Dynamical emergence of a Kosterlitz-Thouless transition in a disordered Bose gas following a quench

We study the dynamical evolution of a two-dimensional Bose gas after a disorder potential quench. Depending on the initial conditions, the system evolves either to a thermal or a superfluid state. Using extensive quasiexact numerical simulations, we show that the two phases are separated by a Kosterliz-Thouless transition. The thermalization time is shown to be longer in the superfluid phase, but no critical slowing down is observed at the transition. The long-time phase diagram is well reproduced by a simple theoretical model. The spontaneous emergence of Kosterlitz-Thouless transitions following a quench is a generic phenomenon that should arise both in the context of nonequilibrium quantum gases and in the context of nonlinear classical wave systems.

Figure 1

Coherence function $g_1(x=0,y,t)$ vs time of a 2D Bose gas after a disorder quench for two sets of initial conditions (the first and last time steps are indicated in each plot) (a) $\gamma m=0.09, g{\rho }_0=1$, and $k_0=0$ ($\gamma m/g{\rho }_0\ll 1$): shortly after the quench, $g_1$ decays algebraically, Eq. (3). At a short time, the gas lies in a prethermal regime: the algebraic exponent $\alpha \left(t\right)\simeq \gamma m/4\pi g{\rho }_0$ (dashed line), then decreases slowly, and $g_1$ is limited by a Lieb-Robinson bound. At a long time, thermal equilibrium establishes: $\alpha \left(t\right)\to \alpha \left(\infty \right)$ and $g_1$ is only limited by the system size. The inset shows the full evolution of $\alpha \left(t\right)$. (b) $\gamma m=0.09, g{\rho }_0=0.003$, and $k_0=0.754$ ($\gamma m/g{\rho }_0\gg 1$): shortly after the quench, $g_1$ acquires an exponential shape, Eq. (4), with a correlation length $\mathcal{L}\left(t\right)$ slowly increasing with time. All data are averaged over 128 disorder realizations and over system sizes $L\in [75,100]$ (a) and $L\in [350,400]$ (b). Space and time units are explained in the main text.

Figure 2

Equilibrium phase diagram reached by the 2D Bose gas a long time after the disorder quench, deduced from the spatial decay of $g_1(\mathit{r},t\to \infty )$. Each cross symbol corresponds to a given set $(k_0^2/2m,\gamma m,g{\rho }_0)$ with $\gamma m$ ranging from 0.01 to 0.49 and $k_0$ from 0 to 0.9. Here $g{\rho }_0=0.1$ is fixed, and we use a system size $L=400$ and 64 disorder realizations for averaging. The dashed curves indicate the points of constant $k_0$ values, and the solid curve is the theoretical prediction for the Kosterlitz-Thouless transition line $(\gamma m,T_{\mathrm{KT}})$, Eq. (5).

Figure 3

(a) Inverse of the algebraic exponent $\alpha \left(\infty \right)$ on the superfluid side of the KT transition, extracted from $g_1(\mathit{r},t\to \infty )$ for different temperatures and disorder strengths. The plot suggests $\alpha \simeq 1/4$ near the transition, irrespective of $\gamma$. The dashed line pinpoints ${\alpha }^{-1}\left(\infty \right)=4$. (b) Correlation length $\mathcal{L}\left(\infty \right)$ extracted from $g_1(\mathit{r},t\to \infty )$ on the normal side, for different temperatures and disorder strengths. In both plots parameters are the same as in Fig. 2, namely $g{\rho }_0=0.1, L=400$, and we average over 64 disorder realizations. The critical temperature $T_{\text{KT}}$ is found numerically by taking the average of the two extreme points lying in the critical area in Fig. 2. The dashed curve is a fit to Eq. (6).

Figure 4

Thermalization time vs temperature for three disorder strengths. Parameters are the same as in Fig. 2 except the system size $L=300$.