Microscopic Theory of Superconductivity in Twisted Double

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Microscopic pairing mechanism, order parameter, and disorder sensitivity in moiré superlattices: Applications to twisted double-bilayer graphene

Rhine Samajdar1 and Mathias S. Scheurer1 1Department of Physics, Harvard University, Cambridge, MA 02138, USA Starting from a continuum-model description, we develop a microscopic weak-coupling theory for superconductivity in twisted double-bilayer graphene. We study both electron-phonon and entirely electronic pairing mechanisms. In each case, the leading superconducting instability transforms under the trivial representation, A, of the point group C3 of the system, while the subleading pairing phases belong to the E channel. We explicitly compute the momentum dependence of the associated order parameters and find that the leading state has no nodal points for electron-phonon pairing but exhibits six sign changes on the Fermi surface if the Coulomb interaction dominates. On top of these system-specific considerations, we also present general results relevant to other correlated graphene- based moiré superlattice systems. We show that, irrespective of microscopic details, triplet pairing will be stabilized if the collective electronic fluctuations breaking the enhanced SU(2)+ × SU(2)− spin symmetry of these systems are odd under time reversal, even when the main SU(2)+ × SU(2)−- symmetric part of the pairing glue is provided by phonons. Furthermore, we discuss the disorder sensitivity of the candidate pairing states and demonstrate that the triplet phase is protected against disorder on the moiré scale.

I. INTRODUCTION erties and gap structures of the leading and subleading instabilities, and their doping dependencies are computed and discussed in relation to experiment. We also de- The family of moiré superlattice systems displaying scribe the possibility that the electron-phonon coupling correlated physics has been expanding rapidly [1]. One provides the main pairing glue but the dominant inter- such heterostructure that has generated much interest is actions that distinguish between singlet and triplet stem twisted double-bilayer graphene (TDBG) [2–5], in which from purely electronic processes. Additionally, we exam- two AB-stacked graphene bilayers are twisted relative to ine the disorder sensitivity of the different pairing states. each other. This system stands out not only due to the additional tunability of the band structure via electric This article is organized as follows. As a starting point, fields [6–12], but also because both the gap of the corre- in Sec.II, we introduce the continuum model for TDBG lated insulating state at half-filling and the critical tem- that serves as the basis for our microscopic computations. perature of the superconducting phase can be enhanced Next, in Secs.III andIV, we examine pairing mechanisms by application of a magnetic field, which hints that its based on electron-phonon and electron-electron interac- physics is different from that of twisted bilayer graphene tions, respectively, and evaluate the basis functions for (TBG) [13–16]. The superconducting phase of TDBG is the associated superconducting states. At the end of found to be very fragile [4,5]: it only emerges [2,3] on Sec.IV, we also discuss the interplay of the two pair- the electron-doped side relative to the insulator at half- ing mechanisms. Proceeding further, we also consider filling of the conduction band and quickly weakens away the impact of disorder on these states in Sec.V. Finally, from this optimal doping value. four appendices,A throughD, supplement our discussion in the aforementioned sections with the details of In a recent publication [17], we performed a systematic our calculations. classification of the possible pairing instabilities in TDBG and other related moiré superlattice systems, using general constraints based on symmetries and energetics. Here, we complement Ref. 17 with an explicit microscopic II. MODEL AND BAND STRUCTURE calculation using a continuum model for the underlying graphene sheets, inspired by theoretical studies of su- The TDBG heterostructure is composed of two indi- arXiv:2001.07716v3 [cond-mat.supr-con] 23 Jun 2020 perconductivity in TBG [18–30]. While there have been vidually aligned Bernal-stacked bilayer graphene (BLG) indications pointing towards an electron-phonon based sheets, twisted relative to each other by a small angle pairing mechanism in TBG [31, 32], recent experiments θ 1, forming a moiré superlattice. When the two [33] seem to challenge these conclusions. In any case, BLGs are decoupled, the spectrum of each exhibits a fa- the situation as regards TDBG is, at present, completely miliar pair of linearly dispersing Dirac cones [34, 35]; we open. As a first step towards understanding the pairing denote the location of the Dirac points of the l-th BLG l glue in TDBG, in this work, we analyze both electron- by Kξ, where ξ = ± labels the two valleys. Upon twist- phonon (conventional) and electron-electron (unconven- ing, one moves away from this limit and the superlattice tional) interactions-based pairing. The symmetry prop- structure now manifests itself in the coupling between 2

known to be important for isolating flat bands. More- over, on including terms accounting for trigonal warping [45] and particle-hole asymmetry [46], the first conduction and valence bands, around charge neutrality, always acquire a sizable dispersion and unlike TBG [34, 47, 48], there no longer exists a sharp “magic angle” at which they are almost perfectly flat. For small θ, these bands overlap with each other but they can be separated by a gate voltage applied between the top and bottom layers, or, in other words, a displacement field. The 10–15 meV bandwidth of the band hosting superconductivity, shown in green in Fig.1(a), is still smaller than the interaction scale [9], revealing the system’s strongly correlated nature. Adding to the distinction from TBG is the fact that the physics of TDBG is dominated by a single band (per spin per valley)—rather than two—as a consequence of the broken twofold rotational symmetry C2 [9], which formerly protected the Dirac points in TBG. Bearing in mind the experimental results, in the following, we focus on this narrow band for a specific twist angle, θ = 1.24◦, FIG. 1. (a) Band structure of TDBG for the ξ = + valley at which superconductivity was detected in TDBG sam- at a twist angle θ = 1.24◦ and a displacement field (the elec- ples [3]. The associated Fermi surfaces for different frac- trostatic energy difference between adjacent sheets) of d = 5 tional fillings of the relevant band, and the density of meV; the latter can be varied to controllably tune the dis- states (DOS) are sketched in Figs.1(b) and (c). As is persion. The band hosting superconductivity is indicated in often the case for graphene-based moiré systems [44, 49], green. (b) Fermi surfaces corresponding to five different fill- we note the existence of van Hove singularities in the ings of this band. (c) The density of states per spin per valley noninteracting DOS at the energies of the saddle points as a function of energy. in the band structure [50, 51]. the lower layer of the top BLG and the topmost sheet of III. CONVENTIONAL PAIRING the bottom stack. This hybridizes the eigenstates at a Bloch vector k in the moiré Brillouin zone (MBZ) with those related by reciprocal lattice vectors of the moiré To begin our analysis of superconductivity, we first superlattice. consider the option that the attractive interaction be- The resulting electronic band structure of TDBG is tween electrons is mediated by the phonon modes of the well described by a continuum model [10, 36–39], re- two-dimensional graphene layers [51]. This is a natural viewed in AppendixA. The key simplification achieved extension of the physics of magic-angle TBG in which the by such a continuum model vis-à-vis microscopic lattice phonon-driven electron-electron attraction, in combina- −1 −2 descriptions is that it restores periodicity [40], facilitat- tion with an enhanced DOS ∼ 10 eV nm [nearly ten ing the application of Bloch’s theorem to a system that, times larger than g (E) in Fig.1], has been predicted to for generic twist angles, will only be quasiperiodic. Our induce intervalley pairing [21–23, 52], yielding a critical low-energy Hamiltonian [Eq. (A1)] is comprised of block- temperature Tc ∼ 1 K[14]. Given that a similar enhance- diagonal elements, which are 4×4 matrices describing the ment of the DOS is lacking in TDBG due to the signifi- Bloch waves of individual graphene bilayers at different cantly larger bandwidth, one may question whether such momenta [41, 42]. The interlayer coupling breaks the a mechanism still holds in this system. translation invariance of the (AB-stacked) BLG unit cell and couples these blocks to one another. Nevertheless, since these terms still have the translational symmetry A. Electron-phonon interactions of the moiré supercell, it is convenient to transform the Hamiltonian from real space to a momentum basis. For To this end, we consider, in particular, acoustic phonon k in the MBZ, the eigenenergies Ek,ξ can be computed modes, prompted by previous works on TBG [23, 52]. independently for each ξ as the intervalley coupling is While the layer-symmetric modes (all four layers mov- negligible for small twist angles. ing together) and the modes where the two BLGs move For an accurate description of the band structure, we against each other (corresponding to a shift of the moiré incorporate lattice relaxation effects [43, 44], which are lattice) remain gapless, the phonons that shift any of 3

