Quantum-Geometry-Induced Superconductivity near a Fractional Chern Insulator

Introduction.— Partially filled Chern bands provide a setting where topology, interaction, and band geometry are inseparable. Repulsive interactions in such bands can stabilize fractional Chern insulators (FCIs) Regnault and Bernevig (2011); Sheng et al. (2011); Parameswaran et al. (2013); Tang et al. (2011); Sun et al. (2011); Neupert et al. (2011), which are lattice analogs of fractional quantum Hall states in the lowest Landau level (LLL). However, a Chern band can develop quantum geometry that deviates from the ideal LLL limit Roy (2014); Parameswaran et al. (2012); Jackson et al. (2015); Claassen et al. (2015); Wang et al. (2021). This geometric structure can strongly affect correlation effects and distinguishes Chern-band systems from the LLL. Experimentally, fractional Chern insulators have been observed in moiré and graphene-based materials  Cai et al. (2023); Zeng et al. (2023); Park et al. (2023); Xu et al. (2023); Ji et al. (2024); Redekop et al. (2024); Park et al. (2025); Lu et al. (2024); Kang et al. (2024); Wang et al. (2025); Waters et al. (2025). In related material platforms, experimental evidence has also revealed other correlated phases, including superconductivity (SC) and charge-density-wave (CDW) order Han et al. (2025); Xu et al. (2026); Lu et al. (2025); Waters et al. (2025); Sun et al. (2026, 2026); Aronson et al. (2025). Extensive efforts have also been made to understand the correlated phases in these materials Bernevig et al. (2025); Crépel and Fu (2023); Reddy et al. (2023); Shavit and Oreg (2024); Jia et al. (2024); Herzog-Arbeitman et al. (2024); Kwan et al. (2025); Yu et al. (2025a, 2024); Guo et al. (2024); Dong et al. (2024a, b); Soejima et al. (2024); Zhou et al. (2024); Huang et al. (2024); Tan and Devakul (2024); Sheng et al. (2024); Song et al. (2024); Zeng et al. (2024); Bernevig and Kwan (2025); Li et al. (2025); Xu et al. (2025); Chen et al. (2026a); Shi et al. (2026). Recent numerical studies further suggest that SC and CDW phases can appear near FCI phases Divic et al. (2025); Wang and Zaletel (2025); Guerci et al. (2025); Chen et al. (2026b). Several field-theoretical approaches and parton constructions have also been developed to understand how superconductivity may emerge from fractionalized excitations Shi and Senthil (2025); Pichler et al. (2026); Nosov et al. (2026). Together, these theoretical and experimental developments point to the possibility of a common microscopic setting from which FCI, CDW, and SC tendencies may emerge. This possibility is especially intriguing because intrinsic superconductivity is not known to occur as a competing phase in the conventional LLL setting. One possible clue is the nontrivial quantum geometry of Chern bands. Despite this progress, a microscopic and analytical understanding of the interplay among FCI, SC, and CDW tendencies is still lacking. In this work, we provide such an analysis and show how quantum geometry can promote SC correlations near an FCI phase.

We study a partially filled Chern band in the anisotropic limit. In this limit, the two-dimensional system can be treated as an array of coupled wires and solved analytically using bosonization and the renormalization group (RG) Von Delft and Schoeller (1998); Giamarchi (2003); Cardy (1996). This coupled-wire construction (CWC) has been widely used for LLL and Chern-band systems, where it captures the essential physics of fractional quantum Hall and fractional Chern insulating phases Kane et al. (2002); Fuji and Furusaki (2019); Shavit and Oreg (2024); Teo and Kane (2014); Sagi and Oreg (2014); Meng (2020). Coupled-wire methods have also been applied to conventional correlated systems, such as Hubbard models, where they provide a useful way to study the competition and interplay among correlated phases, including superconductivity Balents and Fisher (1996); Lin et al. (1997); Shelton et al. (1996); Jaefari and Fradkin (2012); Fradkin et al. (2015). We use this framework to analytically study the interplay among FCI, CDW, and superconducting tendencies, and to identify the role of quantum geometry Provost and Vallee (1980); Marzari et al. (2012); Yu et al. (2025b); Törmä (2023).

