The Ising model in an external magnetic field is one of the most outstanding and intriguing unsolved problems. One of the important properties of the Ising model in an external magnetic field is the Yang-Lee edge singularity. The zeros in the complex magnetic-field plane of the grand partition function of the Ising model on a honeycomb lattice in an external magnetic field are exactly calculated by using the exact integer values for the density of states on L × 2L honeycomb lattices (up to L = 14). The critical line of the Yang-Lee edge singularity of the Ising model on a honeycomb lattice in an external magnetic field is accurately investigated from its Yang-Lee zeros.
We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable
u=e−4K, where
K=J/(kBT), for nonzero external magnetic field,
H. Exact results are given for the phase diagram in the
u plane for the model in one dimension and on infinite-length quasi-one-dimensional strips. In the case of real
h=H/(kBT), these results provide new insights into features of our earlier study of this case. We also consider complex
h=H/(kBT) and
μ=e−2h. Calculations of complex-
u zeros of the partition function on sections of the square lattice are presented. For the case of imaginary
h, i.e.,
μ=eiθ, we use exact results for the quasi-1D strips together with these partition function zeros for the model in 2D to infer some properties of the resultant phase diagram in the
u plane. We find that in this case, the phase boundary
Bu contains a real line segment extending through part of the physical ferromagnetic interval
0≤u≤1, with a right-hand endpoint
urhe at the temperature for which the Yang-Lee edge singularity occurs at
μ=e±iθ. Conformal field theory arguments are used to relate the singularities at
urhe and the Yang-Lee edge.
The partition functions of the two- and three-dimensional finite Ising models in the presence of a magnetic field have been calculated with use of a high-speed computer. Periodic boundary conditions were imposed. All the numbers of configurations for the 3×3×3, 5×5, and 4×6 lattices are tabulated. The distribution of zeros of the partition functions in the complex magnetic-field plane has been obtained both for the ferro- and antiferromagnetic cases. The zeros in the ferromagnetic cases are distributed on the unit circle according to the Yang-Lee theorem, and the dependence on temperature of the distribution functions has been studied. In the antiferromagnetic case most of zeros are on the negative real axis, but a certain number of complex roots appear in the left half of the plane. The magnetization and the magnetic susceptibility have been calculated as functions of temperature.
The spin-one Ising model on the honeycomb lattice has never been solved exactly in spite of its simplicity. Even its exact critical temperature is not known. The exact integer values for the density of states of the spin-one Ising model on the L×2L honeycomb lattice are enumerated up to L=14. The partition function zeros in the complex temperature plane of the spin-one Ising model on the L×2L honeycomb lattice are exactly obtained, using the density of states. The properties of the partition function zeros in the complex temperature plane are related to the behaviors of various thermodynamic functions, in particular, their singular behaviors. The unknown properties of the spin-one Ising model on the honeycomb lattice are investigated, based on its partition function zeros in the complex temperature plane.
The Ising ferromagnet, consisting of magnetic spins, is the simplest system showing phase transitions and critical phenomena at finite temperatures. The Ising ferromagnet has played a central role in our understanding of phase transitions and critical phenomena. Also, the Ising ferromagnet explains the gas-liquid phase transitions accurately. In particular, the Ising ferromagnet in a nonzero magnetic field has been one of the most intriguing and outstanding unsolved problems. We study analytically the partition function zeros in the complex magnetic-field plane and the Yang-Lee edge singularity of the infinite-range Ising ferromagnet in an external magnetic field. In addition, we compare the Yang-Lee edge singularity of the infinite-range Ising ferromagnet with that of the square-lattice Ising ferromagnet in an external magnetic field.