Modified generalized Brillouin zone theory with on-site disorder

Hongfang Liu, Ming Lu, Zhi-Qiang Zhang, and Hua Jiang

Phys. Rev. B 107, 144204 – Published 24 April 2023

Abstract

We study the characterization of the non-Hermitian skin effect (NHSE) in non-Hermitian systems with on-site disorders. We extend the applications of generalized Brillouin zone (GBZ) theory to these systems. By proposing a modified GBZ theory, we give a faithful description of the NHSE. For applications, we obtain a unified $β$ for system with long-range hopping and explain the GBZ irrelevance of the magnetic suppression of the NHSE in the previous study.

Received 2 January 2023
Revised 24 March 2023
Accepted 11 April 2023

DOI:https://doi.org/10.1103/PhysRevB.107.144204

Physics Subject Headings (PhySH)

Topological insulators

Disordered systems Topological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Hongfang Liu^1,2,3, Ming Lu^4,*, Zhi-Qiang Zhang^1,2,†, and Hua Jiang^1,2

¹School of Physical Science and Technology, Soochow University, Suzhou 215006, China
²Institute for Advanced Study, Soochow University, Suzhou 215006, China
³Jiangsu Key Laboratory of Thin Films, Soochow University, Suzhou 215006, China
⁴Beijing Academy of Quantum Information Sciences, Beijing 100193, China

^*luming@baqis.ac.cn
^†zhangzhiqiangphy@163.com

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Vol. 107, Iss. 14 — 1 April 2023

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Images

Figure 1
(a) The analytical (red line) and numerical (gray dots) results of $| F_{1}^{1 / N} |$ vs the eigenvalue $E$ for $W = 5$ . (b) $ln [F (E, β)]$ vs $β$ for different eigenvalues with $E \in [- W / 2, W / 2]$ under disorder strengths: $W = 2$ (green), $W = 5$ (blue), and $W = 8$ (purple). Pentagrams mark the right boundary of the plateaus for $E = \frac{W}{2}$ . The red dashed line is the plot of $ln [| F_{2} |]$ for clean samples. Inset is the numerical results for $F (E, β) = | det [E - H_{PBC} (β)] - det [E - H_{OBC}] |$ with $W = 8$ . The real part $Re [E]$ vs the imaginary part $Im [E]$ of the eigenvalues $E$ for (c) $W = 2$ , (d) $W = 5$ , and (e) $W = 8$ . $β$ for $H_{GBZ}$ is marked in (b). Other parameters are set as $γ = 0.99$ and $N = 60$ .
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Figure 2
(a) Schematic diagram of the model with next-nearest-neighbor hopping (green lines). (b) The schematic plots of $Re [β_{min}]$ vs $Im [β_{min}]$ for clean (red solid line) and disordered samples (area in gray). The disorder effect leads to the broadening of $β_{min}$ marked by the error bar. Thus, a unified $| β |$ is available as marked with the blue solid line. (c) $ln [F (E, β)] / min {ln [F (E, β)]}$ vs $β$ for disorder strength $W = 7$ . The red dashed line presents the overlap of $β_{min}$ for different eigenvalues. (d) $Re [E]$ vs $Im [E]$ under periodic boundary conditions (PBC; black dots), open boundary conditions (OBC; blue dots), and generalized Brillouin zone (GBZ; PBC with $β = 0.77$ [red dashed line in (c)]; red circles). Inset is the plot for clean samples with $β \approx 0.854$ , which gives rise to the bulk-boundary correspondence (BBC) for $E \approx - 0.2 \pm 1.6 i$ . We fix $t = 1$ , $γ = 1.4$ , and $N = 60$ .
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Figure 3
(a) The plot of eigenstates $ρ_{x; n} = \sum_{y} {| ψ_{n} (x, y) |}^{2}$ vs site $x$ for $n = 3$ and 101 with $ϕ = 0$ . Inset plots the real eigenvalues ( $Re [E]$ ) for $ϕ = 0$ (blue dots) and $ϕ = 2 π / (N - 1)$ (red dots), which determines the choices of $n$ . (b) and (c) $F (E_{n}, β)$ vs $β$ for $n = 3$ and 101, respectively. (d)–(f) are the same as (a)–(c), except $ϕ = 2 π / (N - 1)$ . We set $t^{+} = 1.2$ , $t^{-} = 0.8$ , $t_{y} = 1$ , and $N_{x} = N_{y} = N = 20$ .
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Figure 4
(a) and (b) Non-Hermitian Su-Schrieffer-Heeger (SSH) model and (c) and (d) non-Hermitian Chern insulator. (a) The evolution of $ln [F (E, β)]$ with the increase of $β$ for the non-Hermitian SSH model with $W = 0$ (red) and $W = 3$ (blue). The green lines are the cases of $n = 4$ (top) and $n = 30$ (bottom) under disorder, and dashed lines are $β = 1$ (black) and $β = \sqrt{(t_{1} - γ) / (t_{1} + γ)}$ (red). (b1) and (b2) Plots of eigenstates $ρ_{n} = {| ψ_{n} |}^{2}$ for $n = 4$ and 30, respectively. (c) and (d) are the same as (a) and (b), except $W = 10$ (blue), $n = 120$ (bottom), and $ρ_{x; n} = \sum_{y} {| ψ_{n} (x, y) |}^{2}$ .
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