Random field Ising model : statistical properties of low-energy excitations and of equilibrium avalanches

Abstract

With respect to usual thermal ferromagnetic transitions, the zero-temperature finite-disorder critical point of the Random-field Ising model (RFIM) has the peculiarity to involve some 'droplet' exponent $\theta$ that enters the generalized hyperscaling relation $2-\alpha =\nu (d-\theta )$. In the present paper, to better understand the meaning of this droplet exponent $\theta$ beyond its role in the thermodynamics, we discuss the statistics of low-energy excitations generated by an imposed single spin-flip with respect to the ground state, as well as the statistics of equilibrium avalanches i.e. the magnetization jumps that occur in the sequence of ground-states as a function of the external magnetic field. The droplet scaling theory predicts that the distribution $dl/l^{1+\theta }$ of the linear-size $l$ of low-energy excitations transforms into the distribution $ds/s^{1+\theta /d_f}$ for the size $s$ (number of spins) of excitations of fractal dimension $d_f$ ($s\sim l^{d_f}$). In the non-mean-field region $d<d_c$, droplets are compact $d_f=d$, whereas in the mean-field region $d>d_c$, droplets have a fractal dimension $d_f=2\theta$ leading to the well-known mean-field result $ds/s^{3/2}$. Zero-field equilibrium avalanches are expected to display the same distribution $ds/s^{1+\theta /d_f}$. We also discuss the statistics of equilibrium avalanches integrated over the external field and finite-size behaviors. These expectations are checked numerically for the Dyson hierarchical version of the RFIM, where the droplet exponent $\theta (\sigma )$ can be varied as a function of the effective long-range interaction $J(r)\sim 1/r^{d+\sigma }$ in $d=1$.

arXiv comment24 pages, 15 figures, v2=final version

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