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arXiv:2607.25256

Anomalous entanglement scaling from eigenvector nonorthogonality in critical non-Hermitian free fermions

Zhenyu Xiao, Shinsei Ryu

cond-mat.mes-hallquant-phSubmitted Jul 28, 2026• Announced Jul 29, 2026• 0 citations
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Abstract

Entanglement carries universal content that labels phases and critical points. We study the entanglement entropy of the steady states of critical non-Hermitian free-fermion chains. It scales logarithmically with subsystem size, but the coefficients vary continuously with the parameters and form a Rényi family that no single central charge can reproduce. We trace this anomaly to an ``imaginary'' Dirac point, a crossing in the imaginary part of the energy where the occupied state switches between two Bloch states. Their nonorthogonality weakens the occupation discontinuity and lowers the logarithmic coefficient. A low-energy expansion yields closed-form coefficients in excellent agreement with lattice numerics in various one-dimensional critical steady states. Remarkably, weak real onsite disorder leaves this logarithmic scaling intact and enhances the entanglement. Our results provide a generic understanding of entanglement in critical non-Hermitian free-fermion steady states.

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Entanglement carries universal content that labels phases and critical points. We study the entanglement entropy of the steady states of critical non-Hermitian free-fermion chains. It scales logarithmically with subsystem size, but the coefficients vary continuously with the parameters and form a Rényi family that no single central charge can reproduce. We trace this anomaly to an ``imaginary'' Dirac point, a crossing in the imaginary part of the energy where the occupied state switches between two Bloch states. Their nonorthogonality weakens the occupation discontinuity and lowers the logarithmic coefficient. A low-energy expansion yields closed-form coefficients in excellent agreement with lattice numerics in various one-dimensional critical steady states. Remarkably, weak real onsite disorder leaves this logarithmic scaling intact and enhances the entanglement. Our results provide a generic understanding of entanglement in critical non-Hermitian free-fermion steady states.
Abstract The symmetry-resolved Rényi entanglement entropy is the Rényi entanglement entropy of each symmetry sector of a density matrix ρ\rho ρ ρ . This experimentally relevant quantity is known to have rich theoretical connections to conformal field theory (CFT). For a family of critical free-fermion chains, we present a rigorous lattice-based derivation of its scaling properties using the theory of Toeplitz determinants. We consider a class of critical quantum chains with a microscopic U(1) symmetry; each chain has a low energy description given byNmassless Dirac fermions. For the density matrix, ρA\rho _AρA​ ρA , of subsystems ofLneighbouring sites we calculate the leading terms in the largeLasymptotic expansion of the symmetry-resolved Rényi entanglement entropies. This follows from a largeLexpansion of the charged moments of ρA\rho _AρA​ ρA ; we derive tr(eiαQAρAn) = aeiα⟨QA⟩(σL)−x(1+O(L−μ))\mathrm {tr}(\mathrm {e}^{\mathrm {i}\alpha Q_A} \rho _A^n)~=~a \mathrm {e}^{\mathrm {i}\alpha \langle Q_A\rangle } (\sigma L)^{-x}(1+O(L^{-\mu }))tr(eiαQA​ρAn​) = aeiα⟨QA​⟩(σL)−x(1+O(L−μ)) tr(eiαQAρAn)=aeiα⟨QA⟩(σL)-x(1+O(L-μ)) , wherea, xand μ\mu μ μ are universal and σ\sigma σ σ depends only on theNFermi momenta. We show that the exponentxcorresponds to the expectation from CFT analysis. The error term O(L−μ)O(L^{-\mu })O(L−μ) O(L-μ) is consistent with but weaker than the field theory prediction O(L−2μ)O(L^{-2\mu })O(L−2μ) O(L-2μ) . However, using further results and conjectures for the relevant Toeplitz determinant, we find excellent agreement with the expansion over CFT operators.
Non-unitary evolution can give rise to novel steady states classified by their entanglement properties. In this work, we aim to understand the effect of long-range hopping that decays with r−α in non-Hermitian free-fermion systems. We first study two solvable Brownian models with long-range non-unitary dynamics: a large- N SYK 2 chain and a single-flavor fermion chain and we show that they share the same phase diagram. When α>0.5 , we observe two critical phases with subvolume entanglement scaling: (i) α>1.5 , a logarithmic phase with dynamical exponent z=1 and logarithmic subsystem entanglement, and (ii) 0.5<α<1.5 , a fractal phase with z=2α−12 and subsystem entanglement SA∝LA1−z , where LA is the length of the subsystem A . These two phases cannot be distinguished by the purification dynamics, in which the entropy always decays as L/T . We then confirm that the results are also valid for the static SYK 2 chain, indicating the phase diagram is universal for general free-fermion systems. We also discuss phase diagrams in higher dimensions and the implication in measurement-induced phase transitions.
