Lattice-level status of the entanglement-spectrum–corner-Hamiltonian correspondence in integrable open spin chains

For open quantum spin chains — the critical SU(2) Heisenberg, SU(3) Uimin-Lai-Sutherland, and SU(4) Kugel-Khomskii chains, the spin-1 bilinear-biquadratic family, and the critical transverse-field Ising chain — the entanglement spectrum (ES) at any cut position $n$ from the boundary was found by Kim, Katsura, Trivedi, and Han (2016) to match, in level spacings and degeneracy structure, the spectrum of the finite corner Hamiltonian $K^{(n)}=\sum_{i=1}^{n-1} i\,h_i$, the lattice descendant of Baxter's corner transfer matrix. The only available explanations are continuum boundary-conformal-field-theory arguments, which do not explain the site-by-site agreement at small $n$ and break down at criticality, where corner-transfer-matrix eigenstates are non-normalizable. The open problem is to establish this correspondence at the lattice level: derive when and why the entanglement Hamiltonian is proportional to $K^{(n)}$ (or to its refined variant with parabolic weight $i(2n-i+1)/(2n)$ known exactly for the free-fermion open XX chain), characterize precisely the class of lattice Hamiltonians for which it holds (integrability is sufficient in the observed cases but not necessary, and the correspondence fails at specific non-integrable points), and determine the exact finite-$n$ spatial weight of the entanglement Hamiltonian in interacting chains. Extensions posed by the same source ask for a Young-tableaux-style account of the period-$N$ entanglement-spectrum oscillations in chains with quantum-group symmetry (the source observes that its heuristic argument already covers spatially inhomogeneous chains as long as the SU($N$) symmetry remains intact), and for experimentally observable consequences in SU($N$) cold-atom realizations. Later work (2017–2025) settled the corresponding questions for free models and gave converging numerical evidence for lattice Bisognano-Wichmann ansätze, but no lattice-level proof, exact interacting entanglement Hamiltonian, or scope characterization exists, so the core remains open.

Background

Take a quantum spin chain of length $L$ with open boundaries in its ground state $|\psi\rangle$ and cut it after site $n$. The reduced density matrix of the first $n$ sites, $\rho^{(n)}$, contains all entanglement across the cut; writing $\rho^{(n)} = e^{-H_{\mathrm{ent}}^{(n)}}$ defines the entanglement Hamiltonian, and the eigenvalues of $H_{\mathrm{ent}}^{(n)}$ form the entanglement spectrum (ES), a standard fingerprint of a quantum many-body state. For lattice models reducible to free fermions the ES is available exactly from the one-particle correlation matrix (Peschel; Peschel and Eisler), but for interacting chains it is obtained mainly through the density matrix renormalization group (DMRG), a variational method whose efficiency is governed precisely by the spectrum of $\rho^{(n)}$.

A lattice object with a closely related structure is Baxter's corner transfer matrix (CTM), the partition-function-like matrix associated with one quadrant of a two-dimensional classical lattice model. Its logarithm, the corner Hamiltonian, has the form $K = \sum_i i\,h_i$, where $h_i$ is the local transfer term at distance $i$ from the corner, so the weight grows linearly with distance from the corner. For models reducible to free fermions the reduced density matrix of a half-system is exactly a power of the CTM, $\rho \propto e^{-\mathrm{const}\cdot K}$ — the observation underlying early density-matrix renormalization ideas. Whether anything analogous holds for interacting chains was historically unclear, especially at a critical point, where the CTM eigenstates are non-normalizable and the CTM itself is ill-defined except in free-fermion-reducible cases.

Kim, Katsura, Trivedi, and Han (2016) used DMRG to compare, for open chains, the ES at the $n$-th cut position from the edge with the spectrum of the finite corner Hamiltonian $K^{(n)} = \sum_{i=1}^{n-1} i\,h_i$ built from the quantum chain's own local terms. For the critical SU(2) spin-1/2 Heisenberg chain — exactly solvable by the Bethe ansatz, the standard exact-solution method for one-dimensional integrable models — the SU(3) Uimin-Lai-Sutherland chain, and the SU(4) Kugel-Khomskii chain (all critical, with low-energy descriptions by the SU($N$)$_1$ Wess-Zumino-Witten conformal field theory, the exactly solvable continuum theory describing SU($N$)-symmetric critical chains), for the spin-1 bilinear-biquadratic family $H=\sum_i \cos\theta\,\mathbf{S}_i\cdot\mathbf{S}_{i+1} + \sin\theta\,(\mathbf{S}_i\cdot\mathbf{S}_{i+1})^2$ at and near its integrable points, and for the critical transverse-field Ising chain, they found that the level spacings and degeneracy pattern of the two spectra agree up to one overall scale factor for every cut position checked. Two further observations sharpen the puzzle. First, the ES oscillates with period $N$ as a function of cut position in the SU($N$) chains, and unlike the boundary oscillations of the entanglement entropy (which decay into the bulk; Laflorencie, Sørensen, Chang, and Affleck), these oscillations persist deep into the chain; a heuristic counting argument based on Young tableaux (the diagrams labeling irreducible SU($N$) representations) reproduces the period-$N$ degeneracy pattern qualitatively. Second, the correspondence survives at some non-integrable bilinear-biquadratic couplings ($\theta \approx -0.648\pi, -0.352\pi, -0.148\pi$) while failing at others ($\theta \approx 0.148\pi$, near the ULS point; the AKLT point at $\tan\theta = 1/3$ — the gapped, exactly solvable valence-bond state; and the first-order transition at $\theta = -3\pi/4$), so integrability is sufficient in the observed cases but is not required for it.

