Exact full counting statistics for the staggered magnetization and the domain walls in the XY spin chain

TL;DR

This paper provides exact calculations of full counting statistics for magnetization and domain walls in the XY spin chain, deriving universal formulas and testing conformal field theory predictions at criticality.

Contribution

It introduces an exact method for full counting statistics in the XY chain, including a universal interpolation formula based on Painlevé V equations and asymptotic analysis.

Findings

Universal interpolation formula for full counting statistics

Exact results for domain formation probabilities

Validation of conformal field theory predictions

Abstract

We calculate exactly cumulant generating functions (full counting statistics) for the transverse, staggered magnetization and the domain walls at zero temperature for a finite interval of the XY spin chain. In particular, we also derive a universal interpolation formula in the scaling limit for the full counting statistics of the transverse magnetization and the domain walls which is based on the solution of a Painlev\'e V equation. By further determining subleading corrections in a large interval asymptotics, we are able to test the applicability of conformal field theory predictions at criticality. As a byproduct, we also obtain exact results for the probability of formation of ferromagnetic and antiferromagnetic domains in both $\sigma^z$ and $\sigma^x$ basis in the ground state. The analysis hinges upon asymptotic expansions of block Toeplitz determinants, for which we formulate and check numerically a new conjecture.