Abstract
The superconducting state and mechanism are among the least understood phenomena in twisted graphene systems. Recent tunneling experiments indicate a transition between nodal and gapped pairing with electron filling, which is not naturally understood within current theory. We demonstrate that the coexistence of superconductivity and flavor polarization leads to pairing channels that are guaranteed by symmetry to be entirely bandoffdiagonal, with a variety of consequences: most notably, the pairing invariant under all symmetries can have Bogoliubov Fermi surfaces in the superconducting state with protected nodal lines, or may be fully gapped, depending on parameters, and the bandoffdiagonal chiral pwave state exhibits transitions between gapped and nodal regions upon varying the doping. We demonstrate that bandoffdiagonal pairing can be the leading state when only phonons are considered, and is also uniquely favored by fluctuations of a timereversalsymmetric intervalley coherent order motivated by recent experiments. Consequently, bandoffdiagonal superconductivity allows for the reconciliation of several key experimental observations in graphene moiré systems.
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Introduction
The fascinating physics^{1,2} of correlated graphene moiré superlattices, such as twistedbilayer (TBG) and twistedtrilayer graphene (TTG), has generated extensive efforts to uncover the mysteries of their phase diagrams. Much progress has been made towards understanding their normalstate physics, including the correlated insulating phases^{3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18} and the reset behavior^{19,20}; the latter, which is believed to be associated with the onset of flavor polarization, appears in the same density range and can coexist with superconductivity^{13,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34}. However, the form and symmetry of the superconducting order parameter and the pairing glue are still unknown, despite significant theoretical efforts^{27,28,29,30,33,35,36,37,38,39,40,41,42,43,44,45,46,47}.
Tunneling conductance measurements taken within the superconducting state reveal a Vshaped density of states (DOS)^{48,49} which can become Ushaped at other electron concentrations^{49}. Setting aside the possibility of thermal fluctuations as origin^{50}, this is most naturally interpreted as a transition from nodal to fully gapped superconductivity. For a consistent microscopic theoretical understanding, this provides the following challenges: (i) electron–phonon coupling—a widely discussed^{33,35,36,37,38,39,40} pairing mechanism in TBG and TTG—will typically mediate an entirely attractive interaction in the Cooper channel, with the leading pairing state that transforms trivially under all symmetries and is thus fully gapped^{51,52}. (ii) Even when the lowenergy interactions favor an irreducible presentation (IR), e.g., E of C_{3}, with nodal basis functions (p or dwave), the generically fully gapped chiral configuration wins over the nodal nematic one within meanfield. (iii) Even if we assume that the nodal state is energetically favored, e.g., due to significant corrections beyond meanfield^{27,53,54,55}, one is still left to explain why there is a transition to another, fully gapped superconductor upon changing the filling.
In this work, we show that the combination of flavor polarization and the representations of the symmetries in the flat bands of TBG and TTG allow for pairing channels that are completely offdiagonal in the flat bands and that such bandoffdiagonal states can naturally reconcile all three key challenges (i–iii). More specifically, we find two distinct bandoffdiagonal states: one of them transforms under the trivial representation A of the system’s point group C_{6} (or one of A_{1,2} of D_{6} if we set the displacement field to zero) but can nonetheless have symmetryprotected nodal lines, akin to Bogoliubov Fermi surfaces discussed in refs. ^{56,57}, see Fig. 1a–c for an intuitive visual explanation. The surprising possibility of the existence of such Bogliubov Fermi surfaces without an external magnetic field is unique to twisted graphene systems in that it follows as a direct consequence of both the symmetry and relative flatness of their normalstate bands. The second offdiagonal state transforms under a twodimensional IR (E_{2} of C_{6}). Its associated chiral state, E_{2}(1, i), which is favored in the meanfield over the nematic one, has the unique property of exhibiting nodal lines or being fully gapped depending on the filling fraction, even when the order parameter is kept fixed. We supplement our general symmetry arguments and phenomenological models with HartreeFock (HF) calculations on the continuum model, studying a variety of different pairing mechanisms. We find that nodal bandoffdiagonal pairing is favored by the optical A_{1} and B_{1} phonon modes and by fluctuations of a timereversal symmetric intervalley coherent (TIVC) state (the TIVC state has Kekulé order on the graphene scale^{58,59,60}). Evidence for the former has been provided by a recent photoemission study^{61} while evidence for the latter comes from recent STM experiments^{7}. Furthermore, also fluctuations of a timereversalsymmetric sublattice polarized state (SLP+) are attractive in the bandoffdiagonal channel (see Table 2 for a formal definition of the order parameters). We also show that fluctuations of both TIVC and of a nematic, timereversal symmetric IVC order^{62} favor either the bandoffdiagonal A or an E_{1} state with banddiagonal components, which may also be nodal; the winner is determined by the relative amount of nematic IVC and TIVC fluctuations.
Results
Possible pairing states
Let us begin by classifying the superconducting instabilities in graphene moiré systems in the limit where the lowenergy bands are spinpolarized but allow for multiple bands. We denote the spinless lowenergy fermionic creation operators by \({c}_{{{{{{{{\boldsymbol{k}}}}}}}},\alpha,\eta }^{{{{\dagger}}} }\) with momentum k in valley η = ± , and of band index α labeling the upper (α = + ) and lower (α = − ) quasiflat bands. As a result of twofold rotational symmetry, C_{2z}, along the outofplane (z) direction or effective spinless timereversal symmetry, Θ, the noninteracting band structure ξ_{k,α,η} obeys ξ_{k,α,η} = ξ_{−k,α,−η} ≡ ξ_{η⋅k,α} and intervalley pairing is expected to dominate. A general pairing order parameter in the intervalley channel couples as
where the order parameter \({{{\Delta }}}_{{{{{{{{\boldsymbol{k}}}}}}}},\eta }={{{\Delta }}}_{{{{{{{{\boldsymbol{k}}}}}}}},\eta }^{T}\) is a matrix in band space. The physical spin texture of the superconductor is entirely determined by the form of the underlying normalstate’s polarization: if the spins are aligned in the two valleys, the superconductor is a nonunitary triplet, while antialignment^{24,28} leads to a singlettriplet admixed state^{13,27,28}. In both cases, all of the following states are well defined, with the aforementioned spin structures and symmetries given by appropriate combinations of spinless operations and spin rotations (see Supplementary Appendix A1).