In our model, the quantum geometry is controlled by the ratio of the two inter-wire hopping amplitudes, $r=t_{x}^{\prime}/t_{x}$, with increasing $r$ enhancing the quantum geometry. Additionally, a finite $r$ leads to an additional anti-FCI (aFCI) scattering channel, which was initially identified in Ref. Shavit and Oreg (2024). This aFCI channel is enhanced by quantum geometric effect and can destabilize the FCI phase. We demonstrate that the cooperation between the aFCI and FCI channels generates an effective Josephson coupling between wires. This coupling produces a superconducting instability near the FCI phase. The same mechanism can also stabilize charge-density-wave order, depending on microscopic parameters. Using a renormalization-group analysis, we derive the resulting phase diagram and identify superconducting and charge-density-wave regimes near the FCI. Moreover, the superconducting transition temperature is generically enhanced by quantum geometry, which amplifies the aFCI interaction. Thus, our work provides an analytical study of the interplay among correlated phases, including FCI, superconductivity, and CDW order, and highlights the nontrivial role of quantum geometry.

Model and quantum geometry.— We take a minimal Chern-band model containing two $s$ orbitals related by inversion symmetry. The model is described by the following non-interacting Hamiltonian (see also Appendix I SM )

Here $\tau_{\mu}$ are Pauli matrices in orbital space, and $c_{\mathbf{k},\alpha}^{\dagger}$ creates an electron with momentum $\mathbf{k}$ and orbital index $\alpha$. We first consider the 1D limit with $t_{y}=M\neq 0$ and $t_{x}=t_{x}^{\prime}=0$. For each $k_{x}$, the spectrum contains two linearly dispersing modes near $k_{y}=0$, while the modes near $k_{y}=\pi/a$ are gapped by $M$. The system can therefore be viewed as an array of one-dimensional wires. On the $j$-th wire, at partial filling, the system has two Fermi points at $k_{y}=\pm k_{F}$. Expanding near these Fermi points gives two low-energy modes with opposite velocities, or chiralities, denoted by $\psi_{j,1}$ and $\psi_{j,2}$ (see Fig. 1 (a)).

We then introduce weak inter-wire hopping between modes of opposite chirality. The two hopping amplitudes are $t_{x}$ and $t_{x}^{\prime}$, as shown in Fig. 1(a). At filling fraction $\nu=1$, the lowest band is fully filled, and these hopping processes stabilize a Chern insulator. The Chern number is $+1$ or $-1$, depending on whether $|t_{x}|>|t_{x}^{\prime}|$ or $|t_{x}|<|t_{x}^{\prime}|$. In this work, we focus on the anisotropic limit with a $C=1$ lowest band, where $t_{y}=M\gg|t_{x}|>|t_{x}^{\prime}|$. A representative dispersion of the lowest band in this limit is shown in Fig. 1(b).

This construction also makes clear how the lattice Chern band differs from the LLL in a magnetic field along the $z$ direction Meng (2020); Shavit and Oreg (2024). In the LLL-like limit, only the $t_{x}$ process is present Meng (2020). In a Chern insulator, however, more hopping processes are allowed, and a finite $t_{x}^{\prime}$ can be introduced. Therefore, we can increase $r=t_{x}^{\prime}/t_{x}$ to drive the system away from the LLL limit. It is worth mentioning that the limit $t_{x}^{\prime}=0$ was referred to as the optimal limit in Ref. Shavit and Oreg (2024). This should not be confused with the ideal limit, since the trace condition is not satisfied even at $t_{x}^{\prime}=0$.

Moreover, we investigate the evolution of the quantum geometry. As $r$ increases, the integrated quantum metric $Q^{xx}$ along the $x$ direction grows, signaling a more delocalized wave function in the $x$ direction, as shown in Fig. 2(a). Analytically, $Q^{xx}$ takes the form

where $\alpha_{x}=\sqrt{8}\pi\sqrt{t_{x}^{2}+t_{x}^{\prime 2}}/t_{y}$ is a small number measuring the anisotropy of the dispersion. As $r$ is increased from $0$, the quantum geometry is enhanced and then diverges logarithmically as $r\rightarrow 1$ ($t_{x}^{\prime}\rightarrow t_{x}$). This behavior is consistent with the gap closing at $r=1$. Additionally, the momentum-space fluctuations of the quantum metric $Q^{xx}(\mathbf{k})$ increase with increasing $r$, as shown in Fig. 2(b) and (c). Thus, this toy model provides a minimal Chern-band setting with quantum geometry tunable by $r$, where stronger and more fluctuating quantum geometry is realized at larger $r$.

In the rest of the manuscript, we focus on the partially filled lowest $C=1$ band at filling fraction $\nu=1/3$. We start from the weakly coupled-wire limit, where the system is described by a sliding Luttinger liquid (SLL) Kane et al. (2002). We then analyze how the interactions destabilize this SLL toward various competing phases, including a fractional Chern insulator and superconductivity. This CWC also allows us to study the interplay among various phases analytically using bosonization (see Appendix II SM for details of the bosonization procedure). Within bosonization, the low-energy electron operator is written as $\psi_{j,\alpha}\sim K_{j,\alpha}e^{-i\Phi_{j,\alpha}}$. Here $K_{j,\alpha}$ is a Klein factor, and $\Phi_{j,\alpha}$ is the corresponding bosonic field. For later use, we also introduce $\theta_{j}=\frac{1}{2}(\Phi_{j,1}+\Phi_{j,2})$ and $\phi_{j}=\frac{1}{2}(\Phi_{j,1}-\Phi_{j,2})$.

