Scaling and chaos in periodic approximations to the two-dimensional Ising spin glass

We approximate a two-dimensional spin glass by tiling an infinite lattice with large identical unit cells. The interactions within the unit cell are chosen at random, just as when one studies finite-size systems with periodic boundary conditions. But here the unit cells are instead connected to form an infinite lattice, so one may examine correlations on all length scales, and the system can have true phase transitions. For such approximations to the Ising spin glass on the square lattice, we apply the free-fermion method of Onsager and the anticommuting operator approach of Kaufman to obtain numerically exact results for each realization of the quenched disorder. Each such sample shows one or more critical points, with the distribution of critical temperatures scaling with the unit cell size, consistent with what is expected from the scaling theory of low-dimensional spin glasses. Due to “chaos,” the correlations between unit cells can change sign with changing temperature. We examine the scaling of this chaos with unit cell size. Many samples have multiple critical points due to the interactions between unit cells changing sign at temperatures within the ordered phases.