We will classify the pairing states according to the irreducible representations (IRs) of the system’s point group D_{6}, which is generated by sixfold rotations (C_{6z}) along the z axis and twofold rotation symmetry (C_{2x}) along the inplane x axis. Note a displacement field (D_{0} ≠ 0) breaks the inplane rotations leading to the point group C_{6}. Importantly, all IRs of D_{6} and C_{6} are either even or odd under C_{2z}. Choosing the phases of the Bloch states such that C_{2z} acts as c_{k,α,η} → c_{−k,α,−η}, it holds
This immediately implies that the pairing states in all IRs even under C_{2z} (A_{1}, A_{2}, E_{2} of D_{6}) must be antisymmetric in band space and, thus, entirely bandoffdiagonal, whereas the order parameters of the other IRs (B_{1}, B_{2}, E_{1}) are symmetric and can contain both banddiagonal and bandoffdiagonal components. While superconducting order parameters with finite bandoffdiagonal components are rather common in multiband systems, the existence of pairing states that are constrained to be entirely bandoffdiagonal is rather unique and follows from the combination of C_{2z} symmetry and the spin polarization in the normal state. Importantly, this is unaffected by strain or nematic order breaking C_{3z} as long as C_{2z} remains, which guarantees that there are IRs with entirely bandoffdiagonal order parameters.
Choosing the phase conventions of the Bloch states such that C_{2x} and C_{3z} act as \({c}_{{{{{{{{\boldsymbol{k}}}}}}}},\alpha,\eta }\to {({\sigma }_{z})}_{\alpha \alpha }{c}_{({k}_{x},{k}_{y}),\alpha,\eta }\) and \({c}_{{{{{{{{\boldsymbol{k}}}}}}}},\alpha,\eta }\to {c}_{{C}_{3z}{{{{{{{\boldsymbol{k}}}}}}}},\alpha,\eta }\), respectively, the resulting candidate order parameters are summarized in Table 1. Note that a momentumindependent representation of C_{2x} must be σ_{z} due to the bands’ eigenvalues at the ΓM line, which in turn are connected to the topological obstruction of the flat bands^{63}. The reality (Hermiticity) constraint in Table 1 on χ, X, and Y (\(\hat{\chi },\, \hat{X}\), and \(\hat{Y}\)) comes from the residual spinless timereversal symmetry Θ of the normal state^{64,65}. The two twodimensional IRs E_{1,2} are each associated with three pairing states—two nematic phases E_{1,2}(1, 0), E_{1,2}(0, 1) and one chiral state E_{1,2}(1, i).
Spectral properties
We here have the rather unique situation that there are pairing channels, associated with the IRs A_{1,2} and E_{2}, where the pairing is constrained by C_{2z} to be entirely bandoffdiagonal. One immediate very unusual consequence is that the superconducting orderparameter transforming under the trivial representation (A_{1}) has a symmetryimposed line of zeros along the ΓM line, and hence a nodal point in the spectrum. This is related to the topologyinduced nontrivial representation of C_{2x} in band space. We refer to ref. ^{39} for the discussion of other topological nodal points for pairing in obstructed TBG bands. As we will show next, bandoffdiagonal pairing leads to additional unusual spectral properties with farreaching consequences for graphene moiré systems. To this end, consider the following effective Hamiltonian, \({{{{{{{{\mathcal{H}}}}}}}}}_{{\sigma }_{y}}={\sum }_{{{{{{{{\boldsymbol{k}}}}}}}}}{c}_{{{{{{{{\boldsymbol{k}}}}}}}},\alpha,\eta }^{{{{\dagger}}} }{c}_{{{{{{{{\boldsymbol{k}}}}}}}},\alpha,\eta }{\xi }_{\eta \cdot {{{{{{{\boldsymbol{k}}}}}}}},\alpha }+{\sum }_{{{{{{{{\boldsymbol{k}}}}}}}}}[{{{\Delta }}}_{{{{{{{{\boldsymbol{k}}}}}}}}}\,{c}_{{{{{{{{\boldsymbol{k}}}}}}}},+}^{{{{\dagger}}} }{\sigma }_{y}{c}_{{{{{{{{\boldsymbol{k}}}}}}}},}^{{{{\dagger}}} }+\,{{\mbox{H.c.}}}\,]\), where the scalar function Δ_{k} describes the form of pairing. We will here study two cases that are conventionally considered to be fully gapped, (i) a momentumindependent “swave state” (A_{2} or A pairing in Table 1) where Δ_{k} = Δ_{0} and (ii) a “chiral pwave” state, or more precisely an E_{2}(1, i) state, where Δ_{k} = Δ_{0}(X_{k} + iY_{k}) with (X_{k}, Y_{k}) being smooth, MBZperiodic functions transforming as (x, y) under C_{3z}. Furthermore, we parameterize the dispersion, ξ_{η⋅k,α}, of the two flat bands (α = ± ) in valley η = ± as ξ_{k,α} = ϵ_{k} − μ + α δ_{k}, where ϵ_{k} and δ_{k} are C_{3z} (and, for D_{0} = 0, C_{2x}) symmetric functions.
