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arXiv:1506.03593 · Physical Review B

Pairing symmetry of the one-band Hubbard model in the paramagnetic weak-coupling limit: A numerical RPA study

Astrid Tranum Rømer, Andreas Kreisel, Ilya Eremin, Mikhail Malakhov, Thomas Maier, P. J. Hirschfeld, Brian M. Andersen

Published Sep 8, 2015• 63 citations
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Abstract

We study the spin-fluctuation-mediated superconducting pairing gap in a weak-coupling approach to the Hubbard model for a two-dimensional square lattice in the paramagnetic state. Performing a comprehensive theoretical study of the phase diagram as a function of filling, we find that the superconducting gap exhibits transitions from $p$-wave at very low electron fillings to ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave symmetry close to half filling in agreement with previous reports. At intermediate filling levels, different gap symmetries appear as a consequence of the changes in the Fermi surface topology and the associated structure of the spin susceptibility. In particular, the vicinity of a Van Hove singularity in the electronic structure close to the Fermi level has important consequences for the gap structure in favoring the otherwise subdominant triplet solution over the singlet $d$-wave solution. By solving the full gap equation, we find that the energetically favorable triplet solutions are chiral and break time-reversal symmetry. Finally, we also calculate the detailed angular gap structure of the quasiparticle spectrum, and show how spin-fluctuation-mediated pairing leads to significant deviations from the first harmonics both in the singlet ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ gap as well as the chiral triplet gap solution.

