0}}3c(\alpha,\tfrac{\alpha+3}{2},1)\int_{0}^{\infty}d^{2}t\,(t_{1}t_{2})^{\frac{w-2-\alpha}{2}}(t_{1}+t_{2})^{\alpha}e^{-\frac{\pi
t_{1}|p_{1}|^{2}}{\tau_{2}}-\frac{\pi t_{2}|p_{2}|^{2}}{\tau_{2}}}$
$\displaystyle+\sum_{\alpha=(s-1)\,{\rm mod}\,2}^{s-1}\,\,\sum_{p\neq
0}\frac{6\pi|p|^{2}}{\tau_{2}}c(\alpha,\tfrac{\alpha}{2},0)\int_{0}^{\infty}d^{2}t\,(t_{1}t_{2})^{\frac{w-1-\alpha}{2}}(t_{1}+t_{2})^{\alpha}e^{-\frac{\pi|p|^{2}(t_{1}+t_{2})}{\tau_{2}}}\,.$
(D.13)

Note that the sum truncates at $\alpha=s-2$ in the second line since (D.11)
implies that $c(s-1,\beta,1)=0$ for $\alpha=s-1$.

Since $\alpha\in\mathbb{N}$ we can use the binomial expansion on
$(t_{1}+t_{2})^{\alpha}$ and integrate term by term, arriving at the sum of
three boundary contributions,

$\displaystyle{\rm{b.t.}}=$ $\displaystyle-3\\!\sum_{\alpha=(s-1)\,{\rm
mod}\,2}^{s-1}\sum_{\delta=\frac{1-\alpha}{2}}^{\frac{\alpha-1}{2}}\alpha{\alpha{-}1\choose\frac{\alpha-1}{2}{+}\delta}c(\alpha,\tfrac{\alpha}{2},0)\Gamma\big{(}\tfrac{w}{2}{+}\delta\big{)}\Gamma\big{(}\tfrac{w}{2}{-}\delta\big{)}\Big{[}E(\tfrac{w}{2}{+}\delta;\tau)E(\tfrac{w}{2}{-}\delta;\tau){-}E(w;\tau)\Big{]}$
$\displaystyle+3\\!\sum_{\alpha=s\,{\rm
mod}\,2}^{s-2}\,\sum_{\delta=\frac{-\alpha}{2}}^{\frac{\alpha}{2}}{\alpha\choose\frac{\alpha}{2}{+}\delta}c(\alpha,\tfrac{\alpha+3}{2},1)\Gamma\big{(}\tfrac{w}{2}{+}\delta\big{)}\Gamma\big{(}\tfrac{w}{2}{-}\delta\big{)}\Big{[}E(\tfrac{w}{2}{+}\delta;\tau)E(\tfrac{w}{2}{-}\delta;\tau){-}E(w;\tau)\Big{]}$
$\displaystyle+6\sum_{\alpha=(s-1)\,{\rm
mod}\,2}^{s-1}\,\sum_{\delta=\frac{1-\alpha}{2}}^{\frac{\alpha+1}{2}}{\alpha-1\choose\frac{\alpha-1}{2}+\delta}c(\alpha,\tfrac{\alpha}{2},0)\Gamma\big{(}\tfrac{w}{2}+\delta\big{)}\Gamma\big{(}\tfrac{w}{2}+1-\delta\big{)}E(w;\tau)\,,$
(D.14)

where we have used the identity

$\displaystyle\sum_{\underset{p_{1}+p_{2}+p_{3}=0}{p_{1},p_{2},p_{3}\neq
0}}\frac{(\tau_{2}/\pi)^{s_{1}+s_{2}}}{|p_{1}|^{2s_{1}}|p_{2}|^{2s_{2}}}$
$\displaystyle=\sum_{p_{1},p_{2}\neq
0}\frac{(\tau_{2}/\pi)^{s_{1}+s_{2}}}{|p_{1}|^{2s_{1}}|p_{2}|^{2s_{2}}}-\sum_{p\in\Lambda^{\prime}}\frac{(\tau_{2}/\pi)^{s_{1}+s_{2}}}{|p|^{2(s_{1}+s_{2})}}$
(D.15) $\displaystyle=E(s_{1};\tau)E(s_{2};\tau)-E(s_{1}+s_{2};\tau)\,.$
(D.16)