The Bogoliubov spectrum of \({{{{{{{{\mathcal{H}}}}}}}}}_{{\sigma }_{y}}\) has four bands, given by \(\pm{\delta }_{{{{{{{{\boldsymbol{k}}}}}}}}}\pm \sqrt{{({\epsilon }_{{{{{{{{\boldsymbol{k}}}}}}}}}\mu )}^{2}+ {{{\Delta }}}_{{{{{{{{\boldsymbol{k}}}}}}}}}{ }^{2}}\). Consequently, the excitation gap at momentum k reads as
which is shown in Fig. 1d, and therefore exhibits nodes where \( {\delta }_{{{{{{{{\boldsymbol{k}}}}}}}}}=\sqrt{{({\epsilon }_{{{{{{{{\boldsymbol{k}}}}}}}}}\mu )}^{2}+ {{{\Delta }}}_{{{{{{{{\boldsymbol{k}}}}}}}}}{ }^{2}}\). As long as the band structure has Dirac points, there are points k_{D} in the Brillouin zone with \({\delta }_{{{{{{{{{\boldsymbol{k}}}}}}}}}_{D}}=0\), associated with the blue cross in Fig. 1d. Furthermore, for a metallic normal state, μ must be within the bandwidth and, hence, there must be a region R in momentum space where ∣δ_{k}∣ > ∣ϵ_{k} − μ∣. For the momentumindependent A_{2} state, Δ_{k} = Δ_{0}, this implies that there exists \({{{\Delta }}}_{0}^{c} > 0\) such that there is k^{*} ∈ R with parameters (such as the blue circle) above the red solid line in Fig. 1d as long as \( {{{\Delta }}}_{0} < {{{\Delta }}}_{0}^{c}\). By continuity, this means that there must be a nodal point on any line connecting k_{D} and k^{*}. Consequently, for μ within the bandwidth and \({\delta }_{{{{{{{{{\boldsymbol{k}}}}}}}}}_{D}}=0\) for some k_{D}, the A_{2} will always have a nodal line if ∣Δ_{0}∣ is sufficiently small, consistent with the intuitive picture based on the Bogoliubov spectrum in Fig. 1a–c.
We illustrate this further in Fig. 1e using a toy model with \({\delta }_{{{{{{{{\boldsymbol{k}}}}}}}}}=t\left1+{e}^{i{{{{{{{{\boldsymbol{a}}}}}}}}}_{1}\cdot {{{{{{{\boldsymbol{k}}}}}}}}}+{e}^{i{{{{{{{{\boldsymbol{a}}}}}}}}}_{2}\cdot {{{{{{{\boldsymbol{k}}}}}}}}}\right\) and \({\epsilon }_{{{{{{{{\boldsymbol{k}}}}}}}}}={t}^{{\prime} }\mathop{\sum }\nolimits_{j=1}^{3}\cos {{{{{{{{\boldsymbol{a}}}}}}}}}_{j}\cdot {{{{{{{\boldsymbol{k}}}}}}}},\, {{{{{{{{\boldsymbol{a}}}}}}}}}_{j}={[{C}_{3z}]\,}^{j1}{(\sqrt{3},0)}^{T}\). This leads to the second unexpected conclusion that, for any pairing mechanism, including conventional electron–phonon coupling, the leading instability either has nodal lines in a finite region below T_{c} or transforms nontrivially under the symmetries of the normal state. For electron–phonon pairing (or pairing mediated by the fluctuations of any timereversalsymmetric order parameter^{52}, such as the TIVC state) this is particularly unexpected since it is generally believed to always lead to a fully gapped state that transforms trivially under all symmetries. In fact, this can be proven in general terms^{51,52}, even for spinorbitsplit Fermi surfaces and beyond meanfield theory^{52}. The crucial difference to these works, however, is that spinfull timereversal is broken in our case such that the FermiDirac constraint is inconsistent with a nonsignchanging, banddiagonal pairing state. This leads to the unique situation that although electron–phonon coupling will lead to entirely attractive interactions in the Cooper channel, the superconducting energetics is frustrated: the dominant pairing state is determined by whether the energetic loss due to nonresonant bandoffdiagonal Cooper pairs (A_{2} pairing) or the costs from sign changes of the order parameter (such as B_{1}) are less harmful. We will demonstrate this explicitly by a model calculation in sec. “electron–phonon coupling below, where either A_{2} or B_{1} is dominant, depending on the form of the electron–phonon coupling”.
Let us first, however, discuss the general spectral properties of the “chiral pwave” state which is canonically expected to be fully gapped as long as the Fermi surfaces do not cross the zeros of X_{k} + iY_{k}. Three of these zeros have to be at the Γ, K, and \({K}^{{\prime} }\) points as a consequence of C_{3z} symmetry. In the absence of finetuning, X_{k} + iY_{k} will have vortices at these points with vorticity v = + 1. As can be seen in Fig. 1f, where we show the phase of X_{k} + iY_{k} using an admixture of the two lowestorder terms, the net vorticity of + 3 at these highsymmetry points has to be compensated by antivortices at generic momenta. The lowest possible number is three C_{3z}related vortices, which appear near the M points in Fig. 1f. If it holds ∣δ_{k}∣ > ∣ϵ_{k} − μ∣ at any of these zeros k = k_{j}, we obtain a point above the red line in Fig. 1d and, thus, a nodal point along any contour between that k_{j} and k_{D}; as opposed to the A_{2} state, this holds irrespective of the value of Δ_{0} and therefore all the way to zero temperature. In summary, we find that also the E_{2}(1, i) “chiral pwave” state is not generically fully gapped but instead will exhibit a nodal line encircling any zero k_{j} of X_{k} + iY_{k} with \( {\delta }_{{{{{{{{{\boldsymbol{k}}}}}}}}}_{j}} >  {\epsilon }_{{{{{{{{{\boldsymbol{k}}}}}}}}}_{j}}\mu \). This leads to an interesting filling dependence of the superconducting gap, as we illustrate in our toy model in Fig. 1g along with the criterion \({D}_{{{{{{{{{\boldsymbol{k}}}}}}}}}_{j}}:= {\delta }_{{{{{{{{{\boldsymbol{k}}}}}}}}}_{j}}  {\epsilon }_{{{{{{{{{\boldsymbol{k}}}}}}}}}_{j}}\mu  > 0\) evaluated at the vortices at Γ, K/K’, and near M. Depending on μ, D_{k} is positive only near the Γ point or only in a region surrounding the vortices close to the M points, leading to nodal lines encircling Γ and near the M points, respectively, as shown in the inset of Fig. 1g. These regimes are separated by a fully gapped region where D_{k} < 0 for all k, which could explain the fully gapped to nodal transition seen in tunneling experiments^{49} when the filling fraction is changed. Note that \({D}_{{{{{{{{{\boldsymbol{k}}}}}}}}}_{j}}= {\epsilon }_{{{{{{{{{\boldsymbol{k}}}}}}}}}_{j}}\mu  \le 0\) for k_{j} at the K and K’ points. In Fig. 1g, \({D}_{K}={D}_{{K}^{{\prime} }}\) vanishes close to the top of the band, which simply means that the Fermi surfaces cross the K, K’ points and the superconductor has nodal points for this finetuned value of the chemical potential.