The Wannier-Stark localization dynamically alters the entanglement behavior of non-Hermitian free fermions. Utilizing the single-particle correlation matrix technique, we analyze the effective Hamiltonian of these fermions with a Wannier-Stark ladder. Under open boundary conditions, we observe steady-state half-chain entanglement entropy (EE) and identify two distinct area-law regions and an algebraic scaling region. Finite-sized scaling analysis reveals critical scaling behavior of the half-chain EE. Notably, the system demonstrates unique entanglement characteristics under periodic boundary conditions, which diverge from the (1+1)d(1+1)d(1+1)d conformal field theory predictions for Anderson localization. Our findings highlight entanglement phases emerging from the interplay between the non-Hermitian skin effect and disorder-free localization.
Abstract We study the entanglement properties of non-Hermitian free fermionic models with translation symmetry using the correlation matrix technique. Our results show that the entanglement entropy has a logarithmic correction to the area law in both one-dimensional and two-dimensional systems. For any one-dimensional one-band system, we prove that each Fermi point of the system contributes exactly 1/2 to the coefficient c of the logarithmic correction. Moreover, this relation between c and Fermi point is verified for more general one-dimensional and two-dimensional cases by numerical calculations and finite-size scaling analysis. In addition, we also study the single-particle and density–density correlation functions.
Abstract The characterization of topological phases in disordered non-Hermitian systems remains a formidable challenge, as the non-Hermitian skin effect (NHSE) disrupts the conventional bulk-boundary correspondence and renders spectral invariants unreliable. In this letter, we propose the real part of the biorthogonal entanglement entropy as a precise, basis-independent diagnostic that cuts through this complexity. Investigating a disordered non-reciprocal Kitaev chain, we demonstrate that this entropic probe provides a robust, boundary-sensitive diagnostic that faithfully tracks the topological phase transition even in regimes where traditional winding numbers fluctuate wildly. Crucially, we uncover an anomalous critical scaling of the entanglement entropy: while the Hermitian limit follows standard conformal scaling , the non-Hermitian critical point exhibits a super-critical effective central charge of . Within the studied system sizes , this enhancement is consistent with skin-effect–driven non-unitary behavior at the critical point. Our findings establish biorthogonal entanglement not only as a robust order parameter for the topological Anderson insulator but also as an experimentally accessible signature in topo-electric circuits.
We study the critical behavior and the ground-state entanglement of a large class of su(1|1) supersymmetric spin chains with a general (not necessarily monotonic) dispersion relation. We show that this class includes several relevant models, with both short- and long-range interactions of a simple form. We determine the low temperature behavior of the free energy per spin, and deduce that the models considered have a critical phase in the same universality class as a (1+1)-dimensional conformal field theory (CFT) with central charge equal to the number of connected components of the Fermi sea. We also study the Rényi entanglement entropy of the ground state, deriving its asymptotic behavior as the block size tends to infinity. In particular, we show that this entropy exhibits the logarithmic growth characteristic of (1+1)-dimensional CFTs and one-dimensional (fermionic) critical lattice models, with a central charge consistent with the low-temperature behavior of the free energy. Our results confirm the widely believed conjecture that the critical behavior of fermionic lattice models is completely determined by the topology of their Fermi surface.