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arXiv:1003.5167 · New Journal of Physics

Quantum quench dynamics of the sine-Gordon model in some solvable limits

Anı́bal Iucci, Miguel A. Cazalilla

Published May 28, 2010• 141 citations
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Abstract

With regard to the thermalization problem in isolated quantum systems, we investigate the dynamics following a quantum quench of the sine-Gordon model (sGM) in the Luther–Emery and the semiclassical limits. We consider the quench from the gapped to the gapless phase, as well as the reverse one. By obtaining analytic expressions for the one- and two-point correlation functions of the order parameter operator at zero-temperature, the manifestations of integrability in the absence of thermalization in the sGM are studied. It is shown that correlations in the long-time regime after the quench are well described by a generalized Gibbs ensemble. We also consider the case where the system is initially in contact with a reservoir at finite temperature. The possible relevance of our results to current and future experiments with ultracold atomic systems is also considered.

DOI
10.1088/1367-2630/12/5/055019

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With regard to the thermalization problem in isolated quantum systems, we investigate the dynamics following a quantum quench of the sine-Gordon model (sGM) in the Luther–Emery and the semiclassical limits. We consider the quench from the gapped to the gapless phase, as well as the reverse one. By obtaining analytic expressions for the one- and two-point correlation functions of the order parameter operator at zero-temperature, the manifestations of integrability in the absence of thermalization in the sGM are studied. It is shown that correlations in the long-time regime after the quench are well described by a generalized Gibbs ensemble. We also consider the case where the system is initially in contact with a reservoir at finite temperature. The possible relevance of our results to current and future experiments with ultracold atomic systems is also considered.
The sine-Gordon model appears as the low-energy effective field theory of various one-dimensional gapped quantum systems. Here we investigate the dynamics of generic, nonintegrable systems belonging to the sine-Gordon family at finite temperature within the semiclassical approach. Focusing on timescales where the effect of nontrivial quasiparticle scatterings becomes relevant, we obtain universal results for the long-time behavior of dynamical correlation functions. We find that correlation functions of vertex operators behave neither ballistically nor diffusively but follow a stretched exponential decay in time. We also study the full counting statistics of the topological current and find that distribution of the transferred charge is non-Gaussian with its cumulants scaling nonuniformly in time.
Complete information on the equilibrium behavior and dynamics of quantum field theory (QFT) is provided by multipoint correlation functions. However, their theoretical calculation is a challenging problem, even for exactly solvable models. This has recently become an experimentally relevant problem, due to progress in cold-atom experiments simulating QFT models and directly measuring higher order correlations. Here we compute correlation functions of the quantum sine-Gordon model, a prototype integrable model of central interest from both theoretical and experimental points of view. Building upon the so-called truncated conformal space approach, we numerically construct higher order correlations in a system of finite size in various physical states of experimental relevance, both in and out of equilibrium. We measure deviations from Gaussianity due to the presence of interaction and analyze their dependence on temperature, explaining the experimentally observed crossover between Gaussian and non-Gaussian regimes. We find that correlations of excited states are markedly different from the thermal case, which can be explained by the integrability of the system. We also study dynamics after a quench, observing the effects of the interaction on the time evolution of correlation functions, their spatial dependence, and their non-Gaussianity as measured by the kurtosis.
We compute the time evolution of correlation functions after quantum quenches in the sine-Gordon model within the semiclassical approximation, which is expected to yield accurate results for small and slow quenches producing slow quasiparticles with low density. We demonstrate this by reproducing results of a recent form-factor calculation of the relaxation of expectation values [B. Bertini, D. Schuricht, and F. H. L. Essler, J. Stat. Mech. (2014) P100351742-546810.1088/1742-5468/2014/10/P10035]. Extending these results, we find that-in the universal limit of vanishingly small quasiparticle velocities-the expectation values of most vertex operators do not decay to zero. We give analytic expressions for the relaxation of dynamic correlation functions and show that they have diffusive behavior for large timelike separation.
We discuss the quench dynamics near a quantum critical point focusing on the sine-Gordon model as a primary example. We suggest a unified approach to sudden and slow quenches, where the tuning parameter $\ensuremath{\lambda}(t)$ changes in time as $\ensuremath{\lambda}(t)\ensuremath{\sim}\ensuremath{\upsilon}{t}^{r}$, based on the adiabatic expansion of the excitation probability in powers of $\ensuremath{\upsilon}$. We show that the universal scaling of the excitation probability can be understood through the singularity of the generalized adiabatic susceptibility ${\ensuremath{\chi}}_{2r+2}(\ensuremath{\lambda})$, which for sudden quenches (r=0)(r=0)(r=0) reduces to the fidelity susceptibility. In turn this class of susceptibilities is expressed through the moments of the connected correlation function of the quench operator. We analyze the excitations created after a sudden quench of the cosine potential using a combined approach of form-factors expansion and conformal perturbation theory for the low-energy and high-energy sector, respectively. We find the general scaling laws for the probability of exciting the system, the density of excited quasiparticles, the entropy and the heat generated after the quench. In the two limits where the sine-Gordon model maps to hard-core bosons and free massive fermions we provide the exact solutions for the quench dynamics and discuss the finite temperature generalizations.
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.
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.