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A Variational Cluster Approximation for the Heisenberg Model

S Filor

Published Jan 1, 2017• 0 citations
Publisher

Abstract

In this thesis we present a novel variational cluster approximation for Heisenberg spin systems. It is based on the self-energy functional theory by Potthoff for fermionic and bosonic models with local interactions. To develop a similar method for spin systems, we derive a free energy functional which is the starting point of the approximation. Within this approximation, we find an analytical expression to evaluate the free energy by tiling the real system into a set of clusters. We describe the technical details of the spin variational cluster approximation and the evaluation of the free energy. The method is tested for the antiferromagnetic spin-1/2 Heisenberg chain with first- and second-neighbour interactions. Thereby, we investigate the dependence of the approximation on cluster size and the choice of variational parameters. The opportunities and limitations as well as future applications of the method are thoroughly discussed.

DOI
10.53846/goediss-6255

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In this thesis we present a novel variational cluster approximation for Heisenberg spin systems. It is based on the self-energy functional theory by Potthoff for fermionic and bosonic models with local interactions. To develop a similar method for spin systems, we derive a free energy functional which is the starting point of the approximation. Within this approximation, we find an analytical expression to evaluate the free energy by tiling the real system into a set of clusters. We describe the technical details of the spin variational cluster approximation and the evaluation of the free energy. The method is tested for the antiferromagnetic spin-1/2 Heisenberg chain with first- and second-neighbour interactions. Thereby, we investigate the dependence of the approximation on cluster size and the choice of variational parameters. The opportunities and limitations as well as future applications of the method are thoroughly discussed.
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.
We set up a new approach to study the physics of spin systems which uses the resolvent method originally proposed by Keiter and Kuramoto. In analogy to the Baym-Kadanoff formalism the latter introduced the partition sum of a system as a functional of the resolvent of the Hamiltonian and its so called generalized self-energy. This functional is the starting point for a variational cluster method which is based on Potthoffs self-energy functional approach. In a first step we apply our ansatz to a Heisenberg spin chain.
We apply the projected Hartree-Fock theory (PHF) for approximating ground states of Heisenberg spin clusters. Spin-rotational, point-group, and complex-conjugation symmetry are variationally restored from a broken-symmetry mean-field reference, where the latter corresponds to a product of local spin states. A fermionic formulation of the Heisenberg model furnishes a conceptual connection to PHF applications in quantum chemistry and detailed equations for a self-consistent field optimization of the reference state are provided. Different PHF variants are benchmarked for ground-state energies and spin-pair correlation functions of antiferromagnetic spin rings and three different polyhedra, with various values of the local spin-quantum number sss. Although PHF is not suitable to study the thermodynamic limit (where it reduces to the conventional HF results), the low computational cost and the compact wave-function representation make PHF a promising complement to existing approaches for ground states of finite spin clusters, particularly for large local spin sss and a moderately large number of sites NNN. The present work may also motivate future explorations of more accurate post-PHF methods for Heisenberg spin clusters.
A general discussion on applying the cluster variation method to non-uniform Ising and Heisenberg models is given. In particular, a general formula is provided for obtaining the linear response to increases of the external fields and the parameters of interactions for the uniform Ising and Heisenberg models in the cluster variation method. As an illustration, it is shown that the expressions of the spin-pair correlation function in the pair and the square approximation are readily derived by the formula, for the Ising models on the square and simple cubic lattices.
Based on the self-energy-functional approach proposed recently [M. Potthoff, Eur. Phys. J. B 32, 429 (2003)], we present an extension of the cluster-perturbation theory to systems with spontaneously broken symmetry. Our method applies to models with local interactions and accounts for both short-range correlations and long-range order. Short-range correlations are accurately taken into account via exact diagonalization of finite clusters. Long-range order is described by variational optimization of a ficticious symmetry-breaking field. In comparison with related cluster methods, our approach is more flexible and, for a given cluster size, less demanding numerically, especially at zero temperature. An application of the method to the antiferromagnetic phase of the Hubbard model at half-filling shows good agreement with results from quantum Monte Carlo calculations. We demonstrate that the variational extension of the cluster-perturbation theory is crucial to reproduce salient features of the single-particle spectrum.
A general perturbation theory of the statistics of spin interactions is developed in the form of a linked cluster expansion with particular emphasis on the Ising model. The theory applies to the evaluation of the expectation value of arbitrary spin functions as well as of the free energy. The thermodynamical consistency of the perturbation expansion is shown to arise from (1) variational principles satisfied by the free energy after a "renormalization procedure" has been carried out and (2) a generalized "Ward Identity" between renormalized quantities. These results are used to discuss inconsistencies in recent high-density theories of ferromagnetism and an improved theory obtained by the summation of all the convolution diagrams (nodal expansion) is briefly presented. The applicability of the method to general quantum mechanical many-body problems, including the theory of the Heisenberg model of ferromagnetism, is shown.