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

Abstract

We establish bounds on the decay of time-dependent multipoint correlation functionals of one-dimensional quasi-free fermions in terms of the decay properties of their two-point function. At a technical level, this is done with the help of bounds on certain bordered determinants and pfaffians. These bounds, which we prove, go beyond the well-known Hadamard estimates. Our main application of these results is a proof of strong (exponential) dynamical localization of spin-correlation functions in disordered XY -spin chains.

Localization properties of the disordered XY spin chain

“…In Section 5 we present results of [55], showing that all eigenstates of the disordered XY chain, as well as thermal states at any inverse temperature, have exponentially decaying correlations. In this context we stress that, in order to be interpreted as an MBL property, it is important to be able to go beyond ground state correlations, as MBL should reflect the existence of a mobility gap without the need of a ground state gap.…”

“…Here we will discuss a broad range of results of this type. In particular, we will describe recent localization results in the disordered XY chain and closely related disordered free Fermion systems from [1,2,29,48,55]. In the last two subsections we will also include several previously unpublished results, where we will provide detailed proofs.…”

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

“…Bru and W. de Siqueira Pedra made us aware of a missing assumption in Theorems 1.1 and 1.2 of [1]. We correct both statements below.…”

“…However, this assumption fails to cover the case of general s, t ∈ I n when ρ(s, t) = e is H e −it H with some state. Similarly, the correct statement of Theorem 1.2 in [1], which pertains to random operators, is given below.…”

“…We correct both statements below. Since neither Theorems 1.1 nor 1.2 have further application in [1], this does not affect other results concerning Pfaffian bounds and the time-dependent correlations of the XY spin chain. The proof of the following statements is included in the original paper.…”