Dynamical Quantum Phase Transitions in Boundary Time Crystals

TL;DR

The paper shows that dynamical quantum phase transitions (DQPTs) occur in a dissipative open quantum system that hosts a boundary time crystal (BTC). By using a fidelity-based Loschmidt echo for mixed states, the authors detect nonanalytic cusps in the dynamical rate function when a single Hamiltonian parameter is quenched or slowly ramped across the BTC boundary; BTC-targeted quenches produce repeated DQPT zeros due to the time-periodic steady state, while quenches to the non-BTC phase yield a single zero followed by vanishing overlap. Finite-size scaling reveals that the first critical time converges to a constant in the thermodynamic limit with distinct exponents for quench and ramp protocols, illustrating that DQPTs extend into genuinely dissipative time-crystalline phases. Overall, the work demonstrates that DQPT behavior persists beyond closed or Floquet systems and provides a quantitative framework for analyzing DQPTs in open, time-crystalline settings.

Abstract

We demonstrate the existence of a dynamical quantum phase transition (DQPT) in a dissipative collective-spin model that exhibits the boundary time crystal (BTC) phase. We initialize the system in the ground state of the Hamiltonian in either the BTC or the non-BTC phase, and drive it across the BTC transition. The driving is done by an abrupt quench or by a finite-time linear ramp of a Hamiltonian control parameter under Markovian Lindblad dynamics. We diagnose DQPTs through zeros of the fidelity-based Loschmidt echo between the initial state and the evolving mixed state, which induce nonanalytic cusp-like features in the associated rate function. For quenches into the BTC phase, the Loschmidt echo exhibits repeated zeros due to the emergent time-periodic steady state, whereas for quenches into the non-BTC phase, the overlap vanishes and remains zero once the dynamics relaxes to a stationary state. We further show that the DQPT persists under the ramp protocol followed by unitary evolution with the final Hamiltonian. Finally, we analyze the finite-size scaling of the first critical time and find convergence to a constant in the thermodynamic limit, with distinct power-law approaches for the quench and the ramp protocols.

Figures (6)

