Physica 104A (1980) 115-125 © North-Holland Publishing Co. REAL SPACE RENORMALIZATION GROUP STUDY OF THE RANDOM BOND ISING MODEL Moshe SCHWARTZ Department of Physics and Astronomy, TeI-Aviv University, Ramat-Aviv, Israel and Shmuel FISHMAN Baker Laboratory, Cornell University, Ithaca, New York 14853, USA Received II June 1980 The effects of configurational, quenched, bond randomness on the critical properties of the nearest-neighbour Ising model are studied in two and three dimensions by a real space renor- realization group method. Correlations between the random bonds generated by the renor- malization group transformation are taken into account approximately. This proves necessary in order to obtain correct scaling behaviour, i.e., crossover exponent A = a/v. The transition temperature is calculated approximately as a function of the bond variance. I. Introduction The effect of quenched randomness on continuous phase transitions has been studied extensively recently~). In the present work, we restrict ourselves to weak, uncorrelated, quenched bond randomness. For such randomness, it has been shown 2) that, if the specific heat exponent ot of the pure system is negative, the critical behaviour should not be altered by the randomness. On the other hand, if a is positive, randomness is expected to lead to new critical behaviour. Moreover, from scaling3) and momentum space renormalization group (RG)4) analysis, the stability exponent of the pure system fixed point is A = a/v (where v is the correlation length exponent). Specifically, for the Ising model in three dimensions, a >0 for the pure system, and one expects different critical behaviour in the presence of randomness. In d =4-E dimensions it was shown 5) that the Ising model critical behaviour is described by a fixed point of order E~/2. According I0 experience with the E-expansion, one is inclined to believe that this fixed point will also describe the critical behaviour of the random Ising model in three dimensions. For the Ising model, high temperature series expansions6) and real space RG calculations have not been successful so far in revealing crossover to this new critical behaviour, although distinct random critical behaviour has been obtained for a I15