Stability of a fixed point in the replica action for the random field Ising model

TL;DR

This paper reexamines the stability of the fixed point in the effective action for the random field Ising model near six dimensions, finding it stable and computing critical exponents consistent with dimensional reduction at leading order.

Contribution

It clarifies the stability of the non-trivial fixed point and computes critical exponents using epsilon expansion, challenging previous instability claims.

Findings

The fixed point in 6−ε dimensions is stable.

Critical exponents ν and η are computed and consistent with dimensional reduction.

Results support stability of the fixed point at leading order.

Abstract

We reconsider stability of the non-trivial fixed point in $6-\epsilon $ dimensional effective action for the random field Ising model derived by Br\'{e}zin and De Dominicis. After expansion parameters of physical observables are clarified, we find that the non-trivial fixed point in $6-\epsilon$ dimensions is stable, contrary to the argument by Br\'{e}zin and De Dominicis. We also computed the exponents $\nu$ and $\eta$ by the $\epsilon$ expansion. The results are consistent with the argument of the dimensional reduction at least in the leading order.