Effects of a random symmetry-breaking field on topological order in two dimensions

Abstract

The effect of a random symmetry-breaking field ($V=\int d\overrightarrow{\mathrm{x}}\left[h_{1p}cosp\theta \right(\overrightarrow{\mathrm{x}})+h_{2p}sinp\theta (\overrightarrow{\mathrm{x}}\left)\right]$, $h_{1p}$ and $h_{2p}$ random) on topological order in two-dimensional systems is studied. Such a field would simulate the interaction of the idealized two-dimensional system with an underlying substrate if the substrate were to contain patches (islands) in which short-range order is retained rather than being a single crystalline surface. These conditions might well pertain when charge-density waves are studied in chemisorption and physisorption experiments. The transition is studied using a sine-Gordon field theory modified to include the effects of disorder. It is shown that the spin-wave behavior of the planar model is stable against such perturbations provided $4<2\pi K_{\mathrm{eff}}<\frac{1}{2}{p}^2$. That is, planar-model critical behavior should be observable for threefold-symmetric ($p=3\mathrm{}$) perturbations.