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arXiv:cond-mat/0511329

Magnetism and superconductivity within extended Hubbard model for a dimer. Exact results

B. Grabiec, S. Krawiec, M. Matlak

Published Nov 14, 2005• 2 citations
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Abstract

We find the eigenvalues $E_α $ and eigenvectors $|E_α>$ of the extended Hubbard dimer and we represent each part $E_α |E_α> < E_α |$ of the dimer Hamiltonian $(α=1,2,...,16)$in the second quantizations with the use of the Hubbard and spin operators. This procedure gives a review of all competitive, intrinsic interactions, deeply hidden in the orginal form of the considered Hamiltonian. Among competitive interactions we can find ferromagnetic and antiferromagnetic interactions (also in the form appearing in the $t-J$ model), hopping of the Cooper pairs between different dimer lattice sites (similar as in Kulik-Pedan, Penson-Kolb models), as well as, intersite Cooper pair interactions. We plot several coupling constants of these interactions vs Coulomb intrasite interaction $U$ to show that the competition between them strongly depends on the model parameters. The thermodynamical activity of each particular interaction, belonging to a given energy level, depends, however, on the occupation of this level. From the presented review of interactions it is evident that the extended Hubbard model is capable to describe the properties of superconductors with magnetic ordering (including also high-$T_C$ superconductors) where a strong competition between magnetism and superconductivity takes place.

DOI
10.48550/arxiv.cond-mat/0511329

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Ê I nst i t ut e of P hysi cs, Si l esian Uni versity Uni werstytec ka 4, 40-007 Ka to wi ce, Pol an d ( Recei ved December 14, 2001)W e consider t he extended H ubb ard mo del for the single cubic lattice and re w rite it in the form of interacting dimers , using the exact solutio n of the dimer problem .W e anal ytical l y deri ve the second quanti zation form of the dimer H amiltoni an elimina tin g from the consideratio ns uno ccupie d dimer energy levels in the large U limit (it is the only assumption ).T he resulting dimer H amiltonian w ritten with the use of the Hubbard op erators and spin operators contains three terms, visuali zin g explicitl y comp eting magnetic interactions (f erromagnetic, antif erromagnetic) as a generaliz atio n of the t { J mo del.T he presented, nonp erturbati ve metho d, can in principl e b e applied to the cluster of any size (e.g. one central atom and z its nearest neighb ours).T he use of the pro j ection technique can further b e appli ed in the case of a crystal to obtain the second quantization form of the extended H ubbard mo del for the sc lattice in the large U limit.

