Critical surface of the ising model with first-neighbor,second-neighbor,and four-spin interactions

Exact values are obtained for the slopesK ₁ ^c (0, 0)/K ₂,K ₁ ^c (0, 0)/K ₄ of the critical surface of paramagnetic-ferromagnetic transitionsK ₁ ^c (K ₂,K ₄) for the two-dimensional Ising model on a square lattice with first-neighbor, second-neighbor, and four-spin couplingsK ₁,K ₂, andK ₄, respectively. The results are obtained using universality arguments to relate the slopes to known spin-spin correlation functions forK ₂=K ₄=0. The equivalence of different expressions for the slopes in terms of correlation functions yields sum rules for the divergent part of certain sums over the second-neighbor and four-spin energy-energy correlation functions.