FIG. 2. Phonon-mediated superconductivity in TDBG. (a) The form factors associated with the projection of the densities on to the lowest band. In the q = 0 plane, the threefold rotational symmetry of the lattice is directly visible. (b) The four largest eigenvalues λ of the matrix M as a function of temperature T ; the subleading eigenvalues are doubly degenerate, as marked by ×2. (c) Dependence of the superconducting transition temperature on the chemical potential; dashed lines indicate the five integer fillings of Fig.1. (d) Basis functions of the leading (left) and subleading (middle and right) superconducting states at 2/3 filling. Note that the extrema of the order parameter trace out the Fermi surface in Fig.1(b). the sheets of the same BLG against each other acquire form of a density-density interaction, a mass. Concentrating on the former modes, we assume 2 2 that their velocity is the same, motivated by the expected 1 X X X s,s0 H = V 0 (q)% ˆ %ˆ 0 0 , (1) weak interlayer phonon coupling [53]. The electron- int N l,l q,l,s −q,l ,s q 0 0 phonon Hamiltonian [52] can, therefore, be written as l,l =1 s,s =1 a coupling of acoustic phonon modes, α , and the elec- q,l where %ˆq,l,s is the electronic density operator in sheet tron density ρˆq,l of the lower (l = 1) and upper (l = 2) s = 1, 2 of the l-th BLG at in-plane momentum q, i.e., BLG, P ρˆq,l = s %ˆq,l,s. For the case of the electron-phonon coupling discussed above, we obtain

s 2 2 0 D ω D X † s,s X q ~ V 0 (q) = −δl,l0 (2) HEP = √ (−iq.eˆq) αq,l + α−q,l ρˆq,l. l,l 2 2 2 2m ¯ ωq 2m ¯ vph Ω ωq + νn NΩ q,l n

with bosonic Matsubara frequencies νn. In order to study superconducting instabilities, we Here, m is the mass density, N and denote the num- ¯ Ω project this interaction to the partially-filled conduction ber and area of moiré unit cells, respectively, e is the ˆq band harboring superconductivity. Within our approxi- displacement unit vector, and ω v |q| the phonon fre- q = ph mations, the projected interaction and the noninteract- quency; we take v × 6 cm/s [52]. The deformation ph = 2 10 ing band structure of the system do not couple the spins potential D (∼ eV) is much larger than the interlayer 25 of electrons in different valleys. This enhances the spin tunneling parameter (∼ 0.1 eV) [52], which allows for ne- symmetry from the usual SU(2) to SU(2) × SU(2) , glecting the effects of the latter. The Debye frequency + − √ corresponding to independent spin rotation in the two of moiré phonon bands [23] is ∼ v |q |/ . meV, ph m 3 = 4 85 valleys. In this limit, singlet and triplet are exactly de- defining q ≡ |GM | with primitive vectors GM of the re- m 1,2 1,2 generate, which enables us to discuss both of them si- ciprocal moiré lattice. Thus, in contrast to TBG [22], the multaneously; the type of singlet or triplet pairing that system is not in the diabatic regime of ω E [54, 55]. q F arises when breaking the enhanced spin symmetry is com- Integrating out the phonon modes, one arrives, within pletely determined by symmetry and has been worked the conventional BCS approximation, at the phonon- out in Ref. [17]. The microscopic features of this projec- l,s mediated electron-electron interaction, which is of the tion are captured by the “form factors” Fk,q,ξ involved in 4 the projection of the density operator to the low-energy B. Results conduction bands, i.e., The linearized mean-field equation (4), which has X 1 %ˆ → c† c F l,s , the form of an eigenvalue problem of the kernel q,l,s N k,σ,ξ k+q,σ,ξ k,q,ξ k,σ,ξ M(T ) ∆ = λ(T ) ∆ in momentum space, is solved on a Monkhorst-Pack grid for the Brillouin zone. Estimating where c annihilates an electron in the conduction 2 2 2 k,σ,ξ D /(4m ¯ mvph) ' 82.3 meV nm [52, 58], we evaluate the band with momentum k and spin σ in valley ξ. In critical temperature Tc of superconductivity by determin- Fig.2(a), we show the combination of form factors ing when the largest of the eigenvalues {λn(T )} of M(T ) P P l,s 2 l | s Fk,q,+| that enters the mean-field equations for reaches 1; for example, at three-quarters filling, we ob- phonon-mediated pairing. tain Tc = 1.90 K [see Fig.2 (b)]. Interestingly, this value The full mean-field theory of superconductivity in is of the same order as the experimental observations and TDBG is developed in AppendixC; while we rele- suggests that electron-phonon coupling is strong enough gate the details to that section, let us highlight here to induce superconductivity in TDBG as well. As seen the salient features thereof. Focusing on the ener- in Fig.2 (c), Tc is peaked slightly above half filling of getically most favorable intervalley pairing, the super- the conduction band; this peak in Tc lines up with the conducting order parameter ∆ˆ becomes a momentum- maximum of the DOS in Fig.1 (c), as expected. dependent 2 × 2 matrix in spin space that couples as More importantly, beyond these numerical values, this † ˆ † ∝ ck,σ,+(∆k)σσ0 c−k,σ0,− to the low-energy electrons. formalism enables us to determine the IRs under which T † T the superconducting instabilities transform. The crys- In the Nambu basis [56], ψk,σ = (ck,σ,+, c−k,σ,−) , the mean-field Hamiltonian can be expressed in the usual talline point group of TDBG is C3. It is well known [59] Bogoliubov-de Gennes (BdG) [57] form as that C3 has two IRs r, both of which are one-dimensional: the trivial one, A, and the complex representation E (and  ˆ  its complex conjugate partner). The leading instability is Ek,+δσ,σ0 ∆k mf X † σ,σ0 found to transform under the trivial IR, r = A, while the H = ψ   ψ 0 . (3) k,σ ˆ † k,σ k ∆k −Ek,+δσ,σ0 subleading pairing state transforms under r = E; the cal- σ,σ0 r culated basis functions, ∆kµ, are displayed in Fig.2 (d). The fact that the dominant basis function respects all Due to the SU(2)+ × SU(2)− spin symmetry, singlet, symmetries of the system and can be chosen to be real ∆ˆ k = iσ2∆k, and triplet, ∆ˆ k = iσ2 σ · d ∆k, are exactly degenerate. Which of the two is realized depends on and positive for all momenta in the MBZ is consistent the sign of the interaction breaking the enhanced spin with the general results in Ref. 60 (that hold beyond symmetry, which we come back to later in Sec.IVC be- the mean-field approximation). Lastly, we note that all low. For details on the form of the triplet vector, the order parameters in Fig.2 (d) are neither even nor odd admixture of singlet and triplet despite the absence of functions of k, as a consequence of broken inversion and spin-orbit coupling, and the behavior of the irreducible two-fold rotation symmetry in the system. representation (IR) E, we direct the interested reader to Ref. 17. For the electron-phonon interaction introduced above, IV. SUPERCONDUCTIVITY DRIVEN BY COULOMB INTERACTIONS which exhibits an exact SU(2)+ × SU(2)− symmetry, we can treat singlet and triplet on equal footing. The momentum dependence of ∆k close to the superconducting On top of reducing the critical temperature in an transition and, hence, its IR can be inferred from the electron-phonon-driven superconductor, the Coulomb re- linearized gap equation, pulsion can also be the central driving force of superconductivity [61, 62], as has, for instance, been discussed X ∆k = Mk,k0 (T ) ∆k0 , k ∈ FBZ, (4a) in the contexts of monolayer graphene [63–66], artificial k0 graphene [67], and TBG [26, 68]. To explore the scenario of an electronic pairing mechanism in TDBG, we first with a kernel in momentum space given by neglect the electron-phonon coupling and focus on the Coulomb interaction. Since the detailed form of the as- 0 Vk,k0 Ek,+ s,s sociated interaction V 0 in Eq. (1) at the energy scale of Mk,k0 = tanh , (4b) l,l 2Ek,+ 2T our single-band description—including any potential dependence on the layers (l, s) and (l0, s0) of the electrons where Vk,k0 [defined in Eq. (C14)] is simply a sym- involved—is presently unclear, we assume that it can s,s0 metrized rewriting of the interaction Vl,l0 , modulated be described as a pure density-density interaction in the −1 P c c with the appropriate form factors. low-energy conduction band: Hint = N q V (q)ρq ρ−q 5

