Exact models of chiral flat-band superconductors

TL;DR

This paper introduces exact models for chiral flat-band superconductors, revealing how local attractive interactions lead to topological superconductivity with tunable pairing modes and superfluid stiffness.

Contribution

The work presents a class of models with exact superconducting ground states in flat-band systems, highlighting the role of quantum geometry and interaction in pairing topology and degeneracy.

Findings

Exact superconducting ground states in flat-band models.

Superfluid stiffness proportional to attractive interaction scale.

Presence of two nearly degenerate, tunable pairing modes.

Abstract

Recent experiments have reported the surprising observation of superconductivity in flavor polarized, nearly flat bands (FBs) of rhombohedral graphene. Motivated by these findings, we introduce a class of models for single-flavor FBs with inversion symmetry, where we show a local attractive interaction between orbitals with opposite parities leads to an exact superconducting ground state. We argue that this model can be relevant to realistic multi-flavor systems including short-range repulsion, since the main effect of such repulsion is to induce flavor polarization leaving possibly attractive residual interaction between different flavorless orbitals. The nature of the pairing is determined by the interplay between the FB quantum geometry and the interaction, and is often topological when the parent FB is so. Interestingly, each such model has two nearly degenerate pairing modes, whose energetic competition can be tuned by a change in the charge transfer gap between the two orbitals or electron density. These modes have the same angular momentum but different pairing amplitude structure and possibly different topology. We show that the superfluid stiffness is proportional to the attractive interaction scale using a combination of analytical variational upper bounds and numerical bootstrap lower bounds. We find empirically that the maximum superfluid stiffness is achieved when the hot spots of quantum geometry in the Brillouin zone are marginally filled.

Equations

$$p_{\alpha }{u}_{\alpha }(\mathbf{k})=u_{\alpha }(-\mathbf{k}){\mathrm{e}}^{\mathrm{i}\xi (\mathbf{k})}$$

$${\hat{H}}_{AB}=$$

$${\hat{H}}_{BA}-{\hat{H}}_{AB}=V\sum _{\mathbf{R}}\left[{\rho }_B^{\text{FB}}{\hat{n}}_{\mathbf{R}A}-{\rho }_A^{\text{FB}}{\hat{n}}_{\mathbf{R}B}\right].$$

$$\mathcal{T}\mathcal{C}\hat{P}{\hat{H}}_{\text{AB}}\hat{P}(\mathcal{T}\mathcal{C}{)}^{-1}=\hat{P}{\hat{H}}_{\text{BA}}\hat{P}$$

$${\hat{\eta }}^{†}\equiv \int \frac{\mathrm{d}\mathbf{k}}{{\Omega }_{\text{BZ}}}\frac{u_A^{\ast }(\mathbf{k})}{u_B(-\mathbf{k})}{\hat{\gamma }}_{\mathbf{k}}^{†}{\hat{\gamma }}_{-\mathbf{k}}^{†}.$$

$$|z⟩\equiv$$

$$C^{\text{maj}}=C^{\text{BdG}}+C^{\text{FB}}+2C^{<}$$

$$\nu (\mathbf{k};\Re z)\equiv \frac{⟨z|{\hat{\gamma }}_{\mathbf{k}}^{†}{\hat{\gamma }}_{\mathbf{k}}|z⟩}{⟨z|z⟩}=\frac{e^{-2\Re z}|u_A(\mathbf{k}){|}^2}{|u_B(\mathbf{k}){|}^2+e^{-2\Re z}|u_A(\mathbf{k}){|}^2}$$

$$⟨O{⟩}_z\equiv \int \frac{\mathrm{d}\mathbf{k}}{{\Omega }_{\text{BZ}}}O(\mathbf{k})\nu (\mathbf{k};\Re z).$$

$${ϵ}_{1,\text{var}}(\mathbf{k};\Re z)=$$

$$D_{\text{s,var}}^{ij}(\Re z)=$$

$$\displaystyle\ \ \ +e^{2\Re z}\left\langle\frac{\partial_{i}(u^{*}_{A}u_{B})^{*}\partial_{j}(u^{*}_{A}u_{B})}{|u_{A}|^{2}}\right\rangle_{z}\langle|u_{B}|^{2}\rangle_{z}\Bigg{]}.$$

