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arXiv:2407.11295 · Physical Review Research

Dynamical quantum phase transition and thermal equilibrium in the lattice Thirring model

Mari Carmen Bañuls, Krzysztof Cichy, Hao-Ti Hung, Ying-Jer Kao, C.-J. David Lin, Amit Singh

Published May 28, 2025• 3 citations
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Abstract

Using tensor network methods, we simulate the real-time evolution of the lattice Thirring model quenched out of equilibrium in both the critical and massive phases and study the appearance of dynamical quantum phase transitions, as nonanalyticities in the Loschmidt rate. Although the presence of a dynamical quantum phase transition in the model does not correspond to quenches across the critical line of the equilibrium phase diagram at zero temperature, we identify a threshold in the energy density of the initial state, necessary for a dynamical quantum phase transition to be present. Moreover, in the case of the gapped quench Hamiltonian, we unveil a connection of this threshold to a transition between different regions in the finite-temperature phase diagram.

DOI
10.1103/physrevresearch.7.023194

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Using tensor network methods, we simulate the real-time evolution of the lattice Thirring model quenched out of equilibrium in both the critical and massive phases and study the appearance of dynamical quantum phase transitions, as nonanalyticities in the Loschmidt rate. Although the presence of a dynamical quantum phase transition in the model does not correspond to quenches across the critical line of the equilibrium phase diagram at zero temperature, we identify a threshold in the energy density of the initial state, necessary for a dynamical quantum phase transition to be present. Moreover, in the case of the gapped quench Hamiltonian, we unveil a connection of this threshold to a transition between different regions in the finite-temperature phase diagram.
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.
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We investigate dynamical quantum phase transitions (DQPTs) in finite-dimensional systems prepared in thermal equilibrium states and subjected to a sudden quench. A mixed-state Loschmidt amplitude is constructed from first principles within a metric-stationary pseudo-Hermitian framework, providing a self-contained derivation of the finite-temperature quench dynamics. Applying this framework to an NNN-level model consisting of a two-level sector coupled to N−2N-2N−2 spectator states, we find that temperature controls the DQPTs through the redistribution of thermal weights among the eigenstates. This mechanism leads to a dimensionality-dependent threshold temperature that becomes finite when the Hilbert-space dimension reaches five, above which the Loschmidt amplitude loses all real zeros and the DQPTs are fully suppressed. The thermal suppression mechanism suggests a general principle for controlling dynamical criticality through thermal occupation, while the quasi-Hermitian framework provides the self-consistent foundation for its rigorous derivation.
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Finite-Temperature Dynamical Phase Diagram of the 2+12+12+1D Quantum Ising Model
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The recently discovered dynamical phase transition denotes nonanalytic behavior in the real-time evolution of quantum systems in the thermodynamic limit, and it has been shown to occur in different systems at zero temperature [Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)]. In this paper, we extend the analysis to nonzero temperature by studying a generalized form of the Loschmidt echo, the work distribution function, of a quantum quench in the transverse field Ising model. Although the quantitative behavior at nonzero temperatures still displays features derived from the zero-temperature nonanalyticities, it is shown that in this model dynamical phase transitions do not exist if T>0T>0T>0. This is a consequence of the system being initialized in a thermal state. Moreover, we elucidate how the Tasaki-Crooks-Jarzynski relation can be exploited as a symmetry relation for a global quench or to obtain the change of the equilibrium free-energy density.