−1 −2 FIG. 3. (a–d) The RPA polarization function χ0 (in keV Å ), and (e–h) the renormalized Coulomb potential (in meV) at quarter, one-third, two-thirds, and three-quarters ﬁlling [from left to right] of the ﬁrst conduction band in TDBG at 1.75 K.

c P † with ρq = k,σ,ξ ck,σ,ξ ck+q,σ,ξ and [9] ods such as the function renormalization group [76] could be implemented to tackle this problem. Nonetheless, it 2 2 V0 |qm| e e θ is encouraging to view that, with regard to electronic or- V (q) = ,V0 = ' , (5) p 2 2 2 q + κ 20 |Ω |qm 4π0a dering instabilities in TBG, RPA calculations [77] were found to yield qualitatively similar results to renormal- where e is the electronic charge and 0 the vacuum per- ization group methods [78]. Besides, given that we sys- mittivity. In Eq. (5), we have already taken into ac- tematically study the pairing states in a weak-coupling count screening from the gate [69, 70], as described by single-band description, this simultaneously shows that the screening length κ−1 = 2 × 10−8 m [9], as well as any deviations from the analysis below (as might even- from the surrounding boron nitride and the higher bands tually be established in future experiments) must result of TDBG, captured by the eﬀective dielectric constant from the strong-coupling or interband nature of super- = 5 [71]. In addition, we next consider the crucial mo- conductivity. mentum and doping dependence resulting from the inter- Using the low-energy continuum model and restricting nal screening processes of the electrons in the low-energy ourselves to the lowest conduction band as before, we bands hosting superconductivity. compute the Lindhard function [79, 80]

2 X fk,ξ − fk+q,ξ Π0(q, ω = 0) = , (6) A. Screening in the random phase approximation N Ek,ξ − Ek+q,ξ k,ξ

With the aim of describing screening in TDBG, we now where the factor of 2 represents the spin degeneracy, and turn to the random phase approximation (RPA) [72]. fk,ξ is the Fermi function evaluated at band energy Ek,ξ. This is primarily motivated by earlier works on TBG, The resultant effective screened electron-electron interac- where such an approach has been extensively employed tion is given by Vrpa(q) = V (q)/(1 − Π0(q) V (q)), which [30, 73–75]. In particular, its utility was underscored is plotted in Fig.3, together with χ0(q) ≡ Π0(q)/Ω. The by Ref. [26], which discussed the applicability of this coupling is strongly renormalized as a result of this in- method to TBG slightly away from the magic angle— ternal screening and exhibits a significant dependence on this suggestively bodes well for its extension to TDBG, electronic filling. We can now straightforwardly apply where there is never a magic angle to begin with, at least the mean-field machinery established above in Sec.IIIA in our continuum model. The important caveat to be to examine the possibility of Coulomb-driven supercon- cognizant of is that RPA does not apply to the strong- ductivity. To be precise, Eq. (4) still holds; the only coupling regime, and, in principle, more unbiased meth- change is in the kernel M in Eq. (4b) as the phononic in- 6

s,s0 teraction Vl,l0 has to be replaced with its RPA-screened C. Interplay of pairing mechanisms and breaking of SU(2) × SU(2)− symmetry counterpart Vrpa. +

So far, we have focused on interactions with an exact SU(2)+ × SU(2)− spin symmetry, rendering singlet and triplet degenerate. We will here consider different ways of B. Resultant order parameters breaking this symmetry and discuss which type of pairing will be favored. We emphasize that the following results are not limited to TDBG, but are more generally relevant The screening of the Coulomb interaction is highly to other graphene moiré superlattice systems as well. anisotropic, leading to a nontrivial texture of Vrpa in mo- We first point out that singlet will generically domi- mentum space. If Vrpa(q) > 0 ∀ q and has a local maxi- nate over triplet if phonon-mediated interactions alone mum at q = 0, one expects that the kernel M(T ) of the are responsible for pairing [60, 82]. If, however, the gap equation does not have a positive eigenvalue, signal- SU(2)+ × SU(2)−-symmetry-breaking interactions are ing the absence of a superconducting instability. This dominated by the Coulomb repulsion, triplet pairing is is indeed what we find below half-filling of the isolated possible; this is true even when the main pairing glue is band, i.e., for hole doping relative to the correlated in- the SU(2)+×SU(2)− symmetric electron-phonon-induced sulator, with Vrpa illustrated by Figs.3 (e–f). However, interaction—a plausible assumption for graphene moiré as Fig.3 (g) conveys, the RPA screening leads to a lo- systems given our analysis of electron-phonon pairing cal minimum at q = 0 for 2/3 filling. A superconduct- presented above, and the additional insights from exper- ing instability now becomes possible as the pairing can iments on TBG that reveal the relevance of phonons for gain energy if there is a relative sign change between pairing [31, 32]. the order parameters ∆k and ∆k0 connected by the mo- To study this scenario further, let us assume that 0 mentum q = k − k =6 0 where Vrpa(q) is maximal. In the dominant contribution to the SU(2)+ × SU(2)−- fact, the deepest local minimum is obtained at approx- symmetry breaking interactions comes from the fluctua- imately µ = 0.016 eV, which roughly coincides with the j tion of a set of collective electronic modes, φq, that couple optimal doping for superconductivity noticed in experi- to the electrons, ck,α, according to ments. Moving away from this filling, on the electron- doped side, the trough quickly disappears [as is visible X † j j Hcφ = ck+q,αλαβ(k + q, k)ck,β φq. (7) in Fig.3 (h)], thereby killing any Coulomb-driven super- k,q conductivity in the process. Here, we allow for a momentum-dependent coupling ma- At the electron concentrations where superconductiv- trix λ that can couple different internal degrees of free- ity is favored, we find attractive interactions both in the dom, captured by the multi-indices α, β (spin, valley, and A and E representations with basis functions shown in potentially also different bands). We show in AppendixD Fig.4; as in the case of electron-phonon coupling, the that, irrespective of the detailed form of λ and the ac- A representation is dominant. However, the associated tion of the collective mode φj , the competition between order parameter A now exhibits sign changes that are q ∆k singlet and triplet is determined by the properties of φj not imposed by the IR of the superconducting state but q under time reversal: if φj is time-reversal even (odd), i.e., result from the repulsive nature of the interactions. It q it corresponds to a nonmagnetic (magnetic) particle-hole displays 6 of these accidental nodes on the Fermi surface order parameter such as charge (spin) fluctuations, sin- [Fig.4 (a)], which is the minimal nonzero number of nodal glet (triplet) will dominate. Due to the proximity of the points for an order parameter transforming as A of C . 3 superconductor in graphene moiré systems to magnetic An estimate for the superconducting critical temperature phases [2–4, 16, 83, 84], this reveals that these systems with this electronic mechanism yields T < 0.1 K, which c can harbor a triplet pairing phase, even when the pairing is significantly lower than the measured value. Since glue is provided by the electron-phonon coupling. higher-order corrections to RPA can enhance superconductivity [81], this might well be a consequence of our weak-coupling approach and does not rule out an elec- V. DISORDER SENSITIVITY tronic pairing mechanism. Despite transforming under the same, trivial, represen- In this section, we analyze the disorder sensitivity of tation, the dominant superconducting states in Fig.2 (d) the dominant superconducting states derived in Secs.III and Fig.4 (a) for phonon-mediated and electronic pairing andIV above for electron-phonon and Coulomb-driven mechanisms, respectively, will have very different ther- pairing. We will see that although both transform un- modynamic properties due to the absence and presence der the IR A, they behave quite differently in the pres- of accidental nodes. ence of impurities. In addition, we discuss an “Ander- 7

