A full quantitative theory of superconductivity in magic-angle twisted bilayer graphene: Coulomb screening, particle–hole asymmetry, and pairing treated on equal footing

Magic-angle twisted bilayer graphene (MA-TBG) is the flagship moiré superconductor, yet the theory of its superconductivity remains incomplete. Wu, MacDonald, and Martin [Phys. Rev. Lett. 121, 257001 (2018)] showed that phonon-mediated attraction in the flat moiré bands can plausibly reproduce the observed transition temperatures, but their treatment was deliberately minimal: local phonon couplings without retardation or momentum dependence, a phenomenological on-site ($U_0$) and nearest-neighbor ($U_1$) Coulomb repulsion with undetermined screening, a linearized mean-field gap equation, and no account of the strong particle–hole asymmetry of the experimental superconducting domes or of the correlated insulating phases. This record states the source-derived open problem: to build a microscopic, experimentally falsifiable theory of superconductivity in MA-TBG in which the momentum-dependent screened Coulomb interaction, correlation effects, and (retarded, momentum-dependent, possibly interlayer) phonon couplings are treated on equal footing, and which explains the particle–hole asymmetry of the superconducting domes, the small correlated-insulating gap scales (~0.3 meV), and the observed sensitivity of superconductivity to electrostatic screening. One deferred sub-question of the source paper — the topological and electromagnetic characterization of the putative chiral $d$-wave state — has since been completed within the phonon-plus-$d$-wave model by the authors' own follow-up work; the mechanism and pairing symmetry of the actual material remain unresolved as of 2026.

Background

Twisted bilayer graphene (TBG) consists of two graphene sheets rotated by a small angle $\theta$, which produces a moiré superlattice — a long-wavelength interference pattern with lattice constant $a_M = a_0/[2\sin(\theta/2)]$, about 13 nm at $\theta \approx 1.1^\circ$ (with graphene lattice constant $a_0 \approx 0.246$ nm). Near this 'magic angle' the interlayer hybridization flattens the low-energy bands to a bandwidth of only a few meV, giving a huge density of states. Experiments in 2018 (Cao et al., Nature 556, 43 and 556, 50) discovered superconductivity with $T_c \sim 1$–2 K and correlated insulating states at integer fillings of the flat bands, making MA-TBG a model strongly correlated system.

Wu, MacDonald, and Martin constructed a microscopic theory of phonon-mediated superconductivity in MA-TBG. They derived the coupling of moiré-band electrons to the optical $E_2$ and $A_1$ phonon modes of the graphene layers, integrated out the phonons to obtain effective attractive interactions, and solved the linearized BCS gap equation (the standard weak-coupling eigenvalue problem for the pairing kernel) in both the $s$-wave and $d$-wave pairing channels. Without Coulomb repulsion the $s$-wave channel wins; a phenomenological on-site repulsion $U_0$ (on the honeycomb lattice of each layer) suppresses $s$-wave pairing, while a nearest-neighbor repulsion $U_1$ suppresses $d$-wave pairing, so the leading instability depends on the (screening-dependent) values of $U_0$ and $U_1$. Fitting the experimental $T_c \approx 1.7$ K requires $U_0 \approx 3.7$ eV or $U_1 \approx 0.5$ eV, values far below the constrained random phase approximation (cRPA, an ab initio method that computes screened interactions by excluding the low-energy bands) estimates of 9.3 eV and 5.5 eV for free-standing monolayer graphene — implying strong screening by the moiré-band density of states, the hexagonal boron nitride (hBN) dielectric encapsulation, and metallic gates 10–30 nm from the bilayer.

The paper itself identifies what its minimal model cannot do. The calculated $T_c(\mu)$ dome is roughly symmetric between electron and hole doping, whereas the experiments show superconductivity only when the lower flat band is near half filling, not at charge neutrality or in the partially filled upper flat band. The model neglects the momentum dependence of the phonon couplings, retardation (the phonon energy, ~0.2 eV, is not small compared to the meV-scale flat-band bandwidth, so Migdal's theorem — the usual justification for ignoring vertex corrections to the electron–phonon interaction — is not obviously applicable), the interlayer phonon modes (layer breathing and shear modes), and correlation effects beyond the linearized mean-field gap equation. It also does not address the experimentally tiny (~0.3 meV) insulating gaps at integer filling, which suggest the Coulomb repulsion is strongly screened and interplays with superconductivity. Finally, the authors defer the topological character and electromagnetic response of the candidate chiral $d+id$ (time-reversal-symmetry-breaking) superconducting state to follow-up work.

