Non-Abelian topological superconductivity in maximally twisted double-layer spin-triplet valley-singlet superconductors

Abstract

Recent studies point to an exotic spin-triplet valley-singlet (STVS) superconducting phase in certain two-valley electron liquids, including rhombohedral trilayer graphene, Bernal bilayer graphene and ZrNCl, which nevertheless admits only trivial topology. Here, we predict that upon twisting two layers of STVS superconductors, a chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$-wave superconducting phase emerges near the ‘maximal’ twist angle of 30^∘ where the system becomes an extrinsic quasi-crystal with 12-fold tiling. The resulting composite hosts an odd number of chiral Majorana edge modes and a single non-Abelian Majorana zero mode (MZM) in the vortex core. Through detailed symmetry analysis and microscopic modelling, we demonstrate that the non-Abelian topological superconductivity (TSC) forms robustly near the maximal twist when the isolated Fermi pockets coalesce into a single connected Fermi surface in the moiré Brillouin zone. Our results establish the large-twist-angle engineering, with distinct underlying moiré physics from magic-angle graphene, as a viable route toward non-Abelian TSC.

Introduction

Being of fundamental interest and potential use for topological qubits, the search for topological superconductors hosting excitations with non-Abelian exchange statistics has been one of the central topics in condensed matter physics over the past two decades^1,2,3,4,5. The simplest such particles—Majorana zero modes (MZMs) – were originally proposed as vortex core states in chiral p-wave superconductors^6,7,8, but the lack of intrinsic p-wave superconductivity in nature has motivated worldwide efforts to engineer synthetic platforms that emulate this behavior using more conventional ingredients^{9,10,11,12,13,14,15,16,17}. Whether such effective p-wave superconductors and MZMs have been realized in recent experiments is still under debate^18,19,20. The quest for an intrinsic topological superconductor with non-Abelian excitations, on the other hand, remains an ongoing grand challenge to the condensed matter community.

Motivated by recent developments in twisted van der Waals materials^{21,22,23,24,25,26,27,28,29}, a new route toward topological superconductivity (TSC) has been proposed recently which takes two monolayers of high-T_c cuprate superconductor with nodal ${d}_{{x}^{2}-{y}^{2}}$ pairing symmetry, such as Bi₂Sr₂CaCu₂O_8+δ, stacked with a relative angular twist θ^30,31. At θ ≃ 45^∘, the bilayer is predicted to enter a fully-gapped, topological chiral $d\pm {{{{{{{\rm{i}}}}}}}}{d}^{{\prime} }$ phase with spontaneously broken time-reversal symmetry ${{{{{{{\mathcal{T}}}}}}}}$, which has been tentatively identified in a recent experimental study³². Despite its promise for realizing high-T_c TSC, a chiral $d\pm {{{{{{{\rm{i}}}}}}}}{d}^{{\prime} }$ superconductor cannot host truly non-Abelian excitations due to its spin-singlet pairing nature, which is always associated with an even number of MZMs that combine to form usual Abelian fermions. Generalizing the scheme to spin-triplet superconductors with nodal p-wave or f-wave order parameters one can create chiral SC phases with an odd number of MZMs in principle³³, while once again facing the scarcity of nodal p-wave and f-wave superconductors in nature.

Recent progress in superconducting two-dimensional materials, however, has uncovered a growing amount of evidence for f-wave spin-triplet superconductivity, albeit in the disguise of a fully gapped phase: in a recent experiment, rhombohedral trilayer graphene (RTG) was found to superconduct in two different gate-tuned regions, where the peculiar SC2 superconducting phase was borne out of a spin-polarized, valley-unpolarized normal metal³⁴. Such an unusual normal-state fermiology strongly hints at its spin-triplet pairing nature, which is further supported by the observation of an in-plane critical field that far exceeds the Pauli paramagnetic limit. Similar results have also been reported in Bernal bilayer graphene (BBG)³⁵, and both experimental observations were interpreted theoretically as a signature of spin-triplet f-wave pairing^36,37. More recently, Crépel and Fu proposed that a ‘three-particle’ mechanism involving virtual excitons generically gives rise to spin-triplet f-wave pairing in a two-valley electron liquid formed in doped insulators such as ZrNCl³⁸, which provides a plausible explanation for the puzzling doping dependence of the gap structure revealed by early specific heat measurements on Li-doped ZrNCl³⁹.

In the scenarios described above, the parent normal-state Fermi surface (FS) consists of disconnected pockets enclosing the + K and − K corners of the hexagonal Brillouin zone (BZ), as illustrated in Fig. 1a. Upon pairing electrons of the same spin and from opposite valleys, fermion exchange statistics require the order parameter to be odd under exchange of the valleys, which entails a spin-triplet valley-singlet (STVS) pairing. As shown schematically in Fig. 1a, such an STVS superconductor has exactly the ${f}_{x({x}^{2}-3{y}^{2})}$-wave symmetry, while the excitation spectrum exhibits a full superconducting gap because the nodes of the f-wave gap function (located along Γ − M lines) never intersect the disconnected FS. The gapped phase respects a spinless time-reversal symmetry ${{{{{{{{\mathcal{T}}}}}}}}}^{{\prime} }$ and particle-hole symmetry ${{{{{{{\mathcal{P}}}}}}}}$ (such that ${{{{{{{{{\mathcal{T}}}}}}}}}^{{\prime} }}^{2}={{{{{{{{\mathcal{P}}}}}}}}}^{2}=+1$) and thus belongs to symmetry class BDI in Altland–Zirnbauer classification⁴⁰. In two space dimensions, BDI class admits only trivial topology implying that the STVS superconductor is topologically trivial.

Fig. 1: Emergence of chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$-wave superconductivity in maximally twisted double-layer spin-triplet valley-singlet (STVS) superconductors.

a Structure of the ${f}_{x({x}^{2}-3{y}^{2})}$-wave order parameter in the spin-triplet valley-singlet (STVS) superconductor. Gap nodes along Γ−M (dashed lines) are avoided by disconnected Fermi pockets around ± K resulting in a fully-gaped phase. b Schematic of a chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ phase formed by twisting two layers of STVS superconductors at θ ≃ 30^∘. Chiral Majorana edge modes (white arrows) emerge on the edge and a single Majorana zero mode denoted by $\hat{\gamma }$ (red dot) forms at the vortex core threaded by a unit flux Φ₀.

Here, we show that stacking two layers of STVS superconductor with an angular twist θ close to 30^∘ creates an intrinsic chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$-wave topological superconductor as the disconnected FSs around K-valleys in each isolated layer coalesce into a connected FS in the twisted double-layer under moiré band folding at large-angle twist. This ${{{{{{{{\mathcal{T}}}}}}}}}^{{\prime} }$-broken phase belongs to symmetry class D and admits nontrivial topology indicated by integer-valued Chern number C. Our results, based on symmetry analysis and detailed microscopic modeling, show that for θ ≃ 30^∘ ± 0.3^∘, the chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ phase occurs robustly throughout a wide range of electron density and, at exactly 30^∘ twist where the system has a high 12-fold quasi-crystalline symmetry, extends up to the native critical temperature of the double-layer superconductor. Within the chemical potential range where the disconnected K-pockets from the two layers merge into a single FS, we find C = ± 3, indicating non-Abelian topology manifested through an odd number of chiral Majorana modes on its edge and a single MZM in the core of its superconducting vortex (Fig. 1b). As θ deviates from 30^∘, the chiral topological phase evolves into a nodal topological ${f}_{x({x}^{2}-3{y}^{2})}$-wave superconductor, in which nodes of opposite chiralities in the bulk are connected by non-dispersive MZMs on one of the system edges, analogous to flat bands present on zigzag edges of monolayer graphene^41,42. Our results establish the large-twist-angle moiré physics, which is absent in twisted cuprates and also fundamentally different from the small-angle moiré physics in magic-angle twisted graphene (see the comparison in Supplementary Note 1), as a viable route toward non-Abelian TSC.

Results

Normal-state fermiology

For the sake of concreteness and simplicity, we describe the normal state of a monolayer STVS superconductor by a triangular lattice tight-binding model with nearest-neighbor electron hopping − t (Fig. 2a). Such Hamiltonians are widely used to model systems with hexagonal symmetry whose FS consist of disconnected segments around K-points^38,43 shown in Fig. 2b. Motivated by the phenomenology observed in RTG and BBG systems^34,35 and by theoretical ideas introduced in ref. ³⁸, we focus here on equal-spin pairing between electrons belonging to opposite valleys and drop the spin index in our discussions. Moreover, we consider hole doping near K-points by setting t > 0 with band maxima located at ±K given by ${E}_{\max }=3t$ in each isolated layer (the case of electron doping can be covered by setting t < 0 with ${E}_{\min }=3t$). Throughout this work, we follow the convention that k, R, G respectively denote the Bloch momenta, real-space lattice vectors, and reciprocal lattice vectors in layer 1, while $\tilde{{{{{{{{\bf{k}}}}}}}}}$, $\tilde{{{{{{{{\bf{R}}}}}}}}}$, $\tilde{{{{{{{{\bf{G}}}}}}}}}$ stand for their counterparts in layer 2. Given lattice vectors R⁽⁰⁾ and reciprocal lattice vectors G⁽⁰⁾ in an unrotated triangular lattice (open circles in Fig. 2a), we have R = R_z(θ/2)R⁽⁰⁾, $\tilde{{{{{{{{\bf{R}}}}}}}}}={R}_{z}(-\theta /2){{{{{{{{\bf{R}}}}}}}}}^{(0)}$ and G = R_z(θ/2)G⁽⁰⁾, $\tilde{{{{{{{{\bf{G}}}}}}}}}={R}_{z}(-\theta /2){{{{{{{{\bf{G}}}}}}}}}^{(0)}$, with R_z(θ) being the rotation of θ about the z-axis.

