Close to a conformal critical point, the Rényi entropies SnS_n of a block in a massive one-dimensional quantum system scale as (1+1/n)(c/12)ln⁡ξ(1+1/n)(c/12)\ln\xi plus corrections, and Calabrese, Cardy and Peschel (2010) showed — for chains diagonalizable by Baxter's corner transfer matrix, through the modular transformation of Virasoro characters — that these corrections include 'unusual' powers ξ−x′/n\xi^{-x'/n}, where x′x' is the scaling dimension of a primary field of the critical conformal field theory and nn is the Rényi index. The open directions left by Calabrese, Cardy and Peschel (2010) are: which primaries and modular S-matrix elements dominate the corrections, and with what non-universal amplitudes; whether the same correction family persists in generic non-integrable systems near a conformal point; how the unusual corrections interpolate for a finite interval of length ℓ\ell in the regime ℓ∼ξ\ell\sim\xi; and whether other entanglement measures that display unusual corrections in gapless phases, such as valence-bond entanglement, inherit the analogous massive-phase structure. Later literature confirms and enriches the mechanism within integrable model families (XYZ chains, restricted solid-on-solid (RSOS) models including the non-unitary Forrester–Baxter family) and provides adjacent numerical evidence for operator-dimension-controlled power-law corrections near quantum critical points including a non-integrable deformation, but no general dominance rule, no closed-form amplitude theory, no analytic non-integrable extension, no crossover characterization, and no valence-bond result is known: the problem remains open.

Background

When an extended quantum system in its ground state is divided into two parts AA and BB, their entanglement is quantified by the Rényi entropies Sn=(1/(1−n))ln⁡Tr⁡ρAnS_n=(1/(1-n))\ln\operatorname{Tr}\rho_A^n of the reduced density matrix ρA\rho_A (the trace over BB of the ground-state projector); the case n=1n=1 is the von Neumann entanglement entropy, and the whole family {Sn}\{S_n\} encodes the eigenvalue spectrum of ρA\rho_A. In a critical one-dimensional system whose continuum limit is a conformal field theory (CFT — a two-dimensional scale-invariant fixed point characterized by the central charge cc), the Rényi entropies of an interval of length ℓ\ell grow logarithmically, Sn≃(1+1/n)(c/6)ln⁡ℓ+constS_n\simeq(1+1/n)(c/6)\ln\ell+\text{const} (Calabrese and Cardy 2004). Cardy and Calabrese (2010) showed that the corrections to this scaling can be 'unusual': besides the corrections from irrelevant bulk operators expected from finite-size scaling, powers ℓ−2x/n\ell^{-2x/n} appear, with xx the scaling dimension of a relevant operator. They originate from the conical singularities that Tr⁡ρAn\operatorname{Tr}\rho_A^n necessarily introduces at the entanglement cuts when it is represented as a path integral on an nn-sheeted Riemann surface: relevant operators are induced at the branch points, and because the Rényi index nn divides the exponent, these corrections decay anomalously slowly and dominate the standard terms for suitable nn.

In a massive (gapped) one-dimensional system with finite correlation length ξ\xi, close to a conformal critical point, the entanglement of a half-line bipartition (an infinite chain cut into two semi-infinite halves, a single entanglement cut) saturates as Sn≃(1+1/n)(c/12)ln⁡ξ+constS_n\simeq(1+1/n)(c/12)\ln\xi+\text{const} — half the critical interval coefficient, since a single cut carries one conical singularity. Calabrese, Cardy and Peschel (2010) asked whether the unusual power-law corrections survive into the massive phase. For one-dimensional models diagonalizable by Baxter's corner transfer matrix (CTM — the factorization of the row transfer matrix of the corresponding two-dimensional integrable classical model into four corner matrices; for the quantum chain, the half-line reduced density matrix is proportional to a CTM), they showed by explicit examples (massive XXZ, Ising in both phases, and XY chains) that the corrections are powers ξ−kx/n\xi^{-kx/n}, with xx the dimension of the relevant operator that drives the system massive (x=2x=2 for the XXZ chain, x=1x=1 for the Ising energy density): for example the XXZ corrections are ξ−2k/n\xi^{-2k/n}. The amplitudes are non-universal, differ between the two Ising phases (in one phase with alternating signs), and the single-copy entanglement S∞S_\infty (the n→∞n\to\infty limit, governed by the largest eigenvalue of ρA\rho_A) acquires instead an unusual logarithmic correction ∼1/ln⁡ξ\sim 1/\ln\xi.

They then identified the structural origin of these powers. In Yang–Baxter-integrable models (exactly solvable lattice models whose local Boltzmann weights satisfy the Yang–Baxter consistency equation) the CTM eigenvalues have the form qaN+bq^{aN+b} with degeneracies that organize into Virasoro characters χΔ(q)\chi_\Delta(q) of the critical CFT — the generating functions χΔ(q)=q−c/24+Δ∑NdNqN\chi_\Delta(q)=q^{-c/24+\Delta}\sum_N d_N q^N counting the descendant levels above a primary field of conformal weight Δ\Delta (a highest-weight field; its scaling dimension is x=2Δx=2\Delta). Consequently Tr⁡ρAn=χΔ(qn)/χΔ(q)n\operatorname{Tr}\rho_A^n=\chi_\Delta(q^n)/\chi_\Delta(q)^n with the nome q=e−2ϵq=e^{-2\epsilon} fixed by the distance from criticality. Characters transform linearly under the modular transformation q→q~=e−2π2/ϵq\to\tilde q=e^{-2\pi^2/\epsilon}, the transformation matrix being the modular SS-matrix SΔΔ′S_\Delta^{\Delta'} of the CFT (the same matrix that classifies boundary states in Cardy's boundary CFT), and near criticality q~∝ξ−2\tilde q\propto\xi^{-2}. Expanding the transformed characters gives the leading (1+1/n)(c/12)ln⁡ξ(1+1/n)(c/12)\ln\xi term, an order-one constant related to the Affleck–Ludwig boundary entropy (a universal boundary contribution to the free energy, here g=(SΔ0)1/2g=(S_\Delta^{0})^{1/2}), corrections ξ−2k\xi^{-2k} from descendant levels, and — from every primary Δ′\Delta' with SΔΔ′≠0S_\Delta^{\Delta'}\neq0 — unusual corrections ξ−x′/n\xi^{-x'/n} and ξ−x′\xi^{-x'} with x′=2Δ′x'=2\Delta'. The mechanism is expected to be 'valid for any massive one-dimensional model close to a conformal critical point', but it is only derived where a CTM construction exists, i.e. for integrable models.

