A test of hyperscaling for the spin-1/2Ising model in four dimensions

Abstract

The validity of the hyperscaling relation 2 Delta -dv- gamma =0 is studied for the four-dimensional spin-1/2 Ising model. High-temperature series expansions are derived for the fourth-field derivative chi 0(2) of the free energy on the four-dimensional hyper-face-centred cubic (HFCC) and hyper-body-centred cubic (HBCC) lattices to order 9 and 11 respectively. These are analysed, together with other series already available for the susceptibility chi 0 and correlation length xi for the HFCC, HBCC and the hyper-simple cubic (HSC) lattices. All these series are found to behave consistently with the asymptotic form t-q mod lnt mod p, where t is a reduced temperature variable and q is the appropriate mean-field exponent (so that hyperscaling is satisfied automatically). The best estimates for p are as follows: p=0.30+or-0.05 (HFCC), 0.32+or-0.05 (HBCC) for xi 0 (with q=1= gamma ) and p=0.33+or-0.05 for xi 2 (with q=1=2v). These estimates are in good agreement with the renormalisation group (RG) prediction of p=1/3. Results for chi 0(2) are more slowly convergent, but are still consistent with p=1/3 for q=4=2 Delta + gamma .