Localization, $\mathcal{PT}$ symmetry breaking, and topological transitions in non-Hermitian quasicrystals

Localization, $PT$ symmetry breaking, and topological transitions in non-Hermitian quasicrystals

Aruna Prasad Acharya, Aditi Chakrabarty, Deepak Kumar Sahu, and Sanjoy Datta

Phys. Rev. B 105, 014202 – Published 7 January 2022

Abstract

The topological phase transition in a Hermitian system is associated with a change in the topological invariant that characterizes the band structure of the two distinct phases. Recently, the delocalization-localization transition in a quasicrystal described by the non-Hermitian $PT$ -symmetric extension of the Aubry-André-Harper (AAH) Hamiltonian has been related to a topological phase transition. Interestingly, the $PT$ symmetry also breaks down at the same point. In this article, we have shown that the delocalization-localization transition and the breaking of $PT$ symmetry are not connected to a topological phase transition. We have studied the non-Hermitian $PT$ -symmetric AAH Hamiltonian in the presence of Rashba spin-orbit coupling. It has been demonstrated that except in some particular cases, the delocalization-localization and the topological transition points are not identical. In fact, it precedes the topological transition point whenever they do not coincide. However, the breaking of $PT$ symmetry occurs simultaneously with the delocalization-localization transition. Moreover, we have discovered that the Lyapunov exponent, which is often used as a primary signature to characterize the delocalization-localization transition, does not always determine it correctly. The Lyapunov exponent rather identifies the topological transition point in quasicrystals.

Received 2 October 2021
Revised 21 December 2021
Accepted 3 January 2022

DOI:https://doi.org/10.1103/PhysRevB.105.014202

Physics Subject Headings (PhySH)

Anderson localization Metal-insulator transition PT-symmetric quantum mechanics Topological phases of matter

Disordered systems Quasicrystals

PT-symmetry Topological tools

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Aruna Prasad Acharya , Aditi Chakrabarty , and Deepak Kumar Sahu

Department of Physics and Astronomy, National Institute of Technology, Rourkela, Odisha 769008, India

Sanjoy Datta ^*

Department of Physics and Astronomy, National Institute of Technology, Rourkela, Odisha 769008, India and Center for Nanomaterials, National Institute of Technology, Rourkela, Odisha 769008, India

^*[email protected]

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Vol. 105, Iss. 1 — 1 January 2022

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Figure 1
The variation of the DL transition point $h_{c}$ with the strength of the RSO coupling. The numerically calculated values of $h_{c}$ have been shown by diamond shaped markers (in blue), and the analytical result of $h_{t}$ from Eq. (24) has been represented by solid lines (in green). The results are for a lattice with 610 sites and obeying the periodic boundary conditions. In panels (a), (b), (d), and (e), we have set $V = t = 1$ (in arb. units), whereas in panels (c) and (f), $V = 1$ and $t = 0$ . [(a) and (b)] Phase diagram of the DL transition points when one of the RSO hopping amplitude ( $α_{y}$ oder $α_{z}$ ) is set to zero, while the other component varies with a nonzero amplitude. [(d) and (e)] In this case, $α_{y} / t$ is set to 0.40 while the component $α_{z}$ varies and vice versa. [(c) and (f)] The DL transition points determined when $α_{y}$ varies as a function of $α_{z}$ and vice versa. Here, in all the cases, $θ$ is considered to be zero.
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Figure 2
Phase transition in the non-Hermitian quasicrystal described by Eq. (3), where $t$ has been set equal to $V$ ( $t = V = 1$ ), and with $θ = 0$ . The upper panel is for the RSO coupling strength set at $α_{y} = 0$ and $α_{z} = 0.3$ , while in the lower panel $α_{y} = 0.4$ and $α_{z} = 0.5$ . [(a) and (e)] The energy spectrum with an increasing value of the control parameter $h$ . [(b) and (f)] The nature of the largest value of $| Im (E) |$ vs. $h$ across the $PT$ symmetry breaking point. [(c) and (g)] Behavior of the largest (in blue) and smallest (in red) values of the inverse participation ratio with an increasing value of $h$ , before and after the DL transition. [(d) and (h)] The numerically determined values of the winding number $w (h)$ (in blue diamonds) with a variation of $h$ , and the analytical result from Eq. (26) represented by the solid red line. In (b), (c), (d), (f), (g), and (h), the vertical black dotted line depicts the DL transition ( $h_{c}$ ) and the $PT$ symmetry breaking point ( $h_{p}$ ), whereas the green dotted line shows the topological transition point $h_{t}$ determined analytically in Eq. (24).
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Figure 3
Phase transitions described by Eq. (3) with $t = 0, V = 1$ , and $θ = 0$ . The upper panel shows the results when the RSO coupling strength is set at $α_{y} = 1.0$ and $α_{z} = 0.4$ . In the lower panel, $α_{y} = 0.4$ and $α_{z} = 1.0$ . [(a) and (e)] The energy spectrum spanned in the complex plane, with an increasing value of the control parameter $h$ . [(b) and (f)] The nature of the largest value of $| Im (E) |$ vs. $h$ across the $PT$ symmetry breaking point. [(c) and (g)] Behavior of the largest (in blue) and smallest (in red) values of the inverse participation ratio with an increasing value of $h$ before and after the DL transition. [(d) and (h)] The variation of numerically determined values of winding number $w (h)$ with $h$ (in blue) and the analytical result from Eq. (26), shown as solid red line. In (b), (c), (d), (f), (g), and (h), the vertical black dotted line depicts the DL transition ( $h_{c}$ ) and the $PT$ symmetry breaking point ( $h_{p}$ ), whereas the green dotted line shows the topological transition point $h_{t}$ determined analytically in Eq. (24).
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Physical Review B

covering condensed matter and materials physics

Localization, $PT$ symmetry breaking, and topological transitions in non-Hermitian quasicrystals

Aruna Prasad Acharya, Aditi Chakrabarty, Deepak Kumar Sahu, and Sanjoy Datta

Phys. Rev. B 105, 014202 – Published 7 January 2022

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Physical Review B

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Localization, PT symmetry breaking, and topological transitions in non-Hermitian quasicrystals

Aruna Prasad Acharya, Aditi Chakrabarty, Deepak Kumar Sahu, and Sanjoy Datta

Phys. Rev. B 105, 014202 – Published 7 January 2022

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Localization, $PT$ symmetry breaking, and topological transitions in non-Hermitian quasicrystals