Complete analysis of the open off-critical RSOS quantum chains: integrable boundaries, open-chain spectra, and reality of the non-Hermitian double-row transfer matrices

The Forrester–Baxter restricted solid-on-solid (RSOS) lattice models realize, in their continuum scaling limit, the $\varphi_{1,3}$-perturbed minimal models $\mathcal{M}(m,m';t)$ — including the nonunitary ones — and the thermally perturbed $\mathbb{Z}_n$ parafermionic theories. Bianchini, Ercolessi, Pearce and Ravanini (2015) constructed the associated periodic quantum chains as the Hamiltonian limit of the single-row transfer matrices, but left the open (double-row) chains analytically unexplored: only the vacuum boundary condition is covered, the empirically observed real, diagonalizable spectra of the non-Hermitian double-row transfer matrices and open-chain Hamiltonians are unexplained (conjecturally a PT-symmetry effect), and no integrable boundary structure, open-chain spectral solution, or boundary scaling functions are known beyond the Lee–Yang case treated by boundary thermodynamic Bethe ansatz and two critical models. This problem asks for the open-chain counterpart of the single-row analysis: construction or classification of integrable boundary conditions for the elliptic face weights, determination of the open-chain spectra (boundary energies, ground-state degeneracies, boundary operators, boundary TBA) for general $\mathcal{M}(m,m')$, and a proof or explicit refutation of spectral reality and diagonalizability. Two sibling directions from the same paper's conclusion — numerical entanglement studies of these chains and the physical meaning of nonunitary entanglement — are outside this problem's completion boundary.

Background

Yang–Baxter integrable two-dimensional lattice models and one-dimensional quantum chains are two faces of the same structure: when the row-to-row transfer matrix $T(u)$ of a lattice model belongs to a commuting one-parameter family (guaranteed by a Yang–Baxter equation for the local Boltzmann weights), the logarithmic derivative at $u=0$ defines an integrable quantum Hamiltonian, $H=-\left.d\log T(u)/du\right|_{u=0}$. The restricted solid-on-solid (RSOS) models are the classic face-model realization: an integer height $a\in\{1,\dots,m'-1\}$ sits on every site of the square lattice, and neighboring heights must differ by exactly one — the adjacency of the $A_{m'-1}$ Dynkin diagram (a path graph). Andrews, Baxter and Forrester introduced the critical versions, and Forrester and Baxter the elliptic, off-critical weights with crossing parameter $\lambda=(m'-m)\pi/m'$ where $2\le m\le m'$ are coprime. In the continuum scaling limit of the ferromagnetic regime III these models realize the Virasoro minimal models $\mathcal{M}(m,m')$ — two-dimensional conformal field theories with finitely many primary fields and central charge $c=1-6(m'-m)^2/(mm')$, unitary for $m'=m+1$ but nonunitary otherwise (for example the Lee–Yang model $\mathcal{M}(2,5)$, whose state space admits no positive-definite inner product) — perturbed by the thermal field $\varphi_{1,3}$, with $t$ measuring the departure from criticality; in the antiferromagnetic regime II they realize the $\mathbb{Z}_n$ parafermionic conformal field theories ($n=m'-2$), whose order parameters carry fractional spin, perturbed by their own thermal operator.

Bianchini, Ercolessi, Pearce and Ravanini constructed the associated quantum chains: using an elliptic Yang–Baxter algebra of planar tiles encoding the allowed face configurations, they obtained the off-critical Hamiltonians as the logarithmic derivative of the periodic single-row transfer matrices. These Hamiltonians act on the vector space of paths on the $A_{m'-1}$ Dynkin diagram (a space whose dimension is counted by generalized Fibonacci numbers), and their spectra are empirically real even though the matrices are not symmetric — a phenomenon familiar since the $\mathbb{Z}_n$-symmetric chains of von Gehlen and Rittenberg.

