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arXiv:2211.13721 · Physical review. B./Physical review. B

From Hermitian critical to non-Hermitian point-gapped phases

Carlos Ortega-Taberner, Maria Hermanns

Published Jun 6, 2023• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.107.235112

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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.
An attempt is made to study the quantum criticality in non-Hermitian system with topological characterization. We use the zero mode solutions to characterize the topological phases and, criticality and also to construct the phase diagram. The Hermitian counterpart of the model Hamiltonian possess quite a few interesting features such as Majorana zero modes (MZMs) at criticality, unique topological phase transition on the critical line and hence these unique features are of an interest to study in the non-Hermitian case also. We observe a unique behavior of critical lines in presence of non-Hermiticity. We study the topological phase transitions in the non-Hermitian case using parametric curves which also reveal the gap closing point through exceptional points. We study bulk and edge properties of the system where at the edge, the stability dependence behavior of MZMs at criticality is studied and at the bulk we study the effect of non-Hermiticity on the topological phases by investigating the behavior of the critical lines. The study of non-Hermiticity on the critical lines revels the rate of receding of the topological phases with respect to the increase in the value of non-Hermiticity. This work gives a new perspective on topological quantum criticality in non-Hermitian quantum system.
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.
This paper explores the novel concept of topological eigenvalues within the realm of non-Hermitian quantum mechanics, focusing on their role in unveiling spectral singularities. Unlike conventional Hermitian systems, non-Hermitian Hamiltonians, which describe open systems with gain and loss, exhibit complex eigenvalues and unique spectral features such as exceptional points (EPs) where eigenvalues and their corresponding eigenvectors coalesce. We introduce the idea that certain eigenvalues in non-Hermitian systems possess an inherent topological character, dictated by the geometry of the parameter space and the winding properties of the characteristic polynomial. These topological eigenvalues are shown to be intimately linked to spectral singularities, which are non-Hermitian generalizations of bound states in the continuum or points of perfect reflection/transmission, characterized by infinite scattering amplitudes. We develop a theoretical framework to identify and characterize these topological eigenvalues, demonstrating how they emerge from the interplay of non-Hermiticity and underlying symmetries. Our findings reveal that the robust nature of these topological eigenvalues provides a new lens through which to understand the exotic phenomena observed in non-Hermitian systems, including the non-Hermitian skin effect and the breakdown of traditional bulk-boundary correspondence. This work opens new avenues for designing robust non-Hermitian devices and understanding fundamental aspects of open quantum systems.
Quantum entanglement can be an effective diagnostic tool for probing topological phases protected by global symmetries. Recently, the notion of nontrivial topology in critical systems has been proposed and is attracting growing attention. In this work, as a concrete example, we explore the quantum entanglement properties of fermionic symmetry-enriched quantum critical points by constructing exactly solvable models based on stacked multiple Kitaev chains. We first analytically establish the global phase diagram using entanglement entropy and reveal three topologically distinct gapped phases with different winding numbers, along with three topologically distinct transition lines separating them. Importantly, we unambiguously demonstrate that two transition lines exhibit fundamentally different topological properties despite sharing the same central charge. Specifically, they display nontrivial topological degeneracy in the entanglement spectrum under periodic boundary conditions, thereby generalizing the Li-Haldane bulk-boundary correspondence to a broader class of fermionic symmetry-enriched criticality. Additionally, we identify a novel Lifshitz multicritical point at the intersection of the three transition lines, which also exhibits nontrivial topological degeneracy. This work provides a valuable reference for investigating gapless topological phases of matter from the perspective of quantum entanglement.
Topological Euler insulators emerge in multiband systems with real Bloch Hamiltonians and wavefunctions. Their fragile topologies are characterized by the Euler class of degenerate bands and protected by the PTPTPT or C2TC_2TC2​T symmetry in two dimensions, which go beyond the tenfold KKK-theory classification of topological matter. In this work, we extend the conception of topological Euler insulators to non-Hermitian systems and propose a theoretical framework to unlock their nontrivial Euler topology. Focusing on two-dimensional, three-band non-Hermitian lattice models with symmetric Hamiltonians, we formulate a comprehensive description of their topological Euler bands, entanglement spectrum and bulk-boundary correspondence. Three typical models of non-Hermitian Euler insulators are constructed and investigated explicitly to illustrate our theory. Unique topological phase transitions and anomalous edge-band overlaps with non-Hermitian origins are further identified. Our study establishes the presence of topological Euler bands in non-Hermitian systems and unveils their intriguing physical characteristics, thereby broadening the existing territory of topological matter in non-Hermitian open systems.