Locking of Commensurate Phases in the Planar Model in an External Magnetic Field

Commensurate configuration locking is known in models like the anisotropic next-nearest-neighbor Ising model and the Frenkel-Kontorova model. We find an analogous scenario in the planar model with competing interactions when an external magnetic field is applied in the plane in which the spins lie. This model falls in the same symmetry class of the Heisenberg model with planar anisotropy. We performed a low-field, low-temperature expansion for the free energy of the model and we find phase locking energy for states with wave vectors of the form G/p where p is an integer and G is a reciprocal-lattice vector. The helix characterized by p=3 is peculiar because the commensuration energy vanishes at zero temperature. The helix corresponding to p=4 is not stable against the switching of a magnetic field that forces the spins into an up-up-down-down configuration analogous to the spin-flop phase of an antiferromagnet. For a generic commensurate value of p>4, we expect locking both at zero and finite temperature as we have verified for p=5 and 6. The consequences of our results are examined for the 3N model (a tetragonal spin lattice with in-plane competitive interactions up to third-nearest neighbors).