The Sherrington–Kirkpatrick model near T_c and near T=0

Abstract

Some recent results concerning the Sherrington–Kirkpatrick model are reported. For T near the critical temperature T _c , the replica free energy of the Sherrington–Kirkpatrick model is taken as the starting point of an expansion in powers of about the replica symmetric solution . The expansion is kept up to fourth order in δ Q where a Parisi solution Q _ab  = Q(x) emerges, but only if one remains close enough to T _c . For T near zero we show how to separate contributions from x ≪ T ≪ 1 where the Hessian maintains the standard structure of Parisi replica symmetry breaking with bands of eigenvalues bounded below by zero modes. For T ≪ x ≤ 1 the bands collapse and only two eigenvalues, a null one and a positive one, are found. In this region the solution stands in what can be called a droplet-like regime.

Keywords:

• disordered systems
• Ising models
• statistical mechanics
• spin glass