Analytical results on the Heisenberg spin chain in a magnetic field

We obtain the ground state magnetization of the Heisenberg and XXZ spin chains in a magnetic field h as a series in , where h _c 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. (paper)