Fluctuationinduced pairing
Having discussed the unique energetics of pairing and spectral properties of the resulting superconductors in spinpolarized quasiflatbands with Dirac cones on a general level, we next study these aspects more explicitly by solving the superconducting selfconsistency equations in the flat bands common to alternatingtwist graphene systems. We will start with pairing induced by fluctuations of a nearby symmetrybroken phase. To this end, we will couple the lowenergy electrons introduced in Eq. (1) to a collective bosonic field \({\phi }_{j}({{{{{{{\boldsymbol{q}}}}}}}})={\phi }_{j}^{{{{\dagger}}} }({{{{{{{\boldsymbol{q}}}}}}}})\) via
where the Hermitian matrices λ^{j} capture the nature of the correlated insulating phase; we here choose and normalize λ^{j} such that \({({\lambda }^{j})}^{2}={\mathbb{1}}\). Both for twisted bi^{9} and trilayer graphene^{14,15,29}, the stable phases emerging out of the U(4) × U(4)^{9} manifold in the chiralflat (decoupled) limit are natural candidates. Integrating out the bosonic modes, we obtain an effective electronic interaction which in the for superconductivity relevant intervalley Cooper channel reads as
with vertex
t_{ϕ} = ± 1 encoding whether the order parameter is even or odd under timereversal, Θϕ_{j}(q)Θ^{†} = t_{ϕ}ϕ_{j}(q), and χ_{q} > 0 denoting the (static) susceptibility of ϕ_{j}.
Before discussing numerical results for the full model, we first focus on perfectly flat bands. In this limit, the leading superconducting instability within meanfield theory is given by the largest eigenvalue of \({{{{{{{\mathcal{V}}}}}}}}\) in Eq. (6) viewed as a matrix in the multiindex (η, α, β). Furthermore, if there is an antisymmetric, valleyoffdiagonal matrix D obeying (see Methods)
the associated leading superconducting order parameter in Eq. (1) is given by \({({{{\Delta }}}_{{{{{{{{\boldsymbol{k}}}}}}}},\eta })}_{\alpha,{\alpha }^{{\prime} }}={\delta }_{{{{{{{{\boldsymbol{k}}}}}}}}}{(D{\eta }_{x})}_{\alpha,\eta ;{\alpha }^{{\prime} }\eta }\) with δ_{k} > 0; here η_{j} denotes Pauli matrices in valley space and the precise form of δ_{k} is determined by χ(q).
TIVC fluctuations
Motivated by recent experiments^{7} providing direct evidence for TIVC order, we start with TIVC fluctuations as a pairing glue. In the U(4) × U(4) symmetric limit, the TIVC state is associated with λ^{j} = σ_{0}η_{j}, j = x, y, within our conventions. Since t_{ϕ} = + 1, we are looking for Dη_{x} that commutes with λ^{j}. Interestingly, there is a unique antisymmetric, valleyoffdiagonal matrix D ∝ σ_{y}η_{x} with that property, implying that the leading pairing state has the form Δ_{k,η} = σ_{y}δ_{k}, δ_{k} > 0. This is exactly the A_{2} state in Table 1, which, as discussed above, will have nodal lines at least in the vicinity of T_{c} when a finite band dispersion is taken into account. Intuitively, the fact that A_{2} pairing is favored can be understood by noticing that the valleyoffdiagonal form of λ^{j} leads to an attractive interaction across the valleys, which penalizes the B_{1} state with its sign change between the two valleys. In fact, it holds \({{{{{{{{\mathcal{V}}}}}}}}}_{(\eta,\alpha,\beta ),({\eta }^{{\prime} },{\alpha }^{{\prime} },{\beta }^{{\prime} })}=(1\eta \,{\eta }^{{\prime} })\mathop{\sum }\nolimits_{\mu=0}^{3}{({\sigma }_{\mu }^{*})}_{\alpha,\beta }{({\sigma }_{\mu })}_{{\alpha }^{{\prime} },{\beta }^{{\prime} }}\) showing explicitly that it is repulsive (attractive) in the B_{1} (A_{2}) channel.
To go beyond the flatband limit, we solve the superconducting meanfield equations numerically. We take the flat TBG bands from the continuum model^{66} as the starting point. To capture the spinpolarized normal state, we supplement it with Coulomb repulsion and perform HF calculation (see Supplementary Appendix A for details). As can be seen in the resulting band structure shown in Fig. 2a with interaction renormalization assuming filling fraction ν = 2, this not only pushes one spin flavor below the Fermi level but also induces significant band renormalizations. For our subsequent study of superconducticity, we project onto the two bands at the Fermi level and associate them with the creation operators c_{k,α} in the interactions in Eqs. (4) and (5). In our numerical computations, we choose \(\chi ({{{{{{{\boldsymbol{q}}}}}}}})=\frac{1}{{A}_{m}}\frac{V}{{\alpha }^{2}+ {{{{{{{\boldsymbol{q}}}}}}}}{ }^{2}/{k}_{\theta }^{2}}\) where A_{m} is the real space area of a moiré unit cell, and take α = 0.05 for concreteness, although we checked our main conclusion do not crucially depend on this form. In all of our numerics, we work at doping ν = 2.5.
As expected, we indeed find that the A_{2} state dominates, both right at the critical temperature T_{c}, obtained from the linearized gap equation, and at T = 0 as we show by iteratively solving the full selfconsistency equation (see Supplementary Appendix C). One crucial effect of the finite dispersion and splitting between the bands is that a finite interaction strength, V > V_{c,1}, is required to stabilize the superconducting phase, as can be seen in the plot of T_{c} in Fig. 2b. Superconductivity ceases to be a weakcoupling instability as the Bloch states (k, α, η) and \(({{{{{{{\boldsymbol{k}}}}}}}},\, {\alpha }^{{\prime} },\eta )\) are not degenerate for \(\alpha \, \ne \, {\alpha }^{{\prime} }\), cutting off the logarithmic divergence known from BCS theory. The quasiparticle spectrum and order parameter of superconductivity from T = 0 numerics are shown in Fig. 2c, d. In accordance with our general discussion above, we observe that the order parameter only has finite components proportional to σ_{y}, which do not mix with the bandeven contributions ∝ σ_{0,x,z} as a result of C_{2z} symmetry. Furthermore, it does not change sign as a function of k and, for sufficiently small V but still with V > V_{c,1}, the nodal lines in the superconducting spectrum persist all the way to T = 0, while the nodal line is gapped out at low T < T_{c} if V > V_{c,2}.
