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Bulletin of the American Physical Society

Finite-temperature dynamical structure factor of the one-dimensional Bose gas: From the Gross-Pitaevskii equation to the Kardar-Parisi-Zhang universality class of dynamical critical phenomena

Austen Lamacraft, Manas Kulkarni

Published Mar 4, 2014• 1 citation
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We study the finite-temperature dynamical structure factor $S(k,\ensuremath{\omega})$ of a one-dimensional Bose gas using numerical simulations of the Gross-Pitaevskii equation appropriate to a weakly interacting system. The line shape of the phonon peaks in $S(k,\ensuremath{\omega})$ has a width $\ensuremath{\propto}$∣k∣3/2{|k|}^{3/2}∣k∣3/2 at low wave vectors. This anomalous width arises from resonant three-phonon interactions, and reveals a remarkable connection to the Kardar-Parisi-Zhang universality class of dynamical critical phenomena.
Finite temperature spin transport in integrable isotropic spin chains is known to be superdiffusive, with dynamical spin correlations that are conjectured to fall into the Kardar-Parisi-Zhang (KPZ) universality class. However, integrable spin chains have time-reversal and parity symmetries that are absent from the KPZ (Kardar-Parisi-Zhang) or stochastic Burgers equation, which force higher-order spin fluctuations to deviate from standard KPZ predictions. We put forward a nonlinear fluctuating hydrodynamic theory consisting of two coupled stochastic modes: the local spin magnetization and its effective velocity. Our theory fully explains the emergence of anomalous spin dynamics in isotropic chains: it predicts KPZ scaling for the spin structure factor but with a symmetric, quasi-Gaussian, distribution of spin fluctuations. We substantiate our results using matrix-product states calculations.
We theoretically investigate the time dependence of the first-order coherence function for a one-dimensional driven dissipative nonequilibrium condensate. Simulations on the generalized Gross-Pitaevskii equation show that the characteristic time scale of exponential decay agrees with the linearized Bogoliubov theory in the regime of large interaction energy. For very weak interactions, the temporal correlation deviates from the linear theory, and instead respects the dynamic scaling of the Kardar-Parisi-Zhang universality class. This nonlinear dynamics is found to be quantitatively captured by a noisy Kuramoto-Sivashinsky equation for the phase dynamics.