Critical Behavior of the Structure Factor for Higher Harmonics in Density Wave Systems

The structure factor S_n(q) = 〈ψ_n(q)ψ_n^*(q)〉 associated with fluctuations in the nth harmonic density wave (DW) order parameter ψ_n of a uniaxial system is affected by its bare q dependence, and by the coupling Re(ψ_n^*ψ₁ⁿ). The latter involves correlations in the secondary order parameter ψ₁ⁿ, characterized by a correlation length ξ_n = ξ_n0[(T-T_c)/T_c]^-ν, with the same XY model exponent ν for all n, and decaying as q^-(2-η_n) for large q. For n>1, η_n is large. The ratio X_n = (ξ_n0/ξ₁₀)² is universal, and X₂ = ε/20-ε²/100+O(ε³) in d = 4-ε. This naturally explains previously puzzling experimental results for S₂ at a nematic–smectic- A₂ transition. Similar predictions apply for other secondary order parameter correlations.