Unifying description of competing chiral and nematic superconducting states in twisted bilayer graphene

Abstract

We reveal a striking correspondence between electron- and phonon-driven pairing in twisted bilayer graphene (TBG) by mapping an atomistic electronically driven pairing model onto an effective inter-valley, intra-Chern description, originally proposed for phonon-mediated superconductivity. Within the unified framework of intra-Chern pairing, we analyze the competition between nematic and chiral superconducting states. The latter corresponds to the extreme Chern-polarized limit and thus hosts unpaired flat bands within the superconducting gap, which generally disfavors it relative to the nematic states. Crucially, nematic order is locally preferred at each momenta, but the optimal nematic directions are incompatible across the Brillouin zone due to the broken rotation symmetry. This momentum-space frustration enables a chiral ground state at large fillings or weak interactions. Our results thereby both provide a unified understanding of superconductivity in TBG, with a natural cooperation of electron- and phonon-mediated pairing, and clarify the microscopic origin of the competition between the chiral and nematic superconducting states.

Figures (10)

• Figure 1: Pairing between opposite valleys $\eta=\pm$ (blue/red) and in same Chern sector $C=\pm$ (dark rim/center). (a) In the nematic state pairing is finite in both Chern sectors, with relative phase $\phi$ setting the nematic direction. (b) In the chiral state one sector remains unpaired (dashed gray), lowering condensation energy. (c) For the nematic state, the preferred direction $\phi_{\star}$ (colormap) varies with momenta (numerical solution within atomistic model, see Sec. \ref{['sec:atomistic']}). Values at $C_{3z}$-related momenta (black markers) are incompatible, leading to frustration.
• Figure 2: Spectrum of normal state TBG. (a) Low-energy band structure with the moiré flat bands highlighted. States at opposite momenta and valley $\eta = \pm$ (blue/red) are related by spinless time-reversal symmetry $\mathcal{T}$. (b) Band energies are parametrized in terms of the momentum-dependent splitting $\tilde{\epsilon}_{\eta}$ between levels of opposite eigenband $b$, and their offset $\tilde{\mu}_{\eta}$ from the chemical potential $\mu$ (dashed black).
• Figure 3: Local $d$-wave pairing symmetry at the center of the $AA$ region. Orange (gray) dots and light (dark) colored lines denote respectively sites and bonds in the top (bottom) layer. Central sites of both layers are perfectly aligned. Bond line width (hue) represents the pairing amplitude $|\hat{\Delta}_{ij}|$ (phase $\arg (\hat{\Delta}_{ij})$). In all cases, the order parameter is $\pi$-shifted between the layers. Nematic states (a) $\hat{\Delta}_{x}$ and (b) $\hat{\Delta}_{y}$ have eigenvalues $+1$ and $-1$, respectively, under $C_{2y}$ transformation (gray dashed line as rotation axis), which swaps the layers. The chiral state (c) $\hat{\Delta}_{+}$ has eigenvalue $e^{i \omega}$, with $\omega=2\pi/3$, under in-plane rotations $C_{3z}$.
• Figure 4: Condensation energy $E_c(\Delta_\star, \alpha, \phi)$ for nearest neighbor pairing with $J\approx1.35$ eV. (a) Nematic $\hat{\Delta}_x$ (green), $\hat{\Delta}_y$ (orange), and chiral $\hat{\Delta}_{\pm}$ (magenta) states as a function of doping $\mu$ measured from half-filling. Inset shows small region with chiral ground state. (b,c) $E_c$ for different pairing configurations $(\alpha, \phi)$ at (b) $\mu\approx0.5$ meV (blue line/diamond in (a)) and (c) $\mu\approx2$ meV (yellow line/star in (c)). At low doping (b) the nematic state $(\alpha,\phi)=(\pi/2,\pi)$ ($\hat{\Delta}_x$) is the ground state, while the chiral state $\alpha=0$ ($\hat{\Delta}_{+}$) wins at large doping (c).
• Figure 5: Spectrum of nematic $\hat{\Delta}_{x}$ (left) and chiral $\hat{\Delta}_{+}$ (right) states for nearest neighbor pairing at fixed pairing strength $\Delta_0 \approx 49$ meV, corresponding to the self-consistent pairing strength for the nematic state at $\mu \approx 0.5$ meV in Fig. \ref{['fig:atomistic:freeenergy']}, with an averaged bond pairing amplitude $\langle |\hat{\Delta}_{ij}| \rangle \approx 0.2$ meV. (a,b) Electronic density of states (DOS) versus chemical potential $\mu$. (c,d) Band structure along high-symmetry lines at $\mu\approx0.5$ meV (green dashed lines in (a,b)) with color according to sublattice-valley polarization $P$.