Some results for the two-dimensional Ising model with competing interactions

The applicability of the n-replica method to an Ising system with positive and negative bonds of equal strength is investigated. It is shown that this method does reproduce the correct low-temperature and low-(negative bond) concentration series for the ground-state energy. It is essential, however, to take the low-temperature limit after the n to 0 limit. For a square lattice, the slope of the critical line near the pure-system transition point is found to be 2 square root 2/ln( square root 2+1). The onset of spin-glass order is conjectured to be associated with a negative bond concentration at which unfrustrated plaquettes no longer percolate the lattice. The estimated value of 16% is in agreement with Monte Carlo results.