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arXiv:1407.6883 · Nuclear Physics B

Inversion identities for inhomogeneous face models

Holger Frahm, Nikos Karaiskos

Published Sep 2, 2014• 7 citations
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Abstract

We derive exact inversion identities satisfied by the transfer matrix of inhomogeneous interaction-round-a-face (IRF) models with arbitrary boundary conditions using the underlying integrable structure and crossing properties of the local Boltzmann weights. For the critical restricted solid-on-solid (RSOS) models these identities together with some information on the analytical properties of the transfer matrix determine the spectrum completely and allow to derive the Bethe equations for both periodic and general open boundary conditions.

DOI
10.1016/j.nuclphysb.2014.08.013

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We derive exact inversion identities satisfied by the transfer matrix of inhomogeneous interaction-round-a-face (IRF) models with arbitrary boundary conditions using the underlying integrable structure and crossing properties of the local Boltzmann weights. For the critical restricted solid-on-solid (RSOS) models these identities together with some information on the analytical properties of the transfer matrix determine the spectrum completely and allow to derive the Bethe equations for both periodic and general open boundary conditions.
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Using the properties of the local Boltzmann weights of integrable interaction-round-a-face (IRF or face) models we express local operators in terms of generalized transfer matrices. This allows for the derivation of discrete functional equations for the reduced density matrices in inhomogeneous generalizations of these models. We apply these equations to study the density matrices for IRF models of various solid-on-solid type and quantum chains of non-Abelian su(²)³ or Fibonacci anyons. Similar as in the six vertex model we find that reduced density matrices for a sequence of consecutive sites can be’factorized’, i.e. expressed in terms of nearest-neighbour correlators with coefficients which are independent of the model parameters. Explicit expressions are provided for correlation functions on up to three neighbouring sites.
Non-Abelian SU(3)k{SU}{(3)}_{k}SU(3)k​ anyons: inversion identities for higher rank face models
The spectral problem for an integrable system of particles satisfying the fusion rules of is expressed in terms of exact inversion identities satisfied by the commuting transfer matrices of the integrable fused interaction round a face model of Jimbo, Miwa and Okado. The identities are proven using local properties of the Boltzmann weights, in particular the Yang–Baxter equation and unitarity. They are closely related to the consistency conditions for the construction of eigenvalues obtained in the separation of variables approach to integrable vertex models.
We consider the L-state cyclic solid-on-solid lattice models under a class of open boundary conditions. The integrable boundary face weights are obtained by solving the reflection equations. Functional relations for the fused transfer matrices are presented for both periodic and open boundary conditions. The eigenspectra of the unfused transfer matrix is obtained from the functional relations using the analytic Bethe ansatz. For a special case of crossing parameter , the finite-size corrections to the eigenspectra of the critical models are obtained, from which the corresponding conformal dimensions follow. The calculation of the surface free energy away from criticality yields two surface specific heat exponents, and , where coprime to L. These results are in agreement with the scaling relations and .
Abstract Interaction-round the face (IRF) models are two-dimensional lattice models of statistical mechanics defined by an affine Lie algebra and admissibility conditions depending on a choice of representation of that affine Lie algebra. Integrable IRF models, i.e. the models the Boltzmann weights of which satisfy the quantum Yang–Baxter equation, are of particular interest. In this paper, we investigate trigonometric Boltzmann weights of integrable IRF models. By using an ansatz proposed by one of the authors in some previous works, the Boltzmann weights of the restricted IRF models based on the affine Lie algebras su ( 2 ) k and su ( 3 ) k are computed for fundamental and adjoint representations for some fixed levels k. New solutions for the Boltzmann weights are obtained. We also study the vertex-IRF correspondence in the context of an unrestricted IRF model based on su ( 3 ) k (for general k) and discuss how it can be used to find Boltzmann weights in terms of the quantum R ^ matrix when the adjoint representation defines the admissibility conditions.
A bstract This paper represents a continuation of our previous work, where the Boltzmann 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)}_k su(3)k​ su 3 k affine Lie algebra. The admissibility conditions are defined by the adjoint representation. We find the BWs by determining the quantum R matrix of the Uq(sl(3)) {U}_q\left(\mathfrak{sl}(3)\right) Uq​(sl(3)) U q 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 R matrix elements.