This paper studies the entanglement properties of free-fermion systems without hermiticity. From the entanglement entropy scaling, the authors identify the non-unitary conformal field theory with c=-2 in a PT-symmetric non-Hermitian lattice system hosting topological phases. The entanglement spectrum also can detect the topological properties in the gapped phases, including the PT-symmetric Su-Schrieffer-Heeger model and the non-Hermitian Chern insulator.
This paper studies the entanglement properties of free-fermion systems without hermiticity. From the entanglement entropy scaling, the authors identify the non-unitary conformal field theory with c=-2 in a PT-symmetric non-Hermitian lattice system hosting topological phases. The entanglement spectrum also can detect the topological properties in the gapped phases, including the PT-symmetric Su-Schrieffer-Heeger model and the non-Hermitian Chern insulator.
Dramatically different from the Hermitian systems, the conventional bulk-boundary correspondence (BBC) is broken in the non-Hermitian systems. In this paper, we use edge entanglement entropy to characterize the topological properties of non-Hermitian Su-Schrieffer-Heeger Hubbard model. For free fermions, we study the scaling behavior of entanglement entropy and demonstrate that the edge entanglement entropy is a good indicator to delimit different phases of non-Hermitian systems. We further generalize the edge entanglement entropy to the non-Hermitian interacting Hubbard chain and obtain the topological phase diagram in the plane of interaction and non-Hermitian hopping amplitudes. It is found that the Hubbard interaction diminishes and weakens the breakdown of the bulk-boundary correspondence, which eventually disappears at some critical value of interaction.
Topological phase transitions are found in a variety of systems and were shown to be deeply related with a thermodynamic description through scaling relations. Here we investigate the entanglement entropy, which is a quantity that captures the central charge of a critical model and the thermodynamics of the nonreciprocal Su-Schrieffer-Heeger (SSH) model. Although this model has been widely studied, the thermodynamic properties reveal interesting physics not explored so far. In order to analyze the boundary effects of the model, we use Hill's thermodynamics to split the grand potential in two contributions: the extensive one, related to the bulk, and the subdivision one, related to the boundaries. Then, we derive the thermodynamic entropy for both the edges and the bulk, and the heat capacity for the bulk at the topological phase transitions. The latter is related to the central charge when the underlying theory is a conformal field theory, whereas the first reveals the resilience of the topological edge states to finite temperatures. The phase transition between phases that are adiabatically connected with the Hermitian SSH model display the well-known behavior of systems within the Dirac universality class, but the transition between phases with complex energies shows an unexpected critical behavior that signals the emergence of an imaginary time crystal.
The entanglement entropy can be an effective diagnostic tool for probing topological phase transitions. In one-dimensional single particle systems, the periodic driving generates a variety of topological phases and edge modes. In this work, we investigate the topological phase transition of the one-dimensional Floquet Su-Schrieffer-Heeger model using entanglement entropy, and construct the phase diagram based on entanglement entropy. The entanglement entropy exhibits pronounced peaks and follows the logarithmic scaling law at the phase transition points, from which we extract the central charge c=1. We further investigate the entanglement spectrum to accurately distinguish the different topological phases. In addition, the coupling between zero and π modes leads to characteristic splittings in the entanglement spectrum, signaling their hybridization under periodic driving. These results remain robust in non-Hermitian regimes and in the presence of next-nearest-neighbor hopping, demonstrating the reliability and universality of entanglement entropy as a diagnostic for topological phase transitions.
Abstract This topical review article reports rapid progress on the generalization and application of entanglement in non-Hermitian free-fermion quantum systems. We begin by examining the realization of non-Hermitian quantum systems through the Lindblad master equation, alongside a review of typical non-Hermitian free-fermion systems that exhibit unique features. A pedagogical discussion is provided on the relationship between entanglement quantities and the correlation matrix in Hermitian systems. Building on this foundation, we focus on how entanglement concepts are extended to non-Hermitian systems from their Hermitian free-fermion counterparts, with a review of the general properties that emerge. Finally, we highlight various concrete studies, demonstrating that entanglement entropy remains a powerful diagnostic tool for characterizing non-Hermitian physics. The entanglement spectrum also reflects the topological characteristics of non-Hermitian topological systems, while unique non-Hermitian entanglement behaviors are also discussed. The review is concluded with several future directions. Through this review, we hope to provide a useful guide for researchers who are interested in entanglement in non-Hermitian quantum systems.
Usually duality process keeps energy spectrum invariant. In this paper, we provide a duality, which keeps entanglement spectrum invariant, in order to diagnose quantum entanglement of non-Hermitian non-interacting fermionic systems. We limit our attention to non-Hermitian systems with a complete set of biorthonormal eigenvectors and an entirely real energy spectrum. The original system has a reduced density matrix \rho_\mathrm{o} ρ o and the real space is partitioned via a projecting operator \mathcal{R}_{\mathrm o} ℛ o . After dualization, we obtain a new reduced density matrix \rho_{\mathrm{d}} ρ d and a new real space projector \mathcal{R}_{\mathrm d} ℛ d . Remarkably, entanglement spectrum and entanglement entropy keep invariant. Inspired by the duality, we defined two types of non-Hermitian models, upon \mathcal R_{\mathrm{o}} ℛ o is given. In type-I exemplified by the "non-reciprocal model'', there exists at least one duality such that \rho_{\mathrm{d}} ρ d is Hermitian. In other words, entanglement information of type-I non-Hermitian models with a given \mathcal{R}_{\mathrm{o}} ℛ o is entirely controlled by Hermitian models with \mathcal{R}_{\mathrm{d}} ℛ d . As a result, we are allowed to apply known results of Hermitian systems to efficiently obtain entanglement properties of type-I models. On the other hand, the duals of type-II models, exemplified by "non-Hermitian Su-Schrieffer-Heeger model’’, are always non-Hermitian. For the practical purpose, the duality provides a potentially computation route to entanglement of non-Hermitian systems. Via connecting different models, the duality also sheds lights on either trivial or nontrivial role of non-Hermiticity played in quantum entanglement, paving the way to potentially systematic classification and characterization of non-Hermitian systems from the entanglement perspective.
We demonstrate three types of transformations that establish connections between Hermitian and non-Hermitian quantum systems at criticality, which can be described by conformal field theories (CFTs). For the transformation preserving both the energy and the entanglement spectra, the corresponding central charges obtained from the logarithmic scaling of the entanglement entropy are identical for both Hermitian and non-Hermitian systems. The second transformation, while preserving the energy spectrum, does not perserve the entanglement spectrum. This leads to different entanglement entropy scalings and results in different central charges for the two types of systems. We demonstrate this transformation using the dilation method applied to the free fermion case. Through this method, we show that a non-Hermitian system with central charge c = -4 c = − 4 can be mapped to a Hermitian system with central charge c = 2 c = 2 . Lastly, we investigate the Galois conjugation in the Fibonacci model with the parameter \phi \to - 1/\phi ϕ → − 1 / ϕ , in which the transformation does not preserve both energy and entanglement spectra. We demonstrate the Fibonacci model and its Galois conjugation relate the tricritical Ising model/3-state Potts model and the Lee-Yang model with negative central charges from the scaling property of the entanglement entropy.