Real spectra in non-Hermitian topological insulators

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Real spectra in non-Hermitian topological insulators

Kohei Kawabata and Masatoshi Sato

Phys. Rev. Research 2, 033391 – Published 10 September 2020

Abstract

Spectra of bulk or edges in topological insulators are often made complex by non-Hermiticity. Here, we show that symmetry protection enables entirely real spectra for both bulk and edges even in non-Hermitian topological insulators. In particular, we demonstrate the entirely real spectra without non-Hermitian skin effects due to a combination of pseudo-Hermiticity and Kramers degeneracy. This protection relies on nonspatial fundamental symmetry and has stability against disorder. As an illustrative example, we investigate a non-Hermitian extension of the Bernevig-Hughes-Zhang model. The helical edge states exhibit oscillatory dynamics due to their nonorthogonality as a unique non-Hermitian feature.

Received 4 April 2020
Accepted 18 August 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.033391

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

PT-symmetric quantum mechanics Topological phases of matter

2-dimensional systems Topological materials

Band structure methods

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & TechnologyAtomic, Molecular & OpticalStatistical Physics & Thermodynamics

Authors & Affiliations

Kohei Kawabata^1,* and Masatoshi Sato²

¹Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
²Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

^*kawabata@cat.phys.s.u-tokyo.ac.jp

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Vol. 2, Iss. 3 — September - November 2020

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Figure 1
Phase diagram of the non-Hermitian Bernevig-Hughes-Zhang model. Topological phase transitions occur at the phase boundaries, at which an energy gap for the real part of the complex spectrum closes. Each gapped phase is characterized by the Chern number $C \in Z$ of $η H (k)$ . A pair of helical edge states appears for $|C| = 1$ , whereas no edge states appear for $C = 0$ .
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Figure 2
Complex spectrum of the non-Hermitian Bernevig-Hughes-Zhang model. The open boundary conditions are imposed in the $y$ direction (30 sites), whereas the periodic boundary conditions are imposed in the $x$ direction, along which the wave number $k_{x}$ is defined. (a, b) Gapped and topologically nontrivial phase ( $t = 1.0$ , $m = - 0.5$ , $γ = 0.8$ ; $C = - 1$ ). A pair of helical edge states appears around $k_{x} = 0$ . (c, d) Gapped and topologically nontrivial phase ( $t = 1.0$ , $m = 0.2$ , $γ = 0.9$ ; $C = + 1$ ). A pair of helical edge states appears around $k_{x} = \pm π$ . (e, f) Gapped and topologically trivial phase ( $t = 1.0$ , $m = - 2.5$ , $γ = 1.0$ ; $C = 0$ ). No edge states appear between the gapped bands. (g, h) Gapless phase ( $t = 1.0$ , $m = - 0.5$ , $γ = 1.5$ ). The spectrum is entirely real in the gapped phases (a–f), but it is complex in the gapless phase (g, h).
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Figure 3
Complex spectrum of the non-Hermitian Bernevig-Hughes-Zhang model with disorder. The open boundary conditions are imposed in both $x$ and $y$ directions ( $30 \times 30$ sites). Even in the presence of disorder, the spectrum is entirely real for both (a) topological phase ( $t = 1.0$ , $m_{x, y} = - 0.5 + 2.0 ε_{x, y}$ , $γ = 0.8$ ) and (b) trivial phase ( $t = 1.0$ , $m_{x, y} = - 2.5 + 2.0 ε_{x, y}$ , $γ = 1.0$ ). Here, $ε_{x, y}$ is a random variable uniformly distributed over $[- 0.5, 0.5]$ .
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Figure 4
Power oscillation at an edge in the non-Hermitian Bernevig-Hughes-Zhang model. An initial state is prepared to be a localized wave function $| ψ (0) 〉 \propto \sum_{x, y} e^{- {(x - 1)}^{2} / 36 - {(y - 1)}^{2}} | x 〉 | y 〉$ , and the evolutions of the amplitude at the edge [i.e., $P_{edge} (t) : = {|〈 y = 1 | e^{- i H_{BHZ} t} | ψ (0) 〉|}^{2}$ ] are shown. The two-dimensional system consists of $30 \times 30$ sites and has periodic boundaries in the $x$ direction and open boundaries in the $y$ direction. The solid red curve shows the dynamics for the non-Hermitian topological phase ( $t = 1.0$ , $m = - 0.5$ , $γ = 0.8$ ), whereas the solid blue curve shows the dynamics for the non-Hermitian trivial phase ( $t = 1.0$ , $m = - 2.5$ , $γ = 1.0$ ); the dotted orange curve shows the dynamics for the Hermitian topological phase ( $t = 1.0$ , $m = - 0.5$ , $γ = 0$ ), whereas the dotted violet curve shows the dynamics for the Hermitian trivial phase ( $t = 1.0$ , $m = - 2.5$ , $γ = 0$ ).
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Figure 5
Complex spectrum of the non-Hermitian Bernevig-Hughes-Zhang model protected by time-reversal symmetry. The open boundary conditions are imposed in the $y$ direction (30 sites), whereas the periodic boundary conditions are imposed in the $x$ direction, along which the wave number $k_{x}$ is defined. (a, b) Gapped and topologically nontrivial phase ( $t = 1.0$ , $m = - 0.5$ , $γ = 0.4$ ). A pair of helical edge states appears around $k_{x} = 0$ . The helical edge states coalesce with each other and form exceptional points, leading to the complex spectrum at the edges. (c, d) Gapped and topologically trivial phase ( $t = 1.0$ , $m = - 2.5$ , $γ = 0.4$ ). No edge states appear between the gapped bands, and the spectrum is entirely real.
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