One-Dimensional Anisotropic Heisenberg Model at Finite Temperatures

Abstract

The thermodynamics of the one-dimensional Heisenberg-Ising model for |{Delta}| << 1 as well as of the X-Y-Z model is reduced to a set of non-linear integral equations under some plausible assumptions. It is remarkable that the number of unknown functions involved in them becomes finite when {pi}/ cos -1{Delta} is a rational number for the Heisenberg-Ising model and when Kl /{zeta} is a rational number for the X-Y-Z model (where coupling constants Jx, Jy and Jz are parametrized by {zeta}, l, and Jz as Jx = Jz cn(2{zeta}, l) and Jy = Jz dn(2{zeta}, l); 1 [≥]l [≥]0, Kl [≥]2{zeta}[≥]0 and Kl is the complete elliptic integral of the first kind of modulus l). The validity of our theory has been confirmed by the high-temperature expansion of the free energy through the second term for a general value of {Delta} and through the fourth them for {Delta}= [1/2].