# Fingerprints of Moiré Interference between Carriers in Graphene
Superlattices
and Analogies to Twisted Graphene Bilayers at Magic Angles

Saúl A. Herrera  Gerardo G. Naumis<EMAIL_ADDRESS>Depto. de Sistemas
Complejos, Instituto de Física,
Universidad Nacional Autónoma de México (UNAM)
Apdo. Postal 20-364, 01000, CDMX, México.

###### Abstract

Motivated by recent experimental findings on the low-energy spectrum of
Kekulé-patterned graphene, the optoelectronic signatures of graphene
superlattices with a spatial modulation that triples the size of the unit cell
and folds the valleys to the center of the Brillouin zone are studied. For
superlattices like those visualized in recent experiments, the optoelectronic
response reveals multiple species of carriers distinguished by their effective
masses or Fermi velocities. Their signatures are similar to those of systems
hosting multifold fermions in which different frequency intervals are
dominated by different types of quasiparticles. Remarkably, the response of
these systems exhibits a characteristic peak in the optical conductivity
suggesting a kind of moiré interference between the different species of
carriers. We also show a relation between these systems and moiré
superlattices by obtaining a minimal model for a graphene superlattice that
exhibits flat bands like those found in twisted graphene bilayers.

††preprint: APS/123-QED

Introduction.— The exotic effects that spatial modulations can induce in the
electronic properties of two-dimensional materials has been the focus of many
theoretical and experimental works in the last years [1, 2, 3, 4, 5, 6]. More
recently, this interest has been further accelerated by the discovery of
intriguing correlation phenomena in twisted bilayer graphene (TBG) [7, 8, 9,
10, 11, 8, 12, 13, 14, 15, 16, 17, 18], where a slight mismatch between two
rotated graphene lattices leads to large-scale spatial modulations, known as
moiré patterns. The study of spatial modulations has also played a main role
in the field of valleytronics [19, 20, 21, 22, 23, 24], which focuses in the
control of the valley degree of freedom to search for novel mechanisms in
quantum transport or for information storage.

One of the most interesting examples of superlattices resulting from spatial
modulation in a 2D material is Kekulé-patterned graphene. This phase was first
proposed as a novel platform hosting fractionally charged topological
excitations [1, 25] and later considered in a mechanism for unconventional
superconductivity in graphite [26]. Very recently, Kekulé ordering has been
predicted to arise in the correlated insulating states of TBG [27, 28],
increasing the interest in the study of Kekulé-patterned superlattices.

There has also been increasing interest in the transport properties of Kekulé-
patterned graphene for applications in valleytronics [29, 30, 31, 32, 33, 34,
35], since the symmetry of the modulation folds the $K$, $K^{\prime}$ valleys
to the center of the BZ and enables intervalley transport for low-energy
carriers [36, 37, 38, 39]. Kekulé ordering has been predicted to arise in
graphene due to multiple mechanisms like the ordering of adatoms [36, 40],
substrate mismatch [41, 42, 43], isotropic strain [44], electron-phonon
coupling [45] and spin-phonon coupling [46]. However, the experimental
realization of Kekulé-patterned graphene and the probing of its electronic
structure was not achieved until very recently [42, 47]. The measurement of
the low-energy density of states [47] supported predictions about the
existence of two Kekulé-ordered phases: one preserving the Dirac point and the
other opening a gap [37]. However, further studies are required to support
other important features, like the valley-momentum locking [47, 37].

There are three main contributions of this work: (1) Focusing on the types of
superlattices that were recently reported in experiments by Eom et al [47], we
discuss the optical signatures that might prove useful in their experimental
characterization by, for example, confirming the momentum-valley locking [37].
(2) We probe the robustness and generality of such signatures by analyzing
multiple superlattices. This is important since multiple phases can be present
[47] and because other factors, like second-neighbor interactions [48] or a
substrate-induced ionic potential [37], might become important. This also
gives information about which signatures are a direct consequence of the
symmetry induced by the modulation. (3) We discuss a more general model for
these superlattices and show that they might be useful platforms to study the
physics of systems like twisted graphene bilayers and other moiré
superlattices, which have been attracting much research interest recently.

