Exponential size scaling of the Liouvillian gap in boundary-dissipated systems with Anderson localization

Bozhen Zhou, Xueliang Wang, and Shu Chen
Phys. Rev. B 106, 064203 – Published 9 August 2022

Abstract

We carry out a systematical study of the size scaling of the Liouvillian gap in boundary-dissipated one-dimensional quasiperiodic and disorder systems. By treating the boundary-dissipation operators as a perturbation, we derive an analytical expression of the Liouvillian gap, which indicates clearly the Liouvillian gap being proportional to the minimum of boundary densities of eigenstates of the underlying Hamiltonian, and thus give a theoretical explanation why the Liouvillian gap has different size scaling relation in the extended and localized phase. While the Liouvillian gap displays a power-law size scaling ΔgL3 in the extended phase, our analytical result unveils that the Liouvillian gap fulfills an exponential scaling relation ΔgeκL in the localized phase, where κ takes the largest Lyapunov exponent of localized eigenstates of the underlying Hamiltonian. By scrutinizing the extended Aubry-André-Harper model, we numerically confirm that the Liouvillian gap fulfills the exponential scaling relation and the fitting exponent κ coincides pretty well with the analytical result of the Lyapunov exponent. The exponential scaling relation is further verified numerically in other one-dimensional quasiperiodic and random disorder models. We also study the relaxation dynamics and show the inverse of the Liouvillian gap giving a reasonable timescale of asymptotic convergence to the steady state.

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  • Received 11 January 2022
  • Revised 28 July 2022
  • Accepted 29 July 2022

DOI:https://doi.org/10.1103/PhysRevB.106.064203

©2022 American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Bozhen Zhou1,2,*, Xueliang Wang1,2,*, and Shu Chen1,2,3,†

  • 1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
  • 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
  • 3Yangtze River Delta Physics Research Center, Liyang, Jiangsu 213300, China

  • *These authors contributed equally to this work.
  • Corresponding author: schen@iphy.ac.cn

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Issue

Vol. 106, Iss. 6 — 1 August 2022

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