Fig. 2: Modeling normal-state fermiology in double-layer two-valley electron liquids with 30^∘ twist.

a Real-space triangular lattices for two layers of spin-triplet valley-singlet (STVS) superconductors stacked with angular twist θ. Dots in red (blue) denote lattice sites R ($\tilde{{{{{{{{\bf{R}}}}}}}}}$) in layer 1 (2) with lattice constant a = 2.46 Å. Hopping parameter t = 1 eV in all figures. b Fermi surface (FS) contours around ± K points of two decoupled layers at θ = 30^∘ for chemical potential 2.4t < μ < 3t. G₂, G₃ denote reciprocal lattice vectors in layer 1. (c) Dual-momentum space lattice points k_m (red dots) and ${\tilde{{{{{{{{\bf{p}}}}}}}}}}_{n}$ (blue dots) for a fixed momentum k₀ in units of 2π/a (see subsection “Dual momentum-space tight-binding model" in Methods). Red (blue) hexagons denote Wigner-Seitz cells in reciprocal lattice of layer 1 (2). Note that the original location of each k_m in the first Brillouin zone is exactly the location measured from the center of the red hexagon that contains it. Bloch momenta near ± K and $\pm \tilde{K}$ in (b) are covered by m, n = 1, 2, . . . , 6 (encircled dots) as k₀ varies in momentum-space. Inset: Fourier transform t_⊥(q) of inter-layer coupling, t_⊥(K) extrapolated at Ka/π = 4/3. (d) Band structure of ${{{{{{{{\mathcal{H}}}}}}}}}_{0,{{{{{{{\rm{eff}}}}}}}}}$ with topmost bands indexed by p = 1, 2, 3. Dashed lines indicate chemical potentials of FS contours shown in (e–h). Units of k_0,x, k_0,y axes are in Å⁻¹. Red (blue) ± symbols indicate signs of pairing in layer 1 (2).

Upon an angular twist θ, the normal-state Hamiltonians of the two decoupled layers are given by

$$\begin{array}{rlr}{{{{{{{{\mathcal{H}}}}}}}}}_{0}^{(1)}&=\mathop{\sum}\limits_{{{{{{{{\bf{k}}}}}}}}}{\xi }_{1}({{{{{{{\bf{k}}}}}}}}){c}_{1}^{{{{\dagger}}} }({{{{{{{\bf{k}}}}}}}}){c}_{1}({{{{{{{\bf{k}}}}}}}}),&\\ {{{{{{{{\mathcal{H}}}}}}}}}_{0}^{(2)}&=\mathop{\sum}\limits_{\tilde{{{{{{{{\bf{k}}}}}}}}}}{\xi }_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}}){c}_{2}^{{{{\dagger}}} }(\tilde{{{{{{{{\bf{k}}}}}}}}}){c}_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}}),\end{array}$$

(1)

where ${\xi }_{1}({{{{{{{\bf{k}}}}}}}})=-2t{\sum }_{j = 1,3,5}\cos ({{{{{{{\bf{k}}}}}}}}\cdot {{{{{{{{\bf{R}}}}}}}}}_{j})-\mu$ is the kinetic energy term in layer 1, and ${\xi }_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}})=-2t{\sum }_{j = 1,3,5}\cos (\tilde{{{{{{{{\bf{k}}}}}}}}}\cdot {\tilde{{{{{{{{\bf{R}}}}}}}}}}_{j})-\mu$ in layer 2, μ is the chemical potential. For μ ∈ (2t, 3t), the triangular lattice model produces disconnected Fermi pockets around K and −K points shown in Fig. 2b.

Studies of twisted 2D materials have established that the interlayer coupling within a twisted bilayer structure has the general form^{21,24,44,45,46}

$$\begin{array}{rlr}{{{{{{{{\mathcal{H}}}}}}}}}_{T}&=\mathop{\sum}\limits_{{{{{{{{\bf{k}}}}}}}},\tilde{{{{{{{{\bf{k}}}}}}}}}}\left[{c}_{1}^{{{{\dagger}}} }({{{{{{{\bf{k}}}}}}}})T({{{{{{{\bf{k}}}}}}}},\tilde{{{{{{{{\bf{k}}}}}}}}}){c}_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}})+{{{{{{{\rm{h.c.}}}}}}}}\right]&\\ T({{{{{{{\bf{k}}}}}}}},\tilde{{{{{{{{\bf{k}}}}}}}}})&=-\mathop{\sum}\limits_{{{{{{{{\bf{G}}}}}}}},\tilde{{{{{{{{\bf{G}}}}}}}}}}{t}_{\perp }({{{{{{{\bf{k}}}}}}}}+{{{{{{{\bf{G}}}}}}}}){\delta }_{{{{{{{{\bf{k}}}}}}}}+{{{{{{{\bf{G}}}}}}}},\tilde{{{{{{{{\bf{k}}}}}}}}}+\tilde{{{{{{{{\bf{G}}}}}}}}}},\end{array}$$

(2)

where t_⊥(q) is the Fourier transform of the inter-layer coupling t_⊥(r) as a function of spatial separation r between two atomic positions in different layers. It is worth noting that t_⊥(q) decays rapidly as a function of ∣q∣ in general and becomes negligibly small on the scale of q ≃ 2π/a where a denotes the monolayer lattice constant. To be concrete, we model t_⊥(r) by an empirical exponential formula describing σ-bonds formed by p_z-orbitals from the two layers (see Supplementary Note 2) and extrapolate an effective interlayer coupling strength of t_⊥(K) ≃ 0.15t for states near K-points (inset of Fig. 2c).

In twisted cuprates, superconductivity is borne out of large Fermi surfaces and Dirac nodes of the d-wave order parameter are located well inside the Brillouin zone of each layer. The leading-order inter-layer coupling according to Eq. (2) is simply the momentum-preserving term with ${{{{{{{\bf{k}}}}}}}}=\tilde{{{{{{{{\bf{k}}}}}}}}}$ and ${{{{{{{\bf{G}}}}}}}}=\tilde{{{{{{{{\bf{G}}}}}}}}}=0$. This allows treating the inter-layer coupling as a constant in the continuum model in which moiré effects are inessential³⁰. In contrast, superconductivity in an STVS superconductor emerges from two disconnected pockets surrounding the + K and − K points, and, as indicated in Fig. 2b, the leading-order inter-layer terms for a Bloch state with momentum k ≃ ± K in layer 1 include three different processes connecting it to states in layer 2 at $\tilde{{{{{{{{\bf{k}}}}}}}}}=({{{{{{{\bf{k}}}}}}}},{{{{{{{\bf{k}}}}}}}}+{{{{{{{{\bf{G}}}}}}}}}_{2},{{{{{{{\bf{k}}}}}}}}+{{{{{{{{\bf{G}}}}}}}}}_{3})$. The momentum transfer processes above are central ingredients in the celebrated Bistrizer-MacDonald (BM) model of small-angle twisted bilayer graphene²¹ and its variants for other twisted materials with two K-valleys^24,46, where the two valleys are modeled separately by valley-dependent low-energy effective Hamiltonians. However, the BM-type continuum model designed for the small twist-angle limit will fail to describe the maximally twisted double-layer STVS superconductor. This is because for θ ≃ 30^∘, the three different momenta $\tilde{{{{{{{{\bf{k}}}}}}}}}$ in layer 2 are located midway between the $+\tilde{K}$ and $-\tilde{K}$ points (Fig. 2b), where the low-energy Hamiltonian defined for a single valley is no longer valid.

To overcome this difficulty we develop our own method based on the dual momentum-space tight-binding (DMSTB) model (see Methods section) introduced in a recent theoretical study of quasi-crystalline electronic bands in 30^∘ twisted bilayer graphene⁴⁵. This approach was motivated and validated by experimental work^47,48. The authors showed that stacking two identical layers with honeycomb lattice geometries at an exact 30^∘ twist results in an extrinsic quasi-crystal with 12-fold tiling but no exact crystalline symmetries. They further argued that owing to the limited number of leading-order inter-layer processes discussed above, it is possible to construct an effective momentum-space Hamiltonian of a relatively small size. We adapted this method to our triangular lattice geometry to construct an effective Hamiltonian ${{{{{{{{\mathcal{H}}}}}}}}}_{0,{{{{{{{\rm{eff}}}}}}}}}$ based on the dual momentum-space lattice sites k_m=0,...,12 and ${\tilde{{{{{{{{\bf{p}}}}}}}}}}_{n = 0,...,12}$ (see subsection “Dual momentum-space tight-binding model" in Methods and Supplementary Note 3 for details) shown in Fig. 2c and generalized it to arbitrary twist angles close to 30^∘. Within this description moiré bands due to the large angular twist θ ≃ 30^∘ can be well defined up to the leading-order approximation, with k₀ serving as the momentum in the moiré Brillouin zone.

The moiré bands of ${{{{{{{{\mathcal{H}}}}}}}}}_{0,{{{{{{{\rm{eff}}}}}}}}}$ at θ = 30^∘ are solved by exact numerical diagonalization (topmost bands labeled by p = 1, 2, 3 shown in Fig. 2d), and the evolution of the Fermi surface upon increasing the doping level is presented in Fig. 2e–h. Due to level repulsion caused by inter-layer coupling, the band maxima at all K-points are shifted to ${E}_{\max }\simeq 3.06t$. At light hole doping μ > 3.01t, the Fermi surface consists of 12 disconnected pockets stemming from the 12 dual momentum-space sites in Fig. 2c, and resembles the disconnected Fermi surfaces in the decoupled limit (Fig. 2b). It is important to note that due to the moiré band-folding effects introduced by large angle twist, the Fermi surface in the twisted double layer quickly undergoes a Liftshitz transition as doping level increases and becomes connected already at μ ≃ 3.01t (Fig. 2f), at which point the Fermi pockets in the decoupled limit would still remain well isolated (Fig. 2b).

Upon further doping, the Fermi surface undergoes a second Liftshitz transition at μ ≃ 2.96t (Fig. 2g) and the system enters a regime with a single connected Fermi surface centered at the Γ point (Fig. 2h). Such FS then remains stable over a wide range of chemical potentials with higher hole doping (Fig. 2d). Crucially, as we show in the next section, at doping levels where a single connected FS exists, the twisted double-layer STVS material at θ ≃ 30^∘ becomes an intrinsic chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ superconductor with non-Abelian excitations.

Chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$-wave superconductivity at θ ≃ 30^∘

While microscopic mechanisms leading to STVS superconductivity may vary across materials such as RTG/BBG^36,37 and ZrNCl³⁸, on general grounds the interaction responsible for STVS pairing boils down to an effective attraction between electrons in the spin-triplet f-wave channel. In the momentum-space representation, the interaction within each isolated layer of STVS superconductor has the form

$$\begin{array}{rlr}{{{{{{{{\mathcal{V}}}}}}}}}^{(1)}&=-{U}_{0}\mathop{\sum}\limits_{{{{{{{{\bf{k}}}}}}}},{{{{{{{{\bf{k}}}}}}}}}^{{\prime} }}{f}_{1}({{{{{{{\bf{k}}}}}}}}){f}_{1}({{{{{{{{\bf{k}}}}}}}}}^{{\prime} }){c}^{{{{\dagger}}} }({{{{{{{\bf{k}}}}}}}}){c}^{{{{\dagger}}} }(-{{{{{{{\bf{k}}}}}}}})c(-{{{{{{{{\bf{k}}}}}}}}}^{{\prime} })c({{{{{{{{\bf{k}}}}}}}}}^{{\prime} }),&\\ {{{{{{{{\mathcal{V}}}}}}}}}^{(2)}&=-{U}_{0}\mathop{\sum}\limits_{\tilde{{{{{{{{\bf{k}}}}}}}}},{\tilde{{{{{{{{\bf{k}}}}}}}}}}^{{\prime} }}{f}_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}}){f}_{2}({\tilde{{{{{{{{\bf{k}}}}}}}}}}^{{\prime} }){c}^{{{{\dagger}}} }(\tilde{{{{{{{{\bf{k}}}}}}}}}){c}^{{{{\dagger}}} }(-\tilde{{{{{{{{\bf{k}}}}}}}}})c(-{\tilde{{{{{{{{\bf{k}}}}}}}}}}^{{\prime} })c({\tilde{{{{{{{{\bf{k}}}}}}}}}}^{{\prime} }),\end{array}$$

(3)

where U₀ denotes the interaction strength, and f_1,2 are the basis functions, ${f}_{1}({{{{{{{\bf{k}}}}}}}})={\sum }_{j = 1,3,5}\sin ({{{{{{{\bf{k}}}}}}}}\cdot {{{{{{{{\bf{R}}}}}}}}}_{j})$ and ${f}_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}})={\sum }_{j = 1,3,5}\sin (\tilde{{{{{{{{\bf{k}}}}}}}}}\cdot {\tilde{{{{{{{{\bf{R}}}}}}}}}}_{j})$, with exact f-wave symmetries shown in Fig. 1a. For isolated layer 1, one can define the self-consistent mean field ${{{\Delta }}}_{1}=-{U}_{0}{\sum }_{{{{{{{{{\bf{k}}}}}}}}}^{{\prime} }}{f}_{1}({{{{{{{{\bf{k}}}}}}}}}^{{\prime} })\left\langle c(-{{{{{{{{\bf{k}}}}}}}}}^{{\prime} })c({{{{{{{{\bf{k}}}}}}}}}^{{\prime} })\right\rangle$, and the corresponding gap function Δ₁(k) = Δ₁f₁(k). Similarly, ${{{\Delta }}}_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}})={{{\Delta }}}_{2}{f}_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}})$ for isolated layer 2. Note that Δ₁(k) and ${{{\Delta }}}_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}})$ are almost constant near ± K and $\pm \tilde{K}$ but exhibit a valley-dependent sign, as ${f}_{1}({{{{{{{\bf{k}}}}}}}}\simeq \pm {{{{{{{\bf{K}}}}}}}}),{f}_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}}\simeq \pm \tilde{{{{{{{{\bf{K}}}}}}}}})\simeq \mp \frac{3\sqrt{3}}{2}$.

As the disconnected K-pockets in two layers merge into a single connected Fermi surface in the twisted double-layer (Fig. 2h), the piecewise constant gap functions with alternating signs are transformed into continuous functions along the single circular Fermi contour in k₀-space. Signs of Δ₁(k) and ${{{\Delta }}}_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}})$, indicated by red and blue “ ± ” symbols in Fig. 2h, are seen to resemble two orthogonal f-wave components superimposed on top of each other. As the Fermi surface gets reconnected in k₀-space via inter-layer coupling, it is easy to see that nodes in each of the f-wave components are recovered – if time-reversal remains unbroken and hence the order parameters are real. This happens because the projected pairing on the Fermi surface from each layer must change continuously and a nodal point is mandated whenever a sign change in real order parameter occurs. In analogy with twisted d-wave superconductors³⁰, this suggests that, in order to avoid node formation and thus lower the overall superconducting free energy, the twisted STVS double-layer may develop a spontaneous complex phase difference between the order parameters of the two layers. This realizes the chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ phase.

In the following we support this intuitive picture of the chiral ${{{{{{{\mathcal{T}}}}}}}}$-broken phase formation with an explicit microscopic calculation. We note that the self-consistency of Δ₁(k) and ${{{\Delta }}}_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}})$ in isolated layer 1 and layer 2 implicitly rests upon the translational invariance within each decoupled layer. Upon introducing the inter-layer coupling, this translational symmetry is strongly modified. This forces us to reformulate the superconducting gap equations in terms of wave functions and energy bands derived from the DMSTB model ${{{{{{{{\mathcal{H}}}}}}}}}_{0,{{{{{{{\rm{eff}}}}}}}}}$ (see subsection “Mean-field gap equation for twisted double-layer STVS superconductors" of Methods section), so that effects from inter-layer coupling are properly incorporated. In the following, we focus on the topmost three bands indexed by p = 1, 2, 3 (inset of Fig. 2d) that are accessible by experimentally relevant doping levels.

In terms of fermionic operators ${a}_{p}^{{{{\dagger}}} }({{{{{{{{\bf{k}}}}}}}}}_{0})$ which create electrons at k₀ in band p, the Bogoliubov-de Gennes (BdG) Hamiltonian for the superconducting state in the twisted double-layer

$${{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{BdG}}}}}}}}} = \mathop{\sum}\limits_{p,{{{{{{{{\bf{k}}}}}}}}}_{0}}{\xi }_{p}({{{{{{{{\bf{k}}}}}}}}}_{0}){a}_{p}^{{{{\dagger}}} }({{{{{{{{\bf{k}}}}}}}}}_{0}){a}_{p}({{{{{{{{\bf{k}}}}}}}}}_{0})\\ + \mathop{\sum}\limits_{p,{{{{{{{{\bf{k}}}}}}}}}_{0}}\left[{{{\Delta }}}_{p}({{{{{{{{\bf{k}}}}}}}}}_{0}){a}_{p}^{{{{\dagger}}} }({{{{{{{{\bf{k}}}}}}}}}_{0}){a}_{p}^{{{{\dagger}}} }(-{{{{{{{{\bf{k}}}}}}}}}_{0})+{{{{{{{\rm{h.c.}}}}}}}}\right],$$

(4)

where ξ_p(k₀) = E_p(k₀) − μ with E_p(k₀) the kinetic energy of band p, and Δ_p(k₀) the pairing in band p: Δ_p(k₀) = Δ_1,pf_1,p(k₀) + Δ_2,pf_2,p(k₀), where f_l,p(k₀) are dimensionless basis functions characterizing the projected pairings in the moiré Brillouin zone for layer l and band p. The relations between f_l,p(k₀) and ${f}_{1}({{{{{{{\bf{k}}}}}}}}),{f}_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}})$ in Eq. (3) are explicitly given in subsection “Mean-field gap equation for twisted double-layer STVS superconductors" of Methods. Note that mean-fields Δ_1,p, Δ_2,p serve as the superconducting order parameters in moiré band p of layer 1 and layer 2, respectively.

To demonstrate that the projected pairings from layer 1 and 2 form two orthogonal f-wave components at θ = 30^∘, we plot the dimensionless basis functions f_l,p(k₀) for projected pairing in band p = 2 along the circular Fermi surface in the moiré Brillouin zone (Fig. 2d) at μ = 2.9t in Fig. 3a. Clearly, f_1,2(k₀) and f_2,2(k₀) have f-wave symmetries with 6 nodes, and the relative phase shift between the two is exactly $\delta {\phi }_{{{{{{{{{\bf{k}}}}}}}}}_{0}}=\pi /6$ as in two orthogonal f-wave components. Thus, the down-folded pairing interaction in the moiré bands of the twisted double-layer leads to the reconstruction of two orthogonal nodal f-wave order parameters in the moiré BZ, which provides the basis for the ${{{{{{{\mathcal{T}}}}}}}}$-broken chiral phase. Next, we solve for Δ_1,p and Δ_2,p by minimizing the free energy density

$$\begin{array}{r}{{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{SC}}}}}}}}}=\mathop{\sum}\limits_{l,p}\frac{| {{{\Delta }}}_{l,p}{| }^{2}}{{U}_{0}}-\frac{1}{V}\mathop{\sum}\limits_{s,{{{{{{{{\bf{k}}}}}}}}}_{0}}\frac{1}{2\beta }\ln (1+{{{{{{{{\rm{e}}}}}}}}}^{-\beta {E}_{s,{{{{{{{{\bf{k}}}}}}}}}_{0}}}),\end{array}$$

(5)

where V is the volume of the system, β = 1/k_BT (T: temperature), ${E}_{s,{{{{{{{{\bf{k}}}}}}}}}_{0}}$ are the eigenvalues of ${{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{BdG}}}}}}}}}$ (Eq. (4)). Given two identical layers of STVS superconductors, we have ∣Δ_1,p∣ = ∣Δ_2,p∣, and the general solution up to an overall phase is given by Δ_1,p = Δ₀, ${{{\Delta }}}_{2,p}={{{\Delta }}}_{0}{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{\varphi }_{p}}$ where we take Δ₀ to be real. To explore the resulting phase diagram in a concrete setting we set U₀ = 0.013t in Eq. (5) and μ = 2.9t. With t = 1 eV this yields T_c ≃ 3K and Δ₀ ≃ 1 meV at T = 0, almost independent of θ.

Fig. 3: Chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ superconductivity at 30^∘ twist.

a Basis functions f_1,p(k₀), f_2,p(k₀) of the projected pair wavefunctions in band p = 2 along the circular Fermi surface of twisted-double layer at μ = 2.9t with momentum radius ∣k₀∣ ≃ 0.11 Å⁻¹ shown in Fig. 2d. ${\phi }_{{{{{{{{{\bf{k}}}}}}}}}_{0}}$ is the polar angle of k₀ in the 2D plane. A relative shift of $\delta {\phi }_{{{{{{{{{\bf{k}}}}}}}}}_{0}}=\pi /6$ is found between f_1,p(k₀) and f_2,p(k₀) due to 30^∘-twist. b Phase diagram of a twisted double-layer spin-triplet valley-singlet (STVS) superconductor in the μ-θ plane. A robust chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ phase (regions depicted in blue) is found over the entire chemical potential range for θ ≃ 30^∘. c Evolution of phase dependence of the superconducting free energy ${{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{SC}}}}}}}}}(\varphi )$ as a function of twist angle θ at μ = 2.9t. Unit of y-axis set in eV. d Phase diagram in the T-θ plane obtained at μ = 2.9t and coupling constant U₀ = 0.013t, corresponding to critical temperature T_c ≃ 3K. e–g Bulk Bogoliubov excitation gap at (e) θ = 29. 5^∘, (f) θ = 30. 0^∘, and (g) θ = 30. 5^∘, corresponding to dots A, B, and C in (b), respectively. Δ₀ = 1 meV is used here in line with values of μ, U₀ used in (d).

