Dirac equation with scalar potential is quasi-exactly solved and the Bethe ansatz equations are derived in order to obtain the energy eigenvalues and eigenfunctions.
Dirac Equation and Some Quasi-Exact Solvable Potentials in the Turbiner's Classification
In this paper quasi-exact solvability (QES) of Dirac equation with some scalar potentials based on sl(2) Lie algebra is studied. According to the quasi-exact solvability theory, we construct the configuration of the classes II, IV, V, and X potentials in the Turbiner's classification such that the Dirac equation with scalar potential is quasi-exactly solved and the Bethe ansatz equations are derived in order to obtain the energy eigenvalues and eigenfunctions. (general)
Bethe Vectors of Quantum Integrable Models with GL(3 Trigonometric R-Matrix
Samuel Belliard
2013-10-01
Full Text Available We study quantum integrable models with GL(3 trigonometric $R$-matrix and solvable by the nested algebraic Bethe ansatz.Using the presentation of the universal Bethe vectors in terms of projections of products of the currents of the quantum affine algebra $U_q(widehat{mathfrak{gl}}_3$ onto intersections of different types of Borel subalgebras, we prove that the set of the nested Bethe vectors is closed under the action of the elements of the monodromymatrix.
Multireference Coupled Cluster Ansatz
Jeziorski, Bogumil
2010-01-01
Abstract The origin of the multireference coupled cluster Ansatz for the wave function and the wave operator, discovered in Quantum Theory Project in 1981, is presented from the historical perspective. Various methods of obtaining the cluster amplitudes - both state universal and state selective are critically reviewed and further prospects of using the multireference coupled cluster Ansatz in electronic structure theory are briefly discussed.
π- and K-meson Bethe-Salpeter amplitudes
Independent of assumptions about the form of the quark-quark scattering kernel K, we derive the explicit relation between the flavor-nonsinglet pseudoscalar-meson Bethe-Salpeter amplitude ΓH and the dressed-quark propagator in the chiral limit. In addition to a term proportional to γ5, ΓH necessarily contains qualitatively and quantitatively important terms proportional to γ5γ·P and γ5γ·kk·P, where P is the total momentum of the bound state. The axial-vector vertex contains a bound state pole described by ΓH, whose residue is the leptonic decay constant for the bound state. The pseudoscalar vertex also contains such a bound state pole and, in the chiral limit, the residue of this pole is related to the vacuum quark condensate. The axial-vector Ward-Takahashi identity relates these pole residues, with the Gell-Mann endash Oakes endash Renner relation a corollary of this identity. The dominant ultraviolet asymptotic behavior of the scalar functions in the meson Bethe-Salpeter amplitude is fully determined by the behavior of the chiral limit quark mass function, and is characteristic of the QCD renormalization group. The rainbow-ladder Ansatz for K, with a simple model for the dressed-quark-quark interaction, is used to illustrate and elucidate these general results. The model preserves the one-loop renormalization group structure of QCD. The numerical studies also provide a means of exploring procedures for solving the Bethe-Salpeter equation without a three-dimensional reduction. copyright 1997 The American Physical Society
Algebraization of difference eigenvalue equations related to $U_q(sl_2)$
Wiegmann, P. B.; Zabrodin, A. V.
1995-01-01
A class of second order difference (discrete) operators with a partial algebraization of the spectrum is introduced. The eigenfuncions of the algebraized part of the spectrum are polinomials (discrete polinomials). Such difference operators can be constructed by means of $U_q(sl_2)$, the quantum deformation of the $sl_2$ algebra. The roots of polinomials determine the spectrum and obey the Bethe Ansatz equations. A particular case of difference equations for $q$-hypergeometric and Askey-Wilso...
Kp and Toda Tau Functions in Bethe Ansatz
Takasaki, Kanehisa
2011-10-01
Recent work of Foda and his group on a connection between classical integrable hierarchies (the KP and 2D Toda hierarchies) and some quantum integrable systems (the 6-vertex model with DWBC, the finite XXZ chain of spin 1/2, the phase model on a finite chain, etc.) is reviewed. Some additional information on this issue is also presented.
KP and Toda tau functions in Bethe ansatz
Takasaki, Kanehisa
2010-01-01
Recent work of Foda and his group on a connection between classical integrable hierarchies (the KP and 2D Toda hierarchies) and some quantum integrable systems (the 6-vertex model with DWBC, the finite XXZ chain of spin 1/2, the phase model on a finite chain, etc.) is reviewed. Some additional information on this issue is also presented.
Quasi-exactly solvable Fokker-Planck equations
Ho, Choon-Lin; Sasaki, Ryu
2007-01-01
We consider exact and quasi-exact solvability of the one-dimensional Fokker-Planck equation based on the connection between the Fokker-Planck equation and the Schr\\"odinger equation. A unified consideration of these two types of solvability is given from the viewpoint of prepotential together with Bethe ansatz equations. Quasi-exactly solvable Fokker-Planck equations related to the $sl(2)$-based systems in Turbiner's classification are listed. We also present one $sl(2)$-based example which i...
Hutsalyuk, A; Pakuliak, S Z; Ragoucy, E; Slavnov, N A
2016-01-01
We study scalar products of Bethe vectors in integrable models solvable by nested algebraic Bethe ansatz and possessing $\\mathfrak{gl}(2|1)$ symmetry. Using explicit formulas of the monodromy matrix entries multiple actions onto Bethe vectors we obtain a representation for the scalar product in the most general case. This explicit representation appears to be a sum over partitions of the Bethe parameters. It can be used for the analysis of scalar products involving on-shell Bethe vectors. As a by-product, we obtain a determinant representation for the scalar products of generic Bethe vectors in integrable models with $\\mathfrak{gl}(1|1)$ symmetry.
