Localization transition in three dimensions: Monte Carlo simulation of a nonlinear σ model

Abstract

We present a combination of analytical and numerical calculations for the critical behavior of a supersymmetric non-linear -model. This model is expected to describe at least qualitatively the localization transition of a disordered one-electron system. As a result, we obtain a localization length exponent and a set of inverse participation numbers in three dimensions. We nd a continuous phase transition with the features of one-parameter scaling and multifractality at the critical point.

Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model

“…With increasing temperature T (or decreasing field stiffness β) the model in d = 3 is expected to undergo an Anderson-type transition to the phase of unbroken symmetry. This phase transition was studied numerically in [4], where the critical value of β was found to be β c ≈ 0.04. The transition has also been investigated in detail using the Migdal-Kadanoff renormalization scheme [5].…”

“…The importance of the regularization ε, which was omitted from the present argument, becomes evident from the saddle point discussed below. 4. The model at hand describes a disordered quantum system at zero temperature.…”

Phase transition in a model with non-compact symmetry on Bethe lattice and the replica limit

“…ordered type"(16) is not contained in the Eq. (18) or, rather, it corresponds to trivial solution P = 0.The qualitative behaviour of the solution of the Eq.…”

Spin systems with hyperbolic symmetry: a survey

“…For the regular tree (Bethe lattice), further remarkable critical behaviour was observed, in part numerically, in [73]. This reference concerns a more complicated sigma model, but the main predictions also apply to the H 2|2 model [36]. On Z 2 the VRJP is believed to be positive recurrent, i.e., exponentially localised, but this important conjecture about the H 2|2 model and the VRJP remains open.…”

“…Is there an upper critical dimension? For d = 3 aspects of this question were studied numerically in [36], and evidence was found for the existence of a multifractal structure in the H 2|2 model. Multifractal structure is also expected near the Anderson transition for random Schrödinger operators.…”