The Aubry–André model as a hobbyhorse for understanding the localization phenomenon

Domínguez-Castro, G. A.; Paredes, R.

doi:10.1088/1361-6404/ab1670

Cited by 37 publications

(16 citation statements)

References 45 publications

(97 reference statements)

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“…We observe that in the extended regime (W 1) the asymptotic Q 0 scales as 1/N . This behavior is consistent with the more standard definition of IPR in a delocalize regime [13,26]. It shows that the initial excitation can spread all over the chain.…”

Section: Scalingsupporting

confidence: 88%

“…This model, from now on HHAA, was generalized by Aubry and André [17,18] who found that eigenstates are extended under a weak incommensurate potential but become localized at certain potential strength. This transition could be observed through the convergence of a perturbative expansion [19,18], through the exponential decay of the Landauer conductance, [20,21] or by evaluating the inverse participation ratio (IPR) of the eigenstates [22,23,24,25,26,27]. Consequently, 1d incommensurate models have been frequently studied to mimic the Anderson transition of high dimensional disordered systems [17].…”

Section: Introductionmentioning

confidence: 99%

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Ergodicity breaking in an incommensurate system observed by OTOCs and Loschmidt Echoes: From quantum diffusion to sub-diffusion

Lozano-Negro,

Zangara,

Pastawski

2021

Preprint

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The metal-insulator transition (MIT), which includes Anderson localization and Mott insulators as extreme regimes, has received renewed interest as the manybody effects often constitute a limitation for the handling of quantum interference. This resulted in the field dubbed many-body localization (MBL), intensively studied theoretically and experimentally as understanding the appearance of equilibration and thermalization becomes relevant in dealing with finite systems. Here, we propose a new observable to study this transition in a spin chain under the "disorder" of a Harper-Hofstadter-Aubry-André on-site potential. This quantity, which we call zeroth-order gradient entanglement (ZOGE) is extracted from the fundamental Fourier mode of a family of out-of-time-ordered correlators (OTOCs). These are just Loschmidt Echoes, where a field gradient is applied before the time reversal. In the absence of many-body interactions, the ZOGE coincides with the inverse participation ratio of a Fermionic wave function. By adding an Ising interaction to an XY Hamiltonian, one can explore the MBL phase diagram of the system. Close to the critical region, the excitation dynamics is consistent with a diffusion law. However, for weak disorder, quantum diffusion prevails while for strong disorder the excitation dynamics is sub-diffusive.

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Section: Scalingsupporting

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Ergodicity breaking in an incommensurate system observed by OTOCs and Loschmidt Echoes: From quantum diffusion to sub-diffusion

Lozano-Negro,

Zangara,

Pastawski

2021

Preprint

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“…Similar to the KR system, the existence of localizationdelocalization transition (LDT) in the dynamics has been investigated in the kicked Harper model (KHM), whose classical counterpart is a non-integrable system. [7][8][9][10][11][12][13] There are three main dynamical states of the quantum wave packet, localized, normal diffusion, and ballistic spread, corresponding to the change of the potential strength. In that respect, it is the same as the Harper model without the kicks.…”

Section: Introductionmentioning

confidence: 99%

Localization and delocalization properties in quasi-periodically perturbed Kicked Harper and Harper models

Yamada¹,

Ikeda²

2021

Preprint

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We numerically study the single particle localization and delocalization phenomena of an initially localized wave packet in the kicked Harper model (KHM) and Harper model subjected to quasiperiodic perturbation composed of M −modes. Both models are localized in the monochromatically perturbed case M = 1. KHM shows localization-delocalization transition (LDT) above M ≥ 2 as increase of the perturbation strength ǫ. In a time-continuous Harper model with the perturbation, it is confirmed that the localization persists for M = 2 and the LDT occurs for M ≥ 3. In addition, we investigate the diffusive property of the delocalized wave packet for ǫ above the critical strength ǫc (ǫ > ǫc). We also introduce other type systems without localization, which takes place a ballistic to diffusive transition in the wave packet dynamics as the increase of ǫ. In both systems, the ǫ−dependence of the diffusion coefficient well coincide with that in the other type system for large ǫ(> 1).

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“…The Aubry-André model has quasiperiodic disorder [23][24][25][26][27], so is contrary to the Anderson model where the disorder is random, in one-dimension (1D), and for a single particle, it can present both localized and delocalized regimes. All states in the Aubry-André model become localized only above a critical disorder strength, while in the one-particle 1D infinite Anderson model, all states are localized for any disorder strength [28][29][30][31].…”

Section: Introductionmentioning

confidence: 99%

Dynamical Detection of Level Repulsion in the One-Particle Aubry-André Model

Torres-Herrera

Santos

2020

Condensed Matter

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The analysis of level statistics provides a primary method to detect signatures of chaos in the quantum domain. However, for experiments with ion traps and cold atoms, the energy levels are not as easily accessible as the dynamics. In this work, we discuss how properties of the spectrum that are usually associated with chaos can be directly detected from the evolution of the number operator in the one-dimensional, noninteracting Aubry-André model. Both the quantity and the model are studied in experiments with cold atoms. We consider a single-particle and system sizes experimentally reachable. By varying the disorder strength within values below the critical point of the model, level statistics similar to those found in random matrix theory are obtained. Dynamically, these properties of the spectrum are manifested in the form of a dip below the equilibration point of the number operator. This feature emerges at times that are experimentally accessible. This work is a contribution to a special issue dedicated to Shmuel Fishman. arXiv:2001.07683v1 [cond-mat.dis-nn]

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The Aubry–André model as a hobbyhorse for understanding the localization phenomenon

Cited by 37 publications

References 45 publications

Ergodicity breaking in an incommensurate system observed by OTOCs and Loschmidt Echoes: From quantum diffusion to sub-diffusion

Ergodicity breaking in an incommensurate system observed by OTOCs and Loschmidt Echoes: From quantum diffusion to sub-diffusion

Localization and delocalization properties in quasi-periodically perturbed Kicked Harper and Harper models

Dynamical Detection of Level Repulsion in the One-Particle Aubry-André Model

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