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

Abstract

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 $R=J_2/J_1=\pm 1/2$, 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 $R=\pm 1/2$ are evaluated on $L\times 2L$ 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 $2^{288}\approx 5\times 10^{86}$ for $L=12$, and counting all $2^{288}$ states requires enormous computational work. The thermal scaling exponent $y_t=1(=1/\nu )$ (where $\nu$ is the correlation-length critical exponent) of the square-lattice $J_1-J_2$ Ising model is obtained for $J_2>0$ 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 $J_2<0$ and $R=\pm 1/2$, suggesting a first-order phase transition. The shift exponent $\lambda =1.1$ is obtained for $J_2<0$ and $R=\pm 1/2$ and is different from the thermal scaling exponent $y_t$.