# Chiral excitonic order, quantum anomalous Hall effect, and superconductivity
from twofold van Hove singularities in kagome metals

Harley D. Scammell<EMAIL_ADDRESS>School of Physics, University of
New South Wales, Sydney 2052, Australia  Australian Research Council Centre of
Excellence in Future Low-Energy Electronics Technologies, University of New
South Wales, Sydney 2052, Australia  Julian Ingham  Physics Department, Boston
University, Commonwealth Avenue, Boston, MA 02215, USA  Tommy Li  Dahlem
Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität
Berlin, Arnimallee 14, 14195 Berlin, Germany  Oleg P. Sushkov  School of
Physics, University of New South Wales, Sydney 2052, Australia  Australian
Research Council Centre of Excellence in Future Low-Energy Electronics
Technologies, University of New South Wales, Sydney 2052, Australia

###### Abstract

Recent experiments on kagome metals AV3Sb5 (A=K,Rb,Cs) [M. Kang et al.,
arXiv:2105.01689 (2021)] identify twofold van Hove singularities (TvHS) with
opposite concavity near the Fermi energy, generating two approximately
hexagonal Fermi surfaces—one electron-like and the other hole-like. Here we
propose that the TvHS are responsible for the novel charge and superconducting
order observed in kagome metals. We introduce a model of a quasi-two
dimensional system hosting a TvHS, and investigate the interaction induced
many-body instabilities via the perturbative renormalisation group technique.
The TvHS generates a topological $d$-wave excitonic condensate, arising due to
bound pairs of electrons and holes located at opposite concavity vHS. Varying
the chemical potential, detuning from the TvHS, and tuning the bare
interaction strength, we construct a phase diagram which features the
excitonic condensate alongside $d$-wave superconductivity and spin density
wave order. The chiral excitonic state supports a Chern band giving rise to a
quantum anomalous Hall conductance, providing an appealing interpretation of
the observed anomalous Hall effect in kagome metals. Possible alternative
realisations of the TvHS mechanism in bilayer materials are also discussed. We
suggest that TvHS open up interesting possibilities for correlated phases,
enriching the set of competing ground states to include excitonic order.

##  I Introduction

Kagome systems have been a major focus of theoretical and experimental
investigation; due to their ability to realise Dirac points, flat bands and
van Hove singularities, they have been predicted to host a range of novel
correlated phases of matter Johnston1990 ; Schulenburg2002 ; Tan2011 ; Sun2011
; Green2010 ; Guo2009 ; Kiesel2013 ; Wang2013 ; Wen2010 ; Kiesel2012 ; Li2020
; Kang2020 ; Yin2019 ; Baidya2020 ; Ye2018 ; Mielke2021 . Recently, a new
class of materials AV3Sb5 (A=K,Rb,Cs) have attracted a great deal of attention
due to their demonstration of unconventional superconductivity alongside
competing density wave order, spatially modulated superconducting order and
possible signatures of Majorana states in superconducting vortices Ortiz2019 ;
Ortiz2020 ; Ortiz2021 ; Yin2021 ; Xu2021 ; Gupta2021 ; Duan2021 ; Yang2021 ;
Jiang2021 ; Li2021b ; Zhao2021 ; Li2021c ; Ratcliff2021 ; Shumiya2021 ;
Mielke2021b ; Miao2021 ; Du2021 ; Ni2021 ; Chen2021 ; Liang2021 ; Zhu2021 ;
Chen2021b ; Du2021b ; Zhao2021b ; Zhang2021 ; Du2021c ; Tsirlin2021 ; Du2021d
; Qian2021 ; Liu2021 ; Oey2021 ; Yang2021b ; Ortiz2021b ; Yuo2021 ; Chen2021c
; Kang2021 ; Hu2021 ; Zhao2021c ; Tan2021 ; Neupert2021 . Unusually, the
materials exhibit time-reversal symmetry breaking with an anomalous Hall
conductivity in spite of the absence of magnetic ordering; the origins and
relationship between superconductivity, competing order, and the anomalous
Hall effect remain an open question.

The materials consist of a stack of two dimensional layers—a kagome lattice of
vanadium and antimony alternating with a hexagonal lattice of antimony and
triangular lattice of the alkali metal K/Rb/Cs—with electrical transport
predominantly in-plane, as demonstrated by the large ratio between the out-of-
and in- plane resistivity $R_{c}/R_{ab}\approx 600$. The Fermi surface of
these materials consists of several distinct contours, including nearly
circular contours centered at the $\Gamma$ and $K$ points as well as two
approximately hexagonal contours Ortiz2021b . Systems with hexagonal Fermi
surfaces, corresponding to saddle points in the electronic dispersion, have
been predicted to give rise to chiral superconductivity and competing density
wave order, due to the effects of Fermi surface nesting Nandkishore2012 .
However, ARPES and DFT results reveal that the hexagonal Fermi surfaces in the
vanadium metals exhibit an unusual feature—two-fold van Hove singularities
(TvHS), for which the saddle points at each Fermi surface possess opposite
concavity, resulting in one electron-like Fermi surface and one hole-like
Fermi surface Kang2021 .

We argue that doping a system to a TvHS has an ineluctable influence on the
low-energy physics. A single vHS results in a tendency towards density wave
ordering and superconductivity. The appearance of TvHS introduces an
additional tendency towards excitonic order—corresponding to a condensation of
electron-hole pairs—owing to the coupling between an electron-like and hole-
like Fermi surface. We introduce a low energy model which incorporates the
TvHS: featuring an electron-like and hole-like Fermi surface, each doped near
their respective vHS. To understand the interplay and competition between the
various many-body instabilities, we employ the perturbative renormalisation
group (RG) method to determine the dominant ground state order Shankar1994 ;
Maiti2013 ; Schulz1987 ; Dzyaloshinskii1987 ; Furukawa1998 ; LeHur2009 ;
Vavilov2010 ; Fernandes2010 ; Nandkishore2012 . We find that for a range of
parameters, a chiral excitonic order dominates, which breaks time-reversal
symmetry and exhibits a quantum anomalous Hall effect. Allowing for combined
and relative detuning of the Fermi surfaces from the perfect nesting generates
a phase diagram which includes $d$-wave singlet superconductvity and density
wave order. We suggest that the phenomenology produced by this model accounts
for key features observed in the vanadium-based kagome metals, and could
further motivate TvHS engineering in van der Waals heterostructures.

In Section II we introduce our model, and describe key aspects of the RG
method. In Section III we discuss the solution to the RG equations, generating
a phase diagram, and discuss aspects of the excitonic phase. In Section IV we
discuss our findings, and draw links to existing and future experiments.

##  II Model

In this section we first introduce a toy model to provide simple understanding
of the single particle physics of a TvHS, and to suggest its possible
engineering in bilayer systems. We then introduce a patch model to understand
interaction-induced instabilities for hexagonal systems exhibiting a TvHS.

###  II.1 TvHS toy model

A TvHS consists of two approximately hexagonal Fermi surfaces with opposite
concavity vHS so that one surface is electron-like and the other hole-
like—e.g. arising from doping near the vHS of an electronic dispersion at the
$M$-point of a 2D hexagonal Brillouin zone as shown in Fig LABEL:f:Modela. For
the purposes of demonstration, we construct a toy lattice model comprising of
a honeycomb-kagome bilayer with interlayer coupling a site on the honeycomb
lattice to the the three nearest sites on the kagome lattice,

$\displaystyle H$ $\displaystyle=H_{\text{H}}+H_{\text{K}}+H_{\text{T}},$ (1)
$\displaystyle H_{\text{H}}$
$\displaystyle=-\sum_{i}\left\\{t_{+}\left(a_{i}^{\dagger}b_{i}+b_{i}^{\dagger}a_{i}\right)+\gamma_{0}\left(a_{i}^{\dagger}a_{i}+b_{i}^{\dagger}b_{i}\right)\right\\},$
$\displaystyle H_{\text{K}}$
$\displaystyle=-t_{-}\sum_{\braket{i,j}}\left\\{A_{i}^{\dagger}B_{j}+A_{i}^{\dagger}C_{j}+B_{i}^{\dagger}C_{j}\right\\}+\text{h.c.},$
$\displaystyle H_{\text{T}}$
$\displaystyle=\gamma_{1}\sum_{i}(a_{i}^{\dagger}+b_{i}^{\dagger})(A_{i}+B_{i}+C_{j})+\text{h.c.}$

Here $a_{i}^{\dagger},b_{i}^{\dagger}$ create electrons on the ($a,b$)
sublattices of the honeycomb layer at site $i$,
$A_{i}^{\dagger},B_{i}^{\dagger},C_{i}^{\dagger}$ create electrons on the
kagome ($A,B,C$) sublattices at site $i$, and the sum over $\braket{i,j}$
enumerates nearest neighbours. The interlayer coupling $\gamma_{1}$ connects a
given sublattice site of the honeycomb lattice to the nearest three sublattice
sites of the kagome lattice, and $\gamma_{0}$ is an onsite energy shift which
acts to increase the chemical potential in one layer relative to the other,
aligning the valence and conduction bands associated to each layer. A
schematic of the lattice geometry and further discussion of the model is
presented in the Supplementary Material. This model exhibits a TvHS in the
bandstructure, shown in Fig. LABEL:f:Modelb.

We note that for the vanadium metals, the TvHS arises due to different
$d$-orbitals within the kagome layer Kang2021 ; our toy model is not intended
to capture the microscopic origin of the TvHS in these materials.

