Off-Critical Logarithmic Minimal Models

We consider the integrable minimal models $\mathcal{M}(m,m^{\mathrm{\prime }};t)$, corresponding to the ${\phi }_{1,3}$ perturbation off-criticality, in the {\it logarithmic limit,} $m,m^{\mathrm{\prime }}\to \mathrm{\infty }$, $m\mathrm{/}{m}^{\mathrm{\prime }}\to p\mathrm{/}{p}^{\mathrm{\prime }}$ where $p,p^{\mathrm{\prime }}$ are coprime and the limit is taken through coprime values of $m,m^{\mathrm{\prime }}$. We view these off-critical minimal models $\mathcal{M}(m,m^{\mathrm{\prime }};t)$ 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 ${\cal LM}(p,p’;t)$correspondingtothe$\varphi_{1,3}$ perturbation of the critical logarithmic minimal models $\mathcal{L}\mathcal{M}(p,p^{\mathrm{\prime }})$. 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 ${\cal LM}(p,p’)$. 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 ${\beta }_{r,s}=(2-\alpha ){\mathrm{\Delta }}_{r,s}^{p,p^{\mathrm{\prime }}}$ produce all conformal dimensions $\Delta_{r,s}^{p,p’}<{(p’-p)(9p-p’)\over 4pp’}$intheinfinitelyextendedKactable\mathrm{.}ThecorrespondingKaclabels$(r,s)$satisfy$(p s-p’ r)^2< 8p(p’-p)$\mathrm{.}Theexponent$2-\alpha ={p’\over 2(p’-p)}$ is obtained from the logarithmic limit of the free energy giving the conformal dimension $\Delta_t={1-\alpha\over 2-\alpha}={2p-p’\over p’}=\Delta_{1,3}^{p,p’}$fortheperturbingfield$t$. As befits a non-unitary theory, some observables ${\mathcal{O}}_{r,s}$ diverge at criticality.