From time crystals to time quasicrystals: Exploring quasiperiodic phases in transverse field Ising chains

TL;DR

This work demonstrates that a disordered ITF spin chain exhibits a robust discrete time crystal under periodic driving, persisting against interactions and imperfect rotations and strengthening with system size. Under quasiperiodic driving, the system hosts a long-lived time quasicrystal phase manifested as coherent oscillations at incommensurate frequencies that form a prethermal regime before heating to infinite temperature. The prethermal TQC lifetime is largely set by drive parameters and frequency scales, and its stability depends on the disorder structure, with symmetric exchange distributions potentially suppressing the TQC. The results suggest experimental viability in cold atom quantum simulators and offer avenues to explore slow thermalization, prethermal dynamics, and possible topological edge features in quasiperiodically driven, disordered spin systems.

Abstract

Time quasicrystals (TQCs) represent a compelling extension of the concept of time crystals (TCs). While TCs break discrete time-translation symmetry by exhibiting a periodic response at a subharmonic of the driving frequency, TQCs display a more complex temporal order. They respond at multiple incommensurate frequencies, values that are not integer multiples of the fundamental driving frequency, resulting in quasiperiodic dynamics. In this work, we investigate the emergence of a TQC in a disordered quantum Ising chain subjected to a quasiperiodic transverse field. Using exact diagonalization, we find that the transverse magnetization exhibits quasiperiodic oscillations which persist over extended prethermal timescales before eventual decay. This indicates that the TQC exists as a long-lived, prethermal dynamical phase rather than a true equilibrium state. We further assess the robustness of this prethermal TQC against interactions and driving imperfections, confirming its stability under realistic experimental conditions. Finite-size analysis reveals that the prethermal TQC lifetime exhibits minimal dependence on system size. Additionally, we explore the emergence of TQCs in the same chain under symmetric sampling of exchange couplings. Our results demonstrate that the TQC phase is highly sensitive to both the choice of coupling distribution and the values of the driving frequencies. These findings highlight promising experimental prospects for realizing TQCs in cold atomic systems and quantum simulators, providing valuable insights into their stability, dynamical properties, and potential for exploring novel non-equilibrium quantum phases.

Figures (11)

• Figure 1: Time evolution of the magnetizations along $x, y$, and $z$ directions for a chain of 12 sites. For $J=5.5$ and $h=0.3$ (up), the magnetization along the $z$ axis exhibits periodic oscillations with a period of $2T$, while the $x$ and $y$ components remain zero. The inset plot illustrates the fast Fourier transform (FFT) of $m^z(t)$, highlighting a pronounced peak at $\omega=\pi/2$, which signifies the periodic nature of the magnetization. Conversely, for $J=0.3$ and $h=4.5$ (down), the magnetization oscillations are characterized by a lack of periodicity and stability. The peak value in the FFT is multiplied by 0.01, and this adjustment is applied to all other plots as well.
• Figure 2: Time evolution of the magnetization along the $z$-direction for a chain of length $L=12$ subjected to an interaction $\lambda$ with varying strengths. The exchange couplings are randomly selected from the interval $[J/2, 3J/2]$ with $J = 5.5$, while the magnetic fields are drawn from the range $[-h, h]$ with $h = 0.3$. The envelope of the magnetization depends on the strength of the interaction $\lambda$. Increasing $\lambda$, leads to the instability of the oscillations with time period of $2T$, and consequently the destruction of the TC phase. FFT of the magnetization $m^z(t)$ exhibits a subharmonic peak at $\omega=\pi/2$. Increasing $\lambda$ results in a splitting of the Fourier peak, reflecting the beating of magnetization oscillations.
• Figure 3: Time evolution of the magnetization along the $z$-direction for chains of varying lengths subjected to an interaction strength of $\lambda=1.5$. The exchange couplings are randomly selected from the interval $[J/2, 3J/2]$ with $J = 5.5$, while the magnetic fields are drawn from the range $[-h, h]$ with $h = 0.3$. As the system size increases, a significant enhancement in the stability of the magnetization oscillations is observed, characterized by a time period of $2T$. In the inset, the FFT of the magnetization $m^z(t)$ is presented. For smaller systems, such as with length $L=4$, the FFT, illustrated by the black symbols, reveals no sharp peak, indicating the absence of stable oscillations in $m^z(t)$. Conversely, as the system size increases, a pronounced peak emerges at the frequency $\omega=\pi/2$, signifying the establishment of stable oscillations in the magnetization.
• Figure 4: (Top: The absolute value of the magnetization $m^z(t)$ for a chain of length $L=12$ for different values of $\lambda$, with a coupling constant $J=5.5$ and an external field $h=0.3$. The TC remains stable for interaction strengths weaker than a critical value of approximately $0.4 <\lambda_c < 0.5$. Bottom: Finite-size scaling of the critical interaction strength $\lambda_c$ as a function of $1/L$, where $L$ denotes the system size. The data points represent numerical values, and the solid line corresponds to a linear fit. This extrapolation suggests that $\lambda_c$ approaches a finite, nonzero value as $L\to\infty$. Specifically, the thermodynamic limit value of the critical interaction strength is estimated to be $\lambda_c(\infty) \approx 0.67 \pm 0.03$, indicating that the TC phase remains stable for interaction strengths below this threshold.
• Figure 5: Time evolution of the magnetization along the $z$-direction for various values of $\epsilon$ in a chain of length $L=12$ with coupling constant $J=5.5$ and external field $h=0.3$. The discrete TC remains stable against a finite degree of imperfect rotation. The inset shows the FFT of the magnetization. As imperfection increases, the peak at $\omega=\pi/2$ diminishes and ultimately vanishes for larger $\epsilon$, signaling the disappearance of the discrete TC phase.