Note that if the eigenvalue $s$ in (D.14) is even then the sum in the first
line is over odd values of $\alpha$ since otherwise $c(\alpha,\beta,0)$
vanishes from (D.11), so $\delta$ must be an integer. Similarly, if $s$ is
even the $\alpha$ sum in the second line is over even values since otherwise
$c(\alpha,\beta,1)$ vanishes again from (D.11), and once more $\delta$ is an
integer. Conversely for $s$ odd in the first line the sum over $\alpha$ is
over even integers, so $\delta$ is half-integer. For $s$ odd in the second
line the $\alpha$ sum is over odd values and the $\delta$ sum is again over
half-integers. We conclude that for even eigenvalue $s$ the bilinears terms in
both sums are of the form
$E(\frac{w}{2}+\delta;\tau)E(\frac{w}{2}-\delta;\tau)$ with integer $\delta$.
The third line gives terms proportional to non-holomorphic Eisenstein series
with integer indices. Thus we have reproduce the claim (4.25) stated in the
main text.

#### Comparison with modular graph functions

In the case of modular graph functions even eigenvalue $s$ is accompanied by
an even trascendental weight $w\geq 4$ such that $2\leq s\leq w-2$, hence in
this case the sources are always of the form $E(s_{1};\tau)E(s_{2};\tau)$ with
$s_{1},s_{2}\in\mathbb{N}^{\geq 2}$. Conversely, for the generalised
Eisenstein of present interest even eigenvalue $s$ is accompanied by an odd
trascendental weight $w$ such that $s\geq w+1$. In this case the bilinear
sources are always of the form $E(s_{1};\tau)E(s_{2};\tau)$ with
$s_{1},s_{2}\in\mathbb{Z}+\frac{1}{2}$ such that $s_{1}+s_{2}=w$.

Mutatis mutandis, modular graph functions with odd eigenvalue $s$ must have
odd trascendental weight $w\geq 5$ such that $3\leq s\leq w-2$, hence again
the sources are always of the form $E(s_{1};\tau)E(s_{2};\tau)$ with
$s_{1},s_{2}\in\mathbb{N}^{\geq 2}$. While again the generalised Eisenstein
series’ of present interest and odd eigenvalue $s$ are accompanied by an even
weight $w$ and again the bilinear sources are of the form
$E(s_{1};\tau)E(s_{2};\tau)$ with $s_{1},s_{2}\in\mathbb{Z}+\frac{1}{2}$ such
that $s_{1}+s_{2}=w$.

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# Lattice Model For The Quantum Anomalous Hall Effect in Moiré Graphene

Ahmed Khalifa  Department of Physics & Astronomy, University of Kentucky,
Lexington, KY 40506, USA  Ganpathy Murthy  Department of Physics & Astronomy,
University of Kentucky, Lexington, KY 40506, USA  Ribhu K. Kaul  Department of
Physics, The Pennsylvania State University, University Park, PA 16802, USA

###### Abstract

Inspired by experiments on magic angle twisted bilayer graphene, we present a
lattice mean-field model for the quantum anomalous Hall effect in a moiré
setting. Our hopping model thus provides a simple route to a moiré Chern
insulator in commensurately twisted models. We present a study of our model in
the ribbon geometry, in which we demonstrate the presence of thick chiral edge
states that have a transverse localization that scales with the moiré lattice
spacing. We also study the electronic structure of a domain wall between
opposite Chern insulators. Our model and results are relevant to experiments
that will image or manipulate the moiré quantum anomalous Hall edge states.

