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We schematically compare ordinary single-orbital, one-band case (here for a d-wave SC; leftmost column) and multi-orbital, multi-band case (here for s±; second column from left), where the nesting vectors (yellow arrows) connecting the specific “hot spots” designate how pairs (blue and orange arrows) hop. These are contrasted with flat band systems for single-orbital, one-band case (second from right) and single-orbital, multi-band case (rightmost), where yellow arrows represent pair-scattering channels. The top row depicts k-space, while the bottom row displays pairs in real space
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One novel arena for designing superconductors with high TC is the flat band system. A basic idea is that flat bands, arising from quantum mechanical interference, give unique opportunities for enhancing TC with (i) many pair-scattering channels between the dispersive and flat bands, and (ii) an even more interesting situation when the flat band is...
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... These phases are often discussed in connection to characteristic features in the electronic structure such as the VHSs or Dirac fermions, attributed to the inherent topology of the kagome lattice. However, the anticipated quantum states induced by the kagome FBs remain elusive, despite significant theoretical predictions [43][44][45][46][47][48][49][50]. A noteworthy development in this context is the discovery of Ni 3 In, which is reported to host a partial FB at the E F by DFT predictions and simultaneously exhibit non-Fermi liquid transport behaviors [51]. ...
In the quest for novel quantum states driven by topology and correlation, kagome lattice materials have garnered significant interest due to their distinctive electronic band structures, featuring flat bands (FBs) arising from the quantum destructive interference of the electronic wave function. The tuning of the FBs to the chemical potential would lead to the possibility of liberating electronic instabilities that lead to emergent electronic orders. Despite extensive studies, direct evidence of FBs tuned to the chemical potential and their participation in emergent electronic orders have been lacking in bulk quantum materials. Here using a combination of Angle-Resolved Photoemission Spectroscopy (ARPES) and Density Functional Theory (DFT), we reveal that the low-energy electronic structure of the recently discovered Cr-based kagome metal superconductor {\Cr} is dominated by a pervasive FB in close proximity to, and below the Fermi level. A comparative analysis with orbital-projected DFT and polarization dependence measurement uncovers that an orbital-selective renormalization mechanism is needed to reconcile the discrepancy with the DFT calculations, which predict the FB to appear 200 meV above the Fermi level. Furthermore, we observe the FB to shift away from the Fermi level by 20 meV in the low-temperature density wave-ordered phase, highlighting the role of the FB in the emergent electronic order. Our results reveal {\Cr} to stand out as a promising platform for further exploration into the effects of FBs near the Fermi level on kagome lattices, and their role in emergent orders in bulk quantum materials.
... Among the abundant kagome materials, AV 3 Sb 5 (A = K, Rb, and Cs) have attracted great attention due to their intriguing emergent properties 19,20 , such as the interplay/competition between superconductivity and unconventional CDW [20][21][22][23] , time-and/or rotational-symmetry broken phases [24][25][26][27][28][29][30][31] , giant anomalous Hall effect related to the chirality of the charge order 27,32 , the observation of pair-density wave 33,34 , and putative loop current order [35][36][37] . In the exploration of AV 3 Sb 5 materials, it has been a common consensus that the vHS near E F plays a key role in the CDW and possibly also in the superconductivity 38 to CsV 3 Sb 5 , the band width of the dispersive bands just above E F is strongly suppressed, leaving flat bands slightly above E F in CsCr 3 Sb 5 (Fig. 1f), which are believed to play a key role in the unconventional superconductivity 44,47,48 . in the Supplementary Information. The energy bands originated from Sb p orbitals are well captured by the calculation. ...
... Secondly, Hund's coupling induces the localization of the magnetic moments and strongly increases the imaginary part of the electron self-energy, giving rise to the incoherent Cr 3d states, which is further enhanced by the antiferromagnetic fluctuation effect, as observed in our temperature-dependent measurements. Finally, the incipient flat band is believed to be crucial for unconventional superconductivity 44,47,48 . It has been shown that the application of pressure can effectively tune the portions and width of the incipient flat bands 45 , which may be important for the pressurized superconductivity in CsCr 3 Sb 5 41 . ...
... By substituting vanadium with chromium, the sibling compound CsCr 3 Sb 5 provides an ideal platform to investigate the impact of magnetism and electronic correlation effect in the AV 3 Sb 5 -type kagome systems41 . Indeed, recent investigations reveal many novel properties of this new kagome material, including pressurized superconductivity at 6.4 K, frustrated altermagnetism, CDW-like order, and non-Fermi liquid behaviour in the normal state[41][42][43] .Compared to the non-magnetic and weakly correlated AV 3 Sb 5 , incipient flat bands (IFB, flat bands that extend in a small portion of Brillouin zone and locate away from E F )44 originated from correlated Cr 3d orbitals appear near E F , which may dominate the correlated electronic properties of the system. These novel Interestingly, we observe a flat Cr 3d band at about 80 meV below E F , confirming the existence of IFB close to E F . ...
