Rigorous corner transfer matrix-Virasoro character correspondence for Yang-Baxter integrable models, and unusual corrections to scaling for valence bond entanglement in massive spin chains

Near a conformal critical point, the Renyi entropies of a semi-infinite block in a massive one-dimensional integrable system scale as $S_n=\frac{c}{12}(1+\frac{1}{n})\ln\xi+C'_n+B_n\xi^{-x/n}$, where $\xi$ is the correlation length and $x$ the scaling dimension of a relevant operator of the critical conformal field theory. Calabrese, Cardy and Peschel derived this from the organization of Baxter corner transfer matrix (CTM) eigenvalue degeneracies into Virasoro characters of the critical theory, and left two questions open: (i) the CTM-character correspondence - an observation going back to the lattice Virasoro algebra of the 1980s - is verified model by model but lacks a rigorous derivation for the general class of Yang-Baxter integrable models (with fused models producing affine branching coefficients rather than simple characters); (ii) it is unknown whether the unusual $\xi^{-x/n}$ corrections appear in the valence bond entanglement entropy, one of the measures for which such corrections were seen in gapless phases, and if so whether they can be computed exactly. The problem asks for the generality of the CTM-character mechanism, across models (rigorous derivation or delimitation) and across entanglement measures (valence bond entanglement in massive phases).

Background

Dividing an infinite one-dimensional quantum system in its ground state into a subsystem $A$ and its complement, the entanglement between the parts is quantified by the Renyi entropies $S_n=\frac{1}{1-n}\ln\operatorname{Tr}\rho_A^n$ of the reduced density matrix $\rho_A$ (the partial trace of the ground-state density matrix over the complement); the limit $n\to1$ is the von Neumann entanglement entropy, and large $n$ probes the largest eigenvalue ('single-copy entanglement'). At a critical point, whose continuum limit is a conformal field theory (CFT - the field theory invariant under local scale transformations, labeled by its central charge $c$), the entropies of an interval of length $\ell$ grow as $\frac{c}{6}(1+\frac{1}{n})\ln\ell$, and Cardy and Calabrese showed that this leading behavior is obscured by slowly decaying 'unusual' corrections $b_n\ell^{-2x/n}$, where $x<2$ is the scaling dimension of a relevant operator; they originate from the conical singularities where the sheets of the replicated geometry meet.

Away from but near criticality the system is massive: the gap is finite and the correlation length $\xi$ (the inverse gap) is large but finite. For the bipartition of an infinite chain into two semi-infinite halves, Calabrese, Cardy and Peschel showed that in integrable massive models $S_n=\frac{c}{12}\Bigl(1+\frac{1}{n}\Bigr)\ln\xi+C'_n+B_n\,\xi^{-x/n}+\dots$ with further corrections $\xi^{-kx/n}$ and $\xi^{-kx}$, $k$ integer, and $1/\ln\xi$ corrections in the single-copy limit: the massive analogue of the unusual gapless corrections, with half the exponent because only one conical singularity remains at the single boundary point between the two halves.

The exact results rest on Baxter's corner transfer matrix (CTM). In the two-dimensional classical lattice model equivalent to the quantum chain, the CTM $\hat A$ is the transfer matrix of one quadrant ('corner') of the lattice. For the semi-infinite bipartition $\rho_A$ is proportional to $\hat A^4$, so the CTM spectrum is the entanglement spectrum: $-\ln\rho_A$ is a free-fermion-like 'entanglement Hamiltonian' with equally spaced single-particle levels whose spacing encodes $\xi$, and the Renyi entropies become one-dimensional sums $\sum_N d_N q^{anN}$ over the CTM levels. Baxter's CTM theory shows that in integrable models - models whose Boltzmann weights satisfy the Yang-Baxter consistency equation, the structural origin of exact solvability - the eigenvalues of $\hat A^4$ all have the form $q^{aN+b}$, $N=0,1,2,\dots$, with computable integer degeneracies $d_N$.