Gomes, Adriano Doff Sotta [Universidade Tecnologica Federal do Parana (UTFPR), Pato Branco, PR (Brazil)
2011-07-01
Full text: Scalar composite boson masses have been computed in QCD and Technicolor theories with the help of the homogeneous Bethe-Salpeter equation (BSE), resulting in a scalar mass that is twice the dynamically generated fermion or technifermion mass (m{sub dyn}). In the A. Doff, A. A. Natale and P. S. Rodrigues da Silva, Phys. Rev. D 80, 055005 (2009) we study the effect of the normalization condition on the determination of scalar boson masses in dynamically broken gauge theories and verify that the normalization condition does not modify the value of the scalar boson mass when its wave function has the asymptotic behavior exactly as predicted by the OPE. However in walking (or quasi-conformal) gauge theories the asymptotic behavior of fermionic self-energies and the wave function of scalar bound states are dominated by higher order interactions and are characterized by a much harder decrease with the momentum, therefore, in this case, we show that the normalization condition of the BSE do constrain the scalar masses. In this work we apply some results obtained in the cited reference to the model described in A. Doff, Phys. Rev. D 81, 117702 (2010), in particular we compute the Higgs boson masses generated in the model assuming the effects of mixing in the wave function of scalar bound states due to the U(1){sub x} interaction of U' and D' techniquarks. (author)
The predictive power of the ab initio Bethe-Salpeter equation (BSE) approach, rigorously based on many-body Green’s function theory but incorporating information from density functional theory, has already been demonstrated for the optical gaps and spectra of solid-state systems. Interest in photoactive hybrid organic/inorganic systems has recently increased and so has the use of the BSE for computing neutral excitations of organic molecules. However, no systematic benchmarks of the BSE for neutral electronic excitations of organic molecules exist. Here, we study the performance of the BSE for the 28 small molecules in Thiel’s widely used time-dependent density functional theory benchmark set [Schreiber et al., J. Chem. Phys. 128, 134110 (2008)]. We observe that the BSE produces results that depend critically on the mean-field starting point employed in the perturbative approach. We find that this starting point dependence is mainly introduced through the quasiparticle energies obtained at the intermediate GW step and that with a judicious choice of starting mean-field, singlet excitation energies obtained from BSE are in excellent quantitative agreement with higher-level wavefunction methods. The quality of the triplet excitations is slightly less satisfactory
Malik, G P
2016-01-01
Given the Debye temperature of an elemental superconductor (SC) and its Tc, BCS theory enables one to predict the value of its gap 0 at T = 0, or vice versa. This monograph shows that non-elemental SCs can be similarly dealt with via the generalized BCS equations (GBCSEs) which, given any two parameters of the set {Tc, 10, 20 > 10}, enable one to predict the third. Also given herein are new equations for the critical magnetic field and critical current density of an elemental and a non-elemental SC — equations that are derived directly from those that govern pairing in them. The monograph includes topics that are usually not covered in any one text on superconductivity, e.g., BCS-BEC crossover physics, the long-standing puzzle posed by SrTiO3, and heavy-fermion superconductors — all of which are still imperfectly understood and therefore continue to avidly engage theoreticians. It suggests that addressing the Tcs, s and other properties (e.g., number densities of charge carriers) of high-Tc SCs via GBCSE...
The interplay of hadron properties and their modification in an ambient nuclear medium on the one hand and spontaneous chiral symmetry breaking and its restoration on the other hand is investigated. QCD sum rules for D and B mesons embedded in cold nuclear matter are evaluated. We quantify the mass splitting of D- anti D and B- anti B mesons as a function of the nuclear matter density and investigate the impact of various condensates in linear density approximation. The analysis also includes Ds and D*0 mesons. QCD sum rules for chiral partners in the open-charm meson sector are presented at nonzero baryon net density or temperature. We focus on the differences between pseudo-scalar and scalar as well as vector and axial-vector D mesons and derive the corresponding Weinberg type sum rules. Based on QCD sum rules we explore the consequences of a scenario for the ρ meson, where the chiral symmetry breaking condensates are set to zero whereas the chirally symmetric condensates remain at their vacuum values. The complementarity of mass shift and broadening is discussed. An alternative approach which utilizes coupled Dyson-Schwinger and Bethe-Salpeter equations for quark-antiquark bound states is investigated. For this purpose we analyze the analytic structure of the quark propagators in the complex plane numerically and test the possibility to widen the applicability of the method to the sector of heavy-light mesons in the scalar and pseudo-scalar channels, such as the D mesons, by varying the momentum partitioning parameter. The solutions of the Dyson-Schwinger equation in the Wigner-Weyl phase of chiral symmetry at nonzero bare quark masses are used to investigate a scenario with explicit but without dynamical chiral symmetry breaking.
Hilger, Thomas Uwe
2012-04-11
The interplay of hadron properties and their modification in an ambient nuclear medium on the one hand and spontaneous chiral symmetry breaking and its restoration on the other hand is investigated. QCD sum rules for D and B mesons embedded in cold nuclear matter are evaluated. We quantify the mass splitting of D- anti D and B- anti B mesons as a function of the nuclear matter density and investigate the impact of various condensates in linear density approximation. The analysis also includes D{sub s} and D{sup *}{sub 0} mesons. QCD sum rules for chiral partners in the open-charm meson sector are presented at nonzero baryon net density or temperature. We focus on the differences between pseudo-scalar and scalar as well as vector and axial-vector D mesons and derive the corresponding Weinberg type sum rules. Based on QCD sum rules we explore the consequences of a scenario for the ρ meson, where the chiral symmetry breaking condensates are set to zero whereas the chirally symmetric condensates remain at their vacuum values. The complementarity of mass shift and broadening is discussed. An alternative approach which utilizes coupled Dyson-Schwinger and Bethe-Salpeter equations for quark-antiquark bound states is investigated. For this purpose we analyze the analytic structure of the quark propagators in the complex plane numerically and test the possibility to widen the applicability of the method to the sector of heavy-light mesons in the scalar and pseudo-scalar channels, such as the D mesons, by varying the momentum partitioning parameter. The solutions of the Dyson-Schwinger equation in the Wigner-Weyl phase of chiral symmetry at nonzero bare quark masses are used to investigate a scenario with explicit but without dynamical chiral symmetry breaking.
Bethe subalgebras in affine Birman–Murakami–Wenzl algebras and flat connections for q-KZ equations
Isaev, A. P.; Kirillov, A. N.; Tarasov, V. O.
2016-05-01
Commutative sets of Jucys–Murphy elements for affine braid groups of {A}(1),{B}(1),{C}(1),{D}(1) types were defined. Construction of R-matrix representations of the affine braid group of type {C}(1) and its distinguished commutative subgroup generated by the {C}(1)-type Jucys–Murphy elements are given. We describe a general method to produce flat connections for the two-boundary quantum Knizhnik–Zamolodchikov equations as necessary conditions for Sklyanin's type transfer matrix associated with the two-boundary multicomponent Zamolodchikov algebra to be invariant under the action of the {C}(1)-type Jucys–Murphy elements. We specify our general construction to the case of the Birman–Murakami–Wenzl algebras (BMW algebras for short). As an application we suggest a baxterization of the Dunkl–Cherednik elements {Y}\\prime {{s}} in the double affine Hecke algebra of type A. Dedicated to Professor Rodney Baxter on the occasion of his 75th Birthday.
