The continuous-spin Ising model, g0∶φ4∶d field theory, and the renormalization group

Abstract

We have used the method of high-temperature series expansions to investigate the critical point properties of a continuous-spin Ising model and g0∶φ4∶d Euclidean field theory. We have computed through tenth order the hightemperature series expansions for the magnetization, susceptibility, second derivative of the susceptibility, and the second moment of the spin-spin correlation function on eight different lattices. Our analysis of these series is made using integral and Padé approximants. In three dimensions we find that hyperscaling fails for sufficiently Ising-like systems; the strong coupling limit of g0∶φ4∶3 depends on how the ultraviolet cutoff is removed. The level contours of the renormalized coupling constant for this model in theg 0, correlation-length plane exhibit a saddle point. If the ultraviolet cutoff is removed beforeg 0→ ∞, the usual field theory results and the renormalization-group fixed point with hyperscaling is obtained. If the order of these limits is reversed, the Ising model limit where hyperscaling fails and the field theory is trivial is obtained. In four dimensions, we find that hyperscaling fails completely; g0∶φ4∶4 is trivial for all g0 when the ultraviolet cutoff is removed.