Supplemental Material forSpin-fluctuation-induced pairing in twisted bilayer graphene

Ammon Fischer,1 Lennart Klebl,2 Carsten Honerkamp,1 and Dante M. Kennes2, 31Institute for Theoretical Solid State Physics, RWTH Aachen University,

and JARA Fundamentals of Future Information Technology, 52062 Aachen, Germany2Institute for Theory of Statistical Physics, RWTH Aachen University,

and JARA Fundamentals of Future Information Technology, 52062 Aachen, Germany3Max Planck Institute for the Structure and Dynamics of Matter,Center for Free Electron Laser Science, 22761 Hamburg, Germany

S I. ATOMIC STRUCTURE OF TWISTEDBILAYER GRAPHENE

The atomic structure of twisted bilayer graphene(TBG) consists of two super-imposed graphene layersrotated by an angle θ and separated by a distance ofd0 = 0.335 nm. The axis of rotation is chosen to inter-sect two vertically aligned carbon atoms when startingfrom purely AA-stacked bilayer, i.e. two perfectly over-lapping honeycomb lattices. Labelling the upper (lower)monolayer of TBG by u(l), their two-dimensional latticevectors are

al1 = a0(cos(π/6),−sin(π/6))

al2 = a0(cos(π/6), sin(π/6))

aui = R(θ)ali,

(1)

where R(θ) represents a rotation by an angle θ and a0 =0.246 nm is the lattice constant of graphene, which shouldnot be confused with the carbon-carbon bond length ofacc = a0/

√3. The so defined primitive cell is di-atomic

and contains two inequivalent sites with basis vectors

bu(l)1 = (0, 0) (A site),

bu(l)2 =

1

3au(l)1 +

1

3au(l)2 (B site).

(2)

The choice of basis vectors and the type of rotationaround an AA site restricts the symmetry of TBG topoint group D3. The latter contains a threefold in-planerotation C3 = C3z around the z-axis and a twofold out-of-plane rotation C2 = C2y.The lattice geometry of TBG, as defined so far, is notperiodic in general [1–3] since the periods of the twographene layers are incommensurate for arbitrary twistangles. A finite unit cell can only be constructed for somediscrete angles satisfying the condition

cos(θ) =1

2

m2 + n2 +mn

m2 + n2 +mn(3)

with (m,n) being positive integers. In this case, thetwisted bilayer graphene forms a moiré pattern, see Fig. 1(a), containing N(m,n) = 4(m2 + n2 +mn) atoms with

superlattice vectors

L1 = mal1 + nal2 = nau1 +mau2

L2 = R(π/3)L1.(4)

Magic-angle twisted bilayer graphene has a twist angleof θ = 1.05◦ corresponding to the the integers (m,n) =(31, 32). The so defined structure contains N = 11908carbon atoms in the moiré unit cell. When discussingthe geometric structure of TBG, atomic relaxation ef-fects play an important role and may modify the low-energy physics of the system significantly. From exper-iments using transmission electron microscopy (TEM)[4] as well as from structural optimization studies us-ing density functional theory [5] it is known that the in-terlayer distance between the two layers varies over themoiré unit cell. The interlayer spacing takes its maximumvalue dAA = 0.360 nm in the AA regions and its minimalvalue dAB = 0.335 nm in the AB regions. Intermediatespacings d(r) are obtained by using an interpolation sug-gested by [1, 5]

d(r) = d0 + 2d1

3∑i=1

cos (Gi · r) , (5)

where the vector r points to a carbon atom in the moiréunit cell and G are the reciprocal lattice vectors obtainedfrom Eq. (4). Furthermore, the constants d0 = 1

3 (dAA +2dAB) and d1 = 1

9 (dAA−dAB) are defined such to matchthe distances in the AA and AB regions. In order topreserve the D3 symmetry of the system, the corrugationmust be applied symmetric to both layers as depicted inFig. 1 (c).

