Off-critical logarithmic minimal models

Abstract

We consider the integrable minimal models ℳ(m,m′;t), corresponding to the φ1,3 perturbation off-criticality, in the logarithmic limit m,m′ → ∞, m/m′ → p/p′ where p,p′ are coprime and the limit is taken through coprime values of m,m′. We view these off-critical minimal models ℳ(m,m′;t) as the continuum scaling limit of the Forrester–Baxter restricted solid-on-solid (RSOS) models on the square lattice. Applying corner transfer matrices to the Forrester–Baxter RSOS models in regime III, we argue that taking first the thermodynamic limit and second the logarithmic limit yields off-critical logarithmic minimal models ℒℳ(p,p′;t) corresponding to the φ1,3 perturbation of the critical logarithmic minimal models ℒℳ(p,p′). Specifically, in accord with the Kyoto correspondence principle, we show that the logarithmic limit of the one-dimensional configurational sums yields finitized quasi-rational characters of the Kac representations of the critical logarithmic minimal models ℒℳ(p,p′). We also calculate the logarithmic limit of certain off-critical observables \U0001d4aar,s related to one-point functions and show that the associated critical exponents produce all conformal dimensions in the infinitely extended Kac table. The corresponding Kac labels (r,s) satisfy (ps−p′r)2 < 8p(p′ − p). The exponent 2 − α = p′/2(p′ − p) is obtained from the logarithmic limit of the free energy giving the conformal dimension for the perturbing field t. As befits a non-unitary theory, some observables \U0001d4aar,s diverge at criticality.