Abstract
Recent studies point to an exotic spintriplet valleysinglet (STVS) superconducting phase in certain twovalley 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 quasicrystal with 12fold tiling. The resulting composite hosts an odd number of chiral Majorana edge modes and a single nonAbelian Majorana zero mode (MZM) in the vortex core. Through detailed symmetry analysis and microscopic modelling, we demonstrate that the nonAbelian 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 largetwistangle engineering, with distinct underlying moiré physics from magicangle graphene, as a viable route toward nonAbelian TSC.
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Introduction
Being of fundamental interest and potential use for topological qubits, the search for topological superconductors hosting excitations with nonAbelian 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 pwave superconductors^{6,7,8}, but the lack of intrinsic pwave 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 pwave superconductors and MZMs have been realized in recent experiments is still under debate^{18,19,20}. The quest for an intrinsic topological superconductor with nonAbelian 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 highT_{c} cuprate superconductor with nodal \({d}_{{x}^{2}{y}^{2}}\) pairing symmetry, such as Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}, stacked with a relative angular twist θ^{30,31}. At θ ≃ 45^{∘}, the bilayer is predicted to enter a fullygapped, topological chiral \(d\pm {{{{{{{\rm{i}}}}}}}}{d}^{{\prime} }\) phase with spontaneously broken timereversal symmetry \({{{{{{{\mathcal{T}}}}}}}}\), which has been tentatively identified in a recent experimental study^{32}. Despite its promise for realizing highT_{c} TSC, a chiral \(d\pm {{{{{{{\rm{i}}}}}}}}{d}^{{\prime} }\) superconductor cannot host truly nonAbelian excitations due to its spinsinglet pairing nature, which is always associated with an even number of MZMs that combine to form usual Abelian fermions. Generalizing the scheme to spintriplet superconductors with nodal pwave or fwave order parameters one can create chiral SC phases with an odd number of MZMs in principle^{33}, while once again facing the scarcity of nodal pwave and fwave superconductors in nature.
Recent progress in superconducting twodimensional materials, however, has uncovered a growing amount of evidence for fwave spintriplet 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 gatetuned regions, where the peculiar SC2 superconducting phase was borne out of a spinpolarized, valleyunpolarized normal metal^{34}. Such an unusual normalstate fermiology strongly hints at its spintriplet pairing nature, which is further supported by the observation of an inplane 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 spintriplet fwave pairing^{36,37}. More recently, Crépel and Fu proposed that a ‘threeparticle’ mechanism involving virtual excitons generically gives rise to spintriplet fwave pairing in a twovalley electron liquid formed in doped insulators such as ZrNCl^{38}, which provides a plausible explanation for the puzzling doping dependence of the gap structure revealed by early specific heat measurements on Lidoped ZrNCl^{39}.
In the scenarios described above, the parent normalstate 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 spintriplet valleysinglet (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 fwave gap function (located along Γ − M lines) never intersect the disconnected FS. The gapped phase respects a spinless timereversal symmetry \({{{{{{{{\mathcal{T}}}}}}}}}^{{\prime} }\) and particlehole symmetry \({{{{{{{\mathcal{P}}}}}}}}\) (such that \({{{{{{{{{\mathcal{T}}}}}}}}}^{{\prime} }}^{2}={{{{{{{{\mathcal{P}}}}}}}}}^{2}=+1\)) and thus belongs to symmetry class BDI in Altland–Zirnbauer classification^{40}. In two space dimensions, BDI class admits only trivial topology implying that the STVS superconductor is topologically trivial.
