Unconventional superconductivity due to interband polarization

We analyze in detail the superconductivity that arises in an extended Hubbard model describing a multiband system with repulsive interactions. We show that virtual interband processes induce an effective attractive interaction for small momentum transfers, a situation not found in most models of superconductivity from repulsion. This attraction can be traced back, in real space, to the presence of correlated hopping terms induced by interband polarization. We reveal this physics with both strong-coupling expansion and many-body perturbation theory, supplemented by numerical calculations. Finally, we point out interesting similarities with the problem of interacting electrons in twisted bilayer graphene, suggesting the importance of the interband contribution to superconductivity.

Figure 1

Domains of validity of the different analytical methods. The hybridization expansion (HE), interaction expansion (IE), and random phase approximation (RPA) have overlapping regions of validity, allowing for stringent consistency checks. The insets depict the virtual particle-hole processes responsible for pairing, in real space for the HE ($t\ll \mathrm{\Delta }$) and momentum space for the IE ($V\ll \mathrm{\Delta }$).

Figure 2

(a) ${\mathrm{\Delta }}_{\mathrm{H}}/\mathrm{\Delta }$ and (b) the relative deviation $(t_{\mathrm{F}}-t)/t$ obtained for $\mathrm{\Delta }=8t$ and $V/t=2,3$, as a function of doping.

Figure 3

Values of the critical temperature as a function of doping, obtained by solving numerically Eq. (39), for $\mathrm{\Delta }=8t$ and $V/t=2,2.5,3$. We use $N=2700$ unit cells and include the HF corrections to the band structure and the eigenfunctions.

Figure 4

Symmetry of the superconducting order parameter in the BZ, obtained for $\mathrm{\Delta }=8t$, $V=3t$, and four different values of doping. The continuum black lines represent the Fermi surface.

Figure 5

The effective Cooper interaction $V_{n^{\prime },k^{\prime }}^{n,k}$ [see Eq. (36)] in the upper band $n=n^{\prime }=+$ is attractive, as shown here using the incoming momentum $k^{\prime }=K$ as reference, with $\mathrm{\Delta }=8t$ and two different doping concentrations. Squares show the analytical prediction $g_0$ [Eq. (35)] obtained by neglecting intraband contributions.

Figure 6

Some matrix elements ${\mathrm{\Gamma }}_{G,G^{\prime }}^{\mathrm{RPA}}(q\to 0)$ of the screened potential for twisted bilayer graphene, obtained for the twist angle $\theta =1.{05}^{\circ }$ and for $x=-1$ electrons per moiré unit cell. The index $G$ denotes moiré reciprocal lattice vectors, as defined on the right.

Figure 7

On the left, primitive lattice vectors $a_{j=1,2,3}$ and nearest-neighbor vectors $u_{j=1,2,3}$ on the honeycomb lattice. Highlighted in blue is the triangular $B$ lattice and the upper triangle centered on an $A$ site summed over in the effective HE model [see Eq. (6)]. On the right, first Brillouin zone and high-symmetry points $\mathrm{\Gamma }$, $K$, and $K^{\prime }$.

Figure 8

Some diagrams which describe the scattering of Cooper pairs, and appear in the superconducting kernel in Eq. (38). The band labels $A$ and $B$ stand for the sublattice where the band is mostly localized. States $|k,A\rangle ,|k^{\prime },A\rangle$ lie near the Fermi surface of one valley, and states $|-k,A\rangle ,-|k^{\prime },A\rangle$ lie near the Fermi surface of the other valley. The states within the bubbles in diagrams (b) and (d) belong to the same valley, either $K$ or $K^{\prime }$. The overall momentum transfer is such that $\left|q\right|=|k-k^{\prime }|\ll k_F$.