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Kluwer Academic Publishers eBooks

Statistical Mechanics of Quantum Integrable Systems

Miki Wadati, Go Kato, Toshiaki Iida

Published Mar 1, 2006• 3 citations
Publisher

Abstract

Recent developments in statistical mechanics of quantum integrable systems are reviewed. Those studies are fundamental and have a renewed interest related to newly developing fields such as atomic Bose-Einstein condensations, photonic crystals and quantum computations. After a brief summary of the basic concepts and methods, the following three topics are discussed. First, by the thermal Bethe ansatz (TBA), a hard-core Bose gas is exactly solved. The model includes fully the effect of excluded volume and is identified to be a c=1 conformal field theory. Second, the cluster expansion method based on the periodic boundary condition for the Bethe wave function, which we call the Bethe ansatz cluster expansion (BACE) method, is developed for a δ-function gas and the XXX Heisenberg chain. This directly proves the TBA and reveals intrinsic properties of quantum integrable systems. Third, for a δ-function gas, the integral equations for the distribution functions of the quasi-momentum and the quasi-particle energy are solved in the form of power series. In the weak coupling case, the results reproduce those of Bogoliubov theory.

DOI
10.1007/1-4020-2190-9_17

Related papers

Bethe Ansatz Cluster Expansion Method for Quantum Integrable Particle Systems

Go Kato, Miki Wadati

Presented is a method, which we call Bethe ansatz cluster expansion (BACE) method, to investigate statistical mechanics of quantum integrable systems. By use of only the Bethe ansatz equation, n -particle partition functions are exactly calculated. From the expression for the partition function, the n -particle cluster integrals are derived. The results completely agree with those of the thermal Bethe ansatz (TBA) by Yang and Yang. Thus, the BACE method directly proves the validity of the TBA. Partition theory and graphical representation are used to make the presentation simple and transparent.

Journal of the Physical Society of Japan•May 15, 2004•5 citations

Statistical Mechanics of a One-Dimensional δ-function Bose Gas

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Statistical mechanics of a one-dimensional system of bosons with repulsive delta-function interaction is studied. Explicit calculations of the partition function and the cluster integrals are carried out. The results coincide with those by the thermal Bethe ansatz (TBA) method. Thermodynamic quantities such as the pressure, the chemical potential and the energy are given both for large and small coupling limits.

Journal of the Physical Society of Japan•Jul 15, 2001•13 citations

Introduction to the thermodynamic Bethe ansatz

Stijn Jurrien van Tongeren

Abstract 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.

arXiv:1606.02951•Journal of Physics A Mathematical and Theoretical•Jul 20, 2016•47 citations

Exact integrability in quantum field theory and statistical systems

H. B. Thacker

The properties of exactly integrable two-dimensional quantum systems are reviewed and discussed. The nature of exact integrability as a physical phenomenon and various aspects of the mathematical formalism are explored by discussing several examples, including detailed treatments of the nonlinear Schr\"odinger (delta-function gas) model, the massive Thirring model, and the six-vertex (ice) model. The diagonalization of a Hamiltonian by Bethe's Ansatz is illustrated for the nonlinear Schr\"od\'{\i}nger model, and the integral equation method of Lieb for obtaining the spectrum of the many-body system from periodic boundary conditions is reviewed. Similar methods are applied to the massive Thirring model, where the fermion-antifermion and bound-state spectrum are obtained explicitly by the integral equation method. After a brief review of the classical inverse scattering method, the quantum inverse method for the nonlinear Schr\"odinger model is introduced and shown to be an algebraization of the Bethe Ansatz technique. In the quantum inverse method, an auxiliary linear problem is used to define nonlocal operators which are functionals of the original local field on a fixed-time string of arbitrary length. The particular operators for which the string is infinitely long (free boundary conditions) or forms a closed loop around a cylinder (periodic boundary conditions) correspond to the quantized scattering data and have a special significance. One of them creates the Bethe eigenstates, while the other is the generating function for an infinite number of conservation laws. The analogous operators on a lattice are constructed for the symmetric six-vertex model, where the object which corresponds to a solution of the auxiliary linear problem is a string of vertices contracted over horizontal links (arrows). The relationship between the quantum inverse method and the transfer matrix formalism is exhibited. The inverse Gel'fand-Levitan transform which expresses the local field operator as a functional of the quantized scattering data is formulated for the nonlinear Schr\"odinger equation, and some interesting properties of this transformation are noted, including its reduction to a Jordan-Wigner transformation in the limit of infinitely repulsive coupling.

Reviews of Modern Physics•Apr 1, 1981•513 citations

Statistical Mechanics of Soliton Systems

Toshio Tsuzuki, Kazuo Sasaki

The statistical mechanics of the soliton-bearing nonlinear Klein-Gordon systems in one dimension is reviewed. Several different theoretical approaches are discussed: the transfer-integral method, the soliton-gas approach, the Bethe-ansatz method, and a few methods of computing the quantum correction. The recent development in the low-temperature double-series expansion for the free energy is the main subject.

