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Journal of Physics Conference Series

A self-energy functional approach for spin systems

S Filor, Thomas Pruschke

Published Jan 1, 2010• 4 citations
Publisher

Abstract

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.

DOI
10.1088/1742-6596/200/2/022007

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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.
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 develop a general quantum many-body theory in configuration space by extending the traditional coupled-cluster method to a variational formalism. A set of distribution functions are introduced to evaluate the expectation value of the Hamiltonian. An algebraic technique for calculations of these distribution functions via a set of self-consistent set of equations is given. As a demonstration, we apply this method to a quantum antiferromagnetic spin model. It is shown that one of the lowest-order approximations within this new technique agrees with the traditional coupled-cluster method. Comparison with the method of correlated basis functions is also made.
We derive the self-energy functional theory for bosonic lattice systems with broken U(1) symmetry by parametrizing the bosonic Baym-Kadanoff effective action in terms of one- and two-point self-energies. The formalism goes beyond other approximate methods such as the pseudoparticle variational cluster approximation, the cluster composite boson mapping, and the Bogoliubov+U theory. It simplifies to bosonic dynamical-mean-field theory when constraining to local fields, whereas when neglecting kinetic contributions of noncondensed bosons, it reduces to the static mean-field approximation. To benchmark the theory, we study the Bose-Hubbard model on the two- and three-dimensional cubic lattice, comparing with exact results from path integral quantum Monte Carlo. We also study the frustrated square lattice with next-nearest-neighbor hopping, which is beyond the reach of Monte Carlo simulations. A reference system comprising a single bosonic state, corresponding to three variational parameters, is sufficient to quantitatively describe phase boundaries and thermodynamical observables, while qualitatively capturing the spectral functions, as well as the enhancement of kinetic fluctuations in the frustrated case. On the basis of these findings, we propose self-energy functional theory as the omnibus framework for treating bosonic lattice models, in particular, in cases where path integral quantum Monte Carlo methods suffer from severe sign problems (e.g., in the presence of nontrivial gauge fields or frustration). Self-energy functional theory enables the construction of diagrammatically sound approximations that are quantitatively precise and controlled in the number of optimization parameters but nevertheless remain computable by modest means.
Based on a functional-integral formalism, a generalization of the self-energy-functional theory (SFT) is proposed which is applicable to systems of interacting electrons with disorder. Similar to the pure case without disorder, a variational principle is set up which gives the physical (disorder) self-energy as a stationary point of the (averaged) grand potential. Although the resulting self-energy functional turns out to be more complicated, the formal structure of the theory can be retained since the unknown part of the functional is universal. This allows one to construct nonperturbative and thermodynamically consistent approximations via searching for a stationary point on a restricted domain of the functional. The theory and the possible approximations are worked out for models with local interactions and local disorder. This results in a derivation of different mean-field approaches and various cluster extensions, including well-known concepts such as the statistical dynamical mean-field theory, the molecular coherent-potential approximation, and the dynamical cluster approximation. Due to the common formal framework provided by the SFT, one achieves a general systematization of dynamical approaches, i.e., approaches based on the spectrum of one-particle excitations. Different mean-field and different cluster schemes naturally appear in this framework and complement the existing ones. Their prospects for future applications are discussed.
A general quantum many-body theory in configuration space is developed by extending the traditional coupled-cluster method (CCM) to a variational formalism. Two independent sets (destruction and creation sets) of distribution functions are introduced to evaluate the Hamiltonian expectation. An algebraic technique for calculating these distribution functions via two self-consistent set of equations is given. By comparing with the traditional CCM and with Arponen's extension, it is shown that the former is equivalent to a linear approximation to one set of distribution functions and the latter is equivalent to a (generalized) random-phase approximation to it. In addition to these two approximations, other higher-order approximation schemes within the formalism are also discussed. As a demonstration, we apply this technique to a quantum antiferromagnetic spin model.