Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0012159 · Journal of Physics A Mathematical and General

The excitations of the symplectic integrable models and their applications

Marcio Jose Martins

Published May 3, 2002• 4 citations
arXivHTMLPDF

Abstract

The Bethe ansatz equations of the fundamental Sp (2 N ) integrable model are solved by a peculiar configuration of roots leading us to determine the nature of the excitations. They consist of N elementary generalized spinons and N -1 composite excitations made by special convolutions between the spinons. This fact is essential to determining the low-energy behaviour which is argued to be described in terms of 2 N Majorana fermions. Our results have practical applications to spin-orbital systems and also shed new light on the connection between integrable models and Wess-Zumino-Witten field theories.

DOI
10.1088/0305-4470/35/19/103

Related papers

Quelques applications de l'Ansatz de Bethe (Some applications of the Bethe Ansatz)

Paul Zinn-Justin

The Bethe Ansatz is a method that is used in quantum integrable models in order to solve them explicitly. This method is explained here in a general framework, which applies to 1D quantum spin chains, 2D statistical lattice models (vertex models) and relativistic field theories with 1 space dimension and 1 time dimension. The connection with quantum groups is expounded. Several applications are then presented. Finite size corrections are calculated via two methods: The Non-Linear Integral Equations, which are applied to the study of the states of the affine Toda model with imaginary coupling, and their interpolation between the high energy (ultra-violet) and low energy (infra-red) regions; and the Thermodynamic Bethe Ansatz Equations, along with the associated Fusion Equations, which are used to determine the thermodynamic properties of the generalized multi-channel Kondo model. The latter is then studied in more detail, still using the Bethe Ansatz and quantum groups, so as to characterize the spectrum of the low energy excitations.

arXiv:solv-int/9810007•Oct 11, 1998•3 citations

Observations on the Bethe ansatz solutions of the spin-1/2 isotropic anti-ferromagnetic Heisenberg chain

Bernie G. Nickel

Evidence is presented that the solutions of the Bethe ansatz equations for spin-½ isotropic Heisenberg chains in fixed total spin and momentum sectors are the roots of single variable polynomials with integer (or integer based) coefficients. Such solutions are used as a starting point for investigation of long chain (critical region) properties. In the total spin S = 0 sector I conjecture explicit formulae for the Bethe string configuration labelling of all left and right tower excitations in the k = 1, SU(2) Wess-Zumino-Witten model.

Journal of Physics Communications•Jan 30, 2019•1 citation

Fonctions d’onde et produits scalaires dans l’ansatz de Bethe

B. Vallet

Integrable models are physical models for which some quantities can be exactly obtained, without use of perturbation theory. Those very special models are source of an increasing interest in theoretical physics. The direct applications in condensed matter physics and the subtle links evidenced more recently with some supersymmetric gauges theories motivated the development of complex mathematical tools. Among these, Bethe ansatz played an important role, and provides an efficient approach for diagonalizing a lot of models of various nature. The first chapter of this thesis is devoted to the introduction to the two approaches of the Bethe ansatz, said “coordinate” and “algebraic”, in the context of the XXX Heisenberg spin chain and a continuous spin generalization of the Totally Asymmetric Simple Exclusion Process, the so called Zero-range Chipping model with factorized steady state (ZCM). The second chapter is devoted to the Modified Algebraic Bethe Ansatz in the context of the periodic XXX chain. This modified ansatz is proposed for solving the spectral problem of the open spin chain, for which the usual ansatz fails. The scalar product of the obtained modified Bethe states is studied. The third chapter concerns the resolution of the identity and the inverse functional problem. An expression for the spin states in terms of Bethe states est presented for the ZCM, and an expression for the resolution of the identity in term of Bethe states for the infinite XXZ chain is proved, involving in both cases the contribution of bound states. At last, the fourth chapter concerns determinant representations in the Bethe ansatz. An expression for the “matrix elements of the particle number operator” for the delta-Bose gas in terms of a determinant is proved, and some integral representations for the Izergin-Korepin and Slavnov determinants are investigated, then establishing a new formal link between these two determinant representations.

