Temporal Entanglement Transitions in the Periodically Driven Ising Chain

Abstract

Periodically driven quantum systems can host non-equilibrium phenomena without static analogs, including in their entanglement dynamics. Here, we discover $temporal$ $entanglement$ $transitions$ in a Floquet spin chain, which correspond to a quantum phase transition in the spectrum of the entanglement Hamiltonian and are signaled by dynamical spontaneous symmetry breaking. We show that these transitions are entanglement-driven, i.e., they require initially entangled states and remain invisible to conventional local observables. Intriguingly, we find these transitions across a broad range of driving frequencies (from adiabatic to high-frequency regime) and independently of drive details, where they manifest as periodic, sharp entanglement spectrum reorganizations marked by the Schmidt-gap closure, a vanishing entanglement echo, and symmetry-quantum-number flips. At high frequencies, the entanglement Hamiltonian acquires an intrinsic timescale decoupled from the drive period, rendering the transitions genuine steady-state features. Finite-size scaling reveals universal critical behavior with correlation-length exponent $ν=1$, matching equilibrium Ising universality despite its emergence from purely dynamical mechanisms decoupled from static criticality. Our work establishes temporal entanglement transitions as novel features in Floquet quantum matter.

Figures (3)

• Figure 1: Temporal entanglement transitions in the driven TFIM. (a) Schmidt gap $\Delta\lambda = \lambda_0 - \lambda_1$ closing periodically at critical times. (b) Entanglement echo $|E(t)|^2$ vanishing at odd critical times $t_c^{(k=\text{odd})}$, as the dominant Schmidt vector $|\lambda_0\rangle$ orthogonalizes into a distinct symmetry sector. (c) Subsystem parity expectations showing dynamical spontaneous $\mathbb{Z}_2$ symmetry breaking in the entanglement ground state (green) and complementary behavior in the first-excited state (orange). The alternating parity pattern provides consistent physical explanation of the echo's behavior. Red vertical lines mark critical times of symmetry breaking, and overlaid across all panels to provide a synchronized, consistent picture. Parameters: $L=24$, $L_A=9$, $J=1.0$, $h_0=2.0$, $\omega=5.0$, $dt=0.01$.
• Figure 2: Finite-size scaling analysis of temporal entanglement transitions. (a) Critical time scaling $t^*/L_A \propto L_A^{-1/\nu}$ with $\nu = 1.00$. (b) Critical entropy density scaling $\epsilon_0/L_A \propto L_A^{-a}$ with $a = 1.00$. (c) Universal data collapse using scaling ansatz in Eq. \ref{['eq:scaling ansatz']}. (d) Raw data before scaling collapse. The exponent $\nu = 1$ establishes the same universality class for our driven non-equilibrium entanglement spectrum as equilibrium 2D classical Ising/1D TFIM. Parameters: $L = 24$, $J = 1.0$, $h_0 = 2.0$, $\omega = 5.0$, $dt=0.01$.
• Figure 3: Frequency dependence of the first critical time $t^*$ across subsystem sizes $L_A = 4-12$. At low frequencies, $t^* \propto \omega^{-1}$ (adiabatic regime), while at high frequencies ($\omega \gtrsim 10$), $t^*$ saturates to frequency-independent values determined solely by subsystem size, indicating a crossover to Floquet steady-state behavior where the EH develops an intrinsic timescale. The universal saturation indicates that temporal entanglement transitions become intrinsic properties of the effective time-independent dynamics rather than driven phenomena, validating the Floquet-Magnus description and establishing characteristic timescales independent of the external driving protocol. Parameters: $L=24$, $J=1.0$, $h_0=2.0$. Time steps are chosen as $dt=0.01$ for $\omega=[0.1, 10.0]$, $dt=0.002$ for $\omega=\{30, 50\}$ and $dt=0.001$ for $\omega=\{70,100\}$. A total of 153 data points in this plot show excellent collapse (see Fig. 9 of SM Supplemental).