###  II.2 Two-flavour patch model

Independent of the origin of the TvHS, our aim is to consider interaction
driven instabilities for a system doped near a TvHS. The problem of
interaction driven instabilities on a single hexagonal Fermi surface (i.e.
single vHS) has been studied using a three patch model Nandkishore2012 ,
whereby the full Brillouin zone is restricted to three momentum space patches
near the vHS at the $M$-points, since they dominate the density of states.
Following this approach, we define a three patch model and further introduce a
flavour degree of freedom to account for the two opposite concavity Fermi
surfaces—fermions of one flavour (created by $c^{\dagger}$) are electron-like,
while fermions of the other flavour (created by $d^{\dagger}$) are hole-like.
In terms of a spinor
$\psi_{\bm{k}}^{\dagger}=(c_{\bm{k}}^{\dagger},d_{\bm{k}}^{\dagger})$, we
write

$\displaystyle H_{0}$
$\displaystyle=\sum^{\Lambda}_{\bm{k},\alpha}\psi_{\bm{k},\alpha}^{\dagger}((\varepsilon_{\bm{k}}+\delta)\sigma_{z}-\mu)\psi_{\bm{k},\alpha}.$
(2)

The Pauli matrices $\sigma_{i}$ act on the two flavour degrees of freedom, and
$\alpha=1,2,3$ index the patch labels which indicate a fermionic excitation
within a cut-off $\Lambda$ of the momentum $\bm{M}_{\alpha}$. Patches on
opposite sides of the hexagon are related by reciprocal lattice vectors and
are therefore equivalent. Fermions at patches $\alpha\neq\beta$ are connected
by the nesting vector $\bm{Q}_{\alpha\beta}=\bm{M}_{\alpha}-\bm{M}_{\beta}$,
for which $\varepsilon_{\bm{k}+\bm{Q}_{\alpha\beta}}=-\varepsilon_{\bm{k}}$;
the nesting condition is responsible for an enhancement of interaction effects
for fermions on opposite sides of the Fermi surface. For $\delta=\mu=0$, both
flavours are exactly tuned to the vHS saddle point of the dispersion, while
for $\delta,\mu\neq 0$ nesting is only approximate; $\delta$ shifts the
electron and hole Fermi surfaces equally, while $\mu$ acts to change their
relative area.

In the model (2), we treat both vHS saddle points as approximately quadratic
(e.g. near $\bm{M}_{1}$, we have $\varepsilon_{\bm{k}}\propto
k_{x}^{2}-\sqrt{3}k_{x}k_{y}$) although it has been suggested that ARPES data
on the vanadium metals are better fit by a quartic saddle-point for the hole-
like Fermi surface denoted as ‘$p$-type’ Kang2021 . Accounting for the quartic
saddle-point would introduce a stronger divergence in the density of states at
the $p$-type patches, but a weakening of the nesting effect as the $p$-type
Fermi contour is distorted from the hexagonal shape. While interesting, we do
not explore the higher-order nature of the $p$-type vHS in our idealised
setup.

We now consider the possible couplings between the fermions. Due to the large
density of states near the TvHS we expect the Coulomb repulsion to be strongly
screened and therefore model the interactions as short-ranged. The most
general set of interactions between patches/flavours allowed by momentum
conservation are

$\displaystyle V$
$\displaystyle=\frac{1}{2}\sum_{\alpha,\beta}\left[V_{g}+V_{h}+V_{j}+V_{l}+\left(c\leftrightarrow
d\right)\right],$ (3) $\displaystyle V_{g}$
$\displaystyle=g_{1}c^{\dagger}_{\alpha}c^{\dagger}_{\beta}c_{\alpha}c_{\beta}+g_{2}c^{\dagger}_{\alpha}c^{\dagger}_{\beta}c_{\beta}c_{\alpha}$
$\displaystyle\hskip
28.45274pt+g_{3}c^{\dagger}_{\alpha}c^{\dagger}_{\alpha}c_{\beta}c_{\beta}+\frac{1}{2}g_{4}c^{\dagger}_{\alpha}c^{\dagger}_{\alpha}c_{\alpha}c_{\alpha},$
$\displaystyle V_{h}$
$\displaystyle=h_{1}c^{\dagger}_{\alpha}d^{\dagger}_{\beta}d_{\alpha}c_{\beta}+h_{2}c^{\dagger}_{\alpha}d^{\dagger}_{\beta}d_{\beta}c_{\alpha}$
$\displaystyle\hskip
28.45274pt+h_{3}c^{\dagger}_{\alpha}d^{\dagger}_{\alpha}d_{\beta}c_{\beta}+\frac{1}{2}h_{4}c^{\dagger}_{\alpha}d^{\dagger}_{\alpha}d_{\alpha}c_{\alpha},$
$\displaystyle V_{j}$
$\displaystyle=j_{1}d^{\dagger}_{\alpha}d^{\dagger}_{\beta}c_{\alpha}c_{\beta}+j_{2}d^{\dagger}_{\alpha}d^{\dagger}_{\beta}c_{\beta}c_{\alpha}$
$\displaystyle\hskip
28.45274pt+j_{3}d^{\dagger}_{\alpha}d^{\dagger}_{\alpha}c_{\beta}c_{\beta}+\frac{1}{2}j_{4}d^{\dagger}_{\alpha}d^{\dagger}_{\alpha}c_{\alpha}c_{\alpha},$
$\displaystyle V_{l}$
$\displaystyle=l_{1}d^{\dagger}_{\alpha}c^{\dagger}_{\beta}d_{\alpha}c_{\beta}+l_{2}d^{\dagger}_{\alpha}c^{\dagger}_{\beta}d_{\beta}c_{\alpha}$
$\displaystyle\hskip
28.45274pt+l_{3}d^{\dagger}_{\alpha}c^{\dagger}_{\alpha}d_{\beta}c_{\beta}+\frac{1}{2}l_{4}d^{\dagger}_{\alpha}c^{\dagger}_{\alpha}d_{\alpha}c_{\alpha},$

where $V_{g}$ are intraflavour couplings, and $V_{h,j,l}$ are interflavour
couplings, all of which include both inter and intrapatch processes. A sum
over spin indices is implicit. A schematic illustration of the
$g_{i},h_{i},j_{i},l_{i}$ couplings, as well as their representation in terms
of Feynman diagrams, is shown in Fig LABEL:f:Modelc and d. The opposing
concavities of the respective saddle points can be seen from the colour plot
in Fig. LABEL:f:Modelc; going from outside to inside the hexagonal Fermi
surface, the energy changes sign, but the sign change is opposite for the two
Fermi surfaces.

###  II.3 RG analysis

Considering the interacting Hamiltonian,

$\displaystyle H$ $\displaystyle=H_{0}+V,$ (4)

we determine which instabilities arise within the framework of RG. The
instability of the metallic phase and onset of an ordered ground state is
signalled by the susceptibility of the associated order parameter; the
strongest ordering tendencies are those with most divergent susceptibility.
The RG method provides an unbiased approach to study competing orders on an
equal footing, by resumming the logarithmically divergent corrections to the
bare couplings Maiti2013 . For contact interactions such as (3), we find the
log-divergent corrections may be reduced to the following particle-particle
and particle-hole susceptibilities,

$\displaystyle\Pi^{ii}_{pp}(0)=\frac{\nu_{0}}{4}\log\frac{\Lambda}{\text{max}(T,\delta,\mu)}\log\frac{\Lambda}{T}\equiv\ell$
(5) $\displaystyle\Pi^{i\neq
j}_{ph}(0)=\frac{\nu_{0}}{4}\log\frac{\Lambda}{\text{max}(T,\delta,\mu)}\log\frac{\Lambda}{\text{max}(T,\frac{1}{2}(\mu+\delta))}\equiv
d_{1}\ell$ $\displaystyle\Pi^{i\neq
j}_{pp}(Q_{\alpha\neq\beta})=\frac{\nu_{0}}{4}\log\frac{\Lambda}{\text{max}(T,\delta,\mu)}\log\frac{\Lambda}{\text{max}(T,\delta)}\equiv
d_{3}\ell,$
$\displaystyle\Pi^{ii}_{ph}(Q_{\alpha\neq\beta})=\frac{\nu_{0}}{4}\log\frac{\Lambda}{\text{max}(T,\delta,\mu)}\log\frac{\Lambda}{\text{max}(T,\mu)}\equiv
d_{2}\ell.$

where $i,j$ index flavour. The presence of double logarithmic factors is due
to the vHS at the $M$-point, which leads to a log-divergent density of states
$(\nu_{0}/4)\log(\Lambda/\text{max}(T,\delta,\mu))$, as well as a log-
divergence from nesting of the two fermionic states, e.g. particle-particle or
particle-hole. To simplify the analysis, we have treated the vHS logarithmic
contribution as equal in all cases, i.e. as a factor
$\log(\Lambda/\text{max}(T,\delta,\mu))$. The distinguishing feature of each
polarisation operator comes from the ‘nesting’ logarithm, which features a
different combination of $\mu,\delta$ appearing as an infrared cut-off for
each of the cases in (5). As is standard within the patch RG formalism
Maiti2013 , we have chosen the particle-particle susceptibility to define the
RG time $\ell=\Pi^{ii}_{pp}(0)\propto\log^{2}(\Lambda/E)$. The other
susceptibilities then incur a factor $d_{a}$ which account for deviations from
perfect nesting, i.e.
$d_{a}=\\{d_{1},d_{2},d_{3}\\}\in\\{d\Pi_{a}/d\Pi^{ii}_{pp}\\}$—with the
indexing of $\Pi_{a}$ defined in (5). Importantly, the twofold van Hove
singularity introduces the divergent susceptibilities $\Pi^{i\neq j}_{ph}(0)$
and $\Pi^{i\neq j}_{pp}(Q_{\alpha\neq\beta})$, due to the $c$ and $d$ fermions
having opposite concavity dispersion, which strongly modifies the RG analysis
compared to previous studies of a single hexagonal Fermi surface.