In the following, we introduce the models to be studied and discuss the
lattices and their low-energy Hamiltonians, focusing first on superlattices
with the symmetries of those experimentally identified by Eom et al. [47].
Then, we study the dynamic polarizabilities and optical conductivities of
these systems and find the signatures in their optoelectronic response that
might be useful for their experimental characterization, as has been the case
for strained graphene [49, 6]. An emphasis is made here on the characteristic
signature that could help verify the theoretical prediction of a valley-
momentum locking [37]. Afterwards, we discuss a more general model for these
systems and establish a connection to moiré superlattices by showing that one
can use the same hamiltonian to derive a low-energy model similar to that of
chiral TBG exhibiting flat bands, which might allow to study the physics of
TBG and other moiré superlattices with a minimal model based on these types of
systems.

Kekulé-patterned graphene.— We begin our discussion with the model
corresponding to a Kekulé-patterned graphene superlattice in which the
modulation is introduced by a bond-density wave tripling the size of the unit
cell. These bond modulations have been predicted to originate from strain [44,
47], electron-phonon coupling [45], and other mechanisms [36, 46, 41, 42, 43,
50]. The low-energy hamiltonian is given by [37],

$H_{K}=\begin{pmatrix}0&v_{0}k_{-}&\Delta Q_{\nu,+}&0\\\ v_{0}k_{+}&0&0&\Delta
Q_{\nu,-}\\\ \Delta Q^{*}_{\nu,+}&0&0&v_{0}k_{-}\\\ 0&\Delta
Q^{*}_{\nu,-}&v_{0}k_{+}&0\end{pmatrix},$ (1)

where $Q_{\nu,\pm}=v_{0}|\nu|(\nu k_{x}-ik_{y})\pm 3t_{0}(1-|\nu|)$,
$k_{\pm}=k_{x}\pm ik_{y}$, $v_{0}$ is the Fermi velocity and $\nu=0,\pm 1$,
which leads to the Kek-O phase for $\nu=0$ and the Kek-Y phase for $|\nu|=1$.
Recent experiments have supported this model [47]. The band structures for
both Kek-O and Kek-Y exhibit the two valleys folded into the $\Gamma$-point.
As seen in Fig. 1, the Kek-O phase opens a gap while the Kek-Y phase retains
the gapless dispersion [37]. Therefore, the Kek-O phase is not expected to
exhibit optical activity for low frequencies and small doping, and thus our
discussion will be focused on the Kek-Y phase. Nevertheless, as we discuss
below, some results apply to both the Kek-Y and Kek-O phases. The energy
dispersion of the Kek-Y phase is $E_{k\alpha}^{\beta}=\alpha
v_{0}(1+\beta\Delta)k$, with $\alpha,\beta=\pm 1$. The low-energy dispersions
for the Kek-Y and Kek-O phases are shown in Fig. 1c.

Graphene superlattice with a quadratic band crossing.— We introduce now a
model for a graphene superlattice sharing the same symmetry, and thus also
exhibiting a tripled unit cell with the two valleys folded into the
$\Gamma$-point. In this model, however, the superlattice is produced due to
the on-site energies of the atoms being modulated by, for example, the
interaction with a substrate [51, 41]. The study of this second model will
help us to understand how does the optoelectronic response depend on the
physical origin of the modulation and to identify the more robust signatures
that are inherent to the symmetry. This model was used in Ref. [51] to study
the realization of the quantum anomalous Hall effect in graphene introduced by
the influence of a suitable substrate. Also, a similar band structure has been
predicted for graphene-In2Te2 bilayers [41]. Since a substrate-induced ionic
potential or second-neighbor interactions (which might be present in
experiments) produce a similar dispersion in the Kekulé phases [37, 52], these
are additional reasons to consider this model. The lattice is shown in Fig.
1b. It consists of three different onsite energies, with all the bond
strengths being the same. The corresponding low-energy hamiltonian can be
written as [51],

Figure 1:  Superlattices hosting different species of carriers and their low-
energy dispersions. (a) Superlattice associated with $H_{K}$ (Kek-Y phase).
All on-site energies are equal but there are two different bond strengths. The
optical response of this superlattice reveals quasiparticles with velocities
$v_{+}$ and $v_{-}$. (b) Superlattice associated with $H_{Q}$ (Kek-M phase).
All bond strengths are equal but there are three different on-site energies.
The optical response of this superlattice reveals quasiparticles with
effective masses $m_{+}$ and $m_{-}$. In (c) and (d) the low-energy
dispersions of the Kek-Y, Kek-O and Kek-M phases are shown for different
values of the parameters.

