#  Coupled topological flat and wide bands: Quasiparticle formation and
destruction

Haoyu Hu<EMAIL_ADDRESS>Department of Physics & Astronomy, Rice Center for
Quantum Materials, Rice University, Houston, Texas 77005, USA  Qimiao Si
<EMAIL_ADDRESS>Department of Physics & Astronomy, Rice Center for Quantum
Materials, Rice University, Houston, Texas 77005, USA

###### Abstract

Flat bands amplify correlation effects and are of extensive current interest.
When induced by destructive interference such as on kagome lattices, they
provide a platform to explore both topology in correlated settings and
correlation physics enriched by topology. Recent experiments in correlated
kagome metals have found evidence for strange-metal behavior. A major
theoretical challenge is to study the effect of local Coulomb repulsion when
the band topology obstructs a real-space description. We study coupled
topological flat and wide bands in a variant to the kagome lattice.
Exponentially localized and Kramers-doublet Wannier functions are constructed,
which lead to a Kondo lattice description. We identify an orbital-selective
Mott transition from the effective model, and characterize the Kondo-driven
quasiparticles and their destruction. Our work provides a conceptual framework
to address the strange-metal behavior in kagome metals and beyond.

Introduction:  In flat electronic bands, Coulomb interactions are
proportionally enhanced due to their reduced kinetic energy. As such, flat-
band systems serve as a platform for strong correlation physics Paschen and Si
(2021). Heavy fermion metals represent a canonical case of flat bands formed
from highly localized atomic orbitals, and indeed they display rich
correlation physics such as quantum criticality, strange metallicity and
unconventional superconductivity Coleman and Schofield (2005); Kirchner  _et
al._ (2020). Another case, emerging in a growing list of materials,
corresponds to flat bands formed by geometrical interference Mielke (1991); Ye
_et al._ (2018); Yao  _et al._ (2018); Kang  _et al._ (2020); such bands are
often topological. These materials represent a playground to study the
interplay between strong correlations and topology, and have been found to
show unusual properties such as exotic forms of charge-density-wave order
Jiang  _et al._ (2021); III  _et al._ (2022); Zhou and Wang (2021); Teng  _et
al._ (2022); Yin  _et al._ (2022); Setty  _et al._ (2022).

Figure 1: Lattice geometry and qualitative phase diagram. (A) Lattice
geometry. $A,B,C,D,E$ mark the five sites per unit cell. (B) The Wannier
orbitals we construct, which form a triangular lattice (the orange dots). (C)
Illustration of the zero-temperature phase diagram that we determine, for the
Hubbard interaction in between the widths of the flat and wide bands.

Recently, experimental evidence for strange-metal behavior has emerged from
correlated kagome metals Ye  _et al._ (2021); Ekahana  _et al._ (2022). The
observed behavior takes the form of a $T$-linear resistivity or a single-
particle damping rate that is linear in frequency. Strange metals are often
connected with a type of quantum criticality in which the Landau
quasiparticles are destroyed Si  _et al._ (2001); Coleman  _et al._ (2001);
Senthil  _et al._ (2004). In correlated kagome metals, a defining
characteristic is that topological flat bands are coupled to wide bands. Given
the hierarchy in the bandwidths, one might suspect that such systems act as
the analogue of the multiple $3d$ or $4d$ orbitals (not too far from the Fermi
energy) in transition-metal compounds. There, when the orbitals exhibit
different dispersions and yet are kinetically coupled with each other,
orbital-selective Mott behavior is known to develop Huang  _et al._ (2022); Yu
_et al._ (2021). For such coupled-orbital models (and in contrast to models
with fully-decoupled orbitals Anisimov  _et al._ (2002)), the quasiparticles
of the strongly renormalized bands in the metallic regime near the orbital-
selective Mott phase develop in a way akin to how the Kondo effect drives
heavy quasiparticles Hu  _et al._ (2022). However, the topological nature of
the flat bands makes it difficult to describe them in terms of any localized
orbitals. The band topology obstructs the formulation of exponentially
localized and Kramers-doublet Wannier orbitals Marzari  _et al._ (2012). This
poses a challenge for treating the effect of sizable spatially-local Coulomb
repulsion, as well for connecting the behavior of the coupled topological-
flat-wide bands to the orbital-selective Mott physics.

We are thus motivated to study how quasiparticles are formed and,
consecutively, how they are destroyed in a coupled topological flat-wide band
system. To make progress, we consider a lattice with a lower symmetry, which
retains the central issue of topological obstruction while allowing – in a
particularly transparent way – for the construction of exponentially localized
and Kramers-doublet Wannier orbitals. The lattice, as illustrated in Fig. 1A,
is a variant of the kagome lattice. We are able to construct the Wannier
centers, which turn out to form a triangular lattice (Fig. 1B). When the
effective local Coulomb repulsion is larger than the width of the flat band
($D_{\rm flat}$) but smaller than that of the wide band ($D_{\rm wide}$), we
identify a continuous orbital-selective Mott transition [a quantum critical
point (QCP)] (Fig. 1C). The two involved ground states respectively feature a
“large” and a “small” Fermi surface (FS); in an effective Kondo-lattice
formulation that we construct, the large-Fermi surface is Kondo-driven,
whereas the small Fermi surface comes from Kondo destruction Si  _et al._
(2001); Coleman  _et al._ (2001); Senthil  _et al._ (2004). By analogy with
the phase diagram of heavy fermion metals Paschen and Si (2021); Coleman and
Schofield (2005); Kirchner  _et al._ (2020), the existence of the phases with
large and small Fermi surfaces Paschen  _et al._ (2004); Shishido  _et al._
(2005); Park  _et al._ (2006) open up a regime of amplified quantum
fluctuations Si  _et al._ (2001) for beyond-Landau quantum criticality and the
accompanying strange-metal behavior Prochaska  _et al._ (2020). Thus, our work
provides a conceptual framework to address the aforementioned strange-metal
behavior in the flat-band-based metals. The connection with a Kondo lattice
description is also being pursued in moiré systems Ramires and Lado (2021);
Song and Bernevig (2022); Kumar  _et al._ (2022); Guerci  _et al._ (2022). The
approach taken here is expected to shed new light there as well.

Hubbard Model in the Original Lattice: The lattice contains five sublattices
marked as A, B, C, D, E (Fig. 1A). The model is written as
$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{1}$, which contains the on-site
Hubbard interactions,

$\displaystyle\mathcal{H}_{1}={U}\sum_{{\bm{r}},i}n^{\eta}_{{\bm{r}},i,\uparrow}n^{\eta}_{{\bm{r}},i,\downarrow}\,,$
(1)

as well as the noninteracting Hamiltonian that is written in the tight binding
form,

$\displaystyle\mathcal{H}_{0}$ $\displaystyle=$
$\displaystyle\sum_{\langle{\bm{r}},{\bm{r}}^{\prime}\rangle,i,j,\sigma}t\eta_{{\bm{r}},i,\sigma}^{\dagger}\eta_{{\bm{r}},j,\sigma}-\mu\sum_{{\bm{r}},i,\sigma}\eta_{{\bm{r}},i,\sigma}^{\dagger}\eta_{{\bm{r}},i,\sigma}$
$\displaystyle+\sum_{{\bm{r}},\sigma,i\in\\{C,D,E\\}}m\eta_{{\bm{r}},i,\sigma}^{\dagger}\eta_{{\bm{r}},i,\sigma}+\sum_{{\bm{r}},\sigma,i\in\\{D,E\\}}\gamma\eta_{{\bm{r}},i,\sigma}^{\dagger}\eta_{{\bm{r}},i,\sigma}\,.$

Here, $\eta_{{\bm{r}},i,\sigma}^{\dagger}$ creates an electron at site
${\bm{r}}$, sublattice $i\in\\{A,B,C,D,E\\}$ with spin $\sigma$. In addition,
$t$ denotes the strength of nearest-neighbor hopping between two sites that
are connected by a solid line in Fig. 1A, and $\mu$ is the chemical potential.
Finally, $m$ denotes the energy splitting between sublattices C, D, E and
sublattices A, B, while $\gamma$ represents an additional energy splitting
between sublattice C and sublattices D, E. We consider the case when the flat
band is close to the Fermi energy.

