Dynamical topological invariant for the non-Hermitian Rice-Mele model

R. Wang, X. Z. Zhang, and Z. Song

Phys. Rev. A 98, 042120 – Published 18 October 2018

Abstract

We study a non-Hermitian Rice-Mele model without breaking time-reversal symmetry, with the non-Hermiticity arising from imbalanced hopping rates. The Berry connection, Berry curvature, and Chern number are introduced in the context of biorthonormal inner product. It is shown that for a bulk system, although the Berry connection can be complex numbers, the Chern number is still quantized, as a topological invariant. For an opened chain system, the midgap edge modes are obtained exactly, obeying the bulk-edge correspondence. Furthermore, we also introduce a local current in the context of biorthonormal inner product to measure the pumping charge generated by a cyclic adiabatic evolution. Analytical analysis and numerical simulation of the time evolution of the midgap states show that the pumping charge can be a dynamical topological invariant in correspondence with the Chern number. It indicates that the geometric concepts for Hermitian topological insulator can be extended to the non-Hermitian regime.

Received 25 April 2018

DOI:https://doi.org/10.1103/PhysRevA.98.042120

Physics Subject Headings (PhySH)

Edge states Geometric & topological phases

1-dimensional systems

Bloch wave theory Numerical approximation & analysis PT-symmetry Tight-binding model

General Physics

Authors & Affiliations

R. Wang¹, X. Z. Zhang², and Z. Song^1,*

¹School of Physics, Nankai University, Tianjin 300071, China
²College of Physics and Materials Science, Tianjin Normal University, Tianjin 300387, China

^*songtc@nankai.edu.cn

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 98, Iss. 4 — October 2018

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) Schematics of $V δ$ plane. Energy spectrum for the system with parameters at the sides of triangle PQROP for (b) periodic boundary condition and (c) open boundary condition. The spectrum is independent of the imbalance factor $λ$ . It indicates that two bands touch at point (0,0). There are two midgap states for (a) with $δ > 0$ , which are shown to be edge states with identical wave functions for the standard SSH chain with $λ = 1$ and $V = 0$ . Here the energy is in units of $J$ , where $J$ is the scale of the Hamiltonian and we take $J = 1$ .
Reuse & Permissions
Figure 2
(a) Plot of the spectrum of $H_{κ}$ in Eq. (72) with $N = 50, δ = 0.5, κ = 0.05,$ and $λ = 1.5$ . Comparing to that in Fig. 1, the level crossing becomes avoided level crossing around $V = 0$ , at which two types of edge modes are hybridized. Panels (b) and (d) are schematics for two edge modes. (c) The superpositions of the two become midgap states when $V$ turns to zero. When varying $V$ from $- 1$ to 1 adiabatically, (b) will evolve to (c), then (d), resulting an adiabatic particle transport from the leftmost of the chain to the rightmost. Here the energy is in units of $J$ , where $J$ is the scale of the Hamiltonian and we take $J = 1$ .
Reuse & Permissions
Figure 3
(a) Schematics of a simple loop represented in Eq. (47). For an arbitrary loop, it can be decomposed into many rectangular loops. The Chern number of an arbitrary loop is the sum of all the Chern numbers of the rectangles. (b) Six types of segments which can be used to construct any kinds of loop. It turns out that only the vertical one passing through the red dot has nonzero charge transport across the two ends of the chain [see text and Figs. 2–2].
Reuse & Permissions
Figure 4
Numerical simulations of particle transport across two ends of the chain current obtained by exact diagonalization, driven by the time-dependent parameter $V = 1 - 2 ω t, t \in [0, T = ω^{- 1}]$ . Plots of the current (a1)–(d1) and charge accumulation as function of time (a2)–(d2) for several typical speed $ω$ , indicated in the figures. The total charge transfer during the interval $[0, ω^{- 1}]$ is also indicated in each figure. We note that the result accords with our prediction well when the speed is slow enough in (a1) and (b1). As $ω$ increases, the deviation becomes larger and larger. In the case of (a4) and (b4), the accumulated charge becomes positive, which is qualitatively different from the adiabatic process. The parameters of the system are $N = 10, δ = 0.5, κ = 0.05$ , and $λ = 1.5$ . Here the $j_{2 N}$ is in units of $J$ , and $t$ is in units of $J^{- 1}$ where $J$ is the scale of the Hamiltonian and we take $J = 1$ .
Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information