DOI
10.1103/physrevb.92.104505

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We study the spin-fluctuation-mediated superconducting pairing gap in a weak-coupling approach to the Hubbard model for a two-dimensional square lattice in the paramagnetic state. Performing a comprehensive theoretical study of the phase diagram as a function of filling, we find that the superconducting gap exhibits transitions from ppp-wave at very low electron fillings to dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​-wave symmetry close to half filling in agreement with previous reports. At intermediate filling levels, different gap symmetries appear as a consequence of the changes in the Fermi surface topology and the associated structure of the spin susceptibility. In particular, the vicinity of a Van Hove singularity in the electronic structure close to the Fermi level has important consequences for the gap structure in favoring the otherwise subdominant triplet solution over the singlet ddd-wave solution. By solving the full gap equation, we find that the energetically favorable triplet solutions are chiral and break time-reversal symmetry. Finally, we also calculate the detailed angular gap structure of the quasiparticle spectrum, and show how spin-fluctuation-mediated pairing leads to significant deviations from the first harmonics both in the singlet dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ gap as well as the chiral triplet gap solution.
The leading superconducting instabilities of the two-dimensional extended repulsive one-band Hubbard model within spin-fluctuation pairing theory depend sensitively on electron density, band, and interaction parameters. We map out the phase diagrams within a random-phase-approximation spin- and charge-fluctuation approach, and find that while B1g{B}_{1g}B1g​ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) and B2g{B}_{2g}B2g​ (dxy{d}_{xy}dxy​) pairing dominates in the absence of repulsive longer-range Coulomb interactions VNN{V}_{\mathrm{NN}}VNN​, the latter induces pairing in other symmetry channels, including, e.g., A2g{A}_{2g}A2g​ (ggg-wave), nodal A1g{A}_{1g}A1g​ (extended sss-wave), or nodal Eu{E}_{u}Eu​ (ppp-wave) spin-triplet superconductivity. At the lowest temperatures, transition boundaries in the phase diagrams between symmetry-distinct spin-singlet orders generate complex time-reversal symmetry broken superpositions. By contrast, we find that boundaries between singlet and triplet regions are characterized by first-order transitions. Finally, motivated by recent photoemission experiments, we have determined the influence of an additional explicitly attractive nearest-neighbor interaction, VNN<0{V}_{\mathrm{NN}}<0VNN​<0, on the superconducting gap structure. Depending on the electronic filling, such an attraction boosts Eu{E}_{u}Eu​ (ppp-wave) spin-triplet or B1g{B}_{1g}B1g​ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) spin-singlet ordering.
We determine the symmetry of Cooper pairs, on the basis of the perturbation theory in terms of the Coulomb interaction U , for the two-dimensional Hubbard model on the square lattice. The phase diagram is investigated in detail. The Hubbard model for small U is mapped on to an effective Hamiltonian with the attractive interaction using the canonical transformation: H eff = e S H e − S . The gap equation of the weak coupling formulation is solved without numerical ambiguity to determine the symmetry of Cooper pairs. The superconducting gap crucially depends on the position of the van Hove singularity. We show the phase diagram in the plane of the electron filling n e and the next nearest-neighbor transfer t '. The d-wave pairing is dominant for the square lattice in a wide range of n e and t '. The d-wave pairing is also stable for the square lattice with anisotropic t '. The three-band d–p model is also investigated, for which the d-wave pairing is stable in a wide range of n e and t pp (the transfer between neighboring oxygen atoms). In the weak coupling analysis, the second-neighbor transfer parameter - t ' could not be so large so that the optimum doping rate is in the range of 0.8 < n e < 0.85.
One promising way to enhance superconductivity is to have coexisting wide and incipient narrow bands, where the Fermi level intersecting the wide band lies just above the narrow band, by which finite-energy spin fluctuations act as glue to mediate pair scattering. As an extreme case of the narrow band dispersion, we investigate spin-fluctuation-mediated superconductivity in two-dimensional Hubbard models with an incipient flat band. For all of the systems investigated in this study, the Kagome, Lieb, and bilayer square lattices with a flat band, we find that spin-singlet pairing superconductivity is enhanced when the flat band is nearly fully filled, due to the interband pair scattering even when the flat band becomes dispersive by correlation effects. Among these models, enhancement of superconductivity is weak in the Lieb lattice, possibly because the density of states of the wide band goes to zero at the Dirac point where the flat and wide bands intersect. Also, when the electron density is smaller so that the flat band approaches half-filling, ferromagnetic spin fluctuations and spin-triplet pairing arises, which does not develop strongly compared to the case of the spin-singlet pairing for the incipient band situation.
In order to explore how superconductivity arises when charge fluctuations and spin fluctuations coexist, we have obtained a phase diagram against the off-site repulsion VVV and band filling nnn for the extended, repulsive Hubbard model on the square lattice with the fluctuation exchange approximation. We have found the existence of (i) a transition between dxy{d}_{xy}dxy​ and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing symmetries, (ii) fff-pairing in between the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ and CDW phases for intermediate 0.5<n<1.00.5<n<1.00.5<n<1.0 and large VVV, and (iii) for anisotropic cases the pairing symmetry changing, in agreement with a previously proposed ``generic phase diagram,'' as $d\ensuremath{\rightarrow}f\ensuremath{\rightarrow}s$ when VVV (hence the charge fluctuations) are increased. All these are consequences of the structure in the charge and spin susceptibilities, which have peaks habitating at common or segregated positions in kkk space.
The Hubbard model provides a simple framework in which one can study how certain aspects of the electronic structure of strongly interacting systems can be tuned to optimize the superconducting pairing correlations and how these changes affect the mechanisms giving rise to them. Here we use a weak-coupling random phase approximation to study a two-dimensional Hubbard model with a unidirectional modulation of the hopping amplitudes as the system evolves from the uniform square lattice to an array of weakly coupled two-leg ladders. We find that the pairing correlations retain their dominant ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wavelike structure and that they are significantly enhanced for a slightly modulated lattice. This enhancement is traced backed to an increase in the strength of the spin-fluctuation pairing interaction due to favorable Fermi surface nesting in the modulated system. We then use a random-phase approximation BCS framework to examine the evolution of the neutron resonance in the superconducting state. We find that it changes only weakly for moderate modulations, but breaks up into two distinct resonances at incommensurate wave vectors in the limit of weakly coupled ladders.
The Rashba-Hubbard model on the square lattice is the paradigmatic case for studying the effect of spin-orbit coupling, which breaks spin and inversion symmetry, in a correlated electron system. We employ a truncated-unity variant of the functional renormalization group which allows us to analyze magnetic and superconducting instabilities on equal footing. We derive phase diagrams depending on the strengths of Rasbha spin-orbit coupling, real second-neighbor hopping, and electron filling. We find commensurate and incommensurate magnetic phases which compete with ddd-wave superconductivity. Due to the breaking of inversion symmetry, singlet and triplet components mix; we quantify the mixing of ddd-wave singlet pairing with fff-wave triplet pairing.