At $\gamma=0$, the system has a $C_{3z}$ rotational symmetry, an $M_{x}$
mirror symmetry and also an $SU(2)$ spin rotational symmetry. This leads to a
purely flat band as shown in the supplementary material (SM) (Fig. S1B). There
is a quadratic band touching between the flat and a dispersive band at the
center of the Brillouin zone, $\Gamma=(0,0)$. The crossing is protected by
both the $M_{x}$ and $C_{3z}$ symmetry.

Figure 2: Noninteracting bandstructure. The full bandstructure (A) and zoomed-
in bandstructure (B) at $\gamma/t=-0.1,\mu/t=1.2$. In panel (B), we label the
$M_{x}$-symmetry eigenvalues of the band at $M_{x}$-invariant momenta:
$\Gamma=(0,0),K=(0,4\pi/3)$.

Our focus is on the case of a non-zero $\gamma$. Here, the $C_{3z}$ is broken
and the middle band is no longer purely flat, as seen in Figs. 2A, B. However,
for small $\gamma$ (which illustrate for $\gamma=-0.1t$), the middle band
remains relatively flat (the bandwidth is about $0.06t$). Thus, we still call
it a flat band. Near $\Gamma$, there is now a linear crossing between the flat
and dispersive bands along the $\Gamma$-$K$ line. The node is located near the
$\Gamma$ point and is protected by the $M_{x}$ symmetry. When the spin-orbit
coupling is further included (see the SM, Fig. S1E), such a node would be
gapped out and the flat band acquires $\pm 1$ spin Chern number.

In this original lattice basis, the flat band comes from the linear
superposition of the atomic states located in the five different sublattices.
A single-atomic representation of the flat band is needed to make progress,
which we now turn to.

Construction of Wannier Orbitals: We will consider the Hubbard interaction $U$
that is large compared to the width of the flat band, but small compared to
the width of the wide bands (Fig. 1C). Such a range is of interest to the
experimentally-studied systems mentioned earlier. Since there is an energy gap
between the top three bands and the bottom two, we focus on the top three
bands that include the flat band.

To express a band in terms of exponentially localized and Kramers-doublet
Wannier orbitals, the Bloch states must have the same $M_{x}$-symmetry
eigenvalue at all high symmetry points Cano and Bradlyn (2021). In our case,
the topological obstruction can be recognized by noticing that the different
high symmetry points of the flat band have different eigenvalues of the
$M_{x}$-symmetry. As shown in the SM (band $3$ in Table I), and illustrated in
Fig. 2B, the eigenvalue changes from $+1$ at $K$ and $M^{\prime}$ to $-1$ at
$\Gamma$. A similar changes of $M_{x}$-symmetry eigenvalues are also observed
for the two wide bands (bands $1$ and $2$ in Table I, SM). We therefore
consider a combination of Bloch states so that the $+1$-eigenvalue portion of
the flat band combines with the $+1$-eigenvalue portion of the wide band (band
$2$, Table I, SM, and illustrated in Fig. 2).

Through explicit construction, we indeed show the validity of the procedure.
We find the exponentially localized Wannier functions, which are located on a
triangular lattice (Fig. 1B), with the $M_{x}$-symmetry eigenvalue being $+1$.
We call such a Wannier function a $d$ orbital. The hopping integral of the two
neighboring $d$ orbitals is indeed weak in our construction: It turns out to
be on the order of $0.03t$.

The remaining degrees of freedom can be represented by two Wannier orbitals
with $M_{x}$-symmetry eigenvalue $+1$,$-1$ respectively. These two orbitals
mainly come from the top two dispersive bands, and we label them as $c$
orbitals.

Effective Multi-orbital Hubbard Model on the Triangular Lattice: We have thus
succeeded in constructing a single atomic state per unit cell to represent the
flat band. The model, expressed in terms of the $d$ and $c$ orbitals, contains
the kinetic term $H_{0}$ and the interaction term $H_{I}$.

The kinetic term, $H_{0}$, is given in detail in the SM, Eq. S3. It involves
two types of electrons, with $d_{{\bm{R}},\sigma}^{\dagger}$ creating a
$d$-electron (electron in the orbital $d$), which represents the flat-band’s
degrees of freedom, at site ${\bm{R}}$ with spin $\sigma$. Similarly,
$c_{{\bm{R}},a,\sigma}^{\dagger}$ creates a conduction $c$-electron(electron
in orbital $c$), which represents the itinerant degrees of freedom, at site
${\bm{R}}$ with spin $\sigma$ and orbital $a=1,2$. $H_{0}$ contains the part
$H_{d}$, describing the energy level ($E_{d}$) and their negligibly small
hopping parameters; the part $H_{c}$, describing the inter-$c$-electron
hopping parameters, $t^{c}_{{\bm{R}}-{\bm{R}}^{\prime},aa^{\prime},\sigma}$
and the energy levels $E_{a}$ ($a=1,2$); and the part $H_{dc}$, the $d$-$c$
hybridization term with matrix element
$V_{{\bm{R}}-{\bm{R}}^{\prime},a,\sigma}$. These parameters are specified in
the SM, Tab. 2. As is seen there, the dominant hybridization is with one of
the two conduction electron bands, $a=1$. The corresponding hybridization is
off-site because $c_{{\bm{R}},1,\sigma}$ is mirror odd.

The most important interactions here include the Hubbard interactions of the
$d$ electrons ($H_{u}$), and the Hund’s coupling between the $d$ and $c$
electrons ($H_{Hund}$). The interactions are labeled by $u$ and $J_{1,2}$,
respectively. The specific forms of the interactions are given in the SM, Eq.
S4.

Mapping to a Kondo Lattice Model: In the limit of
$u\gg|t_{{\bm{R}}-{\bm{R}}^{\prime}}^{d}|$, the Hubbard interactions suppress
the charge fluctuations of the $d$ electrons and turn them into quantum spins.
By integrating out the high energy degrees of freedom (in the presence of the
Hund’s coupling Ding  _et al._ (2019); Meetei  _et al._ (2013)), we reach a
Kondo-Heisenberg model with the Hund’s coupling. The effective Hamiltonian is

$\displaystyle H_{KH}$ $\displaystyle=$ $\displaystyle
H_{c}+H_{H}+H_{K}+H_{Hund}$ $\displaystyle H_{H}$ $\displaystyle=$
$\displaystyle\sum_{{\bm{R}},{\bm{R}}^{\prime}}J^{H}_{{\bm{R}},{\bm{R}}^{\prime}}\bm{S}_{{\bm{R}}}\cdot\bm{S}_{{\bm{R}}^{\prime}}$
$\displaystyle H_{K}$ $\displaystyle=$
$\displaystyle\sum_{{\bm{R}},{\bm{R}}_{1},a_{1},{\bm{R}}_{2},a_{2}}J^{K}_{{\bm{R}},{\bm{R}}_{1}a_{1},{\bm{R}}_{2}a_{2}}\bm{S}_{{\bm{R}}}\cdot
S^{c}_{{\bm{R}}_{1}a_{1},{\bm{R}}_{2}a_{2}}\,.$

