Introduction

Bardeen–Cooper–Schriffer (BCS) theory1 of superconductivity stands as one of the most successful and influential theories in modern physics. It offers a mean-field, yet non-perturbative and microscopic framework for understanding superconductivity. It has been very successful in describing a large number of superconductors2,3,4. Deviations from the BCS theory are not unusual, which are often attributed to strong interaction effects5,6. Recently, the observations of superconductivity in twisted bilayer graphene7,8,9,10,11 and related graphene family12,13,14 hinted that a new theory is needed to describe superconductors with nearly flat bands. It was observed in a recent experiment11 that some important physical quantities deviate greatly from BCS predictions and the microscopic origins behind them are not yet clear.

For example, the BCS superconducting coherence length ξBCS is expressed as ℏvF/Δ, where vF is the Fermi velocity and Δ is the pairing gap. When the moiré band of twisted bilayer graphene is nearly flat with vF ≈ 103 m/s and Δ ≈ 0.2 meV, ξBCS is estimated to be around 3 nm which is more than one order of magnitude shorter than the values measured using upper critical field measurements11. Furthermore, the low Fermi velocity (or equivalently, large effective mass) should lead to a low superfluid stiffness. This results in an expected Berezinskii–Kosterlitz–Thouless transition temperature much lower than the transition temperature measured at optimal doping11. It had been pointed out by previous works that the quantum metric15,16 of the flat bands, which is the real part of the quantum geometric tensor, is essential in sustaining a supercurrent17,18. Apart from the investigations of superfluid weight19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34, the quantum geometry affects other physical quantities such as the intrinsic nonlinear transport35,36,37 and electron–phonon coupling38.

In recent work, by deriving the Ginzburg–Landau theory for an exactly flat band (with zero bandwidth)39, we pointed out that ξ is determined by the quantum metric of the Bloch wave function which is independent of the interaction strength. This contradicts the intuition that stronger attractive interactions between electrons generally result in a smaller Cooper pair size and shorter coherence length, as described by the BCS theory.

Explicitly, considering the Bloch states of a band represented by \(\left\vert u({{{\boldsymbol{k}}}})\right\rangle\), the quantum geometric tensor \({\mathfrak{G}}\)40,41 is

$${{\mathfrak{G}}}_{ab}=\langle {\partial }_{a}u({{{\boldsymbol{k}}}})| {\partial }_{b}u({{{\boldsymbol{k}}}})\rangle -\langle {\partial }_{a}u({{{\boldsymbol{k}}}})| u({{{\boldsymbol{k}}}})\rangle \langle u({{{\boldsymbol{k}}}})| {\partial }_{b}u({{{\boldsymbol{k}}}})\rangle .$$
(1)

Here, a and b represent the momentum directions. The quantum geometric tensor can be decomposed into real and imaginary parts as \({\mathfrak{G}}={{{\mathcal{G}}}}-i{{{\mathcal{F}}}}/2\), where the real part \({{{\mathcal{G}}}}\) is the quantum metric and the imaginary part \({{{\mathcal{F}}}}\) is the Berry curvature. Berry curvature arises from the phase difference between adjacent Bloch states and characterizes the band topology of materials42,43,44,45,46,47. The study of the physical consequences of the Berry curvature has been one of the central topics in modern physics. On the other hand, the effect of quantum metric, which measures the distance between two quantum states48, is much less studied. It was pointed out that the quantum metric provides the size (or the so-called quadratic spread) ℓqm of the optimally localized Wannier state of a band49, where \({\ell }_{{{{\rm{qm}}}}}=\root 4 \of {\det \overline{{{{\mathcal{G}}}}}}\). Here, \(\overline{{{{\mathcal{G}}}}}\), defined in Eq. (11), is the weighted average of the quantum metric of the Bloch states within a band. The mathematical definition of ℓqm, which we call the quantum metric length, measures the minimal spread of the Wannier functions, is schematically illustrated in Fig. 1. However, the impact of the quantum metric length ℓqm on physical quantities was not clear. Until very recently, the Ginzburg–Landau theory39 shown that at zero temperature, ξ = ℓqm for an exactly flat band, which is independent of the interaction strength.

Fig. 1: The schematic illustration of the quantum metric length ℓqm.
figure 1

In flat-band superconductors with quantum metric, the size (or the quadratic spread) of the optimally localized Wannier functions is ℓqm. ℓqm is also the minimal coherence length (or the minimal size of the Cooper pairs) of the superconductor.

