Defect Energy with Conjugate Boundary Conditions in Spin Glass Models in Two Dimensions

We calculate the naive defect energy $\mathrm{\Delta }E$ of Ising spin glass(SG) models in two dimensions using conjugate boundary conditions. We predict that, in the $\pm J$ model, the averaged value $\overline{\mathrm{\Delta }E}$ converges to some non-zero value in the thermodynamic limit in contrast with $\overline{\mathrm{\Delta }E}=0$ in the Gaussian model. This prediction is incompatible with previous ones but supports a recent Monte Carlo prediction of the presence of the SG phase at finite temperatures in the $\pm J$ Ising model. We also calculate the interface free energy to confirm it.