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arXiv:2608.02451

Spread complexity as a probe in generalized and long-range Aubry-Andre-Harper models

Triyas Sapui, Tanoy Kanti Konar, Subinay Dasgupta, Aditi Sen De

quant-phcond-mat.dis-nncond-mat.stat-mechSubmitted Aug 3, 2026• Announced Aug 4, 2026• 0 citations
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Abstract

We investigate the spread complexity of quantum quenches in generalized and long-range Aubry-Andre-Harper (AAH) models, encompassing regimes with and without mobility edges. In particular, in the generalized AAH models supporting energy-dependent mobility edges, we demonstrate that the long-time averaged spread complexity exhibits nonanalytic behavior when the post-quench quasiperiodic potential crosses the mobility edge associated with the energy of the initial eigenstate, thereby accurately identifying the mobility-edge transition. Such a behavior is supported by the spreading of local density of states. We further derive analytical expressions for the moments and the corresponding Lanczos coefficients for quenches between the limits of vanishing and strong quasiperiodic potentials. The Lanczos coefficients display qualitatively distinct behavior depending on the presence of mobility edges - they exhibit an initial plateau followed by a decay with the Krylov basis index, in contrast to the nearly constant behavior of the conventional AAH model without mobility edges. For LR hopping, the coefficients decay with the Krylov basis index for quenches from the localized to the extended phase, while they coincide with the AAH results for quenches in the opposite direction.

