Two-dimensional axial third-nearest-neighbor Ising model and the S=1/2 anisotropic Heisenberg chain.

Abstract

We consider an axial Ising model with competing interactions up to third neighbors within a low-temperature approximation in terms of interacting and nonreversing domain walls. Assuming small third-neighbor coupling, we find a close relation between the transfer matrix of this classical model and the Hamiltonian of an S=1/2 anisotropic Heisenberg chain in a uniform magnetic field. This connection allows us to obtain the phase boundaries of the ferromagnetic and other low-temperature modulated states. The ground-state energy for arbitrary magnetization values of the Heisenberg chain is shown to provide information about the density of walls or wave number. In contrast with the two-dimensional axial next-nearest-neighbor Ising model, this quantity displays plateaus and reentrant behavior for some relations among the coupling constants. An alternative way to evaluate correlation functions in the antiferromagnetic Heisenberg chain is presented.