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Acta Physica Polonica A

Correlated Electron Systems of Different Dimensionalities

J. Kurzyk, W. Wójcik, Józef Spałek

Published Jul 1, 2008• 0 citations
Publisher

Abstract

The optimized single-particle wave functions contained in the parameters of the Hubbard model (hopping integral t and intraatomic interaction U ) are determined explicitly in the correlated state for electronic systems of various symmetries and dimensions: Hubbard chain, square and triangular lattices, and the three cubic lattices: SC, BCC, and FCC.In effect, the electronic properties of these structures as a function of the interatomic distance R are obtained.In most cases, the model parameters do not scale linearly with the lattice constant.Also, the atomic part of the total ground state energy changes with the U/t ratio and therefore should be (and is) included in the analysis.The solutions of dimensions D > 1 are analyzed by utilizing the approximate Gutzwiller treatment.

DOI
10.12693/aphyspola.114.175

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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.

•Jan 25, 2012•0 citations
The optimized single-particle wave functions contained in the parameters of the Hubbard model (hopping integral t and intraatomic interaction U ) are determined explicitly in the correlated state for electronic systems of various symmetries and dimensions: Hubbard chain, square and triangular lattices, and the three cubic lattices: SC, BCC, and FCC.In effect, the electronic properties of these structures as a function of the interatomic distance R are obtained.In most cases, the model parameters do not scale linearly with the lattice constant.Also, the atomic part of the total ground state energy changes with the U/t ratio and therefore should be (and is) included in the analysis.The solutions of dimensions D > 1 are analyzed by utilizing the approximate Gutzwiller treatment.
The optimized single-particle wave functions contained in the parameters of the Hubbard model (t and U) were determined for an infinite atomic chain. In effect, the electronic properties of the chain as a function of interatomic distance R were obtained and compared for the Lieb - Wu exact solution (LW), the Gutzwiller-Wave-Function approximation (GWF), and the Gutzwiller-ansatz case (GA). The ground state energy and other characteristics for the infinite chain were also compared with those obtained earlier for a nanoscopic chain within the Exact Diagonalization combined with an Ab Initio adjustment of the single-particle wave functions in the correlated state (EDABI method). For the sake of completeness, we briefly characterize also each of the solutions. Our approach completes the Lieb-Wu solution, as it provides the system electronic properties evolution as a function of physically controllable parameter - the interatomic distance.
The optimized single-particle wave functions contained in the parameters of the Hubbard model (t and U ) were determined for an infinite atomic chain.In effect, the electronic properties of the chain as a function of interatomic distance R were obtained and compared for the Lieb-Wu exact solution, the Gutzwiller-wave-function approximation, and the Gutzwiller--ansatz case.The ground state energy and other characteristics for the infinite chain were also compared with those obtained earlier for a nanoscopic chain within the exact diagonalization combined with an ab initio adjustment of the single-particle wave functions in the correlated state (exact diagonalization combined with an ab initio method).For the sake of completeness, we briefly characterize also each of the solutions.Our approach completes the Lieb-Wu solution, as it provides the system electronic properties evolution as a function of physically controlable parameter -the interatomic distance.
One-dimensional atomic chain with a variable-range hopping is described within the extended Hubbard model.The Gutzwiller-ansatz approximation is used to determine the optimized single-particle (Wannier) wave functions in the correlated state.Hopping integral up to the third neighbors is taken into account and the results are compared with those for the infinite hopping range.Ground state energy of the system is compared with that making use of the rigorous Lieb-Wu solution with the optimized wave functions.The evolution of the properties as a function of interatomic distance is discussed.
The dimerized one-dimensional Hubbard model is studied in the framework of lattice density-functional theory (LDFT). The single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ with respect to the lattice sites is considered as the basic variable. The corresponding interaction-energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is defined by Levy's constrained search. Exact numerical results are obtained for $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23}),$ where ${\ensuremath{\gamma}}_{12}={\ensuremath{\gamma}}_{i,i+1}$ for odd i and ${\ensuremath{\gamma}}_{23}={\ensuremath{\gamma}}_{i,i+1}$ for even i are the nearest-neighbor density-matrix elements along the chain. The domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}}$ and the functional dependence of $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23})$ are analyzed. A simple, explicit approximation to $W({\ensuremath{\gamma}}_{12},{\ensuremath{\gamma}}_{23})$ is proposed, which is derived from scaling properties of W, exact dimer results, and known limits. Using this approximation, LDFT is applied to determine ground-state properties and charge-excitation gaps of finite and infinite dimerized chains as a function of the Coulomb-repulsion strength U/tU/tU/t and of the alternation $\ensuremath{\delta}t$ of the hopping integrals tij{t}_{\mathrm{ij}}tij​ ${(t}_{\mathrm{ij}}=t\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}t).$ The accuracy of the method is demonstrated by comparison with available exact solutions and accurate numerical calculations. Goals and limitations of the present approach are discussed particularly concerning its ability to describe the crossover from weak to strong electron correlations.
We discuss the physical properties of strongly correlated electron states for a linear chain obtained with the help of a recently proposed method combining an exact diagonalization in Fock space with an ab initio readjustment of single-particle orbitals in the correlated state. The method extends the current discussion of correlated states, since the properties are obtained with varying lattice spacings. The finite system of N atoms evolves with the increasing interatomic distance from a Fermi-liquid-like state into the Mott insulator. The criteria of the localization are discussed in detail, since the results are already convergent for N> 8.N>~8.N> 8. During this process the Fermi-Dirac distribution is smeared out, the effective band mass increases by \ensuremath{\sim}50%, and the spin-spin correlation functions reduce to those for a Heisenberg antiferromagnet. Values of microscopic parameters such as the hopping and the kinetic-exchange integrals, as well as the magnitude of both intraatomic and interatomic Coulomb and exchange interactions are calculated. We also determine the values of various local electron-lattice couplings, and show that they are comparable to the kinetic exchange contribution in the strong-correlation limit. The magnitudes of the dimerization and the zero-point motion are also discussed. Our results provide a canonical example of a tractable strongly correlated system with a precise, first-principles description as a function of interatomic distance of a model system involving all hopping integrals, all pair-site interactions, and exact one-band Wannier functions.
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.