The complete superconducting phase diagram in the μ − θ space is shown in Fig. 3b. For θ in close vicinity of 30^∘, the system develops a spontaneous ${{{{{{{\mathcal{T}}}}}}}}$-broken phase characterized by 0 < φ_p < π and becomes an intrinsic chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$-wave superconductor. It is noteworthy that the chiral phase persists over almost the entire chemical potential range that produces a connected FS, as shown in blue-shaded regions in Fig. 3b. At θ = 30^∘, the free energy ${{{{{{{{\mathcal{F}}}}}}}}}_{SC}$ is minimized exactly at φ_p = ± π/2 (red solid line in Fig. 3c) with Δ_2,p = ± iΔ_1,p, which corresponds to a perfect $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$-wave symmetry. As θ deviates from 30^∘, φ_p gradually evolves towards 0 or π for $\theta \le {\theta }_{{{{{{{{\rm{c}}}}}}}}}^{-}$ and $\theta \ge {\theta }_{{{{{{{{\rm{c}}}}}}}}}^{+}$, respectively (Fig. 3c), and the two layers of STVS superconductors eventually form a 0(π)-phase junction.

The bulk Bogoliubov excitation energy gaps along the circular Fermi surface at μ = 2.9t in different superconducting phases are shown in Fig. 3e–g. We find that for $\theta \, < \, {\theta }_{{{{{{{{\rm{c}}}}}}}}}^{-}$ and $\theta \, > \, {\theta }_{{{{{{{{\rm{c}}}}}}}}}^{+}$, the system in the ${{{{{{{\mathcal{T}}}}}}}}$-preserving phase is a nodal f-wave superconductor with 6 nodes along the Fermi surface. While in the ${{{{{{{\mathcal{T}}}}}}}}$-broken chiral regime, the system exhibits a full superconducting gap. In the next section, we demonstrate the nontrivial topological properties of the chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ phase as well as the nodal f-wave SC phase by studying their boundary and vortex core excitations.

Ginzburg–Landau theory

To understand why a robust chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ phase emerges at θ ≃ 30^∘, we construct a phenomenological Ginzburg–Landau (GL) theory in terms of the reconstructed ψ₁ ≡ Δ_1,p, ψ₂ ≡ Δ_2,p in moiré band p. Note that due to the reconstruction of pairing interactions in the moiré bands, the symmetry properties of ψ₁, ψ₂, as summarized in Table 1, do not directly follow from the f-wave symmetries in each isolated layer, but need to be derived from the defining equations for the basis functions f_1,2(k₀) and f_2,2(k₀) formulated in the moiré bands (Eq. (16) in the “Mean-field gap equation for twisted double-layer STVS superconductors" in Methods section). For a general twist angle θ, the double-layer has D₆ point group symmetry, which dictates the form of the GL free energy

$${{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{GL}}}}}}}}}[{\psi }_{1},{\psi }_{2}]=\mathop{\sum}\limits_{l=1,2}{\alpha }_{0}| {\psi }_{l}{| }^{2}+\frac{{\beta }_{0}}{2}| {\psi }_{l}{| }^{4}+{a}_{0}| {\psi }_{1}{| }^{2}| {\psi }_{2}{| }^{2}\\ +{b}_{0}({\psi }_{1}{\psi }_{2}^{* }+{{{{{{{\rm{c.c.}}}}}}}})+{c}_{0}({\psi }_{1}^{2}{\psi }_{2}^{* 2}+{{{{{{{\rm{c.c.}}}}}}}})$$

(6)

where the coefficients b₀, c₀ characterize the coherent tunneling of single and double Cooper pairs between the layers, respectively. Taking ψ ≡ ∣ψ₁∣ = ∣ψ₂∣ (see detailed analysis on the validity of this choice in Supplementary Note 4 and 5), this becomes ${{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{GL}}}}}}}}}(\varphi )={{{{{{{{\mathcal{F}}}}}}}}}_{0}+2{b}_{0}{\psi }^{2}\cos (\varphi )+2{c}_{0}{\psi }^{4}\cos (2\varphi )$, where φ is the phase difference between ψ₁ and ψ₂, and ${{{{{{{{\mathcal{F}}}}}}}}}_{0}=2{\alpha }_{0}{\psi }^{2}+({\beta }_{0}+{a}_{0}){\psi }^{4}$.

Table 1 Irreducible representations (ireps) of point group D₆ (θ ≠ 30^∘) and D_6d (θ = 30^∘).

At the maximal twist θ = 30^∘, the twisted double-layer becomes a quasi-crystal with 12-fold tiling as discussed in the Normal-state fermiology subsection. The symmetry of the quasi-crystal is described by the non-crystallographic D_6d point group, which includes an extra improper rotation S₁₂ ≡ σ_h ⊗ C₁₂, i.e., a 12-fold rotation about the z-axis combined with the reflection about the horizontal mirror plane lying mid-way between the two layers (see Fig. 2a). We deduce that under S₁₂: ψ₁ → − ψ₂, ψ₂ → ψ₁. For the GL free energy in Eq. (6) to be invariant under S₁₂, the single-pair tunneling term must vanish: b₀ = 0, and the only φ-dependent term is proportional to coefficient c₀ and exhibits $\cos (2\varphi )$ dependence. As argued in the twisted cuprate case, the coefficient c₀ associated with the double-pair tunneling is generally positive. In Supplementary Note 4 we further verify that c₀ > 0 generally holds in the case of twisted STVS superconductors by expanding the microscopic free energy Eq. (5) in terms of Δ_1,2 and Δ_2,2 using the imaginary-time path integral formalism. Thus, the quasi-crystalline D_6d symmetry dictates that the free energy is always minimized for ${\varphi }_{\min }=\pm \pi /2$ implying a state with spontaneously broken ${{{{{{{\mathcal{T}}}}}}}}$-symmetry.

The distinctive quasi-crystalline D_6d point group at θ = 30^∘ has important consequences for the temperature dependence of the chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ phase. It is worth noting that by construction for general θ (Fig. 2a-b), the basis functions f₁(k₀) and f₂(k₀) are always symmetric about the k_0,y = 0 axis when ${\phi }_{{{{{{{{{\bf{k}}}}}}}}}_{0}}=0,\pi$, and anti-symmetric about k_0,x = 0 when ${\phi }_{{{{{{{{{\bf{k}}}}}}}}}_{0}}=\pm \pi /2$ (see Fig. 3a). As such, the φ = 0 phase is essentially the ${f}_{x({x}^{2}-3{y}^{2})}$ pair function formed by the real combination ${f}_{1}({{{{{{{{\bf{k}}}}}}}}}_{0})+{f}_{2}({{{{{{{{\bf{k}}}}}}}}}_{0})\propto {k}_{x,0}({k}_{x,0}^{2}-3{k}_{y,0}^{2})$, which belongs to the 1D irreducible representation (irep) of D₆ labeled B₁ in Table 1. On the other hand, the φ = π phase corresponds to the ${f}_{y(3{x}^{2}-{y}^{2})}$ pair function formed by an orthogonal real linear combination ${f}_{1}({{{{{{{{\bf{k}}}}}}}}}_{0})-{f}_{2}({{{{{{{{\bf{k}}}}}}}}}_{0})\propto {k}_{y,0}(3{k}_{x,0}^{2}-{k}_{y,0}^{2})$, which belongs to another 1D irep of D₆ labeled B₂. Thus, for θ away from 30^∘, the two phases belonging to distinct ireps of D₆ have different ground state energies (see free energy landscapes at T = 0 in Fig. 3c for θ ≠ 30^∘) and correspond to different T_c.

At θ = 30^∘, however, the two orthogonal f-wave states with φ = 0 and φ = π are degenerate (see free energy landscape in Fig. 3c for θ = 30^∘), and form a 2D representation of D_6d, labeled E₃ in Table 1, with both components having the same T_c. Accordingly, as shown in the T − θ phase diagram Fig. 3d, obtained by minimizing ${{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{SC}}}}}}}}}$ at finite T, the chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ phase extends all the way to T = T_c for θ = 30^∘ because both orthogonal f-wave components condense simultaneously when superconductivity sets in at T_c. For θ away from 30^∘, the component with higher T_c (either φ = 0 or φ = π) sets in first, and one needs to further lower the temperature to access the other orthogonal component with lower T_c to form the chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ phase.

In the zero-temperature limit, the chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ phase extends between critical angles ${\theta }_{{{{{{{{\rm{c}}}}}}}}}^{-}\simeq 29.{7}^{\circ }-29.{8}^{\circ }$ and ${\theta }_{{{{{{{{\rm{c}}}}}}}}}^{+}\simeq 30.{2}^{\circ }-30.{3}^{\circ }$ (Fig. 3b-d). The overall twist angle range of δθ ≃ 0.4 − 0.6^∘ is well within reach of twist angle engineering precision ~ 0.1^∘ now common in the state-of-the-art sample fabrication technique for twisted materials^{22,23,25,26,27}. We note that the twist angle range predicted here is narrower than the chiral $d\pm {{{{{{{\rm{i}}}}}}}}{d}^{{\prime} }$ phase found in twisted cuprates which can span several degrees³⁰. As we explain in Supplementary Note 6, the relatively narrow twist angle range originates from the nontrivial θ-dependence of the reciprocal lattice vectors ${\tilde{{{{{{{{\bf{G}}}}}}}}}}_{m},{{{{{{{{\bf{G}}}}}}}}}_{n}$, for m, n ≠ 0 that determine the interlayer coupling strength, as opposed to the coupling dominated by ${\tilde{{{{{{{{\bf{G}}}}}}}}}}_{0}={{{{{{{{\bf{G}}}}}}}}}_{0}={{{{{{{\boldsymbol{0}}}}}}}}$ in cuprates which is, to good approximation, θ-independent.

