Quantum anomaly, non-Hermitian skin effects, and entanglement entropy in open systems

Nobuyuki Okuma and Masatoshi Sato

Phys. Rev. B 103, 085428 – Published 19 February 2021

Abstract

We investigate the roles of non-Hermitian topology in spectral properties and entanglement structures of open systems. In terms of spectral theory, we give a unified understanding of two interpretations of non-Hermitian topology: quantum anomaly and non-Hermitian skin effects, in which the bulk spectra extremely depend on the boundary conditions. In this context, the fact that the intrinsic higher-dimensional skin effects under the full open boundary condition need the presence of the topological defects is understood in terms of the anomalous fermion production such as the Rubakov-Callan effect in the presence of the magnetic monopole. By using the unified interpretation, we classify the symmetry-protected and higher-dimensional skin effects. In terms of the entanglement structure, we investigate steady states of fermionic open systems whose Liouvillian (rapidity) spectra host non-Hermitian topology. We analyze dissipation-driven Majorana steady states in zero-dimensional open systems and relate them to the Majorana edge modes of topological superconductors by using the entanglement entropy. We also analyze a steady state of a one-dimensional open Fermi system with a non-Hermitian topological spectrum and relate it to the chiral edge states of the Chern insulator on the basis of the trace index defined from the entanglement spectrum. This correspondence indicates that the entanglement generates circular nonreciprocal currents under the periodic boundary condition and the skin-effect voltage with fermion accumulation under the open boundary condition. Finally, we discuss several related topics such as pseudospectral behaviors of Liouvillian dynamics and skin effects in interacting systems.

Received 24 November 2020
Revised 9 January 2021
Accepted 14 January 2021

DOI:https://doi.org/10.1103/PhysRevB.103.085428

Physics Subject Headings (PhySH)

Anomalies Chern insulators Entanglement entropy Majorana fermions Open quantum systems Topological defects

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Nobuyuki Okuma^* and Masatoshi Sato

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

^*okuma@hosi.phys.s.u-tokyo.ac.jp

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Vol. 103, Iss. 8 — 15 February 2021

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Images

Figure 1
Rapidity (left) and Liouvillian (right) spectra of $n = 1$ open Fermi system. In the third quantization, $2^{2 n}$ Liouvillian eigenstates are created by $2 n$ complex fermions ${\bar{β}}_{i}^{†}$ .
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Figure 2
Non-Hermitian topology and its physical interpretations. (a) Correspondence between the non-Hermitian topology and the number of anomalous gapless modes ( $Re E = 0$ ). The PBC spectral curve with the winding number counts the topological charge of anomalous gapless modes. (b) Topological aspect of the non-Hermitian skin effect. The PBC spectral curve (blue) with nonzero winding indicates that the OBC spectrum (red) is drastically different from the PBC one. (c) Spectra and wave functions at energy $E = 1.948$ of one-dimensional $Z_{2}$ skin effect protected by the $A I I^{†}$ time-reversal symmetry, adopted from [44]. The model is given by Eq. (25) ( $t = 1$ , $g = 0.3$ , $Δ = 0.2$ , $δ = (1, 1, 1) \times 10^{- 2}$ , $δ h = 10^{- 3}$ , $L = 100$ ). For the $Z_{2}$ index $ν = 1$ , the OBC spectrum (red) is drastically different from the PBC one (blue), while for $ν = 0$ , the skin effect does not occur. Also, the $Z_{2}$ skin effect for $ν = 1$ is suppressed even by a very small time-reversal breaking term ( $δ h = 10^{- 3}$ ). These results indicate the $Z_{2}$ nature and symmetry-protected nature of the Hamiltonian (25). Each $Z_{2}$ skin mode has the Kramers counterpart localized at the other edge. (d) Spectra and wave functions of two-dimensional $Z_{2}$ skin effect protected by the $A I I^{†}$ time-reversal symmetry, adopted from the Supplemental Material of [44]. The model is given by Eq. (28) ( $Γ = 0.5$ and $L = 20$ ). Only $O (L)$ modes in $O (L^{2})$ modes are localized at the boundary and the topological defect ( $π$ flux).
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Figure 3
Spectrum and skin wave functions at $E = - 0.00041 i$ of three-dimensional class-A skin effect under the monopole gauge configuration on lattice with $n = 14$ . The model is given by Eq. (29). The model parameters are $t = \pm 1$ and $Γ = 0.5$ . The skin wave function is localized at the surface/monopole for $t = \pm 1$ , while the spectrum does not depend on the sgn $t$ .
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Figure 4
(a) Rapidity and (b) Liouvillian spectra of steady-state Majorana fermion system. In the rapidity spectrum, the topological phase transition occurs at the exceptional point. In the Liouvillian spectrum, the strong dissipation induces the steady-state quasidegeneracy. (c) Dissipation dependence of entanglement entropy in a zero-dimensional Majorana open Fermi system.
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Figure 5
(a) Schematic picture of one-dimensional open Fermi system with class-A non-Hermitian topology. One environment is a thermal bath with distribution function $f (ε)$ , and the other environment is the momentum-dependent dissipator whose chemical potential is much lower than that of the system. The rapidity spectrum is described by the Hatano-Nelson model. (b) Entanglement spectrum (ES) of $n = 100$ one-dimensional open Fermi system (58). (c) Dispersion, ES, and trace of ES of Chern insulator with $C = + 1$ . The discontinuity of the ES detects the nontrivial Chern number. (d) Schematic picture of the spin-momentum-locked band and spin-dependent dissipation. Even the coupling constant $g_{↑}$ is $k$ independent, dissipation is effectively $k$ dependent owing to the spin-momentum locking. (e) Schematic picture of the two-dimensional class- $A I I^{†}$ skin effect in surface states of a topological insulator.
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