Fractal dimension of domain walls in two-dimensional Ising spin glasses

Abstract

We study domain walls in two-dimensional Ising spin glasses in terms of a minimum-weight path problem. Using this approach, large systems can be treated exactly. Our focus is on the fractal dimension ${d}_{f}$ of domain walls, which describes via $⟨\ensuremath{\ell}⟩\ensuremath{\sim}{L}^{{d}_{f}}$ the growth of the average domain-wall length with system size $L\ifmmode\times\else\texttimes\fi{}L$. Exploring systems up to $L=320$ we yield ${d}_{f}=1.274(2)$ for the case of Gaussian disorder, i.e., a much higher accuracy compared to previous studies. For the case of bimodal disorder, where many equivalent domain walls exist due to the degeneracy of this model, we obtain a true lower bound ${d}_{f}=1.095(2)$ and a (lower) estimate ${d}_{f}=1.395(3)$ as upper bound. Furthermore, we study the distributions of the domain-wall lengths. Their scaling with system size can be described also only by the exponent ${d}_{f}$, i.e., the distributions are monofractal. Finally, we investigate the growth of the domain-wall width with system size (``roughness'') and find a linear behavior.