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

Tartaglia and Pearce have argued that the nonunitary $n\times n$ fused Forrester-Baxter $RSOS(m,m^{\mathrm{\prime }})$ models are described, in the continuum scaling limit, by the minimal models $\mathcal{M}(M,M^{\mathrm{\prime }},n)$ constructed as the higher-level conformal cosets $(A^{(1)}_1)_k\otimes (A^{(1)}_1)_n/(A^{(1)}1){k+n}$atintegerfusionlevel$n\ge 1$ and fractional level $k=nM\mathrm{/}(M^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}M)-2$ with $(M,M^{\mathrm{\prime }})=(nm-(n\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}1)m^{\mathrm{\prime }},m^{\mathrm{\prime }})$. These results rely on Yang-Baxter integrability and are valid in Regime III for models determined by the crossing parameter $\lambda =(m^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}m)\pi \mathrm{/}{m}^{\mathrm{\prime }}$ in the interval $0<\lambda <\pi \mathrm{/}n$. Here we consider the $2\times 2$ $RSOS(m,m^{\mathrm{\prime }})$ 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 $\mathcal{M}(m,m^{\mathrm{\prime }},1)$ with $m^{\mathrm{\prime }}>2m$. 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 $2\times 2$ $RSOS(m,m^{\mathrm{\prime }})$ models thus realize new Yang-Baxter integrable models in the universality classes of the minimal models $\mathcal{M}(m,m^{\mathrm{\prime }},1)$. 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 ${\chi }_{r,s}^{m,m^{\mathrm{\prime }};(N)}(q)$ for the logarithmic minimal models $\mathcal{L}\mathcal{M}(p,p^{\mathrm{\prime }},1)$ are also obtained for $p^{\mathrm{\prime }}\ge 2p$ by taking the logarithmic limit $m,m^{\mathrm{\prime }}\to \mathrm{\infty }$ with $m^{\mathrm{\prime }}\mathrm{/}m\to p^{\mathrm{\prime }}\mathrm{/}p+$.