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arXiv:2112.13411 · Physical review. B./Physical review. B

Quantum entanglement of non-Hermitian quasicrystals

Limei Chen, Yao Zhou, Shuai A. Chen, Peng Ye

Published Mar 29, 2022• 36 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.105.l121115

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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.
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.
The scaling law of entanglement entropy could undergo qualitative changes during the nonunitary evolution of a quantum many-body system. In this work, we uncover such entanglement phase transitions for free fermions in one-dimensional non-Hermitian quasicrystals (NHQCs). We identify two types of entanglement transitions with different scaling laws and critical behaviors due to the interplay between non-Hermitian effects and quasiperiodic potentials. The first type represents a typical volume-law to area-law transition, which happens together with a PT-symmetry breaking and a localization transition. The second type features an abnormal log-law to area-law transition, which is mediated by a critical phase with a volume-law scaling in the steady-state entanglement entropy. These entangling phases and transitions are demonstrated in two representative models of noninteracting NHQCs. Our results thus advance the study of entanglement transitions in non-Hermitian disordered systems and further disclose the rich entanglement patterns in NHQCs.
One-dimensional non-Hermitian quasicrystals with parity and time-reversal (PT) symmetry can simultaneously exhibit localization-delocalization transition, topological phase transition, and PT-symmetry-breaking transition. This motivates this work to investigate how the absence of PT symmetry impacts topological metal-insulator transitions in non-Hermitian quasicrystals. We propose a non-Hermitian quasiperiodic model that generally does not preserve PT symmetry and demonstrate that, in most parameter regions, such a system supports triple phase transitions that encompass localization, topology, and degeneracy-breaking. The system may also exhibit a particular type of localization-delocalization transition analogous to the Hermitian case, namely, without activating topological phase transitions or degeneracy-breaking transitions. Our work extends the topological metal-insulator transitions previously studied in PT-symmetric systems to a more general class of non-Hermitian setting, and further reveals that non-Hermitian systems can host distinct types of localization behavior.
Many-body localization at nonzero temperature is a quantum-phase transition that is of very fundamental interest to both many-body quantum physics and statistical mechanics. Non-Hermitian systems possess rich and unique phenomena that never are found in their Hermitian counterparts, such as unidirectional invisibility and enhanced sensitivity. In the present study, we explore many-body localization in periodically driven non-Hermitian lattices. The lattices are one-dimensional, and the drive is designed in a piecewise two-step form that consists of a uniform coherent tunneling and incommensurate onsite gain and loss. We find that the system can be in localized, delocalized, or mixed phases depending on the driving frequency and the phase shift of the complex potential. Two critical driving frequencies are identified; one corresponds to the largest phase shift of the complex potential that keeps all the quasienergy spectra real and the eigenstates extended, and the other one is established by the vanishing of all real spectra. We also find that a very weak complex potential can lead the system to localization, when the driving frequency is lower than the critical frequency. We show that in the high-frequency limit, the critical phase shift that separates the two regions with respectively a real and complex spectrum tends to a constant, and the constant can be determined by an effective non-Hermitian Hamiltonian.
Understanding the relationship between many-body localization and spectra in non-Hermitian many-body systems is crucial. In a one-dimensional clean, long-range interaction-induced non-Hermitian many-body localization system, we have discovered the coexistence of static and dynamic spectral real-complex phase transitions, along with many-body ergodic-localized phase transitions. The phase diagrams of these two types of transitions show similar non-monotonic boundary trends but do not overlap, highlighting properties distinct from conventional disorder-induced non-Hermitian many-body localization. We also propose a potential experimental realization of this model in cold-atom systems. Our findings provide valuable insights for further understanding the relationship between non-Hermitian many-body localization and non-Hermitian spectra in long-range interacting systems.
The use of quantum entanglement to study condensed matter systems has been flourishing in critical systems and topological phases. Additionally, using real-space entanglement one can characterize localized and delocalized phases of disordered fermion systems. Here we instead propose the momentum-space entanglement spectrum as a means of characterizing disordered models. We show that localization in one dimension can be characterized by the momentum space entanglement between left and right movers and illustrate our methods using explicit models with spatially correlated disorder that exhibit phases which avoid complete Anderson localization. The momentum space entanglement spectrum clearly reveals the location of delocalized states in the energy spectrum, can be used as a signature of the phase transition between a delocalized and localized phase, and only requires a single numerical diagonalization to yield clear results.