S II. ATOMISTIC TIGHT-BINDINGHAMILTONIAN

The eigenenergies and eigenfunctions of non-interacting TBG are obtained using a single-orbitaltight-binding Hamiltonian for the pz-orbitals of the

2

Figure 1. Atomic structure of twisted bilayer graphene. (a) Atomic structure of twisted bilayer graphene with twistangle θ = 5.09◦ corresponding to (m,n) = (6, 7). The blue labels indicate characteristic stacking patterns emerging throughoutthe moiré unit cell. (b) Downfolding of the (mini-) Brillouin zone of TBG (brown hexagon). (c) Corrugation effects in TBG.(d) Band structure of magic-angle TBG with twist angle θ = 1.05◦ corresponding to (m,n) = (31, 32). The low-energy windowaround charge neutrality (red line) is modified significantly when corrugation of the two graphene sheets is taken into account(left panel). This is reflected in the formation of four flat bands (two-fold spin degeneracy) around charge neutrality that havea bandwidth of ≈ 15meV and are separated from the rest of the spectrum.

carbon atoms [1, 2, 6]:

H =∑R,R′

∑i,j,σ

t(R + ri −R′ − rj)c†R,ri,σ

cR′,rj ,σ. (6)

For our microscopic approach we account for the full π-band spectrum of magic-angle TBG, keeping all N =11908 bands under consideration. In the following, welabel the supercell vector with R and the position vec-tor of site i in the corresponding moiré unit cell with ri.Hence, the operator c†R,ri,σ

creates an electron with spinσ = ↑, ↓ in the pz-orbital of site i, whereas cR,ri,σ de-stroys an electron with the same quantum numbers. Thetransfer integral between orbitals at site i and j, sep-arated by the vector d can be written in Slater-Kosterform [1]

t(d) = t‖(d) + t⊥(d)

t‖(d) = −V 0ppπexp

(−d− acc

δ0

)[1−

(dz

d

)2]

t⊥(d) = −V 0ppσexp

(−d− d0

δ0

)[dz

d

]2.

(7)

Here, dz = d · ez points perpendicular to the graphenesheets and d0 = 1.362 a0 is the vertical spacing ofgraphite. The term Vppσ = 0.48 eV describes the inter-layer hopping between atoms in different monolayers ofTBG, while Vppπ = −2.7 eV models the intralayer hop-ping amplitude between neighboring atoms in a singlegraphene sheet. The parameters are fitted to data of first

principle calculations to match the dispersion of mono-and bilayer graphene [1]. This ensures in particular thatTBG behaves locally similar to graphene with the over-all structure being modulated by the moiré pattern. Theparameter δ0 = 0.184 a0 determines the decay length ofthe transfer integral and is chosen such that the nearest-neighbor intralayer hopping reduces to 0.1Vppπ. For nu-merical calculations it is therefore sufficient to truncatehopping terms for rij > 4 a0 as t(rij) < 10−4 in theseregimes.

To construct the full non-interacting Hamiltonian ofthe periodic system, we define the Bloch wave basis by aFourier transform to mini-Brillouin zone (MBZ) momen-tum k, which is depicted in Fig. 1 (b),

ck,r =1√N

∑R

eikRcR,r

cR,r =1√N

∑k

e−ikRck,r.

(8)

The spin index is omitted for simplicity. Note that themoiré Fourier transform as defined above couples mo-menta k and superlattice vectors R. The latter describethe spatial extent of the moiré unit cell, which in the caseof magic-angle TBG is given by L = |L1,2| = 13.42 nm.

Inserting these expressions into Eq.(6) renders theunperturbed Hamiltonian block-diagonal in momentum

3

space

H0 =∑k

∑R−R′

∑i,j,σ

t(R + ri −R′ − rj)

× e−ik·(R−R′)c†k,ri,σ

ck,rj ,σ

=∑k

∑i,j,σ

[H0(k)]ri,rjc†k,ri,σ

ck,rj ,σ.

(9)

The matrix [H0(k)]ri,rjcan be diagonalized in orbital

space (i, j) for each k to obtain the bandstructure εb(k)and orbital-to-band transformation ubr(k), b = 1..N :

H0 =∑k,b

εb(k)γ†k,bγk,b with γk,b = ubr(k)ck,r. (10)

Since magic-angle TBG contains N = 11908 atoms inthe moiré unit cell, care must be taken when treating thesystem numerically.