(a) (b) (c) The sensitivity parameter can be written as [85]

PFS h † i k,k0 tr Ck,k0 Ck,k0 ζ = , (10) PFS h † i 4 k tr ∆e k∆e k

PFS where k Fk ≡ hFkiFS represents an average of a func- A E E ∆k ∆k,1 ∆k,2 tion Fk over the Fermi surfaces of the system (normalized

-0.10-0.05 0 0.05 0.10 0.15 -0.2-0.1 0 0.1 -0.2-0.1 0 0.1 such that h1i = 1). After some algebra, we are led to

h|∆A|2i − | h∆Ai |2 FIG. 4. Profiles of the different order parameters that trans- ζ k FS k FS (11) = A 2 form under the A (dominant channel) and E (subleading) 2 h|∆k | iFS representations when superconductivity is mediated by the repulsive Coulomb interaction alone. for singlet pairing. We see that, independent of the ratio of intra- (w1) to intervalley (w2 and w3) scattering, the sensitivity of the singlet state is just the variance of the A son theorem” for triplet pairing that naturally emerges basis function ∆k on the Fermi surface. For the basis in graphene-based moiré superlattice systems as a result function of the conventional pairing state illustrated in of the valley degree of freedom. the first panel of Fig.2 (d), the variance is small, so one In Ref. [85], a general expression for the disorder- would expect ζ 1. Evaluating Eq. (11) with the ex- induced suppression, δTc = Tc − Tc,0, of the transition plicit form of our order parameter, we indeed find a small temperature Tc of a superconductor with respect to its value, ζ ' 0.058, and the state is quite robust against clean value Tc,0 was derived: for weak scattering and/or nonmagnetic disorder, similar to the conventional “An- low disorder concentrations, τ −1 → 0, it holds that derson theorem” [87, 88]. Contrarily, despite transforming under the trivial rep- π −1 resentation, the order parameter in Fig.4 (a) has many δTc ∼ − τ ζ. (8) 4 A sign changes and hence, a small value of h∆k iFS com- A All nonuniversal details of the system and the impurity pared to h|∆k |iFS. Accordingly, ζ should be close to the potential are encoded in the sensitivity parameter ζ that maximal possible value of 1/2 that is typically associated r can be written as a Fermi-surface average of the trace only with nontrivial representations, where h∆kiFS = † 0 by symmetry. From the momentum dependence in tr[C 0 C 0 ] where C 0 is the (anti)commutator k,k k,k k,k Fig.4 (a) we obtain ζ ' 0.498 and the state is fragile † † against disorder. This—or the subtle doping dependence C 0 = ∆ T W 0 − t W 0 ∆ 0 T . (9) k,k e k k,k W k,k e k of the electronic pairing mechanism in Fig.3—among Here, T is the unitary part of the time-reversal operator other reasons, could make superconductivity challenging to observe [4,5]. (in our case iσ2τ1 with Pauli matrices σj and τj in spin Next, we analyze triplet pairing. From Eq. (10), we and valley space, respectively), Wk,k0 is the Fourier transform of the impurity potential (in our low-energy descrip- find, tion, a matrix in spin and valley space), and tW = +1 h|∆A|2i − (w2 − w2 − w2)| h∆Ai |2 ζ k FS 1 2 3 k FS . (12) (tW = −1) for nonmagnetic (magnetic) disorder. Fur- = A 2 2 h|∆k | iFS thermore, ∆e k is the superconducting order parameter, now expressed in the full valley-spin space, i.e., we have Consequently, if intravalley scattering dominates, i.e., † † ∆e kT = ∆k σ0τ0 and ∆e kT = ∆k σ · d τ3 for singlet and w1 w2,3, the triplet and singlet states behave identi- triplet pairing, respectively. As follows from our analy- cally. Intuitively, this results from the fact that the rel- sis above, the leading superconducting instabilities have ative sign change of the triplet order parameter between A ∆k = ∆k , transforming trivially under the point group the two valleys is irrelevant if intervalley scattering is C3 of the system (IR A), and its explicit form is given by weak. Note that the values of w1,2,3 parametrizing the the left panel in Fig.2 (d), for electron-phonon pairing, impurity potential are immaterial for the basis function A A and by Fig.4 (a) for Coulomb-driven superconductivity. in Fig.4 (a) for which, | h∆k iFS | h|∆k |iFS, so ζ ' 0.5, To be concrete, let us focus on spinless, nonmagnetic as in the singlet case. The only difference between singlet (tW = +1) local impurities, i.e., Wk,k0 = σ0(w1τ0+w2τ1+ and triplet arises for the phonon-mediated state, where P 2 A 2 A 2 2 2 w3τ2), wj ∈ R, normalized such that j wj = 1. As in h|∆k | iFS ' | h∆k iFS | , leading to ζ ' w2 +w3. Here, the Ref. [85], the normalization is chosen so as to ensure that stability of superconductivity is determined by the ratio spin-magnetic disorder, Wk,k0 = σ3(w1τ0 + w2τ1 + w3τ2), of intra- to intervalley pairing. yields ζ = 1 for a momentum-independent singlet order While, in general, impurities can mediate scattering parameter, i.e., in the Abrikosov-Gorkov limit [86]. between different valleys, the intervalley scattering am- 8 plitudes will be suppressed as long as the impurity po- pairing mechanisms. In both cases, the leading supercon- tential only varies on the length scales of the moiré lat- ducting instability transforms under the IR A, i.e., is in- tice, but is smooth on the atomic scale. Thus, the triplet variant under all point symmetries of the system, and its state with basis function in Fig.2 (d), which can be stabi- order parameter can be found in Fig.2(d), left panel, and lized by a combination of electron-phonon and electron- Fig.4(a), respectively. We expect the explicit form of the electron coupling (see Sec.IVC), can also exhibit an ana- derived superconducting order parameters to be useful logue of the Anderson theorem and be protected against for comparison with future experimental investigations nonmagnetic disorder on the moiré scale. To see this ex- of the system such as quasiparticle-interference experi- plicitly, let us start from the tight-binding description of ments. On the theoretical side, further first-principles the individual graphene layers. We denote the annihi- computations of phononic properties [89, 90] of TDBG lation operator of an electron on Bravais lattice site R, are needed to obtain a refined description of electron- † sublattice τ, spin σ, and layer ` by aR,σ,`,τ and consider phonon pairing. spinless impurities of the general form We also discussed the scenario that the dominant pairing glue, which preserves the enhanced SU(2)+ × SU(2)− X † ``0 Himp = aR,σ,`,τ vττ 0 (R) aR,σ,`0,τ 0 . (13) spin symmetry of the system, is provided by the electron- R phonon coupling while the residual interactions breaking this symmetry are associated with fluctuating bosonic In order to connect to the continuum-model operators, modes φj. It is shown that time-reversal even (odd) φj e.g., in Eq. (B1), we first Fourier transform to get generically favor singlet (triplet) pairing. In combination

