Domain-wall excitations in the two-dimensional Ising spin glass

The Ising spin glass in two dimensions exhibits rich behavior with subtle differences in the scaling for different coupling distributions. We use recently developed mappings to graph-theoretic problems together with highly efficient implementations of combinatorial optimization algorithms to determine exact ground states for systems on square lattices with up to $10\,000\times 10\,000$ spins. While these mappings only work for planar graphs, for example for systems with periodic boundary conditions in at most one direction, we suggest here an iterative windowing technique that allows one to determine ground states for fully periodic samples up to sizes similar to those for the open-periodic case. Based on these techniques, a large number of disorder samples are used together with a careful finite-size scaling analysis to determine the stiffness exponents and domain-wall fractal dimensions with unprecedented accuracy, our best estimates being $\theta = -0.2793(3)$ and $d_\mathrm{f} = 1.273\,19(9)$ for Gaussian couplings. For bimodal disorder, a new uniform sampling algorithm allows us to study the domain-wall fractal dimension, finding $d_\mathrm{f} = 1.279(2)$. Additionally, we also investigate the distributions of ground-state energies, of domain-wall energies, and domain-wall lengths.