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

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 ⟨ℓ⟩∼L^d_f the growth of the average domain-wall length with system size L×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.