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In realistic materials such as twisted bilayer graphene and related moiré flat-band superconductors, the bands are nearly flat, but the dispersion is still finite. One fundamental question arises: What is the interplay between the quantum metric effect and the finite dispersion of the band? In this work, we demonstrate that

$$\xi =\sqrt{{\xi }_{{{{\rm{BCS}}}}}^{2}+{\ell }_{{{{\rm{qm}}}}}^{2}}\,.$$
(2)

In other words, there is an anomalous quantum metric contribution to the superconducting coherence length (recall that ξBCS = ℏvF/Δ). In the flat-band limit with vanishing vF, the quantum metric effect can be significant and even dominant. We show that this is indeed the case for several moiré superconductors with nearly flat bands11,12,13,14. Our result gives a possible explanation for why the observed superconducting coherence length in the recent experiment11 is much larger than expected. It is worth noting that the coherence length is lattice-geometry independent while the quantum metric is lattice-geometry dependent31,50. To resolve the discrepancy, we apply the uniform pairing condition when evaluating the pair correlators and then demonstrate that ℓqm is related to the quantum metric of the minimal trace31. We delineate the physical picture that, in the presence of the quantum metric, increasing the attractive interaction strength between electrons can only reduce the BCS part of the coherence length and squeeze the Cooper pair size down to the quantum metric length ℓqm, but not further, as demonstrated in Fig. 2.

Fig. 2: Bound of the coherence length by quantum metric.
figure 2

a For a conventional superconductor with a dispersive band (as illustrated by the insert) without quantum metric, the coherence length ξ = ℏvF/Δ decreases as Δ (Δ is the superconducting pairing gap) increases and ξ is not bounded from below. b In the presence of quantum metric, the superconducting coherence length ξ has a lower bound of ℓqm. For a superconductor with a narrow band (as illustrated in the insert), the conventional contribution can be suppressed as Δ increases.

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Additionally, for a topological flat band with nontrivial (spin) Chern number, \(\xi \ge a\sqrt{| C| /4\pi }\), where a is the lattice constant. At the end of this work, we show that the quantum metric length ℓqm is important for the superconducting moiré graphene family. As the new length scale ℓqm defined by the quantum metric is a fundamental property of the band structure and its importance should be manifested beyond superconducting phenomena, we expect that ℓqm also plays a crucial role in other interaction-driven ordered states (such as the magnetic or density-wave states33,51) in flat-band systems.

Results

Quantum metric and coherence length

We investigate the interplay between quantum metric and band dispersion in superconductors where superconductivity appears within an isolated narrow band. To begin with, we describe our formalism from a multi-orbital Hamiltonian with two components: the non-interacting part H0 and the attractive interacting part Hint, which read

$${H}_{0}={\sum}_{ij,\alpha \beta ,\sigma }{h}_{ij,\alpha \beta }^{\sigma }{a}_{i\alpha \sigma }^{{{\dagger}} }{a}_{j\beta \sigma },$$
(3)
$${H}_{{{{\rm{int}}}}}=-{\sum}_{i,\alpha }U{a}_{i\alpha \uparrow }^{{{\dagger}} }{a}_{i\alpha \downarrow }^{{{\dagger}} }{a}_{i\alpha \downarrow }{a}_{i\alpha \uparrow },$$
(4)

where \({h}_{ij,\alpha \beta }^{\sigma }\) is the hopping integral and U denotes the on-site attractive interaction strength. aiασ annihilates a fermion with spin σ in the orbital α at the site i (we may call aiασ orbital fermions). Considering an isolated band near the Fermi energy separated from other bands with a large band gap, we can have an effective one-band description. For s-wave superconducting phase, it is common to introduce orbital-dependent order parameters Δα = −U〈aiα↓aiα↑〉. The mean-field ground state has been extensively investigated, particularly with regard to the superfluid weight determined by the quantum metric17. It is possible to project the orbital fermion aiασ onto the fermion cσ of the isolated band, which is referred as the band fermion. In particular, we employ the following projection scheme

$${a}_{i\alpha \sigma }\to \frac{1}{\sqrt{N}}{\sum}_{{{{\boldsymbol{k}}}}}{e}^{i{{{\boldsymbol{k}}}}\cdot ({{{{\boldsymbol{r}}}}}_{i}+{{{{\mathbf{\delta }}}}}_{\alpha })}{u}_{\alpha \sigma }^{* }({{{\boldsymbol{k}}}}){c}_{\sigma }({{{\boldsymbol{k}}}}),$$
(5)

where we explicitly keep the orbital positions {δα} within a unit cell. The Bloch state uασ(k) of the isolated band with energy ϵσ(k) satisfies the time-reversal symmetry \({u}_{\alpha }({{{\boldsymbol{k}}}})\equiv {u}_{\alpha \uparrow }({{{\boldsymbol{k}}}})={u}_{\alpha \downarrow }^{* }(-{{{\boldsymbol{k}}}})\). The projection in Eq. (5) yields an effective one-band mean-field Hamiltonian Hmf,