$$t(\mathbf{k})$$

$$k\to [k_0\hat{\zeta }(\mathbf{k}){]}^{-1}$$

Full text

Recent experiments have reported evidence of chiral, time-reversal-breaking superconductivity (SC) in rhombohedral $L$-layer graphene (R$L$G) with $L=4,5$ Han et al. (2025), $6$ Nguyen et al. (2025) and twisted MoTe2 (tMoTe2) Xu et al. (2025). While SC is, by now, rather common in moiré and more generally flat band (FB) systems, a remarkable feature of these superconductors is that their parent band is likely spin and valley polarized. This poses a major challenge to standard theoretical approaches to superconductivity, which typically rely on the weak-coupling instability of Fermi surfaces of time-reversal related flavors Lian et al. (2019); You and Vishwanath (2019); Yu et al. (2023); Christos et al. (2023); Putzer and Scheurer (2025); Chen et al. (2025a); Schrade and Fu (2024); Jiang et al. (2025a); Chau et al. (2024). Together with evidence of other strongly correlated phenomena in these materials, this suggests that a strong coupling perspective, where the dispersion is considered small compared to interaction, is a better starting point. In such a FB limit, the physics is mostly determined by the interplay between the interaction and the FB wavefunctions, known as the quantum geometry (QG) 111Here, we use QG to refer to the full gauge-invariant information encoded in wavefunction overlaps between arbitrary momenta rather than the long wavelength quantities such as the quantum metric.. As a result, QG is expected to control many properties of the SC phase including the pairing symmetry, single-particle gap, and phase stiffness; the latter two together limit the critical temperature of the phase. Several recent works have aimed to study the effects of QG on pairing in single-flavor bands Geier et al. (2024); Guerci et al. (2025); Wang et al. (2024a); Christos et al. (2025); Patri and Franz (2025); Shavit and Alicea (2025); Yang and Zhang (2025); May-Mann et al. (2025); Dong and Lee (2025); Crépel and Fu (2021); Guerci and Fu (2025); Chen et al. (2025b); Hu et al. (2025) using various approaches. However, an analytically controlled theory beyond the weak coupling limit remains lacking.

On the other hand, the development of exactly solvable models, while rare in realistic setups, can play a pivotal role in understanding complex phases of matter in FB systems. A classic example is the Haldane-Trugman-Kivelson pseudopotential model Haldane (1983); Trugman and Kivelson (1985) for the fractional quantum Hall effect in the lowest Landau level, which energetically justifies the Laughlin wavefunction Laughlin (1983) in the presence of short-ranged repulsions. This has recently been extended to a wider class of FBs, called ideal or vortexable Tarnopolsky et al. (2019); Ledwith et al. (2020); Ozawa and Mera (2021); Mera and Ozawa (2021a, b); Mera et al. ; Wang et al. (2021); Ledwith et al. (2022, 2023), allowing the development of criteria for realizing fractional Chern insulators Regnault and Bernevig (2011); Neupert et al. (2011); Sheng et al. (2011); Qi (2011); Parameswaran et al. (2012); Wu et al. (2013); Liu and Bergholtz (2022); Parameswaran et al. (2013); Bergholtz and Liu (2013) in these systems. Other examples include attractive Hubbard models with spinful time-reversal symmetry Tovmasyan et al. (2016); Huhtinen et al. (2022); Herzog-Arbeitman et al. (2022a) for FB superconductivity Peotta and Törmä (2015); Julku et al. (2016); Törmä et al. (2022); Peotta et al. (2023); Jiang et al. (2025b); Jiang and Barlas (2024); Chen and Law (2024), where an exact $s$-wave pairing ground state is stabilized by the local attraction, as well as models for FB ferromagnetism with repulsive interactions Mielke and Tasaki (1993); Kang and Vafek (2019); Seo et al. (2019); Repellin et al. ; Bultinck et al. ; Lian et al. ; Bernevig et al. (2021); Ledwith et al. (2021); Kang et al. (2024). Recently, two of us have introduced a set of compatibility conditions of the FB wavefunctions, called ‘quantum geometric nesting’ (QGN) Han et al. (2024a) conditions (see also Zhang et al. (2025); Sun et al. (2024)), which enable systematic constructions of infinitely many solvable locally interacting models for certain symmetry broken states that unify the previous examples. One major drawback of the QGN models, however, is that the constructed order parameters are uniform in momentum space. This rules out their applicability to SC phases in a flavor-polarized FB, where the order parameter must vanish at certain points in the Brillouin zone (BZ) by the Pauli exclusion principle.

