The software package was developed that implements the variational cluster approximation for solving the generalized Hubbard model taking into account systems of various dimensionality, with several orbitals, and various lattice geometries. The developed package 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 resulting code 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.
The canonical one-band Hubbard model is studied using a computational method that mixes the Monte Carlo procedure with the mean field approximation. This technique allows us to incorporate thermal fluctuations and the development of short-range magnetic order above ordering temperatures, contrary to the crude finite-temperature Hartree-Fock approximation, which incorrectly predicts a N\'eel temperature
TN that grows linearly with the Hubbard
U/t. The effective model studied here contains quantum and classical degrees of freedom. It thus belongs to the ``spin fermion'' model family widely employed in other contexts. Using exact diagonalization, supplemented by the traveling cluster approximation, for the fermionic sector, and classical Monte Carlo for the classical fields, the Hubbard
U/t vs temperature
T/t phase diagram is studied employing large three- and two-dimensional clusters. We demonstrate that the method is capable of capturing the formation of local moments in the normal state without long-range order, the nonmonotonicity of
TN with increasing
U/t, the development of gaps and pseudogaps in the density of states, and the two-peak structure in the specific heat. Extensive comparisons with determinant quantum Monte Carlo results suggest that the present approach is qualitatively, and often quantitatively, accurate, particularly at intermediate and high temperatures. Finally, we study the Hubbard model including plaquette diagonal hopping (i.e., the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}$ Hubbard model) in two dimensions and show that our approach allows us to study low-temperature properties where determinant quantum Monte Carlo fails due to the fermion sign problem. Future applications of this method include multiorbital Hubbard models such as those needed for iron-based superconductors.
Various methods and approximation schemes are used to study many-electron interacting systems. Two important many-particle models, the Anderson model and the Hubbard model, and their electromagnetic properties have been investigated in many parameter regimes, and applied to physical systems. An Anderson single-impurity model Hamiltonian based calculation of the magnetic susceptibility is performed for YbN in the presence of crystal fields using an alteration of the Non-Crossing Approximation proposed by Zwicknagl et.al., incorporating parameters obtained from ab initio band structure calculations. It yields good agreement with experimental data. For the Anderson lattice model, a variational scheme which uses specific many-electron wavefunctions as basis is applied to both one- and two-dimensional systems represented by symmetric Anderson lattice Hamiltonians. Without much computational effort, the ground state energy is well approximated, especially in strong-coupling limit. Some electronic properties are examined using the variational ground state wavefunction. The one-dimensional Hubbard model has been solved exactly for small-size clusters by diagonalizing the Hamiltonian in the basis of many-electron Bloch states. The results for the energy spectrum and eigenfunctions of the ground state and low-lying excited states are presented. Also, mean field calculations of the two-dimensional single-band Hubbard model and Cu-O lattice model (three-band Hubbard model) are carried out for various physical quantities including the energy, occupation probability, staggered magnetization, momentum distribution Fermi surface and density of states, by using a projection operator formalism. To develop a systematic approach to solving many-electron problems, the many-particle partition function for the free electron gas system is explored using a cumulant expansion scheme. Starting from the ground state, the partition function can be approximated to any order in terms of excitation energy. Its application to interacting systems such as the Anderson model and the Hubbard model is briefly discussed.
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.
We develop the variational-cluster-approximation method based on the thermal-pure-quantum-state approach and apply the method to the calculations of the thermodynamic properties of the Hubbard model, thereby obtaining the temperature dependence of the single-particle excitation spectra, entropy and specific heat, and order parameter of the antiferromagnetic phase efficiently. We thus demonstrate that the method developed here has a potential power for elucidating finite-temperature properties of strongly correlated electron systems.