The precise knowledge of these corrections is not academic: at criticality, unusual corrections must be accounted for to converge numerically extracted universal Rényi data (Sahoo et al 2016), and the same is expected in massive systems, especially non-integrable ones where numerics is the only access. The source paper left the following four directions open: the determination of which primaries dominate and the closed-form computation of the amplitudes (the mechanism nominally allows contributions from all primaries, while the exact solutions contain only a subset of the allowed powers — even in the solved Ising examples a σ\sigma-primary contribution ξ−1/(8n)\xi^{-1/(8n)} is allowed by a non-vanishing SS-matrix element but absent from the exact result); the validity beyond integrability; the ℓ∼ξ\ell\sim\xi crossover for finite intervals; and the extension to other entanglement measures such as valence-bond entanglement (an alternative measure defined on the singlet basis, where one counts the number NcN_c of valence bonds crossing the cut; its mean grows logarithmically at criticality with exactly known coefficient in solvable cases).

Problem Statement

Complete the theory of the unusual finite-correlation-length corrections to block entanglement in massive one-dimensional quantum systems near conformal critical points, as posed by the open directions of Calabrese, Cardy and Peschel (2010). For the Rényi entropies SnS_n of a bipartition of an infinite chain (the half-line cut being the base case), with nn the Rényi index, ξ\xi the correlation length, cc the central charge of the critical conformal field theory, x′=2Δ′x'=2\Delta' the scaling dimensions of the critical theory's primary fields (where Δ\Delta is the conformal weight realized by the ground state, in terms of which Tr⁡ρAn=χΔ(qn)/χΔ(q)n\operatorname{Tr}\rho_A^n=\chi_\Delta(q^n)/\chi_\Delta(q)^n for Virasoro characters χ\chi and nome qq), the established prediction is

Sn  ≃  (1+1n)c12ln⁡ξ  +  Cn  +  ∑x′Bn(x′) ξ−x′/n  +  …S_n\;\simeq\;\Bigl(1+\frac{1}{n}\Bigr)\frac{c}{12}\ln\xi\;+\;C_n\;+\;\sum_{x'}B_n(x')\,\xi^{-x'/n}\;+\;\ldots

with corrections ξ−x′/n\xi^{-x'/n} (and ξ−x′\xi^{-x'}) from every primary x′x' for which the modular matrix element SΔΔ′≠0S_\Delta^{\Delta'}\neq0, alongside descendant powers ξ−2k\xi^{-2k}. The problem asks:

(1) Dominant primaries and amplitudes. Establish a general criterion, valid for the integrable families diagonalizable by the corner transfer matrix, identifying which modular SS-matrix elements SΔΔ′S_\Delta^{\Delta'} vanish or are numerically small, and hence which primary dimensions x′x' control the dominant finite-ξ\xi corrections; and compute the corresponding correction amplitudes — including their signs and alternating phase factors — beyond the explicitly solved XXZ/Ising/XY examples (e.g. for the XYZ chains and the RSOS and non-unitary Forrester–Baxter families), together with their dependence on CTM normalizations and boundary conditions. That this is genuinely open even in solved models is visible in the source's own Ising analysis: the character mechanism nominally allows a σ\sigma-primary contribution ξ−1/(8n)\xi^{-1/(8n)} in the Ising chain (the Ising modular matrix element S1σ=1/2S_{\mathbb{1}}^{\sigma}=1/\sqrt{2} does not vanish), yet the exact expansion contains only the integer-power family ξ−k/n\xi^{-k/n} of the energy operator.

(2) Validity beyond integrability. Establish whether the same universal ξ−x′/n\xi^{-x'/n} family of corrections (and with what amplitudes) persists in generic non-integrable one-dimensional systems close to a conformal critical point, where no corner-transfer-matrix construction exists — by analytic means, or by controlled numerical treatments able to separate the unusual powers from standard, lattice and finite-entanglement corrections.

(3) Crossover. Characterize the crossover for a finite interval of length ℓ\ell in the regime ℓ∼ξ\ell\sim\xi: the full interpolation of the unusual corrections between the critical behavior for ℓ≪ξ\ell\ll\xi and the massive behavior for ℓ≫ξ\ell\gg\xi.

(4) Other entanglement measures. Determine whether other entanglement measures that display unusual corrections in gapless phases — in particular valence-bond entanglement — exhibit the analogous ξ−x′/n\xi^{-x'/n} structure in massive phases, and whether these corrections can be computed exactly.

These four deliverables jointly support one overarching objective: a model-independent characterization of how the conformal data of the critical point (primary dimensions, modular data) enters the finite-ξ\xi scaling of block entanglement, so that the corrections can be predicted and used as a diagnostic rather than merely observed in integrable examples. All four deliverables are required: settling any one of them alone does not complete the overarching objective.

References

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