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 Lx2L square lattices with free boundary conditions in the L-direction and periodic boundary conditions in the 2L-direction up to L = 12 using an exact enumeration method. The total number of states is 2(288) approximate to 5 x 10(86) for L = 12, and counting all 2(288) states requires enormous computational work. The thermal scaling exponent y(t) = 1(= 1/nu) (where v 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 lambda = 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 lambda = 1.1 is obtained for J(2) < 0 and R= +/- 1/2 and is different from the thermal scaling exponent y(t).