Geometric Superconductivity in 3D Hofstadter Butterfly

Geometric Superconductivity in 3D Hofstadter Butterﬂy

Moon Jip Park,1, ∗ Yong Baek Kim,2, 3, † and SungBin Lee1, ‡ 1Department of Physics, KAIST, Daejeon 34141, Republic of Korea 2Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada 3School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea

Electrons on the lattice subject to a strong mag- 4.1, 6.0, 13.8 A˚)[21]. Hence, the theoretical description netic field exhibit the fractal spectrum of elec- of such re-entrant superconductivity would be insufficient trons, which is known as the Hofstadter but- without the full consideration of the Landau quantization terfly. In this work, we investigate unconven- as well as the lattice effect. In addition, there are other tional superconductivity in a three-dimensional superconductors where the quantum oscillation and the Hofstadter butterfly system. While it is gener- superconductivity coexist in the presence of the magnetic ally difficult to achieve the Hofstadter regime, fields. The examples include the organic superconduc-

we show that the quasi-two-dimensional materi- tor, κ − (BEDT − TTF)2Cu(NCS)2[22–26] and high Tc als with a tilted magnetic field produce the large- superconductors [5, 27–29]. In order to understand these scale superlattices, which generate the Hofstadter experiments, it is important to unveil the generic nature butterfly even at the moderate magnetic field of the superconductivity in the Hofstadter regime. strength. We first show that the van-Hove sin- In the current work, we investigate the universal fea- gularities of the butterfly flat bands greatly el- tures of the Hofstadter superconductivity. First of all, evate the superconducting critical temperature, to illustrate the general applicability of our work, we offering a new mechanism of field-enhanced su- demonstrate that the tilted magnetic field applied to a perconductivity. Furthermore, we demonstrate three-dimensional system can generate the Hofstadter that the quantum geometry of the Landau mini- butterfly even with the moderate field strength. The bands plays a crucial role in the description of the van-Hove singularity of the flat bands in the Hofstadter superconductivity, which is shown to be clearly butterfly is shown to enhance the superconducting criti- distinct from the conventional superconductors. cal temperature. Furthermore, the superfluid density of Finally, we discuss the relevance of our results to the superconductors arising from the Hofstadter butterfly the recently discovered re-entrant superconduc- is described by the quantum geometry of the electronic tivity of UTe2 in strong magnetic fields. wave function. It turns out that the quantum geomet- The pairing instability of the conventional supercon- ric properties also determine the superconducting nodal ductivity is dictated by the Bloch electronic states near structures and the supercurrent. Our discovery of the the Fermi surface. These states are described by the quantum geometric superconductivity can be applied to semiclassical equations of the motion[1–6], since the wave generic quasi-two dimensional materials. Finally, based functions are well-extended over many lattice sites. In on our theory, we also discuss the case of the field-induced contrast, the wave packets of the electrons subject to re-entrant superconducting phase in UTe2. a magnetic field are localized with the length scale of 3D Butterfly Spectrum- Our discussion starts the magnetic length. Especially, if the magnetic field is with a generic tight-binding model of the quasi-two- strong enough such that the magnetic length becomes dimensional cubic lattice with the nearest neighbor hop- comparable to the lattice length scale, the fractal pat- pings Tx,y,z in each direction(Tx,y Tz). We only as- terns of the electron spectrum emerge, known as the Hof- sume that the system preserves the time-reversal and the stadter butterfly[7]. The electronic states associated with inversion symmetry, which is dictated by the relation, ∗ the Hofstadter butterfly have no classical analogs, thus Tx,y,z = Tx,y,z. Upon this model, we consider the ap- the conventional semiclassical theories are inapplicable. plication of the tilted magnetic field, B, parallel to the In this extreme quantum limit, what kinds of supercon- yz-plane. The corresponding vector potential can be gen- ductivity would arise? erally written as, A = x|B|(0, cos θ, − sin θ), where θ is arXiv:2007.16205v1 [cond-mat.supr-con] 31 Jul 2020 Recent experiments shed light on this issue and offer the tilt angle from thez ˆ-axis. Considering the Peierls the possible candidate materials for unconventional su- substitution on the lattice model, we derive the three- perconductivity in strong magnetic fields. A prominent dimensional version of the Harper’s model with the tilted example is the discovery of the field-boosted supercon- field[30, 31], ductivity in UTe2[8–20]. The re-entrant superconducting X † X † phase occurs in the presence of the tilted field and sus- H(ky, kz)3D = Txcx0 cx + V0(ky, kz)cxcx, (1) tains the field strength up to 60T[11]. The correspond- hx,x0i x ing magnetic length roughly corresponds to ∼ 3.3nm that is comparable to the lattice length scales(a, b, c = where h, i represents the nearest neighbor hopping. The 2

