Rottoli, Fossati and Calabrese (J. Stat. Mech. 2024) computed numerically the entanglement Hamiltonian of the PT-symmetric non-Hermitian Su-Schrieffer-Heeger chain — a free-fermion chain with alternating hoppings and a staggered imaginary on-site potential — in the gapped phase and at its critical point, where the continuum limit is the non-unitary c=−2c=-2 bcbc-ghost conformal field theory. They conjectured a lattice Bisognano-Wichmann form with a triangular entanglement temperature of unknown slope in the gapped phase, and at criticality a continuum entanglement Hamiltonian containing a novel imaginary ghost-number chemical-potential term that is responsible for the negative entanglement entropy. Their paper leaves three analytic problems open: deriving the corner transfer matrix of the gapped phase, which would fix the exact slope and quantitatively explain finite-size oscillations; deriving the critical entanglement Hamiltonian analytically from the full lattice kernel, as is known for Hermitian free massless fermions; and establishing whether the conjectured structure survives relevant perturbations and interactions, or extends to non-Hermitian systems with genuinely complex spectra. None of these components has been resolved in the literature to date. The problem remains open.

Background

For a subsystem AA of a quantum many-body system with reduced density matrix ρA\rho_A, the entanglement Hamiltonian (EH) is defined as KA=−log⁡ρAK_A=-\log\rho_A; its spectrum is the entanglement spectrum, and it gives the most complete operator-level characterisation of the entanglement of AA. In relativistic unitary quantum field theory, the Bisognano-Wichmann theorem — a foundational result of algebraic quantum field theory — states that the EH of a half-space is proportional to the generator of Lorentz boosts, KA=2π∫x T00(x) dxK_A=2\pi\int x\,T_{00}(x)\,dx, where T00T_{00} is the local energy density; for an interval of length ℓ\ell in a conformal field theory (CFT), the same logic gives KA=∫0ℓβ(x) T00(x) dxK_A=\int_0^\ell \beta(x)\,T_{00}(x)\,dx with a parabolic 'entanglement temperature' β(x)=x(ℓ−x)/ℓ\beta(x)=x(\ell-x)/\ell (Casini and Huerta, 2009). On the lattice, integrable models display an analogous 'lattice Bisognano-Wichmann' structure: the EH equals the lattice Hamiltonian with local couplings weighted by a factor that grows linearly with the distance from the boundary, producing a triangular entanglement temperature. This structure is explained by the corner transfer matrix (CTM) of Baxter — a standard object of two-dimensional integrable statistical mechanics whose fourth power equals, up to normalisation, the reduced density matrix of a half-line, so that KA=−log⁡A^4K_A=-\log\widehat{A}^{4}. For the Hermitian dimerized hopping chain, knowledge of the exact CTM fixes analytically the proportionality constant — the slope of the triangular entanglement temperature (Eisler, Di Giulio, Tonni and Peschel, 2020).

The model studied here is the non-Hermitian Su-Schrieffer-Heeger (nH-SSH) chain with PTPT symmetry, where PP denotes spatial inversion and TT time reversal realised as complex conjugation; their product leaves the Hamiltonian invariant. It is a free-fermion chain on L=2NL=2N sites with alternating real nearest-neighbour hoppings v,w>0v,w>0 and a staggered imaginary on-site potential +iu+iu on even sites and −iu-iu on odd sites, which breaks Hermiticity. For ∣v−w∣>u|v-w|>u the spectrum is real (PTPT-unbroken phase); for ∣v−w∣<u|v-w|<u it is complex (PTPT-broken phase); and the critical lines w−v=±uw-v=\pm u are exceptional points where the momentum-space two-band kernel becomes non-diagonalizable at k=πk=\pi. Because the Hamiltonian is non-Hermitian, the ground state must be defined biorthogonally through the left and right eigenvectors, and the density matrix ρ=∣L⟩⟨R∣\rho=|L\rangle\langle R| is itself non-Hermitian. The single-particle EH kernel is obtained from the correlation matrix CC via kA=log⁡[(1−C)/C]k^{A}=\log[(1-C)/C], with CC built from the biorthogonal state; its eigenvalues can then leave the interval [0,1][0,1]. At the critical point w−v=uw-v=u the continuum limit is the c=−2c=-2 bcbc-ghost CFT — a logarithmic (non-semisimple) conformal field theory of anticommuting ghost fields b,cb,c carrying a conserved ghost-number current J(x)=: ⁣cb ⁣:(x)J(x)=:\!cb\!:(x) — and the bipartite entanglement entropy is negative, scaling with central charge c=−2c=-2 (Chang, You, Wen and Ryu, 2020; Tu, Tzeng and Chang, 2022; Fossati, Ares and Calabrese, 2023).

