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arXiv:2004.12287 · Physical review. B./Physical review. B

Dynamical crossover in the transient quench dynamics of short-range transverse-field Ising models

Ceren B. Dağ, Kai Sun

Published Jun 1, 2021• 13 citations
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Abstract

Dynamical detection of quantum phases and phase transitions (QPTs) in quenched systems with experimentally convenient initial states is a topic of interest from both theoretical and experimental perspectives. Quenched from polarized states, longitudinal magnetization decays exponentially to zero in time for the short-range transverse-field Ising model and hence has a featureless steady state regime, which prevents it from exhibiting dynamical phase transitions of type I. In this paper, we ask whether the transient regimes of such nonequilibrium processes probed by single-site observables, that is magnetization per site, could encode information about the underlying QPT. The decay rates of time-dependent and single-site observables exhibit a dynamical crossover that separates two dynamical regions, ordered and disordered, both of which have distinct nonequilibrium responses. We construct a dynamical order parameterlike quantity that exhibits a scaling law in the vicinity of the crossover. Our results reveal that scaling law exponent in short times in the close vicinity of the dynamical crossover is significantly different than the one predicted by analytical theory for long times. When integrability is strongly broken, the crossover boundary turns into a region that separates two other dynamical regions that act like dynamically ordered and disordered regimes.

