Wess-Zumino-Witten Terms in Twisted Bilayer Graphene

arXiv:2007.00007

Superconductivity, correlated insulators, and Wess-Zumino-Witten terms in twisted bilayer graphene

Maine Christos, Subir Sachdev, and Mathias S. Scheurer Department of Physics, Harvard University, Cambridge MA 02138, USA

Recent experiments on twisted bilayer graphene have shown a high-temperature parent state with massless Dirac fermions and broken electronic flavor symmetry; superconductivity and correlated insulators emerge from this parent state at lower temperatures. We propose that the superconducting and correlated insulating orders are connected by Wess-Zumino-Witten terms, so that defects of one order contain quanta of another order and skyrmion fluctuations of the correlated insulator are a ‘mechanism’ for superconductivity. We present a comprehensive listing of plausible low-temperature orders, and the parent flavor symmetry breaking orders. The previously characterized topological nature of the band structure of twisted bilayer graphene plays an important role in this analysis.

I. INTRODUCTION tures in the order parameter complementary to superconductivity are then electrically charged. The mechanism of associating electric charge with a topological texture A number of recent experimental studies of twisted allows for electrical transport coming from skyrmion de- bilayer graphene (TBG) [1–7] have explored its phase fects; this has been discussed, in particular, in quantum diagram as a function of electron density and temper- Hall ferromagnets [23–25], which could also be relevant ature, and found correlated insulating states at integer for the description of TBG [26]. The skyrmion fluctu- filling fractions separating the superconducting domes ations of the complementary order are then a ‘mecha- at low temperatures. Complementary information has nism’ for superconductivity, analogous to skyrmion fluc- emerged from scanning probe measurements [8,9], show- tuations (i.e. hedgehogs) in the Néel order being a mech- ing a cascade of phase transitions with ‘Dirac revivals’ at anism for valence bond solid order in square lattice anti- the integer filling fractions ν: while the bare flat bands of ferromagnets [19]. TBG only exhibit Dirac cones around charge neutrality TBG has massless Dirac fermions at charge neutrality (ν = 0), additional flavor symmetry breaking is argued [27–29], and these extend all the way to a ‘chiral limit’ by the authors to lead to the re-emergence of Dirac cones [30] when the bands are exactly flat and Landau-level at non-zero integer ν; this defines the high-temperature like. Interestingly, WZW terms can also be obtained from “parent state out of which the more fragile superconduct- exactly flat Landau levels [31]. The arguments of Yao ing and correlated insulating ground states emerge” [9]. and Lee [32] show that the same quantized WZW term Here we propose a common origin for the supercon- is obtained from a theory which focuses on the vicinity of ducting and correlated insulating states. We will con- dispersing Dirac nodes, as would be obtained from a the- nect these orders by Wess-Zumino-Witten (WZW) terms ory which considers the flat (or nearly flat) band across [10, 11] with quantized co-efficients. The WZW term the entire moiré Brillouin zone. We will choose to use associates a Berry phase with spatiotemporal textures the first method here, and employ the theory of linearly- of the different order parameters. Textures or defects dispersing Dirac fermions at all momenta, while imposing in one order parameter contain quanta of the other or- the symmetry constraints arising from their embedding der, leading to proximate phases in which different order in the moiré Brillouin zone. This approach will allow parameters condense and break associated symmetries. us to account for the ‘Dirac revivals’ observed in recent In condensed matter systems in two spatial dimensions, experiments [8,9] in a relatively straightforward manner. WZW terms first appeared [12–18] in studies of the in- We will begin in Sec.II by introducing the Dirac terplay between the antiferromagnetic Néel and valence fermion model of TBG, and discuss its symmetry and bond solid order parameters on the square lattice [19]. arXiv:2007.00007v4 [cond-mat.str-el] 11 Dec 2020 topological properties. Sec.III will list possible spin- They also appeared earlier in the interplay between these singlet superconducting states. Sec.IV will introduce the orders in one dimension [20]. Indeed, studies of field the- partner order parameters mj, which combine in WZW ories with WZW and related terms have been crucial to terms with the superconducting orders. Without addi- our global understanding of the phase diagrams of quan- tional flavor symmetry breaking by a parent (or ‘high tum spin systems in both one and two spatial dimensions temperature’) order M, these mj characterize the corre- [20, 21]. lated insulators near ν = 0. A discussion of the parent Grover and Senthil [22] extended these ideas to include orders M, which are responsible for the Dirac revivals the superconducting order for fermions with Dirac disper- [8,9], appears in Sec.V. These M can combine with suit- sion on the honeycomb lattice, and this will be relevant able mj to form correlated insulators near ν = ±2. The for our analysis here. They showed that skyrmion tex- extension to superconductors with triplet pairing appears 2 in Sec.VI.

While our work was in progress, we learnt of the work C3 of Khalaf et al. [33], which contains some related ideas; we discuss the connections to their work further in Ap- pendixD.

II. MODEL AND SYMMETRIES

To construct the superconducting order parameters, C2y we could, in principle, start with a tight-binding model (b) Action of exact [29, 34–36] of the quasi-flat (and necessary auxiliary) (a) TBG lattice at small twist spatial symmetries bands, write down pairing terms on the lattice, and then angle θ. C3 and C2y. project onto the Dirac cones. Since we, however, do C3 b+ not have a clear understanding (other than symmetry) b+ how the pairing states should look like in real space, KM we here proceed differently by working entirely in mo- K'M KM b mentum space. Denoting the projection (implemented - C2x by operator P ) of the non-interacting Hamiltonian, H0, θ K'M onto the quasi-flat bands by HFB = PH0P , neglecting C2 any coupling between the different valleys of the original graphene layers (associated with index v = ±), and ne- C2y b- glecting spin-orbit coupling, the Hamiltonian must be of the form (d) Action of exact and emergent symmetries in X † HFB = ck,σ,v,s [δss0 kv + gv(k) · ρss0 ] ck,σ,v,s0 . (1) the moiré reciprocal k (c) Brillouin zones for layer 1 lattice with reciprocal (blue) and layer 2 (red) and lattice vectors√ Here σ denotes the spin of the electrons, s and ρx,y,z 2π mini Brillouin zone. b± = √ (1, ± 3). are indices and Pauli matrices in “generalized sublattice a 3 space” that gives rise to the Dirac cones to be discussed below. FIG. 1: Lattice geometry and symmetries. As discussed As can be seen in Fig.1, the lattice does not have an in the main text, we impose C2 as an emergent exact C2 symmetry, but we will impose it as it emerges symmetry. The primitive vectors of the moiré Bravais approximately at small twist angles; this follows natu- lattice will be denoted by ar = a(cos π/6, r sin π/6), rally from the fact that, at small twist angles, the dif- r = ±, with moiré lattice constant a, which we will set ference between the twist axis going through an AA-site to a = 1. or through the center of a hexagon, which leads to an exact C2 symmetry, vanishes asymptotically. To specify due to the C2 symmetry. A topological aspect of twisted the basis for the ρx,y,z matrices in Eq. (1), we choose bilayer graphene, which is crucial for the structure of the the representation RC2 = ρxτxσ0 with τj and σj acting WZW terms, is that it has two Dirac cones per spin and in valley and spin space, respectively; as required, C2 flips the valley and we choose it to flip generalized sub- valley at the KM and K’M points (in the following referred to as “mini valley”) of the moiré Brillouin zone with the lattice as well (to resemble the representation of C2 in same i.e. gxy := (gx(k), gy(k)) single-layer graphene). This fixes the representation, Θ, chirality [28, 29, 37], , v v v of time-reversal: it has to act between different valleys vanishes at these momentum points and winds around once for any contour surrounding KM or K’M. Due to (∝ τx,y) and we want it to be “on-site” in generalized Eq. (3), the chirality must be opposite in the other (non- sublattice space (∝ ρ0,z); out of these four options, only mini) valley. We, thus, consider the following low-energy Θ = σyρ0τxK (where K is the complex conjugation op- 2 1 theory where we only keep the Dirac cones at KM (p = +) erator) is consistent with Θ = − and [RC2 , Θ] = 0. and at K’M (p = −): Combining these two symmetries, we get RC2 Θ = σyρxτ0K, which acts locally in k-space and forces Eq. (1) Λ h to have the form X † pv HLE = fq,σ,v,s,p v νx qxρx X † x y q (4) HFB = ck,σ,v,s [ρ0kv + gv (k)ρx + gv (k)ρy]ss0 ck,σ,v,s0 , k pv i + νy qyρy + pv fq,σ,v,s0,p, (2) ss0 where pv where the velocities νx,y and pv only depend on the prod- x x y y gv (k) = g−v(−k), gv (k) = −g−v(−k), kv = −k−v, uct p·v = ± and momenta q (cut-off as |q| < Λ) are mea- (3) sured relative to the respective Dirac point. Using µx,y,z 3

TABLE I: Here we show the representations of the superconductivity, we here use the notation relevant symmetries in the basis used in Eq. (4), with τi, X † † µi, σi, and ρi, i = 0, x, y, z, acting in valley, mini-valley, HSC = fq ∆qT f−q + H.c. (6) spin, and sublattice space, respectively. For convenience q of the reader, we show more than a minimal set of generators. Here, Tar denotes moiré-lattice translation and refer to ∆q as the superconducting order parameter. by ar deﬁned in Fig.1, SU(2) s and U(1)v are Here, T = iσyτxµx is the unitary part of the anti-unitary conventional spin rotation and valley-U(1). The 2D time-reversal operator, Θ = T K, and the superconduct- space group of the full system is p6mm and that of a ing order parameter ∆q is a matrix in spin, valley, mini- single valley is the magnetic space group 183.188 [38]. valley, and (generalized) sublattice space; it must satisfy

T †∆T T = ∆ , g q = (qx, qy) Rg consequences in Eq. (4) −q q (7) C2 −q ρxτxµx — 2π T −i ρz τz vp vp ∆ C3 C3q e 3 νx = νy due to Fermi-Dirac statistics, with the superscript in + − C2x (qx, −qy) ρxµx νx,y = νx,y, + = − representing matrix transpose. Θ −q σyτxµxK — To organize the discussion and narrow down the mul- C2Θ q σyρxK — titude of possible superconducting order parameters, we 2πi iqar 3 µz r Tar q e e — will first concentrate on singlet pairing, i.e., we have iϕσ s s SU(2)s q e — ∆q ∝ ∆qσ0, where ∆q is a matrix only in valley, mini- iϕτz U(1)v q e — valley, and (generalized) sublattice space; we will come back to triplet pairing in Sec.VI below. Let us focus on pairing of electrons at opposite momenta k and −k in the moiré-Brillouin zone, thus, pre- to denote Pauli matrices in the mini-valley space, the serving moiré translational symmetry, and allowing us representations of all physical symmetries of the system to classify the pairing states accoring to the IRs of the are summarized in TableI. Note that these symmetries point group only. At least in the presence of time-reversal pv pv symmetry, this is expected to be energetically most fa- further imply ν = ν = ν and pv = , independent of x y p −p pv. Suppressing indices and setting = 0 without loss of vorable; it corresponds to pairing between and only generality, Eq. (4) can thus be written as (inter-mini-valley pairing). By the same token, it seems natural to focus on intervalley pairing. Note that the Λ U(1)v symmetry, associated with valley-charge conserva- X † tion, forbids mixing between intra- and intervalley pair- HLE = ν f (qxγx + qyγy) f , (5) q q ∆s q ing [39]. Taken together, we choose q such that only matrix elements with p0 = −p and v0 = −v are non-zero. Furthermore, let us restrict the discussion to the leading- where ν is the velocity of the moiré Dirac cones and γx = s order expansion of ∆q in q, i.e., just the constant term, τzρx and γy = ρy are 16×16 matrices with τi acting on s s ∆q → ∆ , since we are interested in the vicinity of the valley, µi on mini-valley, σi on spin, and ρi on generalized sublattice space. Dirac points. While this seems like a lot of constraints, there are, in The choices of γx,y in Eq. (5), and the symmetry trans- fact, still eight different independent pairing terms which formations in TableI, are sufficient to account for the can realize almost all irreducible representations (only topological character of the TBG band structure for our E1 is missing) of the point group D6 of the system, see purposes. Specifically, the Dirac chiralities of the 2 mini- TableII. valleys in Eq. (5) are the same, and this will play central The fact that we have two different pairs of ∆q that role in the structure of the WZW term. transform under E2 means that they will, in general, mix. In other words, the superconducting partner func- E2 tions for IR E2 have the form χk,1 = aρx + bµzρy and III. POSSIBLE SPIN-SINGLET PAIRING E2 χk,2 = −aτzρy + bτzµzρx, where a and b are some unde- STATES termined, real parameters that depend on microscopic details. Since E2 is a two-dimensional IR, the asso- We will begin our analysis by listing the possible su- ciated superconducting order parameter has the form P E2 perconducting order parameters within the low-energy ∆ = µ=1,2 ηµχk,µ, where the ηµ are constrained by theory introduced above, which we organize according symmetry and can only assume discrete values. to the irreducible representations (IRs) of the symmetry We also point out that it was previously shown [39] group. For a detailed classification of the pairing states that the singlet states odd under C2 cannot give rise to a in the full Brillouin zone and the consequences associ- finite gap at generic momentum points, where the (poten- ated with emergent symmetries and the behavior once tially spin and valley degenerate) bands are separated— these are weakly broken, we refer to Ref. 39. To describe this is related to the fact that C2 C2 simply flips the 4

