Wigner distribution, Wigner entropy and Quantum Refrigerator of a One-Dimensional Off-diagonal Quasicrystal

TL;DR

This work analyzes a one-dimensional off-diagonal quasicrystal with simultaneous diagonal and off-diagonal quasiperiodic modulations to map its localization diagram via fractal dimension $D$ and to classify phases using phase-space tools. It demonstrates that $D$ identifies extended, critical, and localized states, while the Wigner distribution and the derived Wigner entropy $W_S$ distinguish these phases, with $W_S$ peaking in the critical phase. By employing the quasicrystal as the working medium in a quantum Otto cycle, the authors show that the extended phase supports a quantum heat-engine mode, the localized phase favors a heater mode, and, under adiabatic evolution, a refrigerator mode can emerge, highlighting tunable thermodynamic functionality in quasiperiodic systems. These results advance localization theory in quasiperiodic media and broaden their potential applications in quantum thermodynamics and device design.

Abstract

We investigate an off-diagonal quasicrystal featuring simultaneous off-diagonal and diagonal quasiperiodic modulations. By analyzing the fractal dimension, we map out the delocalization-localization phase diagram. We demonstrate that delocalized and localized states can be distinguished via the Wigner distribution, while extended, critical, and localized phases are separated using the Wigner entropy. Furthermore, we explore the quantum thermodynamic properties, revealing that localized states facilitate the emergence of a quantum heater mode, alongside the appearance of a refrigerator mode. These findings enhance our understanding of localization phenomena and expand the thermodynamic applications of quasiperiodic systems.

Figures (6)

• Figure 1: (Color Online) The phase diagram shows how fraction dimension $D$ changes with the ratios $V/t$ and $t_{1}/t$. The color corresponds to the numerical values of $D$. The red dashed line denotes the phase boundary, which serves to separates the extended phase ($D \sim 1$) from the localized ($D \sim 0$) and critical phase ($D \sim 0.5$). The involved parameter is $L=2584$.
• Figure 2: (Color Online)(a) Fractal dimension $D$ as functions of $r$ under $\theta=0$, $\pi/4$, $\pi/3$, and $5\pi/6$. (b) Finite size analysis of $D$ under different $(r,\theta)$, with $1/m$ the inverse Fibonacci index.
• Figure 3: (Color Online) (a) Wigner distribution $W(x,p)$ of an extended state under $r=0.5t$ and $\theta=\frac{\pi}{4}$; (b) Wigner distribution $W(x,p)$ of a critical state under $r=1.5t$ and $\theta=0$; (c) Wigner distribution $W(x,p)$ of a localized state under $r=2t$ and $\theta=\frac{\pi}{6}$; (d) The mean Wigner entropy $\overline{W_{S}}$ as a function of $r$ under various $\theta$. Other involved parameter is $L=400$.
• Figure 4: (Color Online) Schematic illustration of the four-stroke quantum heat cycle process. $T_{h}$ and $T_{c}$ stand for the high-temperature and low-temperature heat sources, respectively. The working medium employed is the generalized AA model with diagonal and off-diagonal quasiperiodic modulations. $r^{i}$ and $r^{f}$ represent the systematic parameters of the corresponding Hamiltonians. The system size is $L=610$.
• Figure 5: (Color Online) Working modes of the four-stroke cycle in the near-adiabatic case. (a) $r^{f}=0.5t$ and $\theta=\pi/6$. (b) $r^{f}=0.5t$ and $\theta=\pi/4$. (c) $r^{f}=0.9t$ and $\theta=\pi/6$. (d) $r^{f}=0.9t$ and $\theta=\pi/4$. (e) $r^{f}=1.1t$ and $\theta=\pi/6$. (f) $r^{f}=1.1t$ and $\theta=\pi/4$. The blue regions represent the Heat engine. The yellow regions denote the Accelerator and the brown regions denote the Heater. Other parameters are $L=610$ and $k_{b}T_{c}=0.1t$.