Emergent flat band and topological Kondo semimetal driven by orbital-selective correlations

Flat electronic bands are expected to show proportionally enhanced electron correlations, which may generate a plethora of novel quantum phases and unusual low-energy excitations. They are increasingly being pursued in d-electron-based systems with crystalline lattices that feature destructive electronic interference, where they are often topological. Such flat bands, though, are generically located far away from the Fermi energy, which limits their capacity to partake in the low-energy physics. Here we show that electron correlations produce emergent flat bands that are pinned to the Fermi energy. We demonstrate this effect within a Hubbard model, in the regime described by Wannier orbitals where an effective Kondo description arises through orbital-selective Mott correlations. Moreover, the correlation effect cooperates with symmetry constraints to produce a topological Kondo semimetal. Our results motivate a novel design principle for Weyl Kondo semimetals in a new setting, viz. d-electron-based materials on suitable crystal lattices, and uncover interconnections among seemingly disparate systems that may inspire fresh understandings and realizations of correlated topological effects in quantum materials and beyond.


Introduction
Certain crystalline lattices feature flat bands, via frustration caused by destructive interference in electron motion 1 , which are increasingly being explored in d-electron-based systems 2,3 .The reduced bandwidth correspondingly enhances the effect of electron correlations.In addition, such flat bands are often topologically nontrivial.As such, these systems represent a new platform to uncover novel physics for both correlation and topology as well as their interplay 4 .For example, kagome metals may host flat bands and have been the subject of considerable recent interest for realizing unusual forms of charge-density-wave order [5][6][7][8][9][10] .They have also been implicated to exhibit a type of strange metal behavior [11][12][13][14] that resembles what has been extensively studied in quantum critical heavy fermion metals 4,15,16 .
In order to strongly influence the low-energy physics, the flat bands need to be placed near the Fermi energy.However, this typically is not the case at the level of bare (noninteracting) electron bandstructure.There have been considerable recent experimental efforts to tune the bare flat bands to the vicinity of the Fermi energy.With the rare exception coming from materials search 13 , the tuning study has met with only a limited success 17,18 .Because the flat bands are associated with d-electrons, the energy scales that determine the flat-band placement (with respect to the Fermi energy) are relatively large and, as a result, it is challenging to achieve the required tuning.We are thus motivated to ask the following important questions: Can electron correlations generate emergent flat bands at the Fermi energy in d-electron-based systems?And, if so, to what extent do the resulting phases display nontrivial correlation and/or topological physics?
We address both issues in a Hubbard model in which the noninteracting limit features a topologically nontrivial flat band that is far away from the Fermi energy.Due to a well-separated hierarchy in the widths of the flat band and wide bands that it is coupled to, and through the formation of compact molecular orbitals 19 , orbital-selective Mott correlations develop 11 .We show that such orbital-selective correlations lead to emergent flat bands that are pinned to the Fermi energy.Moreover, using symmetry constraints in interacting settings 20 , which are based on Green's function eigenstates (as opposed to Bloch states [21][22][23][24][25] ), we demonstrate that the emergent flat bands lead to a topological Kondo semimetal.The latter is in the same family as Weyl Kondo semimetals that appear in topological Kondo-lattice models 26,27 and materials 28,29 of both existing and designed 20,30 heavy fermion systems.The qualitative physics is illustrated in Figs.1a-c.Importantly, our approach is based on an exact construction of molecular orbitals and an exact mapping to a heavy fermion description; this is in addition to the exact constraints that symmetry places, which will also come into our analysis.Our results motivate a design principle for the Weyl Kondo semimetals in the new setting of d-electron-based systems, and point to the realization of fractional Chern insulators 31 in transition-metal compounds.

Results
One-orbital Hubbard model on the clover lattice For a proof-of-principle demonstration, we consider a variant of the kagome lattice, the twodimensional (2D) clover lattice.As shown in Fig. 1d, it contains five sublattices per unit cell.
Leaving the details of the model to be given in the Methods and in Supplementary Note 1, we note that this lattice features a flat band (Supplementary Note 2).As a case study, the model is simplified while preserving the topological nature of the flat band; we do so by removing the C 3 symmetry of the clover lattice (Methods), leaving only a mirror symmetry M x .There is one orbital per site.The Hubbard model takes the form H = H 0 + H 1 , where H 0 is the kinetic term and H 1 represents the onsite Hubbard interaction.We consider the generic setting that has not been analyzed before, namely with the flat band of the noninteracting Hamiltonian being far away from the Fermi energy, as illustrated in Fig. 1a and shown in Fig. 2a.
The lattice can be divided into two groups of sublattices (denoted by blue and yellow dots in Fig. 1d), which contain different numbers of sites per unit cell.The flat-band formation can be seen by considering only the nearest-neighbor hopping between the blue and yellow sites, reflecting a destructive interference of the electronic wavefunction on the lattice 32 (see Supplementary Note 2).
The flat band overlaps with the wide bands.

