Mobility Crossover in Two-Dimensional Berry Crystals

Abstract

A Berry crystal is a random superposition of N plane waves of equal amplitude and fixed wavevector magnitude, propagating in different directions. Using numerical simulations of wavepacket dynamics, spectral analysis based on autocorrelation functions, and scaling of Inverse Participation Ratio, the nature of eigenstates across the energy spectrum of a two-dimensional Berry crystal is characterized. It exhibits Anderson localization and critical (extended but non-ergodic) states, reminiscent of quasicrystals, which sit in the middle ground between periodic and disordered systems and can host critical states. However, in contrast to quasicrystals that display sharp mobility edges separating extended and localized phases, the Berry crystal exhibits an extended regimes of critical states. We name this a "mobility crossover". At weak potential strength, low-energy states are extended while higher-energy states near the backscattering momentum are critical. As the potential strength increases to become comparable with the recoil energy, these critical states evolve into localized states, yielding a transition from extended non-ergodic to localized behavior near the backscattering momentum. An estimate for the boundaries of ergodic extended regimes is given by the mapping onto an effective Anderson model on a Bethe Lattice. The results shed light on the relation between backscattering and Anderson localization in continuous two-dimensional aperiodic systems.

Figures (4)

• Figure 1: (a) Spectrum obtained from $\beta = 0.60, N=100$ using a Gaussian wavepacket as initial state. The simulation time is taken to be $T'=11000$. The red solid line represents the energy spectrum with integration time window $T'/3$, while the blue solid line corresponds to full time window $T'$. Ergodic states live in the part of spectrum that is invariant as the size of time window increases. The dashed line at $E_c = 0.7$ is a guide for eye for the transition from the ergodic regime to the non-ergodic regime. The inset shows part of the 2D random potential $V(x,y)$ generated by the superposition of the $N=100$ plane waves, in a $20a \times 20a$ window. Energy eigenstates obtained from wavepacket dynamics at energies of $\epsilon = 0.66, 1.00$ are plotted in (b) and (c), and the bars have lengths of $10a$. They correspond to extended and critical states respectively. Critical state has regions with high probability density, but they do not spread over the whole window. Extended state has regions with high probability density spread uniformly. The resolved eigenstates from Gaussian wavepacket have probability concentrated near origin and fade near boundaries because of memory of initial state and absorbing boundary condition.
• Figure 2: Localization properties of states with different energies for different potential strength $\beta$ can be inferred from scaling of IPRs. $\ln(\overline{P^{-1}})$ are plotted agaist $\ln L$ with $L$ ranging from 200 to 600. Top row shows scaling relation for $\beta=0.4$ and $N=100$. (a)Extended, target energy $E=0.85$ scaling exponent $\gamma=2.01\pm0.03$. (b)Critical, target energy $E=1.0$ scaling exponent $\gamma=1.52\pm0.08$. (c)Extended, target energy $E=1.4$ scaling exponent $\gamma=1.97\pm0.02$. Bottom row shows scaling relation for $\beta=0.8$ and $N=100$. (d)Extended, target energy $E=0.5$ scaling exponent $\gamma=1.89\pm0.04$. (e)Localized, target energy $E=0.9$ scaling exponent $\gamma=-0.06\pm0.14$. (f)Critical, target energy $E=1.0$ scaling exponent $\gamma=1.11\pm0.15$.
• Figure 3: Eigenstates in various energy ranges and potential strength are plotted, and the bars have lengths of $10a$. Top row shows the probability density of energy eigenstates for $\beta=0.4$ and $N=100$. (a) States are extended at $\epsilon=0.5$. (b) States are critical at $\epsilon=0.85$. There are regions with high probability density, but they do not spread over the whole space. Top row shows that $\beta=0.8$ and $N=100$. (c) States are localized at $\epsilon=0.9$. Probability density is concentrated in small regions, which do not percolate. (d) States are critical at $\epsilon=1.2$.
• Figure 4: Phase diagram of the localization transition in the energy-disorder plane ($E$, $\beta$). The color map displays the fractal dimension $D_2$, extracted from the finite-size scaling of the IPR, $P^{-1} \sim L^{-D_2}$. The extended phase is characterized by $D_2 \approx 2$ (yellow regions), while $D_2 < 2$ (blue to green gradients) indicates the onset of critical or localized behavior. For weak potential ($\beta \leq 0.6$), a transition from low energy extended states to critical states occurs near $E = 0.9$. For $\beta \geq 0.8$, a transition from low energy extended states to critical states occurs near $E = 0.7$. The superimposed red dashed line marks the theoretical estimate of the ergodic transition based on the mapping onto a Cayley tree in momentum space. The V-shaped dip centered at $E \approx 1$ highlights the back-scattering resonance, where the system is most susceptible to localization. The red circles are eigenstates presented in figure \ref{['fig3']}.