•Oct 10, 2019•0 citations

Elementary excitations for the integrable spin-SHeisenberg chain with an impurity spin

Yosuke Morita, Sei-ichiro Suga, A. Okiji

Elementary excitations for the integrable spin-S antiferromagnetic Heisenberg chain with an impurity spin are investigated by using the Bethe ansatz solution. Dressed holes are introduced in order to describe the elementary excitation. The characteristic energy dependence of the spectral density for the elementary excitation at zero temperature is discussed in connection with behaviour patterns of physical quantities. We extend our calculation for the elementary excitation to the case of finite temperatures, applying the method developed by Yang and Yang for one-dimensional interacting boson systems. At finite temperatures, in the case where , the spectral density shows a divergence property at zero excitation energy irrespective of the temperature, whereas in the case where , the peak structure develops in the low-energy region as the temperature is decreased.

Journal of Physics Condensed Matter•Mar 11, 1996•2 citations

Off-Diagonal Bethe Ansatz and Exact Solution of a Topological Spin Ring

Junpeng Cao, Wen‐Li Yang, Kangjie Shi +1 more

A general method is proposed for constructing the Bethe ansatz equations of integrable models without U(1) symmetry. As an example, the exact spectrum of the XXZ spin ring with a Möbius-like topological boundary condition is derived by constructing a modified T-Q relation based on the functional connection between the eigenvalues of the transfer matrix and the quantum determinant of the monodromy matrix. With the exact solution, the elementary excitations of the topological XX spin ring are discussed in detail. It is found that the excitation spectrum indeed shows a nontrivial topological nature.

arXiv:1305.7328•Physical Review Letters•Sep 23, 2013•147 citations

Physical combinatorics and quasiparticles

Giovanni Feverati, Paul A. Pearce, Nicholas S. Witte

We consider the physical combinatorics of critical lattice models and their associated conformal field theories arising in the continuum scaling limit. As examples, we consider A-type unitary minimal models and the level-1 Wess–Zumino–Witten (WZW) model. The Hamiltonian of the WZW model is the invariant XXX quantum spin chain. For simplicity, we consider these theories only in their vacuum sectors on the strip. Combinatorially, fermionic particles are introduced as certain features of RSOS paths. They are composites of dual-particles and exhibit the properties of quasiparticles. The particles and dual-particles are identified, through a conjectured energy-preserving bijection, with patterns of zeros of the eigenvalues of the fused transfer matrices in their analyticity strips. The associated ( m , n ) systems arise as geometric packing constraints on the particles. The analyticity encoded in the patterns of zeros is the key to the analytic calculation of the excitation energies through the thermodynamic Bethe ansatz (TBA). As a by-product of our study, in the case of the WZW or XXX model, we find a relation between the location of the Bethe root strings and the location of the transfer matrix 2-strings.