$${H}_{{{{\rm{mf}}}}}={\sum}_{{{{\boldsymbol{k}}}}}{\epsilon }_{\sigma }({{{\boldsymbol{k}}}}){c}_{\sigma }^{{{\dagger}} }({{{\boldsymbol{k}}}}){c}_{\sigma }({{{\boldsymbol{k}}}})+[\Delta {c}_{\uparrow }^{{{\dagger}} }({{{\boldsymbol{k}}}}){c}_{\downarrow }^{{{\dagger}} }(-{{{\boldsymbol{k}}}})+h.c.]$$
(6)

with Δ = 1/N∑αkΔα∣uα(k)∣2. The projected mean-field Hamiltonian Hmf is independent of the choice of orbital positions {δα}. To facilitate the theoretical analysis, we can adopt the uniform pairing condition and the minimal quantum metric31. The former assumes that the pairing potentials are the same for different orbitals, and the latter is specific to orbital positions corresponding to the minimal trace of quantum metric. Then we can define the Cooper pair operator \(\hat{\Delta }({{{\boldsymbol{q}}}})=\frac{1}{N}{\sum }_{i\alpha }{e}^{-i{{{\boldsymbol{q}}}}\cdot ({{{{\boldsymbol{r}}}}}_{i}+{{{{\mathbf{\delta }}}}}_{\alpha })}{a}_{i\alpha \downarrow }{a}_{i\alpha \uparrow }\) which is formulated after projection as

$$\hat{\Delta }({{{\bf{q}}}})\to \frac{1}{N}{\sum}_{{{{\boldsymbol{k}}}}}{{\Lambda }}({{{\boldsymbol{k}}}}+{{{\boldsymbol{q}}}},{{{\boldsymbol{k}}}}){c}_{\downarrow }(-{{{\boldsymbol{k}}}}){c}_{\uparrow }({{{\boldsymbol{k}}}}+{{{\boldsymbol{q}}}}).$$
(7)

Here the form factor \({{\Lambda }}({{{\boldsymbol{k}}}}+{{{\boldsymbol{q}}}},{{{\boldsymbol{k}}}})={\sum }_{\alpha }{u}_{\alpha }^{* }({{{\boldsymbol{k}}}}+{{{\boldsymbol{q}}}}){u}_{\alpha }({{{\boldsymbol{k}}}})\) appears as the overlap between two Bloch states. Then we can evaluate the pairing correlator \({{{\mathcal{C}}}}({{{\boldsymbol{r}}}})={\sum }_{{{{\boldsymbol{q}}}}}{e}^{-i{{{\boldsymbol{q}}}}\cdot {{{\boldsymbol{r}}}}}\langle \hat{\Delta }({{{\boldsymbol{q}}}}){\hat{\Delta }}^{{{\dagger}} }({{{\boldsymbol{q}}}})\rangle\) to deduce the coherence length. The pairing correlator \({{{\mathcal{C}}}}({{{\boldsymbol{r}}}})\) is expected to decay exponentially as a function of ∣r∣ at zero temperature for an isotropic system. In other words, \({{{\mathcal{C}}}}({{{\boldsymbol{r}}}}) \sim {e}^{-| {{{\boldsymbol{r}}}}| /\xi }\) and the decay length ξ is the superconducting coherence length52. As shown in Supplementary Note 2, \({{{\mathcal{C}}}}({{{\boldsymbol{r}}}})\equiv {\sum }_{{{{\boldsymbol{q}}}}}{e}^{-i{{{\boldsymbol{q}}}}\cdot {{{\boldsymbol{r}}}}}{{{\mathcal{M}}}}({{{\boldsymbol{q}}}})\), where

$${{{\mathcal{M}}}}({{{\boldsymbol{q}}}})=\frac{T}{N}{\sum}_{n{{{\boldsymbol{k}}}}}| {{\Lambda }}({{{\boldsymbol{k}}}}+{{{\boldsymbol{q}}}},{{{\boldsymbol{k}}}}){| }^{2}{G}_{0}(i{\omega }_{n},{{{\boldsymbol{k}}}}+{{{\boldsymbol{q}}}}){G}_{0}(-i{\omega }_{n},-{{{\boldsymbol{k}}}}).$$
(8)