Kagome materials exhibit many novel phenomena emerging from the interplay between lattice geometry, electronic structure, and topology. A prime example is the vanadium-based kagome materials AV3Sb5 (A = K, Rb, and Cs) with superconductivity and unconventional charge-density wave (CDW). More interestingly, the substitution of vanadium by chromium further introduces magnetism and enhances the correlation effect in CsCr3Sb5 which likewise exhibits superconductivity under pressure and competing density-wave state. Here we systematically investigate the electronic structure of CsCr3Sb5 using high-resolution angle-resolved photoemission spectroscopy (APRES) and ab-initio calculations. Overall, the measured electronic structure agrees with the theoretical calculation. Remarkably, Cr 3d orbitals exhibit incoherent electronic states and contribute to incipient flat bands close to the Fermi level. The electronic structure shows a minor change across the magnetic transition at 55 K, suggesting a weak interplay between the local magnetic moment and itinerant electrons. Furthermore, we reveal a drastic enhancement of the electron scattering rate across the magnetic transition, which is relevant to the semiconducting-like transport property of the system at high temperatures. Our results suggest that CsCr3Sb5 is a strongly correlated Hund's metal with incipient flat bands near the Fermi level, which provides an electronic basis for understanding its novel properties in comparison to the non-magnetic and weakly correlated AV3Sb5.
... We note in passing that the square-ladder lattice does not support SLPS due to N 12 1 = 1. On the other hand, the three-sublattice diamond-chain lattice [40] also hosts zero-energy SLPSs at X. Since the destructive interference applies to all three sublattices, the triple degeneracy leads to a three-band-crossing point. There are also cases where destructive interference does not apply to all sublattices. ...
We show that sublattice-polarized states (SLPSs) appear ubiquitously on the common lattices. We first establish the destructive-interference scenario for the SLPSs, which is further systematized by a point-group-symmetry interpretation. The examples on common one-, two-, and three-dimensional lattices are then demonstrated. We also deduce the symmetry-protected robustness of SLPSs against further-neighbor hoppings. Moreover, the destructive-interference scenario can be generalized to the multi-SLP. The important effects on interaction-driven phases are briefly discussed.
... The study of flat band (FB) physics has come up as one of most intriguing field of research in condensed matter physics and material science in recent years [1][2][3]. FBs offer highly degenerate manifold of single particle states and provide an ideal platform to study a wide range of strongly correlated phenomena, such as ferromagnetism [4][5][6][7], superconductivity [8][9][10], quantum Hall effect [11][12][13], and inverse Anderson transition [14,15], to name a few of them. In tight-binding lattice models, FB states originate from destructive quantum interference of electron hoppings between neighboring sites. ...
We present the Stagome lattice, a variant of the Kagome lattice, where one can make any of the bands completely flat by tuning an externally controllable magnetic flux. This systematically allows the energy of the flat band (FB) to coincide with the Fermi level. We have analytically calculated the compact localized states (CLS) associated to each of these flat bands appearing at different values of the magnetic flux. We also show that, this model features nontrivial topological properties with distinct integer values of the Chern numbers as a function of the magnetic flux. We argue that this mechanism for making any of the bands exactly flat could be of interest to address the FB superconductivity in such a system. Furthermore, we believe that the phenomenon of photonic flat band localization could be studied in the Stagome lattice structure, designed for instance using femtosecond laser induced single-mode waveguide arrays.
... Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. rise to novel phenomena [7,[14][15][16][17][18][19][20]. Specifically, the discovery of superconductivity in twisted bilayer graphene in 2018 fueled considerable attention in the field [7]. ...
... The electronic bands obtained from the HamiltonianĤ T (dashed black, blue, and red lines) are compared with the low-energy approximation using the Hamiltonian H ξ,η (solid black, blue, and red lines) for (a) UCHG(5, 5, 3), c(5, 3) = 6 5 , |E 1 (κ ξ,η )| ≈ 0.5680 eV, |a 2 (5, 3)| ≈ 0.0903 eV nm, (b) UCHG(7,7,3), c(7, 3) = 6 5 , |E 1 (κ ξ,η )| ≈ 0.3780 eV, |a 2 (7, 3)| ≈ 0.1031 eV nm, (c) UCHG(9, 9, 3), c(9, 3) = 7 5 , |E 1 (κ ξ,η )| = 0 eV, |a 2 (9, 3)| = 0 eV nm, (d) UCHG(7, 7, 5.2), c(7, 5.2) = 3 4 , |E 1 (κ ξ,η )| ≈ 0.4209 eV, |a 2 (7, 5.2)| ≈ 0.1739 eV nm and (f) UCHG(9, 9, 5.2), c(9, 5.2) =6 5 , |E 1 (κ ξ,η )| = 0 eV, |a 2 (9, 5.2)| = 0 eV nm. See equations (11)-(16). An excellent agreement between the low energy Hamiltonian and the tight-binding Hamiltonian is seen. ...