Nodal topological f-wave phase

As we discussed in previous sections, for $\theta \, < \, {\theta }_{c}^{-}$ ($\theta \, > \, {\theta }_{c}^{+}$), the twisted double-layer favors the φ = 0 (φ = π) phase and becomes a nodal ${f}_{x({x}^{2}-3{y}^{2})}$-wave (${f}_{y(3{x}^{2}-{y}^{2})}$-wave) superconductor. This nodal phase is topologically nontrivial in the sense that the f-wave nodes are characterized by chirality numbers⁴⁹, and, in a geometry with edges, nodes of opposite chirality are connected by protected non-dispersive Majorana edge modes.

To understand the nontrivial topological property of the nodal f-wave phase, we consider the specific case with θ = 29.5^∘ and φ = 0 in band p = 2 corresponding to Fig. 3e. In the Nambu basis $\psi ({{{{{{{{\bf{k}}}}}}}}}_{0})={\left({a}_{2}({{{{{{{{\bf{k}}}}}}}}}_{0}),{a}_{2}^{{{{\dagger}}} }({{{{{{{{\bf{k}}}}}}}}}_{0})\right)}^{T}$, the bulk BdG Hamiltonian is written as ${{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{BdG}}}}}}}}}={\sum }_{{{{{{{{{\bf{k}}}}}}}}}_{0}}{\psi }^{{{{\dagger}}} }({{{{{{{{\bf{k}}}}}}}}}_{0}){H}_{{{{{{{{\rm{BdG}}}}}}}}}({{{{{{{{\bf{k}}}}}}}}}_{0})\psi ({{{{{{{{\bf{k}}}}}}}}}_{0})$, where H_BdG(k₀) = ξ₂(k₀)τ₃ + Δ₂(k₀)τ₁ with τ_α=1,2,3 as Pauli matrices acting on particle-hole space, and Δ₂(k₀) = Δ₀(f_1,2(k₀) + f_2,2(k₀)).

The Hamiltonian H_BdG respects a chiral symmetry ${{{{{{{\mathcal{C}}}}}}}}{H}_{{{{{{{{\rm{BdG}}}}}}}}}{{{{{{{{\mathcal{C}}}}}}}}}^{-1}=-{H}_{{{{{{{{\rm{BdG}}}}}}}}}$ where ${{{{{{{\mathcal{C}}}}}}}}={\tau }_{2}$. It can be represented in the eigenbasis of ${{{{{{{\mathcal{C}}}}}}}}$, by translating τ₃ ↦ τ₁, τ₁ ↦ τ₂. Then, in the vicinity of a nodal point k_0,N, the excitations can be described as 2D massless Dirac fermions

$${h}_{{{{{{{{\rm{N}}}}}}}}}({{{{{{{{\bf{p}}}}}}}}}_{0})\simeq {v}_{1}{p}_{0,1}{\tau }_{1}+{v}_{2}{p}_{0,2}{\tau }_{2},$$

(7)

where p_0,1 and p_0,2 denote the normal and tangential components along the Fermi surface of a momentum p₀ = k₀ − k_0,N. Dirac velocities ${v}_{1}\equiv {\nabla }_{{{{{{{{{\bf{k}}}}}}}}}_{0}}{\xi }_{2}({{{{{{{{\bf{k}}}}}}}}}_{0})\cdot {\hat{{{{{{{{\bf{n}}}}}}}}}}_{1}$ and ${v}_{2}\equiv {\nabla }_{{{{{{{{{\bf{k}}}}}}}}}_{0}}{{{\Delta }}}_{2}({{{{{{{{\bf{k}}}}}}}}}_{0})\cdot {\hat{{{{{{{{\bf{n}}}}}}}}}}_{2}$ are evaluated at k₀ = k_0,N. The chirality at k_0,N can then be defined as $C({{{{{{{{\bf{k}}}}}}}}}_{0,{{{{{{{\rm{N}}}}}}}}})={{{{{{{\rm{sgn}}}}}}}}({v}_{1}{v}_{2})\hat{{{{{{{{\bf{z}}}}}}}}}\cdot ({\hat{{{{{{{{\bf{n}}}}}}}}}}_{1}\times {\hat{{{{{{{{\bf{n}}}}}}}}}}_{2})$.

The chiralities C(k_0,N) of the 6 nodes in the φ = 0 phase are calculated from H_BdG(k₀) and indicated in Fig. 4a. Clearly, nodes with opposite chiralities come in three pairs, reflecting the underlying ${f}_{x({x}^{2}-3{y}^{2})}$-symmetry. It is worth noting that by projecting the bulk spectrum onto the edges oriented along certain high-symmetry directions, e.g. the y-direction, nodes with opposite chiralities do not cancel out. In this situation we expect protected non-dispersive edge modes to appear in analogy with the flat bands on zigzag edges of monolayer graphene⁴².

Fig. 4: Nontrivial topology of the nodal f-wave phase.

a Locations of f-wave nodes in the momentum space of k₀ = (k_0,x, k_0,y). Chiralities of nodes indicated by “ ± " signs in the φ = 0 phase with ${f}_{x({x}^{2}-3{y}^{2})}$-wave pairing symmetry. b Edge spectrum of the Bogoliubov-de-Gennes (BdG) Hamiltonian H_BdG with phase difference φ = 0 in a strip geometry as a function of k_y,0 along edges oriented in the y-direction, calculated using the lattice model in Eq. (8)–(9) with Δ_NN = 0 such that only the ${f}_{x({x}^{2}-3{y}^{2})}$-wave component is present. Inset: Nontrivial topology of nodal f-wave phase is characterized by the BDI invariant N_BDI as a function of k_y,0 (see Supplementary Note 7). The regions with N_BDI = + 1( − 1) correspond to regions with Majorana zero modes (MZMs) highlighted in red (blue) in (b). k_0,x, k_0,y are given in units of Å⁻¹.

To demonstrate the existence of edge states in the nodal f-wave phase, we now introduce a simplified triangular lattice model with lattice constant a₀ (shown schematically in Fig. 5a) for band p = 2 derived from the DMSTB model above (Fig. 2d). The lattice model captures both the parabolic dispersion near Γ and the f-wave pairing symmetry, thus facilitating explicit calculcations of edge states as well as vortex excitations of the chiral superconductor presented below. The lattice model is defined by

$${{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{LAT}}}}}}}}}=\mathop{\sum}\limits_{{{{{{{{\bf{q}}}}}}}}}\left[{\xi }_{{{{{{{{\bf{q}}}}}}}}}{c}_{{{{{{{{\bf{q}}}}}}}}}^{{{{\dagger}}} }{c}_{{{{{{{{\bf{q}}}}}}}}}+({{{\Delta }}}_{{{{{{{{\bf{q}}}}}}}}}{c}_{{{{{{{{\bf{q}}}}}}}}}^{{{{\dagger}}} }{c}_{-{{{{{{{\bf{q}}}}}}}}}^{{{{\dagger}}} }+{{{{{{{\rm{h.c.}}}}}}}})\right],$$

(8)

where ${\xi }_{{{{{{{{\bf{q}}}}}}}}}=-2{t}_{0} \, {\sum }_{j = 1,3,5}\cos ({{{{{{{\bf{q}}}}}}}}\cdot {{{{{{{{\boldsymbol{\delta }}}}}}}}}_{j})-{\mu }_{0}$ denotes the band energy from effective electron hopping − t₀, where we set t₀ ≃ 0.01t by fitting the parabolic dispersion at Γ in Fig. 2d. The gap function is given by

$${{{\Delta }}}_{{{{{{{{\bf{q}}}}}}}}}=2\mathop{\sum}\limits_{j=1,3,5}\left[{{{\Delta }}}_{{{{{{{{\rm{N}}}}}}}}}\sin ({{{{{{{\bf{q}}}}}}}}\cdot {{{{{{{{\boldsymbol{\delta }}}}}}}}}_{j})+{{{{{{{\rm{i}}}}}}}}{{{\Delta }}}_{{{{{{{{\rm{NN}}}}}}}}}\sin ({{{{{{{\bf{q}}}}}}}}\cdot {{{{{{{{\boldsymbol{\delta }}}}}}}}}_{j}^{{\prime} })\right]$$

(9)

with Δ_N (Δ_NN) denoting the first (second) nearest-neighbor pairing amplitudes as shown in Fig. 5a, while δ_j and ${{{{{{{{\boldsymbol{\delta }}}}}}}}}_{j}^{{\prime} }$ are the corresponding bond vectors indicated by red and purple lines, respectively. It is straightforward to check that in the small q expansion, the two pairing terms produce two orthogonal ${f}_{x({x}^{2}-3{y}^{2})}$- and ${f}_{y(3{x}^{2}-{y}^{2})}$-wave components, respectively.

Fig. 5: Exact diagonalization results for edge and vortex core Majorana modes.

a Pairing amplitudes for the f-wave order parameter on the simplified triangular lattice model with lattice constant a₀. Δ_N and Δ_NN denote the nearest-neighbor and second-nearest-neighbor pairing amplitudes, respectively. b Spectral function A(q_x, ω) evaluated at the edges and in the bulk of an infinite strip with width of L = 200 sites. ω is shown in units of t₀. c Local density of states ρ(ω, r) computed on a hexagon-shaped sample with N = 3169 sites and a vortex located at its center. d Real-space probability distribution of zero-energy eigenstates ${{{\Psi }}}_{0}^{\pm }=({{{\Psi }}}_{0}\pm {{{\Psi }}}_{0}^{{\prime} })/\sqrt{2}$. The color bar indicates the normalized value of probability distribution to the maximum site amplitude for each state. In all panels parameters $({\mu }_{0},{{{\Delta }}}_{N},{{{\Delta }}}_{NN})=(0.8{t}_{0},0.5{t}_{0},{{{\Delta }}}_{N}/3\sqrt{3})$ with t₀ = − 0.01 eV were used.

