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By Johannes Berg, Gerold Busch

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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 . It turns out, different realizations of Jij have the same thermodynamics. If an intensive quantity is self-averaging, N1 H β,Jij = H β,Jij . Jij Look at the different averages: H β,Jij = Jij = Tr s e−βH[s,J] H Tr s e−βH[s,J] −∂β ln Z(β, J) = −∂β = −∂β ln Z(β, J) Jij Jij Jij −βH ln Tr s e There is a distinction between si and Jij variables: Jij in numerator and denominator are the same, Jij average after ln Z is taken.

The 2d model has to stable and one unstable fixed point. This implies the existence of a phase transition. If we start at exactly x = 1, we will stay there, otherwise, we will go to x = 0. This RG-flow of the coupling constant implies that at increasing length scales, the system looks increasingly disordered. There are two fixed points which map couplings J onto themselves. x = 0 is a stable fixed point, x = 1 is an unstable fixed point. At any temperature T > 0, the system is on long scales described by x(∞) = 0.

For the susceptibility, we yield ∂m 1 ∂ 2 ln Z = ∂h V ∂h2 1 ∂ 2 ln Z = b−d (b1+d/2 )2 V ∂h 2 χ(t, h) = = b2 χ(t , h ) ∝ t−1 = t−γ The critical exponent is γ = 1. • The heat capacity is given by c(t, h) = 1 ∂ 2 ln Z = b−d (b2 )2 V ∂t2 = b4−d c(t , h ) ∝ t− with the critical exponent α = 4−d d=3 = 2 4−d 2 1 ∂ 2 ln Z V ∂t 2 = t−α 1/2. • Finally, we want to determine the field dependence of the magnetization at t = 0. Recapitulate m(t, h) = b1−d/2 m(b2 t, b1+d/2 h) 1 Choosing b now such that b1+d/2 h = 1, we get b = h 1+d/2 and 1−d/2 m(t, h) − 1+d/2 = h t=0 ∝ with a critical exponent δ = d/2+1 d=3 d/2−1 = 1−d/2 − 1+d/2 h 2 − 1+d/2 m h t, 1 = h1/δ 5.

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