Solvable Models In Statistical Mechanics Research Articles

Randomness, Integrability and Universality

In spring 2022, the Galileo Galilei Institute for Theoretical Physics hosted a seven-week Workshop on ‘Randomness, Integrability and Universality’. The Workshop addressed a series of questions on exactly solvable models of statistical mechanics, having numerous ties and overlaps with various problems in random matrix models, probability, representation theory, and combinatorics. Much recent progress in these areas exploits the underlying notion of quantum integrability. Here we report on the scientific motivations and background of this activity and on its main outputs.

Generalised Onsager Algebra in Quantum Lattice Models

The Onsager algebra is one of the cornerstones of exactly solvable models in statistical mechanics. Starting from the generalised Clifford algebra, we demonstrate its relations to the graph Temperley--Lieb algebra, and a generalisation of the Onsager algebra. We present a series of quantum lattice models as representations of the generalised Clifford algebra, possessing the structure of a special type of the generalised Onsager algebra [1]. The integrability of those models is presented, analogous to the free fermionic eight-vertex model. We also mention further extensions of the models and physical properties related to the generalised Onsager algebras, hinting at a general framework that includes families of quantum lattice models possessing the structure of the generalised Onsager algebras.

Hyperbolic and trigonometric hypergeometric solutions to the star-star equation

We construct the hyperbolic and trigonometric solutions to the star-star relation via the gauge/YBE correspondence by using the three-dimensional lens partition function and superconformal index for a certain mathcal N=2 supersymmetric gauge dual theories. This correspondence relates supersymmetric gauge theories to exactly solvable models of statistical mechanics. The equality of partition functions for the three-dimensional supersymmetric dual theories can be written as an integral identity for hyperbolic and basic hypergeometric functions.

Lens partition function, pentagon identity, and star-triangle relation

We study the three-dimensional lens partition function for $\mathcal N=2$ supersymmetric gauge dual theories on $S^3/\mathbb{Z}_r$ by using the gauge/YBE correspondence. This correspondence relates supersymmetric gauge theories to exactly solvable models of statistical mechanics. The equality of partition functions for the three-dimensional supersymmetric dual theories can be written as an integral identity for hyperbolic hypergeometric functions. We obtain such an integral identity which can be written as the star-triangle relation for Ising type integrable models and as the integral pentagon identity. The latter represents the basic 2-3 Pachner move for triangulated 3-manifolds. A special case of our integral identity can be used for proving orthogonality and completeness relation of the Clebsch-Gordan coefficients for the self-dual continuous series of $U_q(osp(1|2))$.

The Bethe ansatz for the six-vertex and XXZ models: An exposition

In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector $\psi$ and energy $\Lambda$, which satisfy $V \psi = \Lambda \psi$, where $V$ is the the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights $a= b=1$ and $c > 0$. We also show that the same vector $\psi$ satisfies $H \psi = E \psi$, where $H$ is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value $E$ computed explicitly. Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors that amounts to proving that the random cluster model on $\mathbb{Z}^2$ with cluster weight $q >4$ exhibits a first-order phase transition.

Quantum Cohomology via Vicious and Osculating Walkers

We relate the counting of rational curves intersecting Schubert varieties of the Grassmannian to the counting of certain non-intersecting lattice paths on the cylinder, so-called vicious and osculating walkers. These lattice paths form exactly solvable statistical mechanics models and are obtained from solutions to the Yang–Baxter equation. The eigenvectors of the transfer matrices of these models yield the idempotents of the Verlinde algebra of the gauged \({\mathfrak{\hat{u}}(n)_{k}}\)-WZNW model. The latter is known to be closely related to the small quantum cohomology ring of the Grassmannian. We establish further that the partition functions of the vicious and osculating walker model are given in terms of Postnikov’s toric Schur functions and can be interpreted as generating functions for Gromov–Witten invariants. We reveal an underlying quantum group structure in terms of Yang–Baxter algebras and use it to give a generating formula for toric Schur functions in terms of divided difference operators which appear in known representations of the nil-Hecke algebra.

Functional relations and the Yang-Baxter algebra

Functional equations methods are a fundamental part of the theory of Exactly Solvable Models in Statistical Mechanics and they are intimately connected with Baxter's concept of commuting transfer matrices. This concept has culminated in the celebrated Yang-Baxter equation which plays a fundamental role for the construction of quantum integrable systems and also for obtaining their exact solution. Here I shall discuss a proposal that has been put forward in the past years, in which the Yang-Baxter algebra is viewed as a source of functional equations describing quantities of physical interest. For instance, this method has been successfully applied for the description of the spectrum of open spin chains, partition functions of elliptic models with domain wall boundaries and scalar product of Bethe vectors. Further applications of this method are also discussed.

Announcement: Special issue on “Integrable Quantum Systems and Solvable Statistical Mechanics Models”

First Page

Announcement: Special issue on “Integrable Quantum Systems and Solvable Statistical Mechanics Models”

First Page

An elementary approach to 6j-symbols (classical, quantum, rational, trigonometric, and elliptic)

Elliptic 6j-symbols first appeared in connection with solvable models of statistical mechanics. They include many interesting limit cases, such as quantum 6j-symbols (or q-Racah polynomials) and Wilson’s biorthogonal 10 W 9 functions. We give an elementary construction of elliptic 6j-symbols, which immediately implies several of their main properties. As a consequence, we obtain a new algebraic interpretation of elliptic 6j-symbols in terms of Sklyanin algebra representations.

