# Chiral Kondo Lattice in Doped MoTe2/WSe2 Bilayers

Daniele Guerci  Center for Computational Quantum Physics, Flatiron Institute,
New York, New York 10010, USA  Jie Wang  Center for Computational Quantum
Physics, Flatiron Institute, New York, New York 10010, USA  Jiawei Zang
Department of Physics, Columbia University, 538 West 120th Street, New York,
NY 10027  Jennifer Cano  Department of Physics and Astronomy, Stony Brook
University, Stony Brook, New York 11794, USA  Center for Computational Quantum
Physics, Flatiron Institute, New York, New York 10010, USA  J. H. Pixley
Department of Physics and Astronomy, Center for Materials Theory, Rutgers
University, Piscataway, New Jersey 08854, USA  Center for Computational
Quantum Physics, Flatiron Institute, New York, New York 10010, USA  Andrew
Millis  Center for Computational Quantum Physics, Flatiron Institute, New
York, New York 10010, USA  Department of Physics, Columbia University, 538
West 120th Street, New York, NY 10027

###### Abstract

We theoretically study the interplay between magnetism and a heavy Fermi
liquid in the AB stacked transition metal dichalcogenide bilayer system
MoTe2/WSe2 in the regime in which the $Mo$ layer supports localized magnetic
moments coupled by interlayer electron tunnelling to a weakly correlated band
of itinerant electrons in the $W$ layer. We show that the interlayer electron
transfer leads to a chiral Kondo exchange, with consequences including a
strong dependence of the Kondo temperature on carrier concentration and
anomalous Hall effect due to a topological hybridization gap. The theoretical
model exhibits two phases, a small Fermi surface magnet and a large Fermi
surface heavy Fermi liquid; at the mean-field level the transition between
them is first order. Our results provide concrete experimental predictions for
ongoing experiments on MoTe2/WSe2 bilayer heterostructures and introduces a
controlled route to observe a topological selective Mott transition.

_Introduction.—_ Transition metal dichalcogenide (TMD) moiré devices created
by stacking two TMD monolayers have recently emerged as a highly tunable
platform to realize strongly correlated and topological states [1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. This experimental
flexibility has opened the door to control and observe phenomena that has been
out of reach in conventional solid-state platforms such as a continuous Mott
transition [4, 5] in a single sample. Forming heterobilayers with distinct
chemical composition allows one to effectively tune the density and the
interaction within an individual layer which can open the door to
synthetically realize orbital selective Mott transitions [22, 23, 24, 25, 26]
to emulate several strongly correlated electron systems of interest.

Figure 1: Model and qualitative phase diagram. (a) Representation of real-
space structure of MoTe2/WSe2 bilayer showing the two triangular sublattices
of the honeycomb moiré lattice. Red and blue represent the centers of the
Wannier functions describing the relevant states in the bottom ($Mo$) and top
($W$) layers, respectively. The nearest neighbor vector $\bm{\delta}_{i}$, the
$Mo-W$ energy difference $\Delta$ and the intralayer $t_{W}/t_{Mo}$ and
interlayer $t_{h}$ hopping matrix elements are also indicated. (b) Schematic
density of states for filling near $n=1$ representing the $Mo$ layer as a
narrow-band strongly correlated system with lower (LHB) and upper (UHB)
Hubbard bands separated by an energy gap $U$ while the $W$ layer is shown as a
wide-band weakly correlated system. Upper panel: positive $\Delta$; showing
filled $Mo$ LHB and carriers added beyond $n=1$ in the $W$ layer. Lower panel:
negative $\Delta$; mixed valent situation with potential for simultaneous
occupancy of $Mo$\- and $W$-layer states at $n=1$. (c) Qualitative phase
diagram of the AB stacked MoTe2/WSe2 bilayer as a function of hole carrier
density $n$ and displacement field $\Delta$ showing expected phases: at
carrier density $n=1$ as $\Delta$ is decreased the insulating 120∘
antiferromagnet (AFM) gives way to a QAH insulator and then to a Fermi liquid
(FL). For $n>1$ the the system can be described by a Kondo lattice (KL) model.
At other carrier concentrations different metallic phases occur including
moderate mass (FL) and heavy mass (HFL) Fermi liquids; a hole-doped single
band antiferromagnet (AFM) and a Kondo lattice antiferromagnet (KLAFM) with
ordered magnetism on the $Mo$ layer coupled to mobile carriers in the $W$
layer. The HFL is connected to a topological Kondo semimetal (TKSM) at $n=2$.

In this paper we focus on the AB-stacked MoTe2/WSe2 bilayer heterostructure of
recent experimental [1] and theoretical [13, 14, 15, 16, 17, 18] interest and
show it realizes a chiral Kondo lattice. In this system the lattice mismatch
between the two materials leads to a hexagonal moiré lattice with a moiré
lattice constant of $a_{M}\sim 5$nm. As shown in Fig. 1(a) the two sublattices
of the hexagonal moiré lattice correspond to the MoTe2 ($Mo$) and WSe2 ($W$)
layers, with the two sublattices connected by the interlayer hopping.

The combination of the strong spin-momentum locking of the monolayer materials
and the AB stacking configuration reduces the magnitude of the interlayer
tunnelling to a value much smaller than the intra-layer hopping of the WSe2
moiré band [27]. The small interlayer coupling means that the two layers can
be discussed separately and then the effects of the interlayer coupling
considered. The atomic physics of the monolayer materials determines that the
MoTe2 layer has a narrower moiré conduction band than does the WSe2 layer, so
that the MoTe2 layer may be regarded as strongly correlated with an upper and
lower Hubbard band while the WSe2 layer has a wider bandwidth and a carrier
concentration typically far from $n=1$ and may be regarded as weakly
correlated [27, 14]. The resulting density of states (DOS) is sketched in Fig.
1(b). The energy offset between the bands, defined here as $\Delta$, and the
total chemical potential $\mu$ can be controlled in situ by appropriate gate
voltages.

Fig. 1(c) shows a qualitative phase diagram in the plane of band offset
($\Delta$) and total density ($n=1+x$). At large positive $\Delta$ the MoTe2
band is well separated from the bottom of the WSe2 band. The first carriers
added to the system go into the MoTe2 layer, forming a correlated metal which
at $n=1$ becomes a triangular-lattice Mott insulator with $120^{\circ}$
antiferromagnetic (AFM) order. At carrier concentration $n=1$, decreasing
$\Delta$ is predicted [27, 14, 15, 16, 17, 18] and observed [1, 2] to cause a
transition to a quantum anomalous Hall (QAH) state, followed by a transition
to a conventional metallic state. At $\Delta>0$, carriers in excess of the
Mott concentration at half-filling ($n=1$ per moiré unit cell) go into the
WSe2 band (if, as we assume, $U>\Delta$) and are coupled to the spins of the
Mott insulator via an exchange coupling $J_{K}\sim t_{h}^{2}/\Delta$ derived
perturbatively from the interlayer hybridization $t_{h}$, so that the
MoTe2/WSe2 system can be described by an effective Kondo lattice model on the
moiré scale whose study is the central topic of this paper.

Synthetic Kondo lattice models in moiré systems have been previously discussed
in the context of the interplay between localized orbital and delocalized
electrons in twisted bilayer [28] and trilayer graphene [29], and in relation
to the orbital selective Mott transition in a two-band moiré TMD model [30].
More recently, a gate-tunable Kondo interaction in trilayer TMDs has been
predicted to realize heavy fermion quantum criticality [31]. Here, we focus on
how the combination of strong spin-orbit coupling and the non-local structure
of the interlayer hybridization substantially enriches the physics relative to
the standard Kondo-lattice/orbitally selective Mott transition picture.

_Derivation of the Kondo moiré lattice model.—_ The low-energy properties of
the moiré system are described [14] by a Hubbard model on the honeycomb
lattice shown in Fig. 1(a). The two sublattices of the honeycomb lattice give
the centers of Wannier states formed from the $Mo$ (red) and $W$ (blue) sites
respectively. Wannierization of band structure calculations [14] give same-
sublattice hopping parameters $t_{W}\simeq 9$meV, $t_{Mo}\simeq 4.5$meV and an
interlayer hopping $t_{h}\simeq 2$meV shown as solid arrows in 1(a). The
monolayer Ising spin-orbit coupling (SOC) implies that $t_{W}$ has a spin-
dependent complex phase factor $\pm 2\pi/3$ placing the W band minima at the
Dirac points $\bm{\kappa}$ (spin up) and $\bm{\kappa}^{\prime}$ (spin down)
respectively; for details see Ref. [27, 14] or the SM [32]. The local
interaction $U$ which is taken to be the same on both layers for simplicity is
believed to be large: $U/t_{Mo}\gg 1$ [14, 13] and for $\Delta>0$ gives rise
at filling $n=1$ per moiré unit cell to a 120∘ AFM charge transfer insulator
[14]. At nonzero electron doping $n=1+x$ $(x>0)$ the extra carriers go into
the $W$ charger-transfer band (blue shaded region of DOS in Fig. 1(b)) and
$Mo$ sites remain singly occupied. Due to the large bandwidth and the small
doping $x$ we assume that the correlation effects in the $W$ band can be, at
first approximation, ignored. The hybridization term $t_{h}$ induces an
effective spin-exchange Kondo coupling [33, 34, 32] between the dispersive
electrons in the conduction band from the $W$ and the local moments
$\bm{S}_{\mathbf{r}}$ from the $Mo$ layer. The resulting moiré Kondo-lattice
(spin-fermion) model reads:

$\begin{split}\bar{H}=&\sum_{\mathbf{k}\sigma}\xi_{\mathbf{k}\sigma}c^{\dagger}_{\mathbf{k}\sigma}c_{\mathbf{k}\sigma}\\\
+&\sum_{\langle\mathbf{r},\mathbf{r}^{\prime}\rangle_{\text{Mo}}}J_{H}\left(S^{z}_{\mathbf{r}}S^{z}_{\mathbf{r}^{\prime}}+\gamma
S^{+}_{\mathbf{r}}S^{-}_{\mathbf{r}^{\prime}}+h.c.\right)+D(\bm{S}_{\mathbf{r}}\times\bm{S}_{\mathbf{r}^{\prime}})_{z}\\\
+&\frac{1}{2N}\sum_{\mathbf{r}\in\text{Mo}}\sum_{\mathbf{k},\mathbf{p}}e^{-i(\mathbf{k}-\mathbf{p})\cdot\mathbf{r}}J_{\mathbf{k},\mathbf{p}}\bm{S}_{\mathbf{r}}\cdot
c^{\dagger}_{\mathbf{k}\sigma}\,\bm{\sigma}_{\sigma\sigma^{\prime}}\,c_{\mathbf{p}\sigma^{\prime}},\end{split}$
(1)

where
$\xi_{\mathbf{k}\sigma}=-2t_{W}\sum^{3}_{j=1}\cos(\mathbf{k}\cdot\bm{a}_{i}+2\pi
s_{\sigma}/3)-\epsilon_{F}$ is the electron dispersion for the $W$ band,
$\bm{a}_{1}=\sqrt{3}a_{M}(1,0)$,
$\bm{a}_{2,3}=\sqrt{3}a_{M}\left(-1/2,\pm\sqrt{3}/2\right)$ are the lattice
vectors with $a_{M}=5$nm is the moiré-lattice constant,
$s_{\uparrow,\downarrow}=\pm 1$, and $\epsilon_{F}$ is the Fermi energy fixing
the filling $x$ of electrons in the conduction band
$\sum_{\mathbf{r},\sigma}\langle
c^{\dagger}_{\mathbf{r}\sigma}c_{\mathbf{r}\sigma}\rangle/N=x$. The exchange
$J_{H}$ arises from a combination of the spin rotational invariant Heisenberg
exchange in the $Mo$ layer and virtual excitations in the $W$ band mediated by
$t_{h}$. The reduced spin rotational symmetry in the $W$ layer induces the XXZ
anisotropy $\gamma\neq 1/2$ and also Dzyaloshinskii–Moriya $D$ interactions
[8]. For simplicity we use the Heisenberg form $\gamma=1/2$ in Eq. (1) in our
calculations and we treat $J_{H}$ as a phenomenological parameter.