Finite-gap equations for strings on AdS_3 x S^3 x T^4 with mixed 3-form flux
Babichenko, Andrei; Sax, Olof Ohlsson
2014-01-01
We study superstrings on AdS_3 x S^3 x T^4 supported by a combination of Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz three form fluxes, and construct a set of finite-gap equations that describe the classical string spectrum. Using the recently proposed all-loop S-matrix we write down the all-loop Bethe ansatz equations for the massive sector. In the thermodynamic limit the Bethe ansatz reproduces the finite-gap equations. As part of this derivation we propose expressions for the leading order dressing phases. These phases differ from the well-known Arutyunov-Frolov-Staudacher phase that appears in the pure Ramond-Ramond case. We also consider the one-loop quantization of the algebraic curve and determine the one-loop corrections to the dressing phases. Finally we consider some classical string solutions including finite size giant magnons and circular strings.
Bethe vectors of quantum integrable models based on Uq( gl-hat N)
We study quantum Uq( gl-hat N) integrable models solvable by the nested algebraic Bethe ansatz. Different formulas are given for the right and left universal off-shell nested Bethe vectors. It is shown that these formulas can be related by certain morphisms of the positive Borel subalgebra in Uq( gl-hat N) into analogous subalgebra in Uq−1( gl-hat N). (paper)
We proposed the algorithm for umerical solving a boundary value problem for two-quark bound states described by the Salpeter equation with potential V0r2-αs/r which is coupled integro-differential equations depending on physical parameters m0 and αs. In this algorithm an iteration scheme of the continuous analogy of Newton's method, with corresponding choice of the iteration parameter, is realized. It is shown that using the continuation over parameter (m0, αs) method allows one to extend considerably a region of convergence of the iteration method. The solutions of the Salpeter equation for set of parameters m0 and αs are obtained, which reproduce the results are available (when m0=αs=0). 17 refs.; 1 fig.; 2 tabs
CEval-Ansatz zur Wirkungsevaluation / Stockmann'scher Ansatz
Silvestrini, Stefan; Reade, Nicolà
2008-01-01
Der im Rahmen zahlreicher Evaluationsstudien erprobte Ansatz zur Wirkungsevaluation von Projekten und Programmen wurde von Reinhard Stockmann am Centrum für Evaluation (CEval) entwickelt. Der Ansatz, der hier in seiner Grundstruktur vorgestellt wird, ist so gestaltet, dass er flexibel an verschiedenartige Themenfelder sowie an jeweilige organisatorische oder institutionelle Rahmenbedingungen angepasst werden kann. Aufgrund seiner Flexibilität eignet sich der Stockmann'sche Ansatz zur Wirkungs...
On the central quadric ansatz: integrable models and Painleve reductions
Ferapontov, E V; Zhang, A
2012-01-01
It was observed by Tod and later by Dunajski and Tod that the Boyer-Finley (BF) and the dispersionless Kadomtsev-Petviashvili (dKP) equations possess solutions whose level surfaces are central quadrics in the space of independent variables (the so-called central quadric ansatz). It was demonstrated that generic solutions of this type are described by Painleve equations PIII and PII, respectively. The aim of our paper is threefold: -- Based on the method of hydrodynamic reductions, we classify integrable models possessing the central quadric ansatz. This leads to the five canonical forms (including BF and dKP). -- Applying the central quadric ansatz to each of the five canonical forms, we obtain all Painleve equations PI - PVI, with PVI corresponding to the generic case of our classification. -- We argue that solutions coming from the central quadric ansatz constitute a subclass of two-phase solutions provided by the method of hydrodynamic reductions.
Quantisation of Kadomtsev-Petviashvili equation
Kozlowski, Karol K; Torrielli, Alessandro
2016-01-01
A quantisation of the KP equation on a cylinder is proposed that is equivalent to an infinite system of non-relativistic one-dimensional bosons carrying masses $m=1,2,\\ldots$ The Hamiltonian is Galilei-invariant and includes the split $\\Psi^\\dagger_{m_1}\\Psi^\\dagger_{m_2}\\Psi_{m_1+m_2}$ and merge $\\Psi^\\dagger_{m_1+m_2}\\Psi_{m_1}\\Psi_{m_2}$ terms for all combinations of particles with masses $m_1$, $m_2$ and $m_1+m_2$, with a special choice of coupling constants. The Bethe eigenfunctions for the model are constructed. The consistency of the coordinate Bethe Ansatz, and therefore, the quantum integrability of the model is verified up to the mass $M=8$ sector.
Exact Polynomial Solutions of Schrödinger Equation with Various Hyperbolic Potentials
The Schrödinger equation with hyperbolic potential V(x) = −V0sinh2q(x/d)/cosh6(x/d) (q = 0,1,2,3) is studied by transforming it into the confluent Heun equation. We obtain general symmetric and antisymmetric polynomial solutions of the Schrödinger equation in a unified form via the Functional Bethe ansatz method. Furthermore, we discuss the characteristic of wavefunction of bound state with varying potential strengths. Particularly, the number of wavefunction's nodes decreases with the increase of potential strengths, and the particle tends to the bottom of the potential well correspondingly. (general)
Gottfried, Kurt
2005-01-01
"There are a handful of people who soar, whose accompalishments are so off-scale as to nearly defy belief. Hans Bethe (2 July 1906 - 6 March 2005) was of that caliber. As just one measure of his stature, imagine the task of copying his published opus by hand, for that is how he wrote most of it" (2 pages)
An Improved Harmonic Map Ansatz
Ioannidou, T A; Zakrzewski, W J; Ioannidou, Theodora; Kleihaus, Burkhard; Zakrzewski, Wojtek
2004-01-01
The rational map ansatz of Houghton et al is generalised by allowing the profile function, usually a function of $r$, to depend also on $z$ and $\\bar{z}$. It is shown that, within this ansatz, the energies of the lowest B=2,3,4 field configurations of the SU(2) Skyrme model are closer to the corresponding values of the true solutions of the model than those obtained within the original rational map ansatz. In particular, we present plots of the profile functions which do exhibit their dependence on $|z|^2$.
