Theory of Two-Dimensional Magnets

Abstract

Phase transitions in two-dimensional (2-d) magnets are associated with fluctuational excitations in these 2-d systems. Excitations in the well-known Ising model (Section 2), whose discrete symmetry differs from the continuous symmetries of the XY (Section 3) and Heisenberg (Section 4) models, are linear defects (domain walls). In continuously degenerate systems (XY and Heisenberg models) the low- energy excitations are spin-waves. The energy of such a fluctuation depends quadratically on its wavenumber, resulting in the absence of long-range order in 2-d systems (the so-called Landau—Peierls theorem). But the low-temperature phase of the XY model corresponds to the quasi-long-range order which denotes an algebraic decay of spin correlation functions. This quasi-order results from spin-waves and vortices, which are topological excitations confined below a certain temperature. The system transforms into a paramagnetic state via the vortex unbinding transition. In the Heisenberg model vortices are prohibited; however, an other type of topological excitation called ‘skyrmions’ is allowed. Owing to the non-Abelian symmetry group of the order parameter the quasi-long-range order disappears in the Heisenberg model.