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Physical review. B./Physical review. B

Superconductivity in the bilayer Hubbard model: Two Fermi surfaces are better than one

Seher Karakuzu, Steven Johnston, Thomas Maier

Published Dec 6, 2021• 20 citations
Publisher

Abstract

Fully occupied or unoccupied bands in a solid are often considered inert and irrelevant to a material's low-energy properties. But the discovery of enhanced superconductivity in heavily electron-doped FeSe-derived superconductors poses questions about the possible role of incipient bands (those laying close to but not crossing the Fermi level) in pairing. To answer this question, researchers have studied pairing correlations in the bilayer Hubbard model, which has an incipient band for large interlayer hopping ${t}_{\ensuremath{\perp}}$, using many-body perturbation theory and variational methods. They have generally found that superconductivity is enhanced as one of the bands approaches the Lifshitz transition and even when it becomes incipient. Here we address this question using the nonperturbative quantum Monte Carlo (QMC) dynamical cluster approximation (DCA) to study the bilayer Hubbard model's pairing correlations. We find that the model has robust ${s}^{\ifmmode\pm\else\textpm\fi{}}$ pairing correlations in the large ${t}_{\ensuremath{\perp}}$ limit, which can become stronger as one band is made incipient. While this behavior is linked to changes in the effective interaction, we further find that it is counteracted by a suppression of the intrinsic pair-field susceptibility and does not translate to an increased ${T}_{c}$. Our results demonstrate that the highest achievable transition temperatures in the bilayer Hubbard model occur when the system has two bands crossing the Fermi level.