Introduction: In a remarkable burst of experimental progress, magic angle
twisted bilayer graphene has been found to host a wide range of novel quantum
phenomena, from band topology to superconductivity [1, 2, 3, 4, 5, 6, 7].
Notable among the states of matter discovered is the quantum anomalous Hall
effect (QAHE), in which time reversal symmetry is spontaneously broken
resulting in a quantized Hall conductance [5, 4, 7, 8]. This phenomenon, which
will be the focus of this work, brings together in a new setting the two
pillars of the contemporary study of quantum materials: strong correlations
and band topology, all in a structure made simply of Carbon.

The iconic lattice description for the QAHE is Haldane’s honeycomb model [9].
Although the model explicitly breaks time reversal symmetry, it may be viewed
as a useful mean field model when time reversal symmetry is broken
spontaneously by strong interactions. The model starts with spinless electrons
hopping on the honeycomb lattice, which results in a linear dispersion at
half-filling and at low energies. This is graphene’s celebrated realization of
the Dirac equation in which the masslessness is protected by a combination of
${\cal C}_{2}{\cal T}$ symmetry where ${\cal C}_{2}$ denotes rotation by $\pi$
around the $z$ axis and ${\cal T}$ denotes time reversal. Haldane’s
perturbation consists of an imaginary second neighbor hopping chosen
specifically to break ${\cal T}$ but preserves ${\cal C}_{2}$. Such a
perturbation can be shown to guarantee a non-zero Chern number, giving rise to
the QAHE. While the $k$-space picture is convenient to establish bulk
topology, the chiral edge states of the Chern insulator are best studied in
the real-space ribbon geometry. These edge modes have a transverse
localization length on the scale of the spacing $a$ of the underlying
honeycomb lattice. Our goal in this work is to construct a mean field model
for the QAHE in moiré graphene, formulated on a real-space lattice analogous
to Haldane’s model. Such a model would allow studies of the electronic
structure, including the chiral models, of the QAHE state with inhomogeneity
in real space, such as that arising from the presence of edges, internal
domain walls, electrostatic potentials due to external gates, density
gradients, and disorder all of which play an important role in experiments on
moiré graphene.

Figure 1:  The $\lambda_{\rm mh}$ perturbation of Eq. (1) on a single sheet of
graphene. (a) the pattern of imaginary second neighbor hoppings of the
$\lambda_{\rm mh}$ term that breaks ${\cal C}_{2}$ and ${\cal T}$ individually
but preserves their combination. This is a modification of the Haldane model
in which the two sub-lattices have the imaginary hopping in the same sense; in
this “modified Haldane” model they are in the opposite sense. Since the
$\lambda_{\rm mh}$ preserves ${\cal C}_{2}T$, it does not act as a mass term.
(b) the resulting dispersion of $H_{\rm mh}$ in a section of the graphene
Brillouin zone showing that the perturbation preserves the masslessness of the
Dirac equation in each valley but raises the energy of one Dirac point with
respect to the other.

We first review a popular physical picture for the QAHE in moiré graphene [10,
11, 12, 13, 14, 15, 16, 17, 18, 14, 19, 20, 21, 22, 23, 24, 25]. The starting
point is the continuum model [26, 27] in which moiré reconstructions of the
band structure at the K and K′ valleys are treated independently. Close to the
magic angle the band structure consists of two almost flat bands in each
valley that touch at the corners of the moiré BZ, a touching protected by
${\cal C}_{2}{\cal T}$. Upon applying sub-lattice masses on the two layers
(via the alignment with an HBN substrate [28, 29, 30, 31, 32, 33, 34]), ${\cal
C}_{2}$ is broken, the two bands in each valley are gapped, and carry $\pm 1$
Chern number [12, 13]. By time reversal the opposite valley has exactly the
same band structure but with opposite Chern numbers $\mp 1$. The QAHE occurs
when the band structure has just enough electrons to fill up one of these four
bands, with electron-electron interactions spontaneously breaking time
reversal symmetry, resulting in a valley-polarized Chern insulator. In a mean
field (Hartree-Fock) picture of the continuum model, we may posit that the
energy for the electrons in one valley is raised with respect to the other
valley resulting in valley polarization. We note here that it is widely
believed that the QAHE state in moiré graphene is spin polarized (in addition
to the valley polarization just discussed). Since it also does not play a
crucial role for the physics we are looking at, in our manuscript we will
henceforth ignore the electron spin, and focus on the orbital route to
magnetism.