Here, G0(iωn, k) is the normal Gor’kov’s Green function of the band fermions cσ and ωn = (2n + 1)πT is the Matsubara frequency, as defined in “Methods” section. Then we can extract the coherence length by \({\xi }^{2}=-\frac{1}{2{{{\mathcal{M}}}}(0)}{\left.\frac{{d}^{2}{{{\mathcal{M}}}}({{{\boldsymbol{q}}}})}{d{q}^{2}}\right\vert }_{q = 0}\) with q = ∣q∣ at zero temperature. It is essential to emphasize that the validity of the expression in Eq. (8) hinges on the uniform pairing condition, specifically in relation to the Bloch states of the minimal quantum metric31. Without these conditions, the coherence length calculated using Eq. (8) will be overestimated, and additional details can be found in Supplementary Note 2. To see how the quantum metric affects the coherence length ξ, the form factor Λ enters \({{{\mathcal{M}}}}({{{\boldsymbol{q}}}})\) such that

$$| \Lambda ({{{\boldsymbol{k}}}}+{{{\boldsymbol{q}}}},{{{\boldsymbol{k}}}}){| }^{2}=1-{\sum}_{ab}{{{{\mathcal{G}}}}}_{ab}({{{\boldsymbol{k}}}}){q}_{a}{q}_{b}+{{{\mathcal{O}}}}({{{{\boldsymbol{q}}}}}^{2})\,.$$
(9)

The matrix \({{{{\mathcal{G}}}}}_{ab}\) is the quantum metric of Bloch states, namely the real part of quantum geometric tensor \({\mathfrak{G}}\) in Eq. (1),

$${{{{\mathcal{G}}}}}_{ab}({{{\boldsymbol{k}}}})={{{\rm{Re}}}}\left[{{\mathfrak{G}}}_{ab}({{{\boldsymbol{k}}}})\right].$$
(10)

By theoretically evaluating the pairing correlator, we can obtain the coherence length in Eq. (2) as \(\xi =\sqrt{{\xi }_{{{{\rm{BCS}}}}}^{2}+{\ell }_{{{{\rm{qm}}}}}^{2}}\). In fact, the structure of the coherence length in Eq. (2) is general and works regardless of the uniform pairing condition and the minimal quantum metric. The anomalous coherence length is \({\ell }_{{{{\rm{qm}}}}}= \root 4 \of {\det {\overline{{{{\mathcal{G}}}}}}_{ab}}\), where \({\overline{{{{\mathcal{G}}}}}}_{ab}\) is the weighted average of the quantum metric of the band, which is defined by

$${\overline{{{{\mathcal{G}}}}}}_{ab}=\frac{{\sum}_{{{{\boldsymbol{k}}}}}{{{{\mathcal{G}}}}}_{ab}({{{\boldsymbol{k}}}})/\varepsilon ({{{\boldsymbol{k}}}})}{{\sum }_{{{{\boldsymbol{k}}}}}1/\varepsilon ({{{\boldsymbol{k}}}})},$$
(11)

where ε(k) is the dispersion of the Bogoliubov quasiparticle. In the limit of a flat dispersion ε(k), the quantum metric length ℓqm is reduced to the length scale of the minimal quantum metric. The above discussions on the coherence length in Eq. (2) can be easily generalized to an anisotropic system with a non-circular Fermi surface where the quantum metric length becomes spatially dependent due to finite off-diagonal elements in the quantum metric.

To understand the physical consequence of the anomalous coherence length, we note that for a conventional superconductor, ξ = ξBCS decreases as the interaction strength (or equivalently Δ) increases, as schematically shown in Fig. 2a. However, in the presence of the quantum metric, ξ decreases as Δ increases, but approaches the quantum metric length ℓqm (see Fig. 2b). In the flat-band limit, the coherence length (at zero temperature) is independent of the interaction strength and given by ℓqm. In the presence of finite quantum metric, interactions cannot squeeze the Cooper pairs to a size smaller than ℓqm.

Topologically trivial flat-band model

To support the analytical results mentioned above, we employ the mean-field theory on a microscopic model, which features exactly flat bands without dispersion33,53. The normal state Hamiltonian hs(k) for electrons with spin index s reads

$${h}_{s}({{{\boldsymbol{k}}}})=-t[{\lambda }_{x}\sin ({\alpha }_{{{{\boldsymbol{k}}}}})+s{\lambda }_{y}\cos ({\alpha }_{{{{\boldsymbol{k}}}}})].$$
(12)

