Devil's staircase in a one-dimensional model

A one-dimensional model of a chain of canted arrows in an external field is proposed to describe some quasi-one-dimensional systems, e.g., spin-, charge-, and mass-density-wave systems, and helical polymers. The chain has a natural cantedness measured by an angle $\alpha$, and is also subject to an externally applied aligning field of strength $\gamma$. At $T=0\mathrm{}$, the $\alpha -\gamma$ phase diagram has an infinite number of commensurate phases, and the variation with $\alpha$ of the mean angle between nearest-neighbor arrows has the form of a devil's staircase for all $\gamma$. The transfer-integral technique is used to calculate the Helmholtz free energy and the order parameter of the system as functions of temperature. No sharp transitions occur at $T>\mathrm{}0\mathrm{}$, but some of the features of the $T=0\mathrm{}$ phase diagram persist at low temperatures. Transitions occur from one phase to another when the energy required to create a kink in one phase becomes zero. The equations determining the minimum-energy configurations are rewritten so as to define a two-dimensional area-preserving mapping; fixed points and invariant curves are found for this mapping. Invariant curves are either smooth and continuous or chaotic. The relation of the nature of the invariant curves to the question of the completeness of the devil's staircase is discussed.