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WSEAS TRANSACTIONS ON APPLIED AND THEORETICAL MECHANICS

Quadrilateral quantum chain hamiltonian cast in positive semidefinite form containing non-linear fermionic contributions

György Kovács, Konstantin E. Glukhov, Zsolt Gulácsi

Published Jan 1, 2015• 0 citations
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Abstract

The transformation of a Hamiltonian (connected to a quantum mechanical many-body system) in positive semidefinite form represents an important procedure in a powerful tool used in deducing exact results for nonintegrable systems.Motivated by this fact, we report in this paper the exact transformation of a Hubbard type of Hamiltonian (describing in the presented case an itinerant fermionic quadrilateral chain) in positive semidefinite form, in a such a way that the kinetic and interaction energy terms of the Hamiltonian are treated similarly together, and are provided by the same positive semidefinite expression.For this reason, non-linear fermionic contributions are used in block operators from which the positive semidefinite forms are constructed.The transformation is possible to be done when a system of nonlinear equations, called matching equations, are satisfied.The procedure of solving the matching equations is also indicated.

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arXiv:1305.5774•International Journal of Modern Physics B•May 13, 2013•16 citations
The transformation of a Hamiltonian (connected to a quantum mechanical many-body system) in positive semidefinite form represents an important procedure in a powerful tool used in deducing exact results for nonintegrable systems.Motivated by this fact, we report in this paper the exact transformation of a Hubbard type of Hamiltonian (describing in the presented case an itinerant fermionic quadrilateral chain) in positive semidefinite form, in a such a way that the kinetic and interaction energy terms of the Hamiltonian are treated similarly together, and are provided by the same positive semidefinite expression.For this reason, non-linear fermionic contributions are used in block operators from which the positive semidefinite forms are constructed.The transformation is possible to be done when a system of nonlinear equations, called matching equations, are satisfied.The procedure of solving the matching equations is also indicated.
Quantum mechanical many-body systems described by an arbitrary Hamiltonian ( ) are analyzed. It is shown how positive semidefinite operator ( ) properties are able to lead in this case to exact results related to the ground state and the low lying part of the excitation spectrum. This is done independent on dimensionality and integrability. The technique first casts the Hamiltonian in a positive semidefinite form in exact terms ( = + C , where C is a scalar). Second, the ground state is deduced by constructing the most general Hilbert space vector, on which applying , one obtains zero as a result. It is underlined, that the procedure, if applied for variable total number of particles N, allows to obtain information also related to the low lying part of the excitation spectrum. The uniqueness of the ground states can be demonstrated via the study of the kernel of ' = − C . The physical properties of the obtained phases are deduced based on ground state expectation values calculated in terms of the constructed ground states. Since for a fixed structure of , usually the transformation = + C is exact only in a restricted parameter space domain ( ), the deduced ground states are present only in . A global view on the phase diagram is obtained by different transformations of in positive semidefinite form.
A self-contained treatment of the Bogoliubov-Valatin transformation for homogeneous fermionic Hamiltonians is presented. The aim is to provide a quick reference that may also serve as supplementary material for a graduate-level course, and that can be understood with quantum mechanics knowledge up to the level of the second quantization's rules. The objective of the transformation is to cast a quadratic Hamiltonian into a diagonal form that resembles the Hamiltonian of a system of non-interacting particles. To obtain this, the first step consists in putting its coefficient matrix into its canonical form; the transformation can always be performed on fermionic Hamiltonians, only some care must be taken when this form is singular. Having explained how to cast a general matrix into its standard form, a complete description of the transformation is provided; a novel procedure is proposed here for the singular matrix case.
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 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.