The Six and Eight-Vertex Models Revisited

Elliott Lieb's ice-type models opened up the whole field of solvable models in statistical mechanics. Here we discuss the ``commuting transfer matrix'' $T, Q$ equations for these models, writing them in a more explicit and transparent notation that we believe offers new insights. The approach manifests the relationship between the six-vertex and chiral Potts models, and between the eight-vertex and Kashiwara-Miwa models.

Residue Symbols and Jantzen–Seitz Partitions

Jantzen–Seitz partitions are thosep-regular partitions ofnwhich labelp-modular irreducible representations of the symmetric groupSnwhich remain irreducible when restricted toSn−1; they have recently also been found to be important for certain exactly solvable models in statistical mechanics. In this article we study their combinatorial properties via a detailed analysis of their residue symbols; in particular thep-cores of Jantzen–Seitz partitions are determined.

Solvable models in statistical mechanics, from Onsager onward

There is now a whole field in mathematical physics concerned with solvable models in statistical mechanics, field theory, and related areas. We indicate the influence that Onsager's solution of the planar Ising model has had, and continues to have, on this field.

Search for different links with the same Jones' type polynomials: Ideas from graph theory and statistical mechanics

We describe in this talk three methods of constructing different links with the same Jones type invariant. All three can be thought as generalizations of mutation. The first combines the satellite construction with mutation. The second uses the notion of rotant, taken from the graph theory, the third, invented by Jones, transplants into knot theory the idea of the Yang-Baxter equation with the spectral parameter (idea employed by Baxter in the theory of solvable models in statistical mechanics). We extend the Jones result and relate it to Traczyk’s work on rotors of links. We also show further applications of the Jones idea, e.g. to 3-string links in the solid torus. We stress the fact that ideas coming from various areas of mathematics (and theoretical physics) has been fruitfully used in knot theory, and vice versa. 0. Introduction. Exactly ten year ago, at spring of 1984, Vaughan Jones introduced his (Laurent) polynomial invariant of links, VL(t). He checked immediately that it distinguishes many knots which were not taken apart by the Alexander polynomial, e.g. the right handed trefoil knot from the left handed trefoil knot, and the square knot from the granny knot; Fig. 0.1. 1991 Mathematics Subject Classification: Primary: 57M25. Secondary: 82B.

Anisotropic extension of 2D solvable models in statistical mechanics to 3D

Here we generalize the method proposed by Borovick et al., to construct 3D anisotropic statistical models solvable via the Bethe ansatz, starting from the solvable 2D vertex model. These 3D models have positive Boltzmann weights and nearest neighbour interactions. The source model must possess a charge conservation condition. As an example, the 6-vertex and Isergin-Korepin 19-vertex models are considered.

Solvable statistical mechanics models on group lattices

We define a new type of lattice which is associated with a group G. Field theories on group lattices are considered and in certain cases the models are exactly solvable. The statistical mechanics of such systems is analyzed: the partition function and correlation functions are computed. The solutions utilize the irreducible matrix representations of G.

A canonical form of an algebraic equation for genus 3 Reimann surfaces

We propose a canonical form of an algebraic equation describing each conformal equivalence class of genus 3 nonhyperelliptic Riemann surface. The equation yields a 3-sheet covering with nine ramification points. This result can have applications in string theory and in the theory of solvable models in statistical mechanics.

Knot invariants and cellular automata

There is a deep connection between the theory of invariants of knots in three dimensions and certain classes of solvable models in statistical mechanics. We study the consequences of this for a class of cellular automaton systems. Knot invariants furnish both a non-perturbative framework for the classification and analysis of such models and a primary tool with which to study the emergence of thermodynamic behavior in extended systems.

Quantum groups and representations of monoidal categories

This paper is intended to make explicit some aspects of the interactions which have recently come to light between the theory of classical knots and links, the theory of monoidal categories, Hopf-algebra theory, quantum integrable systems, the theory of exactly solvable models in statistical mechanics, and quantum field theories. The main results herein show an intimate relation between representations of certain monoidal categories arising from the study of new knot invariants or from physical considerations and quantum groups (that is, Hopf algebras). In particular categories of modules and comodules over Hopf algebras would seem to be much more fundamental examples of monoidal categories than might at first be apparent. This fundamental role of Hopf algebras in monoidal categories theory is also manifest in the Tannaka duality theory of Deligne and Mime [8a], although the relationship of that result and the present work is less clear than might be hoped.

BRAIDS, LINK POLYNOMIALS AND TRANSFORMATIONS OF SOLVABLE MODELS

It is shown that braid matrices and link polynomials can be systematically constructed from exactly solvable models in statistical mechanics. Through symmetry breaking transformations, different braid matrices are derived from a solvable model. By associating the Markov traces with multi-variable representations, multi-variable link polynomials are obtained. Infinitesimal operators for braid matrices are constructed. Connection of our approach to the conformal field theories and the topological quantum field theory is discussed.