The dominant instabilities are those whose susceptibilities receive
corrections containing the double logarithmic divergences (5), which we
enumerate now. We first establish some notation: $\alpha,\beta$ index patch,
and $\sigma_{i}$ act on flavour, which shall be indexed by Latin characters
$i=c,d$, and define $\sigma_{c}=\tfrac{1}{2}(\sigma_{0}+\sigma_{z})$,
$\sigma_{d}=\tfrac{1}{2}(\sigma_{0}-\sigma_{z})$,
$\sigma_{\pm}=\tfrac{1}{2}(\sigma_{x}\pm i\sigma_{y})$, and $s_{\mu}$ to be
Pauli matrices acting on spin. The order parameters of interest are then
intraflavour spin density wave $\mathcal{S}_{\alpha
i}=\langle\psi^{\dagger}_{\alpha}\sigma_{i}\vec{s}\psi_{\bar{\alpha}}\rangle$
(SDW) and charge density wave $\mathcal{C}_{\alpha
i}=\langle\psi^{\dagger}_{\alpha}\sigma_{i}\psi_{\bar{\alpha}}\rangle$ (CDW),
since the electron-like and hole-like Fermi surfaces are individually nested
and the associated susceptibilities therefore receive corrections involving
$\Pi^{ii}_{ph}(Q_{\alpha\neq\beta})$. Secondly, we have two types of
superconducting instability, intraflavour pairing $\Delta_{\alpha
i}=\langle\psi_{\alpha}\sigma_{i}\psi_{\alpha}\rangle$ from the ordinary
Cooper instability via $\Pi^{ii}_{pp}(0)$, and also from an equivalent
susceptibility pairing fermions in different flavours and at different
patches, a pair density wave ${\cal
P}_{\alpha\pm}=\langle\psi_{\alpha}\sigma_{\pm}\psi_{{\bar{\alpha}}}\rangle$
(PDW) due to the susceptibility $\Pi^{i\neq j}_{pp}(Q_{\alpha\neq\beta})$,
with $\bar{\alpha}$ denoting the patch connected to $\alpha$ by a nesting
vector. Thirdly, we can also pair electrons of one flavour and holes of the
other, since they are relatively nested, giving us an excitonic order
parameter which can be both spin singlet
$\Phi^{C}_{\alpha\pm}=\langle\psi^{\dagger}_{\alpha}\sigma_{\pm}\psi_{\alpha}\rangle$
or spin triplet
$\Phi^{S}_{\alpha\pm}=\langle\psi^{\dagger}_{\alpha}\sigma_{\pm}\vec{s}\psi_{\alpha}\rangle$.
The susceptibilities for these last order parameters receive corrections
involving $\Pi^{i\neq j}_{ph}(0)$.

Hence, the two-fold van Hove singularity introduces the new divergent
susceptibilities $\Pi^{i\neq j}_{ph}(0)$ and $\Pi^{i\neq
j}_{pp}(Q_{\alpha\neq\beta})$, describing the interaction of fermions at
opposite concavity Fermi surfaces (ie opposite flavours), which in turn
introduces the excitonic and PDW instabilities ${\cal P}_{\alpha\pm}$,
$\Phi^{C}_{\alpha\pm}$ and $\Phi^{S}_{\alpha\pm}$, alongside with the usual
superconducting and density wave orders.

We turn now to the RG treatment which identifies the leading instabilities.
Firstly, we compute the leading log2 corrections to the bare couplings defined
in (3). The full RG equations for our model are lengthy, since they involve
sixteen independent coupling constants, so we state their general form here
and reserve explicit expressions for the Supplementary Material (S2.1). The RG
equations describing the flow of the couplings $g_{i}$, $h_{i}$, $j_{i}$,
$l_{i}$ take the form

$\displaystyle\frac{\partial g_{i}}{\partial\ell}$
$\displaystyle=\beta_{g_{i}}(g,j,h,l),\ \frac{\partial
h_{i}}{\partial\ell}=\beta_{h_{i}}(g,j,h,l),$ $\displaystyle\frac{\partial
j_{i}}{\partial\ell}$ $\displaystyle=\beta_{j_{i}}(g,j,h,l),\ \frac{\partial
l_{i}}{\partial\ell}=\beta_{l_{i}}(g,j,h,l)$ (6)

where $\beta_{g_{i}}$, $\beta_{h_{i}}$, $\beta_{j_{i}}$, $\beta_{l_{i}}$ are
functions of all sixteen couplings. Secondly, we compute the leading log2
corrections to the order parameter susceptibilities, which generates the
linear set of gap equations,

$\displaystyle\frac{\partial}{\partial\ell}{\cal C}_{\alpha,c}$
$\displaystyle=2d_{1}\Big{\\{}(g_{2}-2g_{1}){\cal C}_{\alpha,c}-g_{3}{\cal
C}_{\bar{\alpha},c}$ (7) $\displaystyle+(l_{2}-2h_{1}){\cal
C}_{\alpha,d}+(l_{3}-2h_{3}){\cal C}_{\bar{\alpha},d}\Big{\\}}$
$\displaystyle\frac{\partial}{\partial\ell}{\cal S}_{\alpha,c}$
$\displaystyle=2d_{1}\Big{\\{}g_{2}{\cal S}_{\alpha,c}+g_{3}{\cal
S}_{\bar{\alpha},c}+l_{2}{\cal S}_{\alpha,d}+l_{3}{\cal
S}_{\bar{\alpha},d}\Big{\\}}$
$\displaystyle\frac{\partial}{\partial\ell}{\Delta}_{\alpha,c}$
$\displaystyle=-2\sum_{\beta\neq\alpha}\Big{\\{}g_{4}{\Delta}_{\alpha,c}+g_{3}{\Delta}_{\beta,c}+j_{4}{\Delta}_{\alpha,d}+j_{3}{\Delta}_{\beta,d}\Big{\\}}$
$\displaystyle\frac{\partial}{\partial\ell}{\cal P}_{\alpha,+}$
$\displaystyle=-2d_{3}\Big{\\{}h_{2}{\cal P}_{\alpha,+}+h_{1}{\cal
P}_{\alpha,-}+l_{1}{\cal P}_{\bar{\alpha},+}+l_{2}{\cal
P}_{\bar{\alpha},-}\Big{\\}}$
$\displaystyle\frac{\partial}{\partial\ell}{\Phi}^{C}_{\alpha+}$
$\displaystyle=2d_{2}\sum_{\beta\neq\alpha}\Big{\\{}(h_{4}-2l_{4}){\Phi}^{C}_{\alpha+}-j_{4}{\Phi}^{C}_{\alpha-}$
$\displaystyle+(h_{1}-2l_{2}){\Phi}^{C}_{\beta+}+(j_{1}-2j_{2}){\Phi}^{C}_{\beta-}\Big{\\}}$
$\displaystyle\frac{\partial}{\partial\ell}{\Phi}^{S}_{\alpha+}$
$\displaystyle=2d_{2}\sum_{\beta\neq\alpha}\Big{\\{}h_{4}{\Phi}^{S}_{\alpha+}+j_{4}{\Phi}^{S}_{\alpha-}+h_{1}{\Phi}^{S}_{\beta+}+j_{1}{\Phi}^{S}_{\beta-}\Big{\\}}$

with indices as defined previously—$c$, $d$, $\pm$ referring to flavour,
$\alpha$ to patch—and $\bar{\alpha}$ denoting the patch connected to $\alpha$
by a nesting vector. Note, we have taken $\partial_{\ell}d_{i}=0$ to obtain
these simple flow equations. The couplings entering the gap equations are
understood to inherit scale-dependence from the coupling RG equations (II.3).
Diagonalising the gap equations, and integrating over the RG time $\ell$, one
identifies the leading eigenvalue $\lambda_{i}(\ell)$ which diverges fastest
with $\ell$. The associated eigenvector is the order parameter with the
largest critical temperature $T_{c}=\Lambda
e^{-1/(\nu_{0}\lambda_{i})^{1/2}}$, and is therefore the ground state at
$T\lesssim T_{c}$. Moreover, the patch dependence of the eigenvector gives the
spatial structure of the order parameter.

Figure 2: (a) Phase diagram: Vertical axis represents the relative strength of
the bare interflavour couplings – taken equal for simplicity
$h_{i}=j_{i}=l_{i}$ – to the bare intraflavour couplings $g_{i}$. (a)
Horizontal axis is $\delta$, with $\mu=0$. (b) Horizonal axis is $\mu$, with
$\delta=1$. The dominant instabilities are the chiral $d+id$ exciton
$\Phi^{C}$, the chiral $d+id$ superconductor $\Delta$, and the spin density
wave ${\cal S}$, which are found from the leading (diverging) susceptibilities
in the gap equations (7). Edge modes: (c) The 1D dispersion of a
honeycomb/kagome bilayer in the $d+id$ excitonic insulator for an infinite
ribbon of width 120 unit cells. The edge states propagating along the
top/bottom of the ribbon are plotted in red/blue. Specific details of the
lattice model are provided in the Supplement S3.