Rottoli, Fossati and Calabrese (2024) computed the EH of this model numerically. In the PTPT-unbroken gapped phase they found, for an interval in a large system, a kernel whose nearest-neighbour and staggered imaginary chemical-potential parts both follow the triangular entanglement temperature expected from a conjectured half-space EH proportional to the lattice Hamiltonian with linearly growing weights — the first observation of Bisognano-Wichmann-like behaviour in a non-Hermitian model — but with an unknown proportionality constant, and with finite-size oscillations of the imaginary chemical potential that the conjecture does not capture. At the critical point they found, besides the expected parabolic temperature multiplying the energy density, a novel term: an imaginary chemical potential interpolating linearly between 2πi2\pi i and 00 across the interval, which does not scale with the system size and is responsible for the alternating sign of the entanglement spectrum and ultimately for the negative entanglement entropy. They thus conjectured the continuum EH of the ghost CFT to be KA=∫0ℓx(ℓ−x)ℓ T00(x) dx+2πi∫0ℓ(1−xℓ)J(x) dx,K_A = \int_0^\ell \frac{x(\ell-x)}{\ell}\,T_{00}(x)\,dx + 2\pi i \int_0^\ell \left(1-\frac{x}{\ell}\right) J(x)\,dx, where the second term has no analogue in unitary CFTs. All of these results are numerical or conjectural, and the authors explicitly pose as open problems the analytic derivations that would establish them, together with the question of their robustness.

Problem Statement

Establish the entanglement Hamiltonian of the PTPT-symmetric non-Hermitian Su-Schrieffer-Heeger chain on analytic footing, converting the numerical and conjectural results of Rottoli, Fossati and Calabrese (2024) into derivations and determining the scope of their validity. Three linked components, all stated by the source as open problems about the same object, jointly define the objective. (1) Gapped phase: construct analytically the corner transfer matrix (CTM) of the nH-SSH chain in the PTPT-unbroken gapped phase, and from it derive the exact proportionality constant — the slope of the triangular entanglement temperature — in the conjectured half-space entanglement Hamiltonian KA∝∑jj hjK_A\propto\sum_j j\,h_j (with the staggered weights of the source), together with a quantitative explanation of the finite-size oscillations of the staggered imaginary chemical potential that the conjecture does not capture. (2) Critical point: derive analytically the entanglement Hamiltonian at the critical line w−v=uw-v=u, starting from the full lattice kernel — including the longer-range couplings generated at criticality, which in the continuum limit must be resummed into the energy density T00T_{00} — and demonstrate that the controlled continuum limit reproduces the conjectured ghost-CFT form KA=∫0ℓ[x(ℓ−x)/ℓ] T00(x) dx+2πi∫0ℓ(1−x/ℓ) J(x) dxK_A=\int_0^\ell [x(\ell-x)/\ell]\,T_{00}(x)\,dx+2\pi i\int_0^\ell(1-x/\ell)\,J(x)\,dx with the imaginary ghost-number term at weight interpolating linearly between 2πi2\pi i and 00, in the same sense in which the corresponding continuum results are established for Hermitian free massless fermions. (3) Robustness and scope: determine the regime of validity of the two conjectured forms, feature by feature and regime by regime. (3a) Gapped-phase robustness: determine whether the lattice Bisognano-Wichmann locality and the triangular entanglement temperature of component (1) survive relevant PTPT-symmetric perturbations of the gapped phase and its interacting extensions, for at least one explicitly specified relevant perturbation and one interacting extension. (3b) Critical-point robustness: determine whether the parabolic temperature and the imaginary ghost-number term of component (2) survive PTPT-symmetric deformations and interacting extensions that keep the system at the c=−2c=-2 critical point. (3c) PTPT-broken scope: extend the analysis to the PTPT-broken regime, where the Hamiltonian spectrum is genuinely complex, at least at the level of the single-particle entanglement kernel obtained from the biorthogonal density matrix. A certified refutation of a conjectured form — for instance, a demonstration that the gapped-phase kernel does not follow the conjectured weights outside the known crossover and finite-size effects, or that the critical-point kernel does not reduce to the conjectured ghost-CFT form — counts as an answer at the same level as a derivation for the corresponding component (1) or (2) and for the corresponding survival sub-claim (3a) or (3b), but settles only the specific claim it refutes: refuting the structure under one specified perturbation or interaction resolves that sub-claim only for that perturbation or interaction, and the constructive PTPT-broken single-particle extension (3c) is a separate deliverable that no refutation of the gapped- or critical-phase weights discharges. The objective is complete when component (1), component (2), and all three of (3a), (3b), (3c) have each been settled — by a derivation or a refutation for (1), (2), (3a) and (3b), and by the explicit single-particle extension for (3c).

References

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