We use the lattice model in Eq. (8) with Δ_NN = 0 to calculate the edge spectrum of the nodal ${f}_{x({x}^{2}-3{y}^{2})}$-wave superconductor as a function of k_0,y (Fig. 4b), where we identify k_0,y as q_y in Eq. (8)–(9). As anticipated the bulk nodes with opposite chiralities are connected by non-dispersive zero energy modes on the edge (highlighted by red and blue lines in Fig. 4b). As we explain in Supplementary Note 7, for each fixed k_0,y, H_BdG(k_0,x) describes a one-dimensional BDI class topological superconductor oriented in the x-direction, with its topological property characterized by a winding number N_BDI^40,50,51 (inset of Fig. 4b). The bulk-edge correspondence between N_BDI and the number of zero energy modes allows us to establish the edge state associated with each k_y,0 as a MZM. We note that as long as the chiral symmetry ${{{{{{{\mathcal{C}}}}}}}}$ is respected, chiralities of nodes are well defined and act as topological charges that only annihiliate when opposite charges merge in the bulk. Therefore, the large number of non-dispersive MZMs on the edge are protected by the nontrivial bulk topology against ${{{{{{{\mathcal{C}}}}}}}}$-preserving perturbations such as charge disorder^52,53,54,55.

Majorana modes in chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ phase

In the ${{{{{{{\mathcal{T}}}}}}}}$-broken $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ phase, that we shall model by the lattice Hamiltonian Eq. (8) with Δ_NN ≠ 0, the 6 Dirac nodes are gapped out by alternating mass terms produced by the imaginary ${f}_{y(3{x}^{2}-{y}^{2})}$-wave component of the order parameter. This gapped phase belongs to symmetry class D in Altland-Zirnbauer classification^40,50,51 and its topology is therefore characterized by the Chern number C. In analogy with the $d+{{{{{{{\rm{i}}}}}}}}{d}^{{\prime} }$ phase in cuprates we expect each gapped Dirac point to contribute $\frac{1}{2}{{{{{{{\rm{sgn}}}}}}}}({{{\Delta }}}_{{{{{{{{\rm{NN}}}}}}}}})$ to the total Chern number, suggesting that the system will have C = ± 3 in the gapped chiral phase. Because the BdG representation of the spinless superconductor is redundant, the Chern number here determines the number of chiral Majorana edge modes with central charge 1/2 (as opposed to complex fermion modes whose central charge would be 1).

We can now confirm the existence of edge states by placing ${{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{LAT}}}}}}}}}$ on a strip geometry with periodic boundary conditions in the x-direction, and L rows of atoms in the y-direction. The spectral function in 1D momentum space, plotted in Fig. 5b, allows us to visualize the excitations present in each row of the strip. It is defined as

$$A({q}_{x},\omega )={{{{{{{\rm{Im}}}}}}}}{[(\omega +{{{{{{{\rm{i}}}}}}}}\eta )-{{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{LAT}}}}}}}}}({q}_{x})]}^{-1}$$

(10)

where η is a positive infinitesimal, and ${{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{LAT}}}}}}}}}({q}_{x})$ is the L × L Hamiltonian on the strip with q_x the lattice momentum along x. The spectral function reveals a fully gapped bulk and three distinct edge modes traversing the bulk gap, propagating in opposite directions at each edge. This confirms the Chern number C = 3 deduced above from general considerations.

In addition to gapless edge modes, chiral p-wave superconductors threaded with unit magnetic flux are predicted to host unpaired MZMs obeying non-Abelian exchange statistics, which are localized at vortex cores^7,8. To model the effect of an Abrikosov vortex in the $f+{{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ superconductor, we adopt a real-space representation of the lattice model in Eq. (8). We consider a hexagonal domain with open boundary conditions, and place a vortex at the origin. This induces a phase winding on the order parameter for each bond

$${{{\Delta }}}_{{{{{{{{\bf{r}}}}}}}},{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{j}} ={(-1)}^{j-1}{{{\Delta }}}_{{{{{{{{\rm{N}}}}}}}}}\exp (-{{{{{{{\rm{i}}}}}}}}n{\theta }_{{{{{{{{\bf{r}}}}}}}},{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{j}})\\ {{{\Delta }}}_{{{{{{{{\bf{r}}}}}}}},{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{j}^{{\prime} }} ={(-1)}^{j-1}{{{\Delta }}}_{{{{{{{{\rm{NN}}}}}}}}}\exp (-{{{{{{{\rm{i}}}}}}}}n{\theta }_{{{{{{{{\bf{r}}}}}}}},{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{j}^{{\prime} }})$$

(11)

where r denote lattice sites in real-space; n is the vorticity; and ${\theta }_{{{{{{{{\bf{r}}}}}}}},{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{j}}$ is the angle subtended by the midpoint of the bond (${{{{{{{\bf{r}}}}}}}}+\frac{1}{2}{{{{{{{{\boldsymbol{\delta }}}}}}}}}_{j}$), the origin, and the x-axis. We then numerically diagonalize the 2N × 2N matrix representing ${{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{LAT}}}}}}}}}$, where N is the number of lattice sites.

For a single-quantum vortex solution with n = 1, we indeed find a single zero-energy mode, which manifests as a zero-bias peak in the local density of states (LDOS) at the vortex core. Its partner lives at the sample edge, as shown in Fig. 5c. The LDOS is given by

$$\rho (\omega ,{{{{{{{\bf{r}}}}}}}})=\mathop{\sum}\limits_{{E}_{n} > 0}\left\{| {u}_{n}({{{{{{{\bf{r}}}}}}}}){| }^{2}\delta (\omega -{E}_{n})+| {v}_{n}({{{{{{{\bf{r}}}}}}}}){| }^{2}\delta (\omega +{E}_{n})\right\}$$

(12)

where E_n are eigenvalues of ${{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{LAT}}}}}}}}}$ with eigenstates ${{{\Psi }}}_{n}({{{{{{{\bf{r}}}}}}}})={({u}_{n}({{{{{{{\bf{r}}}}}}}}),{v}_{n}({{{{{{{\bf{r}}}}}}}}))}^{T}$. To confirm the nature of the two zero-energy states, denoted Ψ₀ and ${{{\Psi }}}_{0}^{{\prime} }$, we plot their real-space wavefunctions in Fig. 5d. The symmetric and anti-symmetric linear combination of these states are self-conjugate eigenstates of the charge conjugation operator, and represent Majorana zero modes localized at the sample edge and vortex core, respectively.

Discussion

Here we discuss how the exotic non-Abelian TSC phase can be realized in twisted double layers formed by the promising candidate materials, rhombohedral graphene^34,35 and ZrNCl³⁸, which are thought to be STVS superconductors. It is worth noting that the spinless-fermion triangular lattice model used for our theoretical considerations captures most of the essential features of the spin-triplet SC2 phase in rhobohedral trilayer graphene (RTG): (i) the SC2 phase emerges under strong displacement fields which polarize the layer and sublattice degrees of freedom, such that electrons involved in superconducting pairing actually live on an effective triangular lattice formed by the A (or B) sublattice³⁵; (ii) the Fermi surface of the parent spin-polarized valley-unpolarized normal state underlying the SC2 phase is well reproduced by our model (Fig. 2).

To realize the topological phase with non-Abelian excitations, it is crucial that the isolated K-pockets from each layer coalesce into a single connected Fermi surface in the moiré Brillouin zone. Our results based on the DMSTB model suggest that a minimal Fermi momentum k_Fa ~ 0.2 (a = 2.46 Å for graphene) measured from $\pm K(\tilde{K})$ is required for a single connected Fermi surface to emerge (see Fig. 2f). The typical value of Fermi momentum in RTG, corresponding to a low doping level with carrier density n_2D ≈ 0.5 × 10¹² cm⁻², was found to be of order k_Fa ~ 0.1³⁴, which is almost on par with the minimal requirement according to our theory. We note that higher doping levels, up to n_2D ~ 1 × 10¹² cm⁻², are indeed accessible through electrostatic gating^34,35, and the non-Abelian topological superconductivity could thus be achieved by further raising the doping level in maximally twisted double-layer RTG.

The triangular lattice model also captures the parabolic dispersion near ± K of the doped band insulator ZrNCl^56,57 and was in fact used to study the STVS pairing³⁸. In particular, ZrNCl superconducts within a wide range of electron doping x ~ 0.01 − 0.3^58,59,60 (x: number of electrons per unit cell). The simple parabolic dispersion near ± K allows us to extract a Fermi momentum ${k}_{{{{{{{{\rm{F}}}}}}}}}a=\sqrt{x\pi }\simeq 0.2-1.0$ with lattice constant a = 3.663 Å for ZrNCl, which suggests that the condition for a single connected FS in maximally twisted double-layer is readily fulfilled.

The f-wave pairing interaction in the non-magnetic ZrNCl may generally involve all three triplet channels given its spin-degenerate band structure, with the spinor part of the pair function characterized by a three-component d-vector ${\hat{{{\Delta }}}}_{t}=({{{{{{{\bf{d}}}}}}}}\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}){{{{{{{\rm{i}}}}}}}}{\sigma }_{y}$ (σ_α=x,y,z: Pauli matrices for spins). Given the D₆-point group of the twisted double-layer, however, the doublet formed by equal-spin states $\{\left\vert \uparrow \uparrow \right\rangle ,\left\vert \downarrow \downarrow \right\rangle \}$ and the anti-parallel state $\left\vert \uparrow \downarrow \right\rangle +\left\vert \downarrow \uparrow \right\rangle$ will in general be distinct in energies as they belong to different E₁ and A₂ ireps of D₆, respectively. In the E₁ phase with a two-component order parameter d = (d_x, d_y, 0), the total BdG Hamiltonian is decomposed into two independent spin sectors, each one described by the spinless model discussed in this work. Thus, our analysis directly applies and the system near the maximal twist becomes a spinful chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ superconductor. As degrees of freedom are doubled, there would be two species of chiral Majorana modes on the edge corresponding to three complex fermions; as well as two MZMs, one from each spin sector, localized at the single-vortex core. It is worth noting, however, as the two-component d-vector can rotate around the vortex core, the spinful chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ state admits half-quantum vortex (HQV) solutions trapping a π-flux. Following the analysis developed for spinful chiral p-wave superconductors in the context of Sr₂RuO₄ and ³He-A phase^7,61, the HQV is equivalent to a single-quantum vortex in one of the effective spin sectors and thus hosts a non-Abelian MZM.

In obtaining the phase diagrams in Fig. 3, we considered a relatively strong pairing interaction which yields a native T_c ≃ 3K and sizable pairing amplitude Δ₀ ≃ 1 meV for the twisted double-layer. While such temperature and energy scales are directly relevant to ZrNCl⁶⁰, the spin-triplet superconductivity in RTG and BBG is found with a much lower T_c ≃ 50 mK^34,35. It is important to note, however, that the emergence of the chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ phase follows from general symmetry principles as illustrated in our Ginzburg-Landau analysis. The proposed mechanism should therefore be largely insensitive to microscopic details and remains applicable to RTG and BBG. We further note that phase diagrams in Fig. 3 do not change qualitatively when the inter-layer coupling strength is varied, as long as it remains on the scale of t_⊥(K) ~ 0.1t. For weaker interlayer coupling, t_⊥(K) ≲ 0.01t, the energy bands become dense in energy space and the parameter regime with a single connected Fermi surface in Fig. 3b is reduced.

