Level repulsion and dynamics in the finite one-dimensional Anderson model

E. Jonathan Torres-Herrera, J. A. Méndez-Bermúdez, and Lea F. Santos

Phys. Rev. E 100, 022142 – Published 29 August 2019

Abstract

This work shows that dynamical features typical of full random matrices can be observed also in the simple finite one-dimensional (1D) noninteracting Anderson model with nearest-neighbor couplings. In the thermodynamic limit, all eigenstates of this model are exponentially localized in configuration space for any infinitesimal on-site disorder strength $W$ . But this is not the case when the model is finite and the localization length is larger than the system size $L$ , which is a picture that can be experimentally investigated. We analyze the degree of energy-level repulsion, the structure of the eigenstates, and the time evolution of the finite 1D Anderson model as a function of the parameter $ξ \propto {(W^{2} L)}^{- 1}$ . As $ξ$ increases, all energy-level statistics typical of random matrix theory are observed. The statistics are reflected in the corresponding eigenstates and also in the dynamics. We show that the probability in time to find a particle initially placed on the first site of an open chain decays as fast as in full random matrices and much faster than when the particle is initially placed far from the edges. We also see that at long times, the presence of energy-level repulsion manifests in the form of the correlation hole. In addition, our results demonstrate that the hole is not exclusive to random matrix statistics, but emerges also for $W = 0$ , when it is in fact deeper.

Received 30 April 2019

DOI:https://doi.org/10.1103/PhysRevE.100.022142

Physics Subject Headings (PhySH)

Anderson localization Single-particle dynamics

Random & disordered media

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

E. Jonathan Torres-Herrera ^1,*, J. A. Méndez-Bermúdez¹, and Lea F. Santos²

¹Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla, 72570, Mexico
²Department of Physics, Yeshiva University, New York City, New York 10016, USA

^*Corresponding author: etorresh@ifuap.buap.mx

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Vol. 100, Iss. 2 — August 2019

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Images

Figure 1
Level repulsion parameter $β$ (a) and average ratio of spacings between consecutive levels $〈 \tilde{r} 〉$ as a function of $ξ$ . Four system sizes are considered: $L = 512, 1024, 2048, 4096$ , which are only distinguishable for very large $ξ$ (the size increases from bottom to top). Crosses indicate fitting curves: the linear functions are written in the main panel and the inset of (a), while the function for (b) is given by Eq. (13). Averages over 100 disorder realizations.
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Figure 2
Scaled average participation ratio for eigenstates in the middle of the spectrum (a) and for basis vectors (b) as a function of $ξ$ . System sizes are (from top to bottom) $L = 256, 512, 1024, 2048$ . Circles in (a) are the RMT predictions for $β = 1, 2, 4$ , while crosses (orange) represent Eq. (16). Circles in (b) are obtained from Eq. (17) with $κ = 1 / 3$ . For the averages we collected $10^{4}$ statistical data obtained from disorder realizations and states around to the middle of the spectrum.
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Figure 3
Local density of states (left) and survival probability (right) for GSE-like statistics; $n_{0} = 1, 2, 3, 8, 20, L / 2$ from top to bottom. Right panels: Dashed (blue) and dot-dashed (green) lines correspond to the asymptotic decays $t^{- 3}$ and $t^{- 1}$ , respectively. Horizontal double dotted-dashed (red) lines correspond to the infinite-time average. In addition to the numerical curves for $W \neq 0$ , we show with orange lines the corresponding analytical expressions (26) for $W = 0$ . All panels are for $L = 16 384$ ; averages over $10^{3}$ disorder realizations; and $ξ = 6.31$ , which implies $〈 \tilde{r} 〉 \sim {〈 \tilde{r} 〉}_{GSE}$ .
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Figure 4
Survival probability for initial states with $ε_{n_{0}} \approx 0$ for Poissonian (turquoise) and GOE-like nearest-neighbor energy-level spacing distributions (a), GUE-like (b), GSE-like (c) and the picket-fence spectrum with $W \approx 0$ ( $ξ = 1000$ ) (d). Dotted-dashed lines indicate the $t^{- 1}$ decay. Horizontal double dotted-dashed lines correspond to the infinite-time average. The insets in (a), (b), and (c) zoom in the correlation hole for the GOE-, GUE-, and GSE-like statistics. $L = 16 384$ ; averages over initial states and disorder realizations that give a total of $10^{4}$ data. An additional average is done over small windows of time to further reduce fluctuations.
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Figure 5
Spread in time of the probability to find the particle on any site. Three initial states are shown: particle on $n_{0} = 1$ (a), particle on $n_{0} = 2$ (b), and particle on $n_{0} = L / 2$ (c). Parameters: $β \approx 4, L = 1000$ , and one disorder realization.
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