Analytical results on the Heisenberg spin chain in a magnetic field

Abstract

We obtain the ground state magnetization of the Heisenberg and XXZ spin chains in a magnetic field h as a series in , where hc is the smallest field for which the ground state is fully polarized. All the coefficients of the series can be computed in closed form through a recurrence formula that involves only algebraic manipulations. For some values of the anisotropy parameter the expansion is numerically observed to be convergent in the full range .To that end we express the free energy at mean magnetization per site as a series in whose coefficients can be similarly recursively computed in closed form. This series converges for all . The recurrence is nothing but the Bethe equations when their roots are written as a double series in their corresponding Bethe number and in . It can also be used to derive the corrections in finite size, that correspond to the spectrum of a free compactified boson whose Luttinger parameter can be expanded as a similar series.The method presumably applies to a large class of models: it also successfully applies to a case where the Bethe roots lie on a curve in the complex plane.