Devil's staircases and Manhattan profile in an exact model for an incommensurate structure in an electric field

TLDR: In this article, the ground state of a one-dimensional model with two sublattices is calculated, which is a variation of the piecewise parabola Frenkel-Kontorova model, and the phase diagram of this model as a function of the chemical potential and the electric field is explicitly calculated.

Abstract: The ground state of a one-dimensional model with two sublattices which is a variation of the piecewise parabola Frenkel-Kontorova model. is explicitly calculated. This model involves an additional parameter (the electric field) which breaks a (non-symmorphic) symmetry element when it is non-zero. At a fixed commensurability ratio :. it is shown that the polarisation curve as a function of the electric field is the sum of a linear part and a staircase, This staircase is either a harmless staircase (: rational) with true first-order transitions or a Devil's staircase ( :irrational). The plateaus of these staircases are obtained when an unusual condition (called subcommensurability condition) which involves the relative phase shift between the two sublattices is fulfilled. The phase diagram of this model as a function of the chemical potential and the electric field is explicitly calculated. It is entirely filled by the domain of stability of phases which are characterised both by a rational commensurability ratio and a polarisation fulfilling the subcommensurability condition. These domains are polygons with an infinite number of edges. At fixed electric field. the commensurability ratio {varies as a function of the chemical potential as a Devil's staircase with plateaus at rational values of :but these ones have widths submitted to selection rules related to the model symmetry. The corresponding curve for the variation of the polarisation is a new 'devilish' function called a Manhattan profile. This curve exhibits infinitely many plateaus but unlike a Devil's staircase is alternatively increasing and decreasing infinitely many times by discontinuities. The results obtained on this model are favorably compared with experimental results in thiourea. Predictions are also given and in particular a scintil- lating variation of the polarisation when the temperature or the pressure varies (which is the consequence of the Manhattan profile). Finally. it is noted that the method used for the exact solution of this model can be extended to a wider class of other models: (1) with several sublattices. (2) with long-range interactions. (3) with lattices at any dimension. which should in the future allow more complex models, which approach more closely the real systems. to be solved exactly.