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arXiv:2008.10561 · Physical review. B./Physical review. B

Fraction of delocalized eigenstates in the long-range Aubry-André-Harper model

Nilanjan Roy, Auditya Sharma

Published Feb 11, 2021• 59 citations
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Abstract

We uncover a systematic structure in the single-particle phase diagram of the quasiperiodic Aubry-Andr\'e-Harper (AAH) model with power-law hoppings ($\ensuremath{\sim}\frac{1}{{r}^{\ensuremath{\sigma}}}$) when the quasiperiodicity parameter is chosen to be a member of the metallic mean family of irrational Diophantine numbers. In addition to the fully delocalized and localized phases, we find a coexistence of multifractal (localized) states with the delocalized states for $\ensuremath{\sigma}<1$ ($\ensuremath{\sigma}>1$). The fraction of delocalized eigenstates in these phases can be obtained from a general sequence, which is a manifestation of a mathematical property of the metallic mean family. The entanglement entropy of the noninteracting many-body ground states respects the area law if the Fermi level belongs in the localized regime while logarithmically violating it if the Fermi level belongs in the delocalized or multifractal regimes. The prefactor of the logarithmically violating term is significantly larger in the delocalized phase in comparison to that in the multifractal phase. Entanglement entropy shows the area law even in the delocalized regime for special filling fractions, which are related to the metallic means.

