Superconductivity in imbalanced bilayer Hubbard model: enhanced $d$-wave and weakened $s^\pm$-wave pairing

TL;DR

The study shows that density imbalance in an imbalanced bilayer Hubbard model, accessible via an interlayer site-energy $\epsilon_2$, can markedly modify superconducting pairing, driving a transition from $d$-wave to $s^{\pm}$-wave as interlayer coupling $t_\perp$ increases. By employing dynamical cluster approximation with a CT-AUX solver and solving the Bethe-Salpeter equation in the particle-particle channel, the authors map a phase diagram where moderate imbalance can maximize $T_c$ for $d$-wave pairing, while large $t_\perp$ favors interlayer $s^{\pm}$-wave pairing with suppressed $T_c$ under strong imbalance. A key finding is that an underdoped layer can sustain a large pairing scale while an overdoped layer provides phase stiffness, enabling $T_c$ enhancement beyond single-layer values and even allowing SC on a single layer under certain conditions. These results provide a concrete mechanism for Tc enhancement via layer differentiation and have relevance for bilayer materials and quantum simulators, including ultracold atoms.

Abstract

We investigate the bilayer model with two layers of imbalanced densities coupled by the interlayer hybridization. Using the large-scale dynamical cluster quantum Monte Carlo simulation, we discovered that increased hybridization induces a transition in the superconducting pairing from $d$-wave to $s^{\pm}$-wave and the superconducting $T_c$ of $d$-wave pairing exhibits a non-monotonic dependence on the density imbalance. Remarkably, the optimal superconductivity(SC) occurs at a moderate imbalance. Our results support the possibility of $T_c$ enhancement in composite picture where the underdoped layer provides the pairing strength while the overdoped layer promotes the phase coherence. In addition, the SC can be possibly hosted by a single layer, which is reminiscent of our recent exploration on the trilayer Hubbard model. Our present study thus provides new insight that the SC can be enhanced via the layer differentiation.

Figures (12)

• Figure 1: Schematic diagram of the imbalanced bilayer Hubbard model. The blue and yellow arrows represent nearest-neighbor and next-nearest-neighbor hoppings respectively and the dark arrow indicates the interlayer hybridization. The onsite energy $\epsilon_2>0$ is introduced to tune the density imbalance between layers.
• Figure 2: The density versus interlayer hybridization phase diagram of imbalanced bilayer Hubbard model consists of Fermi liquid (FL), $d$-wave on both layers ($d$-BL), $d$-wave on single layer ($d$-SL) with lower doping, and $s^{\pm}$-wave SC phases. The estimated phase boundaries are denoted by dashed green lines. The magnitude of $T_c$ is indicated by the size and color of circles. The fixed average electron density is $n=0.875$.
• Figure 3: Temperature evolution of $1-\lambda_{s^\pm}(T)$ for varying density distribution at the characteristic $t_\perp/t =1.0$.
• Figure 4: The density evolution of $s^\pm$-wave paring $T_c$ at the three characteristic hybridization $t_\perp/t=0.8,1.0$ and $2.0$.
• Figure 5: The eigenvector $\phi_{s^{\pm}}(\mathbf{K},\pi T)$ of bonding ($\mathbf{K}_z=0$) and anti-bonding ($\mathbf{K}_z=\pi$) at the lowest simulated temperature $T/t = 0.04$ for various density distributions at $t_\perp/t = 1.0$.