An Investigation of the Electron Transfer and Repulsive Coulomb Interactions in the One-, Two-, Three-Dimensional Isotropic Hubbard Models II: Extension to a Wannier and Higher Angular Momentum Basis
In the previous paper we have investigated the isotropic Hubbard model, with S-orbits on each site, for one-, two- and three-dimensional isotropic lattices. The method involved the determination of the one-electron energies and eigenfunctions of the transfer interaction, the latter being utilised to form determinantal eigenfunctions for the many-electron system as a whole. In this paper we have examined the consequences of using an atomic basis by introducing Wannier functions and studied physical consequences of this more exact approach. To reduce the transformed two-body interaction to manageable form we have used the rotational properties of individual atomic orbitals incorporating them via irreducible tensorial methods. We find that, although additional components due to Wannier "tails" are small they may modify the nearest neighbor transfer terms significantly and also introduce next-nearest and next-next nearest neighbor transfer interactions. We have also investigated the effect of using atomic orbitals with a higher angular momentum on each site. Both p-orbitals and d-orbitals have been studied and the broad conclusion is that the corresponding Hubbard model reduces to a series of S-like orbital models, one for each component, m, of the degenerate set on each site. The transfer interaction in each case becomes, in general, m dependent, each m component being effectively decoupled from others.
In the previous paper we have investigated the isotropic Hubbard model, with S-orbits on each site, for one-, two- and three-dimensional isotropic lattices. The method involved the determination of the one-electron energies and eigenfunctions of the transfer interaction, the latter being utilised to form determinantal eigenfunctions for the many-electron system as a whole. In this paper we have examined the consequences of using an atomic basis by introducing Wannier functions and studied physical consequences of this more exact approach. To reduce the transformed two-body interaction to manageable form we have used the rotational properties of individual atomic orbitals incorporating them via irreducible tensorial methods. We find that, although additional components due to Wannier "tails" are small they may modify the nearest neighbor transfer terms significantly and also introduce next-nearest and next-next nearest neighbor transfer interactions. We have also investigated the effect of using atomic orbitals with a higher angular momentum on each site. Both p-orbitals and d-orbitals have been studied and the broad conclusion is that the corresponding Hubbard model reduces to a series of S-like orbital models, one for each component, m, of the degenerate set on each site. The transfer interaction in each case becomes, in general, m dependent, each m component being effectively decoupled from others.
In the previous paper we have investigated the isotropic Hubbard model, with S-orbits on each site, for one-, two- and three-dimensional isotropic lattices. The method involved the determination of the one-electron energies and eigenfunctions of the transfer interaction, the latter being utilised to form determinantal eigenfunctions for the many-electron system as a whole. In this paper we have examined the consequences of using an atomic basis by introducing; Wannier functions and studied physical consequences of this more exact approach. To reduce the transformed two-body interaction to manageable form we have used the rotational properties of individual atomic orbitals incorporating them via irreducible tensorial methods. We find that, although additional components due to Wannier "tails" are small they may modify the nearest neighbor transfer terms significantly and also introduce next-nearest and next-next nearest neighbor transfer interactions. We have also investigated the effect of using atomic orbitals with a higher angular momentum on each site. Both p-orbitals and d-orbitals have been studied and the broad conclusion is that the corresponding Hubbard model reduces to a series of S-like orbital models, one for each component, m, of the degenerate set on each site. The transfer interaction in each case becomes, in general, m dependent, each m component being effectively decoupled from others.
In this paper we investigate the Hubbard model, with S-orbits on each site, for one-, two- and three-dimensional isotropic lattices with arbitrary values of transfer and on-site repulsion parameters. The method of diagonalization is based on equivalences due to Stevens for number conserving operators and a unitary transformation utilizing permutational symmetries of the problem. We obtained simple analytical formulas for single particle eigenstates and somewhat more involved expression for determinantal multi-electron states. Numerical calculations have been performed which demonstrate the boundaries of the lowest band for various filling factors using examples in one- and two-dimensions. The states within the band have also been identified and degeneracies evaluated numerically. These computations were based on a Monté Carlo technique in conjunction with particle conservation requirements. Agreement with some well-known limiting cases has been found.
In this paper we investigate the Hubbard model, with S-orbits on each site, for one-, two- and three-dimensional isotropic lattices with arbitrary values of transfer and on-site repulsion parameters. The method of diagonalization is based on equivalences due to Stevens for number conserving operators and a unitary transformation utilizing permutational symmetries of the problem. We obtained simple analytical formulas for single particle eigenstates and somewhat more involved expression for determinantal multi-electron states. Numerical calculations have been performed which demonstrate the boundaries of the lowest band for various filing factors using examples in one- and two-dimensions. The states within the band have also been identified and degeneracies evaluated numerically. These computations were based on a Monte Carlo technique in conjunction with particle conservation requirements. Agreement with some well-known limiting cases has been found.
