By Johannes Berg, Gerold Busch
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Extra info for Advanced Statistical Physics: Lecture Notes (Wintersemester 2011/12)
In low-energy configuration, some bond will be unsatisfied. There is often a large number of ways of placing the unsatisfied bonds which leads to an unstable state. A rough energy landscape can lead to complex low-T behaviour. 1 A mean-field model of spin glasses In spin glasses not all bounds have the same couplings. This leads to interesting properties. Mezard et al. give an easy example: Consider three characters (A,B,C). In a tragedy there is a fight between two groups and the characters have to choose sides.
Nevertheless, as the couplings Jij are all different, si will be sometimes positive, sometimes negative. We therefore decide to square it. The parameter is named after its inventors S. Edwards and P. Andersen. There are several possibilities to investigate the SK-model: • numerical simulation (exercise) • set up self-consistent equations for mi , given Jij . • model the disorder by random variables and average over the Jij . 1 Modeling the disorder by random variables So far, we considered a specific choice of Jij .
This spin-glass model was proposed by D. Sherrington and S. Kirkpatrick in 1975 and solved by Giorgio Parisi in 1979. The proof of the correctness of this solution was done by Tallagrand in 2005! For the Weiss ferromagnet, we use the magnetization m as order parameter and have a symmetry breaking through a magnetic field h: If there is a freezing transition, what would be the order parameter? The symmetry breaking for ferromagnet lim h+ →0+ 1 N →∞ N lim si h+ = m+ i For spin glasses, the approach is quite similar: 1 →0 N →∞ N lim lim si 2 σ = qEA i with hi = σi and σi ∈ ±1.