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We find the eigenvalues EαE_α Eα​ and eigenvectors ∣Eα>|E_α>∣Eα​> of the extended Hubbard dimer and we represent each part Eα∣Eα><Eα∣E_α |E_α> < E_α |Eα​∣Eα​><Eα​∣ of the dimer Hamiltonian (α=1,2,...,16)(α=1,2,...,16)(α=1,2,...,16)in the second quantizations with the use of the Hubbard and spin operators. This procedure gives a review of all competitive, intrinsic interactions, deeply hidden in the orginal form of the considered Hamiltonian. Among competitive interactions we can find ferromagnetic and antiferromagnetic interactions (also in the form appearing in the t−Jt-Jt−J model), hopping of the Cooper pairs between different dimer lattice sites (similar as in Kulik-Pedan, Penson-Kolb models), as well as, intersite Cooper pair interactions. We plot several coupling constants of these interactions vs Coulomb intrasite interaction UUU to show that the competition between them strongly depends on the model parameters. The thermodynamical activity of each particular interaction, belonging to a given energy level, depends, however, on the occupation of this level. From the presented review of interactions it is evident that the extended Hubbard model is capable to describe the properties of superconductors with magnetic ordering (including also high-TCT_CTC​ superconductors) where a strong competition between magnetism and superconductivity takes place.
Abstract We express the Hubbard dimer Hamiltonian $ H_{d} = {\textstyle \sum \nolimits ^{16} _{\alpha = 1} }E_{\alpha}\vert E_{\alpha}\rangle \langle E_{\alpha}\vert $ in the second quantization with theuse of the Hubbard and spin operators. We consider the case of positive and negative U. We decompose the resulting Hamiltonian into several parts collecting all the terms belonging to the same energy level. Such a decomposition visualizes explicitely all intrinsic interactions competing together and deeply hidden in the original form of the dimer Hamiltonian. Among them are competitive ferromagnetic and antiferromagnetic interactions. There are also hopping terms present which describe Cooper pairs hopping between sites 1 and 2 with positive and negative coupling constants (similar as in Kulik–Pedan, Penson–Kolb models). We show that the competition between intrinsic interactions strongly depends on the model parametrs and the averaged occupation number of electrons n ∈ [0, 4] resulting in different regimes of the model (as e.g. t–J model regime, etc.). (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
We investigate two-site electronic correlations within generalized Hubbard model, which incorporates the conventional Hubbard model (parameters: t (hopping between nearest neighbours), U (Coulomb repulsion (attraction))) supplemented by the intersite Coulomb interactions (parameters: J (1) (parallel spins), J (2) (antiparallel spins)) and the hopping of the intrasite Cooper pairs (parameter: V ).As a first step we find the eigenvalues Eα and eigenvectors |E α of the dimer and we represent each partial Hamiltonian Eα|Eα Eα| (α = 1, 2, . . ., 16) in the second quantization with the use of the Hubbard and spin operators.Each dimer energy level possesses its own Hamiltonian describing different two-site interactions which can be active only in the case when the level will be occupied by the electrons.A typical feature is the appearance of two generalized t-J interactions ascribed to two different energy levels which do not vanish even for U = J (1) = J (2) = V = 0 and their coupling constants are equal to ±t in this case.In the large U -limit for J (1) = J (2) = V = 0 there is only one t-J interaction with coupling constant equal to 4t 2 /|U | as in the case of a real lattice.The competition between ferromagnetism, antiferromagnetism and superconductivity (intrasite and intersite pairings) is also a typical feature of the model because it persists in the case U = J (1) = J (2) = V = 0 and t = 0.The same types of the electronic, competitive interactions are scattered between different energy levels and therefore their thermodynamical activities are dependent on the occupation of these levels.It qualitatively explains the origin of the phase diagram of the model.We consider also a real lattice as a set of interacting dimers to show that the competition between magnetism and superconductivity seems to be universal for fermionic lattice models.
The phase diagrams and superconducting properties of the extended Hubbard model with pair hopping interaction, i.e., the Penson-Kolb-Hubbard model are studied. The analysis of the model is performed for d-dimensional hypercubic lattices, including d=1d=1d=1 and $d=\ensuremath{\infty},$ by means of the (broken symmetry) Hartree-Fock approximations and, for $d=\ensuremath{\infty},$ by the slave-boson mean-field method. For d=1,d=1,d=1, at half filling the phase diagram is shown to consist of nine different phases including two superconducting states with center-of-mass momentum q=0q=0q=0 and q=Qq=Qq=Q $(\ensuremath{\eta}$ pairing), site and bond-located antiferromagnetic and charge-density wave states as well as three mixed phases with coexisting site and bond orderings. The stability range of the bond-type orderings shrinks with increasing lattice dimensionality d and for $d=\ensuremath{\infty}$ the corresponding diagram consists of four phases only, involving exclusively site-located orderings. Comparing the pair hopping model with the attractive Hubbard model we found in both cases gradual evolution from the BCS-like limit to the tightly bound pairs regime and a monotonic increase of the gap in the excitation spectrum with increasing coupling. However, the dynamics of electron pairs in both models is qualitatively different, which results in different dependences of condensation energies and critical temperatures on interaction parameters as well as in different electrodynamic properties, especially in a strong coupling regime.
An extended Hubbard model for a single band, including Coulomb repulsion and correlated hopping between nearest neighbors, is studied using a generalized mean-field approach. Antiferromagnetism and superconductivity are probed for arbitrary occupation number, near and away from half filling. Binding of pairs in the superconducting state of this purely repulsive model is mediated by the correlated hopping in the form of a covalent-bond configuration, with partial intrasite and intersite pairings. A region of coexistence is conjectured, the superconductivity being suppressed by the saturation of the staggered magnetic moment. Singlet superconducting nonmagnetic states are obtained for the almost-empty- or full-band cases. On the other hand, antiferromagnetism induces mixed s- and p-type superconductivities in the neighborhood of half filling.
A microscopic theory of superconductivity in the extended Hubbard model which takes into account the intersite Coulomb repulsion and electron-phonon interaction is developed in the limit of strong correlations. The Dyson equation for normal and pair Green functions expressed in terms of the Hubbard operators is derived. The self-energy is obtained in the noncrossing approximation. In the normal state, antiferromagnetic short-range correlations result in the electronic spectrum with a narrow bandwidth. We calculate superconducting Tc by taking into account the pairing mediated by charge and spin fluctuations and phonons. We found the d-wave pairing with high-Tc mediated by spin fluctuations induced by the strong kinematic interaction for the Hubbard operators. Contributions to the d-wave pairing coming from the intersite Coulomb repulsion and phonons turned out to be small.
Ê I nst i t ut e of P hysi cs, Si l esian Uni versity Uni werstytec ka 4, 40-007 Ka to wi ce, Pol an d ( Recei ved December 14, 2001)W e consider t he extended H ubb ard mo del for the single cubic lattice and re w rite it in the form of interacting dimers , using the exact solutio n of the dimer problem .W e anal ytical l y deri ve the second quanti zation form of the dimer H amiltoni an elimina tin g from the consideratio ns uno ccupie d dimer energy levels in the large U limit (it is the only assumption ).T he resulting dimer H amiltonian w ritten with the use of the Hubbard op erators and spin operators contains three terms, visuali zin g explicitl y comp eting magnetic interactions (f erromagnetic, antif erromagnetic) as a generaliz atio n of the t { J mo del.T he presented, nonp erturbati ve metho d, can in principl e b e applied to the cluster of any size (e.g. one central atom and z its nearest neighb ours).T he use of the pro j ection technique can further b e appli ed in the case of a crystal to obtain the second quantization form of the extended H ubbard mo del for the sc lattice in the large U limit.