Anisotropic Exponents for Avalanche Correlation Lengths in Self-Affine Growth of Magnetic Domains

Driven interfaces in a wide variety of systems undergo a critical depinning transition as the driving force is increased to a critical value, $F_c$. Near this transition, growth consists of discrete avalanches with a power law distribution of sizes and a diverging length scale along the interface $\xi_\|\sim|F_c-F|^{\nu_\|}$. Scaling theories often assume that correlations perpendicular to the interface diverge with an exponent $\nu_\perp = \alpha\nu_\|$, where $\alpha$ is the self-affine roughness exponent \footnote{O. Narayan, D. Fisher PRB 82 (1993)}. We simulate depinning of a self-affine domain wall in the 3D random field Ising model to determine the ratio $\chi \sim \nu_\perp/\nu_\|$. Analyzing individual avalanches show that the height $l_\perp$ and width along the interface $l_\|$ scale as $l_\perp\sim l_\|^\chi$ with $\chi=0.9\pm0.05$ over 3 decades in systems of $10^{10}$ spins. This value of $\chi$ is significantly greater than $\alpha\sim0.67$. Finite size scaling was used to confirm the value of $\chi$. The probability of reaching the top of a system of width $L$ and height $L^\chi$ as a function of $|F-F_c|L^{1/\nu_\|}$ collapses for $\chi=0.9\pm0.03$. We discuss the implications for other scaling relations and the conditions where $\chi$ and $\alpha$ should differ.