Correlation functions of the integrable higher-spin XXX and XXZ spin chains through the fusion method

TL;DR

This paper derives multiple-integral formulas for correlation functions in integrable higher-spin XXX and XXZ spin chains, explicitly including the emptiness formation probability, using fusion methods and Bethe ansatz techniques.

Contribution

It provides a novel single-term multiple-integral representation for correlation functions in higher-spin chains, extending previous results to the massless regime.

Findings

Derived explicit multiple-integral formulas for correlation functions.

Explicitly calculated the emptiness formation probability.

Extended the fusion method to higher-spin XXZ chains in the gapless regime.

Abstract

For the integrable higher-spin XXX and XXZ spin chains we present multiple-integral representations for the correlation function of an arbitrary product of Hermitian elementary matrices in the massless ground state. We give a formula expressing it by a single term of multiple integrals. In particular, we explicitly derive the emptiness formation probability (EFP). We assume $2s$-strings for the ground-state solution of the Bethe ansatz equations for the spin-$s$ XXZ chain, and solve the integral equations for the spin-$s$ Gaudin matrix. In terms of the XXZ coupling $\Delta$ we define $\zeta$ by $\Delta=\cos \zeta$, and put it in a region $0 \le \zeta < \pi/2s$ of the gapless regime: $-1 < \Delta \le 1$ ($0 \le \zeta < \pi$), where $\Delta=1$ ($\zeta=0$) corresponds to the antiferromagnetic point. We calculate the zero-temperature correlation functions by the algebraic Bethe ansatz, introducing the Hermitian elementary matrices in the massless regime, and taking advantage of the fusion construction of the $R$-matrix of the higher-spin representations of the affine quantum group.