Unusual corrections to the entanglement entropy of the massive XYZ chain and the character structure of its corner transfer matrix

The bipartite Renyi entropy of the spin-1/2 XYZ chain is known exactly for a cut into two semi-infinite half-chains through Baxter's corner transfer matrix (CTM). In the scaling limit this entropy, written as a function of the correlation length, coincides with that of a bulk Ising model, while at finite lattice spacing the relation between the mass parameter and the correlation length generates path-dependent 'unusual' corrections - including a term with a continuously varying exponent $\xi^{-2\mu_0/\pi}$ tied to sine-Gordon bound states and a $1/\ln\xi$ term along zero-slope approach paths - whose field-theoretical origin is unknown, alongside the question of whether the decomposition of the CTM spectrum into $c=1/2$ Ising characters reflects an underlying Virasoro or extended algebraic structure. Ercolessi, Evangelisti, Franchini and Ravanini (Phys. Rev. B 85, 115428 (2012)) posed a five-part program around one objective, a complete theory of the near-criticality corrections: a massive sine-Gordon derivation of these corrections by ultraviolet counterterm analysis; an algebraic characterization of the CTM-character correspondence; extensions to finite intervals (testing a predicted exponent doubling) and to excited states; a classification of approach paths and of when correction coefficients vanish; and the parallel analysis in terms of the mass gap. All five parts remain open: parallel exact results exist only in bound-state-free restricted solid-on-solid (RSOS) families, two competing but unproven interpretations exist for the character structure, and the authors' promised follow-up papers never appeared.

Background

For a quantum system in a pure state, the entanglement between a subsystem $A$ and its complement is measured by the Renyi entropies $S_\alpha=\frac{1}{1-\alpha}\ln\mathrm{Tr}\,\rho_A^\alpha$ (with $\alpha>0$; the limit $\alpha\to1$ is the von Neumann entropy), where $\rho_A$ is the reduced density matrix obtained by tracing out the complement. For a one-dimensional critical system described by a conformal field theory (CFT), the Renyi entropy of a block of length $\ell$ (a bipartition with two boundary points) grows as $S_\alpha=\frac{c+\bar c}{12}(1+\frac{1}{\alpha})\ln\ell+\text{const.}$, where $\ell$ is the subsystem length, $c$ and $\bar c$ are the central charges of the two chiral halves of the CFT, and the lattice spacing $a_0$ provides the ultraviolet cutoff; a half-chain (a bipartition with a single boundary point) carries half of this coefficient. In a gapped (massive) one-dimensional system the entropy of a half-chain instead saturates to a value controlled by the correlation length $\xi$ (the inverse of the lightest excitation mass), $S_\alpha\simeq\frac{c}{12}(1+\frac{1}{\alpha})\ln\xi+A_\alpha+\text{corrections}$; for integrable lattice models this follows exactly from Baxter's corner transfer matrix (CTM) construction - the transfer matrix describing one quadrant of the two-dimensional classical lattice, four of whose copies multiply into a row of the model, and whose eigenvalue spectrum generates the reduced density matrix of the associated quantum chain (see the review by Peschel).

The structure of the subleading corrections is subtle and matters for interpreting numerical simulations. Cardy and Calabrese showed that at criticality 'unusual corrections' arise from relevant operators located at the conical singularity of the replicated Riemann surface (the surface built from $\alpha$ cyclically glued copies of the system on which $\mathrm{Tr}\,\rho_A^\alpha$ is computed): their exponents carry a factor $1/\alpha$ that depends on the Renyi index, unlike ordinary corrections. Calabrese, Campostrini, Essler and Nienhuis, and Xavier and Alcaraz, found that in the critical XXZ chain the first correction scales as $\ell^{-2K/\alpha}$, where $K$ is the Luttinger parameter (which measures the compactification radius of the $c=1$ free boson describing the critical line), and a naive scaling argument would then predict terms of the form $\xi^{-K/\alpha}$ in the massive phase.

