Exact Solution of Frenkel-Kontorova Models with a Complete Devil's   Staircase in Higher Dimensions
  We solve exactly a class of Frenkel-Kontorova models with piecewise parabolic potential, which has $d$ sub-wells in a period. With careful analysis, we show that the phase diagram of the minimum enthalpy configurations exhibits the structure of a complete $d$-dimensional devil's staircase. The winding number of a minimum enthalpy configuration is locked to rational values, while the fraction of atoms in each sub-well is locked to values which are sub-commensurable with the winding number. 
Hsien-chung Kao, Shih-Chang Lee, and Wen-Jer Tzeng