Honeycomb-lattice antiferromagnetic Ising model in a magnetic field

The exact number of states Ω(E,M)Ω(E,M) of the Ising model only with the nearest-neighbor interaction J   on L×2LL×2L honeycomb lattices (up to L=12L=12), as a function of energy E and magnetization M  , is evaluated for the first time. For L=12L=12, the total number of states is 2288 (≈5×1086≈5×1086). Classifying all states 2288 according to their E and M   values is an enormous work. Given the number of states Ω(E,M)Ω(E,M), the exact partition function Z(a,x)=∑E,MΩ(E,M)aExMZ(a,x)=∑E,MΩ(E,M)aExM is obtained, where a=e2βJa=e2βJ (β=1/kBTβ=1/kBT) and x=e−2βHx=e−2βH (H  : magnetic field). The properties of the honeycomb-lattice antiferromagnetic (J<0J<0) Ising model in a magnetic field is discussed based on the exact partition function. The precise distributions of the partition function zeros in the complex temperature (a=e2βJa=e2βJ) plane of the honeycomb-lattice Ising model for real H≠0H≠0 are also obtained for the first time.