Fields below their lower critical dimension: Applications to liquid crystals

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Abstract
In systems with a complex order parameter, eiϕ(x), the correlation function C(x)=〈ei[ϕ(x)−ϕ(0)]〉 can be expanded in cumulants g(2n)(x)=〈[ϕ(x)−ϕ(0)]2n〉c. For large x, g(2n)(x)∼xωn where ωn depends on dimensionality, d, and the Hamiltonian. We introduce two important dimensions, dL* and dL, associated with ωn. For d>dL*, ω1>ω2>⋯>ωn, and for dL>d>dL*, ω1>0. ω1 becomes zero when d=dL and thus dL corresponds to the usual lower critical dimension at which long-range order in eiϕ(x) disappears. For dL≥d>dL*, the large x behavior of C(x) is therefore determined by g(2)(x). At d=dL* all ωn are equal, and all cumulants are needed. After introducing these concepts with a nonlinear spin-wave model, we consider applications to correlations in liquid crystals where the physical order parameter ψ is related to the order parameter ψSC is a gauge where phase fluctuations are a minimum via ψ=ψSCe−iq0L where q0L is the phase associated with a gauge transformation. We show that for q0L, dL*≤2 in all cases. Thus the large x behavior of R(x)=−ln〈eiq0[L(x)−L(0)]〉 is determined by the second cumulant of [L(x)−L(0)]. We evaluate R(x) for 2<d<4. At the anisotropic critical point [ν∥=(5−d)ν⊥] in three dimensions, R(x)∼(lnx)2 rather than Inx as previosuly reported. In addition, we show that the decoupling approximation, G(x)=〈ψ(x)ψ*(0)〉∼〈ψSC(x)ψSC*(0)〉e−R(x), is valid in the smectic-A and nematic phases and at the critical point when there is isotropic scaling (ν∥=ν⊥) for 2<d<4 and when there is anisotropic scaling for 3<d<4. The decoupling approximation breaks down except in the smectic-A phase when there is anisotropic scaling for d≤3.