TABLE II: Summary of different singlet pairing states capture the superconducting states (with Ns real compo- in the low-energy Dirac theory (5). The fact that there nents). Our goal is to systematically find the remaining are two different sets of order parameters transforming Ma, a = Ns + 3,..., 7, in the particle-hole channel, i.e., under E2 means that the corresponding basis functions of the form can mix. We also indicate, in the last column, whether X † X † Oj = Ψ M2+j+N Ψ = f mjf , (10) the pairing states will gap the Dirac cones, and refer the q s q q q q q reader to Ref. 39 for a discussion of the gap structure in the full Brillouin zone at non-integer ν. that will give rise to a joint WZW term for the unit length field na conjugate to the order parameters; na is s Order parameter ∆ transform as IR of D6 gap assumed constant in (9). We will refer to the associated 2 m j = 1,... 5 − N partner order parameters 1 const., z A1 j, s, as the of τzµz z A2 the superconducting state. These will be our candidates 2 2 τzµzρz x(x − 3y ) B1 for the correlated insulators found in experiment. 2 2 ρz y(3x − y ) B2 We know from Refs. 14 and 32 that a WZW will be 2 2 (ρx, −τzρy) (x − y , 2xy) E2 generated if 2 2 (µzρy, τzµzρx) (x − y , 2xy) E2 tr [Γi1 Γi2 Ma1 Ma2 Ma3 Ma4 Ma5 ] = 8N i1i2a1a2a3a4a5 , (11) with non-zero N . The integer N determines the co- sign of momenta (and valley here) in 2D, exactly as time- efficient of the WZW term. The WZW term can be writ- reversal does [40]. We here see that this also holds around ten in an explicit form preserving all symmetries only B the Dirac cones, since the states transforming under 1 by extending the field na to 4 dimensional spacetime and B2 in TableII will not induce a gap. (u, τ, x, y) with an additional dimension u: 5 2πN Z 1 Z X IV. WESS-ZUMINO-WITTEN TERMS SWZW = i du dτdxdy abcde Ω4 WITHOUT ADDITIONAL ORDERS 0 abcde=1 × na∂unb∂τ nc∂xnd∂yne , (12) Having established the notation, the non-interacting 2 where Ω4 = 8π /3 is the surface area of a unit sphere model, and the different superconducting states, we are in 5 dimensions. We are assuming here that the com- now in a position to look for natural WZW terms of those bined order parameters have 5 components. In models e.g. superconducting states with other order parameters, , with larger symmetry, there could be additional order associated with the correlated insulator. There will be no parameter components which would combine to yield a M ‘high temperature’ or ‘parent’ orders in this section. sum of terms like those in (12) but with a larger overall symmetry [41]. While the Nambu basis in Eq. (8) allows to bring all A. Procedure for finding WZW terms pairing states in TableII in the form of the mass terms in Eq. (9), it constrains the possible partner orders we can WZW terms have previously been studied in the con- study: we will only be able to write down mj that are text of Dirac theories [14, 32] and we will make use of diagonal in spin (∝ σ0, σz). One straightforward way to these results here. To this end, let us first define the generalize the analysis proceeds by considering several al- Nambu spinor ternative choices of non-redundant Nambus spinors, such as fq↑ Ψq = † , (8) f f f−q↓ q,p=+ q,v=+ Ψq = † , Ψq = † . (13) f−q,p=− f−q,v=− which is non-redundant, i.e., a complex rather than a Majorana fermion and the results of Refs. 14 and 32 ap- The first option allows to write down any inter-mini- ply. With the new field in Eq. (8), we can write the action valley pairing (singlet and triplet), which again in- associated with the above superconducting Dirac theory cludes all of the pairing states we are interested in. as Moreover, partner order parameters in the particle-hole Z Z channel with arbitrary spin polarization (only restricted 2 h † S = dt d q Ψq (∂t + q1Γ1 + q2Γ2)Ψq to intra-mini-valley, which means moiré-translation- invariant states) can be captured. The second choice 7 X i will still allow to write down all of our pairing terms, + n Ψ† M Ψ a−2 q a q (9) the inter-valley pairing order parameters; as for the part- a=3 ner order parameters, we can now write down density- where Γ1 = τzρx, and Γ2 = ηzρy, and ηi are Pauli matri- wave terms, that break the moiré translational sym- ces in Nambu space. In Eq. (9), Ma, a = 3,...Ns + 2, metry, but cannot write down any inter-valley-coherent 5 states. Clearly, many more choices are possible, such as IRs remain, leading to Ns = 2, as mentioned above. We † T Ψq = (fq,ρ=1, f−q,ρ=2) ; since the kinetic terms in our note that for non-integer fillings, such that the chemi- Hamiltonian are off diagonal in sublattice space, for the cal potential does not go through the Dirac nodes of the sublattice Nambu spinor a unitary transformation must normal-state bandstructure, the A1 pairing state can re- first be applied to the Hamiltonian to bring it to a form main gapless, while the A2 state will necessarily have 6 where the pairing term is off diagonal in sublattice space nodal points on any Fermi surface enclosing the Γ point i π ρ τ i π ρ τ µ and the kinetic terms are diagonal (e 4 x x and e 4 x y y [39]. for A1 and A2 pairings respectively). For each of A1 and A2, we have derived the com- However, a more efficient criterion that is equivalent plete list of mathematically possible sets of partner order to Eq. (11) for any such choice of Nambu spinor can be parameters satisfying Eq. (14); these are listed in Ap- derived, see AppendixB: the partner orders mj, Dirac pendixE. However, only a small fraction of them are matrices γi, and the superconducting order parameter physically meaningful options if we assume that none must obey of the symmetries in TableI are broken in the parent Hamiltonian for superconductivity and the partner order T γi∆T = −∆T γi 6= 0, i = 1, 2, (14a) parameters. T To understand the reduction of possibilities resulting mj∆T = ∆T mj 6= 0, j = 1, 2, 3, (14b) from symmetries, consider the mathematically possible [γ γ m m m ] ∝ . tr i1 i2 j1 j2 j3 i1i2j1j2j3 (14c) choice of

m1 = τxρx, m2 = τyµxρx, m3 = µxρz (15) Anticipating that this will be the only relevant case below, we have here already assumed that Ns = 2, i.e., for the partner order parameters in Eq. (10). As long only one-component complex superconducting order pa- as we have U(1)v, τxρx must “fluctuate with” τyρx; more rameters (two fluctuating real components) play a role. precisely, any low-energy field theory containing a field Note that the third condition requires the partner or- coupling to m1 must also contain another field that de- ders to anti-commute with the kinetic terms in our Dirac scribes fluctuations of τyρx. However, we have already Hamiltonian, implying they will gap out the Dirac cones. exhausted the number of three particle-hole order pa- We finally note that, although we began by considering rameters forming a WZW term with superconductivity. a non-redundant basis, the conditions which account for This would already be enough to discard this choice of every possible non-redundant Nambu basis are equivalent partner order parameters as it is incomplete from a sym- to Eq. (11) in a redundant extended Nambu basis. This metry perspective. We note that it is also incomplete and the criterion 14, are derived in AppendixB. due to translational symmetry which requires (at least to quadratic order) that, e.g., µxρz fluctuates with µyρz. This means that mj in Eq. (15) constitute a valid set of B. Possible partner orders partner orders only if both moiré translation and U(1)v are broken. Using the procedure outlined above, we can system- Applying such an analysis to all of the mathematically atically study all possible partner order parameters for possible sets of partner order parameters, we find the re- the different superconducting states. All of these or- maining, physically relevant options summarized in Ta- ders will have N = 2 in Eqs. (11) and (12). The value bleIII. We note that the symmetries leading to the reduc- N = 2 implies that a skyrmion in the partner order has tion of possibilities for mj, such as translation and U(1)v charge ±4e [22]. The anisotropies in the free energy of for Eq. (15), do not involve the spatial rotation symmetry the partner orders can allow stable half skyrmions (i.e. C6. Consequently, the presence of lattice strain and/or merons) of charge ±2e [33], and condensation of merons nematic order [6, 42–46] will not lead to additional op- or skyrmions leads to superconductivity. tions. Out of the pairing states in TableII, only those transforming under A1 or A2 allow for partner order parameters with WZW terms given that our alternative cri- V. HIGH-ENERGY SYMMETRY BREAKING terion (14) requires that the pairing multiplied with T AT HALF-FILLING must commute with the anti-symmetric γy = ρy and anti-commute with the symmetric γx = ρxτz. Of the Next, let us take into account the additional symmetry possible pairings, only A1 and A2 satisfy this condi- breaking, associated with an order M, that is believed to tion. In fact, it is no coincidence that this is corre- set in at much higher temperatures than superconduc- lated with whether these superconducting order param- tivity and the correlated insulator, as found in a recent eters will lead to a gap around the Dirac cones. Equa- experiments [8,9]. At present, the microscopic form of tion (14a) implies that the superconducting order param- the underlying order parameters is not known and so eters anti-commute with the kinetic terms in the Nambu we will systematically analyze different possibilities. For Hamiltonian and, as such, gap out the spectrum. Conse- concreteness, we focus here on the vicinity of half filling quently, only states transforming under one-dimensional of the conduction or valence band, i.e., ν = ±2. 6

TABLE III: Possible partner order parameters, mj, j = 1, 2, 3, see Eq. (10), for singlet pairing, assuming that all symmetries in TableI of the bare model (5) are preserved. We further deﬁned µ± = (µx ± µy)/2, τ± = (τx ± τy)/2, ω± = exp(±2πi/3), and provide the associated IRs of the point group D6. In the second to last column, we indicate how we denote these states in this work, including quantum spin Hall (QSH), a moiré density wave, which breaks moiré translation invariance and is even (MDW+) or odd (MDW−) under time-reversal, time-reversal even/odd intervalley-coherent phases (IVC±), a sublattice polarized state (SP), and a valley-polarized state (VP). In the last column, we denote the high-temperature orders M of Sec.IV (see also TableIV) for which the order will survive projection to one of the energy eigenspaces. M’s which will lead to additional Fermi surfaces as described in TableIV and are thus less likely are denoted with brackets. The partner order parameters listed here are possible at ν = 0 (without high-temperature orders M) and at ν = ±2 (given one of the listed M is present ).

Pairing mj IR Θ Tar U(1)v SU(2)s Type M A1 (τ+, τ−)ρx; ρz A1/B1; B2 + 1 m = 1; m = 0 1 IVC+; SP µx;[µzσz] ∗ A1 (µ+, µ−)ρz; ρzτzµz B2/A1; B1 + (ωr, ωr ); 1 m = 0 1 MDW+; VP τxρyµz; τzσz A1 (σx, σy, σz)τzρz A2 + 1 m = 0 3 QSH µx; τxρyµz;[τzµz] A2 (τ+, τ−)µzρx; ρz B2/A2; B2 −; + 1 m = 1; m = 0 1 IVC−; SP τzµx;[µzσz] ∗ A2 (µ+, µ−)τzρz; ρzτzµz A2/B1; B1 −; + (ωr, ωr ); 1 m = 0 1 MDW−; VP τzσz; τxρy A2 (σx, σy, σz)τzρz A2 + 1 m = 0 3 QSH τzµx; ρyτx;[τzµz]

To define the different options for this symmetry- TABLE IV: All possible high-temperature symmetry broken high-temperature state, we will consider adding breaking orders and how they transform. We take only momentum-independent quadratic terms to Eq. (5), i.e., one representative of states which are related by a the parent Hamiltonian HLE is replaced by U(1)v or SU(2)s rotation. We denote any state which breaks translation symmetry by MDW, any state which Λ X † breaks U(1)v conservation by IVC and indicate whether HeLE = fq [qxγx + qyγy + M] fq . (16) it also breaks time-reversal symmetry. "FM" denotes q ferromagnet, "Sp" denotes spin, "Mv" denotes mini-valley, and "V" stands for valley, one or more of Here M—a 16 × 16 matrix in valley, mini-valley, spin, which can be polarized ("P"). For example, we call the and sublattice space—is the high-temperature order pa- order τzµz "MvVP". We also indicate which states will rameter. As the data of Ref.9 indicates that Dirac cause bands to cross and yield additional Fermi surfaces. cones re-emerge around integer fillings as a consequence of these high-energy orders, we want M to commute with the Dirac matrices γx,y. Additionally, we require it to M SU(2)s Θ Tar U(1)v Extra FSs Type have only two different (and, hence, 8-fold degenerate) τzµz MvVP eigenspaces to correctly reproduce the reduction of de- µx MDW+ τ ρ µ IVC generacy of the Dirac cones by a factor of two at ν = ±2, x y z + τ σ SpVP see Fig.2(a,b). Because of this reduced degeneracy, the z z µzσz SpMvP WZW terms in this Section will have N = 1 in Eqs. (11) τzµxσz MDW+ and (12), and skyrmions in the partner orders will have ρyτxσz IVC+ charge 2e [22]. Finally, to further reduce the number ρyµxτxσz IVC-MDW+ of possibilities, we will focus on order parameter config- τzµx MDW− urations of M that are minima of symmetry-restricted ρyτx IVC− free-energy expansions and, as such, can be reached by a ρyτxµx IVC-MDW− second order transitions from the high-temperature phase σz FM without M. µxσz MDW− While the complete list of remaining M is provided τzµzσz SpMvVP in TableIV, we next discuss the different classes of order ρyτxµzσz IVC− parameters separately, organized by whether they respect time-reversal and/or spin-rotation symmetry.

