Entanglement-entropy program for Forrester–Baxter RSOS models beyond regime III: universality classes of the remaining regimes, logarithmic-limit entropy normalization, finite-interval lattice methods, and integrable extensions to coset models and multipartite measures

Bianchini and Ravanini (2016) computed the Rényi entanglement entropy of the half-infinite quantum chains associated with the Forrester–Baxter restricted-solid-on-solid (FBRSOS) models in regime III, using Baxter's corner transfer matrix (CTM) to obtain the reduced density matrix exactly. They found the scaling $S_A^{(n)}=\frac{c_{\rm eff}}{12}\frac{n+1}{n}\log\xi+A^{(n)}+\dots$ with the correlation length $\xi$, governed by the effective central charge of the non-unitary minimal models and with correction exponents built from effective dimensions shifted by the lowest conformal weight. Their paper explicitly acknowledges four linked open problems: (i) identification of the universality classes of the FBRSOS regimes other than regime III, in particular whether the critical lines for $s\neq1$ (the analogue of the ABF regime I–II line, a unitary $\mathbb{Z}_{r-2}$ parafermionic theory for $s=1$) are described by non-unitary parafermionic conformal field theories; (ii) a principled derivation of the ad hoc renormalization $\bar A^{(n)}=\lim_{r,s\to\infty}(A^{(n)}-\log\frac{1}{r})$ of the non-universal entropy constant in the off-critical logarithmic minimal-model limit; (iii) a lattice procedure to compute entanglement entropies of finite intervals in interacting integrable spin chains away from criticality; and (iv) extension to non-unitary coset and $W_n$-type lattice realizations and to multipartite measures such as logarithmic negativity within an integrable lattice approach. Adjacent results bear on parts of this program — a $c_{\rm eff}$-irreversibility theorem for PT-invariant non-unitary quantum field theory, fused-RSOS constructions of non-unitary cosets, Hamiltonian-truncation and form-factor methods for intervals in continuum field theory, and negativity results in free and continuum settings — while the stated lattice-integrable and classification questions remain open.

Background

Restricted-solid-on-solid (RSOS) models are two-dimensional classical lattice models of interaction-round-a-face type: an integer height $\ell_i\in\{1,\dots,r-1\}$ lives on each site, and neighboring heights must differ by exactly one, so that admissible heights sit on adjacent nodes of the $A_{r-1}$ Dynkin diagram. The Andrews–Baxter–Forrester (ABF) model is the $s=1$ case; its elliptic Boltzmann weights satisfy the Yang–Baxter equation (the consistency condition that makes the model exactly solvable) and lead to several regimes (regions of the temperature- and anisotropy-parameter space), of which regime III is a massive phase whose continuum scaling limit is the unitary minimal conformal field theory $\mathcal{M}_{r-1,r}$ perturbed by its thermal operator $\Phi_{1,3}$ (the primary field coupled to the temperature-like parameter; the critical $\mathcal{M}_{r-1,r}$ itself is reached at the regime III–IV boundary). Forrester and Baxter extended the model to coprime pairs $(r,s)$ with crossing parameter $\lambda=s\pi/r$, finding ten regimes; regime III then realizes the non-unitary minimal model $\mathcal{M}_{r-s,r}$ perturbed by $\Phi_{1,3}$ (Riggs; Nakanishi). The central charge is $c=1-\frac{6s^2}{r(r-s)}$, which is negative for most coprime pairs $(r,s)$; physical quantities are governed instead by the effective central charge $c_{\rm eff}=c-24\Delta_{\min}$, where $\Delta_{\min}$ is the lowest conformal weight, so that $c_{\rm eff}=1-\frac{6}{r(r-s)}>0$.

Quantizing such a two-dimensional classical model along one direction gives a one-dimensional quantum chain whose entanglement entropy measures quantum correlations across a bipartition. For a subsystem $A$ with reduced density matrix $\rho_A$, the Rényi entropy is $S_A^{(n)}=\frac{1}{1-n}\log\operatorname{Tr}\rho_A^n$ (the limit $n\to1$ gives the von Neumann entropy). Baxter's corner transfer matrix (CTM) — the row-to-row transfer matrix of one quadrant of the lattice — factorizes the partition function into four corner contributions and, for integrable models in a massive regime, expresses the reduced density matrix of a half-infinite chain as a simple Boltzmann-weight operator on one-dimensional height configurations, making $S_A^{(n)}$ exactly computable. De Luca and Franchini applied this to the ABF model ($s=1$), recovering unitary minimal scaling on one critical line and $\mathbb{Z}_{r-2}$ parafermionic scaling (a $\mathbb{Z}_N$-symmetric conformal field theory with fractional-spin currents) on the other. Bianchini and Ravanini did the same for general $(r,s)$ in regime III, obtaining

