Critical properties of Dyson's hierarchical model

Abstract

Critical properties of Dyson's hierarchical Ising model in one dimension with 'potential' falling off like r-(1+ sigma ) are examined in the range 0< sigma <1 where a phase transition is known to occur. A new exact renormalisation group recursion relation is derived for a 'dual' spin probability density function. Together with a scaling-type assumption the authors obtain eta =2- sigma and delta =(1+ sigma )/(1- sigma ) for all 0< sigma <1. Independent evidence, however, suggests that delta =3 for 0< sigma <or=1/2, along with other classical critical exponents in this region. In the non-classical region 1/2< sigma <1 accurate numerical values are obtained for the critical exponents nu and gamma and various Delta sigma = sigma -1/2 expansions to third order in Delta sigma .