terms, recall that all relevant pairing terms transform un- A. Preserving spin-rotation and time-reversal der A1 or A2 and, hence, are described by a single complex number (Ns = 2 real numbers). We are, thus, left Let us start with states that preserve both spin- with three partner order parameters, and SU(2)s is the rotation invariance and time-reversal symmetry. To see only symmetry with a three-dimensional IR. Therefore, that this is a particularly important class of M for WZW we will be able to ﬁnd cases without an anisotropy term 7

valley 1 = valley 2 = μ

K K' E M M (a) Degeneracy for ν = 0. (a) Bands without an M.

valley 1 = valley 2 =

E KM K'M E

(b) Bands with M = τzµz. (b) Degeneracy for ν = −2. FIG. 3: How additional Fermi surfaces emerge for the high-temperature order M = τzµz. Their absence requires the valleys v = ± to mix away from the KM (p = +) and K’M (p = −) points. The same is true for most M ∝ µz in TableIV. (a) shows the bare band structure around the KM and K’M points without M and how they connect along a one-dimensional E E momentum cut. Part (b) is the same with M = τzµz added, clearly exhibiting additional Fermi surfaces. For

μ μ simplicity, we have depicted the bandstructure here to be the same for either valley for all momenta, although they are only required to be mirror images of each (c) Degeneracy (d) Degeneracy other. Including this splitting away from the for ν = −3 with for ν = −2 with high-symmetry points does not alter our argument. eigenvalues eigenvalues {+1, 0, 0, −1}. {+1, 0, 0, −1}. where we grouped together symmetry-related choices, FIG. 2: Dirac cones (each cone is 2-fold degenerate) and with respect to translation and U(1)v symmetry. their filling (blue). (a) ν = 0 with no additional Intuitively, the first one in Eq. (17) simply corresponds symmetry breaking; (b) Dirac revival at ν = −2 due to to pushing down (up) in energy those states where val- a high-temperature order parameter M with two ley and mini-valley are identical (opposite). The sec- 8-dimensional eigenspaces; (c) ν = −3 with an M with ond one can be thought of as an “inter-mini-valley- 3 eigenspaces labeled by eigenvalues {+1, 0, 0, −1}; (d) coherent state” or a time-reversal symmetric moiré den- ν = −2 M also with an with 3 eigenspaces labeled sity wave (MDW+), breaking moiré-translation symme- {+1, 0, 0, −1}—there are no active Dirac cones and an try. Note that the actual system only has a discrete M with this structure will only work for ν = ±3. For translational symmetry, corresponding to a discrete ro- simplicity, we show only half of the 8 Dirac cones tational symmetry of the vector (µx, µy), see TableI; (associated with spin, valley, and mini-valley) and do therefore, it is associated with a discrete set of symmetry- not explicitly display that the Dirac cones are part of a inequivalent√ configurations—in this case, M = µx and bandstructure in the moiré Brillouin zone with finite M = µx + 3µy, as can be derived by minimizing the bandwidth, as in Fig.3. free energy (see AppendixA). Despite being inequivalent from the point of view of the symmetries of the microscopic model, we can focus only on one of these two op- between the different fluctuating partner order parame- tions, say M = µx, as they are related by the continuous iϕµ ters only if SU(2)s is present. Taking into account the symmetry, e z , which is an emergent symmetry of our constraints mentioned above to correctly reproduce the low-energy model (5), including all the superconducting Dirac revival, we are left with three classes of options states we consider. Finally, the third term in Eq. (17) is an “inter-valley-coherent” (IVC) state, that preserves M = τzµz,M = (µx, µy),M = (τx, τy)ρyµz, (17) translational symmetry. As a result of the continuous 8

U(1)v symmetry, we can choose M = τxρyµz without sulator is the only correlated insulator that can provide loss of generality. a WZW term for singlet pairing and both at ν = 0 as There is one additional restriction concerning these well as ν = ±2 with high-temperature M = τzµz. high-temperature orders, which is related to the connec- Rather than projecting the orders from the full space, tivity of the bands in the moiré Brillouin zone away from a simpler and a more general approach is to repeat the the Dirac cones, that we have not taken into account procedure of Sec.IVA to find WZW terms directly in the yet. This is most clearly illustrated by way of an exam- reduced (8 × 8) eigenspaces of M. First of all, this repro- ple: as illustrated in Fig.3, M = τzµz requires mixing duces the above finding that the QSH order parameter of the valleys away from the high-symmetry points, since remains a partner order parameter. Second, it also shows otherwise additional Fermi surfaces will necessarily ap- that M = τzµz allows for further partner order parame- pear in some parts of the Brillouin zone. In this sense, ters that were not included already in TableIII: for in- this choice of M and all other high-temperature order stance, as can be seen in first and seventh line in TableV, parameters that require additional mixing of the bands, the projection to one of the eigenspaces of M = τzµz al- which we indicate in TableIV, are less natural candidate lows for a set of partner order parameters consisting of orders to explain the behavior seen in experiment [8,9]. a (one-component) sublattice polarized (SP), ρz, and a However, for completeness, we study all of them. When (two-component) state, ρxµx(τ+, τ−), which can be inter- analyzing whether a state will give rise to extra Fermi preted as either and IVC or MDW in the full space. In surfaces, we allow for arbitrary mixing of the bands that the full space, this set of mj is incomplete due to trans- is not prohibited by the symmetries of the system. For lational symmetry, similar to the example in Eq. (15) instance, while one might think that τxρyµz will lead to discussed above. In either of the two eigenspaces of the same band structure as shown in Fig.3, the bands can M = τzµz, however, the action of translation can be rep- hybridize since U(1)v is broken (so is the emergent valley resented on the three components ρxµx(τ+, τ−); ρz since iϕρy τx symmetry, with e , away from the Dirac cones) and Mρxµx(τx, τy) = ρxµy(−τy, τx). unwanted Fermi surfaces can be avoided. In a similar way, the other two M in Eq. (17) can Next, we discuss the resulting possible WZW terms be- be analyzed. As for the mj in TableIII, which are al- tween superconducting orders and correlated insulators ready possible without any M, these two M only work born out of the high-temperature parent Hamiltonian for A1 pairing but are both associated with two different M (16) for the different possible . Note that there are partner orders. As can be seen in TableV, M = µx/y two crucial consequences of having an additional high- (M = ρyτx,yσx,y,z) makes one (two) additional set(s) of temperature order parameter: first, it can remove some of mj possible at ν = ±2. When breaking spin-rotation the options in TableIII of partner order parameters that symmetry in the next subsection below, we will see were possible without M since these order parameters that high-temperature orders M can stabilize many more vanish upon projection to the low- (relevant for ν = −2) partner orders with WZW terms. or high-energy (ν = +2) eigenspace of M. However, by Before turning to this, we mention there is some am- virtue of reducing the number of active degrees of free- biguity in presenting the partner order in the presence dom and by breaking certain symmetries, M can also of a given M. This follows from the observation that provide additional options that were not possible with- there are several partner orders in the full 16-dimensional out it. space that project to the same orders in the relevant 8- Because the physics will be the easiest, let us begin dimensional eigenspaces of M. E.g., both the regular by illustrating this with the first high-temperature or- 0 QSH insulator with mj = σjτzρz as well as mj = Mmj = der, M = τzµz, in Eq. (17). It is readily seen that σjµzρz are equally valid for M = τzµz. Here and in the it transforms under A2 and, hence, reduces D6 to C6. main text, we always only show one of these equivalent We can write down an effective model that only con- options. To this end, we will always show the unique form tains the 4 “active” Dirac cones, see Fig.2(b), by re- of the order parameter that will have N= 6 0 in Eq. (11) placing fq,σ,v,s,p → δp,vfeq,σ,v,s. The low-energy the- and, hence, can give rise to a WZW term in the full ory is now given by the four Dirac cones described by space (but, in some cases, will require additional broken PΛ † q feq [qxγex + qyγey]feq , with 8 × 8 reduced Dirac ma- symmetries). For completeness, we provide in TableXII trices γex = τzσ0ρx and γey = τ0σ0ρy (note: no µ0- a complete list that also contains these alternative and matrix anymore); the superconducting states become redundant choices explicitly. P † † HSC = ∆ q feq iσyτxfe−q + H.c., for both A1 and A2. They become identical upon projection, as expected since M = τzµz transforms under A2. It is straightforward to B. Breaking spin-rotation invariance project the partner order parameters in TableIII and one finds that only one set, (σx, σy, σz)τzρz, the three component QSH order parameter, survives projection; this is Let us next generalize our discussion of high- due to the fact that it is the only set of mj in TableIII temperature orders M to include the breaking of spin- that commutes with M = τzµz. We indicate this in the rotation invariance, while keeping time-reversal symme- last column of TableIII and conclude that the QSH in- try. In this case, we are left with the following combina- 9

TABLE V: Partner orders for singlet pairing which are not included in TableIII and are candidates for ν = ±2. We only list partner orders which are disallowed without the additional symmetry breaking of an M. We indicate the M for which the partner order mj is a candidate and note that the partner orders are only deﬁned up to multiplication by M in the projected space. For instance, the orders mj = ρzτzµy(σx, σy) with corresponding M = µzσz can also be expressed as mj = ρzτzσx(µx, µy), mj = ρzτzσy(µx, µy), or mj = ρzτzµx(σx, σy); all of these anti-commute and are orders that survive projection. M which will lead to additional Fermi surfaces as described in TableIV and are thus less likely are denoted with square brackets. The full set of orders for singlet pairing, including those in TableIII (up to projection) can be found in TableXII and TableXV. We only include those orders for which at least two components are related by a valley, mini-valley, or spin rotation. In the last column that indicates the type of partner order, "Sp" denotes spin, "S" denotes sublattice, "V" denotes valley, one or more of which can be polarized ("P"). "SBO" denotes spin-bond ordering, and "AFM" denotes antiferromagnetism (see Sec.VII for more info). We note that the labeling of the symmetries of each mj are only well deﬁned up to multiplication by the corresponding M. Also note for the cases in this table only, we distinguish between µx and µy for orders which have at least two distinct M’s proportional to both µx and µy.

M Partner Orders mj Partner SC SU(2)s Θ Tar U(1)v Type µx; τzµyσz;[τzµz] ρxµx(τ+, τ−); ρz A1 + IVC-MDW+; SP τzµxσz; τzσx/y;[µzσz] ρxµzσz(τ+, τ−); ρz A1 + IVC+; SP ρyτxσz; τzσz;[µzσx/y] τzρzσz(µ+, µ−); ρzτzµz A1 + MDW+; VP ⊥ τzµxσz; ρyµxτxσz; µy;[µzσz] ρzτzµy(σx, σy); ρzτzσz A1 + AFM+; SpVSP ⊥ ρyµxτxσz; τzσz; ρyτyσz; τyρyµz τyµzρx(σx, σy); ρzτzσz A1 + SBO-AFM+; SpVSP ⊥ ρyτxσz; τzµxσz;[τzµz] ρzµz(σx, σy); ρzτzσz A1 + SBO-AFM+; SpVSP τzµy; µxσz;[τzµz] ρxµx(τ+, τ−); ρz A2 + IVC-MDW+; SP µxσz; τzσx/y;[µzσz] ρxσz(τ+, τ−); ρz A2 −; + IVC−;SP τzσz;[ρyτxµzσz; µzσx/y] ρzσz(µ+, µ−); ρzτzµz A2 −; + MDW−; VP ⊥ µxσz; ρyµyτxσz; τzµy;[µzσz] ρzµy(σx, σy); ρzτzσz A2 −; + AFM−; SpVSP ⊥ ρyµxτxσz; τzσz; τyρy;[ρyτxµzσz] τxρx(σx, σy); ρzτzσz A2 −; + SBO-AFM−; SpVSP ⊥ µxσz;[τzµz; ρyτxµzσz] ρzµz(σx, σy); ρzτzσz A2 + SBO-AFM−; SpVSP