$H_{Q}=\begin{pmatrix}m_{0}v_{0}^{2}&v_{0}k_{-}&2t_{0}\Delta_{0}&0\\\
v_{0}k_{+}&-m_{0}v_{0}^{2}&0&0\\\
2t_{0}\Delta_{0}&0&m_{0}v_{0}^{2}&-v_{0}k_{+}\\\
0&0&-v_{0}k_{-}&-m_{0}v_{0}^{2}\end{pmatrix}.$ (2)

The parameter $t_{0}$ is the hopping integral defining the energy scale,
$\Delta_{0}$ introduces a valley coupling and $m_{0}$, an effective mass,
breaks the sublattice symmetry. The low energy dispersion for this hamiltonian
is $E_{k\alpha}^{\beta}=\beta\Delta
m_{0}v_{0}^{2}+\alpha\sqrt{v_{0}^{2}k^{2}+(m_{0}+\beta\Delta
m_{0})^{2}v_{0}^{4}}$ with $\alpha,\beta=\pm$ and after scaling the coupling
parameter as $\Delta=t_{0}\Delta_{0}/m_{0}v_{0}^{2}$ for convenience. In the
following we refer to this as the Kek-M phase. The energy dispersion is shown
in Fig. 1d.

Polarizability.— The optoelectronic response, within linear response theory,
is given by the dynamical polarizability [53, 54, 55], which can be written as

$\Pi(\omega,q)=-g_{s}\sum_{\alpha\alpha^{\prime}\beta\beta^{\prime}}\int\frac{d^{2}k}{4\pi^{2}}\frac{f_{k\alpha}^{\beta}-f_{k^{\prime}\alpha^{\prime}}^{\beta^{\prime}}}{E_{k\alpha}^{\beta}-E_{k^{\prime}\alpha^{\prime}}^{\beta^{\prime}}+\omega^{+}}F_{\alpha\alpha^{\prime}}^{\beta\beta^{\prime}}(\bm{k},\bm{k^{\prime}})$
(3)

where $f_{k\alpha}^{\beta}=[\exp(E_{k\alpha}^{\beta}-\mu)/k_{B}T+1]^{-1}$ is
the Fermi-Dirac distribution, $g_{s}=2$ is the spin degeneracy and
$\omega^{+}=\omega+i\eta_{0}$ is the frequency with an infinitesimally small
imaginary part added for convergence. The scattering probability is given by
the form factor
$F_{\alpha\alpha^{\prime}}^{\beta\beta^{\prime}}(\bm{k},\bm{k^{\prime}})=|\langle\Psi_{k\alpha}^{\beta}|\Psi_{k^{\prime}\alpha^{\prime}}^{\beta^{\prime}}\rangle|^{2}$
with $\bm{k^{\prime}}=\bm{k}+\bm{q}$.

In the following sections we discuss some of the general results and
signatures of $\Pi(\omega,q)$ for the low-energy hamiltonians $H_{Q}$ and
$H_{K}$ introduced above. For a more detailed discussion of the calculation we
direct the reader to the Supplementary Material.

Figure 2: Moiré interference between different species of carriers as
exhibited in the optical conductivity. (a) Optical conductivity of the Kek-Y
phase for a coupling of $\Delta=0.1$. The inset shows different activation
frequencies for carriers with velocities $v_{\pm}$ (b) Optical conductivity of
the Kek-M phase for the parameters $\Delta_{0}=0.2$ and
$m_{0}v_{0}^{2}/t_{0}=0.3$. The inset shows different activation frequencies
for carriers with effective masses $m_{\pm}$. The conductivity is shown in
units of $4e^{2}/h$. (c) The relation between the intravalley activation
frequencies $\omega_{\pm}$ and the intervalley resonance frequency at
$\omega_{M}$ is that of a Moiré interference pattern [see Eq. (4)]. (d)
Summary of the signatures of each phase (estimations of $\omega_{M}$
correspond to $m_{0}v_{0}^{2}/t_{0}\sim 0.2$ and $\mu\sim 0.5$ eV).

