Gauge-dependent critical properties of the nematic to smectic-A phase transition in the 1/N-expansion

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Abstract
A generalized de Gennes model for the nematic to smectic-A transition with an N complex component order parameter, Ψ, is studied in the 1/N expansion. This model is similar to the Landau-Ginzburg model for the superconducting transition with deviations, δn, of the director from its uniform equilibrium value, n0, playing the role of the vector potential. It is, however, not gauge invariant, and properties of Ψ change under gauge transformations : ψ = Ψ eiqoL, A = δ n + ∇L. Phase fluctuations in ψ are a maximum in the physical (LC) gauge with δn ⟩ n0 and a minimum in the gauge (SC) with ∇.A = 0. A continuum of gauges parameterized by an angle θ with θ = 0(θ = π/2) corresponding to the LC (SC) gauge is introduced. Isotropic and anisotropic critical behaviour is found as in the ε-expansion. Thermal critical exponents ν and α are independent of gauge. Magnetic exponents γ and η depend continuously on 0 for 0 < θ < π/2. The susceptibility χ LC in the LC gauge is related to that, χSC, in the SC gauge via χLC = χsc(exp g) where g is negative and more than logarithmically singular in reduced temperature, t, and wave number q near criticality because the splay elastic constant K0 1 behaves as a dangerous irrelevant variable. Precursors of the smectic phase Landau-Peierls instability in three dimensions appear in g in the nematic phase. It is argued that this unusual behaviour of χ LC may be responsible for mosaicity seen in X-ray experiments