Chiral and topological superconductivity in isospin polarized multilayer graphene

Domain: physics
Published: 2025-12-01
Editorial outcome: major_revision

Abstract
A microscopic mechanism for chiral p-wave superconductivity from Coulomb repulsion is proposed for spin- and valley-polarized state of rhombohedral multilayer graphene. The superconducting instability arises when strong Thomas-Fermi screening of the Coulomb potential allows Friedel oscillations to take over–leading to an effective attraction on length scales below the Fermi wavelength. The superconducting critical temperature is largest at low density below a Lifshitz transition to an annular Fermi sea, where the additional pocket strongly enhances Thomas-Fermi screening. The Lifshitz transition also marks a topological phase transition from a trivial to a topological superconducting phase hosting Majorana fermions. The chirality of the superconducting order parameter is selected by the chirality of the valley-polarized Bloch electrons. Our results are in reasonable agreement with observations in a recent experiment on tetralayer graphene.

Introduction
Chiral superconductivity, characterized by spontaneous time-reversal symmetry breaking and finite-angular momentum Cooper pairing1, is a long-sought quantum phase of matter with unusual superconducting and magnetic properties. Interest in chiral superconductors is further fueled by their potential for hosting topological phases and Majorana fermions2,3. While previous material candidates, such as Sr2RuO4 and UTe25–7, showed initial signs of chiral superconductivity, recent experiments strongly suggest single-component superconducting order parameters that are non-chiral8–15.

Very recently, signatures of chiral superconductivity have been observed in rhombohedral-stacked tetralayer graphene under electron doping16. While superconductivity has been previously discovered and intensively studied in crystalline trilayer17–30 and bilayer graphene31–36, the newly found superconducting state in tetralayer graphene at low density is remarkably distinctive in that it exhibits large spontaneous anomalous Hall effect above Tc and magnetic hysteresis in resistance below Tc. These observations demonstrate time-reversal-breaking superconductivity in a pure carbon system. Its pairing symmetry and pairing mechanism are open questions for investigation.

A key feature of rhombohedral multilayer graphene is the flat band dispersion near K and K0 point, leading to a strong correlation effect19,23. As a result, spin and valley isospin symmetry breaking occurs at low temperature, giving rise to half and quarter metal phases23. Interestingly, the superconducting state in tetralayer graphene at low density borders the quarter metal and its phase boundary shows no or little change with the applied magnetic field, indicating that the superconducting state is likely fully spin and valley polarized16. Thus, tetralayer graphene provides a rare opportunity for investigating Cooper pairing of single-flavor electrons in a solid-state platform.

In this work, we study the microscopic mechanism and pairing symmetry of superconductivity in multilayer graphene that develops from the spin- and valley-polarized quarter metal normal state [Fig. 1a, b]. Our mechanism is based on the overscreening of Coulomb interaction due to charge fluctuations, which leads to an effective attraction at length scales on a fraction of the Fermi wavelength, driving Cooper pairing. Using a minimal model for the band dispersion, our theory predicts that superconductivity occurs at low densities with enhanced density of states due to the annular Fermi surface and displacement field-induced flatness of the band. Based on our analysis, we further propose that a Wigner crystalline insulating state occurs at the Lifshitz transition from simply-connected to annular Fermi sea that splits the superconducting dome into two. This provides an explanation for the two proximate superconducting domes SC1 and SC2 with similar properties observed in the experiment16.

Results
Band dispersion
In rhombohedral multilayer graphene, low-energy bands come from sublattice-polarized states in the top and bottom layers. An out-of-plane electric field induces a potential bias equal to 2D between these layers and opens up an energy gap while flattening the dispersion near K and K0 points. Therefore, the Fermi energy is only a few meV above the band bottom for a small electron density of around 5 × 1011 cm−2, where time-reversal-breaking superconductivity is observed.

The low-energy band dispersion is highly tunable by the electric field. As the displacement field D increases, the curvature at K and K0 changes from positive to negative19,23 as shown in Fig. 1c for ABCA tetralayer graphene, resulting in a Mexican-hat shaped dispersion. In this case, a Lifshitz transition from simply-connected to annular Fermi surface occurs when the Fermi energy crosses the top of the hat as the electron density is reduced.

We capture the essential features of the electric-field-tuned conduction band in rhombohedral n-layer graphene with a minimal band

εk = D 1 + (k/k0)2n + ℏ2k2 / 2m (1)

where D is the displacement field, k0 is a momentum scale, and m is the effective mass. The dispersion (1) is circularlysymmetric. The inclusion of additional hopping terms leads to trigonal warping. For now, we neglect trigonal warping and Berry curvature effects and will treat them perturbatively later.

Rytova-Keldysh potential
The density-density interaction can be written as

Hint = 1/2 sum_{k1,k2,q} V(q) ψ^y_{k1+q} ψ^y_{k2-q} ψ_{k2} ψ_{k1} (2)

where ψ^y_k (ψ_k) are the annihilation (creation) operators of spin-polarized electrons in a single valley. Importantly, in a 2D material surrounded by a dielectric with a lower dielectric permittivity, the Coulomb interaction between two charges can be described by the Rytova-Keldysh potential40–42, taking the form:

V(q) = e^2 / 2ϵ|q|(1 + rK|q|) (3)

where ϵ = 5ϵ0 is the dielectric permittivity of the surrounding hBN (with ϵ0 the vacuum permittivity), and rK is the Rytova-Keldysh parameter. The Rytova-Keldysh parameter depends on the difference between the dielectric response of the 2D material under study and that of the surrounding insulator. Since the 2D dielectric screening is determined by interband transitions and thereby depends on the band gap, rK of multilayer graphene is affected by the displacement field43. We will find that the superconducting pairing strength depends sensitively on rK.

Electron pairing from screened Coulomb repulsion
We study superconductivity in the spin- and valley-polarized state. This is justified because (i) valley polarization is observed to occur at a much higher temperature than superconductivity, and (ii) the primary superconducting state (SC1 in ref. 16) was observed within the quarter-metal state, and (iii) the primary superconducting states (SC1 and SC2) survive to very high in-plane magnetic fields.

Theoretically, we also expect that the energy scales of spin- and valley polarization to be much larger than superconductivity: Estimating the energy scale of isospin polarization in the Stoner model23 as Δpol ≈ ν(EF)V(2kF,2)EF ≈ 2 to 3 meV in the relevant range of density and displacement field, it is an order of magnitude larger than the pairing strength [see numerical results below].

Since total spin S2 and valley imbalance NK − NK0 are conserved quantum numbers, there are no isospin fluctuations at zero temperature that can drive superconductivity in the fully-polarized quarter metal state. This motivates us to consider a mechanism for superconductivity in the spin- and valley-polarized state based on screening of electron-electron interactions by charge fluctuations44–46. The screening is described by the charge susceptibility χRe(q,τ) = −1/A <T τ ρe,q(τ)ρe,-q(0)>, which in random phase approximation (RPA) is determined by the expression

χRe(q,iΩn)RPA = χ0e(q,iΩn) / 1 - Vq χ0e(q,iΩn) (4)

in terms of the charge susceptibility of the non-interacting electron gas

χ0e(q,iΩn) = 1/A sum_k (nF(εk) - nF(εk+q)) / (εk - εk+q + iℏΩn) (5)

where nF is the Fermi-Dirac distribution. The screened interaction potential is then

V~(q) = Vq / 1 - Vq χ0e(q,0) = Vq ϵ^-1(q,0)RPA (6)

The charge susceptibility and screened interaction potential for the annular Fermi pocket are shown in Fig. 2a, b, respectively. The q→0 limit of the denominator describing the screened interaction in Eq. (6) lim_{q→0} Vq χ0e(q,0) = e^2 / 2πℓTF defines the Thomas-Fermi screening length ℓTF = 4πϵ/e^2ν(EF).

Due to the large density of states, the charge susceptibility becomes large for momenta below twice the Fermi momentum kF,2 of the outer Fermi surface. This leads to a suppression of the electron-electron interaction at small momenta and a peak at 2kF,2. This renormalized interaction potential promotes a p-wave superconducting instability.

In real space, the screening strongly reduces the repulsion above the Thomas-Fermi screening length ℓTF. Furthermore, Friedel oscillations appear with period 2kF,1, 2kF,2 due to the two Fermi surfaces. When the Thomas-Fermi screening length is much shorter than the wavelengths π/kF,1, π/kF,2, Friedel oscillations dominate over Coulomb repulsion and lead to an effective attraction at short length scales, as obtained from our calculations shown in Fig. 2c. We find ℓTF ≈ 1 to 2 nm, for typical parameters where we observe superconductivity as discussed below, while 2π/kF,1, 2π/kF,2 is typically around 20 nm to 30 nm. This mechanism benefits from a Mexican-hat-shaped dispersion because the additional pocket contributes a large density of states, leading to a short Thomas-Fermi screening length. This enables superconductivity at a larger density.

To determine the superconducting order parameter and its critical temperature, we solve the self-consistency equations

Δ(k) = -1/A sum_{k'} V~(k-k') <ψ^y_{-k'} ψ^y_{k'}> (7)

<ψ^y_{-k} ψ^y_{k}> = - Δ(k) / 2Ek tanh(Ek / 2kBT)

where Ek is the quasiparticle energy

Ek = sqrt((εk - μ)^2 + |Δ(k)|^2) (8)

Here, the interaction V~(k-k') scatters a pair of electrons at opposite momenta (k', -k') to (k, -k).

Due to the rotational symmetry in our model, we decompose the pairing interaction into angular harmonics:

V~l(k,k') = 1/2π integral_0^{2π} dϑ e^{ilϑ} V~(ϑ,k,k') (9)

where we wrote V~(k-k') = V~(ϑ,k,k') with ϑ the angle between k and k', and k, k' the their magnitude. The order parameter can also be expanded into angular harmonics Δ(k) = sum_{l=0} ηl(k) e^{ilθ} with (kx, ky) = k(cosθ, sinθ). The equations for the critical temperature Tc for different angular harmonics decouple.

Around Tc, we linearize the self-consistency equation and find for the individual angular harmonics,

ηl(k,Tc) = - 1/8π^2 integral_0^∞ dk' k' V~l(k,k') tanh(|εk' - μ| / 2kBTc) / |εk' - μ| ηl(k',Tc) (10)

For the superconducting order parameter at zero temperature, the self-consistency relation reads

ηl(k,0) = - 1/8π^2 integral_0^∞ dk' k' V~l(k,k') ηl(k',0) / 2 sqrt((εk' - μ)^2 + |ηl(k',0)|^2) (11)

Both equations can be solved iteratively, as detailed in SM section B. This angular decomposition applies to a circularly symmetric interaction potential, where the circular symmetry is preserved for the screened potential when the dispersion is circularly symmetric.

Importantly, the screened interaction V~ is positive and large at momentum transfer q = k-k' around 2kF,2, compared to small q. Such q-dependent interaction favors an order parameter that takes opposite signs at opposite ±k points on the Fermi surface, i.e., it favors p-wave pairing.

For most superconductors, the pairing interaction is weak and therefore the pairing potential Δ(k) is small compared to the Fermi energy and only appreciable in the vicinity of the Fermi surface. For this reason, in solving the gap equation, it suffices to use “on-shell” pairing interaction V~l(k,k') at Fermi wave vectors k, k'. In contrast, in multilayer graphene, the combination of low electron density and flat band bottom leads to a larger ratio of interaction to the Fermi energy. As a consequence, we will show that Δ and Tc can be on the order of the Fermi energy. Indeed, the recent experiment on tetralayer graphene16 reports an unusually high upper critical field at low density, indicating a strong-coupling superconductor with coherence length comparable to interparticle distance. For strong-coupling superconductors, the superconducting gap Δ(k) can be large even away from the Fermi momentum. Therefore, in solving the gap equation, it is necessary to use the interaction V~l(k,k') with full k, k' dependence.