We focus our attention on the excitonic condensate, which emerges due to the
TvHS. We find that the leading excitonic state is of a $d$-wave nature,
hosting the doubly-degenerate eigenvectors,

$\displaystyle\Phi^{C}_{\pm,a}$ $\displaystyle=\Phi_{a}e^{\pm
i\phi_{a}}\tfrac{1}{\sqrt{6}}\left(1,-2,1\right),$
$\displaystyle\Phi^{C}_{\pm,b}$ $\displaystyle=\Phi_{b}e^{\pm
i\phi_{b}}\tfrac{1}{\sqrt{2}}\left(1,0,-1\right)$ (8)

with eigenvectors/values for all the other order parameters presented in the
Supplementary Material. Here $\Phi_{a},\Phi_{b}$ are real scalars,
$\phi_{a},\phi_{b}$ are distinct U(1) phases, and the row vectors enumerate
patch indices, therefore encoding the spatial structure of the two order
parameters. Continuing to the full Fermi surface, the spatial vectors behave
as $\sim\cos(2\theta_{\bm{k}})$, $\sin(2\theta_{\bm{k}})$, with
$\theta_{\bm{k}}$ the momentum angle. Similar $d$-wave eigenvectors appear for
the superconducting states $\Delta$, which are the two-flavour analogues of
the superconducting states found in Nandkishore2012 . The reason for this
structure owes to the patches: both $\Delta$ and $\Phi^{C}$ form from fermions
on the same patch; the gap equation connects $\Delta$ or $\Phi^{C}$ of a given
patch to $\Delta$ or $\Phi^{C}$ on the other two patches. With this structure,
the eigenvectors must have zero or four nodes when continued around the Fermi
surface—four nodes corresponds to the $d$-wave nature for both $\Delta$ and
$\Phi_{C/S}$. This does not occur for density wave states—charge (CDW), spin
(SDW) and pair (PDW)—which form as interpatch order parameters.

##  III Results

###  III.1 Phase diagram

Having established the RG recipe for determining the leading instabilties, we
now present the resulting phase diagram obtained from integrating the coupling
flow equations and solving the gap equations. We allow for two physically
pertinent tuning handles coming from the noninteracting model (2)—the chemical
potential $\mu$, which acts to shift the relative area of the two flavours of
Fermi surfaces, and the detuning $\delta$, which acts to shift the Fermi
surfaces of each flavour while keeping their areas equal. Moreover, we
consider variations of the initial values of the interflavour couplings (3),
which could be modified for e.g. via application of pressure.

Tuning these three parameters, and determining the leading instability via the
RG scheme outlined above, we arrive the the phase diagram Figure 2. For the
parameter range of interest, we find spin density wave, superconducting and
excitonic phases. We note that, since our analysis relies on existence of both
electron and hole-like Fermi surfaces, we must restrict to $\delta>\mu$. We
also note that our RG analysis does not take into account possible coexistence
with nearby, subleading phases. We therefore do not draw sharp phase
boundaries. The possibility of coexistence can be determined via a Landau-
Ginzburg analysis which we do not present here.

Increasing $\mu$ changes the relative area of the two Fermi surfaces, which
suppresses the tendency to excitonic order. Therefore increasing $\mu$ while
keeping a vHS close to the Fermi level should suppress excitonic order
relative to superconductivity and SDW, as seen in the phase diagram Fig. 2a.
On the other hand, varying $\delta$ keeps the area of the two Fermi surfaces
the same while distancing the vHS from the Fermi level, so increasing $\delta$
disfavours SDW and superconductivity relative to excitonic order, as can be
seen in the phase diagram Fig. 2b.

The bottom right of the phase diagram Fig. 2b corresponds to two hexagonal
Fermi surfaces with weak interflavour coupling. In the limit of zero
interflavour coupling, one obtains two decoupled hexagonal Fermi surfaces,
which naturally reproduces the result of Nandkishore, Levitov and Chubukov
Nandkishore2012  i.e. $d$-wave superconductivity dominates. This regime is
reached for smaller values of the interflavour coupling than we present, which
is why it does not appear in the phase diagram.

###  III.2 Properties of the chiral excitonic condensate

We now focus on the excitonic order. As seen in the previous section, there
exist degenerate excitonic states (8), which appear as solutions to the linear
gap equations (7). To determine the coexistence or competition between these
degenerate states, we calculate the Landau-Ginzburg free energy. To begin, we
account for the excitonic states (8) within the mean field Hamiltonian,

$\displaystyle\mathcal{H}_{MF}=H_{0}+\sum_{\nu=a,b}\tfrac{1}{4\lambda}|\Phi_{\nu}|^{2}+$
(9) $\displaystyle\sum_{\begin{subarray}{c}\nu=a,b\\\
\alpha=1,2,3\end{subarray}}\Phi_{\nu}\left(e^{-i\phi_{\nu}}c^{\dagger}_{\alpha}\left(\hat{u}_{\nu}\right)_{\alpha,\alpha^{\prime}}d_{\alpha^{\prime}}+e^{i\phi_{\nu}}d^{\dagger}_{\alpha}\left(\hat{u}_{\nu}\right)_{\alpha,\alpha^{\prime}}c_{\alpha^{\prime}}\right)+\text{h.c.},$

where $\lambda=2\left(h_{1}-h_{4}+j_{1}-2j_{2}+j_{4}-2l_{2}+2l_{4}\right)$ is
the leading, doubly-degenerate eigenvalue for the $d$-wave excitonic
susceptibilities, and the patch structure of the corresponding eigenvectors
appear here as diagonal matrices (in patch space)
$u_{a}=\text{diag}\left(1,-2,1\right)/\sqrt{6}$,
$u_{b}=\text{diag}\left(1,0,-1\right)/\sqrt{2}$.

Near the critical temperature, the free energy of (9) can expanded in powers
of the degenerate order parameters,

$\displaystyle\mathcal{F}[\Phi]=\mathcal{F}_{0}+\left(\tfrac{1}{4\lambda}-a(T)\right)(|\Phi_{a}|^{2}+|\Phi_{b}|^{2})$
(10)
$\displaystyle+b(T)\left(\Phi_{a}^{4}+\Phi_{b}^{4}+\tfrac{4}{3}\Phi_{a}^{2}\Phi_{b}^{2}+\tfrac{2}{3}\Phi_{a}^{2}{\Phi}_{b}^{2}\cos(2(\phi_{a}-\phi_{b}))\right)$

where $\mathcal{F}_{0}$ is the free energy of the free fermions, and
$a(T),b(T)>0$ Sigrist2005 . The free energy is minimised by coexisting order
parameters, with $\Phi_{a}=\Phi_{b}=\Phi_{0}$ and $\phi_{a}-\phi_{b}=\pi/2$
(mod $\pi$). The coexisting states form a single order parameter of the form
$\Phi=\Phi_{0}e^{\pm i\theta_{\alpha}}$, $\theta_{\alpha}=\\{a,b,c\\}$.
Continuing around the Fermi surface, the combined order parameter becomes
$\Phi\sim\Phi_{0,\bm{k}}(\cos(2\theta_{\bm{k}})\pm i\sin(2\theta_{\bm{k}}))$,
which is a chiral $d\pm id$ order. The chirality $\pm$ is spontaneously
selected by the ground state, which therefore breaks time-reversal symmetry.
We note that this framework allows one to additionally consider the question
of coexistence of $\Delta\neq 0$, $\Phi\neq 0$, which is particularly relevant
near the phase boundaries of these two ordered states. However, we do not
carry out this analysis here.

To understand the topological properties of chiral excitonic condensate, we
may introduce a simplified two-band Hamiltonian

$\displaystyle
H=\sum_{\bm{k}}\left(\begin{array}[]{cc}c^{\dagger}_{\bm{k}}&d^{\dagger}_{\bm{k}}\end{array}\right)\left(\begin{array}[]{cc}\varepsilon_{\bm{k}}+\delta-\mu&|\Phi_{\bm{k}}|e^{\mp
2i\theta_{\bm{k}}}\\\ |\Phi_{\bm{k}}|e^{\pm
2i\theta_{\bm{k}}}&-\varepsilon_{\bm{k}}-\delta-\mu\end{array}\right)\left(\begin{array}[]{c}c_{\bm{k}}\\\
d_{\bm{k}}\end{array}\right)$ (16)

which approximates the hexagonal Fermi surface as a circle. The chiral order
parameter winds twice along the Fermi surface and vanishes away from the Fermi
surface, and the two-band model may be described by a first quantised
Hamiltonian of the form
$\mathcal{H}(\bm{k})=\widetilde{\varepsilon}_{\bm{k}}\bm{d}_{\bm{k}}\cdot\bm{\sigma}$
where the unit vector $\bm{d}_{\bm{k}}$ possesses a skyrmion configuration
with topological charge
$C=\frac{1}{4\pi}\int{\bm{d}_{\bm{k}}\cdot\left(\nabla_{k_{x}}\bm{d}_{\bm{k}}\times\nabla_{k_{y}}\bm{d}_{\bm{k}}\right)dk_{x}dk_{y}}=\pm
2$, which is equal to the Chern number.

The nontrivial topological invariant implies a quantised anomalous Hall
conductivity $\sigma_{xy}=\tfrac{e^{2}}{2\pi}C$ which is carried by two chiral
edge modes. In order to demonstrate the non-trivial topology, we diagonalise a
mean-field Hamiltonian defined on a lattice, which provides a completion of
the low-energy description we have so far considered to the entire Brillouin
zone. We consider the honeycomb/kagome bilayer in an infinite ribbon geometry
with zigzag edges. The 1D dispersion of the ribbon is plotted in Fig. 2c for
the $d+id$ phase, which exhibits two chiral edge modes, with the left/right-
movers propagating along the top/bottom of the ribbon. Full details of the
lattice model are provided in the Supplement S3. Anticipating future
realisations in van der Waals materials, Supplement S3 also presents exact
diagonalisation results for a honeycomb-honeycomb bilayer system.

