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Introduction to Special Issue: Integrable Quantum Systems and Solvable Statistical Mechanics Models

Alexander Rudol'fovich Its, Michio Jimbo, Jean Michel Maillet

Journal of Mathematical PhysicsAug 27, 20091 citation

Quantum Integrable Models in Discrete 2 + 1 Dimensional Space-Time: Auxiliary Linear Problem on a Lattice, Zero-Curvature Representation, Isospectral Deformation of the Zamolodchikov-Bazhanov-Baxter Model

С. М. Сергеев

An invariant approach to quantum integrable models in wholly discrete 2 + 1-dimensional spacetime is considered. An auxiliary linear problem on two-dimensional lattices generalizing quantum chains is formulated. A method of constructing a complete set of

Jan 1, 200411 citations

Quantum and Classical Statistical Mechanics of the Integrable Models in 1 + 1 Dimensions

R. Bullough, D. J. Pilling, JUSSI TIMONEN +2 more

Research reports in physicsJan 1, 19900 citations

Ising model

Belal Ehsan Baaquie

The simplest possible quantum degree of freedom has only two discrete possible values and is called an Ising spin. This could be a spin system having only two possible values, namely “up” or “down,” or this could be a quantum particle that can occupy only two possible positions. To simplify the problem to its bare essentials, a two-state system is studied in zero-dimension space and in Euclidean time; the two-state system occupies a single point and, as it evolves in time, it can point either up or down. Furthermore, time evolution is simplified by discretizing time into a finite time lattice. The two-state degree of freedom in zero-dimension space propagating on a time lattice is called the Ising model. The one-dimensional Ising model is a toy model that is studied to develop the essential ideas of quantum mechanics. The concepts of state space, Hamiltonian, evolution kernel, and path integration have a discrete and simple representation in the Ising model. The Ising model can be viewed as a statistical mechanical system in thermal equilibrium, with the Ising spins occupying a one-dimensional space lattice, and hence is called the one-dimensional Ising model. The Ising model can be defined on higher dimensional lattices and forms one of the bedrocks of theoretical statistical mechanics. In Sections 8.1 to 8.5 the degree of freedom, state space, Hamiltonian, path integral and correlator for the Ising model are discussed. In Section 8.6, the technique of spin decimation is illustrated using an Ising model with coupling depending on the lattice site.The simplest possible quantum degree of freedom has only two discrete possible values and is called an Ising spin. This could be a spin system having only two possible values, namely “up” or “down,” or this could be a quantum particle that can occupy only two possible positions. To simplify the problem to its bare essentials, a two-state system is studied in zero-dimension space and in Euclidean time; the two-state system occupies a single point and, as it evolves in time, it can point either up or down. Furthermore, time evolution is simplified by discretizing time into a finite time lattice. The two-state degree of freedom in zero-dimension space propagating on a time lattice is called the Ising model. The one-dimensional Ising model is a toy model that is studied to develop the essential ideas of quantum mechanics. The concepts of state space, Hamiltonian, evolution kernel, and path integration have a discrete and simple representation in the Ising model. The Ising model can be viewed as a statistical mechanical system in thermal equilibrium, with the Ising spins occupying a one-dimensional space lattice, and hence is called the one-dimensional Ising model. The Ising model can be defined on higher dimensional lattices and forms one of the bedrocks of theoretical statistical mechanics. In Sections 8.1 to 8.5 the degree of freedom, state space, Hamiltonian, path integral and correlator for the Ising model are discussed. In Section 8.6, the technique of spin decimation is illustrated using an Ising model with coupling depending on the lattice site.

Cambridge University Press eBooksMar 27, 20140 citations

Quantum Integrable Models on 1 + 1 Discrete Space Time

L. D. Faddeev, A. Yu. Volkov

NATO ASI series. Series B : PhysicsJan 1, 19973 citations

A master solution of the quantum Yang–Baxter equation and classical discrete integrable equations

Vladimir V. Bazhanov, С. М. Сергеев

We obtain a new solution of the star-triangle relation with positive Boltzmann weights, which contains as special cases all continuous and discrete spin solutions of this relation, that were previously known.This new master solution defines an exactly solvable two lattice model of statistical mechanics, which involves continuous spin variables, living on a circle, and contains two temperature-like parameters.If one of the these parameters approaches a root of unity (corresponds to zero temperature), the spin variables freezes into discrete positions, equidistantly spaced onWe obtain a new solution of the star-triangle relation with positive Boltzmann weights, which contains as special cases all continuous and discrete spin solutions of this relation, that were previously known.This new master solution defines an exactly solvable two lattice model of statistical mechanics, which involves continuous spin variables, living on a circle, and contains two temperature-like parameters.If one of the these parameters approaches a root of unity (corresponds to zero temperature), the spin variables freezes into discrete positions, equidistantly spaced on

arXiv:1006.0651Advances in Theoretical and Mathematical PhysicsJan 1, 201277 citations

Complex of three-dimensional solvable models

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