Fields below their lower critical dimension: Applications to liquid crystals

In systems with a complex order parameter, ei(x), the correlation function C(x)=ei[(x)-(0)] can be expanded in cumulants g(2n)(x)=[(x)-(0)]2nc. For large x, g(2n)(x)xn where n depends on dimensionality, d, and the Hamiltonian. We introduce two important dimensions, dL* and dL, associated with n. For d>dL*, 1>2>>n, and for dL>d>dL*, 1>0. 1 becomes zero when d=dL and thus dL corresponds to the usual lower critical dimension at which long-range order in ei(x) disappears. For dLd>dL*, the large x behavior of C(x) is therefore determined by g(2)(x). At d=dL* all n 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 is related to the order parameter SC is a gauge where phase fluctuations are a minimum via =SCe-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)=-lneiq0[L(x)-L(0)] is determined by the second cumulant of [L (x)-L (0)]. We evaluate R (x) for 2