Zhang, L; Weng, M -H
2016-01-01
The matrix element of the weak transition {\\Lambda}_b\\rightarrow{\\Lambda}_c can be expressed in terms of six form factors. {\\Lambda}_Q(Q = b;c) can be regarded as composed of a heavy quark Q(Q = b;c) and a diquark which is made up of the remaining two light quarks. In this picture, we express these six form factors in terms of Bethe-Salpeter wave functions to second order in the 1/m_Q expansion. With the kernel containing both the scalar confinement and the one-gluon-exchange terms we calculate the form factors and the decay widths of the semileptonic decay {\\Lambda}_b\\rightarrow{\\Lambda}_clv as well as nonleptonic decays {\\Lambda}_b\\rightarrow{\\Lambda}_cP(V) numerically. We also add QCD corrections since they are comparable with 1/m_Q corrections.
An improved harmonic map ansatz
Ioannidou, Theodora [Maths Division, School of Technology, University of Thessaloniki, Thessaloniki 54124 (Greece)]. E-mail: ti3@auth.gr; Kleihaus, Burkhard [Institut fuer Physik, Universitaet Oldenburg, Postfach 2503, D-26111 Oldenburg (Germany)]. E-mail: kleihaus@theorie.physik.uni-oldenburg.de; Zakrzewski, Wojtek [Department of Mathematical Sciences, University of Durham, Durham DH1 3LE (United Kingdom)]. E-mail: w.j.zakrzewski@durham.ac.uk
2004-09-16
The rational map ansatz of Houghton et al. [Nucl. Phys. B 510 (1998) 507] is generalised by allowing the profile function, usually a function of r, to depend also on z and z-bar . It is shown that, within this ansatz, the energies of the lowest B=2,3,4 field configurations of the SU(2) Skyrme model are closer to the corresponding values of the true solutions of the model than those obtained within the original rational map ansatz. In particular, we present plots of the profile functions which do exhibit their dependence on z and z-bar .The obvious generalisation of the ansatz to higher SU(N) models involving the introduction of more projectors is briefly mentioned.
An improved harmonic map ansatz
The rational map ansatz of Houghton et al. [Nucl. Phys. B 510 (1998) 507] is generalised by allowing the profile function, usually a function of r, to depend also on z and z-bar . It is shown that, within this ansatz, the energies of the lowest B=2,3,4 field configurations of the SU(2) Skyrme model are closer to the corresponding values of the true solutions of the model than those obtained within the original rational map ansatz. In particular, we present plots of the profile functions which do exhibit their dependence on z and z-bar .The obvious generalisation of the ansatz to higher SU(N) models involving the introduction of more projectors is briefly mentioned
Hedgehog ansatz and its generalization for self-gravitating Skyrmions
Canfora, Fabrizio
2013-01-01
The hedgehog ansatz for spherically symmetric spacetimes in self-gravitating nonlinear sigma models and Skyrme models is revisited and its generalization for non-spherically symmetric spacetimes is proposed. The key idea behind our construction is that, even if the matter fields depend on the Killing coordinates in a non-trivial way, still the corresponding energy-momentum tensor can be compatible with spacetime symmetries. Our generalized hedgehog ansatz reduces the Skyrme equations to coupled differential equations for two scalar fields together with several constraint equations between them. Some particular field configurations satisfying those constraints are presented in several physically important spacetimes including stationary and axisymmetric spacetimes. Incidentally, a new exact solution is obtained under the standard hedgehog ansatz, which represents a global monopole inside a black hole with the Skyrme effect.
Hedgehog ansatz and its generalization for self-gravitating Skyrmions
Canfora, Fabrizio; Maeda, Hideki
2013-04-01
The hedgehog ansatz for spherically symmetric spacetimes in self-gravitating nonlinear sigma models and Skyrme models is revisited and its generalization for nonspherically symmetric spacetimes is proposed. The key idea behind our construction is that, even if the matter fields depend on the Killing coordinates in a nontrivial way, the corresponding energy-momentum tensor can still be compatible with spacetime symmetries. Our generalized hedgehog ansatz reduces the Skyrme equations to coupled differential equations for two scalar fields together with several constraint equations between them. Some particular field configurations satisfying those constraints are presented in several physically important spacetimes, including stationary and axisymmetric spacetimes. Incidentally, new exact solutions are obtained under the standard hedgehog ansatz, one of which represents a global monopole inside a black hole with the Skyrme effect.
Obituary: Hans Albrecht Bethe, 1906-2005
Wijers, Ralph
2007-12-01
now call the "Bethe Ansatz." Soon after his acceptance of an assistant professorship at Tübingen in 1932, he had to flee Hitler's Germany because his mother was Jewish. Bethe went to the Bragg Institute in Manchester, England, where he worked again with Peierls. In 1934, Cornell University unexpectedly offered him a position as part of R. Clifton Gibbs's expansion of the physics department; he accepted and stayed there for the rest of his life. Right from the start, Bethe enjoyed America and its atmosphere very much. His first activity there was to write the "Bethe Bible": three articles in Reviews of Modern Physics to educate his colleagues in theoretical nuclear physics. Then he did the work that astrophysicists will still appreciate him most for, and which brought him the 1967 Nobel Prize. Having worked with George Gamow's student Charles Critchfield (at Gamow's suggestion) on the proton-proton chain for nuclear fusion in the Sun (published in 1938), Bethe was initially a bit discouraged with Arthur Eddington's estimates of the Solar core temperature; their calculations did not agree well with the observed solar luminosity. However, at the Washington conference in 1937, he heard of Strömgren's new estimates of the solar interior, which brought his and Critchfield's theory into much better agreement with the data. Fairly soon after the meeting, Bethe also worked out the process whereby more massive stars must accomplish hydrogen fusion, in what we now call the CNO cycle. Curiously, Bethe held up its publication briefly in order to compete for a prize for the best unpublished paper on energy production in stars. He did win, and used the money in part to bring his mother to the United States; eventually, the paper appeared in Physics Review in 1939, and founded a whole branch of astrophysics. The war brought Bethe to the Manhattan project, of which he became one of the intellectual leaders. He ploughed through problems theoretical and practical by attacking them
Coordinate Bethe ANSÄTZE for Non-Diagonal Boundaries
Ragoucy, Eric
2013-11-01
Bethe ansatz goes back to 1931, when H. Bethe invented it to solve some one-dimensional models, such as XXX spin chain, proposed by W. Heisenberg in 1928. Although it is a very powerful method to compute eigenvalues and eigenvectors of the corresponding Hamiltonian, it can be applied only for very specific boundary conditions: periodic boundary ones, and so-called open-diagonal boundary ones. After reviewing this method, we will present a generalization of it that applies also to open-triangular boundary conditions. This short note presents only the basic ideas of the technique, and does not attend to give a general overview of the subject. Interested readers should refer to the original papers and references therein.