X † ``0 with the fact that there are indications of magnetism in Himp = ak,σ,`,τ veττ 0 (q) ak+q,σ,`0,τ 0 , (14) the phase diagram of several graphene-based moiré su- k,q perlattice systems [2–4, 16, 83, 84], triplet pairing seems

0 0 like a natural possibility, even if electron-phonon cou- `` −1 P `` iR·q where veττ 0 (q) = N R vττ 0 (R)e is the Fourier pling provides the key ingredient for the value of the crit- transform of the impurity potential. Clearly, if the po- ical temperature. In such a scenario, it is possible that tential of the local perturbation is smooth on the atomic ``0 0 the superconducting transition temperature of the triplet length scales, veττ 0 (q) with q connecting the K and K is only weakly aﬀected by a reduction of the Coulomb points (in the original Brillouin zone of graphene) is neg- interaction, which could, however, induce a transition ligibly small. Transforming back to real space, Eq. (14) from triplet to singlet pairing. We have also demon- can, thus, be approximately written as strated that, contrary to common wisdom, the associated Z triplet state will enjoy protection from an Anderson the- † ``0 Himp ' dr aσ,`,τ,ξ(r) vττ 0 (ξ; r) aσ,`0,τ 0,ξ(r), (15) orem against nonmagnetic impurities at the moiré length scales. in the continuum description. Most importantly, it is diagonal in valley space. This property will remain un- changed upon projection onto the bands of the system ACKNOWLEDGMENTS (see AppendixB). We ﬁnally note that the resulting protection of the We thank Darshan Joshi, Peter Orth, Subir Sachdev, triplet state against this type of disorder is not altered Harley Scammell, and Yanting Teng for valuable discus- by the fact that, in general, the impurity potential will sions. This research was supported by the National Sci- (1) (2) be momentum dependent, Wk,k0 = τ0σ0fk,k0 + τ3σ0fk,k0 ence Foundation under Grant No. DMR-1664842, and the with scalar functions f (1,2) (constrained to respect time- computing resources for the numerics in this paper were A kindly provided by Subir Sachdev. reversal symmetry): as long as the basis function ∆k is (almost) momentum independent, the commutator in Eq. (9) will vanish (approximately), wherefore ζ = 0 (ζ 1). Appendix A: Continuum model for TDBG

In this section, we brieﬂy review the continuum model for TDBG introduced by Koshino [10]. Each unro- VI. CONCLUSION AND OUTLOOK tated bilayer graphene (BLG)√ sheet has lattice vectors a1 = a(1, 0) and a2 = a(1/2, 3/2), with lattice constant In this work, we have analyzed the dominant and a ' 0.246 nm; their√ reciprocal lattice counterparts√ are subleading superconducting instabilities for TDBG us- b1 = (2π/a)(1, −1/ 3) and b2 = (2π/a)(0, 2/ 3). Once ing a microscopic continuum model, and discussed both a relative rotation is applied between the two BLGs, the electron-phonon and RPA-screened purely electronic lattice vectors diﬀer from the untwisted case, and, for 9

(l) (a) (b) the l-th BLG, are given by ai = R(∓θ/2)ai (with ∓ for l = 1, 2), R(θ) being the matrix for rotations by an- 60 gle θ. Correspondingly, the reciprocal lattice vectors are 60 (l) modiﬁed to bi = R(∓θ/2) bi. The moiré pattern thus 40 formed is characterized in the limit of small θ by the 40 M (1) (2) reciprocal lattice vectors Gi = bi − bi (i = 1, 2). In valley ξ = ±1, the Dirac points of graphene are located 20 (l) (l) (l) 20 at Kξ = −ξ [2b1 + b2 ]/3 for the l-th BLG. Denoting by A`,B` the sublattice on layer ` = 1, 2, 3, 4 [labeled by the double-index (l, s) in Sec.III], the contin- 0 0 uum Hamiltonian for TDBG at small twist angles θ ( 1) can be expressed in the Bloch basis of carbon’s pz or- -20 -20 bitals, (A1,B1,A2,B2,A3,B3,A4,B4) as

 †  H0(k1) S (k1) -40 0 † -40  S(k1) H0(k1) U  HAB-AB =  †  + V,  UH0(k2) S (k2) 0 S(k2) H0(k2) (A1)

(l) ◦ where kl = R(±θ/2)(k −Kξ ) with ± for l = 1, 2. Using FIG. 5. Band structure of TDBG at (a) θ = 1.28 , and (b) ◦ the shorthand k± = ξkx ± iky, the other building blocks θ = 1.33 , which are the twist angles used in the devices are where Shen et al. [2] observed signatures of superconducting behavior. Note the overall similarity of the dispersions to 0 0 −~vk− 0 d −~vk− that in Fig.1(a) at θ = 1.24◦, for which superconductivity H0(k) = 0 ,H0(k) = , −~vk+ d −~vk+ 0 was reported by Liu et al. [3]. v k γ S(k) = ~ 4 + 1 . (A2) ~v3k− ~v4k+ corrugation effect [10]: such lattice relaxation not only 0 H0 and H0 above are the Hamiltonians of monolayer expands (shrinks) AB (AA) stacking regions but also en- graphene, where the band velocity v is ~v/a = 2.1354 eV hances the energy gaps separating the lowest-energy and [44, 91]. At so-called dimer sites, the A1 atoms of the first excited bands [43, 44, 47]. In momentum space, the cou- layer sit atop the B2 atoms of the second, and this results pling U hybridizes the eigenstates at a Bloch vector k in in the small additional on-site potential d0 = 0.050 eV [92] the moiré Brillouin zone with those at q = k + G, where M M with respect to nondimer sites. The interlayer coupling G = m1G1 + m2G2 for m1, m2 ∈ Z. of AB-stacked BLG is captured by the matrix S, wherein Lastly, another important tuning knob in the experi- γ1 = 0.4 eV is the coupling between the abovementioned mental setup is the potential difference between the top dimer√ sites, while the parameters v3 and v4 are related by and bottom graphene layers. This lends another useful vi = ( 3/2)γia/~ (i = 3, 4) [92] to the diagonal hopping degree of controllability to the system, enabling one to elements γ3 = 0.32 eV and γ4 = 0.044 eV. In AB-stacked change the band separations by adjusting the gate volt- BLG, v3 and v4 account for trigonal warping of the en- age difference. In Eq. (A1), it is parametrized by V , the ergy band and electron-hole asymmetry, respectively. interlayer asymmetric potential, The long-wavelength moiré potential stemming from 3 d d d 3 d the angular twist between the two BLGs also induces V = diag 1, 1, − 1, − 1 , (A4) couplings between the atoms in the second and third lay- 2 2 2 2 ers. This moiré interlayer hopping is effectively described 1 by the matrix U in Eq. (A1), which is given by [37, 44, 91] where stands for the 2 × 2 unit matrix, and d connotes the electrostatic energy difference between adjacent lay- 0 0 −ξ u u u u ω i ξ GM·r ers, arising from a constant perpendicular field. U = + e 1 u0 u u0ωξ u Putting all these ingredients together, we numerically 0 ξ diagonalize the Hamiltonian (A1) in momentum space u u ω i ξ (GM+GM)·r 2πi/3 + e 1 2 ; ω = e , (A3) u0ω−ξ u (with only a limited number of wavevectors q inside the (1) (2) M cutoff circle |q−(Kξ +Kξ )/2| < 4|G1 |) to obtain the where u = 0.0797 eV and u0 = 0.0975 eV [44]; crucially, low-energy eigenstates and energy bands. Despite this u =6 u0. This difference between the diagonal (u) and truncation, one can ensure satisfactory convergence of the off-diagonal (u0) amplitudes represents the out-of-plane electronic property of interest by choosing a sufficiently 10

