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:solv-int/9912001

Construction of variable mass sine-Gordon and other novel inhomogeneous quantum integrable models

Anjan Kundu

Published Nov 26, 1999• 0 citations
arXivHTMLPDF

Abstract

The inhomogeneity of the media or the external forces usually destroy the integrability of a system. We propose a systematic construction of a class of quantum models, which retains their exact integrability inspite of their explicit inhomogeneity. Such models include variable mass sine-Gordon model, cylindrical NLS, spin chains with impurity, inhomogeneous Toda chain, the Ablowitz-Ladik model etc.

DOI
10.48550/arxiv.solv-int/9912001

Related papers

Algebraic Construction of Quantum Integrable Models Including Inhomogeneous Models

Anjan Kundu

Exploiting the quantum integrability condition we construct an ancestor model associated with a new underlying quadratic algebra. This ancestor model represents an exactly integrable quantum lattice inhomogeneous anisotropic model and at its various realizations and limits can generate a wide range of integrable models. They cover quantum lattice as well as field models associated with the quantum $R$-matrix of trigonometric type or at the undeformed $q \\to 1$ limit similar models belonging to the rational class. The classical limit likewise yields the corresponding classical discrete and field models. Thus along with the generation of known integrable models in a unifying way a new class of inhomogeneous models including variable mass sine-Gordon model, inhomogeneous Toda chain, impure spin chains etc. are constructed.

•Jan 1, 1999•3 citations

1 Yang-Baxter algebra and generation of quantum integrable models

Anjan Kundu

An operator defoemed quantum algebra is discovered exploiting the quantum Yang-Baxter equation with trigonometric R-matrix. This novel Hopf algebra along with its q → 1 limit seem to be the most general Yang-Baxter algebra underlying quantum integrable systems. Three different directions of application of this algebra in integrable systems depending on different sets of values of deforming operators are identified. Fixed values on the whole lattice yield subalgebras linked to standard quantum integrable models, while the associated Lax operators generate and classify them in an unified way. Variable values construct a new series of quantum integrable inhomogeneous models. Fixed but different values at different lattice sites can produce a novel class of integrable hybrid models including integrable matter-radiation models and quantum field models with defects, in particular, a new quantum integrable sine-Gordon model with defect. PACS numbers 02.30.Ik, 02.20.Uw, 05.50+q, 03.65.Fd 1

•Jan 1, 2008•6 citations

Integrable models: dynamical mass generation

Alexei M. Tsvelik

In our discussion of one-dimensional physics, more than once we have encountered situations when some relevant interaction scales to strong coupling. In such a situation the original description becomes inapplicable at low energies and must be replaced by some other description. In all previous cases I have restricted the discussion by some qualitative analysis, promising to provide more details later. Now the time is ripe to fulfil the promise. In fact, almost all our understanding of strong coupling physics comes from exact solutions of just a few models. The most ubiquitous among them is the sine-Gordon model, which we have already encountered in this book many times. Anyone who thoroughly understands this model and possibly a few others (such as the O(3) nonlinear sigma model and the off-critical Ising model) may consider themself an expert in the area of strongly correlated systems. The sine-Gordon model belongs to a category of integrable field theories. This means that it has an infinite number of constants of motion. On the one hand this fact makes it possible to solve this model exactly and describe its thermodynamics and correlation functions; on the other hand it makes the results less general than one might wish. First of all, not all interesting models are integrable, and one may wonder how and in what respect their behaviour may differ from the behaviour of integrable models. Integrable models possess some exceptional physical properties (such as ballistic transport) which are destroyed when the integrability is violated. There are several important review articles which one can read to familiarize oneself with the subject.

Cambridge University Press eBooks•Nov 20, 2003•0 citations

Construction of Integrable Spin Systems with Long-Range Interactions

Kazuhiro Hikami, P. P. Kulish, Miki Wadati

Presented is a general method to construct quantum spin Hamiltonians with long-range pairwise interactions. The Hamiltonian is given explicitly in terms of the R -matrices on an inhomogeneous lattice. Applications to spin-1/2 chains, higher spin chains and Z n models are shown.

Journal of the Physical Society of Japan•Sep 15, 1992•48 citations

Introduction to Integrable Many-Body systems III

Zoltán Bajnok, Ladislav Šamaj

Introduction to Integrable Many-Body systems III This is the third part of a three-volume introductory course about integrable systems of interacting bodies. The emphasis is put onto the method of Thermodynamic Bethe ansatz. Two kinds of integrable models are studied. Systems of itinerant electrons, forming a part of Condensed Matter Physics, involve the Hubbard lattice model of electrons with short-ranged one-site interactions (Sect. 20) and the s-d exchange Kondo model (Sect. 21), describing the scattering of conduction electrons on a spin-s impurity. Methods and basic concepts used in Quantum Field Theory are explained on the integrable (1 + 1)-dimensional sine-Gordon model. We start with the classical description of the model in Sect. 22, analyze its finite energy field configurations (soliton, anti-soliton and breathers) and show its classical integrability. The model is quantized by using two schemes: the conformal (Sect. 23) and Lagrangian (Sect. 24) quantizations. The scattering matrix of the sine-Gordon theory is derived at the full quantum level in the bootstrap scheme and is compared to its classical limit in Sect. 25. The parameters of the scattering matrix are related to those of the Lagrangian by calculating the ground-state energy in an applied magnetic field in two ways: Conformal perturbation theory and Thermodynamic Bethe ansatz (Sect. 26). The relation of the sine-Gordon theory to the XXZ Heisenberg model, which provides a complete solution of the sine-Gordon model in a finite volume, is pointed out in Sect. 27. The obtained results are applied in Sect. 28. to the derivation of the exact thermodynamics for the (symmetric) two-component Coulomb gas; this is the first classical two-dimensional fluid with exactly solvable thermodynamics.