Field-Enhanced Superconductivity- After estab- lishing the three-dimensional Hofstadter butterfly system, we now consider the generic form of the pairing interaction in the BCS channel, which can be written as, 1 X H = V (k, k0)c† (k)c† (−k)c (−k0)c (k0), pairing 2 αβγδ α β γ δ k,k0 (3) where the Greek letters represent the spin and the sites 0 in the magnetic unit cell. Vαβγδ(k, k ) describes the microscopic interaction. The superconducting critical temperature, Tc, can be formally evaluated by solving the self-consistent gap equation[33, 34] with the superconducting pairing kernel, K, FIG. 1. Superlattice in the presence of the tilted mag- X 0 0 netic field (a) the field direction is normal to xy-plane and ∆γδ(k) = Kγδ,λµ(k, k )∆λµ(k ), (4) (b) the field direction is tilted towardsy ˆ-axis. In the tilted k0,λµ field, the projected two-dimensional unit cell forms a superlattice structure (bottom). The superlattice structure can where the order parameter is defined as, ∆γδ(k) ≡ greatly enhance the effective magnetic flux per plaquette, and P 0 † 0 † 0 k0,αβ Vαβγδ(k, k )hcα(k )cβ(−k )i. By diagonalizing it realizes the Hofstadter butterfly spectrum with the moder- the full tight-binding model in Eq. (1), one can solve the ate strength of the magnetic field. gap equation. Fig.2 (b) shows Tc of the singlet pairing with the on-site interaction as a function of the chemical potential. We find that the band flattening greatly en- effective potential, V0, is explicitly given as, hances Tc compared to the case without a magnetic field. V0(ky, kz) = 2Ty cos(2πΦzx + ky) + 2Tz cos(2πΦyx + kz). Especially, Tc diverges near the vHSs. We observe that (2) the enhancement of Tc near the vHS of the Landau minibands is a generic feature, independent of the specific where Φy(z) is the flux penetrating the xz (xy) planes of form of the interactions. Notice that, in the weak field the unit cell respectively. regime, the Chandrasekhar-Clogston limit[35, 36] would If the field direction is normal to the xy-plane, the restrain the superconducting Tc. Therefore, this field- Landau minibands overlap with each other through the induced superconductivity would be most likely realized z-directional dispersion. In this case, the Hofstadter but- as a re-entrant phase, which arises when the Fermi level terfly cannot exist unless the field strength is huge. On crosses the vHSs of the Landau bands. We dub this form the other hand, if the field direction is tilted from the of the superconductivity as the three-dimensional Hofs- plane, the projected lattice normal to the field direction tadter butterfly superconductor(3D HBSC). In the fol- forms a superlattice structure, which enlarges the effec- lowing sections, we study the quantum geometrical char- tive flux(See Fig.1). The effect of the tilted field can be acters of the 3D HBSC. more intuitively understood by analyzing V0(ky, kz) of Geometric Supercurrent- The superfluid weight, Eq. (2)[31, 32]. To do so, we only consider the potential which measures the change of the free energy, F (q), due Ty first. In the presence of the flux, Φz, the low en- to the spatially modulating order parameter, ∆(r, q) = ergy Landau bands form well-localized eigenstates along |∆|e2iq·r, characterizes the superfluid behavior of 3D thex ˆ-direction at the minima of Ty cos(2πΦzx + ky). HBSC. The superfluid weight, Ds, can be formally eval- The effectivex ˆ-directional lattice length is enhanced by uated as, Φ . Then, we consider the perturbation of the poten- z 1 ∂2F (q) tial, Tz. The flux, Φy, introduces the additional effec- [Ds]i,j = (5) tive potential to the localized Landau bands, which is V ∂qi∂qj given by, Veff = 2Tz cos(2πΦeffnx − Φeffky + kz), where where V is the volume of the system and the subscripts Φeff = Φy/Φz is the effective flux, and nx indicates the i, j indicate the x, y directions. The superfluid weight, enlarged superlattice sites. We find, owing to the super- Ds = Dconv + Dgeo, can be decomposed into the two 2 lattice structure, the effective flux is modified and given independent contributions: Dconv ∝ ∂k(k), which is by the relative ratio between the two fluxes. As a re- proportional to the curvature of the normal bands, (k), sult, the Hofstadter butterfly spectrum can be achieved and Dgeo, which is the anomalous contribution from the by tuning the tilt angle θ even at the moderate strength quantum geometry of the wave functions[37–44]. of the magnetic field (See supplementary material for Furthermore, Dgeo is independent of the quasiparticle the explicit numerical demonstration of the flux magni- dispersions, and it is the only contribution to the super- fication with the full tight-binding model.). fluid weigth if the bands are flat (k = µ). The situation 3

