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arXiv:1305.5774 · International Journal of Modern Physics B

EXACT GROUND STATES OF CORRELATED ELECTRONS ON PENTAGON CHAINS

Zsolt Gulácsi

Published May 13, 2013• 16 citations
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Abstract

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.

DOI
10.1142/s0217979213300090

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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.
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.
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.
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 consider the standard repulsive Hubbard model with a flat lowest-energy band for two one-dimensional lattices (diamond chain and ladder) as well as for a two-dimensional lattice (bilayer) at half filling of the flat band. The considered models do not fall in the class of Mielke-Tasaki flat-band ferromagnets, since they do not obey the connectivity conditions. However, the ground-state ferromagnetism can emerge, if the flat band becomes dispersive. To study this kinetic-energy-driven ferromagnetism we use perturbation theory and exact diagonalization of finite lattices. We find as a typical scenario that small and moderate dispersion may lead to a ferromagnetic ground state for sufficiently large on-site Hubbard repulsion U>UcU>{U}_{c}U>Uc​, where Uc{U}_{c}Uc​ increases monotonically with the acquired bandwidth. However, we also observe for some specific parameter cases, that (i) ferromagnetism appears at already very small Uc{U}_{c}Uc​, (ii) ferromagnetism does not show up at all, (iii) the critical on-site repulsion Uc{U}_{c}Uc​ is a nonmonotonic function of the bandwidth, or that (iv) a critical bandwidth is needed to open the window for ground-state ferromagnetism.
Based on a positive semidefinite operator technique, exact ground states are deduced for the non-integrable conducting polymers possessing pentagon type of unit cell. The study is done in the presence of many-body spin-orbit interaction (SOI), local and nearest neighbor Coulomb repulsion (NNCR) and presence of external EEE electric and BBB magnetic fields, such that the effects of BBB on both orbital and spin degrees of freedom is considered. The SOI, NNCR, and presented external field configurations presence in exact conducting polymer ground states is a novelty, so the development of the technique for the treatment possibility of such strongly correlated cases is presented in details. The deduced ground states show a broad spectrum of physical characteristics ranging from charge density waves, metal-insulator transitions, to interesting external field driven effects as e.g. modification possibility of a static charge distribution by a static external magnetic field.