Here, \({\alpha }_{{{{\boldsymbol{k}}}}}=\chi [\cos ({k}_{x}a)+\cos ({k}_{y}a)]\) and λi are the Pauli matrices in orbital basis. s = ±1 denotes the spins ↑ and ↓. The hs(k) has a pair of perfectly flat bands at energies ϵk = ±t which are depicted in Fig. 3a (solid lines) and the corresponding wave functions are \(\left\vert {u}_{\pm }\right\rangle =1/\sqrt{2}{(\pm 1,is{e}^{is{\alpha }_{{{{\boldsymbol{k}}}}}})}^{T}\) for the upper band (+) and the lower band (−). The flat band is topologically trivial with the Berry curvature vanishing over the whole Brillouin zone. We can tune the quantum metric by altering the parameter χ in αk. It is straightforward to obtain the quantum metric for  + band with components \({{{{\mathcal{G}}}}}_{ab}({{{\boldsymbol{k}}}})={\chi }^{2}{a}^{2}\sin ({k}_{a})\sin ({k}_{b})/4\), which is the minimal quantum metric since the orbitals are located at high-symmetry positions. The averaged quantum metric defined by Eq. (11) is given by \({\overline{{{{\mathcal{G}}}}}}_{ab}={\delta }_{ab}{\chi }^{2}/8\) which is related to the quantum metric length \({\ell }_{{{{\rm{qm}}}}}=\sqrt{2}\chi /4\). In Fig. 3b, we plot the distribution of \({{{\rm{Tr}}}}[{{{\mathcal{G}}}}({{{\boldsymbol{k}}}})]\) that respects the C4 symmetry and that \({{{\rm{Tr}}}}[{{{\mathcal{G}}}}({{{\boldsymbol{k}}}})]\) reaches its maximum at M/2. Since we are interested in a superconducting phase, we do not include other possible ground state ansatz. In the Nambu basis \({\Psi }_{{{{\boldsymbol{k}}}}}={({a}_{A,{{{\boldsymbol{k}}}}\uparrow },{a}_{B,{{{\boldsymbol{k}}}}\uparrow },{a}_{A,-{{{\boldsymbol{k}}}}\downarrow }^{{{\dagger}} },{a}_{B,-{{{\boldsymbol{k}}}}\downarrow }^{{{\dagger}} })^T}\) with an attractive interaction as Eq. (4), we have the mean-field Hamiltonian Hmf

$${H}_{{{{\rm{mf}}}}}={\sum}_{{{{\boldsymbol{k}}}}}{\Psi }_{{{{\boldsymbol{k}}}}}^{{{\dagger}} }\left[\begin{array}{cc}{h}_{\uparrow }({{{\boldsymbol{k}}}})-\mu &\hat{\Delta }\\ {\hat{\Delta }}^{{{\dagger}} }&-{h}_{\downarrow }^{* }(-{{{\boldsymbol{k}}}})+\mu \end{array}\right]{\Psi }_{{{{\boldsymbol{k}}}}}.$$
(13)

Here, \(\hat{\Delta }={{{\rm{diag}}}}[{\Delta }_{A},{\Delta }_{B}]\) is the mean-field pairing order parameters. The Fermi energy μ is chosen such that the  + band is half-filled. The solutions of the order parameters yield ΔA = ΔB = U/4, which satisfy the uniform pairing condition.

Fig. 3: Quantum metric and coherence length for the topological trivial flat-band model.
figure 3

a The energy spectrum of the flat-band model in ref. 33. The solid (dashed) lines denote t2 = 0 and t2 = 0.02t, respectively. t2 denotes the nearest hopping which makes the band dispersive. b The profiles of the quantum metric \({{{\rm{Tr}}}}[{{{\mathcal{G}}}}]\) of the conduction band in the first Brillouin zone. The color bar denotes the magnitude of \({{{\rm{Tr}}}}[{{{\mathcal{G}}}}]\). c The calculated coherence length ξ for χ = 5 as a function of the attractive interaction U/t. The red, purple, and blue denote the cases of t2 = 0, t2 = 0.01t and t2 = 0.02t, respectively. The theoretical bound ℓqm is indicated by a dashed green line which coincides with the red one. d The quantum metric dependence of ξ as the parameter χ varies when U = 0.4t. The dashed light blue line marks the length scale ℓqm. All calculations are conducted at kBT = 0.001t and at half-filling μ = t.

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Due to the absence of band dispersion, the coherence length \(\xi =\sqrt{2}\chi /4\) depends solely on the quantum metric. This is illustrated in Fig. 3d, where the numerical results [Eq. (8)] of pair correlation functions align with ℓqm. To incorporate the finite band dispersion, one can introduce an additional nearest-hopping term \(\delta h=-2{t}_{2}[\cos ({k}_{x}a)+\cos ({k}_{y}a)]{\lambda }_{0}\) to hs(k), where λ0 is the 2 × 2 identity matrix. This term gives rise to a band dispersion as well as the conventional contribution ξBCS to the total coherence length ξ. In Fig. 3c, the total coherence length gradually decreases for t2 = 0.01t, 0.02t when the attractive interaction strength U increases. In particular, ξ approaches ℓqm in the flat-band limit due to the suppression of ξBCS, as expected from Eq. (2).