The interactionstrengthdependence of the superconducting gap can be more clearly seen in Fig. 2e, where we show the DOS for the selfconsistent solution at T = 0. For large V, the superconductor becomes fully gapped at T = 0, leading to a Ushaped DOS. With smaller V, the magnitude of the orderparameter decreases and the superconductor eventually exhibits nodal lines, as explained above. In the regime just before these nodal lines appear, there is an increase in the DOS near the Fermi level, roughly when the order parameter and the maximal band splitting are comparable, leading to a Vshaped DOS (green line). The lifetime parameter used to compute the DOS is 0.3 meV; this choice was based on our kgrid spacing. While it is not necessarily small with respect to the tunneling gap (which vanishes at V_{c,2}), it is small with respect to Δ(k), which is of order 5 meV just as the state is becoming fully gapped for our choice of normal state. This behavior of the DOS with interaction strength may offer a natural explanation for the Ushaped tunneling conductance measurements near ν = 2 and Vshaped tunneling conductance measurements near ν = 3 observed in TTG^{49}; if we are considering TIVC fluctuations of the insulator at ν = 2, then it may be reasonable to expect the coupling to these fluctuations could grow weaker as we dope towards ν = 3, in line with the experimentally observed ν dependence.
Note that the regime we call Vshaped here is strictly speaking fully gapped. However, the crucial difference to the BCS state is that the gap is much smaller than the orderparameter magnitude as a result of the different Bogoliubov spectrum in Eq. (3). This is why, depending not only on the magnitude of the pairing but also on the precise form of the normal state, the resulting tunneling spectra can resemble those observed experimentally^{48,49}, such as the green curve in Fig. 2e, making the A_{2} state an attractive candidate. The regime of small V where stable superconductivity with true Bogoliubov Fermi surfaces is observed can further exhibit a peak at ω = 0 which is due to a Van Hove singularity crossing the Fermi level, see blue curve in Fig. 2e; while this peak has not been observed experimentally, its presence crucially depends on details of the normalstate band structure and is only found to be energetically favored in a very small regime of V in our model.
Electron–phonon coupling
To illustrate that the offdiagonal A_{2} state is more generally favored beyond just TIVC fluctuations, we next discuss electron–phonon coupling, which is frequently considered a plausible pairing mechanism for twisted moiré systems^{33,35,36,37,38,39}. Similar to ref. ^{35}, we use that the optical A_{1}, B_{1}, and E_{2} phonon modes are known^{67} to dominate the electron–phonon coupling in singlelayer graphene. As these are optical phonons, we further assume that the impact of the interlayer coupling on the phonons can be neglected and arrive at
for the electron–phonon coupling, where v_{μ} encode the layer structure of the modes (see Methods). Symmetry dictates that the vertices Λ_{g} are given by \({{{\Lambda }}}_{{A}_{1}}={\eta }_{x}{\rho }_{x},\, {{{\Lambda }}}_{{B}_{1}}={\eta }_{y}{\rho }_{x}\), and \({{{{{{{{\boldsymbol{\Lambda }}}}}}}}}_{{E}_{2}}=({\eta }_{z}{\rho }_{y},{\rho }_{x})\) where ρ acts on the microscopic sublattice basis. Integrating out the phonons and projecting to the flat bands, we obtain an effective electronelectron interaction (see Methods)
where the coupling constants V_{g} of the three different phonon modes g = A_{1}, B_{1}, E_{2} are estimated to obey \({V}_{{A}_{1}}={V}_{{B}_{1}}\simeq 1.33{V}_{{E}_{2}}\) for parallel spins in the two valleys, while \({V}_{{A}_{1}}={V}_{{B}_{1}}=0\) for antiparallel spins. From Eq. (9), it is clear that the induced interaction would be always completely attractive if we focused on intraband pairing, \(\alpha={\alpha }^{{\prime} }=\beta={\beta }^{{\prime} }\), which in spinful systems generically favors the trivial pairing channel^{51,52}. In our case, the combination of two energetically close bands and the trivial pairing being purely bandoffdiagonal leads to competition between different superconductors, even with electron–phonon coupling alone.
To demonstrate this, we study intravalley pairing within the meanfield approximation and parametrize the relative strength of the different phonon modes with an angle variable θ_{ph} according to \({V}_{{A}_{1}}={V}_{{B}_{1}}={V}_{0}\cos {\theta }_{{{{{{{{\rm{ph}}}}}}}}},{V}_{{E}_{2}}={V}_{0}\sin {\theta }_{{{{{{{{\rm{ph}}}}}}}}}\). The results of the meanfield calculation are summarized in Fig. 3. We see that the A_{2} pairing state is favored by the intervalley phonons (θ_{ph} = 0) inspite of its bandoffdiagonal nature leading to a suppressed gap [see Fig. 3a]. This is natural as these phonons mediate an attractive interaction between the two valleys which disfavors the B_{1} state, similar to TIVC fluctuations. In fact, focusing on the leading, momentumindependent term, \({\lambda }_{{{{{{{{\boldsymbol{k}}}}}}}},{{{{{{{{\boldsymbol{k}}}}}}}}}^{{\prime} }}^{g,+}\to {\lambda }^{g,+},g={A}_{1},{B}_{1}\), symmetry dictates \({\lambda }^{{A}_{1},+}\propto {\sigma }_{0}{\eta }_{1}\) and \({\lambda }^{{B}_{1},+}\propto {\sigma }_{0}{\eta }_{2}\) in the chiral limit (see Supplementary Appendix D3). This maps the problem exactly to that of TIVC fluctuations, immediately explaining why the order parameter has a fixed sign in Fig. 3b. As θ_{ph} is increased, the B_{1} state is favored (roughly for θ_{ph} > π/4) as can be seen in Fig. 3c. This is expected since the intravalley E_{2} phonon mediates an attractive interaction within each valley such that the energy gain due to the enhanced gap [Fig. 3d], associated with the banddiagonal matrix elements of the B_{1} state, will overcompensate the energetic loss due to the sign change of B_{1}’s orderparameter between the two valleys. This picture is consistent with the dominant and nonsignchanging nature of the banddiagonal components of the B_{1} state, see Fig. 3e–g. Finally, this behavior can also be understood by applying the commutator criterion in Eq. (7) in the microscopic sublattice basis, see Supplementary Appendix D1.
