Anisotropic Heisenberg Linear Chain at Nonzero Temperature

The statistical mechanics of the anisotropic Heisenberg chain is treated by the Bogolyubov variational method, taking the anisotropic $X-Y$ model as trial Hamiltonian. The free energy of the antiferromagnet is well behaved for all values of the temperature and anisotropy, reducing in the isotropic case to results obtained previously by Katsura and Bulaevskii. The behavior of the ferromagnet in the isotropic case is similar to that found by Katsura.