On a Renormalization Group Approach to Dimensional Crossover

A recently proposed renormalization group approach to dimensional crossover in quasi-one-dimensional quantum antiferromagnets is improved and then shown to give identical results, in some cases, to those obtained earlier. Recently, [1] a renormalization group (RG) approach was proposed to study a quasione dimensional quantum antiferromagnet, in which the ratio of inter-chain to intra-chain couplings, R → 0. The approach works equally well for crossover to two or three dimensional behavior. It is based on a non-linear σ-model representation. In D space-time dimensions, using the imaginary time formalism, and rescaling time so that the spin-wave velocity is one, the action is written: S = (Λ/2g) ∫ d~x(∇~ φ). (1) Here ~ φ is the unit normalized n-component order parameter. It represents the threecomponent Néel order parameter for the Heisenberg antiferromagnet in (D-1) spatial dimensions at T = 0. Alternatively, it could present a classical n-component magnetic order parameter in D spatial dimensions at finite temperature. Λ is an ultraviolet cut-off and g is the dimensionless coupling constant. In the spin-s quantum antiferromagnet, g ≈ 2/s and increases with frustration. For the classical system, g is proportional to the temperature, T . There are actually two quite distinct cases, corresponding to integer or half-integer spin in the quantum system. For integer spin the (1+1) dimensional system has a finite correlation