The available explanations are continuum ones. In relativistic quantum field theory, the Bisognano-Wichmann theorem identifies the entanglement Hamiltonian of a half-space bipartition with the boost generator, whose local weight grows linearly with distance from the entangling surface; in two-dimensional conformal field theory (Cardy and Tonni), the entanglement Hamiltonian of a finite interval is the energy density weighted by a parabola. For gapped Lorentz-invariant one-dimensional field theories, Cho, Ludwig, and Ryu proved that the low-lying entanglement spectrum of a half-space bipartition equals the spectrum of the underlying conformal field theory on a strip of width $\log(\xi/a)$, with $\xi$ the correlation length and $a$ the lattice spacing — a continuum counterpart of the ES–corner-spectrum correspondence. These scaling-limit statements, however, do not explain the site-by-site agreement observed at small $n$, do not delimit which lattice Hamiltonians inherit the property, and leave a specific tension unresolved: the exact solution of the open XX chain (Peschel; Peschel and Eisler) gives the local part of the entanglement Hamiltonian the parabolic weight $i(2n-i+1)/(2n)$ rather than the linear weight $i$ of the corner Hamiltonian, so which form, if either, is correct at finite $n$ in the interacting chains is unknown.

Problem Statement

Establish the lattice-level status of the entanglement-spectrum–corner-Hamiltonian correspondence observed by Kim, Katsura, Trivedi, and Han for integrable open spin chains. For an open chain of length $L$ with local Hamiltonian $H = \sum_i h_i$ and ground state $|\psi\rangle$, the reduced density matrix of the first $n$ sites defines $H_{\mathrm{ent}}^{(n)} = -\ln\rho^{(n)}$; the associated corner Hamiltonian is $K^{(n)} = \sum_{i=1}^{n-1} i\,h_i$. The observed fact is that the entanglement spectrum equals the corner spectrum up to one overall scale for every cut position $n$ checked, in the critical SU(2)/SU(3)/SU(4) Heisenberg, Uimin-Lai-Sutherland, and Kugel-Khomskii chains, the spin-1 bilinear-biquadratic family, and the critical transverse-field Ising chain. The problem has three linked components, bound by the single question of whether this equality is an exact structural property of the lattice models or a continuum approximation whose corrections are invisible at DMRG precision:

(1) Derivation. Prove — or derive with sharp, controlled error bounds — when and why $H_{\mathrm{ent}}^{(n)}$ is proportional to $K^{(n)}$ (or to its refined variant $\sum_{i=1}^{n-1} i(2n-i+1)/(2n)\,h_i$), for arbitrary cut position $n$, by a lattice-level argument rather than a continuum boundary-conformal-field-theory one. The derivation must confront the critical point, where the corner-transfer-matrix eigenstates are non-normalizable and the classical CTM construction is ill-defined except in free-fermion-reducible models.

(2) Scope. Characterize precisely the class of lattice Hamiltonians for which the correspondence holds. Integrability is sufficient in the observed cases but not necessary: the correspondence survives at the non-integrable bilinear-biquadratic points $\theta \approx -0.648\pi, -0.352\pi, -0.148\pi$ and fails near the ULS point at $\theta \approx 0.148\pi$, at the AKLT point, and at the first-order transition $\theta = -3\pi/4$. The candidate criterion — a low-energy description by a Lorentz-invariant (conformal or gapped) field theory — must be converted into a testable lattice-level condition that predicts this pattern of successes and failures.

(3) Exact weight. Determine the exact spatial weight of the local terms in $H_{\mathrm{ent}}^{(n)}$ at finite $n$ for the interacting chains: the linear $i$ of the corner Hamiltonian, the parabolic $i(2n-i+1)/(2n)$ established for the open XX chain, or another profile; in particular, whether the parabolic form is the general finite-$n$ behavior across models, as the free-fermion evidence suggests but the interacting numerics cannot confirm.

The problem comprises three linked parts — (1) a lattice-level derivation, (2) a scope criterion, and (3) a finite-$n$ weight profile — jointly posed by the single question above. The same source also poses two further questions: (4) extend the Young-tableaux counting argument for the period-$N$ oscillation of the entanglement spectrum to chains with quantum-group symmetry — a one-parameter deformation of ordinary Lie-algebra symmetry, with $U_q(sl_2)$ the deformed analogue of SU(2) — such as the $U_q(sl_2)$-symmetric spin-1/2 XXZ chain with boundary magnetic fields, and firm up its heuristic form for spatially inhomogeneous chains, to which the source notes it applies as long as the SU($N$) symmetry remains intact; (5) identify experimentally observable consequences of the persistent period-$N$ entanglement-spectrum oscillation in cold-atom realizations of SU($N$) chains. Resolution of the free-fermion case alone does not answer this problem: exact free results were already available to the source, and the correspondence it reports is observed in interacting chains.

References

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