Topological bound of the coherence length

In the previous subsection, we have demonstrated how the quantum metric gives a lower bound for the superconducting coherence length. We now consider a system which possesses topological flat bands. As pointed out previously49,54, the quantum metric has a lower bound which is proportional to the Chern number. Therefore, we expect that there is a finite quantum metric length which serves as the lower bound of the superconducting coherence length for a superconductor with nontrivial spin Chern numbers.

Specifically, the quantum geometric tensor is a positive semidefinite matrix, and in two spatial dimensions, we have the inequality \(\sqrt{\det {{{\mathcal{G}}}}({{{\boldsymbol{k}}}})}\ge | {{{{\mathcal{F}}}}}_{xy}({{{\boldsymbol{k}}}})| /2\), which implies that a topological band must necessarily possess a finite quantum metric. According to Eq. (2), this indicates that there is a lower bound on the coherence length ξ which is determined by the topology of the band such that

$$\xi \ge {\ell }_{{{{\rm{qm}}}}}\ge a\sqrt{| C| /4\pi }\,,$$
(14)

where C denotes the (spin) Chern number of a band with a lattice constant a. For demonstration, we consider a two-orbital square lattice with short- and long-range hoppings (Fig. 4a) with a finite spin Chern number55,56,57. Under the basis \({a}_{{{{\boldsymbol{k}}}}\sigma }={({a}_{A{{{\boldsymbol{k}}}}\sigma },{a}_{B{{{\boldsymbol{k}}}}\sigma })}^{T}\), the non-interacting Hamiltonian is \({H}_{0}={\sum }_{{{{\boldsymbol{k}}}},\sigma }{a}_{{{{\boldsymbol{k}}}}\sigma }^{{{\dagger}} }{H}_{{{{\boldsymbol{k}}}}}{a}_{{{{\boldsymbol{k}}}}\sigma }\), where Hk = ∑ihi(k)λi. Here \({h}_{0}({{{\boldsymbol{k}}}})= (\sqrt{2}-1) \cos (2{k}_{x}a)\cos (2{k}_{y}a)/2\), \({h}_{x}({{{\boldsymbol{k}}}})=-\sqrt{2}[\cos ({k}_{x}a)+ \cos ({k}_{y}a)]/2\), \({h}_{y}({{{\boldsymbol{k}}}})= \sqrt{2}[\cos ({k}_{x}a) -\cos ({k}_{y}a)]/2\), and \({h}_{z}({{{\boldsymbol{k}}}})=-\sqrt{2}\sin ({k}_{x}a)\sin ({k}_{y}a)\). The λi are the Pauli matrices on the orbital basis. Importantly, the lowest band is nearly flat with a spin Chern number C = 2 (see Fig. 4b). The bandwidth is ~1% of the total band gap.

Fig. 4: Quantum metric and coherence length for topological flat-band model.
figure 4

a A two-orbital square lattice with short- and long-range hoppings, b the electronic band structure, c the quantum metric distribution of the lower flat band, and d coherence length ξv.s. pairing gap Δ. In (a), the inter-orbital nearest hopping, intra-orbital next-nearest-neighbor and fifth-nearest-neighbor hoppings are labeled. In (b), the lower band (purple) has nearly zero bandwidth with the Chern number C = 2. In (d), the coherence length is extracted from the pair correlation function and it is bounded by a Chern number, which is guided by the dashed green line.

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In Fig. 4c, we depict the distribution of \({{{\rm{Tr}}}}[{{{\mathcal{G}}}}({{{\boldsymbol{k}}}})]\), which exhibits C4 symmetry and has a large quantum metric at X/2 and points connected by symmetry. To demonstrate the effect of the nontrivial Chern number, in the mean-field calculations, we assume the flat band is half-filled for simplicity. The uniform pairing condition is also satisfied as ΔA = ΔB = Δ. Furthermore, we have calculated the Cooper pair correlation functions and extracted the coherence length from Eq. (8), which exhibits a decreasing trend as the band pairing potential Δ increases, as shown in Fig. 4d. Especially, in the limit of large Δ, the coherence length ξ converges to ~\(\root 4 \of {\det \overline{{{{\mathcal{G}}}}}}\) which is larger than \(\sqrt{| C| /4\pi }a\) as predicted by Eq. (14). This result clearly demonstrates how the superconducting coherence length is related to the quantum geometry (both the quantum metric and the topology) of the relevant band.

