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arXiv:2208.08406 · Physical review. B./Physical review. B

Finite-temperature dynamics in gapped one-dimensional models in the sine-Gordon family

Márton Kormos, Dániel Vörös, Gergely Zaránd

Published Nov 29, 2022• 17 citations
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Abstract

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.

DOI
10.1103/physrevb.106.205151

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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.
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.
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.
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 study Family-Vicsek scaling of charge, spin, and energy fluctuations in the one-dimensional Hubbard model at infinite temperature. Using a quantum generating function approach, we compute time-dependent cumulants of transferred conserved quantities and analyze how the corresponding roughness depends on subsystem size and time. We start by focusing on a single interface at half the chain and determine the transport exponents. Then we turn to fluctuations of a small finite interval and study the Family-Vicsek universality of fluctuations over an extended timescale. We find that the long-time scaling behavior is controlled by integrability. In the free limit, charge, spin, and energy all display ballistic transport. In the interacting integrable Hubbard chain, charge and spin cross over to a KPZ scaling regime, while the energy sector remains ballistic. Once integrability is broken by a next-nearest-neighbor density interaction, the long-time dynamics becomes diffusive in all sectors. In every case we also observe a short-time microscopic regime with apparently universal ballistic growth before the hydrodynamic scaling window sets in. The Family-Vicsek setup allows us to determine the growth, the saturation as well as the dynamical exponents.
We study the dissipative dynamics of one-dimensional fermions, described in terms of the sine-Gordon model in its free massive boson or semi-classical limit, while keeping track of forward scattering processes. The system is prepared in the gapped ground state, and then coupled to environment through local currents within the Lindblad formalism. The heating dynamics of the system is followed using bosonization. The single particle density matrix exhibits correlations between the left and right moving particles. While the density matrix of right movers and left movers is translationally invariant, the left-right sector is not, corresponding to a translational symmetry breaking charge density wave state. Asymptotically, the single particle density matrix decays exponentially with exponent proportional to -\gamma t|x|\Delta^2 − γ t | x | Δ 2 where \gamma γ and \Delta Δ are the dissipative coupling and the gap, respectively. The charge density wave order parameter decays exponentially in time with an interaction independent decay rate. The second R'enyi entropy grows linearly with time and is essentially insensitive to the presence of the gap.