Engineering Anderson Localization in Arbitrary Dimensions with Interacting Quasiperiodic Kicked Bosons

TL;DR

This work addresses realizing Anderson localization and its quantum critical behavior in higher effective dimensions by combining a minimal two-boson Lieb–Liniger system with quasiperiodic kicking in a Floquet setting. By tuning incommensurate driving frequencies, the authors induce synthetic dimensions so that the effective dimensionality is $d = N + N_\omega$, enabling localization transitions in $d \ge 3$. Finite-time scaling of the energy growth reveals Anderson transitions in $d=3$ and $d=4$ with critical exponents $\nu$ and $s$ matching the orthogonal universality class and satisfying Wegner’s scaling, demonstrating universality across dimensions. The approach provides a versatile, experimentally accessible platform to study Anderson localization and quantum criticality in arbitrary dimensions, with potential extensions to other symmetry classes and to multifractal physics at criticality.

Abstract

We study the interplay of interactions and quasiperiodic driving in the Lieb-Liniger model of one-dimensional bosons subjected to a sequence of delta kicks. Building on the known mapping between the kicked rotor and the Anderson model, we show that both interparticle interactions and quasiperiodic modulations of the kicking strength can independently and simultaneously generate synthetic dimensions. In the absence of modulation, interactions between two bosons already promote an effective two-dimensional Anderson model. Introducing one or two additional incommensurate frequencies further extends the system to three and four effective dimensions, respectively. Through extensive numerical simulations of the two-body dynamics and finite-time scaling analysis, we observe Anderson localization and the associated critical behavior characteristic of the orthogonal universality class. This combined use of interactions and quasiperiodic driving thus provides a versatile framework for emulating Anderson localization and its transition in arbitrary dimensions.

Figures (5)

• Figure 1: Time evolution of the total energy of the two particles for different values of the stochasticity parameter $K$ at finite interaction strength $c=10$ and fixed $\varepsilon=0.3$ with a) $N_\omega=0$ and b) $N_\omega=2$. In the former case a), no transition is observed and energy always saturates with different plateau depending on $K$ (here $K/\hbar_e=0.7,\,1.3,\,2.3$) as expected from scaling theory of localization. In the latter case b) a transition is observed. Close to the critical point $K/\hbar_e=1.3$ (green curve) anomalous diffusion is observed with exponent $1/2$ (black dashed line) consistent with four dimensional scaling. For $K=0.7\hbar_e<K_c$ (blue curve) the energy growth saturates as expected in the localized regime. Above the critical point $K=2.3\hbar_e>K_c$ (red curve) the dynamics is diffusive.
• Figure 2: The rescaled quantity $\ln\Lambda=\ln\left[\langle E(t)\rangle/t^{2/d}\right]$ as a function of $K$ for different times between $t=10$ and $t=200$. All curves intersect approximately at the critical point demonstrating the existence of a metal-insulator transition and allowing to locate the critical point $K_c$. Here $c = 10$, $\varepsilon = 0.3$ and $N_\omega=1$ for the left figure ($d=3$) and $N_\omega=2$ for the right one ($d=4$).
• Figure 3: Finite time scaling applied to the numerical results with $c=10$, $N=2$ and $\varepsilon=0.3$ for $N_\omega=0$ (a), $N_\omega=1$ (b) and $N_\omega=2$ (c). The time evolution of $\langle E \rangle$ is computed as a function of time, from 5 to 150 kicks, for several values of $K$ (see color bars). The scaling function $\ln\Lambda$, with $\Lambda=\langle E\rangle/t^{2/d}$, displays a lower branch (blue) associated with the localized regime and an upper branch (red) associated with the diffusive regime for $N_\omega\ge 1$ as a function of $\ln(\xi/t^{1/d})$. The continuous curves are fits using a Taylor expansion \ref{['eq:expansion_loglambda']} of the scaling function up to fourth order with the corresponding critical exponents (see text). Dashed lines are the expected asymptotic behaviors.
• Figure 4: Determination of the critical exponents $\nu$ and $s$ by fitting $(\ln\Lambda)'(K_c,t)$ as function of $t$ in log-log scale. Here $c=10$, $\varepsilon=0.3$ and $N_\omega=1,2$ for a) and b).
• Figure 5: Phase diagram of the dynamical phases in the $K$-$\varepsilon$ plane for two bosons with $c=10$. The color code corresponds to the derivative of $\ln\Lambda(t)$ at large time $t=170$ for a) $N_\omega=1$ and b) $N_\omega=2$. The red curve is the critical line where this derivative is zero.