Here, $S_{\bm{R}}$ is the emergent spin-$1/2$ local moment formed by the
localized $d$ electrons, and $\bm{S}^{c}$ is the corresponding spin of the
conduction electrons. In the large $u$ limit, the hoppings of $d$ electrons
induce a Heisenberg interactions of strength
$J^{H}_{{\bm{R}},{\bm{R}}^{\prime}}=2|t_{{\bm{R}},{\bm{R}}^{\prime}}^{d}|^{2}/\widetilde{u}$,
where $\widetilde{u}=4u\phi_{0}/3$ with $\phi_{0}$ being of order unity (see
the SM, section on the “Kondo model”). The non-local hybridization terms
between the $d$ and $c$ electrons lead to non-local Kondo couplings of
strength
$J^{K}_{{\bm{R}},{\bm{R}}_{1},{\bm{R}}_{2},a_{1},a_{2}}=4V_{{\bm{R}}-{\bm{R}}_{2},a_{2}}^{*}V_{{\bm{R}}-{\bm{R}}_{1},a_{1}}/\widetilde{u}$.
Here, the local moment of the $d$ electron is Kondo coupled to a spin operator
$\bm{S}^{c}_{{\bm{R}}_{1}a_{1},{\bm{R}}_{2}a_{2}}=\frac{1}{2}c_{{\bm{R}}_{1}a_{1}}^{\dagger}\bm{\sigma}c_{{\bm{R}}_{2}a_{2}}$
that are formed by two electron operators from the sites $R_{1},R_{2}$ and the
bands $a_{1},a_{2}$ respectively. Finally, the Hamiltonians for the $c$
electrons and for the Hund’s coupling remain unchanged from those given in the
effective multi-orbital Hubbard model. In particular, the Hund’s coupling acts
between the $d$ and $c$ electrons of the same site (see the SM).

Competition between magnetic fluctuations and Kondo effect For convenience in
notation, we now treat $\widetilde{u}=4u\phi_{0}/3$ as our tuning parameter,
but we note that tuning $\widetilde{u}$ is equivalent to varying the original
Hubbard interaction $U$. At smaller $\widetilde{u}=4u\phi_{0}/3$, the Kondo
effect dominants. Increasing $\widetilde{u}=4u\phi_{0}/3$ enhances the
relative effect of the antiferromagnetic exchange interactions between the
local moments, which favors the correlations between the local moments and is
detrimental to the development of the Kondo effect. A more detailed analysis
of this competition is described in the SM.

Large-$N$ limit and the phase diagram To observe the quantum phase transition,
we introduce the psuedo-fermion representation of the spin operators
$\bm{S}_{{\bm{R}}}=\frac{1}{2}\sum_{\sigma,\sigma^{\prime}}f_{{\bm{R}},\sigma}^{\dagger}\bm{\sigma}_{\sigma,\sigma^{\prime}}f_{{\bm{R}},\sigma^{\prime}}$
and solve the model in the large-$N$ limit (with a generalization of $SU(2)$
to $SU(N)$ and a suitable rescaling of the coupling constants in terms of
$1/N$; see the SM). The ground state is characterized by the bond fields
$\chi_{{\bm{R}},{\bm{R}}^{\prime}}$ Affleck and Marston (1988) and the
hybridization fields $\zeta_{{\bm{R}},{\bm{R}}^{\prime}a}$:

$\displaystyle\chi_{{\bm{R}},{\bm{R}}^{\prime}}=\frac{1}{N}\sum_{\sigma}\langle
f_{{\bm{R}},\sigma}^{\dagger}f_{{\bm{R}}^{\prime},\sigma}\rangle$
$\displaystyle\zeta_{{\bm{R}},{\bm{R}}^{\prime}a}=\frac{1}{N}\sum_{\sigma}\langle
f_{{\bm{R}},\sigma}^{\dagger}c_{{\bm{R}}^{\prime},a,\sigma}\rangle\,,$ (4)

and their maximum amplitudes
$\zeta=\text{max}_{{\bm{R}},{\bm{R}}^{\prime},a}\\{|\zeta_{{\bm{R}},{\bm{R}}^{\prime}a}|\\}$
and
$\chi=\text{max}_{{\bm{R}},{\bm{R}}^{\prime}}\\{|\chi_{{\bm{R}},{\bm{R}}^{\prime}}|\\}$.
We illustrate the phase diagram in Fig. 3.

Figure 3:  The zero-temperature phase diagram calculated in the large-$N$
limit. Here, $\chi=\text{max}\\{|\chi_{{\bm{R}},{\bm{R}}^{\prime}}|\\}$
denotes the largest amplitude of the bond operators.
$\zeta=\text{max}\\{|\zeta_{{\bm{R}}a_{1}}|\\}$ denotes the largest amplitude
of the hybridization fields. As the interaction $\widetilde{u}$ is increased,
$\eta$ gradually decrease and goes to zero at $\widetilde{u}=0.35$ (marked by
the red dot), which signals a continuous orbital-selective Mott transition
between a Kondo-driven and Kondo-destroyed phases, with a large and a small
Fermi surface respectively.

At small $\tilde{u}$, we realize a heavy Fermi liquid phase via the
condensation of the hybridization field $\zeta$. The pseudo-fermions, ${f}$,
are then endowed with a $U(1)$ charge in the presence of the nonzero
hybridization $\zeta_{{\bm{R}}_{1}a_{2}}$. They are thus converted into
composite electrons, which hybridize with the conduction electrons. The
composite electrons unambiguously carry a $+1$ $M_{x}$-symmetry eigenvalue,
which is the opposite to the eigenvalue of the dominantly hybridizing
conducting $c_{1}$ band. Thus, the renormalized heavy bands contain a Dirac
node, as shown in Fig. 4A. The node is protected by $M_{x}$ and the $SU(2)$
symmetry: Because of the formation of the hybridized bands, the Fermi surface
encloses the electrons residing on both the conduction and flat bands. This is
the large Fermi surface, which is shown in Fig. 4B.

Increasing $\widetilde{u}$ leads to a Kondo destruction Si  _et al._ (2001);
Coleman  _et al._ (2001); Senthil  _et al._ (2004): The local moments couple
to each other, and they no longer form a Kondo singlet with the spins of the
conduction electrons. Consequently, the Fermi surface is derived from the
dispersion of the conduction electrons, plotted in Fig. 4C. It is shown in
Fig. 4D, and is small in the sense that Fermi surface encloses only the
conduction electrons. The frustrated nature of the effective lattice (Fig. 1B)
has led to the Kondo-destroyed phase via the bond variables
$\chi_{{\bm{R}},{\bm{R}}^{\prime}}$. However, the physical mechanism leading
to the Kondo destruction is more universal, as characterized by a global phase
diagram for the competition between the Kondo and RKKY/Heisenberg interactions
Paschen and Si (2021).

Figure 4:  Quantum phases with large and small Fermi surfaces. Bandstructure
(A) and the Fermi surface (B) in the phase with a large Fermi surface. Shown
here are the results at $\tilde{u}=0.27$. For a larger value of
$\tilde{u}=0.35$, the corresponding bandstructure (C) and the Fermi surface
(D) are shown for the phase with a small Fermi surface. In panel (A) and panel
(B), the brown dot marks the Dirac nodes. The dashed line in panel (A) and its
counterpart in panel (C) mark the Femi energy. The red lines in panel (B) and
panel (D) denote the Fermi surface.