Reformulating the TBA equations for the quark anti-quark potential and their two loop expansion
The boundary thermodynamic Bethe Ansatz (BTBA) equations introduced in http://dx.doi.org/10.1007/JHEP08(2012)134http://dx.doi.org/10.1007/JHEP10(2013)135 to describe the cusp anomalous dimension contain imaginary chemical potentials and singular boundary fugacities, which make its systematic expansion problematic. We propose an alternative formulation based on real chemical potentials and additional source terms. We expand our equations to double wrapping order and find complete agreement with the direct two-loop gauge theory computation of the cusp anomalous dimension
Bernstein, Jeremy
2012-10-01
In 1937, two years after he moved to the US to escape Nazi persecution, the physicist Hans Bethe sent a letter to his mother in Germany. In it, he wrote, "I think I am about the leading theoretician in America. [Eugene] Wigner is certainly better and [Robert] Oppenheimer and [Edward] Teller probably just as good. But I do more and talk more and that counts too."
The Tamm-Dancoff Approximation as the boson limit of the Richardson-Gaudin equations for pairing
A connection is made between the exact eigen states of the BCS Hamiltonian and the predictions made by the Tamm-Dancoff Approximation. This connection is made by means of a parametrised algebra, which gives the exact quasi-spin algebra in one limit of the parameter and the Heisenberg-Weyl algebra in the other. Using this algebra to construct the Bethe Ansatz solution of the BCS Hamiltonian, we obtain parametrised Richardson-Gaudin equations, leading to the secular equation of the Tamm-Dancoff Approximation in the bosonic limit. An example is discussed in depth.
Combined Exp-Function Ansatz Method and Applications
Gui Mu
2013-01-01
Full Text Available Our aim is to present a combined Exp-function ansatz method. This method replaces the traditional assumptions of multisolitons by a combination of the hyperbolic functions and triangle functions in Hirota bilinear forms of nonlinear evolution equation. Using this method, we can obtain many new type analytical solutions of various nonlinear evolution equations including multisoliton solutions as well as breath-like solitons solutions. These solutions will exhibit interesting dynamic diversity.
Ising spin glass on Bethe-like lattices
Ising spin glass on Bethe-like lattices is studied focusing on the replica symmetry breaking near the spin glass transition temperature. To see the frustration effects of small loops, spin glass order parameter functions and the de Almeida-Thouless (AT) lines in small magnetic fields are obtained for the Bethe-like cactus lattices. As approximations for realistic short range models, they are compared with the results for the Bethe lattice without small loops to see the effects of the loops. Triangular, tetrahedral and square cactus lattices are studied. The slope of the spin glass order parameter function for a cactus lattice is smaller than the corresponding one for the Bethe lattice. The replica symmetry breaking region in fields for the cactus lattice is larger than that for the corresponding Bethe lattice except for the smallest number of connectivity of the loops in the triangular and tetrahedral cactus lattices. To obtain the results, an equation among quantities that are related to the spin glass order parameter is used. This equation is shown to be related to an equation derived within a cluster approximation without using replicas. (author)
Generalized Hill-Wheeler ansatz
Nunez, J.; Esebbag, C.; Martin, M.T.; Rebollo, L.; Plastino, A.
1984-09-01
The Hill-Wheeler ansatz for the total wave function, within the Generator Coordinate Method framework, is generalized by recourse to the theory of distributions. The ensuing approach allows one to obtain a basis that spans the collective subspace, without having to deal explicitly with the eigenvectors and eigenvalues of the overlap kernel. Applications to an exactly soluble model and anharmonic vibrations illustrate the present treatment. 37 refs.
Regge behaviour within the Bethe-Salpeter approach
Kubrak, Stanislav; Williams, Richard
2014-01-01
We present a calculation of the spectrum of light and heavy quark bound states in the rainbow-ladder truncation of Dyson-Schwinger/Bethe-Salpeter equations. By extending the formalism include the case of total angular momentum J=3, we are able to explore Regge trajectories and make prediction of tensor bound states for light and heavy quarkonia.
Parameter-free ansatz for inferring ground state wave functions of even potentials
Flego, S. P.; A. Plastino; Plastino, A.R.
2011-01-01
Schr\\"odinger's equation (SE) and the information-optimizing principle based on Fisher's information measure (FIM) are intimately linked, which entails the existence of a Legendre transform structure underlying the SE. In this comunication we show that the existence of such an structure allows, via the virial theorem, for the formulation of a parameter-free ground state's SE-ansatz for a rather large family of potentials. The parameter-free nature of the ansatz derives from the structural inf...
Pauli principle for one-dimensional bosons and the algebraic Bethe ansatz
To construct the physical vacuum in completely integrable models of one-dimensional boson fields, it is essential that the momenta of all particles be distinct. A proof is present that they actually are distinct
We consider quantum quenches in the so-called q-boson lattice model. We argue that the Generalized Eigenstate Thermalization Hypothesis holds in this model, therefore the Generalized Gibbs Ensemble (GGE) gives a valid description of the stationary states in the long time limit. For a special class of initial states (which are the pure Fock states in the local basis) we are able to provide the GGE predictions for the resulting root densities. We also give predictions for the long-time limit of certain local operators. In the q → ∞ limit the calculations simplify considerably, the wave functions are given by Schur polynomials and the overlaps with the initial states can be written as simple determinants. In two cases we prove rigorously that the GGE prediction for the root density is correct. Moreover, we calculate the exact time dependence of a physical observable (the one-site Emptiness Formation Probability) for the quench starting from the state with exactly one particle per site. In the long-time limit the GGE prediction is recovered. (paper)
Fermionic Basis in Conformal Field Theory and Thermodynamic Bethe Ansatz for Excited States
Hermann Boos
2011-01-01
Full Text Available We generalize the results of [Comm. Math. Phys. 299 (2010, 825-866] (hidden Grassmann structure IV to the case of excited states of the transfer matrix of the six-vertex model acting in the so-called Matsubara direction. We establish an equivalence between a scaling limit of the partition function of the six-vertex model on a cylinder with quasi-local operators inserted and special boundary conditions, corresponding to particle-hole excitations, on the one hand, and certain three-point correlation functions of conformal field theory (CFT on the other hand. As in hidden Grassmann structure IV, the fermionic basis developed in previous papers and its conformal limit are used for a description of the quasi-local operators. In paper IV we claimed that in the conformal limit the fermionic creation operators generate a basis equivalent to the basis of the descendant states in the conformal field theory modulo integrals of motion suggested by A. Zamolodchikov (1987. Here we argue that, in order to completely determine the transformation between the above fermionic basis and the basis of descendants in the CFT, we need to involve excitations. On the side of the lattice model we use the excited-state TBA approach. We consider in detail the case of the descendant at level 8.
