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arXiv:2110.09798 · Journal of Physics Communications

The crossing multiplier for solvable lattice models

Alexander A. Belavin, Doron Gepner, J. Ramos Cabezas

Published Jul 22, 2022• 3 citations
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Abstract

Abstract We study the large class of solvable lattice models, based on the data of conformal field theory. These models are constructed from any conformal field theory. We consider the lattice models based on affine algebras described by Jimbo et al, for the algebras ABCD and by Kuniba et al for G 2. We find a general formula for the crossing multipliers of these models. It is shown that these crossing multipliers are also given by the principally specialized characters of the model in question. Therefore we conjecture that the crossing multipliers in this large class of solvable interaction round the face lattice models are given by the characters of the conformal field theory on which they are based. We use this result to study the local state probabilities of these models and show that they are given by the branching rule, in regime III.

DOI
10.1088/2399-6528/ac8392

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Abstract We study the large class of solvable lattice models, based on the data of conformal field theory. These models are constructed from any conformal field theory. We consider the lattice models based on affine algebras described by Jimbo et al, for the algebras ABCD and by Kuniba et al for G 2. We find a general formula for the crossing multipliers of these models. It is shown that these crossing multipliers are also given by the principally specialized characters of the model in question. Therefore we conjecture that the crossing multipliers in this large class of solvable interaction round the face lattice models are given by the characters of the conformal field theory on which they are based. We use this result to study the local state probabilities of these models and show that they are given by the branching rule, in regime III.
We consider the lattice models based on affine algebras described by Jimbo et al., for the algebras 𝐴𝐵𝐶𝐷 and by Kuniba et al. for 𝐺2. We find a general formula for the crossing multipliers of these models, with the result that these crossing multipliers are given by the principally specialized characters of the model in question. Therefore we conjecture that the crossing multipliers in a large class of solvable interaction round the face lattice models are given by the characters of the conformal field theory on which they are based. We also elaborate on some details of the computation of the Local state probability of these models, conjecturing an expression for it.
Local State Probabilities of Solvable Lattice Models: Relatives of An(1)A_{n}^{(1)}An(1)​ Family
We present the results for the local state probabilities (LSP) of the solvable lattice models, constructed around rational conformal field theory given by WZW model on SO(3)4R=SU(2)4R/Z2SO(3)_{4 R}=SU(2)_{4 R} / Z_{2}SO(3)4R​=SU(2)4R​/Z2​ together with primary field ϕ1ϕ_{1}ϕ1​(symmetric tensor of degree 2). Some conjectures for the LSP for some higher rank relatives of An(1)A_{n}^{(1)}An(1)​ face models are also presented.
We conjecture the inversion relations for thermalized solvable interaction round the face (IRF) two dimensional lattice models. We base ourselves on an ansatz for the Baxterization described in the 90's. We solve these inversion relations in the four main regimes of the models, to give the free energy of the models, in these regimes. We use the method of Baxter in the calculation of the free energy of the hard hexagon model. We believe these results to be quite general, shared by most of the known IRF models. Our results apply equally well to solvable vertex models. Using the expression for the free energy we calculate the critical exponent α, and from it the dimension of the perturbing (thermal) operator in the fixed point conformal field theory (CFT). We show that it matches either the coset O/G or G/O, where O is the original CFT used to define the model and G is some unknown CFT, depending on the regime. This agrees with known examples of such models by Huse and Jimbo et al.
This paper represents a continuation of our previous work, where the Bolzmann weights (BWs) for several Interaction-Round-the Face (IRF) lattice models were computed using their relation to rational conformal field theories. Here, we focus on deriving solutions for the Boltzmann weights of the Interaction-Round the Face lattice model, specifically the unrestricted face model, based on the su(3)k\mathfrak{su}(3)_ksu(3)k​ affine Lie algebra. The admissibility conditions are defined by the adjoint representation. We find the BWs by determining the quantum RRR matrix of the Uq(sl(3))U_q(\mathfrak{sl} (3))Uq​(sl(3)) quantum algebra in the adjoint representation and then applying the so-called Vertex-IRF correspondence. The Vertex-IRF correspondence defines the BWs of IRF models in terms of RRR matrix elements.
The Yang-Baxter-Zamolodchikov-Faddeev (YBZF) algebras and their many applications are the subject of this reivew. I start by the solvable lattice statistical models constructed from YBZF algebras. All two-dimensional integrable vertex models follow in this way and are solvable via Bethe Ansatz (BA) and their generalizations. The six-vertex model solution and its q(2q−1) vertex generalization including its nested BA construction are exposed. YBZF algebras and their associated physical models are classified in terms of simple Lie algebras. It is shown how these lattice models yield both solvable massive quantum field theories (QFT) and conformal models in appropriated scaling (continuous) limits within the lattice light-cone approach. The method of finite-size calculations from the BA is described as well as its applications to derive the conformal properties of integrable lattice models. It is conjectured that all integrable QFT and conformal models follow in a scaling limit from these YBZF algebras. A discussion on braid and quantum groups concludes this review.
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.