Abstract
This paper shows that the Anderson transition on random graphs has two critical localization lengths, which control the critical behavior of specific observables, and are associated with two different critical exponents (the known ${\ensuremath{\nu}}_{\ensuremath{\parallel}}=1$ for the average localization length ${\ensuremath{\xi}}_{\ensuremath{\parallel}}$ and the new ${\ensuremath{\nu}}_{\ensuremath{\perp}}=0.5$ for the typical localization length ${\ensuremath{\xi}}_{\ensuremath{\perp}}$). The behavior we find for ${\ensuremath{\xi}}_{\ensuremath{\perp}}$ is identical to the recent predictions for the many-body localization transition, strongly suggesting that both transitions belong to the same universality class.
Highlights
There has been a huge interest recently in the nonergodic properties of many-body states [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16], in particular related to many-body localization (MBL) [17,18]
Our results clearly show that there exist two different localization lengths in the Anderson transition on random graphs, ξ describing rare branches and ξ⊥ describing the bulk, which control the critical behavior of different physical observables and are associated with distinct critical exponents ν ≈ 1 and ν⊥ ≈ 0.5
This clarifies the nature of the Anderson transition in the limit of infinite dimensionality, which remains, for the bulk properties, a continuous, secondorder phase transition, while rare events, characteristic of random graphs, are responsible for the discontinuous properties described up to now
Summary
We present a full description of the nonergodic properties of wave functions on random graphs without boundary in the localized and critical regimes of the Anderson transition. We find that they are characterized by two critical localization lengths: the largest one describes localization along rare branches and diverges with a critical exponent ν = 1 at the transition. Our results are identical to the theoretical predictions for the typical localization length in the many-body localization transition, with the same critical exponent This strongly suggests that the two transitions are in the same universality class and that our techniques could be directly applied in this context
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