Spin-spin correlation functions in the frustrated two-dimensional planar model

The low-temperature limit of the frustrated 2-D x-y model is studied via the generalized Villain model. The spin-spin correlation functions are calculated for two limits of the density of frustrated plaquettes "xf." In the dilute case the model is described in terms of the usual spin waves, vortex and symmetry-breaking excitations, plus the "curved" or frustrated half-integer static charges. In this limit, to lowest order in xf, the infinite susceptibility phase of the ferromagnetic model remains. The stability of this result is shown to hold including higher excited states for the static vortices although there seems to be a lower critical temperature. In the special limit xf~1/2 a model with only vortex, symmetry-breaking, and frustrated plaquettes is studied. The spin-spin correlation functions are calculated for large separation distances. There, the susceptibility has an exponential behavior to lowest order in an expansion in the density of thermal vortices. We discuss under which conditions this result would lead us to the nonexistence of spin-glass ordering for the planar model in two dimensions.