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

The effect of a random symmetry-breaking field ($V={∫}d{{→}}{x}[{h}₁ₚcosp{θ}({{→}}{x})+{h}₂ₚsinp{θ}({{→}}{x})]$, ${h}₁ₚ$ 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<2πKeff<1/2p². That is, planar-model critical behavior should be observable for threefold-symmetric ($p=3$) perturbations.