Commensurate-incommensurate transitions and a floating devil's staircase

Renormalization-group equations for the uniaxial commensurate-incommensurate (C-IC) transition in two dimensions are derived. The soliton density $\rho$ is a nonanalytic function of the misfit parameter $\mu$ even at high temperatures where only a floating phase (i.e., algebraic correlations with exponent $\eta$) is possible. The singularity at $\mu \to 0$ is $\rho -T_{\mu }\sim {\mu }^{\eta -3\mathrm{}}$, where $T$ is temperature. In the ($T$,$\mu$) plane the floating phase is singular therefore on all lines where its density (relative to the substrate) is rational; this is the remnant of the low-temperature devil's staircase. At low temperatures a matching procedure with the fermion approach is obtained.