Anomalous transport in quasiperiodic lattices: emergent exceptional points at band edges and log-periodic oscillations

Abstract

Quasiperiodic systems host exotic transport regimes that are distinct from those found in periodic or disordered lattices. In this work, we study quantum transport in the Aubry-André-Harper lattice in a two-terminal setup coupled to zero-temperature reservoirs, where the conductance is evaluated via the nonequilibrium Green's function method. In the extended phase, we uncover a universal subdiffusive transport when the bath chemical potential aligns with the band edges. Specifically, the typical conductance displays a scaling of $\mathcal{G}_{\text{typ}}\sim L^{-2}$ with system size $L$. We attribute this behavior to the emergence of an exceptional point (Jordan normal form) in the transfer matrix in the thermodynamic limit. In the localized phase, the conductance shows exponential decay governed by the Lyapunov exponent. Intriguingly, in the critical phase, we identify pronounced log-periodic oscillations of the conductance as a function of system size, arising from the discrete scale invariance inherent to the singular-continuous spectrum. We further extend our analysis to the generalized Aubry-André-Harper model and provide numerical evidence suggesting that the exact mobility edge resides within a finite spectral gap. This results in a counter-intuitive exponential suppression of conductance precisely at the mobility edge. Our work highlights the distinct transport behaviors in quasiperiodic systems and elucidates how they are rigorously dictated by the underlying local spectral structure.

Figures (6)

• Figure 1: Transport in the extended phase of the AAH model. (a)--(c) Typical conductance $\mathcal{G}_{\text{typ}}$ as a function of system size $L$ (up to Fibonacci number $F_{24}$) for the AAH model with $V=0.5$. The panels correspond to energies within the band ($E_{\text{in}}$), in the gap ($E_{\text{out}}$), and at the band edge ($E_{\text{edge}}$), respectively. Two representative energies are shown for each case, and data are geometrically averaged over 500 samples. Dashed lines indicate analytical fits using Eq. \ref{['G_loc']} in (b) and Eq. \ref{['G_edge']} in (c), where slopes are presented. Target energies at band edges were identified using the periodic approximation ($b \approx F_{20}/F_{21}$). The specific energy values are: $E_{\text{in}} \in \{0, 1.8398\}$, $E_{\text{out}} \in \{0.8163, 1.6816\}$, and $E_{\text{edge}} \in \{0.3351, 1.7433\}$. (d1)--(d2) Linear growth of the off-diagonal transfer matrix element $T_{12}$ of the transfer matrix [see its definition in Eq. (\ref{['tmatrix']})] at the two selected band-edge energies. The insets display the half-trace of the transfer matrix for $L=F_{24}$, with the positions of $E_{\text{edge}}$ marked by vertical dashed lines.
• Figure 2: Transport in the localized phase of the AAH model. Finite-size scaling of the typical conductance $\mathcal{G}_{\text{typ}}$ as a function of system size $L$ (up to $F_{24}$) is performed. $V=1.5$. (a) Energies within the spectrum ($E_{\text{in}}$). The data for different energies are slightly shifted vertically for clarity. The decay rate is governed by the analytic Lyapunov exponent $\lambda = \ln V$. (b) Energies in the gaps ($E_{\text{out}}$). The decay rate is governed by the energy-dependent Lyapunov exponent calculated numerically. Two representative energies are presented for each case, and data are geometrically averaged over 500 samples. Dashed lines denote exponential fits according to Eq. \ref{['G_loc']}. For $E_{\text{out}}$, the Lyapunov exponents are computed via the transfer matrix method with 5000 samples and $\sigma=0.0003$. The specific energy values used in the plot are: $E_{\text{in}} \in \{-0.0001, 0.3834\}$ and $E_{\text{out}} \in \{1.3179, 2.7339\}$.
• Figure 3: Transport in the critical phase of the AAH model. Finite-size scaling of the typical conductance $\mathcal{G}_{\text{typ}}$ at the critical point ($V=1$) is shown. Four representative in-band energies ($E_{\text{in}}$) are analyzed. Data are geometrically averaged over 5000 samples. The maximum system size is $L=F_{24}$ for $E_{\text{in},1}$ and $L=F_{21}$ for the other cases. Data points corresponding to Fibonacci lengths ($L=F_n$) are highlighted with yellow dots. For $E_{\text{in},4}$, a power-law fit yields an exponent $\alpha \approx 2.24$ (dashed line). The specific energies and their corresponding symbolic codes are: (a) $E_{\text{in},1}=0$ ($\{000000\dots\}$); (b) $E_{\text{in},2}=0.1719$ ($\{010101\dots\}$); (c) $E_{\text{in},3}=0.1874$ ($\{011011\dots\}$); and (d) $E_{\text{in},4}=0.1891$ ($\{011111\dots\}$).
• Figure 4: Stroboscopic analysis and gap transport in the critical phase. (a) Stroboscopic finite-size scaling of the typical conductance for the band-center state $E_{\text{in},1}$. The conductance is plotted for selected subsequences of system sizes that share the same inflation factor ($\lambda \approx \phi^3$): $L \in \{F_{3n}\}$, $\{F_{3n+1}\}$, and a geometric progression $L \approx 11\phi^{3n}$. Dashed lines indicate power-law fits according to Eq. \ref{['G_crit']}. Inset: Probability distribution of the scaling exponent $\alpha$, obtained from an ensemble of length subsequences satisfying the $\phi^3$ scaling relation. Each fit utilizes at least three data points. The distribution centers around a mean value $\langle \alpha \rangle \approx 1.03$. (b) Exponential decay of the typical conductance for energies within spectral gaps ($E_{\text{out}}$). Dashed lines represent fits to Eq. \ref{['G_loc']}, where the Lyapunov exponents are calculated numerically with $\sigma=0.0003$ and 5000 samples. The specific gap energies are $E_{\text{out},1}=1.0316$ and $E_{\text{out},2}=2.1580$.
• Figure 5: Spectrum and mobility edge transport in the GAAH model. (a) Energy spectrum as a function of the quasiperiodic potential strength $V$, calculated via exact diagonalization. Parameters are set to $U=0.5$, $\theta=0$, and $b=(\sqrt{5}-1)/2$. Blue regions correspond to the absolutely continuous spectrum hosting extended states, while red regions correspond to the pure point spectrum hosting localized states. The solid black line marks the analytical mobility edge. (b) Finite-size scaling of the typical conductance $\mathcal{G}_{\text{typ}}$ for three distinct energies located exactly at the mobility edge. Results are geometrically averaged over 500 samples. Dashed lines indicate exponential fits according to Eq. \ref{['G_out']}, using Lyapunov exponents computed numerically ($\sigma=0.0003$, 5000 samples). Inset: Finite-size scaling of the typical spectral gap $\Delta_{\text{typ}}$ in the vicinity of the selected mobility edge energies. The specific energy values are $E_{c,1}=1.84$, $E_{c,2}=0.24$, and $E_{c,3}=-2.32$.