Progress of Theoretical Physics Supplement•Jan 1, 1988•23 citations

The path integral approach to thermodynamics

Fabian H. L. Eßler, Holger Frahm, Frank Göhmann +2 more

A very curious situation arises in the context of the calculation of the partition function from the spectrum of an integrable Hamiltonian. Despite the validity of the Bethe ansatz equations for all energy eigenvalues of the model the direct evaluation of the partition function is rather difficult. In contrast to ideal quantum gases the eigenstates are not explicitly known: the Bethe ansatz equations provide just implicit descriptions that pose problems of their own kind. Yet, knowing the behaviour of quantum chains at finite temperatures is important for many reasons. As a matter of fact, the groundstate is strictly inaccessible due to the very fundamentals of thermodynamics. Therefore the study of finite temperatures is relevant for theoretical as well as experimental reasons. At high temperatures, quantum systems show only trivial static properties without correlations. Lowering the temperature, the systems enter a large regime with non-universal correlations and finally approach the quantum critical point at exactly zero temperature showing universal, non-trivial properties with divergent correlation lengths governed by conformal field theory [51]. In Chapter 5 of this book we have discussed the traditional Thermodynamical Bethe Ansatz (TBA) as developed for the Heisenberg model and the Hubbard model [155, 433–435] on the basis of a method [496] invented for the Bose gas.

Cambridge University Press eBooks•Feb 7, 2005•1 citation
Recent developments in statistical mechanics of quantum integrable systems are reviewed. Those studies are fundamental and have a renewed interest related to newly developing fields such as atomic Bose-Einstein condensations, photonic crystals and quantum computations. After a brief summary of the basic concepts and methods, the following three topics are discussed. First, by the thermal Bethe ansatz (TBA), a hard-core Bose gas is exactly solved. The model includes fully the effect of excluded volume and is identified to be a c=1 conformal field theory. Second, the cluster expansion method based on the periodic boundary condition for the Bethe wave function, which we call the Bethe ansatz cluster expansion (BACE) method, is developed for a δ-function gas and the XXX Heisenberg chain. This directly proves the TBA and reveals intrinsic properties of quantum integrable systems. Third, for a δ-function gas, the integral equations for the distribution functions of the quasi-momentum and the quasi-particle energy are solved in the form of power series. In the weak coupling case, the results reproduce those of Bogoliubov theory.
Presented is a method, which we call Bethe ansatz cluster expansion (BACE) method, to investigate statistical mechanics of quantum integrable systems. By use of only the Bethe ansatz equation, n -particle partition functions are exactly calculated. From the expression for the partition function, the n -particle cluster integrals are derived. The results completely agree with those of the thermal Bethe ansatz (TBA) by Yang and Yang. Thus, the BACE method directly proves the validity of the TBA. Partition theory and graphical representation are used to make the presentation simple and transparent.
Abstract 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.
The properties of exactly integrable two-dimensional quantum systems are reviewed and discussed. The nature of exact integrability as a physical phenomenon and various aspects of the mathematical formalism are explored by discussing several examples, including detailed treatments of the nonlinear Schr\"odinger (delta-function gas) model, the massive Thirring model, and the six-vertex (ice) model. The diagonalization of a Hamiltonian by Bethe's Ansatz is illustrated for the nonlinear Schr\"od\'{\i}nger model, and the integral equation method of Lieb for obtaining the spectrum of the many-body system from periodic boundary conditions is reviewed. Similar methods are applied to the massive Thirring model, where the fermion-antifermion and bound-state spectrum are obtained explicitly by the integral equation method. After a brief review of the classical inverse scattering method, the quantum inverse method for the nonlinear Schr\"odinger model is introduced and shown to be an algebraization of the Bethe Ansatz technique. In the quantum inverse method, an auxiliary linear problem is used to define nonlocal operators which are functionals of the original local field on a fixed-time string of arbitrary length. The particular operators for which the string is infinitely long (free boundary conditions) or forms a closed loop around a cylinder (periodic boundary conditions) correspond to the quantized scattering data and have a special significance. One of them creates the Bethe eigenstates, while the other is the generating function for an infinite number of conservation laws. The analogous operators on a lattice are constructed for the symmetric six-vertex model, where the object which corresponds to a solution of the auxiliary linear problem is a string of vertices contracted over horizontal links (arrows). The relationship between the quantum inverse method and the transfer matrix formalism is exhibited. The inverse Gel'fand-Levitan transform which expresses the local field operator as a functional of the quantized scattering data is formulated for the nonlinear Schr\"odinger equation, and some interesting properties of this transformation are noted, including its reduction to a Jordan-Wigner transformation in the limit of infinitely repulsive coupling.
A very curious situation arises in the context of the calculation of the partition function from the spectrum of an integrable Hamiltonian. Despite the validity of the Bethe ansatz equations for all energy eigenvalues of the model the direct evaluation of the partition function is rather difficult. In contrast to ideal quantum gases the eigenstates are not explicitly known: the Bethe ansatz equations provide just implicit descriptions that pose problems of their own kind. Yet, knowing the behaviour of quantum chains at finite temperatures is important for many reasons. As a matter of fact, the groundstate is strictly inaccessible due to the very fundamentals of thermodynamics. Therefore the study of finite temperatures is relevant for theoretical as well as experimental reasons. At high temperatures, quantum systems show only trivial static properties without correlations. Lowering the temperature, the systems enter a large regime with non-universal correlations and finally approach the quantum critical point at exactly zero temperature showing universal, non-trivial properties with divergent correlation lengths governed by conformal field theory [51]. In Chapter 5 of this book we have discussed the traditional Thermodynamical Bethe Ansatz (TBA) as developed for the Heisenberg model and the Hubbard model [155, 433–435] on the basis of a method [496] invented for the Bose gas.