We close with several important remarks. Firstly, our Wannier construction
procedure also works equally well for the case with a spin-orbit coupling.
Accordingly, our Kondo-lattice construction and the results for the
quasiparticle formation and destruction readily apply in this case. Secondly,
by emphasizing the interplay and coupling between the flat and wide bands, our
work brings out a new degree of analogies between the correlated quantum
materials that host topological flat-bands Ye  _et al._ (2018); Yao  _et al._
(2018); Kang  _et al._ (2020); Teng  _et al._ (2022); Yin  _et al._ (2022);
Setty  _et al._ (2022) and moiré systems Balents  _et al._ (2020); Ramires and
Lado (2021); Song and Bernevig (2021); Kumar  _et al._ (2022); Guerci  _et
al._ (2022), in which the strange-metal behavior has also been indicated Jaoui
_et al._ (2022). Compared to the topology of the graphene-based moiré flat
bands Song and Bernevig (2021); Po  _et al._ (2019), the bands in our case are
also topological but are simpler. This has been crucial in allowing us to
provide a proof-of-principle demonstration of both the emergence and
destruction of proper quasiparticles in systems with coupled topological flat-
wide bands through the Kondo route. Our explicit construction of the Kondo
lattice also sets the stage for the study of a global quantum phase diagram
Paschen and Si (2021) in such topological flat-band-anchored systems.

In conclusion, we have advanced a realistic model to study the effect of local
Coulomb repulsion for a system of coupled topological flat and wide bands. By
constructing exponentially localized and Kramers-doublet Wannier functions for
these bands, we are able to construct a Kondo lattice description. Analysis of
this effective model leads to an orbital-selective Mott transition, and allows
us to characterize the development of Kondo-driven quasiparticles and their
destruction. The quantum phase transition involves phases with large and small
Fermi surfaces. Our work provides a conceptual framework to describe the
amplified quantum fluctuations of multi-band systems with topological flat
bands, and sets the stage to understand the strange-metal behavior that is
emerging in kagome metals and other flat-band systems.

Acknowledgements.  We thank Lei Chen, Silke Paschen, Chandan Setty, Ming Yi
and, especially, Gabriel Aeppli for useful discussions. The work was primarily
supported by the U.S. DOE, BES, under Award No. DE-SC0018197 and additionally
supported by the Robert A. Welch Foundation Grant No. C-1411. The majority of
the computational calculations have been performed on the Shared University
Grid at Rice funded by NSF under Grant EIA-0216467, a partnership between Rice
University, Sun Microsystems, and Sigma Solutions, Inc., the Big-Data Private-
Cloud Research Cyberinfrastructure MRI-award funded by NSF under Grant No.
CNS-1338099 and by Rice University, and the Extreme Science and Engineering
Discovery Environment (XSEDE) by NSF under Grant No. DMR160057. The work of
Q.S. was performed in part at the Aspen Center for Physics, which is supported
by the NSF grant No. PHY-1607611.

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## Supplementary Material

###  .1 Band topology and Wannier construction

The space group is generated by the translational symmetry and the mirror
$M_{x}$ symmetry. At $\gamma=0$, the model has an additional $C_{3z}$
symmetry. We observe a completely flat band and a quadratic band touching
between the flat and wide bands, as shown in Figs. S1 B, C. By introducing a
non-zero $\gamma$, the flat band is no longer completely flat, but its
bandwidth remains relatively narrow at small $\gamma$. Near $\Gamma$ there is
a linear crossing (Dirac node) between the flat band and wide band as shown in
Fig. S1 D. Such Dirac node is protected by the $M_{x}$ and $SU(2)$ spin
symmetry. If we introduce spin-orbit coupling that breaks the $SU(2)$ spin
symmetry, the Dirac node will be gapped out and the flat band acquires $\pm 1$
Chern number. In Fig. S1 E, we show the band structure of the following spin-
orbit coupling

$\displaystyle
H_{soc}=\sum_{{\bm{k}},\sigma}i\lambda\sigma\bigg{[}\eta_{{\bm{k}},C,\sigma}^{\dagger}\eta_{{\bm{k}},D,\sigma}+\eta_{{\bm{k}},D,\sigma}^{\dagger}\eta_{{\bm{k}},E,\sigma}+\eta_{{\bm{k}},E,\sigma}^{\dagger}\eta_{{\bm{k}},C,\sigma}\bigg{]}+\text{h.c.}$
(S1)

We can also observe the non-trivial topology by calculating the $M_{x}$
eigenvalue of each band at high symmetry points. We focus on the case of
$\gamma\neq 0$ and show the result in Tab. 1 and also in Fig. S1 D,E. We note
that, at $\gamma\neq 0$, each band forms a one-dimensional (1d) irreducible
representation of the little group at each high-symmetry point, which is
characterized by the $M_{x}$ eigenvalue. However, in the case of zero
$\gamma=0$, we have both $C_{3z}$ and $M_{x}$ symmetries, and therefore, a
two-dimensional (2d) irreducible representation could be formed by the bands
at high symmetry points. The case of $\gamma\neq 0$ simplifies the
construction of the Wannier orbitals but still keeps the non-trivial topology
of the band structure.

The high symmetry points are
$\Gamma=(0,0),M^{\prime}=(2\pi/\sqrt{3},0),K=(0,4\pi/3)$ and we label the
bands from the top to the bottom as band $1,2,3,4,5$, with band $3$ being the
flat band. We note that a non-zero but small $\lambda$ will not change the
$M_{x}$ eigenvalue of each band. If we only consider the flat band (band 3),
its $M_{x}$ eigenvalues at $\Gamma,M,K$ are $-1,+1,+1$ respectively, which do
not admit an Wannier-orbital representation if we preserve all the symmetry.
However, by taking band 2 at the $\Gamma$ point and band $3$ at the
$M^{\prime}$ and $K$ points, we have the $M_{x}$ eigenvalue $+1,+1,+1$ which
can be represented by an $M_{x}$-even Wannier orbitals. This indicates that we
can construct the Wannier orbitals by combining bands 2 and 3. A similar
procedure has been taken for moiré systems Song and Bernevig (2021). As we
show later, the constructed Wannier orbital would have a large overlap with
the flat band and can be used to represent the flat-band degrees of freedom.
After removing the $+1,+1,+1$ eigenvalues at $\Gamma,M^{\prime},K$, the
remaining eigenvalues are $+1,-1$ for $\Gamma$ point, $+1,-1$ for $M^{\prime}$
point and $+1,-1$ for $K$ point. These combinations admit two Wannier orbitals
that are mirror even and mirror odd respectively. These two Wannier orbitals
mainly describe wide-band degrees of freedom.

Momentum | $\Gamma$ | $M^{\prime}$ | $K$
---|---|---|---
$M_{x}$ eigenvalue of band 1 | +1 | -1 | +1
$M_{x}$ eigenvalue of band 2 | +1 | +1 | -1
$M_{x}$ eigenvalue of band 3 | -1 | +1 | +1
Table 1: $M_{x}$ symmetry eigenvalues of the top three bands. We label the
bands from the top to the bottom as 1,2,3,4,5.