and respectively.) It is particularly important to point out that the positive semi-definiteness of the QGT puts an important lower bound to the diagonal part of the metric such that [g(k)]xx,yy ≥ Im[g(k)xy] if Tx = Ty [37]. To elaborate this feature, we explicitly calculate the QGT of the lowest Landau minibands. Fig.3(a) shows the momentum distribution of gxx(k) as a function of the magnetic flux, Φeff. In general, the QGT is a function of the momentum. However, we find that the momentum fluctuation of the QGT diminishes as the flux decreases(Fig.3(a)). In the low flux limit, the momentum distribution of the QGT becomes uniform. As such, the black line in Fig.3(b) shows the asymptotic conver- gence of the QGT to its theoretical lower bounds, which is nothing more than the Chern number of the Landau minibands. Accordingly, the integrand of Ds in Eq. (6) reaches to the theoretical lower bound set by the QGT(red line in Fig.3(b)). As a result, the superfluid weight also saturates to the lower bound. This result holds regardless of the pairing symmetry of the order parameter since the momentum distribution of the QGT becomes uniform. In conclusion, 3D HBSC realizes the peculiar superfluidity where the supercurrent transport is solely mediated through its quantum geometric channel. In contrast to the conventional phenomenology of FIG. 2. Enhanced critical temperature in the three dimensional Hofstadter butterfly spectrum (a) Lan- ordinary superconductors[45], the geometric superfluid dau minibands spectrum of the three-dimensional Hofstadter weight, Dgeo, in Eq. (6) realizes the nonlinear current- butterfly as a function of the tilt angle(0◦ is normal to xy- density consecutive relation as, plane). (b)-(c) The normalized density of states(black line) 2 and Tc(red line) as a function of the chemical potential (b) e|∆| 2e 3 4 j = (∇χ − A), (8) when (Φy, Φz) = ( , ) and (c) without the field. We find that geo p 5 5 µ2 + |∆|2 c Tc is enhanced near the vHS of the Landau minibands as compared to the case without the field. We used T = T = 1, x y where χ is the phase of the superconducting order pa- T = 0.05, and the onsite interaction strength V = 0.6. z rameter. The current density becomes linearly proportional to the order parameter amplitude, as µ decreases. The nonlinear relation in Eq. (8) is the consequence of of the 3D HBSC fits to this case. More precisely, Dgeo is dictated by the quantum geometric tensor(QGT)[37] the flatband superfluidity. We also note that the inclu- (also known as Fubini-Study metric), g(k), through the sion of this geometric current, jgeo, introduces a finite following expression, Meissner effect, which is in sharp contrast to the previ- ous predictions[34]. Universal Nodal Structure- The quantum geom- Z d3k etry of the Landau minibands also plays an important [Ds]ij ≈ [Dgeo]ij = 3 M(k)Re[g(k)]ij, (6) BZ (2π) role in the gap structure of the BdG quasiparticle. To see this, we consider the generic form of the pairing po- 2 where M(k) = √ 2|∆(k)| is a form factor. ∆(k) is the tential between the Landau minibands, which is written µ2+|∆(k)|2 as |∆(k)|eiφ(k)a (k)a (−k) + h.c., where a (k) is the an- superconducting order parameter of the Landau mini- s s¯ s nihilation operator of the Landau minibands near the bands. The QGT is given in terms of the Bloch wave Fermi level with the spin s. We also define the corre- function u(k) of the Landau miniband as, sponding velocity field of the pairing operator[46, 47], † v(k) ≡ ∇ φ(k) − β(k) + β(−k). This velocity field is [g(k)]i,j = ∂iu (k)∂ju(k) − βi(k)βj(k). (7) k a gauge invariant quantity as it is invariant under the † −iΛ(k) Here, βj(k) = iu (k)∂ju(k) corresponds to the Berry U(1) transformation, u(k) → u(k)e , where Λ(k) connection, and the imaginary component of the QGT is a smooth scalar function. We can consider a wind- corresponds to the Berry curvature[39]. (For the detailed ing number along a loop on the two-dimensional normal derivation of Eq. (6), see the supplementary material Fermi surface, which counts the vorticity of the velocity 4

FIG. 3. Distribution of the quantum geometric tensor and the superfluid weight (a) the fluctuation about the av- erage of the QGT of the lowest landau band for Φ = 1 (left) eff 3 FIG. 4. Geometry induced universal superconducting and Φ = 1 (right) respectively. In the low flux limit, the eff 5 nodal structure (a) map of the velocity field v(k) (red ar- fluctuation decreases, and the distribution asymptotically be- rows) and the Fermi surfaces(blue contours) when Φy = 1/2, comes uniform. (b) Comparison of QGT and the normalized ¯ Φz = 1/4. The QGT ensures the non-zero windings of the ve- superfluid weight, Ds,(the normalization is divided by M(k) locity field along the Fermi surface. (b) Examples of the filling in Eq. (6)). In the low flux limit, the contribution of the enforced semimetal in 3D Hofstadter butterfly. We find the QGT and the superfluid weight converges due to the band protected two-fold degeneracy, which forms the Weyl nodes. flattening. In this regime, the superfluid weight asymptoti- (c) Superconducting gap structure of (b). The blue surface cally saturates the lower bound. represents the normal Weyl nodes, which is the monopole of the Berry curvature. The superconducting nodes(tips of the red surface) occur at the location of the vortices of the velocity field(See Fig.4 (a)). The vortical configurations accom- field on the Fermi surfaces. pany the singularity of the velocity field, which is nothing more than the superconducting nodes(|∆(k)| = 0). In- terestingly, the QGT of the normal band encapsulates lation and gives rise to the additional factor of −1 at the information of the vorticity of the superconductivity the kx = π plane. The role of the symmetry is analo- as, gous to the nonsymmorphic time-reversal symmetry in the context of the filling enforced semimetal phases[48– X I dk Z · v(k) = 2Im dkidkj[g(k) − g(−k)]ij, 50]. The physical consequence is the modified Kramer’s π π Ci 2π FS degeneracy at (π, ± , ± )[48–50]. The two-fold point Ci 2 2 (9) degeneracy in three-dimensions is the Weyl nodes which become the source of the vorticity in Eq. (9). The vor- where FS represents the Fermi surface, and Ci represents tices in normal Fermi surface realize as the Weyl nodes the one-dimensional loops encircling each vortex. If the in the BdG spectrum(See Fig.4 (c)), each of which is Fermi surface possesses a non-zero Chern number, the protected by the Chern number, C ∈ Z, in accordance right hand side of Eq. (9) is equal to the difference of the with the class D in the well-known Altland-Zirnbauer Chern number between the two Fermi surfaces at k and classification[51, 52]. As a result, we expect the emer- −k. gence of the universal Weyl superconductivity control- For example, we can consider the generic tilt angles lable by the tilt angle. The Weyl superconductivity is with the flux, Φy,z = py,z/q, of an even integer denomi- expected to be robust even in the existence of the pair nator, q ∈ 2Z. The magnetic unit cell contains the even density wave instability and mixed phases[52–54]. integer sites, and we can define a half-magnetic unit cell Application to UTe2- Finally, we discuss the possi- translation with the complex conjugation, Tˆ q K, and ble connection of the field polarized re-entrant phase of x, 2 the additional momentum translation ky,z → ky,z ± π, UTe2 with the 3D HBSC[11]. UTe2 is the orthorhombic the product of which forms an emergent anti-unitary paramagnet, possessing the quasi-two-dimensional Fermi symmetry of the Hamiltonian in Eq. (1). This sym- surfaces[17, 55, 56]. Two re-entrant phases appear near metry squares to eikx due to the half-unit cell trans- the boundary of the metamagnetic phase transition line. 5

