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Global Journal of Pure and Applied Sciences

Comparison of a two electron with a two charged boson variational hubbard interactions

Godfrey Ejiroghene Akpojotor, Sunny A Ojobor

Published Jun 15, 2006• 0 citations
Publisher

Abstract

There is growing belief that both the conventional Bose Einstein condensation (CBEC) and non-conventional Bose Einstein condensation (NBEC) can be obtained from repulsive and attractive interactions of bosons respectively. However, there is still no generally accepted model to obtain both condensates. Since results in the literature show that the Hubbard model two electron interactions has both repulsive and attractive regions depending on the interaction strength, we are considering its extension to charged bosons to investigate the possibility of obtaining the condensates from it. As a preliminary investigation, we have compared in this present study, the two electron variational Hubbard Hamiltonian with that modified for charged bosons in N = 2, N x N = 4 x 4 and N x N x N = 5 x 5 x 5 lattices. The results show only slight differences between the cases of the electrons and bosons. The implication that there is possibility of obtaining both condensates from the Hubbard model is then discussed in relation to superconductivity. Keywords: Bose Einstein condensation, superconductivity, Hubbard model, electrons, bosons Global Journal of Pure and Applied Sciences Vol. 12(1) 2006: 119-123

DOI
10.4314/gjpas.v12i1.16577

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There is growing belief that both the conventional Bose Einstein condensation (CBEC) and non-conventional Bose Einstein condensation (NBEC) can be obtained from repulsive and attractive interactions of bosons respectively. However, there is still no generally accepted model to obtain both condensates. Since results in the literature show that the Hubbard model two electron interactions has both repulsive and attractive regions depending on the interaction strength, we are considering its extension to charged bosons to investigate the possibility of obtaining the condensates from it. As a preliminary investigation, we have compared in this present study, the two electron variational Hubbard Hamiltonian with that modified for charged bosons in N = 2, N x N = 4 x 4 and N x N x N = 5 x 5 x 5 lattices. The results show only slight differences between the cases of the electrons and bosons. The implication that there is possibility of obtaining both condensates from the Hubbard model is then discussed in relation to superconductivity. Keywords: Bose Einstein condensation, superconductivity, Hubbard model, electrons, bosons Global Journal of Pure and Applied Sciences Vol. 12(1) 2006: 119-123
We present a phase diagram for a dilute two-dimensional Bose gas on a lattice. For one sort of boson we consider a realistic case of the van der Waals interaction between particles with a strong hard-core repulsion U and a van der Waals attractive tail V. For V<2t,V<2t,V<2t, t being a hopping amplitude, the phase diagram of the system contains regions of usual one-particle Bose-Einstein condensation (BEC). However for V>2tV>2tV>2t we have total phase separation on a Mott-Hubbard Bose solid and a dilute Bose gas. For two sorts of structureless bosons described by the two-band Hubbard model, an s-wave pairing of the two bosons of different sort $〈{b}_{1}{b}_{2}〉\ensuremath{\ne}0$ is possible. The results we obtained should be important for different Bose systems, including submonolayers of 4He,{}^{4}\mathrm{He},4He, excitons in semiconductors, Schwinger bosons in magnetic systems and holons in HTSC. In the HTSC case a possibility of two-holon pairing in the slave-boson theories of superconductivity can restore a required charge 2e2e2e of a Cooper pair.
The understanding of the mechanisms responsible for superconductivity in strongly correlated systems is an interesting and important subject in condensed matter physics. Several theoretical proposals were considered for these systems. The Coulomb interaction between electrons allow a new approach to study this problem. In this paper, we use a usual Hubbard model with a local repulsive interaction to describe a 2D system. The system of equations are solved using the Green's functions method, within a Hubbard-I mean field approximation, which allows to treat the strong interaction limit. We consider both cases of attractive and repulsive interactions and obtain the zero temperature phase diagram of the model. Our results show, in the repulsive case, the existence of a superconducting ground state mediated by the kinetic electronic energy and described by a non-local order parameter. A minimum value of the repulsive interaction UminU_{min}Umin​ is required to create a pairing state. At finite temperatures, for strong interactions, the critical temperature TcT_cTc​ shows a saturation similar to the Bose-Einstein condensation observed for strong attractive interactions.
The two-site version of the Bose-Hubbard model can be used as a simplified model of N bosons confined to two quantum wells. We study the behavior of low-temperature Bose-Einstein condensates as a function of the strength of the repulsive interactions between the bosons and of the size of the quantum-mechanical tunneling between the quantum wells. The ground-state of weakly interacting bosons tends to be one in which almost all particles are in the same quantum-mechanical state (‘Bose-Einstein Condensate’, or ‘BEC’). It is found that even very small amounts of interwell tunneling lead to a BEC, but that zero tunneling leads to two separate BECs, one in each well (‘fragmentation’). We study systematically the transition between a single BEC comprised of bosons in both wells, and a fragmented condensate consisting of two BECs, one for each well. Our analysis will use matrix computational methods to solve the Schrödinger equation of the two-well boson system, and obtain the energies and wavefunctions as eigenvalues and eigenvectors of the Hamiltonian matrix. We will compare the results obtained with previous analyses, and against analytical approximations developed for the limiting case of a large number of bosons (large N).
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A phase of matter in which fermion quartets form a superconducting condensate, rather than the paradigmatic Cooper pairs, is a recurrent subject of experimental and theoretical studies. However, a comprehensive microscopic understanding of charge-4e4e4e superconductivity as a quantum phase is lacking. Here, we study a two-orbital tight-binding model with attractive Hubbard-type interactions. Such a model naturally provides the Bose-Einstein condensate as a limit for electron quartets and supports charge-4e4e4e superconductivity, as we show by mapping it to a spin-1/2 chain in this perturbative limit. Using density matrix renormalization group calculations for the one-dimensional case, we further establish that the ground state is indeed a superfluid phase of 4e4e4e charge carriers and that this phase can be stabilized well beyond the perturbative regime. Importantly, we demonstrate that 4e4e4e condensation dominates over 2e2e2e condensation even for nearly decoupled orbitals, which is a more likely scenario in electronic materials. Our model paves the way for both experimental and theoretical exploration of 4e4e4e superconductivity and provides a natural starting point for future studies beyond one dimension or more intricate 4e4e4e states.