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Journal of the Physical Society of Japan

Theorems on the Partition Functions of the Heisenberg Ferromagnets

Taro Asano

Published Aug 1, 1970• 97 citations
Publisher

Abstract

It is shown that all the zeros of the partition function lie on the unit circle in the complex `fugacity' plane and the first inequality of Griffiths, Kelly and Sherman for the spin correlation functions also holds for the anisotropic Heisenberg ferromagnets as in the Ising model. The isotropic Heisenberg ferromagnets and the Ising ferromagnets are involved as special cases and thus the methods of this article provide alternative proof of the corresponding theorems in the Ising model.

DOI
10.1143/jpsj.29.350

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It is shown that all the zeros of the partition function lie on the unit circle in the complex `fugacity' plane and the first inequality of Griffiths, Kelly and Sherman for the spin correlation functions also holds for the anisotropic Heisenberg ferromagnets as in the Ising model. The isotropic Heisenberg ferromagnets and the Ising ferromagnets are involved as special cases and thus the methods of this article provide alternative proof of the corresponding theorems in the Ising model.
The theorem of Lee and Yang has been extended to the ferromagnetic Ising model with arbitrarily mixed spin values of Sj = ½, 1, and 32, including the case of equal spin values as a special one. Namely, it has been proved that the zeros of the partition function for the above Ising model with higher spin values lie on the unit circle in the fugacity plane (or complex magnetic-field plane). Expressions for general correlation functions in Ising ferromagnets with higher spin values have been derived in terms of the above generalized theorem. By the use of these expressions, the relations among the critical indices are discussed and the same results are obtained as those predicted by the scaling-law approach.
The distribution of energy eigenvalues is discussed in the Heisenberg model. In particular, the first and second moments of energy eigenvalues are formulated. A method to reduce the partition function of the Heisenberg model with higher spin to that of spin ½ is presented. It yields the theorem which states if all the zeros of the partition function in the case of spin ½ are on the unit circle of the complex magnetic field plane, then it is true in the case of general spin. Furthermore, it is proved at low temperature that the zeros of the partition function of the ferromagnetic Heisenberg model with spin ½ are on the unit circle of the complex field plane. A theorem is given also on the distribution of zeros in antiferromagnetic case. Finally, an idea of coarse-graining of energy eigenvalues is proposed on the complex temperature plane in the Heisenberg model.
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 Hz-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 Hij=−(JijzSizSjz+JijxSixSjx+JijySiySjy)in an arbitrary transverse field (Hx, Hy) under the ``ferromagnetic'' condition Jijz≥|Jijx| and Jijz≥|Jijy|. 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 Hz. 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 Tc.
The distribution of zeros of the partition function for one-dimensional lattice system of generalized Ising spins with magnitude unity is studied, where it is assumed that the component spins interact with the nearest neighbours through a ferromagnetic coupling and each spin state has a certain degeneracy. The spin states are denoted by variable σ which can take either of three values, plus and minus unities and zero with the respective degeneracies, D(1), D(0) and D(-1), where it is assumed D(1) = D(-1). zeros of the partition function Z(ζ) of the system as exposed to an external magnetic field are studied on the ζ-plane, where ζ=exp(H) indicates the exponential function of the external field divided by temperature H and it is regarded as a complex number. The distribution of zeros is classified into three typical patterns, corresponding to the respective three regions on the plane of temperature(T) versus degneracy ratio (ρ=D(0)/D(1)): Zeros never lie on the unit circle in the region of larger ρ, all on the unit circle in the region of smaller ρ and only a part on the unit circle in the remaining region. On the basic of this result, it is expected in the case of two- or three-dimensional system that phase trasitions of first and second orders occur in tha cases of ρ larger and smaller than acritical calue ρc, respectively.