One of the re-entrant phases occurs with the field di- ACKNOWLEDGMENTS rection perpendicular to the magnetic easy axis(a-axis), ◦ and the other occurs with the tilted field angle ∼ 25 M.J.P. is grateful to Gil Young Cho and GiBaik Sim for on the b-c plane[11]. In these two re-entrant phases, it fruitful and enlightening discussions. M.J.P. and S.B.L. is expected that the superconducting order parameters are supported by National Research Foundation Grants are distinct[57], and our theory may explain the latter (NRF-2020R1F1A1073870, NRF-2020R1A4A3079707). re-entrant phase with tilted field angle. Y.B.K. is supported by the NSERC of Canada, the Cen- ter for Quantum Materials at the University of Toronto, and the Killam Research Fellowship of the Canada Coun- Unlike the other uranium-based ferromag- cil for the Arts. netic superconductors(URhGe[58], UCoGe[59], and UGe2[60]), the field-induced ferromagnetic fluctuation[61] is a less compatible explanation of the re-entrant superconductivity. On the other hand, COMPETING INTERESTS the metamagnetic transition is shown to accompany several Lifshitz transitions with small hysteresis[14]. It The authors declare no conflict of interest. may signal the Fermi surface reconstruction associated with the Landau quantization. In such cases, we speculate that the crossing of the vHSs of Landau bands AUTHOR CONTRIBUTIONS and the Fermi level plays a crucial role in enhancing the superconductivity, thereby realizes as the re-entrant superconductivity at the tilted field. If so, we further S.B.L. and Y.B.K. conceived and supervised the re- speculate that another re-entrant phase dome would search. M.J.P. performed the calculations in this work. appear at higher fields where the next crossing of the All authors contributed to writing the manuscript. Landau level occurs. Moreover, we propose that the effect of the Landau quantization can be experimentally confirmed by the Hall effect measurement near the METHODS metamagnetic transition line. Numerical Calculation of Superconducting Kernel Matrix Discussions- We have studied possible superconducting states in a three-dimensional Hofstadter butterfly sys- The superconducting critical temperature, Tc, can be tem. Using the quasi-two-dimensional system subject to formally evaluated by solving the self-consistent gap the tilted magnetic field, we show that the Hofstadter equation[33, 34], butterfly spectrum arises from the resulting superlattice structure. It is demonstrated that the superconductiv- X 0 0 ity in the Hofstadter butterfly system possesses a num- ∆γδ(k) = Kγδ,λµ(k, k )∆λµ(k ), (10) 0 ber of intriguing properties that are in stark contrast to k ,λµ the conventional superconductivity. Unlike the supercon- where the order parameter is defined as, ∆ (k) ≡ ductivity in the semi-classical regime, the vHSs of the γδ P 0 † 0 † 0 Landau mini-bands in presence of the attractive interac- k0,αβ Vαβγδ(k, k )hcα(k )cβ(−k )i. The superconduct- tion greatly enhance the superconducting critical temper- ing pairing kernel, K, is evaluated as, ature. Once the superconductivity is induced in the Hofs- 0 tadter Landau bands, the superconducting properties are Kγδ,λµ(k, k ) = (11) dictated by the quantum geometry of the Landau bands, − T P V (k, k0)[G(k0, ω )] [G(−k0, −ω )] . α,β,k,ωn αβγδ n αλ n βµ which is reflected in the geometric supercurrent and universal nodal structure. These behaviors can be exper- Here ωn is the Matsubara frequencies and G(k, ωn) is the imentally confirmed by the London penetration-depth Matsubara Green function of the normal Hamiltonian in measurement and thermal Hall measurement[54, 62]. We Eq. (1). (See supplementary material for the derivation also note that our results are not restricted to the or- of K). Eq. (10) takes the form of an eigenvalue problem thorhombic lattice. The tilted field can induce the Hof- of the Kernel matrix, where the onset of the supercon- stadter butterfly, and the enhancement of Tc can also ducting instability occurs if the largest eigenvalue of K is occur in other layered materials such as graphite and reached 1. In the calculation of the critical temperature, bulk transition metal dichalcogenides. Constructing the we determine Tc by investigating the largest eigenvalue microscopic theory of the superconductivity in these ma- of the Kernel matrix, while gradually decreasing the tem- terials would be an interesting direction for future study. perature. 6

