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arXiv:1905.13255 · Physical review. B./Physical review. B

Study of counterintuitive transport properties in the Aubry-André-Harper model via entanglement entropy and persistent current

Nilanjan Roy, Auditya Sharma

Published Nov 25, 2019• 30 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.100.195143

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Comment on “Study of counterintuitive transport properties in the Aubry-André-Harper model via entanglement entropy and persistent current”

Francisco Marcelo Fernández, Diego R. Alcoba, Alicia Torre +3 more

In a recent study of the transport properties in the Aubry-Andr\'e-Harper model [Phys. Rev. B 100, 195143 (2019)], the authors point out the failure of the Hellmann-Feynman theorem (HFT) in the case of degenerate states. They try to support such a statement from earlier papers that prove exactly the opposite. It appears that the application of the HFT to degenerate states has not been well understood. For this reason, in this Comment we illustrate the correct application of the HFT to the model just mentioned.

Physical review. B./Physical review. B•May 19, 2020•0 citations
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 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.
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.
We numerically investigate the link between the delocalization-localization transition and entanglement in a disordered long-range hopping model of spinless fermions by studying various static and dynamical quantities. This includes the inverse participation ratio, level statistics, entanglement entropy, and number fluctuations in the subsystem along with quench and wave-packet dynamics. Finite systems show delocalized, quasilocalized, and localized phases. The delocalized phase shows strong area-law violation, whereas the (quasi)localized phase adheres to (for large subsystems) the strict area law. The idea of ``entanglement contour'' nicely explains the violation of area law and its relationship with ``fluctuation contour'' reveals a signature at the transition point. The relationship between entanglement entropy and number fluctuations in the subsystem also carries signatures for the transition in the model. Results from the Aubry-Andre-Harper model are compared in this context. The propagation of charge and entanglement are contrasted by studying quench and wave-packet dynamics at the single-particle and many-particle levels.
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.
Observing critical phases in lattice models is challenging due to the need to analyze the finite time or size scaling of observables. We study how the computational topology technique of persistent homology can be used to characterize phases of a generalized Aubry-Andr\'e-Harper model. The persistent entropy and mean squared lifetime of features obtained using persistent homology behave similarly to conventional measures (Shannon entropy and inverse participation ratio) and can distinguish localized, extended, and critical phases. However, we find that the persistent entropy also clearly distinguishes ordered from disordered regimes of the model. The persistent homology approach can be applied to both the energy eigenstates and the wave packet propagation dynamics.
In a recent study of the transport properties in the Aubry-Andr\'e-Harper model [Phys. Rev. B 100, 195143 (2019)], the authors point out the failure of the Hellmann-Feynman theorem (HFT) in the case of degenerate states. They try to support such a statement from earlier papers that prove exactly the opposite. It appears that the application of the HFT to degenerate states has not been well understood. For this reason, in this Comment we illustrate the correct application of the HFT to the model just mentioned.