Two-dimensional-lattice spin models with long-range antiferromagnetic interactions

We consider a classical system, consisting of m-component unit vectors (m=2,3), associated with a two-dimensional lattice {uₖ{}k{∈}openZ²} and interacting via translationally and rotationally invariant antiferromagnetic pair potentials of the long-range form W=Wⱼₖ={ε}{}xⱼ-xₖ{{{}}}^{{{-}}p}uⱼ{·}uₖ, p>2, where {ε} is a positive quantity, setting energy and temperature scales (i.e., T*=kBT/{ε}), and xₖ are the coordinates of the lattice sites. A spin-wave approach predicts orientational disorder (in the thermodynamic limit) at all finite temperatures and for all p>2; this agrees with available rigorous results for p{≥}4, whereas no such theorems are known in the literature when 2p4. We also report here Monte Carlo simulation results for the model defined by m=2, p=3; calculations were carried out between T*=0.1 and T*=1, and sample-size effects were investigated at the lowest temperature examined. Energy and specific-heat results showed a rather smooth behavior, and no sample-size effect; on the other hand, we found a significant amount of finite-size order, and its pronounced decrease with increasing sample size. In the absence of more stringent rigorous results, we conjecture that the present potential models are disordered at all finite temperatures, for all p>2.