Fractional Chern insulator.— At $\nu=1/3$, the FCI is stabilized by the following effective correlated hopping process, as illustrated schematically in Fig. 3(a),

where the particle-hole operator is defined as $O_{j}(r)=\psi_{j,1}^{\dagger}(r)\psi_{j,2}(r)$. The displacement $a$ denotes a small real-space separation. As initially pointed out in Ref. Shavit and Oreg (2024), there is another closely related scattering process, denoted aFCI, which takes the form (see also Fig. 3(a))

Microscopically, the FCI interaction is induced by the combination of on-site inter-orbital repulsion with strength $U$ and inter-wire hopping characterized by $t_{x}$. The aFCI interaction is generated by the analogous process with the inter-wire hopping replaced by the $t_{x}^{\prime}$ process. The coupling strengths obtained from perturbation theory read (see Appendix V SM for details of the derivation)

where $E_{H}$ denotes the energy of the high-energy electrons. $Q^{yy}(k_{F})$ is the quantum geometry along the $y$ direction at the Fermi point in the 1D limit. Enhanced quantum geometry along $y$ therefore enhances the overall coupling strength. Here, we are mostly interested in the relative strength between FCI and aFCI scattering. This relative strength controls the stability of the FCI phase and plays an important role in stabilizing the superconducting phase discussed below. As shown in Eq. 5, the relative coupling strength between FCI and aFCI is $r$, which is also directly related to the quantum geometry $Q^{xx}$. Intuitively, when the electronic wave functions become more delocalized along the $x$ direction, as characterized by the enhancement of $Q^{xx}$, the additional correlated hopping process represented by the aFCI channel becomes more important.

Finally, within the bosonization framework, the interactions given in Eqs. 3 and 4 correspond to the interaction vertices $g_{FCI/aFCI}e^{i\Theta_{j}^{FCI/aFCI}(r)}$, with

Superconducting (and charge-density-wave) instability.— We now show that superconducting and charge-density-wave correlations are naturally generated when the FCI and aFCI scattering processes are both present. This follows from the operator product expansion (OPE) Cardy (1996),

where $::$ denotes normal ordering and $\Delta^{\lambda}$ denotes the scaling dimension of $\Theta_{j}^{\lambda}$ with $\lambda\in\{FCI,aFCI,SC,CDW\}$ Cardy (1996) (see Appendix VI SM for the derivation). From Quantum-Geometry-Induced Superconductivity near a Fractional Chern Insulator, we observe that the fusion between $e^{i\Theta^{FCI}}$ and $e^{\pm i\Theta^{aFCI}}$ naturally leads to interaction vertices that we denote as SC and CDW, with the corresponding bosonic fields defined as

To identify the physical content of $\Theta_{j}^{SC}$ and $\Theta_{j}^{CDW}$, we transform back to the electronic fields, where we find

Thus $e^{i\Theta_{j}^{SC}(r)}$ describes a Josephson coupling between pairing fields on neighboring wires, which promotes coherent pairing correlations across the wire array and stabilizes an SC phase. By contrast, $e^{i\Theta_{j}^{CDW}(r)}$ locks particle-hole correlations between wires and stabilizes a CDW phase.

Qualitatively, the emergence of the SC channel is illustrated in Fig. 3 (b). We start by combining the $e^{i\Theta^{FCI}_{j}}$ process with the conjugate aFCI process $e^{-i\Theta^{aFCI}_{j}}$. After canceling electron and hole operators within the same wire and chirality, the remaining operator contains a pair of electron operators on wire $j$ and a pair of hole operators on wire $j+1$, corresponding to a Josephson coupling between the two wires. Intuitively, the OPE shows that correlations in the FCI and aFCI channels induce correlations in the SC and CDW channels.