Moiré Interference in the Optical Conductivity.— In the following we discuss
the signatures in the optical conductivity, which can be obtained directly
from Eq. (3) in the limit of $q\rightarrow 0$ [53, 39]. In Fig. 2 we plot the
optical conductivities obtained for the Kek-Y and the Kek-M phases using the
low-energy models introduced above. Two interesting features are seen to
appear in the optical conductivities of both superlattices: (1) While the
intervalley conductivity of graphene starts at an onset frequency of
$\omega_{0}=2\mu$ (due to Pauli blocking), for both superlattices two onset
frequencies $\omega_{\pm}\approx 2\mu(1\pm\Delta)$ are seen instead. (2) An
absorption peak arises at low frequencies in the optical conductivity of both
superlattices. The resonance occurs at a frequency given by

$\omega_{M}=\frac{\omega_{+}-\omega_{-}}{2}.$ (4)

Interestingly, this last relation coincides with the definition of a Moiré
interference pattern, originating from the interference of two slightly
mismatching scales defined by frequencies $\omega_{+}$ and $\omega_{-}$. The
resulting Moiré pattern has a larger scale defined by
$\frac{1}{2}(\omega_{+}-\omega_{-})$, in coincidence with Eq. (4). We make the
remark that the relation in Eq. (4) holds for both models regardless of the
fact that $H_{K}$ and $H_{Q}$ describe modulations with different physical
origins, have different energy dispersions, and that the expressions for
$\omega_{\pm}$ and $\omega_{M}$ as a function of the valley coupling are
different in each case. This points to the signature being originated from the
symmetry alone. In terms of coupling parameters, the resonance peak for the
Kek-M phase is given by $\omega_{M}\approx 2\Delta_{0}t_{0}$ (at high doping).
For the case of the Kek-Y phase, the peak occurs at $\omega_{M}\approx
2\Delta\mu$ (see Supplementary Material for more details). Since the resonance
at $\omega_{M}$ corresponds to intervalley transitions, its observation would
provide evidence for the predicted valley-momentum locking [37, 47].

Remarkably, the features at $\omega_{\pm}$ in the optical conductivity belong
to the response of two species of quasiparticles in each superlattice:
massless Dirac fermions with Fermi velocities $v_{\pm}=v_{0}\pm\Delta v_{0}$
in the Kek-Y phase and Dirac fermions with masses $m_{\pm}=m_{0}\pm\Delta
m_{0}$ for the Kek-M phase. We refer to the Dirac quasiparticles with
velocities $v_{+}$ and $v_{-}$ in the Kek-Y phase as “fast” and “slow”
fermions and to the Dirac quasiparticles with masses $m_{+}$ and $m_{-}$ in
the Kek-M as “heavy” and “light” fermions, respectively. In order to see why,
we use the fact that in a previous work [39] it was noted that the scattering
probability for Kekulé-patterned graphene is composed of three terms: two
corresponding to each intravalley scattering plus an additional term related
to intervalley transitions,

$F_{\alpha\alpha^{\prime}}^{\beta\beta^{\prime}}(\bm{k},\bm{q})=\delta_{\beta,\beta^{\prime}}F_{\alpha\alpha^{\prime}}^{\beta}-\beta\beta^{\prime}\Big{(}\frac{q\sin\varphi}{2|\bm{k+q}|}\Big{)}^{2}.$
(5)

This expression allows the polarizability to be conveniently separated into
intravalley and intervalley contributions. For the superlattices introduced
above, this property allows to separate $\Pi(\omega,q)$ into three
contributions when summing over the $\beta,\beta^{\prime}$ indices. Since
these indices essentially label the valleys, this separation allows to better
understand the contribution from each valley to the total optoelectronic
response of the superlattice. Here we make an emphasis on the fact that each
intravalley contribution in the polarizability can be attributed to a
different specie of quasiparticle, while the intervalley contribution can be
attributed to a kind of “moiré interference” between the quasiparticles, which
produces the signature at $\omega_{M}$.

