Coexistence of extended and localized states in the one-dimensional non-Hermitian Anderson model

Cem Yuce and Hamidreza Ramezani

Phys. Rev. B 106, 024202 – Published 18 July 2022

Abstract

In one-dimensional Hermitian tight-binding models, mobility edges separating extended and localized states can appear in the presence of properly engineered quasiperiodical potentials and coupling constants. On the other hand, mobility edges do not exist in a one-dimensional Anderson lattice since localization occurs whenever a diagonal disorder through random numbers is introduced. Here we consider a nonreciprocal non-Hermitian lattice and show that the coexistence of extended and localized states appears with or without diagonal disorder in the topologically nontrivial region. We discuss that the mobility edges appear basically due to the boundary condition sensitivity of the nonreciprocal non-Hermitian lattice.

Received 27 February 2022
Revised 19 April 2022
Accepted 5 July 2022

DOI:https://doi.org/10.1103/PhysRevB.106.024202

Physics Subject Headings (PhySH)

Anderson localization Bound states Edge states Metal-insulator transition Topological insulators

Disordered systems Non-Hermitian systems Random & disordered media Topological materials

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalGeneral Physics

Authors & Affiliations

Cem Yuce ¹ and Hamidreza Ramezani ²

¹Department of Physics, Eskisehir Technical University, Eskisehir 26000, Turkey
²Department of Physics and Astronomy, University of Texas Rio Grande Valley, Edinburg, Texas 78539, USA

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Vol. 106, Iss. 2 — 1 July 2022

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Images

Figure 1
(a) A representation for a lattice with asymmetrical couplings under MBC with $N = 14$ and $p = 7$ . The lattice has one edge and one circular ring. In the ring, the couplings are $J_{R}$ in the clockwise direction and $J_{L}$ in the counterclockwise direction. (b) and (d) The energy spectra in the complex plane, where extended states are placed on the loop and localized states are placed on the real axis inside the loop. (c) and (e) The sudden jump from almost zero to a large IPR value indicates the coexistence of extended and localized eigenstates. The numerical parameters are $J_{L} = 1$ , $J_{R} = 0.2$ , $N = 200$ , and $p = 100$ [(b) and (c)] and $p = 18$ [(d) and (e)]. The PBC (OBC) spectrum is denser for a (smaller) larger $p$ . The total number of the states with almost zero IPR values are equal to $N - p$ .
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Figure 2
IPR values and their corresponding energy eigenvalues in the complex plane at various potential strengths for Anderson (a) and (b) and Aubry-André models (c) and (d), respectively, when $J_{L} = 1, J_{R} = 0.2$ , and $N = 2 p = 200$ . At strong on-site potentials (in red), all eigenvalues lie on the real axis, indicating that all eigenstates are localized. Contrarily, at weak on-site potentials (in black), almost half of the eigenstates are extended while the other half are localized. In the intermediate case (blue), there are still a few extended eigenstates. Note that $V_{0} = 2$ is the phase transition point for the quasiperiodical potential. One can see a few extended states with small complex eigenvalues due to the finite number of the lattice sites (localization length is large and it practically becomes extended). (If the lattice is much longer, then one would see its localization character.)
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Figure 3
The butterfly spectra under MBC in three dimensions at $V_{0} = 1$ (a) and $V_{0} = 2$ (b), where $E_{R}$ and $E_{I}$ are real and imaginary parts of the energy eigenvalues. In the two-dimensional complex energy plane with fixed $β$ , the spectrum determines a loop and a line inside the loop as in Fig. 1. In three dimensions, where $β$ is the vertical axis, the butterfly shape appears. The parameters are given by $J_{L} = 1$ and $J_{R} = 0.2$ and $N = 2 p = 100$ .
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Figure 4
The critical strengths for the Anderson (a) and Aubry-Andre (b) models under MBC as a function of $J_{R}$ at $J_{L} = 1$ and $N = 2 p = 200$ . The shaded area has winding number $ω = 1$ and complex spectra. The top unshaded area has zero winding number and real spectra. In the topologically nontrivial region (shaded area), localized and extended states coexist. On the other hand, in the topologically trivial region, only localized states exist.
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