Evaluation of Quantum geometric Tensor Mark Zic, Hyunsoo Kim, Johnpierre Paglione, and Nicholas P. Butch, “Nearly ferromagnetic spin-triplet superconductivity,” Science 365, 684–687 (2019), In general, the definition of the QGT in Eq. (7) is https://science.sciencemag.org/content/365/6454/684.full.pdf. not practical as it requires the gauge fixing to take the [9] Dai Aoki, Ai Nakamura, Fuminori Honda, DeXin Li, derivative of the eigenstates[63]. To avoid this, we utilize Yoshiya Homma, Yusei Shimizu, Yoshiki J. Sato, Georg the following relation, Knebel, Jean-Pascal Brison, Alexandre Pourret, Daniel Braithwaite, Gerard Lapertot, Qun Niu, Michal Valika, † 0 = ∂µ(ui (k)H(k)uj(k)) Hisatomo Harima, and Jacques Flouquet, “Unconven- = ( (k) − (k))(∂ u†(k))u (k) + u†(k)∂ H(k)u (k), tional superconductivity in heavy fermion ute2,” Jour- j i µ i j i µ j nal of the Physical Society of Japan 88, 043702 (2019), † † ui (k)∂µH(k)uj(k) https://doi.org/10.7566/JPSJ.88.043702. ∂µui (k)uj(k) = . 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SUPPLEMENTARY MATERIAL FOR “3D HOFSTADTER BUTTERFLY SUPERCONDUCTOR”

CONTENTS

acknowledgments 5

Competing interests 5

Author contributions 5

Methods 5 Numerical Calculation of Superconducting Kernel Matrix5 Evaluation of Quantum geometric Tensor6

Data availability 6

References 6

Supplementary Material for “3D Hofstadter Butterﬂy Superconductor”9

spectrum of 3D Hofstadter Model 9

Superconducting Kernel Matrix 10

Quantum Geometric Tensor 12

Calculation of Superﬂuid Weights 13

Geometric supercurrent 14

SPECTRUM OF 3D HOFSTADTER MODEL

In this section, we study the energy spectrum of the three-dimensional Hofstadter model in detail. For the clarity, we start our discussion by writing Eq. (1) again here.

X † † H(ky, kz)3D = Txcx0 cx + V3D(x, ky, kz)cxcx, (S1) hx,x0i

V3D(x, ky, kz) = 2Ty cos(2πΦzx + ky) + 2Tz cos(2πΦyx + kz). where h, i represents the nearest neighbor hopping and Φy(z) is the flux penetrating the xz (xy) planes of the unit cell respectively. Ignoring Tz first, the application of Φz generates the two-dimensional Landau bands. The low Landau bands form the well-localized eigenstates at the minima of the potential 2Ty cos(2πΦzx + ky). Fig. S1 (a) shows the wave function of the lowest Landau levels localized at the minima of the potential(red solid line) . The finite tunneling between the localized envelopes of the Landau level gives rise to the effective band width of the mini-Landau bands, Teff. Now, if we turn on Tz, the Landau bands feel the effective potential 2Tz cos(2πΦeffnx − Φeffky + kz) (red dashed line in Fig. S1). As a result, we can write down the effective Hamiltonian of the Landau bands as,

X † † H(ky, kz)eff = TeffJx0 Jx + Veff(x, ky, kz)JxJx, (S2) hx,x0i

Veff(x, ky, kz) = 2Tz cos(2πΦeffnx − Φeffky + kz).

Φy where Φeff = is the enhanced flux due to the formation of the superlattice and Jx is the annihilation operator of Φz the Landau level. Since Φeff is the function of the tilt angle, the Hofstadter butterfly spectrum can be realized even if the magnetic length scale does not reach the primitive lattice lengths. The correspondence between Eq. (S1) and (S2) is explicitly confirmed in the full tight-binding model calculation in Fig. S1 (b). 10

(a) 4 0.06

2 0.04

0 Potential 0.02

0 -2 0 500 1000 1500 2000 Sites (b) -2.1

0.1 -2.2

0 -2.3

-0.1

-2.4

0 0.05 0.1 0.15 0.2 0 0.25 0.5 0.75 1

FIG. S1. (a) Illustration of the effective superlattice structure. The black line shows the wave functions of the four lowest Landau levels of Eq. (S1) with Φz = 4/2000, Φy = 2/2000. The wave functions are localized at the minima of the potential Ty(red solid line). The localized states are separated each other by 500 lattice sites, which form the superlattice structure. the 1 potential Tz(red dashed line) acting on the Landau level now effectively have the flux Φeff = 2 . (b)-(c) Comparison of the energy spectrum of Eq. (S1) of the lowest Landau bands and Eq. (S2). The application of the small tilted flux Φy(left) shows the full Butterfly spectrum which has correspondence with the magnified effective flux Φz(right). We set Tx = 1,Ty = 0.5,Tz = 0.005.

