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Quantum-to-semiclassical Husimi dynamics of non-Hermitian localization transitions

Pallabi Chatterjee, Bhabani Prasad Mandal, Ranjan Modak

Abstract

The localization transition in the Hermitian Aubry-André model is known to have a clear classical origin, with the critical point being exactly predictable from an analysis of classical phase-space trajectories. Motivated by this correspondence, we investigate whether a similar classical origin exists for localization transitions in non-Hermitian quasiperiodic Hamiltonians. Using semiclassical Husimi dynamics together with a detailed phase-space stability analysis, we show that localization transitions persist even in the semiclassical limit of such non-Hermitian models. However, in sharp contrast to the Hermitian Aubry-André case, the transition point inferred from classical phase-space analysis does not coincide with the quantum critical point. Instead, we find that the semiclassical transition depends sensitively on the choice of the irrational parameter defining the quasiperiodic potential, indicating the absence of a universal classical-quantum correspondence for the localization transition in the non-Hermitian setting. Nonetheless, we identify a suitable parameter regime in which the classical dynamics can faithfully mimic the quantum dynamics over a finite but appreciable time window.

Quantum-to-semiclassical Husimi dynamics of non-Hermitian localization transitions

Abstract

The localization transition in the Hermitian Aubry-André model is known to have a clear classical origin, with the critical point being exactly predictable from an analysis of classical phase-space trajectories. Motivated by this correspondence, we investigate whether a similar classical origin exists for localization transitions in non-Hermitian quasiperiodic Hamiltonians. Using semiclassical Husimi dynamics together with a detailed phase-space stability analysis, we show that localization transitions persist even in the semiclassical limit of such non-Hermitian models. However, in sharp contrast to the Hermitian Aubry-André case, the transition point inferred from classical phase-space analysis does not coincide with the quantum critical point. Instead, we find that the semiclassical transition depends sensitively on the choice of the irrational parameter defining the quasiperiodic potential, indicating the absence of a universal classical-quantum correspondence for the localization transition in the non-Hermitian setting. Nonetheless, we identify a suitable parameter regime in which the classical dynamics can faithfully mimic the quantum dynamics over a finite but appreciable time window.
Paper Structure (17 sections, 56 equations, 13 figures)

This paper contains 17 sections, 56 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Model I
  4. Model II
  5. Lattice dynamics
  6. Quantum Husimi Distribution of the Continuum Model
  7. Semiclassical limit of the Husimi distribution
  8. Formalism
  9. Trajectory analysis
  10. Trajectory analysis of Model I
  11. Trajectory analysis of Model II
  12. Time evolution of the semiclassical Husimi distribution
  13. Semiclassical Husimi dynamics of Model I
  14. Semiclassical Husimi dynamics of Model II
  15. Special $\beta$ limit
  16. ...and 2 more sections

Figures (13)

  • Figure 1: Velocity of the excitation propagation $v$ vs. $V$ at time $t=100$. Results are for Model I, Model II, and the Hermitian AA model. Results are with a starting coherent initial state centred at the centre of the lattice. System size is taken as $L=601$.
  • Figure 2: Quantum Husimi distribution at different time instances starting from coherent state $|0\rangle$ and evolving under the Hamiltonian [Eq. (\ref{['continuous_model_I']}) ] continuous version of the Model I. Data is for $V=0.2, 0.6, 0.85,1.5.$ The color code represents the Husimi values.
  • Figure 3: Quantum Husimi distribution at different time instances starting from coherent state $|0\rangle$ and evolving under the Hamiltonian [ Eq. (\ref{['continuous_model_II']})] continuous version of the Model II. Data is for $V=0.2, 0.6, 1.5$. The color code represents the Husimi values.
  • Figure 4: Upper panel: Comparison of $\sigma^2$ vs. $t$ plots for non-Hermitian quantum continuous Model I [Eq. (\ref{['continuous_model_I']}) ] and its corresponding lattice model, for $\beta=\frac{\sqrt{5}-1}{2}$. The symbols represent the quantum continuous model, and dashed lines represent the data for its lattice version. The lower panel represents the same for Model II\ref{['continuous_model_II']} and its lattice version.
  • Figure 5: Trajectory evolution goverened by the semiclassical Hamiltonian of Model I [Eq. (\ref{['semi_H_model1']})] at different $V$ values. Trajectory evolution shows a transition near $V_c= \frac{1}{2}\sqrt{(J_L+J_R)^2 + \frac{4\Delta^2}{(4\pi\beta)^2}} \sim 0.752$. Data are for $J_L=1, J_R=0.5$. $\beta=\frac{\sqrt{5}-1}{2}$.
  • ...and 8 more figures