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 Kvalleys in each isolated layer coalesce into a connected FS in the twisted doublelayer under moiré band folding at largeangle twist. This \({{{{{{{{\mathcal{T}}}}}}}}}^{{\prime} }\)broken phase belongs to symmetry class D and admits nontrivial topology indicated by integervalued 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 12fold quasicrystalline symmetry, extends up to the native critical temperature of the doublelayer superconductor. Within the chemical potential range where the disconnected Kpockets from the two layers merge into a single FS, we find C = ± 3, indicating nonAbelian 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 nondispersive MZMs on one of the system edges, analogous to flat bands present on zigzag edges of monolayer graphene^{41,42}. Our results establish the largetwistangle moiré physics, which is absent in twisted cuprates and also fundamentally different from the smallangle moiré physics in magicangle twisted graphene (see the comparison in Supplementary Note 1), as a viable route toward nonAbelian TSC.
Results
Normalstate fermiology
For the sake of concreteness and simplicity, we describe the normal state of a monolayer STVS superconductor by a triangular lattice tightbinding model with nearestneighbor electron hopping − t (Fig. 2a). Such Hamiltonians are widely used to model systems with hexagonal symmetry whose FS consist of disconnected segments around Kpoints^{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 equalspin pairing between electrons belonging to opposite valleys and drop the spin index in our discussions. Moreover, we consider hole doping near Kpoints 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, realspace 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 zaxis.
Upon an angular twist θ, the normalstate Hamiltonians of the two decoupled layers are given by
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}
where t_{⊥}(q) is the Fourier transform of the interlayer 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 σbonds 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 Kpoints (inset of Fig. 2c).
In twisted cuprates, superconductivity is borne out of large Fermi surfaces and Dirac nodes of the dwave order parameter are located well inside the Brillouin zone of each layer. The leadingorder interlayer coupling according to Eq. (2) is simply the momentumpreserving term with \({{{{{{{\bf{k}}}}}}}}=\tilde{{{{{{{{\bf{k}}}}}}}}}\) and \({{{{{{{\bf{G}}}}}}}}=\tilde{{{{{{{{\bf{G}}}}}}}}}=0\). This allows treating the interlayer coupling as a constant in the continuum model in which moiré effects are inessential^{30}. 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 leadingorder interlayer 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 BistrizerMacDonald (BM) model of smallangle twisted bilayer graphene^{21} and its variants for other twisted materials with two Kvalleys^{24,46}, where the two valleys are modeled separately by valleydependent lowenergy effective Hamiltonians. However, the BMtype continuum model designed for the small twistangle limit will fail to describe the maximally twisted doublelayer 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 lowenergy Hamiltonian defined for a single valley is no longer valid.
To overcome this difficulty we develop our own method based on the dual momentumspace tightbinding (DMSTB) model (see Methods section) introduced in a recent theoretical study of quasicrystalline electronic bands in 30^{∘} twisted bilayer graphene^{45}. 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 quasicrystal with 12fold tiling but no exact crystalline symmetries. They further argued that owing to the limited number of leadingorder interlayer processes discussed above, it is possible to construct an effective momentumspace 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 momentumspace lattice sites k_{m=0,...,12} and \({\tilde{{{{{{{{\bf{p}}}}}}}}}}_{n = 0,...,12}\) (see subsection “Dual momentumspace tightbinding 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 leadingorder approximation, with k_{0} 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 interlayer coupling, the band maxima at all Kpoints 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 momentumspace 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 bandfolding 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 doublelayer STVS material at θ ≃ 30^{∘} becomes an intrinsic chiral \(f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }\) superconductor with nonAbelian 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 ZrNCl^{38}, on general grounds the interaction responsible for STVS pairing boils down to an effective attraction between electrons in the spintriplet fwave channel. In the momentumspace representation, the interaction within each isolated layer of STVS superconductor has the form