Molecular orbitals, effective extended Hubbard model and the solution method
A flat band that is topologically nontrivial cannot by itself be represented by exponentially localized symmetry-preserving (Kramers-doublet) Wannier orbitals 25 .Such a Wannierization only becomes possible when other bands are considered along with the flat band.In addition, a flat band coming from destructive interference comprises states from multiple (inequivalent) atomic sites.If one Wannier orbital is to primarily capture this flat band, this Wannier orbital (and, by extension, the others accompanying it) must involve multiple atomic orbitals.In other words, in this case, the Wannier orbitals are necessarily molecular orbitals.We again stress that the mapping we use is exact.
In our case, we can restrict to three bands (see Supplementary Note 3).We find the centers of the three localized Kramers-doublet Wannier orbitals to be located near the geometric center of the unit cell, which forms a triangular lattice (see Fig. 1e).Importantly, one Wannier orbital primarily captures the flat band 11 ; it is the most localized and is denoted as the d orbital.The other two Wannier orbitals are dominated by the wide bands; they decay much more slowly and are marked as c orbitals.The large difference in the width of the flat band (D flat ) and the wide bands (D wide ) opens up a range of interactions that are in between.In this range, the electron correlations are strongly orbital-selective and the system affords a Kondo/Anderson-lattice model description 11 .
We project the Hubbard model of the original lattice to the Wannier basis.This leads to the effective model expressed in terms of the d and c Wannier orbitals with H eff = H 0 + H int .The kinetic term is specified in the Methods.For the interaction terms, we keep the most dominant interactions on the Wannier basis.They include the onsite Hubbard interaction among the d electrons and the density-density interactions between the d and c electrons: where n a i,σ = a † iσ a iσ , with a = d, c α , α = 1, 2, and n a i = σ n a σ .The onsite Hubbard interaction on the d-orbital, u, is the most dominant one, given the much more localized nature of this Wanner orbital.The density-density interactions between the d and c electrons, F α , are weaker but also sizable: F 1 /u ≈ 0.3 and F 2 /u ≈ 0.25.These effective interaction parameters are determined by those of the original Hubbard model (see Supplementary Note 3).The interactions among the c electrons are relatively small compared to their bandwidths and, accordingly, will be unimportant.
To take into account the effect of the interactions, we use the U(1) slave spin (SS) method 33 .
Given that only the onsite interaction of the d orbital is important, we need to introduce a SS representation for the d orbital only: where the auxiliary bosonic and fermionic operators, o † and f † σ , carry the charge and spin degrees of freedom, respectively.We treat the SS formulation at the saddle-point level and self-consistently solve the corresponding Hamiltonians for the SS and the auxiliary fermion parts.The SS method is also used to obtain the contributions to the single-electron excitations from the (interaction-driven) incoherent part of the spectrum.
The details are found in the Methods and Supplementary Note 6.

Emergent flat band at the Fermi energy
We are now in position to discuss the effect of interactions on the single-electron excitations.
Consider first the density of states (DOS).In the noninteracting case, as shown in Fig. 2a, the d electron DOS (the red curve) has a sharp peak compared with the background (purple color) c electron component.This reflects the d electrons as primarily describing the flat band.As can be seen, the peak is located far away from the Fermi level, which also reflects its origin from the noninteracting flat band (see Supplementary Note 1).
Importantly, under the influence of electron correlations, a new flat band emerges.This is demonstrated in Fig. 2b with u = 1.6.The emergent flat band is pinned to the Fermi energy, as captured by the coherent peak (the red solid lines) in the DOS.The background DOS associated with the conduction c electrons is largely unchanged from its noninteracting counterpart.Varying the interaction strongly influences the spectral weight of the emergent flat band (see Supplementary Note 7): This part of the spectral weight is reduced as the interaction increases (comparing Fig. 2b and Supplementary Fig. S5a); the reduced spectral weight is transferred to the incoherent part (the red dashed lines).This form persists until the weight of the coherent peak is completely lost and the system goes through an orbital-selective Mott transition.When that happens, the incoherent parts of the single-electron excitations develop into the full-fledged lower and upper Hubbard bands (see Supplementary Note 7 and Fig. S5b).