In the last term of Eq. (1),
$J_{\mathbf{k},\mathbf{p}}=J_{K}V^{*}_{\mathbf{k}}V_{\mathbf{p}}$ with
$J_{K}=2t^{2}_{h}[1/\Delta+1/(U_{\text{Mo}}-\Delta)]$ is the Kondo exchange
and the hopping $t_{h}$ between opposite sublattices gives the form factor
$V_{\mathbf{k}}=\sum_{j=1}^{3}e^{i\bm{\delta}_{j}\cdot\mathbf{k}}$ with
$\bm{\delta}_{j}$ displayed as green arrows in Fig. 1(a). In proximity to the
high-symmetry points $\bm{\kappa}$ and $\bm{\kappa}^{\prime}$, centers of the
spin up and down Fermi sea at low-doping, respectively, the form factor takes
the $p$-wave chiral form
$V_{\bm{\kappa}+\mathbf{k}}\simeq-3a_{M}(k_{x}-ik_{y})/2$ and due to time-
reversal symmetry
$V_{\bm{\kappa}^{\prime}+\mathbf{k}}=V^{*}_{\bm{\kappa}-\mathbf{k}}$. The
displacement field dependence [31] and chirality will be seen to have
important physics consequences.

To quantify the Kondo coupling at the Fermi energy we introduce a Fermi
surface (FS) averaged Kondo exchange $\bar{J}_{K}$ as:

$\bar{J}_{K}=\frac{J_{K}}{\rho_{\epsilon_{F}}}\frac{3\sqrt{3}a_{M}^{2}}{8\pi^{2}}\oint_{\rm
FS}dk_{t}\frac{|V_{\mathbf{k}}|^{2}}{|\nabla_{\mathbf{k}}\epsilon_{\mathbf{k}\sigma}|}\simeq\frac{xJ_{K}}{4\rho_{\epsilon_{F}}t_{W}},$
(2)

where the line integral is over the conduction electron FS, $k_{t}$ is the
component of $\mathbf{k}$ along the tangent to the FS curve,
$\rho_{\epsilon_{F}}$ is the DOS at the Fermi energy and in obtaining the
explicit analytic form the $W$ band dispersion was expanded to leading
(quadratic) order around its minimum.

_Results and Methods.—_ In this section, we employ the mean-field theory of
Abrikosov fermions [35, 36, 37] to study the competition between the magnetic
and the HFL phase. For this purpose we factorize the local magnetic spin into
charge neutral spinons
$\bm{S}_{\bm{r}}=\chi^{{\dagger}}_{\bm{r}\alpha}\bm{\sigma}_{\alpha\beta}\chi_{\bm{r}\beta}/2$
subject to a constraint
$\sum_{\sigma}\chi^{{\dagger}}_{\bm{r}\sigma}\chi_{\bm{r}\sigma}=1$. In the
following we adopt $\bm{\sigma}$ for the spin degrees of freedom. In our
notation, $c_{\bm{r}}$ is a spin-doublet
$(c_{\bm{r}\uparrow},c_{\bm{r}\downarrow})^{T}$ and same for
$\chi_{\mathbf{r}}$. We treat the Kondo interaction with an unrestricted
Hartree Fock ansatz that is equally split across the hybridization order
parameter $\Phi$ and magnetic order $\bm{M}$ in the $Mo$ layer, which can
induce a non-zero polarization in the $W$ layer $\bm{m}$. The HFL is captured
by a non-vanishing amplitude of the hybridization
$\Phi_{\bm{r}}\equiv-\sum_{j=1}^{3}\langle
c^{{\dagger}}_{\bm{r}+\bm{\delta}_{j}}\sigma_{0}\chi_{\bm{r}}\rangle/2$; the
magnetic order is characterized by the variational parameters
$\bm{M}_{\mathbf{r}}=(M^{\parallel}\cos\mathbf{Q}\cdot\mathbf{r},M^{\parallel}\sin\mathbf{Q}\cdot\mathbf{r},M^{z})$
and
$\bm{m}_{\mathbf{r}}=(m^{\parallel}\cos\mathbf{Q}\cdot\mathbf{r},m^{\parallel}\sin\mathbf{Q}\cdot\mathbf{r},m^{z})$
with $\mathbf{Q}=\bm{\kappa}-\bm{\kappa}^{\prime}$,
$M^{\parallel}=\sum_{\mathbf{k}}[\langle\chi^{\dagger}_{\mathbf{k}}\sigma_{+}\chi_{\mathbf{k}+\mathbf{Q}}\rangle+c.c.]/2N$,
$M^{z}=\sum_{\mathbf{k}}\langle\chi^{\dagger}_{\mathbf{k}}\sigma_{z}\chi_{\mathbf{k}}\rangle/2N$,
$m^{\parallel}=\sum_{\mathbf{k}}[V^{*}_{\mathbf{k}}\langle
c^{\dagger}_{\mathbf{k}}\sigma_{+}c_{\mathbf{k}+\mathbf{Q}}\rangle
V_{\mathbf{k}+\mathbf{Q}}+c.c.]/2N$ and
$m^{z}=\sum_{\mathbf{k}}V^{*}_{\mathbf{k}}\langle
c^{\dagger}_{\mathbf{k}}\sigma_{z}c_{\mathbf{k}}\rangle V_{\mathbf{k}}/2N$. To
obtain the values of the variational parameters we minimize the mean-field
free-energy as detailed in the SM [32].

Figure 2: Competition between HFL and magnetic order. (a) Phase diagram in
plane of WSe2 filling $x$ vs $J_{H}/t_{W}$ calculated from solution of mean
field equations. Blue does (white background) indicate regions of small Fermi
surface metallic phase coexisting with 120∘ antiferromagnetic metallic order;
red points (orange background) indicate regions of large Fermi surface heavy
Fermi liquid phase. (b) Line cut at $J_{H}/t_{W}=0.025$ showing $x$-dependence
of the mean-field order parameters: hybridization $\Phi$, local moment
staggered magnetization $M^{\parallel}$ and itinerant electron staggered
magnetization $|m^{\parallel}|$. $m^{\parallel}$ is opposite to
$M^{\parallel}$; we plot the absolute value of both quantities. The inset
shows the effective Kondo exchange as a function of the doping $x$. A first
order transition happens at the critical doping $x_{c}\simeq 0.45$
corresponding to $\rho_{\epsilon_{F}}\bar{J}_{K}\simeq 0.14$. (c) Energy of
the two different phases as a function of $x$. (d) Evolution of the average
quasiparticle mass as a function of $x$ in logarithmic scale. The data are
obtained setting $J_{K}/t_{W}=1$ (bandwidth $9t_{W}$).

Fig. 2(a) shows the calculated mean-field phase diagram in the plane $x$
versus $J_{H}/t_{W}$. The evolution of the mean-field parameters along the
line cut, red line in Fig. 2(a), is displayed in Fig. 2(b). In the small
doping regime $x<x_{c}$ the mean-field minimum describes a magnetic solution
with in-plane 120∘ AFM (AFM-xy). The antiferromagnetic sign of $J_{K}$ means
that the conduction electron staggered magnetization $m^{\parallel}$ is
directed oppositely to $\bm{M}$ shown as the orange line in Fig. 2(b). As
doping $x$ increases, $\rho_{\epsilon_{F}}\bar{J}_{K}$ grows as shown in the
inset of Fig. 2(b), driving a transition of the general type discussed by
Doniach [38] to a non-magnetic Kondo lattice state in which the system becomes
paramagnetic $\bm{M}=0$ and the localised moments hybridize with the
conduction electrons giving rise to a large Fermi surface of heavy
quasiparticles.

The transition between the AFM-xy phase and the HFL is first order: both the
AFM-xy order parameter and the hybridization change discontinuously and the
computed energies cross [Fig. 2(c)]. Across the phase transition the topology
of the Fermi surface changes from electron like in the magnetic phase to hole
like in the HFL phase. To gain insight on the nature of the transition, it is
instructive to compare the low-doping behaviour
$\rho_{\epsilon_{F}}\bar{J}_{K}\ll 1$ of the two energy scales in the Kondo
lattice model: the Kondo temperature $T_{K}$ and the magnetic energy
$E_{AFM}$. Expanding to close to the bottom of the conduction band we have
$T_{K}\simeq\epsilon_{F}\exp\left[-1/(\rho_{\epsilon_{F}}\bar{J}_{K})\right]$
where for a quadratic dispersion the DOS is constant
$\rho_{\epsilon_{F}}\simeq\rho_{0}=3\sqrt{3}a^{2}_{M}m_{W}/(4\pi h^{2})$,
$\epsilon_{F}\simeq x/(2\rho_{0})$ and $m_{W}=\hbar^{2}/(9t_{W}a^{2}_{M})$. On
the other hand, in the magnetic state the Kondo coupling leads to a staggered
polarization of conduction electrons which lowers the energy of the magnetic
state: $E_{AFM}\simeq-3J_{H}M^{2}/2-\bar{J}^{2}_{K}\rho_{\epsilon_{F}}M^{2}/2$
($M=1/2$). The linear dependence of $\bar{J}_{K}$ on doping $x$ [Eq. (2)]
implies that the scaling of the magnetic to paramagnetic transition is
different from the standard Doniach scaling [38]. Finally, Fig. 2(c) shows the
evolution of the average over the FS of the quasiparticle mass $\bar{m}^{*}$
in logarithmic scale, where $m^{*}$ is defined as
$m^{*}=\hbar|\mathbf{k}_{F}|/|\bm{v}_{F}|$,
$\bar{m}^{*}=\oint_{FS}dk_{t}m^{*}/(2\pi k_{F})$ with $\bm{v}_{F}$ and
$\mathbf{k}_{F}$ Fermi velocity and momentum, respectively. The transition is
signalled by a drastic change of the quasiparticle mass. Interestingly, we
find a diminution of $\bar{m}^{*}$ increasing the doping $x$ and a splitting
of the quasiparticle mass in the magnetic regime.

