One-Dimensional Sums and Finitized Characters of $2\times 2$ Fused RSOS Models

Abstract

Tartaglia and Pearce have argued that the nonunitary $n\times n$ fused Forrester-Baxter $\mbox{RSOS}(m,m')$ models are described, in the continuum scaling limit, by the minimal models $\mathcal{M}(M,M^{\prime },n)$ constructed as the higher-level conformal cosets $(A_1^{(1)}{)}_k\otimes (A_1^{(1)}{)}_n/(A_1^{(1)}{)}_{k+n}$ at integer fusion level $n\ge 1$ and fractional level $k=nM/(M^{\prime }-M)-2$ with $(M,M^{\prime })=(nm-(n-1)m^{\prime },m^{\prime })$. These results rely on Yang-Baxter integrability and are valid in Regime III for models determined by the crossing parameter $\lambda =(m^{\prime }-m)\pi /m^{\prime }$ in the interval $0<\lambda <\pi /n$. Here we consider the $2\times 2$ $\mbox{RSOS}(m,m')$ models in the interval $\frac{\pi }{2}<\lambda <\pi$ and investigate the associated one-dimensional sums. In this interval, we verify that the one-dimensional sums produce new finitized Virasoro characters $c{h}_{r,s}^{(N)}(q)$ of the minimal models $\mathcal{M}(m,m^{\prime },1)$ with $m^{\prime }>2m$. We further conjecture finitized bosonic forms and check that these agree with the ground state one-dimensional sums out to system sizes $N=12$. The $2\times 2$ $\mbox{RSOS}(m,m')$ models thus realize new Yang-Baxter integrable models in the universality classes of the minimal models $\mathcal{M}(m,m^{\prime },1)$. For the series $\mathcal{M}(m,2m+1,1)$ with $m\ge 2$, the spin-1 one-dimensional sums were previously analysed by Jacob and Mathieu without the underlying Yang-Baxter structure. Finitized Kac characters ${\chi }_{r,s}^{m,m^{\prime };(N)}(q)$ for the logarithmic minimal models $\mathcal{LM}(p,p^{\prime },1)$ are also obtained for $p^{\prime }\ge 2p$ by taking the logarithmic limit $m,m^{\prime }\to \infty$ with $m^{\prime }/m\to p^{\prime }/p+$.