We now construct the Wannier orbitals for the case of $\gamma=-0.1,\lambda=0$.
However, the same procedure applies to the case with a small but non-zero
$\lambda$. We introduce the following three trial wavefunctions (six trial
wavefunctions if we account for the spin degeneracy) and construct the Wannier
orbitals of the top three bands via the projection method:

$\displaystyle|\text{Trial};d,{\bm{R}},\sigma\rangle=\sum_{{\bm{r}},i}e^{-\frac{({\bm{r}}-{\bm{R}})^{2}}{2\sigma}}\alpha^{f}_{i}\eta_{{\bm{r}},i,\sigma}|0\rangle$
$\displaystyle|\text{Trial};c,1,{\bm{R}},\sigma\rangle=\sum_{{\bm{r}},i}e^{-\frac{({\bm{r}}-{\bm{R}})^{2}}{2\sigma}}r_{x}\eta_{{\bm{r}},i,\sigma}|0\rangle$
$\displaystyle|\text{Trial};c,2,{\bm{R}},\sigma\rangle=\sum_{{\bm{r}},i}e^{-\frac{({\bm{r}}-{\bm{R}})^{2}}{2\sigma}}r_{y}\eta_{{\bm{r}},i,\sigma}|0\rangle$

where $\sigma=0.25a_{0}$, with $a_{0}$ being the distance between the two
nearest atoms, describes the decaying rate.
$\alpha^{d}_{A,B,C,D,E}(=0.5,0.5,1.4,1,1)$ captures the imbalance of weight
distributions between different sublattices. After the projection procedure,
we find three exponentially-localized Wannier orbitals
$|d,{\bm{R}},\sigma\rangle,|c,{\bm{R}},1,\sigma\rangle,|c,{\bm{R}},2,\sigma\rangle$,
which come from the three trial wavefunctions respectively. We then define the
electron operators in the Wannier basis via

$\displaystyle
d_{{\bm{R}},\sigma}^{\dagger}|0\rangle=|d,{\bm{R}},\sigma\rangle\quad,\quad
c_{{\bm{R}},1,\sigma}^{\dagger}|0\rangle=|c,{\bm{R}},1,\sigma\rangle\quad,\quad
c_{{\bm{R}},2,\sigma}^{\dagger}|0\rangle=|c,{\bm{R}},2,\sigma\rangle\,.$

By construction, both $d_{{\bm{R}},\sigma}$ and $c_{{\bm{R}},2,\sigma}$ are
even under $M_{x}$ and $c_{{\bm{R}},1,\sigma}$ is odd under $M_{x}$.

Figure S1: Brillouin zone and the noninteracting bandstructure. The first
Brillouin zone (A). Full band structure (B) and zoomed-in band structure (C)
at $\gamma=\lambda=0$ with a completely flat band near the Fermi energy. (D)
Band structure at non-zero $\gamma$ and $\lambda=0$. We observe a relatively
flat band and a Dirac node between the flat and wide bands. (E) Band structure
at a non-zero $\gamma$ and a non-zero $\lambda$, the Dirac node is gapped by
the spin-orbit coupling and the flat band acquires $\pm 1$ Chern number. In
(D) and (E), we label the $M_{x}$ eigenvalue of the relevant bands at high
symmetry points.

In addition, we also calculate the orbital weight of each band, which
characterizes the overlapping between each band and each Wannier orbital. The
overlapping between the Wannier orbitals
$d_{{\bm{k}},\sigma},c_{{\bm{k}},1,\sigma}$ and $c_{{\bm{k}},2,\sigma}$ and
the $i$-th bands is defined as

$\displaystyle W^{d}_{i}=\frac{1}{2N}\sum_{{\bm{k}}}\sum_{\sigma}|\langle
u_{{\bm{k}},i,\sigma}|d,{\bm{k}},\sigma\rangle|^{2}$ $\displaystyle
W^{c,1}_{i}=\frac{1}{2N}\sum_{{\bm{k}}}\sum_{\sigma}|\langle
u_{{\bm{k}},i,\sigma}|c,{\bm{k}},1,\sigma\rangle|^{2}$ $\displaystyle
W^{c,2}_{i}=\frac{1}{2N}\sum_{{\bm{k}}}\sum_{\sigma}|\langle
u_{{\bm{k}},i,\sigma}|c,{\bm{k}},2,\sigma\rangle|^{2}$

where $|u_{{\bm{k}},i,\sigma}\rangle$ is the Bloch function of the $i$-th band
with spin index $\sigma$,
$|d,{\bm{k}},\sigma,\rangle,|c,{\bm{k}},1,\sigma\rangle,|c,{\bm{k}},2,\sigma\rangle$
are the Fourier transformation of
$|d,{\bm{R}},\sigma\rangle,|c,{\bm{R}},1,\sigma\rangle,|c,{\bm{R}},2,\sigma\rangle$
respectively. Because of the $SU(2)$ spin symmetry, we average over the
contributions between the spin up and spin down. From numerical calculations,
we have

$\displaystyle W^{d}_{1}=0\%\quad,\quad W^{d}_{2}=12\%\quad,\quad
W^{d}_{3}=88\%\quad,\quad W^{d}_{4}=0\%\quad,\quad W^{d}_{5}=0\%$
$\displaystyle W^{c,1}_{1}=34\%\quad,\quad W^{c,1}_{2}=57\%\quad,\quad
W^{c,1}_{3}=9\%\quad,\quad W^{c,1}_{4}=0\%\quad,\quad W^{c,1}_{5}=0\%$
$\displaystyle W^{c,2}_{1}=66\%\quad,\quad W^{c,2}_{2}=31\%\quad,\quad
W^{c,2}_{3}=3\%\quad,\quad W^{c,2}_{4}=0\%\quad,\quad W^{c,2}_{5}=0\%$

where we label the bands from the top to the bottom as $1,2,3,4,5$; band $3$
is the flat band. We observe that the flat bands are mainly represented by the
$d$ orbitals, and the two dispersive bands ($1$ and $2$)are mainly captured by
the two $c$ orbitals.

###  .2 Parameters of the model

After constructing the Wannier orbitals, we project the hopping and
interaction terms to the Wannier basis, which leads to an interacting model.
In this section, we provide the parameters of the interacting model. The
parameters of the hopping and hybridization are shown in Tab. 2. We also
determine the dominant local interactions as follows,

$\displaystyle u/U=0.149\quad,\quad F_{1}/U=0.044\quad,\quad
F_{2}/U=0.036\quad,\quad J_{1}/U=0.088\quad,\quad J_{2}/U=0.072$ (S2)

Here, we drop the pairing hopping terms (i.e.
$d^{\dagger}d^{\dagger}cc,c^{\dagger}c^{\dagger}dd$). Because we focus on the
case, where the Hubbard interaction $u$ turns the $d$ electron into a quantum
spin, such terms are suppressed. In addition, the interactions between the $c$
electrons are also dropped. We consider the parameter region where the
bandwidth of the $c$ electrons is much larger than the interactions, and the
interactions between the $c$ electrons only renormalize the bandwidth of the
$c$ electrons.