_Physical properties.—_ The HFL is characterized by the quasiparticle band
structure in Fig. 3(a). In this regime the local moments in $Mo$ layer
participate to the total volume [39] enclosed by the FS and give rise to a
large hole-like FS which encircles the $\bm{\kappa}^{\prime}(\bm{\kappa})$
point for $\uparrow(\downarrow)$ electrons in the moiré Brillouin zone as
shown by the solid red line in Fig. 3(b). We also show the variation of the
quasiparticle mass around the Fermi surface in Fig. 3(c). In addition, we find
quite unconventional properties that trace back to the form factor
$V_{\mathbf{k}}$ in the Kondo coupling $J_{\mathbf{k},\mathbf{p}}$. The chiral
nature of $V_{\mathbf{k}}$ gives rise to a chiral hybridization order
parameter whose amplitude is proportional to $\langle
c^{\dagger}_{\bm{\kappa}+\mathbf{k}\uparrow}\chi_{\bm{\kappa}+\mathbf{k}\uparrow}\rangle\propto\Phi(k_{x}-ik_{y})$
for spin up and $\langle
c^{\dagger}_{\bm{\kappa}^{\prime}+\mathbf{k}\downarrow}\chi_{\bm{\kappa}^{\prime}+\mathbf{k}\downarrow}\rangle\propto-\Phi(k_{x}+ik_{y})$
for spin down and results into the topological character of the hybridization
gap. The color code in Fig. 3(b) shows the Berry curvature
$\Omega_{\mathbf{k}\uparrow}=-2\imaginary\innerproduct{\partial_{k_{x}}u_{\mathbf{k}\uparrow}}{\partial_{k_{y}}u_{\mathbf{k}\uparrow}}$
of the lower HFL band which is characterized by bright peaks at the position
of the bare conduction electron FS. The opposite winding of spin $\uparrow$
and $\downarrow$ hybridization gap results in an opposite Berry curvature for
the two spin $\Omega_{\mathbf{k}\uparrow}=-\Omega_{-\mathbf{k}\downarrow}$.
The Berry curvature originates from the chiral interlayer hybridization
between $\chi$ and $c$ fermions. Expanding the heavy Fermi liquid Hamiltonian
around the electron pocket at $\bm{\kappa}$ for $\uparrow$ gives in the basis
$\Psi=(\chi,c)$:

$H_{\uparrow}(\bm{\kappa}+\mathbf{k})\simeq\left(\begin{matrix}\lambda&-\Delta_{K}(k_{x}-ik_{y})\\\
-\Delta^{*}_{K}(k_{x}+ik_{y})&\hbar^{2}k^{2}/(2m_{W})-\mu\end{matrix}\right),$
(3)

with $\Delta_{K}=3J_{K}a_{M}\Phi/2$ and due to time-reversal symmetry
$H_{\downarrow}(\bm{\kappa}^{\prime}+\mathbf{k})=H^{*}_{\uparrow}(\bm{\kappa}-\mathbf{k})$.
The HFL is adiabatically connected to a topological compensated Kondo
semimetal at total filling $n=2$ with a non-quantized spin Hall effect [40].
The topological gap found here is distinct from the topological Kondo gap in
bulk heavy fermion systems [41, 42] such as SmB6 [43, 44, 45, 46], which is
induced by the the strong spin-orbit coupling and the opposite parity of $d$
and $f$ orbitals. In the system considered here the orbitals of conduction
electrons and local moments have identical parity character and the
topological hybridization gap originates from the non-local exchange involving
nearest neighbour sites and the Ising SOC.

Figure 3: Properties of the HFL. (a) Band structure of the heavy Fermi liquid
along high-symmetry directions where the chiral Kondo coupling opens a
topological hybridization gap between the upper (orange) and lower (blue)
bands. The energy is measured with respect to the chemical potential. (b)
Berry curvature of the lower heavy Fermi liquid band, where the red circles
denote the hole Fermi surface. The flux of $\Omega_{\mathbf{k}\uparrow}$ in
the entire Brillouin zone gives Chern number $C_{\uparrow}=1=-C_{\downarrow}$.
(c) Quasiparticle effective mass $m^{*}/m_{W}$ as a function of the angle
$\phi$ on the heavy electrons Fermi surface. The calculation has been
performed at $x=0.46$ and $J_{K}/t_{W}=1$ (bandwidth $9t_{W}$).

In the opposite regime, at $x<x_{c}$ the ground state has a 120∘ AFM order
with a small Fermi surface. The spin-flip scattering processes mediated by the
modulation $\mathbf{Q}$ connecting the spin $\uparrow$ and $\downarrow$ FSs
give rise to a SOC term in the conduction electron Hamiltonian. As detailed in
the SM [32] the expansion of the Hartree-Fock Hamiltonian close to the origin
of the magnetic Brillouin zone gives:

$\begin{split}\mathcal{H}_{c}(\mathbf{k})=\left(\frac{\hbar^{2}k^{2}}{2m_{W}}-\epsilon_{F}\right)\sigma_{0}+\frac{9J_{K}a^{2}_{M}}{8}\sum_{a}d_{a}(\mathbf{k})\sigma_{a},\end{split}$
(4)

where $d_{z}(\mathbf{k})=M^{z}(k^{2}_{x}+k^{2}_{y})$,
$d_{x}(\mathbf{k})=-M^{\parallel}(k^{2}_{x}-k^{2}_{y})$ and
$d_{y}(\mathbf{k})=2M^{\parallel}k_{x}k_{y}$. The SOC splits the conduction
electron FS giving rise to two different Fermi momenta $k^{\pm}_{F}$. A
nonzero in-plane $M^{\parallel}$ component in the Hamiltonian (4) induces a
$2\pi$-Berry phase winding around the origin of the magnetic Brillouin zone.
In the presence of a nonzero Zeeman field we have a Berry curvature whose
momentum space distribution depends on the value of the parameters in the
Hamiltonian (4).

Figure 4: Phase diagram and transport in magnetic field. (a) Phase diagram
plotted as a function of the $W$-layer carrier concentration $x$ and the out-
of-plane magnetic field $h_{W}$. The blue region denotes the HFL regime where
the hybridization order parameters is finite. The color code before the green
line shows the value of the in-plane antiferromagnetic order parameter
$M^{\parallel}$. The solid green line shows the first order phase transition
line. On the other hand, the magneta one shows the second order transition
from the canted AFM-xy to the FM-z. (b) Evolution of the variational parameter
along the line cut at $x=0.52$ as a function of $h_{W}/t_{W}$. (c) Intrinsic
contribution to the anomalous Hall conductivity for three different values of
the Zeeman field. The vertical dashed lines denote the critical filling where
the first order transition takes place. (d) Intrinsic AHE for different values
of the filling $x$ as a function of the Zeeman field. The vertical dashed line
denotes the first order transition between the magnetic and the HFL phase,
while the solid one the second order transition from a canted-AFM to the FM-z.
Above this line the AHE vanishes. The calculation has been performed at
$J_{K}/t_{W}=1$, $J_{H}/t_{W}=0.05$ (bandwidth $9t_{W}$).

Finally, we study the evolution of the mean-field solution in an external out-
of-plane magnetic field. The spin-valley locking in the AB-stacked
configuration implies that the magnetic field $B$ acts as a spin-valley Zeeman
field [3] $H_{z}=-h_{Mo}\sum_{\mathbf{r}\in
Mo}\chi^{\dagger}_{\mathbf{r}}\sigma^{z}\chi_{\mathbf{r}}/2+h_{W}\sum_{\mathbf{r}\in
W}c^{\dagger}_{\mathbf{r}}\sigma_{z}c_{\mathbf{r}}/2$ where
$h_{\alpha}=g_{\alpha}\mu_{B}B$ and $g_{\alpha}$ is the gyromagnetic ratio for
the two different TMDs (here we use use $g_{Mo}=g_{W}=10$ [3, 47]). The phase
diagram in the plane $x$ vs $B$ is shown in Fig. 4(a). At low-doping a small
magnetic field introduces an out-of-plane magnetization $M^{z}$ giving rise to
a canted AFM-xy. Above the second order transition line (magenta line in Fig.
4(a)), that in the limit of vanishing doping takes the value $9J_{H}/2$, the
canted AFM-xy turns into a ferromagnetic solution along $z$ (FM-z). For larger
fillings, blue region in Fig. 4(a) starting at $x_{c}\simeq 0.488$ in the
limit $h_{W}\to 0$, the small-field solution is the HFL. The line cut Fig.
4(b), red line at $x=0.52$ in Fig. 4(a), shows the evolution of the
variational parameters in magnetic field. In the HFL a small magnetic field
induces a net magnetization $M^{z}$ in the localized orbital $\chi$. On the
other hand, the Kondo hybridization is slightly affected by the external
field. At a critical field displayed as a solid green line in Fig. 4(a) we
find a first-order transition that is accompanied by an abrupt jump in the
magnetization as well as by a discontinuous change in the conduction electron
Fermi surface. The state above the first order line is a canted AFM-xy for
$x<0.56$ and a FM-z for $x>0.56$.

_Transport in magnetic field.—_ We use a Boltzmann equation approach to
describe the transport of electrons in the different regimes of the phase
diagram in Fig. 4(a), for details we refer to SM [32]. The Hall conductivity
has contribution from an Ohmic part $\sigma_{xy}^{\rm Ohm}$ which depends on
the extrinsic impurity scattering rate, and also has an intrinsic geometric
contribution [48, 49] $\sigma_{xy}^{\rm AH}$ determined by the Berry
curvature. In the HFL regime the non-vanishing hybridization implies that the
neutral $\chi$-spinon also contributes to the charge transport properties [50,
51]. Approximating the HFL FS shown in Fig. 3(b) with a circular FS with
effective mass $\bar{m}^{*}$ the Ohmic contribution takes the simple form
$\sigma_{xx}=e^{2}\tau(1-x)/\bar{m}^{*}$ and $\sigma_{xy}^{\rm
Ohm}=-e^{3}B\tau^{2}(1-x)/{\bar{m}^{*}}{}^{2}$. Conversely, in the magnetic
regime only the concentration $x$ of conduction electrons contributes to the
charge transport properties. Assuming a single momentum- and band-independent
transport time we find $\sigma_{xx}=e^{2}\tau
x/\left(\sum_{\lambda}m_{\lambda}/2\right)$ and $\sigma^{\rm
Ohm}_{xy}=e^{3}\tau^{2}Bx/\left(m_{W}\sum_{\lambda}m_{\lambda}/2\right)$ with
$m_{\lambda}=m_{W}/[1-\lambda J_{K}|M|/(4t_{W})]\,(\lambda=\pm)$. As a result,
increasing the magnetic field across the first order transition line [green
line in Fig. 4(a)-(b)] we find a drastic jump in electrical conductivities,
while in the HFL we have $1-x$ hole-like carriers in the AFM-xy phase a
density of $x$ electrons. We observe that the normal Hall effect has been
employed in heavy-Fermi liquid systems [52, 53, 54] as a proxy for the Fermi
surface change.