Newman-Janis Ansatz in conformastatic spacetimes
Gutiérrez-Piñeres, Antonio C
2016-01-01
The Newman-Janis Ansatz was used first to obtain the stationary Kerr metric from the static Schwarzschild metric. Many works have been devoted to investigate the physical significance of this Ansatz, but no definite answer has been given so far. We show that this Ansatz can be applied in general to conformastatic vacuum metrics, and leads to stationary generalizations which, however, do not preserve the conformal symmetry. We investigate also the particular case when the seed solution is given by the Schwarzschild spacetime and show that the resulting rotating configuration does not correspond to a vacuum solution, even in the limiting case of slow rotation. In fact, it describes in general a relativistic fluid with anisotropic pressure and heat flux. This implies that the Newman-Janis Ansatz strongly depends on the choice of representation for the seed solution. We interpret this result as as a further indication of its applicability limitations.
Exact solution of a new class of Hubbard-type models with open boundary conditions
A new class of Hubbard-type models with open boundary conditions in one dimension is studied in the framework of coordinate Bethe ansatz method. The energy spectrum, integrable boundary conditions and the corresponding Bethe ansatz equations are derived. (authors)
Hans Bethe and the Global Energy Problems
Ioffe, B. L.
2005-01-01
Bethe's view-point on the global energy problems is presented. Bethe claimed that the nuclear power is a necessity in future. Nuclear energetic must be based on breeder reactors. Bethe considered the non-proliferation of nuclear weapons as the main problem of long-range future of nuclear energetics. The solution of this problem he saw in heavy water moderated thermal breeders, using uranium-233, uranium-238 and thorium as a fuel.
Tetraquark bound states in a Bethe-Salpeter approach
Heupel, Walter; Eichmann, Gernot; Fischer, Christian S.
2012-01-01
We determine the mass of tetraquark bound states from a coupled system of covariant Bethe-Salpeter equations. Similar in spirit to the quark-diquark model of the nucleon, we approximate the full four-body equation for the tetraquark by a coupled set of two-body equations with meson and diquark constituents. These are calculated from their quark and gluon substructure using a phenomenologically well-established quark-gluon interaction. For the lightest scalar tetraquark we find a mass of the o...
Convexifying the Bethe Free Energy
Meshi, Ofer; Globerson, Amir; Friedman, Nir
2012-01-01
The introduction of loopy belief propagation (LBP) revitalized the application of graphical models in many domains. Many recent works present improvements on the basic LBP algorithm in an attempt to overcome convergence and local optima problems. Notable among these are convexified free energy approximations that lead to inference procedures with provable convergence and quality properties. However, empirically LBP still outperforms most of its convex variants in a variety of settings, as we also demonstrate here. Motivated by this fact we seek convexified free energies that directly approximate the Bethe free energy. We show that the proposed approximations compare favorably with state-of-the art convex free energy approximations.
Bethe-Salpeter bound-state structure in Minkowski space
Gutierrez, C.; Gigante, V.; Frederico, T.; Salmè, G.; Viviani, M.; Tomio, Lauro
2016-08-01
The quantitative investigation of the scalar Bethe-Salpeter equation in Minkowski space, within the ladder-approximation framework, is extended to include the excited states. This study has been carried out for an interacting system composed by two massive bosons exchanging a massive scalar, by adopting (i) the Nakanishi integral representation of the Bethe-Salpeter amplitude, and (ii) the formally exact projection onto the null plane. Our analysis, on one hand, confirms the reliability of the method already applied to the ground state and, on the other one, extends the investigation from the valence distribution in momentum space to the corresponding quantity in the impact-parameter space, pointing out some relevant features, like (i) the equivalence between Minkowski and Euclidean transverse-momentum amplitudes, and (ii) the leading exponential fall-off of the valence wave function in the impact-parameter space.
Bethe-Salpeter bound-state structure in Minkowski space
Gutierrez, C; Frederico, T; Salmè, G; Viviani, M; Tomio, Lauro
2016-01-01
The quantitative investigation of the scalar Bethe-Salpeter equation in Minkowski space, within the ladder-approximation framework, is extended to include the excited states. This study has been carried out for an interacting system composed by two massive bosons exchanging a massive scalar, by adopting (i) the Nakanishi integral representation of the Bethe-Salpeter amplitude, and (ii) the formally exact projection onto the null plane. Our analysis, on one hand, confirms the reliability of the method already applied to the ground state and, on the other one, extends the investigation from the valence distribution in momentum space to the corresponding quantity in the impact-parameter space, pointing out some relevant features, like (i) the equivalence between Minkowski and Euclidean transverse-momentum amplitudes, and (ii) the leading exponential fall-off of the valence wave function in the impact-parameter space.
ODE/IM correspondence for modified $B_2^{(1)}$ affine Toda field equation
Ito, Katsushi
2016-01-01
We study the massive ODE/IM correspondence for modified $B_2^{(1)}$ affine Toda field equation. Based on the $\\psi$-system for the solutions of the associated linear problem, we obtain the Bethe ansatz equations. We also discuss the T-Q relations, the T-system and the Y-system, which are shown to be related to those of the $A_3/{\\bf Z}_2$ integrable system. We consider the case that the solution of the linear problem has a monodromy around the origin, which imposes nontrivial boundary conditions for the T-/Y-system. The high-temperature limit of the T- and Y-system and their monodromy dependence are studied numerically.
