Multi-colour braid-monoid algebras

Birman–Murakami–Wenzl (BMW) algebra was introduced in connection with knot theory. We treat here interaction round the face solvable (IRF) lattice models. We assume that the face transfer matrix obeys a cubic polynomial equation, which is called the three block case. We prove that the three block theories all obey the BMW algebra. We exemplify this result by treating in detail the SU(2)2×2 fused models, and showing explicitly the BMW structure. We use the connection between the construction of solvable lattice models and conformal field theory. This result is important to the solution of IRF lattice models and the development of new models, as well as to knot theory.

Here we present a statistical mechanical lattice model which is exactly solvable using a matrix method and allows treatment of adsorption induced gate opening structural transformations of metal-organic frameworks which are nanoporous materials with exceptional adsorption properties. Modelling of these structural changes presents a serious theoretical challenge when the solid and gas species are treated in an even handed way. This exactly solvable model complements other simulation based approaches. The methodology presented here highlights the competition between the potential for adsorption and the energy required for structural transition as a driving force for the features in the adsorption isotherms.

Introduction to algorithms for Monte Carlo simulations and their application to QCD

Lattice models in statistical mechanics for N=2 superconformal field theory

Applications of the mathematical formalism put forward in the previous paper (see ibid., p.3471 (1987)) are made to a number of well known lattice models in statistical mechanics. Sequences of algebraic curves are obtained from the branches of an algebraic function Lambda 1, where Lambda +1 at real temperatures is the branch of Lambda 1 which is the partition function per site of an m* infinity lattice section. These sequences are viewed as approximations to parts of the limiting locus Cinfinity of partition function zeros. Approximations to critical points can be obtained in a natural extension of these curves beyond the branch points of Lambda +1.

Lattice models in statistical mechanics for N = 2 superconformal field theory

Solvable lattice models with broken [formula omitted] symmetry and Hecke's indefinite modular forms

Coupled map lattices (CML) are a useful tool to study properties of nonlinear spatially extended systems such as spatial patterns, space-time chaos, space-time intermittence, etc. The purpose of this paper is to present the first rigorous results on the subject. Our approach is based on the representation of the correspondent infinite-dimensional dynamical systems as lattice models of statistical mechanics. This allows to prove (Bunimovich and Sinai, 1988) that in CML generated by expanding one-dimensional maps with diffusion-like coupling for small enough coupling there exists a unique Gibbs distribution that is invariant with respect to dynamics and to space translations and mixing. It means that under these conditions the system exhibits turbulent (chaotic in space and time) behaviour.

Kitaev’s quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. The model was originally based on finite groups, and was later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized quantum double model. These ribbon operators are important tools to understand quasi-particle excitations. It turns out that there are some subtleties in defining the operators in contrast to what one would naively think of. In particular, one has to distinguish two classes of ribbons which we call locally clockwise and locally counterclockwise ribbons. Moreover, we point out that the issue already exists in the original model based on finite non-abelian groups, but it seems to not have been noticed in the literature. We show how certain common properties would fail even in the original model if we were not to distinguish these two classes of ribbons. Perhaps not surprisingly, under the new definitions ribbon operators satisfy all properties that are expected. For instance, they create quasi-particle excitations only at the end of the ribbon, and the types of the quasi-particles correspond to irreducible representations of the Drinfeld double of the input Hopf algebra. However, the proofs of these properties are much more complicated than those in the case of finite groups. This is partly due to the complications in dealing with general Hopf algebras rather than group algebras.

These notes provide an introduction to recent work by Kevin Costello in which integrable lattice models of classical statistical mechanics in two dimensions are understood in terms of quantum gauge theory in four dimensions. This construction will be compared to the more familiar relationship between quantum knot invariants in three dimensions and Chern-Simons gauge theory. (Based on a Whittaker Colloquium at the University of Edinburgh and a lecture at Strings 2016 in Beijing.)

A series of solvable lattice models with face interaction are introduced on the basis of the affine Lie algebraXn(1)=An(1),Bn(1),Cn(1),Dn(1). The local states taken on by the fluctuation variables are the dominant integral weights ofXn(1) of a fixed level. Adjacent local states are subject to a condition related to the vector representation ofXn. The Boltzmann weights are parametrized by elliptic theta functions and solve the star-triangle relation.

This paper provides a comprehensive review of our previously obtained results on exactly solvable lattice models, with a primary focus on the Abelian sandpile model, dimer model, loop-erased random walks and their connections to the enumeration of spanning trees.

2p-D-15 Solvable lattice models and N=2 superconformal algebra

We introduce and study twelve multivariable theta functions defined by pfaffians with elliptic function entries. We show that, when the crossing parameter is a cubic root of unity, the domain wall partition function for the eight-vertex-solid-on-solid model can be written as a sum of two of these pfaffians. As a limit case, we express the domain wall partition function for the three-colour model as a sum of two Hankel determinants. We also show that certain solutions of the TQ-equation for the supersymmetric eight-vertex model can be expressed in terms of elliptic pfaffians.

The behaviour of the finite-temperature C-function, defined by Neto and Fradkin (1993 Nucl. Phys. B 400 525), is analysed within a d -dimensional exactly solvable lattice model, recently considered by Vojta (1996 Phys. Rev. B 53 710), which is of the same universality class as the quantum nonlinear O(n) sigma model in the limit n. The scaling functions of C for the cases d = 1 (absence of long-range order), d = 2 (existence of a quantum critical point), d = 4 (existence of a line of finite-temperature critical points that ends up with a quantum critical point) are derived and analysed. The locations of regions where C is monotonically increasing (which depend significantly on d) are exactly determined. The results are interpreted within the finite-size scaling theory that has to be modified for d = 4.