Orbital-selective Mott correlations
To expound the origin of the emergent flat band, we further analyze the orbital-selective Mott correlations in the regime of interactions of our interest, viz.D flat < u < D wide .As shown in Fig. 3, the metal-to-insulator transition of the d electrons occurs at u c = 2.4.We reiterate that the range of the interactions being considered here is weaker than the width of the wide bands associated with the c electrons (see Supplementary Fig. S3).Thus, the c electrons are fully itinerant.This justifies the neglect of the interactions among the c electrons, as we have done, so that the quasiparticle weight of the c-electrons remains to be 1 as seen in Fig. 3.The existence of an orbital-selective Mott transition is further illustrated in the nature of the Fermi surface.As shown in Fig. 3 (insets), the Fermi surface undergoes a dramatic change across the transition.This change parallels the electron localization-delocalization (Kondo destruction) physics of heavy fermion systems [34][35][36][37][38] .
The phase with the d-electrons being itinerant corresponds to the Kondo-screened phase, in which the local moment is converted into (fragile, or heavy) electronic excitations that hybridize with the conduction electrons to form the quasiparticles.By contrast, the orbital-selective Mott phase (OSMP) is analogous to the Kondo-destroyed phase of the heavy fermion systems, in which the Fermi surface is formed entirely from the conduction electrons.The fact that our noninteracting flat band is, to begin with, far away from the Fermi energy makes the parallel with the heavy fermion systems especially clear.Our analysis of this flat-band system provides a realization of the Kondo physics in a one-band Hubbard model that physically describes a d-electron-based system.
More specifically, the value of the interaction illustrated in Fig. 2b is marked by an arrow in Fig. 3.The differentiation between the quasiparticle weights of the d and c electrons at this interaction characterizes the orbital-selective nature of the electron correlations.This is reflected in the d-electron spectral weight: as seen in Fig. 2b, the incoherent peaks (the red dashed lines) are well formed, which corresponds to the precursor of the lower and upper Hubbard bands of the OSMP (see Supplementary Fig. S5b).The coherent spectral weight, i.e. the central peak (the red solid line of Fig. 2b) is thus described in terms of the Kondo resonance of a Kondo lattice model, in which the local moments correspond to the effective spin degrees of freedom associated with the lower and upper Hubbard bands.This description makes precise the notion that the flat band at the Fermi energy is emergent, driven by the orbital-selective Mott correlations.

Topological Kondo semimetal
The energy dispersion of the electronic states is shown in Fig. 4. From the dispersion of the interacting (u = 1.6) case, we again see that a Fermi-energy-bound flat band emerges in the interacting case.
We are then in position to analyze the symmetry constraints [21][22][23][24][25] .In the noninteracting limit, As shown in Fig. 4d, a relatively small Zeeman coupling (illustrated here with m = 0.03, which is small compared to the width of the emergent flat band of ∼ 0.1) causes a substantial separation of the nodes.These nodes now have two-fold degeneracy.
The orbital-selective Mott correlations are caused by local correlations.While we have provided a case study of how such correlations give rise to emergent flat bands in a particular 2D model, a similar conclusion is expected in general cases, including for models in three dimensions (3D).There is an important distinction though.In 3D, topological nodes develop in the presence of SOC under symmetry constraints 25,39 .For noncentrosymmetric systems, or for centrosymmetric systems with the breaking of time-reversal symmetry, we can then expect the emergent flat bands to feature Weyl nodes leading to a Weyl Kondo semimetal.
To further expound on the generality of our theoretical results, we note that the metallic regime with strong orbital-selective correlations can be viewed through the Kondo analogy.From this perspective, the emergent flat band describes low energy coherent electronic excitations associated with the Kondo-driven composite fermions.Because low energy electronic excitations are always Fermi-energy bound, and also based on the well-established understanding that Kondodriven composite fermions occur in the immediate vicinity of the Fermi energy, the emergent flat bands that develop through our proposed mechanism must be pinned near the Fermi energy.This represents a general principle.To explicate on this generality, we have mapped out a phase diagram to show that the proposed mechanism operates over an extended region in the u-ϵ 0 d parameter space (region "II" of the phase diagram given in Supplementary Note 8 and Fig. S6).Furthermore, we have carried out related calculations in a more general setting and find a similar development of an emergent flat band when the noninteracting flat band is located substantially away from the Fermi energy; the details of this analysis appear in Supplementary Note 9 and Fig. S7.