Ercolessi, Evangelisti, Franchini and Ravanini (2012) tested this expectation exactly for the XYZ spin-1/2 chain, $H=-J\sum_n(\sigma_n^x\sigma_{n+1}^x+J_y\sigma_n^y\sigma_{n+1}^y+J_z\sigma_n^z\sigma_{n+1}^z)$ (with $J>0$ a positive overall energy scale, so that the model is parametrized by the anisotropy couplings $(J_y,J_z)$ of the plane in which its phase diagram is drawn), the archetypal interacting integrable quantum chain, solvable through its exact mapping to Baxter's eight-vertex model (the two-dimensional classical vertex model whose commuting transfer matrices underlie the solution). Baxter's elliptic parametrization of the couplings by two moduli (denoted $l$ and $\mu$, with $0\le\mu\le\pi$) gives, for the bipartition into two semi-infinite half-chains, a closed expression for the Renyi entropy in terms of infinite $q$-products ($q$-Pochhammer symbols $(a;q)_\infty=\prod_{k\ge0}(1-aq^k)$). In the scaling limit the massive XYZ chain renormalizes to the sine-Gordon field theory (a massive interacting relativistic theory of one compactified boson) with $\mu$ proportional to the compactification radius: for $\mu<\pi/2$ the low-energy excitations are free spinons, while for $\pi/2<\mu<\pi$ stable bound states (breathers) appear, the hallmark of the genuinely interacting attractive regime.

Two exact findings define the problem. First, the CTM partition function $\mathcal{Z}(q)=\prod_{j\ge1}(1+q^j)$ entering the entropy is formally equal to the character (the generating series of a representation) of the $c=1/2$ Ising CFT spin field, so that in the scaling limit the Renyi entropy as a function of $\xi$ takes the bulk-Ising form, modulo a multiplicative redefinition of the correlation length:

$$S_\alpha=\frac{1+\alpha}{12\alpha}\ln \frac{\xi}{a_0}-\frac{1}{2}\ln 2-\frac{1}{1-\alpha}\sum_{n\ge1}\sigma_{-1}(n)\Big[\Big(\frac{\xi}{a_0}\Big)^{-2n/\alpha}-\Big(\frac{\xi}{a_0}\Big)^{-4n/\alpha}\Big]+\frac{\alpha}{1-\alpha}\sum_{n\ge1}\sigma_{-1}(n)\Big[\Big(\frac{\xi}{a_0}\Big)^{-2n}-\Big(\frac{\xi}{a_0}\Big)^{-4n}\Big],$$

where $\sigma_{-1}(n)=\sum_{d|n}d^{-1}$ is the divisor function: the subleading terms carry the operator content of the $c=\bar c=1/2$ Ising theory (the leading correction $\xi^{-2/\alpha}$ is the 'unusual' form $\xi^{-2h/\alpha}$ with $h=1$, the Ising energy field) rather than that of the $c=1$ free boson describing the critical line, so that the naive scaling prediction fails for this interacting model. Consistently, the reduced density matrix can be written as $\rho\propto e^{-H_{\mathrm{CTM}}}$ with $H_{\mathrm{CTM}}=\sum_j 2\epsilon_j\eta_j^\dagger\eta_j$ a free-fermion (Majorana) Hamiltonian built from the CTM eigenvalues. Second, once the lattice spacing is kept finite, the relation between the mass parameter and the correlation length depends on the path by which the critical line is approached in the $(J_z,J_y)$ plane, and this generates additional path-dependent corrections of the forms $\xi^{-2h}$, $\xi^{-2h/\alpha}$, $\xi^{-(2-h)}$, $\xi^{-2(1+1/\alpha)}$ and even $1/\ln\xi$, where $h$ is the scaling dimension of a relevant operator of the critical bulk theory. Most strikingly, along straight approach paths in the coupling plane that meet the critical line at $\mu=\mu_0$ a correction with a continuously varying exponent $\xi^{-2\mu_0/\pi}$ appears - a member of the $\xi^{-(2-h)}$ family with $h$ the dimension of the sine-Gordon vertex operator $\mathrm{e}^{i\beta\phi}$ - and along curves that meet the critical line with zero slope a $1/\ln\xi$ term emerges, of the type associated with marginal operators. The coefficient of the $\xi^{-2\mu_0/\pi}$ term is controlled by the bound states: it vanishes both on the special slope $m_0=-2/\cos\mu_0$ (with $m_0$ the slope of the straight approach path in the $(J_z,J_y)$ plane; this is the slope followed by the renormalization-group flow, i.e. by paths that keep $\mu$ fixed) and at the free/bound-state crossing point $\mu=\pi/2$, so that in the authors' words this term is 'turned on by the existence of a bound state' as 'a clear signature of a truly interacting theory'. The authors state that 'no fundamental reason is known' why CTM spectra of integrable models decompose into characters in the mass parameter and that any connection with an underlying Virasoro algebra (the symmetry algebra of the generators of conformal transformations in a two-dimensional conformal field theory) 'remains to be discovered', while the lattice-generated terms 'still lack a field theoretical interpretation'.

