Robust Correlation-Induced Localization Under Time-Reversal Symmetry Breaking

Abstract

We study Anderson localization in a one-dimensional disordered system with long-range correlated hopping decaying as $1/r^{a}$ with complex hopping amplitudes that break time-reversal symmetry in a tunable fashion by varying their argument. We find analytically a corelation-induced algebraic localization that is robust to a finite strength of the time-reversal-symmetry-breaking parameter, beyond which all states delocalize. This establishes a localization--delocalization transition driven by the interplay between long-ranged correlated hopping and time-reversal symmetry breaking. In addition to obtaining the static localization phase diagram, we also investigate the dynamical phase diagram through the lens of wavepacket spreading. We find that the growth in time of the mean-squared displacement of a wavepacket, which is subdiffusive for the time-reversal symmetric case, becomes diffusive for any finite value of the time-reversal-symmetry-breaking parameter.

Figures (10)

• Figure 1: Phase diagram in the $\theta$–$a$ parameter plane. The system exhibits a localization-delocalization transition with the delocalized phase (I) appearing for $|\theta| \geqslant \pi a/2$, $a<1$. Further, the localized phase exhibits two flavors [labeled (II) and (III)] characterized by different power-law decays of the eigenstates wavefunctions. The color map indicates the power-law exponent $s$ from Eq. \ref{['eq:decay-profile']} across the parameter space.
• Figure 2: Spatial decay profile of eigenstates in the bulk of the spectrum. The eigenstates exhibit algebraic localization, as specified in Eq. \ref{['eq:decay-profile']}, (a-c) which collapses for all system sizes(darker colors represent larger system sizes), and (d) become delocalized for $\theta > \pi a/2$ and $a<1$. Black dashed lines show the analytical predictions of Eq. \ref{['eq:decay-profile']}. The data are averaged over 500 disorder realizations for all system sizes.
• Figure 3: Spectrum of the momentum-space hopping term $\tilde{E}_p$ for $a=0.5$. (a) For $\theta > \pi a/2$, the spectrum diverges in both directions, whereas (b) for $\theta < \pi a/2$, it diverges in only one direction. (Inset) Spectrum of the $M$ matrix, which is bounded from both directions.
• Figure 4: Dynamics of the width of a wave-packet starting from an initially localized state. The panels show the wave-packet dynamics for $a = 0.25$ at different values of $\theta$, with the darkest color corresponding to $N=8192$: panel (a) (Regime-II with $\theta = 0$) displays an intermediate-time regime with $2\beta = \frac{1}{2-a}|_{a=0.25} \approx 0.58$, panels (b) and (c) (Regime-II with $0<\theta<\pi a/2$) exhibit diffusive dynamics between the ballistic front and saturation, and panel (d) ($\theta = 2\pi a/3$) corresponds to the delocalized phase (I), where the intermediate regime is also diffusive. The data is averaged over 500 disorder realizations for $N \leq 1024$ and 100 otherwise.
• Figure 5: Schematic illustration of the dynamical windows (top) and their associated scaling exponents (bottom). The top panel shows a schematic diagram of the different regimes in the dynamics of the wave-packet width: the red, black, and blue dashed lines represent the ballistic, intermediate (sub-)diffusive, and saturation regimes, respectively, while the green and blue vertical dashed lines mark the crossover timescales $\tau \sim N^{\lambda}$ and $t_{\rm sat} \sim N^{\delta}$ between these windows. The table reports the corresponding exponents $\Omega$, $2\beta$, $\gamma$, $\lambda$, and $\delta$ governing the dynamics in different windows.