Abstract The entanglement entropy encodes fundamental characteristics of quantum many‐body systems, and is particularly subtle in non‐Hermitian settings where eigenstates generically become non‐orthogonal. In this work, it is found that negative biorthogonal entanglement generically arises from topologically protected non‐orthogonal edge states in free fermion systems, especially for flat‐band edge states. Departing from previous literature which associated negative entanglement with exceptional gapless points, it is showed that robustly negative entanglement can still occur in gapped systems. Gapless 2D flat‐band edge states, however, exhibit novel entanglement behavior which scales quadratically with the transverse dimension L y , independent of system parameters. This dramatically negative scaling can be traced to a new mechanism known as non‐Hermitian critical skin compression (nHCSC), where topological and skin localization in one direction produces a hierarchy of extensively many probability non‐conserving entanglement eigenstates across a cut in another direction. This discovery sheds light on new avenues where topology interplays with criticality and non‐Hermitian localization, unrelated to traditional notions of topological entanglement entropy. This topologically protected negative entanglement also manifests in the second Rényi entropy, which can be measured through SWAP operator expectation values.
Abstract The entanglement entropy encodes fundamental characteristics of quantum many‐body systems, and is particularly subtle in non‐Hermitian settings where eigenstates generically become non‐orthogonal. In this work, it is found that negative biorthogonal entanglement generically arises from topologically protected non‐orthogonal edge states in free fermion systems, especially for flat‐band edge states. Departing from previous literature which associated negative entanglement with exceptional gapless points, it is showed that robustly negative entanglement can still occur in gapped systems. Gapless 2D flat‐band edge states, however, exhibit novel entanglement behavior which scales quadratically with the transverse dimension L y , independent of system parameters. This dramatically negative scaling can be traced to a new mechanism known as non‐Hermitian critical skin compression (nHCSC), where topological and skin localization in one direction produces a hierarchy of extensively many probability non‐conserving entanglement eigenstates across a cut in another direction. This discovery sheds light on new avenues where topology interplays with criticality and non‐Hermitian localization, unrelated to traditional notions of topological entanglement entropy. This topologically protected negative entanglement also manifests in the second Rényi entropy, which can be measured through SWAP operator expectation values.
This Letter introduces a new class of robust states known as exceptional bound (EB) states, which are distinct from the well-known topological and non-Hermitian skin boundary states. EB states occur in the presence of exceptional points, which are non-Hermitian critical points where eigenstates coalesce and fail to span the Hilbert space. This eigenspace defectiveness not only limits the accessibility of state information but also interplays with long-range order to give rise to singular propagators only possible in non-Hermitian settings. Their resultant EB eigenstates are characterized by robust anomalously large or negative occupation probabilities, unlike ordinary Fermi sea states whose probabilities lie between 0 and 1. EB states remain robust after a variety of quantum quenches and give rise to enigmatic negative entanglement entropy contributions. Through suitable perturbations, the coefficient of the logarithmic entanglement entropy scaling can be continuously tuned. EB states represent a new avenue for robustness arising from geometric defectiveness, independent of topological protection or nonreciprocal pumping.
This work introduces a new class of robust states known as Exceptional Boundary (EB) states, which are distinct from the well-known topological and non-Hermitian skin boundary states. EB states occur in the presence of exceptional points, which are non-Hermitian critical points where eigenstates coalesce and fail to span the Hilbert space. This eigenspace defectiveness not only limits the accessibility of state information, but also interplays with long-range order to give rise to singular propagators only possible in non-Hermitian settings. Their resultant EB eigenstates are characterized by robust anomalously large or negative occupation probabilities, unlike ordinary Fermi sea states whose probabilities lie between zero and one. EB states remain robust after a variety of quantum quenches and give rise to enigmatic negative entanglement entropy contributions. Through suitable perturbations, the coefficient of the logarithmic entanglement entropy scaling can be continuously tuned. EB states represent a new avenue for robustness arising from geometric defectiveness, independent of topological protection or non-reciprocal pumping.
Quantum entanglement is one essential element to characterize many-body quantum systems. However, the entanglement measures are mostly discussed in Hermitian systems. Here, we propose a natural extension of entanglement and Rényi entropies to non-Hermitian quantum systems. There have been other proposals for the computation of these quantities, which are distinct from what is proposed in the current paper. We demonstrate the proposed entanglement quantities which are referred to as generic entanglement and Rényi entropies. These quantities capture the desired entanglement properties in non-Hermitian critical systems, where the low-energy properties are governed by the non-unitary conformal field theories (CFTs). We find excellent agreement between the numerical extrapolation of the negative central charges from the generic entanglement/Rényi entropy and the non-unitary CFT prediction. Furthermore, we apply the generic entanglement/Rényi entropy to symmetry-protected topological phases with non-Hermitian perturbations. We find the generic n-th Rényi entropy captures the expected entanglement property, whereas the traditional Rényi entropy can exhibit unnatural singularities due to its improper definition.
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.
Quantum matter emerges from the interplay of fluctuations, topology, and entanglement, which - in equilibrium - governs quantized transport, universal criticality, and topological classification. Non-Hermitian systems, widely explored in platforms ranging from electric circuits to photonics, are intrinsically out-of-equilibrium, and display fundamentally new phenomena, including complex spectra, spectral winding, exceptional topology, and non-unitary dynamics. A central challenge is understanding how the complex single-particle spectrum governs universal many-body behavior. We introduce a symmetry-protected dynamical topological index derived directly from the complex spectrum. Through the lens of algebraic topology, more specifically Morse theory, we identify critical points in the spectrum with topological defects, whose curvature and stability are protected under continuous deformations. This links spectral geometry to many-body observables, unifying non-Hermitian band topology, entanglement, and transport. We demonstrate that non-Hermitian quantum criticality in non-interacting systems is controlled by gain-and-loss-selected non-equilibrium steady states, which dynamically generate an emergent imaginary Fermi surface whose Fermi points host scale-invariant gapless modes with logarithmic entanglement scaling and algebraic correlations. Our work establishes a unified framework for non-Hermitian quantum matter, connecting spectral topology to Morse theory, revealing a topological foundation of non-equilibrium quantum criticality.
We study the properties of the entanglement spectrum in gapped non-interacting non-Hermitian systems, and its relation to the topological properties of the system Hamiltonian. Two different families of entanglement Hamiltonians can be defined in non-Hermitian systems, depending on whether we consider only right (or equivalently only left) eigenstates or a combination of both left and right eigenstates. We show that their entanglement spectra can still be computed efficiently, as in the Hermitian limit. We discuss how symmetries of the Hamiltonian map into symmetries of the entanglement spectrum depending on the choice of the many-body state. Through several examples in one and two dimensions, we show that the biorthogonal entanglement Hamiltonian directly inherits the topological properties of the Hamiltonian for line gapped phases, with characteristic singular and energy zero modes. The right (left) density matrix carries distinct information on the topological properties of the many-body right (left) eigenstates themselves. In purely point gapped phases, when the energy bands are not separable, the relation between the entanglement Hamiltonian and the system Hamiltonian breaks down.