##  IV Discussion

Topological excitonic states have been explored theoretically in a number of
previous proposals Yu2014 ; YuenZhou2014 ; Huang2019 ; Efimkin2013 ; Gon2017 ,
with various scenarios for producing topological condensates both due to
intrinsic interaction effects Chen2017 ; Seradjeh2009 ; Seradjeh2012 ;
Budich2014 , and coupling to external probes Wan2019 ; Bardyn2015 ; Sun2021 .
Possible experimental signatures of a topological excitonic state have been
identified in InAs/GaSb bilayers Du2017  and transition metal dichalcogenides
Varsano2020 . Our analysis illustrates that $d$-wave excitonic condensation
arises naturally as the ground state of a broad class of systems—hexagonal
systems doped near a twofold van Hove singularity—suggesting new candidate
materials for topological excitonic ground states. Unlike in many previous
works, the topology of the condensate is not inherited from the Berry
curvature at the $K$-points or from spin-orbit coupling, but appears at the
$M$-point intrinsically due to interaction driven, spontaneous time-reversal
symmetry breaking. Existing experiments have explored excitonic condensation
in bilayers with circular Fermi surfaces; exploring the regime where the two
Fermi surfaces are hexagonal would be an interesting variation, hosting
different phenomenology.

In the case of the vanadium metals AV3Sb5, we discuss a few relevant aspects
of experiment. Superconductivity exists alongside density wave order which
sets in at a higher temperature $T_{*}\approx 90$ K than superconductivity
($T_{c}\approx 3.5$ K) and appears to compete with superconductivity Yang2021
; Jiang2021 ; Li2021b ; Zhao2021 ; Li2021c ; Ratcliff2021 ; Shumiya2021 ;
Mielke2021b ; Miao2021 ; Du2021 . The Fermi surface is reconstructed by the
presence of the density wave order, but DFT calculations and magnetotransport
indicate that the roughly hexagonal Fermi contours hosting the twofold van
Hove singularities remain in the reconstructed band structure, which mainly
undergoes changes near the $K$ points Ortiz2021b . The TvHS therefore appears
both in the presence and absence of density wave order. A significant
anomalous Hall effect is observed at temperatures near to $T_{*}$ despite the
lack of magnetic ordering. The presence of chiral excitonic order would offer
an appealing interpretation of the anomalous Hall effect and broken time-
reversal symmetry.

The application of $c$-axis pressure suppresses $T_{*}$ and increases $T_{c}$
Zhu2021 ; Chen2021b ; Du2021b ; Zhao2021b ; Zhang2021 ; Du2021c ; Tsirlin2021
; Du2021d . Applied pressure should change the relative value of the
intra/interflavour couplings by changing the separation of the two-dimensional
layers, and also shifts the effective values of $\mu$ and $\delta$
Consiglio2021 . First principles calculations could relate our phase diagram
to these experiments by determining $\delta$, $\mu$ and couplings as a
function of pressure. Variation of doping and pressure indicates the presence
of two distinct superconducting phases Liu2021 ; Oey2021 ; Yang2021b ; it is
possible that the TvHS dominates the physics in one doping regime, while other
interaction physics could be important elsewhere in the phase diagram Wu2021 ;
Denner2021 ; Park2021 ; Christensen2021 ; Li2020b ; Scammell2021 —e.g. it has
been shown that sublattice interference near the $K$-points can result in
higher-order topological multigap $s$\- or $p$-wave states Li2020 ; Li2020b ;
Scammell2021 .

Shifting $\mu$, i.e. moving the Fermi level closer to one of the vHS, creates
a mismatch in the Fermi surface areas, which disfavours the excitonic phase.
By contrast, shifting $\delta$ moves both vHS away from the Fermi level while
keeping both Fermi surfaces areas the same, which should disfavour
SDW/superconductivity relative to excitonic order. Experiments could measure
how the anomalous Hall signal evolves with pressure, chemical potential, and
strain—which vary $\mu$ and $\delta$ differently—probing for the phase
boundary of the excitonic ordered phase and its relation to the other ground
states present. Additionally, we suggest that the existence of an exciton
condensate should lead to Andreev-like interband tunneling, and effect which
has been used to probe excitonic order in bilayer systems.

Another possible scenario for topological excitonic condensation is to
engineer TvHS in van der Waals heterostructures made from materials with
hexagonal symmetry such as graphene or transition metal dichalcogenides
(TMDCs) Yoo2019 ; Tritsaris2020 ; Zhu2020 ; Zhang2019 . In moiré systems, the
reduced bandwidth of the bands near charge neutrality brings the opposite
concavity vHS of the valence and conduction bands closer in energy, so that an
bias field could feasibly result in a TvHS. The valence and conduction bands
can be further aligned through spin splitting the bands via a TMDC layer
Zaletel2020 ; siriviboon2021abundance , the effect of which can be tuned via
twist angle David2019 . Finally, valley polarisation is observed in twisted
layered systems LiuDaiReview ; lin2021zerofield ; scammell2021theory , which
could also be exploited to align the valence and conduction bands, e.g.
through methods discussed in Ying2021 ; Li2020c . In the context of layered
van der Waals materials, a possible experimental probe would be the enhanced
tunneling between layers induced by excitons, e.g. BurgPRL2018 ;
EfimkinPRB2020 .

## Acknowledgements

We thank Brenden Ortiz, Michael Denner, Colin Nancarrow and Dmitry Efimkin for
discussions and comments. HS and OPS acknowledge funding from ARC Centre of
Excellence FLEET. TL acknowledges support from the Deutsche
Forschungsgemeinschaft.

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Supplementary Material

##  S1 Lattice model realisation

Here we provide additional details of the toy model introduced in Section II,
Eq. (1). The Hamiltonian density in momentum space is

$\displaystyle{\cal H}_{\bm{k}}$
$\displaystyle=\left(\begin{array}[]{ccccc}-\gamma_{0}&-t_{+}\sum_{l_{i}}e^{i\bm{l}_{i}\cdot\bm{k}}&\gamma_{1}e^{i\bm{h}_{A}\cdot\bm{k}}&\gamma_{1}e^{i\bm{h}_{B}\cdot\bm{k}}&\gamma_{1}e^{i\bm{h}_{C}\cdot\bm{k}}\\\
-t_{+}\sum_{l_{i}}e^{-i\bm{l}_{i}\cdot\bm{k}}&-\gamma_{0}&\gamma_{1}e^{-i\bm{h}_{A}\cdot\bm{k}}&\gamma_{1}e^{-i\bm{h}_{B}\cdot\bm{k}}&\gamma_{1}e^{-i\bm{h}_{C}\cdot\bm{k}}\\\
\gamma_{1}e^{-i\bm{h}_{A}\cdot\bm{k}}&\gamma_{1}e^{i\bm{h}_{A}\cdot\bm{k}}&0&-2t_{-}\cos\left(\bm{d}_{3}\cdot\bm{k}\right)&-2t_{-}\cos\left(\bm{d}_{2}\cdot\bm{k}\right)\\\
\gamma_{1}e^{-i\bm{h}_{B}\cdot\bm{k}}&\gamma_{1}e^{i\bm{h}_{B}\cdot\bm{k}}&-2t_{-}\cos\left(\bm{d}_{3}\cdot\bm{k}\right)&0&-2t_{-}\cos\left(\bm{d}_{1}\cdot\bm{k}\right)\\\
\gamma_{1}e^{-i\bm{h}_{C}\cdot\bm{k}}&\gamma_{1}e^{i\bm{h}_{C}\cdot\bm{k}}&-2t_{-}\cos\left(\bm{d}_{2}\cdot\bm{k}\right)&-2t_{-}\cos\left(\bm{d}_{1}\cdot\bm{k}\right)&0\\\
\end{array}\right),$ (S6)

where $\bm{d}_{i}$ and $\bm{l}_{i}$ are the lattice vectors for kagome and
hexagonal layers respectively, and $\bm{h}_{A},\bm{h}_{B},\bm{h}_{C}$ are the
lattice vectors connecting a site of the hexagon $(a,b)$ to the three nearest
kagome sites $(A,B,C)$, as depicted in Fig S1a. We have defined the
Hamiltonian matrix (S6) to act on the basis,

$\displaystyle\hat{\Psi}_{\bm{k}}=(a_{\bm{k}},b_{\bm{k}},A_{\bm{k}},B_{\bm{k}},C_{\bm{k}})^{T}.$
(S7)

Taking $t_{+}=t_{-}=1,\gamma_{0}=3,\gamma_{1}=0.25$, the bandstructure and
Fermi surfaces are shown in Fig. S1b & c.

(a)(b)(c)

Figure S1: Bilayer tight binding toy model: (a) Lattice geometry; black dots
indicate the honeycomb sites connected by dashed lines, coloured dots indicate
kagome sites coloured by sublattice. Inset shows nearest neighbour tunneling
vectors $\bm{h}_{\sigma}$ connecting honeycomb and kagome sites. (b)
Bandstructure and (c) Fermi surface for $\gamma_{0}=3,\gamma_{1}=0.25$. The
colour scaling indicates the wavefunction weight coming from either the kagome
or the honeycomb lattice. The chemical potential is shown as the dotted black
line in (b), and is chosen to correspond to doping near the $\bm{M}$-point.
Corresponding Fermi surface is plotted in (c), with the Brillouin zone
boundary shown in dashed black.

##  S2 Renormalisation group equations

The divergence of particle-particle and particle-hole susceptibilities for
coupling within and between flavours gives rise to an instability of the Fermi
surface towards an ordered ground state. The RG approach allows a systematic
determination of the instability which dominates, by determining the
susceptibility which diverges most strongly. In this Supplement we present the
details of this calculation.