To the conventional Peierls-Hubbard model, involving both on-site (U) and nearest-neighbor (V) Coulomb repulsions, we add ``off-diagonal'' terms, not expressible purely in terms of site densities, representing bond-bond (W) and mixed bond-site (X) electron-electron repulsive interactions involving nearest neighbors. We review earlier analyses of these interactions and discuss relative magnitudes of the parameters in applications to real materials. As a specific illustration, we investigate the effects of the off-diagonal W and X terms on dimerization in the one-dimensional, half-filled-band Peierls-Hubbard models, which have been widely applied to conjugated polymers (such as trans-polyacetylene) and to related quasi-one-dimensional charge-density wave (CDW) systems. Using both weak- and strong-coupling perturbation theory for large systems and exact diagonalizations of small systems, we investigate thoroughly the nature of the ground state of the model. For a broad range of the site-diagonal Hubbard parameters (U,V), including the values believed to be relevant to trans-polyacetylene, we find that the off-diagonal terms (W,X) initially enhance dimerization, thereby stabilizing the dimerized [or bond-order-wave (BOW)] ground state. For (unphysically) large values of W relative to U and V, dimerization is destroyed, and the BOW ground state goes over to a ferromagnetic ground state or a CDW ground state, depending on the relative sizes of U, V, and W. We conclude with a general discussion of the applicability of the Peierls-Hubbard models to quasi-one-dimensional materials, including the potential importance of the breaking of charge conjugation (``particle-hole'') symmetry by the X term.
In this Thesis, we develop and apply an accurate method to determine the ground-state properties of strongly-correlated electrons in the framework of lattice model Hamiltonians. In lattice density-functional theory (LDFT) the basic variable is the one-particle density matrix γ. From a generalized HK theorem, the ground state energy Egs[γgs] = minγ E[γ] is obtained by minimizing the energy-functional E[γ] over all the physical or representable γ. The energy functional can be divided into two contributions: the kinetic-energy functional T [γ], which linear dependence on γ is exactly known, and the correlation-energy functional W [γ], which dependence on γ is not explicitly known. Finding accurate approximations for W [γ] constitutes the actual challenge of this thesis. Part of this work is based on previous studies which derive an approximation of W [γ] for the Hubbard Hamiltonian based on scaling hypothesis and an exact analytical results for the Hubbard dimer. However, this approach is limited to spin independent and homogeneous systems. In order to extend the scope of LDFT, we have developed three different approaches to derive W [γ] in order to study broken symmetry problems. First we extend the previous scaling functional to system having charge transfer. By a systematic study of the dependence of W [γ] on the charge distribution we find similar scaling properties as for the homogeneous case. An extension to the Hubbard model on bipartite lattices is then derived and applied to finite and infinite systems with repulsive and attractive interactions. The high accuracy of this functional is demonstrated. Nevertheless, this approach is difficult to transfer to more complex systems, since the calculation of W [γ] applies to the system as a whole. To overcome this problem, we derive another approximation based on local scaling properties. This functional is formulated on-site so that it can be applied to any ordered or disordered Hamiltonian with on-site interaction. As applications we study the metal-insulator transition in the ionic Hubbard model in oneand two-dimensions, as well as the one-dimensional Hubbard chain with first and second nearest neighbor hoppings. Finally, we develop a numerical approach of W [γ], on the basis on exact diagonalizations of an effective many-body Hamiltonian corresponding to a cluster surrounded by an effective medium. This effective Hamiltonian depends on the density matrix γ and allows to derive approximation to W [γ] which improve systematically with increasing cluster size. The formulation is spin-dependent and allows a straightforward generalization to multi-orbital correlated systems such spd-Hamiltonians. In addition it take into account the effects of short-range charge and spin fluctuations in the functional. The accuracy of the method is demonstrated for the Hubbard model by comparison with the Bethe-Ansatz solution (1D) and quantum Monte Carlo simulations (2D). Finally, an outlook or relevant future developments of the theory is provided.
Originally, the Hubbard model was derived for describing the behavior of strongly correlated electrons in solids. However, for over a decade now, variations of it have also routinely been implemented with ultracold atoms in optical lattices, allowing their study in a clean, essentially defect-free environment. Here, we review some of the vast literature on this subject, with a focus on more recent non-standard forms of the Hubbard model. After giving an introduction to standard (fermionic and bosonic) Hubbard models, we discuss briefly common models for mixtures, as well as the so-called extended Bose-Hubbard models, that include interactions between neighboring sites, next-neighbor sites, and so on. The main part of the review discusses the importance of additional terms appearing when refining the tight-binding approximation for the original physical Hamiltonian. Even when restricting the models to the lowest Bloch band is justified, the standard approach neglects the density-induced tunneling (which has the same origin as the usual on-site interaction). The importance of these contributions is discussed for both contact and dipolar interactions. For sufficiently strong interactions, the effects related to higher Bloch bands also become important even for deep optical lattices. Different approaches that aim at incorporating these effects, mainly via dressing the basis, Wannier functions with interactions, leading to effective, density-dependent Hubbard-type models, are reviewed. We discuss also examples of Hubbard-like models that explicitly involve higher p orbitals, as well as models that dynamically couple spin and orbital degrees of freedom. Finally, we review mean-field nonlinear Schrödinger models of the Salerno type that share with the non-standard Hubbard models nonlinear coupling between the adjacent sites. In that part, discrete solitons are the main subject of consideration. We conclude by listing some open problems, to be addressed in the future.