tions of Pauli matrices simplest minima with the correct eigenspectrum for each M and discuss any other possibilities in AppendixA. We M = τzσ,M = µzσ,M = τz(µx, µy)σ, (18) note, in passing, that no WZW terms are possible start- M = ρy(τx, τy)σ,M = ρy(τx, τy)(µx, µy)σ. ing from a parent theory with an M of this form that corresponds to filling ν = ±3: in this case, the effec- As before, we have already grouped them together as tive low-energy theory will be a theory with 4 × 4 Dirac multi-component order parameters such that different matrices. Since the maximal number of anti-commuting components transform into each other under the sym- 4×4 Hermitian matrices is 5, this is not compatible with metries of the system. While for the first two options Eq. (11). in Eq. (18) all possible orientations of these vector or- ν = ±2 der parameters are symmetry-equivalent, we have to an- Returning to , we conclude that Eq. (18) only alyze the possible stable phases for the remaining three leads to five different high-temperature orders to con- 4 8 choices; these are matrix- and third-rank-tensor-valued sider, which are summarized in line to in TableIV. order parameters. This analysis can be performed sys- We not only list their symmetries, but also whether they tematically by writing down the most general free-energy require additional mixing away from the KM and K’M expansion in terms of these components, see AppendixA. point, to avoid unwanted Fermi surfaces. We find that of the multitude of options, only some of We analyze these terms in the same way as above. As the configurations for each of the last three order param- before, we find that some of the partner orders which eters in Eq. (18) will have the correct eigenspace degen- are already possible at ν = 0 (without any M) remain, eracy needed for four degenerate Dirac cones at ν = 2 (or as indicated in the last column of TableIII. In addition, ν = −2). For example, for the high-temperature order the presence of these high-temperature orders leads to ρy(τx, τy)σ, both ρyτxσz and ρy(τxσx + τyσy) are sta- additional options, summarized in TableV (the full list ble minima of the most general free energy. While the of redundant options is given in TableXII); these latter first of these two options, does have only two eigenvalues, cases are, thus, only possible around ν = ±2. As antic- ±1, (each 8-fold degenerate) and, hence, shifts the Dirac ipated above, for all of the WZW terms with M break- cones as shown in Fig.2(b), the second one has eigenval- ing spin-rotation symmetry, the lack of three-dimensional ues ±1 (4-fold each) and 0 (8-fold) and, hence, can only IRs implies that not all three partner orders can trans- work for filling ν = ±3, see Fig.2(c). Here we take the form under the same IR and anisotropy terms between 10 the two distinct classes of partner orders are generically points: the A1 and A2 triplets do not induce a gap in expected. In fact, for M = ρyµxτxσz, a WZW term is our Dirac theory either. We also point out that triplet possible with all three particle-hole partner orders trans- pairing cannot be ruled out a priori due to the presence forming under different IRs (see TableXII). Since this of disorder, such as variations of the local twist angles, as requires more fine-tuning, we do not include this option triplet pairing can be protected by an Anderson theorem, in TableV. special to graphene moiré superlattices, as has recently been shown [47]. In TableVI, we have focused on the regular SU(2) s spin C. Breaking time-reversal symmetry symmetry and neglected the admixture of spin-singlet and triplet, possible due to the proximity to an enhanced Finally, we can also repeat the same analysis for high- spin symmetry [39]. Contrary to the case of singlet pair- temperature order parameters that are odd under time- ing, the IR of the complete symmetry group is thus three- reversal symmetry. We find that all of these terms are dimensional for A1,2 and B1,2: as is well known, there are incompatible with the A1 pairing term, i.e., the projec- two distinct types of stable triplet vectors, which we will tion of the A1 pairing term onto the eigenspaces of any choose as of these order parameters vanishes. For A2, the following d = (1, 0, 0)T , d = (0, 1, i)T four classes of time-reversal odd M are possible (20) and refer to as “unitary” and “non-unitary” triplets, re- τz(µx, µy), ρy(τx, τy), (µx, µy)σ, ρyµzσ(τx, τy). (19) spectively. E A discussion of all stable configurations of these multi- In the case of the IR 1, there are two different component orders can be found in AppendixA. But we forms of momentum-independent order parameters with find, as before, that the additional options which involve the same symmetries and the associated basis func- χE1 = (aρ + bµ ρ )σ linear combinations of the different components do not tions are superpositions, k,1,j y z x j and E1 have the correct degeneracies of eigenspaces required for χk,2,j = (aτzρx − bτzµzρy)σj, j = 1, 2, 3, a, b ∈ R. ν = ±2. The four M associated with Eq. (19) together Here, the superconducting order parameter has the form P E1 with the additional possible M with the right degenera- ∆k = µ=1,2,j=1,2,3 ηµ,jχk,µ,j. Since it transforms as cies to describe ν = ±2, but that lead to vanishing pair- the product of a two- and a three-dimensional IR, the ing, can be found in the last 7 lines of TableIV. The set of symmetry-inequivalent order parameters becomes different partner orders can be read off from TableIII quite rich and has been discussed in detail in Ref. 39 for and TableV as before. twisted bilayer graphene. Here, we will not need further details about these triplet phases since only τzσj (B1) and µzσj (B2) sat- T VI. GENERALIZATION TO TRIPLET PAIRING isfy the condition of ∆T γj = −γj ∆T with T = iσyτxµx, T T for γy = −γy = ρy, and γx = γx = τzρx which is the In this section, we extend the previous discussion to criterion (14a) for anti-commuting with the kinetic terms also include triplet pairing. in Nambu space. The unitary triplet in Eq. (20) corresponds to Ns = 2 × 3 real components; as this is already more than the A. Possible triplets states five components forming the WZW term in Eq. (12), it cannot give rise to WZW terms as long as spin-rotation We can repeat the same procedure of determining pos- invariance is preserved. Similarly, the manifold SO(3) of sible WZW partners for triplet pairing. To this end, the non-unitary triplet is not consistent with the WZW let us begin by discussing the different possible triplet term in Eq. (12) either. This is different if spin-rotation M states. These are characterized by an order-parameter invariance is broken by high-temperature orders , as we will discuss next. ∆q in Eq. (6) involving the spin Pauli matrices σx,y,z. As in singlet pairing, we restrict possible pairing terms to those which pair electrons with opposite momenta, and between opposite valleys and mini-valleys, i.e., only B. High-temperature orders and WZW terms the off-diagonal matrix elements of ∆q in valley and mini- valley space, p0 = −p and v0 = −v, are non-zero. Keeping We repeated the same analysis discussed in detail in only the momentum-independent terms around the KM Sec.V above for singlet pairing, but now for the two t and K’M points, ∆q → ∆ , we obtain the different triplet unitary and non-unitary triplets transforming under B1 states listed in TableVI according to the IRs of the spa- and B2; we went through all M that lead to a Dirac tial point group D6. Similar to the singlet case above, we revival at ν = ±2, investigated whether the respective see that the property derived in Ref. 39 of triplet states pairing states survive projection to their eigenspaces, and even under C2 not giving rise to a gap in isolated (valley- searched for all partner order parameters in this reduced and/or spin-degenerate) bands, carries over to the Dirac space which will give rise to joint WZW terms (12), with 11

TABLE VI: Summary of the triplet pairing states a symmetry, also the unitary triplets transforming under according to the IRs of the spatial point group D6. The B1,2 in TableVII are distinct from the singlets A1,2 with last column indicates whether the superconducting state WZW terms studied above. can gap out the Dirac cones. The allowed triplet vectors We also note that the partner orders discussed for the one-dimensional IRs are given in Eq. (20), while in the spinless model in Ref. 33—in our notation we refer to Ref. 39 for E1 and the gap structure of the µzρxσx(τx, τy); ρz, see AppendixD for more details—are pairing states in the entire Brillouin zone at generic ν. among our possibilities for triplet pairing. As can be read off from TableVII, these partner orders are possible t Order parameter ∆ Transform as IR of D6 Gap for unitary triplet pairing, with high-temperature order 2 µzρzd · σ const., z A1 M = µzσx and for non-unitary pairing with M = σx. τzρzd · σ z A2 Out of these two different M, only the first one will lead 2 2 τzd · σ x(x − 3y ) B1 to additional Fermi surfaces if no further mixing between 2 2 µzd · σ y(3x − y ) B2 the bands occurs far away from the KM and K’M points. (ρy, τzρx)σj (xz, yz) E1 As can be seen in TableVII, our analysis reveals that µz(ρx, −τzρy)σj (xz, yz) E1 there are many more options for triplet pairing and associated partner orders in the presence of M. We finally note that all of the partner order parameters for the triplets which are not spin polarized were already N = 1. The results are summarized in TableVII and present in TableIII above and can, thus, also be pos- will be discussed next. sible partner states for both singlet and triplet phases, First, as explained above, only M that break SU(2)s and both at ν = 0 (without M, singlet only) as well as are possible. In principle, there are two different ways of ν = ±2 (with the appropriate M). Clearly, the QSH breaking it: using the conventions for the triplet vectors state, τzρzσj in TableIII, can only provide the partner in Eq. (20), M could correspond to a polarization along order parameter for singlet superconductivity, as triplet σz. Then, as a consequence of the residual spin-rotation will necessarily require broken spin-rotation symmetry. If symmetry along the σz axis, both the unitary and non- we also take into account the partner orders for singlets unitary triplet have three independent real components. in TableV, which only work for ν = ±2, we see that For instance, the unitary triplet can be parametrized in all the partner orders in TableVII that are possible for this case as both unitary and non-unitary triplet pairing also work iϕ for singlet. d = ∆e (cos θex + sin θey) (21) = (n1 + i n2)ex + (n3 + i n4)ey, VII. SUMMARY AND DISCUSSION where we introduce the unit vectors ej and, in the second line, a redundant parameterization with the four na associated with mass terms Ma in the Nambu-Dirac Experimental studies of the low-temperature phase di- theory (9). Even if we ignore the additional constraint agram for TBG show superconducting domes separated (n1/n2 = n3/n4) accounting for the fact that only three by correlated insulators at integer filling fractions [2–7]. of them are independent, this does not correspond to the We connected spin-singlet superconductivity to possible scenario we are interested where skyrmions in the three- order parameters for correlated insulators, referred to as component partners orders carry electric charge and form partner orders mj, by WZW terms in Sec.IV. More re- the Cooper pairs. This is why we will not further discuss cent STM observations [8,9] have argued for further sym- this spin polarization of M. metry breaking in a high-temperature parent state with The second way of breaking spin-rotation symmetry Dirac fermions at each integer filling fraction. Only at by M corresponds to having M ∝ σx, i.e., along d for ν = 0 is no symmetry breaking required in this parent the unitary and perpendicular to it for the non-unitary state for the Dirac fermions to appear, as illustrated in triplet state in Eq. (20). While the non-unitary triplet Fig.2, and so the results in Sec.IV and the order param- iϕ transforms as d → e d under the residual spin rota- eters mj in TableIII can be applied at this filling. We tion (by ϕ along σx here) and, thus, will remain distinct considered order parameters M for the high-temperature from any of the singlets when introducing M, the uni- symmetry breaking in the vicinity of ν = ±2 in Sec.V tary triplet is explicitly invariant under the residual spin and TableIV; these additional orders have two conse- rotation. For this reason, one might be tempted to con- quences for WZW terms. First, for a given M, they clude that it becomes equivalent to one of the singlets in can rule out certain combinations of partner orders and TableII, mix with it, and will not have to be discussed superconductivity, since some of these order parameters separately. This is, however, not the case and again re- vanish upon projection to one of the eigenspaces of M. lated to the special role of C2 symmetry in two spatial di- However, all of the candidate orders for ν = 0 still remain mensions [39, 40]: we have seen that only singlet (triplet) possible for ν = ±2, if an appropriate M is present, as even (odd) under C2 can give rise to a gap and fulfill the indicated in the last column in TableIII. Second, the criteria for WZW terms. Consequently, as long as C2 is high-temperature order parameter will reduce the num- 12

a TABLE VII: Possible partner order parameters, mj, j = 1, 2, 3, see Eq. (10), for unitary (B1,2) and non-unitary b triplet pairing pairing (B1,2) with triplet vectors deﬁned in Eq. (20). The ﬁnal two columns correspond to the high-temperature orders required to break the spin-rotation symmetry, i.e., all of these options only work at ν = ±2. The second to last column, M a, refers to unitary and the last column, M b, to non-unitary triplet pairing.M a’s which will lead to additional Fermi surfaces as described in TableIV and are thus less likely are denoted with square brackets. The full set of orders for triplet pairing, including including those related by multiplication by M are listed in Table XIV and TableXV.