To detect the ${{{{{{{\mathcal{T}}}}}}}}$-broken chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ phase a suite of spectroscopic and transport experiments proposed for the chiral $p+{{{{{{{\rm{i}}}}}}}}{p}^{{\prime} }$ and $d+{{{{{{{\rm{i}}}}}}}}{d}^{{\prime} }$ phases in Sr₂RuO₄ and twisted cuprates can be applied. The nonzero orbital angular momentum L_z = ± 3 in the chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ phase can be probed by polar Kerr effect measurements^62,63, and the two-minimum free energy landscape near θ = 30^∘ shown in Fig. 3c will give rise to anomalous π-periodic inter-layer Josephson current-phase relation ${I}_{{{{{{{{\rm{J}}}}}}}}}=(2e/\hslash )\partial {{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{SC}}}}}}}}}/\partial \varphi \propto \sin (2\varphi )$³⁰. Upon tuning θ and T, the transition from fully gapped chiral phase to nodal f-wave phase (Fig. 3d–h) can be detected by a change from U-shaped to V-shaped spectra in the bulk LDOS, which can be probed by scanning tunneling microscopy (STM) measurements⁶⁴. Moreover, STM can be used to detect and resolve the spatial profile of the zero bias peaks induced by the MZM localized at the vortex core^16,17, as well as the non-dispersive edge MZMs in the nodal f-wave phase.

Conclusions

We established an avenue through twist-angle engineering toward intrinsic chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ TSC with non-Abelian excitations. In particular, the emergence of non-Abelian TSC in maximally twisted STVS superconductors relies on a special type of large-angle moiré physics which is absent in twisted cuprates and is fundamentally different from the moiré physics in small-angle twisted graphene (see detailed comparison between these systems in Supplementary Note 1). Our Ginzburg-Landau analysis reveals that the energetics leading to the exotic chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ phase are governed by a non-crystallographic D_6d symmetry, which emerges generically in the 12-fold quasi-crystalline structure formed at 30^∘ twist. By virtue of adiabatic continuity we expect the gapped topological phase to persist in a finite range δθ of twist angles around 30^∘ and our microscopic model indeed indicates stability for δθ ≃ 0. 4^∘ − 0. 6^∘, well within the capability of current sample fabrication techniques. We note that the possibility of chiral $f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }$ pairing was also suggested in a recent work on high-angular momentum-superconductivity in large-angle-twisted homobilayer systems⁶⁵.

The proposed mechanism applies in general to any two-valley material with hexagonal symmetry that exhibits gapped f-wave superconductivity in its monolayer form. The formation of a single connected FS in the twisted double-layer - an important prerequisite for non-Abelian excitations - requires states from the two different valleys $\pm K(\pm \tilde{K})$ to hybridize (Fig. 2c–h). Therefore, the large-angle moiré physics necessarily violates the fundamental valley conservation symmetry U_v(1) that underpins the well-established moiré physics in small-angle-twisted graphene and transition-metal dichalcogenides. This provides an alternative symmetry perspective on why the BM-type continuum models applicable in the small-angle limit, in which the U_v(1) is built-in by construction, fail to describe the moiré physics at maximal twist. As we discuss in detail in Supplementary Note 1, the U_v(1)-violation in maximally twisted double-layer is crucial for non-Abelian TSC because any description respecting the U_v(1) symmetry cannot support a single unpaired non-Abelian Majorana mode in a zero-momentum-paired superconductor, regardless of its pairing symmetry. This U_v(1)-based criterion reveals a profound connection between the large-angle moiré physics established in this work and ${{{{{{{\mathcal{T}}}}}}}}$-broken non-Abelian TSC.

While our assumption of a dominant pairing interaction in the f-wave channel for spin-triplet SC2 phase of RTG/BBG is supported by proposals based on acoustic phonons^36,37 and renormalization group analysis⁶⁶, some recent works suggest an alternative possibility of chiral $p\pm {{{{{{{\rm{i}}}}}}}}{p}^{{\prime} }$ pairing symmetry^67,68,69, which is also compatible with the phenomenology of the SC2 phase. In Supplementary Note 8 we present a detailed analysis of pairing instabilities in all possible pairing channels in maximally twisted double-layer RTG/BBG. Our microscopic calculations reveal that the chiral f-wave phase takes up the vast majority of the superconducting phase diagram, which lends strong support to the central idea developed in this work. In particular, we find that the chiral p-wave phase is favored only when the p-wave coupling constant is overwhelmingly larger than the coupling constant in the f-wave channel.

Our detailed calculations in addition show that under a dominant chiral p-wave interaction, the superconducting free energy and inter-layer Josephson current at θ = 30^∘ exhibits a 2π-periodicity in its φ-dependence, as opposed to the anomalous π-periodic dependence in the case of f-wave pairing (see Fig. 3c and Fig. S5 in Supplementary Note 8). These two contrasting behaviors can serve as experimental signatures discriminating between f-wave and chiral p-wave order parameters proposed for the SC2 phase in RTG. These results suggest potential applications of large-twist-angle engineering in probing the pairing symmetries of unconventional superconductors.

While the exact nature of the pairing symmetry of the SC2 spin-triplet phase remains to be settled by future experiments, it is important to note that even under the topological chiral $p\pm {{{{{{{\rm{i}}}}}}}}{p}^{{\prime} }$ pairing symmetry, an isolated monolayer still cannot support non-Abelian excitations due to the disconnected nature of its Fermi surface. On the other hand, maximally twisted double layer favors a configuration with the same p-wave chiralities in both layers. The resulting composite system then becomes a standard spinless chiral $p\pm {{{{{{{\rm{i}}}}}}}}{p}^{{\prime} }$ superconductor hosting non-Abelian MZMs when the disconnected FS in each layer coalesce into a single FS – a detailed discussion of this is given in Supplementary Note 8. Thus, the alternative assumption of chiral p-wave pairing does not alter our conclusion that a maximal twist is required to turn the system into a non-Abelian topological phase, which further fortifies the connection between the relatively unexplored large-angle moiré physics and non-Abelian topological superconductivity.

Methods

Dual momentum-space tight-binding (DMSTB) model

Here we briefly outline the basic idea behind our generalized DMSTB model and present a detailed derivation in Supplementary Note 3.

The DMSTB model is rooted in the observation that for any given momentum k₀, the sets of Bloch states involved in the inter-layer coupling Hamiltonian Eq. (2) are ${{{{{{{{\mathcal{S}}}}}}}}}_{1}({{{{{{{{\bf{k}}}}}}}}}_{0})=\{\left\vert {{{{{{{{\bf{k}}}}}}}}}_{0}+\tilde{{{{{{{{\bf{G}}}}}}}}},1\right\rangle ,\forall \tilde{{{{{{{{\bf{G}}}}}}}}}\}$ in layer 1, and ${{{{{{{{\mathcal{S}}}}}}}}}_{2}({{{{{{{{\bf{k}}}}}}}}}_{0})=\{\left\vert {{{{{{{{\bf{k}}}}}}}}}_{0}+{{{{{{{\bf{G}}}}}}}},2\right\rangle ,\forall {{{{{{{\bf{G}}}}}}}}\}$ in layer 2. By viewing the Bloch states $\left\vert {{{{{{{{\bf{k}}}}}}}}}_{m}={{{{{{{{\bf{k}}}}}}}}}_{0}+{\tilde{{{{{{{{\bf{G}}}}}}}}}}_{m},1\right\rangle \in {{{{{{{{\mathcal{S}}}}}}}}}_{1}({{{{{{{{\bf{k}}}}}}}}}_{0})$ and $\left\vert {\tilde{{{{{{{{\bf{k}}}}}}}}}}_{n}={{{{{{{{\bf{k}}}}}}}}}_{0}+{{{{{{{{\bf{G}}}}}}}}}_{n},2\right\rangle \in {{{{{{{{\mathcal{S}}}}}}}}}_{2}({{{{{{{{\bf{k}}}}}}}}}_{0})$ as “Wannier orbitals" localized at the dual momentum-space lattice sites k_m and ${\tilde{{{{{{{{\bf{p}}}}}}}}}}_{n}\equiv {{{{{{{{\bf{k}}}}}}}}}_{0}-{\tilde{{{{{{{{\bf{k}}}}}}}}}}_{n}\equiv -{{{{{{{{\bf{G}}}}}}}}}_{n}$, the inter-layer coupling in Eq. (2) can be regarded as ‘inter-site hopping’ between k_m and ${\tilde{{{{{{{{\bf{p}}}}}}}}}}_{n}$ with ‘hopping strength’ ${t}_{\perp ,mn}({{{{{{{{\bf{k}}}}}}}}}_{0})={t}_{\perp } ({{{{{{{{\bf{k}}}}}}}}}_{m}-{\tilde{{{{{{{{\bf{p}}}}}}}}}}_{n})$ determined precisely by the geometric distance $| {{{{{{{{\bf{k}}}}}}}}}_{m}-{\tilde{{{{{{{{\bf{p}}}}}}}}}}_{n}|$ (see Supplementary Note 3). The rapidly decaying character of t_⊥(q) implies weak hybridization among Wannier orbitals (Bloch states) at k_m and ${\tilde{{{{{{{{\bf{p}}}}}}}}}}_{n}$, and states in the twisted double-layer live predominantly only on a small number of k_m and ${\tilde{{{{{{{{\bf{p}}}}}}}}}}_{n}$ points. Thus, there exists an approximately closed finite subspace over which an effective Hamiltonian ${{{{{{{{\mathcal{H}}}}}}}}}_{0,{{{{{{{\rm{eff}}}}}}}}}$ can be constructed and the mapping from k₀ to the set of eigenvalues E_p(k₀) (p: band index) of ${{{{{{{{\mathcal{H}}}}}}}}}_{0,{{{{{{{\rm{eff}}}}}}}}}({{{{{{{{\bf{k}}}}}}}}}_{0})$ then defines the band structure and the Fermi surface reformulated in the k₀-space.

