Soluble renormalization groups and scaling fields for low-dimensional Ising systems

A variety of one-dimensional Ising spin systems, including staggered and parallel magnetic fields, alternating and second neighbor interactions, four-spin coupling, etc., are discussed in terms of renormalization group theory. A continuous range of distinct renormalization groups is constructed in exact closed form, analyzed in detail, and compared with exactly calculated thermodynamic properties. Fixed point linearization yields relevant, irrelevant, and marginal operators. All groups yield identical “critical” behavior (at T = 0) with η = 1, δ = ∞, γ = ν = 2 − α, and with identical linear scaling fields. A generalization of Wegner's analysis to discrete groups yields explicit power series for the nonlinear scaling fields; these are seen to depend on the particular renormalization group and, hence, are physically nonunique. A planar, multiconnected “truncated tetrahedron” model of effective dimensionality log2 3 is analyzed via a dedecoration and star-triangle group revealing highly singular behavior as T → Tc = 0.