Emptiness Formation Probability

TL;DR

This paper establishes rigorous bounds on the decay rate of the emptiness formation probability in the ground state of the Heisenberg XXZ model across different dimensions, confirming some predictions and introducing new bounds for higher dimensions.

Contribution

It provides the first rigorous bounds for the emptiness formation probability in dimensions two and higher, using reflection positivity and path integral methods.

Findings

In 1D, bounds confirm previous integrable system predictions.

In dimensions ≥ 2, new bounds are established.

Decay rate of the probability is of order exp(-c L^{d+1}).

Abstract

We present rigorous upper and lower bounds on the emptiness formation probability for the ground state of a spin-$1/2$ Heisenberg XXZ quantum spin system. For a $d$-dimensional system we find a rate of decay of the order $\exp(-c L^{d+1})$ where $L$ is the sidelength of the box in which we ask for the emptiness formation event to occur. In the $d=1$ case this confirms previous predictions made in the integrable systems community, though our bounds do not achieve the precision predicted by Bethe ansatz calculations. On the other hand, our bounds in the case $d \geq 2$ are new. The main tools we use are reflection positivity and a rigorous path integral expansion which is a variation on those previously introduced by Toth, Aizenman-Nachtergaele and Ueltschi.