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Physical Review

High-Temperature Series Expansions for the Spin-½ Heisenberg Model by the Method of Irreducible Representations of the Symmetric Group

George Allen Baker Jr., George Stanley Rushbrooke, H.E. Gilbert

Published Aug 31, 1964• 505 citations
Publisher

Abstract

We show how the partition functions for finite clusters with spin-\textonehalf{} Heisenberg interactions may be computed efficiently and generally to any desired number of powers in reciprocal temperature. As an example, we have expanded the zero-magnetic-field free energy to the twenty-first power for the linear Heisenberg model and for nonzero magnetic field give an expression good through the tenth power. We introduce the concept of the two-point Pad\'e approximant and use it to analyze the energy for the linear Heisenberg model.

DOI
10.1103/physrev.135.a1272

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We show how the partition functions for finite clusters with spin-\textonehalf{} Heisenberg interactions may be computed efficiently and generally to any desired number of powers in reciprocal temperature. As an example, we have expanded the zero-magnetic-field free energy to the twenty-first power for the linear Heisenberg model and for nonzero magnetic field give an expression good through the tenth power. We introduce the concept of the two-point Pad'e approximant and use it to analyze the energy for the linear Heisenberg model.
Exact expressions for the partition function, spin pair correlation function, and susceptibility of the onedimensional isotropic classical Heisenberg model are obtained in zero external field with cyclic boundary conditions. It is shown that the methods used to derive these results enable the partition functions and susceptibilities of finite clusters of interacting classical spins to be evaluated in terms of the $3n\ensuremath{-}j$ symbols of Wigner. Exact results in one dimension are also obtained for the partition function and susceptibility of a "planar" classical Heisenberg model. In this model the spin vectors interact via a Heisenberg coupling but each spin vector is restricted to lie in a plane.The anisotropic classical Heisenberg model described by the Hamiltonian $\mathcal{H}=\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{(\mathrm{ij})}2({{J}_{\mathrm{ij}}}^{x}{{s}_{i}}^{x}{{s}_{j}}^{x}+{{J}_{\mathrm{ij}}}^{y}{{s}_{i}}^{y}{{s}_{j}}^{y}+{{J}_{\mathrm{ij}}}^{z}{{s}_{i}}^{z}{{s}_{j}}^{z})\ensuremath{-}mH\ensuremath{\Sigma}\stackrel{N}{j=1}{{s}_{i}}^{z},$ where six{{s}_{i}}^{x}si​x, siy{{s}_{i}}^{y}si​y, and siz{{s}_{i}}^{z}si​z are components of the unit vector si{\mathbf{s}}_{i}si​, is also considered. A perturbation series for the zero-field free energy of the anisotropic model in one dimension with nearest-neighbor interactions Jijx=Jijy=J{{J}_{\mathrm{ij}}}^{x}={{J}_{\mathrm{ij}}}^{y}=JJij​x=Jij​y=J and ${{J}_{\mathrm{ij}}}^{z}=\ensuremath{\gamma}J$ is developed in powers of $\ensuremath{\gamma}\ensuremath{-}1$ using the isotropic model as the unperturbed system. Detailed calculations are performed to third order in $\ensuremath{\gamma}\ensuremath{-}1$. It is found that the perturbation series for the energy per spin breaks down as $T\ensuremath{\rightarrow}0$. A high-temperature series expansion for the anisotropic model, which is valid for a general interaction potential and lattice, is derived by generalizing the methods developed by Horwitz and Callen for the Ising model. This series is rearranged to give a simplified diagram expansion. Finally, a practical technique for calculating the high-temperature series expansions of the zero-field free energy and susceptibility of the isotropic classical Heisenberg model is presented.
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This work presents an algorithm for calculating high temperature series expansions (HTSE) of Heisenberg spin models with spin S=1/2 S = 1 / 2 in the thermodynamic limit. This algorithm accounts for the presence of a magnetic field. The paper begins with a comprehensive introduction to HTSE and then focuses on identifying the bottlenecks that limit the computation of higher order coefficients. HTSE calculations involve two key steps: graph enumeration on the lattice and trace calculations for each graph. The introduction of a non-zero magnetic field adds complexity to the expansion because previously irrelevant graphs must now be considered: bridged graphs. We present an efficient method to deduce the contribution of these graphs from the contribution of sub-graphs, that drastically reduces the time of calculation for the last order coefficient (in practice increasing by one the order of the series at almost no cost). Previous articles of the authors have utilized HTSE calculations based on this algorithm, but without providing detailed explanations. The complete algorithm is publicly available, as well as the series on many lattice and for various interactions.
The eigen-functions and eigen-values of finite three-dimensional Heisenberg models with anisotropy constant as a parameter (spin=1/2) were obtained exactly with the aid of group theoretical technique and by using a high speed computer. The calculated eigenvalues are identical with those obtained by Serber and Dresselhaus for the isotropic simple cubic 2×2×2 lattice. However, it was found that some terms in their full expressions were to be corrected. The zeros of the partition function of the above 2×2×2 lattice have been investigated as functions of the anisotropy parameter both in the complex temperature plane and in the complex magnetic field plane. Thermodynamic functions such as energy, specific heat, magnetization, and susceptibility have been numerically computed for these finite system.
An improved Padé method is applied to the entropy and the reduced susceptibility χ R (=χ· k B T / N ( g µ) 2 ) expanded in powers of the internal energy in order to calculate the specific heat c ( T ) for S ≤2 and the zero-field susceptibility for S =1/2 of a quantum Heisenberg spin chain, respectively. The exponents α( c ( T )∼ T α ) obtained for an antiferromagnetic exchange and 1≤ S ≤2 are 1.6, 1.4 and 1.3, which are much different from the value (α=1) given by the spin-wave theory (SW). It is also shown that the polynomials with exponents given by SW cannot fit in with the Padé approximants for these cases. This disagreement with the behaviors of SW is ascribed, at least for S =1, to the nonzero gap found by Botet and Jullien.
We derive a variational cluster approximation for Heisenberg spin systems at finite temperature based on the ideas of the self-energy functional theory by Potthoff for fermionic and bosonic systems with local interactions. Partitioning the real system into a set of clusters, we find an analytical expression for the auxiliary free energy, depending on a set of variational parameters defined on the cluster, whose stationary points provide approximate solutions from which the thermodynamics of spin models can be obtained. We explicitly describe the technical details of how to evaluate the free energy for finite clusters and remark on specific problems and possible limitations of the method. To test the approximation we apply it to the antiferromagnetic spin chain and compare the results for varying cluster sizes and choices of variational parameters with the exact Bethe ansatz solution.