Critical behavior of the structure factor for higher harmonics in density wave systems

The structure factor Sn(q)=ψn(q)ψn*(q) associated with fluctuations in the nth harmonic density wave (DW) order parameter ψn of a uniaxial system is affected by its bare q dependence, and by the coupling Re(ψn*ψ1n). The latter involves correlations in the secondary order parameter ψ1n, characterized by a correlation length ξn=ξn0[(T-Tc)/Tc]-ν, with the same XY model exponent ν for all n, and decaying as q-(2-ηn) for large q. For n>1, ηn is large. The ratio Xn=(ξn0/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.