Zeros of the Partition Function for the Heisenberg, Ferroelectric, and General Ising Models

Abstract

The Lee‐Yang theorem for the zeros of the partition function of a ferromagnetic Ising model with real pair spin interactions is extended to general Ising models with complex many‐spin interactions (satisfying appropriate ``ferromagnetic'' and spin inversion symmetry conditions). When many‐spin interactions are present, all zeros lie on the imaginary H<sup>z</sup>‐axis for sufficiently low (but fixed) T, but, in general, some leave the imaginary axis as T → ∞. The extended Ising theorem is used to prove the same result for a Heisenberg system of arbitrary spin with the real anisotropic pair interaction Hamiltonian $${\mathcal{H}}_{\mathrm{ij}}{\mathrm{=-(J}}_{\mathrm{ij}}^z{S}_i^z{S}_j^z{\mathrm{+J}}_{\mathrm{ij}}^x{S}_i^x{S}_j^x{\mathrm{+J}}_{\mathrm{ij}}^y{S}_i^y{S}_j^y)$$ in an arbitrary transverse field (H<sup>x</sup>, H<sup>y</sup>) under the ``ferromagnetic'' condition $J_{\mathrm{ij}}^z{\mathrm{\ge |J}}_{\mathrm{ij}}^x|$ and $J_{\mathrm{ij}}^z{\mathrm{\ge |J}}_{\mathrm{ij}}^y\mathrm{|.}$ The analyticity of the limiting free energy of such a Heisenberg ferromagnet and the absence of a phase transition are thereby established for all (real) nonzero magnetic fields H^z. The Ising theorem is also applied to hydrogen‐bonded ferroelectric models to prove, in particular, that the zeros for the KDP model lie on the imaginary electric field axis for all T below the transition temperature T_c.