Phase diagram for the generalized Villain model.

Abstract

Berge et al. have generalized Villain's fully frustrated XY model on a square lattice by multiplying the antiferromagnetic exchange constant by a factor \ensuremath{\eta}. Using the Monte Carlo method, they find that the specific heat displays both Ising-type and Kosterlitz-Thouless-type phase transitions with ${T}_{\mathrm{I}<{T}_{\mathrm{KT}}}$, where ${T}_{\mathrm{I}(\mathrm{\ensuremath{\eta}})}$\ensuremath{\rightarrow}${T}_{\mathrm{KT}(\mathrm{\ensuremath{\eta}})}$ as \ensuremath{\eta}\ensuremath{\rightarrow}1, thus implying a multicritical point. Using mean-field theory we find a phase diagram in good qualitative agreement with that found by Berge et al., explicitly producing the tetracritical point at \ensuremath{\eta}=1, and providing a physical picture for the structure of the phases. The nonferromagnetic, collinear phase for \ensuremath{\eta}>1 is found to possess antiferromagnetic order. When a magnetic field H is included, the paramagnetic and ferromagnetic phases coalesce to a single collinear phase, and the antiferromagnetic and noncollinear phases coalesce to a single noncollinear phase. The critical surface ${H}_{c}$(T,\ensuremath{\eta}) separating these phases (which should be characterized by a divergence in the staggered susceptibility) has been determined, again within mean-field theory. A phase-only mode-fluctuation analysis is also presented, yielding results consistent with the mean-field analysis, as well as explicitly revealing the fluctuating modes that become unstable at the transitions; with these modes one can explain the presence (and absence) of susceptibility peaks for the four phase transitions found by Berge et al. For ${T}_{\mathrm{I}<\mathrm{T}<{T}_{\mathrm{KT}}}$, one and only one mode condenses, leading to a standard KT phase transition.