Domain walls, fluctuations and the incommensurate-commensurate transition in two and three dimensions

A low temperature calculation for the incommensurate-commensurate transition in a system with a single axis of modulation is performed. The theory is formulated in terms of wall and phonon degrees of freedom. Phonons renormalize the domain wall interaction to larger values. By means of a simple self-consistent calculation scheme the square root law for the domain wall density in 2 dimensions is reproduced. For d = 3 the classical logarithmic law is found to be valid, in contrast to previous investigations of the author. The results can be interpolated by an exponent β= 3 - d / 2(d - 1) for 1 < d ≤ 3.