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arXiv:2011.05673 · Physical review. B./Physical review. B

Dynamical quantum phase transition in a bosonic system with long-range interactions

Marvin Syed, Tilman Enss, Nicolò Defenu

Published Feb 12, 2021• 21 citations
arXivHTMLPDF

Abstract

In this paper, we investigate the dynamical quantum phase transitions appearing in the Loschmidt echo and the time-dependent order parameter of a quantum system of harmonically coupled degenerate bosons as a function of the power-law decay $\ensuremath{\sigma}$ of long-range interactions. Following a sudden quench, the nonequilibrium dynamics of this system are governed by a set of nonlinear coupled Ermakov equations. To solve them, we develop an analytical approximation valid at late times. Based on this approximation, we show that the emergence of a dynamical quantum phase transition hinges on the generation of a finite mass gap following the quench, starting from a massless initial state. In general, we can define two distinct dynamical phases characterized by the finiteness of the post-quench mass gap. The Loschmidt echo exhibits periodical nonanalytic cusps whenever the initial state has a vanishing mass gap and the final state has a finite mass gap. These cusps are shown to coincide with the maxima of the time-dependent long-range correlations.

DOI
10.1103/physrevb.103.064306

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Interacting many-body systems that are driven far away from equilibrium can exhibit phase transitions between dynamically emerging quantum phases, which manifest as singularities in the Loschmidt echo. Whether and under which conditions such dynamical transitions occur in higher-dimensional systems with spontaneously broken continuous symmetries is largely elusive thus far. Here, we study the dynamics of the Loschmidt echo in the three-dimensional O($N$) model following a quantum quench from a symmetry-breaking initial state. The O($N$) model exhibits a dynamical transition in the asymptotic steady state, separating two phases with a finite and vanishing order parameter, that is associated with the broken symmetry. We analytically calculate the rate function of the Loschmidt echo and find that it exhibits periodic kink singularities when this dynamical steady-state transition is crossed. The singularities arise exactly at the zero crossings of the oscillating order parameter. As a consequence, the appearance of the kink singularities in the transient dynamics is directly linked to a dynamical transition in the order parameter. Furthermore, we argue, that our results for dynamical quantum phase transitions in the O($N$) model are general and apply to generic systems with continuous symmetry breaking.

