Wall wandering and the dimensionality dependence of the commensurate-incommensurate transition

Abstract

The effect of the fluctuation-induced wandering of the domain walls (or "solitons") on the nature of the uniaxial commensurate-incommensurate phase transition at low temperature in a $d$-dimensional system is treated didactically using simple phenomenological arguments which are checked by lattice calculations for $d=2\mathrm{}$. It is found that the domain wall density, or incommensurability, $\overline{q}\left(\delta \right)$, measuring the deviation from the commensurate wave vector, vanishes with the driving potential (or temperature), $\delta$, as ${(\delta -{\delta }_c)}^{\overline{\beta }}$ with $\overline{\beta }=\frac{\mathrm{}(3-d)\mathrm{}}{2(d-1\mathrm{})\mathrm{}}$ for $1<d<~3$. For $d=2\mathrm{}$ this reproduces the result of Pokrovsky and Talapov, and of others; for $d=1\mathrm{}$ no transition occurs; for $d>~3$ the classical result $\overline{q}\sim {\frac{1}{\ln }(\delta -{\delta }_c)}^{-1\mathrm{}}$ always applies (in disagreement with a calculation by Nattermann suggesting $\overline{\beta }=\frac{1}{2}$ for $d\ge 2$).