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 $$J_20$$ and $$R=\pm 1/2$$ , in agreement with the Ising universality class. The shift exponent $$\lambda =1.00$$ is obtained for $$J_2>0$$ and $$R=\pm 1/2$$ , 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