Off-Critical Logarithmic Minimal Models

Abstract

We consider the integrable minimal models $\mathcal{M}(m,m^{\prime };t)$, corresponding to the ${\phi }_{1,3}$ perturbation off-criticality, in the {\it logarithmic limit\,} $m,m^{\prime }\to \infty$, $m/m^{\prime }\to p/p^{\prime }$ where $p,p^{\prime }$ are coprime and the limit is taken through coprime values of $m,m^{\prime }$. We view these off-critical minimal models $\mathcal{M}(m,m^{\prime };t)$ 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 $\mathcal{LM}(p,p^{\prime };t)$ corresponding to the ${\phi }_{1,3}$ perturbation of the critical logarithmic minimal models $\mathcal{LM}(p,p^{\prime })$. 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 $\mathcal{LM}(p,p^{\prime })$. We also calculate the logarithmic limit of certain off-critical observables ${\mathcal{O}}_{r,s}$ related to One Point Functions and show that the associated critical exponents ${\beta }_{r,s}=(2-\alpha ){\Delta }_{r,s}^{p,p^{\prime }}$ produce all conformal dimensions ${\Delta }_{r,s}^{p,p^{\prime }}<\frac{(p^{\prime }-p)(9p-p^{\prime })}{4p{p}^{\prime }}$ in the infinitely extended Kac table. The corresponding Kac labels $(r,s)$ satisfy $(ps-p^{\prime }r{)}^2<8p(p^{\prime }-p)$. The exponent $2-\alpha =\frac{p^{\prime }}{2(p^{\prime }-p)}$ is obtained from the logarithmic limit of the free energy giving the conformal dimension ${\Delta }_t=\frac{1-\alpha }{2-\alpha }=\frac{2p-p^{\prime }}{p^{\prime }}={\Delta }_{1,3}^{p,p^{\prime }}$ for the perturbing field $t$. As befits a non-unitary theory, some observables ${\mathcal{O}}_{r,s}$ diverge at criticality.

18 pages, 5 figures; version 3 contains amplifications and minor typographical corrections

References in corpus (3)

• A modular invariant bulk theory for the c=0 triplet model
• Field theory of Ising percolating clusters
• Coset Graphs in Bulk and Boundary Logarithmic Minimal Models

Cited by in corpus (4)

• Fusion hierarchies, T-systems and Y-systems of logarithmic minimal models
• Coset construction of logarithmic minimal models: branching rules and branching functions
• Kac boundary conditions of the logarithmic minimal models
• Infinitely extended Kac table of solvable critical dense polymers