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arXiv:2601.10056

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

Jinyuan Shang, Haiping Hu

cond-mat.mes-hallcond-mat.dis-nnquant-phSubmitted Jan 15, 2026• Announced Jan 16, 2026
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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.

Comments
11 pages, 6 figures

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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 Gtyp∼L−2\mathcal{G}_{\text{typ}}\sim L^{-2}Gtyp​∼L−2 with system size LLL. 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.
We study the single-particle properties of two-dimensional quasicrystals where the underlying geometry of the tight-binding lattice is crystalline but the on-site potential is quasicrystalline. We will focus on the two-dimensional (2D) generalized Aubry-André model which has a varying form to its quasiperiodic potential, through a deformation parameter and varied irrational periods of cosine terms, which allows a continuous family of on-site quasicrystalline models to be studied. We show that the 2D generalized Aubry-André model exhibits many single-particle mobility edges which we confirm for finite systems and supports critical states across large parameter regions. Critical states are neither fully localized nor extended. We observe that diagonal quasicrystalline models can support many energy intervals of critical states in the spectrum while stabilizing both localized and extended states in other energy intervals; we refer to these as anomalous mobility edges. We show that critical states are present independent of system size through a scaling analysis of the inverse participation ratio and that they are present in spectra that also contain extended and localized states, confirming that at least one anomalous mobility edge is present. Due to this, these models exhibit anomalous diffusion of initially localized states across the majority of parameter regions, including deep in the normally localized regime. The presence of critical states in large parameter regimes and throughout the spectrum will have consequences for the many-body properties of quasicrystals, including the formation of the Bose glass and the potential to host a many-body localized phase. Published by the American Physical Society 2024
Localization in non-Hermitian quasicrystals can differ fundamentally from its Hermitian counterpart when non-reciprocity is spatially disordered. Here we study a one-dimensional non-Hermitian Aubry-André-Harper chain with a Bernoulli imaginary gauge field and quasiperiodic onsite modulation. In the nearest-neighbor limit, we identify an anomalous transition from a fully erratic non-Hermitian skin effect (ENHSE) phase to a fully localized phase. Although the fractal dimension vanishes in both regimes, the Lyapunov exponent and the fluctuation of the eigenstate center of mass sharply distinguish them. For generic finite-size realizations, this transition is further accompanied by a complex-to-real spectral change under periodic boundary conditions and a change of spectral winding from nontrivial to trivial. With weak next-nearest-neighbor hopping, we uncover an anomalous mobility edge at the same location as in the Hermitian generalized Aubry-André-Harper model, but separating Anderson-localized states from ENHSE-type macroscopic-accumulation states rather than extended states. We further show that this anomalous localization structure is reflected in spectral winding and wave-packet dynamics: single realizations exhibit winding-dependent drift, winding-resolved averaging preserves opposite directional responses, and full disorder averaging largely restores Hermitian-like transport. Our results establish practical diagnostics of anomalous localization and mobility edges in non-Hermitian quasicrystals.
We study dephasing enhanced transport in boundary-driven quasiperiodic systems. Specifically, we consider dephasing modeled by current-preserving Lindblad dissipators acting on the noninteracting Aubry-Andr\'e-Harper and Fibonacci bulk systems. The former is known to undergo a critical localization transition with a suppression of ballistic transport above a critical value of the potential. At the critical point, the presence of nonergodic extended states yields anomalous subdiffusion. The Fibonacci model, on the other hand, yields anomalous transport with a continuously varying exponent depending on the potential strength. By computing the covariance matrix in the nonequilibrium steady state, we show that sufficiently strong dephasing always renders the transport diffusive. The interplay between dephasing and quasiperiodicity gives rise to a maximum of the diffusion coefficient for finite dephasing, which suggests that the combination of quasiperiodic geometries and dephasing can be used to control noise enhanced transport.
We investigate and map out the nonequilibrium phase diagram of a generalization of the well known Aubry-Andr\'e-Harper (AAH) model. This generalized AAH (GAAH) model is known to have a single-particle mobility edge which also has an additional self-dual property akin to that of the critical point of the AAH model. By calculating the population imbalance, we get hints of a rich phase diagram. We also find a fascinating connection between single particle wave functions near the mobility edge of the GAAH model and the wave functions of the critical AAH model. By placing this model far from equilibrium with the aid of two baths, we investigate the open system transport via system size scaling of nonequilibrium steady state (NESS) current, calculated by fully exact nonequilibrium Green's function (NEGF) formalism. The critical point of the AAH model now generalizes to a `critical' line separating regions of ballistic and localized transport. Like the critical point of the AAH model, current scales subdiffusively with system size on the `critical' line ($I\ensuremath{\sim}{N}^{\ensuremath{-}2\ifmmode\pm\else\textpm\fi{}0.1}$). However, remarkably, the scaling exponent on this line is distinctly different from that obtained for the critical AAH model (where $I\ensuremath{\sim}{N}^{\ensuremath{-}1.4\ifmmode\pm\else\textpm\fi{}0.05}$). All these results can be understood from the above-mentioned connection between states near the mobility edge of the GAAH model and those of the critical AAH model. A very interesting high temperature nonequilibrium phase diagram of the GAAH model emerges from our calculations.
We investigate localization phenomena and exact mobility edges in a quasiperiodic Aharonov-Bohm chain, where the 1D canonical diagonal and off-diagonal Aubry-André-Harper models are laterally coupled to an auxiliary sublattice threaded by a synthetic magnetic flux. By analytically computing the Lyapunov exponent via Avila's global theory of one-frequency Schrödinger operators, we derive exact expressions for mobility edges. In the diagonal limit, the coupling to the auxiliary sublattice generates hyperbolic mobility edges that exhibit a sign-changing divergence and can be continuously tuned by the magnetic flux. In the off-diagonal regime, the interference between quasiperiodic hopping and indirect tunneling through the auxiliary sublattice gives rise to exact anomalous mobility edges that separate critical and extended states. Critical states and anomalous mobility edges emerge even in the absence of incommensurately distributed zeros in the hopping modulation, a behavior distinct from that of conventional off-diagonal models. Our results establish a rigorous theoretical framework for engineering controllable mobility edges, with the synthetic flux providing a tunable experimental knob that paves the way for realizations in platforms such as superconducting quantum circuits and photonic waveguides.
We study nonequilibrium steady-state transport in a boundary driven one-dimensional fermionic lattice setup which is further subjected to particle loss. We analyze the system size scaling of conductance at zero temperature for different values of the chemical potential of the boundary reservoirs. We consider a variety of loss channel configurations: (i) single loss at the middle site of the lattice, (ii) multiple but nonextensive lossy channels, and (iii) extensive lossy channels. For the cases (i) and (ii), the conductance scaling with system size remains robust (i.e., same as the case with no loss) for chemical potential within and outside the lattice band, while at the band edge rich anomalous conductance scaling emerges. For case (iii), the conductance scaling becomes ballistic in the thermodynamic limit for any value of chemical potential. We explain the emergence of these different system size scalings of conductance by analyzing the spectral properties of the associated non-Hermitian transfer matrices of the underlying lattice. We demonstrate that the emergence of anomalous scaling is deeply connected to the existence of exceptional points of transfer matrices. Our study unravels that by carefully optimizing the loss mechanism configurations, one can in principle realize systems with rich transport properties in low-dimensional open quantum systems.