Experimental Realization of the 1D Random Field Ising Model

We have measured magnetic-field-induced avalanches in a square artificial spin ice array of interacting nanomagnets. Starting from the ground state ordered configuration, we imaged the individual nanomagnet moments after each successive application of an incrementally increasing field. The statistics of the evolution of the moment configuration show good agreement with the canonical one-dimensional random field Ising model. We extract information about the microscopic structure of the arrays from our macroscopic measurements of their collective behavior, demonstrating a process that could be applied to other systems exhibiting avalanches.

Figure 1

(a) Atomic force microscope image of the rotated square lattice. (b) Magnetic force microscope (MFM) image of the ground state of $L=70$ array, sample C. (c) Normalized magnetization measured with a Quantum Design SQUID magnetometer (MPMS 3) as a function of applied field for an $L=10000$ array (blue circles), and the fields (red stars) corresponding to the values of $H_{\mathrm{ext}}$ used in the MFM studies. (d) Schematic of the lattice with arrows indicating initial moment configuration, and the yellow arrow indicating the direction of $H_{\mathrm{ext}}$. The lattice constant for the structure is also indicated as $a$. (e) Schematic of the lattice with arrows indicating the polarized state moment configuration.

Figure 2

Digitized moment maps for the field evolution of avalanche growth for an $L=70$ array, sample C. The blue stripes represent columns of moments that were aligned with the $H_{\mathrm{ext}}$ direction in the initial state of the system. The white points represent moments that were anti-aligned with $H_{\mathrm{ext}}$ in the initial configuration. The red points represent moments that flipped to align with the field.

Figure 3

Averaged distribution, $D_{\mathrm{avg}}(S)$ for clusters smaller than the system size ($S<L$), averaging over all samples with a given $L$. In an infinite system this quantity is expected to follow a $S^{-1}$ power law (dashed line) as derived in the Supplemental Material (Sec. SI-7). In a finite system it is modified by a fitted exponential cutoff (Supplemental Material, Sec. SI-5). A fit to this form for $L=80$ is shown (dash-dot line).

Figure 4

(a) Complementary cumulative distribution functions (CCDFs) of cluster sizes for a single $L=70$ array, (sample A), and $H_{\mathrm{ext}}=430.5$, 434.5, 439, 444, 448, 452, 456, 460, 466, and 470 Oe. The dotted lines are fitted model predictions. (b) The experimentally recovered values of $p(H_{\mathrm{ext}})$ calculated from the data and fits in (a). The blue line is predicted from Eq. (7) with fitted values of $R=19.2\pm 1.0$ and $h^{\prime }=426.6\pm 0.7\mathrm{Oe}$. (c) Gaussian distributions of random fields [Eq. (3)] using the fitted values of $R$ and $h^{\prime }$ for all four arrays with $L=70$ (samples A, B, C, and D). (d) Moments of the cluster size distribution plotted as a function of $1\text{−}p$, for the single $L=70$, sample A array, where dashed lines show the model prediction as discussed in the text. Here, we include fields in which spanning clusters are present, so that the deviation from the model due to finite size at smaller values of $1\text{−}p$ is apparent.

Figure 5

(a) Distribution of cluster sizes measured for various values of $H_{\mathrm{ext}}$, averaging data over all three arrays with $L=100$ (A, B, and C). Here, we plot this distribution using values of $H_{\mathrm{ext}}$ for which there is substantial island moment flipping, but do not plot the distribution for fields in which spanning clusters dominate. (b) The same data after scaling the axes compared to the predicted scaling function $Ax{e}^{-x}$ in black, as described in the text.