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arXiv:2602.04792

Dynamical Quantum Phase Transitions in Boundary Time Crystals

Sukrut Mondkar, Priya Ghosh, Ujjwal Sen

quant-phcond-mat.othercond-mat.stat-mechSubmitted Feb 4, 2026• Announced Feb 5, 2026• 0 citations
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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.

Comments
10 pages, 6 figures, 3 tables

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We explore the possibility of dynamical quantum phase transitions (DQPTs) occurring during the temporal evolution of a quenched transverse field Ising chain coupled to a particle loss type of bath (local in Jordan-Wigner fermion space) using two versions of the Loschmidt overlap (LO), namely, the fidelity induced LO and the interferometric phase induced LO. The bath, on the one hand, dictates the dissipative evolution following a sudden quench and on the other, plays a role in dissipative mixed state preparation in the later part of the study. During a dissipative evolution following a sudden quench, no trace of DQPTs are revealed in both the fidelity and the interferometric phase approaches; however, remarkably the interferometric phase approach reveals the possibility of inter-steady state DQPTs in passage from one steady state to the other when the system is subjected to a quench after having reached the first steady state. We further probe the occurrences of DQPTs when the system evolves unitarily after being prepared in a mixed state of engineered purity by ramping the transverse field in a linear fashion in the presence of the bath. In this case though the fidelity approach fails to indicate any DQPT, the interferometric approach indeed unravels the possibility of occurrence of DQPTs which persists even up to a considerable loss of purity of the engineered initial state as long as a constraint relation involving the dissipative coupling and ramping time (rate) is satisfied. This constraint relation also marks the boundary between two dynamically inequivalent phases; in one the LO vanishes for the critical momentum mode (and hence DQPTs exist) while in the other no such critical mode can exist and hence the LO never vanishes.
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The notion of a dynamical quantum phase transition (DQPT) was recently introduced [Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)] as the nonanalytic behavior of the Loschmidt echo at critical times in the thermodynamic limit. In this work the quench dynamics in the ground state sector of the two-dimensional Kitaev honeycomb model is studied regarding the occurrence of DQPTs. For general two-dimensional systems of BCS type it is demonstrated how the zeros of the Loschmidt echo coalesce to areas in the thermodynamic limit, implying that DQPTs occur as discontinuities in the second derivative. In the Kitaev honeycomb model DQPTs appear after quenches across a phase boundary or within the massless phase. In the 1d limit of the Kitaev honeycomb model it becomes clear that the discontinuity in the higher derivative is intimately related to the higher dimensionality of the nondegenerate model. Moreover, there is a strong connection between the stationary value of the rate function of the Loschmidt echo after long times and the occurrence of DQPTs in this model.
Abstract Dynamical quantum phase transitions (DQPTs) can occur following quenches in quantum systems when the rate function, a dynamical analogue of the free energy, becomes non-analytic at critical times. Here we exhaustively investigate in an exemplary model how the dynamically evolving state responds to a second quench. We demonstrate that for quenches where the initial and final Hamiltonian belong to different phases always result in DQPTs, irrespective of the intermediate quench and dynamics or the time of the second quench. However, if the initial and final Hamiltonian belong to the same equilibrium phase then the intermediate Hamiltonian must belong to a different phase. In this case, the second quench time in relation to the critical times of the first quench becomes crucial to the existence of DQPTs.
We investigate the entanglement dynamics along quantum trajectories during the seeding of time-crystalline order in a boundary time crystal (BTC). Specifically, how entanglement spreads among different spin ensembles when a BTC attempts to seed its time-crystalline behavior onto otherwise static spin ensembles, through a collective dissipative channel. We analyse both the dynamical growth of entanglement in time and the steady-state properties of the system. Our results reveal two fundamentally distinct regimes. In the seeded BTC phase, the steady-state entanglement entropy between the ensembles grows with system size NNN, accompanied by macroscopic fluctuations along the trajectories. In contrast, in the non-seeded static phase, both the steady-state entanglement and its fluctuations decay exponentially with NNN. The model thus features a measurement-induced phase transition (MIPT) driven by the seeding mechanism. Furthermore, these findings establish dissipative seeding as a powerful mechanism for controlling quantum correlations in open many-body systems, with direct experimental relevance to this class of model without a postselection barrier.
Driven-dissipative many-body systems support nontrivial quantum phases absent in equilibrium. As a prominent example, the interplay between coherent driving and collective dissipation can lead to a dynamical quantum phase that spontaneously breaks time-translation symmetry. This so-called boundary time crystal (BTC) is fragile in the presence of local dissipation, which can easily relax the system to a stationary state. In this Letter, we demonstrate a robust BTC that is intrinsically induced by local dissipation. We provide extensive numerical evidence to support the existence of the BTC and study its behaviors in different regimes. In particular, with decreasing interaction range, we identify a transition from mean-field limit cycles to correlated BTCs featuring sizable quantum correlations. Our studies significantly broaden the scope of nonequilibrium phases and shed new light on the experimental search for dynamical quantum matter.