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Functional Relations in Solvable Lattice Models I: Functional Relations and Representation Theory

Atsuo Kuniba, Tomoki Nakanishi, Junji Suzuki

Published Sep 26, 1993• 222 citations
OpenAlex

Abstract

Abstract. Reported are two applications of the functional relations (T-system) among a commuting family of row-to-row transfer matrices proposed in the previous paper Part I. For a general simple Lie algebra Xr, we determine the correlation lengths of the associated massive vertex models in the anti-ferroelectric regime and central charges of the RSOS models in two critical regimes. The results reproduce known values or even generalize them, demonstrating the efficiency of the T-system.

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Algebraic analysis of solvable lattice models

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Finite-size corrections and scaling dimensions of solvable lattice models: An analytic method

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Physical Review Letters•Feb 25, 1991•109 citations
Abstract. Reported are two applications of the functional relations (T-system) among a commuting family of row-to-row transfer matrices proposed in the previous paper Part I. For a general simple Lie algebra Xr, we determine the correlation lengths of the associated massive vertex models in the anti-ferroelectric regime and central charges of the RSOS models in two critical regimes. The results reproduce known values or even generalize them, demonstrating the efficiency of the T-system.
Abstract. We study a system of functional relations among a commuting family of row-to-row transfer matrices in solvable lattice models. The role of exact sequences of the finite dimensional quantum group modules is clarified. We find a curious phenomenon that the solutions of those functional relations also solve the so-called thermodynamic Bethe ansatz equations in the high temperature limit for sl(r + 1) models. Based on this observation, we propose possible functional relations for models associated with all the simple Lie algebras. We show that these functional relations certainly fulfill strong constraints coming from the fusion procedure analysis. The application to the calculations of physical quantities will be presented in the subsequent publication.
We study a system of functional relations among a commuting family of row-to-row transfer matrices in solvable lattice models. The role of exact sequences of the finite-dimensional quantum group modules is clarified. We find a curious phenomenon where the solutions of those functional relations also solve the so-called thermodynamic Bethe ansatz equations in the high temperature limit for sl(r+1) models. Based on this observation, we propose possible functional relations for models associated with all the simple Lie algebras. We show that these functional relations certainly fulfill strong constraints coming from the fusion procedure analysis. The application to the calculations of physical quantities will be presented in the subsequent paper.
Determinantal functional equations satisfied by the row transfer matrix eigenvalues of critical A–D–E lattice spin models are presented. These are obtained for models associated with the Lie algebras [Formula: see text], [Formula: see text], AL, DL and E6,7,8 by exploiting connections with functional equations satisfied by the row transfer matrix eigenvalues of the six-vertex model at rational values of the crossing parameter λ=sπ/h where h is the Coxeter number. In addition, fusion is used to derive special functional equations, called inversion identity hierarchies, which provide the key to the direct calculation of finite-size corrections, central charges and conformal weights for the critical A–D–E lattice models.
Background of the problem The spin 1/21/21/2 XXZ model for Δ<−1\Delta <-1Δ<−1 The six-vertex model in the anti-ferroelectric regime Solvability and symmetry Correlation functions-physical derivation Level one modules and bosonization Vertex operators Space of states-mathematical picture Traces of vertex operators Correlation functions and form factors The XXXXXXXXX limit q→−1q\rightarrow-1q→−1 Discussions List of formulas.
A general analytic method for calculating finite-size corrections, central charges, and scaling dimensions of solvable lattice models is presented. The approach is to solve the special functional equations or inversion identities satisfied by the commuting row transfer matrices of these lattice models at criticality. For purposes of illustration, the method is applied to calculate the central charge c=4/5 and leading magnetic scaling dimension x=2/15 of hard hexagons. These numbers are rational due to special values of Rogers dilogarithms.