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arXiv:0806.0289 · Progress of Theoretical Physics Supplement

Exact Many-Electron Ground States on Diamond and Triangle Hubbard Chains

Zsolt Gulácsi, A. P. Kampf, D. Vollhardt

Published Jan 1, 2008• 31 citations
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Abstract

We construct exact ground states of interacting electrons on triangle and diamond Hubbard chains. The construction requires (i) a rewriting of the Hamiltonian into positive semidefinite form, (ii) the construction of a many-electron ground state of this Hamiltonian, and (iii) the proof of the uniqueness of the ground state. This approach works in any dimension, requires no integrability of the model, and only demands sufficiently many microscopic parameters in the Hamiltonian which have to fulfill certain relations. The scheme is first employed to construct exact ground state for the diamond Hubbard chain in a magnetic field. These ground states are found to exhibit a wide range of properties such as flat-band ferromagnetism and correlation induced metallic, half-metallic or insulating behavior, which can be tuned by changing the magnetic flux, local potentials, or electron density. Detailed proofs of the uniqueness of the ground states are presented. By the same technique exact ground states are constructed for triangle Hubbard chains and a one-dimensional periodic Anderson model with nearest-neighbor hybridization. They permit direct comparison with results obtained by variational techniques for f-electron ferromagnetism due to a flat band in CeRh3B2.

DOI
10.1143/ptps.176.1

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We construct exact ground states of interacting electrons on triangle and diamond Hubbard chains. The construction requires (i) a rewriting of the Hamiltonian into positive semidefinite form, (ii) the construction of a many-electron ground state of this Hamiltonian, and (iii) the proof of the uniqueness of the ground state. This approach works in any dimension, requires no integrability of the model, and only demands sufficiently many microscopic parameters in the Hamiltonian which have to fulfill certain relations. The scheme is first employed to construct exact ground state for the diamond Hubbard chain in a magnetic field. These ground states are found to exhibit a wide range of properties such as flat-band ferromagnetism and correlation induced metallic, half-metallic or insulating behavior, which can be tuned by changing the magnetic flux, local potentials, or electron density. Detailed proofs of the uniqueness of the ground states are presented. By the same technique exact ground states are constructed for triangle Hubbard chains and a one-dimensional periodic Anderson model with nearest-neighbor hybridization. They permit direct comparison with results obtained by variational techniques for f-electron ferromagnetism due to a flat band in CeRh3B2.
We model conducting pentagon chains with a multi-orbital Hubbard model and prove that well below half filling, exact ferromagnetic ground states appear. The rigorous method we use is based on the transformation of original hamiltonian into positive semidefinite form. This technique is independent of the spatial dimension and does not require integrability of the model. The obtained ferromagnetism is connected to dispersionless bands but in a much broader sense than flat-band ferromagnetism requires, where on every site a Hubbard term is present. In our case, only a small percentage of, even randomly distributed, sites are only interacting.
We construct a class of exact ground states for correlated electrons on pentagon chains in the high density region and discuss their physical properties. In this procedure the Hamiltonian is first cast in a positive semidefinite form using composite operators as a linear combination of creation operators acting on the sites of finite blocks. In the same step, the interaction is also transformed to obtain terms which require for their minimum eigenvalue zero at least one electron on each site. The transformed Hamiltonian matches the original Hamiltonian through a nonlinear system of equations whose solutions place the deduced ground states in restricted regions of the parameter space. In the second step, nonlocal product wave functions in position space are constructed. They are proven to be unique ground states which describe non-saturated ferromagnetic and correlated half metallic states. These solutions emerge when the strength of the Hubbard interaction Ui is site-dependent inside the unit cell. In the deduced phases, the interactions tune the bare dispersive band structure such to develop an effective upper flat band. We show that this band flattening effect emerges for a broader class of chains and is not restricted to pentagon chains. For the characterization of the deduced solutions, uniqueness proofs, exact ground state expectation values for long-range hopping amplitudes and correlation functions are also calculated. The study of physical reasons which lead to the appearance of ferromagnetism has revealed a new mechanism for the emergence of an ordered phase, described here in detail. This works as follows: starting from a completely dispersive bare band structure, the interactions quench the kinetic energy, hence the ordered phase is obtained solely by a drastic decrease of the interaction energy. Since Ui are site dependent, this determinative decrease is obtained by a redistribution of the double occupancy di such to attain small di where the on-site Coulomb repulsion Ui is high, and vice versa. The kinetic energy quench leads to the upper effective flat band, whose role is to enhance by its degeneracy the switching to the ordered phase dictated and stabilized by the interactions present. It is shown that for this phenomenon to occur, a given degree of complexity is needed for the chain, and the mechanism becomes inactive when the Ui interactions are homogeneous, or are missing from the ground state wave function.
Using a positive semidefinite operator technique we deduce exact ground states for a zig-zag hexagon chain described by a non-integrable Hubbard model with on-site repulsion. Flat bands are not present in the bare band structure, and the operators , introducing the electrons into the ground state, are all extended operators and confined in the quasi-1D chain structure of the system. Consequently, by increasing the number of carriers, the operators become connected, i.e. touch each other on several lattice sites. Hence the spin projection of the carriers becomes correlated in order to minimize the ground-state energy by reducing as much as possible the double occupancy leading to a ferromagnetic ground state. This result demonstrates in exact terms in a many-body frame that the conjecture made at the two-particle level by G. Brocks et al., [Phys. Rev. 93 (2004) p.146405.] that the Coulomb interaction is expected to stabilize correlated magnetic ground states in acenes, is clearly viable, and opens up new directions in the search for routes in obtaining organic ferromagnetism. Due to the itinerant nature of the obtained ferromagnetic ground state, the systems under discussion may also have direct application possibilities in spintronics.
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.
Using the Hubbard chain at quarter filling as a model system, we study the ground state properties of highly doped antiferromagnets. In particular, the Hubbard chain at quarter filling is unstable against 2kF2{k}_{F}2kF​- and 4kF4{k}_{F}4kF​-periodic potentials, leading to a large variety of charge and spin ordered ground states. Employing the density matrix renormalization group method, we compare the energy gain of the ground state induced by different periodic potentials. For interacting systems the lowest energy is found for a 2kF2{k}_{F}2kF​-periodic magnetic field, resulting in a band insulator with spin gap. For strong interaction, the 4kF4{k}_{F}4kF​-periodic potential leads to a half-filled Heisenberg chain and thus to a Mott insulating state without spin gap. This ground state is more stable than the band insulating state caused by any non-magnetic 2kF2{k}_{F}2kF​-periodic potential. Adding more electrons, a cluster-like ordering is preferred.