In this work, we introduce a new class of exact models for SC in a single FB which is assumed to be isolated from other bands in the system. We find that an inversion symmetry suffices to guarantee a perfect compatibility between the wavefunctions at $\pm\bm{k}$ through a generalized nesting condition, so that any local attraction between a pair of orbitals with opposite parities gives rise to exact SC ground states. The primary effects of Coulomb repulsion are presumed to induce the flavor polarization, such that an effective attraction resulting from the coupling to collective bosons (e.g. phonons Han et al. (2024b)) can overcome the residual repulsion between distinct orbitals 222We also note that in the FB limit the retardation effects of the bosons are not significant, so the interactions can be regarded instantaneous.. In fact, a spinful FB model with a strong on-site Hubbard repulsion plus a subleading inter-orbital density attraction of our sort would still be exactly solvable by combining the previous results on flavor ferromagnetism Mielke and Tasaki (1993); Kang and Vafek (2019); Seo et al. (2019); Repellin et al. ; Bultinck et al. ; Lian et al. ; Bernevig et al. (2021); Ledwith et al. (2021); Kang et al. (2024) and ours. This justifies the relevance of our models to realistic systems. Our construction reveals a very simple condition for a single flavor FB to host SC in the presence of short-ranged attraction. Furthermore, it goes beyond previous works by allowing arbitrarily complex order parameters including the topological ones supporting Majorana edge modes Read and Green (2000); Volovik (2003). Our construction enables us to obtain various properties of the SC states including analytical variational bounds on the quasi-particle gap and the superfluid stiffness based on QG. Furthermore, we numerically establish a non-zero lower bound of the superfluid stiffness, using quantum many-body bootstrap lower bound Gao et al. (2025a), which is proportional to the interaction strength. One of the most surprising implications of our construction is the existence of two almost degenerate, particle-hole conjugate pairing modes which may have different Majorana Chern numbers in two dimensions. This suggests that the existence of two distinct SC domes switchable by tuning electron density and/or displacement field is possibly a general feature of QG-stabilized single-flavor FB SC and may explain the appearance of two distinct SC domes in R$L$G with $L=4,5,6$ Han et al. (2025); Nguyen et al. (2025). We illustrate our results with a minimal two-band model that reproduces many features of the FBs of R$L$G with $L=5$, suggesting our formalism may be relevant to the observed chiral SC in this system.

**The model. ** We begin by discussing the general construction before presenting a concrete model and discussing its possible relevance to physical systems. We consider a single isolated FB with inversion symmetry. The interaction scale is assumed to be much smaller than the band gap to the remote bands, allowing us to project the interaction onto the FB subspace. We express the electronic structure with an inversion-symmetric set of local orbitals $\{\phi_{\bm{R}\alpha}(\bm{r})\}$ in unit cell $\bm{R}$, which are eigenstates of the inversion operation $\mathcal{I}$ in the sense that $\mathcal{I}\phi_{\bm{R}\alpha}(\bm{r})=p_{\alpha}\phi_{-\bm{R}\alpha}(\bm{r})$ where $p_{\alpha}=\pm 1$ is the orbital parity. We emphasize that these are not Wannier orbitals describing the FB, which could be non-local due to the topology of the FB. Instead, they are the microscopic, local basis states on which the FB wavefunctions have support. The corresponding electron annihilation operator on orbital $\alpha$ and momentum $\bm{k}$ satisfies $\mathcal{I}\hat{c}_{\bm{k}\alpha}\mathcal{I}=p_{\alpha}\hat{c}_{-\bm{k}\alpha}$ 333Here we implicitly embed all the orbital positions to the inversion center.. The FB electron operators $\hat{\gamma}_{\bm{k}}\equiv\sum_{\alpha}u^{*}_{\alpha}(\bm{k})\hat{c}_{\bm{k}\alpha}$ are defined by the Bloch wavefunction of the FB, $\vec{u}(\bm{k})$. Inversion assures they obey a ‘nesting’ relation

