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arXiv:2410.00682 · Journal of Statistical Mechanics Theory and Experiment

Relaxation dynamics of integrable field theories after a global quantum quench

Emanuele Di Salvo, Dirk Schuricht

Published Jan 1, 2025• 5 citations
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Abstract

Abstract We apply the linked cluster expansion as well as the quench action approach to study the time evolution of one-point functions after a quantum quench in integrable field theories. We argue that the relaxation towards the stationary value fundamentally differs depending on the locality properties of the considered observable: while for local operators both exponential and power-law decaying terms are present in the dynamics, for semi-local operators the latter are absent. We explicitly confirm this for the Ising field theory, the sinh-Gordon model and the repulsive sine-Gordon model.

DOI
10.1088/1742-5468/ad9f4e

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Abstract We apply the linked cluster expansion as well as the quench action approach to study the time evolution of one-point functions after a quantum quench in integrable field theories. We argue that the relaxation towards the stationary value fundamentally differs depending on the locality properties of the considered observable: while for local operators both exponential and power-law decaying terms are present in the dynamics, for semi-local operators the latter are absent. We explicitly confirm this for the Ising field theory, the sinh-Gordon model and the repulsive sine-Gordon model.
We consider the time evolution in the repulsive sine-Gordon quantum field theory after the system is prepared in a particular class of initial states. We focus on the time dependence of the one-point function of the semi-local operator exp⁡(i β Φ(x)/2)\exp\big(i\ β\ Φ(x)/2\big)exp(i β Φ(x)/2). By using two different methods based on form-factor expansions, we show that this expectation value decays to zero exponentially, and we determine the decay rate by analytical means. Our methods generalise to other correlation functions and integrable models.
Abstract. We consider quantum quenches to the sinh-Gordon integrable quantum field theory from a particular class of initial states. Our analysis includes the case of mass and interaction quenches starting from a non-interacting theory. By means of the recently developed quench action method, we fully characterize the stationary state reached at long times after the quench in terms of the corresponding rapidity distribution. We also provide exact results for the expectation values of arbitrary vertex operators in the post-quench stationary state by proposing a formula based on the analogy with the standard thermodynamic Bethe ansatz. Finally, we comment on the behavior of the post-quench stationary state under the mapping between the sinh-Gordon field theory and the one-dimensional Lieb–Liniger model.
We implement numerical linked cluster expansions (NLCEs) to study dynamics of lattice systems following quantum quenches, and focus on a hard-core boson model in one-dimensional lattices. We find that, in the nonintegrable regime and within the accessible times, local observables exhibit exponential relaxation. We determine the relaxation rate as one departs from the integrable point and show that it scales quadratically with the strength of the integrability breaking perturbation. We compare the NLCE results with those from exact diagonalization calculations on finite chains with periodic boundary conditions, and show that NLCEs are far more accurate.
We study the dynamics of the quantum Ising chain following a zero-temperature quench of the transverse field strength. Focusing on the behavior of two-point spin correlation functions, we show that the correlators of the order parameter display an effective asymptotic thermal behavior; i.e., they decay exponentially to zero with a phase coherence rate and a correlation length dictated by the equilibrium law with an effective temperature set by the energy of the initial state. On the contrary, the two-point correlation functions of the transverse magnetization or the density-of-kinks operator decay as a power law and do not exhibit thermal behavior. We argue that the different behavior is linked to the locality of the corresponding operator with respect to the quasiparticles of the model: nonlocal operators, such as the order parameter, behave thermally, while local ones do not. We study which features of the two-point correlators are a consequence of the integrability of the model by analyzing their robustness with respect to a sufficiently strong integrability-breaking term.
We show, using the quench action approach (Caux and Essler 2013 Phys. Rev. Lett. 110 257203 ), that the whole post-quench time evolution of an integrable system in the thermodynamic limit can be computed with a minimal set of data which are encoded in what we denote the generalized single-particle overlap coefficient This function can be extracted from the thermodynamically leading part of the overlaps between the eigenstates of the model and the initial state. For a generic global quench the shape of in the low momentum limit directly gives the exponent for the power law decay to the effective steady state. As an example we compute the time evolution of the static density–density correlation in the interacting Lieb–Liniger gas after a quench from a Bose–Einstein condensate. This shows an approach to equilibrium with power law t −3 which turns out to be independent of the post-quench interaction and of the considered observable.
We consider dynamic (non-equal-time) correlation functions of local observables after a quantum quench. We show that, in the absence of long-range interactions in the final Hamiltonian, the dynamics is determined by the same ensemble that describes static (equal-time) correlations. For many integrable models, static correlation functions of local observables after a quantum quench relax to stationary values, which are described by a generalized Gibbs ensemble. The same generalized Gibbs ensemble then determines dynamic correlation functions, and the basic form of the fluctuation dissipation theorem holds, although the absorption and emission spectra are not simply related as in the thermal case. For quenches in the transverse field Ising chain, we derive explicit expressions for the time evolution of dynamic order parameter correlators after a quench.