Microscopic foundation of the entanglement-spectrum-corner-Hamiltonian correspondence in integrable open spin chains

The low-lying entanglement spectrum of an open integrable quantum spin chain at a cut $n$ sites from the edge empirically matches the spectrum of the corner Hamiltonian $K^{(n)}=\sum_{i=1}^{n-1} i\,h_i$ — the generator of Baxter's corner transfer matrix — including at criticality and at cuts close to the physical boundary, but this correspondence has never been established at the lattice level. Kim, Katsura, Trivedi and Han (Phys. Rev. B 94, 195110 (2016)) posed the linked open questions of a microscopic derivation of the entanglement-Hamiltonian/corner-Hamiltonian relation valid for arbitrary cut position and at criticality (existing arguments are continuum boundary conformal field theory); the precise finite-$n$ spatial weight profile, linear in the distance from the cut as in the corner Hamiltonian versus the parabolic $i(2n-i+1)/(2n)$ suggested by exact XX-model and boundary-conformal results; and the class of lattice Hamiltonians for which the correspondence holds, given that it survives at some nonintegrable points while failing at AKLT-type and first-order points. Later work settled the free-fermion analogues (exact commuting operators with the parabolic weight; quantified lattice Bisognano-Wichmann approximations; exact gapped half-infinite corner-transfer-matrix results) and formalized the gapped Lorentz-invariant side via boundary conformal field theory, but for interacting critical chains the microscopic basis, the weight profile at large $n$ and the validity conditions remain open, as does the extension to quantum-group-deformed ($q$-deformed) symmetry.

Background

When a one-dimensional quantum system in a pure ground state is split into a subsystem and its complement, all bipartite entanglement is encoded in the reduced density matrix $\rho_A$. Expressing the ground state in its Schmidt decomposition, the eigenvalues of $\rho_A$ are the squared Schmidt coefficients $\lambda_\alpha^2$, and the operator $H_{\mathrm{ent}}=-\ln\rho_A$ — the entanglement (or modular) Hamiltonian — has eigenvalues $\varepsilon_\alpha=-\ln\lambda_\alpha^2$ whose collection is the entanglement spectrum (ES). Except in special cases, $H_{\mathrm{ent}}$ is not known in closed form even when the physical Hamiltonian is exactly solvable.

The classical archetype of a solvable entanglement Hamiltonian is Baxter's corner transfer matrix (CTM) of exactly solved two-dimensional lattice models: the transfer matrix of a quadrant of the lattice can be written as $C=e^{-\kappa K}$, where the corner Hamiltonian $K$ is a sum of the local interaction terms $h_i$ of the associated one-dimensional quantum chain, each weighted by its distance from the quadrant corner. For a half-infinite chain this identifies the reduced density matrix with (a power of) the CTM, so the entanglement spectrum literally becomes the corner-Hamiltonian spectrum. This identification is exact for gapped half-infinite chains — the transverse-field Ising chain via the two-dimensional Ising CTM, the XY chain, the XXZ chain in its massive phase $\Delta>1$ via the six-vertex CTM, the XYZ chain via the eight-vertex CTM, and the coupled harmonic-oscillator chain — with the local energy-density term carrying a linear weight in the distance from the entangling cut and exactly equidistant single-particle entanglement levels (reviewed by Dalmonte, Eisler, Falconi and Vermersch, 2022).

Kim, Katsura, Trivedi and Han (2016) tested this correspondence far outside its proven domain. Using the density matrix renormalization group (DMRG — a variational method that delivers the Schmidt data of chain ground states), they computed the ES at every cut position $n$ from the open edge of critical SU($N$) spin chains ($2\le N\le 4$; the SU($N$) chain is the higher-symmetry generalization of the spin-1/2 Heisenberg chain, with one spin per site transforming under the fundamental representation of SU($N$)), of the critical transverse-field Ising chain, and of integrable points of the spin-1 bilinear-biquadratic family $H=\sum_i[\cos\theta\,\mathbf{S}_i\cdot\mathbf{S}_{i+1}+\sin\theta\,(\mathbf{S}_i\cdot\mathbf{S}_{i+1})^2]$, and compared it with the spectrum of the corner Hamiltonian $K^{(n)}=\sum_{i=1}^{n-1} i\,h_i$ of a chain of length $n$, with $i$ the distance of the local term from the entangling cut (the spectra are unchanged under reflecting the chain, so the opposite orientation convention is immaterial for spectra; the orientation matters only when reading the weight profile). The level spacings and degeneracy patterns of the low-lying ES and corner spectra agreed for all $n$ checked, including cuts close to the open edge, where continuum arguments do not obviously apply. The ES also oscillates with period $N$ as a function of cut position, persisting deep into the bulk — explained heuristically via Young tableaux (the combinatorial arrays encoding irreducible SU($N$) representations) — and the correspondence survives at some nonintegrable bilinear-biquadratic points while failing at the AKLT point (an exactly solvable gapped valence-bond-solid point) and at a first-order transition point.

