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arXiv:1908.09852 · SciPost Physics

Entanglement spectrum and symmetries in non-Hermitian fermionic non-interacting models

Loïc Herviou, Nicolas Regnault, Jens Hjorleifur Bardarson

Published Nov 28, 2019• 110 citations
arXivHTMLPDF

Abstract

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.

DOI
10.21468/scipostphys.7.5.069

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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.
The entanglement spectrum is a useful tool to study topological phases of matter, and contains valuable information about the ground state of the system. Here, we study its properties for free non-Hermitian systems for both point-gapped and line-gapped phases. While the entanglement spectrum only retains part of the topological information in the former case, it is very similar to Hermitian systems in the latter. In particular, it not only mimics the topological edge modes, but also contains all the information about the polarization, even in systems that are not topological. Furthermore, we show that the Wilson loop is equivalent to the many-body polarization and that it reproduces the phase diagram for the system with open boundaries, despite being computed for a periodic system.
Much has been learned about universal properties of the eigenstate entanglement entropy for one-dimensional lattice models, which is described by a Hermitian Hamiltonian, while much less has been understood for non-Hermitian systems. In the present work we study a non-Hermitian, noninteracting model of fermions which is invariant under combined PTPTPT transformation. Our models show a phase transition from a PTPTPT unbroken phase to broken phase as we tune the Hermiticity-breaking parameter. Entanglement entropy of such systems can be defined in two different ways, depending on whether we consider only right (or equivalently, only left) eigenstates or a combination of both left and right eigenstates which form a complete set of biorthonormal eigenstates. We demonstrate that the entanglement entropy of the ground state and also of the typical excited states shows some unique features in both of these phases of the system. Most strikingly, entanglement entropy obtained taking a combination of both left and right eigenstates shows an exponential divergence with system size at the transition point. While in the PTPTPT-unbroken phase, the entanglement entropy obtained from only the right (or equivalently, left) eigenstates shows identical behavior to an equivalent Hermitian system which is connected to the non-Hermitian system by a similarity transformation.
Exotic singular objects, known as exceptional points, are ubiquitous in non-Hermitian physics. They might be spectral singularities in energy bands that produce anomalous effects and defectiveness. The quantum entanglement of a generic non-Hermitian model with two different types of spectral exceptional points (SEPs) is systematically investigated in this paper. We discovered a relationship between non-unitary conformal field theories and the kkk-linear-type SEPs, which is typically associated with PT\mathcal{PT}PT-symmetry or pesdo-Hermicity spontaneous breaking. The underlying association between kkk-square-root-type SEPs, which arise concurrently with real (imaginary) gap closing in the complex spectrum, mimicking first-order-phase-transition criticalities, and complex conformal field theories (cCFTs) is addressed through the calculation of complex central charges. From the entanglement spectrum, zero-energy exceptional modes are found to be distinct from normal zero modes or topological boundary modes. Finally, we include a brief discussion of analogous non-Hermitian quantum spin models and endeavor to establish an intuitive understanding of exceptional points through the spin picture in various scenarios.
Recent years have seen a growing interest in topological phases beyond the standard paradigm of gapped isolated systems. One recent direction is to explore topological features in non-Hermitian systems that are commonly used as effective descriptions of open systems. Another direction explores the fate of topology at critical points, where the bulk gap collapses. One interesting observation is that both systems, though very different, share certain topological features. For instance, both systems can host half-integer quantized winding numbers and have very similar entanglement spectra. Here we make this similarity explicit by showing the equivalence of topological invariants in critical systems with non-Hermitian point-gap phases, in the presence of sublattice symmetry. Also, the corresponding entanglement spectra show the same topological features. This correspondence may carry over to other features and even be helpful to deepen our understanding of non-Hermitian systems using our knowledge of critical systems and vice versa.
Crystalline topological insulators owe their topological character to the protection that certain boundary states acquire because of certain point-group symmetries. We first show that a Hermitian operator obeying supersymmetric quantum mechanisms (SUSY QM) delivers the entanglement spectrum. We then show that such an entanglement spectrum that is compatible with a certain point-group symmetry obeys a certain local spectral symmetry. The latter result is applied to the stability analysis of four fermionic non-interacting Hamiltonians, the last of which describes graphene with a Kekule distortion. All examples have the remarkable property that their entanglement spectra inherit a local spectral symmetry from either an inversion or reflection symmetry that guarantees the stability of gapless boundary entangling states, even though all examples fail to support protected gapless boundary states at their physical boundaries.
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.