On the asymptotic expansion of the correlators in the XX spin chain

TL;DR

This paper investigates the asymptotic behavior of correlators in the XX spin chain, revealing that only the first two terms align with Luttinger liquid theory, while higher-order terms involve states far from Fermi points.

Contribution

It demonstrates that the asymptotic expansion of correlators in the XX spin chain extends beyond Luttinger liquid predictions, including contributions from states with quantum numbers far from Fermi points.

Findings

Only the first two terms match Luttinger liquid theory.

Higher-order terms involve states with particles and holes far from Fermi points.

The full expansion cannot be captured by Luttinger liquid theory.

Abstract

Using the XX- spin chain as an example we show that in general the asymptotic expansion of the correlator at large distances is not given by the sum over the low-lying particle-hole intermediate states. Only the first two terms in the expansion agree with the predictions of the Luttinger liquid theory. The other terms are in general given by the intermediate states with the particles and holes with the quantum numbers far away from the Fermi- points (at the distances of order of the length of the chain). We argue that the whole expansion cannot be described by the Luttinger liquid theory.