The incommensurate-commensurate transition in a random-field model

The effect of random-field-induced domain wall wandering on the nature of the uniaxial incommensurate-commensurate (IC) transition is studied in the framework of a d-dimensional 'random-rod' model. In this model, the random field h(r) couples linearly to the wall elongations. Both short- and long-range field correlations (h(r)h(0)) approximately mod r mod -D+1 are considered. It is found that the domain wall density l -1 vanishes with the driving parameter delta (e.g. temperature, pressure) as ( delta - delta c ) beta with beta =(5-D)/2(D-3) for 3<D<5. For D>d, D has to be replaced by d. If D(d) is linear in d then beta -(d R -d)/2(d-d c - ) (*) for d c - <or=d<d R -where d R and d c - denote the upper critical dimensionality for the roughening transition and the lower critical dimensionality for the occurrence of a conventionally ordered phase, respectively-D(d R )=5, D(d c - )=3. For D>or=5 or d>or=d R the classical result l -1 approximately ln -1 ( delta - delta c ) -1 applies; for D<or=3 or d<or=d c - there is no IC transition. The results for a corresponding bulk random-field model follow from the 'random-rod' model by means of the Villain relation D(d)=(5+2d)/3 for all d, i.e. from (*) with d R =5, d c -= 2. It is argued that the formula (*) applies for a variety of IC transitions with d R and d c - specifying different universality classes.