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

1982; American Physical Society; Volume: 25; Issue: 5 Linguagem: Inglês

10.1103/physrevb.25.3192

1095-3795

Michael E. Fisher, Daniel S. Fisher,

Nonlinear Dynamics and Pattern Formation

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 $d=2$. It is found that the domain wall density, or incommensurability, $\overline{q}(\ensuremath{\delta})$, measuring the deviation from the commensurate wave vector, vanishes with the driving potential (or temperature), $\ensuremath{\delta}$, as ${(\ensuremath{\delta}\ensuremath{-}{\ensuremath{\delta}}_{c})}^{\overline{\ensuremath{\beta}}}$ with $\overline{\ensuremath{\beta}}=\frac{(3\ensuremath{-}d)}{2(d\ensuremath{-}1)}$ for $1<d<~3$. For $d=2$ this reproduces the result of Pokrovsky and Talapov, and of others; for $d=1$ no transition occurs; for $d>~3$ the classical result $\overline{q}\ensuremath{\sim}{\frac{1}{\mathrm{ln}}(\ensuremath{\delta}\ensuremath{-}{\ensuremath{\delta}}_{c})}^{\ensuremath{-}1}$ always applies (in disagreement with a calculation by Nattermann suggesting $\overline{\ensuremath{\beta}}=\frac{1}{2}$ for $d\ensuremath{\ge}2$).