For θ ≃ 30^∘, states near ± K$(\tilde{K})$ in layer 1 (2) (Fig. 2b) are well covered by the 12 dual momentum-space lattice points k_m, m = 1, 2, . . . , 6 and ${\tilde{{{{{{{{\bf{p}}}}}}}}}}_{n},\,n=1,2,...,6$ indicated in Fig. 2c. This motivates us to consider ${{{{{{{{\mathcal{H}}}}}}}}}_{0,{{{{{{{\rm{eff}}}}}}}}}({{{{{{{{\bf{k}}}}}}}}}_{0})$ which includes these 12 sites together with leading-order corrections from their nearest-neighboring points ${{{{{{{{\bf{k}}}}}}}}}_{m},{\tilde{{{{{{{{\bf{p}}}}}}}}}}_{n}$ with m, n = 0, 7, . . . , 12 as illustrated in Fig. 2c. As further justified in Supplementary Note 3, the leading nearest-neighbor hopping terms between ${{{{{{{{\bf{k}}}}}}}}}_{m},{\tilde{{{{{{{{\bf{p}}}}}}}}}}_{n}$ in such approximation scheme accounts exactly for the leading-order inter-layer terms depicted in Fig. 2b. The effective normal-state Hamiltonian of a near-30^∘-twisted double-layer then reads ${{{{{{{{\mathcal{H}}}}}}}}}_{0,{{{{{{{\rm{eff}}}}}}}}}={\sum }_{{{{{{{{{\bf{k}}}}}}}}}_{0}}{{{{{{{{\mathcal{H}}}}}}}}}_{0}({{{{{{{{\bf{k}}}}}}}}}_{0})$ with

$${{{{{{{{\mathcal{H}}}}}}}}}_{0}({{{{{{{{\bf{k}}}}}}}}}_{0})= \mathop{\sum }\limits_{m=0}^{12}{\xi }_{1}({{{{{{{{\bf{k}}}}}}}}}_{m}){c}_{1}^{{{{\dagger}}} }({{{{{{{{\bf{k}}}}}}}}}_{m}){c}_{1}({{{{{{{{\bf{k}}}}}}}}}_{m})\\ +\mathop{\sum }\limits_{n=0}^{12}{\xi }_{2}({\tilde{{{{{{{{\bf{k}}}}}}}}}}_{n}){c}_{2}^{{{{\dagger}}} }({\tilde{{{{{{{{\bf{k}}}}}}}}}}_{n}){c}_{2}({\tilde{{{{{{{{\bf{k}}}}}}}}}}_{n})\\ -\mathop{\sum }\limits_{m,n=0}^{12}\left[{t}_{\perp }({{{{{{{{\bf{k}}}}}}}}}_{m}-{\tilde{{{{{{{{\bf{p}}}}}}}}}}_{n}){c}_{1}^{{{{\dagger}}} }({{{{{{{{\bf{k}}}}}}}}}_{m}){c}_{2}({\tilde{{{{{{{{\bf{k}}}}}}}}}}_{n})+{{{{{{{\rm{h.c.}}}}}}}}\right].$$

(13)

To verify that ${{{{{{{{\mathcal{H}}}}}}}}}_{0,{{{{{{{\rm{eff}}}}}}}}}$ in Eq. (13) provides an accurate description of the normal-state fermiology near the maximal twist, we note that the approach above applies to any twist angle close to 30^∘, including commensurate twist angles where the bilayer forms a periodic moiré superlattice and the band structure can be computed exactly via a real-space lattice model. As a convenient test case we consider commensurate angle ${\theta }_{c}=2{\sin }^{-1}(\sqrt{3}/(2\sqrt{13}))\approx 27.{8}^{\circ }$ which gives rise to a moiré unit cell with 26 sites. As we demonstrate in Supplementary Note 3, k₀ becomes exactly the crystal momentum at θ_c, and ${{{{{{{{\mathcal{H}}}}}}}}}_{0,{{{{{{{\rm{eff}}}}}}}}}$ reproduces the electronic bands of the moiré lattice model with excellent accuracy. We further note that the summation over k₀ in Eq. (13) should be restricted to within an area the size of the moiré Brillouin zone at θ = θ_c to avoid over-counting of the degrees of freedom.

We further note that inter-layer spatial displacements between two layers due to angular twist can generally introduce nonzero phases in the inter-layer tunneling term in Eq. (13)^24,46. As we explain in Supplementary Note 9, such phases cancel out in the total phase of Cooper pairs due to the spinless time-reversal symmetry ${{{{{{{{\mathcal{T}}}}}}}}}^{{\prime} }$ and thus do not affect our analysis on the superconducting phase.

Mean-field gap equation for twisted double-layer STVS superconductors

To derive the mean-field gap equations for the twisted double-layer, we first rewrite the total interaction ${{{{{{{{\mathcal{V}}}}}}}}}_{{{{{{{{\rm{tot}}}}}}}}}={{{{{{{{\mathcal{V}}}}}}}}}^{(1)}+{{{{{{{{\mathcal{V}}}}}}}}}^{(2)}$ in the band basis. Note that the fermionic operator creating an electron at k₀ in band p is given by

$${a}_{p}^{{{{\dagger}}} }({{{{{{{{\bf{k}}}}}}}}}_{0})=\mathop{\sum}\limits_{m}{u}_{pm}({{{{{{{{\bf{k}}}}}}}}}_{0}){c}^{{{{\dagger}}} }({{{{{{{{\bf{k}}}}}}}}}_{m})+\mathop{\sum}\limits_{n}{u}_{pn}({{{{{{{{\bf{k}}}}}}}}}_{0}){c}^{{{{\dagger}}} }({\tilde{{{{{{{{\bf{k}}}}}}}}}}_{n}),$$

(14)

where the coefficients u_pm(k₀), u_pn(k₀) can be found by exact numerical diagonalization of ${{{{{{{{\mathcal{H}}}}}}}}}_{0,{{{{{{{\rm{eff}}}}}}}}}$ in Eq. (13). Fermionic operators in ${{{{{{{{\mathcal{V}}}}}}}}}^{(1)}$ and ${{{{{{{{\mathcal{V}}}}}}}}}^{(2)}$ can be rewritten as ${c}^{{{{\dagger}}} }({{{{{{{{\bf{k}}}}}}}}}_{m})={\sum }_{p}{u}_{mp}^{* }({{{{{{{{\bf{k}}}}}}}}}_{0}){a}_{p}^{{{{\dagger}}} }({{{{{{{{\bf{k}}}}}}}}}_{0})$, ${c}^{{{{\dagger}}} }({\tilde{{{{{{{{\bf{k}}}}}}}}}}_{n})={\sum }_{p}{u}_{np}^{* }({{{{{{{{\bf{k}}}}}}}}}_{0}){a}_{p}^{{{{\dagger}}} }({{{{{{{{\bf{k}}}}}}}}}_{0})$. Using the one-to-one correspondence ${{{{{{{{\bf{k}}}}}}}}}_{m}\equiv {{{{{{{{\bf{k}}}}}}}}}_{0}+{\tilde{{{{{{{{\bf{G}}}}}}}}}}_{m}$ and ${\tilde{{{{{{{{\bf{k}}}}}}}}}}_{n}\equiv {{{{{{{{\bf{k}}}}}}}}}_{0}+{{{{{{{{\bf{G}}}}}}}}}_{n}$, the pairing interaction in the band basis becomes

$${{{{{{{{\mathcal{V}}}}}}}}}_{{{{{{{{\rm{eff}}}}}}}}}=-{U}_{0}({F}_{1}^{{{{\dagger}}} }{F}_{1}+{F}_{2}^{{{{\dagger}}} }{F}_{2}),$$

(15)

where ${F}_{l}^{{{{\dagger}}} }\equiv {\sum }_{{{{{{{{{\bf{k}}}}}}}}}_{0},p} \, {f}_{l,p} ({{{{{{{{\bf{k}}}}}}}}}_{0}){a}_{p}^{{{{\dagger}}} }({{{{{{{{\bf{k}}}}}}}}}_{0}){a}_{p}^{{{{\dagger}}} }(-{{{{{{{{\bf{k}}}}}}}}}_{0})$ is the pair creation operator in layer l = 1, 2, with

$$\begin{array}{rlr}{f}_{1,p}({{{{{{{{\bf{k}}}}}}}}}_{0})&=\mathop{\sum}\limits_{m}{{{\Lambda }}}_{m,p}({{{{{{{{\bf{k}}}}}}}}}_{0}){f}_{1}({{{{{{{{\bf{k}}}}}}}}}_{0}+{\tilde{{{{{{{{\bf{G}}}}}}}}}}_{m}),&\\ {f}_{2,p}({{{{{{{{\bf{k}}}}}}}}}_{0})&=\mathop{\sum}\limits_{n}{{{\Lambda }}}_{n,p}({{{{{{{{\bf{k}}}}}}}}}_{0}){f}_{2}({{{{{{{{\bf{k}}}}}}}}}_{0}+{{{{{{{{\bf{G}}}}}}}}}_{n}).\end{array}$$

(16)

Here, ${{{\Lambda }}}_{m,p}({{{{{{{{\bf{k}}}}}}}}}_{0})\equiv {u}_{m,p}^{* }({{{{{{{{\bf{k}}}}}}}}}_{0}){u}_{-m,p}^{* }(-{{{{{{{{\bf{k}}}}}}}}}_{0})$ and ${{{\Lambda }}}_{n,p}({{{{{{{{\bf{k}}}}}}}}}_{0})\equiv {u}_{n,p}^{* }({{{{{{{{\bf{k}}}}}}}}}_{0}){u}_{-n,p}^{* }(-{{{{{{{{\bf{k}}}}}}}}}_{0})$ denote the form factors arising generally from projecting the interaction onto the band basis, and we introduced the shorthand notation ${u}_{-m,p}^{* }(-{{{{{{{{\bf{k}}}}}}}}}_{0})$ to denote the coefficient associated with c^†( − k_m) and ${a}_{p}^{{{{\dagger}}} }(-{{{{{{{{\bf{k}}}}}}}}}_{0})$. The standard mean-field reduction for Eq. (15) leads to the gap equation ${{{\Delta }}}_{l,p}\equiv -{U}_{0}\left\langle {F}_{l,p}\right\rangle$ for the pairing Δ_l,p in the BdG Hamiltonian in Eq. (4) for the superconducting state in the twisted double-layer, and Δ_l,p are obtained by minimizing ${{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{SC}}}}}}}}}$ in Eq. (5).