[TABLE]

where $\xi(\bm{k})$ is the gauge-dependent phase (sewing matrix Alexandradinata et al. (2014); Fang et al. (2012)) that is odd in $\bm{k}$ (SM Sec. III A). This is a generalized QGN condition differing from the original one Han et al. (2024a). A more general discussion for other FB systems can be found in SM Sec. II.

The central result of this work is the following: for any pair of orbitals $A,B$ with opposite parities $p_{A}p_{B}=-1$, an attractive interaction between them is exactly solvable (after projecting onto the FB subspace):

[TABLE]

where $V>0$, $\hat{n}_{\bm{R}\alpha}\equiv\hat{c}^{\dagger}_{\bm{R}\alpha}\hat{c}_{\bm{R}\alpha}$, and $\hat{H}_{AB,2}=V\sum_{\bm{R}}\left[\rho^{\text{R}}_{B}(\hat{n}_{\bm{R}A}+\hat{n}_{\bm{R}B})+\rho^{\text{FB}}_{A}\hat{n}_{\bm{R}B}\right]$ is a single-particle counter term ensuring that the projected Hamiltonian has no bare dispersion, i.e. it only contains fermion quartic terms $\sim\hat{\gamma}^{\dagger}\hat{\gamma}\hat{\gamma}^{\dagger}\hat{\gamma}$. $\rho^{\text{R}}_{\alpha}$ and $\rho_{\alpha}^{\text{FB}}$ are, respectively, the average electron density on $\alpha$ orbital in the occupied (lower) remote bands and the FB, e.g. $\rho_{\alpha}^{\text{FB}}\equiv\int\frac{\mathrm{d}\bm{k}}{\Omega_{\text{BZ}}}|u_{\alpha}(\bm{k})|^{2}$ where the integration is over the first BZ with volume $\Omega_{\text{BZ}}$. This counter term is a simple modification to the charge transfer gaps of the single-particle Hamiltonian and admits an apparent simplification when the FB is the lowest or the highest in the spectrum.

**A particle-hole duality. ** The counter term in the construction is asymmetric for orbitals $A,B$ although the interaction is symmetric. Thus, exchanging $A\leftrightarrow B$ will lead to a different exactly solvable Hamiltonian, $\hat{H}_{BA}$, that has the same interaction but differs in values of the relative potentials (charge transfer gaps):

[TABLE]

The ground states of $\hat{H}_{BA}$ and $\hat{H}_{AB}$ also have different SC orders with possibly different topology as we will discuss below. The presence of two competing SC phases is thus a fundamental feature of this problem.

In the FB subspace, the two Hamiltonians are related by the combination of a FB particle-hole transformation $\mathcal{C}:\hat{\gamma}\rightarrow\hat{\gamma}^{\dagger}$ (implying $\nu\rightarrow 1-\nu$) and a time-reversal transformation $\mathcal{T}:\bm{k}\rightarrow-\bm{k},\mathrm{i}\rightarrow-\mathrm{i}$ (SM Sec. III C):

[TABLE]

and thus all the properties of the two models, including the excitation spectra and the stiffness, are related accordingly; see Fig. 1 for an illustration. In the below discussions we will focus on $\hat{H}_{AB}$ without loss of generality.

