# Decay of Complex-time Determinantal and Pfaffian\ Correlation Functionals in Lattices

## TL;DR

This paper extends bounds on many-body localization of quasi-free fermions to high dimensions and complex times, using analyticity and the Hadamard three-line theorem to show decay of correlation functions.

## Contribution

It generalizes previous one-dimensional, real-time results to high-dimensional, complex-time correlation functionals using complex analysis techniques.

## Key findings

- Dynamical localization for one-particle systems implies localization for many-point fermionic correlations.
- Decay bounds are established for complex-time determinantal and Pfaffian correlation functionals.
- Results are relevant for understanding weakly interacting fermions in high-dimensional systems.

## Abstract

We supplement the determinantal and Pfaffian bounds of Sims and Warzel (Commun Math Phys 347:903--931, 2016) 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 Sims and Warzel (2016), 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.