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Physical review. B, Solid state

Distribution of Zeros of the Partition Function for the Finite Two-Dimensional Heisenberg Model

Vinod Mubayi, Chanchal K. Majumdar, Kanwar Krishan

Published Oct 1, 1973• 2 citations
Publisher

Abstract

The exact excitation spectrum of the 3 \ifmmode\times\else\texttimes\fi{} 3 square and the 3 \ifmmode\times\else\texttimes\fi{} 4 rectangular lattices interacting via the nearest-neighbor anisotropic Heisenberg interaction $H=J\ensuremath{\Sigma}{(\mathrm{ij})}^{}[S_{i}^{}{}_{}{}^{z}S_{j}^{}{}_{}{}^{z}+\ensuremath{\gamma}(S_{i}^{}{}_{}{}^{x}S_{j}^{}{}_{}{}^{x}+S_{i}^{}{}_{}{}^{y}S_{j}^{}{}_{}{}^{y})]$ has been obtained for various values of the anisotropy constant $\ensuremath{\gamma}$. For $0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}1$, the zeros of the partition function in the complex $\ensuremath{\mu}=\mathrm{exp}(\ensuremath{-}\frac{m{H}_{e}}{{k}_{B}T})$ plane obey the generalized Lee-Yang theorem for ferromagnetic coupling. The usual ferromagnetic transition seems to persist up to $\ensuremath{\gamma}\ensuremath{\simeq}0.6$. For $0.6\ensuremath{\lesssim}\ensuremath{\gamma}\ensuremath{\le}1$, nothing definite about a ferromagnetic transition can be inferred from the behavior of the zeros. When $\ensuremath{\gamma}>1$, the ferromagnetic zeros do not obey the generalized Lee-Yang theorem at sufficiently low temperatures. For antiferromagnetic coupling, the zeros lie on the negative real axis for all values of $\ensuremath{\gamma}$ and of the temperature of the studied lattices.

DOI
10.1103/physrevb.8.3305

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The exact excitation spectrum of the 3 \ifmmode\times\else\texttimes\fi{} 3 square and the 3 \ifmmode\times\else\texttimes\fi{} 4 rectangular lattices interacting via the nearest-neighbor anisotropic Heisenberg interaction H=J\ensuremathΣ(ij)[SizSjz+\ensuremathγ(SixSjx+SiySjy)]H=J\ensuremath{\Sigma}{(\mathrm{ij})}^{}[S_{i}^{}{}_{}{}^{z}S_{j}^{}{}_{}{}^{z}+\ensuremath{\gamma}(S_{i}^{}{}_{}{}^{x}S_{j}^{}{}_{}{}^{x}+S_{i}^{}{}_{}{}^{y}S_{j}^{}{}_{}{}^{y})]H=J\ensuremathΣ(ij)[Si​​zSj​​z+\ensuremathγ(Si​​xSj​​x+Si​​ySj​​y)] has been obtained for various values of the anisotropy constant \ensuremathγ\ensuremath{\gamma}\ensuremathγ. For 0\ensuremath≤\ensuremathγ\ensuremath≤10\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}10\ensuremath≤\ensuremathγ\ensuremath≤1, the zeros of the partition function in the complex \ensuremathμ=exp(\ensuremath−mHekBT)\ensuremath{\mu}=\mathrm{exp}(\ensuremath{-}\frac{m{H}_{e}}{{k}_{B}T})\ensuremathμ=exp(\ensuremath−kB​TmHe​​) plane obey the generalized Lee-Yang theorem for ferromagnetic coupling. The usual ferromagnetic transition seems to persist up to \ensuremathγ\ensuremath≃0.6\ensuremath{\gamma}\ensuremath{\simeq}0.6\ensuremathγ\ensuremath≃0.6. For 0.6\ensuremath≲\ensuremathγ\ensuremath≤10.6\ensuremath{\lesssim}\ensuremath{\gamma}\ensuremath{\le}10.6\ensuremath≲\ensuremathγ\ensuremath≤1, nothing definite about a ferromagnetic transition can be inferred from the behavior of the zeros. When \ensuremathγ>1\ensuremath{\gamma}>1\ensuremathγ>1, the ferromagnetic zeros do not obey the generalized Lee-Yang theorem at sufficiently low temperatures. For antiferromagnetic coupling, the zeros lie on the negative real axis for all values of \ensuremathγ\ensuremath{\gamma}\ensuremathγ and of the temperature of the studied lattices.
The exact excitation spectrum of a 3×3 two dimensional square lattice interacting via the nearest neighbour anisotropic Heisenberg Hamiltonian H = J ∑ < i 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) and antiferromagnetic ( J >0) coupling. For 0⩽γ⩽10{\leqslant}\gamma{\leqslant}10⩽γ⩽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.
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.
The eigenfunctions of a two-dimensional finite 3×3 lattice of Heisenberg model ( s =1/2) are obtained exactly with the aid of group theoretical technique and by using a computer. The eigenvalues of the anisotropic exchange Hamiltonian have been computed. The zeros of the partition function for the two-dimensional finite 3×3 lattice lie on the same unit circle in the complex magnetic field plane as those obtained in the three-dimensional finite 2×2×2 lattice. The zeros of the partition function obtained by coarse-graining of eigenvalues distribute on a line in the complex temperature plane. The thermodynamic quantities–specific heat, entropy, magnetization–has been numerically computed for ferro- and antiferromagnetic coupling. The critical index of the magnetization versus field is discussed on the basis of these numerical results.
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.
To study the relations between Yang-Lee's theory of the phase transitions and the distribution of the zeros of the partition functions in the antiferromagnetic case, the partition functions of finite (4×4 and 4×6) Ising model with the nearest and the next nearest neighbor interactions are calculated. The distributions of the zeros in the fugacity z plane of the partition function for various combinations of the values of J / k T - J '/ k T or tanh ( J /2 k T )-tanh ( J '/2 k T ) are investigated. The patterns of the distribution of the zeros show several types corresponding to ferromagnetic state, antiferromagnetic state and superantiferromagnetic state. In the antiferromagnetic ordered state the locus of the zeros is found to be nearly two concentric circles and to cross the positive real axis at two points z c and 1/ z c when the next neighbor ferromagnetic interaction is introduced. This means that our system shows the phase transitions at finite critical fields H c and - H c . The locus of zeros of Fisher's model of super-exchange antiferromagnet is also shown to cross the positive real axis at z c and 1/ z c below the Néel temperature.