Renormalization Group Calculation for Critical Points of Higher Order with General Propagator*

Recently, Horrireich, Luban and Shtrikman have / used renormalization group techniques to discuss the ~ d~/,,= 1. (3) onset of helical okder inmagnetic systems [1, 2]. lii i 1 particular, the existence of new types of critical beFor o~= 2 these conditions reduce to those given prehavior has been p~stulatedfor the “Lifshitz” point viously [5, 6]. In some cases (such as isotropic propawhere the transition from a uniformly ordered to a gators [1—3])dmin may be larger than three. A more helically ordered state occurs. At such a point, the interesting physical case is obtained if only one cornpropagator differs from the usual Wilson form, G 1 ponent of kenters G1 as k2L and the remaining = k2 + r. Refs. [1—2] consider propagators of the components have k2 dependence. Eqs. (2)—(3) then form G~= + k~+r wherek~is ad 1-dimensional give db =(30 — 1)1(0 — 1)— 1/Land dmin = 3 wave vector, and d1 + d2 = d, the dimension of the Thus, we haye db >3 > dmin for all 0 <2L + 1. lattice. For anisotropic systems, the critical point expoHere we consider critical propagators (r= 0) of the nents {r~}are defined by examining the behavior of form the critical two-poii~it function for a wave-vector lying J G~ =~ 1k11°’, (1) entirely in one of the d1-dimensional subspaces: