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Acta Physica Polonica A

Spin-Triplet Pairing Induced by Hund's Rule Exchange in Orbitally Degenerate Systems: Hartree-Fock Approximation

M. Zegrodnik, Józef Spałek

Published May 1, 2012• 2 citations
Publisher

Abstract

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.

DOI
10.12693/aphyspola.121.1051

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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 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.
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 studied ferromagnetism in the one-dimensional Hubbard model with doubly degenerate atomic orbitals by means of the density-matrix renormalization-group method and obtained the ground-state phase diagrams. It was found that ferromagnetism is stable from low to high (0<n<1.75)(0<n<1.75)(0<n<1.75) electron density when the interactions are sufficiently strong. The quasi-long-range order of triplet superconductivity coexists with the ferromagnetic order for a strong Hund coupling region, where the interorbital interaction ${U}^{\ensuremath{'}}\ensuremath{-}J$ is attractive. At quarter-filling (n=1),(n=1),(n=1), the insulating ferromagnetic state appears, accompanying orbital quasi-long-range order. For low densities (n<1),(n<1),(n<1), ferromagnetism occurs owing to the ferromagnetic exchange interaction caused by virtual hoppings of electrons, the same as in the quarter-filled system. This comes from separation of the charge and spin-orbital degrees of freedom in the strong-coupling limit. This ferromagnetism is fragile against variation of band structure. For high densities (n>1),(n>1),(n>1), the phase diagram of the ferromagnetic phase is similar to that obtained in infinite dimensions. In this case, the double exchange mechanism is operative for stabilizing the ferromagnetic order and this long-range order is robust against variation of the band dispersion. A partially polarized state appears in the density region $1.68\ensuremath{\lesssim}n\ensuremath{\lesssim}1.75$ and phase separation occurs for n just below the half-filling (n=2).(n=2).(n=2).
We study the effect of the nearest neighbor anisotropic exchange interaction on superconductivity in extended Hubbard model in a square lattice. This Hamiltonian is one of a base model to describe various effects on triplet superconductors, e.g. Sr 2 RuO 4 , because the ferromagnetic exchange is advantagous for triplet pairing. Within a one-band meanfield approximation, we calculate the critical temperatures of spin singlet and triplet superconductivity. We obtain a phase diagram in the longitudinal vs transverse exchange couplings and we show how the stable pairing state is controlled by the anisotropy of the exchange interaction.