Mobility edges and fractal states in quasiperiodic Gross-Pitaevskii chains

TL;DR

The paper investigates localization in a nonlinear Aubry-André chain realized as a Gross-Pitaevskii lattice, revealing branching mobility edges and fractal states in the Bogoliubov-de Gennes spectrum. Ground states are computed via constrained gradient descent, and BdG excitations are analyzed; in the high-density limit, the BdG problem maps to an extended Harper model, enabling a tight-binding interpretation with density-dependent hopping. Key findings include nontrivial mobility-edge branching caused by delocalized phase tongues penetrating localized regions and the robust delocalization of low-energy phonons, mediated by an effective low-energy theory that renormalizes the Aubry-André potential. The work highlights rich nonlinear and interaction-driven modifications of localization in quasiperiodic lattices, with implications for optical lattices and many-body localization scenarios in interacting bosonic systems.

Abstract

We explore properties of a Gross-Pitaevskii chain subject to an incommensurate periodic potential, i.e., a nonlinear Aubry-Andre model. We show that the condensate crucially impacts the properties of the elementary excitations. In contrast to the conventional linear Aubry-Andre model, the boundary between localized and extended states (mobility edge) exhibits nontrivial branching. For instance, in the high-density regime, tongues of extended phases at intermediate energies penetrate the domain of localized states. In the low-density case, the situation is opposite, and tongues of localized phases emerge. Moreover, intermediate critical (fractal) states are observed. The low-energy phonon part of the spectrum is robust against the incommensurate potential. Our study shows that accounting for interactions, already at the classical level, lead to highly nontrivial behavior of the elementary excitation spectrum.

Figures (7)

• Figure 1: The ground state amplitudes (we use blue lines instead of discrete dots for presentation purposes) for the chain size $N=2584$. Here $a=3$ with (a) $W=1$, (b) $W=3$, and (c) $W=10$. In the insets, we compare $G_l$'s and the linear response estimations \ref{['gs']} (dashed orange lines). Upon $W$ growth, deviations from the linear response become apparent.
• Figure 2: In the nonlinear Aubry-André model \ref{['ham1']}, $W_\textrm{NL}$ serves as an indicator when localized modes emerge upon $W$ growth. It depends on the particle density $a$. Here we plot it for low- (a) and high- (b) density regimes. In (a), blue and red curves correspond to the emergence of the localized modes at the edge ($\lambda \approx 5$) and the middle ($\lambda \approx 2.5$) of the spectrum, respectively. In (b), only $W_{\textrm{NL}}$ for the edge is presented. The dashed lines are theoretical predictions \ref{['LowDEst']} and \ref{['HighDEst']}.
• Figure 3: Overview of the nonlinear Aubry-André model \ref{['ham1']} properties. In panels (a)-(d), we show elementary excitations spectra for $a = 0.01, 1, 5, 15$ and $N=2584$, respectively, and $\tau$, which indicates the corresponding participation number scaling properties. In panel (e), we present the scaling exponent and participation number $\lambda$-dependencies for $W=15$ [vertical section of panel (d)], which reveals multiple mobility edges. For the scaling analysis we used three system sizes ($N = 987, 1597, 2584)$, and $\tau$ averaged over a small energy window $\delta\lambda = 0.01$. Further insights into the model properties can be obtained using reciprocal space wave functions. In panels (f) and (g), we contrast spectra for $a=3$ with respective $W=3$ and $W=10$. In the former case, all the modes are delocalized, having well-resolved $k$-space peaks, whereas in the latter case, upon $\lambda$ growth, the localization emerges through an intermediate fractal regime. The corresponding wave functions -- extended, critical, and localized -- for $\lambda = 0.051, \, 7.9, \, \text{and} \, 11$ are shown in panels (i)-(k), respectively ($N = 2584$).
• Figure 4: For a given density $a$ (here, $a=18$) the phase diagram on the $W-\lambda$ plane has nontrivial branching due to the tongues. In the high-density regime, the tongues of the delocalized phase penetrate the localized phase domain. At the tongue ends, fractal states (green dots) can be observed. The inset shows the scaling of the participation ratio for the states indicated by labels (i)-(iii). Exponent $\tau$ reveals their respective extended, fractal, and localized nature.
• Figure 5: The ground state amplitudes for the density $a = 0.3$ and $N=2584$ (low-density regime), for different potential strengths $W = 1, 3, 10$ from (a)-(c). respectively. All notations are identical to those of Fig. \ref{['fig:gs']}.