Perturbative RG.— After establishing the connections among FCI, aFCI, SC, and CDW instabilities, we perform a perturbative RG calculation to further identify their interplay. We focus on a two-wire limit, where the system consists of two wires with $j=1,2$. This limit is analytically tractable and is also sufficient to capture the competition and intertwining among these channels. Within the two-wire limit, the free-boson part is described by

where $\theta_{e/o}=\frac{1}{2}(\theta_{1}\pm\theta_{2})$ and $\phi_{e/o}=\frac{1}{2}(\phi_{1}\pm\phi_{2})$. $K_{i}$ are the corresponding Luttinger parameters, $u_{i}$ is the velocity, $v_{1}$ and $v_{2}$ are symmetry-allowed couplings in the time-reversal-breaking system (see Appendix VI SM for the corresponding values derived from the microscopic model). We consider interactions $g_{\lambda}e^{i\Theta_{j}^{\lambda}}+\text{h.c.}$ with $\lambda\in\{FCI,aFCI,SC,CDW\}$ as perturbations. Here $g_{\lambda}$ is the corresponding coupling constant. To facilitate the RG calculation, we also introduce dimensionless couplings $y_{\lambda}\propto g_{\lambda}$. The RG flows of the coupling constants are

where $l$ denotes the RG step. The full RG equations and their derivation are provided in Appendix VI SM .

Equation (Quantum-Geometry-Induced Superconductivity near a Fractional Chern Insulator) shows that SC and CDW couplings are generated under RG once both FCI and aFCI scatterings are present with $y_{FCI}y_{aFCI}\neq 0$. Thus, even if the microscopic Hamiltonian has no bare $y_{SC}$ or $y_{CDW}$, a finite SC coupling is induced, consistent with the OPE in Quantum-Geometry-Induced Superconductivity near a Fractional Chern Insulator. The scaling dimensions $\Delta^{SC}=2/K_{o}$ and $\Delta^{CDW}=18K_{e}$ control whether the generated SC or CDW coupling becomes the leading instability.

We solve the full set of RG equations in the absence of microscopic couplings in the SC and CDW channels, while keeping finite bare couplings in the FCI and aFCI channels. The resulting phase diagram is shown in Fig. 4(a), where the ground state is determined by which dimensionless coupling $y_{\lambda}$ first reaches $1$ during the RG flow. A gapless region remains when none of the scattering processes becomes relevant. Both SC and CDW instabilities naturally appear near the FCI region. In addition, we also find that a relatively strong single-particle dispersion, or bandwidth, favors SC over CDW when the FCI phase is unstable.

Finally, we note that the superconducting transition temperature increases as the system develops more pronounced quantum geometry. Within the RG flow, the SC transition temperature can be estimated from $T_{SC}=T_{0}e^{-l^{*}_{SC}}$. Here $T_{0}$ is the ultraviolet energy scale, which is approximately the noninteracting bandwidth. $l^{*}_{SC}$ is the RG step at which the dimensionless coupling $y_{SC}$ first reaches $1$, indicating that SC correlations become relevant. Enhancing quantum geometry by increasing $r$ enhances the $y_{aFCI}$ coupling. As a consequence, the SC tendency is strengthened by the FCI/aFCI product term in the RG flow (Quantum-Geometry-Induced Superconductivity near a Fractional Chern Insulator). In Fig. 4(b), we illustrate how the SC temperature scale increases as $r$ is increased.

Filling $\nu=2/3$.— We finally mention that a similar mechanism can also drive an SC instability at filling $\nu=2/3$. At this filling, the bosonized fields stabilizing the FCI phase can be obtained by particle-hole conjugation Fuji and Furusaki (2019). We consider both the FCI interaction and the related aFCI interaction, characterized by the bosonic fields (see Appendix VII SM and Ref. Fuji and Furusaki (2019) for further discussion of particle-hole conjugation)

Combining the two scattering processes again generates SC and CDW correlations, with

whose fermionic representations are

Thus, SC and CDW correlations can again be induced by the cooperation of FCI and aFCI scattering. However, the nature of the resulting phases is different from the $\nu=1/3$ case. This is because the FCI-favoring term at $\nu=2/3$ involves three wires. Consequently, the resulting Josephson coupling and CDW locking involve next-nearest-neighbor wires.

Summary and discussion.— In summary, we have provided an analytical study of competing phases in a partially filled interacting Chern band. In our model, the quantum geometry is controlled by the parameter $r$, and increasing $r$ enhances the quantum geometry. As the quantum geometry is enhanced, a competing aFCI scattering process emerges and becomes stronger. The cooperation between the aFCI and FCI channels naturally produces correlations in the SC and CDW channels. Using a perturbative RG analysis, we derive the phase diagram of the model and find SC and CDW phases near the FCI phase. We further demonstrate that the estimated SC transition temperature is enhanced by the quantum-geometry effect. We show that SC and CDW correlations can also be induced by the cooperation of FCI and aFCI scattering at filling fraction $\nu=2/3$. Therefore, our work provides an analytical study of the interplay among FCI, SC, and CDW phases, and identifies the crucial role of quantum geometry in organizing these instabilities.

Supplementary Materials
Haoyu Hu