where U_{0} 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 fwave symmetries shown in Fig. 1a. For isolated layer 1, one can define the selfconsistent 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 Δ_{1}(k) = Δ_{1}f_{1}(k). Similarly, \({{{\Delta }}}_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}})={{{\Delta }}}_{2}{f}_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}})\) for isolated layer 2. Note that Δ_{1}(k) and \({{{\Delta }}}_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}})\) are almost constant near ± K and \(\pm \tilde{K}\) but exhibit a valleydependent 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 Kpockets in two layers merge into a single connected Fermi surface in the twisted doublelayer (Fig. 2h), the piecewise constant gap functions with alternating signs are transformed into continuous functions along the single circular Fermi contour in k_{0}space. Signs of Δ_{1}(k) and \({{{\Delta }}}_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}})\), indicated by red and blue “ ± ” symbols in Fig. 2h, are seen to resemble two orthogonal fwave components superimposed on top of each other. As the Fermi surface gets reconnected in k_{0}space via interlayer coupling, it is easy to see that nodes in each of the fwave components are recovered – if timereversal 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 dwave superconductors^{30}, this suggests that, in order to avoid node formation and thus lower the overall superconducting free energy, the twisted STVS doublelayer 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 selfconsistency of Δ_{1}(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 interlayer 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 “Meanfield gap equation for twisted doublelayer STVS superconductors" of Methods section), so that effects from interlayer 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 k_{0} in band p, the Bogoliubovde Gennes (BdG) Hamiltonian for the superconducting state in the twisted doublelayer
where ξ_{p}(k_{0}) = E_{p}(k_{0}) − μ with E_{p}(k_{0}) the kinetic energy of band p, and Δ_{p}(k_{0}) the pairing in band p: Δ_{p}(k_{0}) = Δ_{1,p}f_{1,p}(k_{0}) + Δ_{2,p}f_{2,p}(k_{0}), where f_{l,p}(k_{0}) 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}(k_{0}) and \({f}_{1}({{{{{{{\bf{k}}}}}}}}),{f}_{2}(\tilde{{{{{{{{\bf{k}}}}}}}}})\) in Eq. (3) are explicitly given in subsection “Meanfield gap equation for twisted doublelayer STVS superconductors" of Methods. Note that meanfields 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 fwave components at θ = 30^{∘}, we plot the dimensionless basis functions f_{l,p}(k_{0}) 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}(k_{0}) and f_{2,2}(k_{0}) have fwave 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 fwave components. Thus, the downfolded pairing interaction in the moiré bands of the twisted doublelayer leads to the reconstruction of two orthogonal nodal fwave 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
where V is the volume of the system, β = 1/k_{B}T (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} = Δ_{0}, \({{{\Delta }}}_{2,p}={{{\Delta }}}_{0}{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{\varphi }_{p}}\) where we take Δ_{0} to be real. To explore the resulting phase diagram in a concrete setting we set U_{0} = 0.013t in Eq. (5) and μ = 2.9t. With t = 1 eV this yields T_{c} ≃ 3K and Δ_{0} ≃ 1 meV at T = 0, almost independent of θ.
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 blueshaded 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 fwave 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 fwave 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 ψ_{1} ≡ Δ_{1,p}, ψ_{2} ≡ Δ_{2,p} in moiré band p. Note that due to the reconstruction of pairing interactions in the moiré bands, the symmetry properties of ψ_{1}, ψ_{2}, as summarized in Table 1, do not directly follow from the fwave symmetries in each isolated layer, but need to be derived from the defining equations for the basis functions f_{1,2}(k_{0}) and f_{2,2}(k_{0}) formulated in the moiré bands (Eq. (16) in the “Meanfield gap equation for twisted doublelayer STVS superconductors" in Methods section). For a general twist angle θ, the doublelayer has D_{6} point group symmetry, which dictates the form of the GL free energy
where the coefficients b_{0}, c_{0} characterize the coherent tunneling of single and double Cooper pairs between the layers, respectively. Taking ψ ≡ ∣ψ_{1}∣ = ∣ψ_{2}∣ (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 ψ_{1} and ψ_{2}, and \({{{{{{{{\mathcal{F}}}}}}}}}_{0}=2{\alpha }_{0}{\psi }^{2}+({\beta }_{0}+{a}_{0}){\psi }^{4}\).