Application to Moiré materials

The graphene-based moiré systems provide versatile platforms to explore the exotic phenomena related to the flat bands58,59,60,61,62,63,64,65,66,67. In the superconducting graphene-based moiré family14, the quantum metric effect is indeed very crucial. Particularly, the quantum metric plays a significant role in determining the coherence length in magic-angle twisted bilayer graphene (MATBG) with twisted angle θ ≈ 1.08°. To provide a qualitative estimation of the impact of the quantum metric, we employ the Bistritzer–MacDonald model to elucidate the significance of the quantum metric in the context of graphene-based moiré materials68. We also assume the presence of an s-wave superconducting phase. As shown in Fig. 5, the quantum metric length ℓqm = 1.2LM ≈ 13 nm. Here, LM ≈ a0/θ represents the moiré lattice constant. By employing the self-consistent mean-field study (in Supplementary Note 4), we calculate the total coherence length using Eq. (8) to take into account the band dispersion. Using the interaction strength U = 0.6 meV, which gives Tc ≈ 1.7 K, we obtain a conventional contribution of  ~3 nm and ℓqm ~ 13 nm at θ = 1.08°. Therefore, the total superconducting coherence length given by Eq. (2), is indeed dominated by the quantum metric contribution.

Fig. 5: A list of the moiré band structures, quantum metric, and geometric contributions of the coherence length ℓqm for magic-angle twisted bilayer graphene (MATBG), magic angle twisted trilayer graphene (MATTG) and twisted double bilayer graphene (TDBG).
figure 5

For both MATBG and MATTG, the quantum metric is plotted for the highest valence band, and it exhibits divergence near the K points. In TDBG, an electric field potential of V = 40 meV is applied, leading to flat bands near charge neutrality with Chern number C = ±2. The quantum metric is plotted for the lowest conduction band. In evaluating ℓqm, we ignore the band dispersion.

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A large family of moiré systems exhibit superconductivity, such as magic-angle twisted trilayer graphene (MATTG)13 and twisted double-bilayer graphene (TDBG)12. Similar to MATBG, the quantum metric effects cannot be neglected, as shown in Fig. 5. For MATTG, ℓqm = 1.2LM, and for TDBG, ℓqm = 0.5LM. The calculations of ℓqm in Fig. 5 are made by averaging the quantum metric over the moiré Brillouin zone without considering the quasiparticle energy in Eq. (11). Notably, the flat band in TDBG carries a non-zero valley Chern number C = 2, leading to a topology-bound coherence length, as discussed previously. We focus on the quantum metric within a single band, while the generalization to multiple nearly degenerate flat bands consists of replacing the one-band quantum metric with the non-abelian quantum metric69,70. The quantum metric length calculated for the moiré systems is a qualitative estimation because of the limitations of the continuum model and the simple s-wave pairing assumption. It will be an open question of the role that quantum metric plays in unconventional superconductivity for moiré systems.

Conclusion

In this work, we highlight that an intrinsic length scale, ℓqm, derived from the quantum metric, gives rise to an anomalous contribution of the coherence length in superconductors. Particularly in the case of flat bands, ℓqm plays a dominant role in determining the length scale of physical quantities, such as the superconducting coherence length. This length scale is likely also related to the size of vortices, Andreev bound states etc. We propose that our theory may also be applicable to quantum ordered phase in flat-band systems, since ℓqm is derived from the quantum geometry of the band and is independent of the interaction-driven order parameter. Furthermore, it would also be interesting to explore potential extensions of ℓqm to the physical properties of other ordered states (such as ferromagnetic and antiferromagnetic states) with flat bands and quantum metric.

Methods

Mean-field theory and Gor’kov Green function. For a mean-field study, we can decouple the interaction term Hint in Eq. (4) with pairing order parameters \({\Delta }_{\alpha }=-U\langle {\hat{a}}_{i\alpha \downarrow }{\hat{a}}_{i\alpha \uparrow }\rangle\) to yield a mean-field Hamiltonian Hmf

$${H}_{{{{\rm{mf}}}}}={\sum}_{{{{\boldsymbol{k}}}}}{\Psi }_{{{{\boldsymbol{k}}}}}^{{{\dagger}} }(\hat{h}\otimes {\tau }_{z}+{{{\rm{Re}}}}\hat{\Delta }\otimes {\tau }_{x}+{{{\rm{Im}}}}\hat{\Delta }\otimes {\tau}_{y}){\Psi }_{{{{\boldsymbol{k}}}}}$$
(15)