SUPERCONDUCTING KERNEL MATRIX

In this section, we describe the detailed numerical implementation of the superconducting Kernel matrix. To start our discussion, we would like to deﬁne the superconducting kernel matrix in a general setting. We deﬁne the Matusbara Green function of the full BdG Hamiltonian for imaginary time, τ, as,

1 X G (τ) ≡ −hT c (τ)c†(0)i = e−iωnτ G (iω ), (S3) ij τ i j β ij n ωn∈Z where β = 1/T and i, j represents the orbital index. ωn = (2n + 1)π/β is the Matsubara frequency(n ∈ Z). We note that the anomalous Green function is not separately deﬁned. Tτ is the imaginary time-ordering such that,

( † † ci(τ)cj(0) if τ > 0 Tτ ci(τ)cj(0) ≡ † (S4) −cj(0)ci(τ) if τ < 0.

The equal-time correlation function can be written in terms of the Green function in the frequency space as,

† − X hcj(0)ci(0)i = Gij(τ → 0 ) = T Gij(iωn). (S5) ωn∈Z 11

We now aim to evaluate the Green function in the frequency space. In the Nambu basis, we can decompose the BdG Hamiltonian into the normal and the pairing parts respectively. h0(k) 0 0 ∆(k) HBdG ≡ H0(k) + Hpairing(k) = T + † , (S6) 0 −h0 (−k) ∆ (k) 0 where h0(k) and ∆(k) are the normal state Hamiltonian and the pairing potential with the momentum k respectively. To further elaborate the above expansion, we conceive that G0(iωn) has the following matrix structure, 0 G0(k, k , iωn) = δk,k0 G0(k, iωn) 1 0 ! iωn−h0(k) g0(k, iωn) 0 = 1 ≡ T . (S7) 0 T 0 −g0 (−k, −iωn) iωn+h0 (−k) The Green function of the BdG Hamiltonian can be decomposed into the normal part and the pairing part as well, 1 G(iωn) = iωn − (H0 + Hpairing) 1 = −1 G0 (iωn)(I − G0(iωn)Hpairing) 1 = G0(iωn), (S8) I − G0(iωn)Hpairing −1 where G0(iωn) ≡ (iωn − H0) is the Matsubara Green function of H0. In the weak pairing limit, we can expand the 1 above expression, using the relation, ( 1−M ≈ 1 + M), as,

G(iωn) ≈ (I + G0(iωn)Hpairing)G0(iωn)

= G0(iωn) + G0(iωn)HpairingG0(iωn). (S9)

The second term in Eq. (S9) is the first-order correction from the Hpairing to the Green function. By plugging Eq. (S9) to Eq. (S7), the first-order correction is given as, g0(k, iωn) 0 0 ∆(k) g0(k, iωn) 0 G0(iωn)HpairG0(iωn) = T † T 0 −g0 (−k, −iωn) ∆(k) 0 0 −g0 (−k, −iωn) T 0 −g0(k, iωn)∆(k)g0 (−k, −iωn) = T † . (S10) −g0 (−k, −iωn)∆ (k)g0(k, iωn) 0 From the above result, we can evaluate the equal-time correlation function in Eq. (S5) as, † † X X T hci (k)cj(−k)i ≡ T Gi,(j+Norb)(k, iωn) = −T [g0(k, iωn)∆(k)g0 (−k, −iωn)]ij (S11) ωn ωn where Norb is the size of the matrix h0. To derive the self-consistent gap equation, we also equate the left hand side of the equation to the superconducting order parameter. To do so, we consider the general form of the interaction as, 1 X H = V (q)c† (k + q)c† (k0 − q)c (k0)c (k), (S12) int N αβγδ α β γ δ k,k0,q where N = NxNyNz is the number of the total three-dimensional lattice sites. Taking only the BCS channel where the center of the mass momentum is zero(k + k0 = 0), Eq. (S12) reduces as, 1 X H = V (k0 − k)c† (k0)c† (−k0)c (−k)c (k). (S13) BCS N αβγδ α β γ δ k,k0 We perform the BCS mean-field decomposition of the above Hamiltonian as, 1 X H = V (k0 − k)c† (k0)c† (−k0)c (−k)c (k) (S14) pairing N αβγδ α β γ δ k,k0 1 X = V (k0 − k)[hc† (k0)c† (−k0)ic (−k)c (k) + c† (k0)c† (−k0)hc (−k)c (k)i] N αβγδ α β γ δ α β γ δ k,k0 1 X + V (k0 − k)[c† (k0)c† (−k0) c (k)c (−k) − hc† (k0)c† (−k0)ihc (k)c (−k)i] N αβγδ α β γ δ α β γ δ k,k0 X ∗ † † ≈ ∆γδ(k)cγ (−k)cδ(k) + ∆αβ(k)cα(k)cβ(−k) + constants. k 12 where O ≡ Oˆ − hOî indicates the fluctuation. Using Eq. (S11), we derive the generic linearized gap equation for the multi-band system as following,