Moreover, the AFM-xy and the HFL are characterized by different intrinsic
anomalous Hall conductivity $\sigma^{\mathrm{AH}}_{xy}$. Figs. 4(c) shows the
evolution of $\sigma^{\mathrm{AH}}_{xy}$ along different line cuts at constant
Zeeman field as a function of doping $x$ in the phase diagram Fig. 4(a). In
the AFM regime, $\sigma^{\mathrm{AH}}_{xy}$ originates from the
$d_{x}(\mathbf{k})$ and $d_{y}(\mathbf{k})$ terms in Eq. (4) induced by the
in-plane component of the magnetic ordering which gives rise to the anomalous
Hall effect (AHE) shown in Fig. 4(c). As we increase the doping the AHE
decreases due to the smaller Berry curvature flux imbalance between the two
Fermi surfaces with momenta
$k^{\lambda}_{F}=\sqrt{2m_{\lambda}\epsilon_{F}/\hbar^{2}}$. The anomalous
Hall effect is enhanced by interaction effects between conduction electrons
which favours the out-of-plane canting of the AFM spin texture. Finally, above
a critical density shown by the solid green line in the phase diagram Fig.
4(a) a first order transition from the canted AFM-xy to the HFL occurs. The
transition is signalled by a jump in the AHE effect as displayed in Fig. 4(c).
A non-quantized jump of the AHE across the heavy Fermi liquid transition can
also originate from a chiral spin liquid state [55]. In the HFL regime, the
anomalous Hall effect originates from the chiral hybridization gap with
opposite sign for spin up and down electrons which results in a non-quantized
spin Hall effect. See SM [32] for details. A nonzero Zeeman field induces a
flux imbalance between $\uparrow$ and $\downarrow$ which gives a finite AHE in
the HFL as you can see in Fig. 4(c). In Fig. 4(d) we consider different
concentration in the $W$-layer and we study the evolution as a function of
$h_{W}$. We highlight that in the large field regime we enter in the FM-z
regime where the AHE vanishes as displayed in the regions above the solid
vertical lines in Fig. 4(c). The transition from the HFL to the FM-z is
characterized transition by a drastic jump in the AHE effect which can reach
values of the order of $\sim e^{2}/h$ depending on the flux imbalance between
the two Fermi surfaces. On the other hand, in the small Fermi surface regime
the AHE evolves smoothly vanishing when the local moments are aligned to the
external field.

_Discussion.—_ We present a microscopic theory to explain the competition
between AFM and the HFL in MoTe2/WSe2. A crucial finding is that the Kondo
exchange is chiral, which gives rise to predictions that can be directly
tested in experiments. Among them we emphasise the topological character of
the hybridization gap in the heavy Fermi liquid regime and the Berry phase
winding induced by spin-flip processes in the magnetic one. Both effects give
rise to AHE which can be measured by transport experiments in magnetic field.
We show how transport measurements can clearly distinguish the various phases.

We find two different first order transitions: either as a function of the
doping $x$ at zero magnetic field, or by tuning a magnetic field
(metamagnetism). The first order character of either transition is generically
robust for a finite region of fluctuation corrections to mean field theory.
However, while symmetry implies the magnetic field tuned transition is first
order, there is no symmetry argument known to us that constrains the order of
the doping tuned transition. Our theory for the field tuned transition is in
agreement with recent experimental observations [56] where at a critical field
$B_{c}$ the heavy Fermi liquid undergoes a metamagnetic transition with a
sharp jump in the carrier concentration from $x-1$ to $x$ and an abrupt
variation of the quasiparticle mass at a critical magnetic field $B_{c}$. We
also find that at $B_{c}$ the transition between the magnetic state and the
heavy Fermi liquid is signalled by a drastic jump of the AHE. Additional
experiments are needed to unveil the nature of the transition as a function of
the doping at zero field.

We observe that the paramagnetic HFL solution becomes at filling $n=2$ a
compensated topological Kondo semimetal with non-quantized quantum spin Hall
effect. We also notice that in the magnetic regime the AHE can be further
enhanced by spontaneous ferromagnetism induced by accounting the on-site
interaction between conduction electrons. Finally, the effect of quantum
fluctuations beyond the mean-field approach as well as the role of anisotropic
effects in the Kondo exchange coupling are important questions for future
research.

These results highlight several concrete experimental predictions relevant for
current and future experimental studies in TMDs bilayers. More, broadly this
work provides a controlled route to realize a topological selective Mott
transition.

###### Acknowledgements.

_Acknowledgements.—_ We thank Kin Fai Mak and Jie Shan for sharing with us
experimental data prior to its publication and for insightful discussions. We
have benefited from discussions with Elio König, Piers Coleman, John Sous,
Raquel Queiroz, Corentin Bertrand and Roberto Raimondi. This work was
partially supported by the Air Force Office of Scientific Research under Grant
No. FA9550-20-1-0260 (J.C.) and Grant No. FA9550-20-1-0136 (J.H.P.) and the
Alfred P. Sloan Foundation through a Sloan Research Fellowship (J.H.P.).
J.H.P. acknowledges the Aspen Center for Physics, where some of this work was
completed, which is supported by National Science Foundation grant
PHY-1607611. J.Z. and A.J.M. acknowledges support from the NSF MRSEC program
through the Center for Precision-Assembled Quantum Materials (PAQM) -
DMR-2011738. Flatiron Institute is a division of the Simons Foundation.
_Author contributions.—_ DG, JW, JZ, JC JP and AM conceptualized the research.
DG and JW developed the theoretical model and performed the numerical
calculations. DG, JW, JZ, JC JP and AM analysed the results and edited the
first draft. _Statement on competing interests.—_ All authors declare that
they have no competing interests. _Data and materials availability.—_ All data
needed to evaluate the conclusions in the paper are present in the paper
and/or the Supplementary Materials.

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Supplementary material for “ ”

Daniele Guerci§, Jie Wang§, Jiawei Zhang, Jennifer Cano, J. H. Pixley and
Andrew Millis

§These authors contributed equally.

These supplementary materials contain the details of analytic calculations as
well as additional numerical details supporting the results presented in the
main text.

##  Appendix A The Kondo lattice Hamiltonian

In this section we detail the connection between the tight-binding model and
the Schrieffer-Wolff transformation to obtain the Kondo lattice Hamiltonian
given in the main text.

###  A.1 The tight-binding Hamiltonian

Assuming $W$-layer weakly correlated the tight-binding Hamiltonian that
describes the low-energy properties of the system reads $H=H_{0}+H_{W}+H_{t}$:

$\begin{split}H_{0}=&-\frac{\Delta}{2}(N_{Mo}-N_{W})+U\sum_{\mathbf{r}\in
Mo}n_{\mathbf{r}\uparrow}n_{\mathbf{r}\downarrow}\\\
H_{W}=&-t_{W}\sum_{\langle\mathbf{r},\mathbf{r}^{\prime}\rangle\in
W}e^{-i\nu_{\mathbf{r},\mathbf{r}^{\prime}}2\pi\sigma/3}c^{\dagger}_{\mathbf{r}\sigma}c_{\mathbf{r}^{\prime}\sigma},\\\
H_{t}=&-t_{Mo}\sum_{\langle\mathbf{r},\mathbf{r}^{\prime}\rangle\in
Mo}f^{\dagger}_{\mathbf{r}\sigma}f_{\mathbf{r}^{\prime}\sigma}-t_{h}\sum_{\langle\mathbf{r},\mathbf{r}^{\prime}\rangle}f^{\dagger}_{\mathbf{r}}c_{\mathbf{r}^{\prime}}.\end{split}$
(1)

We have introduced the operator $f_{\mathbf{r}\sigma}$ for the $Mo$ and
$c_{\mathbf{r}\sigma}$ for $W$-layer, respectively. We notice that differently
from $H_{0}$ and $H_{W}$ the contribution $H_{t}$ changes the local
configuration in the $Mo$ layer.

###  A.2 The Schrieffer-Wolff transformation

We assume that $U$ is the largest energy scale, $U\gg t_{Mo}$, and
$\Gamma(\epsilon_{F})/\Delta<1$ where $\Gamma(\epsilon_{F})$ is the
hybridization function:

$\Gamma(\epsilon_{F})=\pi t^{2}_{h}\rho(\epsilon_{F})\langle
V^{*}_{\mathbf{k}}V_{\mathbf{k}}\rangle_{FS}=\pi
t^{2}_{h}\left(\frac{3\sqrt{3}a^{2}_{M}}{8\pi^{2}}\right)\oint_{FS}dk_{t}\frac{V^{*}_{\mathbf{k}}V_{\mathbf{k}}}{|\nabla_{\mathbf{k}}\epsilon_{\mathbf{k}\sigma}|}.$
(2)

Fig. S1 show the evolution of $\Gamma(\epsilon_{F})$ as a function of the
filling $x$ for the value of $t^{2}_{h}/\Delta$ corresponding to
$J_{K}/t_{W}=1$ value used for the numerical calculations shown in the
manuscript. Within these assumptions valence fluctuations in $Mo$ layer are
suppressed. We now observe that the tunneling $H_{t}$ can be decomposed as:

$H_{t}=\sum_{q=-1}^{1}\sum_{d=-1}^{1}T_{q,d},$ (3)

where $T_{q,d}$ gathers all tunneling events that change the number of
electrons unbalance between $Mo$ and $W$ by $q$ and the double occupancies in
$Mo$ layer by $d$. We list the $T_{q,d}$ operators below:

$\begin{split}&T_{+1,+1}=-t_{h}\sum_{\mathbf{r}\in
Mo}\sum_{\sigma}\sum_{j=1}^{3}n_{\mathbf{r}\bar{\sigma}}f^{\dagger}_{\mathbf{r}\sigma}c_{\mathbf{r}+\bm{\delta}_{j}\sigma},\\\
&T_{+1,0}=-t_{h}\sum_{\mathbf{r}\in
Mo}\sum_{\sigma}\sum_{j=1}^{3}h_{\mathbf{r}\bar{\sigma}}f^{\dagger}_{\mathbf{r}\sigma}c_{\mathbf{r}+\bm{\delta}_{j}\sigma},\\\
&T_{0,+1}=-t_{Mo}\sum_{\mathbf{r}\in
Mo}\sum_{\sigma}\sum_{j=1}^{3}\left[n_{\mathbf{r}\bar{\sigma}}f^{\dagger}_{\mathbf{r}\sigma}f_{\mathbf{r}+\bm{\gamma}_{j}\sigma}h_{\mathbf{r}+\bm{\gamma}_{j}\bar{\sigma}}+n_{\mathbf{r}+\bm{\gamma}_{j}\bar{\sigma}}f^{\dagger}_{\mathbf{r}+\bm{\gamma}_{j}\sigma}f_{\mathbf{r}\sigma}h_{\mathbf{r}\bar{\sigma}}\right],\\\
&T_{0,0}=-t_{Mo}\sum_{\mathbf{r}\in
Mo}\sum_{\sigma}\sum_{j=1}^{3}\left[n_{\mathbf{r}\bar{\sigma}}f^{\dagger}_{\mathbf{r}\sigma}f_{\mathbf{r}+\bm{\gamma}_{j}\sigma}n_{\mathbf{r}+\bm{\gamma}_{j}\bar{\sigma}}+h_{\mathbf{r}\bar{\sigma}}f^{\dagger}_{\mathbf{r}\sigma}f_{\mathbf{r}+\bm{\gamma}_{j}\sigma}h_{\mathbf{r}+\bm{\gamma}_{j}\bar{\sigma}}+h.c.\right],\end{split}$
(4)

we notice that $T_{-q,-d}=T^{\dagger}_{q,d}$,
$n_{\mathbf{r}\sigma}=f^{\dagger}_{\mathbf{r}\sigma}f_{\mathbf{r}\sigma}$ and
$h_{\mathbf{r}\sigma}=1-n_{\mathbf{r}\sigma}$. Moreover, the terms $T_{-1,+1}$
and $T_{+1,-1}$ vanish. Before moving on, we notice that the intralayer
hopping $H_{Mo}$ does not change the charge imbalance $N_{Mo}-N_{W}$ ($q=0$),
i.e. $H_{Mo}$ commutes with $N_{Mo}-N_{W}$ ($[N_{Mo}-N_{W},H_{Mo}]=0$).
Moreover, $[N_{Mo}-N_{W},T_{q,d}]=2qT_{q,d}$ and

$\sum_{\mathbf{r}\in
Mo}[n_{\mathbf{r}\uparrow}n_{\mathbf{r}\downarrow},T_{q,d}]=dT_{q,d}.$ (5)

We seek an unitary transformation $U=\exp(-iS)$ which eliminates hops between
states with different numbers of doubly occupied sites and interlayer charge
imbalance [33, 34, 57]:

$\bar{H}\equiv e^{iS}He^{-iS}=H+[iS,H]+\frac{[iS,[iS,H]]}{2!}+\cdots,$ (6)

where we applied the Baker-Campbell-Haussdorf formula. We now notice that to
the lowest order in the expansion we have:

$[H_{0},iS]=\sum_{q,d}^{(q,d)\neq 0}T_{q,d},$ (7)

where in previous sum we exclude the term with $q=0$ and $d=0$ in shorthand
notation $(q,d)\neq 0$. We readily realize that the solution of Eq. (7) reads:

$S=-i\sum_{q,d}^{(q,d)\neq 0}\frac{T_{q,d}}{dU-q\Delta}.$ (8)

From the latter expression we find:

$\begin{split}\bar{H}=&H_{0}+T_{0,0}+H_{W}+\frac{1}{2}\sum_{q,d}^{(q,d)\neq
0}\sum_{q^{\prime},d^{\prime}}^{(q^{\prime},d^{\prime})\neq
0}\frac{[T_{q,d},T_{q^{\prime},d^{\prime}}]}{-q\Delta+dU}+\sum_{q,d}^{(q,d)\neq
0}\frac{[T_{q,d},H_{W}+T_{0,0}]}{-q\Delta+dU}+O([H_{W}+T_{0,0}]S^{2}).\end{split}$
(9)

Figure S1: Hybridization function. Evolution of $\Gamma(\epsilon_{F})/\Delta$
as a function of the filling $x$ of the conduction band.

Projecting the model in the low-energy subspace with one-electron in $Mo$
layer, i.e. $n_{\mathbf{r}}=1$ for $\mathbf{r}\in Mo$, we find that the latter
term vanishes. Furthermore, the projection constraints $q^{\prime}=-q$ and
$d^{\prime}=-d$:

$\bar{H}=-t_{W}\sum_{\langle\mathbf{r},\mathbf{r}^{\prime}\rangle\in
W}e^{-i\nu_{\mathbf{r},\mathbf{r}^{\prime}}2\pi
s_{\sigma}/3}c^{\dagger}_{\mathbf{r}\sigma}c_{\mathbf{r}^{\prime}\sigma}-\mu
N_{W}+\frac{1}{2}\sum_{q,d}^{(q,d)\neq
0}\frac{[T_{q,d},T_{-q,-d}]}{dU-q\Delta},$ (10)

where the constant energy term $\Delta N_{W}/2$ has been absorbed in the
chemical potential shift and $T_{0,0}$ vanishes in the low-energy subspace. By
performing straightforward calculations we find that the commutator in Eq.
(10) gives:

$\begin{split}\frac{1}{2}\sum_{q,d}^{(q,d)\neq
0}\frac{[T_{q,d},T_{-q,-d}]}{dU-q\Delta}=&t^{2}_{h}\left(\frac{1}{\Delta}+\frac{1}{U-\Delta}\right)\sum_{\mathbf{r}\in
Mo}\sum_{jl=1}^{3}\bm{S}_{\mathbf{r}}\cdot
c^{\dagger}_{\mathbf{r}+\bm{\delta}_{j}\sigma}\,\bm{\sigma}_{\sigma\sigma^{\prime}}\,c_{\mathbf{r}+\bm{\delta}_{l}\sigma^{\prime}}+\frac{4t^{2}_{Mo}}{U}\sum_{\langle\mathbf{r},\mathbf{r}^{\prime}\rangle\in
A}\bm{S}_{\mathbf{r}}\cdot\bm{S}_{\mathbf{r}^{\prime}}\\\
&+t^{2}_{h}\left(\frac{1}{\Delta}-\frac{1}{U-\Delta}\right)\sum_{\mathbf{k}}|V_{\mathbf{k}}|^{2}c^{\dagger}_{\mathbf{k}}c_{\mathbf{k}}.\end{split}$
(11)

Expanding close to the bottom of the conduction band
[$\kappa(\kappa^{\prime})$ for $\uparrow(\downarrow)$]
$|V_{\kappa/\kappa^{\prime}+\mathbf{k}}|^{2}\simeq 9(a_{M}k)^{2}/4$ we realize
that the latter terms can be simply accounted by a redefinition of the bare
mass $m_{W}\to m_{W}/[1+(t_{h}/\Delta-t_{h}/(U-\Delta))t_{h}/(2t_{W})]$.
Finally, we find the effective spin-fermion Hamiltonian:

$\bar{H}=\sum_{\mathbf{k}}\xi_{\mathbf{k}\sigma}c^{\dagger}_{\mathbf{k}\sigma}c_{\mathbf{k}\sigma}+J_{H}\sum_{\langle\mathbf{r},\mathbf{r}^{\prime}\rangle\in
A}\bm{S}_{\mathbf{r}}\cdot\bm{S}_{\mathbf{r}^{\prime}}+\frac{1}{2N}\sum_{\mathbf{r}\in\text{Mo}}\sum_{\mathbf{k},\mathbf{p}}e^{-i(\mathbf{k}-\mathbf{p})\cdot\mathbf{r}}J_{\mathbf{k},\mathbf{p}}\bm{S}_{\mathbf{r}}\cdot
c^{\dagger}_{\mathbf{k}\sigma}\,\bm{\sigma}_{\sigma\sigma^{\prime}}\,c_{\mathbf{p}\sigma^{\prime}},$
(12)

where $J_{H}=4t^{2}_{Mo}/U$,
$J_{\mathbf{k},\mathbf{p}}=J_{K}V^{*}_{\mathbf{k}}V_{\mathbf{p}}$,
$J_{K}=2t^{2}_{h}[1/\Delta+1/(U-\Delta)]$,
$\xi_{\mathbf{k}\sigma}=\epsilon_{\mathbf{k}\sigma}-\epsilon_{F}$,
$\epsilon_{\mathbf{k}\sigma}$ is the electron dispersion and $\epsilon_{F}$
fixes the number of electron in the conduction band. The dispersion relation
reads
$\epsilon_{\mathbf{k}\sigma}=-2t_{W}\sum^{3}_{j=1}\cos(\mathbf{k}\cdot\bm{a}_{i}+2\pi
s_{\sigma}/3)$, $\bm{a}_{1}=\sqrt{3}a_{M}(1,0)$,
$\bm{a}_{2,3}=\sqrt{3}a_{M}\left(-1/2,\pm\sqrt{3}/2\right)$ are the lattice
vectors with $a_{M}=5$nm the moiré cell lattice constant. The form factor is
$V_{\mathbf{k}}=\sum_{j=1}^{3}e^{i\mathbf{k}\cdot\bm{\delta}_{j}}$. Despite
$U$ is large and $J_{H}$ small the particular form of the exchange interaction
$J_{\mathbf{k},\mathbf{p}}=J_{K}V^{*}_{\mathbf{k}}V_{\mathbf{p}}$ gives rise
to a non-trivial competition between a low-density magnetic phase and a
paramagnetic heavy Fermi liquid. We observe that in the following we will
measure the value of $\Delta$ with respect to the bottom of conduction band
$-6t_{W}$.

####  A.2.1 On the higher-order corrections

Higher order corrections in the expansion lower the symmetry of the spin
interaction in the $Mo$-layer. This can be simply realized noticing that the
phase factor in the Hamiltonian $H_{W}$ lower the spin-symmetry of the model
to $U(1)$ around $z$. Including higher-order hopping processes mediated by
$t_{h}$ gives an XXZ spin model with Dzyaloshinskii–Moriya interactions:

$H_{S}=J_{H}\sum_{\langle\mathbf{r},\mathbf{r}^{\prime}\rangle\in\text{Mo}}\left(S^{z}_{\mathbf{r}}S^{z}_{\mathbf{r}^{\prime}}+\gamma
S^{+}_{\mathbf{r}}S^{-}_{\mathbf{r}^{\prime}}+h.c.\right)+D\sum_{\langle\mathbf{r},\mathbf{r}^{\prime}\rangle\in\text{Mo}}\left(\bm{S}_{\mathbf{r}}\times\bm{S}_{\mathbf{r}^{\prime}}\right)_{z},$
(13)

we refer to Ref. [8] for additional details. For simplicity we consider the
isotropic limit $\gamma=1/2$ and $D=0$ in our calculations.