At the maximal twist θ = 30^{∘}, the twisted doublelayer becomes a quasicrystal with 12fold tiling as discussed in the Normalstate fermiology subsection. The symmetry of the quasicrystal is described by the noncrystallographic D_{6d} point group, which includes an extra improper rotation S_{12} ≡ σ_{h} ⊗ C_{12}, i.e., a 12fold rotation about the zaxis combined with the reflection about the horizontal mirror plane lying midway between the two layers (see Fig. 2a). We deduce that under S_{12}: ψ_{1} → − ψ_{2}, ψ_{2} → ψ_{1}. For the GL free energy in Eq. (6) to be invariant under S_{12}, the singlepair tunneling term must vanish: b_{0} = 0, and the only φdependent term is proportional to coefficient c_{0} and exhibits \(\cos (2\varphi )\) dependence. As argued in the twisted cuprate case, the coefficient c_{0} associated with the doublepair tunneling is generally positive. In Supplementary Note 4 we further verify that c_{0} > 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 imaginarytime path integral formalism. Thus, the quasicrystalline 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 quasicrystalline 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. 2ab), the basis functions f_{1}(k_{0}) and f_{2}(k_{0}) are always symmetric about the k_{0,y} = 0 axis when \({\phi }_{{{{{{{{{\bf{k}}}}}}}}}_{0}}=0,\pi\), and antisymmetric 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 D_{6} labeled B_{1} 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 D_{6} labeled B_{2}. Thus, for θ away from 30^{∘}, the two phases belonging to distinct ireps of D_{6} 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 fwave 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 E_{3} 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 fwave 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 zerotemperature 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. 3bd). 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 stateoftheart 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 degrees^{30}. As we explain in Supplementary Note 6, the relatively narrow twist angle range originates from the nontrivial θdependence 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, θindependent.
Nodal topological fwave phase
As we discussed in previous sections, for \(\theta \, < \, {\theta }_{c}^{}\) (\(\theta \, > \, {\theta }_{c}^{+}\)), the twisted doublelayer 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 fwave nodes are characterized by chirality numbers^{49}, and, in a geometry with edges, nodes of opposite chirality are connected by protected nondispersive Majorana edge modes.
To understand the nontrivial topological property of the nodal fwave 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}(k_{0}) = ξ_{2}(k_{0})τ_{3} + Δ_{2}(k_{0})τ_{1} with τ_{α=1,2,3} as Pauli matrices acting on particlehole space, and Δ_{2}(k_{0}) = Δ_{0}(f_{1,2}(k_{0}) + f_{2,2}(k_{0})).
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 τ_{3} ↦ τ_{1}, τ_{1} ↦ τ_{2}. Then, in the vicinity of a nodal point k_{0,N}, the excitations can be described as 2D massless Dirac fermions
where p_{0,1} and p_{0,2} denote the normal and tangential components along the Fermi surface of a momentum p_{0} = k_{0} − 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 k_{0} = 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}(k_{0}) 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 highsymmetry directions, e.g. the ydirection, nodes with opposite chiralities do not cancel out. In this situation we expect protected nondispersive edge modes to appear in analogy with the flat bands on zigzag edges of monolayer graphene^{42}.
To demonstrate the existence of edge states in the nodal fwave phase, we now introduce a simplified triangular lattice model with lattice constant a_{0} (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 fwave 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
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 − t_{0}, where we set t_{0} ≃ 0.01t by fitting the parabolic dispersion at Γ in Fig. 2d. The gap function is given by
with Δ_{N} (Δ_{NN}) denoting the first (second) nearestneighbor 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.
