Equilibrium avalanches in spin glasses

We study the distribution of equilibrium avalanches (shocks) in Ising spin glasses which occur at zero temperature upon small changes in the magnetic field. For the infinite-range Sherrington-Kirkpatrick (SK) model, we present a detailed derivation of the density $\rho \left(\Delta M\right)$ of the magnetization jumps $\Delta M$. It is obtained by introducing a multicomponent generalization of the Parisi-Duplantier equation, which allows us to compute all cumulants of the magnetization. We find that $\rho \left(\Delta M\right)\sim \Delta M^{-\tau }$ with an avalanche exponent $\tau =1$ for the SK model, originating from the marginal stability (criticality) of the model. It holds for jumps of size $1\ll \Delta M<N^{1/2}$, being provoked by changes of the external field by $\delta H=O\left(N^{-1/2}\right)$ where $N$ is the total number of spins. Our general formula also suggests that the density of overlap $q$ between initial and final states in an avalanche is $\rho \left(q\right)\sim 1/(1-q)$. These results show interesting similarities with numerical simulations for the out-of-equilibrium dynamics of the SK model. For finite-range models, using droplet arguments, we obtain the prediction $\tau =(d_{\mathrm{f}}+\theta )/d_{\mathrm{m}}$ where $d_{\mathrm{f}},d_{\mathrm{m}}$, and $\theta$ are the fractal dimension, magnetization exponent, and energy exponent of a droplet, respectively. This formula is expected to apply to other glassy disordered systems, such as the random-field model and pinned interfaces. We make suggestions for further numerical investigations, as well as experimental studies of the Barkhausen noise in spin glasses.

Figure 1

The Parisi-function $q\left(x\right)$, with its two plateaus for $x<x_m$ and $x>x_c$. Note that this gives two $\delta$-function contributions to the derivative of the inverse function $\frac{dx\left(q\right)}{dq}=x_m\delta (q-q_m)+(1-x_c)\delta (q-q_c)+\text{smooth}\text{part}$.

Figure 2

Power-law density of jump sizes for the SK model. The power law receives contributions from all overlaps $1-q$. The curves in the lower part show the contributions from $(1-q)=2^{-k}$, $k=1,...,12$, each of which takes the form of the jump density in mean-field glasses with one-step replica symmetry breaking. The three nearly coinciding lines on the top show $\rho \left(\Delta m\right)$ evaluated from Eq. (57), for external fields $H=0,0.25$, and $0.5$, respectively, using approximations for $q\left(\widehat{x}\right)$ described in the text. The increase of $H$ decreases the cutoff at large $\Delta m$, while the avalanche distribution for $\Delta m\ll 1$ is a universal power law, not affected by $H$.