a b Pairing mj D6 Θ Tar U(1)v SU(2)s Type M M a b B1 /B1 (τ+, τ−)µzρx; ρz B2/A2; B2 −; + 1 m = 1; m = 0 IVC−; SP τzµxσx;[µzσx] σx a b ∗ B1 /B1 (µ+, µ−)ρz; τzρzµz B2/A1; B1 + (ωr, ωr ); 1 m = 0 MDW+; VP τzσx;[ρyτxµzσx] σx a b B1 /B2 (τ+, τ−)ρxσx; ρz A1/B1; B2 −; + 1 m = 1; m = 0 spIVC−; SP [µzσx] σx a b ∗ B1 /B2 (µ+, µ−)τzρzσx; τzρzµz A2/B1; B1 + (ωr, ωr ); 1 m = 0 MDW+; VP τzσx σx a ⊥ B1 (σy, σz)ρz; τzρzσx B2/B2; A2 −;+ 1 m = 0 AFM−; SpVSP [ρyτxµzσx] - a ⊥ B1 (σy, σz)ρzτzµz; τzρzσx B1/B1; A2 −;+ 1 m = 0 AFM−; SpVSP τzµxσx - a ∗ ⊥ B1 ρzµx(σy, σz); ρzτzσx B2/B2; A2 −; + (ωr, ωr ); 1 m = 0 MDW-AFM−; SpVSP [µzσx] - a ⊥ B1 ρxµzτx(σy, σz); ρzτzσx B2/B2; A2 + 1 m = 1 IVC-SBO+; SpVSP τzσx - a b B2 /B2 (τ+, τ−)ρx; ρz A1/B1; B2 + 1 m = 1; m = 0 IVC+; SP µxσx;[µzσx] σx a b ∗ B2 /B2 (µ+, µ−)τzρz; ρzτzµz A2/B1; B1 −; + (ωr, ωr ); 1 m = 0 MDW−; VP ρyτxσx; τzσx σx a b B2 /B1 (τ+, τ−)ρxσxµz; ρz B2/A2; B2 + 1 m = 1; m = 0 IVC+; SP [µzσx] σx a b ∗ B2 /B1 (µ+, µ−)ρzσx; τzρzµz B2/A1; B1 −; + (ωr, ωr ); 1 m = 0 MDW−; VP τzσx σx a ⊥ B2 (σy, σz)ρz; τzρzσx B2/B2; A2 −; + 1 m = 0 AFM−; SpSVP ρyτxσx - a ⊥ B2 (σy, σz)ρzτzµz; τzρzσx B1/B1; A2 −; + 1 m = 0 AFM−; SpVSP µxσx - a ⊥ B2 ρxτx(σy, σz); ρzτzσx B2/B2; A2 −; + 1 m = 1 IVC-SBO−; SpVSP τzσx - a ∗ ⊥ B2 ρzµxτz(σy, σz); ρzτzσx A2/A2; A2 + (ωr, ωr ); 1 m = 0 MDW-AFM+; SpVSP [µzσx] -

ber of low-energy degrees of freedom and break a certain • MDW+: time-reversal even, density modulations subset of the symmetries of the system, which will allow on the moiré lattice scale. for additional combinations of mj and superconductivity with a WZW term; these options, which are thus only • MDW−: as in MDW+, but time-reversal odd. possible at ν = ±2, are listed TableV. ρ τ µ In Sec.VI, we have repeated the same analysis for • VP ( z z z): valley, mini-valley, and moiré sublat- triplet pairing. Here, spin-rotation invariance has to be tice polarized state which is partner to either an m explicitly broken in the high-temperature phase to ob- MDW± as j. tain a WZW term. Therefore, the proposed connection ⊥ • AFM+: time-reversal even, in-plane, two- between correlated insulators and a triplet superconduc- sublattice antiferromagnet on the moiré lattice ν = 0 M tor will not be possible around . Additional scale. around ν = ±2, however, can reduce the spin symmetry ⊥ and lead to the various possible combinations of triplet • AFM−: as in AFM+, but time-reversal odd. pairing and mj summarized in TableVII. ⊥ Our comprehensive discussion of allowed combinations • SBO+: time-reversal even spin-bond ordering on of superconductivity and correlated insulators in the ab- the moiré lattice scale. sence or presence of possible M involves a variety of ⊥ different order parameters. Recalling that our starting • SBO−: as in SBO+, but time-reversal odd. point is the low-energy Dirac theory (5) with γx,y repre- • SpVSP (ρzτzσz): valley, spin, and moiré sublattice senting 16 × 16 matrices in valley (τi), mini-valley (µi), polarized state which is partner to either an AFM± spin (σi), and generalized sublattice space (ρi), we stud- m ied the following types of orders: or SBO± as j. • QSH: quantum spin Hall, leading to opposite Chern • IVC+: time-reversal even intervalley coherent number bands for spin up and down. state, which has density modulations on the graphene lattice scale. We finally make a few remarks on the structure and • IVC−: as in IVC+, but time-reversal odd. implications of our central results in TablesIII-V and VII. Let us first note that only the superconducting states • SP (ρz): moiré sublattice polarized state which is transforming under one-dimensional IRs of D6 can give partner to either an IVC+ or IVC− as mj. rise to WZW terms, irrespective of M and filling. In fact, 13 for singlet only A1 or A2 and for (both unitary or non- that M = σx around ν = ±2 is realized (not realized), unitary) triplet only B1 or B2 are possible. If, indeed, the the superconducting state will have to be (cannot be) a correlated insulators and superconductors are intimately non-unitary triplet. related by a WZW term, the number of pairing states is We point out that our key results—the sets of part- thus fairly constrained, as the two-dimensional IRs give ner orders and high-temperature order parameters M— rise to the majority of different superconducting order are not altered when three-fold rotation symmetry, C3, parameters [39]. On top of this, the superconducting is broken due to the presence of strain and electronic domes closest to charge neutrality will have to be singlet nematic order [6, 42–46]; the broken C3 symmetry can in that scenario. If the superconductor is due to electron- also explain the observed Landau-level degeneracy near phonon coupling, we know from the general analysis of charge neutrality [49, 50]. To see why it does not affect [48] and [47] that the superconducting order parameter our results, first note that removing the C3 symmetry must be spin singlet and transform trivially under all will allow the Dirac cones to move away from the K and symmetries. Consequently, only the particle-hole order K0 points, but we can still write down a low-energy the- parameters in the lines with pairing A1 in TableIII and ory as in Sec.II by expanding around the shifted Dirac TableV are possible. One would then view the electron- cones. The only modifications are anisotropic Dirac ve- phonon coupling having “tipped the balance” towards a locities and that the momentum transfers and, hence, particular type of superconductivity. The WZW term, the MDW states become incommensurate with the moiré which is a Berry phase term independent of a specific lattice. However, because none of the relevant supercon- Hamiltonian [32], will continue to apply and constrain ducting states transform non-trivially under C3 and we the partner insulating orders. did not use this symmetry to exclude further partner or- It is also worth pointing out that, while we have iden- ders or high-temperature orders, TablesIII-V andVII tified 15 possible high-temperature orders, four of them still apply when C3, is broken (with the sole exception have to be regarded as less natural choices: they require of the transformation behavior of the MDW states under additional symmetry breaking away from the KM and Tar in TablesIII andVII). While our mechanism thus K’M points to avoid spurious Fermi surfaces coexisting still applies when electronic nematic order (and strain) with the Dirac points (see TableIV). This also has impli- breaks C3, nematic order itself cannot be a partner order cations for the partner orders as it, e.g., makes M = µzσx parameter for any superconductor, as it is inconsistent ⊥ and, hence, the MDW-AFM± and spIVC− partner orders with Eq. (14c). less plausible for unitary triplet pairing. A recent Monte-Carlo study [51] has found evidence of Furthermore, we emphasize that our relation between the VP state, with order parameter ρzτzµz, around ν = 0 M and the associated sets of superconducting and part- (referred to as quantum valley Hall state in Ref. 51). As ner order parameters could give crucial insights. For in- can be seen in TableIII, this order together with MDW ± stance, if future experiments establish that the parent can provide the three partner order parameters for both state around ν = ±2 is characterized by the MDW+ or- singlet pairing states around charge neutrality. However, der parameter M = µx, the pairing state must be the we caution that the two mini-valley Dirac nodes have A1 singlet and the partner order parameters have to be opposite chiralities in Refs. 51, while those in Eq. (5) have either the IVC+ and SP phases, mj = (τxρx, τyρx; ρz), the same chiralities. It is not clear whether the short- or the QSH state with mj = τzρzσj or the IVC-MDW+ range non-local interactions in their models are sufficient and SP phases, mj = (ρxµxτx, ρxµxτy; ρz). Furthermore, to include the effects of the WZW terms of the same if a correlated insulating state mj breaks time-reversal chirality Dirac nodes that we have investigated here. symmetry then the pairing cannot be singlet pairing and The WZW connection between the superconductivity transform under A1; in that case, electron-phonon cou- and the correlated insulator order also has interesting pling alone cannot be responsible for superconductivity consequences for the structure of the core of a supercon- as mentioned above. However, if the high-temperature ducting vortex which could be explored in scanning tun- order M breaks only translational symmetry, but pre- neling microscopic experiment. By analogy to vortices in serves all others in TableI, the pairing must be the A1 the valence bond solid order of insulating antiferromag- singlet. nets carrying unpaired spins [52, 53], superconducting We note that QSH is the only example of a set of vortices would carry quanta of the partner order. partner order parameters where all three components Taken together, we have proposed a mechanism by are related by symmetry and, as such, requires the least which superconductivity and the correlated insulators are amount of fine tuning of all mj. As can be read off in intimately related in TBG (see also the work of Khalaf TableIII, it is relevant to both A1 and A2 singlet pairing et al. [33] discussed in AppendixD). While future ex- at ν = 0 and ν = ±2 with five possible M (four of which periments will have to establish whether this is realized will not give rise to extra Fermi surfaces); for all other in the system or not, we believe that our systematic dis- partner orders, two different IRs have to be energetically cussion of the different microscopic realizations of this close in energy for the connection of correlated insula- physics can help constrain the order parameters of su- tor and superconductivity to be physically plausible. To perconductivity, the correlated insulators, and the high- provide another example, if future experiments establish temperature parent in TBG and, potentially, also related 14 moiré superlattices. Numerical studies of models with distinct solutions for this option which cannot be related WZW terms [17] perturbed by symmetry-breaking and by translation: chemical potential terms will also be useful. √ 1 3 m = µx m = µx + µy (A5) 2 2 Acknowledgements As second example, we study M = ρy(τx, τy)σi, and M = P P3 v ρ τ σ i.e. We acknowledge useful discussions with S. Chatter- write i=x,y j=1 ij y i j, , parametrize it v jee, E. Berg, P. Jarillo-Herrero, E. Khalaf, Shang Liu, by the matrix-valued real order parameter . Deriving R. Samajdar, T. Senthil, A. Yazdani, O. Vafek, and the action of all symmetries in TableI, it is easy to show F A. Vishwanath. This research was supported by the that the most general expression for the free energy National Science Foundation under Grant No. DMR- up to quartic order reads as 2002850. T T 2 T T F = a tr v v + b1 tr v v + b2tr v vv v , (A6) a b b Appendix A: Free-energy expansions for M with unknown real-valued coefficients , 1, and 2. Min- imizing (conveniently done via singular-value decompo- sition) and taking the symmetry-inequivalent minima Some of the high-temperature order parameters M, yields the options in the first row of TableVIII. The anal- given in Eqs. (17), (18), and (19) of the main text, have ysis M = τz(µx, µy)σ is similar. However, note that one several components and are vectors, matrices, or third- has to go to sixth order to find all symmetry in-equivalent rank tensors. Assuming that these phases are reached states in order to determine which ground states are not by a single, second order phase transition, each of them equivalent by translation. can only assume certain discrete configurations. We here Finally, we briefly summarize how we obtained the derive these configurations by writing down the most gen- phases for the more complicated, tensor-valued case eral free-energy expansions and minimizing them. ρy(τx, τy)(µx, µy)σ. We again parametrize as As this is a standard procedure and the analysis is very similar for the different cases, we illustrate it with M = vlmnρyµlτmσn, vlmn ∈ R, (A7) a few instructive examples and collect the final results in TableVIII. with summation over repeated indices assumed and P We begin with M = i=x,y viµi, vi ∈ R, which is just rewrite a vector-valued order parameter. For convenience, we M = ρ (v µ τ σ + v µ τ σ ) rewrite our real vector v as a complex scalar: y xmn x m n ymn y m n 1 1 1 = (vxmn + ivymn)(µx − iµy) M = (µ + iµ )(v − iv ) + (µ − iµ )(v + iv ) 2 2 x y 1 2 2 x y 1 2 (A8) 1 ≡ µ+v∗ + µ−v + (v − iv )(µ + iµ ) τ σ ρ 2 xmn ymn x y m n y ∗ + − so that under our discrete translation symmetry, the ≡ v µ + vmnµ τmσnρy iφ 2π mn phase φ of v = e |v| transforms as φ → φ + 3 . To quartic order, the most general form the free energy may and write the most general rotationally invariant free en- take that obeys translation symmetry is ergy we can construct from a complex rank 2 tensor that 2 3 3 4 obeys the symmetries of our system. F ∝ a|v| + b1Re[v ] + b2Im[v ] + c|v| (A1) † † 2 † † F = atr v v + b1(tr v v ) + b2tr v vv v In minimizing this free energy, we note that only the third † ∗ T † T ∗ † ∗ 2 order terms proportional to b1 and b2 here will fix the + b3tr v v v v + b4tr v vv v + b5|tr v v | phase φ. We constrain these two coefficients by consid- (A9) ering how v transforms under our remaining point group b symmetries: or equivalently with re-definition of couplings i: ∗ F = a v†v + b ( v†v)2 + b0 (v∗ v )2 C2 : v → v (A2) tr 1 tr 2 mn mp npq 0 ∗ 2 0 ∗ 2 +b3(vnmvpmnpq) + b4(mklnpqvmnvkp) (A10) 0 ∗ ∗ c3 : v → v (A3) +b5(ablrsqmklnpqvmnvkpvarvbs)