This shows that, as opposed to the conventional scenario^{51,52}, there are two possible leading superconducting states and the superconducting pairing state does not transform trivially under the symmetries of the system even when phonons alone provide the pairing glue. We have checked in our T = 0 numerics that a 60–70 meV ⋅ (nm)^{2} coupling to A_{1} and B_{1} phonons (based on ref. ^{67}) is roughly of the order needed to stabilize the A_{2} pairing, assuming the normal state is the flat bands of the unrenormalized continuum model, which in our case has a bandwidth of 2 meV. However, we note that if the interactionrenormalized band splitting is much larger than the continuummodel bandwidth, or if the normal state has antiparallel spins in either valley, additional particlehole fluctuations, such as those of TIVC order, will also be required for pairing. An interesting scenario arises for antiparallel spins in the two valleys as a magnetic field will cant the spins and, hence, increase the projection of the intervalley phonon matrix elements to the flat bands. At least in TTG, with the suppressed orbital coupling, this could give rise to reentrant superconductivity at high fields^{25}.
Other particlehole fluctuations
Finally, we discuss pairing induced by fluctuations of other particlehole instabilities. In Table 2, we list the resulting leading superconductors taking λ^{j} in Eq. (4) to be any of the different strongcoupling candidate order parameters^{9,13,14,15,29}. In particular, in addition to the TIVC, we will consider the timereversalodd Kramers intervalley coherent state (KIVC), and timereversalodd and even sublattice polarized states (SLP− and SLP+). To analyze how sensitive our conclusions are to the precise form of the coupling of the strongcoupling fluctuating orders to the electrons, we also perform numerics by projecting momentumindependent coupling vertices in the microscopic basis with the correct symmetries (see, e.g., Table II in ref. ^{13}), listed as \({\bar{\lambda }}^{j}\) in Table 2, to the flat bands. In the band basis, this leads to momentumdependent coupling vertices, cf. Eq. (9). Motivated by recent experiments^{7}, we will also consider fluctuations of an additional nematic, timereversal symmetric, layerodd, intervalley coherent state (NIVC)^{62} which is not a candidate ground state in the strongcoupling limit; unlike the other strongcoupling ground states, the NIVC has no momentumindependent representation in the flatband basis but does have a momentumindependent matrix order parameter in the sublattice basis which takes the form \({\lambda }^{(j,{j}^{{\prime} })}={({\eta }_{x},\, {\eta }_{y})}_{j}{({\rho }_{0},\, {\rho }_{z})}_{{j}^{{\prime} }}\). The results for fluctuations of the projected strongcoupling orders \({\bar{\lambda }}^{j}\) in Table 2 and of the projected NIVC state are shown in Fig. 4, where we use the angle θ_{fluc.} to tune the relative strength between TIVC and any of the other type of fluctuationinduced interactions by multiplying the TIVC interaction potential with \(\cos ({\theta }_{{{{{{{{\rm{fluc}}}}}}}}.})\) and the other fluctuation potential with \(\sin ({\theta }_{{{{{{{{\rm{fluc}}}}}}}}.})\). In our microscopic numerics, we have taken a potential form \(\chi ({{{{{{{\boldsymbol{q}}}}}}}})=\frac{1}{{A}_{m}}\frac{V}{{\alpha }^{2}+ {{{{{{{\boldsymbol{q}}}}}}}}{ }^{2}/{k}_{\theta }^{2}}\) again with α = 0.2 and with V = 4200 meV ⋅ (nm)^{2}. We chose the value of V such that the transitions between the different pairing states are clearly visible in Fig. 4 when varying θ_{fluc.}. In accordance with the prediction for \({\bar{\lambda }}^{j}\) in Table 2, SLP + fluctuations further stabilize the A_{2} superconductor, see Fig. 4a. As such, the banddiagonal B_{1} superconducting channel, where SLP + fluctuations are also attractive, can become the leading channel (favored over A_{2} as a result of the finite bandwidth) only very close to θ_{fluc.} = π/2. KIVC fluctuations, however, are repulsive for A_{2} pairing and favor the B_{1} state more strongly.
So far, the strongcoupling (λ^{j}) and sublattice (\({\bar{\lambda }}^{j}\)) form of the couplings in Table 2 lead to the same conclusions. This is different for SLP− fluctuations (Fig. 4d), where the projectioninduced momentum dependence in the band basis can stabilize the E_{1} superconductor. This can be understood by applying Eq. (7) in a sublattice basis (see Supplementary Appendix D1). We also find the E_{1} state when fluctuations of the NIVC state of ref. ^{62} dominate. Examples of the E_{1} nematic and B_{2} order parameters that emerge for SLP− fluctuations or NIVC fluctuations are shown in Supplementary Appendix F. We point out that the nematic E_{1} pairing is also an interesting candidate given that despite having nonzero pairing in the σ_{0}, σ_{x}, σ_{z} channels, it will be nodal as long as the σ_{x} components do not gap out the nodes in the banddiagonal parts.
Discussion
Taken together, we see that the proposed bandoffdiagonal A_{2} superconductor is an especially attractive candidate for TBG and TTG: first, it can lead to both Vshaped or Ushaped DOS, depending on lifetime parameters, the normal state, and the coupling strength V, see Fig. 2e. As these parameters might vary from sample to sample and within a sample (e.g., V is expected to decrease upon doping further away from the insulator), this can naturally explain the tunneling data of^{48,49}. We emphasize however that at least at the level of our meanfield numerics, we only expect a Vshape in the regime where the superconducting pairing is of the order of the bandwidth; this is the regime, where although the pairing is finite and can be quite large, the gap in the superconducting spectrum is either just closing or very small relative to the pairing. Increasing the pairing further will lead to an evolution from V to Ushaped while decreasing the pairing will eventually lead to a nodal Fermi surface and presumably a peak at zero energy in the DOS. Second, despite its interband nature, A_{2} is the unique pairing state that is favored by fluctuations of two out of the four strongcoupling candidates we consider for the correlated insulator, see Fig. 4a–c. What is more, this includes the TIVC state, signatures of which are observed in recent experiments^{7}. Finally, it is also favored by the likely dominant^{61,68} optical intervalley phonon modes. We emphasize that, both in the case of fluctuating correlated insulators and phonons, the minimum attractive coupling needed to stabilize a purely bandoffdiagonal state depends on the energy splitting between the two flat bands in the normal state; if the bands of our normal state are closer to degenerate, irrespective of the total bandwidth, the needed coupling to stabilize the A_{2} pairing in meanfield will decrease.