X 0 † 0 † 0 ∆γδ(k) = Vαβγδ(k − k)hcα(k )cβ(−k )i k0,αβ X 0 0 0 T 0 = −T Vαβγδ(k − k)[g0(k , iωn)∆(k )g0 (−k , −iωn)]α,β 0 k ,αβ,ωn X 0 0 0 = − T Vαβγδ(k − k)[g0(k , iωn)]αλ[g0(−k, −iωn)]βµ ∆λµ(k ). (S15) 0 k ,αβ,ωn We notice that the above equation is a form of tensorial multiplication, and it can be simpliﬁed by introducing the superconducting pairing kernel, which is given as,

0 X 0 0 Kγδ,λµ(k, k ) = −T Vαβγδ(k − k)[g0(k , iωn)]αλ[g0(−k, −iωn)]βµ (S16)

k,ωn The linearized self-consistent equation in Eq. (S15) reduces to

X 0 0 ∆γδ(k) = Kγδ,λµ(k, k )∆λµ(k ) (S17) k0,λµ

We numerically diagonalize the Kernel matrix and derive the eignevalues λn as a function of the temperature. The onset of the superconducting instability is indicated by the condition, max[λn](T ) ≥ 1. The superconducting critical temperature is derived by the condition, max[λn](Tc) = 1.

QUANTUM GEOMETRIC TENSOR

Before going to the details of the superfluid weight, we first introduce the quantum geometric tensor(QGT) and several generic properties. We mainly follow the discussion of Ref. [39]. For the generic Bloch Hamiltonian H(k) with the momentum k, we can diagonalize the Hamiltonian as, H(k) = U(k)D(k)U †((k), (S18) where D = diag(1, 2, ...n) is the diagonal matrix containing energies. U(k) = (u1(k), u2(k), ...un(k)) is the unitary matrix, where the column vectors are the eigenstates. Then, we define the QGT of the i-th band as,

† † † ∗ gµν (k) ≡ ∂µui (k)∂ν ui(k) − [∂µui (k)ui(k)][ui (k)∂ν ui(k)] = gνµ X † † = ∂µui (k)uj(k)uj(k)∂ν ui(k). (S19) j6=i Furthermore, the QGT can be decomposed into the real and imaginary parts as, 1 Reg (k) = (g + g ) µν 2 µν νµ 1 = (∂ u†(k)∂ u (k) + ∂ u†(k)∂ u (k) + 2[u†(k)∂ u (k)][u†(k)∂ u (k)]) (S20) 2 µ i ν i ν i µ i i µ i i ν i 1 Img (k) = (g − g ) µν 2i µν νµ i 1 = − (∂ u†(k)∂ u (k) − ∂ u†(k)∂ u (k)) = − F (S21) 2 µ i ν i ν i µ i 2 µν where Fµν is the Berry curvature. An important property of the QGT is the positive semideﬁniteness such that, for a complex vector c, the following relations is satisﬁed,

X † cµgµν (k)cν ≥ 0. (S22) µ,ν=x,y If we choose c = (1, i), we derive the relation that

gxx(k) + gyy(k) + i(gxy(k) − gyx(k)) ≥ 0. (S23)

In the isotropic system(Tx = Ty), we ﬁnd that gxx,yy(k) ≥ Imgxy(k). 13

CALCULATION OF SUPERFLUID WEIGHTS

In this section, we now explain the relationship between the superﬂuid weigth and the QGT of the Hofstadter butterﬂy. We consider the following decomposition of the generic BdG Hamiltonian,

† h0(k) ∆(k) U(k)D(k)U (k) ∆(k) HBdG = † T = † ∗ T ∆ (k) −h0 (−k) ∆ (k) −U (−k)D(−k)U (−k) U(k) D(k) U †(k)∆(k)U ∗(−k) U †(k) = . (S24) U ∗(−k) U T (−k)∆†(k)U(k) −D(−k) U T (−k)

Here we have utilized the diagonalization in Eq. (S18). If the Fermi level is near n-th band and the interband gap is suﬃciently larger than the interaction strength, we can project the Hamiltonian to the n-th band. In such cases, we can generally write the form of the unitary matrix and the order parameter matrix as,

† ∗ T [U (k)∆(k)U (−k)]i,j = δinδjn∆proj(k), [∆(k)]i,j = [U(k)]i,n[U (−k)]n,j∆proj(k) (S25) where ∆proj(k) is a complex order parameter acting on the projected band. We now consider the case where the non-zero supercurrent ﬂows with the superconducting order parameter gradient, ∆(r) = |∆|e2iq·r. The corresponding BdG Hamiltonian couples the electron sector of k + q and the hole sector of −k + q like the case of the Fulde-Ferrell states. The Hamiltonian is now written as, h0(k + q) ∆(k) HBdG(q) ≡ † T . ∆ (k) −h0 (−k + q) We now take the similar decomposition in Eq. (S25),

U(k + q)D(k + q)U †(k − q) ∆(k) H (q) = (S26) BdG ∆†(k) −U ∗(−k + q)D(−k + q)U T (−k + q)