Lattice Model: It is not obvious a priori how to write a mean field lattice
model that leads to the the QAHE, with the same mechanism as in the continuum
mean field picture. Nonetheless, inspired by the continuum picture, we seek a
perturbation to the monolayer graphene Hamiltonian that raises the energy in
one valley with respect to the other, but maintains the masslessness of the
Dirac points in each valley. The perturbation must break both ${\cal T}$ and
${\cal C}_{2}$ so that the band structures in the two valleys can be
different, but must preserve ${\cal C}_{2}{\cal T}$ so that the Dirac equation
is left massless. Such a term can be constructed by modifying the honeycomb
Haldane model, arranging for the sense of the imaginary hopping on the A and B
sub-lattices to run oppositely (in the Haldane model they run in the same
direction). This is shown pictorially in Fig. (1),

$H_{\rm mh}=-t\sum_{\langle
i,j\rangle}c_{i}^{\dagger}c_{j}+\lambda_{mh}\sum_{\langle\langle
i,j\rangle\rangle}\mathrm{i}\nu_{ij}c_{i}^{\dagger}c_{j},$ (1)

where $c_{i}$ denotes the electron’s annihilation operator at honeycomb site
$i$ and $\nu_{ij}=\pm 1$. The A-A hopping and B-B hopping change sign under
both ${\cal C}_{2}$ and ${\cal T}$, but are left invariant under the ${\cal
C}_{2}{\cal T}$. The band structure of monolayer graphene perturbed with
$\lambda_{\rm mh}$, shown in the right panel of Fig. (1), displays the
expected behavior, two massless Dirac cones at K and K′ but with touching
points that are displaced in energy. This model has been introduced
previously, and was dubbed the “modified Haldane model” in the context of
studying anti-chiral edge states in two dimensional fermion systems [35, 36].
While the $\lambda_{mh}$ term does not lead to a Chern insulator in monolayer
graphene, we now show that when added to twisted moiré graphene it results in
the QAHE.

We now incorporate this perturbation into a lattice model of twisted bilayer
graphene. In order to work on a moiré system on the lattice, we use
commensurate twist angles with large moiré triangular lattice vectors ${\bf
A}_{1}$ and ${\bf A}_{2}$ ($A_{M}\equiv|{\bf A}_{1}|=|{\bf A}_{2}|$) and a
smaller moiré Brillouin zone [26, 37, 38]. To demonstrate that $\lambda_{\rm
mh}$ indeed results in the QAHE in the moiré graphene, we construct a tight-
binding model on the commensurate lattice [39, 40, 41, 42] for the twisted
bilayer graphene system,

$H_{\rm TBG}=\sum_{\mu}H_{\mu,\rm
mh}+\sum_{\mu,i}m_{\mu}\sigma_{z}c_{\mu,i}^{\dagger}c_{\mu,i}+\sum_{i,j}t_{ij}^{\perp}c_{t,i}^{\dagger}c_{b,j},$
(2)

where $c_{\mu,i}$ is annihilation operator for an electron in layer $\mu=t,b$
and at lattice position $i$. The first “intralayer” term, defined in Eq. (1),
consists of the usual graphene hopping as well as the $\lambda_{mh}$
perturbation term. The second term corresponds to the effect of the HBN
substrate which is a staggered mass on each layer with $\sigma_{z}$ denoting
the sublattice degree of freedom. The last term corresponds to the interlayer
hopping. We take a simple interlayer hopping model that is a function of the
in-plane distance between two atoms in different layers and falls off
exponentially with increasing distance. This gives the following form for the
interlayer hopping, $t^{\perp}(r)=t_{v}\exp(-r/\eta)$, where $r$ is the in-
plane distance between the two atoms, $\eta$ controls the range of the hopping
and $t_{v}$ is the hopping amplitude. Lattice relaxations reduce the A-A
hopping between the layers in contrast to A-B hopping [43, 44, 45, 46] which
we include in the model by the parameter $\kappa$ which defines the ratio
between the two.