Our calculation of the density of states, ν and critical temperature Tc, as a function of electron density and displacement field, is shown in Fig. 3a, b, respectively. Both the displacement field D and the electron density n together determine the Fermi surface size and topology. For small D < Dc ≈ 40 meV, the dispersion increases monotonically with k and the Fermi surface is a single circle with kF = sqrt(4πn), whereas at large displacement field (D > Dc), a Lifshitz transition from annular to simply-connected Fermi sea occurs with increasing electron density, which is accompanied by a large jump in the normal-state density of states at the Fermi level [Fig. 3a]. Correspondingly, superconducting properties depend strongly on the displacement field. For small D, a superconducting state is found at a small density where the chemical potential lies close to the relatively flat band bottom with a large density of states. For large D, the superconducting state sets in at low densities where the Fermi sea is annular, and Tc is largest close to the Lifshitz transition.

It should be noted that the value of Tc is sensitive to the dielectric screening of Coulomb repulsion by the multilayer graphene, which, in turn, depends on the displacement field-induced band gap. As a function of the Rytova-Keldysh parameter rK [Fig. 3c], the typical Tc at D = 60 meV ranges from Tc ≈ 3 K at rK = 1 nm to a suppression of Tc to below 0.1 K above rK = 10 nm. With bare Coulomb repulsion, a maximal Tc ≈ 8 K is obtained, see SM section H. The decrease of Tc with strong dielectric screening is consistent with electron pairing by Coulomb repulsion. Without knowing the value of rK for tetralayer graphene, we cannot make a quantitative prediction of Tc. Nonetheless, for the reasonable range of rK considered here, superconductivity always onsets at n < 0.7 × 1012 cm−2 in agreement with experimental observation, and the calculated Tc is acceptable compared to the experimental value, especially considering that the mean-field theory generally predicts higher values of Tc.

Going beyond our mean-field treatment, we estimate the critical temperature for a BKT transition47–49 kBTBKT = π/2 J determined by the superfluid stiffness. An order-of-magnitude estimate is obtained using the superfluid stiffness expression J = ℏ^2n/4m for a quadratic band whose mass m ≈ 2me is matched so that its density of states matches the typical density of state ν(EF) ≈ 2νe at the Fermi energy of our model [compare Fig. 3a]. Based on this estimate, we find a typical BKT transition temperature kBTBKT ≈ 0.9 K at n = 0.5 × 1012 cm−2. This indicates that the critical temperature in this system may be limited by vortex proliferation in the BKT transition.

Strong-coupling superconductivity
For the chiral px ± ipy pairing, the calculated pairing potential Δ(k) at T = 0 can reach a few tenths of the Fermi energy. Notably, Δ(k) is large not only close to the Fermi surface, but also extends to several times the Fermi wavevector with a sign change at around 2kF,2 [Fig. 4a]. Its functional form closely follows the interaction potential up to a proportionality factor setting the pairing strength [see SM section C]. The presence of substantial pairing potential away from the Fermi surfaces is a consequence of the strong-coupling nature of this superconducting state—because kBTc and Δ are of the order of the Fermi energy, pairing between electrons away from the Fermi surface is relevant. This is captured by our direct solution of Eqs. (10), (11).

Due to the large pairing potential, the chemical potential changes appreciably in the superconducting state. This can be estimated by taking into account the pairing potential in relating electron density to chemical potential

n = 1/2πℏ^2 integral d^2k (1 - (εk - μ) / Ek) (12)

At zero temperature, the change in chemical potential approaches 0.1 meV. The change in electron density is taken into account in Fig. 3a. The change in chemical potential δμ ∝ |Δ|^2 is expected to increase linearly with decreasing temperature.

Topological superconductivity and Lifshitz transition
The quasiparticle gap Δ0 = min_k Ek as a function of density n and displacement field D is shown in Fig. 4b. Our p±ip superconductor generally has a full gap, except at the Lifshitz transition from simply-connected to annular Fermi sea, where the system has a point node at the Fermi point k = 0 at which the pairing potential vanishes. The closing of the quasiparticle gap marks a topological quantum phase transition from a topological superconductor with unity Chern number in the region with a single Fermi pocket to a topologically trivial state in the annular region at large displacement fields50,51. The topological superconductor at small displacement fields hosts chiral Majorana edge modes and Majorana zero modes in the vortex2. In contrast, the trivial state with annular Fermi sea is adiabatically connected to the Bose-Einstein limit μ → −∞, because the pairing potential Δ(k) is finite at the band bottom of the Mexican hat dispersion, which forms a ring at k ≠ 0. Note that in our theory, even though the superconducting gap vanishes at k = 0 at the Lifshitz transition, the gap at the outer Fermi surface Δ(kF2) and critical temperature remains large throughout the transition – therefore the quasiparticle gap closure is not visible in the critical temperature calculations Fig. 3b.

Competing state
Since the p-wave superconductivity found in our RPA calculation occurs at relatively low density, it is important to consider its competition with the Wigner crystal state, which generally appears in 2D Coulomb systems at sufficiently low density. To estimate the transition to the Wigner crystal, we compute the gas parameter rs = Eint/Ekin given by the ratio of interaction Eint to kinetic energy Ekin52. In our calculations, we find that the superconducting state is robust against the Wigner crystal state for rs < 40. The transition to the Wigner crystal depends sensitively on the gate screening as the interparticle distances at low density approach tens of nm.

In Fig. 4, the region where rs > 40 is encircled by the cyan line; in a homogeneous electron gas, the transition to a Wigner crystal occurs around rs ≈ 30–4052–56. This region is where the band bottom is most flat, so that the kinetic energy per particle is low. It coincides with the region where the density of states is largest, see Fig. 3a. We expect crystalline order is likely to dominate there, so that the superconducting region is divided into two, separated by an insulating Wigner crystal state.

Trigonal warping and Berry curvature effects
Our minimal model neglects trigonal warping, which arises from electron hoppings beyond nearest-neighbor atoms in multilayer graphene. With trigonal warping, the band dispersion becomes asymmetric εk ≠ ε-k, which weakens intravalley pairing between ±k states. Fortunately, the energy scale of trigonal warping is small in tetralayer graphene in the range of density and displacement field of interest, as evidenced by the nearly symmetric band dispersion in Fig. 1. For the range of parameters where we observe superconductivity, the pairing is driven by the interaction energy, and therefore is robust against the pair breaking effect of trigonal warping.

Up to now, we have neglected the effect of electron Bloch wave functions |uk> within the unit cell. The momentum dependence of the complex-valued wavefunction |uk> gives rise to Berry curvature, which in the presence of time-reversal symmetry breaking, leads to an orbital magnetic moment in the polarized valley57. We have shown that the chirality of the Bloch wave functions is responsible for the chirality of the superconducting order parameter. We also verified that our mechanism strongly favors p-wave pairing over higher angular momenta, see SM section G.