• Figure 1: Illustration of BTC and non-BTC phases in the model of Eq. \ref{['eq:master-eqn']}. Shown is the average magnetization $\langle S^z \rangle/N$ for $N=100$ as a function of time in (a) the BTC phase and (b) the non-BTC phase. In the BTC regime, the dynamics of the order parameter $\langle S^z \rangle/N$ relaxes to a time-periodic steady state with persistent oscillations, whereas in the non-BTC regime, the average magnetization approaches a time-independent stationary value. The parameters corresponding to the BTC phase are chosen as $\omega_0 = -1$, $\omega_x = 0$, $\omega_z = -0.25$, and $\kappa = 0.1$. For the non-BTC phase, we set $\omega_0 = 0$, while keeping all other parameters unchanged. The quantities plotted along the horizontal and vertical axes in each panel are dimensionless.
• Figure 2: Dynamical quantum phase transition under a quench from the non-BTC to the BTC phase.(a) The fidelity-based Loschmidt echo $\mathcal{L}_F(t)$ is shown as a function of time following an abrupt quench of $\omega_0$ from $\omega_{0,i} = 0$ (non-BTC) to $\omega_{0,f} = -1$ (BTC) for system sizes $N=50$ (red), $N=60$ (blue), and $N=70$ (purple). The system is initially prepared in the ground state of the pre-quench Hamiltonian and subsequently evolves under the Lindblad dynamics with the post-quench Hamiltonian. The first vanishing of $\mathcal{L}_F(t)$ defines the first critical time $t_{c}^{(1)}$, signalling the onset of a DQPT. The inset magnifies the region near the $t$-axis to highlight the numerical zeros. The numbers smaller than machine precision ($10^{-15}$) are concurrent with zero. We obtain $t_{c}^{(1)}$ as the midpoint of the first times at which $\mathcal{L}_F$ drops below and rises above $10^{-15}$, denoted respectively as $t_{\downarrow}^{(1)}$ and $t_{\uparrow}^{(1)}$. The values of the first critical times, $t_{c}^{(1)}$, along with the respective values of $t_{\downarrow}^{(1)}$ and $t_{\uparrow}^{(1)}$ for each of the three $N$ are provided in Tab. \ref{['tab:FLE-quench']}. The data is generated with the time-step of $\Delta t = 0.001$. (b) Plot of $\mathcal{L}_F(t)$ for $N=250$ with same parameters as in (a) demonstrating that $\mathcal{L}_F(t)$ periodically becomes zero. (c) The rate function $f_F(t) = -(1/N) \ln \mathcal{L}_F(t)$ corresponding to (a) is shown with the curves of the same colors for the respective $N$ values as in (a). At the first critical time $t_{c}^{(1)}$, where $\mathcal{L}_F$ vanishes, $f_F$ exhibits a cusp-like singularity, signalling a DQPT. Solid curves show numerically accessible values of $f_F$, while dashed curves are extrapolations inside the region where $\mathcal{L}_F$ falls below machine precision.
• Figure 3: Finite-size scaling of the first critical time under a quench from the non-BTC to the BTC phase. The first critical time $t_{c}^{(1)}$ is plotted as a function of system size $N$. The red dots are the numerical data, while the blue curve is a power-law fit of the form $t_{c}^{(1)}(N) = a N^{-b} + c$ in the range $N \in [1045,2900]$, indicating convergence to a constant value in the thermodynamic limit. The best-fit values of fitting parameters are $a = -1.194 \pm 0.053$, $b=0.354 \pm 0.009$, and $c=3.225 \pm 0.002$. Since the data is generated with a time resolution of $\Delta t = 0.001$, values of the parameters up to the third decimal place are meaningful. Therefore, we have truncated the errors to the third decimal place. The RMS fitting error is less than 0.001. The values of both pre- and post-quench parameters are the same as in Fig. \ref{['fig:FLE-quench']}.
• Figure 4: DQPT under a quench from the BTC to the non-BTC phase. The fidelity-based Loschmidt echo $\mathcal{L}_F(t)$ is shown for $N=100$ following a quench of $\omega_0$ from $\omega_{0,i} = -1$ (BTC) to $\omega_{0,f} = 0$ (non-BTC). After the first vanishing of $\mathcal{L}_F$ at $t_{c}^{(1)} = 20.778$, the overlap remains zero as the dynamics relaxes to a stationary steady state. The inset highlights the behavior near the $t$-axis. Unlike quenches into the BTC phase, no revivals of the overlap occur at later times.
• Figure 5: DQPT under a finite-time ramp from the non-BTC to the BTC phase.(a) The fidelity-based Loschmidt echo $\mathcal{L}_F$ is shown during the post-ramp unitary evolution for $N=70$ (red), $N=80$ (blue), and $N=90$ (purple). The parameter $\omega_0$ is ramped linearly from $\omega_{0,i} = 0$ to $\omega_{0,f} = -1$ over a duration $\tau = 5$ according to Eq. \ref{['eq:ramp-omega0']} under Lindblad dynamics, after which dissipation is switched off and the system evolves unitarily. $\mathcal{L}_F$ is computed with respect to the mixed state at the end of the ramp. The first zero of $\mathcal{L}_F$ defines the critical time $t_{c}^{(1)}$. The inset magnifies the region near the $t$-axis. Numerical zeros are identified using a threshold of $10^{-13}$. The values of $t_{c}^{(1)}$ are listed in Tab. \ref{['tab:FLE-ramp']}. (b) The rate function $f_F$ corresponding to (a) is shown with the curves of the same colors for the respective $N$ values as in (a). At the first critical time $t_{c}^{(1)}$, where $\mathcal{L}_F$ vanishes, $f_F$ exhibits a cusp-like singularity, confirming the presence of a DQPT under the ramp protocol. Solid curves are generated for numerically accessible values, while dashed curves show extrapolations inside the numerically inaccessible region.