S III. MAGNETIC INSTABILITIES ANDSTONER CRITERION

A. Stoner Criterion

In the manuscript, we study magnetic instabilities inTBG described by short-ranged Coulomb interactions.To this end, we follow Ref. [7] and employ a repulsiveHubbard term for electrons with opposite spin σ withσ = −σ residing on the same carbon site

Hint =1

2U∑R,i,σ

nR,ri,σnR,ri,σ. (11)

To treat the interacting system in a pertubative manner,we define the free Matsubara Green’s function in orbital-momentum space as

gr,r′(iω,k) =∑b

ubr(k)(iω − eb(k))−1ub∗r′ (k). (12)

We then calculate the renormalized interaction in thespin channel within the random-phase approximation(RPA) to analyze the electronic instabilities mediated byspin-fluctuation exchange between electrons to high or-der in the bare coupling U . Since the initial short-rangedinteraction vertex U has no momentum and frequencydependence, the full susceptibility of the system can beapproximated by χRPA(q) = χ0(q)/[1 + Uχ0(q)]. Here,we use the multi-index qunatum number q = (q, iω) andindicate matrices of dimension N ×N with an hat sym-bol, e.g. χ = χr,r′ . Magnetic instabilities may sub-sequently be classified according to a generalized Stonercriterion: The effective (RPA) interaction diverges, whenthe smallest eigenvalue λ0 of χ(q) reaches −1/U , markingthe onset of magnetic order for all interaction strengths

U ≥ Ucrit. = −1/λ0. The corresponding eigenvectorv(0)(q) is expected to dominate the spatial structure oforbital magnetization.

In this letter, we study magnetic instabilities with em-phasis on the static, long-wavelength limit (q, iω → 0)on the moiré scale. The latter limit proves to containthe relevant physics when starting with local repulsiveinteraction. We stress again that momenta q are relatedvia moiré Fourier transform Eq. (8) to the superlatticevectors R. The RPA susceptibility predicts spin correla-tions at length scales intermediate to the c-c bond scaleand moiré length scale, thus being described by order-ings at q = 0. The system shows the same order in allmoiré unit cells with variable correlations present on thec-c bond scale.

B. Spin susceptibility

For analyzing the magnetic properties of the systemon the RPA level, it is therefore sufficient to computethe free polarization function χ0(q) defined as

χ0r,r′ (q, iω) =1

Nβ

∑k,ω′

gr,r′(iω′,k)gr′,r (i(ω′ + ω) ,k+q).

(13)The Matsubara summation occuring in Eq. (13) can beevaluated analytically yielding the well-known Lindhardfunction for multi-orbital systems

χ0r,r′ (q, iω) =1

N

∑k,b,b′

nF (εb′(k))− nF (εb(k + q))

iω + εb′(k)− εb(k + q)

× ub′

r (k)ub′∗

r′ (k)ub∗r (k + q)ubr′(k + q),

(14)

where nF (ε) = (1 + eβε)−1 is the Fermi function. Whilethe analytical evaluation of the Matsuabra sum occuringin Eq. (13) is the standard procedure for systems con-taining only few atoms in the unit cell, this approach isdestined to fail in our atomistic approach as it scaleslike O(N4). For magic-angle TBG with N = 11908atoms in the moiré unit cell, it is more efficient to com-pute the Matsubara sum in Eq. (13) numerically over aproperly chosen frequency grid and in each step computeHadamard products of band-summed non-local Green’sfunctions gr,r′(iω′,k)gTr,r′ (i(ω′ + ω) ,k+q). The expres-sion then scales like O(N3Nω) with Nω being the numberof fermionic frequencies needed to achieve proper conver-gence. To this end, non-linear mixing schemes [8] areproven to outperform any linear summation such thatwe only need to sum over Nω ≈ 1000 frequencies whenaccessing temperatures down to T = 0.03meV. The mo-mentum sum occuring in Eq. (13) is evaluated over 24 kpoints in the MBZ using a momentum meshing proposedby Cunningham et al. [9]. In particular, we checked

4

that the results are sufficiently converged when takingan denser mesh into account.

