Interaction induced Anderson transition in a kicked one dimensional Bose gas

Abstract

We investigate the Lieb-Liniger model of one-dimensional bosons subjected to periodic kicks. In both the non-interacting and strongly interacting limits, the system undergoes dynamical localization, leading to energy saturation at long times. However, for finite interactions, we reveal an interaction-driven transition from an insulating to a metallic phase at a critical kicking strength, provided the number of particles is three or more. Using the Bethe Ansatz solution of the Lieb-Liniger gas, we establish a formal correspondence between its dynamical evolution and an Anderson model in $N$ spatial dimensions, where $N$ is the number of particles. This theoretical prediction is supported by extensive numerical simulations for three particles, complemented by finite-time scaling analysis, demonstrating that this transition belongs to the orthogonal Anderson universality class.

Figures (3)

• Figure 1: Upper panel: time evolution of the total energy of the three particles for different values of the stochasticity parameter $K$ at finite interaction strength $c=10$. Close to the critical point $K_c/\hbar_e\simeq 2.1$ (green curve) anomalous diffusion is observed with exponent $2/3$ (black dashed line). For $K=1.5\hbar_e<K_c$ (blue curve) the energy growth saturates as expected in the localized regime. Above the critical point $K=3.2\hbar_e>K_c$ (red curve) the dynamics is diffusive up to some time where the evolution is sensitive to finite size effects. Lower panel: phase diagram of the dynamical phases in the $c$-$K$ plane. The circles correspond to numerical determination of the critical line while the color code is just a guide to the eyes based on a fit of the data. Here $\hbar_e=2.89$, $N=3$ and $N_s=29$. The white arrow indicates the path we follow to analyse the transition at $c=10$.
• Figure 2: Finite time scaling applied to the numerical results with $c=10$ and $N=3$. The time evolution of $\langle E \rangle$ is computed as a function of time, from 20 to 200 kicks, for several values of $K/\hbar_e$ between $K/\hbar_e=0.4$ to $K/\hbar_e=2.9$. The scaling function $\ln\Lambda$ (panel a), with $\Lambda=\langle E\rangle/t^{2/3}$, displays a lower branch (blue) associated with the localized regime and an upper branch (red) associated with the diffusive regime. The continuous curve is a fit using a Taylor expansion of the scaling function up to fourth order and a critical exponent $\nu=1.56$. The dependence of the scaling parameter $\xi(K)$ is displayed on panel b) and shows a divergent behavior around the critical point $K_c/\hbar_e=2.13$. The continuous light-blue curve is a fit using a critical exponent $\nu=1.56$ (see text).
• Figure 3: (a) The rescaled quantity $\ln\Lambda=\ln[\langle E(t) \rangle t^{-2/3}]$ as a function of $K$ for different times between $t=5$ (light blue) and $t=150$ (dark blue). All curves intersect approximately at the critical point $(K_c/\hbar_e\simeq 2.1,\ln\Lambda_c\simeq 1.54)$ demonstrating the existence of a metal-insulator transition. (b) Determination of the critical exponent $\nu$ by fitting $(\ln\Lambda)'(K_c,t)\sim t^{1/3\nu}$ in log-log scale. Parameters are the same as in Fig. \ref{['fig_scaling']}.