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

TL;DR

This paper investigates the one-dimensional sums of 2x2 fused RSOS models in a specific parameter regime, revealing new finitized Virasoro characters for minimal models and proposing conjectured bosonic forms, thus expanding the understanding of integrable models in conformal field theory.

Contribution

It introduces new Yang-Baxter integrable models in the minimal model universality classes and conjectures finitized bosonic forms matching ground state sums for system sizes up to 12.

Findings

Verification of finitized Virasoro characters from one-dimensional sums.

Proposal of conjectured bosonic forms for these characters.

Extension of analysis to logarithmic minimal models via the logarithmic limit.

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 ${\cal M}(M,M',n)$ 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 $k=nM/(M'\!-\!M)-2$ with $(M,M')=\big(nm-(n\!-\!1)m',m'\big)$. These results rely on Yang-Baxter integrability and are valid in Regime III for models determined by the crossing parameter $\lambda=(m'\!-\!m)\pi/m'$ in the interval $0<\lambda<\pi/n$. Here we consider the $2\times 2$ $\mbox{RSOS}(m,m')$ models in the interval $\tfrac{\pi}{2}<\lambda<\pi$ and investigate the associated one-dimensional sums. In this interval, we verify that the one-dimensional sums produce new finitized Virasoro characters $ch_{r,s}^{(N)}(q)$ of the minimal models ${\cal M}(m,m',1)$ with $m'>2m$. 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 ${\cal M}(m,m',1)$. For the series ${\cal M}(m,2m+1,1)$ 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 $\chi_{r,s}^{m,m';(N)}(q)$ for the logarithmic minimal models ${\cal LM}(p,p',1)$ are also obtained for $p'\ge 2p$ by taking the logarithmic limit $m,m'\to\infty$ with $m'/m\to p'/p+$.