DOI
10.1103/physrevb.103.075124

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We uncover a systematic structure in the single-particle phase diagram of the quasiperiodic Aubry-Andr'e-Harper (AAH) model with power-law hoppings (\ensuremath∼1r\ensuremathσ\ensuremath{\sim}\frac{1}{{r}^{\ensuremath{\sigma}}}\ensuremath∼r\ensuremathσ1​) when the quasiperiodicity parameter is chosen to be a member of the metallic mean family of irrational Diophantine numbers. In addition to the fully delocalized and localized phases, we find a coexistence of multifractal (localized) states with the delocalized states for \ensuremathσ<1\ensuremath{\sigma}<1\ensuremathσ<1 (\ensuremathσ>1\ensuremath{\sigma}>1\ensuremathσ>1). The fraction of delocalized eigenstates in these phases can be obtained from a general sequence, which is a manifestation of a mathematical property of the metallic mean family. The entanglement entropy of the noninteracting many-body ground states respects the area law if the Fermi level belongs in the localized regime while logarithmically violating it if the Fermi level belongs in the delocalized or multifractal regimes. The prefactor of the logarithmically violating term is significantly larger in the delocalized phase in comparison to that in the multifractal phase. Entanglement entropy shows the area law even in the delocalized regime for special filling fractions, which are related to the metallic means.
The single-particle eigenstates of the Aubry-Andr\'e-Harper model are known to show a delocalization-localization transition at a finite strength of the quasiperiodic disorder. In this work, we point out that an intimate relationship exists between the subband structure of the spectrum and transport properties of the model. To capture the transport properties, we have not only used a variety of single-particle measures like inverse participation ratio, and von Neumann entropy, but also many-particle measures such as persistent current and its variance, and many-body entanglement entropy. The many-particle measures are very sensitive to the subband structure of the spectrum. Even in the delocalized phase, surprisingly the entanglement entropy is substantially suppressed when the Fermi level is in the band gaps, whereas the persistent current is vanishingly small for the same locations of the Fermi level. The entanglement entropy seems to follow area law exclusively for these special locations of Fermi level or filling fractions of free fermions. A study of the standard deviation of persistent current offers further distinguishing features for the special fillings. In the delocalized phase, the standard deviation vs mean persistent current curves are discontinuous for the nonspecial values of filling fractions and continuous (closed) for the special values of filling fractions, whereas in the localized phase, these curves become straight lines for both types of filling fractions. We have also discussed how the results depend on the system size. Our results, especially on the persistent current, can potentially be tested experimentally using the present day setups based on ultracold atoms.
We investigate the nonequilibrium dynamics of entanglement entropy and out-of-time-order correlator (OTOC) of noninteracting fermions at half-filling starting from a product state to distinguish the delocalized, multifractal (in the limit of nearest neighbor hopping), localized and mixed phases hosted by the quasiperiodic Aubry-André-Harper (AAH) model in the presence of long-range hopping. For sufficiently long-range hopping strength a secondary logarithmic behavior in the entanglement entropy is found in the mixed phases whereas the primary behavior is a power-law the exponent of which is different in different phases. The saturation value of entanglement entropy in the delocalized, multifractal and mixed phases depends linearly on system size whereas in the localized phase (in the short-range regime) it is independent of system size. The early-time growth of OTOC shows very different power-law behaviors in the presence of nearest neighbor hopping and long-range hopping. The late time decay of OTOC leads to noticeably different power-law exponents in different phases. The spatial profile of OTOC and its system-size dependence also provide distinct features to distinguish phases. In the mixed phases the spatial profile of OTOC shows two different dependences on space for small and large distances respectively. Interestingly the spatial profile contains large fluctuations at the special locations related to the quasiperiodicity parameter in the presence of multifractal states.
As opposed to random disorder, which localizes single-particle wave functions in one dimension (1D) at arbitrarily small disorder strengths, there is a localization-delocalization transition for quasiperiodic disorder in the 1D Aubry-Andr\'e model at a finite disorder strength. On the single-particle level, many properties of the ground state at criticality have been revealed by applying a real-space renormalization-group scheme; the critical properties are determined solely by the continued-fraction expansion of the incommensurate frequency of the disorder. Here, we investigate the many-particle localization-delocalization transition in the Aubry-Andr\'e model with and without interactions. In contrast to the single-particle case, we find that the critical exponents depend on a Diophantine equation relating the incommensurate frequency of the disorder and the filling fraction which generalizes the dependence, in the single-particle spectrum, on the continued-fraction expansion of the incommensurate frequency. This equation can be viewed as a generalization of the resonance condition in the commensurate case. When interactions are included, numerical evidence suggests that interactions may be irrelevant at at least some of these critical points, meaning that the critical exponent relations obtained from the Diophantine equation may actually survive in the interacting case.
In this work, we map the phase diagrams of one-dimensional quasiperiodic models using artificial neural networks. We observe that the multiclass classifier precisely distinguishes the various phases, namely the delocalized, multifractal, and localized phases, when trained on the eigenstates of the long-range Aubry-Andr\'e Harper (LRH) model. Additionally, when this trained multilayer perceptron is fed with the eigenstates of the Aubry-Andr\'e Harper (AAH) model, it identifies various phases with reasonable accuracy. We examine the resulting phase diagrams produced using a single disorder realization and demonstrate that they are consistent with those obtained from the conventional method of fractal dimension analysis. Interestingly, when the neural network is trained using the eigenstates of the AAH model, the resulting phase diagrams for the LRH model are less exemplary than those previously obtained. Further, we study binary classification by training the neural network on the probability density corresponding to the delocalized and localized eigenstates of the AAH model. We are able to pinpoint the critical transition point by examining the metric ``accuracy'' for the central eigenstate. The effectiveness of the binary classifier in identifying a previously unknown multifractal phase is then evaluated by applying it to the LRH model.
We investigate the electric-field driven power-law random banded matrix (PLRBM) model where a variation in the power-law exponent $\ensuremath{\alpha}$ yields a delocalization-to-localization phase transition. We examine the periodically driven PLRBM model with the help of the Floquet operator. The level spacing ratio and the generalized participation ratio of the Floquet Hamiltonian reveal a drive-induced weak multifractal (fractal) phase accompanied by diffusive (subdiffusive) transport on the delocalized side of the undriven PLRBM model. On the localized side, the time-periodic model remains localized---the average level-spacing ratio corresponds to Poisson statistics and logarithmic transport is observed in the dynamics. Extending our analysis to the aperiodic Thue-Morse driven system, we find that the aperiodically driven clean long-range hopping model (clean counterpart of the PLRBM model) exhibits the phenomenon of exact dynamical localization on tuning the drive parameters at special points. The disordered time-aperiodic system shows diffusive transport followed by relaxation to the infinite-temperature state on the delocalized side, and a prethermal plateau with subdiffusion on the localized side. Additionally, we compare this with a quasiperiodically driven Aubry-Andr\'e-Harper model that also undergoes a localization-delocalization transition. Unlike the disordered long-range model, it features a prolonged prethermal plateau followed by subdiffusion to the infinite temperature state, even on the delocalized side.
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.