Geometric superfluid stiffness of Kekulé superconductivity in magic-angle twisted bilayer graphene

Abstract

Superconductivity in twisted graphene is probed by tunneling spectroscopy and superfluid stiffness, two observables that access the same order parameter from complementary perspectives. We show that a finite-momentum pair-density-wave (PDW) state, consistent with reported Kekulé signatures, reconciles substantial low-energy tunneling weight with an approximately $T^2$ suppression of the low-temperature superfluid stiffness. The PDW order produces a Bogoliubov Fermi surface and finite zero-bias conductance. The same gapless quasiparticles also enter the geometric superfluid response, yielding a low-temperature stiffness suppression that persists in the flat-band limit. We further predict that, under density or displacement-field tuning, enhanced residual zero-bias conductance should accompany reduced low-temperature stiffness, providing a direct experimental link between tunneling spectroscopy and phase rigidity in twisted graphene.

Figures (4)

• Figure 1: Schematic of pairing processes in Kekulé superconductivity in twisted bilayer graphene (TBG). The blue and orange curves represent the two flat bands of a single valley along a closed trajectory in the moiré Brillouin zone (mBZ). The gray region denotes bands remote from the flat-band manifold.
• Figure 2: Superfluid stiffness at zero temperature, $D_{s,0}$, as a function of the superconducting order parameter $\Delta$ (blue curve). The dashed gray line indicates the crossover boundary between the $U$ and $V$ regimes. The strong pairing $U$-regime is more appropriate for tri-layer systems.
• Figure 3: (a) Lowest positive BdG eigenvalue in the active band at the order parameter $\Delta=0.31\,\mathrm{meV}$, showing a small gapless Bogoliubov Fermi surface (in dark blue). (b) Comparison between theory and experiment. The experimental data are adapted from Fig. 3(a) of Ref. Tanaka2025, which shows the normalized superfluid stiffness in the hole-doped regime. In the theoretical curve, $D_s$ is computed at $\Delta=0.31\,\mathrm{meV}$, with reference temperature $T_{\mathrm{base}}\simeq 0.1\,\mathrm{K}$. The dashed lines denote power-law fits: the theoretical result gives an exponent $n\simeq 2.1$, while the experimental data are consistent with $n\simeq 2$. (c) Temperature dependence of the inverse bare superfluid stiffness, $D_s^{-1}(T)$, for representative gap values $|\Delta|=0.40$, $0.36$, and $0.31\,\mathrm{meV}$. (d) Corresponding zero-energy density of states, $\mathrm{DOS}(E=0)$, which is proportional to the zero-bias tunneling conductance, for the same parameters. Over the superconducting temperature range shown, $D_s^{-1}(T)$ closely tracks the evolution of $\mathrm{DOS}(E=0)$.
• Figure 4: Plot of the BKT transition temperature $T_{\text{BKT}}$ versus the zero-temperature superfluid stiffness $D_{s,0}$, motivated by Fig. 2h in Ref. Tanaka2025. The experimental data are adapted from the hole-doped data in Fig. 2h of Ref. Tanaka2025, using the authors' definition of $T_c^{(0.5)}$ for points with $D_{s,0}\leq 0.42$ meV.