Lee-Yang Theorem Research Articles

Heisenberg Model for the 3×3 Square Lattice

The exact excitation spectrum of a 3×3 two dimensional square lattice interacting via the nearest neighbour anisotropic Heisenberg Hamiltonian H = J ∑ { S i z S j z +γ( S i x S j x + S i y S j y )}, has been obtained for various values of the anisotropy constant γ. The zeros of the partition function in the complex µ=exp (- m H e / k B T ) plane are calculated for different γ for both ferromagnetic ( J 0) coupling. For \(0{\leqslant}\gamma{\leqslant}1\), the ferromagnetic zeros obey the generalized Lee-Yang theorem at all temperatures, but, for γ>1, they violate it at sufficiently low temperatures. For antiferromagnetic coupling, the zeros lie on the (unphysical) negative real axis for all values of γ and temperature. The zero-field susceptibility, for antiferromagnetic coupling, does not display a local maximum as a function of temperature in contrast to Kawabata's results on the same lattice.

Feynman Rules and Factor Ordering in Derivative Coupling and Spin-One Theories

Several methods may be employed to derive the Feynman rules for canonical theories. With emphasis on vector particle theories, we recast the theorem of Lee and Yang into functional integral and Hori operator formulations. For the class of Lagrangians considered, equivalent results are obtained. The canonical generating functional is used to relate T and T* products to the Hamiltonian and Lagrangian versions of theories with dependent fields. With the aid of Hori operators, we show that there is a wide class of ordering prescriptions compatible with the Lee-Yang theorem. Of these, the only familiar candidates are normal ordering and symmetrization of momenta and coordinates. It is argued that the latter is preferred, thus confirming a result of Suzuki and Hattori. Finally, the expanded Lee-Yang theorem for general ordering prescriptions is outlined, and ordering for a Yang-Mills theory is discussed.

Theory of monomer-dimer systems

We investigate the general monomer-dimer partition function, P(x), which is a polynomial in the monomer activity, x, with coefficients depending on the dimer activities. Our main result is that P(x) has its zeros on the imaginary axis when the dimer activities are nonnegative. Therefore, no monomer-dimer system can have a phase transition as a function of monomer density except, possibly, when the monomer density is minimal (i.e. x = 0). Elaborating on this theme we prove the existence and analyticity of correlation functions (away from x = 0) in the thermodynamic limit. Among other things we obtain bounds on the compressibility and derive a new variable in which to make an expansion of the free energy that converges down to the minimal monomer density. We also relate the monomer-dimer problem to the Heisenberg and Ising models of a magnet and derive Christoffell-Darboux formulas for the monomer-dimer and Ising model partition functions. This casts the Ising model in a new light and provides an alternative proof of the Lee-Yang circle theorem. We also derive joint complex analyticity domains in the monomer and dimer activities. Our considerations are independent of geometry and hence are valid for any dimensionality.

Density of Zeros on the Lee-Yang Circle for Two Ising Ferromagnets

Extrapolations of high-field and high-temperature series expansions have been used to construct numerical approximations to the density of zeros $g(\ensuremath{\theta})$ on the Lee-Yang circle for the Ising ferromagnets on a two-dimensional square and a three-dimensional diamond lattice. For temperatures above the critical temperature the density is zero for $|\ensuremath{\theta}|<{\ensuremath{\theta}}_{G}$ and then varies as ${(\ensuremath{\theta}\ensuremath{-}{\ensuremath{\theta}}_{G})}^{\ensuremath{\mu}}$, with $\ensuremath{\mu}\ensuremath{\simeq}\ensuremath{-}0.1 \mathrm{and} +0.1$ for the square and diamond lattices, respectively.

Extension of the Lee-Yang Circle Theorem

We study the zeros of the partition function of a classical spin system and compute a region of the complex fugacity plane where they necessarily lie. We recover the Lee-Yang circle theorem as a special example; we also find that a spin system with finite-range interaction has no phase transition at high temperature.

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

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.

Generalizations of the Lee-Yang theorem

The zeros of the partition function of the classical Heisenberg model are shown to occur for imaginary magnetic field. The same result holds for a two-sublattice antiferromagnet in terms of the staggered magnetic field.

Statistical Thermodynamics of Finite Ising Model. II

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.

Generalization of the Lee-Yang Theorem

Generalization of the Lee-Yang Theorem

Generalized Lee-Yang's Theorem

Lee-Yang's theorem which states that all the roots of the partition function of the ferromagnetic Ising model of spin 1/2 lie on the unit circle in the complex `fugacity' plane is generalized to the case of S =1 and 3/2, where S is the magnitude of the spin.