$(i,j)$ | $(0,1)$ | $(-1,0)$ | $(1,0)$ | $(0,-1)$ | $\pm(-2,2)$ | $(-2,0)$ | $(0,2)$ | $(2,0)$ | $(0,-2)$ | $(-1,2)$ | $(1,-2)$
---|---|---|---|---|---|---|---|---|---|---|---
$t_{i{a}_{1}+ja_{2}}^{d}/t$ | -0.023 | -0.023 | -0.023 | -0.023 | 0.025 | 0.019 | 0.019 | 0.019 | 0.019 | -0.017 | -0.017
| $(1,-2)$ | $(-1,2)$ |  |  |  |  |  |  |  |  |
| -0.017 | -0.017 |  |  |  |  |  |  |  |  |
$(i,j)$ | $(1,0)$ | $(0,-1)$ | $(1,1)$ | $(-1,-1)$ | $(-1,2)$ | $(-2,1)$ | $(1,-2)$ | $(2,-1)$ | $(2,0)$ | $(0-2)$ |
$V_{i{a}_{1}+ja_{2},1}/t$ | 0.066 | -0.066 | -0.025 | 0.025 | 0.047 | -0.047 | 0.040 | -0.040 | 0.032 | -0.032 |
| $(-3,2)$ | $(-2,3)$ | $(-3,1)$ | $(-1,3)$ |  |  |  |  |  |  |
| -0.028 | 0.028 | 0.020 | -0.020 |  |  |  |  |  |  |
$(i,j)$ | (0,0) | $(0,1)$ | $(-1,0)$ | $(1,0)$ | $(0,-1)$ | $(1,-1)$ | $(-1,1)$ | $(1,1)$ | $(-1,-1)$ | $(2,-2)$ |
$V_{i{a}_{1}+ja_{2},2}/t$ | -0.04 | 0.0481 | 0.0481 | 0.0457 | 0.0457 | -0.0758 | -0.0726 | -0.0377 | -0.0377 | 0.0384 |
| (0,0) | $(-1,2)$ | $(-2,1)$ | $(2,0)$ | $(0,-2)$ | $(-2,2)$ |  |  |  |  |
| 0.0365 | 0.0365 | 0.0275 | 0.0275 | -0.0242 |  |  |  |  |  |
$(i,j)$ | $(0,1)$ | $(0,-1)$ | $(-1,0)$ | $(1,0)$ | $(-1,1)$ | $(1,-1)$ | $(-1,-1)$ | $(1,1)$ | $(2,-2)$ | $(-2,2)$ |
$t_{ia_{1}+ja_{2},11}^{c}$ | -0.1768 | -0.1768 | -0.1768 | -0.1768 | -0.0789 | -0.0789 | -0.0378 | -0.0378 | -0.0403 | -0.0403 |
$(i,j)$ | $(0,-1)$ | $(0,1)$ | $(-1,0)$ | $(1,0)$ | $(-1,-1)$ | $(1,1)$ | $(2,0)$ | $(-2,0)$ | $(0,-2)$ | $(0,2)$ |
$t_{ia_{1}+ja_{2},22}^{c}$ | 0.1389 | 0.1389 | 0.1389 | 0.1389 | 0.0292 | 0.0292 | -0.0383 | -0.0383 | -0.0383 | -0.0383 |
$(i,j)$ | $(0,0)$ | $(0,-1)$ | $(0,1)$ | $(-1,0)$ | $(1,0)$ | $(-1,-1)$ | $(1,1)$ | $(2,0)$ | $(-2,0)$ | $(0,-2)$ | $(0,2)$
$t_{ia_{1}+ja_{2},12}^{c}$ | 2.3657 | 0.1389 | 0.1389 | 0.1389 | 0.1389 | 0.0292 | 0.0292 | -0.0383 | -0.0383 | -0.0383 | -0.0383
| $E_{d}/t$ | $E_{1}/t$ | $E_{2}/t$ |  |  |  |  |  |  |  |
| 0.01 | 0.92 | 1.37 |  |  |  |  |  |  |  |
Table 2: Parameters of the kinetic term in the Wannier basis.

###  .3 Effective Multi-orbital Hubbard Model

The model, expressed in terms of the $d$ and $c$ orbitals, contains the
kinetic term $H_{0}$ and the interaction term $H_{I}$. The kinetic term takes
the following form:

$\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle H_{d}+H_{c}+H_{dc}$
$\displaystyle H_{d}$ $\displaystyle=$
$\displaystyle\sum_{{\bm{R}},{\bm{R}}^{\prime},\sigma}\bigg{[}t^{d}_{{\bm{R}}-{\bm{R}}^{\prime},\sigma}+(E_{d}-\mu)\delta_{{\bm{R}},{\bm{R}}^{\prime}}\bigg{]}d_{{\bm{R}},\sigma}^{\dagger}d_{{\bm{R}}^{\prime},\sigma}$
$\displaystyle H_{c}$ $\displaystyle=$
$\displaystyle\sum_{{\bm{R}},{\bm{R}}^{\prime},a,a^{\prime},\sigma}\bigg{[}t^{c}_{{\bm{R}}-{\bm{R}}^{\prime},aa^{\prime},\sigma}-\mu\delta_{{\bm{R}},{\bm{R}}^{\prime}}\delta_{a,a^{\prime}}\bigg{]}c_{{\bm{R}},a,\sigma}^{\dagger}c_{{\bm{R}}^{\prime},a^{\prime},\sigma}$
$\displaystyle+\sum_{{\bm{R}},a,\sigma}E_{a}c_{{\bm{R}},a,\sigma}^{\dagger}c_{{\bm{R}},a,\sigma}$
$\displaystyle H_{dc}$ $\displaystyle=$
$\displaystyle\sum_{{\bm{R}},{\bm{R}}^{\prime},a,\sigma}\bigg{[}V_{{\bm{R}}-{\bm{R}}^{\prime},a,\sigma}d_{{\bm{R}},\sigma}^{\dagger}c_{{\bm{R}}^{\prime},a^{\prime},\sigma}+\text{h.c.}\bigg{]}\,.$
(S3)

The dominant hybridization is between $d_{{\bm{R}},a,\sigma}$ and the new
conduction-electron band $1$, $c_{{\bm{R}},1,\sigma}$, which is mirror odd;
correspondingly, the hybridization is off-site. For interactions, it suffices
to keep the ones between two $d$-electrons and those between the $d$\- and
$c$-electrons. The interactions between the $c$-electrons are omitted, since
they are small compared to the corresponding bandwidth.

The interactions we consider include the Hubbard interactions of the $d$
electrons ($H_{u}$), the density-density interactions between the $d$ and $c$
electrons $(H_{F})$ (which is unimportant in the local-moment regime our case
corresponds to), and the Hund’s coupling between the $d$ and $c$ electrons
($H_{Hund}$). The interactions are labeled by $u,F_{1,2},J_{1,2}$,
respectively. The full interaction Hamiltonian then takes the form of

$\displaystyle H_{I}=H_{u}+H_{F}+H_{Hund}$ $\displaystyle
H_{u}=\sum_{{\bm{R}}}\frac{u}{2}(n^{d}_{{\bm{R}}}-1)^{2}\quad,\quad
H_{F}=\sum_{{\bm{R}},a}W_{a}n^{f}_{{\bm{R}}}n^{c}_{{\bm{R}},a},$
$\displaystyle
H_{Hund}=-\sum_{{\bm{R}},a}J_{a}\bm{S}_{{\bm{R}}}\cdot\bm{S}^{c}_{{\bm{R}},a}\,,$
(S4)

where
$\bm{S}_{{\bm{R}}}=\frac{1}{2}d^{\dagger}_{{\bm{R}}}\bm{\sigma}d_{{\bm{R}}}$,
$\bm{S}^{c}_{{\bm{R}},a}=\frac{1}{2}c_{{\bm{R}},a}^{\dagger}\bm{\sigma}c_{{\bm{R}},a}$
are spin operators of $d$\- and $c$-electrons respectively.
$n^{d}_{{\bm{R}},\sigma}=d_{{\bm{R}},\sigma}^{\dagger}d_{{\bm{R}},\sigma},n^{c}_{{\bm{R}},a,\sigma}=c_{{\bm{R}},a,\sigma}^{\dagger}c_{{\bm{R}},a,\sigma}$
are the density operators of $d$ and $c$ electrons.
$n^{d}_{{\bm{R}}}=\sum_{\sigma}n^{d}_{{\bm{R}},\sigma},n^{c}_{{\bm{R}},a}=\sum_{\sigma}n^{c}_{{\bm{R}},a,\sigma}$.