Acta Physica Slovaca Reviews and Tutorials•Apr 1, 2011•11 citations

Algebraic Approach in Unifying Quantum Integrable Models

Anjan Kundu

A novel algebra underlying integrable systems is shown to generate and unify a large class of quantum integrable models with given $R$ matrix, through reductions of an ancestor Lax operator and its different realizations. Along with known discrete and field models a new class of inhomogeneous and impurity models is obtained.

arXiv:hep-th/9810220•Physical Review Letters•May 17, 1999•27 citations
The inhomogeneity of the media or the external forces usually destroy the integrability of a system. We propose a systematic construction of a class of quantum models, which retains their exact integrability inspite of their explicit inhomogeneity. Such models include variable mass sine-Gordon model, cylindrical NLS, spin chains with impurity, inhomogeneous Toda chain, the Ablowitz-Ladik model etc.
Exploiting the quantum integrability condition we construct an ancestor model associated with a new underlying quadratic algebra. This ancestor model represents an exactly integrable quantum lattice inhomogeneous anisotropic model and at its various realizations and limits can generate a wide range of integrable models. They cover quantum lattice as well as field models associated with the quantum RRR-matrix of trigonometric type or at the undeformed qto1q \\to 1qto1 limit similar models belonging to the rational class. The classical limit likewise yields the corresponding classical discrete and field models. Thus along with the generation of known integrable models in a unifying way a new class of inhomogeneous models including variable mass sine-Gordon model, inhomogeneous Toda chain, impure spin chains etc. are constructed.
An operator defoemed quantum algebra is discovered exploiting the quantum Yang-Baxter equation with trigonometric R-matrix. This novel Hopf algebra along with its q → 1 limit seem to be the most general Yang-Baxter algebra underlying quantum integrable systems. Three different directions of application of this algebra in integrable systems depending on different sets of values of deforming operators are identified. Fixed values on the whole lattice yield subalgebras linked to standard quantum integrable models, while the associated Lax operators generate and classify them in an unified way. Variable values construct a new series of quantum integrable inhomogeneous models. Fixed but different values at different lattice sites can produce a novel class of integrable hybrid models including integrable matter-radiation models and quantum field models with defects, in particular, a new quantum integrable sine-Gordon model with defect. PACS numbers 02.30.Ik, 02.20.Uw, 05.50+q, 03.65.Fd 1
In our discussion of one-dimensional physics, more than once we have encountered situations when some relevant interaction scales to strong coupling. In such a situation the original description becomes inapplicable at low energies and must be replaced by some other description. In all previous cases I have restricted the discussion by some qualitative analysis, promising to provide more details later. Now the time is ripe to fulfil the promise. In fact, almost all our understanding of strong coupling physics comes from exact solutions of just a few models. The most ubiquitous among them is the sine-Gordon model, which we have already encountered in this book many times. Anyone who thoroughly understands this model and possibly a few others (such as the O(3) nonlinear sigma model and the off-critical Ising model) may consider themself an expert in the area of strongly correlated systems. The sine-Gordon model belongs to a category of integrable field theories. This means that it has an infinite number of constants of motion. On the one hand this fact makes it possible to solve this model exactly and describe its thermodynamics and correlation functions; on the other hand it makes the results less general than one might wish. First of all, not all interesting models are integrable, and one may wonder how and in what respect their behaviour may differ from the behaviour of integrable models. Integrable models possess some exceptional physical properties (such as ballistic transport) which are destroyed when the integrability is violated. There are several important review articles which one can read to familiarize oneself with the subject.
Introduction to Integrable Many-Body systems III This is the third part of a three-volume introductory course about integrable systems of interacting bodies. The emphasis is put onto the method of Thermodynamic Bethe ansatz. Two kinds of integrable models are studied. Systems of itinerant electrons, forming a part of Condensed Matter Physics, involve the Hubbard lattice model of electrons with short-ranged one-site interactions (Sect. 20) and the s-d exchange Kondo model (Sect. 21), describing the scattering of conduction electrons on a spin-s impurity. Methods and basic concepts used in Quantum Field Theory are explained on the integrable (1 + 1)-dimensional sine-Gordon model. We start with the classical description of the model in Sect. 22, analyze its finite energy field configurations (soliton, anti-soliton and breathers) and show its classical integrability. The model is quantized by using two schemes: the conformal (Sect. 23) and Lagrangian (Sect. 24) quantizations. The scattering matrix of the sine-Gordon theory is derived at the full quantum level in the bootstrap scheme and is compared to its classical limit in Sect. 25. The parameters of the scattering matrix are related to those of the Lagrangian by calculating the ground-state energy in an applied magnetic field in two ways: Conformal perturbation theory and Thermodynamic Bethe ansatz (Sect. 26). The relation of the sine-Gordon theory to the XXZ Heisenberg model, which provides a complete solution of the sine-Gordon model in a finite volume, is pointed out in Sect. 27. The obtained results are applied in Sect. 28. to the derivation of the exact thermodynamics for the (symmetric) two-component Coulomb gas; this is the first classical two-dimensional fluid with exactly solvable thermodynamics.
A novel algebra underlying integrable systems is shown to generate and unify a large class of quantum integrable models with given RRR matrix, through reductions of an ancestor Lax operator and its different realizations. Along with known discrete and field models a new class of inhomogeneous and impurity models is obtained.