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arXiv:1309.1673 · Physical Review B

Dynamical phase transitions, time-integrated observables, and geometry of states

James M. Hickey, Sam Genway, Juan P. Garrahan

Published Feb 10, 2014• 61 citations
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Abstract

We show that there exist dynamical phase transitions (DPTs), as defined by Heyl et al. [Phys. Rev. Lett. 110, 135704 (2013)], in the transverse-field Ising model (TFIM) away from the static quantum critical points. We study a class of special states associated with singularities in the generating functions of time-integrated observables as found by Hickey et al. [Phys. Rev. B 87, 184303 (2013)]. Studying the dynamics of these special states under the evolution of the TFIM Hamiltonian, we find temporal nonanalyticities in the initial-state return probability associated with dynamical phase transitions. By calculating the Berry phase and Chern number we show the set of special states have interesting geometric features similar to those associated with static quantum critical points.

DOI
10.1103/physrevb.89.054301

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We show that there exist dynamical phase transitions (DPTs), as defined by Heyl et al. [Phys. Rev. Lett. 110, 135704 (2013)], in the transverse-field Ising model (TFIM) away from the static quantum critical points. We study a class of special states associated with singularities in the generating functions of time-integrated observables as found by Hickey et al. [Phys. Rev. B 87, 184303 (2013)]. Studying the dynamics of these special states under the evolution of the TFIM Hamiltonian, we find temporal nonanalyticities in the initial-state return probability associated with dynamical phase transitions. By calculating the Berry phase and Chern number we show the set of special states have interesting geometric features similar to those associated with static quantum critical points.
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.
Dynamical phase transitions (DPTs) occur after quenching some global parameters in quantum systems, and are signalled by the nonanalytical time evolution of the dynamical free energy, which is calculated from the Loschmidt overlap between the initial and time evolved states. In a recent Letter [M. Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)], it was suggested that DPTs are closely related to equilibrium phase transitions (EPTs) for the transverse field Ising model. By studying a minimal model, the XY chain in a transverse magnetic field, we show analytically that this connection does not hold generally. We present examples where DPT occurs without crossing any equilibrium critical lines by the quench, and a nontrivial example with no DPT but crossing a critical line by the quench. Although the nonanalyticities of the dynamical free energy on the real time axis do not indicate the presence or absence of an EPT, the structure of Fisher lines for complex times reveals a qualitative difference.
Recent research into time-integrated observables has revealed a special class of states which cap- ture the singular features of the generating functions of those observables, as estimated by full counting statistics (FCS). In this work we extend the results of [Phys. Rev. B 87 184303 (2013)] to the 1d anisotropic XY -model and find a set of FCS critical points associated with the time-integrated transverse magnetization and anisotropy. We show dynamical phase transitions (DPTs) as defined in [Phys. Rev. Lett. 110 135704 (2013)] do not emerge on quenching the states associated with the time-integrated anisotropy across FCS critical points. We also study the timescales required to prepare the associated special states of the transverse magnetization using appropriate Markovian baths and find they are independent of the number of spins in the chain. However the probability to evolve to such a state decreases drastically with increasing system size due to dissipation. Thus when preparing such states it is preferable to use few body systems and to reach the thermodynamic limit it is necessary to use an ancillary system in conjunction with the Markovian baths.
We investigate the dynamics following sudden quenches across quantum critical points belonging to different universality classes. Specifically, we use matrix product state methods to study the quantum Ising chain in the presence of two additional terms which break integrability. We find that in all models the rate function for the return probability to the initial state becomes a nonanalytic function of time in the thermodynamic limit. This so-called ``dynamical phase transition'' was first observed in a recent work by Heyl, Polkovnikov, and Kehrein [Phys. Rev. Lett. 110, 135704 (2013)] for the exactly-solvable quantum Ising chain, which can be mapped to free fermions. Our results for ``interacting theories'' indicate that nonanalytic dynamics is a generic feature of sudden quenches across quantum critical points. We discuss potential connections to the dynamics of the order parameter.
Using an infinite matrix product state (iMPS) technique based on the time-dependent variational principle (TDVP), we study two major types of dynamical phase transitions (DPT) in the one-dimensional transverse-field Ising model (TFIM) with long-range power-law ($\ensuremath{\propto}1/{r}^{\ensuremath{\alpha}}$ with rrr interspin distance) interactions out of equilibrium in the thermodynamic limit---DPT-I: based on an order parameter in a (quasi-)steady state, and DPT-II: based on nonanalyticities (cusps) in the Loschmidt-echo return rate. We construct the corresponding rich dynamical phase diagram, while considering different quench initial conditions. We find a nontrivial connection between both types of DPT based on their critical lines. Moreover, and very interestingly, we detect a new DPT-II dynamical phase in a certain range of interaction exponent $\ensuremath{\alpha}$, characterized by what we call anomalous cusps that are distinct from the regular cusps usually associated with DPT-II. Our results provide the characterization of experimentally accessible signatures of the dynamical phases studied in this work.
Dynamical quantum phase transitions (DQPTs) manifested in the nonanalyticities in the temporal evolution of a closed quantum system generated by the time-independent final Hamiltonian, following a quench (or ramping) of a parameter of the Hamiltonian, is an emerging frontier of nonequilibrium quantum dynamics. We, here, introduce the notion of a dynamical topological order parameter (DTOP) that characterizes these DQPTs occurring in quenched (or ramped) two-dimensional closed quantum systems; this is quite a nontrivial generalization of the notion of DTOP introduced in Budich and Heyl [Phys. Rev. B 93, 085416 (2016)] for one-dimensional situations. This DTOP is obtained from the ``gauge-invariant'' Pancharatnam phase extracted from the Loschmidt overlap, i.e., the modulus of the overlap between the initially prepared state and its time-evolved counterpart reached following a temporal evolution generated by the time-independent final Hamiltonian. This generic proposal is illustrated considering DQPTs occurring in the subsequent temporal evolution following a sudden quench of the staggered mass of the topological Haldane model on a hexagonal lattice where it stays fixed to zero or unity and makes a discontinuous jump between these two values at critical times at which DQPTs occur. What is remarkable is that while the topology of the equilibrium model is characterized by the Chern number, the emergent topology associated with the DQPTs is characterized by a generalized winding number.