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arXiv:cond-mat/0703818 · Physical Review Letters

Exact Many-Electron Ground States on the Diamond Hubbard Chain

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

Published Jul 13, 2007• 62 citations
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Abstract

Exact ground states of interacting electrons on the diamond Hubbard chain in a magnetic field are constructed which exhibit a wide range of properties such as flat-band ferromagnetism and correlation-induced metallic, half-metallic, or insulating behavior. The properties of these ground states can be tuned by changing the magnetic flux, local potentials, or electron density.

DOI
10.1103/physrevlett.99.026404

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Exact ground states of interacting electrons on the diamond Hubbard chain in a magnetic field are constructed which exhibit a wide range of properties such as flat-band ferromagnetism and correlation-induced metallic, half-metallic, or insulating behavior. The properties of these ground states can be tuned by changing the magnetic flux, local potentials, or electron density.
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 investigate a mechanism to establish ground-state ferromagnetism in flat-band Hubbard systems based on a kind of order-from-disorder effect driven by dispersion. As a paradigm we consider a frustrated diamond chain, where for ideal diamond geometry the lowest one-electron band is flat, but the ground state remains paramagnetic for arbitrary on-site repulsion UUU. We focus on half-filling of the flat band. By using numerical and analytical arguments we present the ground-state phase diagram for a distorted diamond chain, i.e., the former flat band becomes dispersive. Driven by the interplay of dispersion and interaction the ground state is ferromagnetic if the interaction exceeds a critical value Uc{U}_{c}Uc​.
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.
Exact-diagonalization and quantum Monte Carlo methods are used to study the occurrence of a metal-insulator transition and the distribution of charges and magnetic moments in the ground state of a one- and two-dimensional half-filled system of correlated electrons, described by a one-band simplified Hubbard model extended with bond-charge--site-charge interactions. Numerical results for the electrical conductivity, the density of states, and the spin- and charge-density correlation functions strongly suggest that the bond-charge--site-charge interaction plays a fundamental role in the occurrence of a metal-insulator transition and of ferromagnetic order.
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.
The exact phase diagrams of the one-dimensional Hubbard model, both attractive and repulsive, in the presence of an arbitrary magnetic field h for various electron concentrations n and on-site interaction strengths U <0 or U >0 are calculated and investigated. The exact ground-state properties, namely, the ground-state energy, the average spin (magnetization), the concentration of the doubly occupied sites, the kinetic energy, the chemical potential, the spin (magnetic) susceptibility and the charge compressibility, are calculated and examined over a wide range of interaction strengths U for various h and n . It is found that the spin susceptibility at half-filling is non-analytic and changes discontinuously as U →0. The exact theory shows the absence of a charge energy gap in the U ⩽0 region for all n and provides, for the chemical potential, the rigorous upper and lower bounds for half-filled and empty bands respectively. The analytical results derived for the weak-coupling limit and asymptotic expansions for strong coupling are in full agreement with the numerical calculations.