~ † ~ large cutoff parameter [42]. Since the intervalley coupling Recognizing that Ψk,σ,ξ(r) Ψk0,σ0,ξ0 (r) ∝ δσ,σ0 and inde- is negligible at small twist angles, this calculation can be pendent of σ, the total density can be split into two com- carried out separately for each valley. Samples of the ponents as band structures obtained in this fashion are presented in Figs.1(a) and5. ρ (r) = ρintra(r) + ρinter(r), with intra 1 X ~ † ~ † ρ (r) = Ψ (r) Ψ 0 (r) c c 0 , N k,ξ k ,ξ k,σ,ξ k ,σ,ξ k,k0,σ,ξ Appendix B: Derivation of the form factors (B4) inter 1 X ~ † ~ † In TDBG, there are four main degrees of freedom for ρ (r) = Ψ (r) Ψ 0 (r) c c 0 . N k,ξ k ,−ξ k,σ,ξ k ,σ,−ξ our continuum fields, which are indexed by σ (spin), ` = k,k0,σ,ξ (l, s) (layer), τ (sublattice), and ξ (valley), so the elec- Within the continuum description used in this work, the tronic creation operators at (continuum) position r ∈ R2 will be denoted by a r . Although the following overlap between the Bloch states for different ξ and, σ,`,τ,ξ( ) inter framework can be applied to any observable, let us be- hence, ρ , vanishes identically. This eventually gives gin with the total electronic density at a two-dimensional rise to the enhanced SU(2)+ ×SU(2)− spin-rotation sym- inter position r, given by metry. In reality, ρ is nonzero (but small), which breaks this symmetry. However, as has been shown in X † X Ref. 17, the behavior when this symmetry is weakly bro- ρˆ(r) = a (r) a (r) ≡ a† (r) a (r), σ,`,τ,ξ σ,`,τ,ξ α α ken is completely determined by symmetry, which is why σ,`,τ,ξ α (B1) we do not have to consider it separately in this work. Once again turning to Bloch’s theorem, we can write where α is a multi-index encompassing all four individual ~ ik.r ~ ik.r X ~ iG.r indices. Using the completeness relation Ψk,ξ(r) = e Uk,ξ(r) = e Uk,ξ(G) e , (B5) G FBZ X X where ~ is lattice periodic, but with the periodicity |α, ri = |Ψk,ni hΨk,n | α, ri (B2) Uk,ξ n k of the moiré superlattice. The Fourier transform of the density is simply for the Bloch states | i—where the sum involves all Ψk,n Z momenta of the first moiré Brillouin zone (FBZ) and all −iq.r ρq = dr e ρ (r) (B6) bands n of the system—we can write, at the level of the field operators FBZ Z " 1 X X ~ † ~ 0 † = dr U (G) U 0 (G ) c c 0 N k,ξ k ,ξ k,σ,ξ k ,σ,ξ FBZ k,k0 G,G0,σ,ξ 1 X X ~ aα(r) = √ Ψk,n(r) ck,n. (B3) # N α 0 0 n k × exp i(k − k + G − G − q).r

In this equation, we have introduced the vector notation " # ~ r ≡ hα, r| i, and c are the electronic op- 1 X † X ~ † ~ (Ψk,n( ))α Ψk,n k,n = c c 0 U (G) U (G) , N k,σ,ξ k+q−G ,σ,ξ k,ξ k+q,ξ erators in band n with momentum k. k,G0,σ,ξ G When all bands are taken into account, Eq. (B3) is just a change of basis and, hence, exact. However, in such that k, k+q−G0 ∈ FBZ. This condition is satisﬁed the following we will project the theory on the single by exactly one G0, therefore allowing us to eliminate the conduction band (per spin and valley)—colored green in summation over G0 in the last equation above. Note that Fig.1(a)—that hosts superconductivity; within the cur- in the penultimate step, we have made use of the relation ~ ~ rent notation, this subspace is associated with four values Uk+G1 (G2) = Uk(G1 +G2), which we can always enforce; ~ ~ of the band index n that we will, from now on, relabel this is equivalent to asserting Ψk+G(r) = Ψk(r) ∀ G. We as n → (σ, ξ). The projection amounts to restricting ﬁnally have the sum on the right-hand side of Eq. (B3) to only these 1 X four values. Plugging this truncated form of Eq. (B3) ρ = c† c F , (B7) q N k,σ,ξ FBZ(k+q),σ,ξ k,q,ξ into Eq. (B1), the projected electronic density operator, k,σ,ξ ρ (r), reads as where we have introduced the form factors 1 X X ~ † ~ † ρ (r) = Ψk,σ,ξ(r) Ψk0,σ0,ξ0 (r) ck,σ,ξ ck0,σ0,ξ0 . X ~ † ~ N Fk,q,ξ ≡ Uk,ξ(G) Uk+q,ξ(G), (B8) k,k0 σ,σ0,ξ,ξ0 G 11

X ~ † ~ = Uk,ξ(G) Uk+FBZ(q),ξ(G + Gq), (B9) can be projected in a similar fashion, %ˆq,l,s → %q,l,s as G 1 X † l,s with G q − q , i.e., the reciprocal lattice vector % = c c F (B10) q = FBZ( ) q,l,s N k,σ,ξ FBZ(k+q),σ,ξ k,q,ξ which folds the wavevector q back into the first Brillouin k,σ,ξ zone. The last form in Eq. (B9) shows most clearly, why the form factors are expected to decay with large |q|. with Finally, let us elucidate how the U~ entering the form X k F l,s ≡ U~ † G P l,s U~ G , (B11) factors are computed from the continuum model in Ap- k,q,ξ k,ξ( ) k+q,ξ( ) pendixII. For notational simplicity, let us recast both G spin and valley blocks of Eq. (A1) in the form of a sin- where P l,s is a square diagonal matrix that projects on gle Hamiltonian HeAB-AB = h(−i∇) + T (r), where h, to layer (l, s). By definition, it necessarily holds that which is a matrix in α-space, has no explicit spatial de- l,s F = P F ≡ P F ` . pendence whereas T encapsulates the interlayer coupling k,q,ξ l,s k,q,ξ ` k,q,ξ and is moiré-lattice periodic. Appealing to translational symmetry and decomposing the latter in a Fourier se- P ries as T (r) = TG exp (iG.r), it is straightforward G Appendix C: Linearized mean-field gap equation to show that the Schrödinger equation HeAB-AB |Ψn,ki = E |Ψn,ki translates to n,k Having established the procedure for the projection of X 0 the density operators to the low-energy subspace formed h(k + G) U~ (G) + T 0 U~ (G ) = E U~ (G) n,k G−G n,k n,k n,k by the single conduction band (per spin and valley), we G0 are now well-positioned to construct a mean-field the- ~ ory of superconductivity in TDBG. We start from the ∀ k ∈ FBZ. This provides a prescription to read off Uk from the eigenvectors obtained upon diagonalizing the density-density interaction given by Eq. (1) and keep s,s0 Hamiltonian HAB-AB in Eq. (A1). Vl,l0 (q) general for now. We also emphasize that q in Eq. (1) extends over all two-dimensional momenta but On a side note, we remark that the density, %ˆq,l,s, in a s,s0 particular layer ` = (l, s), rather than the total density, Vl,l0 (q) need not be periodic in the moiré Brillouin zone.

