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arXiv:2101.03465

Dual-mapping and quantum criticality in off-diagonal Aubry-André models

Tong Liu, Xu Xia

Published Jan 10, 2021• 4 citations
arXivHTMLPDF

Abstract

We study a class of off-diagonal quasiperiodic hopping models described by one-dimensional Su-Schrieffer-Heeger chain with quasiperiodic modulations. We unveil a general dual-mapping relation in parameter space of the dimerization strength $λ$ and the quasiperiodic modulation strength $V$, regardless of the specific details of the quasiperiodic modulation. Moreover, we demonstrated semi-analytically and numerically that under the specific quasiperiodic modulation, quantum criticality can emerge and persist in a wide parameter space. These unusual properties provides a distinctive paradigm compared with the diagonal quasiperiodic systems.

DOI
10.48550/arxiv.2101.03465

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We study a class of off-diagonal quasiperiodic hopping models described by one-dimensional Su-Schrieffer-Heeger chain with quasiperiodic modulations. We unveil a general dual-mapping relation in parameter space of the dimerization strength λλλ and the quasiperiodic modulation strength VVV, regardless of the specific details of the quasiperiodic modulation. Moreover, we demonstrated semi-analytically and numerically that under the specific quasiperiodic modulation, quantum criticality can emerge and persist in a wide parameter space. These unusual properties provides a distinctive paradigm compared with the diagonal quasiperiodic systems.
We study a class of off-diagonal quasiperiodic hopping models described by a one-dimensional Su-Schrieffer-Heeger chain with quasiperiodic modulations. With regard to these models, we unveil a general dual-mapping relation in parameter space of the dimerization strength $\ensuremath{\lambda}$ and the quasiperiodic modulation strength VVV, regardless of specific details of quasiperiodic modulations. Moreover, we demonstrated semianalytically and numerically that under the specific quasiperiodic modulation, quantum criticality, namely the criticality of wave functions, can emerge and persist in a wide parameter space. These unusual results provide a distinctive paradigm compared with the diagonal quasiperiodic systems and perspectives for future investigations on quasiperiodic disorder.
We propose a general framework for constructing self-dual one-dimensional quasiperiodic lattice models with arbitrary-range hoppings and multifractal behaviors. Our framework generates a broad spectrum of one-dimensional quasicrystals, ranging from the off-diagonal Aubry-Andr\'e-Harper models on one end, to those featuring long-range hoppings with varied quasiperiodic modulations on another. Focusing on models with off-diagonal quasiperiodic hoppings with power-law decay, we exploit the fact that, when the self-dual condition is satisfied, the system must be in the critical state with multifractal properties. This enables the engineering of models with competing extended, critical, and localized phases, with richly structured mobility edges separating them. As an outstanding example, we show that a limiting case of our family of self-dual quasicrystals can be implemented using Rydberg-atom arrays. Our work offers a systematic route toward critical phases from self-duality considerations, and would facilitate the experimental simulation of these exotic states.
We investigate the quantum criticality and universality in Aubry-Andr\'e-Harper (AAH) model with ppp-wave superconducting pairing $\mathrm{\ensuremath{\Delta}}$ in terms of the generalized fidelity susceptibility (GFS). We show that the higher order GFS is more efficient in spotlighting the critical points than lower order ones, and thus the enhanced sensitivity is propitious for extracting the associated universal information from the finite-size scaling in quasiperiodic systems. The GFS obeys power-law scaling for localization transitions and thus scaling properties of the GFS provide compelling values of critical exponents. Specifically, we demonstrate that the fixed modulation phase $\ensuremath{\phi}=\ensuremath{\pi}$ alleviates the odd-even effect of scaling functions across the Aubry-Andr\'e transition with $\mathrm{\ensuremath{\Delta}}=0$, while the scaling functions for odd and even numbers of system sizes with a finite $\mathrm{\ensuremath{\Delta}}$ cannot coincide irrespective of the value of $\ensuremath{\phi}$. A thorough numerical analysis with odd number of system sizes reveals the correlation-length exponent $\ensuremath{\nu}\ensuremath{\simeq}$ 1.000 and the dynamical exponent $z\phantom{\rule{4pt}{0ex}}\ensuremath{\simeq}$ 1.388 for transitions from the critical phase to the localized phase, suggesting the unusual universality class of localization transitions in the AAH model with a finite ppp-wave superconducting pairing lies in a different universality class from the Aubry-Andr\'e transition. The results may be testified in near term state-of-the-art experimental settings.