Subsequent work has addressed pieces of this program but not the whole. Choi and Choi [Phys. Rev. B 98, 241412 (2018)] computed the electron–phonon coupling atomistically for a ~10,000-atom moiré supercell and found $\lambda > 1$ near half filling, with strong electron–hole asymmetry after atomic relaxation and a genuinely nonadiabatic regime. Schrodi, Aperis, and Oppeneer [Phys. Rev. Research 2, 012066 (2020)] solved full-bandwidth multiband Eliashberg equations (the frequency-dependent generalization of BCS theory) and found that pairing of electrons far from the Fermi level produces particle–hole asymmetric superconducting domes. Pizarro et al. [Phys. Rev. B 100, 161102 (2019)] computed the internal RPA screening of the Coulomb interaction and proposed dielectric engineering. Kang and Vafek [Phys. Rev. B 107, 075408 (2023)] identified intrinsic particle-hole-symmetry-breaking terms in ab initio-derived continuum Hamiltonians. Hou et al. [Phys. Rev. B 111, 125140 (2025)] studied correlated phases in an extended eight-orbital Hubbard model via Hartree–Fock. Zhu and Devereaux [Phys. Rev. B 113, 035446 (2026)] developed a first-principles momentum-space continuum theory of the electron–phonon coupling plus a generalized Eliashberg–McMillan treatment beyond the adiabatic approximation. On the experimental side, device engineering (Yankowitz et al., Science 2019; Saito et al., Nat. Phys. 2020; Arora et al., Nature 2020) showed that superconductivity and the correlated insulators are partially independent and device-dependent, and Barrier et al. [Phys. Rev. X 16, 031040 (2026)] demonstrated that electrostatic screening by an adjacent graphene bilayer completely suppresses both superconductivity and the correlated insulator. Yet, as the 2026 literature itself states, whether superconductivity in TBG is phonon-driven or electron-driven remains unresolved, and no theory treats screening, correlations, and pairing on equal footing.

Problem Statement

Construct a full quantitative, microscopic theory of superconductivity in magic-angle twisted bilayer graphene in which Coulomb screening, electron correlations, and the electron–phonon interaction are treated on equal footing — meaning that all three are computed from the same microscopic Hamiltonian and combined self-consistently to comparable accuracy, rather than any one of them being fixed phenomenologically and fed into the others — replacing the minimal model of Wu, MacDonald, and Martin (local phonon couplings, phenomenological on-site and nearest-neighbor repulsions $U_0$ and $U_1$ with undetermined screening, and a linearized mean-field gap equation). The theory must in particular determine:

1. Coulomb screening. The effective, momentum-dependent Coulomb interaction in realistic devices — including screening by the enhanced moiré-band density of states, the dielectric encapsulation (hBN), and nearby metallic gates at 10–30 nm — and hence the effective $U_0$ and $U_1$ (and longer-range components) that decide whether the leading pairing instability is $s$-wave or $d$-wave. The source paper explicitly leaves 'a calculation of the momentum dependent dielectric function' for future work.

2. Particle–hole asymmetry. Whether the experimentally observed asymmetry of the superconducting domes (superconductivity when the lower flat band is near half filling, generally not at charge neutrality or in the partially filled upper flat band, though later devices softened this) is intrinsic — arising from the particle–hole-asymmetric band structure, interaction-driven symmetry breaking, or frequency-dependent (Eliashberg-type) pairing physics — or extrinsic, due to disorder or device-specific factors; and, if intrinsic, to reproduce it quantitatively.

3. Previously neglected interactions and correlations. The quantitative effect on $T_c$ and the gap structure of (a) the momentum dependence of the phonon-mediated interaction, (b) retardation and the nonadiabatic regime in which Migdal's theorem does not protect against vertex corrections, (c) interlayer phonon couplings (layer breathing and shear modes), and (d) correlation effects beyond the linearized mean-field gap equation.

4. Coexistence with correlated insulating states. A consistent account of the very small (~0.3 meV) insulating gaps at integer fillings, the strength and screening of the Coulomb interactions they imply, and the competition or independence of insulating and superconducting phases, consistent with the device-engineering experiments (pressure, WSe2 substrates, dual-gate screening) and with the observation that electrostatic screening can completely suppress superconductivity.

5. Pairing symmetry and observable signatures. The actual pairing symmetry of the material and its experimental signatures. Within the phonon-plus-$d$-wave model, the topological character and electromagnetic response of the chiral $d+id$ state (spontaneous bulk supercurrent, vortex–antivortex lattice, topology) have already been established by the authors' follow-up work, so the open part of this component is the experimental determination of the realized pairing state (including alternatives proposed since — nematic pairing, which spontaneously breaks the moiré rotational symmetry; Kekulé or pair-density-wave pairing, in which the pair wave function also modulates in space; or other unconventional states) and its directly measurable consequences.

These five components serve one overarching objective: a single experimentally falsifiable account of the superconducting phase of magic-angle twisted bilayer graphene — its mechanism, pairing symmetry, and dependence on screening and dielectric environment — that is quantitative rather than phenomenological. Here 'quantitative rather than phenomenological' means that the transition temperature and the gap structure are predicted from the microscopic parameters and the device geometry rather than from quantities fitted to the superconducting data. The five components are jointly posed as a single objective: one account that treats them together rather than as independent sub-problems.

References

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