Design principle for Weyl Kondo semimetals in physical d-electron systems
Weyl Kondo semimetals have so far been explored in f -electron-based materials, Ce 3 Bi 4 Pd 3 28,29 and several newly proposed Ce-, Pr-and U-based compounds 20,30 .The present work leads us to propose a design principle for realizing Weyl Kondo semimetals in a new setting.Importantly, our theoretical results are expected to be robust against the effect of the residual interactions among the quasiparticles.This is so because the origin of the topological nodes lies in the symmetry constraints, which have recently been shown to operate on the eigenvectors of the matrix associated with the exact single-electron Green's function of an interacting system 20 .Accordingly, our proof-of-principle demonstration enables us to advance a new materials design procedure for Weyl Kondo semimetals in the setting of d-electron-based systems.The procedure would start from 3D lattices that can host flat bands from quantum interference.Examples include the pyrochlore lattice 40 , the perovskite lattices 41 , and other 3D versions of the bipartite crystalline lattices 42 .We seek materials with d-elements, and utilize orbital-selective correlations to drive interacting flat bands that are Fermi-energy-bound.Symmetry constraints can then lead to either Dirac or (with the breaking of inversion or time-reversal symmetry) Weyl nodes in these emergent flat bands.The latter case corresponds to a Weyl Kondo semimetal.
The procedure for this materials identification approach goes beyond that for Weyl Kondo semimetals in f -electron-based systems 30 .In addition to the requirement for both correlations and crystalline symmetry constraints, it also involves the crystal lattice conditions for the formation of flat bands in the bare dispersion.We reiterate that the noninteracting flat bands are not required to be near the Fermi energy.This is an important feature in the proposed materials design principle, given that the noninteracting flat bands in relevant materials are generically away from their Fermi energy.

Implications for fractional Chern insulators
Fractional Chern insulators, with a fractional quantum Hall effect and the associated fractional charge in a lattice setting, have been proposed in correlated models with an appropriate (such as 1/3) filling ratio of a flat band when the latter crosses the Fermi energy [43][44][45] .Experimental evidence has recently been identified in twisted bilayer graphene 31 , in which the moiré bands are located near the Fermi energy, in a small external magnetic field.Our results on the Fermienergy-bound emergent flat bands raise the possibility of another potential platform to realize the fractional Chern insulators, namely in d-electron-based 2D systems.Indeed, when a spin-orbit coupling is included in the 2D model, the Dirac node is gapped leading to flat Z 2 topological bands (see Supplementary Note 4).The residual interactions (which develop beyond the saddlepoint analysis in the slave-spin approach that we have carried out) could be ferromagnetic (see Supplementary Note 5).In that case, the flat band can develop a nonzero Chern number and can be analyzed for a lattice realization of fractional quantum Hall effect 46 .Indeed, the combination of the flatness of the associated bands (see Supplementary Fig. S4) and the aforementioned residual interactions among the heavy quasiparticles represents a condition that is similar to what happens in the moiré systems 31 ; however, the Z 2 nature of the flat bands makes them distinct and rare 47 .
Accordingly, with appropriate fillings, our results suggest that the corresponding d-electron-based 2D materials provide a new setting for realizing a fractional Chern insulator.The naturalness of the emergent flat band crossing the Fermi energy makes our proposal robust.Thus, this represents a promising new direction for a systematic examination.

