Incommensurate and commensurate phases in asymmetric clock models

When the ordinary nearest-neighbor $p$-state clock model (discrete $\mathrm{xy}$ 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.

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