For open rather than periodic chains, the standard tool, introduced by Sklyanin for vertex models, is the double-row transfer matrix, built from the Yang–Baxter algebra together with boundary weights satisfying a reflection (boundary Yang–Baxter) equation; the resulting double-row family again commutes and yields an open-chain Hamiltonian. For face models the boundary structure is subtler because the heights at the two ends of the strip themselves carry boundary conditions. The 2015 paper established the open off-critical chains only for the simplest (vacuum) boundary conditions and left their complete analysis explicitly open, recording the empirical facts that the double-row transfer matrices and open chains, though non-Hermitian, have real spectra, are diagonalizable, and show a characteristic pattern of asymptotically degenerate ground states ($2n$ in regime II, $2m-2$ in regime III) with a finite gap that closes at criticality where the expected conformal spectra reappear — facts conjectured, but not proven, to follow from a PT symmetry, meaning invariance under an antilinear symmetry combining a spatial reflection with complex conjugation, whose unbroken phase forces real spectra as in Bender and Boettcher's non-Hermitian quantum mechanics.

The question is also tied to the corner-transfer-matrix route to entanglement. Corner transfer matrices (transfer matrices acting on a quadrant of the lattice) generate the reduced density matrix of a half-chain, and the observed scaling $S\sim\frac{1}{3}c_{\mathrm{eff}}\log\xi$ of the entanglement entropy with the correlation length $\xi$, governed by the effective central charge $c_{\mathrm{eff}}=c-24\Delta_{\min}$ (with $\Delta_{\min}$ the lowest conformal dimension) rather than by $c$ itself, was the first lattice confirmation of the conformal results for nonunitary theories.

Problem Statement

Carry out the complete analysis of the open off-critical RSOS quantum chains associated with the $\varphi_{1,3}$-perturbed minimal models $\mathcal{M}(m,m';t)$ (Forrester–Baxter regime III) and the thermally perturbed $\mathbb{Z}_n$ parafermionic chains (regime II), left open by Bianchini–Ercolessi–Pearce–Ravanini (2015). This comprises three linked deliverables that jointly constitute the open-chain counterpart of that paper's single-row analysis:

(a) Boundary integrability. Construct the commuting double-row transfer matrices of the elliptic Forrester–Baxter RSOS models — including the vacuum boundary condition as a special case and, more generally, nontrivial integrable boundary conditions obtained by solving (or classifying the solutions of) the boundary Yang–Baxter (reflection) equations for the elliptic face weights — and derive the open-chain Hamiltonians as their logarithmic derivatives.

(b) Open-chain spectra. Determine the spectra of these open chains: the analytic structure of the double-row eigenvalues as functions of the spectral parameter (inversion identities — functional relations of the type $X(u)X(-u)\propto\mathbf{1}$ relating eigenvalues at reflected spectral parameter values — or equivalent functional relations), the finite-size and boundary energies, the pattern of asymptotically degenerate lowest levels ($2n$ in regime II and $2m-2$ in regime III, observed empirically), the boundary operator content in the scaling limit, and the boundary scaling functions (boundary thermodynamic Bethe ansatz, the integral-equation method that derives finite-size and boundary energies from the scaling limit of the lattice functional relations) for general $m,m'$, including the recovery of the expected conformal boundary spectra as $t\to0$.

(c) Spectral reality. Prove — or refute by an explicit counterexample — that the double-row transfer matrices and the open-chain Hamiltonians, although non-Hermitian (real but not symmetric, and with no positive-definite metric in the nonunitary cases), have real spectra and are diagonalizable; in particular, identify the conjectured PT-like antilinear symmetry on the path spaces and establish that it is unbroken over the full domain stated by the problem — all coprime $2\le m\le m'$, off-critical $t$ in both regimes II and III, and every integrable boundary condition constructed in (a) — or exhibit another mechanism such as an explicit similarity transformation to a symmetric matrix. A refutation must exhibit a concrete member of this domain — a specific pair $m,m'$, regime, $t$, and boundary condition — at which reality or diagonalizability fails.

Deliverable (a) supplies the commuting double-row family, (b) is its spectral solution, and (c) explains the structural fact on which the empirical analysis rests; a complete answer addresses all three. Within (a) and (b) the enumerated items are all required — in (b) the scope 'for general $m,m'$' applies to every listed item — while (c) is satisfied by any one of the routes it states: a proof, by the conjectured PT-like symmetry or another mechanism, or an explicit counterexample. The source's two further suggested directions — systematic numerical studies of entanglement of finite intervals in these chains (e.g. by the density-matrix renormalization group, DMRG, the standard variational method for one-dimensional quantum chains), and the physical interpretation of entanglement in nonunitary conformal theories — are not part of this problem's completion boundary.

References

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