Methods

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In this paper, the authors propose chiral p-wave superconductivity in tetralayer graphene. Considering a quarter metal state with a nearly flat band, they obtain a chiral p-wave superconducting state from the screened Coulomb interaction. They also find the Fermi surface of the system shows the Lifshitz transition, which results in a topological phase transition. The topic is timely, and the result is interesting. However, the following points need to be clarified before making a recommendation, as they are crucial for understanding and accepting the research.\n1. The authors only consider a particular multilayer graphene, ABCA tetralayer graphene. The author should clarify why they chose this specific multilayer graphene. For example, is it possible to obtain a similar result in trilayer graphene or differently stacked graphene\n2. The authors assume a spin and magnetic gap deriving the quarter metal, then calculate the superconducting critical temperature independently from a screened Coulomb interaction. However, the magnetic order itself also originates from the Coulomb interaction. The author should justify this treatment.\n3. A minor point: When D is introduced in the left column on page 2 for the first time, the author should explicitly mention its definition because 2D is often used to represent “two dimensions.”", "reviewer_2": "The authors proposed that the screened Coulomb interaction can realize chiral superconductivity in rhombohedral multilayer graphene, motivated by the recent experiment on the ABCA tetralayer showing possible spin- and valley-polarized superconductivity. They approximated the single-particle band structure by a circularly symmetric minimal model with parameters fitted by the k.p band model. Using RPA and the minimal band model, they found that p-wave superconductivity (which the authors suggested to be chiral p-wave, e.g., p+ip) can emerge with T_c varying from 100 mK to a few K, depending on the screening parameters of the Coulomb interaction. They further pointed out the topological transition within the superconducting region and argued that the results may explain the SC1 and SC2 regions in the experiment.\nThe subject of this manuscript is important as the new ABCA tetralayer graphene experiment may open up a new direction for realizing chiral topological superconductors and Majorana physics. Overall, the manuscript is well-written, and the qualitative results are interesting. This work shows the first microscopic calculation of the superconductivity driven by the screened Coulomb interaction using a simplified band structure. While the analysis seems appropriate at first glance, there are several outstanding issues that need to be addressed. As a result, I cannot recommend publication. I will be happy to review the manuscript again after the authors resolve all the issues. The questions/comments are as follows:\n(1) The Kohn-Luttinger mechanism here (and in other graphene-based systems) seems to overestimate T_c. Unlike the phonon-mediated pairing, there is no Migdal theorem for the screened Coulomb interaction. As a result, the vertex correction and other related contributions may modify the estimated T_c significantly. The authors should discuss this issue and provide some quantitative estimates if possible.\n(2) It needs to be clarified how reliable the minimal band model is based on the current results. The authors argued that ''the energy scale of trigonal warping is small in tetralayer graphene in the range of density and displacement field of interest, as evidenced by the nearly symmetric band dispersion shown in Fig. 1.'' However, it’s not evident from Fig. 1, which is a plot of E vs. k_x, that trigonal warping is indeed weak.\nThe k.p band structure is NOT circularly symmetric, and the dispersion along k_x and k_y (relative to the valley momentum) can be very different. The fitting in Appendix A and Fig. 5 need to be discussed more carefully. The authors may compare the density of states of the minimal model and the k.p model, which should provide some hints about the validity of the minimal model.\n(3) The conclusion of chiral pairing is based on the Berry curvature argument without any quantitative calculations. I would like to know how large this splitting between two chiral order parameters is due to the effect of Berry curvature. Conventionally, one needs to check the quartic order of the order parameter in free energy to determine the chiral or nodal pairing, as shown in Ref. 19. The authors may want to check the quartic term.\n(4) The authors may want to discuss other potential pairing mechanisms. What has been achieved here is that the screened Coulomb interaction may provide a way to explain the putative spin- and valley-polarized superconductivity in ABCA graphene. However, it does not rule out other possibilities. It is possible that several different pairing glues result in the experimentally observed superconductivity. Some discussions about the possible mechanisms will be useful.\nMinor issues:\n(5) The authors employ the Rytova-Keldysh potential, with the 2D screening length r_K as a tuning parameter, based on its inverse dependence on the band gap of multilayer graphene, and thereby on the displacement field D. On the contrary, the dual-gated Coulomb potential is considered in many other existing literature. Why did the author pick this particular one over the other one? The authors did say that the choice of the Coulomb potential (before RPA) does not affect the qualitative results. However, the should authors explain their choice when introducing the Rytova-Keldysh potential. The authors mentioned that the Rytova-Keldysh potential is often used for 2D systems of which the dielectric constant is much larger than the dielectric environment. Based on the current manuscript, it is unclear that the dielectric constant of multilayer graphene is much larger than hBN, so relevant experiments should be cited.\n(6) Fig. 2(a) shows the single-particle static polarization functon(Eq. 5) by setting the frequency to zero. I have a few clarifying questions: Does Fig. 2(a) only include intraband transitions when the Fermi level aligns with the annular Fermi pocket? Are interband transitions, which are gap-dependent, effectively incorporated through the parameter r_K?\nAdditionally, which value of r_K is used for Fig. 2(b) and (c)? In Fig. 2(c), the attraction length scale is about 1 nm—comparable to the multilayer graphene thickness. Why, then, is the Rytova-Keldysh potential valid? The authors should clarify these points in the manuscript.\n(7) In Fig. 3(c), the highest Tc is achieved at r_K = 1 nm, approximately matching the thickness of tetralayer graphene. Does using the Keldysh potential with r_K = 1 nm yield a Tc comparable to using the bare 3D Coulomb potential with RPA?\n(8) According to Fig. 2(a), k_{F,2} is approximately 0.3 nm^-1, corresponding to a Fermi wavelength of around 20 nm, which greatly exceeds the 1 nm attraction length scale indicated in Fig. 2(d). How do the authors conclude that: “Our mechanism is based on the overscreening of Coulomb interaction due to charge fluctuations, which leads to an effective attraction at length scales of a few Fermi wavelengths, driving Cooper pairing.” and “The screening strongly reduces the repulsion on long length scales compared to the Fermi wavelength, and leads to an overscreening with an effective attraction at distances of a few Fermi wavelengths, as demonstrated in Fig. 2(c).”\n(9) This work shows the first microscopic calculations for the screened-Coulomb-driven superconductivity in the tetralayer graphene system. However, this possibility was also suggested in Ref. [62]. The authors cited Ref. [62] in an obscure way -- They put a very short footnote [36], stating \"For a mean field study of superconductivity from attraction, see Ref. [62]).'' Since the screened-Coulomb mechanism was also suggested in Ref. [62], the footnote is not accurate.\n(10) Could the authors comment on the role of the WSe_2 layer in the tetralayer graphene experiment, particularly within the context of the Kohn-Luttinger mechanism and the screening effects of the Rytova-Keldysh potential?\n(11) Typo in Eq. (6): The frequency omega should be zero.\n(12) In the section “Topological superconductivity and Lifshitz transition”: “The quasiparticle gap ∆0 ≡ min_k Ek as a function of density n and displacement field D is shown in Fig. 4(c).” There is no Fig. 4(c).", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "The authors performed a calculation of a Coulomb repulsion originated theory for spin-valley polarized chiral superconductivity in tetralayer graphene, aimed at explaining the recently observed superconductivity in such systems. The theoretical calculations follow the standard mean field theory procedures and seemed to lead to desired results, and this is a timely study of the very interesting recent experiment. It is however difficult to claim how reliable the mean field calculations are for such a strongly interacting system. I also find the paper not presented clearly, with unjustified or ambiguous points in the authors calculations and arguments. I therefore cannot recommend the current paper draft to Nature Communication. My detailed questions and comments are given below.\n1. The form of band dispersion in Eq. (1) is proposed in a strange way: if k_0 is fixed, the dispersion will not reach k^n (the well-known dispersion of n-layer graphene) at D=0, so D is not an independent tuning parameter. After checking App. A I can understand that k_0 and m are functions of D, and probably k_0 is proportional to D^{1/n} such that k^n dispersion can be reached at D=0. It would be more natural to redefine the first part as \\sqrt{D^2+k^2/k_0^2} with k_0 roughly constant. Or at least, in the current form, the authors should explain in the main text that k_0 and m are functions of D (and k_0 proportional to D^{1/n}).\n2. Related to the above question, it seems the theory proposed in this paper is largely independent of the number of graphene layers n. Are all the calculations specific for n=4, and if so, anything special about n=4? In other words, does the theory provide an understanding why the chiral superconductivity is observed in tetra-layer graphene?\n3. Is the mass term m in Eq. (1) the particle-hole symmetry breaking term? Can the authors understand from their theory why the superconductivity occurs in the conduction band?\n4. Some setups of the model considered are unclear. In writing down the interaction in Eq. (2) and later, are the electron operators defined for one particular spin and valley? It is difficult to find in the paper whether the pairing is intra-valley or intervalley. Also, I don’t think k_{F,1} and k_{F,2} are defined before they first appear.\n5. The effective attractive interaction near r=1nm in Fig. 2(c) is counterintuitive. RPA, which can be understood as the dynamical generalization of Thomas-Fermi screening, usually only reduces the repulsion due to opposite charges accumulated nearby. It seems unlikely to have “over-screening” that the repulsion is flipped into attractions. It is more likely that the interaction is attractive in non-s wave channels, instead of being globally attractive. Although s-wave here is forbidden by spin-valley polarization, the RPA calculations seem to be independent from this assumption. Can the authors provide an intuitive picture for understanding the attraction? Also, RPA is a perturbative method for interactions much smaller than the Fermi energy, how is it justified here, where interaction is much larger than the kinetic energy (as indicated by r_s the authors calculated)?\n6. A related question, is the effective RPA interaction above sufficient to induce spin-valley polarization, i.e., the starting point of the current theory? The interaction in Fig. 2(c) looks much weaker and localized than the bare interaction; the highly local interaction may resemble the Hubbard model which gives anti-ferromagnetism instead of (orbital) ferromagnetism.\n7. The form factor or the interaction in Eq. (13) looks incorrect. Shouldn’t it be the inner product of u_{k+q} with u_k, and u_{k’-q} with u_{k’}?\n8. I do not understand what “charge-ordered state” in the discussion mean. Does it mean Wigner Crystal? The authors may want to clarify. Also, there is (almost) no mentioning of the Lifshitz transition of Fermi surface in the main text (probably the cyan line region in Fig. 4?), which the authors should explain better.\n9. The electron-phonon coupling is completely ignored in the study here. Is there a particular reason for ignoring the effect of phonons? Also, how would the authors propose to tell experimentally whether the superconductivity is from Coulomb repulsion instead of phonons? For instance, if the screening of Coulomb interaction is enhanced (by metallic gate, etc), would the authors predict an enhancement or decrease of superconducting Tc?\n10. The authors only focused on the time-reversal breaking superconducting phases in the experiment. The experiment also observes more conventional superconductivity in two other parameter regimes. Do the authors have comments on the other superconductivity regimes? Is the claim that Coulomb interaction is better for realizing chiral superconductivity (in contrast to phonons which may favor s-wave superconductivity more)?\n11. A minor question: I believe \\omega in Eq. (6) should be set to zero already."}, "author_response": "NCOMMS-24-62309A\nResponse to referees for “Chiral and topological superconductivity in\nisospin polarized multilayer graphene”\nDear Editors, dear Referees,\nWe hereby resubmit our manuscript titled “Chiral and topological superconductivity in\nisospin polarized multilayer graphene” to be considered for publication in Nature\nCommunications.\nWe appreciate the thoughtful and constructive comments from all Referees and your\nconsideration of our work. We provide a detailed response to all comments below and\nhave improved our manuscript in accordance with all comments.