###  S2.1 Flow equations for the couplings

We define $g_{i},h_{i},j_{i},l_{i}$ as the bare couplings and enumerate the
leading logarithmic corrections, which in our case diverge with
$\ell\sim\log^{2}\Lambda$. The Feynman diagrams representing these corrections
are shown in Fig S2. Summing the corrections for each respective coupling, we
gather the corrections into the following coupled RG flow equations,

$\displaystyle\frac{\partial g_{1}}{\partial\ell}$
$\displaystyle=-2d_{1}\left(g_{1}^{2}-g_{2}g_{1}-h_{1}l_{2}+h_{3}\left(h_{3}-l_{3}\right)+h_{1}^{2}\right)$
$\displaystyle\frac{\partial g_{2}}{\partial\ell}$
$\displaystyle=d_{1}\left(g_{2}^{2}+g_{3}^{2}+l_{2}^{2}+l_{3}^{2}\right)$
$\displaystyle\frac{\partial g_{3}}{\partial\ell}$
$\displaystyle=2d_{1}\left(-g_{1}g_{3}+2g_{2}g_{3}+h_{3}l_{2}+h_{1}l_{3}-2h_{1}h_{3}+l_{2}l_{3}\right)-g_{3}^{2}-2g_{4}g_{3}-j_{3}\left(j_{3}+2j_{4}\right)$
$\displaystyle\frac{\partial g_{4}}{\partial\ell}$
$\displaystyle=-2g_{3}^{2}-g_{4}^{2}-2j_{3}^{2}-j_{4}^{2}$ $\displaystyle\ \
\frac{\partial h_{1}}{\partial\ell}$
$\displaystyle=2\left(d_{1}\left(g_{1}\left(l_{2}-2h_{1}\right)+g_{3}\left(l_{3}-h_{3}\right)+g_{2}h_{1}\right)+d_{2}\left(h_{1}h_{4}+j_{1}j_{4}\right)-d_{3}\left(h_{1}h_{2}+l_{1}l_{2}\right)\right)$
$\displaystyle\frac{\partial h_{2}}{\partial\ell}$
$\displaystyle=-d_{3}\left(h_{1}^{2}+h_{2}^{2}+l_{1}^{2}+l_{2}^{2}\right)$
$\displaystyle\frac{\partial h_{3}}{\partial\ell}$
$\displaystyle=2d_{1}\left(g_{3}\left(l_{2}-h_{1}\right)+g_{1}\left(l_{3}-2h_{3}\right)+g_{2}h_{3}\right)$
$\displaystyle\frac{\partial h_{4}}{\partial\ell}$
$\displaystyle=d_{2}\left(2h_{1}^{2}+h_{4}^{2}+2j_{1}^{2}+j_{4}^{2}\right)$
$\displaystyle\ \ \frac{\partial j_{1}}{\partial\ell}$
$\displaystyle=2d_{2}\left(h_{4}j_{1}+h_{1}\left(j_{1}+j_{4}\right)\right)$
$\displaystyle\frac{\partial j_{2}}{\partial\ell}$
$\displaystyle=2d_{2}\left(h_{4}j_{2}+h_{1}\left(j_{2}+j_{4}\right)-2j_{2}l_{2}-2j_{2}l_{4}+j_{1}l_{2}-j_{4}l_{2}+j_{1}l_{4}\right)$
$\displaystyle\frac{\partial j_{3}}{\partial\ell}$
$\displaystyle=-2\left(g_{4}j_{3}+g_{3}\left(j_{3}+j_{4}\right)\right)$
$\displaystyle\frac{\partial j_{4}}{\partial\ell}$
$\displaystyle=d_{2}\left(4h_{1}\left(j_{1}+j_{2}\right)+4h_{4}j_{4}+4j_{1}l_{2}-8j_{2}l_{2}-2j_{4}l_{4}\right)-4g_{3}j_{3}-2g_{4}j_{4}$
$\displaystyle\ \ \frac{\partial l_{1}}{\partial\ell}$
$\displaystyle=-2d_{3}\left(h_{2}l_{1}+h_{1}l_{2}\right)$
$\displaystyle\frac{\partial l_{2}}{\partial\ell}$
$\displaystyle=2d_{1}\left(g_{2}l_{2}+g_{3}l_{3}\right)+2d_{2}\left(h_{1}l_{2}+h_{4}l_{2}+h_{1}l_{4}-j_{2}^{2}-j_{4}j_{2}+j_{1}\left(j_{2}+j_{4}\right)-l_{2}^{2}-2l_{2}l_{4}\right)-2d_{3}\left(h_{1}l_{1}+h_{2}l_{2}\right)$
$\displaystyle\frac{\partial l_{3}}{\partial\ell}$
$\displaystyle=2d_{1}\left(g_{3}l_{2}+g_{2}l_{3}\right)$
$\displaystyle\frac{\partial l_{4}}{\partial\ell}$
$\displaystyle=d_{2}\left(4h_{1}l_{2}+2h_{4}l_{4}-4j_{2}^{2}+4j_{1}j_{2}-j_{4}^{2}-4l_{2}^{2}-2l_{4}^{2}\right)$
(S8)

Figure S2: Flow equations for the couplings: The Feynman diagrams representing
leading logarithmic corrections to the couplings (3) are shown. The resulting
coupled RG equations governing the flow of the couplings
$g_{i},h_{i},j_{i},l_{i}$ are given by (S2.1).

###  S2.2 Solutions to the gap equations

To calculate the susceptibility of the order parameters ${\cal C}_{\alpha
i},{\cal S}_{\alpha i},{\cal P_{\alpha\pm}},\Delta_{\alpha
i},\Phi^{C}_{\alpha\pm},\Phi^{S}_{\alpha\pm}$, we calculate the
renormalisation of the quantities $\mathcal{C}_{\alpha
i}=\langle\psi^{\dagger}_{\alpha}\sigma_{i}\psi_{\bar{\alpha}}\rangle$ (CDW),
$\mathcal{S}_{\alpha
i}=\langle\psi^{\dagger}_{\alpha}\sigma_{i}\vec{s}\psi_{\bar{\alpha}}\rangle$
(SDW), and ${\cal
P}_{\alpha\pm}=\langle\psi_{\alpha}\sigma_{\pm}\psi_{{\bar{\alpha}}}\rangle$
(PDW), with $\bar{\alpha}$ denoting the patch connected to $\alpha$ by a
nesting vector, and $\Delta_{\alpha
i}=\langle\psi_{\alpha}\sigma_{i}\psi_{\alpha}\rangle$ (SC),
$\Phi^{C}_{\alpha\pm}=\langle\psi^{\dagger}_{\alpha}\sigma_{\pm}\psi_{\alpha}\rangle$
(spin-singlet excitons) and
$\Phi^{S}_{\alpha\pm}=\langle\psi^{\dagger}_{\alpha}\sigma_{\pm}\vec{s}\psi_{\alpha}\rangle$
(spin-triplet excitons). The relevant Feynman diagrams are shown in Fig S3.
The resulting RG equations for the flow of these composite operators, which
constitute the gap equations for the order parameters, are as presented in Eq.
(7).

In this section we shall step through each order parameter type and
diagonalise the associated gap equation (7), giving the eigenvectors—which
represent the possible ground states—and eigenvalues—which determine the
favored ground state.

Figure S3: Gap equations: The Feynman diagrams representing the leading
logarithmic corrections to the order parameter fields are shown. The result is
a set of linear equations for the susceptibilities, which give the gap
equations for the various possible order parameters, in Eq. (7).

Spin-singlet excitonic order. The charge exciton order parameter can be
written in the form

$\displaystyle\Phi^{C}=\sum_{\alpha,\pm}u_{\alpha,\pm}\langle\psi^{\dagger}_{\alpha}\sigma_{\pm}\psi_{\alpha}\rangle$
(S9)

where $u_{\alpha,\pm}$ are eigenvector components of the gap equation (7) (in
the $\Phi^{C}$ sector). Explicitly, the eigenvalues and eigenvectors are
elements of the vector ${\cal E}_{\Phi}$, and columns of the matrix
$U_{\Phi}$, respectively,

$\displaystyle\lambda_{i}\in{\cal E}_{\Phi}$
$\displaystyle=\left(\begin{array}[]{c}2\left(2h_{1}+h_{4}+2j_{1}-4j_{2}-j_{4}-4l_{2}-2l_{4}\right)\\\
-2\left(2h_{1}+h_{4}-2j_{1}+4j_{2}+j_{4}-4l_{2}-2l_{4}\right)\\\
-2\left(h_{1}-h_{4}+j_{1}-2j_{2}+j_{4}-2l_{2}+2l_{4}\right)\\\
-2\left(h_{1}-h_{4}+j_{1}-2j_{2}+j_{4}-2l_{2}+2l_{4}\right)\\\
2\left(h_{1}-h_{4}-j_{1}+2j_{2}-j_{4}-2l_{2}+2l_{4}\right)\\\
2\left(h_{1}-h_{4}-j_{1}+2j_{2}-j_{4}-2l_{2}+2l_{4}\right)\\\
\end{array}\right),\ \ u_{\alpha,\pm}\in
U_{\Phi}=\frac{1}{\sqrt{6}}\left(\begin{array}[]{cccccc}1&-1&-\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}&\sqrt{\frac{3}{2}}&\frac{1}{\sqrt{2}}\\\
1&-1&0&\sqrt{2}&0&-\sqrt{2}\\\
1&-1&\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}&-\sqrt{\frac{3}{2}}&\frac{1}{\sqrt{2}}\\\
1&1&-\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}&-\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}\\\
1&1&0&\sqrt{2}&0&\sqrt{2}\\\
1&1&\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}&\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}\\\
\end{array}\right)$ (S22)

where the upper (lower) three elements correspond to $+$ ($-$), and the three
elements within an upper/lower block index the three patches $\alpha$. The
first two eigenvectors are $s$-wave since there is no sign change of the order
parameter as one moves around the Fermi surface by permuting the patches. The
remaining four have $d$-wave character, with two sign changes along the Fermi
surface.