**Exact solvability. ** The pairing operator that generate the exact many-body ground states (GS) of the FB-subspace projected (implemented by $\hat{P}$) Hamiltonian, $\hat{P}\hat{H}_{\text{AB}}\hat{P}$, is:

[TABLE]

Following Eq. 1 it is apparent that the form factor $f(\bm{k})=u_{A}^{*}(\bm{k})/u_{B}(-\bm{k})$ is antisymmetric in $\bm{k}$. Note that $f(\bm{k})$ may have singularities in the BZ where $u_{B}(-\bm{k})=0$, but we will show that this does not lead to any divergence in any physical observables. Exchanging $A\leftrightarrow B$ leads to the particle-hole conjugate form factor, which has the same phase winding about the origin (and thus the same angular momentum) but inverse amplitude. The GS in the $2N$-particle sector is $(\hat{\eta}^{\dagger})^{N}|\text{vac}\rangle$ 444The vacuum state here is defined as the lower remote bands being occupied but the FB and the higher remote bands being empty. which has zero energy. A detailed proof of this claim is presented in SM Sec. II. We sketch the strategy here: upon projection onto the FB subspace, the interactions will become (up to unimportant constant and chemical potential shifts) $\hat{P}\hat{H}_{AB}\hat{P}=V\sum_{\bm{R}}\hat{P}\hat{c}^{\dagger}_{\bm{R}B}\hat{c}_{\bm{R}A}\hat{P}\hat{c}^{\dagger}_{\bm{R}A}\hat{c}_{\bm{R}B}\hat{P}$ which is positive semi-definite (PSD). Then one can check that $[\hat{P}\hat{c}^{\dagger}_{\bm{R}A}\hat{c}_{\bm{R}B}\hat{P},\hat{\eta}^{\dagger}]=0$ therefore $\hat{P}\hat{H}_{AB}\hat{P}(\hat{\eta}^{\dagger})^{N}|\text{vac}\rangle=0$ since the combination $\hat{P}\hat{c}^{\dagger}_{\bm{R}A}\hat{c}_{\bm{R}B}\hat{P}$ can move to the right of $(\hat{\eta}^{\dagger})^{N}$ and annihilate the vacuum. Using the PSD property of the projected Hamiltonian, we conclude that these states are GSs.

Several technical remarks are in order: (1) the terms at different $\bm{R}$s generally do not commute or form a closed algebra, therefore this model is not a commuting projector Hamiltonian or a ‘generalized mean-field Hamiltonian Somma et al. (2006); Nussinov and Ortiz (2009)’ (which consists of terms that form a polynomially small algebra) but only belong to the frustration-free (FF) family; (2) Different from the QGN models Han et al. (2024a), here $\hat{\eta}^{\dagger}$ does not necessarily commute with the projected Hamiltonian, so that more degrees of freedom for the possible GSs are liberated at the expense of no pseudospin ${\rm SU}(2)$ symmetry being exact in this model; (3) Crucially, the (unprojected) interactions here are strictly local in real space, distinct from those with ‘local’ structures only in momentum space Phillips et al. (2020); Ibañez et al. (2009); Lerma H. et al. (2011); Dukelsky et al. (2001, 2004); Ortiz et al. (2005).

**The ground states. ** Since all the GSs in different number sectors are degenerate, any linear combination of them is still a GS. In particular it is convenient to define a coherent state with an arbitrary complex number $z$:

[TABLE]

which can be verified to be precisely a BCS wavefunction, given by an occupied Bogoliubov-de-Gennes (BdG) band with wavefunction : $\vec{w}^{T}(\bm{k})=\left(e^{-z}u^{*}_{A}(\bm{k})\vec{u}^{T}(\bm{k})\ ,\ u_{B}(\bm{k})\vec{u}^{\dagger}(-\bm{k})\right)^{T}$ (SM Sec. III B). This simple result suggests that certain BCS MF theories of the current model could yield exact result Wang et al. (2017) despite the strong coupling nature of the problem.

With this mapping, in two dimensional cases, one can also define the majorana Chern number of the SC GS:

[TABLE]

where $C^{\text{FB}}$ and $C^{<}$ are the Chern numbers of the FB and the occupied remote bands, respectively, and $C^{\text{BdG}}$ is the Chern number of the wavefunction $\vec{w}(\bm{k})$, which is independent of $z$. In two-orbital systems, we prove in SM Sec. III D that $C^{\text{BdG}}$ is always [math]. If the FB is the lower band, $C^{<}=0$, implying that $C^{\text{maj}}=C^{\text{FB}}$ is directly determined by the topology of the parent FB ($C^{\text{maj}}=-C^{\text{FB}}$ for the upper-band case). Exchanging $A\leftrightarrow B$, the conjugate SC order may have a different $C^{\text{maj}}$, as we show with an example in App. LABEL:app:_L_band.

