Localization on Certain Graphs with Strongly Correlated Disorder

Abstract

Many-body localisation in interacting quantum systems can be cast as a disordered hopping problem on the underlying Fock-space graph. A crucial feature of the effective Fock-space disorder is that the Fock-space site energies are strongly correlated -maximally so for sites separated by a finite distance on the graph. Motivated by this, and to understand the effect of such correlations more fundamentally, we study Anderson localisation on Cayley trees and random regular graphs, with maximally correlated disorder. Since such correlations suppress short distance fluctuations in the disorder potential, one might naively suppose they disfavour localisation. We find however that there exists an Anderson transition, and indeed that localisation is more robust in the sense that the critical disorder scales with graph connectivity K as √ K, in marked contrast to K ln K in the uncorrelated case. This scaling is argued to be intimately connected to the stability of many-body localisation. Our analysis centres on an exact recursive formulation for the local propagators as well as a self-consistent mean-field theory; with results corroborated using exact diagonalisation.

Localization in quasiperiodic chains: A theory based on convergence of local propagators

“…We obtain the distributions using both the continued fractions as well as self-consistently; both of which show excellent mutual agreement, as well as with results obtained from exact diagonalisation. One main result here is that the distributions of y(ω) in the localised phase have Lévy tails ∝ y −3/2 ; which likewise arise in localised phases of disordered systems, for both uncorrelated [36] and correlated disorder [39], suggesting they are rather universal in localised systems. It is also of course because of these fat-tails that the geometric mean (y typ ) is Lévy-tailed distribution for y(ω) CF for y(ω) convergent y typ (ω) finite ∆ typ (ω) finite…”

“…which have their own continued-fraction representations. Physically, ∆ 0 (n) is the contribution to ∆ 0 of a process where the particle goes out to site n from the root site 0 and retraces its path back to the root [39]. Such path contributions are also present in the forward-scattering approximation [42], albeit in an un-renormalised fashion.…”

Self-consistent theory of mobility edges in quasiperiodic chains

“…Two notable points to take away from the analytic expressions are, however, that (i) the distributions have a ∝y −3/2 power-law (Lévy) tail and (ii) the support of the distributions have a sharp lower cutoff, which arises because the j 's have a bounded distribution. We add that the Lévy tail in P y (y) seems quite universal, as it arises also in Anderson localization in disordered systems in the presence of both uncorrelated [19] as well as maximally correlated disorder [32]. In summary, we have introduced a self-consistent theory for MEs in quasiperiodic chains with nearest-neighbor hoppings.…”

“…(12) and (14) show that y typ diverges as y typ ∼ (ω − ω ME ) −1 on approaching the ME from the localized side [with y typ ∼ (V − 2) −1 as V → 2 + in the ω-independent AAH limit]. As this divergence is proportional to that of the localization length ξ (ω) [32], ξ (ω) thus diverges with a critical exponent of ν = 1, which likewise agrees with the exactly known Lyapunov exponents for the mosaic [11] and AAH [33] models.…”

How periodic driving stabilizes and destabilizes Anderson localization on random trees

“…While the problem of many-body localization (driven or otherwise) can also be equivalently viewed as one of singleparticle localization on the high-dimensional Fock-space graph, strong correlations in the latter render it qualitatively different from conventional Anderson localization on such graphs [20,21]. Nevertheless, a natural step towards a theoretical understanding of the mechanism governing localization in Floquet systems is to analytically study a simpler problem in a more controlled setting-the fate of Anderson transitions and localization on high-dimensional graphs.…”