Localization challenges quantum chaos in the finite two-dimensional Anderson model

Abstract

It is believed that the two-dimensional (2D) Anderson model exhibits localization for any nonzero disorder in the thermodynamic limit and it is also well known that the finite-size effects are considerable in the weak disorder limit. Here we numerically study the quantum-chaos to localization transition in the finite 2D Anderson model using standard indicators used in the modern literature such as the level spacing ratio, spectral form factor, variances of observable matrix elements, participation entropy and the eigenstate entanglement entropy. We show that many features of these indicators may indicate emergence of robust single-particle quantum chaos at weak disorder. However, we argue that a careful numerical analysis is consistent with the single-parameter scaling theory and predicts the breakdown of quantum chaos at any nonzero disorder value in the thermodynamic limit. Among the hallmarks of this breakdown are the universal behavior of the spectral form factor at weak disorder, and the universal scaling of various indicators as a function of the parameter u = (W ln V ) −1 where W is the disorder strength and V is the number of lattice sites.

Eigenstate entanglement entropy in the integrable spin- 12 XYZ model

“…Highly-excited eigenstates of integrable interacting Hamiltonians, such as the spin- 1 2 XXZ chain [19] and the spin- 1 2 Heisenberg (XXX) chain [20], on the other hand have been found to exhibit an average entanglement entropy whose leading volume-law term is not maximal. It is actually close to the one of translationally invariant quadratic models [21][22][23], and of quantum-chaotic quadratic models [24][25][26][27] and typical (Haar-random) Gaussian pure states in the Hilbert space [14,28]. Since the previously mentioned studies of the average eigenstate entanglement entropy of integrable interacting Hamiltonians were carried out in the presence of U(1) [19] and SU (2) [20] symmetry, our goal in this work is to ex-plore what happens in the absence of those symmetries as well as when the model exhibits supersymmetry (in short, SUSY).…”

Thermalization and localization in a discretized quantum field theory

“…The point at which the transition occurs, however, is known to drift logarithmically to lower disorder strengths as one increases the system size. In the bottom, right panel, we next plot the gap ratio as a function of ∆ scaled by ln N instead, as discussed in [10], for different system sizes, and see that the graphs intersect at the point (∆ ln N ) * ≈ 15.2. This suggests that in the limit of N → ∞, the transition from GOE to Poisson distributed eigenvalues would occur at ∆ → 0 and the system would, therefore, again localize for arbitrarily small disorder.…”

“…In contrast to thermalizing systems, those in the localized phase do not develop long-range correlations, and subregions retain memory of their initial entanglement structure. Localization is also characterized by unique spectral statistics in measures such as the spectral gap ratio and spectral form factor [6][7][8][9][10], that have interesting connections to random matrix theory as well [11,12].…”

Dynamical localization in a non-Hermitian Floquet synthetic system

“…[14][15][16][17][18][19][20][21][22] For example, the dynamical localization (DL) occurring after the Ehrenfest time demonstrates the lack of quantum-classical correspondence in Floquet driven systems due to the quantum coherence, which has received extensive investigations in the past few decades. [23][24][25][26] The mechanism governing chaotic diffusion plays a key role in quantum scrambling and non-equilibrium dynamics. It has been reported that the chaotic dynamics induced by Floquet driven potential dominates the exponentially-fast information scrambling, [27][28][29] the quantum thermalization process [30,31] and entanglement.…”