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

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

2003-02-26

arxiv.org/abs/cond-mat/0302538v2

Eur. Phys. J. B 33, 203-207 (2003)

Physics

Condensed Matter

Disordered Systems and Neural Networks

6 pages, 2 figures

Scientific paper

10.1140/epjb/e2003-00157-8

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 $\tau=T_c-T$ 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 $\tau$. The transition becomes smoother for infinitesimally small field while it is third order at strictly zero field.

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