For pristine graphene, the valleys are degenerated and separated in momentum
space. Therefore, the total polarizability for low-energy carriers in graphene
is simply given by two times (accounting for valley degeneracy) the single
valley polarizability, $\Pi^{g}_{v_{0}}(\omega,q)$ [54, 55, 53]. That is,

$\Pi(\omega,q)=2\times\Pi^{g}_{v_{0}}(\omega,q)\text{\qquad(graphene),}$ (6)

where the subscript stands for carriers with a Fermi velocity $v_{0}$. This is
not the case when a modulation like the Kekulé bond order breaks the valley
degeneracy, since the response of each valley will then be different.
Furthermore, the possibility for intervalley transport indicates that the
total polarizability is not merely the sum of the responses of each valley.
Indeed, it can be shown by using Eqs. (3) and (5) that the total
polarizability of the Kek-Y phase, $\Pi_{Y}(\omega,q)$, can be written as

$\Pi_{Y}(\omega,q)=\Pi^{g}_{v_{+}}(\omega,q)+\Pi^{g}_{v_{-}}(\omega,q)+\Pi^{M}_{v_{M}}(\omega,q),$
(7)

where the first two terms in the right side of the last equation correspond to
the well-known single-valley polarizabilities of Dirac fermions in graphene
[54, 55, 53], only with the Fermi velocity $v_{0}$ replaced by a different
velocity $v_{\pm}=v_{0}\pm\Delta v_{0}$ for each valley. On the other hand,
the last term corresponds to the intervalley response and is responsible for
the resonance at $\omega_{M}$ in the optical conductivity.

For the Kek-M phase, although the physical origin of the modulation and the
energy spectrum are different, a completely analogous result is obtained. We
find that the total polarizability can be written as

$\Pi_{Q}(\omega,q)=\Pi^{g}_{m_{+}}(\omega,q)+\Pi^{g}_{m_{-}}(\omega,q)+\Pi^{M}_{m_{M}}(\omega,q).$
(8)

where first two terms in the right side of the last equation correspond to the
well-known single-valley polarizabilities of massive Dirac fermions in a
graphene lattice with a gap induced by a broken sublattice symmetry [56], only
with the effective mass $m_{0}$ replaced by $m_{\pm}=m_{0}\pm\Delta m_{0}$ for
each valley. In this case too, the last term corresponds to the intervalley
response and is responsible for the resonance at $\omega_{M}$ in the optical
conductivity.

Therefore, even when the nature of the modulations that give rise to the
superlattice in both models are different, in both cases the polarizability
can be separated into single-valley components plus an intervalley term that
gives rise to signatures related to moiré interference effects. The different
kinds of valley-coupling modulations split a dynamical property in each valley
(the Fermi velocity $v_{0}\rightarrow v_{\pm}$ or the effective mass
$m_{0}\rightarrow m_{\pm}$) and additionally introduce an intervalley
response.

In more general terms, the valley coupling can introduce a valley-dependent
renormalization of the dynamical properties in the polarizability of the
superlattice of the form

$2\times\Pi^{g}_{\mu_{0}}\xrightarrow{\text{{\hbox{\Delta>0}}}}\Pi^{g}_{\mu_{+}}+\Pi^{g}_{\mu_{-}}+\Pi^{M}_{\mu_{M}}$
(9)

where $\mu_{0}$ is a dynamical property of the low-energy excitations (here,
it can be the Fermi velocity $v_{0}$ or the effective mass $m_{0}$), which
splits as $\mu_{0}\rightarrow\mu_{\pm}=\mu_{0}\pm\Delta\mu_{0}$ when the
valley coupling $\Delta$ is introduced.