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 nondispersive 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 onedimensional BDI class topological superconductor oriented in the xdirection, with its topological property characterized by a winding number N_{BDI}^{40,50,51} (inset of Fig. 4b). The bulkedge 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 nondispersive 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 AltlandZirnbauer 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 xdirection, and L rows of atoms in the ydirection. 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
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 pwave superconductors threaded with unit magnetic flux are predicted to host unpaired MZMs obeying nonAbelian 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 realspace 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
where r denote lattice sites in realspace; 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 xaxis. We then numerically diagonalize the 2N × 2N matrix representing \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{LAT}}}}}}}}}\), where N is the number of lattice sites.
For a singlequantum vortex solution with n = 1, we indeed find a single zeroenergy mode, which manifests as a zerobias 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
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 zeroenergy states, denoted Ψ_{0} and \({{{\Psi }}}_{0}^{{\prime} }\), we plot their realspace wavefunctions in Fig. 5d. The symmetric and antisymmetric linear combination of these states are selfconjugate 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 nonAbelian TSC phase can be realized in twisted double layers formed by the promising candidate materials, rhombohedral graphene^{34,35} and ZrNCl^{38}, which are thought to be STVS superconductors. It is worth noting that the spinlessfermion triangular lattice model used for our theoretical considerations captures most of the essential features of the spintriplet 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) sublattice^{35}; (ii) the Fermi surface of the parent spinpolarized valleyunpolarized normal state underlying the SC2 phase is well reproduced by our model (Fig. 2).
To realize the topological phase with nonAbelian excitations, it is crucial that the isolated Kpockets 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_{F}a ~ 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 × 10^{12} cm^{−2}, was found to be of order k_{F}a ~ 0.1^{34}, which is almost on par with the minimal requirement according to our theory. We note that higher doping levels, up to n_{2D} ~ 1 × 10^{12} cm^{−2}, are indeed accessible through electrostatic gating^{34,35}, and the nonAbelian topological superconductivity could thus be achieved by further raising the doping level in maximally twisted doublelayer 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 pairing^{38}. 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 \({k}_{{{{{{{{\rm{F}}}}}}}}}a=\sqrt{x\pi }\simeq 0.21.0\) with lattice constant a = 3.663 Å for ZrNCl, which suggests that the condition for a single connected FS in maximally twisted doublelayer is readily fulfilled.
The fwave pairing interaction in the nonmagnetic ZrNCl may generally involve all three triplet channels given its spindegenerate band structure, with the spinor part of the pair function characterized by a threecomponent dvector \({\hat{{{\Delta }}}}_{t}=({{{{{{{\bf{d}}}}}}}}\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}){{{{{{{\rm{i}}}}}}}}{\sigma }_{y}\) (σ_{α=x,y,z}: Pauli matrices for spins). Given the D_{6}point group of the twisted doublelayer, however, the doublet formed by equalspin states \(\{\left\vert \uparrow \uparrow \right\rangle ,\left\vert \downarrow \downarrow \right\rangle \}\) and the antiparallel 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 E_{1} and A_{2} ireps of D_{6}, respectively. In the E_{1} phase with a twocomponent 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 singlevortex core. It is worth noting, however, as the twocomponent dvector can rotate around the vortex core, the spinful chiral \(f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }\) state admits halfquantum vortex (HQV) solutions trapping a πflux. Following the analysis developed for spinful chiral pwave superconductors in the context of Sr_{2}RuO_{4} and ^{3}HeA phase^{7,61}, the HQV is equivalent to a singlequantum vortex in one of the effective spin sectors and thus hosts a nonAbelian 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 Δ_{0} ≃ 1 meV for the twisted doublelayer. While such temperature and energy scales are directly relevant to ZrNCl^{60}, the spintriplet 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 GinzburgLandau 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 interlayer 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 Sr_{2}RuO_{4} 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 twominimum free energy landscape near θ = 30^{∘} shown in Fig. 3c will give rise to anomalous πperiodic interlayer Josephson currentphase 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 fwave phase (Fig. 3d–h) can be detected by a change from Ushaped to Vshaped spectra in the bulk LDOS, which can be probed by scanning tunneling microscopy (STM) measurements^{64}. 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 nondispersive edge MZMs in the nodal fwave phase.