Discussion
Our work opens a new bridge between topological flat bands and correlation physics.The interaction effect tends to localize the molecular orbital that has the most overlap with the flat band.
As a result, these molecular orbitals play the role of local moments, by analogy with the local spins of Kondo systems.Our work provides a rare non-perturbative way to address the interplay between correlations and topology effects in such flat band systems and a variety of correlated materials 48 .As such, it promises to elucidate the physics of correlated kagome transition-metal compounds 2,3,13,14 and related materials.We also expect that our analysis will inspire new understandings of the correlation effects in moiré structures, which are increasingly being viewed from a Kondo perspective [49][50][51][52][53] , as well as in other flat band systems 54 .Our work has also allowed us to advance a new materials design principle to identify Weyl Kondo semimetals in the new setting of d-electron-based systems.We expect the interconnections that our work reveals among seemingly disparate systems to inspire new realizations and understandings of correlated topological effects in a wide variety of quantum materials and structures.Finally, we note that our theoretical result for the emergent flat band is now supported by experiment: In a frustrated-lattice material, an emergent flat band has been observed by angle resolved photoemission spectroscopy at the Fermi energy, even though the ab initio noninteracting band structure predicts a flat band that is considerably away from the Fermi energy 12 .

Hubbard model on the clover lattice
The clover lattice has been discovered in real materials such as the van der Waals system Fe 5 GeTe 2 32,55 .As shown in Fig. 1d, it contains five sites in each unit cell, which are reclassified into two groups as marked by the yellow and blue colors.For an illustrative purpose, we restrict our model to have only a d z 2 orbital on each site.The case with other d-orbitals, such as d xz /d yz , have a similar realization of the geometry-induced flat bands 32 .We consider the Hubbard model written as H = H 0 + H 1 , where H 0 is the kinetic term that connects the two different groups of sublattices and H 1 represents the onsite one-orbital Hubbard interaction.We label the orbitals based on the group of sublattices to be A/B and C/D/E respectively.For each site, we consider the onsite Hubbard interaction, where η α (α = 1 ∼ 5) goes through all the five orbitals in each unit cell.The kinetic Hamiltonian is written as Here t denotes the nearest-neighbor hopping between the two sites that are connected by the solid lines shown in Fig. 1d, and µ 0 is the chemical potential.In addition, m denotes the energy splitting between the two groups of sublattices, which have a different local environment and thus generically have different energy levels.Finally, γ represents an additional energy splitting between C and D/E, which breaks the C 3 rotational symmetry.As mentioned earlier, we work with the case that breaks C 3 symmetry to simplify the symmetry characterization and, thus, the Wannier construction.It is possible that this C 3 symmetry breaking spontaneously appears as a result of interactions that drive a nematic order, although, for our illustrative purpose, we do not pursue this route specifically.A detailed analysis of the dispersion is shown in Supplementary Note 1.

Effective extended Hubbard model
We project the Hubbard model of the original lattice to the Wannier basis.This leads to the effective model expressed in terms of the d and c Wannier orbitals with H eff = H 0 + H int .The kinetic term takes the following form: Here, d † iσ (c † iασ ) creates a heavy (light) electron at the position i with spin σ (and orbital α).In addition, t ij (t αβ ij ) denotes the hopping parameter between the d (c) electrons at the positions i and j (orbitals α and β ).Moreover, V α represents the hybridization between the light orbital α and the heavy orbital.Finally, ∆ α describes the difference in the energy levels between the d orbital and c orbitals, and µ specifies the chemical potential.We will focus on the case with the d-electron level being deep below the Fermi energy and will show that the interaction effect creates a heavy band near the Fermi energy.The noninteracting dispersion is shown in Fig. 4a, where there is a Dirac crossing between the flat band and lower wide band located deep below the Fermi energy; the crossing is protected by the M x lattice symmetry 25 .

Slave spin method and self-consistent equations
We describe the U(1) slave spin approach 33 .Because the bandwidth of the heavy orbital is much smaller than those of the light orbitals, the interaction effect is most pronounced on the d orbital.
We therefore only introduce the SS representation on the d orbital: The auxiliary bosonic field o † σ = P + S + σ P − is represented by the spin operator accompanied with the projection operators suitable for a system that is away from half filling.We treat the SS formulation at the saddle-point level by fully decoupling the SS and auxiliary fermion operators.
This leads to the decoupled Hamiltonian: where and ϵ f (k) and V α k are the Fourier transforms of t ij and . Finally, we introduce the Lagrangian multiplier λ σ to remove the unphysical Hilbert space (see Supplementary Note 6).The pseudo-spin carries the U(1) charge degree of freedom; the quasiparticle weight associated with the coherent part near the Fermi level is described by Z = ⟨O σ ⟩⟨O † σ ⟩.An (orbital-selective) Mott localization transition happens when some quasiparticle weight Z goes to zero.
In addition to the coherent quasiparticle peak, the SS method also calculates the contributions from the incoherent excitations.The Green's function of the d electron G d is obtained by the convolution of the SS and the auxiliary fermions, with where H k is the hopping matrix between the two groups of sublattices.The localized wavefunction takes the following form 32 :