Spin-triplet excitonic order. Analogously, the spin exciton order parameter
can be written in the form

$\displaystyle\Phi^{S}=\sum_{\alpha,\pm}u_{\alpha,\pm}\langle\psi^{\dagger}_{\alpha}\sigma_{\pm}\psi_{\alpha}\rangle$
(S23)

In this case the solution to the gap equations gives the eigenvalues and
eigenvectors as elements of the vector ${\cal E}_{\Phi}$ and columns of the
matrix $U_{\Phi}$, respectively,

$\displaystyle\lambda_{i}\in{\cal E}_{\Phi}$
$\displaystyle=\left(\begin{array}[]{c}2\left(2h_{1}+h_{4}+2j_{1}-4j_{2}-j_{4}-4l_{2}-2l_{4}\right)\\\
-2\left(2h_{1}+h_{4}-2j_{1}+4j_{2}+j_{4}-4l_{2}-2l_{4}\right)\\\
-2\left(h_{1}-h_{4}+j_{1}-2j_{2}+j_{4}-2l_{2}+2l_{4}\right)\\\
-2\left(h_{1}-h_{4}+j_{1}-2j_{2}+j_{4}-2l_{2}+2l_{4}\right)\\\
2\left(h_{1}-h_{4}-j_{1}+2j_{2}-j_{4}-2l_{2}+2l_{4}\right)\\\
2\left(h_{1}-h_{4}-j_{1}+2j_{2}-j_{4}-2l_{2}+2l_{4}\right)\\\
\end{array}\right),\ \ u_{\alpha,\pm}\in
U_{\Phi}=\frac{1}{\sqrt{6}}\left(\begin{array}[]{cccccc}1&-1&-\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}&\sqrt{\frac{3}{2}}&\frac{1}{\sqrt{2}}\\\
1&-1&0&\sqrt{2}&0&-\sqrt{2}\\\
1&-1&\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}&-\sqrt{\frac{3}{2}}&\frac{1}{\sqrt{2}}\\\
1&1&-\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}&-\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}\\\
1&1&0&\sqrt{2}&0&\sqrt{2}\\\
1&1&\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}&\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}\\\
\end{array}\right).$ (S36)

Intrapatch superconductivity. We write the intrapatch superconducting order
parameter in the form

$\displaystyle\Delta=\sum_{\alpha,\pm}u_{\alpha,i}\langle\psi^{\dagger}_{\alpha}\sigma_{i}\psi^{\dagger}_{\alpha}\rangle.$
(S37)

In this case the eigenvalues and eigenvectors are

$\displaystyle\lambda_{i}\in{\cal
E}_{\Delta}=\left(\begin{array}[]{c}-2\left(2g_{3}+g_{4}+2j_{3}+j_{4}\right)\\\
2\left(-2g_{3}-g_{4}+2j_{3}+j_{4}\right)\\\
2\left(g_{3}-g_{4}+j_{3}-j_{4}\right)\\\
2\left(g_{3}-g_{4}+j_{3}-j_{4}\right)\\\
2\left(g_{3}-g_{4}-j_{3}+j_{4}\right)\\\
2\left(g_{3}-g_{4}-j_{3}+j_{4}\right)\\\ \end{array}\right),\ \
u_{\alpha,i}\in
U_{\Delta}=\frac{1}{\sqrt{6}}\left(\begin{array}[]{cccccc}1&-1&-\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}&\sqrt{\frac{3}{2}}&\frac{1}{\sqrt{2}}\\\
1&-1&0&\sqrt{2}&0&-\sqrt{2}\\\
1&-1&\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}&-\sqrt{\frac{3}{2}}&\frac{1}{\sqrt{2}}\\\
1&1&-\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}&-\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}\\\
1&1&0&\sqrt{2}&0&\sqrt{2}\\\
1&1&\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}&\sqrt{\frac{3}{2}}&-\frac{1}{\sqrt{2}}\\\
\end{array}\right).$ (S50)

As with the excitonic eigenvectors, the upper/lower three elements correspond
to $i=c,d$, while the three elements in each upper/lower half correspond to
the three patches.

Pair density wave. The interpatch superconducting (PDW) order parameter takes
the form

$\displaystyle{\cal
P}=\sum_{\alpha\neq\alpha^{\prime},\pm}v_{\alpha,\pm}\langle\psi_{\alpha}\sigma_{\pm}\psi_{\alpha^{\prime}}\rangle.$
(S51)

Here, the eigenvalues and eigenvectors are

$\displaystyle\lambda_{i}\in{\cal E}_{\cal P}$
$\displaystyle=\left(\begin{array}[]{c}-2\left(h_{1}+h_{2}+l_{1}+l_{2}\right)\\\
2\left(h_{1}-h_{2}+l_{1}-l_{2}\right)\\\
2\left(h_{1}-h_{2}-l_{1}+l_{2}\right)\\\
2\left(-h_{1}-h_{2}+l_{1}+l_{2}\right)\\\ \end{array}\right),\ \
v_{\alpha,\pm}\in U_{\cal
P}=\frac{1}{2}\left(\begin{array}[]{cccc}1&1&-1&-1\\\ 1&-1&-1&1\\\
1&-1&1&-1\\\ 1&1&1&1\\\ \end{array}\right)$ (S60)

The upper and lower two elements correspond to the index of $\sigma_{\pm}$,
i.e. flavours $c$ and $d$.

Spin and charge density wave order. For CDW and SDW, we write the order
parameters as

$\displaystyle{\cal
C}=\sum_{\alpha\neq\alpha^{\prime},i}u_{\alpha,\pm}\langle\psi^{\dagger}_{\alpha}\sigma_{i}\psi_{\alpha^{\prime}}\rangle,\
\ {\cal
S}=\sum_{\alpha\neq\alpha^{\prime},i}v_{\alpha,\pm}\langle\psi^{\dagger}_{\alpha}\sigma_{i}\vec{s}\psi_{\alpha}\rangle.$
(S61)

Solving the gap equations gives us the eigenvalues and eigenvectors

$\displaystyle{\cal E}_{\cal C}$
$\displaystyle=\left(\begin{array}[]{c}-2\left(2g_{1}-g_{2}+g_{3}-2h_{1}-2h_{3}+l_{2}+l_{3}\right)\\\
2\left(-2g_{1}+g_{2}+g_{3}-2h_{1}+2h_{3}+l_{2}-l_{3}\right)\\\
2\left(-2g_{1}+g_{2}+g_{3}+2h_{1}-2h_{3}-l_{2}+l_{3}\right)\\\
2\left(-2g_{1}+g_{2}-g_{3}-2h_{1}-2h_{3}+l_{2}+l_{3}\right)\\\
\end{array}\right),\ \ U_{\cal C}=\left(\begin{array}[]{cccc}-1&-1&1&1\\\
-1&1&-1&1\\\ 1&-1&-1&1\\\ 1&1&1&1\\\ \end{array}\right).$ (S70)

Similarly for SDW,

$\displaystyle{\cal E}_{\cal S}$
$\displaystyle=\left(\begin{array}[]{c}2\left(g_{2}+g_{3}-l_{2}-l_{3}\right)\\\
2\left(g_{2}-g_{3}+l_{2}-l_{3}\right)\\\
2\left(g_{2}-g_{3}-l_{2}+l_{3}\right)\\\
2\left(g_{2}+g_{3}+l_{2}+l_{3}\right)\\\ \end{array}\right),\ \ U_{\cal
S}=\left(\begin{array}[]{cccc}-1&-1&1&1\\\ -1&1&-1&1\\\ 1&-1&-1&1\\\
1&1&1&1\\\ \end{array}\right).$ (S79)

In Fig. S4 we show the RG flow of the order parameter eigenvalues for three
represented choices of bare coupling values, $\mu$, $\delta$, and interactions
$g_{i},h_{i},j_{i},l_{i}$. The leading instability is identified as the fasted
diverging eigenvalue.

Figure S4: Leading divergent susceptibilities. Blue, gray, red, dashed gray
correspond to the (leading) $\Phi^{C}$, $\Delta$, $\Phi^{S}$ and SDW; all
other order parameters are subdominant in the parameters space considered.
Everywhere $g_{i}=0.1$. Three representative cases are presented: (a)
$\delta=\mu=0$, with $h_{i}=j_{i}=l_{i}=0.025$, (b) $\delta/\Lambda=0.1$,
$\mu/\Lambda=0.05$, with $h_{i}=j_{i}=l_{i}=0.025$, and (c)
$\delta/\Lambda=\mu/\Lambda=0.1$, with $h_{i}=j_{i}=l_{i}=0.05$.

##  S3 Properties of the chiral excitonic condensate

In this section we elaborate on the properties of the chiral excitonic
condensate. The excitonic order parameter winds by a phase $\pm 4\pi$ around
the Fermi surface and fully gaps the bulk dispersion (c.f. Fig S5). As a
result, the mean-field Hamiltonian for the excitonic condensate describes a
Chern insulator with Chern number $C=\pm 2$. In order to demonstrate the non-
trivial topology, we diagonalise a mean-field Hamiltonian defined on a
lattice, which provides a completion of the low-energy description we have so
far considered to the entire Brillouin zone. Following on from the toy model
of Section S1, we shall consider a honeycomb/kagome bilayer in an infinite
ribbon geometry with zigzag edges. In addition, we shall present results for a
honeycomb-honeycomb bilayer systems.