\nIn the time since our first submission of our article, some of our predictions have\nbeen confirmed by experiments:\nBased on our estimate of the ratio between interaction and kinetic energy, we\npredicted a Wigner crystalline phase that separates two superconducting domes (Fig\n4b). This prediction has subsequently been confirmed by our colleagues: In the\nresistive phase separating the superconducting domes SC1 and SC2 the resistance\nincreases with temperature, indicating an insulating state consistent with our predicted\nWigner crystal (see Extended Data Figure 2(d, e) in the revised manuscript of the\nexperiment arXiv:2408.15233).\nAdditionally, our prediction indicated that superconductivity is likely to occur in\nrhombohedral pentalayer graphene as well, which has been subsequently observed, as\nis shown in the revised manuscript of the experiment arXiv:2408.15233.\nFinally, we would like to highlight that in answering the questions of all referees, we\ncame up with an intuitive explanation for the mechanism for superconductivity in spin-\npolarized metals such as rhombohedral graphene: When Thomas-Fermi screening is\nsufficiently strong so that the screening length becomes shorter than the Fermi\nwavelength, then Friedel oscillations take over the repulsion and lead to an effective\nattraction on length scales of the Fermi wavelength.\nGoing beyond our mean-field treatment, we estimated the BKT transition temperature\nto be around 0.9 K, indicating that the BKT transition likely limits the upper critical\ntemperature of the superconducting phase.\nYours sincerely,\nMax Geier, Margarita Davydova, and Liang Fu\nREVIEWER COMMENTS\nReviewer #1 (Remarks to the Author):\nIn this paper, the authors propose chiral p-wave superconductivity in tetralayer\ngraphene. Considering a quarter metal state with a nearly flat band, they obtain a chiral\np-wave superconducting state from the screened Coulomb interaction. They also find\nthe Fermi surface of the system shows the Lifshitz transition, which results in a\ntopological phase transition. The topic is timely, and the result is interesting. However,\nthe following points need to be clarified before making a recommendation, as they are\ncrucial for understanding and accepting the research.\nOur response: We thank the Referee for their interest in our article and appreciating\nthe timeliness of our results. We have clarified the comments and questions raised by\nthe Referee, which have improved our manuscript, and hope that with these\nimprovements the manuscript can be recommended for publication.\nReviewer #1: 1. The authors only consider a particular multilayer graphene, ABCA\ntetralayer graphene. The author should clarify why they chose this specific multilayer\ngraphene. For example, is it possible to obtain a similar result in trilayer graphene or\ndifferently stacked graphene\nOur response:\nRecent experimental observations [Han et al arXiv 10.48550/arXiv.2408.15233] indicate\nthat, unlike in other similar few-layer stacked graphene systems, ABCA tetralayer\ngraphene possesses signatures a chiral superconducting phase in spin- and valley-\npolarized region of the phase diagram specifically in tetralayer graphene. This was the\nmain motivation for choosing this particular stacking in our work.\nOther rhomobohedrally stacked multilayer graphene systems, such as AB, ABC\nstacking etc., have a similar dispersion to that of ABCA graphene. Non-rhombohedral\nstackings, such as ABA, ABAB, have a fundamentally different dispersion, with the\nmain distinguishing feature that for rhombohedral stacking a displacement field opens\na band gap proportional to the interlayer potential difference, while this is not the case\nfor non-rhombohedral stacking (compare to e.g. [Ghazaryan PRB 107, 104502 (2023)]).\nWe expect that our theory can also similar superconducting properties in the spin- and\nvalley- polarized phase of rhombohedrally stacked graphene with different number of\nlayers as well. The main difference, is, therefore, quantitative. Increase in the number of\nlayers is conducive to increase in coupling strength as the band bottom flattens, thus,\nour theory would favor larger number of layers. Indeed, it appears that, going from 3-\nlayer ABC graphene that has been studied experimentally in the past, to tetralayer\nABCA graphene, indicators of the chiral superconducting phase appear. Our paper is\naimed to verify whether chiral superconducting order is indeed plausible in experiment\nin experiment and why the strength of superconducting coupling seems to be rather\nstrong.\nWe added a comment in the conclusion:\n“We expect that our mechanism may also apply to isospin-polarized phases of\nrhombohedral graphene with a different number of layers because all these systems\nshare the feature of a large density of states at the band bottom under an applied\ndisplacement field \\cite{Ghazaryan2023Mar}. Crucially, calculations of\nRef.\\cite{Ghazaryan2023Mar} indicate that tetralayer graphene reaches a significantly\nlarger density of states close to the band bottom under a displacement field compared\nto smaller number of layers. Our theory shows that the large density of states induces a\nshort Thomas-Fermi screening length that enables superconductivity from repulsion via\noverscreened Friedel oscillations. This significant quantitative difference may explain\nwhy superconductivity has been observed in rhombohedral tetralayer and pentalayer\ngraphene \\cite{Han2024Aug}, but not for smaller layer number up to now.”\nReviewer #1: 2. The authors assume a spin and magnetic gap deriving the quarter\nmetal, then calculate the superconducting critical temperature independently from a\nscreened Coulomb interaction. However, the magnetic order itself also originates from\nthe Coulomb interaction. The author should justify this treatment.\nOur response: It is correct that the magnetic order arises from Coulomb interactions. It\ncan be obtained in a Stoner model as performed in Ref. [Ghazaryan et al., PRB 107,\n104502 (2023)]. We estimated the energy scale of Stoner polarization as $\\Delta_{\\rm\npol} \\approx \\nu(E_F) V(2 k_F) E_F \\approx 2 $ to $ 3\\ {\\rm meV}$ in the relevant range\nof density and displacement field, which is about an order of magnitude larger than our\nobtained pairing strength of up to 0.3 meV. This separation of energy scales justifies\nthe treatment of superconductivity as a perturbation on top of the spin- and valley\npolarized state.\nWe modified the introduction to the section on “Electron pairing from Coulomb\nrepulsion”:\n“We study superconductivity in the spin- and valley-polarized state. This is justified\nbecause (i) valley polarization is observed to occur at a much higher temperature than\nsuperconductivity, and (ii) the primary superconducting state (SC1 in\nRef.\\cite{Han2024Aug}) was observed within the quarter-metal state, and (iii) the\nprimary superconducting states (SC1 and SC2) survive to very high in-plane magnetic\nfield.\nTheoretically, we also expect that the energy scales of spin- and valley polarization to\nbe much larger than superconductivity: Estimating the energy scale of isospin\npolarization in the Stoner model \\cite{Ghazaryan2023Mar} as $\\Delta_{\\rm pol} \\approx\n\\nu(E_F) V(2 k_{F,2}) E_F \\approx 2 $ to $ 3\\ {\\rm meV}$ in the relevant range of density\nand dispacement field, it is an order of magnitude larger than the obtained pairing\nstrength [Fig.\\ref{fig:4}].\nSince total spin $S^2$ and valley imbalance $N_{K} - N_{K'}$ are conserved quantum\nnumbers, there are no isospin fluctuations at zero temperature that can drive\nsuperconductivity in the fully-polarized quarter metal state. This motivates us to\nconsider a} mechanism for superconductivity in the spin- and valley-polarized state\nbased on screening of electron-electron interactions {\\color{blue} by charge\nfluctuations} \\cite{Kohn1965Sep,Chubukov1993Jul,Maiti2013Aug}.”\nReviewer #1: 3. A minor point: When D is introduced in the left column on page 2 for\nthe first time, the author should explicitly mention its definition because 2D is often\nused to represent “two dimensions.”\nOur response: We thank the Referee for the catch, we now defined D at its first\nmention.\nReviewer #2 (Remarks to the Author):\nThe authors proposed that the screened Coulomb interaction can realize chiral\nsuperconductivity in rhombohedral multilayer graphene, motivated by the recent\nexperiment on the ABCA tetralayer showing possible spin- and valley-polarized\nsuperconductivity. They approximated the single-particle band structure by a circularly\nsymmetric minimal model with parameters fitted by the k.p band model. Using RPA and\nthe minimal band model, they found that p-wave superconductivity (which the authors\nsuggested to be chiral p-wave, e.g., p+ip) can emerge with T_c varying from 100 mK to\na few K, depending on the screening parameters of the Coulomb interaction. They\nfurther pointed out the topological transition within the superconducting region and\nargued that the results may explain the SC1 and SC2 regions in the experiment.\nThe subject of this manuscript is important as the new ABCA tetralayer graphene\nexperiment may open up a new direction for realizing chiral topological\nsuperconductors and Majorana physics. Overall, the manuscript is well-written, and the\nqualitative results are interesting. This work shows the first microscopic calculation of\nthe superconductivity driven by the screened Coulomb interaction using a simplified\nband structure. While the analysis seems appropriate at first glance, there are several\noutstanding issues that need to be addressed. As a result, I cannot recommend\npublication. I will be happy to review the manuscript again after the authors resolve all\nthe issues. The questions/comments are as follows:\nOur response: We thank the Referee for their thoughtful comments and clear interest\nin our article. We appreciate the comments and questions that we now have addressed\nin detail and incorporated in our article. The comments have improved our manuscript.\nWe hope that our response is satisfactory and that the improved article is now ready\nfor publication.\nReviewer #2: (1) The Kohn-Luttinger mechanism here (and in other graphene-based\nsystems) seems to overestimate T_c. Unlike the phonon-mediated pairing, there is no\nMigdal theorem for the screened Coulomb interaction. As a result, the vertex correction\nand other related contributions may modify the estimated T_c significantly. The authors\nshould discuss this issue and provide some quantitative estimates if possible.\nOur response: We agree with the Referee that vertex corrections beyond RPA can be\nrelevant and may alter the quantitative estimate of T_c. We included a discussion of\nrelevant literature in the text. We leave more quantitative estimates of Tc and pairing\nstrength for future work, because such quantitative estimates are known to be\nunreliable for strongly coupled superconductors as we study in our work. We\nemphasize that the recent work [X. Cie et al.,106, L220502 (2022)] has argued for the\nKohn-Luttinger mechanism (including only second-order perturbation theory) to be\nnegligible in comparison to that of (dynamic) screening of Coulomb interaction.\nWe added to the discussion section in the manuscript:\n“The RPA is expected to be qualitatively correct, but corrections beyond RPA are\nexpected to significantly alter the quantitative predictions because of the absence of\nMigdal's theorem \\cite{Migdal1958Jun} for superconductivity from screened electron-\nelectron repulsion \\cite{Takada1993Mar}. %Ref.\\cite{Grabowski1984Jun} argued that\nhigher-order corrections to RPA generally reduce the critical temperature. Different\ncorrections contribute to an enhancement or reduction of $T_c$. For p-wave pairing,\nRef.\\cite{Takada1993Mar} identified an enhanced $T_c$ by a factor of $\\approx 3$. A\ndetailed study of corrections beyond RPA is left for future work.”\nReviewer #2: (2) It needs to be clarified how reliable the minimal band model is based\non the current results. The authors argued that ''the energy scale of trigonal warping is\nsmall in tetralayer graphene in the range of density and displacement field of interest, as\nevidenced by the nearly symmetric band dispersion shown in Fig. 1.'' However, it’s not\nevident from Fig. 1, which is a plot of E vs. k_x, that trigonal warping is indeed weak.\nThe k.p band structure is NOT circularly symmetric, and the dispersion along k_x and\nk_y (relative to the valley momentum) can be very different. The fitting in Appendix A\nand Fig. 5 need to be discussed more carefully. The authors may compare the density\nof states of the minimal model and the k.p model, which should provide some hints\nabout the validity of the minimal model.\nOur response: We agree with the Referee that trigonal warping leads to relevant\ndistortions of the Fermi surface. By the suggestion of the Referee, we compare the\ndensity of states of our model with the results from the 8-band model as shown in Fig.\n1(c) Ref. [Ghazaryan et al., PRB 107, 104502 (2023)], see below for our summary.\nInterestingly, our results show that the superconducting pairing potential is not pinned\nto the Fermi surface, instead it is largest far away from the Fermi surface, as seen in\nFig. 4(a). This implies that deformations of the Fermi surface due to trigonal warping\nare less relevant for the pairing strength and critical temperature.\nOur comparison is summarized and added to Appendix A of our manuscript:\n“To validate our fit, we compare our density of states with the density of states\nobtained from the 8-band model in Ref.\\cite{Ghazaryan2023Mar}. Translating the\nresult of Fig.~1(c) of Ref.\\cite{Ghazaryan2023Mar} to the spin- and valley polarized\ncase (which requires to include a factor 1/4 in density of states and density), the\ndensity of states of the 8-band model with $D = 60\\ {\\rm meV}$ peaks at $n \\approx\n0.2\\ 10^{12} {\\rm cm}^{-2}$ with value $\\nu \\approx 5.5\\ \\nu_e$ and falls off to $\\nu\n\\approx 1.2\\ \\nu_e$ at $n \\approx 0.5\\ 10^{12} {\\rm cm}^{-2}$. This behavior is\napproximately reproduced by the density of states of our circularly symmetric model,\ncompare to Fig.\\ref{fig:3}(a) in the main text. ”\nReviewer #2: (3) The conclusion of chiral pairing is based on the Berry curvature\nargument without any quantitative calculations. I would like to know how large this\nsplitting between two chiral order parameters is due to the effect of Berry curvature.