##  Appendix B The mean-field approach

In this section we detail the mean-field of Abrikosov fermions discussed in
the main text. We perform the decomposition of the spin-$1/2$ into spinons
$\bm{S}_{\mathbf{r}}=\chi^{\dagger}_{\mathbf{r}\alpha}\bm{\sigma}_{\alpha\beta}\chi_{\mathbf{r}\beta}/2$
and performing the mean-field decomposition in the magnetic and excitonic
channels we obtain the mean-field Hamiltonian:

$\begin{split}\bar{H}_{\rm
mf}=&\lambda\sum_{\mathbf{k}}\chi^{\dagger}_{\mathbf{k}}\chi_{\mathbf{k}}+\frac{h^{z}}{2}\sum_{\mathbf{k}}\chi^{\dagger}_{\mathbf{k}}\sigma^{z}\chi_{\mathbf{k}}+\frac{h^{\parallel}}{2}\sum_{\mathbf{k}}\left[\chi^{\dagger}_{\mathbf{k}+\mathbf{Q}}\sigma^{-}\chi_{\mathbf{k}}+h.c.\right]+\sum_{\mathbf{k}\sigma}\xi_{\mathbf{k}\sigma}c^{\dagger}_{\mathbf{k}\sigma}c_{\mathbf{k}\sigma}\\\
&+\frac{M^{z}}{2}\sum_{\mathbf{k}}J_{\mathbf{k},\mathbf{k}}c^{\dagger}_{\mathbf{k}}\sigma^{z}c_{\mathbf{k}}+\frac{M^{\parallel}}{2}\sum_{\mathbf{k}}\left[J_{\mathbf{k}+\mathbf{Q},\mathbf{k}}c^{\dagger}_{\mathbf{k}+\mathbf{Q}}\sigma^{-}c_{\mathbf{k}}+h.c.\right]\\\
&+J_{K}\sum_{\mathbf{k}}\left[V^{*}_{\mathbf{k}}\Phi^{*}c^{\dagger}_{\mathbf{k}}\chi_{\mathbf{k}}+h.c.\right].\end{split}$
(14)

In the previous expression $M^{z}$ and $M^{\parallel}$ are the out-of-plane
and in-plane components of the magnetization of the local moments, the
magnetic field $h^{z}$ and $h^{\parallel}$ are given by:

$\begin{split}&h^{z}=6M^{z}+\sum_{\mathbf{p}}J_{\mathbf{p},\mathbf{p}}{\langle
c^{\dagger}_{\mathbf{p}}\sigma^{z}c_{\mathbf{p}}\rangle}/N,\\\
&h^{\parallel}=-3M^{\parallel}+\sum_{\mathbf{p}}\left(J_{\mathbf{p},\mathbf{p}+\mathbf{Q}}{\langle
c^{\dagger}_{\mathbf{p}}\sigma^{+}c_{\mathbf{p}+\mathbf{Q}}\rangle}+h.c.\right)/(2N).\end{split}$
(15)

The excitonic order parameter reads:

$\Phi=-\sum_{\mathbf{k}}V^{*}_{\mathbf{k}}\langle
c^{\dagger}_{\mathbf{k}}\chi_{\mathbf{k}}\rangle/(2N).$ (16)

The mean-field free-energy reads:

$\begin{split}F_{\rm
mf}=&-k_{B}T\sum_{\mathbf{k}}\sum_{\lambda}\log\left[1+e^{-E_{\mathbf{k}\lambda}/k_{B}T}\right]+2J_{K}N|\Phi|^{2}+\frac{3J_{H}N}{2}{M^{\parallel}}^{2}\\\
&-3J_{H}N{M^{z}}^{2}-J_{K}M^{z}Nm^{z}-J_{K}NM^{\parallel}m^{\parallel}-\lambda
N+N\mu x,\end{split}$ (17)

where $E_{\mathbf{k}\lambda}$ is the mean-field dispersion relation. The mean-
field solution is obtained by minimizing $F_{\rm mf}$ with respect to the
variational parameters $M^{z},M^{\parallel},\Phi,m^{z},m^{\parallel}$. The
Lagrange multiplier $\lambda$ imposes the Gutzwiller constraint
$\sum_{\mathbf{k}}\langle\chi^{\dagger}_{\mathbf{k}}\chi_{\mathbf{k}}\rangle/N=1$,
while the chemical potential $\mu$ fixes the number of particle in conduction
band $\sum_{\mathbf{k}}\langle
c^{\dagger}_{\mathbf{k}}c_{\mathbf{k}}\rangle/N=x$. Taking the saddle point of
Eq. (17) with respect to the variational parameters gives a set of self-
consistency equations that are solved by find-root algorithm.

###  B.1 Energetics of the HFL and AFM states

In this section we detail the weak coupling expansion to determines the
characteristic energy scales of the AFM and HFL phases that are the RKKY
energy and the Kondo temperature, respectively. We will also introduce the
effective model describing the quasiparticle excitations in the two different
regimes.

####  B.1.1 AFM

In the magnetic regime the local moments $\bm{S}_{\mathbf{r}}$ form a 120∘ AFM
order with
$\langle\bm{S}_{\mathbf{r}}\rangle=\left(M^{\parallel}\cos\mathbf{Q}\cdot\mathbf{r},M^{\parallel}\sin\mathbf{Q}\cdot\mathbf{r},M^{z}\right)$,
$\mathbf{Q}=\kappa-\kappa^{\prime}$, correspondingly the conduction electron
Hamiltonian reads:

$\begin{split}\bar{H}^{c}_{\rm
mf}=&\sum_{\mathbf{k}}\xi_{\mathbf{k}\sigma}c^{\dagger}_{\mathbf{k}\sigma}c_{\mathbf{k}\sigma}+\frac{J_{K}M^{z}}{2}\sum_{\mathbf{k}}|V_{\mathbf{k}}|^{2}c^{\dagger}_{\mathbf{k}}\sigma^{z}c_{\mathbf{k}}+\frac{J_{K}M^{\parallel}}{2}\sum_{\mathbf{k}}\left[V^{*}_{\mathbf{k}}V_{\mathbf{k}+\mathbf{Q}}c^{\dagger}_{\mathbf{k}}\sigma^{+}c_{\mathbf{k}+\mathbf{Q}}+h.c.\right].\end{split}$
(18)

In the limit of low-doping the Fermi surface is a small electron pocket around
$\kappa(\uparrow)$ and $\kappa^{\prime}(\downarrow)$ that are folded into the
origin of the magnetic Brillouin zone $\gamma_{m}$, depicted in Fig. S4(b).
Expanding close to quadratic order around $\gamma_{m}$ and keeping only the
two lowest energy bands we find the continuum model:

$\bar{\mathcal{H}}^{c}_{\rm
mf}=\sum_{\mathbf{k}}c^{\dagger}_{\mathbf{k}}\left[\left(\frac{\hbar^{2}k^{2}}{2m_{W}}-\epsilon_{F}\right)\sigma_{0}+\frac{9J_{K}a^{2}_{M}}{8}\sum_{a}d_{a}(\mathbf{k})\sigma_{a}\right]c_{\mathbf{k}},$
(19)

where $d_{z}(\mathbf{k})=M^{z}k^{2}$,
$d_{x}(\mathbf{k})=-M^{\parallel}(k^{2}_{x}-k^{2}_{y})$ and
$d_{y}(\mathbf{k})=2M^{\parallel}k_{x}k_{y}$. We readily realize that due to
the SOC term the spin is no longer a good quantum number, the eigenstates
$\ket{u_{\mathbf{k}\lambda}}$ are labeled by $\lambda=\pm$ and the
corresponding eigenvalues are
$\epsilon_{\mathbf{k}\lambda}=(\hbar^{2}k^{2}/2m_{W}-\epsilon_{F})+\lambda
9J_{K}a^{2}_{M}k^{2}|M|/8$ with $|M|=\sqrt{(M^{z})^{2}+(M^{\parallel})^{2}}$.
We observe that the theory is O$(3)$ invariant under rotation of the
magnetization $M^{a}\to R_{ab}M^{b}$. The resulting stabilization energy does
not depend on the orientation of the local moments. The Fermi momentum
obtained by setting $\epsilon_{\mathbf{k}\lambda}=0$ reads
$k^{\pm}_{F}=\sqrt{2m_{\lambda}\epsilon_{F}/\hbar^{2}}$ where
$m_{\lambda}=m_{W}/[1+\lambda(J_{K}/8t_{W})]$ and the Fermi energy is
$\epsilon_{F}\simeq x/\left(\sum_{\lambda}\rho_{\lambda}\right)$ with
$\rho_{\lambda}=\rho_{0}m_{\lambda}/m_{W}$. We notice that in the Kondo regime
$J_{K}$ is smaller than the bandwidth of conduction electrons $9t_{W}$ so that
the mass is always positive. The kinetic energy variation with respect to the
normal state reads:

$\delta\epsilon^{c}_{\rm mf}=\epsilon^{c}_{\rm
mf}-\epsilon^{c}_{J_{K}=0}=-\rho_{0}\bar{J}^{2}_{K}|M|^{2}/2,$ (20)

where $\bar{J}_{K}$ is the average over the FS of the Kondo exchange. We
conclude that the total energy per site in the magnetic regime is given by:

$\epsilon_{\rm mf}=-3J_{H}|M|^{2}/2-\rho_{0}\bar{J}^{2}_{K}|M|^{2}/2.$ (21)

The energy gain from the coupling between the conduction band and the local
moments goes quadratically in the electron density $x$. We conclude observing
that interaction effects between conduction electrons in the B sublattice
introduce the tendency to develop a finite out-of-plane ferromagnetic
polarization. The analysis of the effect of interaction between conduction
electrons is left to future studies.

