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

TL;DR

This paper develops a scaling theory for the Ising spin glass in zero magnetic field near six dimensions, highlighting nonperturbative aspects of the replica symmetry broken phase and addressing infrared divergence issues.

Contribution

It introduces a nonperturbative scaling framework based on Parisi's ultrametric order parameter, resolving infrared divergence problems by keeping the free propagator unexpanded.

Findings

Infrared divergences arise in perturbative calculations of the spin glass phase.

The ta-expansion of critical exponents remains well-behaved despite divergences.

Unexpanded free propagator with two mass scales avoids infrared divergent integrals.

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 w^2. Infrared divergent integrals inevitably appear when we try to compute amplitudes perturbatively, nevertheless the \epsilon-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 w^2 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.