These states have simple structures that allow for analytical computations of various important properties. For later convenience, we define the occupation function

[TABLE]

where $\Re$ means the real part. In terms of this function we define the FB electron-averaged expectation for any function $O(\bm{k})$:

[TABLE]

For example, $\nu(\Re z)\equiv\langle 1\rangle_{z}$ is the FB filling fraction of the state $|z\rangle$, which is monotonically tuned by $\Re z$. The off-diagonal-long-range-order $\langle z|\hat{\eta}^{\dagger}|z\rangle=\nu(\Re z)\mathrm{e}^{z}$ is nonzero and has an arbitrary phase, confirming that this state breaks the charge $U(1)$ symmetry in the ground states.

**Analytical upper bounds on excitation gap and superfluid stiffness. ** With the exact GSs one can infer several important properties. First, we use the single mode approximation to upper bound and estimate the excitation spectrum in the charge-$1$ sector. The strategy is to construct simple trial state $\hat{\gamma}^{\dagger}_{\bm{k}}|z\rangle$ (or equivalently $\hat{\gamma}_{-\bm{k}}|z\rangle$) and evaluate its energy (SM Sec. V):

[TABLE]

We note that taking $\Re z\rightarrow\pm\infty$ produces the exact result for the spectrum in the single-paritcle/hole sector of the problem: $\epsilon_{1\text{p}}(\bm{k})=V\rho^{\text{FB}}_{A}|u_{B}(\bm{k})|^{2}$ and $\epsilon_{1\text{h}}(\bm{k})=V\rho^{\text{FB}}_{B}|u_{A}(\bm{k})|^{2}$. This bound can serve as an estimate of the pairing gap. It can be systematically improved with more complicated trial states (e.g. trions) or through numerical calculations. It should be noted that even if $\epsilon_{1,\text{var}}$ has a gapped spectrum, this does not necessarily mean the real charge-$1$ spectrum is gapped; however, there can at most be coincidental gap closing points, since the BdG wavefunction $\vec{w}(\bm{k})$ is patch-wise continuously well-defined so that no nodes necessarily arise.

Next we turn to upper bounding the superfluid stiffness. The approach is to consider a flat gauge connection $\bm{A}$ and then optimize the energy response over all possible BCS states. Surprisingly, we prove in SM Sec. IV that the GS state at $\bm{A}=0$ is already the lowest-energy state within this ansatz space for an infinitesimal $\bm{A}$ due to the inversion symmetry, yielding a simple, gauge-invariant diamagnetic bound Scalapino et al. (1993); Mao and Chowdhury (2023) on the stiffness (in the sense that $D_{\text{s,var}}-D_{\text{s}}\succeq 0$ is PSD):

[TABLE]

where $\Omega_{\text{UC}}$ is the unit cell volume. This remarkable tightness of the diamagnetic bound within this broad variational space should be contrasted with the case of QGN models where the form factor must be varied with $\bm{A}$ in order to produce the tightest bound (see SM Sec. IV F and Refs. Gao et al. (2025a); Huhtinen et al. (2022)). This expression contains a filling-weighted average of the orbital-selective quantum metric Yu et al. (2024a) and can be further bounded by conventional geometric quantities (App. A). Intuitively, it measures on average how much mismatch between the wavefunctions at opposite occupied momenta is introduced by $\bm{A}$. We note that in the $\Re z\rightarrow\pm\infty$ limit this result relates to the bound-state mass in the two-particle/hole sector, $m_{2\text{p}/2\text{h}}$, e.g. $D^{ij}_{\text{s,var}}(\Re z\rightarrow+\infty)=\frac{\nu\rho^{\text{FB}}_{A}}{2\Omega_{\text{UC}}}[m_{2\text{p}}^{-1}]_{ij}$. See App. LABEL:app:_bound_state for discussions on the exact spectrum of the bound state.

