Geometric expansion of fluctuations and average shadows

TL;DR

This paper develops a geometric method to analyze the asymptotic behavior of cumulants of local observables, revealing their dependence on the shape and topology of the region, with applications to quantum Hall systems.

Contribution

It introduces a systematic approach linking cumulant expansion coefficients to geometric and topological features of regions, including convex moments and Euler characteristics.

Findings

Cumulant expansion coefficients relate to convex geometric moments.

Odd cumulants in 2D depend on the Euler characteristic.

Monte Carlo calculations demonstrate skewness in quantum Hall states.

Abstract

Fluctuations of observables provide unique insights into the nature of physical systems, and their study stands as a cornerstone of both theoretical and experimental science. Generalized fluctuations, or cumulants, provide information beyond the mean and variance of an observable. In this paper, we develop a systematic method to determine the asymptotic behavior of cumulants of local observables as the region becomes large. Our analysis reveals that the expansion is closely tied to the geometric characteristics of the region and its boundary, with coefficients given by convex moments of the connected correlation function: the latter is integrated against intrinsic volumes of convex polytopes built from the coordinates, which can be interpreted as average shadows. A particular application of our method shows that, in two dimensions, the leading behavior of odd cumulants of conserved quantities is topological, specifically depending on the Euler characteristic of the region. We illustrate these results with the paradigmatic strongly-interacting system of two-dimensional quantum Hall state at filling fraction $1/2$, by performing Monte-Carlo calculations of the skewness (third cumulant) of particle number in the Laughlin state.