DOI
10.1103/physrevb.104.245109

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Fully occupied or unoccupied bands in a solid are often considered inert and irrelevant to a material's low-energy properties. But the discovery of enhanced superconductivity in heavily electron-doped FeSe-derived superconductors poses questions about the possible role of incipient bands (those laying close to but not crossing the Fermi level) in pairing. To answer this question, researchers have studied pairing correlations in the bilayer Hubbard model, which has an incipient band for large interlayer hopping t\ensuremath⊥{t}_{\ensuremath{\perp}}t\ensuremath⊥​, using many-body perturbation theory and variational methods. They have generally found that superconductivity is enhanced as one of the bands approaches the Lifshitz transition and even when it becomes incipient. Here we address this question using the nonperturbative quantum Monte Carlo (QMC) dynamical cluster approximation (DCA) to study the bilayer Hubbard model's pairing correlations. We find that the model has robust s\ifmmode±\else\textpm\fi{s}^{\ifmmode\pm\else\textpm\fi{}}s\ifmmode±\else\textpm\fi pairing correlations in the large t\ensuremath⊥{t}_{\ensuremath{\perp}}t\ensuremath⊥​ limit, which can become stronger as one band is made incipient. While this behavior is linked to changes in the effective interaction, we further find that it is counteracted by a suppression of the intrinsic pair-field susceptibility and does not translate to an increased Tc{T}_{c}Tc​. Our results demonstrate that the highest achievable transition temperatures in the bilayer Hubbard model occur when the system has two bands crossing the Fermi level.
Fully occupied or unoccupied bands in a solid are often considered inert and irrelevant to a material's low-energy properties. But the discovery of enhanced superconductivity in heavily electron-doped FeSe-derived superconductors poses questions about the possible role of incipient bands (those laying close to but not crossing the Fermi level) in pairing. To answer this question, researchers have studied pairing correlations in the bilayer Hubbard model, which has an incipient band for large interlayer hopping t⊥t_\perpt⊥​, using many-body perturbation theory and variational methods. They have generally found that superconductivity is enhanced as one of the bands approaches the Liftshiz transition and even when it becomes incipient. Here, we address this question using the nonperturbative quantum Monte Carlo (QMC) dynamical cluster approximation (DCA) to study the bilayer Hubbard model's pairing correlations. We find that the model has robust s±s_\pms±​ pairing correlations in the large t⊥t_\perpt⊥​ limit, which can become stronger as one band is made incipient. While this behavior is linked to changes in the effective interaction, we further find that it is counteracted by a suppression of the intrinsic pair-field susceptibility and does not translate to an increased TcT_cTc​. Our results demonstrate that the highest achievable transition temperatures in the bilayer Hubbard model occur when the system has two bands crossing the Fermi level.
It has been proposed that the superconducting transition temperature Tc{T}_{\text{c}}Tc​ of an unconventional superconductor with a large pairing scale but strong phase fluctuations can be enhanced by coupling it to a metal. However, the general efficacy of this approach across different parameter regimes remains an open question. Using the dynamical cluster approximation, we study this question in a system composed of an attractive Hubbard layer in the intermediate coupling regime, where the magnitude of the attractive Coulomb interaction ∣U∣|U|∣U∣ is slightly larger than the bandwidth WWW, hybridized with a noninteracting metallic layer. We find that while the superconducting transition becomes more mean-field-like with increasing interlayer hopping, the superconducting transition temperature Tc{T}_{\text{c}}Tc​ exhibits a nonmonotonic dependence on the strength of the hybridization ${t}_{\ensuremath{\perp}}$. This behavior arises from a reduction of the effective pairing interaction in the correlated layer that outcompetes the growth in the intrinsic pair-field susceptibility induced by the coupling to the metallic layer. We find that the largest Tc{T}_{\text{c}}Tc​ inferred here for the composite system is comparable to the maximum value currently estimated for the isolated negative-UUU Hubbard model.
The bilayer Hubbard model with an intralayer hopping ttt and an interlayer hopping ${t}_{\ensuremath{\perp}}$ provides an interesting testing ground for several aspects of what has been called unconventional superconductivity. One can study the type of pair structures which arise when there are multiple Fermi surfaces. One can also examine the pairing for a system in which the structure of the spin-fluctuation spectral weight can be changed. Using a dynamic cluster quantum Monte Carlo approximation, we find that near half filling, if the splitting between the bonding and antibonding bands ${t}_{\ensuremath{\perp}}/t$ is small, the gap has B1g{B}_{1g}B1g​ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave) symmetry, but when the splitting becomes larger, A1g{A}_{1g}A1g​ (${s}^{\ifmmode\pm\else\textpm\fi{}}$-wave) pairing is favored. We also find that in the ${s}^{\ifmmode\pm\else\textpm\fi{}}$ pairing region, the pairing is driven by interlayer spin fluctuations and that Tc{T}_{c}Tc​ is enhanced.
The bilayer Hubbard model with an intra-layer hopping ttt and an inter-layer hopping t⊥t_\perpt⊥​ provides an interesting testing ground for several aspects of what has been called unconventional superconductivity. One can study the type of pair structures which arise when there are multiple Fermi surfaces. One can also examine the pairing for a system in which the structure of the spin-fluctuation spectral weight can be changed. Using a dynamic cluster quantum Monte Carlo approximation, we find that near half-filling, if the splitting between the bonding and anti-bonding bands t⊥/tt_\perp/tt⊥​/t is small, the gap has B1gB_{1g}B1g​ (dx2−y2d_{x^2-y^2}dx2−y2​-wave) symmetry but when the splitting becomes larger, A1gA_{1g}A1g​ (s±s^\pms±-wave) pairing is favored. We also find that in the s±s^\pms± pairing region, the pairing is driven by inter-layer spin fluctuations and that TcT_cTc​ is enhanced.
Coupling two strongly correlated layers opens a distinct route to repulsion-driven superconductivity. Yet the same interlayer coupling makes the optimization of superconductivity highly nontrivial. Using the dynamical vertex approximation, we determine the superconducting landscape in the enlarged parameter space spanned by filling and interlayer hopping. We identify an optimal regime at intermediate interlayer hopping and moderate doping away from half filling, where the system exhibits high superconducting transition temperatures (TcT_{\rm c}Tc​) with an interband s±s_{\pm}s±​-wave gap structure. Toward half filling, however, strong self-energy effects can suppress the superconducting tendency by reducing the low-energy spectral weight. We further show that a Fermi-surface distortion induced by longer-range hoppings can weaken this suppression and enhance TcT_{\rm c}Tc​. Our results establish how filling, interlayer hopping, and Fermi-surface geometry jointly optimize the superconducting regime in the strongly correlated bilayer Hubbard model.
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.