The available theoretical support is continuum. Boundary conformal field theory (BCFT — conformal field theory on domains with boundaries, in which the entangling cut acts as a conformal boundary) predicts that the entanglement Hamiltonian of a critical system is a weighted integral of the energy density $T_{00}$: for a half-infinite subsystem the weight is linear in the distance from the cut (the continuum avatar of the Bisognano-Wichmann theorem of relativistic quantum field theory, which states that the modular Hamiltonian of a half-space is the Lorentz-boost generator, proportional to $\int x\,T_{00}(x)\,dx$), while for a finite interval of $n$ sites adjacent to a physical boundary the weight is parabolic, $\propto i(2n-i)/(2n)$ with $i$ the distance from the cut (Cardy and Tonni, 2016). Exact free-fermion results for the XX model give this parabolic envelope in its lattice form, $i(2n-i+1)/(2n)$ (Peschel, 2004; Peschel and Eisler, 2009; Eisler and Peschel, 2018), which differs from the boundary-conformal continuum weight only by a relative $O(1/n)$ lattice-convention term. For free fermions the entanglement Hamiltonian is exactly reducible to a function of the subsystem correlation matrix, and a closed-form operator commuting with it carries precisely this trigonometric-to-parabolic weight profile; but the exact entanglement Hamiltonian itself at criticality contains longer-range terms and deviates from any purely local profile, so the correspondence can only be a statement about low-lying spectra rather than an operator identity.

For genuinely interacting chains — the SU($N$) chains of the source work — no exact entanglement Hamiltonian is known and the evidence remains numerical. What is missing is a microscopic, lattice-level account of the correspondence: a derivation valid for arbitrary cut position and at criticality, the precise finite-$n$ weight profile, and the class of lattice Hamiltonians for which the correspondence holds.

Problem Statement

For an open quantum spin chain with a unique ground state, the low-lying entanglement spectrum at a cut $n$ sites from the open edge empirically coincides with the spectrum of the corner Hamiltonian $K^{(n)}=\sum_{i=1}^{n-1} i\,h_i$ (with $h_i$ the chain's local interaction term and $i$ its distance from the entangling cut) for integrable models, both gapped and critical, and even at some nearby nonintegrable points, while failing at other points such as the AKLT and first-order transition points. Establish this entanglement-spectrum/corner-Hamiltonian correspondence on a microscopic, lattice level. Concretely:

(1) Derive the correspondence — the equality of the low-lying spectrum of the entanglement Hamiltonian $H_{\mathrm{ent}}^{(n)}$ with that of $K^{(n)}$, or whatever exact corrected form it takes — for arbitrary finite cut position $n$ and for critical models, in particular for the critical SU($N$) chains where it was observed, by a calculation on the lattice model rather than through continuum boundary conformal field theory.

(2) Determine the precise spatial weight profile of the finite-$n$ correspondence, that is, of the operator realizing it ($K^{(n)}$ or its exact corrected form): decide whether the correct profile is the linear weight $i$ of the corner Hamiltonian or the parabolic profile $i(2n-i+1)/(2n)$ (with $i$ the distance from the cut) suggested by exact XX-model results and boundary conformal field theory — at minimum for interacting chains, where exact diagonalization of large-$n$ corner Hamiltonians is currently prohibitive.

(3) Characterize the class of lattice Hamiltonians for which the correspondence holds, in the direction of necessary and sufficient conditions (candidates include integrability, low-energy Lorentz invariance, gapped versus critical spectrum, and symmetry class), so as to explain the known anchor cases: critical SU($N$) chains and other integrable points where it holds, nonintegrable bilinear-biquadratic points where it nevertheless holds, and AKLT-type and first-order points where it fails. A characterization by conditions that account for these anchor cases is sufficient here; a complete necessary-and-sufficient classification is not required. As a required component of this deliverable, establish whether the Young-tableaux counting of the degeneracy structure extends to quantum-group-deformed symmetries such as $U_q(sl_2)$ (the $q$-deformed enveloping algebra of $\mathrm{SU}(2)$, which governs the spin-$1/2$ XXZ chain, with ordinary $\mathrm{SU}(2)$ recovered at $q\to 1$).

These three deliverables are components of a single objective — a microscopic theory of when and why the many-body entanglement spectrum of an open chain is encoded in the corner Hamiltonian of the associated two-dimensional integrable model. The source work also speculated, as a further direction rather than a posed question, on experimental detection of the persistent period-$N$ entanglement-spectrum oscillation in cold-atom SU($N$) chains; that direction does not belong to the completion boundary of this problem.

References

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