A Supersymmetric Hierarchical Model for Weakly Disordered 3 d Semimetals
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nnales Henri Poincar´ e
A Supersymmetric Hierarchical Model for Weakly Disordered 3d Semimetals Giovanni Antinucci, Luca Fresta
and Marcello Porta
Abstract. In this paper, we study a hierarchical supersymmetric model for a class of gapless, three-dimensional, weakly disordered quantum systems, displaying pointlike Fermi surface and conical intersections of the energy bands in the absence of disorder. We use rigorous renormalization group methods and supersymmetry to compute the correlation functions of the system. We prove algebraic decay of the two-point correlation function, compatible with delocalization. A main technical ingredient is the multiscale analysis of massless bosonic Gaussian integrations with purely imaginary covariances, performed via iterative stationary phase expansions.
1. Introduction An important conjecture in mathematical quantum mechanics is that disordered, noninteracting, 3d quantum systems display a localization/delocalization transition as a function of the disorder strength [1,7]. The simplest model that is expected to give rise to such transition is the Anderson model, described by a random Schr¨ odinger operator Hω = −Δ + γVω ,
on 2 (Z3 )
(1.1)
with −Δ the lattice Laplacian and Vω a random potential, e.g., (Vω ψ)(x) = ω(x)ψ(x) with {ω(x)}x∈Z3 i.i.d. random variables with variance O(1). From a mathematical viewpoint, a lot is known about this problem for strong disorder, |γ| 1. There, one expects wave packets not to spread in time, and transport to be suppressed (zero conductivity). This phenomenon has been rigorously understood for general d-dimensional models starting from the seminal work [41], where a KAM-type multiscale analysis approach to localization was developed, and later via the fractional moments method [3]. See [6] for a pedagogical review of mathematical results on Anderson localization.
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Instead, for small disorder much less is known from a rigorous viewpoint. In three dimensions, one expects nontrivial transport and an emergent diffusive behavior of the quantum dynamics. Unfortunately, so far no fully satisfactory rigorous result is available on this problem. Results have been obtained for tree graphs and similar structures [4–6,13,14,40,51,52,69]. The analogous problem for random matrix models is much better understood; see [32] for a review of recent results. Concerning short-ranged lattice models, important progress has been obtained in [33–35], where diffusion for the Anderson model has been proven in the scaling limit, and in [27,28], where a localization/delocalization transition for a supersymmetric effective model has been established. (See also [24] for more recent extensions.) The starting point of [27,28] is a mapping of the disorder-averaged correlations of the Anderson model into those of an interacting supersymmetric quantum field theory model. This mapping was first introduced in physics in [30] (see also [75], for a related approach based on the replica trick) and allows to import field-theoretic metho
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