Abstract.
We study the ferromagnetic random-field Ising model on random graphs of fixed connectivity z (Bethe lattice) in the presence of an external magnetic field H. We compute the number of single-spin-flip stable configurations with a given magnetization m and study the connection between the distribution of these metastable states in the H-m plane (focusing on the region where the number is exponentially large) and the shape of the saturation hysteresis loop obtained by cycling the field between -∞ and +∞ at T=0. The annealed complexity ΣA(m,H) is calculated for z=2,3,4 and the quenched complexity ΣQ(m,H) for z=2. We prove explicitly for z=2 that the contour ΣQ(m,H)=0 coincides with the saturation loop. On the other hand, we show that ΣA(m,H) is irrelevant for describing, even qualitatively, the observable hysteresis properties of the system.
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Detcheverry, F., Rosinberg, M. & Tarjus, G. Metastable states and T=0 hysteresis in the random-field Ising model on random graphs. Eur. Phys. J. B 44, 327–343 (2005). https://doi.org/10.1140/epjb/e2005-00132-5
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DOI: https://doi.org/10.1140/epjb/e2005-00132-5