D(k + q) U †(k + q)∆(k)U ∗(−k + q) = U(k, q) U(k, q)†, (S27) U T (−k + q)∆†(k)U(k + q) −D(−k + q) where U(k, q) ≡ diag(U(k + q),U ∗(−k + q)). Now the projected Hamiltonian on the n-th band can be written as, (k + q) ∆proj(k, q) Hproj,nn(q) ≈ † (S28) ∆proj(k, q) −(−k + q) where (k) is the normal state energy of the projected band. The expression of ∆proj can be derived using Eq. (S25),

† ∗ ∆proj(k, q) = U (k + q)∆(k)U (−k + q) X † T ∗ = [U (k + q)]n,N1 [U(k)]N1,n[U (−k)]n,N2 [U (−k + q)]N2,n∆proj(k). (S29)

N1,N2 We have now derive the generic expression of the superconducting order parameter with the finite center of mass momentum 2q. Using this expression, we are now ready to calculate the superfluid weight. In the flat band limit, we 2 2 1/2 can approximate (k) = µ, the resulting BdG quasiparticle energy is given as, EBdG(k) = ±(µ + |∆proj(k, q)| ) . Then, the zero temperature superfluid weight is given by,

To evaluate the above expression, we ﬁrst need to calculate the q derivative of ∆proj(k, q) (We use the red color to denote q and label the momentum just by a subscript for the clarity).

2 X † T ∗ T ∗ † 2 |∆proj(k, q)| = [Uk+q]n,N1 [Uk]N1,N [U−k]n,N2 [U−k+q]N2,n [U−k+q]n,N3 [U−k]N3,n[Uk ]n,N4 [Uk+q]N4,n |∆proj(k)| . N1,2,3,4 (S31) 14

We find that the first order derivative vanishes. Therefore, the second order term vanishes in Eq. S30. We calculate the second derivative of q as, 2 2 ∂ |∆proj(k, q)| 2 = (F4 + F12)|∆proj(k)| , (S32) ∂qi∂qj q=0 which we decompose into the two parts. The first part, F4, is given as, 2 † T 2 ∗ 2 T ∗ † 2 F4 = [∂ijUkUk]n,n + [U−k∂ijU−k]n,n + [∂ijU−kU−k]n,n + [Uk∂ijUk]n,n, (S33) and the second part, F12 is given as, † T ∗ † T ∗ † † F12 = −[∂iUkUk]n,n[U−k∂jU−k]n,n − [∂iUkUk]n,n[∂jU−kU−k]n,n + [∂iUkUk]n,n[Uk ∂jUk]n,n (S34) T ∗ T ∗ T ∗ † T ∗ † + [U−k∂iU−k]n,n[∂jU−kU−k]n,n − [U−k∂iU−k]n,n[Uk∂jUk]n,n − [∂iU−kU−k]n,n[Uk∂jUk]n,n + (i ↔ j). In the above expression, the first two terms and the last two terms cancel each other. The expression is further reduced to † † T ∗ T ∗ = [∂iUkUk]n,n[Uk ∂jUk]n,n + [U−k∂iU−k]n,n[∂jU−kU−k]n,n (S35) † † T ∗ T ∗ +[∂jUkUk]n,n[Uk ∂iUk]n,n + [U−k∂jU−k]n,n[∂iU−kU−k]n,n. Summing up the contributions of Eq. (S33) and Eq. (S36), we find that the total contribution is given by, 2 2 X ∂ |∆proj(k, q)| X 1 † † † † 2 = −4Re ([∂iUk∂jUk]n,n + [∂jUk∂iUk]n,n) + [Uk∂iUk]n,n[Uk ∂jUk]n,n |∆proj(k)| ∂qi∂qj 2 k q=0 k X 2 = −4 Regij(k)|∆proj(k)| , (S36) k In the last equality, we have used Eq. (S20). Accordingly, we find that 2 1 X ∂i∂j|∆proj(k, q)| [Ds]i,j = − 2V EBdG(k) k 2 2 X |∆proj(k)| = Regij(k). (S37) V EBdG(k) k Finally, we arrive at Eq. (6) in the main text.

GEOMETRIC SUPERCURRENT

We consider the gradient coupling to the order parameter in the free energy of the Hofstadter superconductor. According to Eq. (6) in the main text, we can write down the gradient term as (assuming no momentum dependence), 1 2e f ≡ F /V = |(−i ∇ − A)∆|2. (S38) grad grad 2γ ~ c where γ ≡ pµ2 + |∆|2. We now consider the variation of the free energy under the change of the gauge ﬁeld, A → A + δA. It is given as, δf j e 2e 2e = geo − (∆(i ∇ − A)∆∗ + ∆∗(−i ∇ − A)∆) = 0. (S39) δA c cγ ~ c ~ c We derive the expression of the geometric supercurrent as, e|∆|2 2e jgeo = (∇χ − A). (S40) pµ2 + |∆|2 c which is Eq. (8) in the main text. We point out that the free energy is calculated in the zero temperature limit, and the argument does not rely on the basic assumption of the Landau-Ginzburg theory, which assumes the small order parameter amplitude.