Kekulé Superconductivity in Twisted Magic Angle Bilayer Graphene

TL;DR

This work proposes a microscopic mechanism for superconductivity in twisted bilayer graphene based on intra-valley Kekulé PDW order, implemented within the Bistritzer–MacDonald continuum model with a short-range attractive interaction. The PDW is characterized by a nontrivial quantum texture, a form-factor matrix that couples bands and momentum, and a stable phase with equal-layer condensates and pairing momentum at a high-symmetry point, favoring spin-triplet pairing. The theory predicts nematic order from C3 symmetry breaking, a U-to-V transition in tunneling DOS linked to Bogoliubov Fermi surfaces, and an intrinsic finite zero-bias conductance in the gapless regime, all of which align with experimental observations and coherence-length trends. Overall, Kekulé/PDW superconductivity emerges as a robust, microscopic candidate for unconventional superconductivity in the twisted graphene family, providing a cohesive framework that connects STM Kekulé patterns, DOS evolution, and transport signatures.

Abstract

While it has been one of the most important new physics discoveries in the last decade, the nature of superconductivity in the twisted graphene family remains an unsolved problem. Motivated by recent scanning tunneling experiments that report Kekulé ordering in moiré graphene superconductors, we develop a microscopic theory of this superconductivity for the twisted bilayer system. The pairing we find is an intra-valley, finite-momentum pair-density wave (PDW) that intrinsically carries a Kekulé modulation. This state exhibits four salient features: (i) spontaneous breaking of $C_3$ rotation symmetry, producing nematic order (ii)with triplet pairing; and (iii) a quasiparticle density of states that evolves from a V-shaped profile to a fully gapped, U-shaped spectrum as the attraction increases which is accompanied by (iv) systematic behavior of the temperature dependent zero bias conductance. These features align with key experimental signatures. We find, as well, that with only modest interaction strengths, the state is near to a BEC-like phase, consistent with the observed extremely short coherence lengths. Taken together, these results identify a microscopic intra-valley Kekulé PDW as a compelling candidate for unconventional superconductivity in the twisted graphene family.

Figures (8)

• Figure 1: Cartoon of the twisted bilayer graphene geometry and Cooper Pairs. This serves to contrast intravalley and intervalley pairing. Two graphene Brillouin zones (BZs) are rotated by $\pm\theta$ (gray/red). Two mini-moiré BZs associated with $K/K'$ valleys are represented by orange and blue; the $\Gamma/M$-point is shown in the right/left mini-BZ. Intravalley pairing involves pairs within each valley and associated mini-BZ, while intervalley pairing involves two different mini-BZs.
• Figure 2: Flat-band dispersion and form factors from the Bistritzer--MacDonald (BM) model in the mini-Brillouin zone. Parameters: $w_{AA}=0.08$ eV, $w_{AB}=0.11$ eV, twist angle $\theta\simeq 0.94$, and $\hbar v_F=0.684$ eV$\cdot$nm. (a,b) Band energies $E_1(\mathbf{k})$ and $E_2(\mathbf{k})$ measured from half-filling; each flat band has bandwidth $W \approx 0.5$ meV and respects three-fold rotation $C_3$. (c,d) Magnitudes of the intra-band and inter-band form factors (taken from layer $L=1$) with pairing momentum $\mathbf{Q}=M$, which enter into the gap equation and are central to superconductivity.
• Figure 3: The figure shows the grand-canonical thermodynamic potential $\Omega$ of the singlet and triplet pairing states as a function of interaction strength $V \equiv V_0(q=0)$. Results are computed at near-zero temperature $T=0.001~\mathrm{meV}$ and chemical potential $\mu=-0.59~\mathrm{meV}$. The plotted quantity is $\Omega-\Omega_N$, measured relative to the normal-state value $\Omega_N$. The inset shows the corresponding superconducting order-parameter magnitude $|\Delta|$ for the two states. For the interaction strengths shown, the triplet state has a lower $\Omega$ than the singlet state, indicating that triplet pairing is thermodynamically favored.
• Figure 4: Order-parameter amplitude $|\Delta|$ (left axis) and chemical potential $\mu$ (right axis) versus coupling strength $V$. The dashed line marks the flat-band bottom. For attractive $V \approx -1.7\,\mathrm{meV}$ ($\Delta\simeq 0.73$meV), the system crosses into the BEC-like regime.
• Figure 5: (a) Density of states (DOS) versus energy $\omega$ (meV) for three different gap amplitudes, with curves labeled by $|\Delta| \simeq 0.49, 0.36,$ and $0.32$ meV. As the order parameter is reduced, the bottom of the characteristic U-shaped gap narrows towards a V-shape. Notably, a finite zero-bias conductance emerges in the latter case (b) The lowest positive Bogoliubov-de Gennes eigenvalue $E_1(M, \mathbf{k})$ plotted across the mini-Brillouin zone for $|\Delta| = 0.49$ meV. At this amplitude (still within the BCS-like regime), the spectrum remains fully gapped, corresponding to the U-shaped DOS. (c). At a smaller amplitude of $|\Delta| = 0.32$ meV, a small Bogoliubov Fermi surface appears and the (V-shaped) DOS spectrum becomes gapless.