####  B.1.2 HFL

In the paramagnetic regime electrons are described by the mean-field
Hamiltonian reads:

$\bar{H}_{\rm
mf}=\sum_{\mathbf{k}\sigma}\xi_{\mathbf{k}\sigma}c^{\dagger}_{\mathbf{k}\sigma}c_{\mathbf{k}\sigma}+\lambda\sum_{\mathbf{k}}\chi^{\dagger}_{\mathbf{k}}\chi_{\mathbf{k}}+J_{K}\sum_{\mathbf{k}\sigma}\left[V^{*}_{\mathbf{k}}\Phi^{*}c^{\dagger}_{\mathbf{k}}\chi_{\mathbf{k}}+h.c.\right].$
(22)

We easily realize that the Green’s function of the problem reads:

$\bm{G}^{-1}_{\sigma}(\mathbf{k},i\epsilon)=\left(\begin{matrix}i\epsilon-\lambda&-J_{K}\Phi
V_{\mathbf{k}}\\\
-J_{K}\Phi^{*}V^{*}_{\mathbf{k}}&i\epsilon-\xi_{\mathbf{k}\sigma}\end{matrix}\right),$
(23)

so that the saddle-point equation for the bosonic amplitude $\Phi$ can be
written as:

$\Phi=-\frac{T}{2N}\sum_{\mathbf{k}\sigma}\sum_{i\epsilon}V^{*}_{\mathbf{k}}\mathcal{G}_{\chi,\sigma}(\mathbf{k},i\epsilon)J_{K}V_{\mathbf{k}}\Phi
G_{c,\sigma}(\mathbf{k},i\epsilon),$ (24)

where
$\mathcal{G}_{\chi,\sigma}(\mathbf{k},i\epsilon)=(i\epsilon-\lambda)^{-1}$ is
the bare Green’s function and
$G_{c,\sigma}(\mathbf{k},i\epsilon)=[i\epsilon-\xi_{\mathbf{k}\sigma}-\Sigma_{c,\sigma}(\mathbf{k},i\epsilon)]^{-1}$
with

$\Sigma_{c,\sigma}(\mathbf{k},i\epsilon)=J^{2}_{K}|\Phi|^{2}|V_{\mathbf{k}}|^{2}/(i\epsilon-\lambda).$
(25)

From the latter expression we readily find the quasiparticle residue:

$Z_{\mathbf{k}}=\frac{1}{1-\partial_{z}\Sigma_{c,\sigma}(\mathbf{k},z)|_{z=0}}=\frac{1}{1+J^{2}_{K}|\Phi|^{2}V^{*}_{\mathbf{k}}V_{\mathbf{k}}/\lambda^{2}}.$
(26)

The latter quantity evaluated at the Fermi surface of the heavy Fermi liquid
gives the mass enhancement of the quasiparticles. Discarding the $\Phi=0$
solution the equation reduces to

$\frac{1}{J_{K}}=\frac{1}{2N}\sum_{\mathbf{k}\sigma}\frac{f(E_{\mathbf{k}-\sigma})-f(E_{\mathbf{k}+\sigma})}{\sqrt{(\xi_{\mathbf{k}\sigma}-\lambda)^{2}+4J^{2}_{K}|\Phi|^{2}|V_{\mathbf{k}}|^{2}}}=\frac{1}{2N}\sum_{\mathbf{k}\sigma}\frac{\theta(-E_{\mathbf{k}-\sigma})}{\sqrt{(\xi_{\mathbf{k}\sigma}-\lambda)^{2}+4J^{2}_{K}|\Phi|^{2}|V_{\mathbf{k}}|^{2}}},$
(27)

where
$E_{\mathbf{k}\pm}=(\xi_{\mathbf{k}\sigma}+\lambda)/2\pm\sqrt{(\xi_{\mathbf{k}\sigma}-\lambda)^{2}+4J^{2}_{K}|\Phi|^{2}|V_{\mathbf{k}}|^{2}}/2$
and the RHS is obtained taking the we took the zero temperature limit and
considering the case $x<1$ where only the lower band is filled. In addition we
also have the self-consistent equation for $\lambda$:

$\frac{T}{N}\sum_{\mathbf{k}\sigma}\sum_{i\epsilon}\frac{1}{i\epsilon-\lambda-\Sigma_{\chi,\sigma}(\mathbf{k},i\epsilon)}=1\implies\frac{1}{2N}\sum_{\mathbf{k}\sigma}\theta(-E_{\mathbf{k}-\sigma})\left(1+\frac{\xi_{\mathbf{k}\sigma}-\lambda}{\sqrt{(\xi_{\mathbf{k}\sigma}-\lambda)^{2}+4J^{2}_{K}|\Phi|^{2}|V_{\mathbf{k}}|^{2}}}\right)=1.$
(28)

The onset of the HFL instability is determined looking at the instability
condition of the normal state to interlayer hybridization. In this case the
solution of Eq. (28) is $\lambda=0$, i.e. the local moments are pinned at the
Fermi level, and Eq. (24) becomes:

$\frac{1}{J_{K}}+\Pi^{0}_{\chi c}(\mathbf{q}\to 0,i\Omega=0)=0,$ (29)

where $\Pi^{0}_{\chi c}(\mathbf{q}\to 0,\tau)=-\langle
T_{\tau}(\hat{V}_{\mathbf{q}=0}(\tau)\hat{V}^{\dagger}_{\mathbf{q}=0})\rangle/N$,
$\hat{V}_{\mathbf{q}=0}=\sum_{\mathbf{k}}V^{*}_{\mathbf{k}}c^{\dagger}_{\mathbf{k}}\chi_{\mathbf{k}}/\sqrt{2}$
and $\Pi^{0}_{\chi c}(\mathbf{q}\to
0,i\Omega)=\int^{\beta}_{0}e^{i\Omega\tau}\Pi^{0}_{\chi c}(\mathbf{q}\to
0,\tau)$. Expanding close to the bottom of the band Eq. (29) becomes:

$\frac{1}{J_{K}}-\frac{\rho_{0}}{2}\int^{E_{\Lambda}}_{\epsilon_{F}}d\epsilon\frac{|V_{\epsilon}|^{2}}{\epsilon-\epsilon_{F}}-\frac{\rho_{0}}{2}\int^{\epsilon_{F}}_{0}d\epsilon\frac{|V_{\epsilon}|^{2}}{\epsilon_{F}-\epsilon}=0,$
(30)

where $|V_{\epsilon}|=9a^{2}_{M}k^{2}_{\epsilon}/4$ with
$k_{\epsilon}=\sqrt{2m_{W}\epsilon/\hbar^{2}}$.The integral is characterized
by a log singularity at $\epsilon_{F}$. We introduce the IR cutoff $T_{K}$,
$\epsilon\to\epsilon\pm T_{K}$, that regularize the divergence. Finally, by
performing simple calculations we find:

$T_{K}\simeq\epsilon_{F}e^{-1/(\rho_{0}\bar{J}_{K})}.$ (31)

Since the average over the Fermi surface $\bar{J}_{K}$ goes linearly with the
doping $x$ in conduction band we find that the Kondo temperature is
exponentially suppressed in the limit $x\to 0^{+}$. Finally, we notice that
the Fermi energy is proportional to the filling factor in the WSe2 layer,
$\epsilon_{F}\propto x$. Away from the low-doping regime of exponential
suppression we have $T_{K}\propto x$ which is different from the conventional
result $T_{K}\propto\sqrt{x}$. The behavior $m/m^{*}\sim T_{K}\propto x$ is
consistent with experimental results in Ref. [56].

##  Appendix C Transport properties

In this section we detail the evaluation of the transverse and longitudinal
conductivities in the various phases of the phase diagram.

###  C.1 Charge transport in the HFL

Figure S2: Intrinsic topological response of the HFL. Spin Hall effect (bottom
panel) and anomalous Hall effect (top panel) in the HFL as a function of the
Zeeman field. The calculation is performed at $J_{K}/t_{W}=1$, $x=0.65$.
Figure S3: Longitudinal conductivity. Ratio between the numerical evaluation
of Eq. (32) and $\sigma_{xx}=e^{2}\tau(1-x)/\bar{m}^{*}$ obtained assuming a
circular Fermi surface with average velocity.

In the paramagnetic regime the local moments fractionalize giving rise to a
finite density of holes in MoTe2 layer. In this regime the field
$\Phi_{\mathbf{r}}$ is equivalent to the holon operator carrying physical
charge $-1$ [50], i.e. opposite to the electron charge. As a result both
conduction electrons $c$ and spinon $\chi$ contribute to the charge current.
Within the semiclassical Boltzmann equation approach and in the relaxation
time approximation the transport properties are simply obtained as integrals
over the quasiparticle Fermi surface [48, 49]:

$\sigma_{xx}=\frac{3\sqrt{3}a^{2}_{M}e^{2}\tau}{8\pi^{2}}\sum_{\sigma}\oint_{FS}dk_{t}\frac{\bar{v}^{x}_{F,\sigma}\bar{v}^{x}_{F,\sigma}}{\hbar|\bar{\bm{v}}_{F,\sigma}|},$
(32)

and

$\sigma_{xy}\equiv\sigma^{\rm Ohm}_{xy}+\sigma^{\rm
AH}_{xy}=\frac{3\sqrt{3}a^{2}_{M}}{8\pi^{2}}\frac{e^{3}\tau^{2}B}{\hbar}\sum_{\sigma}\oint_{FS}dk_{t}\frac{\bar{v}^{y}_{F,\sigma}\left[\bar{v}^{y}_{F,\sigma}\partial_{k_{x}}-\bar{v}^{x}_{F,\sigma}\partial_{k_{y}}\right]\bar{v}^{x}_{F,\sigma}}{\hbar|\bar{\bm{v}}_{F,\sigma}|}+\frac{e^{2}}{2\pi
h}\sum_{\sigma}\oint_{FS}dk_{t}\,\bm{t}\cdot\bm{A}_{\sigma}(\mathbf{k}),$ (33)

where the anomalous Hall contribution $\sigma^{\rm AH}_{xy}$ arises from the
circulation of the Berry connection
$\bm{A}_{\sigma}(\mathbf{k})=i\innerproduct{u_{\mathbf{k}\sigma}}{\partial_{k_{a}}u_{\mathbf{k}\sigma}}$,
with $\ket{u_{\mathbf{k}\sigma}}$ occupied eigenstate of the Hamiltonian in
Eq. (22), along the FS. In Eqs. (32) and (33) $k_{t}$ is the component of
$\mathbf{k}$ along the tangent $\bm{t}$ to the FS curve. We notice that the FS
of the heavy quasiparticle is obtained by the set of $\mathbf{k}$ points
solution of the equation:

${\text{heavy
FS}}:\xi_{\mathbf{k}\sigma}-J^{2}_{K}|\Phi|^{2}|V_{\mathbf{k}}|^{2}/\lambda=0,$
(34)

where the second term comes from the conduction electron self-energy
$\Sigma_{c,\sigma}(\mathbf{k},0)$ introduced in Eq. (25) computed at
$i\omega=0$. The Fermi velocity $\bm{v}_{F,\sigma}$ is obtained replacing
$\mathbf{k}\to\mathbf{k}+e\mathbf{A}(t)/\hbar$ in
$\bar{\xi}_{\mathbf{k}\sigma}=Z_{\mathbf{k}\sigma}[\xi_{\mathbf{k}\sigma}-J^{2}_{K}|\Phi|^{2}|V_{\mathbf{k}}|^{2}/\lambda]$
and expanding to linear order around the FS we find
$\bar{\xi}_{\mathbf{k}_{F}+\frac{e}{\hbar}\mathbf{A}(t)\sigma}\simeq
e\bar{\bm{v}}_{F}\cdot\mathbf{A}(t)$ where:

