Symmetry constraints on superconductivity in twisted bilayer graphene: Fractional vortices, $4e$ condensates, or nonunitary pairing

When two graphene sheets are twisted relative to each other by a small angle, enhanced correlations lead to superconductivity whose origin remains under debate. Here, we derive some general constraints on superconductivity in twisted bilayer graphene (TBG), independent of its underlying mechanism. Neglecting weak coupling between valleys, the global symmetry group of TBG consists of independent spin rotations in each valley in addition to valley charge rotations, $\mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right)\times \mathrm{U}\left(1\right)$. This symmetry is further enhanced to a full SU(4) in the idealized chiral limit. In both cases, we show that any charge $2e$ pairing must break the global symmetry. Additionally, if the pairing is unitary the resulting superconductor admits fractional vortices. This leads to the following general statement. Any superconducting condensate in either symmetry class has to satisfy one of three possibilities: (i) the superconducting pairing is nonunitary, (ii) the superconducting condensate has charge $2e$ but admits at least half quantum vortices which carry a flux of $h/4e$, or (iii) the superconducting condensate has charge $2me, m>1$, with vortices carrying $h/2me$ flux. The latter possibility can be realized by a symmetric charge $4e$ superconductor ($m=2$). Nonunitary pairing (i) is expected for superconductors observed in the vicinity of flavor polarized states. On the other hand, in the absence of flavor polarization, e.g., in the vicinity of charge neutrality, one of the two exotic possibilities (ii) and (iii) is expected. We sketch how all three scenarios can be realized in different limits within a strong coupling theory of superconductivity based on skyrmions. Finally we discuss the effect of symmetry lowering anisotropies and experimental implications of these scenarios.

Figure 1

Schematic illustration of the three possibilities for superconducting pairing with the global symmetry $G=\mathrm{U}{\left(1\right)}_V\times \mathrm{SU}{\left(2\right)}_K\times \mathrm{SU}{\left(2\right)}_{K^{\prime }}$ or $G_{\mathrm{high}}=\mathrm{SU}\left(4\right)$. Nonunitary pairing (i) is expected in TBG for superconductivity in the vicinity of flavor symmetry breaking, whereas unitary charge $2e$ superconductivity with fractional vortices (ii) or higher charge superconductivity (iii) is expected in the absence of flavor symmetry breaking. The unitary charge $2e$ superconductor always breaks the global ($G$ or $G_{\mathrm{high}}$) symmetry and is thus only possible at $T=0$. The possible order parameter manifolds for the unitary charge $2e$ superconductor and its associated vortices are summarized in Table 1.