Off-Critical Logarithmic Minimal Models

Abstract

We consider the integrable minimal models ${\cal M}(m,m';t)$, corresponding to the $\varphi_{1,3}$ perturbation off-criticality, in the {\it logarithmic limit\,} $m, m'\to\infty$, $m/m'\to p/p'$ where $p, p'$ are coprime and the limit is taken through coprime values of $m,m'$. We view these off-critical minimal models ${\cal M}(m,m';t)$ 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 {\it logarithmic limit\,} yields off-critical logarithmic minimal models ${\cal LM}(p,p';t)$ corresponding to the $\varphi_{1,3}$ perturbation of the critical logarithmic minimal models ${\cal LM}(p,p')$. 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 ${\cal LM}(p,p')$. We also calculate the logarithmic limit of certain off-critical observables ${\cal O}_{r,s}$ related to One Point Functions and show that the associated critical exponents $\beta_{r,s}=(2-\alpha)\,\Delta_{r,s}^{p,p'}$ produce all conformal dimensions $\Delta_{r,s}^{p,p'}<{(p'-p)(9p-p')\over 4pp'}$ in the infinitely extended Kac table. The corresponding Kac labels $(r,s)$ satisfy $(p s-p' r)^2< 8p(p'-p)$. The exponent $2-\alpha ={p'\over 2(p'-p)}$ is obtained from the logarithmic limit of the free energy giving the conformal dimension $\Delta_t={1-\alpha\over 2-\alpha}={2p-p'\over p'}=\Delta_{1,3}^{p,p'}$ for the perturbing field $t$. As befits a non-unitary theory, some observables ${\cal O}_{r,s}$ diverge at criticality.