SUPPLEMENTARY NOTE 3: EFFECTIVE MULTI-ORBITAL HUBBARD MODEL
For the kinematic hoppings of the effective molecular orbitals, we refer to Table II in the SM of Ref. 11.However, we focus on the case such that, in the noninteracting case, the flat band lies considerably below the Fermi energy.In Fig. S3, we present the DOS for the d and c electrons in the absence of any hybridization between the two orbitals.For the interaction part, the ratio between the Hubbard interaction of the effective d electron and the one on the original lattice is u/U = 0.15 11 .For the same reason that the onsite interaction among the c-orbitals is unimportant, the Hund's coupling does not play an important role here 11

SUPPLEMENTARY NOTE 4: ANALYSIS OF THE EFFECTIVE MULTI-ORBITAL HUB-BARD MODEL WITH A SPIN-ORBIT COUPLING
In this section, we discuss the influence of the spin-orbit coupling (SOC).For our purpose, it is adequate to consider a SOC that is small compared to the other energy scales.We can then analyze the effect of the SOC perturbatively, by using the same Wannier orbitals described above.We thus project the SOC of the original lattice onto the Wannier orbital basis.We consider the SOC along the z direction of the original clover lattice, which takes the form as Here, t soc is the strength of the SOC, and σ = ± represents spin up and spin down, respectively.
The hopping parameters on the basis of the three Wannier orbitals are shown in Supplementary Because the value of the SOC is small, we use the same saddle-point parameters as we obtained in the case without the SOC.The renormalization factors Z for the different orbitals will then renormalize the SOC accordingly.
As shown in Fig. S4, the SOC generically gaps out the Dirac nodes (located along Γ − K) for our 2D model, both for the cases with and without interactions.This leads to flat Z 2 topological bands.We stress that, the flatness of the emergent bands close to the Fermi energy is robust against the SOC.

SUPPLEMENTARY NOTE 5: RESIDUAL INTERACTIONS AND THEIR COMPETITION
We have so far considered the dominant terms of the interactions that are projected into the Wannier basis: the on-site Coulomb interaction among the most localized molecular (Wannier) orbitals (d) and the other on-site interactions between the d orbitals and those of the more extended  The third type of residual interactions consists of the direct exchange among the d-electrons.
They can be constructed from the projection of the original Hubbard interaction to the Wannier basis.Although the emergent molecular orbitals are exponentially localized, even the most compact (d) orbital still extends over several lattice sites.The shared sites between the nearby molecular orbitals contribute to this ferromagnetic direct exchange interaction.These typically ferromagnetic interactions between the neighboring sites have the form of the Hund's-like two-body interactions.
We have so far focused on spin exchange interactions.The processes leading to the first and third types of residual interactions naturally lead to density-density interactions between neighboring sites as well.
In summary, the forms of the residual interactions are of the spin-spin and density-density interactions among neighboring sites.An appropriate study of the dominant residual interactions should include the above terms, and the competition between them could lead to different phases.
For example, if the antiferromagnetic interactions prevail over the ferromagnetic terms, they promote antiferromagnetic-ordering tendencies and the associated quantum criticality.If the ferromagnetic interactions dominate, they may split the Kramers degeneracy, leading to bands with a non-zero Chern number.With the help of other short-range interactions, including the short-range density-density interactions, this regime has the potential to yield fractional Chern insulating (FCI) states.
The interaction term can be expressed by the SS operators as described by Eq. 6 in the main text.
The noninteracting part of the Hamiltonian now takes the following form: We decouple the f and o at the saddle-point level with We take the single site decoupling ⟨o i o † j ⟩ ≈ ⟨o i ⟩⟨o † j ⟩, and Taylor expand o † iσ at the saddle point as follows: where We notice that ⟨o σ ⟩ = ⟨O σ ⟩.Combining the above equations and introducing the Lagrangian multipler λ σ to enforce the constraint in Eq.S4, we obtain the self-consistent equations as described in Eq. 5 of the main text.
The Green's function of the d electron is obtained by the convolution of the SS and the auxiliary fermion propagators, with After an analytical continuation, we have the spectral function In this section, we illustrate that the conclusion we have reached is applicable to other types of geometry-induced flat band systems.(For an overall discussion, see the main text.)The example presented here is a model defined on the kagome lattice, which has recently been found to display a Kondo-destruction quantum critical point 19 .Here, we consider the case when the energy dispersion in the non-interacting limit is shown in Fig. S7a, where the flat band is situated away from the Fermi energy.The dispersion for the interacting case (u = 2.1) is displayed in Fig. S7b, where once again, we observe the emergence of a flat band close to the Fermi energy.Due to the inclusion of spin-orbital coupling, the obtained solution is a topological insulator (TI), featuring a hybridization gap as depicted in the zoomed-in plot in Fig. S7c.Like in the canonical Kondo systems, the system is highly tunable.As shown in Fig. S7d the three Wannier orbitals have different M x eigenvalues.The flat d orbital has the M x eigenvalue +1, while the two c orbitals have the M x eigenvalues of −1 and +1 respectively 11 .In the presence of time-reversal symmetry, along the Γ − K line, the flat band from the d orbital has a symmetryprotected Dirac crossing with the c orbital of the opposite mirror eigenvalue.This same symmetry constraint also applies to the Kondo-driven flat band.The Dirac node for the emergent flat band is shown in Fig. 4b.A node from the flat band close to the Fermi energy allows a high tunability.