Multifold fermions.— Some of the signatures discussed so far, namely,
different coexisting quasiparticles characterized by different activation
frequencies, low-frequency sharp absorption peaks and (in the case of the
Kek-Y phase) a multi-step conductivity with a dependence
$\sigma\sim\omega^{d-2}$ (where $d$ is the spatial dimension), are quite
similar to those found in the optical conductivity of multifold fermions [57,
58]. Multifold fermions are the generalization of Weyl fermions to a higher
effective spin representation that exhibit a remarkable optoelectronic
response, including exotic circular photogalvanic effects [58, 59, 60, 61,
62]. A number of crystals have recently been shown to exhibit multiple species
of these quasiparticles coexisting at low energies and, particularly, the
study of their optical conductivity has been the focus of multiple theoretical
and experimental works [61, 63, 58, 64, 65, 66, 67, 68, 69]. It has been noted
in previous works [41, 51, 39] that, due to the folding of the $K$ and
$K^{\prime}$ valleys into the $\Gamma$-point, the resulting low-energy band
structure in some Kekulé-modulated superlattices can be described by higher
pseudospin representations of the Dirac equation [70]. As can be seen in Fig.
1, the dispersion of the Kek-Y is very similar to that of a pseudospin-$3/2$
system [39], while the dispersion of Kek-M phase resembles that of a
pseudospin-$1$ system [41, 51] (notice the threefold crossing shown in Fig.
1d). Therefore, it could be expected that the optical signatures of the
superlattices studied here would share some similarities with those found in
systems hosting multifold fermions. Indeed, in systems hosting multifold
fermions the optical conductivity exhibits multiple linear steps
(characteristic of linearly-dispersive bands $\sigma\sim\omega^{d-2}$)
indicating different activation frequencies for each type of multifold fermion
[65, 68, 66]. Here we find that for both the Kek-Y and Kek-M phase the
different species of carriers exhibit distinct activation frequencies
(features at $\omega_{\pm}$ in Fig. 2a,b). Furthermore, the optical
conductivity of materials hosting multifold fermions like CoSi [66], RhSi [68]
and other Weyl semimetals like NbP [71] exhibit low-frequency narrow peaks
originating from transitions between SOC-split bands. Also, the SOC splits a
threefold node into a spin-$3/2$ fermion and a twofold Weyl fermion and the
energetic splitting (and therefore the position of the peak) is a measure of
the SOC strength [65, 66]. On the other hand, we find that very similar sharp
peaks appear (around $\omega_{M}\sim\Delta$) in the conductivity of the Kek-Y
and Kek-M phases at low frequencies due to transitions between bands that are
split by the valley coupling $\Delta$ introduced by the Kekulé modulation
(which also introduces the different species of quasiparticles). In both cases
the frequency of the sharp peak is given by the coupling amplitude. This
suggests that the Kekulé modulation in these systems might play a role in the
optical response similar to that played by the SOC in systems hosting
multifold fermions. These remarks might lead to interesting connections to
multifold fermions and deserve further study.

Figure 3:  Merging of the Dirac cones leading to flat bands in a superlattice
with a modulation tripling the unit cell. (a) Superlattice with a structure
similar to that of a moiré superlattice exhibiting flat bands due to the
merging of Dirac cones. The atoms in one sublattice have been labeled like
those in an effective moiré superlattice. (b) Band flattening and gap
maximization in the superlattice model. For parameters $\alpha U_{\pm}=1.5\pm
0.3$, the Dirac cones are almost completely separated (blue, solid lines). As
the value of $\alpha U_{+}$ increases and that of $\alpha U_{-}$ decreases,
the Dirac cones further merge (green, dashed lines) until the bands become
almost flat (red, solid lines) and the gap is maximized. The parameters used
for the dispersions shown are $\alpha U_{\pm}=1.5\pm 0.3,3.35\pm 2.65,15.05\pm
14.95$, respectively.