Figure 2:  The band structure of the commensurately twisted lattice model
$H_{\rm TBG}$, Eq. (2) in the moiré BZ computed with periodic boundary
conditions (without edges). (a) Without the perturbation term, $\lambda_{\rm
mh}=0$, ${\cal T}$ is preserved and a Chern insulator is not possible. The two
isolated flat bands in the middle are doubly degenerate, which can be
understood from the continuum model as arising due to the valley degeneracy.
The remote bands in gray do not play an important role in our work. (b)
Turning on $\lambda_{\rm mh}$ breaks ${\cal T}$ and allows the moiré system to
have bands that have well defined Chern numbers, the arrows point to the value
of the Chern number of the bands. Chern numbers are computed by the standard
lattice method [47]. This spinless Moire system would be a Chern insulator at
1/4 and 3/4 filling of the flat band system. We used $\theta\approx
1.08^{\circ}$, $t=2.7eV$, $t_{v}=0.686eV$, $\kappa=0.0$, $\eta=0.3a$,
$m_{t}=m_{b}=10meV$ and $\lambda_{\rm mh}=8meV$.

In Fig. (2) we show the band structure of the model first with $\lambda_{\rm
mh}=0$ and then with $\lambda_{\rm mh}\neq 0$. For $\lambda_{\rm mh}=0$, there
are two flat bands that are each doubly degenerate. The degeenracy can be
understand in the continuum model, as arising from the two valleys. Since
${\cal T}$ is preserved, no Chern insulator is possible. For $\lambda_{\rm
mh}\neq 0$ we observe that the flat bands split into four isolated bands (one
pair moves up and the other pair moves down in energy). As we would expect
from the continuum model picture, since the $\lambda_{\rm mh}$ perturbation
raises the energy of one valley’s Dirac touching with respect to the other in
monolayer graphene, the two flat bands in one valley move up, while those in
the other move down. The Chern numbers of the pair that move up are $\pm 1$,
while those of the pair that move down have $\mp 1$, consistent with the fact
that they arose from two valleys connected by time reversal symmetry when
$\lambda_{\rm mh}=0$. Evidently, if we fill the flat band manifold with only
enough electrons to fill one band (or three bands), we obtain a Chern
insulator. This demonstrates that the $\lambda_{\rm mh}$ perturbation can turn
the moiré system into a quantum anomalous Hall insulator. We note here for
completeness that for a single sheet of graphene, $H_{\rm mh}$ supplemented
with a sublattice mass term breaks ${\cal T}$ but nonetheless gives rise only
to a trivial insulator; the Chern insulator that we find here for the twisted
case essentially requires the subtle moiré band reconstructions.

Edge States: Having established that the $\lambda_{\rm mh}$ perturbation
creates a QAH state from the bulk $k$-space topology, we now study the model
in a ribbon geometry to access the edge states. We construct a ribbon by
taking the system to be infinite along the “longitudinal” $\mathbf{A}_{2}$
direction (with the momentum parallel to $\mathbf{A}_{2}$ being a good quantum
number) and finite along the “transverse” $\mathbf{A}_{1}$ direction which
defines the width of the ribbon. Due to the large number of atoms in the unit
cell we use the Lanczos algorithm to obtain the bands near charge neutrality
where the Chern bands and hence the edge states reside. Fig. (3)(a) shows the
band structure of the Hamiltonian in Eq. (2) of a ribbon with a width of 20
moiré unit cells. We see the projection of the four bulk bands near charge
neutrality but most importantly we see two pairs of gap crossing modes that
are localized on the upper and lower edges of the ribbon that correspond to
the edge states of the Chern insulator. The location of the support of the
wavefunctions in the transverse direction is determined by computing the
position expectation value of the wave function along the width of the ribbon
[48]. Note that there are also edge states that lie at energies above and
below the four flat bands which are reminiscent of the “moiré edge states”
that exist before adding the $H_{\rm mh}$ perturbation [42, 49, 50, 51, 52,
48, 53]. These states are counter propagating along each edge which makes them
susceptible to back scattering in contrast with the robust edge states of the
Chern insulator discussed here. Furthermore, they do not cross the band gap
between the flat bands and the remote bands.

