Exact solution in an external magnetic field of Ising models with three-spin interactions

The three-spin interaction Hamiltonian ${\mathcal{H}}^{\mathrm{}\left(3\right)\mathrm{}}\equiv -J_3\Sigma {〈\mathrm{ijk}〉}^{}{s}_i{\sigma }_j{\sigma }_k-H\Sigma i^{}{s}_i-H^{\prime }\Sigma j^{}{\sigma }_j$ (the $s$ and $\sigma$ spins belong to different sublattices) is solved for some two-dimensional lattices by a generalized star-triangle transformation. The $H^{\prime }=0\mathrm{}$ internal energy, specific heat, and magnetization are explicitly calculated, and the singularity structures of other functions (e.g.$x$ are studied. Although the actual critical-point exponents remain the same as for systems with two-spin interaction, the three-spin interaction gives rise to different amplitude and background terms. In particular, the $H^{\prime }=0\mathrm{}$ magnetization is nonzero for all finite $T$, because of the lack of spin-reversal symmetry; the diameter is seen explicitly to have an energy singlularity, with critical exponent $1-\alpha$.