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

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 onsite 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 ∝ (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 that 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. arXiv:1904.11989v2 [cond-mat.dis-nn]

Speck of chaos

“…We note that correlations of other kinds, such as those caused by the Shnirelmann's peak observed in some integrable models, may lead to correlation holes of different forms and at different time scales [12], but they are easily distinguished by the holes generated by quantum chaos, which can be described using random matrix theory [68].…”

“…The term quantum chaos, as used in this work, refers to properties of the spectrum and eigenstates that are similar to those found in full random matrices, such as strongly correlated eigenvalues [1][2][3] and eigenstates close to random vectors [4][5][6][7][8][9][10]. Level statistics as in random matrices are found also in some integrable models, but they are caused by finite-size effects [11,12] or change abruptly upon tiny variations of the Hamiltonian parameters [13,14]. Other definitions of quantum chaos include the short-time exponential growth of out-of-time order correlators [15][16][17][18][19][20] and diffusive transport [21][22][23], although exponential behaviors of fourpoint correlation functions appear also near critical points of integrable models [24][25][26][27][28] and ballistic transport has been observed in the chaotic single-defect X X Z model [29].…”

Onset of universality in the dynamical mixing of a pure state

“…In addition to the previous discussion, let us note that for the corresponding disorder strength h used in each case, HF (h = 0.5) and OE (h = 0.1) subspaces, the level spacing distribution of Hamiltonian ( 23) is GOE-like; however, there is an essential distinction between both cases, related to the presence or absence of interactions, namely for the HF subspace an infinitesimal disorder strength could bring the system to a chaotic regime in the thermodynamic limit (L → ∞) 37 , meanwhile for the OE subspace in the same limit, an infinitesimal disorder strength will induce single-particle-like localization and the level spacing distribution will be the corresponding to uncorrelated random variables from a Poisson process 38,39 . Interestingly, this fact could be indicating that the distribution P(λ ) of eigenvalues of the random density matrix (2) is more sensitive for the characterization of the chaotic nature of a given finite system than the level spacing distribution of the associated Hamiltonian.…”

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

“…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]. Despite this difference, when the systems are finite and have small disorder strengths, they present similar level spacing distributions; namely, they show distributions as in RMT, the so-called Wigner-Dyson distributions [32,33]. This is a finite-size effect, not a signature of chaos.…”