What is the transverse localization length of the edge states? Given the
microscopic form of the mean field perturbation in Eq. (1) (illustrated in
Fig. 1(a)) one may naively assume that the edge states are localized on the
scale of a graphene lattice spacing. On the other hand, in the continuum model
the moiré lattice spacing $A_{M}$ is the only length scale that appears, and
hence we might expect that the edge state wave function to be localized on
this larger scale. We plot the edge state wave function squared amplitude in
Fig. 3(b). We see that the edge mode wavefunctions extend up to approximately
one moiré unit cell length $A_{M}$, in agreement with the expectation from the
continuum model. The fundamental graphene lattice constant $a$ is about 50
times smaller than $A_{M}$. Furthermore, we have checked that the edge state
localization length depends linearly on $A_{M}$ demonstrating that it is tied
to this larger length scale (See supplemental material).

Figure 3:  Band structure and wavefunctions of $H_{\rm TBG}$ in a ribbon
geometry (with edges) with the same parameters as in Fig. 2 and with a
transverse width of 20 moiré unit cells and infinite longitudinal direction.
(a) Shows the bands in a window of energy that includes the four flat bands of
Fig. 2(b). The color of the eigenstates represents the expectation value of
the transverse co-ordinate of the ribbon, in accordance with the shown color
bar. The bulk bands appear green since their average transverse co-ordinate is
in the middle. Likewise edge states localized on the upper (lower) edge appear
blue (red). We observe that chiral mid-gap states show up between the first
and second and between the third and fourth flat bands, corresponding to the
edge states of a Chern insulator at 1/4 and 3/4 filling. In addition, counter-
propagating edge states not associated with the QAHE appear below the first
band and above the fourth band. A possible fermi level for a Chern insulator
at bulk filling of 1/4 is shown as a dashed line. (b) Shows the detail of
$|\Psi|^{2}$ for the two edge states’ wave functions at the Fermi level marked
in (a). The system diagonalized had a transverse width of 20 moiré unit cells,
but for clarity most of the bulk has been omitted as represented by the broken
$y$-axis. For visual clarity we have shown three repeats of the unit cell
along the longitudinal direction. Black arrows indicate the chirality of the
edge modes inferred from their group velocities in (a).

Domain Walls: The QAHE phase in moiré graphene has been argued to be an
orbital ferromagnet, which breaks time reversal symmetry spontaneously [10,
11, 12, 13, 14, 15, 16, 17, 18, 14, 19, 20, 21, 22, 23, 24, 25]. This discrete
symmetry breaking results in two ground states corresponding to clockwise or
anti-clockwise chiral edge states ($C=\pm 1$) connected by time reversal
symmetry, which can be accessed in our model by choosing the mean field term
$\lambda_{mh}$ either positive or negative. Experiments on twisted bilayer
graphene aligned with hBN have reported the co-existence of ferromagnetic
domains in the system[5, 4, 7, 8, 54], with a possible mesoscale pattern of
opposite domains separated by domain walls. The domain walls themselves are
fascinating objects that provide an experimentally tunable realization of the
zero modes trapped by defects in topological band systems, a fundamental
problem in physics [55, 56]. Our model allows us to explicitly compute the
wavefunctions of the chiral modes localized at the domain walls. To set up a
domain wall, we study a configuration where the order parameter $\lambda_{mh}$
varies smoothly across the width of the ribbon between two regions of opposite
Chern number. We use $\lambda_{mh}=\lambda_{\rm mh_{0}}\tanh{(y/\xi)}$, with
$\xi$ a characteristic length scale of the domain wall. In Fig. 4(a) we show
the resulting band structure. Focusing at a Fermi level corresponding to 1/4
filling (shown as a dashed line), we find four localized states: two states at
the center of the sample (trapped by the domain wall), and one each on the
upper and lower edges. The two parallel chiral states in the domain wall are
consistent with the behavior of a domain wall separating regions with $C=\pm
1$ Chern numbers. The corresponding wave function for one of the domain wall
states is shown in Fig. 4(b) exhibiting a state that is localized in the
middle of the ribbon with a localization length controlled by $A_{M}$, the
moiré length scale. Also shown in this figure is the upper and lower chiral
edge state.