Comments
11 pages, 8 figures

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We investigate the spread complexity of quantum quenches in generalized and long-range Aubry-Andre-Harper (AAH) models, encompassing regimes with and without mobility edges. In particular, in the generalized AAH models supporting energy-dependent mobility edges, we demonstrate that the long-time averaged spread complexity exhibits nonanalytic behavior when the post-quench quasiperiodic potential crosses the mobility edge associated with the energy of the initial eigenstate, thereby accurately identifying the mobility-edge transition. Such a behavior is supported by the spreading of local density of states. We further derive analytical expressions for the moments and the corresponding Lanczos coefficients for quenches between the limits of vanishing and strong quasiperiodic potentials. The Lanczos coefficients display qualitatively distinct behavior depending on the presence of mobility edges - they exhibit an initial plateau followed by a decay with the Krylov basis index, in contrast to the nearly constant behavior of the conventional AAH model without mobility edges. For LR hopping, the coefficients decay with the Krylov basis index for quenches from the localized to the extended phase, while they coincide with the AAH results for quenches in the opposite direction.
In this study, we investigate the localization transition and quantum criticality {in the ground state of the} disordered Aubry-André-Harper (AAH) model, where a quasiperiodic potential is hybridized with a disordered potential. In the clean limit, the AAH model undergoes a localization transition from an extended phase to a localized phase via an intermediate critical phase as the strength of the quasiperiodic potential is varied. While the staggered potential merely shifts the critical point to a lower value, Fibonacci and Thue-Morse potentials induce immediate localization. This contrast reveals the sensitivity of localization behavior to the structural complexity of the potential, with the onset of localization correlating with the sequence's complexity. More specifically, the system follows a hierarchy defined by the complexity measures of the applied potentials. In addition, the typical fidelity susceptibility exhibits a power-law scaling behavior at the localization transition, enabling reliable extraction of the critical exponent. We focus on the AAH model with the Fibonacci potential due to its minimal finite-size effects compared to other cases. For the disordered AAH model with the Fibonacci potential, we determine critical exponents that differ from those of the AAH model without disorder and the Anderson model. Moreover, despite differences in localization behavior, we find that the disordered AAH models with the staggered potential and the Fibonacci potential share the same correlation-length critical exponent. These findings provide a unified framework for understanding localization transitions in quasiperiodic systems and are amenable to experimental validation using emerging techniques.
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.
The Anderson localization phase transition in the Aubry–André–Harper (AAH) model with p -wave superconducting (SC) pairing is numerically investigated by suddenly changing the on-site potential from zero to various finite values which fall into the extended, critical and localized phase regimes shown in this model. The time evolution of entanglement entropy (EE), mean width of wave packets and Loschmidt echo of the system exhibit distinct but consistent dynamical signatures in those three phases. Specifically, the EE grows as a power function of time with the exponent of which varies in the extended phase but keeps almost unchanged in the critical phase for different quench parameters. However, if the system is in the localized phase after a quench, the EE grows much slower and will soon get saturated. The time-dependent width of wave packets in the system shows similar behaviors as the EE. In addition, from the perspective of the dynamical phase transition, we find that the Loschmidt echo oscillates and always keeps finite when the system is quenched in the extended phase. In contrast, in the critical or localized phase, the echo will reach to zero at some time intervals or will decay almost to zero after a long-time evolution. The universal features of these quantities in the critical phase of the system with various SC pairing amplitudes are also observed.
We relate the probability distribution of the work done on a statistical system under a sudden quench to the Lanczos coefficients corresponding to evolution under the postquench Hamiltonian. Using the general relation between the moments and the cumulants of the probability distribution, we show that the Lanczos coefficients can be identified with physical quantities associated with the distribution, e.g., the average work done on the system and its variance, as well as the higher order cumulants. In a sense this gives an interpretation of the Lanczos coefficients in terms of experimentally measurable quantities. Consequently, our approach provides a way towards understanding spread complexity, a quantity that measures the spread of an initial state with time in the Krylov basis generated by the postquench Hamiltonian, from a thermodynamical perspective. We illustrate these relations with two examples. The first one involves quench done on a harmonic chain with periodic boundary conditions and with nearest neighbor interactions. As a second example, we consider mass quench in a free bosonic field theory in ddd spatial dimensions in the limit of large system size. In both cases, we find out the time evolution of the spread complexity after the quench and relate the Lanczos coefficients with the cumulants of the distribution of the work done on the system.
One-dimensional quasiperiodic systems with power-law hopping, 1/r^{a}, differ from both the standard Aubry-André (AA) model and from power-law systems with uncorrelated disorder. Whereas in the AA model all single-particle states undergo a transition from ergodic to localized at a critical quasidisorder strength, short-range power-law hops with a>1 can result in mobility edges. We find that there is no localization for long-range hops with a≤1, in contrast to the case of uncorrelated disorder. Systems with long-range hops rather present ergodic-to-multifractal edges and a phase transition from ergodic to multifractal (extended but nonergodic) states. Both mobility and ergodic-to-multifractal edges may be clearly revealed in experiments on expansion dynamics.
We investigate localization transitions and spectral topology in a one-dimensional non-Hermitian generalization of the Aubry-André model in which both the nearest-neighbor and the next-nearest-neighbor hopping amplitudes are nonreciprocal. By extending the Fermi-surface point-matching method to nonreciprocal hopping, we derive a closed-form expression for the energy-dependent mobility edge in which the two nonreciprocity parameters are absorbed into exponentially renormalized effective hopping amplitudes. The mobility edge forms a single parabola in the energy--potential plane: nearest-neighbor nonreciprocity rigidly shifts the localization boundary toward stronger potentials, whereas next-nearest-neighbor nonreciprocity reduces the curvature of the boundary and thereby broadens the energy window in which extended and localized states coexist. Exact diagonalization confirms the analytical boundary for purely nearest-neighbor, purely next-nearest-neighbor, and combined nonreciprocity, and recovers the known Hermitian mobility edge in the reciprocal limit. We further analyze the spectral topology under periodic boundary conditions and show that the spectral winding numbers evaluated at base energies near the two band edges directly bracket the mixed phase: the winding number at the lower band edge drops when the mobility edge enters the spectrum and the first localized states appear, while the winding number at the upper band edge drops when the last extended states localize, delineating the full potential-strength window over which extended and localized states coexist. These results provide a compact analytical framework that connects energy-dependent localization, spectral topology, and nonreciprocity in quasiperiodic lattices, and they are directly testable in photonic, atomic, and electrical-circuit platforms.