Connection with Moiré Superlattices.— Moiré patterns are a characteristic
feature of spatially modulated materials. Recently, there have been a number
of works studying superlattice models exhibiting flat bands [72, 73, 74, 75,
76, 77, 78]. We make the remark that low-energy hamiltonians for TBG
essentially describe two Dirac cones (one corresponding to each layer) coupled
through some (angle-dependent) lattice modulation. On the other hand, the
graphene superlattices presented here also describe two Dirac cones
(corresponding to a single layer) coupled through some lattice modulation.
Therefore, let us explore the connection between these systems and moiré
superlattices. More precisely, we explore whether it is possible to produce a
model for a graphene superlattice which can reproduce, at least partially, the
phenomenology of TBG. This would be of interest since, due to the large scale
modulations in TBG, even the simplest model is quite complicated [8]. As a
consequence, its analytical studies have been mainly focused to the zero modes
[8]. For the purpose of studying the relation between graphene superlattices
and TBG, we introduce the more general form of $H_{Q}$ [51]. This model was
recently used in Ref. [79] to propose a valley switch based on pseudo-Andreev
reflection. As in the case of $H_{Q}$, the superlattice is produced due to the
on-site energies of the atoms in the lattice being affected by the interaction
with the substrate. The general hamiltonian can be written as,

$H=\begin{pmatrix}m_{0}v_{0}^{2}&v_{0}k_{-}&0&t_{0}\Delta_{A}\\\
v_{0}k_{+}&-m_{0}v_{0}^{2}&-t_{0}\Delta_{B}^{*}&0\\\
0&-t_{0}\Delta_{B}&-m_{0}v_{0}^{2}&v_{0}k_{-}\\\
t_{0}\Delta_{A}^{*}&0&v_{0}k_{+}&m_{0}v_{0}^{2}\end{pmatrix}.$ (10)

Notice that the hamiltonian $H_{Q}$ is a particular case of $H$ (after a
suitable reordering of the basis [79]). We take $m_{0}\rightarrow 0$ and
consider that the coupling between the Dirac cones is introduced as a
perturbation bias field $U_{\pm}$ turned on by a parameter $\alpha$ so we
rewrite $\Delta_{A}^{*}=\alpha U_{-}$ and $\Delta_{B}=-\alpha U_{+}$. After
re-shuffling the third and fourth elements of the basis, one gets

$\mathcal{H}=\begin{pmatrix}0&\mathcal{D}_{-}^{*}\\\
\mathcal{D}_{+}&0\end{pmatrix},\quad\mathcal{D}_{r}=\begin{pmatrix}-2i\overline{\partial}&\alpha
U_{r}\\\ \alpha U_{-r}&-2i\overline{\partial}\end{pmatrix},$

where $r=\pm$ and we have used $k_{j}\rightarrow-i\partial_{r_{j}}$ so
$k_{+}\rightarrow-i(\partial_{x}+i\partial_{y})\equiv-2i\overline{\partial}$.
This hamiltonian has the same form that the chiral model for TBG [8] with the
field $U(\pm r)$ reduced to a two-parameter effective field $U_{\pm}$. It can
be shown that it exhibits the bands of two Dirac cones merging and flattening
for certain values of the parameters. For real values of $U_{+}$ and $U_{-}$
one gets a superlattice like that shown in Fig. 3a. Due to the
$\sqrt{3}\times\sqrt{3}$ increased size of the lattice, one can assign three
different on-site energies to the atoms in one of the sublattices. As is shown
in Fig. 3b, this simple model is enough to obtain flat bands from the merging
of two Dirac cones. This minimal model might prove useful to study the
signatures in the optoelectronic response of moiré superlattices [80] and to
further study the role of the Kekulé ordering in their correlated states [27,
28, 81]. Further investigation of these possibilities is left to future work.

Conclusion.— We studied graphene superlattices with a tripled unit cell and
folded Dirac cones. To study superlattices like those seen in recent
experiments, we used linear response theory to find signatures that could aid
in the experimental confirmation of recent theoretical predictions as, for
example, the valley-momentum locking. We analyzed the robustness of such
signatures and their origin. Its optical response suggests two species of
carriers with signatures similar to those of multifold fermions. Finally, we
presented a minimal model for a superlattice that develops flat bands
mimicking twisted graphene bilayers at magic angles. Since two of the graphene
superlattices we studied have been recently visualized in experiments (the
Kek-Y and Kek-O phases) [47], we hope that some of the signatures discussed
here can serve to further validate the predicted electronic properties of
these systems.

We thank UNAM-DGAPA project IN102620 and CONACyT project 1564464. S. A. H. was
supported by the Consejo Nacional de Ciencia y Tecnología (CONACyT).

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