###  .4 Kondo model

In this section, we derive the low-energy effective model for sufficiently
large values of $u$. The action of the original Hamiltonian reads

$\displaystyle
S=\int_{\tau}\sum_{{\bm{R}},\sigma}d_{{\bm{R}},\sigma}^{\dagger}\partial_{\tau}d_{{\bm{R}},\sigma}+\int_{\tau}\sum_{{\bm{R}},a,\sigma}c_{{\bm{R}},a,\sigma}^{\dagger}\partial_{\tau}c_{{\bm{R}},a,\sigma}+\int_{\tau}(H_{d}+H_{c}+H_{dc}+H_{U}+H_{F}+H_{Hund})d\tau\,.$

We perform a Hubbard-Stratonovich transformation to decouple the Hubbard
interaction and Hund’s coupling term

$\displaystyle Z$ $\displaystyle=$
$\displaystyle\int_{d,d^{\dagger},c,c^{\dagger}}e^{-S}$ $\displaystyle=$
$\displaystyle\int_{d,d^{\dagger},c,c^{\dagger},\phi}\exp\bigg{\\{}-\int_{\tau}\sum_{{\bm{R}},\sigma}d_{{\bm{R}},\sigma}^{\dagger}\partial_{\tau}d_{{\bm{R}},\sigma}-\int_{\tau}\sum_{{\bm{R}},a,\sigma}c_{{\bm{R}},a,\sigma}^{\dagger}\partial_{\tau}c_{{\bm{R}},s,\sigma}-\int_{\tau}(H_{f}+H_{c}+H_{dc}+H_{W})$
$\displaystyle-\int_{\tau}\frac{u}{3}\bm{\phi}_{{\bm{R}}}\cdot\bm{\phi}_{{\bm{R}}}-\int_{\tau}\frac{2u}{3}\sum_{{\bm{R}}}\bm{\phi}_{\bm{R}}\cdot(\bm{S}^{d}_{\bm{R}}+\sum_{a}\frac{3J_{a}}{4u}\bm{S}_{{\bm{R}}a}^{c})\bigg{\\}}$

where we have dropped the interactions between the conduction electrons. We
then introduce the special unitary matrix $U_{\bm{R}}$ that satisfies

$\displaystyle\phi_{0}U_{\bm{R}}^{\dagger}\sigma^{z}U_{\bm{R}}=\sum_{\mu}\phi_{\bm{R}}^{\mu}\sigma^{\mu}$
(S5)

and the new fermions operators

$\displaystyle\psi_{{\bm{R}},\sigma}=\sum_{\sigma^{\prime}}[U_{\bm{R}}^{\dagger}]_{\sigma\sigma^{\prime}}d_{{\bm{R}},\sigma^{\prime}}\,.$
(S6)

We parametrize $U_{\bm{R}}$ with

$\displaystyle
U_{\bm{R}}=\begin{bmatrix}z_{{\bm{R}},1}&-z_{{\bm{R}},2}^{\dagger}\\\
z_{{\bm{R}},2}&z_{{\bm{R}},1}^{\dagger}\end{bmatrix}$ (S7)

with $\sum_{a}z_{{\bm{R}},a}^{\dagger}z_{{\bm{R}},a}=1$ Now we have the
following new action

$\displaystyle S$ $\displaystyle=$ $\displaystyle
S_{\psi}+S_{B}+S_{t}+S_{V}+S_{F}+S_{c}$ $\displaystyle S_{\psi}$
$\displaystyle=$
$\displaystyle\int_{\tau}\sum_{{\bm{R}},\sigma}\psi_{{\bm{R}},\sigma}^{\dagger}\partial_{\tau}\psi_{{\bm{R}}\sigma}-\int_{\tau}\frac{2u}{3}\sum_{{\bm{R}}}\phi_{0}\psi_{{\bm{R}}}^{\dagger}\sigma^{z}\psi_{{\bm{R}}}$
$\displaystyle S_{B}$ $\displaystyle=$
$\displaystyle\int_{\tau}\sum_{{\bm{R}}}\psi_{\bm{R}}^{\dagger}(U_{\bm{R}}^{\dagger}\partial_{\tau}U_{\bm{R}})\psi_{\bm{R}}$
$\displaystyle S_{t}$ $\displaystyle=$
$\displaystyle\int_{\tau}\sum_{{\bm{R}},{\bm{R}}^{\prime}}t_{{\bm{R}},{\bm{R}}^{\prime}}\psi_{\bm{R}}^{\dagger}U_{\bm{R}}^{\dagger}U_{{\bm{R}}^{\prime}}\psi_{{\bm{R}}^{\prime}}$
$\displaystyle S_{V}$ $\displaystyle=$
$\displaystyle\int_{\tau}\sum_{{\bm{R}},{\bm{R}}^{\prime},a}V_{{\bm{R}},{\bm{R}}^{\prime}a}\psi_{{\bm{R}}}^{\dagger}U_{\bm{R}}^{\dagger}c_{{\bm{R}}^{\prime},a,\sigma}$
$\displaystyle S_{c}$ $\displaystyle=$
$\displaystyle\int_{\tau}\sum_{{\bm{R}},a,\sigma}c_{{\bm{R}},a,\sigma}^{\dagger}\partial_{\tau}c_{{\bm{R}},s,\sigma}+\int_{\tau}\frac{2u}{3}\sum_{{\bm{R}}}\phi_{0}U_{\bm{R}}^{\dagger}\sigma^{z}U_{\bm{R}}\cdot(\sum_{a}\frac{3J_{a}}{4u}\bm{S}_{{\bm{R}}a}^{c})+\int_{\tau}\sum_{{\bm{R}},a}F_{a}\sum_{\sigma}\psi_{{\bm{R}},\sigma}^{\dagger}\psi_{{\bm{R}},\sigma}n_{{\bm{R}},c}^{c}+H_{c}$

The large $U$ leads to a local moment formation, which gives $\phi_{0}\neq 0$
at the saddle point level. This would then gap out the $\psi$ fermions. We
integrate out the gapped $\psi$ fermion and find

$\displaystyle S_{Berry}=\langle
S_{B}\rangle_{0}=-\int_{\tau}\sum_{{\bm{R}},a}z_{{\bm{R}},a}\partial_{\tau}z_{{\bm{R}},a}^{\dagger}$
$\displaystyle S_{Heisenberg}=-\frac{1}{2}\langle
S_{t}^{2}\rangle_{0}=\int_{\tau}\sum_{{\bm{R}},{\bm{R}}^{\prime}}\frac{2|t_{{\bm{R}},{\bm{R}}^{\prime}}|^{2}}{4u\phi_{0}/3}\bm{S}_{\bm{R}}\cdot\bm{S}_{{\bm{R}}^{\prime}}$
$\displaystyle S_{Kondo}=-\frac{1}{2}\langle
S_{V}^{2}\rangle_{0}=\int_{\tau}\sum_{{\bm{R}},{\bm{R}}_{1}a_{1},{\bm{R}}_{2}a_{2}}\frac{2V_{{\bm{R}}_{,}{\bm{R}}_{2}a_{2}}^{*}V_{{\bm{R}},{\bm{R}}_{1}a_{1}}}{4u\phi_{0}/3}\bm{S}_{\bm{R}}\cdot\bm{S}_{{\bm{R}}_{1}a_{1},{\bm{R}}_{2}a_{2}}$

where for a given operator $O$, $\langle
O\rangle_{0}=\bigg{[}\int_{\psi,\psi^{\dagger}}Oe^{-S_{\psi}}\bigg{]}/\bigg{[}\int_{\psi,\psi^{\dagger}}e^{-S_{\psi}}\bigg{]}$
and the spin operator $\bm{S}_{\bm{R}}$ is defined as
$\bm{S}_{{\bm{R}}}=\frac{1}{2}z^{\dagger}_{\bm{R}}\bm{\sigma}z_{\bm{R}}$,
which is the spin moment of a $d$ electron. $S_{Berry}$ is the Berry phase
term of the spin operator $\bm{S}_{\bm{R}}$, $S_{Heisenberg}$ and $S_{Kondo}$
are the Heisenberg interaction and Kondo interaction term respectively.
Finally, we transform the action to the Hamiltonian and reach the Kondo-
Heisenberg Hamiltonian with the Hund’s coupling as shown in the main text.

