Evidence for a trivial ground-state structure in the two-dimensional Ising spin glass

We study how the ground state of the two-dimensional Ising spin glass with Gaussian interactions in zero magnetic field changes on altering the boundary conditions. The probability that relative spin orientations change in a region far from the boundary goes to zero with the (linear) size of the system L like $L^{-\lambda },$ where $\lambda =-0.70\pm \mathrm{0.08.}$ We argue that $\lambda$ is equal to $d-d_f$ where $d\hspace{0.5em}\mathrm{}(=2\mathrm{})\mathrm{}$ is the dimension of the system and $d_f$ is the fractal dimension of a domain wall induced by changes in the boundary conditions. Our value for $d_f$ is consistent with earlier estimates. These results show that, at zero temperature, there is only a single pure state (plus the state with all spins flipped) in agreement with the predictions of the droplet model.