\nConventionally, one needs to check the quartic order of the order parameter in free\nenergy to determine the chiral or nodal pairing, as shown in Ref. 19. The authors may\nwant to check the quartic term.\nOur response: We expect that the dominant contribution selecting chiral px \\pm i py\nover nodal px / py pairing originates from the orbital angular momentum of the\nelectrons in the valley. Because we assume that the superconductivity develops on top\nof a fully isospin-polarized phase, this provides an explicit form of the free energy\nfavoring chiral superconducting order. This effect enters via the form factor already in\nquadratic order in the calculations of condensation energy, charge susceptibility, and\nself-consistency equations for the pairing term. In contrast, the mechanism discussed\nin Ref. 19 assumes spontaneous symmetry breaking towards chiral order.\nIn the revised version of the manuscript, we added a new Appendix E where\ncalculations of critical temperature and pairing gap including the form factor are\ndetailed. Taking into account only the 4\\pi Berry phase effect from the microscopic\ntheory without trigonal warping, we find a difference in around 20% between the Tc of l\n= 1 and l = - 1 pairing, see the new Appendix E for details. We expect this splitting at\nquadratic order to dominate over contributions from quartic order so that chiral pairing\nis favored.\nReviewer #2: (4) The authors may want to discuss other potential pairing mechanisms.\nWhat has been achieved here is that the screened Coulomb interaction may provide a\nway to explain the putative spin- and valley-polarized superconductivity in ABCA\ngraphene. However, it does not rule out other possibilities. It is possible that several\ndifferent pairing glues result in the experimentally observed superconductivity. Some\ndiscussions about the possible mechanisms will be useful.\nOur response: We thank the Referee for the suggestion and added to the discussion\nsection:\n“Alternative mechanisms for superconductivity in an isospin-polarized metal include\nphonons and fluctuations of the isospin polarization. We assume that the\nsuperconductivity develops on top of a fully spin- and valley-polarized state, which is in\nline with experimental observation. For such case, the Pauli principle requires the\nsuperconducting state to be odd-parity. However, phonons typically lead to an\nisotropic attraction that favors the $s$-wave channel and is averaged out for higher\nangular momenta, and thus, we expect that this is likely not an important mechanism\nfor pairing for the system considered here. Fluctuations of the isospin order parameter\ncan indeed mediate strong pairing in the proximity to the phase transition in the isospin\norder \\cite{Fay1980Oct,Lohneysen2007Aug,Dong2023May,Dong2023Oct}. We remark\nthat these fluctuations are not present in the fully polarized phase that we consider\nhere.“\nReviewer #2: Minor issues:\n(5) The authors employ the Rytova-Keldysh potential, with the 2D screening length r_K\nas a tuning parameter, based on its inverse dependence on the band gap of multilayer\ngraphene, and thereby on the displacement field D. On the contrary, the dual-gated\nCoulomb potential is considered in many other existing literature. Why did the author\npick this particular one over the other one? The authors did say that the choice of the\nCoulomb potential (before RPA) does not affect the qualitative results. However, the\nshould authors explain their choice when introducing the Rytova-Keldysh potential. The\nauthors mentioned that the Rytova-Keldysh potential is often used for 2D systems of\nwhich the dielectric constant is much larger than the dielectric environment. Based on\nthe current manuscript, it is unclear that the dielectric constant of multilayer graphene is\nmuch larger than hBN, so relevant experiments should be cited.\nOur response: The Rytova-Keldysh potential suppresses the interactions on short\ndistances of the order of the r_K screening length while dual-gate screening\nsuppresses interactions on long distances of the order of the gate distance. Thus,\nthese two mechanisms work on different scales.\nBoth mechanisms are generally present. Our study shows that Rytova-Keldysh\ndielectric screening can strongly suppress superconductivity, while gate screening on\ntypical gate distances of around 30 nm and above has almost no effect. The latter is\ndemonstrated by our new numerical results summarized in the new Appendix F of the\nresubmitted manuscript.\nIn our case, gate screening is irrelevant to the superconducting state because the\nThomas-Fermi screening length from RPA is of order of 1-2 nm, much shorter than\ntypical distance of around 30nm to the gates. In contrast, the pairing strength on short\ndistances is relevant for the superconductivity, therefore, the Rytova-Keldysh dielectric\nscreening is important to consider.\nWe adapted the discussion on screening in the main text, page 3 left column:\n“In real space, the screening strongly reduces the repulsion above the Thomas-Fermi\nscreening length $\\ell_{\\rm TF}$. Furthermore, Friedel oscillations appear with period\n$2 k_{F,1},\\ 2 k_{F,2}$ due to the two Fermi surfaces. When the Thomas-Fermi\nscreening length is much shorter than the wavelengths $\\pi/k_{F,1},\\ \\pi/k_{F,2}$,\nFriedel oscillations dominate over Coulomb repulsion and lead to an effective attraction\nat short length scales, as obtained from our calculations shown in Fig.\\ref{fig:2}(c). We\nfind $\\ell_{\\rm TF} \\approx 1$ to $2\\ {\\rm nm}$, for typical parameters where we\nobserve superconductivity as discussed below, while $2\\pi/k_{F}$ is typically around\n$20 \\ {\\rm nm} $ to $ 30\\ {\\rm nm}$. This mechanism benefits from a mexican-hat\nshaped dispersion because the additional pocket contributes a large density of states\nleading to a short Thomas-Fermi screening length. This enables superconductivity at\nlarger density.”\nWe further added a discussion in the discussion section:\n“We note that, besides possible suppression of a charge-ordered competing state,\nscreening from metallic gates becomes relevant to the superconducting state only\nwhen the distance to the metallic gates becomes of order of $\\ell_{\\rm TF}$. Because\ntypical distances to metallic fields in experiment are of order $30\\ {\\rm nm}$, much\nlarger than the typical $\\ell_{\\rm TF} \\approx 1$ to $2 \\ {\\rm nm}$ obtained from our\ncalculations, gate-screening is irrelavant to the superconducting state. Only when\nreducing the gate distance to the order of $\\ell_{\\rm TF}$ the superconducting order\ngets suppressed, see App.\\ref{app:gate_screening} for details. ”\nFinally, we added a new Appendix F containing numerical results on how the\nsuperconducting state responds to gate screening.\nReviewer #2: (6) Fig. 2(a) shows the single-particle static polarization functon(Eq. 5) by\nsetting the frequency to zero. I have a few clarifying questions: Does Fig. 2(a) only\ninclude intraband transitions when the Fermi level aligns with the annular Fermi pocket?\nAre interband transitions, which are gap-dependent, effectively incorporated through\nthe parameter r_K? Additionally, which value of r_K is used for Fig. 2(b) and (c)? In Fig.\n2(c), the attraction length scale is about 1 nm—comparable to the multilayer graphene\nthickness. Why, then, is the Rytova-Keldysh potential valid? The authors should clarify\nthese points in the manuscript.\nOur response: Thank you for this question. Yes, Fig. 2(a) includes only intraband\ncontributions from the conduction band. Also yes, interband transitions to the valence\nband are effectively incorporated through the Rytova-Keldysh parameter r_K. The\nparameter r_K is chosen to be r_K = 3nm, same for all calculations shown in the\nmanuscript. We added a specification of r_K in the caption of Fig 2.\nWe discovered a missing factor of 2pi in the Fourier transform of the interaction\npotential from reciprocal to real space. With the correction, the attraction length scale\nis around 5nm, as shown in the updated paned 2(c). We apologize for the confusion.\nWith the corrected value of 5nm for the attraction length scale, there is no contradiction\nwith the validity of the Rytova Keldish potential anymore due to the hierarchy\n“graphene thickness < Rytova-Keldysh parameter < attraction length scale”.\nWe corrected Fig. 1 and added a specification of r_K in the caption of Fig 2.\nReviewer #2: (7) In Fig. 3(c), the highest Tc is achieved at r_K = 1 nm, approximately\nmatching the thickness of tetralayer graphene. Does using the Keldysh potential with\nr_K = 1 nm yield a Tc comparable to using the bare 3D Coulomb potential with RPA?\nOur response: With bare Coulomb repulsion, a maximal Tc of around 8K is obtained.\nThe observation that the superconducting properties depend sensitively on dielectric\nscreening parameterized by r_K indicates that a detailed modeling of the Coulomb\ninteraction on short length scales compared to the film thickness is necessary for an\naccurate prediction of Tc. The conclusion brings attention to an important aspect of\ninteractions in layered graphene systems which suggests a direction for future studies.\nWe highlight this result in bottom left paragraph of page 4 the main text and added a\nnew Appendix H that shows the numerical data for bare Coulomb repulsion.\nReviewer #2: (8) According to Fig. 2(a), k_{F,2} is approximately 0.3 nm^-1,\ncorresponding to a Fermi wavelength of around 20 nm, which greatly exceeds the 1 nm\nattraction length scale indicated in Fig. 2(d). How do the authors conclude that: “Our\nmechanism is based on the overscreening of Coulomb interaction due to charge\nfluctuations, which leads to an effective attraction at length scales of a few Fermi\nwavelengths, driving Cooper pairing.” and “The screening strongly reduces the\nrepulsion on long length scales compared to the Fermi wavelength, and leads to an\noverscreening with an effective attraction at distances of a few Fermi wavelengths,\nas demonstrated in Fig. 2(c).”\nOur response: We thank the Referee for the careful reading. As mention in our\nresponse to comment (6) of Referee 2, we corrected a missing factor of 2pi in the\nFourier transform of the interaction potential from reciprocal to real space. With the\ncorrection, the attraction length scale is around 5nm.\nWe further corrected:\n“Our mechanism is based on the overscreening of Coulomb interaction due to charge\nfluctuations, which leads to an effective attraction at length scales on a fraction of the\nFermi wavelength, driving Cooper pairing.”\nWe thank the Referee for the useful comment. The second point mentioned by the\nreferee is reformulated in a discussion now highlighting the Thomas-Fermi screening\nlength as the relevant length scale for the screening and its relation to the attraction\nfrom Friedel oscillations:\n“In real space, the screening strongly reduces the repulsion above the Thomas-Fermi\nscreening length $\\ell_{\\rm TF}$. Furthermore, Friedel oscillations appear with period\n$2 k_{F,1},\\ 2 k_{F,2}$ due to the two Fermi surfaces. When the Thomas-Fermi\nscreening length is much shorter than the wavelengths $\\pi/k_{F,1},\\ \\pi/k_{F,2}$,\nFriedel oscillations dominate over Coulomb repulsion and lead to an effective attraction\nat short length scales, as obtained from our calculations shown in Fig.\\ref{fig:2}(c). We\nfind $\\ell_{\\rm TF} \\approx 1$ to $2\\ {\\rm nm}$, for typical parameters where we\nobserve superconductivity as discussed below, while $2\\pi/k_{F}$ is typically around\n$20 \\ {\\rm nm} $ to $ 30\\ {\\rm nm}$. This mechanism benefits from a mexican-hat\nshaped dispersion because the additional pocket contributes a large density of states\nleading to a short Thomas-Fermi screening length. This enables superconductivity at\nlarger density.” (See also our response to comment (5) of Reviewer #2 above).\nWe verified that the screened interaction potential exhibits Friedel oscillations with\nperiod of half a Fermi wavelength, as typical for Friedel oscillations.\nReviewer #2: (9) This work shows the first microscopic calculations for the screened-\nCoulomb-driven superconductivity in the tetralayer graphene system. However, this\npossibility was also suggested in Ref. [62]. The authors cited Ref. [62] in an obscure\nway -- They put a very short footnote [36], stating \"For a mean field study of\nsuperconductivity from attraction, see Ref. [62]).'' Since the screened-Coulomb\nmechanism was also suggested in Ref. [62], the footnote is not accurate.\nOur response: We changed the citation and now write after the acknowledgements:\n“During final stages of writing of the current manuscript, Ref.\\cite{Chou2024Sep}\nappeared, where a charge fluctuation mechanism for p-wave superconductivity in\nrhombohedral graphene has been discussed as well.”\nReviewer #2: (10) Could the authors comment on the role of the WSe_2 layer in the\ntetralayer graphene experiment, particularly within the context of the Kohn-Luttinger\nmechanism and the screening effects of the Rytova-Keldysh potential?\nOur response: The experiment measured 3 devices: Device 1 has a bilayer WSe2\nbeneath the tetra-layer graphene. Device 2 has a monolayer WSe2 on top of the tetra-\nlayer graphene. Device 3 is a bare tetra-layer graphene without WSe2. For Device 1 (2),\nthe displacement field was applied so that the electrons we pushed towards (away\nfrom) the WSe2 layers. The three devices yield qualitatively similar, but quantitively\ndistinct results, in particular the critical temperatures of the time-reversal symmetry\nbreaking phase were measured to be 160 mK, 210 mK and 300 mK for device 1, 2 and\n3 respectively.\nThe quantitative differences can be consistent with a different dielectric screening due\nto the different configurations of the WSe2 layers. In our theory, the different\nconfigurations change the r_K parameter of the Rytova-Keldysh potential. However, we\ncannot exclude other effects on the normal-state dispersion due to hybridization with\nthe layer that may also cause a quantitative change. Because we cannot disentangle\nthese effects from the current experimental data, we prefer not to comment on these in\nthe manuscript.