∗ We note under the point group symmetries: C2x : v → v (A4) z ∗ C2 : vmn → −σmm0 vm0n (A11) Allowing us to set b2 = 0. Then the phase of v is ﬁxed by maximizing (minimizing) cos 3φ which has max- π 2πn ima (minima) at φ = 0, 3 + 3 . Then we have two C3 : vmn → vmn (A12) 15

∗ C2x : v → −v , (A13) Appendix B: Alternative condition for WZW term all of which leave the free energy invariant. We can con- We here provide a derivation of the set of conditions sider what vmn will minimize the energy depending on in Eq. (14) of the main text for a WZW term, the values of the coefficients and obtain the following T c.1 For γi we have γi∆T = −∆T γ 6= 0, i = 1, 2, ground states: i T c.2 For mj we have mj∆T = ∆T mj 6= 0, j = 1, 2, 3, 1 i 0 iφ 1 0 0 iφ v = e v = e c.3 tr[γi1 γi2 mj1 mj2 mj3 ] ∝ j1,j2,i1,i2,i3 , 1 0 0 0 2 i 0 0 by showing that they are equivalent to Eq. (11). To recall 1 i 0 iφ 1 0 0 iφ v = e v = e our notation, γi denote the Dirac matrices, ∆iσyτxµx = 3 i −1 0 4 0 0 0 ∆T is our pairing order parameter, ∆, multiplied by the (A14) 1 0 0 iφ 1 0 0 iφ unitary part of the time-reversal operator T = iσyτxµx, v5 = e v6 = e 0 1 0 0 i 0 and mj, j = 1, 2, 3, are the partner order candidates. a 0 i a ib 1 We can construct all possible partners in the non- v = eiφ v = eiφ 7 0 b 0 8 ia −b −i redundant Nambu bases specified by Eq. (8) and Eq. (13) by first considering all possible orders in a redundant Nambu space given by the spinor: b The final two ground states, depend on the couplings i and and have a, b 6= 0 (however we note for most possible fq Ψq = † (B1) couplings bi, the ground states are one of the first six op- f−q tions). We verify the above states are ground states by scanning the space of couplings and verifying that min- provided that all partner orders we can construct in a imizing the free energy over many points in the phase redundant Nambu basis will survive projection to some P = 1 (1 + space does not yield any new minima. Therefore, while non-redundant Nambu basis via projection C 2 η C) η the above states are true ground states for some region z where z acts here on the redundant Nambu space, C is real and symmetric, and ηzC commutes with all of the parameter space of the bi, it is possible that there are additional ground states in some small region of pa- the terms in our Hamiltonian in the redundant Nambu A rameter space that is missed by this minimization proce- basis. For example, for singlet pairing 1 and the first (τ , τ )ρ ; ρ C = µ dure. The phase φ is fixed by adding a cubic or sixth (or option in TableIII + − x z, we may choose z C = σ higher) order term to the free energy. The most general or z for the definition of our projection operator. C third order term we can add to fix φ is: These two choices of correspond to the spin-Nambu and mini-valley-Nambu spinor, given by Eq. (8) and by the first choice in Eq. (13), respectively. The second ∆F = v v v 3 abc ijk ai bj ck (A15) Nambu spinor in Eq. (13), corresponding to C = τz, would not work in this case, as the IVC state under con- However, we note this term is 0 always since our tensor is sideration does not commute with τz. a 2×3 matrix. We then consider the most general sixth In the redundant basis, our kinetic terms take the form: order term we can add that will depend on the phase φ, Λ X † Hkin = Ψq[ρxτzqx + ρyηzqy]Ψq (B2) T 3 T T T T T T ∆F6 = b5tr[v v] +b6tr[v vv v]tr[v v]+b7tr[v vv vv v] q (A16) Our pairing term is given by: We find the states v1, v2, and v3 can be made independent of φ via a valley or spin rotation. For v4, v5, and v7 Λ X † the 6th order contribution requires minimizing (or max- Hpair = Ψq[Re[δ](−σyτxµx∆ηy) imizing) cos(6φ), v6 and v8 require a 12th order term q (B3) which minimizes (maximizes) cos(12φ), yielding the 17 + Im[δ](−σyτxµx∆ηx)]Ψq orders listed for this option in TableVIII. We note that the options which are relevant for our analysis at ν = 2 where δ = eiφ captures the phase of the superconducting are those which have an eigenspectrum {+1, +1, −1, −1} order parameter. as shown in Fig.2. Of the options in Table√ VIII, the We assume conditions (c.1), (c.2), (c.3) and show only ones with this spectrum are ρyσxτx( 3µx +µy) and that they imply Eq. (11). The transformation which ρy(σxτxµx + σyτyµy). Therefore in the rest of the text takes partners mj, kinetic terms γi and ∆T to a (non- and in particular in TableV, M = ρyτxµxσx and cor- redundant) Nambu basis is as follows: responding orders mj will also have additional distinct options for mj which may be obtained by applying the 1 T 1 T π π m → M = (m + m )η + (m − m ) P −iµz −iτz µz σz j j j j z j j C rotations e 12 and e 4 to the mj for this M 2 2 only. (B4) 16

TABLE VIII: Ground states of free energy expansion for high-temperature orders. We omit states which are related

to another state in the table by a spin or U(1)v rotation or by Tar .

M Possible Ground States ρy(τx, τy)σ ρyτxσz ρy(τxσx + τyσy) τz(µx, µy)σ τzµxσz τz(µx√σx + µyσy) τzσx( 3µx + µy) ρy(τx, τy)(µx, µy)σ ρyτx(µxσx + µyσy) ρyσx(µxτx + µyτy) ρy(τxµxσx + τxµyσy − τyµxσy + τyµyσx) ρyτ√xσxµx ρyτxσx( 3µx + µy) ρyµx(τxσx +√τyσy) ρy(τxσx + τyσy)( 3µx + µy) √ ρy(µxτxσx + µyτyσy) √ √ ρy(µx( √3τxσx − τyσy) + µy(τx√σx + 3τyσy)) √ ρyµx((1 +√ 3)σxτx − √( 3 − 1)τyσy) + ρyµy(( √3 − 1)σxτx + ( √3 + 1)τyσy) ρyµx((1 + 3)σxτx + ( 3 − 1)τyσy) + ρyµy(−( 3 − 1)σxτx + ( 3 + 1)τyσy) √ ρy(µx(aτxσx + bτyσy) + µy(τxσz)) √ ρy(µx( 3(aτxσx + bτyσy) − τxσz) + µy(((aτxσx + bτyσy) + 3τxσz)) ρy(µx(aτxσx − bτ√yσy + τxσz) + µy(aτyσx + bτxσy − τyσz)) √ ρy((aτxσx − bτyσ√y + τxσz)( 3√µx + µy) + (aτyσx + bτxσy − τyσz)(−µx +√ 3µy)) √ ρy((aτxσx − bτyσy + τxσz)(µx( √3 + 1) + µy( √3 − 1)) + (aτyσx + bτxσy − τyσz)(−µx(√ 3 − 1) + µy√( 3 + 1))) ρy((aτxσx − bτyσy + τxσz)(µx( 3 + 1) − µy( 3 − 1)) + (aτyσx + bτxσy − τyσz)(µx( 3 − 1) + µy( 3 + 1))) τz(µx, µy) τzµx ρy(µx, µy)(τx, τy) ρyτxµx ρy(µx√τx + µyτy) ρyτx( 3µx + µy) (µx, µy)σ σxµx µx√σx + µyσy σx( 3µx + µy) ρyµz(τx, τy)σ ρyµzτxσz ρyµz(τxσz + τyσy)

1 T 1 T mj will commute with all the γi and superconducting γi → Γi = (γi + γ ) + (γi − γ )ηz PC (B5) 2 i 2 i term in an extended Nambu basis, which we may see by applying the above transformations to the γi, mj and pairing term, without the projection operator Pc. Finally ∆T → M4,5 = i∆T ηx/yPC (B6) we show that the trace in Eq. (11) is nonzero. Note that where ηz acts on redundant Nambu space and ηzC in the in the non-redundant Nambu basis our two components 1 M M ∝ 1 (1 + projector PC = 2 (1 + ηzC) is chosen to commute with of the superconducting order satisfy 4 5 2 all γi and mj and to anti-commute with ∆T . Further- ηzC)ηz and we have: more, C must be real and symmetric as already stated γ γ m m m [∆T ] [∆T ] → Γ Γ M M M M M above. Such a choice C always exists for each of our 1 2 1 2 3Re Im 1 2 1 2 3 4 5 C = σ µ τ 1 1 candidate pairings for some 0/x 0/z 0/z (assuming = (1 + η C) (γ γ m m m + γT γT mT mT mT )η for our triplet states that they are polarized along the 2 z 2 1 2 1 2 3 1 2 1 2 3 z σx-direction). This is guaranteed by (c.3) which requires 1 T T T T T m1m2m3 ∝ ρzτz, indicating that an even number of mj + (γ1γ2m1m2m3 − γ γ m m m ) ηz 2 1 2 1 2 3 must anti-commute with σx, µz, and τz respectively and (B7) that we then may always ﬁnd some product of µzτzσx that commutes with the mj. Given the anti-commutation relations of the mj and 1 tr[Γ1Γ2M1M2M3M4M5] ∝ tr (1 + ηzC)γ1γ2m1m2m3 γi in (c.3), we see the Mj j = 1, 2, 3 and Γi will anti- 2 commute and that (c.1) and (c.2) similarly imply that (B8) M4 and M5 will anti-commute with M1,2,3 and Γ1,2. We note that (c.1), (c.2), and (c.3) imply that any partner Where the anti-symmetric part of Γ1Γ2M1M2M3 does 17 not contribute to the trace and we have used the condi- fermions. The latter are important for our analysis as T T T T T tion tr[γ1γ2m1m2m3] = tr[γ1 γ2 m1 m2 m3 ] which holds they automatically emerge when using redundant Nambu given the anti-commutation relations of the γi and mj. spinors. Our beginning action is: Then we see given tr[γ1γ2m1m2m3] ∝ Id that: Z Z " 5 # 2 † i X 1 S = dt d q fq ∂t + γiq + m na−2ma−2 fq tr[Γ1Γ2M1M2M3M4M5] ∝ tr (1 + ηzC)γ1γ2m1m2m3 2 a=3 † ∗ † ∗ 1 + mfqRe[δ]∆T f−q + imfqIm[δ]∆T f−q + H.c., = tr γ1γ2m1m2m3 ∝ Id 2 (C1) (B9) where δ ∈ C captures the complex phase of the supercon- We have, thus, shown that (c.1), (c.2), and (c.3) imply ducting order parameter and n4 = Re[δ] and n5 = Im[δ]. Eq. (11). The matrices mj square to unity, n is a 5 component To verify the converse of this, we assume the condition unit vector and m is the magnitude of the orders mj and Eq. (11) superconducting orders which transform under SO(5). We then may deﬁne a redundant Majorana spinor as:

tr [Γi1 Γi2 Mi1 Mi2 Mi3 Mi4 Mi5 ] ∝ j1,j2,i1,i2,i3,i4,i5 (B10) η1q 1 ∗ i ∗ η = , η1q = √ (fq+f−q), η2q = √ (fq−f−q). η2q 2 2 and show that (c.1), (c.2), and (c.3) follow. We note (C2) M Γ that the anti-commutation relations of i and i im- We introduce the Pauli matrices ηi which act on redun- ply that Eq. (c.1) and Eq. (c.2) are satisfied and also dant Majorana space. We define γ0 = ρzτz and the fol- that mj and γi must anti-commute. Note that this is lowing additional operators: where the requirement that CT = C and C be real become relevant. It can be verified that M1 and M2 anti- iγ γ˜ = γ η γ˜ = − 0 ((γ + γT )η − (γ − γT )) commuting yields the conditions (1 + C){m1, m2} = 0 0 0 y i i i y i i T T 2 and (1 − C){m1 , m2 } = 0, which requires C be sym- γ0 T T m˜ 1,2,3 = − ((mj + m ) − (mj − m )ηy)m ˜ 4,5 = −γ0η ∆T metric to insure {m1, m2} = 0. Similar arguments hold 2 j j x/z for the Γi and M4 and M5 with the requirement that (C3) C∗ = C. We then establish the trace in Eq. (c.3) is nonzero by: So that our action may be re-written as: [Γ Γ M M M M M ] ∝ I tr 1 2 1 2 3 4 5 d Z " 7 # 3 1 µ X 1 S = d q ηq iγ˜µq + m na−2m˜ a−2 η−q =⇒ tr (1 + ηzC)(γ1γ2m1m2m3 (B11) 2 4 a=3 (C4) Z T T T T T 1 3 T +γ γ m m m ) ∝ Id ≡ d qη Mη 1 2 1 2 3 2 q −q m γ The anti-commutation relations of j and i ensure h i [γ γ m m m + γT γT mT mT mT ] = 2 [γ γ m m m ] µ P7 tr 1 2 1 2 3 1 2 1 2 3 tr 1 2 1 2 3 . where M =γ ˜0 iγ˜µq + m a=3 na−2m˜ a−2 and ηq = Then we have: T ηq γ˜0 and the time coordinate is included in the measure R 3 1 d q. When integrating out fermion fields, we have for tr (1 + ηzC)γ1γ2m1m2m3 ∝ tr [γ1γ2m1m2m3] ∝ Id 2 Majorana fields: (B12) Z m γ − 1 ηT Mη Then with the anti-commutation relations of j and i Pf(M) = dηe 2 (C5) we have that Eq. (11) implies:

tr[γi1 γi2 mj1 mj2 mj3 ] ∝ i1,i2,j1,j2,j3 (B13) It may be veriﬁed that M is a skew-symmetric matrix. We may then re-exponentiate to get an eﬀective action and (c.1), (c.2) and (c.3) are true as desired. with the help of the identity:

n2 1 tr[ln((η ⊗I )T M)] Pf(M) = i e 2 y n (C6) Appendix C: Derivation of WZW term for Majorana

