Anderson localization on random regular graphs (RRG) attracts much attention as a toy model of many-body localization. Here, the critical behavior at the localization transition on RRG, related to that on infinite Bethe lattices, is studied numerically by means of a population dynamics approach. Proceeding up to correlation volumes as large as ${10}^{19}$ , the authors obtain the value of one half for the correlation-length critical exponent, in agreement with analytical predictions. They find very well pronounced corrections to scaling, similar to those in models with high spatial dimensionality and with many-body localization transitions.Anderson localization on random regular graphs (RRG) attracts much attention as a toy model of many-body localization. Here, the critical behavior at the localization transition on RRG, related to that on infinite Bethe lattices, is studied numerically by means of a population dynamics approach. Proceeding up to correlation volumes as large as ${10}^{19}$, the authors obtain the value of one half for the correlation-length critical exponent, in agreement with analytical predictions. They find very well pronounced corrections to scaling, similar to those in models with high spatial dimensionality and with many-body localization transitions.

arXiv:1903.07877Physical review. B./Physical review. BJun 11, 201979 citations

Statistics of eigenstates near the localization transition on random regular graphs

Konstantin S. Tikhonov, A. D. Mirlin

Anderson localization on random regular graphs (RRG) attracts much attention in view of its relation to many-body localization. Here, the authors study the statistics of eigenfunctions and energy levels on RRG, focussing on the delocalized (ergodic) phase and the critical regime. Results of exact diagonalization are in outstanding agreement with analytical theory based on the saddle-point analysis of the effective supersymmetric action supplemented by numerical solution of the corresponding self-consistency equation.Anderson localization on random regular graphs (RRG) attracts much attention in view of its relation to many-body localization. Here, the authors study the statistics of eigenfunctions and energy levels on RRG, focussing on the delocalized (ergodic) phase and the critical regime. Results of exact diagonalization are in outstanding agreement with analytical theory based on the saddle-point analysis of the effective supersymmetric action supplemented by numerical solution of the corresponding self-consistency equation.

arXiv:1810.11444Physical review. B./Physical review. BJan 7, 2019110 citations

Many-body localization and 3D Anderson transition

Kaya, Furkan

Jan 1, 20250 citations

Subdiffusion in the Anderson model on the random regular graph

Giuseppe De Tomasi, Soumya Bera, Antonello Scardicchio +1 more

We study the finite-time dynamics of an initially localized wave packet in the Anderson model on the random regular graph (RRG) and show the presence of a subdiffusion phase coexisting both with ergodic and putative nonergodic phases. The full probability distribution $\mathrm{\ensuremath{\Pi}}(x,t)$ of a particle to be at some distance $x$ from the initial state at time $t$ is shown to spread subdiffusively over a range of disorder strengths. The comparison of this result with the dynamics of the Anderson model on ${\mathbb{Z}}^{d}$ lattices, $d>2$ , which is subdiffusive only at the critical point implies that the limit $d\ensuremath{\rightarrow}\ensuremath{\infty}$ is highly singular in terms of the dynamics. A detailed analysis of the propagation of $\mathrm{\ensuremath{\Pi}}(x,t)$ in space-time $(x,t)$ domain identifies four different regimes determined by the position of a wave front ${X}_{\text{front}}(t)$ , which moves subdiffusively to the most distant sites ${X}_{\text{front}}(t)\ensuremath{\sim}{t}^{\ensuremath{\beta}}$ with an exponent $\ensuremath{\beta}<1$. Importantly, the Anderson model on the RRG can be considered as proxy of the many-body localization transition (MBL) on the Fock space of a generic interacting system. In the final discussion, we outline possible implications of our findings for MBL.We study the finite-time dynamics of an initially localized wave packet in the Anderson model on the random regular graph (RRG) and show the presence of a subdiffusion phase coexisting both with ergodic and putative nonergodic phases. The full probability distribution $\mathrm{\ensuremath{\Pi}}(x,t)$ of a particle to be at some distance $x$ from the initial state at time $t$ is shown to spread subdiffusively over a range of disorder strengths. The comparison of this result with the dynamics of the Anderson model on ${\mathbb{Z}}^{d}$ lattices, $d>2$, which is subdiffusive only at the critical point implies that the limit $d\ensuremath{\rightarrow}\ensuremath{\infty}$ is highly singular in terms of the dynamics. A detailed analysis of the propagation of $\mathrm{\ensuremath{\Pi}}(x,t)$ in space-time $(x,t)$ domain identifies four different regimes determined by the position of a wave front ${X}_{\text{front}}(t)$, which moves subdiffusively to the most distant sites ${X}_{\text{front}}(t)\ensuremath{\sim}{t}^{\ensuremath{\beta}}$ with an exponent $\ensuremath{\beta}<1$. Importantly, the Anderson model on the RRG can be considered as proxy of the many-body localization transition (MBL) on the Fock space of a generic interacting system. In the final discussion, we outline possible implications of our findings for MBL.

arXiv:1908.11388Physical review. B./Physical review. BMar 3, 202071 citations

Quantum thermalization and many-body Anderson localization I & II notes

David A. Huse

Jun 1, 20160 citations

{10}^{19}

, the authors obtain the value of one half for the correlation-length critical exponent, in agreement with analytical predictions. They find very well pronounced corrections to scaling, similar to those in models with high spatial dimensionality and with many-body localization transitions.

x

from the initial state at time

t

is shown to spread subdiffusively over a range of disorder strengths. The comparison of this result with the dynamics of the Anderson model on

{\mathbb{Z}}^{d}

lattices,

d>2

, which is subdiffusive only at the critical point implies that the limit $d\ensuremath{\rightarrow}\ensuremath{\infty}$ is highly singular in terms of the dynamics. A detailed analysis of the propagation of $\mathrm{\ensuremath{\Pi}}(x,t)$ in space-time

(x,t)

domain identifies four different regimes determined by the position of a wave front

{X}_{\text{front}}(t)

, which moves subdiffusively to the most distant sites ${X}_{\text{front}}(t)\ensuremath{\sim}{t}^{\ensuremath{\beta}}$ with an exponent $\ensuremath{\beta}<1$. Importantly, the Anderson model on the RRG can be considered as proxy of the many-body localization transition (MBL) on the Fock space of a generic interacting system. In the final discussion, we outline possible implications of our findings for MBL.