Griffiths singularity in the random Ising ferromagnet

TL;DR

This paper derives the explicit form of the Griffiths singularity in the random ferromagnetic Ising model, showing an essential singularity in free energy near zero magnetic field due to replica instantons.

Contribution

It provides a detailed derivation of the Griffiths singularity in the random Ising model using the replica formalism and identifies the role of instanton excitations.

Findings

Free energy exhibits an essential singularity as h -> 0 in the paramagnetic phase.

The singularity has the form exp[-(const)h^{-D/3}] for 1<D<4.

Replica instantons are responsible for the Griffiths singularity.

Abstract

The explicit form of the Griffiths singularity in the random ferromagnetic Ising model in external magnetic field is derived. In terms of the continuous random temperature Ginzburg-Landau Hamiltonian it is shown that in the paramagnetic phase away from the critical point the free energy as the function of the external magnetic field h in the limit h -> 0 has the essential singularity of the form $\exp[-(const)h^{-D/3}]$ (where 1<D<4 is the space dimensionality). It is demonstrated that in terms of the replica formalism this contribution to the free energy comes due to off-perturbative replica instanton excitations.