\nReviewer #2: (11) Typo in Eq. (6): The frequency omega should be zero.\nOur response: We thank the Referee for pointing to this typo and corrected it.\nReviewer #2: (12) In the section “Topological superconductivity and Lifshitz transition”:\n“The quasiparticle gap ∆0 ≡ min_k Ek as a function of density n and displacement field\nD is shown in Fig. 4(c).” There is no Fig. 4(c).\nOur response: We thank the Referee for pointing to this typo, the correct figure\nreference is Fig 4b. We corrected it in the resubmitted manuscript.\nReviewer #3 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports.\nThis is part of the Nature Communications initiative to facilitate training in peer review\nand to provide appropriate recognition for Early Career Researchers who co-review\nmanuscripts.\nOur response: We thank the Referee for providing training and guidance in the\nreviewing process for the contributing early career researcher and ensuring high-quality\nreferee reports.\nReviewer #4 (Remarks to the Author):\nThe authors performed a calculation of a Coulomb repulsion originated theory for spin-\nvalley polarized chiral superconductivity in tetralayer graphene, aimed at explaining the\nrecently observed superconductivity in such systems. The theoretical calculations\nfollow the standard mean field theory procedures and seemed to lead to desired\nresults, and this is a timely study of the very interesting recent experiment. It is however\ndifficult to claim how reliable the mean field calculations are for such a strongly\ninteracting system. I also find the paper not presented clearly, with unjustified or\nambiguous points in the authors calculations and arguments. I therefore cannot\nrecommend the current paper draft to Nature Communication. My detailed questions\nand comments are given below.\nOur response: We agree with the Referee that mean-field calculations can only\nqualitatively capture the physics, while it is well-known that predictions from mean-field\ntheory for critical temperature or pairing strength are quantitatively inaccurate (see also\nour response to Referee 2, comment 1, and Referee 4, comment 5). However, it is also\nwell-know that a quantitative theory of strong-coupling superconductivity does not\nexist, and the best one can achieve with current methods is qualitative correctness and\npredictions within rough orders of magnitude of relevant quantities.\nApart from making qualitative predictions, or theory makes an important physical point:\na coexistence of the isospin-polarizing order (with energy scale that is larger than the\ntypical superconducting pairing) and favorable conditions for superconducting\ninstability (such as flat dispersion/large density of states and proximity to the charge-\nordering transition) provides a very natural route for chiral superconductivity where\nexplicit symmetry breaking is inherited from the isospin order. It is a very simple\nprinciple which can become a powerful approach in the future theoretical and\nexperimental search for unconventional superconductivity.\nWe have provided detailed answers to the Referee’s comments below and clarified the\nmanuscript in all places where a potential misunderstanding of our arguments may\nhave occurred. We hope that our answers and revisions resolved any perceivable\nambiguity or unjustified arguments. We hope that our clarifications are convincing for\nthe Referee to reconsider their recommendation for publication in Nature\nCommunications.\nReviewer #4: 1. The form of band dispersion in Eq. (1) is proposed in a strange way: if\nk_0 is fixed, the dispersion will not reach k^n (the well-known dispersion of n-layer\ngraphene) at D=0, so D is not an independent tuning parameter. After checking App. A I\ncan understand that k_0 and m are functions of D, and probably k_0 is proportional to\nD^{1/n} such that k^n dispersion can be reached at D=0. It would be more natural to\nredefine the first part as \\sqrt{D^2+k^2/k_0^2} with k_0 roughly constant. Or at least, in\nthe current form, the authors should explain in the main text that k_0 and m are\nfunctions of D (and k_0 proportional to D^{1/n}).\nOur response: We apologize for the confusion and thank the Referee for the careful\nreading. We now emphasize in the main text that m and k_0 are D-dependent fit\nparameters. When using using a parameterization \\sqrt(D + k^2 / k_0^2) for the first\nterm in the dispersion, the quantity k_0 does not have units of momentum, and in\naddition assuming k_0 to be roughly constant would not be accurate. For this reason\nwe prefer to stick to our previous notation, and simply clarify the dependence of\nparameters on D.\nReviewer #4: 2. Related to the above question, it seems the theory proposed in this\npaper is largely independent of the number of graphene layers n. Are all the calculations\nspecific for n=4, and if so, anything special about n=4? In other words, does the theory\nprovide an understanding why the chiral superconductivity is observed in tetra-layer\ngraphene?\nOur response: See also our response to Comment 1 of Referee #1. Because all\nrhombohedrally stacked graphene systems have qualitatively similar dispersion, our\nmodel is indeed not qualitatively sensitive to the number of layers. However,\ncalculations from Ref. [Ghazaryan et al., PRB (2023)] show that the rhombohedral\ntetralayer graphene reaches a significantly larger density of states close to the band\nbottom compared to smaller number of layers. Our theory shows that the large density\nof states is crucial for the emergence of superconductivity, because it leads to a short\nThomas Fermi screening length enabling an effective attraction from overscreened\nFriedel oscillations. Therefore, this significant quantitative difference in the density of\nstates of different layer number may explain why superconductivity has been observed\nin tetralayer rhombohedral graphene, but not (yet) for other layer number.\nWe added a comment in the conclusion:\n“We expect that our mechanism may also apply to isospin-polarized phases of\nrhombohedral graphene with a different number of layers because all these systems\nshare the feature of a large density of states at the band bottom under an applied\ndisplacement field \\cite{Ghazaryan2023Mar}. Crucially, calculations of\nRef.\\cite{Ghazaryan2023Mar} indicate that tetralayer graphene reaches a significantly\nlarger density of states close to the band bottom under a displacement field compared\nto smaller number of layers. Our theory shows that the large density of states induces a\nshort Thomas-Fermi screening length that enables superconductivity from repulsion via\noverscreened Friedel oscillations. This significant quantitative difference may explain\nwhy superconductivity has been observed in rhombohedral tetralayer and pentalayer\ngraphene \\cite{Han2024Aug}, but not for smaller layer number up to now. ”\n(See also our response to Reviewer #1 Comment 1. above)\nReviewer #4: 3. Is the mass term m in Eq. (1) the particle-hole symmetry breaking\nterm? Can the authors understand from their theory why the superconductivity occurs\nin the conduction band?\nOur response: Yes, the mass term in Eq. (1) breaks the particle-hole symmetry\nbetween valence and conduction band. The particle-hole symmetry breaking arises\nfrom higher-order hopping terms and an interlayer potential difference induced, e.g., by\nthe displacement field, see Ref. [Koshino 2009] for a detailed discussion.\nOur theory favors superconductivity in the conduction band, because due to the\nparticle-hole symmetry breaking terms, the conduction band has a much larger density\nof states at the band bottom compared to the density of states in at the valence band\ntop. As large density of states is required for our charge-fluctuation mechanism (this\ncan be seen because the charge susceptibility at q = 0 is equal to the density of\nstates), we expect superconductivity only in the conduction, but not in the valence\nband.\nWe added a comment in the conclusion:\n“Related, we remark that multilayer rhombohedral graphene under a displacement field\nexhibits a larger density of states close to the band bottom in the conduction\ncompared to the valence band \\cite{Ghazaryan2023Mar}, which favors the charge-\nfluctuation mechanism for superconductivity in the conduction but not in the valence\nband.”\nReviewer #4: 4. Some setups of the model considered are unclear. In writing down the\ninteraction in Eq. (2) and later, are the electron operators defined for one particular spin\nand valley? It is difficult to find in the paper whether the pairing is intra-valley or\nintervalley. Also, I don’t think k_{F,1} and k_{F,2} are defined before they first appear.\nOur response: We apologize for the confusion. In the revised version, we have clarified\nbelow Eq. (2) that the electron operators are for spin-polarized electrons within a single\nvalley. To further clarify that the pairing is intra-valley, we added a specification in the\nsummarizing paragraph at the end of our introduction. We further clarified that k_F1\nand k_F2 are the momenta of the inner and outer Fermi surfaces, respectively, in the\ncaption of Fig 1 where they first appear.\nReviewer #4: 5. The effective attractive interaction near r=1nm in Fig. 2(c) is\ncounterintuitive. RPA, which can be understood as the dynamical generalization of\nThomas-Fermi screening, usually only reduces the repulsion due to opposite charges\naccumulated nearby. It seems unlikely to have “over-screening” that the repulsion is\nflipped into attractions. It is more likely that the interaction is attractive in non-s wave\nchannels, instead of being globally attractive. Although s-wave here is forbidden by\nspin-valley polarization, the RPA calculations seem to be independent from this\nassumption. Can the authors provide an intuitive picture for understanding the\nattraction? Also, RPA is a perturbative method for interactions much smaller than the\nFermi energy, how is it justified here, where interaction is much larger than the kinetic\nenergy (as indicated by r_s the authors calculated)?\nOur response: It is correct that the RPA calculations are independent from any\nassumptions on the pairing symmetry. Indeed, we find that the RPA screened potential\nbecomes attractive on a length scale of a fraction of the Fermi wavelength. Intuitively,\nthis arises from Friedel oscillations of the electrostatic potential of a charge placed in\nan electron gas: In the presence of strong screening, the strong repulsion of the bare\nCoulomb interaction is sufficiently suppressed so that Friedel oscillations become\nlarger than the screened Coulomb repulsion, leading to an effective attraction.\nIndependent of spin-polarization, this screened potential generally favors finite angular\nmomentum pairing over s-wave, because a strong repulsion at very short distances\nremains.\nThe RPA calculation here is only expected to be only qualitatively correct, because it’s\nquantitative accuracy is not justified for strong interactions as the Referee points out. It\nis well-known that RPA tends to overestimate pairing strength, see also our response\nto comment 1 of Referee #2. However, it has been argued that RPA captures the\nqualitatively correct features, see for example Ref. [X. Can et al., PRB 106, L220502\n(2022)].\nWe clarified in the main text:\n“In real space, the screening strongly reduces the repulsion above the Thomas-Fermi\nscreening length $\\ell_{\\rm TF}$. Furthermore, Friedel oscillations appear with period\n$2 k_{F,1},\\ 2 k_{F,2}$ due to the two Fermi surfaces. When the Thomas-Fermi\nscreening length is much shorter than the wavelengths $\\pi/k_{F,1},\\ \\pi/k_{F,2}$,\nFriedel oscillations dominate over Coulomb repulsion and lead to an effective attraction\nat short length scales, as obtained from our calculations shown in Fig.\\ref{fig:2}(c). We\nfind $\\ell_{\\rm TF} \\approx 1$ to $2\\ {\\rm nm}$, for typical parameters where we\nobserve superconductivity as discussed below, while $2\\pi/k_{F}$ is typically around\n$20 \\ {\\rm nm} $ to $ 30\\ {\\rm nm}$. This mechanism benefits from a mexican-hat\nshaped dispersion because the additional pocket contributes a large density of states\nleading to a short Thomas-Fermi screening length. This enables superconductivity at\nlarger density.”\nReviewer #4: 6. A related question, is the effective RPA interaction above sufficient to\ninduce spin-valley polarization, i.e., the starting point of the current theory? The\ninteraction in Fig. 2(c) looks much weaker and localized than the bare interaction; the\nhighly local interaction may resemble the Hubbard model which gives anti-\nferromagnetism instead of (orbital) ferromagnetism.\nOur response: It is correct that isospin polarization and superconductivity both occur\nfrom Coulomb repulsion. However, they occur at different energy scales, and we treat\nsuperconducting order perturbatively with respect to the isospin order. Isospin\npolarization can be obtained in a Stoner model where the energy scale is \\nu(E_F) V(2\nk_F) E_F \\approx 2 meV where \\nu(E_F) is the density of states at the Fermi energy E_F\nand V(2 k_F) is a measure for the bare Coulomb interaction energy at the (outer) Fermi\nsurface. This energy scale is about an order of magnitude larger than the pairing\nstrength (compare to manuscript Fig. 4). A detailed discussion on how the isospin\npolarization is obtained from the Stoner model applied to rhombohedral graphene is\ncontained in Ref. Ghazaryan et al., PRB 2023.\nPlease see also our response to Referee 1, comment 2.\nReviewer #4: 7. The form factor or the interaction in Eq. (13) looks incorrect.\nShouldn’t it be the inner product of u_{k+q} with u_k, and u_{k’-q} with u_{k’}?\nOur response: We thank the Referee for the careful reading and correcting the typo.\nReviewer #4: 8. I do not understand what “charge-ordered state” in the discussion\nmean. Does it mean Wigner Crystal? The authors may want to clarify. Also, there is\n(almost) no mentioning of the Lifshitz transition of Fermi surface in the main text\n(probably the cyan line region in Fig. 4?), which the authors should explain better.