Exact four-spinon dynamical correlation function of the Heisenberg model

TL;DR

This paper derives the exact four-spinon contribution to the dynamical correlation function of the S=1/2 Heisenberg model, providing new analytical expressions and studying various limits with implications for understanding quantum spin chains.

Contribution

It presents the first exact expression for the four-spinon contribution using quantum affine symmetry, extending previous two-spinon results and analyzing isotropic and Ising limits.

Findings

Exact four-spinon contribution derived

Series representation in isotropic limit obtained

Explicit first-order Ising limit expression provided

Abstract

In this paper we derive the exact expression of the four-spinon contribution to the dynamical correlation function of the spin S= 1/2 anisotropic (XXZ) Heisenberg model in the antiferromagnetic regime. We extensively study its isotropic (XXX) limit and derive perturbatively the Ising one. Our method relies on the quantum affine symmetry of the model, which allows for a systematic diagonalization of the Hamiltonian in the thermodynamic limit and for an exact calculation of matrix elements of local spin operators. In fact, we argue that the familiar criticism of this method related to the complication of these matrix elements is not justified. First, we give, in the form of contour integrals, an exact expression for the n-spinon contribution. After we compile recently found results concerning the two-spinon contribution, we specialize the n-spinon formula to the new case n=4. Then we give an explicit series representation of this contribution in the isotropic limit. Finally, after we show that this representation is free of divergences, we discuss the Ising limit in which a simple expression is found up to first order in the anisotropy parameter.