C. Leading Instabilities

In the manuscript, we classify the different leadingeigenvectors of the RPA analysis according to their real-space profile in the moiré unit cell following the nomen-clature introduced in Ref. [7]. The three potential groundstates of the interacting system at T = 0.03meV aredepicted in Fig. 2 (b): (i) AFM: moiré-modulated an-tiferromagnetic phase on the carbon-carbon bond scale("Ångström"-scale) with increased weight in the AA re-gions, (ii) DAFM: moiré-modulated antiferromagneticphase with opposite signs between the AA and AB re-gions that becomes visible as a node in the absolute valueof the order parameter, (iii) FM: moiré-modulated fer-romagnetic phase (FM) that exhibits the same overallsign in the moiré unit cell. The choice of the temper-ature T = 0.03meV should provide resolution of theflat energy bands in TBG as extensively discussed inRef. [7]. As long as T � O(1meV) the flat band physicsis not resolved and the system inherits magnetic orderfrom purely AA and AB stacked bilayer graphene. ForT ≈ O(1meV) ≈ 10K, significant deviations due to theflat bands occur, leading to the plethora of magneticphases described above.

S IV. SELF-CONSISTENT BOGOLIUBOVDE-GENNES EQUATIONS FOR TBG

A. Fluctuation-Exchange approximation

For interaction values U < Ucrit the system is in theparamagnetic regime and the magnetic instabilities pre-scribed by the RPA analysis are not strong enough to ac-tually occur. In this regime, spin and charge fluctuationscontained in the transverse and longitudinal spin channelcan give rise to an effective interaction between electronsthat may lead to the formation of Cooper pairs. Theleading RPA diagrams to the irreducible singlet particle-particle scattering vertex Γ2(q, ν) are captured within thefluctuations-exchange approximation (FLEX) [10, 11]

Γ2(q) = U − U2χ0(q)

1 + Uχ0(q)+

U3χ20(q)

1− U2χ20(q)

. (15)

As in the previous section, we only consider the staticlong-wavelength limit (q, iω → 0) and thus focus on thepairing structure on the carbon-carbon bonds within themoiré unit cell. To this end, the real-space profile of theeffective interaction Γ2 for different chemical potentials isshown in Fig. 2 (c). As mentioned in the manuscript, theinteraction vertex is staggered through the moiré unit cell

close to a DAFM/AFM instability, opening the door forunconventional singlet Cooper pairs in the two graphenesheets of TBG. In particular, the interlayer interactionstrength is an order of magnitude smaller than compara-ble intralayer terms. This indicates that the main pairingwill create in-plane Cooper pairs. In the manuscript, wethus only visualize the projection of the superconduct-ing order parameters on the in-plane form factor basis ofeach graphene sheet, i.e. a layer-resolved representation.

B. Self-consistent BdG Formalism

In the next step, we analyze the effective particle-particle scattering vertex Eq. (15) using a mean-field de-coupling to extract pairing symmetries and spatial dis-tribution of the superconducting order parameter. In thestatic long-wavelength limit (q, iω → 0), we may henceneglect the momentum dependence of the gap parameterand effectively solve a one-unit cell system with periodicboundary conditions. While this approach does not takecorrelations between different moiré unit cells into ac-count, it allows for all pairing contributions from withinthe moiré unit cell. Due to the proximity to the anti-ferromagnetic ordered state, we restrict the mean-fielddecoupling to spin-singlet configurations that are sym-metric under the exchange of spatial indices

∆ij = −1

2[Γ2(q = 0)]ij 〈ci↑cj↓ − ci↓cj↑〉MF. (16)

The expectation value 〈·〉MF can be calculated by diag-onalizing the resulting mean-field Hamiltonian HMF inNambu-space using a Bogoliubov de-Gennes transforma-tion [12]

HMF = ψ†(H0 ∆

∆† −H0

)ψ

〈ci↑cj↓ − ci↓cj↑〉MF =∑n

(uni v

n∗j + unj v

n∗i

)tanh

(En

2T

).