In this paper, we explore quantum criticality in the disordered Aubry-Andr\'e (AA) model. For the pure AA model, it is well known that it hosts a critical point separating an extended phase and a localized insulator phase by tuning the strength of the quasiperiodic potential. Here we unearth that the disorder strength $\mathrm{\ensuremath{\Delta}}$ contributes an independent relevant direction near the critical point of the AA model. Our scaling analyses show that the localization length $\ensuremath{\xi}$ scales with $\mathrm{\ensuremath{\Delta}}$ as $\ensuremath{\xi}\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}^{\ensuremath{-}{\ensuremath{\nu}}_{\mathrm{\ensuremath{\Delta}}}}$ with ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\Delta}}}$ a new critical exponent, which is estimated to be ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\Delta}}}\ensuremath{\approx}0.46$. This value is remarkably different from the counterparts for both the pure AA model and the Anderson model. Moreover, rich critical phenomena are discovered in the critical region spanned by the quasiperiodic and the disordered potentials. In particular, in the extended phase side, we show that the scaling theory satisfies a hybrid scaling form as a result of the overlap between the critical regions of the AA model and the Anderson localization.
We explore quantum criticality and Kibble-Zurek scaling (KZS) in the Aubry-Andr\'e-Stark (AAS) model, where the Stark field of strength $\ensuremath{\varepsilon}$ is added onto the one-dimensional quasiperiodic lattice. We perform scaling analysis and numerical calculations of the localization length, the inverse participation ratio (IPR), and the energy gap between the ground and first excited states to characterize critical properties of the delocalization-localization transition. Remarkably, our scaling analysis shows that, near the critical point, the localization length $\ensuremath{\xi}$ scales with $\ensuremath{\varepsilon}$ as $\ensuremath{\xi}\ensuremath{\propto}{\ensuremath{\varepsilon}}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}\ensuremath{\approx}0.3$, a new critical exponent for the AAS model, which is different from the counterparts for both the pure Aubry-Andr\'e (AA) model and the pure Stark model. The IPR I\mathcal{I}I scales as $\mathcal{I}\ensuremath{\propto}{\ensuremath{\varepsilon}}^{s}$ with the critical exponent $s\ensuremath{\approx}0.098$, which is also different from both of the pure models. The energy gap $\mathrm{\ensuremath{\Delta}}E$ scales as $\mathrm{\ensuremath{\Delta}}E\ensuremath{\propto}{\ensuremath{\varepsilon}}^{\ensuremath{\nu}z}$ with the same critical exponent $z\ensuremath{\approx}2.374$ as that for the pure AA model. We further reveal hybrid scaling functions in the overlap between the critical regions of the Anderson and Stark localizations. Moreover, we investigate the driven dynamics of the localization transitions in the AAS model. By linearly changing the Stark (quasiperiodic) potential, we calculate the evolution of the localization length and the IPR, and we study their dependence on the driving rate. We find that the driven dynamics from the ground state is well described by the KZS with the critical exponents obtained from the static scaling analysis. When both the Stark and the quasiperiodic potentials are relevant, the KZS form includes the two scaling variables. This work extends our understanding of critical phenomena on localization transitions and generalizes the application of the KZS to hybrid models.
In this study, we explore the quantum critical phenomena in generalized Aubry-André models, with a particular focus on the scaling behavior at various filling states. Our approach involves using quantum fidelity susceptibility to precisely identify the mobility edges in these systems. Through a finite-size scaling analysis of the fidelity susceptibility, we are able to determine both the correlation-length critical exponent and the dynamical critical exponent at the critical point of the generalized Aubry-André model. Based on the Diophantine equation conjecture, we can determines the number of subsequences of the Fibonacci sequence and the corresponding scaling functions for a specific filling fraction, as well as the universality class. Our findings demonstrate the effectiveness of employing the generalized fidelity susceptibility for the analysis of unconventional quantum criticality and the associated universal information of quasiperiodic systems in cutting-edge quantum simulation experiments.