It is known that a system which exhibits a half filled lowest flat band and the localized one-particle Wannier states on the flat band satisfy the connectivity conditions, is always ferromagnetic. Without the connectivity conditions on the flat band,
the system is non-magnetic. We show that this is not always true. The reason is connected to a peculiar behavior of the band situated just above the flat band.
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 se
midefinite form. This technique is independent of the spatial dimesion 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.
Working in a subspace with dimensionality much smaller than the dimension of the full Hilbert space, we deduce exact 4-particle ground states in 2D samples containing hexagonal repeat units and described by Hubbard type of models. The procedure ident
ifies first a small subspace ${cal{S}}$ in which the ground state $|Psi_grangle$ is placed, than deduces $|Psi_grangle$ by exact diagonalization in ${cal{S}}$. The small subspace is obtained by the repeated application of the Hamiltonian $hat H$ on a carefully chosen starting wave vector describing the most interacting particle configuration, and the wave vectors resulting from the application of $hat H$, till the obtained system of equations closes in itself. The procedure which can be applied in principle at fixed but arbitrary system size and number of particles, is interesting by its own since provides exact information for the numerical approximation techniques which use a similar strategy, but apply non-complete basis for ${cal{S}}$. The diagonalization inside ${cal{S}}$ provides an incomplete image about the low lying part of the excitation spectrum, but provides the exact $|Psi_grangle$. Once the exact ground state is obtained, its properties can be easily analyzed. The $|Psi_grangle$ is found always as a singlet state whose energy, interestingly, saturates in the $U to infty$ limit. The unapproximated results show that the emergence probabilities of different particle configurations in the ground state present Zittern (trembling) characteristics which are absent in 2D square Hubbard systems. Consequently, the manifestation of the local Coulomb repulsion in 2D square and honeycomb types of systems presents differences, which can be a real source in the differences in the many-body behavior.
For a general class of conducting polymers with arbitrary large unit cell and different on-site Coulomb repulsion values on different type of sites, I demonstrate in exact terms the emergence possibility of an upper, interaction created effective fla
t band. This last appears as a consequence of a kinetic energy quench accompanied by a strong interaction energy decrease, and leads to a non-saturated ferromagnetic state. This ordered state clearly differs from the known flat-band ferromagnetism. This is because it emerges in a system without bare flat bands, requires inhomogeneous on-site Coulomb repulsions values, and possesses non-zero lower interaction limits at the emergence of the ordered phase.
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 $U_i$ 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 details (because of lack of space see the continuation in the paper).
Employing a rigorous theoretical method for the construction of exact many-electron ground states we prove that interactions can be employed to tune a bare dispersive band structure such that it develops a flat band. Thereby we show that pentagon cha
in polymers with electron densities above half filling may be designed to become ferromagnetic or half metallic.
Using a positive semidefinite operator technique one deduces 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
$hat B^{dagger}_{mu,sigma}$ introducing the electrons into the ground state, are all extended operators and confined in the quasi 1D chain structure of the system. Consequently, increasing the number of carriers, the $hat B^{dagger}_{mu,sigma}$ 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 two-particle level by G. Brocks et al. [Phys.Rev.Lett.93,146405,(2004)] that the Coulomb interaction is expected to stabilize correlated magnetic ground states in acenes is clearly viable, and opens 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 have also direct application possibilities in spintronics.
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 t
his 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.