$\bar{\bm{v}}_{F,\sigma}=\frac{Z_{\mathbf{k}_{F},\sigma}}{\hbar}\nabla_{\mathbf{k}}[\xi_{\mathbf{k}\sigma}-J^{2}_{K}|\Phi|^{2}|V_{\mathbf{k}}|^{2}/\lambda]\Big{|}_{\mathbf{k}_{F}},$
(35)

and
$Z_{\mathbf{k},\sigma}=[1-\partial_{z}\Sigma_{c,\sigma}(\mathbf{k},z)|_{z=0}]^{-1}$
is the quasiparticle weight. The evaluation of $\sigma_{xx}$ and $\sigma^{\rm
Ohm}_{xy}$ is considerably simplified observing that the Fermi surface
consists of a hole-pocket around $\bm{\kappa}^{\prime}$ for spin $\uparrow$
and $\bm{\kappa}$ for spin $\downarrow$, respectively. Assuming a circular
hole-like Fermi surface with average mass $\bar{m}^{*}$ the longitudinal
contribution becomes

$\sigma_{xx}=2\frac{e^{2}\tau}{\bar{m}^{*}}\left(\frac{\sqrt{3}a^{2}_{M}}{8\pi^{2}}\right)k^{2}_{F}\int^{2\pi}_{0}d\phi\cos^{2}\phi=\frac{e^{2}\tau}{\bar{m}^{*}}\left(2\pi
k^{2}_{F}\frac{\sqrt{3}a^{2}_{M}}{8\pi^{2}}\right)=\frac{e^{2}\tau(1-x)}{\bar{m}^{*}}.$
(36)

As a sanity check we show in Fig. S3 the ratio between Eq. (36) and the
numerical evaluation of Eq. (32) for various concentrations of electrons in
the $W$-layer. By following the same line of reasoning we obtain $\sigma^{\rm
Ohm}_{xy}=-e^{3}\tau^{2}B(1-x)/\bar{m}^{*}{}^{2}$. We now look at the
anomalous contribution $\sigma^{\rm AH}_{xy}$ which can be conveniently
written as $\sigma^{\rm AH}_{xy}=e^{2}(W_{\uparrow}+W_{\downarrow})/h$ where
$W_{\sigma}=\int
d^{2}\mathbf{k}f(\epsilon_{\mathbf{k}\sigma})\Omega_{\sigma}(\mathbf{k})/(2\pi)$
and
$\Omega_{\sigma}(\mathbf{k})=\partial_{k_{x}}A^{y}_{\sigma}(\mathbf{k})-\partial_{k_{y}}A^{x}_{\sigma}(\mathbf{k})$.
Due to the opposite winding of spin $\uparrow$ and $\downarrow$ we find that
in the absence of a magnetic field $W_{\uparrow}=-W_{\downarrow}$. We observe
that the difference $W_{\uparrow}-W_{\downarrow}$ gives a finite spin Hall
(SH) conductivity [40] even in the absence of the external field. Fig. S2
shows $W_{\uparrow}\pm W_{\downarrow}$ as a function of the magnetic field at
doping $x=0.46$ and $J_{K}/t_{W}=1$. The small value of
$W_{\uparrow}+W_{\downarrow}$ at finite $B$ follows from the Berry curvature
distribution which is peaked around the bare FS of the conduction electrons.
As a result we find a small AH contribution and a large SH one.

####  C.1.1 Expansion around the original Fermi surface: the topological Kondo
Hamiltonian

Here we derive the $\mathbf{k}\cdot\mathbf{p}$ Hamiltonian describing the
regions around $\bm{\kappa}$ and $\bm{\kappa}^{\prime}$ where the Fermi
surface of spin $\uparrow$ and $\downarrow$, respectively, conduction
electrons is located. The analysis clarifies the topological origin of the
hybridization gap. To start with we observe that the mean-field heavy Fermi
liquid Hamiltonian reads:

$H_{\sigma}(\mathbf{k})=\left(\begin{matrix}\lambda&J_{K}\Phi
V_{\mathbf{k}}\\\
J_{K}\Phi^{*}V^{*}_{\mathbf{k}}&\xi_{\mathbf{k}\sigma}\end{matrix}\right).$
(37)

Expanding the form factor $V_{\mathbf{k}}$ and the dispersion
$\xi_{\mathbf{k}\sigma}$ around $\bm{\kappa}$ we readily find:

$H_{\uparrow}(\bm{\kappa}+\mathbf{k})=\left(\begin{matrix}\lambda&-\Delta_{K}(k_{x}-ik_{y})\\\
-\Delta_{K}(k_{x}+ik_{y})&\hbar^{2}k^{2}/(2m_{W})-\mu\end{matrix}\right),$
(38)

and
$H_{\downarrow}(\bm{\kappa}^{\prime}+\mathbf{k})=H^{*}_{\uparrow}(\bm{\kappa}-\mathbf{k})$.
The crossing between the local moment $\chi$ and the $c$-electron dispersive
band takes place of a circle with radius $k_{c}=\sqrt{2m_{W}(\mu+\lambda)}$
where the continuum model reduces to
$H_{\uparrow}(\bm{\kappa}+\mathbf{k}_{c})=-\Delta_{K}k_{c}(e^{i\theta}\tau^{-}+e^{-i\theta}\tau^{+})$.
The interlayer Kondo hybridization lifts the degeneracy and induces winding in
the two dimensional space associated with the interlayer degrees of freedom
$\bm{\tau}$. The resulting Berry curvature can be readily obtained from the
Kubo formula observing that the eigenvalues of Eq. (38) are described by the
matrix
$U(\mathbf{k})=\exp[-i\varphi_{\mathbf{k}}(\mathbf{k}\times\bm{\tau})_{z}/2]$
with $\tan\varphi_{\mathbf{k}}=\Delta_{K}k/d_{z}$ and $d_{z}$ projection of
the Hamiltonian along $\tau^{z}$. The integral gives quantized Chern number
$C_{\uparrow}=-C_{\downarrow}=1$. The model gives a topological Kondo metal
which is adiabatically connected to a quantum Spin Hall Kondo insulator at
filling $2$.

###  C.2 The magnetic regime

Figure S4: Bandstructure and Berry curvature of the small Fermi surface
magnetic state. (a) Dispersion of the electrons close to the bottom of the
conduction band. Dashed lines are obtained with the continuum Hamiltonian (19)
while the solid ones are the lowest energy bands of the mean-field Hamiltonian
(18). (b) Berry curvature around the origin of the magnetic Brillouin zone.
Solid lines show the two Fermi surfaces. The calculations has been performed
at doping $x=0.08$, $J_{K}/t_{W}=1$, $J_{H}/t_{W}=0.025$ and magnetic field
$h_{W}/t_{W}=0.08$.

We now turn our attention in the magnetic regime where only the $c$-electrons
contribute to charge transport. The low-energy Hamiltonian is given in Eq.
(19) and describes electrons with dispersion relation
$\epsilon_{\mathbf{k}\lambda}=\hbar^{2}k^{2}/2m_{\lambda}$ shown in Fig. S4(a)
and group velocity $\bm{v}_{\mathbf{k}\lambda}=\hbar\mathbf{k}/m_{\lambda}$
with $m_{\lambda}=m_{W}/[1-\lambda J_{K}|M|/(4t_{W})]$ and $\lambda=\pm$.
Under the assumption of a single transport time, i.e. momentum- and band-
independent, we apply Eqs. (32) and (33) to find $\sigma_{xx}=e^{2}\tau
x/\left(\sum_{\lambda}m_{\lambda}/2\right)$ and $\sigma^{\rm
Ohm}_{xy}=e^{3}\tau^{2}Bx/\left(m_{W}\sum_{\lambda}m_{\lambda}/2\right)$. We
now conclude our analysis considering the AHE contribution coming from the the
Berry phase winding introduced by the $d_{x}(\mathbf{k})$ and
$d_{y}(\mathbf{k})$ terms in Eq. (19). In the absence of an external magnetic
field $M^{z}=0$ and the eigenstates are simply $\ket{u_{\mathbf{k}\pm}}=(1,\pm
e^{-2i\varphi_{\mathbf{k}}})\sqrt{2}$ with $2\pi$-Berry phase around the
origin of the magnetic Brillouin zone. A small Zeeman term opens a gap in the
band structure and gives rise to a finite Berry curvature
$\Omega_{\pm}(\mathbf{k})$:

$\Omega_{\pm}(\mathbf{k})=\pm\frac{64h^{2}k^{2}\eta^{2}}{\left[16k^{4}\eta^{2}+\left(2h+\frac{m_{\uparrow}-m_{\downarrow}}{m_{\uparrow}m_{\downarrow}}k^{2}\right)^{2}\right]^{3/2}},$
(39)

where $\eta=9J_{K}M^{\parallel}/8$ and $m_{\uparrow/\downarrow}=m_{W}/[1\pm
J_{K}M^{z}/(4t_{W})]$. We notice that the Berry curvature is an even function
of $k$, vanishes quadratically at $k=0$ and takes its maximum value at finite
momentum $k$. The maximum is located at $k^{4}_{\rm max}=h^{2}/(8\eta^{2})$
for $M^{z}=0$. The momentum space distribution in the magnetic moiré Brillouin
zone is given in Fig. S4(b). We observe that from Eq. (39) the anomalous Hall
conductivity is obtained as:

$\sigma^{\rm AH}_{xy}=\frac{e^{2}}{2\pi h}\sum_{\lambda}\int
d^{2}\mathbf{k}\Omega_{\lambda}(\mathbf{k})f(\epsilon_{\mathbf{k}\lambda}-\mu),$
(40)

which in the zero temperature limit becomes:

$\sigma^{\rm
AH}_{xy}=\frac{e^{2}}{h}\int^{k^{+}_{F}}_{k^{-}_{F}}dk\,k\,\Omega_{+}(k)=\frac{e^{2}}{h}\frac{h[k^{2}(m_{\downarrow}-m_{\uparrow})-2hm_{\uparrow}m_{\downarrow}]}{\sqrt{16k^{4}\eta^{2}(m_{\uparrow}m_{\downarrow})^{2}+[k^{2}(m_{\uparrow}-m_{\downarrow})+2hm_{\uparrow}m_{\downarrow}]^{2}}}\Bigg{|}^{k^{+}_{F}}_{k^{-}_{F}}.$
(41)

Fig. S5 shows the evolution of the anomalous Hall conductivity in the low-
density regime. Finally, Fig. S6 gives bird’s-eye view over the anomalous Hall
effect in the plane of the $W$-layer concentration and the Zeeman field.

Figure S5: Magnetic field in the small Fermi surface state. Bottom panel:
evolution of the in-plane and out-of-plane magnetization as a function of the
applied field. Top panel: anomalous Hall conductivity. The calculations has
been performed at doping $x=0.05$, $J_{K}/t_{W}=1$, $J_{H}/t_{W}=0.05$.
Figure S6: Colormap of the AHE. Intrinsic contribution to the anomalous Hall
conductivity in the plane of $W$-layer concentration $x$ and $h_{W}$ Zeeman
field.