The spectrum of fluctuations around Sompolinsky's mean field solution for a spin glass

Abstract

The spectrum of the stability matrix associated with Sompolinsky's solution (1981) for a long-range spin glass is studied near Tc in a magnetic field. It is shown that the reparametrisation (or gauge) invariance of the theory locks together not only the order parameters q(x) and Delta (x) but also their fluctuations, and gives rise to a gauge-invariant spectrum which is therefore the same both for the Parisi solution (1979) and for the Sompolinsky solution. In the limit of zero magnetic field, earlier results for the Parisi solution are recovered. The marginal stability of both theories is demonstrated for all fields in the Almeida-Thouless phase.