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arXiv:1612.07046 · Physical Review Letters

Loschmidt Echo Revivals: Critical and Noncritical

Rouhollah Jafari, Henrik Johannesson

Published Jan 5, 2017• 110 citations
arXivHTMLPDF

Abstract

A quantum phase transition is generally thought to imprint distinctive characteristics on the nonequilibrium dynamics of a closed quantum system. Specifically, the Loschmidt echo after a sudden quench to a quantum critical point-measuring the time dependence of the overlap between initial and time-evolved states-is expected to exhibit an accelerated relaxation followed by periodic revivals. We here introduce a new exactly solvable model, the extended Su-Schrieffer-Heeger model, the Loschmidt echo of which provides a counterexample. A parallell analysis of the quench dynamics of the three-site spin-interacting XY model allows us to pinpoint the conditions under which a periodic Loschmidt revival actually appears.

DOI
10.1103/physrevlett.118.015701

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A quantum phase transition is generally thought to imprint distinctive characteristics on the nonequilibrium dynamics of a closed quantum system. Specifically, the Loschmidt echo after a sudden quench to a quantum critical point-measuring the time dependence of the overlap between initial and time-evolved states-is expected to exhibit an accelerated relaxation followed by periodic revivals. We here introduce a new exactly solvable model, the extended Su-Schrieffer-Heeger model, the Loschmidt echo of which provides a counterexample. A parallell analysis of the quench dynamics of the three-site spin-interacting XY model allows us to pinpoint the conditions under which a periodic Loschmidt revival actually appears.
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.
A critically enhanced decay of the Loschmidt echo is characteristic of sudden quench dynamics near a quantum phase transition. Here, we demonstrate that the decay and revival of the Loschmidt echo follows power-law scaling in the system size and the distance from a critical point with equilibrium critical exponents. We reveal such dynamical scaling laws by analyzing relevant perturbations to the Loschmidt echo cast in a scaling invariant form. We confirm the validity and the generality of the predicted dynamical scaling laws with a diverse range of critical models such as Ising spin models with a short and long range interaction, a finite-component system phase transition, and a topological phase transition. Moreover, using the integrability of systems in the thermodynamic limit, we derive such scaling laws analytically from a microscopic analysis. Our finding promotes the Loschmidt echo to a quantitative non-equilibrium probe of criticality and allows for quantitative predictions on the role of criticality on various physical scenarios where the Loschmidt echo is central to describing non-equilibrium dynamics.
We study the complete phase space and the quench dynamics of an exactly solvable spin chain, the cluster-XY model. In this chain, the cluster term and the XY couplings compete to give a rich phase diagram. The phase diagram is studied by means of the quantum geometric tensor. We study the time evolution of the system after a critical quantum quench using the Loschmidt echo. The structure of the revivals after critical quantum quenches presents a nontrivial behavior depending on the phase of the initial state and the critical point.
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 the Loschmidt echo (LE) in a central spin model in which a central spin is globally coupled to an environment (EEE) which is subjected to a small and sudden quench at t=0t=0t=0, so that its state at t=0+t={0}^{+}t=0+ remains the same as the ground state of the initial environmental Hamiltonian before the quench; this leads to a nonequilibrium situation. This state now evolves with two Hamiltonians, the final Hamiltonian following the quench and its modified version which incorporates an additional term arising due to the coupling of the central spin to the environment. Using a generic short-time scaling of the decay rate, we establish that in the early-time limit, the rate of decay of the LE close to the quantum critical point (QCP) of EEE is independent of the quenching. We also study the temporal evolution of the LE and establish the presence of a crossover to a situation where the quenching becomes irrelevant. In the limit of large quench amplitude the nonequilibrium initial condition is found to result in a drastic increase in decoherence at large times, even far away from a QCP. These generic results are verified analytically as well as numerically, choosing EEE to be a transverse Ising chain where the transverse field is suddenly quenched.
In most lattice models, the closing of a band gap typically occurs at high-symmetry points in the Brillouin zone. Differently, in the Creutz model---describing a system of spinless fermions hopping on a two-leg ladder pierced by a magnetic field---the gap closing at the quantum phase transition between the two topologically nontrivial phases of the model can be moved by tuning the hopping amplitudes. We take advantage of this property to examine the nonequilibrium dynamics of the model after a sudden quench of the magnetic flux through the plaquettes of the ladder. For a quench to one of the equilibrium quantum critical points, we find that the revival period of the Loschmidt echo---measuring the overlap between initial and time-evolved states---is controlled by the gap closing zero-energy modes. In particular, and contrary to expectations, the revival period of the Loschmidt echo for a finite ladder does not scale linearly with size but exhibits jumps determined by the presence or absence of zero-energy modes. We further investigate the conditions for the appearance of dynamical quantum phase transitions in the model and find that, for a quench to an equilibrium critical point, such transitions occur only for ladders of sizes which host zero-energy modes. Exploiting concepts from quantum thermodynamics, we show that the average work and the irreversible work per lattice site exhibit a weak dependence on the size of the system after a quench across an equilibrium critical point, suggesting that quenching into a different phase induces effective correlations among the particles.