Conclusions
We established an avenue through twistangle engineering toward intrinsic chiral \(f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }\) TSC with nonAbelian excitations. In particular, the emergence of nonAbelian TSC in maximally twisted STVS superconductors relies on a special type of largeangle moiré physics which is absent in twisted cuprates and is fundamentally different from the moiré physics in smallangle twisted graphene (see detailed comparison between these systems in Supplementary Note 1). Our GinzburgLandau analysis reveals that the energetics leading to the exotic chiral \(f\pm {{{{{{{\rm{i}}}}}}}}{f}^{{\prime} }\) phase are governed by a noncrystallographic D_{6d} symmetry, which emerges generically in the 12fold quasicrystalline 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 highangular momentumsuperconductivity in largeangletwisted homobilayer systems^{65}.
The proposed mechanism applies in general to any twovalley material with hexagonal symmetry that exhibits gapped fwave superconductivity in its monolayer form. The formation of a single connected FS in the twisted doublelayer  an important prerequisite for nonAbelian excitations  requires states from the two different valleys \(\pm K(\pm \tilde{K})\) to hybridize (Fig. 2c–h). Therefore, the largeangle moiré physics necessarily violates the fundamental valley conservation symmetry U_{v}(1) that underpins the wellestablished moiré physics in smallangletwisted graphene and transitionmetal dichalcogenides. This provides an alternative symmetry perspective on why the BMtype continuum models applicable in the smallangle limit, in which the U_{v}(1) is builtin 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 doublelayer is crucial for nonAbelian TSC because any description respecting the U_{v}(1) symmetry cannot support a single unpaired nonAbelian Majorana mode in a zeromomentumpaired superconductor, regardless of its pairing symmetry. This U_{v}(1)based criterion reveals a profound connection between the largeangle moiré physics established in this work and \({{{{{{{\mathcal{T}}}}}}}}\)broken nonAbelian TSC.
While our assumption of a dominant pairing interaction in the fwave channel for spintriplet SC2 phase of RTG/BBG is supported by proposals based on acoustic phonons^{36,37} and renormalization group analysis^{66}, 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 doublelayer RTG/BBG. Our microscopic calculations reveal that the chiral fwave 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 pwave phase is favored only when the pwave coupling constant is overwhelmingly larger than the coupling constant in the fwave channel.
Our detailed calculations in addition show that under a dominant chiral pwave interaction, the superconducting free energy and interlayer Josephson current at θ = 30^{∘} exhibits a 2πperiodicity in its φdependence, as opposed to the anomalous πperiodic dependence in the case of fwave pairing (see Fig. 3c and Fig. S5 in Supplementary Note 8). These two contrasting behaviors can serve as experimental signatures discriminating between fwave and chiral pwave order parameters proposed for the SC2 phase in RTG. These results suggest potential applications of largetwistangle engineering in probing the pairing symmetries of unconventional superconductors.
While the exact nature of the pairing symmetry of the SC2 spintriplet 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 nonAbelian excitations due to the disconnected nature of its Fermi surface. On the other hand, maximally twisted double layer favors a configuration with the same pwave chiralities in both layers. The resulting composite system then becomes a standard spinless chiral \(p\pm {{{{{{{\rm{i}}}}}}}}{p}^{{\prime} }\) superconductor hosting nonAbelian 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 pwave pairing does not alter our conclusion that a maximal twist is required to turn the system into a nonAbelian topological phase, which further fortifies the connection between the relatively unexplored largeangle moiré physics and nonAbelian topological superconductivity.