The other bandoffdiagonal superconductor we identify transforms under the IR E_{2}, i.e., can be thought of as a pwave state. Its spectral properties also agree well with the experiment as the chiral configurations, E_{2}(1, i), which is favored within meanfield theory over a nematic E_{2} state, can also have nodal regions, depending on filling. As can be seen in Fig. 1g, this can lead to a transition from gapped to nodal when increasing the electron filling starting at ν ≃ 2. However, as opposed to the A_{2} state, E_{2} does not naturally appear as leading instability when considering optical phonons or fluctuations of any of the strongcoupling order parameters of the correlated insulator. While this makes it energetically less natural than A_{2}, we cannot exclude it since its phenomenology agrees well with the experiment and since the precise form of the coupling of the dominant lowenergy collective excitations are not known—significant momentum dependencies beyond λ^{j} and \({\bar{\lambda }}^{j}\) in Table 2 could stabilize E_{2} pairing as well. We also find in our numerics a nematic E_{1} state which may be preferred over its chiral version in the presence of sufficient strain or due to fluctuation corrections^{27,53,54,55}. We find the E_{1} state is the leading instability of nematic IVC fluctuations and SLP − fluctuations, and is a subleading instability of TIVC fluctuations. The E_{1} state is interesting in its own right, as it can also be nodal.
As superconductivity might further coexist with TIVC order^{7}, we have checked (see Supplementary Appendix E) that this does not alter our main observation: the preserved C_{2z} symmetry still allows for entirely bandoffdiagonal states, with transitions from nodal to full gapped, which are stabilized (among other fluctuations) by intervalley phonons.
For the future, it will be interesting to go beyond meanfield and analyze the competition of our bandoffdiagonal states with oddfrequency pairing, which we study in a followup work^{69}. It also seems promising to study Andreev reflection^{48,49} for our interband pairing scenario. On a more general level, our work shows that the observation of nodal pairing in twisted graphene systems does not immediately exclude a chiral superconducting state nor an entire electron–phononbased pairing mechanism. It illustrates that a microscopic understanding of the superconducting states in graphene moiré systems requires taking into account their intrinsically multiband nature.
Note added. Just before posting our work, ref. ^{70} appeared online, which discusses pairing induced by A_{1} phonons in spinful TBG bands.
Methods
Flatband limit
To derive Eq. (7), we take the flatband limit, ξ_{k,±} → 0, in the linearized gap equation. For the interaction defined in Eqs. (4)–(6)), we get (with moiré cell area A_{m})
We define \({({\hat{{{\Delta }}}}_{{{{{{{{\boldsymbol{k}}}}}}}}})}_{\alpha,\eta ;{\alpha }^{{\prime} },{\eta }^{{\prime} }}:={({{{\Delta }}}_{{{{{{{{\boldsymbol{k}}}}}}}},\eta })}_{\alpha,{\alpha }^{{\prime} }}{\delta }_{\eta,{\eta }^{{\prime} }}\) and note that finding the leading superconducting state according to Eq. (10) is equivalent to determining \({\hat{{{\Delta }}}}_{{{{{{{{\boldsymbol{k}}}}}}}}}\) that maximizes the functional
Since \({\chi }_{{{{{{{{\boldsymbol{k}}}}}}}}{{{{{{{{\boldsymbol{k}}}}}}}}}^{{\prime} }} \, > \, 0\), the maximum value will be reached if we can maximize \({t}_{\phi }\,{{\mbox{tr}}}\,[{\lambda }^{j}{\hat{{{\Delta }}}}_{{{{{{{{{\boldsymbol{k}}}}}}}}}^{{\prime} }}{({\lambda }^{j})}^{{{{\dagger}}} }{\hat{{{\Delta }}}}_{{{{{{{{\boldsymbol{k}}}}}}}}}^{{{{\dagger}}} }]\) for each \({{{{{{{\boldsymbol{k}}}}}}}},\, {{{{{{{{\boldsymbol{k}}}}}}}}}^{{\prime} },\, j\) separately. As the Frobenius inner product \({\left\langle A,B\right\rangle }_{F}=\,{{\mbox{tr}}}\,[{A}^{{{{\dagger}}} }B]\) reaches its maximum (minimum) at fixed \(\left\langle A,\, A\right\rangle\) and \(\left\langle B,\, B\right\rangle\), if A = cB with c > 0 (c < 0), \({t}_{\phi }\,{{\mbox{tr}}}\,[{\lambda }^{j}{\hat{{{\Delta }}}}_{{{{{{{{{\boldsymbol{k}}}}}}}}}^{{\prime} }}{({\lambda }^{j})}^{{{{\dagger}}} }{\hat{{{\Delta }}}}_{{{{{{{{\boldsymbol{k}}}}}}}}}^{{{{\dagger}}} }]\) is maximized if \({\hat{{{\Delta }}}}_{{{{{{{{\boldsymbol{k}}}}}}}}}={t}_{\phi }{c}_{{{{{{{{\boldsymbol{k}}}}}}}},{{{{{{{{\boldsymbol{k}}}}}}}}}^{{\prime} }}{\lambda }^{j}{\hat{{{\Delta }}}}_{{{{{{{{{\boldsymbol{k}}}}}}}}}^{{\prime} }}{({\lambda }^{j})}^{{{{\dagger}}} }\) with \({c}_{{{{{{{{\boldsymbol{k}}}}}}}},{{{{{{{{\boldsymbol{k}}}}}}}}}^{{\prime} }} \, > \, 0\). For the ansatz \({\hat{{{\Delta }}}}_{{{{{{{{\boldsymbol{k}}}}}}}}}={\delta }_{{{{{{{{\boldsymbol{k}}}}}}}}}D{\eta }_{x}\) (and assuming for now that δ_{k} has a fixed sign for all k), this is obeyed if
We state Eq. (12) as the (anti)commutator condition (7) in the main text [equivalent if \({({\lambda }^{j})}^{2}={\mathbb{1}}\)], not only because it highlights the simple algebraic and basis independent nature of the condition but also since it emphasizes the similarities to the generalized Anderson theorem of^{71,72}.
