Study of the frustrated Ising model on a square lattice based on the exact density of states

The square-lattice Ising model with nearest-neighbor ( J_1 ) and next-nearest-neighbor ( J_2 ) interactions is exactly unsolvable. The square-lattice J_1-J_2 Ising model is frustrated for J_2<0 . For R=J_2/J_1=± 1/2 , the square-lattice J_1-J_2 Ising model for J_2<0 is the most frustrated, and its ground states are infinitely degenerate. The exact integer values for the density of states of the J_1-J_2 Ising model for R=± 1/2 are evaluated on L× 2L square lattices with free boundary conditions in the L -direction and periodic boundary conditions in the 2 L -direction up to L=12 using an exact enumeration method. The total number of states is 2^288≈ 5× 10^86 for L=12 , and counting all 2^288 states requires enormous computational work. The thermal scaling exponent y_t=1(=1/ν ) (where ν is the correlation-length critical exponent) of the square-lattice J_1-J_2 Ising model is obtained for J_2>0 and R=± 1/2 , in agreement with the Ising universality class. The shift exponent λ =1.00 is obtained for J_2>0 and R=± 1/2 , equaling the thermal scaling exponent y_t . On the other hand, the thermal scaling exponent y_t=2.0 of the square-lattice J_1-J_2 Ising model is obtained for J_2<0 and R=± 1/2 , suggesting a first-order phase transition. The shift exponent λ =1.1 is obtained for J_2<0 and R=± 1/2 and is different from the thermal scaling exponent y_t .