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Geometric Superconductivity in 3D Hofstadter Butterfly

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Geometric Superconductivity in 3D Hofstadter Butterfly Geometric Superconductivity in 3D Hofstadter Butterfly Moon Jip Park,1, ∗ Yong Baek Kim,2, 3, y and SungBin Lee1, z 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 = xjBj(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 ^-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 y X y 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 sys- tem, 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)cy (k)cy (−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 mi- croscopic interaction. The superconducting critical tem- perature, Tc, can be formally evaluated by solving the self-consistent gap equation[33, 34] with the supercon- ducting 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 ^-axis. In the tilted k0,λµ field, the projected two-dimensional unit cell forms a super- lattice 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 y 0 y 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 mini- bands 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 j∆je2iq·r, characterizes the superfluid behavior of 3D thex ^-direction at the minima of Ty cos(2πΦzx + ky). HBSC. The superfluid weight, Ds, can be formally eval- The effectivex ^-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].
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