arXiv:0903.0510•Journal of Statistical Mechanics Theory and Experiment•Oct 19, 2009•5 citations
The Bethe ansatz equations of the fundamental Sp (2 N ) integrable model are solved by a peculiar configuration of roots leading us to determine the nature of the excitations. They consist of N elementary generalized spinons and N -1 composite excitations made by special convolutions between the spinons. This fact is essential to determining the low-energy behaviour which is argued to be described in terms of 2 N Majorana fermions. Our results have practical applications to spin-orbital systems and also shed new light on the connection between integrable models and Wess-Zumino-Witten field theories.
The Bethe Ansatz is a method that is used in quantum integrable models in order to solve them explicitly. This method is explained here in a general framework, which applies to 1D quantum spin chains, 2D statistical lattice models (vertex models) and relativistic field theories with 1 space dimension and 1 time dimension. The connection with quantum groups is expounded. Several applications are then presented. Finite size corrections are calculated via two methods: The Non-Linear Integral Equations, which are applied to the study of the states of the affine Toda model with imaginary coupling, and their interpolation between the high energy (ultra-violet) and low energy (infra-red) regions; and the Thermodynamic Bethe Ansatz Equations, along with the associated Fusion Equations, which are used to determine the thermodynamic properties of the generalized multi-channel Kondo model. The latter is then studied in more detail, still using the Bethe Ansatz and quantum groups, so as to characterize the spectrum of the low energy excitations.
Evidence is presented that the solutions of the Bethe ansatz equations for spin-½ isotropic Heisenberg chains in fixed total spin and momentum sectors are the roots of single variable polynomials with integer (or integer based) coefficients. Such solutions are used as a starting point for investigation of long chain (critical region) properties. In the total spin S = 0 sector I conjecture explicit formulae for the Bethe string configuration labelling of all left and right tower excitations in the k = 1, SU(2) Wess-Zumino-Witten model.
Integrable models are physical models for which some quantities can be exactly obtained, without use of perturbation theory. Those very special models are source of an increasing interest in theoretical physics. The direct applications in condensed matter physics and the subtle links evidenced more recently with some supersymmetric gauges theories motivated the development of complex mathematical tools. Among these, Bethe ansatz played an important role, and provides an efficient approach for diagonalizing a lot of models of various nature. The first chapter of this thesis is devoted to the introduction to the two approaches of the Bethe ansatz, said “coordinate” and “algebraic”, in the context of the XXX Heisenberg spin chain and a continuous spin generalization of the Totally Asymmetric Simple Exclusion Process, the so called Zero-range Chipping model with factorized steady state (ZCM). The second chapter is devoted to the Modified Algebraic Bethe Ansatz in the context of the periodic XXX chain. This modified ansatz is proposed for solving the spectral problem of the open spin chain, for which the usual ansatz fails. The scalar product of the obtained modified Bethe states is studied. The third chapter concerns the resolution of the identity and the inverse functional problem. An expression for the spin states in terms of Bethe states est presented for the ZCM, and an expression for the resolution of the identity in term of Bethe states for the infinite XXZ chain is proved, involving in both cases the contribution of bound states. At last, the fourth chapter concerns determinant representations in the Bethe ansatz. An expression for the “matrix elements of the particle number operator” for the delta-Bose gas in terms of a determinant is proved, and some integral representations for the Izergin-Korepin and Slavnov determinants are investigated, then establishing a new formal link between these two determinant representations.
Elementary excitations for the integrable spin-S antiferromagnetic Heisenberg chain with an impurity spin are investigated by using the Bethe ansatz solution. Dressed holes are introduced in order to describe the elementary excitation. The characteristic energy dependence of the spectral density for the elementary excitation at zero temperature is discussed in connection with behaviour patterns of physical quantities. We extend our calculation for the elementary excitation to the case of finite temperatures, applying the method developed by Yang and Yang for one-dimensional interacting boson systems. At finite temperatures, in the case where , the spectral density shows a divergence property at zero excitation energy irrespective of the temperature, whereas in the case where , the peak structure develops in the low-energy region as the temperature is decreased.
A general method is proposed for constructing the Bethe ansatz equations of integrable models without U(1) symmetry. As an example, the exact spectrum of the XXZ spin ring with a Möbius-like topological boundary condition is derived by constructing a modified T-Q relation based on the functional connection between the eigenvalues of the transfer matrix and the quantum determinant of the monodromy matrix. With the exact solution, the elementary excitations of the topological XX spin ring are discussed in detail. It is found that the excitation spectrum indeed shows a nontrivial topological nature.
We consider the physical combinatorics of critical lattice models and their associated conformal field theories arising in the continuum scaling limit. As examples, we consider A-type unitary minimal models and the level-1 Wess–Zumino–Witten (WZW) model. The Hamiltonian of the WZW model is the invariant XXX quantum spin chain. For simplicity, we consider these theories only in their vacuum sectors on the strip. Combinatorially, fermionic particles are introduced as certain features of RSOS paths. They are composites of dual-particles and exhibit the properties of quasiparticles. The particles and dual-particles are identified, through a conjectured energy-preserving bijection, with patterns of zeros of the eigenvalues of the fused transfer matrices in their analyticity strips. The associated ( m , n ) systems arise as geometric packing constraints on the particles. The analyticity encoded in the patterns of zeros is the key to the analytic calculation of the excitation energies through the thermodynamic Bethe ansatz (TBA). As a by-product of our study, in the case of the WZW or XXX model, we find a relation between the location of the Bethe root strings and the location of the transfer matrix 2-strings.