A numerical study of the localization transition of Aubry-André type models

TL;DR

The paper investigates the localization transition in the Aubry–André model and its extension using two MPT-based tools, the geometric Binder cumulant and polarization-amplitude renormalization, with irrational modulation parameters approached via Fibonacci-ratio approximants. It finds that for irrational fillings the critical strength $W_c$ tends to zero in the thermodynamic limit, while for rational fillings $W_c$ converges to $2t$, and that spikes at $0<W/t<2$ appear at certain densities, moving toward irrational limits as system size grows. In the extended model, spikes persist but approach finite $W/t$ values due to finite-gap openings, indicating a distortion of the phase diagram. These results illuminate how rational vs irrational fillings control localization and suggest potential high-to-low conductivity switching in quasi-periodic systems.

Abstract

We use tools based on the modern theory of polarization for a numerical study of the localization transition of the Aubry-André model. In this model the spatial modulation of the potential, $α$, is an irrational number, which we approximate as the ratio of Fibonacci numbers, $F_{n+1}/F_n$, where $F_n=L$ is also the system size. We calculate the phase diagram as a function of particle density (filling) and potential strength $W$. We calculate the geometric Binder cumulant and also apply a renormalization approach. At any given finite system size we find that at many densities the transition occurs at or near $W=2t$ ($t$ denotes the hopping). This is where single particle states are known to localize. However, we also find "spikes", densites at which the transition occurs in the range $0<W<2t$. These spikes occur for densities at which there are no partially filled bands. As the system size (and both $F_n$ and $F_{n+1}$ in $α$) is increased the spikes tend towards zero, but the density at which they occur also changes slightly: they approach irrational numbers which can be written as Fibonacci ratios or sums thereof. For densities which are fixed ratios for all system sizes, the transition occurs at $W=2t$. We also study an extension of the original Aubry-André model with second nearest neighbor hoppings. This model also exhibits a distorted phase diagram compared to the original one, with spikes which do not necessarily tend to zero, but to finite values of $W$, determined by the modifed gap structure.

Figures (8)

• Figure 1: Panel (a) on the left shows the calculated phase diagram for four different system sizes, $L=377,987,1597,4181$. The predicted $W/t=2$ line is found for most fillings, but all system sizes exhibit spikes where the localization transition occurs at $W/t<2$. The right panels ((b), (c), and (d)) show the critical $W$ for three spikes as a function of system size for six different sizes, $L=377,987,1957,4181,6765,17711$. The spikes are labeled in panel (a) as I, II, and III. In the plots for the three spikes, (b), (c), and (d), the scales of all axes are logarithmic.
• Figure 2: Energy levels of the Aubry-André model for a system of size $L=610$ with periodic boundary conditions as a function of $W/t$. Only states with $\epsilon<0$ are shown, since the system is symmetric with respect inversion around $\epsilon=0$. The scale of the $W/t$ axis is logarithmic. Band gaps are found at states whose indices correspond to Fibonacci numbers or sums of Fibonacci numbers.
• Figure 3: $P(X)$, distribution of the total position $X$, for four different calculations of system size $L=987$. Panels (a) and (b) show the results for $N=737$. Panel (a)(panel (b)) shows results for $W/t=1.98$($W/t=2.02$). Panels (c) and (d) show results for $N=738$. Panel (c)(panel (d)) shows results for $W/t=1.98$($W/t=2.02$).
• Figure 4: RG flow lines. Panel (a) shows a calculation for half filling. The iterations were carried out between $L= 2584$ and $L=610$. The dashed line ($W = 2t$) indicates a repulsive fixed point. The flowlines originating below(above) $W=2t$ tend to the attractive fixed point at $W = 0$($W \rightarrow \infty$). Panel (b) shows a calculation for an irrational filling ($\lim_{n \rightarrow \infty} 2F_{n-2}/F_n$). The iterations were carried out between the two systems $L=1597$, $N=1220$ and $L=987$, $N=754$. In this case $W=0$($W \rightarrow \infty$) is a(n) repulsive(attractive) fixed point. Flow lines started at finite $W$ all tend to infinity, indicating localization.
• Figure 5: Energy levels of the extended Aubry-André model for a system of size $L=1597$ with periodic boundary conditions as a function of $W/t$. The $W/t$-axis is plotted on a logarithmic scale. $\epsilon$ on the vertical axis refers to a scaled and normalized energy eigenvalue according to $\epsilon = (E-E_{min})/(E_{max} - E_{min})$. The energy levels of three particular states, with numbers $I_{state} = 1220, 987, 610$ are plotted as thick lines. These filling numbers correspond to spikes I, II, and III in Fig. \ref{['fig:pdt2rat']}.