arXiv:1709.00421•Physical review. B./Physical review. B•Oct 30, 2017•61 citations
In this paper, we investigate the dynamical quantum phase transitions appearing in the Loschmidt echo and the time-dependent order parameter of a quantum system of harmonically coupled degenerate bosons as a function of the power-law decay \ensuremathσ\ensuremath{\sigma}\ensuremathσ of long-range interactions. Following a sudden quench, the nonequilibrium dynamics of this system are governed by a set of nonlinear coupled Ermakov equations. To solve them, we develop an analytical approximation valid at late times. Based on this approximation, we show that the emergence of a dynamical quantum phase transition hinges on the generation of a finite mass gap following the quench, starting from a massless initial state. In general, we can define two distinct dynamical phases characterized by the finiteness of the post-quench mass gap. The Loschmidt echo exhibits periodical nonanalytic cusps whenever the initial state has a vanishing mass gap and the final state has a finite mass gap. These cusps are shown to coincide with the maxima of the time-dependent long-range correlations.
We unveil the role of the long-time average of the Loschmidt echo in the characterization of nonequilibrium quantum phase transitions by studying sudden quench processes across quantum phase transitions in various quantum systems. While the dynamical quantum phase transitions are characterized by the emergence of a series of zero points at critical times during time evolution, we demonstrate that nonequilibrium quantum phase transitions can be identified by nonanalyticities in the long-time average of the Loschmidt echo. The nonanalytic behaviors are illustrated by a sharp change in the long-time average of the Loschmidt echo or the corresponding rate function or the emergence of divergence in the second derivative of the rate function when the driving quench parameter crosses the phase transition points. The connection between the second derivative of the rate function and fidelity susceptibility is also discussed.
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.
Abstract We explore the dynamics of long-range Kitaev chain by varying pairing interaction exponent. It is well known that the distinctive features of the nonequilibrium dynamics of a closed quantum system are closely related to the equilibrium phase transitions. Specifically, the return probability (Loschmidt echo) of the system to its initial state, in the finite-size system, is expected to exhibit periodicity after a sudden quench to a quantum critical point. The time of the first revivals scales inversely with the maximum of the group velocity. We show that, contrary to expectations, the periodicity of the return probability breaks for a sudden quench to the non-trivial quantum critical point. Further, we find that the periodicity of the return probability scales inversely with the group velocity at the gap closing point for a quench to the trivial critical point of truly long-range pairing case. Also, analyzing the effect of averaging quenched disorder shows that the revivals in the short-range pairing cases are more robust against disorder than that of the long-range pairing case. We also study the effect of disorder on the non-analyticities of the rate function of the return probability which introduced as a witness of the dynamical phase transition. We show that the non-analyticities in the rate function of return probability are washed out in the presence of strong disorders.
We study dynamical quantum phase transitions (DQPTs) in the extended Bose-Hubbard model after a sudden quench of the nearest-neighbor interaction strength. Using the time-dependent density matrix renormalization group, we demonstrate that interaction-driven DQPTs can appear after quenches between two topologically trivial insulating phases---a phenomenon that has so far only been studied between gapped and gapless phases. These DQPTs occur when the interaction strength crosses a certain threshold value that does not coincide with the equilibrium phase boundaries, which is in contrast to quenches that involve a change of topology. In order to elucidate the nonequilibrium excitations during the time evolution, we define a new set of string and parity order parameters. We find a close connection between DQPTs and these newly defined order parameters for both types of quenches. In the interaction-driven case, the order parameter exhibits a singularity at the time of the DQPT only when the quench parameter is close to the threshold value. Finally, the timescales of DQPTs are scrutinized and different kinds of power laws are revealed for the topological and interaction-driven cases.
We explore the dynamics of long-range Kitaev chain by varying pairing interaction exponent. It is well known that the distinctive features of the nonequilibrium dynamics of a closed quantum system are closely related to the equilibrium phase transitions. Specifically, the return probability (Loschmidt echo) of the system to its initial state, in the finite-size system, is expected to exhibit periodicity after a sudden quench to a quantum critical point. The time of the first revivals scales inversely with the maximum of the group velocity. We show that, contrary to expectations, the periodicity of the return probability breaks for a sudden quench to the non-trivial quantum critical point. Further, we find that the periodicity of the return probability scales inversely with the group velocity at the gap closing point for a quench to the trivial critical point of truly long-range pairing case. Also, analyzing the effect of averaging quenched disorder shows that the revivals in the short-range pairing cases are more robust against disorder than that of the long-range pairing case. We also study the effect of disorder on the non-analyticities of the rate function of the return probability which introduced as a witness of the dynamical phase transition. We show that the non-analyticities in the rate function of return probability are washed out in the presence of strong disorders. © 2020 IOP Publishing Ltd.
Interacting many-body systems that are driven far away from equilibrium can exhibit phase transitions between dynamically emerging quantum phases, which manifest as singularities in the Loschmidt echo. Whether and under which conditions such dynamical transitions occur in higher-dimensional systems with spontaneously broken continuous symmetries is largely elusive thus far. Here, we study the dynamics of the Loschmidt echo in the three-dimensional O(NNN) model following a quantum quench from a symmetry-breaking initial state. The O(NNN) model exhibits a dynamical transition in the asymptotic steady state, separating two phases with a finite and vanishing order parameter, that is associated with the broken symmetry. We analytically calculate the rate function of the Loschmidt echo and find that it exhibits periodic kink singularities when this dynamical steady-state transition is crossed. The singularities arise exactly at the zero crossings of the oscillating order parameter. As a consequence, the appearance of the kink singularities in the transient dynamics is directly linked to a dynamical transition in the order parameter. Furthermore, we argue, that our results for dynamical quantum phase transitions in the O(NNN) model are general and apply to generic systems with continuous symmetry breaking.