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arXiv:1011.0380 · Physical Review A

Universality and robustness of revivals in the transverse field XY model

Juho Häppölä, Gábor B. Halász, Alioscia Hamma

Published Mar 12, 2012• 68 citations
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Abstract

We study the structure of the revivals in an integrable quantum many-body system, the transverse field $XY$ spin chain, after a quantum quench. The time evolutions of the Loschmidt echo, the magnetization, and the single-spin entanglement entropy are calculated. We find that the revival times for all of these observables are given by integer multiples of ${T}_{\mathrm{rev}}\ensuremath{\simeq}L/{v}_{\mathrm{max}}$, where $L$ is the linear size of the system and ${v}_{\mathrm{max}}$ is the maximal group velocity of quasiparticles. This revival structure is universal in the sense that it does not depend on the initial state and the size of the quench. Applying nonintegrable perturbations to the $XY$ model, we observe that the revivals are robust against such perturbations: they are still visible at time scales much larger than the quasiparticle lifetime. We therefore propose a generic connection between the revival structure and the locality of the dynamics, where the quasiparticle speed ${v}_{\mathrm{max}}$ generalizes into the Lieb-Robinson speed ${v}_{\text{LR}}$.

DOI
10.1103/physreva.85.032114

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We study the structure of the revivals in an integrable quantum many-body system, the transverse field XYXYXY spin chain, after a quantum quench. The time evolutions of the Loschmidt echo, the magnetization, and the single-spin entanglement entropy are calculated. We find that the revival times for all of these observables are given by integer multiples of Trev\ensuremath≃L/vmax{T}_{\mathrm{rev}}\ensuremath{\simeq}L/{v}_{\mathrm{max}}Trev​\ensuremath≃L/vmax​, where LLL is the linear size of the system and vmax{v}_{\mathrm{max}}vmax​ is the maximal group velocity of quasiparticles. This revival structure is universal in the sense that it does not depend on the initial state and the size of the quench. Applying nonintegrable perturbations to the XYXYXY model, we observe that the revivals are robust against such perturbations: they are still visible at time scales much larger than the quasiparticle lifetime. We therefore propose a generic connection between the revival structure and the locality of the dynamics, where the quasiparticle speed vmax{v}_{\mathrm{max}}vmax​ generalizes into the Lieb-Robinson speed vLR{v}_{\text{LR}}vLR​.
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The dynamics of relative entropy and l1{l}_{1}l1​ norm of coherence, as well as, the Wigner-Yanase skew and quantum Fisher information are studied in the one-dimensional XYXYXY spin chain in the presence of a time-dependent transverse magnetic field. We show that, independent of the initial state of the system and while the relative entropy of coherence, the l1{l}_{1}l1​ norm of coherence, and quantum Fisher information are incapable, surprisingly, the Wigner-Yanase skew information dynamic can truly spotlight the equilibrium critical point. We also observe that, when the system is quenched to the critical point, these quantities show suppressions and revivals. Moreover, the first suppression (revival) time scales linearly with the system size and its scaling ratio is unique for all quenches independent of the initial phase of the system. This is the promised universality of the first suppression (revival) time.
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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.
We discuss different quench protocols for Ising and XY spin chains in a transverse magnetic field. With a sudden local magnetic field quench as a starting point, we generalize our approach to a large class of local non-sudden quenches. Using finite time path field theory (FTPFT) perturbative methods, we show that the difference between the sudden quench and a class of quenches with non-sudden switching on the perturbation vanishes exponentially with time, apart from non-substantial modifications that are systematically accounted for. As the consequence of causality and analytic properties of functions describing the discussed class of quenches, this is true at any order of perturbation expansion and thus for the resummed perturbation series. The only requirements on functions describing the perturbation strength switched on at a finite time t=0 are as follows: (1) their Fourier transform f(p) is a function that is analytic everywhere in the lower complex semiplane, except at the simple pole at p=0 and possibly others with ℑ(p)<0; and (2) f(p)/p converges to zero at infinity in the lower complex semiplane. A prototypical function of this class is tanh(ηt), to which the perturbation strength is proportional after the switching at time t=0. In the limit of large η, such a perturbation approaches the case of a sudden quench. It is shown that, because of this new type of universal behavior of Loschmidt echo (LE) that emerges in an exponentially short time scale, our previous results for the sudden local magnetic field quench of Ising and XY chains, obtained by the resummation of the perturbative expansion, extend in the long-time limit to all non-sudden quench protocols in this class, with non-substantial modifications systematically taken into account. We also show that analogous universal behavior exists in disorder quenches, and ultimately global ones. LE is directly connected to the work probability distribution, and the described universal behavior is therefore appropriate in potential concepts of quantum technology related to spin chains.
In a recent letter [J. Cardy, Phys. Rev. Lett. 112, 220401 (2014)], the author made a very interesting observation that complete revivals of quantum states after quantum quench can happen in a period that is a fraction of the system size. This is possible for critical systems that can be described by minimal conformal field theories with central charge c<1c<1c<1. In this paper, we show that these complete revivals are impossible in microscopic realizations of those minimal models. We will prove the absence of the mentioned complete revivals for the critical transverse field Ising chain analytically, and present numerical results for the critical line of the XY chain. In particular, for the considered initial states, we will show that criticality has no significant effect in partial revivals. We also comment on the applicability of quasiparticle picture to determine the period of the partial revivals qualitatively. In particular, we detect a regime in the phase diagram of the XY chain in which one can not determine the period of the partial revivals using the quasiparticle picture.