Parisi's mean-field solution for spin glasses as an analytic continuation in the replica number

Abstract

Parisi's replica symmetry breaking solution for spin glasses is extended to finite replica number n. The free energy Fp(n) obtained this way, as well as its first two derivatives with respect to n, are shown to join the corresponding values in the Sherrington-Kirkpatrick (SK) solution at a characteristic value ns(T), where stability breaks down in the latter. The continuation composed of the SK branch FSK(n) for n>or=ns(T) and the Parisi branch FP(n) for 0<or=n<or=nS(T) fulfils the requirements of convexity, monotonicity and stability for all n.