Correlation functions of the XXZ Heisenberg spin- chain in a magnetic field

Abstract

Using the algebraic Bethe ansatz method, and the solution of the quantum inverse scattering problem for local spins, we obtain multiple integral representations of the n-point correlation functions of the XXZ Heisenberg spin-1 2 chain in a constant magnetic field. For zero magnetic field, this result agrees, in both the massless and massive (anti-ferromagnetic) regimes, with the one obtained from the q-deformed KZ equations (massless regime) and the representation theory of the quantum affine algebra U q (ŝl 2 ) together with the corner transfer matrix approach (massive regime).

Spin–spin correlation functions of the XXZ- Heisenberg chain in a magnetic field

“…A proof of these results together with their extension to non-zero magnetic field has been obtained in 1999 [14,15] for both regimes using algebraic Bethe ansatz [16,17] and the actual resolution of the so-called quantum inverse scattering problem [14,18].…”

“…In the thermodynamic limit, these multiple sums lead to m fold integrals over contours C h j which depend on the value of j, on the regime considered and also on the value of the magnetic field. The answer can be written generically as [15] …”

“…Following the paper [15], we consider below the action of the monodromy matrix elements on the dual state which can be constructed similarly to (2.8) via the operators C(λ) as…”

“…To compute these elementary blocks, the following successive problems have to be addressed [14,15]: (i) determination of the ground state ψ g |, (ii) evaluation of the action of the product of the local operators E j , j j on this ground state, and (iii) computation of the scalar product of the resulting state with |ψ g .…”

“…For the XXZ spin-1 2 Heisenberg chain in a magnetic field, these problems have been solved in [14,15] in the framework of algebraic Bethe ansatz. The central object of this method is the so called quantum monodromy matrix depending on a complex variable λ (the spectral parameter).…”