**Examples and numerical investigations. ** Here we demonstrate our construction with simple FBs in two dimensions. We start from an inversion symmetric continuum two-band model with Hamiltonian

[TABLE]

where $M$ is a positive integer, and $k,\bar{k}\equiv k_{x}\pm\mathrm{i}k_{y}$. When $\delta=1$, this model has an isolated FB at zero energy separated from a higher band with gap $\Delta$, which has wavefunction $\vec{u}^{T}(\bm{k})=(1,-k^{M})/\sqrt{1+|\bm{k}|^{2M}}$ and $C^{\text{FB}}=-M$. When $M$ is odd, we can apply the general construction to construct exact models for chiral SC that contain simply the local attraction between the two orbitals. Taking $A=1,B=2$, we find that the exact pairing order parameter is $\hat{\eta}^{\dagger}=\int_{\bm{k}}\frac{1}{k^{M}}\hat{\gamma}_{\bm{k}}^{\dagger}\hat{\gamma}_{-\bm{k}}^{\dagger}$ (which gives rise to the Moore-Read $p-\mathrm{i}p$ ‘Pfaffian’ state when $M=1$ Wang et al. (2017); Ibañez et al. (2009); Moore and Read (1991)); whereas $A=2,B=1$ amounts to the particle-hole dual $\hat{\eta}^{\dagger}=\int_{\bm{k}}\bar{k}^{M}\hat{\gamma}_{\bm{k}}^{\dagger}\hat{\gamma}_{-\bm{k}}^{\dagger}$. In both cases $C^{\text{maj}}=C^{\text{FB}}=-M$ is odd and thus the SC is topological and supports majorana edge mode.

The above model is defined in open momentum space with an arbitrary length unit. To regularize its QG, we compactify the model onto a hexagonal BZ by simply replacing Ledwith et al. (2025)

[TABLE]

in Eq. 12 and the wavefunction $\vec{u}(\bm{k})$, where $\hat{\zeta}(\bm{k})\equiv\zeta(k)-\pi\bar{k}/\Omega_{\text{BZ}}$ is a non-holomorphic, periodic modification of the Weierstrass $\zeta$ function Haldane (2018), whose period is fixed to $2\pi(1\pm\mathrm{i}/\sqrt{3})$ in the numerical study of this work (i.e. the real-space primitive vectors are taken to be $\bm{a}_{1,2}=\frac{1}{2}(1,\pm\sqrt{3})$). $k_{0}$ is a tunable momentum scale within which the wavefunction is approximately holomorphic. The QG of the FB is mainly concentrated at $|\bm{k}|\lesssim k_{0}$, whose total strength positively depends on $k_{0}$. The corresponding hopping amplitudes exponentially decays with a rate $\sim k_{0}$ in real space (we note that a strictly local lattice realization of Chern band is impossible Chen et al. (2014)).

The Hamiltonian $t(\bm{k})$ is a flattened version of the celebrated BHZ model Bernevig et al. (2006) that is broadly relevant to topological phenomena in two dimensions (see e.g. Tan et al. (2024); Soejima et al. (2025)). One perspective is that this wavefunction $\vec{u}(\bm{k})$ can be regarded as that of a topological band built from $s$ and $p$ orbitals in the $M=1$ case with concentrated charge and Berry curvature distribution Ledwith et al. (2025); Song and Bernevig (2022); Herzog-Arbeitman et al. (2024a), which describes the QG of twisted bilayer graphene. On the other hand, the $M=5$ model is related to a widely used effective two-band model of the flat band bottom of R$5$G Herzog-Arbeitman et al. (2024b); Dong et al. (2024); Tan and Devakul (2024); Patri and Franz (2025); Soejima et al. (2024); Bernevig and Kwan (2025), capturing the essential feature of its QG, such as concentrated Berry curvature in a ring-shaped region. Interpreting the two orbitals as located on the bottom and top layers, their charge transfer gap can be directly tuned by a perpendicular displacement field. We mention that there exist a simple $L$-orbital generalization of the two-orbital $M=1$ model in Eq. 12, which may more faithfully describe the physics of R$L$G for arbitrary $L$ (App. LABEL:app:_L_band). In this model, choosing $A,B=1,2$ or $2,1$ will yield different $C^{\text{maj}}=1$ or $2L-3$.