Using the expression for the densities from Eq. (B10) and the same multi-index notation, ` = (l, s) introduced in AppendixA, we ﬁnd

X `,`0 † † ` `0 Hint = Vq ck,σ,ξ cFBZ(k+q),σ,ξck0,σ0,ξ0 cFBZ(k0−q),σ0,ξ0 Fk,q,ξFk0,−q,ξ0 , (C1) k,k0,q with k, k0 ∈ FBZ. All repeated internal indices (σ, σ0, ξ, ξ0, `, `0) are summed over, and we adopt this convention for the remainder of the discussion as well. Furthermore, we omit the “FBZ” hereafter and implicitly assume that the momentum arguments of the ﬁeld operators are folded back into the ﬁrst Brillouin zone. Retaining only terms in the 0 0 homogeneous intervalley Cooper channel (k = −k , ξ = −ξ ), Hint reduces to

cc X `,`0 ` `0 † † Hint = − Vq Fk,q,ξ F−k,−q,−ξ ck,σ,ξ c−k,σ0,−ξ ck+q,σ,ξ c−k−q,σ0,−ξ. (C2) k,q

` ` ∗ The form factors in this channel can be simpliﬁed further: time-reversal symmetry dictates Fk,q,ξ = (F−k,−q,−ξ) , wherefore ∗ cc X † † ``0 ``0 ` `0 Hint = Ck,q ck,σ,+c−k,σ0,−ck+q,σ,+c−k−q,σ0,−, Ck,q ≡ − Vq + V−q Fk,q,+ Fk,q,+ . (C3) k,q

An advantage of Eq. (C3) is that it now suﬃces to com- main text, one ﬁnds pute the wavefunctions for a single valley, say, ξ = +, only. For the electron-phonon coupling discussed in the 2 2 2 D X ωq X X l,s C = δl,l0 F , (C4) k,q m¯ v2 ω2 + ν2 k,q,+ ph n q n l s 12

whereas for the screened Coulomb interaction, Vrpa(q), Equation (C8) now becomes 2 V 0 E Ck,q = −(Vrpa(q) + Vrpa(−q)) Fk,q,+ . (C5) ˆ X k,k k,+ ˆ ∆k = tanh ∆k0 , σ,σ0 2 E 2T σ,σ0 The mean-ﬁeld decoupling proceeds by deﬁning, as k0 k,+ usual, the expectation value (C13)

0 deﬁning (∆k)σ,σ0 ≡ ck,σ,+c−k,σ0,− , (C6) whereupon we get (neglecting a constant piece) Vk,k0 ≡ Ck,k0−k. (C14)

X † † It is easy to observe that Hmf C c c 0 . .. (C7) int = k,q k,σ,+ −k,σ0,− ∆k+q σ,σ0 + H c k,q ∗ Vk,k0 = Vk0,k = Vk+G,k0+G. (C15) Compactly expressing the q-sum as In Eq. (C13), we now see explicitly that singlet and ˆ X 0 triplet are degenerate as expected due to the pres- ∆k ≡ Ck,q ∆k+q 0 , (C8) σ,σ0 σ,σ q ence of the SO(4) ' SU(2)×SU(2) symmetry of independent spin rotations in each valley: inserting either the mean-field interaction is consolidated into a singlet, (∆ˆ k)σ,σ0 ≡ ∆k (iσy)σ,σ0 , or a triplet ansatz, FBZ ˆ 0 ≡ iσ σ · d 0 , we obtain the same eigen- (∆k)σ,σ ∆k ( y )σ,σ mf X † ˆ † Hint = ck,σ,+ ∆k c−k,σ0,− + H.c.. (C9) value equation (4). σ,σ0 k Obviously, due to Eq. (C15), V is Hermitian (it is, in our case, real and symmetric) which guarantees that the T † T In terms of the Nambu spinor ψk,σ = (ck,σ,+, c−k,σ,−) kernel M only has real eigenvalues. In general, the sum [56], the full Hamiltonian, including both hopping and over k0 in Eq. (4a) involves arbitrarily large momenta. pairing, can be kneaded into the BdG form of Eq. (3), However, due to the decay of the form factors with large taking advantage of the fact that the dispersions in the momentum transfer q = k0 − k in Eq. (C3), we restrict two valleys are related as Ek,+ = E−k,− as a consequence the sum to k0 in the FBZ. of time-reversal symmetry. All the microscopic details At sufficiently large T , Mk,k0 (T ) ∼ 1/T and Eq. (4a) of TDBG are encoded in this mean-field Hamiltonian does not have a solution. If, below a critical temperature through the band energies and the form factors. Tc, a solution exists for the order parameter, its structure For self-consistency, Eq. (C8) is required to hold, in momentum space is, of course, interaction-dependent: where the expectation value on the right-hand side [see when Vk0−k > 0 (< 0), i.e., the interaction is repulsive Eq. (C6)] is calculated within Hmf. We have (attractive), ∆k and ∆k0 are favored to have the opposite D E X (same) sign. ck,σ,+c−k,σ0,− = −T [GBdG(k, ωn)] 1,2 ; σ,σ0 ωn the numbers 1, 2 denote the indices in Nambu space, Appendix D: Selection of singlet or triplet pairing while σ and σ0 are spin indices, and the BdG Green’s −1 function is given by GBdG(k, ωn) = (iωn − hBdG(k)) . In this appendix, we discuss in more detail the pos- We expand in the superconducting order parameter, sibility that the dominant SU(2)+ × SU(2)−-preserving −1 sc−1 sc pairing interactions are provided by phonons while those GBdG = G0 − Σ ∼ G0 + G0Σ G0, where breaking this enhanced spin symmetry—down to the −1 Ek,+ 0 usual total SU(2) spin symmetry—are dominated by G0 = iωn − δσ,σ0 , (C10) 0 −Ek,+ purely electronic physics. Specifically, we show that triplet (singlet) pairing is generically favored if the latter  ˆ  0 ∆k sc 0 class of interactions are dominated by time-reversal odd Σ =  σ,σ  , (C11) ˆ † (even) particle-hole fluctuations. ∆k 0 σ,σ0 To this end, let us assume that, at the energy scales relevant to superconductivity, there is a set of collec- and get, to linear order in ∆ˆ : j tive electronic modes, φq, j = 1, 2,...Nb, that dominates the part of the pairing mechanism that breaks X −T ˆ c c 0 = ∆k k,σ,+ −k,σ ,− 0 the SU(2)+ × SU(2)− symmetry (and, thus, determines (iωn − Ek,+)(iωn + Ek,+) σ,σ ωn whether singlet or triplet will win). We do not want to 1 E = tanh k,+ ∆ˆ . make any assumptions about its microscopic form and, k 0 2 Ek,+ 2T σ,σ thus, take the general form of the coupling to the elec- (C12) trons given in Eq. (7) of the main text. As we will see 13

