Commensurate-incommensurate transitions and a floating devil's staircase

Abstract

Renormalization-group equations for the uniaxial commensurate-incommensurate (C-IC) transition in two dimensions are derived. The soliton density $\ensuremath{\rho}$ is a nonanalytic function of the misfit parameter $\ensuremath{\mu}$ even at high temperatures where only a floating phase (i.e., algebraic correlations with exponent $\ensuremath{\eta}$) is possible. The singularity at $\ensuremath{\mu}\ensuremath{\rightarrow}0$ is $\ensuremath{\rho}\ensuremath{-}{T}_{\ensuremath{\mu}}\ensuremath{\sim}{\ensuremath{\mu}}^{\ensuremath{\eta}\ensuremath{-}3}$, where $T$ is temperature. In the ($T$,$\ensuremath{\mu}$) 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.