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A Multiorbital DMFT Analysis of the Dynamic Hubbard Model

Christopher Polachic

Published Jan 1, 2015• 0 citations
Publisher

Abstract

The conventional, single band Hubbard model captures the dynamics that arise in strongly correlated electron systems in the competition between electron transport and local Coulomb interactions. However, some of the important physics of real lattices is overlooked by its simplicity. The dynamic Hubbard model (DHM) provides additional degrees of freedom for electron states in a lattice by including a higher energy orbital and hybridization parameters for electrons to take advantage of a reduced Coulomb repulsion in the upper orbital when the lattice is at half-filling. This ``relaxation'' of the electronic state is accompanied by an asymmetry between electron and hole quasiparticles in the model. In this thesis we employ the computational tool of multiorbital dynamical mean field theory (MODMFT) to study electron-hole asymmetry in the DHM as evidenced in the behaviour of the quasiparticle weight and optical conductivity of electrons. In agreement with previous exact diagonalization work on the DHM, we find that the relaxation of electronic states is associated with quasiparticle dressing in the hole regime of the lower orbital, and a transfer of optical conductivity spectral weight from low to high frequencies. These observations are evidences of the electron-hole asymmetry in the two-orbital system. In this context, we evaluate the role of two critical variables in the DHM: the local and nonlocal hybridizations which parameterize interorbital electron transitions in the Hamiltonian. We demonstrate that the electron-hole asymmetry of the model has a more complex relationship to these hybridization parameters than previously identified. Mott physics plays an essential role in the quasiparticle dressing due to the local hybridization, while the nonlocal hybridization promotes electron-hole asymmetry by providing a separate channel for electron transport which serves to reduce the quasiparticle weight of holes in the lower band. We also briefly examine the phenomenon of orbital selective Mott transitions in the DHM within the parameter regime of electron state relaxation, and evaluate the role played by the hybridization parameters in this context.

