The numerical method was developed that combines the variational cluster approximation and the exact diagonalization techniques to solve Hubbard models of various dimensionality, number of orbitals, and lattice geometry. The developed method was tested on the one-band Hubbard model; antiferromagnetic ordering and the superconducting phase were observed, in accordance with known results obtained by other techniques. The method may be successfully used for studying properties of high-temperature superconductors.
The Variational cluster approximation (VCA) is an approximate method for finding properties the Hubbard model, the basic model used in the study of strongly correlated electron systems. Such systems include, for instance, high-temperature and organic superconductors.The method involves solving exactly the Hubbard model on a small cluster of lattice sites, after adding fields that represent the effect of the cluster's environment. We present an outline of the method and describe its implementation. In particular, we show how to efficiently carry frequency-wavevector integrals that are central to the method.
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.
In this thesis, quantum cluster methods are used to calculate electronic properties of correlated-electron systems. A special focus lies in the determination of the ground state properties of a 3/4 filled triangular lattice within the one-band Hubbard model. At this filling, the electronic density of states exhibits a so-called van Hove singularity and the Fermi surface becomes perfectly nested, causing an instability towards a variety of spin-density-wave (SDW) and superconducting states. While chiral d+id-wave superconductivity has been proposed as the ground state in the weak coupling limit, the situation towards strong interactions is unclear. Additionally, quantum cluster methods are used here to investigate the interplay of Coulomb interactions and symmetry-breaking mechanisms within the nematic phase of iron-pnictide superconductors. The transition from a tetragonal to an orthorhombic phase is accompanied by a significant change in electronic properties, while long-range magnetic order is not established yet. The driving force of this transition may not only be phonons but also magnetic or orbital fluctuations. The signatures of these scenarios are studied with quantum cluster methods to identify the most important effects. Here, cluster perturbation theory (CPT) and its variational extention, the variational cluster approach (VCA) are used to treat the respective systems on a level beyond mean-field theory. Short-range correlations are incorporated numerically exactly by exact diagonalization (ED). In the VCA, long-range interactions are included by variational optimization of a fictitious symmetry-breaking field based on a self-energy functional approach. Due to limitations of ED, cluster sizes are limited to a small number of degrees of freedom. For the 3/4 filled triangular lattice, the VCA is performed for different cluster symmetries. A strong symmetry dependence and finite-size effects make a comparison of the results from different clusters difficult. The ground state in the weak-coupling limit is superconducting with chiral d+id-wave symmetry, in accordance to previous renormalization group approaches. In the regime of strong interactions SDW states are preferred over superconductivity and a collinaer SDW state with nonuniform spin moments on a quadrupled unit cell has the lowest grand potential. At strong coupling, inclusion of short-range quantum fluctuations turns out to favor this collinear state over the chiral phase predicted by mean-field theory. At intermediate interactions, no robust conclusion can be drawn from the results. Symmetry-breaking mechanisms within the nematic phase of the iron-pnictides are studied using a three-band model for the iron planes on a 4-site cluster. CPT allows a local breaking of the symmetry within the cluster without imposing long-range magnetic order. This is a crucial step beyond mean-field approaches to the magnetically ordered state, where such a nematic phase cannot easily be investigated. Three mechanisms are included to break the fourfold lattice symmetry down to a twofold symmetry. The effects of anisotropic magnetic couplings are compared to an orbital ordering field and anisotropic hoppings. All three mechanisms lead to similar features in the spectral density. Since the anisotropy of the hopping parameters has to be very large to obtain similar results as observed in ARPES, a phonon-driven transition is unlikely.
In order to understand various local phenomena related to electronic phase separation discovered recently in strongly correlated materials, quantum cluster methods are applied to a theoretical model, i.e., Hubbard model, which has successfully predicted basic properties of strongly correlated systems. Our calculations are performed at two levels: exact calculations based on the isolated small clusters via exact diagonalization and approximate calculations applied to large lattices using the variational cluster approximation (VCA).\nWe first study phase separation instabilities with the formation of a pairing gap in the two-dimensional Hubbard model for 8-site Betts clusters. The exact diagonalization method is applied to extract exact ground and excited states of the Hubbard model. The results show that the electronic states with one hole off half filling are unstable and they energetically prefer creating the spatial phase separation. The effect of next nearest hopping is also discussed in this part.\nWe studied the variation of the pairing when the out-of-plane correlation is present by adding an interacting apical site above the two-dimensional cluster plane. The calculations indicate that the out-of-plane correlation can be detrimental to the in-plane pairing effect. When it is not too strong to destroy electron pairs on the plane, the modulation of the apical site can drive the pairing gap to show a sinusoidal variation which is consistent with recent experimental discoveries.\nVariational cluster approximation is introduced to explore local properties of larger lattices. The local nematic state is studied in the square lattice using this method, which indicates that it is possible that the electronic states can locally break the C4 symmetry (90o$ rotational symmetry) to form a pattern where the electronic properties, such as charge and spin correlations, are different in x and y directions.\nThe spatial phase separation in the square and honeycomb lattices is also investigated by the variational cluster approximation. The study shows that Coulomb interaction and lattice geometry are both crucial for electronic phase separation. We propose that electronic phase separation is the result of an instability of the Fermi surface at the boundary of the first Brillouin zone of antiferromagnetic states.
The variational cluster approach (VCA) is applied to the one-dimensional Hubbard model at zero temperature using clusters (chains) of up to ten sites with full diagonalization and the Lanczos method as cluster solver. Within the framework of the self-energy-functional theory (SFT), different cluster reference systems with and without bath degrees of freedom, in different topologies and with different sets of variational parameters, are considered. Static and one-particle dynamical quantities are calculated for half-filling as a function of
U as well as for fixed
U as a function of the chemical potential to study the interaction- and filling-dependent metal-insulator (Mott) transition. The recently developed
Q-matrix technique is used to compute the SFT grand potential. For benchmarking purposes we compare the VCA results with exact results available from the Bethe ansatz, with essentially exact dynamical density-matrix renormalization-group data, with (cellular) dynamical mean-field theory and full diagonalization of isolated Hubbard chains. Several issues are discussed including convergence of the results with cluster size, the ability of cluster approaches to access the critical regime of the Mott transition, efficiency in the optimization of correlated-site vs bath-site parameters, and of multidimensional parameter optimization. We also study the role of bath sites for the description of excitation properties and as charge reservoirs for the description of filling dependencies. The VCA turns out to be a computationally cheap method which is competitive with established cluster approaches.