If we can find a solution to Eq. (12), we know that the maximum (or at least one of the possibly degenerate maxima) of \({{{{{{{\mathcal{F}}}}}}}}[{\hat{{{\Delta }}}}_{{{{{{{{\boldsymbol{k}}}}}}}}}]\) is of the form of \({\hat{{{\Delta }}}}_{{{{{{{{\boldsymbol{k}}}}}}}}}={\delta }_{{{{{{{{\boldsymbol{k}}}}}}}}}D{\eta }_{x}\) where δ_{k} is obtained as the maximum of the reduced functional
or equivalently as the largest eigenvector of \({\chi }_{{{{{{{{\boldsymbol{k}}}}}}}}{{{{{{{{\boldsymbol{k}}}}}}}}}^{{\prime} }}\) viewed as a matrix in k and \({{{{{{{{\boldsymbol{k}}}}}}}}}^{{\prime} }\). As \({\chi }_{{{{{{{{\boldsymbol{k}}}}}}}}{{{{{{{{\boldsymbol{k}}}}}}}}}^{{\prime} }} > 0\) (due to stability), the PerronFrobenium theorem then immediately implies δ_{k} > 0, in line with out assumption above and as stated in the main text.
Electron–phonon coupling
To present more details on the electron–phonon coupling, the associated displacement operators in Eq. (8) can be expressed in terms of canonical bosons, b_{g,α,μ,q},
where j refers to the two components for the E_{2} phonon (is idle for A_{1}, B_{1}), M is the carbon mass, and ω_{g}(q) is the phonon dispersion, characterizing the phononic part of the Hamiltonian, \({{{{{{{{\mathcal{H}}}}}}}}}_{P}={\sum }_{{{{{{{{\boldsymbol{q}}}}}}}}}{\omega }_{g}({{{{{{{\boldsymbol{q}}}}}}}}){b}_{g,j,\mu,{{{{{{{\boldsymbol{q}}}}}}}}}^{{{{\dagger}}} }{b}_{g,j,\mu,{{{{{{{\boldsymbol{q}}}}}}}}}\).
As for \({({{{{{{{{\boldsymbol{v}}}}}}}}}_{\mu })}_{\ell }\) in Eq. (8), ℓ = 1, 2 refers to the physical graphene layer in the case of TBG. One can, in principle, choose any orthonormal basis; we will find it convenient to use the layerexchange even and odd states, \({{{{{{{{\boldsymbol{v}}}}}}}}}_{\pm }={(1,\pm 1)}^{T}/\sqrt{2}\). For TTG, the situation is more involved (see Supplementary Appendix D2), but our arguments about which phonons are attractive in which pairing channels will hold for both systems.
We project \({{{{{{{{\mathcal{H}}}}}}}}}_{EP}\) in Eq. (8) onto the two flat bands (α = ± ) in each valley η of the spinpolarized continuummodel, leading to a coupling term similar to Eq. (4) with momentumdependent coupling matrices, \({\lambda }^{j}\to {\lambda }_{{{{{{{{\boldsymbol{k}}}}}}}},{{{{{{{{\boldsymbol{k}}}}}}}}}^{{\prime} }}^{g,j,\mu }\). Investigating the matrix elements \({\lambda }_{{{{{{{{\boldsymbol{k}}}}}}}},{{{{{{{{\boldsymbol{k}}}}}}}}}^{{\prime} }}^{g,j,\mu }\), we notice that they almost vanish for the layerodd intervalley (A_{1}, B_{1}) phonons, which can be understood as a consequence of chiral and particlehole symmetry (see Supplementary Appendix D3). The situation is the reverse for the intravalley (E_{2}) phonons, where the layereven matrix elements are numerically small and the layerodd matrix elements dominate. We therefore focus on layereven (odd) intervalley (intravalley) phonon couplings.
Neglecting the momentum dependence in the phonon frequencies and retardation effects, the resulting electronelectron interaction in the intervalley Cooper channel obtained by integrating out the phonons is given by Eq. (9). Here, \({V}_{g}={g}_{g}^{2}/(2N{\omega }_{g}^{2}) \, > \, 0\) and \({V}_{{A}_{1}}={V}_{{B}_{1}}\simeq 1.33{V}_{{E}_{2}}\) results from \({g}_{{A}_{1}}={g}_{{B}_{1}}\simeq {g}_{{E}_{2}}\) and the phonon frequencies estimated in ref. ^{67}. Importantly, this only holds for parallel spins in the two valleys. For antiparallel spins, the projection of the coupling matrices to the flat bands vanishes for the intervalley phonon modes A_{1} and B_{1} such that \({V}_{{A}_{1}}={V}_{{B}_{1}}=0\).
Data availability
The data generated in this study are available in the Zenodo database under the accession code https://zenodo.org/record/8381555and in the figshare repository https://doi.org/10.6084/m9.figshare.23897019.
Code availability
The codes used to generate the plots are available from the corresponding author on request.
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Acknowledgements
M.S.S. acknowledges funding from the European Union (ERC2021STG, Project 101040651—SuperCorr). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. M.C. and S.S. acknowledge funding by U.S. National Science Foundation grant No. DMR2002850. M.S.S. Thanks B. Putzer for the discussions. M.C. thanks P. Ledwith, J. Dong, and D. Parker for their helpful discussions.
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Christos, M., Sachdev, S. & Scheurer, M.S. Nodal bandoffdiagonal superconductivity in twisted graphene superlattices. Nat Commun 14, 7134 (2023). https://doi.org/10.1038/s41467023424714
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DOI: https://doi.org/10.1038/s41467023424714
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