Fermions 2 where in is a prefactor that will not change our result. Then we have: While WZW terms have been previously derived for complex fermions in several works, see, e.g., Ref. 41 for a 1 T Seﬀ = − tr[ln((ηy ⊗ In) M)] (C7) recent study, we here provide a derivation for Majorana 2 18

We then expand about a small δn · m˜ and find: where in the final step, we introduce auxiliary coordinate u, and extend our unit vector n to depend on the addi- 1 T Seff = − tr[ln((ηy ⊗ In) M)] tional coordinate u such that n(τ, x, y, 1) = n(τ, x, y) 2 and n(τ, x, y, 0) = (1, 0, 0, 0, 0). Finally, we divide by 4 1 1 ηyγ˜0mδn · m˜ u = − tr[ln(−ηyM)] + tr[ln(1 + )] so that the coordinate will also be anti-symmetrized 2 2 ηyM and take the trace over our m˜ j we obtain: 1 mδn · m˜ M †γ˜ 0 Z Z 1 ' Tr † 16N iµνρσ 3 2 M M δSeff = d x duαβγδtr [nα∂µnβ∂ν nγ ∂ρnδ∂σn] 512π (C8) 0 Z Z 1 4!N i 3 where we ignore the first term in the expansion as it = d x duαβγδtr [nα∂τ nβ∂xnγ ∂ρnδ∂yn] 32π 0 will not contribute to the topological term. Expanding Z Z 1 M †M, the term which is of the correct order in n · m˜ 3N i 3 = 2π 2 d x duαβγδtr [nα∂unβ∂τ nγ ∂xnδ∂yn] , and momenta to contribute to the WZW term we are 8π 0 interested in is: (C13)

1 2 2 2 −1 which is the topological term we wished to derive at level δSeﬀ = tr m δ(−∂ + m ) 2 (C9) N where N is 1 for ν = 2 and 2 for ν = 0. 2 2 −1 3 ×((−∂ + m ) mγ˜µ∂µ(n · m˜ )) n · m˜

Here δSeff is the variation of the action with respect to Appendix D: Connection to Khalaf et al. δn · m˜ . Inserting complete bases of momenta eigenstates and neglecting momenta indices of n·m˜ , in Fourier space The work of Khalaf et al. [33] appeared while our we find: work was in progress. They discuss a specific scenario Z 3 Z 3 Z 3 Z 3 d q d k1 d k2 d k3 for the WZW term in a spinless model of twisted bilayer δSeff = −i τ (2π)2 (2π)2 (2π)2 (2π)2 graphene. Using our labelling of Pauli matrices— for valley, µ for mini-valley, and ρ for sublattice—they con- 1 1 sidered × (k1)µ(k2)ν (k3)ρ 2 2 2 2 q + m (q + k1) + m 1 1 me 1 = τxρy, me 2 = τyρy, me 3 = ρz. (D1) × 2 2 2 2 (q + k1 + k2) + m (q + k1 + k2 + k3) + m 5 However, before being able to connect to our sets of pos- m 3 sible WZW terms, we have to make sure that we use × tr γ˜µγ˜ν γ˜ρδ(n · m˜ )(n · m˜ ) n · m˜ 2 the same conventions (as indicated by the tildes in the (C10) equation above). Comparing our Dirac matrices defining the non-interacting Hamiltonian in Eq. (5), γx = ρxτz, We then take in the denominators k1, k2, k3 → 0 so that γy = ρy, with theirs, γx = µzρx, γy = µzρyτz, we find we may integrate out the variable q: e e that the field operators are related by feq = U2U1fq, Z 3 Z 3 Z 3 Z 3 d q d k1 d k2 d k3 where δSeff ' −i 2 2 2 2 (k1)µ(k2)ν (k3)ρ (2π) (2π) (2π) (2π) µ0 + µz µ0 − µz τ0 + τz τ0 − τz 5 U1 = + ρz,U2 = + ρz. 1 m 3 2 2 2 2 × tr γ˜µγ˜ν γ˜ρδ(n · m˜ )((n · m˜ )) n · m˜ (q2 + m2)4 2 (D2) With this, we can rewrite Eq. (D1) in our conventions, −i Z d3k Z d3k Z d3k = 1 2 3 (k ) (k ) (k ) yielding 64π (2π)2 (2π)2 (2π)2 1 µ 2 ν 3 ρ m = −µ τ ρ , m = µ τ ρ , m = ρ . 1 3 1 z y x 2 z x x 3 z (D3) × tr γ˜µγ˜ν γ˜ρδ(n · m˜ )(n · m˜ ) n · m˜ 2 (C11) The first two components transform into each other under U(1)v and are the IVC part and the third order pa- Then Fourier transforming back we find: rameter is the valley Hall state of Ref. 54. Note that m1, m2 break time-reversal symmetry, while m3 preserves 1 Z 3 C2 δSeff = d xtr [˜γµγ˜ν γ˜ρδ(n · m˜ )(∂µn · m˜ ) it but breaks , in agreement with the statements in 128π Ref. 54. ×(∂ν n · m˜ )(∂ρn · m˜ )n · m˜ ] Since Ref. 33 considers a model without spin, there Z Z 1 is no unique mapping to our spinful description of 1 3 = d x dutr [˜γµγ˜ν γ˜ρδ(n · m˜ )(∂µn · m˜ ) the system. In fact, there are two natural micro- 128π 0 scopic realizations: first, reinserting spin as (m1, m2) = ×(∂ν n · m˜ )(∂ρn · m˜ )∂u(n · m˜ )] µzρxσ(τ+, τ−), m3 = ρz we recover the partner orders in (C12) the 11th line of TableVII and the second line in TableV 19

(in both tables labelled as type IVC+; SP). Our analysis, belled as type IVC−; SP) that also the spinless realiza- thus, shows that it can form WZW terms with both sin- tion, (m1, m2) = µzρx(τ+, τ−), m3 = ρz, can form a glet and triplet pairing (although in the unitary triplet WZW term with the A2 singlet for M = τzµy, i.e., only if case, additional Fermi surfaces will appear), and which translation invariance is broken in the high-temperature M have to be realized in the respective cases. phase (note M = µzσz will lead to unwanted Fermi sur- Second, we see from the fourth line of TableIII (la- faces).

Appendix E: Full set of partner order parameters

We list the full set of possible orders for singlet pairings A1 and A2 in TableIX and triplet pairings B1 and B2 in TableX and TableXI in the full space with no additional symmetry breaking. These include options we eliminated in TableIII and TableVII for breaking translational, valley rotation, or spin rotation symmetry but are still mathematically viable options in that they satisfy Eq. (11) in the full space. We also list which of the above 1 orders survive projection to a subspace deﬁned by a high-temperature order M with projection 2 (1 + M) for singlet pairing in TableXII and TableXIII and for triplet pairing in Table XIV and TableXV.

TABLE IX: Forms of possible orders for singlet pairing. Note that the options which do not appear by the main text either require additional symmetry breaking beyond the options for ν = 2 or have three independently ﬂuctuating options.

Partner Orders Partner 1 Partner 2 Partner 3 SC Partner

1 µx/yρz µy/xτzρzσj µzρzσj A1 2 ρxτx/y τy/xµzρxσj µzρzσj A1 3 ρxτx/y ρxτy/xµx/y ρzµx/y A1 4 τx/yµx,yρx τzµy,xρzσj τx/yµzρxσj A1 5 τx/yµx/yρx τy/xρxσj µx/yρzσj A2 6 τx/yµzρx τy/xρxσj µzρzσj A2

7 ρx(τx, τy) ρz A1 8 ρxµzσj (τx, τy) ρz A1 9 ρxµx/y(τx, τy) ρz A1 10 ρz(µx, µy) ρzτzµz A1 11 τzρzσj (µx, µy) ρzτzµz A1 12 ρxτx,y(µx, µy) ρzτzµz A1 13 τzρzµx,y(σi, σj ) ρzτzσk A1 14 ρzµz(σi, σj ) ρzτzσk A1 15 ρxµzτx,y(σi, σj ) ρzτzσk A1 16 ρxσj (τx, τy) ρz A2 17 ρxµa(τx, τy) ρz A2 18 ρzσj (µa, µb) ρzτzµc A2 19 ρxτx,y(µi, µj ) ρzτzµk A2 20 τzρz(µx, µy) ρzτzµz A2 21 ρzµa(σi, σj ) ρzτzσk A2 22 ρxτx,y(σi, σj ) ρzτzσk A2 20

TABLE X: Forms of possible orders for triplet pairing, choosing a triplet state polarized along σx. Note that the options which do not appear by the main text either require additional symmetry breaking beyond the options for ν = 2 or have three independently ﬂuctuating options.

Partner Orders Partner 1 Partner 2 Partner 3 SC Partner

1 τx/yµzρx τx/yµx/yσxρx τzµy/xρzσx B1 2 τx/yµzρx τy/xσxρx µzρzσx B1 3 τx/yµzρx τy/xµzρxσy/z ρzσy/z B1 4 ρzµx/y τx/yσxρx τy/xµx/yσxρx B1 5 τx/yµx/yσxρx τy/xµzσy/zρx µy/xσz/yρz B1 6 τx/yσxρx τx/yρxσy/zµz τzµzσz/yρz B1 7 ρzµx/y ρzµy/xσy/z τzρzµzσy/z B1 8 ρxµx/y ρzµzσx τzµy/xσxρz B1 9 ρzσy/z τzµzσz/yρz µzρzσx B1 10 τzµx/yσxρz σy/zρz µx/yσz/yρz B1 11 τx/yρx τy/xσxρxµa σxρzµa B2 12 τx/yρx τy/xσy/zρx σy/zρz B2 13 τx/yρxσxµa τx/yσy/zρx σz/yρzτzµa B2 14 ρxσxµa σy/zρz τzσz/yρzµa B2

14 ρxµz(τx, τy) ρz B1 15 ρxσx(τx, τy) ρz B1 16 ρxµzσy/z(τx, τy) ρz B1 17 ρxσxµx/y(τx, τy) ρz B1 18 ρxσxτx/y(µx, µy) ρzτzµz B1 19 ρzτzσx(µx, µy) ρzτzµz B1 20 ρzσy/z(µx, µy) ρzτzµz B1 21 ρz(µx, µy) ρzτzµz B1 22 τx/yµzρx(σy, σz) ρzτzσx B1 23 ρz(σy, σz) ρzτzσx B1 24 ρzµx/y(σy, σz) ρzτzσx B1 25 ρzτzµz(σy, σz) ρzτzσx B1 26 ρx(τx, τy) ρz B2 27 ρxσxµa(τx, τy) ρz B2 28 ρxσy/z(τx, τy) ρz B2 29 ρxτx/yσx(µa, µb) ρzτzµc B2 30 ρzτz(µx, µy) ρzτzµz B2 31 ρzσx(µa, µb) ρzτzµc B2 32 ρzτzσy/z(µa, µb) ρzσxµc B2 33 ρxτx/y(σy, σz) ρzτzσx B2 34 ρz(σy, σz) ρzτzσx B2 35 ρzτzµi(σy, σz) ρzτzσx B2

TABLE XI: Forms of possible orders for non-unitary triplet pairing which do not transform into one another with triplet state proportional to σy + iσz. Note additional symmetry breaking is required for many of these options.