$S_A^{(n)}=\frac{c_{\rm eff}}{12}\,\frac{n+1}{n}\,\log\xi+A^{(n)}+B_{1}^{(n)}\,\xi^{-\frac{2}{n}(\Delta_{1}-\Delta_{\min})}+\dots$

where $\xi$ is the correlation length: the same logarithmic law as in unitary theories but with $c$ replaced by $c_{\rm eff}$, and power-law corrections whose exponents involve the effective dimensions $\Delta_{\mathfrak h}-\Delta_{\min}$ rather than the bare ones. Two further facts frame the open questions. First, in the double limit $r,s\to\infty$ at fixed ratio $r/s=R/S$ the FB RSOS model realizes the off-critical logarithmic minimal models $\mathcal{LM}_{R-S,R}+t\Phi_{1,3}$ (Pearce–Rasmussen–Zuber; Pearce–Seaton) — non-unitary theories whose Hamiltonians develop Jordan blocks (non-diagonalizable degeneracies) at criticality — and the modular $S$-matrix (the matrix governing the transformation of the torus partition function under exchange of its two cycles) degenerates, forcing an ad hoc renormalization of the entropy constant. Second, in massive integrable quantum field theory the entropy of an infinite bipartition is obtained from one-point functions of branch-point twist fields (fields implementing the cyclic replica permutation; Cardy–Castro-Alvaredo–Doyon), while in critical conformal field theory finite intervals are handled by twist-field two-point functions (Calabrese–Cardy); on the lattice, however, no analogue of the CTM is known for finite intervals of interacting chains. For mixed-state and multipartite entanglement the standard measure is the logarithmic negativity $\mathcal{E}=\log\|\rho^{T_B}\|_1$, the logarithm of the trace norm of the partially transposed density matrix, which is computable in conformal and massive field theory but not within the existing lattice transfer-matrix algebra.

Problem Statement

Resolve the four linked open problems acknowledged by Bianchini–Ravanini (2016), which together complete their corner-transfer-matrix entanglement program for Forrester–Baxter RSOS models beyond its regime-III, half-infinite-chain, bipartite core:

1. Universality classes of the FB RSOS regimes. Forrester and Baxter's model has ten regimes; only the regime-III massive phase (continuum limit $\mathcal{M}_{r-s,r}+\Phi_{1,3}$) and the critical line separating regimes III and IV (the non-unitary minimal model $\mathcal{M}_{r-s,r}$ itself) are identified. Identify the universality classes of the remaining regimes, in particular the conformal field theories governing the other critical lines for $s\neq1$ — the analogue of the ABF regime I–II line, which for $s=1$ is the unitary $\mathbb{Z}_{r-2}$ parafermionic theory — and determine whether they are described by the non-unitary parafermionic coset conformal field theories of Ahn–Chung–Tye (coset theories, built as quotients of affine current algebras, containing fractional-spin parafermion currents) or by some other classification scheme. Entanglement entropy itself, via the corner-transfer-matrix construction, is the proposed diagnostic.

2. Normalization in the logarithmic minimal-model limit. In the limit $r,s\to\infty$ at fixed ratio $r/s=R/S$ the model realizes the off-critical logarithmic minimal model $\mathcal{LM}_{R-S,R}+t\Phi_{1,3}$; the non-universal entropy constant must then be defined through the renormalization $\bar A^{(n)}=\lim_{r,s\to\infty}(A^{(n)}-\log\frac{1}{r})$, equivalent to multiplying the partition function by $r$. Derive this prescription from first principles.

3. Finite-interval entanglement in integrable spin chains. Provide a lattice procedure for genuinely interacting integrable spin chains (not only free or non-interacting specializations) that yields Rényi and von Neumann entanglement entropies of a finite interval (finite subsystem size, away from criticality), either exactly or with an accuracy that can be certified against known limits and numerical benchmarks — the analogue, for finite intervals, of what the corner transfer matrix achieves for the infinite bipartition and of the correlation-matrix method for free chains.

4. Integrable extensions and multipartite measures. Extend the lattice density-matrix/CTM construction to non-unitary coset realizations — the fused FB RSOS models (obtained by the fusion procedure, which merges blocks of adjacent lattice rows into a single coarser model) conjectured by Tartaglia–Pearce to realize higher-level non-unitary minimal cosets, and the $W_n$-algebra-based series (the $W_n$ algebra extends the Virasoro algebra by higher-spin currents) and dilute/Pasquier-type generalizations (Pasquier models place the heights on Dynkin diagrams other than $A_{r-1}$, and dilute versions also allow neighbouring heights to be equal) — and compute multipartite entanglement measures, in particular the logarithmic negativity $\mathcal{E}=\log\|\rho^{T_B}\|_1$, within a consistent integrable lattice approach. Extending the construction to any one of the named model families satisfies the extension part of this component; computing a multipartite measure, in particular the logarithmic negativity, for that realization is a separate part that may be resolved independently.

The four components are jointly the paper's remaining program and share one objective — extending exact entanglement computations for exactly solved non-unitary lattice models from the regime-III half-chain result to the full model family and the full set of entanglement questions it motivates. They are independent deliverables: a submission may resolve any one or more of them and record the remainder as open, while completing the program requires all four. For the finite-interval question, continuum conformal-field-theory and form-factor methods already handle the problem in the continuum, whereas a lattice procedure for integrable spin chains is still lacking in general; continuum-only truncation methods, free-chain specializations, and purely numerical (e.g. density-matrix-renormalization-group, a variational method for one-dimensional quantum many-body ground states) results remain relevant as benchmarks for that missing lattice procedure.

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