\nOur response: By charge-ordered state, we imply the general class of states with a\ncharge-density order. This includes charge-density waves and Wigner crystals. These\ntwo charge orders are continuously connected as the Wigner crystal is the extreme\nlimit of a charge density wave where the magnitude of the wave approaches the mean\nelectron density, so that the charge density in the valleys of the wave approaches zero.\nWe further thank the referee for the suggestion to clarify the Lifshitz transition in our\nmanuscript. At the first mentioning of the Lifshitz transition in the main text, we now\nclarify:\n“In this case, a Lifshitz transition from simply-connected to annular Fermi sea occurs\nwhen the Fermi energy crosses the top of the hat as the electron density is reduced.”\nThe Lifshitz transition is depicted by the white dashed line in our numerical results of\nFig. 3(a), 3(b), and 4(b). We now emphasize this fact in the caption of both figures.\nThe cyan-shaded region encircled by the cyan line in Fig 4(b) marks the region where\nthe gas parameter r_s is larger than 40. Qualitatively, a large gas parameter indicates\nthat the system may likely charge-order into a Wigner crystalline, insulating state. We\ncomment on this region in section “Competing state” on page 5 of our manuscript.\nReviewer #4: 9. The electron-phonon coupling is completely ignored in the study here.\nIs there a particular reason for ignoring the effect of phonons? Also, how would the\nauthors propose to tell experimentally whether the superconductivity is from Coulomb\nrepulsion instead of phonons? For instance, if the screening of Coulomb interaction is\nenhanced (by metallic gate, etc), would the authors predict an enhancement or\ndecrease of superconducting Tc?\nOur response: We thank the Referee for the suggestion of investigating the role of\nphonons and how to tell whether the mechanism is from charge fluctuations by\nscreening.\nRegarding phonons, we included the following text in the discussion section (see also\nour response to Referee 2, comment (4)):\n“Alternative mechanisms for superconductivity in an isospin-polarized metal include\nphonons and fluctuations of the isospin polarization. We assume that the\nsuperconductivity develops on top of a fully spin- and valley-polarized state, which is in\nline with experimental observation. For such case, the Pauli principle requires the\nsuperconducting state to be odd-parity. However, phonons typically lead to an\nisotropic attraction that favors the $s$-wave channel and is averaged out for higher\nangular momenta, and thus, we expect that this is likely not an important mechanism\nfor pairing for the system considered here. Fluctuations of the isospin order parameter\ncan indeed mediate strong pairing in the proximity to the phase transition in the isospin\norder \\cite{Fay1980Oct,Lohneysen2007Aug,Dong2023May,Dong2023Oct}. We remark\nthat these fluctuations are not present in the fully polarized phase that we consider\nhere.”\nRegarding the effect of screening from gates, we found that it only plays role when the\ndistance to the metallic gates is on order of the Thomas-Fermi screening length. The\nThomas-Fermi screening length in the superconducting regions is on order of 1-2 nm.\nWhen reducing the gate distance to this length scale, the superconducting pairing and\nthe transition temperature are indeed reduced. This is detailed in our new Appendix F.\nAdditionally, we comment in the discussion section:\n“We note that, besides possible suppression of a charge-ordered competing state,\nscreening from metallic gates becomes relevant to the superconducting state only\nwhen the distance to the metallic gates becomes of order of $\\ell_{\\rm TF}$. Because\ntypical distances to metallic fields in experiment are of order $30\\ {\\rm nm}$, much\nlarger than the typical $\\ell_{\\rm TF} \\approx 1$ to $2 \\ {\\rm nm}$ obtained from our\ncalculations, gate-screening is irrelavant to the superconducting state. Only when\nreducing the gate distance to the order of $\\ell_{\\rm TF}$ the superconducting order\ngets suppressed, see App.\\ref{app:gate_screening} for details. ”\nReviewer #4: 10. The authors only focused on the time-reversal breaking\nsuperconducting phases in the experiment. The experiment also observes more\nconventional superconductivity in two other parameter regimes. Do the authors have\ncomments on the other superconductivity regimes? Is the claim that Coulomb\ninteraction is better for realizing chiral superconductivity (in contrast to phonons which\nmay favor s-wave superconductivity more)?\nOur response: Thank you for your question. At this point, we prefer to keep the focus\nof our manuscript on the novel time-reversal symmetry breaking phases and leave a\nunifying discussion of the full phase diagram for future work.\nReviewer #4: 11. A minor question: I believe \\omega in Eq. (6) should be set to zero\nalready.\nOur response: We thank the Referee for pointing to this typo and corrected it.\nList of changes:\n- All changes are highlighted in blue in the “manuscript difference file” included in the\nresubmission.\n- Appendices were rearranged so they occur in order as mentioned in the text.\nResponse to referee reports\nReviewer #2 (Remarks to the Author):\nReviewer 2 : I have reviewed the responses (from Reviewers #1 and #2) and the\nrevised manuscript. The authors have resolved most of the issues and internal\nconsistency. I am inclined to recommend acceptance, provided that the authors\ncarefully address the following comments.\nOur response : We thank the referee for their positive assessment of our response and\nrevisions and address the referee’s comments below.\nReviewer 2 : (1) The circularly symmetric model in this manuscript indeed predicts the\ndegenerate px and py pairing channels. However, such degeneracy is unclear for a\nrealistic dispersion that admits only C3 rotation. Can the authors comment on this point?\nA related comment: In Appendix E, the authors explore how Berry curvature affects the\nangular momentum of the superconducting phase, focusing on D = 60 meV, which\ncorresponds to a trivial SC with an annular Fermi surface. Are there results for the case\nin the 1P (topologically nontrivial) phase?\nOur response : To protect the degeneracy between px and py pairing, it is sufficient to\nhave C3 rotation symmetry and time-reversal symmetry. However, in our model, the\ntime-reversal symmetry is broken by the Berry curvature of the gapped Dirac cone in\nvalley-polarized state. This time-reversal symmetry breaking favors a chiral px - i py\nsuperconducting state with chirality selected by the valley. This is detailed in our\ndiscussion in the paragraphs headed by “Trigonal warping and Berry curvature effects.”\nin the main text.\nWe also analyzed the effect of Berry curvature on pairing in the 1P topologically\nnon-trivial phase, see below for the critical temperature and Fermi-surface pairing\nstrength computed for D = 35 meV. The critical temperature and pairing strength is\nstrongly suppressed for one of the angular momenta of the chiral superconducting order\nwhile it remains largely unaffected for the other. Again, this demonstrates that the Berry\nphase effect strongly favors one chirality over the other.\nRemarkably, this effect is much stronger for the topological chiral superconducting state\nin the 1P polarized state with simply connected Fermi sea compared to in the trivial\nchiral superconducting state in the system with annular Fermi sea.\nWe included this figure and discussion as additional data in App. E.\nFig. R2.1 : Critical temperature 𝑇 and Fermi-surface pairing strength ∆( 𝑘 ) calculated for 𝐷 = 35 meV,\nc 𝐹\nKeldysh screening parameter 𝑟 = 3 nm and distance to the metallic 𝑑 = 30 nm.\n𝐾\nReviewer 2 : (2) In the cover letter, ``In the resistive phase separating the\nsuperconducting domes SC1 and SC2 the resistance increases with temperature,\nindicating an insulating state consistent with our predicted Wigner crystal (see Extended\nData Figure 2(d, e) in the revised manuscript of the experiment arXiv:2408.15233).'' I\nsuspect that the authors meant ``the resistance increases as the temperature\ndecreases.'' I am unable to find the figures for the temperature dependence of the\ninsulating state in Extended Data Figure 2(d, e) of the experimental paper. There might\nbe some misunderstanding here.\nOur response : We thank the referee for the careful reading and apologize for the\nconfusion. We agree that the statement to indicate the insulating state should read ``the\nresistance increases as the temperature decreases.'' .\nIndeed, the precise figure we had in mind is not contained in the Extended Data Figure\n2(d, e), we apologize for the mistake on our side. We would like to offer the context on\nhow our theoretical proposal of the Wigner crystalline state connects with experimental\nobservation.\nAfter our theoretical work was concluded, we discussed our results with our\nexperimental colleagues, Long Ju’s team, pointing out the possibility of a Wigner\ncrystalline insulating phase separating the two superconducting domes SC1 and SC2.\nAt first, our proposal of this Wigner crystalline state was surprising to the experimental\nteam, because the resistance measured in the resistive state separating the two domes\nis only a factor ~3 larger compared to the neighboring metallic states. However, after\nour paper was posted, they revisited the experimental data and found that the\nresistance of the resistive state separating SC1 and SC2 clearly increases as\ntemperature decreases – thus indicating a weakly insulating state.\nIt is possible that the insulating state has a small gap, which may explain why the\nresistance of the state separating SC1 and SC2 is small, unlike the highly insulating\nstate at low density. It should also be noted that edges of graphene are often\nconducting, which complicates the determination of bulk resistance.\nThe data showing the resistance increase with decreasing temperature in the resistive\nregions separating SC1 and SC2 remains unpublished up to now. (One of us presented\na figure containing the data in a talk at the Kivelson conference in Stanford in March\n2025.) Unfortunately, since this data is not yet published, we cannot present it here in\nthis response.\nHowever, the top panel of Fig. 2(e) in the revised version of the experimental paper\n( arXiv:2408.15233v2) (which is the same as as the Extended Data Fig 2(e) in the\npublished manuscript [Nature 643, 654–661 (2025)]) contains a region around density\nn = 0.78 10 12 cm - 2 that connects to the resistive line separating SC1 and SC2: In this\ne\ndata, one sees that the resistance increases with decreasing temperature (blue T =\n7mK to yellow T = 540 mK) around density n = 0.78 10 1 2 cm - 2 and displacement field\ne\nD/eps = 1.14 V/nm. This datapoint is directly connected to the resistive line separating\n0\nthe two superconducting domes (compare to Extended Data Fig. 1 of [Nature 643,\n654–661 (2025)]). For convenience, we include this figure here below:\nFig. R2.2 : Extended Data Fig. 2(e) from [Nature 643, 654–661 (2025)] showing resistance as a function\nof temperature and density. It displays increasing resistance with lowering temperature around n = 0.78\ne\n10 12 cm - 2 and 𝐷 /ε = 1 . 14 V/nm. This datapoint lies within the region of increased resistance separating\ntemperature ranging from T = 7 mK (blue) to T = 540 mK (yellow).\nReviewer 2 : (3) In the revised manuscript and the cover letter for resubmission, the\nauthors claimed that their predicted Wigner crystal can explain the experiment. I have\nstrong reservations about this claim. First, the original experiment (posted before the\noriginal submission of this work) has reported an insulating state separating SC1 and\nSC2. Second, the authors did not do any concrete calculations related to the Wigner\ncrystal. Wigner crystal is a nice suggestion/speculation but not a concrete prediction by\nthe authors. The authors should modify the discussions/claims related to the Wigner\ncrystal.\nOur response : We double-checked the original submission, and there is no explicit\nmention of an insulating state separating SC1 and SC2. The original submission\ncontains data showing a region with increased resistance between SC1 and SC2, but it\nhad not been identified as an insulating state. This is evidenced because data\ncontaining the resistance as a function of temperature for the region separating SC1\nand SC2 is not shown in the manuscript (neither in the published version). The\nsuggestion that this region is insulating came from our theory and brought to the\nexperimentalists after our theory manuscript was posted on arXiv, please see our\nresponse to Referee 2, Comment (2) for details.\nIt is correct that we did not perform concrete calculations related to the Wigner\ncrystalline state beyond the qualitative estimate based on the\ninteraction-to-kinetic-energy ratio. We therefore weaken our statements in the\nmanuscript. We changed the sentence “Our analysis further predicts that a Wigner\ncrystalline insulating state occurs …” to “Based on our analysis we further propose that\na Wigner crystalline insulating state occurs …”.\nReviewer 2 : (4) Regarding Reviewer #1's point 1: The authors' response is based on\nthe density of states, which affects the Thomas-Fermi screening length. Indeed, the\nexisting Bernal and ABC graphene experiments do not support a sufficiently large\ndensity of states within the quarter metal phase. However, the isospin transitions are\nalso essential and depend on the Coulomb interaction as well as the band structure.\nOne has to compare the energies of the quarter metal state and other isospin states to\nunderstand the complete isospin phase diagram. Thus, the existing argument based on\nthe density of states near the bottom of the band is not rigorous. The authors need to\nclarify this issue.