Stability of commensurate phases near the critical temperature: A renormalization-group calculation

The phase diagram of a modulated system in a field which changes the periodicity is investigated near the critical temperature. For certain values of the field, the system can gain energy by locking into phases where the wave vector is commensurable with the reciprocal-lattice vectors. The widths, Δₖ, of these phases are calculated by renormalization-group theory in 4-ε dimensions. We find Δₖ~[(Tc-T)T]^ξₖ, with ${{ξ}}ₖ=1/2(k{-}1)[1+{{2k}²{-}4k+3}{10(k{-}1)}{ε}{-}{{8k}³{-}{20k}²+6k+1}{100(k{-}1)}{{ε}}²],$ where $2k$ is the order of the commensurability. Near ${T}c$, the wave vector locks into every single commensurate value as the field is varied, thus generating a "devil's staircase"-like behavior.