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

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 zeta , 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 ( zeta rational) with true first-order transitions or a Devil's staircase ( zeta 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. At fixed electric field, the commensurability ratio zeta varies as a function of the chemical potential as a Devil's staircase with plateaus at rotational values of zeta 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.