Boundary Time Crystals: Beyond Mean-Field Theory

TL;DR

This work addresses boundary time crystals (BTC) in finite-size open quantum systems, where conventional mean-field theory (MFT) fails to capture long-time dynamics. It introduces the stroboscopic rotating wave approximation (SRWA), which splits evolution into a long-time decay governed by an effective Lindblad operator $\bar{\mathcal{L}}$ and short-time oscillations described by a reduced quantum dynamical semigroup, providing a beyond-MFT framework. The method reveals that competing dephasing along three directions sustains persistent BTC oscillations and yields analytic expressions for the steady state, oscillation period, and decay rate in the high-frequency driving regime. This SRWA tool offers a practical route to analyze periodically driven open quantum systems and understand time-crystal formation in finite quantum devices.

Abstract

Boundary time crystals are a class of exotic dissipative quantum phases that spontaneously break continuous time-translation symmetry in the thermodynamic limit of open quantum systems. In finite-size systems, the long-time evolution of boundary time crystals exhibits decaying oscillations that cannot be captured by widely used mean-field theory. To address this issue, we develop an effective approach called the stroboscopic rotating wave approximation, which provides a well approximate state for the long-time evolution of boundary time crystals under strong driving. In this approach, the order parameter exhibits both a long-time decaying envelope governed by an effective Lindblad superoperator and short-time oscillations dominated by a reduced quantum dynamical semigroup. Our results reveal that the competition among dephasing processes along three distinct directions induces persistent oscillations, marking the emergence of the boundary time crystal phase. We obtain the analytical expressions for the steady-state density operator, the oscillation period, and the decay rate of the order parameter in the regime where the coherent energy splitting exceeds the dissipation rate. Our work provides a beyond-mean-field theoretical tool for studying the dynamics of periodically driven open quantum systems and understanding the formation of time crystals.

Figures (5)

• Figure 1: Time evolution of the order parameter $\langle S_z \rangle$ in BTC. The initial state for all four panels is chosen as all spins pointing down. (a) and (b) show the evolution at the 100th and 400th driving periods $T_0$, respectively, The blue dashed lines represent the results obtained from the second-order cumulant mean-field approximation, while the black and red solid lines correspond to the numerical simulations of Eq. \ref{['eq:1']} for system sizes $N=40$ and $N=80$, respectively. Panels (c) and (d) show the evolution at the 100th and 400th driving periods, and the purple and green circles are the results obtained from the SRWA approximation.
• Figure 2: Time evolution of the fidelity between the density matrices obtained from the numerical simulations of BTC and SRWA ($N=50$).
• Figure 3: Evolution of the oscillation period of $\langle \hat{S}_z \rangle$ over time. The driving frequency is $\omega_0/\kappa = 40$ and $N = 10$. Brown circles are the results of the oscillation period relative to the driving period, calculated from the peaks obtained in the numerical simulations of Eq. \ref{['eq:1']}, while blue crosses represent the results calculated from the troughs. The black and red solid lines correspond to the theoretical calculations from Eq. \ref{['eq:30']}, where the second term takes the positive and negative signs, respectively.
• Figure 4: Schematic diagram for calculating the oscillation period of the order parameter $\langle \hat{S}{z} \rangle$. The red curve represents the time evolution of $\langle \hat{S}{z} \rangle$. $r_n$ and $r_{n+1}$ denote two consecutive stroboscopic points. $s^{}(r_n)$ and $s^{}(r_{n+1})$ denote two consecutive peak points. $T_0$ and $T$ represent the driving period and the oscillation period, respectively.
• Figure 5: Decay rate of the time evolution of the order parameter $\langle S_{z} \rangle$. The driving frequency is $\omega_0/\kappa = 40$ and $N = 10$. Brown circles and blue crosses indicate $\gamma$ calculated from the peaks and troughs of the numerical simulations of Eq. \ref{['eq:1']}, respectively. Black and red solid lines correspond to the theoretical calculations from Eq. \ref{['eq:34']}, with the second term taken with positive and negative signs, respectively. The theoretical value given by Eq. \ref{['eq:34']} shows a deviation from the numerical results, with an offset of order $(\kappa/\omega_0)^2$.