Exact models with a complete Devil's staircase

Abstract

The author describes two exact models which exhibit a complete Devil's staircase which can both be calculated explicitly with the same method. The first one is a discrete Frenkel-Kontorova model (an elastic chain of atoms submitted to a continuous piecewise parabolic periodic potential). The atomic mean distance of the ground state varies with the model parameters as a complete Devil's staircase. The authors also calculates explicitly the Peierls-Nabarro barrier and the depinning force of the chain which is always locked on the periodic potential. (However, it is noted that the lack of differentiability of this periodic potential always makes the chain locked and the Devil's staircase complete, unlike the case for smooth periodic potentials.) The second model is the Ising chain under an external magnetic field and with long-range antiferromagnetic interactions proposed by Bak and Bruinsma (1982). It is rigorously proved that the ground state of this model is periodically modulated and that its magnetisation varies as a function of the magnetic field which is a complete Devil's staircase. This one is also explicitly calculated.