Incommensurate and commensurate phases in asymmetric clock models

When the ordinary nearest-neighbor p-state clock model (discrete xy model) is generalized to include asymmetric interactions, an incommensurate phase appears for integer p>~3 in addition to the usual liquid and commensurate phases. Aside from being theoretically interesting, it is of practical importance in studies of the commensurate-incommensurate transition where the existence of a discrete nearest-neighbor model with this property gives a computational advantage over further-neighbor and continuum models. For p=3, the incommensurate phase always has a high degree of discommensuration and a Lifshitz point will occur where the incommensurate, liquid, and commensurate phases coincide. For p=2 no incommensurate phase occurs. The system is analyzed at low temperature using a transfer matrix technique recently used by J. Villain and P. Bak to analyze a similar model with further-neighbor interactions.