Transport, multifractality, and the breakdown of single-parameter scaling at the localization transition in quasiperiodic systems

There has been a revival of interest in localization phenomena in quasiperiodic systems with a view to examining how they differ fundamentally from such phenomena in random systems. Motivated by this, we study transport in the quasiperiodic, one-dimensional Aubry-Andre model and its generalizations to two and three dimensions. We study the conductance of open systems, connected to leads, as well as the Thouless conductance, which measures the response of a closed system to boundary perturbations. We find that these conductances show signatures of a metal-insulator transition from an insulator, with localized states, to a metal, with extended states having (a) ballistic transport (one dimension), (b) superdiffusive transport (two dimensions), or (c) diffusive transport (three dimensions); precisely at the transition, the system displays subdiffusive critical states. We calculate the $\beta$ function $\beta \left(g\right)=dln\left(g\right)/dln\left(L\right)$ and show that, in one and two dimensions, single-parameter scaling is unable to describe the transition. Furthermore, the conductances show strong nonmonotonic variations with $L$ and an intricate structure of resonant peaks and subpeaks. In one dimension the positions of these peaks can be related precisely to the properties of the number that characterizes the quasiperiodicity of the potential; and the $L$ dependence of the Thouless conductance is multifractal. We find that, as dimension increases, this nonmonotonic dependence of $g$ on $L$ decreases and, in three dimensions, our results for $\beta \left(g\right)$ are reasonably well approximated by single-parameter scaling.

Figure 1

Length-dependent conductances and multifractality in the Auby-Andre model in one dimension and its 2D and 3D generalizations. (a) Semilog plots vs $L$ of the conductances $g_{\mathrm{T}}$ (Thouless) and $g_{\mathrm{L}}$ (Landauer) at illustrative values of $V$ in the metallic ($V=0.95$) and insulating ($V=1.05$) regimes in one dimension. On the metallic side, both $g_{\mathrm{T}}$ and $g_{\mathrm{L}}$ show nonmonotonic (roughly speaking, small-wavelength) fluctuations about an $L$-independent mean value. On the insulating side, $g_{\mathrm{T}}^2\propto g_{\mathrm{L}}$; both $g_{\mathrm{L}}$ and $g_{\mathrm{T}}$ decay exponentially with $L$, the latter only on average because $g_{\mathrm{T}}$ still displays nonmonotonic fluctuations (enlarged view in the inset). (b) Log-log plots vs $L$ of the Thouless conductance $g_{\mathrm{T}}$, at the $1d$ critical point $V=1$, showing an average decay (dashed red line) with $g_{\mathrm{T}}\propto L^{\alpha }$ and $\alpha \simeq -1.8$, with hierarchically organized peaks, whose heights also decay as a power of $L$ but with different exponents (for notational simplicity denoted generically by $\alpha$), which depend on ${\mathcal{S}}_{L_1,L_2}$, the set of peaks at the lengths $L_{i+1}=L_i+L_{i-1}$, with the seed lengths $L_1$ and $L_2$; for the illustrative sets ${\mathcal{S}}_{1,1}$ (green filled circles) and ${\mathcal{S}}_{2,2}$ (blue filled circles) we obtain the decay exponents $\simeq -0.95$ and $\simeq -1.22$, respectively. (c) Log-log plots vs $L$ of the Landauer conductance $g_{\mathrm{L}}$, at the 1D critical point $V=1$, showing an average decay $g_{\mathrm{L}}\propto L^{\alpha \left(E\right)}$, with energy-dependent exponents $\alpha \left(E\right)$, shown for the representative energies $E_1=0[\alpha \left(E_1\right)\simeq -1.01]$, $E_2=1.98496[\alpha \left(E_2\right)\simeq -1.0]$, and $E_3=0.18906032[\alpha \left(E_3\right)\simeq -2.2$; see text]; note the nonmonotonic fluctuations about these mean-decay lines. The grey curve shows the Thouless conductance ($g_{\mathrm{T}}^{\mathrm{typ}}$) obtained by using the typical level spacing (see text) at $E_1=0$; its overall slope is close to that of the $g_{\mathrm{L}}$; $g_{\mathrm{T}}^{\mathrm{typ}}$ has been scaled to fall on top of $g_{\mathrm{L}}$. (d) This nonmonotonicity in log-log plots of $g_{\mathrm{T}}\left(L\right)$ vs $L$ persists in two and three dimensions (inset), as we show by illustrative data at the the metal-insulator critical points; in two dimensions, the critical $g_{\mathrm{T}}\left(L\right)\sim L^{-0.21}$ exhibits an overall subdiffusive scaling. (e) A fractal analysis of the $L$ dependence of the energy-averaged $g_{\mathrm{T}}$, i.e., $g_{\mathrm{T}}^{\infty }$ (see the Supplemental Material [32] Sec. S1 2 and the main text) reveals multifractal scaling of the nonmonotonic variations of $g_{\mathrm{T}}\left(L\right)$ at the critical point. (f) A plot of the singularity spectrum $f\left(\alpha \right)$ vs $\alpha$ corroborates this multifractality (see main text); note that the singularity spectrum narrows on the metallic side $V<1$.

