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

TL;DR

This paper investigates quantum transport in quasiperiodic Aubry-André-Harper lattices, revealing universal subdiffusive behavior at band edges, emergent exceptional points, and log-periodic oscillations linked to the system's spectral properties.

Contribution

It uncovers the role of exceptional points and discrete scale invariance in transport phenomena of quasiperiodic systems, extending understanding of mobility edges and spectral influences.

Findings

Universal subdiffusive conductance scaling as L^{-2} at band edges

Emergence of exceptional points in transfer matrix in the extended phase

Log-periodic oscillations of conductance in the critical phase

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-Andr\'e-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-Andr\'e-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.