where the Ψk is the spinor with components \({({\Psi }_{{{{\boldsymbol{k}}}}})}_{\alpha \uparrow }={a}_{\alpha \uparrow }({{{\boldsymbol{k}}}})\) and \({({\Psi }_{{{{\boldsymbol{k}}}}})}_{\alpha \downarrow }={a}_{\alpha \downarrow }^{{{\dagger}} }(-{{{\boldsymbol{k}}}})\). Here aασ is a Fermion operator on the orbital basis and τx,y,z are the Pauli matrices. The \(\hat{h}\) is the matrix with elements \({(\hat{h})}_{\alpha \beta }={h}_{\alpha \beta }({{{\boldsymbol{k}}}})-\mu {\delta }_{\alpha \beta }\) and the pairing matrix \(\hat{\Delta }\) has elements \({(\hat{\Delta })}_{\alpha \beta }={\Delta }_{\alpha }{\delta }_{\alpha \beta }\). Within the mean-field Hamiltonian, we can define the Green function \({\hat{G}}_{\alpha {\alpha }^{{\prime} },\sigma {\sigma }^{{\prime} }}(i{\omega }_{n},{{{\boldsymbol{k}}}})=\langle {({\psi }_{{{{\boldsymbol{k}}}}})}_{\alpha \sigma }(i{\omega }_{n}){({\psi }_{{{{\boldsymbol{k}}}}}^{{{\dagger}} })}_{{\alpha }^{{\prime} }{\sigma }^{{\prime} }}(i{\omega }_{n})\rangle\) with

$$\hat{G}(i{\omega }_{n},{{{\boldsymbol{k}}}})=\frac{1}{i{\omega }_{n}-(\hat{h}\otimes {\tau }_{z}+{{{\rm{Re}}}}\hat{\Delta }\otimes {\tau }_{x}+{{{\rm{Im}}}}\hat{\Delta }\otimes {\tau }_{y})}.$$
(16)

where ωn = (2n + 1)πkBT is the Matsubara frequency. Then one may evaluate the pairing correlation function \({{{\mathcal{C}}}}({{{\boldsymbol{r}}}},{{{{\boldsymbol{r}}}}}^{{\prime} })\) with the Green function for a multicomponent fermion aασ. For an s-wave superconductor, we expect an exponential decay behavior in \({{{\mathcal{C}}}}({{{\boldsymbol{r}}}},{{{{\boldsymbol{r}}}}}^{{\prime} })\) as a function of \(| {{{\boldsymbol{r}}}}-{{{{\boldsymbol{r}}}}}^{{\prime} }|\).

On the other hand, we apply a mean-field theory to the effective two-band model after the projection. For a superconducting phase, we can introduce an s-wave pairing order parameter \(\Delta =-\frac{U}{N}{\sum }_{{{{\boldsymbol{k}}}}}\langle {c}_{\downarrow }(-{{{\boldsymbol{k}}}}){c}_{\uparrow }({{{\boldsymbol{k}}}})\rangle ,\) and set Δ to be real via fixing the gauge. Here cσ is a Fermion operator on the flat band. Then we can have a mean field Hamiltonian

$${H}_{{{{\rm{mf}}}}}={\sum}_{{{{\boldsymbol{k}}}}}{\psi }_{{{{\boldsymbol{k}}}}}^{{{\dagger}} }\{[\epsilon ({{{\boldsymbol{k}}}})-\mu ]{\tau }_{z}+\Delta {\tau }_{x}\}{\psi }_{{{{\boldsymbol{k}}}}},$$
(17)

where \({\psi }_{{{{\boldsymbol{k}}}}}={[\begin{array}{c}{c}_{\uparrow }({{{\boldsymbol{k}}}}),{c}_{\downarrow }^{{{\dagger}} }(-{{{\boldsymbol{k}}}})\end{array}]}^{T}\) is the Nambu spinor. One can directly extract the Green’s function G for the band fermions as

$$G(i{\omega }_{n},{{{\boldsymbol{k}}}})=\frac{-i{\omega }_{n}{\tau }_{0}-[\epsilon ({{{\boldsymbol{k}}}})-\mu ]{\tau }_{z}-\Delta {\tau }_{x}}{{\omega }_{n}^{2}+{[\epsilon ({{{\boldsymbol{k}}}})-\mu ]}^{2}+{\Delta }^{2}},$$
(18)

In evaluating physical quantities such as the pairing correlation function, one should first project the observables onto an isolated band, and then apply Wick’s theorem via Gor’kov’s Green function. The projection helps uncover the role of quantum metric in physical quantities such as the coherence length.