Dynamical Quantum Phase Transitions in Boundary Time Crystals

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.

💡 Research Summary

In this work the authors investigate dynamical quantum phase transitions (DQPTs) in an open many‑body system that hosts a boundary time crystal (BTC) phase. The model under study is a collective‑spin system of N spin‑½ particles with Hamiltonian
(H = K\bigl(\omega_{0} S_{x} + \omega_{x} S_{x}^{2} + \omega_{z} S_{z}^{2}\bigr))
and a single Lindblad jump operator (L = \sqrt{\kappa},S_{-}). The dynamics follows the Markovian master equation
(\dot\rho = -i