textit{Holey Graphene} (HG) is a widely used graphene material for the synthesis of high-purity and highly crystalline materials. The electronic properties of a periodic distribution of lattice holes are explored here, demonstrating the emergence of flat bands. It is established that such flat bands arise as a consequence of an induced sublattice site imbalance, i.e., by having more sites in one of the graphene's bipartite sublattice than in the other. This is equivalent to the breaking of a path-exchange symmetry. By further breaking the inversion symmetry, gaps and a nonzero Berry curvature are induced, leading to topological bands. In particular, the folding of the Dirac cones from the hexagonal Brillouin zone (BZ) to the holey superlattice rectangular BZ of HG, with sizes proportional to an integer $n$ times the graphene's lattice parameter, leads to a periodicity in the gap formation such that $n \equiv 0$ (mod $3$). A low-energy hamiltonian for the three central bands is also obtained revealing that the system behaves as an effective $\alpha-\mathcal{T}_{3}$ graphene material. Therefore, a simple protocol is presented here that allows for obtaining flat bands at will. Such bands are known to increase electron-electron correlation effects. Therefore, the present work provides an alternative system that is much easier to build than twisted systems, allowing for the production of flat bands and potentially highly correlated quantum phases.
... In fact, the bilayer Hubbard model has been widely studied from the past [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21], and sAE-wave superconductivity [22] is found to be strongly enhanced near half filling when the vertical electron hopping (t ⊥ ) between the layers is several times larger than the in-plane hopping, and the Fermi level (E F ) lies in the vicinity of the edge of one of the bands [4][5][6][15][16][17][18][19]. Nowadays, a band whose edge lies just below or above E F is often referred to as an incipient band, and has attracted interest in the study of iron-based superconductors [24][25][26][27][28][29][30][31], bilayer and ladder-type lattices [17][18][19][20][32][33][34][35][36], and flat band superconductivity [37][38][39][40][41]. ...
Inspired by a recent experiment showing that La3Ni2O7 exhibits high Tc superconductivity under high pressure, we theoretically revisit the possibility of superconductivity in this material. We find that superconductivity can take place, which is somewhat similar to that of the bilayer Hubbard model consisting of the Ni 3d3z2−r2 orbitals. Although the coupling with the 3dx2−y2 orbitals degrades superconductivity, Tc can still be high enough to understand the experiment thanks to the very high Tc reached in the bilayer Hubbard model.
... Indeed, phenomena such as ferromagnetism [1][2][3][4], superconductivity [5][6][7][8] and Wigner crystallization [9][10][11] are extensively studied within these systems. Not only observable in materials with Kagome-type structures [12][13][14][15][16][17] or twisted bilayer graphene [18][19][20][21][22], flat bands can also be engineered in artificial atomic systems, as represented photonic lattices [23][24][25][26][27][28][29]. ...
In this study, we investigate various geometric aspects of a photonic hexagonal lattice made of triple-leg stripline resonators (TSRs) in a circuit QED system. The inherent two-fold degenerate spatial modes of the TSR act as two distinct orbitals in our 2D lattice system. Remarkably the energy spectra of the system exhibits the dispersive quadratic band-touching to the top and bottom flat bands. Our analysis reveals how the system harnesses destructive interference to establish flat bands via stabilized compact localized states (CLSs). We further explore the real-space topology corresponding to the flat bands by finding proper non-contractible loop states (NLSs). Additionally, in a zigzag-structured hexagonal lattice, we demonstrate the induction of topological flat edge modes at zero energy by analyzing the Zak phase. We also elucidate the quantum geometric origin of other dispersive edge bands that arise from the singular point of the flat band.
... In recent years, the concept of electronic flat bands has gained prominence in the fields of materials science and condensed matter physics [1][2][3][4][5][6][7][8][9][10]. This heightened interest can be attributed to the electronic, geometrical and topological properties inherent to flat bands [10][11][12][13] and their potential to give rise to novel phenomena [7,[14][15][16][17][18][19][20]. Specifically, the discovery of superconductivity in twisted bilayer graphene (TBLG) in 2018 fueled considerable attention in the field [7]. ...