The structural fact underlying the corrections is that these degeneracies organize into characters of the critical CFT. A Virasoro character $\chi_\Delta(q)=q^{-c/24+\Delta}\sum_N d_N q^N$ generates the level degeneracies of the highest-weight representation of the Virasoro algebra (the infinite-dimensional symmetry algebra of two-dimensional CFT) with weight $\Delta=x/2$; which representation appears is fixed by the spin at the origin and the boundary conditions at infinity. The observation goes back to the late 1980s: the lattice Virasoro algebra of Itoyama and Thacker in the eight-vertex model, and the Kyoto school's analysis of local height probabilities in restricted solid-on-solid (RSOS) models - lattice models whose heights are restricted to adjacent values, with critical limits given by the unitary minimal CFTs. It is supported by boundary conformal field theory - at criticality on an annulus of radii $R_>$ and $R_<$ the CTM spectrum has exactly the character form with conjugate nome $R_</R_>$ (the nome is the expansion parameter $q$ of the character; Cardy) - and by the combinatorics of semi-infinite walks on RSOS graphs, on which the Virasoro generators act simply. For the Andrews-Baxter-Forrester (ABF) RSOS family the correspondence is on the firmest ground: Melzer showed that the relevant CTM sums coincide with fermionic character sums, identities whose polynomial counterparts were later proved by Warnaar; but these are model-specific results. For models obtained by fusion - the construction of new integrable models whose local states and Boltzmann weights are composites built from several copies of a fundamental model - the corresponding object is generally a branching coefficient of an affine Lie algebra (a coefficient in the decomposition of the characters of an affine algebra into characters of an affine subalgebra) rather than a simple character (Date, Jimbo, Miwa and Okado), still with simple modular properties. Because the conjugate nome is exponentially small in $\ln\xi$, the modular transformation of the characters - equivalently, Poisson resummation of the CTM configuration sums - produces both the universal $\frac{c}{12}\ln\xi$ term and the unusual corrections; this was verified in detail for the massive Ising chain and the antiferromagnetic XXZ chain. But the correspondence itself has only been verified model by model: a rigorous derivation valid for the general class of Yang-Baxter models is missing.

The valence bond entanglement entropy (VBE) is a different, basis-dependent measure introduced by Alet, Capponi, Laflorencie and Mambrini: the ground state is decomposed in a basis of products of valence bonds (spin singlets shared by pairs of sites), and the measure counts the singlets crossing the bipartition, averaged over the decomposition. It is computable in valence-bond quantum Monte Carlo (a Monte Carlo method that samples the valence bond basis directly) and reproduces the qualitative scaling of the von Neumann entropy; Jacobsen and Saleur computed it exactly at criticality for the XXX chain and the Potts model (for instance the mean number of crossing bonds grows as $\frac{4}{\pi^2}\ln L$), and Alcaraz and Rittenberg studied shared-information functionals of this type in integrable stationary states at criticality, where such unusual corrections appear. Whether the VBE inherits the massive $\xi^{-x/n}$ corrections, and whether it can then be computed exactly, is the second open question.

Problem Statement

The task concerns a single mechanism - the organization of corner transfer matrix spectra into Virasoro characters - and asks for its generality in two linked directions:

(1) Across models. Establish rigorously, for the general class of massive integrable lattice models whose near-critical limit is described by a conformal field theory, that the eigenvalue degeneracies of the corner transfer matrix (equivalently, of the entanglement Hamiltonian of the corresponding semi-infinite quantum chain) coincide with the coefficients of a Virasoro character of that theory, or with the corresponding affine branching coefficients for models obtained by fusion; alternatively, delimit the correspondence by an explicit counterexample, i.e. a model of that class whose degeneracies coincide with the coefficients of neither a Virasoro character nor an affine branching coefficient of the theory. A derivation that covers only a proper subclass does not by itself establish the correspondence for the general class. The correspondence is verified model by model - the eight-vertex/XYZ chain, the Ising chain, the antiferromagnetic massive XXZ chain, and the ABF/RSOS families - and for models obtained by fusion the corresponding object is a branching coefficient of an affine Lie algebra rather than a simple Virasoro character.

(2) Across measures. Determine whether the unusual corrections $B_n\xi^{-x/n}$ (with the accompanying towers $\xi^{-kx/n}$ and $\xi^{-kx}$, $k$ integer) to the massive Renyi entropy saturation - which follow from the modular transformation of the character form of the CTM spectrum - also appear in the valence bond entanglement entropy, which displays analogous unusual corrections in gapless phases, and, if so, whether they can be computed exactly.

Part (1) supplies the rigorous foundation of the mechanism and part (2) tests its observable reach; both remain open.

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