PHASE-DIAGRAM OF A 3-DIMENSIONAL ANISOTROPIC LONG-RANGE ISING-MODEL VERSUS TEMPERATURE AND MAGNETIC-FIELD

The phase diagram of a three-dimensional ising model in a magnetic field with commensurate and incommensurate phases is studied at low temperature within a mean-field approximation. This analysis was motivated by problems in condensed matter physics concerning, for example, bipolaronic charge-density waves, neutral-to-ionic transitions in organic salts, staging in intercalation compounds, magneto-elastic materials, stacking in discotic liquid crystals, etc. The interactions between the ising spins are long-range exponentially decaying, and are antiferromagnetic in one direction and ferromagnetic in the perpendicular planes. The ground state of this model is known to be a commensurate or an incommensurate structure with a wavevector that varies as a complete devil's staircase as a function of the magnetic field. Provided some conditions hold on the model parameters, we prove rigorously that, for small enough temperature, the mean-field variational form of the model still yields a minimum that is a commensurate or an incommensurate structure, the wavevector of which also varies as a complete devil's staircase as a function of the magnetic field or temperature. The explicit computation of the phase diagram is done by using two methods that are consistent with one another: the first is based on the numerical analysis of a transfer matrix at zero degrees kelvin; the second is based on an approximation valid at low temperature, which allows one to map this mean-field model onto a Frenkel-Kontorova model. This second method yields an analytical expression for the transition lines of the phase diagram.