Holey Graphene (HG) is a widely used graphene material for the synthesis of high-purity and highly crystalline materials. In this work, we explore the electronic properties of a periodic distribution of lattice holes, demonstrating the emergence of flat bands with compact localized states. It is shown that the holes break the bipartite sublattice and inversion symmetries, inducing gaps and a nonzero Berry curvature. Moreover, the folding of the Dirac cones from the hexagonal Brillouin zone (BZ) to the holey superlattice rectangular BZ of HG with sizes proportional to an integer n times the graphene's lattice parameter leads to a periodicity in the gap formation such that n ≡ 0 (mod 3). Meanwhile, it is shown that if n ≡ ±1 (mod 3), a gap emerges where Dirac points are folded along the Γ − X path. The low-energy hamiltonian for the three central bands is also obtained, revealing that the system behaves as an effective α − T 3 graphene material. Therefore, a simple protocol is presented here that allows obtaining flat bands at will. Such bands are known to increase electron-electron correlated effects. This work provides an alternative system, much easier to build than twisted systems, to obtain highly correlated quantum phases.
... This connectivity supports 3D CLSs (Fig. 1c) and flat bands that are dispersionless in all spatial directions (Fig. 1g) (mathematically, the pyrochlore is the 3D line graph of the diamond lattice 22 ). The pyrochlore system has attracted attention as a host for various phenomena, including magnetism, superconductivity, topology and correlation physics 12,14,15,22,[26][27][28][29][30][31][32][33][34][35] . Furthermore, within the framework of the single-orbital isotropic band model, this system has been predicted to support a doubly-degenerate flat band and Dirac nodal lines 15 . ...
... The flat bands in CaNi 2 are located well below E F , making them detectable by ARPES, but tempering possible effects on low-energy properties such as thermodynamics and transport quantities. The ability to push the flat bands closer to E F such that one is partially filled would open the possibility of realizing instabilities towards symmetry breaking and exotic phases as noted in earlier theoretical studies 26,28,37 . To investigate the tunability of the pyrochlore flat bands, we synthesized single crystals of the isostructural C15 Laves phase Ca(Rh 1−x Ru x ) 2 . ...
... Further isolation of the flat band within the d-electron spectrum could potentially be achieved by identifying material structures with stronger crystal electric-field effects. At the same time, recent work has predicted that non-isolated flat bands may host unique correlated phenomena, including robust multi-band 26 and chiral 13 superconductivity and may further enable Kondo behaviour observed in other d-electron flat-band systems 30 . These examples suggest that a broad array of exotica may potentially be enabled within this platform. ...
Electronic flat-band materials host quantum states characterized by a quenched kinetic energy. These flat bands are often conducive to enhanced electron correlation effects and emergent quantum phases of matter¹. Long studied in theoretical models2–4, these systems have received renewed interest after their experimental realization in van der Waals heterostructures5,6 and quasi-two-dimensional (2D) crystalline materials7,8. An outstanding experimental question is if such flat bands can be realized in three-dimensional (3D) networks, potentially enabling new materials platforms9,10 and phenomena11–13. Here we investigate the C15 Laves phase metal CaNi2, which contains a nickel pyrochlore lattice predicted at a model network level to host a doubly-degenerate, topological flat band arising from 3D destructive interference of electronic hopping14,15. Using angle-resolved photoemission spectroscopy, we observe a band with vanishing dispersion across the full 3D Brillouin zone that we identify with the pyrochlore flat band as well as two additional flat bands that we show arise from multi-orbital interference of Ni d-electrons. Furthermore, we demonstrate chemical tuning of the flat-band manifold to the Fermi level that coincides with enhanced electronic correlations and the appearance of superconductivity. Extending the notion of intrinsic band flatness from 2D to 3D, this provides a potential pathway to correlated behaviour predicted for higher-dimensional flat-band systems ranging from tunable topological¹⁵ to fractionalized phases¹⁶.
... Lai et al [12] suggest that gold-doped lead apatite may have stronger effects than Cu. Griffin [13], Si and Held [14], and Kurleta et al [15] argue that the flat bands might boost electron-phonon mediated superconductivity; Si and Held [14] also suggest purely electronic flat-band superconductivity [19][20][21] as a possible alternative. ...
We briefly review the status quo of research on the putative superconductor Pb9Cu(PO4)6O also known as LK-99. Further, we provide ab initio derived tight-binding parameters for a two- and five-band model, and solve these in dynamical-mean-field theory. The interaction-to-bandwidth ratio makes LK-99 a Mott or charge transfer insulator. Electron or hole doping (which is different from substituting Pb by Cu and thus differs from LK-99) is required to make it metallic and potentially superconducting.
