Dynamical Quantum Phase Transitions in Boundary Time Crystals

TL;DR

This paper uncovers a dynamical quantum phase transition in a dissipative collective-spin model exhibiting boundary time crystal behavior, characterized by nonanalytic features in the Loschmidt echo during quenches and ramps.

Contribution

It demonstrates the existence of DQPTs in boundary time crystals and analyzes their persistence and finite-size scaling under different driving protocols.

Findings

DQPTs occur in a dissipative boundary time crystal model.

Loschmidt echo shows zeros and nonanalytic cusps during quenches.

Finite-size scaling indicates convergence of critical times in the thermodynamic limit.

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.