j below, it is sufficient to specify the behavior of φq un- second equation is related to time-reversal symmetry der time-reversal symmetry: we will focus on all of these and Hermiticity. Apart from Sφ, the total action S = modes being time-reversal even, tφ = +, or odd, tφ = −, S0 + Sφ + Sint + Scφ contains the free electronic theory, corresponding to Z ˆ ˆ † 0 † 0 † Θ φq Θ = tφ φ−q, Θ λ(−k, −k )Θ = tφ λ(k, k ), (D1) S0 = ck,α −iωn + hαβ(k) ck,β, (D5) k where Θˆ and Θ are the anti-unitary time-reversal operators in Fock and single-particle space, respectively. the fermion-boson coupling, Scφ, analogous to Eq. (7), For the remainder of this section, we use the field- and the contribution Sint describing all other interactions integral description and denote the associated field op- preserving the SU(2)+ ×SU(2)− symmetry [such as those erators of the electrons and collective bosonic modes in Eq. (C1)]. j by the same symbols, ck,α and φq, employing the com- To study superconductivity, we use the following low- bined Matsubara-frequency and momentum notation, energy description. We diagonalize the free Hamiltonian, k ≡ (iωn, k) and q ≡ (iνn, q); here ωn and νn are fermionic and bosonic Matsubara frequencies, respectively. The h(k) ψ = E ψ , (D6) effective low-energy dynamics of the collective bosonic kξσ kξ kξσ modes is described by where we have already taken into account that spin-orbit Z 1 j −1 j0 coupling can be neglected (spin σ is a good quantum Sφ = φq χ (iνn, q) j,j0 φ−q, (D2) 2 q number) and further ignored any intervalley mixing on the level of the noninteracting Hamiltonian h(k). These where R · · · ≡ T P P ... and χ−1 iν , q is the (full) mixing terms are very small leading to the approximate q νn q ( n ) susceptibility in the corresponding particle-hole channels. valley-charge-conservation symmetry, U(1)v, which al- It can be shown that [60] lows us to focus on intervalley pairing [17]. Breaking U(1)v weakly (as is the case for the real system) will T χ (iνn, q) = χ (−iνn, −q), (D3) generate a small admixture of intravalley pairing, which † is, however, not of interest to our discussion here. In χ (iνn, q) = χ (iνn, q). (D4) Eq. (D6) and the following, we concentrate on the eigen- While the first property simply follows from χ being states that give rise to Fermi surfaces and project the P a correlator of twice the same real bosonic field, the electronic fields according to ckα → ξ,σ(ψkξσ)αfkξσ.

j f f f f Integrating out the bosonic modes φq, we obtain an eﬀective action S = S0 + Sint with free contribution S0 = R † f f k fkξσ(−iωn +Ekξ)fkξσ and electron-electron interactions described by Sint. The relevant Cooper channel of Sint can be written as Z Z f σ1σ2 0 0 † † S V k , ξ k, ξ f 0 0 f 0 0 f f , (D7) cc = σ3σ4 ( ; ) k ξ σ1 −k −ξ σ2 −k−ξσ3 kξσ4 k k0 where Vσ1σ2 k0, ξ0 k, ξ Wσ1σ2 k0, ξ0 k, ξ t Vσ1σ2 k0, ξ0 k, ξ is the sum of the SU × SU -preserving σ3σ4 ( ; ) = σ3σ4 ( ; ) + φ ∆ σ3σ4 ( ; ) (2)+ (2)− interactions (W) and the additional contribution (∆V) from the collective bosonic modes. Making use of Eq. (D1) and the fact that time-reversal symmetry implies Θ ψkξσ = sσψ−k−ξσ¯ , where s↑ = +, s↓ = − and ↑¯ =↓, ↓¯ =↑, the latter can be rewritten as

0 σ1σ2 0 0 1 j∗ 0 0 j 0 ∆V (k , ξ ; k, ξ) = − σ σ Λ 0 (k , k)[χ(k − k)] 0 Λ 0 (k , k), (D8) σ3σ4 2 2 3 ξ σ2,ξσ3 j,j ξ σ1,ξσ4

j 0 † j 0 introducing the shorthand Λξσ,ξ0σ0 (k, k ) = ψkξσ λ (k, k ) ψk0ξ0σ0 .

ξ Next, we decouple, as usual, the interaction in Eq. (D7) ∆σσ0 (k) can be written as in the Cooper channel with the help of the Hubbard- ξ Z σσ0 0 0 ξ V 0 (k, ξ; k , ξ ) 0 Stratonovich ﬁelds ∆ 0 (k) and ∆ 0 (k). To linear order ξ σσ ξ 0 σσ σσ ∆ 0 (k) = − ee ∆ 0 (k ). (D9) σσ 2 2 σe σe in these ﬁelds, the saddle-point equation with respect to k0 ωn0 + Ek0ξ0 As the system has SU(2) spin-rotation symmetry and the enhanced SU(2)+ × SU(2)− symmetry is broken, we 14 know that we can discuss spin singlet and triplet sepa- M(T ): upon decreasing temperature, the largest eigen- ξ rately. Beginning with the former, we write ∆σσ0 (k) = value λ(T ) increases and reaches 1 at Tc. Denoting the ξ ξ −ξ s eigenvalue when setting F 0 → by λ T and treating sσδσ,σ¯ 0 ∆k, with ∆k = ∆−k. The saddle-point equation ξk,ξk 0 0( ) s (D9) leads to Fξk,ξ0k0 as a perturbation yields λ(T ) ∼ λ0(T )+tφδλs(T ) at leading order, where

Z 0 s 0 ξ Fξk,ξ0k0 + tφ Fξk,ξ0k0 ξ0 ξ ∗ s ξ Z Z (∆ ) F 0 0 ∆ 0 ∆e k = 1 1 ∆e k0 , (D10) e k ξk,ξ k e k 0 2 2 2 2 2 2 δλ T . (D12) k (ωn + E ) (ω 0 + E 0 0 ) s( ) = 1 1 kξ n k ξ 0 2 2 2 2 2 2 k k (ωn + Ekξ) (ωn0 + Ek0ξ0 )

ξ ξ 2 2 1/2 ξ ξ ξ where we introduced ∆e k = ∆k/(ωn + Ekξ) (to make Similarly, for triplet, with, say, ∆σσ0 (k) = δσ,σ¯ 0 ∆k, ∆k = 0 −ξ the kernel Hermitian). Fξk,ξ0k0 is related to the SU(2)+ × −∆−k, we ﬁnd the same equations as above, only with s t SU(2)−-symmetric W, while the contribution of the col- Fk,k0 replaced by Fk,k0 = lective electronic modes, i.e., ∆V in Eq. (D8), is con- tained in 1 X j∗ 0 0 j0 0 s s 0 Λ 0 0 (k, k )[χ(k − k )]j,j0 Λ 0 0 (k, k ). 2 σ σ ξσ,ξ σ ξσ,ξ σ σ,σ0 s 1 X j∗ 0 0 j0 0 F 0 0 = Λ 0 0 (k, k )[χ(k−k )] 0 Λ 0 0 (k, k ). ξk,ξ k 2 ξσ,ξ σ j,j ξσ,ξ σ σ,σ0 Most importantly, the eigenvalue will behave as λ(T ) ∼ (D11) λ0(T )+tφδλt(T ), where δλt(T ) is given by Eq. (D12) but s t + Note that, due to Eq. (D4), χ(q) has real eigenvalues, with Fk,k0 → Fk,k0 . Since ∆e k > 0 (without loss of general- all of which must be positive as required by stability; ity) if electron-phonon coupling dominates F 0, as proven s s hence, we ﬁnd Fξk,ξk0 > 0. Furthermore, F is symmet- in Ref. 60 and can, for example, be seen in Fig.2 (d), s s ric, Fξk,ξk0 = Fξ0k0,ξk, as follows from Eq. (D3). Like pre- we conclude that δλs(T ) > δλt(T ). Therefore, the eigen- viously, we view the saddle-point equation as a matrix value of singlet (triplet) will be larger and, hence, singlet equation, ∆e λ(T ) = M(T ) ∆e with a Hermitian matrix (triplet) will dominate for tφ = + (tφ = −).

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