Fused RSOS Lattice Models as Higher-Level Nonunitary Minimal Cosets

Abstract

We consider the Forrester-Baxter RSOS lattice models with crossing parameter $\lambda =(m^{\prime }-m)\pi /m^{\prime }$ in Regime~III. In the continuum scaling limit, these models are described by the minimal models $\mathcal{M}(m,m^{\prime })$. We conjecture that, for $\lambda <\pi /n$, the $n\times n$ fused RSOS models with $n\ge 2$ are described by the higher-level coset $(A_1^{(1)}{)}_k\otimes (A_1^{(1)}{)}_n/(A_1^{(1)}{)}_{k+n}$ at fractional level $k=nM/(M^{\prime }-M)-2$ with $(M,M^{\prime })=(nm-(n-1)m^{\prime },m^{\prime })$. To support this conjecture, we investigate the one-dimensional sums arising from Baxter's off-critical corner transfer matrices. In unitary cases ($m=m^{\prime }-1$) it is known that, up to leading powers of $q$, these coincide with the branching functions $b_{r,s,\ell }^{m^{\prime }-n,m^{\prime },n}(q)$. For general nonunitary cases ($m<m^{\prime }-1$), we identify the ground state one-dimensional RSOS paths and relate them to the quantum numbers $(r,s,\ell )$ in the various sectors. For $n=1,2,3$, we obtain the local energy functions $H(a,b,c)$ in a suitable gauge and verify that the associated one-dimensional sums produce finitized forms that converge, as $N$ becomes large, to the fractional level branching functions $b_{r,s,\ell }^{M,M^{\prime },n}(q)$. Extending the work of Schilling, we also conjecture finitized bosonic branching functions $b_{r,s,\ell }^{M,M^{\prime },n;(N)}(q)$ for general $n$ and check that these agree with the one-dimensional sums for $n=1,2,3$ out to system sizes $N=14$. Lastly, the finitized Kac characters ${\chi }_{r,s,\ell }^{P,P^{\prime },n;(N)}(q)$ of the $n\times n$ fused logarithmic minimal models $\mathcal{LM}(p,p^{\prime })$ are obtained by taking the {\em logarithmic limit\/} $m,m^{\prime }\to \infty$ with $m/m^{\prime }\to p/p^{\prime }+$.

References in corpus (3)

• Logarithmic Superconformal Minimal Models
• Finite-Volume Spectra of the Lee-Yang Model
• RSOS Quantum Chains Associated with Off-Critical Minimal Models and ${\mathbb{Z}}_n$ Parafermions

Cited by in corpus (4)

• Three Blocks Solvable Lattice Models and Birman--Murakami--Wenzl Algebra
• On the Algebraic Approach to Solvable Lattice Models
• One-Dimensional Sums and Finitized Characters of $2\times 2$ Fused RSOS Models
• Exactly Solved Models and Beyond: a special issue in honour of R J Baxter's 75th birthday