Figure 2

Open-system conductance in two and three dimensions. (a) $g_{\mathrm{K}}(E=0,L)$ for insulating ($V=1.05$), critical ($V=1$), and metallic ($V=1$) states in two dimensions. The solid black lines are fits of the data for the asymptotic $L$ dependence, exponential decay for the insulating state and power-law scalings for critical and metallic states. The nonmonotonicity of $g_{\mathrm{K}}\left(L\right)$ is shown in the inset. (b) $g_{\mathrm{K}}\left(L\right)$ across localized to diffusive metal transition in three dimensions. A weak nonmonotonic variations, larger than the error bars, can be seen. The solid lines are fit to the data obtained via scaling collapse, shown in (c). The inset in (c) clearly indicates the critical point at $V_c=2.22\pm 0.01$ in terms of a crossing of $g_{\mathrm{K}}$ vs $V$ curves for different $L$. (d) Conductance near $V=1/V_c\simeq 0.45$ follows superdiffusive scaling, $g_{\mathrm{K}}\sim L^{\alpha }$ with $1<\alpha <2$, as shown by the fits (solid lines) to the data points and it asymptotically approaches to the ballistic scaling deep in the metallic side, e.g., $\alpha =1.73$ for $V=0.1$ as shown.

Figure 3

$\beta$ functions in one, two, and three dimensions. (a) $\beta \left(\tilde{g}\right)$ in one dimension extracted from ${\tilde{g}}_{\mathrm{T}}$ across metal-insulator transition for various sequences ${\mathcal{S}}_{L_1,L_2}$, obtained by fitting with exponential decay and power law for $V>1$ and $V\le 1$, respectively. Same color with different symbols represents $\beta \left({\tilde{g}}_{\mathrm{T}}\right)$ calculated for different $\mathcal{S}$ and the same $V$. The red curve at the critical point corresponds to the $\beta$ function extracted from the overall behavior of $g_{\mathrm{T}}^{\mathrm{typ}}\left(L\right)$. On the insulating side to make all the curves fall on the same line we choose different microscopic conductance ($g_{\mathrm{T}}^0$) for different sequence (inset). $\beta \left(\tilde{g}\right)$ extracted in similar manner for (b) ${\tilde{g}}_{\mathrm{L}}$ in one dimension, (c) ${\tilde{g}}_{\mathrm{K}}$ in two dimensions, and (d) ${\tilde{g}}_{\mathrm{K}}$ in three dimensions (solid lines) for values of $V$ indicated in the figure panels. In (b) and (c) the straight lines for $\beta \left({\tilde{g}}_{\mathrm{L}}\right)$ in the insulating side ($V>1$) have been collapsed to a single curve by choosing an appropriate $g_{\mathrm{L}}^0\left(V\right)$, as shown for one dimension in the inset of (b). In (d) the black dashed line is the $\beta$ function calculated from the scaling collapse of Fig. 2.