Universality beyond power laws and the average avalanche shape

The study of critical phenomena and universal power laws has been one of the central advances in statistical mechanics during the second half of the past century, explaining traditional thermodynamic critical points1, avalanche behaviour near depinning transitions2,3 and a wide variety of other phenomena4. Scaling, universality and the renormalization group claim to predict all behaviour at long length and timescales asymptotically close to critical points. In most cases, the comparison between theory and experiments has been limited to the evaluation of the critical exponents of the power-law distributions predicted at criticality. An excellent area for investigating scaling phenomena is provided by systems exhibiting crackling noise, such as the Barkhausen effect in ferromagnetic materials5. Here we go beyond powerlaw scaling and focus on the average functional form of the noise emitted by avalanches—the average temporal avalanche shape4. By analysing thin permalloy films and improving the data analysis methods, our experiments become quantitatively consistent with our calculation for the multivariable scaling function in the presence of a demagnetizing field and finite field-ramp rate. The average temporal avalanche shape has been measured for earthquakes6 and for dislocation avalanches in plastically deformed metals7,8, but the primary experimental and theoretical focus has always been Barkhausen avalanches in magnetic systems5,6,9–11. Theory and experiment agreed well for avalanche sizes and durations, but the strikingly asymmetric shapes found experimentally in ribbons11 disagreed sharply with the theoretical predictions, for which the asymmetry in the scaling shapes under time reversal was at most very small4,6. (We note that the relevant models are not microscopically time-reversal invariant; temporal symmetry is thus emergent.) Doubts about universality4 were resolved when eddy currents were shown to be responsible for the asymmetry, at least on short timescales12, but the exact form of the asymptotic universal scaling function of the Barkhausen avalanche shape still remained elusive. Here, we report an experimental study of Barkhausen noise in permalloy thin films, where a careful study of the average avalanche shapes leads to symmetric shapes, undistorted by eddy currents (which are suppressed by the sample geometry). We provide a quantitative explanation of the experimental results by solving exactly the mean-field theories for two general models of magnetic reversal: a domain-wall dynamics model13 and the random-field Ising model14. The two mean-field theories