DOI
10.7939/r3fn11115

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The conventional, single band Hubbard model captures the dynamics that arise in strongly correlated electron systems in the competition between electron transport and local Coulomb interactions. However, some of the important physics of real lattices is overlooked by its simplicity. The dynamic Hubbard model (DHM) provides additional degrees of freedom for electron states in a lattice by including a higher energy orbital and hybridization parameters for electrons to take advantage of a reduced Coulomb repulsion in the upper orbital when the lattice is at half-filling. This ``relaxation'' of the electronic state is accompanied by an asymmetry between electron and hole quasiparticles in the model. In this thesis we employ the computational tool of multiorbital dynamical mean field theory (MODMFT) to study electron-hole asymmetry in the DHM as evidenced in the behaviour of the quasiparticle weight and optical conductivity of electrons. In agreement with previous exact diagonalization work on the DHM, we find that the relaxation of electronic states is associated with quasiparticle dressing in the hole regime of the lower orbital, and a transfer of optical conductivity spectral weight from low to high frequencies. These observations are evidences of the electron-hole asymmetry in the two-orbital system. In this context, we evaluate the role of two critical variables in the DHM: the local and nonlocal hybridizations which parameterize interorbital electron transitions in the Hamiltonian. We demonstrate that the electron-hole asymmetry of the model has a more complex relationship to these hybridization parameters than previously identified. Mott physics plays an essential role in the quasiparticle dressing due to the local hybridization, while the nonlocal hybridization promotes electron-hole asymmetry by providing a separate channel for electron transport which serves to reduce the quasiparticle weight of holes in the lower band. We also briefly examine the phenomenon of orbital selective Mott transitions in the DHM within the parameter regime of electron state relaxation, and evaluate the role played by the hybridization parameters in this context.
A model to describe electronic correlations in energy bands is considered. The model is a generalization of the conventional Hubbard model that allows for the fact that the wave function for two electrons occupying the same Wannier orbital is different from the product of single-electron wave functions. We diagonalize the Hamiltonian exactly on a four-site cluster and study its properties as a function of band filling. The quasiparticle weight is found to decrease and the quasiparticle effective mass to increase as the electronic band filling increases, and spectral weight in one- and two-particle spectral functions is transferred from low to high frequencies as the band filling increases. Quasiparticles at the Fermi energy are found to be more ``dressed'' when the Fermi level is in the upper half of the band (hole carriers) than when it is in the lower half of the band (electron carriers). The effective interaction between carriers is found to be strongly dependent on band filling becoming less repulsive as the band filling increases and attractive near the top of the band in certain parameter ranges. The effective interaction is most attractive when the single-hole carriers are most heavily dressed, and in the parameter regime where the effective interaction is attractive, hole carriers are found to ``undress,'' hence become more like electrons, when they pair. It is proposed that these are generic properties of electronic energy bands in solids that reflect a fundamental electron-hole asymmetry of condensed matter. The relation of these results to the understanding of superconductivity in solids is discussed.
Dynamic Hubbard models have been proposed as extensions of the conventional Hubbard model to describe the orbital relaxation that occurs upon double occupancy of an atomic orbital. These models give rise to pairing of holes and superconductivity in certain parameter ranges. Here we explore the changes in carrier effective mass and quasiparticle weight and in one- and two-particle spectral functions that occur in a dynamic Hubbard model upon pairing, by exact diagonalization of small systems. It is found that pairing is associated with lowering of effective mass and increase of quasiparticle weight, manifested in transfer of spectral weight from high to low frequencies in one- and two-particle spectral functions. This ``undressing'' phenomenology resembles observations in transport, photoemission, and optical experiments in high-Tc{T}_{c}Tc​ cuprates. This behavior is contrasted with that of a conventional electron-hole symmetric Holstein-like model with attractive on-site interaction, where pairing is associated with ``dressing'' instead of undressing.
We examine whether the Mott transition of a half-filled, two-orbital Hubbard model with unequal bandwidths occurs simultaneously for both bands or whether it is a two-stage process in which the orbital with narrower bandwith localizes first (giving rise to an intermediate ``orbital-selective'' Mott phase). This question is addressed using both dynamical mean-field theory and a representation of fermion operators in terms of slave quantum spins, followed by a mean-field approximation (similar in spirit to a Gutzwiller approximation). In the latter approach, the Mott transition is found to be orbital selective for all values of the Coulomb exchange (Hund) coupling JJJ when the bandwidth ratio is small and only beyond a critical value of JJJ when the bandwidth ratio is larger. Dynamical mean-field theory partially confirms these findings, but the intermediate phase at J=0J=0J=0 is found to differ from a conventional Mott insulator, with spectral weight extending down to arbitrary low energy. Finally, the orbital-selective Mott phase is found, at zero temperature, to be unstable with respect to an interorbital hybridization VVV and replaced at small VVV by a state with a large effective mass (and a low quasiparticle coherence scale) for the narrower band.
We develop here a general formalism for multiorbital Mott systems that can be used to understand dynamical and static spectral weight transfer. We find that the spectral weight transferred from the high-energy scales is greatly increased as a result of the multiorbital structure. As a consequence certainly dynamically generated symmetries are obtained at lower values of doping than in the single-band Hubbard model. For example, in the atomic limit for filling less than one electron per site, the particle-hole symmetric condition in the lower band shifts from the one-band result of x=1/3x=1/3x=1/3 to x=1/(2no+1)x=1/(2{n}_{o}+1)x=1/(2no​+1), where no{n}_{o}no​ is the number of orbitals with an unpaired spin. Transport properties computed from effective low-energy theories that forbid double occupancy of bare electrons, such as the multiorbital ttt-JJJ generalization, should all be be sensitive to this particle-hole symmetric condition. Away from the atomic limit, the dynamical contributions increase the transferred spectral weight. Consequently, any phenomena that are sensitive to an emergent particle-hole symmetry should be obtained at x<[1/(2no+1)]x<[1/(2{n}_{o}+1)]x<[1/(2no​+1)].
The influence of short-range Coulomb correlations on the Mott transition in the single-band Hubbard model at half filling is studied within cellular dynamical mean-field theory for square and triangular lattices. Finite-temperature exact diagonalization is used to investigate correlations within two-, three-, and four-site clusters. Transforming the nonlocal self-energy from a site basis to a molecular-orbital basis, we focus on the interorbital charge transfer between these cluster molecular orbitals in the vicinity of the Mott transition. In all cases studied, the charge transfer is found to be small, indicating weak Coulomb-induced orbital polarization despite sizable level splitting between orbitals. These results demonstrate that all cluster molecular orbitals take part in the Mott transition and that the insulating gap opens simultaneously across the entire Fermi surface. Thus, at half filling we do not find orbital-selective Mott transitions or a combination of band filling and Mott transition in different orbitals. Nevertheless, the approach toward the transition differs greatly between cluster orbitals, giving rise to a pronounced momentum variation along the Fermi surface, in agreement with previous works. The near absence of Coulomb-induced orbital polarization in these clusters differs qualitatively from single-site multiorbital studies of several transition-metal oxides, where the Mott phase exhibits nearly complete orbital polarization as a result of a correlation driven enhancement of the crystal-field splitting. The strong single-particle coupling among cluster orbitals in the single-band case is identified as the source of this difference.
The Dynamic Hubbard Model (DHM) has been introduced as a framework for the electron-hole asymmetry observed in materials like high-temperature superconductors. In this study, we aim to understand the intricate dynamics of hole interactions, particularly focusing on the competition between on-site repulsion and hole-lattice coupling. Our results reveal a striking feature-an effective nearest-neighbor attraction between holes, even in the presence of the on-site repulsion typical of Mott insulators. Notably, the on-site repulsion not only coexists but also enhances the nearest-neighbor coupling. Our exploration of the DHM unravels unexpected and significant characteristics, enriching our understanding of hole interactions in correlated electron systems. The model's capacity to capture both on-site repulsion and effective nearest-neighbor attraction offers a valuable perspective.