Partner Orders Partner 1 Partner 2 Partner 3 SC Partner U 1 τx/yµzρx τy/xµzρxσx ρzσx B1 U 2 ρzµx/y ρzµy/xσx ρzτzµzσx B1 U 3 τx/yρx τy/xσxρx σxρz B2 U 4 ρxµzσx(τx, τy) ρz B1 U 5 ρz(µx, µy) ρzτzµz B1 U 6 ρzσx(µx, µy) ρzτzµz B1 U 7 ρxµz(τx, τy) ρz B1 U 8 ρx(τx, τy) ρz B2 U 9 ρzτzσx(µa, µb) ρzτzµc B2 U 10 ρxσx(τx, τy) ρz B2 U 11 τzρz(µx, µy) ρzτzµz B2 21 0 z σ σ 0 z µ µ z 0 σ σ z 0 z τ τ τ z ρ z 0 ; z µ µ ρ 0 z . When there is more than one ; x τ τ y x z x µ σ σ M z σ and the partners for this symmetry µ y y x z x σ z ρ σ σ ρ ) a ρ ρ x ; s y 3 y z τ ) τ τ , µ y M y x z x τ τ τ , τ ; µ 2 z x ( ) ρ τ A z y z ( y ; ρ x µ µ , µ ρ z x — — — — for z or x τ σ σ j x z µ or x y ( ρ ρ ρ µ m 0 z x x ρ σ σ 0 ; τ z 0 z τ τ µ µ z 0 or ) z x σ σ y j τ σ z x z σ ρ ) , σ ρ ρ z x ; z y z z µ σ µ µ τ ( µ ; 0 z z y σ σ ρ z z 0 , µ ρ µ µ x µ ( ) z y ρ , τ x τ ( x ’s partners, which is why we do not include it explicitly in this table. ρ a s 3 0 z is related by a mini-valley rotation to σ σ M 0 z 0 b x z 0 z s 3 σ σ y µ µ σ σ σ 0 z τ z 0 z 0 z M τ τ x µ µ τ τ ρ ρ z τ z z z ρ τ ρ ; z ; j ; ρ j σ ; σ 0 x y x x z 0 x µ µ z τ σ z σ z y µ µ µ y x 0 z x ρ 0 x z z µ 0 σ z µ y τ τ σ σ pairing (if existing). Note that we write everything in the full basis, that contain both Dirac ρ τ ρ τ z x µ ) µ σ x ρ z ρ y µ ρ x 2 τ ) ) y z ) τ x z z y , τ y τ A y τ ρ ρ τ x σ ; z z τ , σ , µ x ρ ( ρ x ; x σ z z y x or σ and µ ( τ µ τ τ ( or ( z z ; z x 1 z 1 µ σ ρ τ µ z ) y z A A z z z y 0 x x µ µ µ ρ or ρ σ ρ z or µ µ z x y x y , µ ρ can in general mix due to the broken symmetry related to the respective for ρ ρ ρ τ z x µ or or x z y τ z j , for x 0 z µ τ ρ τ τ τ ( 3 σ σ z 0 m {·} ; x , 0 z 0 z z 0 ; ) σ σ ρ τ τ 2 ; τ ) τ σ σ ) y z x , z 0 y 0 z y ρ ρ σ σ , τ µ or µ z z 0 , σ x x , µ µ σ j x τ z µ x = 1 µ z x x z ( σ σ µ µ ρ ρ ρ µ j ( ( ; ; y z z x ; , 0 z x τ z ρ ρ ρ q τ µ µ µ z x 0 z z ; y f 0 z τ z τ τ ρ x σ µ j σ σ 0 z ρ x z z ρ x z 0 z 0 z τ σ σ y τ ρ ρ ρ m σ σ µ µ τ z † q z y τ ) f µ µ ) y q y ) ) y , µ , τ y P x x , µ µ τ , τ ( x = ( x z x µ τ j ρ ( ( ρ x O ρ z z z z z σ σ z z z x x σ σ µ τ τ x σ x y x σ µ x x breaking term will be related by the same mini-valley rotation to z τ z M z µ ρ µ τ y τ µ µ µ x z ρ τ τ y y ρ ρ TABLE XII: The different time-reversal symmetric high-temperature order parameters, their symmetries, and the associated possible partner order cones, and the two options in given in parameters, (symmetry-distinct) option, we separate them by “or”. We note that 22 z 0 σ σ z 0 µ µ 0 x τ τ 0 y ρ ρ z µ z z x τ σ µ z ρ ; z ρ z z 0 τ σ σ ; 0 x z 0 µ µ µ µ 0 x 0 z 0 x τ τ µ τ µ τ 0 0 z y pairings only. ρ σ ρ σ z 2 ρ y 0 ; z z A ρ ρ σ µ 0 x z z 0 x τ τ ρ ρ µ µ ) ) 0 z , for y j y τ τ 3 µ , , µ , σ z z 2 x x ρ 2 µ , z σ µ x τ ( ( A ρ ) = 1 y or or or for j , τ j , x q τ m z y 0 0 x 0 ( f ρ ρ σ µ σ µ j 0 x z 0 0 z τ τ σ µ m µ σ or † 0 x q τ τ j f z 0 y σ ρ q ρ ρ 0 x z ; ρ µ µ P z 0 z z 0 τ τ x τ σ µ µ = z 0 z τ j j σ σ z σ ρ z O ; ρ ) z y τ z 0 , τ σ σ x z 0 τ µ ( µ 0 x x τ τ ρ 0 y x ρ ρ µ ) y , σ x order parameters, σ ( z µ z ρ z σ z z x y σ ρ µ µ x x M z x τ τ τ µ y ρ TABLE XIII: The different high-temperature order parameters which are odd under time reversal, their symmetries, and the associated possible partner 23 0 x σ σ z y z 0 µ µ x 0 0 z µ µ 0 z σ σ τ µ µ τ z y 0 z z τ τ τ τ τ z z x z τ ρ ρ ). For orders which depend z 0 ρ z z x 0 z µ µ ρ ρ x ; y ) σ z x σ σ ; µ ; σ z ρ 0 ρ x x x z z y y µ µ ∝ µ τ σ y x τ τ ; 0 σ σ x 0 z z x x τ τ z y τ τ σ σ σ µ ρ ∆ ρ z 0 y ) x z ) , µ τ τ τ τ y z σ y x ; x z ; ρ µ ρ ρ ) ; , µ ( , µ y x x 0 z x 2 τ ; µ y x µ µ µ , µ ( x B y ( z σ σ x or z x σ σ z 0 x ρ ρ x y τ µ x z τ τ 0 x σ z σ σ ( — — — ρ for ρ τ τ x ρ y z z x 0 x ) τ τ ρ j ρ µ σ z z σ x ) or y m z ρ z ρ or or τ τ , µ ; z x y , σ ) 0 x ρ ρ y µ z 0 z z 0 σ x σ ( σ 0 σ µ z µ z ( σ σ , µ ; z 0 y x 0 y z µ ρ µ y τ τ τ τ τ τ µ µ x z 0 x ( or z ρ τ τ τ x z or ρ z z x σ ρ ; ρ ρ ρ ; z z 0 ρ z σ σ 0 x ; ) x µ 0 z y x σ σ z ρ z σ σ µ µ 0 z y ρ z 0 0 z µ µ , σ τ z τ τ µ µ ) y τ 0 z y σ ) τ τ ( z y ; , τ ρ x ) ; , τ τ z 0 x z ( τ µ µ x 0 x , σ ( x z ρ µ µ y x z 0 ρ ρ y x ρ σ 0 σ x σ ( σ σ τ τ y z z 0 x τ τ ρ σ z ) σ x ) pairing (if existing). We only include one of each set of high-temperature orders 0 x y ρ ρ y 0 x µ µ 2 , τ µ µ 0 z , µ z 0 x x 0 B z 0 x τ τ x τ σ µ σ σ σ µ µ x ( ; z 0 µ z 0 0 z z x ( σ τ µ τ µ τ τ ρ x z ρ and x y τ σ ; σ z z z σ z 1 τ 0 x ρ ρ µ z z y τ τ ; ; z z z z B z ρ x τ µ µ ρ σ µ ρ ρ z 0 y ; y z y 0 ; x y x τ τ τ µ µ µ τ τ µ µ y x ) z x z y , for 0 x ) y τ σ ρ σ σ ) σ 3 ) σ z σ z x x y , z 0 z , µ ρ ρ ρ τ τ 2 ) , σ x , µ , , σ y y µ ; ; x y ) σ ( 1 ( µ σ , µ y ( ( z x y x z = 1 B or z x ρ τ τ τ , µ µ ρ j σ y z ( x x y x z , µ µ σ or — — — σ µ for z ρ ρ or y ( x x q z z j µ z ρ f σ σ ρ µ y ρ j z x x or or x 0 m x σ 0 ρ x ρ µ σ σ m σ x µ µ 0 or x τ † z q 0 z 0 x ; µ ; µ x ρ f µ µ 0 σ σ z z ρ τ 0 z 0 z τ 0 q x ρ x z z x τ τ µ σ µ σ ; τ ; σ µ µ 0 z P z x y y x τ τ x ρ z σ τ σ µ 0 x x ; σ = ρ x x z z τ µ y µ z z ; ρ ρ ρ ρ j x σ ρ ρ 0 x ; σ z ; ) O σ σ 0 x on spin, we include for each order a possibility aligned with the x direction and one in-plane order. x ρ y z y σ σ ρ 0 z z y 0 µ z µ , τ µ µ σ σ µ x µ ; 0 z τ ) τ τ ( ) y ) x z y y ) σ z , τ µ z , τ x x z µ , τ x τ ρ y x σ ( τ , σ τ τ z ( x y ( x τ z ρ z σ z ρ µ ( µ ρ z x x ρ ρ ρ z z z z x z σ σ x x z x x σ x σ σ µ σ τ τ x σ x x y x x µ σ x x z τ τ M z z µ ρ µ µ y τ τ y µ µ x µ z z ρ τ ρ τ y y τ ρ ρ partner order parameters, which are related by valley or mini-valley rotation. Here, our triplet state is aligned in the x direction in spin space ( TABLE XIV: The different high-temperature order parameters which do not break time reversal symmetry, their symmetries, and the associated possible 24 0 z x 0 0 x z 0 σ σ µ µ σ σ σ σ z 0 z y x 0 0 z µ µ τ τ µ µ µ µ 0 z z x τ y τ z ρ ρ τ τ z y x z x 0 z τ z ρ µ ρ µ x z σ σ ρ z 0 σ ρ ; τ τ ; ; ; ) z ). For orders which depend y x z 0 x z y x µ x y σ σ ρ ; µ σ µ σ σ z 0 σ σ x y y z τ τ y z σ τ τ τ τ ) ∝ ; z x , µ z x z ) ρ ρ x ρ ρ y µ ∆ 0 x µ ; ( ) µ µ y , µ ) z z 0 z y x µ τ τ τ ( z µ , σ z 2 ( , µ ρ y τ ) z x B z σ z ρ µ ( ρ or ( , σ ; — — — or for y or or or j σ ; 0 z ( σ σ z 0 z m 0 x 0 σ σ x x ρ y σ σ 0 z µ σ z σ µ 0 x y z µ µ z x ρ or µ µ τ 0 τ z τ ; µ τ τ z x x z ρ ρ ρ ρ z 0 z 0 x z ρ σ σ µ µ ; ρ ; ; 0 z ; τ τ ) x 0 0 x y z y τ x τ σ σ z µ z , τ x 0 σ σ x ρ y x ρ x 0 ρ z ; µ µ τ σ τ µ µ x z ( z 0 ) ρ σ x ) y x τ τ z x τ τ y ρ ρ ) , µ , τ ) y y x z ; τ µ ( , τ ( , µ x x x z y τ ρ σ σ ( µ z y x ( τ µ ρ x x z σ x 0 ρ τ x µ µ z ρ z z z y ρ 0 x τ τ 0 z τ τ z σ z σ z σ x σ z ρ 0 y ρ z ρ z 0 ρ τ τ z y µ 0 z µ µ µ z x y z µ µ σ σ 0 z x ρ ρ τ τ y x τ τ z σ x σ σ z ; ρ ρ pairing (if existing). Again, we only include one of each set of high-temperature orders x z µ σ ρ ; z 2 z ; y ; ; ρ µ B µ z z z x 0 τ z 0 y z y x ρ ρ τ τ ; τ τ µ z µ y σ σ z x z x 0 x 0 z µ ρ µ ρ and ρ ρ τ τ 0 z µ x µ y z x σ σ 0 z 1 σ σ ) ρ ρ τ τ ) z B z ) ; ) y ) , σ , σ z y y y z z , µ σ , σ σ ρ , µ x σ ( , for ( y x 1 z x µ 3 z σ ρ ( µ µ , B or ( µ z ( 2 z ρ , ρ or or — — — for x or or y σ j or τ = 1 x x y z 0 x m µ τ 0 τ x j σ σ 0 z σ z z x 0 µ µ x z 0 , σ σ x τ ρ ρ τ 0 τ z σ σ q ρ z z x σ σ 0 z f ρ z ρ ρ z j µ µ ρ 0 z µ z ; ; τ m τ ) ; ρ † y q ; 0 z y f z y z σ σ z , µ µ ρ τ τ q z y x µ x ; z x on spin, we include for each order a possibility aligned with the x direction and one in-plane order. µ µ τ ) x µ ρ ρ z 0 P x y σ y x σ σ ;( σ z ) σ σ , τ = x ρ y z 0 x ) ρ 0 x j τ µ µ y σ σ ( , µ O 0 z z ; x , τ ) µ µ µ x µ y z y x ( τ z τ τ ( ρ x , τ y µ z x x σ x µ x ρ ρ σ τ τ z ( x x ρ x ρ ρ ρ z x z x σ σ z x z z x σ x µ z σ τ σ z µ x µ µ y x x σ τ M z x µ x ρ τ y τ τ z µ µ order parameters, τ y ρ y which are related by valley or mini-valley rotation. Here, our triplet state is aligned in the x direction in spin space ( ρ ρ TABLE XV: The different high-temperature order parameters which break time reversal symmetry, their symmetries, and the associated possible partner 25

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