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The Philosophical Magazine A Journal of Theoretical Experimental and Applied Physics

Even-parity spin-triplet paired states by combined effect of Hund’s rule and correlations in two-band Hubbard model: a brief overview

M. Zegrodnik

Published Oct 17, 2014• 0 citations
Publisher

Abstract

Universal aspects of the Hund’s rule induced spin-triplet pairing are analysed within the two-band Hubbard model on a square lattice. According to our calculations, this pairing mechanism in conjunction with the correlation effect can result in stability of the paired phase in the so-called purely repulsive interactions regime, in which there is no effectively attractive interaction. Furthermore, even though all of the interaction terms are of intrasite character, the pairing contains both intra- and inter-site components. In effect, the gap parameter has a mixture of s-wave and extended s-wave symmetries. The calculations have been carried out with the use of the Statistically Consistent Gutzwiller Approximation developed by us in recent years.

DOI
10.1080/14786435.2014.965767

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Universal aspects of the Hund’s rule induced spin-triplet pairing are analysed within the two-band Hubbard model on a square lattice. According to our calculations, this pairing mechanism in conjunction with the correlation effect can result in stability of the paired phase in the so-called purely repulsive interactions regime, in which there is no effectively attractive interaction. Furthermore, even though all of the interaction terms are of intrasite character, the pairing contains both intra- and inter-site components. In effect, the gap parameter has a mixture of s-wave and extended s-wave symmetries. The calculations have been carried out with the use of the Statistically Consistent Gutzwiller Approximation developed by us in recent years.
The intrasite and intersite spin-triplet pairing gaps induced by interband Hund's rule coupling and their correlations are analyzed in the doubly degenerate Hubbard Hamiltonian. To include the effect of correlations, the statistically consistent Gutzwiller approximation is used. In this approach the consistency means that the averages calculated from the self-consistent equations and those determined variationally coincide with each other. Emphasis is put on the solution for which the average particle number is conserved when carrying out the Gutzwiller projection. This method leads to a stable equal-spin paired state in the so-called repulsive interactions limit (U > 3J) in the regime of moderate correlations. The interband hybridization introduces an inequivalence of the bands which, above a critical magnitude, suppresses the paired state due to both the Fermi-wavevector mismatch for the Cooper pair and the interband hopping allowed by the Pauli principle.
We demonstrate the stability of the spin-triplet paired s-wave (with an admixture of extended s-wave) state for the limit of purely repulsive interactions in a degenerate two-band Hubbard model of correlated fermions. The repulsive interactions limit represents an essential extension of our previous analysis (2013 New J. Phys. 15 073050), regarded here as I. We also show that near the half-filling the considered type of superconductivity can coexist with antiferromagnetism. The calculations have been carried out with the use of the so-called statistically consistent Gutzwiller approximation (SGA) for the case of a square lattice. We suggest that the electron correlations in conjunction with the Hund's rule exchange play the crucial role in stabilizing the real-space spin-triplet superconducting state. A sizable hybridization of the bands suppresses the homogeneous paired state.
We discuss the spin-triplet pairing mechanism induced by the Hund rule ferromagnetic exchange.We include explicitly the eect of interband hybridization and treat the problem by starting from an extended Hubbard model for a doubly degenerate band, making the simplest HartreeFock approximation for the part involving the pairing and the Hubbard interaction.The conditions of stability of various phases are determined as a function of both band lling and microscopic parameters.The phase diagram contains regions of stability of the spin-triplet superconducting phase coexisting with either saturated or non-saturated ferromagnetism.Phase diagrams for the cases of constant density of states and that of square lattice have been provided.The inuence of hybridization on the stability of considered phases, as well as the temperature dependences of magnetic moment and the superconducting gap are also discussed.
We consider the coexistence of the Hund's-rule-exchange induced spin-triplet paired state with the antiferromagnetic ordering by starting from the extended Hubbard model for a doubly degenerate band.We use the density of states appropriate for the square lattice and treat the problem in the HartreeFock approximation.The temperature dependences of the superconducting gaps, the magnetic moment, and the chemical potential are presented.The free energy in the considered phase is evaluated, as well as the corresponding free energies in four additional phases: paramagnetic, ferromagnetic, superconducting of type A and superconducting of type A1 coexisting with ferromagnetism; they occur in the proper range of parameters: band lling n and the interaction parameters U/W and J/W .The low temperature values of the superconducting gaps and staggered magnetic moment are also analyzed as a function of band lling.
We calculate the effective interaction Weff between two holes added to the ground state of the repulsive three-band Hubbard model. To make contact with Cooper theory and with earlier Hubbard model cluster studies, we first use a perturbative canonical transformation, to generate a two-body Hamiltonian. Then, we extend the results to all orders. The approach is exact in principle, and we obtain a close analytic expression including explicitly the effects of all virtual transitions to 4-body intermediate states. Our scheme naturally lends itself to embody off-site, inter-planar, phonon-mediated and other interactions which are not considered in the Hubbard model but may well be important. The result depends qualitatively on the symmetry of the two-hole state: 1 B2 and 1 A2 pairs are special, because the bare holes do not interact by the on-site repulsion (W=0 pairs). The effective interaction in these channels is attractive and leads to a Cooper-like instability of the Fermi liquid; however Weff is repulsive for Triplet pairs. Bound two-hole states of the same nature were reported earlier in small cluster calculations by exact diagonalisation methods; only symmetric clusters are good models of the plane. Once Weff is known, the pair eigenfunction is determined by an integral equation. We present numerical estimates of the binding energy |∆ | of the pairs, which is in the physically interesting range of tens of meV if unscreened on-site repulsion parameters are used. 1 I.
Excitonic phases described by the quantum condensation of electron-hole pairs (or excitons) are studied in the two-orbital Hubbard model with the Hund's rule coupling J . Using the variational cluster approximation, we calculate the ground-state energy and order parameters, and show that the Hund's rule coupling always stabilizes the excitonic spin-density- wave state and destabilizes the excitonic charge-density-wave state. The pair coherence lengths calculated in the spin-singlet and spin-triplet states indicate that only the spin-triplet excitons are paired more tightly with increasing J . The single-particle spectrum and density of states are also calculated to clarify the characters of these excitonic density-wave states.