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

We study the finite-temperature dynamical structure factor $S(k,\omega )$ 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 $S(k,\omega )$ has a width $\propto$${\left|k\right|}^{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.

Figure 1

Dynamical structure factor $S(k,\omega )$ of a 1D Bose gas described by the Gross-Pitaevskii equation for wave vectors $k=2\pi p/L$, $p=64,32,16$ (right to left). $L=5\times 2^{13}$, and temperature $T=0.005$, with length being measured in units of the healing length, and energy in units of the chemical potential. Inset: Scaling collapse of the phonon peaks using the ansatz Eq. (1).

Figure 2

The characteristic curve $X_{+}\left(t\right)$ giving the path of a right-moving phonon wave packet is not straight due to the influence of the counterpropagating waves.

Figure 3

Dependence of linewidth on wave vector, for system sizes $N=2^{13}$ (red dots) and $N=2^{14}$ (blue crosses), with $T=0.005$. The data are almost perfectly coincident. The blue line is a fit for wave vectors $2\pi p/L$, $p=2,4,...64$ giving $z=1.510\pm 0.018$. Higher wave vectors show a marked deviation from $3/2$ scaling, and $k=2\pi /L$ is omitted because the linewidth lies below resolution. The green dashed line shows a comparison with the result of Ref. [20].