A renormalisation group approach to a quantum spin system

The phenomenological renormalisation group transformation, introduced by Nightingale (1976), is extended to a form suitable for the study of a quantum mechanical system at T=0. The spin-1/2 Ising model in a transverse field Gamma , in one and two dimensions, is considered. The new scaling form, called phenomenological time scaling, is derived by exploiting the connection between a d dimensional quantum mechanical system at T=0 and a (d+1) dimensional classical system in the anisotropic limit of weakly coupled chains. Estimates are made of the critical field Gamma c and the exponent nu associated with the vanishing of the first excitation energy at T=0. These results also provide some information about the thermal critical behaviour of the anisotropic three-dimensional Ising model.