Stability of commensurate phases of a planar model with competing interactions

Abstract

We study a one-dimensional planar model with competing interactions at zero temperature in the limit of low and high external magnetic fields. The stability of commensurate phases with wave number ${q}_{0}=2\ensuremath{\pi}(P/Q)$, where $P/Q$ is an irreducible fraction, is discussed in the continuum approximation. We show that for small fields $H\ensuremath{\rightarrow}0+$, the helical phase with period $Q>4$ has a width proportional to ${H}^{Q/2}$. Similarly, for high fields $H\ensuremath{\rightarrow}{H}_{c}\ensuremath{-}$, where ${H}_{c}$ is the critical field for the parafan transition, the fan phase with period $Q>~4$ has a width proportional to $|H\ensuremath{-}{H}_{c}{|}^{(Q\ensuremath{-}2)/4}$ if $Q$ is an even number and $|H\ensuremath{-}{H}_{c}{|}^{(Q\ensuremath{-}1)/2}$ if $Q$ is an odd number.