###  S3.1 Real space Hamiltonian

Figure S5: Exact diagonalisation results: Left: the bulk dispersion for the
honeycomb/kagome bilayer with parameters $t_{+}=t_{-}=t$, $t_{\perp}=0.1t$,
and $\Delta=0.3t$ (red), $\Delta=0$ (blue). Right: the path in the Brillouin
zone along which the dispersion is plotted.

In this section, we shall switch to more compact notation compared to the main
text and S1, for ease of describing the real space structure of the excitonic
order parameter. We introduce the simplified lattice model

$\displaystyle H$ $\displaystyle=H_{+}+H_{-}+H_{\perp}+H_{\Delta}\ \ ,$
$\displaystyle H_{+}$
$\displaystyle=-t_{+}\sum_{\langle\bm{r},\bm{r}^{\prime}\rangle\in+}{c^{\dagger}_{\bm{r}^{\prime}}c_{\bm{r}}}+\gamma_{0}\sum_{\bm{r}\in+}{c^{\dagger}_{\bm{r}}c_{\bm{r}}}\
\ ,$ $\displaystyle H_{-}$
$\displaystyle=-t_{-}\sum_{\langle\bm{r},\bm{r}^{\prime}\rangle\in\nu}{c^{\dagger}_{\bm{r}^{\prime}}c_{\bm{r}}}\
\ ,$ $\displaystyle H_{\perp}$
$\displaystyle=\gamma_{1}\sum_{\begin{subarray}{c}\bm{r}\in\sigma=\\{a,b\\}\\\
j=\\{A,B,C\\}\end{subarray}}{c^{\dagger}_{\bm{r}+\bm{h}_{j\sigma}}c_{\bm{r}}}+\text{h.c.}\
\ ,$ $\displaystyle H_{\Delta}$
$\displaystyle=\sum_{\bm{r}^{\prime}\in+,\bm{r}\in-}{\Delta(\bm{r}^{\prime},\bm{r})c^{\dagger}_{\bm{r^{\prime}}}c_{\bm{r}}}$
(S80)

consisting of a honeycomb layer stacked on top of a kagome layer, where
$\nu=+,-$ denote the honeycomb and kagome layers, $a,b$ denote the sublattices
of the honeycomb layer, $A,B,C$ denote the sublattices of the kagome layer,
and $\bm{h}_{j\sigma}$ are vectors connecting a site in sublattice $\sigma$ in
the honeycomb layer to its nearest neighbors in sublattice $j$ in the kagome
layer. The Hamiltonian consists of the bilayer toy model – with nearest
neighbor hopping in the honeycomb and kagome planes with hopping energies
$t_{\pm}$, a relative chemical potential $\gamma_{0}$ between the two layers,
as well as tunneling $\gamma_{1}$ – along with an excitonic pairing term
$\Delta(\bm{r}^{\prime},\bm{r})$ from a site in the honeycomb layer to its
three nearest neighbors in the kagome layer. We choose the excitonic pairing
function $\Delta(\bm{r},\bm{r}^{\prime})$ so that the lattice theory possesses
an equivalent continuum limit to our field theory description of the three
patches surrounding the $M$ points. The spatial wavefunctions of the
eigenstates $|\bm{k}\approx\bm{M}_{j},\nu\rangle$ of $H_{\parallel}$ in the
upper and lower layers ($\nu=+,-$) are given explicitly by

$\displaystyle\psi^{\dagger}_{\bm{k},\nu}=\frac{1}{\sqrt{N}}\sum_{\bm{r}\in\nu}{\varphi_{\bm{k},\nu}(\bm{r})c^{\dagger}_{\bm{r}}}$
(S81)

where

$\displaystyle\varphi_{\bm{k},+}(\bm{r})=\tfrac{1}{\sqrt{2}}e^{i\bm{M}_{j}\cdot\bm{r}},\
\ \ \
\varphi_{\bm{k},-}(\bm{r})=\begin{cases}\tfrac{1}{\sqrt{2}}e^{i\bm{M}_{j}\cdot\bm{r}}\
&j\neq\sigma_{j}\\\ 0\ \ \ &j=\sigma_{j}\end{cases}$ (S82)

where $\bm{k}\approx\bm{M}_{j}$, and the sublattice index
$(\sigma_{1},\sigma_{2},\sigma_{3})=(a,b,c)$ corresponds to the sites in the
kagome lattice for which the 2D projection of the bond vector
$\bm{h}_{\sigma_{j}A}$ is parallel to $\bm{M}_{j}$. Near the $M$ points, the
effective Hamiltonian projected onto states near the Fermi surface is given by

$\displaystyle
H_{\Delta}=\sum_{j,\bm{k}\approx\bm{M}_{j}}{\Delta_{j}\psi^{\dagger}_{\bm{k},+}\psi_{\bm{k},-}}+\text{h.c.},$
$\displaystyle\Delta_{j}=\tfrac{1}{2}e^{\frac{i\pi}{6}}\sum_{\sigma\neq
j}{\Delta(\bm{r}_{A}+\bm{h}_{\sigma
A},\bm{r}_{A})-\Delta(\bm{r}_{B}+\bm{h}_{\sigma B},\bm{r}_{B})}$ (S83)

where $\bm{r}_{A},\bm{r}_{B}$ are the coordinates of the A and B sites in the
first unit cell. The direct tunneling Hamiltonian $H_{\perp}$ does not appear
in the effective Hamiltonian near the $M$ points, since the contributions from
hopping processes involving opposite sublattices in the honeycomb layer
interfere destructively. A $d\pm id$ order parameter
$(\Delta_{1},\Delta_{2},\Delta_{3})=(\Delta,\Delta e^{\frac{2\pi
i}{3}\ell},\Delta e^{\frac{4\pi i}{3}\ell})$ for $\ell=\pm 2$ implies that the
real-space pairing functions are given by

$\displaystyle\Delta(\bm{r}_{A}+\bm{h}_{aA},\bm{r}_{A})=-\Delta(\bm{r}_{B}+\bm{h}_{aB},\bm{r}_{A})=e^{-\frac{i\pi}{6}}\Delta$
$\displaystyle\Delta(\bm{r}_{A}+\bm{h}_{bA},\bm{r}_{A})=-\Delta(\bm{r}_{B}+\bm{h}_{bB},\bm{r}_{A})=e^{-\frac{i\pi}{6}+\frac{2\pi
i}{3}\ell}\Delta$
$\displaystyle\Delta(\bm{r}_{A}+\bm{h}_{cA},\bm{r}_{A})=-\Delta(\bm{r}_{B}+\bm{h}_{cB},\bm{r}_{A})=e^{-\frac{i\pi}{6}+\frac{4\pi
i}{3}\ell}\Delta$ (S84)

which we illustrate in Fig. S6.

Figure S6: The real space pairing Hamiltonian (S84) for a $d+id$ excitonic
insulator ($\ell=2$) . The large (small) circles indicate sites in the
honeycomb (kagome) layer, while the colored bonds represent the phases of the
excitonic pairing; blue, green, red, yellow, magenta, and cyan bonds
corresponding to the phases $e^{-\frac{i\pi}{6}},e^{-\frac{5\pi
i}{6}},e^{i\pi},e^{\frac{5i\pi}{6}},e^{\frac{i\pi}{6}},1$.

Figure S7: Right: The 1D dispersion of an AA stacked honeycomb bilayer for an
infinite ribbon with width 120 unit cells, with parameters $t_{+}=t_{-}=t$,
$t_{\perp}=0.1t$, $\Delta=0.3t$. The edge states on the top/bottom of the
ribbon are plotted in red/blue. Left: kagome

###  S3.2 Exact diagonalisation results

We diagonalise the mean-field lattice Hamiltonian for an infinite ribbon
geometry in the $\ell=2$ phase with zigzag edges for parameters $t_{+}=t_{-}$,
$\gamma=0.1t$, $\Delta=0.4t$, and show the 1D dispersion as a function of
momentum $k_{x}$ along the ribbon in Fig. S7 in the main text, with energy in
units of $t_{+}$. The dispersion exhibits two chiral edge modes, with the
left-(right-)movers propagating along the top (bottom) of the ribbon
represented by red (blue). The bulk dispersion is shown in Fig. S5.

For comparison, we have also diagonalised a lattice model consisting of two
$AA$ stacked honeycomb layers. In this case the chiral $d$-wave order
generates pairing between an $A$ site in one layer and its nearest neighboring
$B$ sites in the opposite layer, with the pairing function

$\displaystyle\Delta(\bm{r}_{A}+\bm{d}_{1}+\bm{c},\bm{r}_{A})=-\Delta(\bm{r}_{B}-\bm{d}_{1}+\bm{c},\bm{r}_{B})=\tfrac{1}{2}\Delta$
$\displaystyle\Delta(\bm{r}_{A}+\bm{d}_{1}+\bm{c},\bm{r}_{A})=-\Delta(\bm{r}_{B}-\bm{d}_{1}-\bm{c},\bm{r}_{B})=\tfrac{1}{2}e^{\frac{2\pi
i}{3}\ell}\Delta$
$\displaystyle\Delta(\bm{r}_{A}+\bm{d}_{1}-\bm{c},\bm{r}_{A})=-\Delta(\bm{r}_{B}-\bm{d}_{1}-\bm{c},\bm{r}_{B})=\tfrac{1}{2}e^{\frac{4\pi
i}{3}\ell}\Delta$ (S85)

The 1D dispersion for an infinite ribbon in the $\ell=2$ phase with parameters
$t_{+}=t_{-}=t,\gamma_{0}=2t,\gamma_{1}=0.1t,\Delta=0.3t$ is shown in Fig. S7,
with $t_{\pm},\gamma_{0},\gamma_{1}$ referring (as in the kagome model) to the
intralayer hopping energies, chemical potential shift between the layers and
vertical hopping energies respectively.