###  .5 The RKKY/Heisenberg vs. Kondo competition

There are two emergent energy scales in the Kondo lattice model: the Kondo
energy scale $E_{K}$ and the Heisenberg scale $E_{H}$. The Kondo scale
$E_{K}\sim De^{-1/(\rho J^{K})}$, where
$J^{K}=\text{max}\\{J^{K}_{{\bm{R}},{\bm{R}}_{1}a_{1},{\bm{R}}_{2}a_{2}}\\}$
is the maximum amplitude of the Kondo coupling, $\rho$ is the $c$-electron
density of states at the Fermi energy, and $D$ is the bandwidth of the $c$
bands. The Heisenberg scale $E_{H}$ describes the spin fluctuations, and is
proportional to the
$J^{H}=\text{max}\\{J_{{\bm{R}},{\bm{R}}^{\prime}}^{H}\\}$. When $E_{K}\gg
E_{H}$, the Kondo physics dominates, the $d$ electrons are Kondo-screened by
the $c$ electrons and a heavy Fermi liquid ensues. When $E_{H}\gg E_{K}$, the
spin fluctuations dominate. They favor the formation of a spin singlet between
the two $d$-spin and destroy the Kondo effect. The competition between the two
energy scales leads to a quantum phase transition. Such quantum phase
transition can be realized by tuning $\widetilde{u}$. We note that both
$J^{K}$ and $J^{H}$ are proportional to $\widetilde{u}$. However, as we
increase $\widetilde{u}$, $E_{K}$ decrease exponentially, which is much faster
than the power-law decay of $E_{H}$, which gives the possibility of quantum
phase transition.

We remark that the Hund’s coupling will destabilize the formation of neither
the Kondo singlet nor the spin singlet between the $d$-spins. In the Kondo
phase, the Hund’s coupling and Kondo coupling act differently, with the Hund’s
coupling being on-site and the dominant Kondo coupling being off-site.
Therefore, the two effects can co-exist. In the $E_{H}$ dominating phase, the
coupling $J_{H}$ aligns the spin of the $d$ and $c$ electrons and produce a
larger spin moment on each site. Then, the cooperation between the Hund’s and
Heisenberg interaction favors the spin-singlet formation between the emerged
local moments formed by both the $d$ and $c$ electrons. However, the hopping
amplitude of the $c$ electrons is large.The emergent local moments are mainly
formed by the $d$-electrons and the singlet formed by the emerged local
moments is then adiabatically connected to the singlet of the pure $d$
electrons.

###  .6 Large-$N$ limit

We solve the Kondo-Heisenberg model with Hund’s coupling at the large-$N$
limit. We generalize $SU(2)$ spin symmetry to $SU(N)$ spin symmetry and let
$J^{H}_{{\bm{R}},{\bm{R}}^{\prime}}\rightarrow
J^{H}_{{\bm{R}},{\bm{R}}^{\prime}}/N$ and
$J^{K}_{{\bm{R}},{\bm{R}}_{1}a_{1},{\bm{R}}_{2}a_{2}}\rightarrow
J^{K}_{{\bm{R}},{\bm{R}}_{1}a_{1},{\bm{R}}_{2}a_{2}}/N,J_{a}\rightarrow
J_{a}/N$. We then introduce Abrikosov fermions ${f}_{{\bm{R}},\sigma}$, which
satisfy a local constraint
${f}_{{\bm{R}},\sigma}^{\dagger}{f}_{{\bm{R}},\sigma}=1$, and rewrite the spin
operators as
$\bm{S}_{{\bm{R}}}=\frac{1}{2}\sum_{\sigma,\sigma^{\prime}}{f}_{{\bm{R}},\sigma}^{\dagger}\bm{\sigma}_{\sigma,\sigma^{\prime}}{f}_{{\bm{R}},\sigma^{\prime}}$.

We next perform a Hubbard-Stratonovich transformation

$\displaystyle H_{H}$ $\displaystyle\rightarrow$
$\displaystyle\sum_{{\bm{R}},{\bm{R}}^{\prime}}\frac{J^{H}_{{\bm{R}},{\bm{R}}^{\prime}}}{2}\bigg{[}N\chi_{{\bm{R}}^{\prime},{\bm{R}}}\chi_{{\bm{R}}^{\prime},{\bm{R}}}-\sum_{\sigma}\bigg{(}{f}_{{\bm{R}},\sigma}^{\dagger}{f}_{{\bm{R}}^{\prime},\sigma}\chi_{{\bm{R}}^{\prime},{\bm{R}}}+\text{h.c.}\bigg{)}\bigg{]}$
$\displaystyle H_{J_{K}}$ $\displaystyle\rightarrow$
$\displaystyle\sum_{{\bm{R}},{\bm{R}}_{1},{\bm{R}}_{2},a_{1},a_{2}}\frac{J^{K}_{{\bm{R}},{\bm{R}}_{1}a_{1},{\bm{R}}_{2}a_{2}}}{2}\bigg{[}N\zeta_{{\bm{R}},{\bm{R}}_{2}a_{2}}\zeta_{{\bm{R}},{\bm{R}}_{1}a_{1}}^{*}-\sum_{\sigma}\bigg{(}\zeta^{*}_{{\bm{R}},{\bm{R}}_{1}a_{1}}{f}_{{\bm{R}},\sigma}^{\dagger}c_{{\bm{R}}_{2},a_{2},\sigma}+\eta_{{\bm{R}},{\bm{R}}_{2}a_{2}}c_{{\bm{R}}_{1},a_{1},\sigma}^{\dagger}f_{{\bm{R}},\sigma}\bigg{)}\bigg{]}$
$\displaystyle H_{Hund}$ $\displaystyle\rightarrow$
$\displaystyle\sum_{{\bm{R}},a}\frac{J_{a}}{2}\bigg{[}-N\zeta_{{\bm{R}},{\bm{R}}a}\zeta_{{\bm{R}},{\bm{R}}a}^{*}+\sum_{\sigma}\bigg{(}\zeta_{{\bm{R}},{\bm{R}}a}^{*}{f}_{{\bm{R}},\sigma}^{\dagger}c_{{\bm{R}},a,\sigma}+\text{h.c.}\bigg{)}$
(S8)

where $\chi_{{\bm{R}},{\bm{R}}^{\prime}},\eta_{{\bm{R}},{\bm{R}}_{1}a_{1}}$
are the bosonic fields used in the decoupling procedure. In the large $N$
limit, we have a series of saddle point equations

$\displaystyle\chi_{{\bm{R}},{\bm{R}}^{\prime}}=\frac{1}{N}\sum_{\sigma}\langle{f}_{{\bm{R}},\sigma}^{\dagger}{f}_{{\bm{R}}^{\prime},\sigma}\rangle$
$\displaystyle\zeta_{{\bm{R}},{\bm{R}}^{\prime}a}=\frac{1}{N}\sum_{\sigma}\langle{f}_{{\bm{R}},\sigma}^{\dagger}c_{{\bm{R}}^{\prime},a,\sigma}\rangle.$
(S9)

The local constraints
$\sum_{\sigma}{f}^{\dagger}_{{\bm{R}},\sigma}{f}_{{\bm{R}},\sigma}=1$ are
satisfied on average by introducing a Lagrangian multiplier $\lambda$:
$\lambda(\sum_{\sigma}{f}^{\dagger}_{{\bm{R}},\sigma}{f}_{{\bm{R}},\sigma}-1)$.
$\lambda$ is determined by requiring

$\displaystyle\sum_{\sigma}\langle{f}_{{\bm{R}},\sigma}^{\dagger}{f}_{{\bm{R}},\sigma}\rangle=1$
(S10)

We solved Eqs. S9 and S10 self-consistently and derive the phase diagram at
zero temperature as shown in Fig. 3. In the small Fermi-surface phase, the
Luttinger theorem of the Kondo lattice can be seen as satisfied by the
separate responses of the local moments and conduction electrons Senthil  _et
al._ (2004); Pivovarov and Si (2004) to the adiabatic insertion of an external
flux Oshikawa (2000).