Eur. Phys. J. B 11, 629-634 (1999)
https://doi.org/10.1007/s100510051191

Scaling and infrared divergences in the replica field theory of the Ising spin glass

¹ Institute for Theoretical Physics, Eötvös University, Pázmány Péter sétány 1, Bld. A, 1117 Budapest, Hungary
² Service de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France
³ Department of Physics of Complex Systems, Eötvös University, Pázmány Péter sétány 1, Bld. A, 1117 Budapest, Hungary

Corresponding author: ^a temtam@poe.elte.hu

Received: 23 December 1998
Revised: 23 March 1999
Published online: 15 October 1999

Abstract

Replica field theory for the Ising spin glass in zero magnetic field is studied around the upper critical dimension d=6. A scaling theory of the spin glass phase, based on Parisi's ultrametrically organised order parameter, is proposed. We argue that this infinite step replica symmetry broken (RSB) phase is nonperturbative in the sense that amplitudes of scaling forms cannot be expanded in term of the coupling constant w². Infrared divergent integrals inevitably appear when we try to compute amplitudes perturbatively, nevertheless the ϵ-expansion of critical exponents seems to be well-behaved. The origin of these problems can be traced back to the unusual behaviour of the free propagator having two mass scales, the smaller one being proportional to the perturbation parameter w² and providing a natural infrared cutoff. Keeping the free propagator unexpanded makes it possible to avoid producing infrared divergent integrals. The role of Ward-identities and the problem of the lower critical dimension are also discussed.