FIG. 1 .FIG. 2 .FIG. 3 .FIG. 4 .
FIG. 1. Illustration of the bare and emergent flat bands and lattice geometry.a, In the noninteracting case, a flat band (red solid line) appears far away from the Fermi energy.b, In the presence of orbitalselective correlations, an interaction-driven flat band emerges at the Fermi energy (red solid line), while leaving incoherent excitations far away from the Fermi energy (red dashed lines).c, The emergent flat band crosses a dispersive band, leading to a topological Kondo semimetal with symmetry-protected Dirac/Weyl nodes that are pinned close to the Fermi energy, within an effective Kondo energy scale.d, Geometry of the clover lattice with 5 sublattices per unit cell.The lattice does not have inversion symmetry.This can be seen from the mismatch between the (dark) blue sublattices and their inversion counterparts (dots in light blue).e, The Wannier orbitals are near the geometric centers (shaded blue circles) of the unit cells, which form a triangular lattice.
FIG. S2.Wavefunction for the electronic states of the flat band.Illustrated here is the real space representation of the electronic wavefunction for the flat band in the clover lattice.
FIG. S3.The noninteracting density of states projected to the Wannier orbitals.a,b, The DOS of the d and c electrons, respectively, in the absence of any hybridization between the two orbitals.

V 2 ia 1 +ja 2 −
07542 + i0.00449 −0.07542 − i0.00449 0.07311 + i0.00509 0.07311 − i0.00509 TABLE I. Parameters for the noninteracting Hamiltonian in the Wannier basis -part I. Shown here are the hopping parameters for the d orbitals and the hybridization between the d and c orbitals for spin up when t soc = 0.06.The parameters in the spin-down case are the Hermitian conjugate of their spin-up counterparts.

µ
FIG.S4.The electronic dispersion in the presence of the SOC.Shown here are the electronic dispersions for u = 0 (a) and for u = 1.6 (b), with the SOC in the original lattice to have a strength of t soc = 0.06.
FIG. S5.Evolution of the DOS across the orbital-selective Mott transition.a, The DOS of the d and c electrons for u = 2.2, which is just below the critical value u c = 2.4.The coherent part of the d-electron spectrum corresponds to the central peak (the red solid line), and its incoherent part appears as side peaks (the red dashed lines).The c-electron spectral function is shown in purple.b, The case of u = 2.5, which is larger than u c and, thus, the system lies inside the OSMP.The d-electrons are in a Mott insulating state, and their spectrum contains only the lower and upper Hubbard bands.
FIG. S7.Topological Kondo effect in a general setting.a, The noninteracting band structure.b, The dispersion of the coherent single-electron excitations at u = 2.1.The red solid curve denotes the emergent flat band close to the Fermi energy.The grey lines mark the incoherent single-electron excitations.c, The zoomed-in view of the emergent flat band.d, The band structure at u = 2.1 with a Zeeman splitting m z = 0.05.

Table . I
and Supplementary Table.II.

TABLE II .
Parameters for the noninteracting Hamiltonian in the Wannier basis -part II.Shown here are the hopping parameters between the c electrons for spin up when t soc = 0.06, the chemical potential