Eur. Phys. J. B 33, 203-207 (2003)
https://doi.org/10.1140/epjb/e2003-00157-8

On the Parisi-Toulouse hypothesis for the spin glass phase in mean-field theory

¹ Dipartimento di Fisica, Università di Roma “La Sapienza”, Istituto Nazionale Fisica della Materia Unità di Roma I and SMC, P.le Aldo Moro 2, 00185 Roma, Italy
² SMC-INFM, Dipartimento di Fisica, Università di Roma “La Sapienza”, P.le Aldo Moro 2, 00185 Roma, Italy
³ HAS Research Group for Theoretical Physics, Eötvös University, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary

Corresponding authors: ^a andrea.crisanti@phys.uniroma1.it - ^b tommaso.rizzo@phys.uniroma1.it - ^c temtam@helios.elte.hu

Received: 3 March 2003
Published online: 4 June 2003

Abstract

We consider the spin-glass phase of the Sherrington-Kirkpatrick model in the presence of a magnetic field. The series expansion of the Parisi function q(x) is computed at high orders in powers of and H. We find that none of the Parisi-Toulouse scaling hypotheses on the q(x) behavior strictly holds, although some of them are violated only at high orders. The series is resummed yielding results in the whole spin-glass phase which are compared with those from a numerical evaluation of the q(x). At the high order considered, the transition turns out to be third order on the Almeida-Thouless line, a result which is confirmed rigorously computing the expansion of the solution near the line at finite τ. The transition becomes smoother for infinitesimally small field while it is third order at strictly zero field.