DOI
10.1103/physrevb.103.214402

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Dynamical detection of quantum phases and phase transitions (QPTs) in quenched systems with experimentally convenient initial states is a topic of interest from both theoretical and experimental perspectives. Quenched from polarized states, longitudinal magnetization decays exponentially to zero in time for the short-range transverse-field Ising model and hence has a featureless steady state regime, which prevents it from exhibiting dynamical phase transitions of type I. In this paper, we ask whether the transient regimes of such nonequilibrium processes probed by single-site observables, that is magnetization per site, could encode information about the underlying QPT. The decay rates of time-dependent and single-site observables exhibit a dynamical crossover that separates two dynamical regions, ordered and disordered, both of which have distinct nonequilibrium responses. We construct a dynamical order parameterlike quantity that exhibits a scaling law in the vicinity of the crossover. Our results reveal that scaling law exponent in short times in the close vicinity of the dynamical crossover is significantly different than the one predicted by analytical theory for long times. When integrability is strongly broken, the crossover boundary turns into a region that separates two other dynamical regions that act like dynamically ordered and disordered regimes.
Dynamical phase transitions (DPTs) are generally defined under two different but related categories: DPT-I where the equilibrium value, or long-time average, of the order parameter as a function of the control parameter demonstrates a phase boundary; DPT-II where the Loschmidt return rate shows a cusp singularity in real-time dynamics. Quenched from polarized states, one-point observables have featureless steady state regimes for short-range transverse field Ising models (TFIM), which prevents them from exhibiting DPT-I, unlike long-range TFIM. In this Letter, we incorporate Lieb-Robinson bounds into the definition of DPT by choosing appropriate temporal cutoffs and show that in the lightcone, a quantum phase transition (QPT) far from equilibrium could emerge with an associated scaling law at the vicinity of the transition. More specifically, the observable at the cutoff could exhibit order parameterlike behavior in analogy to equilibrium quantum criticality, however with certain important differences that we discuss in detail. The scaling of the dynamical order parameter at the vicinity is determined via the decay rate of the observable in time, hence rendering the QPT a truly nonequilibrium phase transition. We apply the procedure to both integrable and nonintegrable short-range TFIM and elaborate on how the QPT signature changes with integrability.
Extracting critical behavior in the wake of quantum quenches has recently been at the forefront of theoretical and experimental investigations in condensed matter physics and quantum synthetic matter, with particular emphasis on experimental feasibility. Here, we investigate the potential of single-site observables in probing equilibrium phase transitions and dynamical criticality in short-range transverse-field Ising chains. For integrable and near-integrable models, our exact and mean-field-theory analyses reveal a truly out-of-equilibrium universal scaling exponent in the vicinity of the transition that is independent of the initial state and the location of the probe site so long as the latter is sufficiently close to the edge of the chain. Signature of a dynamical crossover survives when integrability is strongly broken. Our work provides a robust scheme for the experimental detection of quantum critical points and dynamical scaling laws in short-range interacting models using modern ultracold-atom setups.
Recently, single-site observables have been shown to be useful for probing critical slowing down in sudden quench dynamics [Da\ifmmode \breve{g}\else \u{g}\fi{} et al., Phys. Rev. B 107, L121113 (2023)]. Here, we demonstrate the potential of single-site magnetization as a probe of quantum phase transitions in integrable and nonintegrable transverse-field Ising chains (TFIC). We analytically prove the requirement of zero modes for the quasistationary regime to emerge at a probe site near the edge, and show how this regime gives rise to a nonanalytic behavior in the dynamical order profiles. Our t\mathit{t}t-DMRG calculations verify the results of the quench mean-field theory for near-integrable TFIC both with finite-size and finite-time scaling analyses. We find that both finite-size and finite-time analyses suggest a dynamical critical point for a strongly nonintegrable and locally connected TFIC. We finally demonstrate the presence of a quasistationary regime in the power-law interacting TFIC, and extract local dynamical order profiles for TFIC in the long-range Ising universality class with algebraic light cones.
We consider the finite-time quench dynamics in the quantum transverse field Ising model which exhibits a second order phase transition from a paramagnetic to a ferromagnetic phase, as the transverse magnetic field is decreased. These dynamics have been thoroughly investigated in previous studies when the critical point is crossed during the quench; here, we quench the system from deep in the paramagnetic phase to just above the critical field so that the system remains in the gapped phase throughout the quench duration. On linearly quenching the infinitely large system, we find that the behavior of mean longitudinal defect density and mean transverse magnetization at the end of the quench falls into three distinct scaling regimes as the quench time is increased. For sufficiently small quench times, these observables remain roughly constant, but for larger quench times, a crossover occurs from the Kibble-Zurek scaling law to the quadratic quench rate law when the Kibble-Zurek time is of the order of relaxation time at the final quench field. These results are shown analytically using power series and uniform asymptotic expansions of the exact solution of the model, and also compared with an adiabatic perturbation theory in the third regime. We find that the above mentioned scaling regimes hold for quenches within the ferromagnetic phase also, and provide a general scaling argument for crossover from the Kibble-Zurek regime to an adiabatic regime for slow quenches within a gapped phase.
A crossover between different power-law relaxation behaviors of many-body periodically driven integrable systems has come to light in recent years. We demonstrate, using integrable quantum systems, that similar kinds of dynamical transitions may also occur in the relaxation of such systems following a sudden quench. Particularly, we observe two distinct power-law relaxation behaviors following a sudden quench in the integrable XY model, depending upon whether the quenched Hamiltonian lies in the commensurate or the incommensurate phase. The relaxation behavior for quenches at and near the boundary line, called the disorder line (DL), separating these phases is also characterized. The relaxation at the DL shows a scaling exponent different from that for quenches to either of the commensurate or incommensurate phases. The transitions occur through a crossover from the commensurate or incommensurate scaling behavior to the DL scaling behavior. The crossover time diverges as a power law as the parameters of the final quenched Hamiltonian approach the DL. The transitions are also observed to be robust under weak integrability-breaking perturbations but disappear following strongly chaotic quenches.
We theoretically study the dynamics of a transverse-field Ising chain with power-law decaying interactions characterized by an exponent α, which can be experimentally realized in ion traps. We focus on two classes of emergent dynamical critical phenomena following a quantum quench from a ferromagnetic initial state: The first one manifests in the time-averaged order parameter, which vanishes at a critical transverse field. We argue that such a transition occurs only for long-range interactions α≤2. The second class corresponds to the emergence of time-periodic singularities in the return probability to the ground-state manifold which is obtained for all values of α and agrees with the order parameter transition for α≤2. We characterize how the two classes of nonequilibrium criticality correspond to each other and give a physical interpretation based on the symmetry of the time-evolved quantum states.