Problem Statement

Provide a complete theory of the subleading corrections to the bipartite Renyi entropy of the massive XYZ spin-1/2 chain near its critical line, and of the algebraic structure underlying its corner transfer matrix. The problem was posed by Ercolessi, Evangelisti, Franchini and Ravanini (Phys. Rev. B 85, 115428 (2012)) as five interlocking components that together define the completion boundary:

1. Field-theoretical origin. Construct the continuum derivation of the lattice-generated, path-dependent corrections to the half-chain Renyi entropy $S_\alpha(\xi)$ in the massive sine-Gordon scaling limit, generalizing the ultraviolet-regularization and counterterm argument of Cardy and Calabrese from the critical to the massive interacting theory. The derivation must produce, not merely observe, the exponents and coefficients of the full set of terms $\xi^{-2h}$, $\xi^{-2h/\alpha}$, $\xi^{-(2-h)}$, $\xi^{-2(1+1/\alpha)}$ and $1/\ln\xi$ (with $h$ the scaling dimension of the operator responsible for a given term), including the correction with continuously varying exponent $\xi^{-2\mu_0/\pi}$ that appears along straight approach paths meeting the critical line at $\mu=\mu_0$ with $\pi/2<\mu_0<\pi$ (a member of the $\xi^{-(2-h)}$ family in which $h$ is the dimension of the sine-Gordon vertex operator $\mathrm{e}^{i\beta\phi}$) and the logarithmic term that appears along zero-slope approach curves, and it must explain why these terms are turned on by the existence of bound states.

2. Algebraic structure of the CTM. Determine whether the decomposition of the corner transfer matrix spectrum into $c=1/2$ Ising characters in the mass parameter reflects an underlying Virasoro or extended algebraic structure (for example a modular structure in parameter space) that can be made explicit, or prove that it is accidental in the sense of being forced by the analytic structure of the elliptic functions alone; in either case the observed Ising operator content and the Majorana form of the corner Hamiltonian should follow from the characterization.

3. Geometry and excited states. Extend the analysis to a bipartition into a finite interval (two boundary points) and test whether the correlation-length exponent doubles relative to the half-chain, in analogy with critical chains, where the exponent of the unusual correction is multiplied by the number of twist-field insertions (the branch-point operators of the replicated geometry) at the edges of the blocks - the factor-of-two case measured for two disjoint blocks in critical Ising and XY chains; and perform the corresponding analysis for excited states, checking whether the formally conformal structure found for the ground state persists or changes.

4. Path classification. Classify which approach paths in the coupling space (in the elliptic parametrization, which trajectories in the $(l,\mu)$ or $(J_z,J_y)$ plane) generate which types of subleading terms, and give criteria predicting when the coefficients of particular terms vanish - beyond the two cases already known: straight paths of slope $m_0=-2/\cos\mu_0$ (the slope followed by the renormalization-group flow, along which $\mu$ stays fixed) and the free/bound-state crossing point $\mu=\pi/2$.

5. Infrared scale. Redo the correction analysis with the entropy expressed in terms of the mass gap (the energy of the lightest excitation) instead of the correlation length, and characterize the qualitatively different subleading series generated by this choice of physical infrared scale.

These five components share one scientific objective: they are the source authors' own decomposition of the task of understanding how, why and with what universality the entanglement entropy of an interacting massive integrable chain approaches criticality. Component 1 supplies the operator interpretation that components 3, 4 and 5 export to other geometries, states, paths and scales, while component 2 explains the operator content that component 1 must reproduce; an answer that resolves the common objective must address them jointly, although individual components may be settled in sequence.

References

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