Parameter-free ansatz for inferring ground state wave functions of even convex potentials
Schrödinger's equation (SE) and the information-optimizing principle based on Fisher's information measure are intimately linked (Frieden et al 1999 Phys. Rev. E 60 48), which entails the existence of a Legendre transform structure underlying the SE (Flego et al 2011 J. Math. Phys. 52 082103). In this paper, we show that the existence of such a structure allows, via the virial theorem, for the formulation of a parameter-free ground state's SE ansatz for a rather large family of potentials. The parameter-free nature of the ansatz derives from the structural information it incorporates through its Legendre properties. (paper)
Parameter-free ansatz for inferring ground state wave functions of even potentials
Flego, S P; Plastino, A R
2011-01-01
Schr\\"odinger's equation (SE) and the information-optimizing principle based on Fisher's information measure (FIM) are intimately linked, which entails the existence of a Legendre transform structure underlying the SE. In this comunication we show that the existence of such an structure allows, via the virial theorem, for the formulation of a parameter-free ground state's SE-ansatz for a rather large family of potentials. The parameter-free nature of the ansatz derives from the structural information it incorporates through its Legendre properties.
The first part of this book is a literary portrait of the great natural scientist. The book was the result of a number of personal meetings, telephone interviews and letters exchanged, which began in 1977 and lasted two years. Bethes work comprises so many aspects of modern physics and astrophysics that only a fat encyclopedia could do him justice. The author hopes to convey at least an idea of the tremendous scope of this work. But the main theme in the article in 'The New Yorker' and in the resulting book is a discussion about energy. The importance of the energy problem is such that it completely penetrates science and politics. Thus, the third chapter is concerned with energy-political options, the catastrophe of and radioactivity after Chernobyl, and the development of concepts of reactor safety. (orig./HSCH)
Basso, Benjamin, E-mail: bbasso@perimeterinstitute.ca [Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5 (Canada); Rej, Adam, E-mail: arej@ias.edu [School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540 (United States)
2014-02-15
Studying the scattering of excitations around a dynamical background has a long history in the context of integrable models. The Gubser–Klebanov–Polyakov string solution provides such a background for the string/gauge correspondence. Taking the conjectured all-loop asymptotic equations for the AdS{sub 4}/CFT{sub 3} correspondence as the starting point, we derive the S-matrix and a set of spectral equations for the lowest-lying excitations. We find that these equations resemble closely the analogous equations for AdS{sub 5}/CFT{sub 4}, which are also discussed in this paper. At large values of the coupling constant we show that they reproduce the Bethe equations proposed to describe the spectrum of the low-energy limit of the AdS{sub 4}×CP{sup 3} sigma model.
On the central quadric ansatz: integrable models and Painlevé reductions
Ferapontov, E. V.; Huard, B.; Zhang, A.
2012-05-01
It was observed by Tod (1995 Class. Quantum Grav.12 1535-47) and later by Dunajski and Tod (2002 Phys. Lett. A 303 253-64) that the Boyer-Finley (BF) and the dispersionless Kadomtsev-Petviashvili (dKP) equations possess solutions whose level surfaces are central quadrics in the space of independent variables (the so-called central quadric ansatz). It was demonstrated that generic solutions of this type are described by Painlevé equations PIII and PII, respectively. The aim of our paper is threefold: (1) Based on the method of hydrodynamic reductions, we classify integrable models possessing the central quadric ansatz. This leads to the five canonical forms (including BF and dKP). (2) Applying the central quadric ansatz to each of the five canonical forms, we obtain all Painlevé equations PI-PVI, with PVI corresponding to the generic case of our classification. (3) We argue that solutions coming from the central quadric ansatz constitute a subclass of two-phase solutions provided by the method of hydrodynamic reductions.
Zeilberger, Doron
2011-01-01
While it is trivial to multiply two C-finite sequences (just like integers), it is not quite so trivial to "factorize" them, or to decide whether they are "prime". The former is plain linear algebra, while the latter is heavy-duty non-linear algebra, getting hairy systems of algebraic equations that can be solved, in principle, using Gr\\"obner bases and the Buchberger algorithm, but, alas, sooner or later it becomes too hard even for the fastest and largest computers. The main technical novely of this article is a fast "algorithm" (it cheats and uses floating-point arithmetic, please don't tell anyone!) for deciding whether a given C-finite sequence can be written as a product of C-finite sequences of lower order. This article accompanies the Maple package Cfinite available from http://www.math.rutgers.edu/~zeilberg/tokhniot/Cfinite .
A generalized Hill-Wheeler ansatz
Nuñez, J.; Esebbag, C.; Martin, M. T.; Rebollo, L.; Plastino, A.
1984-06-01
The Hill-Wheeler ansatz for the total wave function, within the Generator Coordinate Method framework, is generalized by recourse to the theory of distributions. The ensuing approach allows one to obtain a basis that spans the collective subspace, without having to deal explicitly with the eigenvectors and eigenvalues of the overlap kernel. Applications to an exactly soluble model and anharmonic vibrations illustrate the present treatment.
A generalized Hill-Wheeler ansatz
The Hill-Wheeler ansatz for the total wave function, within the Generator Coordinate Method framework, is generalized by recourse to the theory of distributions. The ensuing approach allows one to obtain a basis that spans the collective subspace, without having to deal explicitly with the eigenvectors and eigenvalues of the overlap kernel. Applications to an exactly soluble model and anharmonic vibrations illustrate the present treatment. (orig.)
Spectra of heavy mesons in the Bethe-Salpeter approach
Fischer, Christian S.; Kubrak, Stanislav; Williams, Richard [Justus-Liebig-Universitaet Giessen, Institut fuer Theoretische Physik, Giessen (Germany)
2015-01-01
We present a calculation of the spectrum of charmonia, bottomonia and B{sub c}-meson states with ''ordinary'' and exotic quantum numbers. We discuss the merits and limitations of a rainbow-ladder truncation of Dyson-Schwinger and Bethe-Salpeter equations and explore the effects of different shapes of the effective running coupling on ground and excited states in channels with quantum numbers J ≤ 3. We furthermore discuss the status of the X(3872) as a potential (excited) quark-antiquark state and give predictions for the masses of charmonia and bottomonia in the tensor channels with J= 2, 3. (orig.)
