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arXiv:2009.00546 · SciPost Physics

Entanglement, non-hermiticity, and duality

Limei Chen, Shuai A. Chen, Peng Ye

Published Jul 9, 2021• 27 citations
arXivHTMLPDF

Abstract

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.

DOI
10.21468/scipostphys.11.1.003

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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.
As a hallmark of a pure quantum effect, quantum entanglement has provided unconventional routes to characterize condensed matter systems. Here, from the perspective of quantum entanglement, we disclose exotic quantum physics in non-Hermitian quasicrystals. We study a class of experimentally realizable models for non-Hermitian quasicrystal chains, in which asymmetric hopping and complex potential coexist. We diagnose the global phase diagram by means of entanglement from both the real-space and momentum-space partitions. By measuring the entanglement entropy, we numerically determine the metal-insulator transition point. We combine real-space and momentum-space entanglement spectra to complementarily characterize the delocalization phase and the localization phase. Inspired by the entanglement spectrum, we further analytically prove that a duality exists between the two phase regions. The transition point is self-dual and exact, further validating the numerical result from diagonalizing non-Hermitian matrices. Finally, we identify the mobility edge by means of entanglement.
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.
Non-Hermitian degeneracies are usually discussed through spectral coalescence, whereas entanglement is a property of eigenvectors and need not be fixed by the eigenvalues alone. We formulate a compact bipartite model that separates these two notions. A Hermitian operator with a degenerate eigenspace is transformed by an invertible non-unitary similarity map. The resulting Hamiltonian is non-Hermitian and retains a non-defective degeneracy at every finite value of the non-Hermiticity parameter. For an exactly solvable two-qubit realization, the right eigenstates evolve continuously from product states to maximally entangled states although the spectrum is unchanged. We distinguish the positive right-state reduced density matrix from the generally non-positive biorthogonal reduction, for which entropy may become complex. The same two-qubit solution gives a closed partial-transpose negativity and a Schmidt-gauged non-local magic. Entanglement grows monotonically with the non-Hermiticity parameter, whereas the non-local magic vanishes for both the product and maximally entangled limits and is largest at an intermediate coupling. In larger bipartite spaces, the Page entropy and Haar-averaged purity provide reference values for eigenstate typicality. These diagnostics separate non-defective degeneracy, exceptional-point sensitivity, Haar-typical entanglement, and non-stabilizer correlations without relying on a proliferation of basis-dependent spectral quantities.
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 description of states and dynamics in non-Hermitian systems is fundamentally linked to the choice of an appropriate theoretical framework -- a point of ongoing debate in the field. This work addresses this issue by proposing a consistent formulation that reconciles existing controversies and establishes a unified theoretical understanding. Our approach rests on two foundational premises: (i) the dynamics of both left and right-vectors of a non-Hermitian system must satisfy the Schrödinger equation; (ii) the theoretical framework must preserve gauge invariance, ensuring that physical quantities are independent of unobservable phase choices. Building on these physically motivated assumptions, we refine the biorthogonal framework, leading to a gauge-invariant non-Hermitian quantum theory. Our framework naturally encompasses the open-system effective non-Hermitian evolution as a special case, and can naturally reduce to standard quantum mechanics in the Hermitian limit. As a concrete application, we analyze the dynamical phase transition in a one-dimensional Su-Schrieffer-Heeger (SSH) model within this gauge-invariant non-Hermitian quantum theory. Notably, our formulation naturally generalizes the known condition for such transitions in Hermitian two-band systems, namely, dki⋅dkf=0\mathbf{d}_{k}^i\cdot\mathbf{d}_{k}^f=0dki​⋅dkf​=0, to the non-Hermitian case, where it takes the form Re[dkidki⋅dkfdkf]=0\mathrm{Re}\Bigl[\frac{\mathbf{d}_{k}^i}{d_{k}^i}\cdot\frac{\mathbf{d}_{k}^f}{d_{k}^f}\Bigr]=0Re[dki​dki​​⋅dkf​dkf​​]=0. Furthermore, we identify entirely new dynamical phase transitions that cannot be characterized by the winding number. We hope that this gauge-invariant non-Hermitian quantum theory will find broad applications in the study of non-Hermitian systems.