\nOur response :\nWe agree with the referee that the isospin polarization is a prerequisite for\nsuperconductivity from the charge fluctuation mechanism considered in this work, and\nthat in addition to large Fermi-surface density of states ν( 𝐸 ) one requires a large bare\n𝐹\ninteraction 𝑉 . At a qualitatively level, this is formalized by the Stoner criterion for isospin\npolarization:\nν( 𝐸 ) 𝑉 > 1\n𝐹\nWe further agree that a complete calculation requires comparing the ground state\nenergies of the different isospin polarized phases. Such a calculation has been\nperformed using a Stoner model, specifically for ABCA graphene in [Ghazaryan PRB\n107, 104502 (2023)]. The authors obtained the following phase diagram, which includes\nthe 1P/A (isospin polarized in a metallic state with single pocket / annular Fermi sea)\nregion in the electron-doped side at low density:\nFig. R2.3 : Fig. 3 taken from Ref. [Ghazaryan PRB 107, 104502 (2023)] showing the isospin-polarization\nphase diagram of ABCA graphene calculated from a Stoner model. Here ∆ is the displacement field that\n1\nequals D in our notation.\nWe trust these previous calculations, as well as the experimental data (in particular\nquantum oscillation) that clearly indicates the isospin polarized state, as sufficient\njustification to take the isospin polarized state as the starting point for our theoretical\nmodel.\nWe added a clarifying sentence in the conclusion:\n“Ref.\\cite{Ghazaryan2023Mar} also demonstrates that the isospin polarization\nemerges in a Stoner model due to large density of states and strong interaction.”\nReviewer #3 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports.\nThis is part of the Nature Communications initiative to facilitate training in peer review\nand to provide appropriate recognition for Early Career Researchers who co-review\nmanuscripts.\nOur response : We thank the Referee for providing training and guidance in the\nreviewing process for the contributing early career researcher and ensuring high-quality\nreferee reports.\nReviewer #4 (Remarks to the Author):\nReviewer 4 : The authors have answered my questions and concerns in details with\nsufficient clarity, which I do appreciate. I additionally note that (as requested by the\neditor) the questions raised by Reviewer 1 roughly coincide with part of my questions,\nwhich are reasonably well addressed by the authors. Generically, I must say I am still\nnot sure how quantitatively well the methods and the calculations are valid, including the\nsingle-particle model, and perturbative RPA calculations, etc. However, the theory the\nauthors propose is indeed a plausible mechanism at this stage, which I think is a timely\ncontribution to the field. Therefore, given that my questions are mostly well addressed, I\ncan recommend the paper for publication. I do have two minor follow up questions:\nOur response : We thank the referee for appreciating our detailed response to their\nearlier questions, judging our mechanism as plausible, recognizing the timeliness of our\nwork, and the recommendation for publication. We provide a detailed response to the\nreferee’s questions below.\nWe agree that the RPA calculations are qualitative, and it is well known that there are\nquantitative inaccuracies associated with the uncontrolled random-phase approximation\nin computing the renormalized interaction potential. Nevertheless, we emphasize that\nour theory yields the important qualitative predictions: The SC1 dome is identified as a\ntopological chiral superconductor while the SC2 dome is chiral but non-topological, and\nthese two domes are separated by an insulating Wigner crystalline state. Our prediction\nof the Wigner crystalline state was confirmed by the experimentalists by\nresistance-as-a-function of temperature measurements after we brought our theoretical\nprediction to them. Further, our work shows how Berry curvature effects in the massive\nDirac cones lift the degeneracy between l = 1 and l = -1 chiral superconducing pairing.\nReviewer 4 : 1. The different energy scales in isospin polarization vs. superconductivity\nis still not fully clear to me. Is there a simple argument for using the bare Coulomb\ninteraction in the Stoner model for polarization, and using screened interaction in the\nsuperconductivity calculation? Is it because their physics happen at different energy\nscales or length scales?\nOur response : Both isospin polarization and superconductivity arise from Coulomb\ninteraction among the electrons. On large energy scales, Coulomb interaction drives the\nisospin polarization among the electrons. The corresponding energy is estimated from a\nStoner model as ν( 𝐸 ) 𝑉( 2 𝑘 ) 𝐸 which we estimate to be around 2 to 3 meV for ABCA\n𝐹 𝐹 𝐹\ngraphene for typical density around 0.5 10 12 cm -2 , see also our discussion in the section\nheaded “Electron pairing from screened Coulomb repulsion” in the main text.\nWe emphasize that the Hamiltonian preserves spin-rotation symmetry. Therefore, there\nare no spin fluctuations in the isospin-fully-polarized phase. Further neglecting\nelectron-phonon coupling, charge fluctuations are the only remaining quantum\nfluctuation that may give rise to emergent superconducting phases.\nOn small energy scales, Coulomb interaction may give rise to further emergent\nphenomena. In our case, we investigate under which conditions superconductivity from\ncharge fluctuations is a possibility. In our proposed mechanism, superconductivity\nemerges from strong quantum correlations among the electrons. This requires\naccounting for diagrams of second- and higher-order perturbation theory, which is\nachieved by the RPA approximation yielding the screened Coulomb interaction. The\nscreened Coulomb potential is sufficient to yield superconductivity from self-consistency.\nThe energy scale for superconductivity is the pairing strength at the (outer) Fermi\nsurface ∆( 𝑘 ) , for which our calculations yield values up to 0.3 meV [see Fig. 4 in the\n𝐹 , 2\nmain text], around ten times smaller than the energy scale for isospin polarization.\nThe different energy scales of Stoner isospin polarization and superconductivity justify\nthe separate treatment of superconductivity in the isospin-polarized phase. A separate\nsolution of self-consistency equations for the isospin polarization is not necessary,\nbecause the isospin-polarization occurs on much larger energy scales.\nPlease see also our response to Referee 2, Comment (4). We added a clarifying\nsentence in the conclusion:\n“Ref.\\cite{Ghazaryan2023Mar} also demonstrates that the isospin polarization\nemerges in a Stoner model due to large density of states and strong interaction.”\nReviewer 4 : 2. The authors predict the chiral superconductivity is valley polarized, thus\nthe pairing would carry a nonzero momentum. Are there ways to detect such nonzero\npairing momentum that the authors can comment?\nOur response : We thank the referee for the suggestion to elaborate on this aspect.\nDetecting the Q = 2K finite momentum pair density wave state is hard, because the\ndensity variations are on atomistic length scales. Detecting these in tunneling\nspectroscopy would require atomic-resolution STM measurements, which are\ninaccessible because the sample is enclosed on both sides with hBN and graphite\ngates.\nIt may be possible to detect signatures of the finite momentum pairing in Josephson\ncoupling experiments such as proposed in [Phys. Rev. Lett. 84, 4970], or in\nside-contacted tunneling conductance measurements [Sci. Adv.10, eadr4817 (2024)].\nWe leave working out the precise fingerprints in such experiments for future work.\nList of changes :\nAll changes are highlighted in blue in the attached\n“manuscript_differences.pdf”.\n- Weakened statement: “Based on our analysis we further propose that a Wigner\ncrystalline insulating state occurs…” on page 2.\n- Added results showing symmetry breaking between chiral superconducting state\nwith angular momentum l = +- 1 in the topological superconducting region SC1 in\nApp. E.\n- Added clarifying sentence in conclusion:\n“Ref. [23] also demonstrates that the isospin polarization emerges in a Stoner\nmodel due to large density of states and strong interaction.”\nResponse to Referee reports - Final resubmission\nNCOMMS-24-62309C\nREVIEWERS' COMMENTS\nReviewer #2 (Remarks to the Author):\nI have reviewed the response and the revised manuscript. The authors have addressed\nall my comments. The revised manuscript is suitable for publication in Nature\nCommunications.\nReviewer #3 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed\nreports. This is part of the Nature Communications initiative to facilitate training in peer\nreview and to provide appropriate recognition for Early Career Researchers who co-\nreview manuscripts.\nReviewer #4 (Remarks to the Author):\nThe authors have addressed my follow up questions, and I have no further questions. I\ncan recommend the manuscript for publication.\nOur response:\nWe thank all reviewers for their recommendation. We would like to express our\ngratitude for their thoughtful comments and questions during the process, these have\nimproved our article."}, {"round": 2, "reviewer_comments": {"reviewer_2": "I have reviewed the responses (from Reviewers #1 and #2) and the revised manuscript. The authors have resolved most of the issues and internal consistency. I am inclined to recommend acceptance, provided that the authors carefully address the following comments.\n(1) The circularly symmetric model in this manuscript indeed predicts the degenerate px and py pairing channels. However, such degeneracy is unclear for a realistic dispersion that admits only C3 rotation. Can the authors comment on this point?\nA related comment: In Appendix E, the authors explore how Berry curvature affects the angular momentum of the superconducting phase, focusing on D = 60 meV, which corresponds to a trivial SC with an annular Fermi surface. Are there results for the case in the 1P (topologically nontrivial) phase?\n(2) In the cover letter, ``In the resistive phase separating the superconducting domes SC1 and SC2 the resistance increases with temperature, indicating an insulating state consistent with our predicted Wigner crystal (see Extended Data Figure 2(d, e) in the revised manuscript of the experiment arXiv:2408.15233).'' I suspect that the authors meant ``the resistance increases as the temperature decreases.'' I am unable to find the figures for the temperature dependence of the insulating state in Extended Data Figure 2(d, e) of the experimental paper. There might be some misunderstanding here.\n(3) In the revised manuscript and the cover letter for resubmission, the authors claimed that their predicted Wigner crystal can explain the experiment. I have strong reservations about this claim. First, the original experiment (posted before the original submission of this work) has reported an insulating state separating SC1 and SC2. Second, the authors did not do any concrete calculations related to the Wigner crystal. Wigner crystal is a nice suggestion/speculation but not a concrete prediction by the authors. The authors should modify the discussions/claims related to the Wigner crystal.\n(4) Regarding Reviewer #1's point 1: The authors' response is based on the density of states, which affects the Thomas-Fermi screening length. Indeed, the existing Bernal and ABC graphene experiments do not support a sufficiently large density of states within the quarter metal phase. However, the isospin transitions are also essential and depend on the Coulomb interaction as well as the band structure. One has to compare the energies of the quarter metal state and other isospin states to understand the complete isospin phase diagram. Thus, the existing argument based on the density of states near the bottom of the band is not rigorous. The authors need to clarify this issue.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "The authors have answered my questions and concerns in details with sufficient clarity, which I do appreciate. I additionally note that (as requested by the editor) the questions raised by Reviewer 1 roughly coincide with part of my questions, which are reasonably well addressed by the authors. Generically, I must say I am still not sure how quantitatively well the methods and the calculations are valid, including the single-particle model, and perturbative RPA calculations, etc. However, the theory the authors propose is indeed a plausible mechanism at this stage, which I think is a timely contribution to the field. Therefore, given that my questions are mostly well addressed, I can recommend the paper for publication. I do have two minor follow up questions:\n1. The different energy scales in isospin polarization vs. superconductivity is still not fully clear to me. Is there a simple argument for using the bare Coulomb interaction in the Stoner model for polarization, and using screened interaction in the superconductivity calculation? Is it because their physics happen at different energy scales or length scales?\n2. The authors predict the chiral superconductivity is valley polarized, thus the pairing would carry a nonzero momentum. Are there ways to detect such nonzero pairing momentum that the authors can comment?"}, "author_response": "Response to Referee reports - Final resubmission\nNCOMMS-24-62309C\nREVIEWERS' COMMENTS\nReviewer #2 (Remarks to the Author):\nI have reviewed the response and the revised manuscript. The authors have addressed\nall my comments. The revised manuscript is suitable for publication in Nature\nCommunications.\nReviewer #3 (Remarks to the Author):\nI co-reviewed this manuscript with one of the reviewers who provided the listed\nreports. This is part of the Nature Communications initiative to facilitate training in peer\nreview and to provide appropriate recognition for Early Career Researchers who co-\nreview manuscripts.\nReviewer #4 (Remarks to the Author):\nThe authors have addressed my follow up questions, and I have no further questions. I\ncan recommend the manuscript for publication.\nOur response:\nWe thank all reviewers for their recommendation. We would like to express our\ngratitude for their thoughtful comments and questions during the process, these have\nimproved our article."}, {"round": 3, "reviewer_comments": {"reviewer_2": "I have reviewed the response and the revised manuscript. The authors have addressed all my comments. The revised manuscript is suitable for publication in Nature Communications.", "reviewer_3": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_4": "The authors have addressed my follow up questions, and I have no further questions. I can recommend the manuscript for publication."}, "author_response": ""}]

Decision letter