Direct Bethe-Salpeter solutions in Minkowski space
Carbonell, J
2016-01-01
We review a method to directly solve the Bethe-Salpeter equation in Minkowski space, both for bound and scattering states. It is based on a proper treatment of the many singularities which appear in the kernel and propagators. The off-mass shell scattering amplitude for spinless particles interacting by a one boson exchange was computed for the first time. Using our Minkowski space solutions for the initial (bound) and final (scattering) states, we calculate elastic and transition (bound to scattering state) electromagnetic form factors. The conservation of the transition electromagnetic current J.q=0, verified numerically, confirms the validity of our solutions.
Tetra quark bound states in a Bethe-Salpeter approach
Heupel, Walter; Eichmann, Gernot [Institut fuer Theoretische Physik, Justus-Liebig-Universitaet Giessen, D-35392 Giessen (Germany); Fischer, Christian S., E-mail: christian.fischer@theo.physik.uni-giessen.de [Institut fuer Theoretische Physik, Justus-Liebig-Universitaet Giessen, D-35392 Giessen (Germany); GSI Helmholtzzentrum fuer Schwerionenforschung GmbH, Planckstr. 1, D-64291 Darmstadt (Germany)
2012-12-05
We determine the mass of tetraquark bound states from a coupled system of covariant Bethe-Salpeter equations. Similar in spirit to the quark-diquark model of the nucleon, we approximate the full four-body equation for the tetraquark by a coupled set of two-body equations with meson and diquark constituents. These are calculated from their quark and gluon substructure using a phenomenologically well-established quark-gluon interaction. For the lightest scalar tetraquark we find a mass of the order of 400 MeV and a wave function dominated by the pion-pion constituents. Both results are in agreement with a meson molecule picture for the f{sub 0}(600). Our results furthermore suggest the presence of a potentially narrow all-charm tetraquark in the mass region 5-6 GeV.
Tetraquark bound states in a Bethe-Salpeter approach
Heupel, Walter; Fischer, Christian S
2012-01-01
We determine the mass of tetraquark bound states from a coupled system of covariant Bethe-Salpeter equations. Similar in spirit to the quark-diquark model of the nucleon, we approximate the full four-body equation for the tetraquark by a coupled set of two-body equations with meson and diquark constituents. These are calculated from their quark and gluon substructure using a phenomenologically well-established quark-gluon interaction. For the lightest scalar tetraquark we find a mass of the order of 400 MeV and a wave function dominated by the pion-pion constituents. Both results are in agreement with a meson molecule picture for the f_0(600). Our results furthermore suggest the presence of a potentially narrow all-charm tetraquark in the mass region 5-6 GeV.
Tetraquark bound states in a Bethe-Salpeter approach
Heupel, Walter; Eichmann, Gernot; Fischer, Christian S.
2012-12-01
We determine the mass of tetraquark bound states from a coupled system of covariant Bethe-Salpeter equations. Similar in spirit to the quark-diquark model of the nucleon, we approximate the full four-body equation for the tetraquark by a coupled set of two-body equations with meson and diquark constituents. These are calculated from their quark and gluon substructure using a phenomenologically well-established quark-gluon interaction. For the lightest scalar tetraquark we find a mass of the order of 400 MeV and a wave function dominated by the pion-pion constituents. Both results are in agreement with a meson molecule picture for the f0 (600). Our results furthermore suggest the presence of a potentially narrow all-charm tetraquark in the mass region 5-6 GeV.
Tetra quark bound states in a Bethe-Salpeter approach
We determine the mass of tetraquark bound states from a coupled system of covariant Bethe-Salpeter equations. Similar in spirit to the quark-diquark model of the nucleon, we approximate the full four-body equation for the tetraquark by a coupled set of two-body equations with meson and diquark constituents. These are calculated from their quark and gluon substructure using a phenomenologically well-established quark-gluon interaction. For the lightest scalar tetraquark we find a mass of the order of 400 MeV and a wave function dominated by the pion-pion constituents. Both results are in agreement with a meson molecule picture for the f0(600). Our results furthermore suggest the presence of a potentially narrow all-charm tetraquark in the mass region 5-6 GeV.
Dhar, Abhishek; Sriram Shastry, B.
2000-09-01
We present a calculation of the lowest excited states of the Heisenberg ferromagnet in 1D for any wave vector. These turn out to be string solutions of Bethe's equations with a macroscopic number of particles in them. They are identified as generalized quantum Bloch wall states, and a simple physical picture is provided for the same.
Dhar, Abhishek; Shastry, B. Sriram
2000-01-01
We present a calculation of the lowest excited states of the Heisenberg ferromagnet in 1-d for any wave vector. These turn out to be string solutions of Bethe's equations with a macroscopic number of particles in them. These are identified as generalized quantum Bloch wall states, and a simple physical picture provided for the same.
Entanglement entropy in quantum many-particle systems and their simulation via ansatz states
Barthel, Thomas
2009-12-10
A main topic of this thesis is the development of efficient numerical methods for the simulation of strongly correlated quantum lattice models. For one-dimensional systems, the density-matrix renormalization-group (DMRG) is such a very successful method. The physical states of interest are approximated within a certain class of ansatz states. These ansatz states are designed in a way that the number of degrees of freedom are prevented from growing exponentially. They are the so-called matrix product states. The first part of the thesis, therefore, provides analytical and numerical analysis of the scaling of quantum nonlocality with the system size or time in different, physically relevant scenarios. For example, the scaling of Renyi entropies and their dependence on boundary conditions is derived within the 1+1-dimensional conformal field theory. Conjectures and analytical indications concerning the properties of entanglement entropy in critical fermionic and bosonic systems are confirmed numerically with high precision. For integrable models in the thermodynamic limit, general preconditions are derived under which subsystems converge to steady states. These steady states are non-thermal and retain information about the initial state. It is shown that the entanglement entropy in such steady states is extensive. For short times, the entanglement entropy grows typically linearly with time, causing an exponential increase in computation costs for the DMRG method. The second part of the thesis focuses on the development and improvement of the abovementioned numerical techniques. The time-dependent DMRG is complemented with an extrapolation technique for the evaluated observables. In this way, the problem of the entropy increase can be circumvented, allowing for a precise determination of spectral functions. The method is demonstrated using the example of the Heisenberg antiferromagnet and results are compared to Bethe-Ansatz data for T=0 and quantum Monte Carlo data