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 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 formulate a finite-temperature scheme of the variational cluster approximation (VCA) particularly suitable for an exact-diagonalization cluster solver. Based on the analytical properties of the single-particle Green's function matrices, we explicitly show the branch-cut structure of logarithm of the complex determinant functions appearing in the self-energy-functional theory (SFT) and whereby construct an efficient scheme for the finite-temperature VCA. We also derive the explicit formulas for entropy and specific heat within the framework of the SFT. We first apply the method to explore the antiferromagnetic order in a half-filled Hubbard model by calculating the entropy, specific heat, and single-particle excitation spectrum for different values of on-site Coulomb repulsion
U and temperature
T. We also calculate the
T dependence of the single-particle excitation spectrum in the strong coupling region, and discuss the overall similarities to and the fine differences from the spectrum obtained by the spin-density-wave mean-field theory at low temperatures and the Hubbard-I approximation at high temperatures. Moreover, we show a necessary and sufficient condition for the third law of thermodynamics in the SFT. On the basis of the thermodynamic properties, such as the entropy and the double occupancy, calculated via the
T and/or
U derivative of the grand potential, we obtain a crossover diagram in the
(U,T) plane, which separates a Slater-type insulator and a Mott-type insulator. Next, we demonstrate the finite-temperature scheme in the cluster-dynamical-impurity approximation (CDIA), i.e., the VCA with noninteracting bath orbitals attached to each cluster, and study the paramagnetic Mott metal-insulator transition in the half-filled Hubbard model. Formulating the finite-temperature CDIA, we first address a subtle issue regarding the treatment of the artificially introduced bath degrees of freedom which are absent in the originally considered Hubbard model. We then apply the finite-temperature CDIA to calculate the finite-temperature phase diagram in the
(U,T) plane. Metallic, insulating, coexistence, and crossover regions are distinguished from the bath-cluster hybridization-variational-parameter dependence of the grand-potential functional. We find that the Mott transition at low temperatures is discontinuous, and the coexistence region of the metallic and insulating states persists down to zero temperature. The result obtained here by the finite-temperature CDIA is complementary to the previously reported zero-temperature CDIA phase diagram.
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.