Within the well-defined lattice model, we plot $D_{\text{s,var}}$ for various choices of $k_{0}$ for the cases of $M=1$ and $M=5$ in Fig. 2. We find that, interestingly, the maximum stiffness is approximately achieved when the hot spot of QG is marginally filled. This reveals a more interesting relation between QG and the stiffness than the cases of QGN models. We note that for all cases studied, the single-particle spectrum $\epsilon_{1,\text{var}}$ always has a much larger gap than the superfluid stiffness, suggesting they are strongly coupled SC.

Although the variational upper bound demonstrates a connection to QG akin to the case of $s$-wave FB superconductivity Peotta and Törmä (2015); Huhtinen et al. (2022); Herzog-Arbeitman et al. (2022a); Mao and Chowdhury (2024); Hazra et al. (2018), it cannot prove the finiteness of the stiffness. We thus performed quantum many-body bootstrap Barthel and Hübener (2012); Baumgratz and Plenio (2012); Lin and Lindsey (2019); Mazziotti (2020); Haim et al. (2020, 2020); Lin and Lindsey (2019); Mazziotti (2023); Wang et al. (2024b); Gao et al. (2025b); Scheer (2024); Khoo and Lindsey (2024) calculations to rigorously lower bound this quantity with results shown in Fig. 3 using the method introduced in Ref. Gao et al. (2025a). We find that the lower bound extrapolates to a finite value in the thermodynamic limit and exhibits qualitatively similar behavior as the upper bounds.

**Discussions. ** Our work establishes that a single polarized FB with attractive short-range interactions can host SC order with properties crucially determined by its QG. In particular, the resulting SC can be topological and its stiffness is set by the interaction scale, leading to a parametrically large transition temperature. We stress that only an effective inversion symmetry structure in the FB QG, i.e. the property Eq. 1, is needed in our construction. This enables the usage of our approach in systems with emergent or approximate inversion symmetry, in e.g. charge concentrated bands, tMoTe2, or the $\bm{k}\cdot\bm{p}$ low-energy model of R$L$G which has an approximate $SO(2)$ rotation symmetry (the symmetry breaking terms in the realistic Hamiltonian primarily affects the dispersion Herzog-Arbeitman et al. (2024b), which is small in the FB region, but not the QG). While in this work we have mainly focused on presenting the general approach, we find it quite remarkable that the general property and the simple two-band model we study already mimic certain observations in R$L$G Han et al. (2025); Nguyen et al. (2025), such as the existence of two different SC domes tunable by displacement field (charge transfer gap) and electron density (Fig. 1). We will defer a more realistic modeling of the R$L$G systems using the solvable limit revealed in this work to future works.

**Acknowledgement. ** We thank Patrick Ledwith, Ashvin Vishwanath, Qingchen Li, Tixuan Tan, Tonghang Han, Maine Christos, Zhihuan Dong and Steven A. Kivelson for the helpful discussions. We thank Felix Desrochers for pointing out a typo in the first version. Z. H. is supported by a Simons Investigator award, the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (Ashvin Vishwanath, 651440). J.H.-A. is supported by a Hertz Fellowship. E. K. is supported by NSF MRSEC DMR-2308817 through the Center for Dynamics and Control of Materials. The authors thank the Harvard FAS Reaserch Computing (FASRC) for computational support.

Appendix A A bound on $D_{\text{s,var}}$

We can obtain an elementary upper bound for the $D_{\text{s,var}}$ as follows. For convenience we define the FB projection matrix $P_{\alpha\beta}(\bm{k})\equiv u_{\alpha}u^{*}_{\beta}(\bm{k})$ and $x\equiv\mathrm{e}^{-2\Re z}$. We also define the BZ average $\langle O\rangle=\int\frac{\mathrm{d}\bm{k}}{\Omega_{\text{BZ}}}O(\bm{k})$.

First we use Cauchy-Schwartz inequality to write

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