(17)

Here, uni (vni ) are the particle (hole) amplitudes of theBdG quasi-particles resulting from the diagonalization ofthe Hamiltonian in Eq. (17)

HMF =(Uψ)†(E 0

0 −E

)(Uψ)

U =

(u v−v∗ u

),

(18)

and ψ = (c1↑, ..., cN↑, c†1↓, ..., c

†N↓)

T is the 2N -componentNambu vector.

∑n denotes a sum over the positive

quasi-particle energies En > 0 and E is the correspondingdiagonal matrix. To solve this set of self-consistent equa-

5

Figure 2. Spin correlations in magic-angle TBG. (a)Magnetic RPA phase diagram showing the critical onsite interactionstrength Ucrit. vs. chemical potential µ in the four flat bands of TBG at T = 0.03meV. The vertical lines indicate the integerfillings ±3,±2,±1 that show an increased magnetic ordering tendency towards a moiré-modulated ferromagnetic state, whileaway from integer fillings weaker antiferromagnetic tendencies dominate. (b) Spatial distribution of the leading eigenvector ofthe RPA analysis on the carbon-carbon bond scale in the moiré unit cell. For simplicity, only the lower layer of TBG is shownas well as a linecut through the unit cell. The leading eigenvector of the DAFM instabilities is staggered throughout the moiréunit cell with strongest weight in the AA regions as depicted in the linecut in (b). (c) Effective spin-fluctuation mediatedpairing vertex Γ2(q = 0) close to a DAFM (upper panel) and FM (lower panel) magnetic instability. In the former case, thepairing vertex is staggered in real-space with strong on-site repulsion and nearest-neighbor attraction. Real-space profiles areshown starting from an atom located in the AA, AB or DW region of TBG, respectively.

tions we start with an initial guess for ∆ij and iterate un-til convergence is achieved using a linear mixing schemeto avoid any bipartite solutions. Since the atomic ar-rangement is highly inhomogeneous in the moiré unit cell,we track the free energy during each self-consistency cy-cle and for different initial configurations to ensure properconvergence of the algorithm into the actual global min-imum. The free energy of the system in the low temper-ature regime reads

F = E − TS ≈ Eg −∑n

En −∑ij

|∆ij |2

Γ2,ij(19)

where Eg = 2∑nEnnF (En) is the excitation energy of

the quasi-particles. The different initial configurationsare chosen to transform according to the irreducible rep-resentations of the D6h point group of the honeycomblattice. This procedure aligns with the insights from theprevious paragraph that the spin-fluctuation mediatedpairing vertex Eq. (15) will create in-plane Cooper pairswith strongest pairing amplitude living on the nearest-

neighbor bonds of the two single graphene sheets. Thephase factors of the different nearest-neighbor pairingchannels are shown in Fig. 3.

C. Supercurrent and magnetic field

To characterize the different superconducting phases ofthe system, we compute the layer resolved quasi-particlebond current in TBG [12]

Jnm =e

i~〈c†ntnmcm − c†mtmncn〉enm. (20)

In the atomistic approach presented here, the quasi-particle current Jnm is only defined between two car-bon atoms residing at sites rn(m) in the moiré unit cell.Therefore, we take an amplitude-weighted average ofneighboring bonds to arrive at a vector field represen-

6

Figure 3. Form factors for different nearest-neighbor (sin-glet) pairing channels on the honeycomb lattice. The complexlinear combination d± id is characterized by the phase factorw = exp(±i 2π/3).

tation J as shown in Fig. 3 in the manuscript

J(rn) =1

3

∑〈m〉

Jnmenm. (21)

Here, enm points from the atom at position rn to its threenearest-neighbors rm on each graphene sheet. In partic-ular, the current amplitude is negligible at distances ex-ceeding nearest-neighbor atoms and thus an average overnearest-neighbors is sufficient.

The spontaneously flowing currents of quasi-particlesinduce a magnetic field that can be calculated by apply-ing the Biot-Savart law

B(r) =µ0

4π

ˆJ(r)× r − r′

|r − r′|3d3r, (22)

where µ0 is the vacuum permeability. Since the cur-rent co-propagates in the two graphene sheets of TBGin the "chiral" phase, the magnetic fields induced by thesupercurrents add constructively making this particularfeature of the TRS breaking phase measurable in exper-iment.

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