The 1/2-XXZ spin-chain at finite magnetic field: Crossover phenomena driven by temperature

Abstract

We investigate the asymptotic behaviour of spin-spin correlation functions for the integrable Heisenberg chain. To this end we use the Quantum Transfer Matrix (QTM) technique developed in \cite{AK} which results in a set of non-linear integral equations (NLIE). In the case of the largest eigenvalue the solution to these equations yields the free energy and by modifications of the paths of integration the next-leading eigenvalues and hence the correlation lengths are obtained. At finite field $h>0$ and sufficiently high temperature $T$ the next-leading eigenvalue is unique and given by a 1-string solution to the QTM taking real and negative values thus resulting into exponentially decaying correlations with antiferromagnetic oscillations. At sufficiently low temperatures a different behaviour sets in where the next-leading eigenvalues of QTM are given by a complex conjugate pair of eigenvalues resulting into incommensurate oscillations. The above scenario is the result of analytical and numerical investigations of the QTM establishing a well defined crossover temperature $T_c(h)$ at which the 1-string eigenvalue to the QTM gets degenerate with the 2-string solution. Among other things we find a simple particle-hole picture for the excitations of the QTM and we make contact with the dressed charge formulation of CFT.