Decay of Complex-Time Determinantal and Pfaffian Correlation Functionals in Lattices

Abstract

We supplement the determinantal and Pfaffian bounds of [4] for many-body localization of quasi-free fermions, by considering the high dimensional case and complex-time correlations. Our proof uses the analyticity of correlation functions via the Hadamard three-line theorem. We show that the dynamical localization for the one-particle system yields the dynamical localization for the many-point fermionic correlation functions, with respect to the Hausdorff distance in the determinantal case. In [4], a stronger notion of decay for many-particle configurations was used but only at dimension one and for real times. Considering determinantal and pfaffian correlation functionals for complex times is important in the study of weakly interacting fermions.

Decay of multi-point correlation functions in $\mathbb{Z}^d$

“…where dist(w, V ) := min v∈V |w − v| is the distance between a point w and a set V . This distance was used in [10,11]. In particular [11] extends some results of [43] to higher dimensions, in the D H distance setting.…”

“…This distance was used in [10,11]. In particular [11] extends some results of [43] to higher dimensions, in the D H distance setting.…”

“…(1.1) Such multi-point correlation functions have been studied in many papers, see e.g. [1,11,14,43]. One of our goals of this paper is to study the decay of multi-point determinant in Z d , and in particular provide answers to open questions proposed in [11,43].…”

“…[1,11,14,43]. One of our goals of this paper is to study the decay of multi-point determinant in Z d , and in particular provide answers to open questions proposed in [11,43].…”

Correction to: Decay of Determinantal and Pfaffian Correlation Functionals in One-Dimensional Lattices

“…On the occasion of the extension of part of our results to complex times and higher dimensions [2], J.-B. Bru and W. de Siqueira Pedra made us aware of a missing assumption in Theorems 1.1 and 1.2 of [1].…”