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physica status solidi (b)

Magnetic and superconducting competition within the Hubbard dimer. Exact solution

M. Matlak, T. Słomska, B. Grabiec

Published Jan 17, 2005• 1 citation
Publisher

Abstract

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)

DOI
10.1002/pssb.200460051

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Método atômico para o modelo de Hubbard

Renan Lira Nunes

The objective of this work is the theoretical study of the one band Hubbard model for solids and the introduction of an alternative methodology to circumvent its limitations. To this objective, a Hamiltonian composed of two atoms (the Hubbard dimmer) is introduced and solved analitically. From this Hamiltonian, the atomic Green’s functions are obtained for the dimmer, which is used as a seed for a cumulant expansion and to obtain the Green’s functions for the lattice. The density of states for several values of interest for the parameters of the problem is calculated, as well as the occupation numbers. The analysis of the dimmer Hamiltonian reveals four important competitive interactions: ferromagnetic, antiferromagnetic and superconducting with positive or negative coupling constants. As a result of using the dimmer solution, the particle-hole simetry for high values of the electronic correlation is recovered, the Mott transition is obtained in a consistent way (the electronic correlation is no longer overestimated) and by the analysis of the ocupation numbers of the model for different values of the chemical potential, one can observe a competition between fundamental states of the model near the energy region Eo + U, as well as a magnetic phase and a Kondo effect region.

•Jan 1, 2016•0 citations
Abstract We express the Hubbard dimer Hamiltonian Hd=∑α=116Eα∣Eα⟩⟨Eα∣H_{d} = {\textstyle \sum \nolimits ^{16} _{\alpha = 1} }E_{\alpha}\vert E_{\alpha}\rangle \langle E_{\alpha}\vertHd​=∑α=116​Eα​∣Eα​⟩⟨Eα​∣ 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 find the eigenvalues $E_α $ 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.
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.
Abstract The interplay and competition of magnetic and superconducting correlations in the weakly interacting two‐dimensional Hubbard Model is investigated by means of the functional renormalization group. At zero temperature the flow of interactions in one‐loop approximation evolves into a strong coupling regime at low energy scales, signalling the possible onset of spontaneous symmetry breaking. This is further analyzed by a mean‐field treatment of the strong renormalized interactions which takes into account magnetic and superconducting order simultaneously. The effect of strong correlations on single‐particle properties in the normal phase is studied by calculating the flow of the self‐energy. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Abstract We analyze a competition between d‐wave superconductivity (SC) and antiferromagnetism (AF) and show the influence of these phenomena on the superfluid stiffness. We deal with the extended Hubbard model on a 2D square lattice with the nn and nnn hopping (t–t′–U–W model) in the BCS‐Hartree–Fock approximation. Thermodynamic analysis shows that both, second or first order phase transition between the AF, SC and mixed AF + SC phases are possible. The homogeneous mixed phase or phase separation can be stabilized depending on interaction parameters, band structure and band filling. The corresponding temperature phase diagrams are determined. In the coexistence region we investigate the influence of the antiferromagnetism on the superfluid properties. In the mixed AF + SC ground state the superfluid density is reduced with growing antiferromagnetic ordering and vanishes at half–filling. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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.
The objective of this work is the theoretical study of the one band Hubbard model for solids and the introduction of an alternative methodology to circumvent its limitations. To this objective, a Hamiltonian composed of two atoms (the Hubbard dimmer) is introduced and solved analitically. From this Hamiltonian, the atomic Green’s functions are obtained for the dimmer, which is used as a seed for a cumulant expansion and to obtain the Green’s functions for the lattice. The density of states for several values of interest for the parameters of the problem is calculated, as well as the occupation numbers. The analysis of the dimmer Hamiltonian reveals four important competitive interactions: ferromagnetic, antiferromagnetic and superconducting with positive or negative coupling constants. As a result of using the dimmer solution, the particle-hole simetry for high values of the electronic correlation is recovered, the Mott transition is obtained in a consistent way (the electronic correlation is no longer overestimated) and by the analysis of the ocupation numbers of the model for different values of the chemical potential, one can observe a competition between fundamental states of the model near the energy region Eo + U, as well as a magnetic phase and a Kondo effect region.