Methods
Dual momentumspace tightbinding (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 k_{0}, the sets of Bloch states involved in the interlayer 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 momentumspace lattice sites k_{m} and \({\tilde{{{{{{{{\bf{p}}}}}}}}}}_{n}\equiv {{{{{{{{\bf{k}}}}}}}}}_{0}{\tilde{{{{{{{{\bf{k}}}}}}}}}}_{n}\equiv {{{{{{{{\bf{G}}}}}}}}}_{n}\), the interlayer coupling in Eq. (2) can be regarded as ‘intersite 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 doublelayer 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 k_{0} to the set of eigenvalues E_{p}(k_{0}) (p: band index) of \({{{{{{{{\mathcal{H}}}}}}}}}_{0,{{{{{{{\rm{eff}}}}}}}}}({{{{{{{{\bf{k}}}}}}}}}_{0})\) then defines the band structure and the Fermi surface reformulated in the k_{0}space.
For θ ≃ 30^{∘}, states near ± K\((\tilde{K})\) in layer 1 (2) (Fig. 2b) are well covered by the 12 dual momentumspace 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 leadingorder corrections from their nearestneighboring 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 nearestneighbor hopping terms between \({{{{{{{{\bf{k}}}}}}}}}_{m},{\tilde{{{{{{{{\bf{p}}}}}}}}}}_{n}\) in such approximation scheme accounts exactly for the leadingorder interlayer terms depicted in Fig. 2b. The effective normalstate Hamiltonian of a near30^{∘}twisted doublelayer then reads \({{{{{{{{\mathcal{H}}}}}}}}}_{0,{{{{{{{\rm{eff}}}}}}}}}={\sum }_{{{{{{{{{\bf{k}}}}}}}}}_{0}}{{{{{{{{\mathcal{H}}}}}}}}}_{0}({{{{{{{{\bf{k}}}}}}}}}_{0})\) with
To verify that \({{{{{{{{\mathcal{H}}}}}}}}}_{0,{{{{{{{\rm{eff}}}}}}}}}\) in Eq. (13) provides an accurate description of the normalstate 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 realspace 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, k_{0} 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 k_{0} in Eq. (13) should be restricted to within an area the size of the moiré Brillouin zone at θ = θ_{c} to avoid overcounting of the degrees of freedom.
We further note that interlayer spatial displacements between two layers due to angular twist can generally introduce nonzero phases in the interlayer 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 timereversal symmetry \({{{{{{{{\mathcal{T}}}}}}}}}^{{\prime} }\) and thus do not affect our analysis on the superconducting phase.
Meanfield gap equation for twisted doublelayer STVS superconductors
To derive the meanfield gap equations for the twisted doublelayer, 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 k_{0} in band p is given by
where the coefficients u_{pm}(k_{0}), u_{pn}(k_{0}) 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 onetoone 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
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
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 meanfield 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 doublelayer, and Δ_{l,p} are obtained by minimizing \({{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{{{\rm{SC}}}}}}}}}\) in Eq. (5).
Data availability
The data that support the findings presented in the main text and the Supplementary Information are available from the corresponding author upon reasonable request.
Code availability
The computer codes that support the findings presented in the main text and the Supplementary Information are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors thank Oguzhan Can, Rafael Haenel, Christine AuYeung, Andrea Damascelli, Andreas Schnyder for illuminating discussions and communications. This work was supported by NSERC and the Canada First Research Excellence Fund, Quantum Materials and Future Technologies Program. B.T.Z. acknowledges the support of the Croucher Foundation.
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B.T.Z. and M.F. conceived the idea. B.T.Z. developed the generalized DMSTB model and performed the major part of the calculations on the normalstate fermiology and superconducting phase diagram. S.E. and D.K. contributed to the edge state and vortex state calculations. B.T.Z. wrote the manuscript with contributions from all authors; M.F. supervised the project.
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Zhou, B.T., Egan, S., Kush, D. et al. NonAbelian topological superconductivity in maximally twisted doublelayer spintriplet valleysinglet superconductors. Commun Phys 6, 47 (2023). https://doi.org/10.1038/s42005023011655
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DOI: https://doi.org/10.1038/s42005023011655
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