Figure 4:  Electronic structure of a QAHE domain wall in the ribbon geometry.
(a) Shows the band structure in the presence of a $C=\pm 1$ domain wall (see
text), colored with the same colorbar as in Fig. 3. At the Fermi level (dashed
line) for 1/4 bulk filling there are now four chiral edge states: two of these
are quasi-degenerate and localized in the middle (and hence appear green) at
the domain wall, and the other two are at the upper and lower edge of the
ribbon. (b) Density for the localized mode wave functions at the 1/4 Fermi
level, showing the chiralities inferred from the group velocities of the band
structure. For clarity we show only one of the two modes trapped on the domain
wall, since they have similar support. We have shown five repeats along the
longitudinal direction. The profile of the domain wall used ($\xi=0.9A_{M}$)
is shown on the right. All other parameetrs are same as in Fig. 2.

Energetics: We have demonstrated that the $\lambda_{\rm mh}$ perturbation of
Eq. (1) does indeed create a mean field QAH state in moiré graphene, and the
chiral edge modes appropriate for both a hard edge and a domain wall. Our mean
field ansatz is also conceptually attractive, because it has only two “moving
parts”: The sublattice masses to gap the flat bands and give them nontrivial
Chern numbers in each valley, and $\lambda_{\rm mh}$ to raise one valley with
respect to the other, which we have made simply arguments has to result in a
Chern insulator. However, it is interesting to step back and ask whether this
mean field ansatz is unique or favored in some sense. From a symmetry point of
view, the only necessary (but not sufficient) requirement to get a Chern
insulator is that the perturbation break ${\cal T}$. Clearly many imaginary
hopping patterns, such as Haldane’s perturbation, or those involving further
neighbor hopping and modulation of the hopping with the moiré periodicity, can
achieve this. However, unlike our simple ansatz, it is a priori unclear which
of these will result in a Chern insulator. A full Hartree-Fock calculation on
the lattice, including all the bands, will by definition lead to the best mean
field Hamiltonian. However, this is computationally prohibitive. To gain
further physics insight into the issue in a limited parameter space, we have
constructed a mean field model which has the sublattice masses, the modified
Haldane term with a coefficient $\lambda_{\rm mh}$, and also a Haldane term
with a coefficient $\lambda_{\rm h}$. Note that the Haldane term by itself
also leads to a Chern insulator for TBG at quarter filling. We variationally
compute the total energy, including the Coulomb interaction screened by gates
[10, 11, 12, 13, 14, 15, 16, 17, 18, 14], as a function of $\lambda_{\rm
mh},\lambda_{\rm h}$ for a fixed set of sublattice masses. As shown in the SM,
we find that the minimum occurs when $\lambda_{\rm h}=0,\lambda_{\rm mh}\neq
0$. Thus, at least in this limited set of variational parameters, the modified
Haldane term is preferred.

Conclusion: In summary we have introduced a lattice mean field model for the
quantum anomalous Hall effect in twisted bilayer graphene. The mechanism that
gives rise to the Chern insulator is physically appealing: The sublattice
masses on the two layers break ${\cal C}_{2}$, gap out the flat bands in each
valley and give them nontrivial Chern numbers, while the modified Haldane term
breaks ${\cal T}$ and raises the states in one valley with respect to the
other. Our mean field model’s conceptual simplicity, its computational
tractability, and its good energetics (albeit in a limited parameter space)
make it attractive for study. By solving our lattice model in a ribbon
geometry, we find that the chiral edge modes of the QAH state have a
transverse localization length that scales with the moiré lattice spacing. We
have also used our model to study the domain walls occurring between regions
where the order parameter changes sign.

A number of interesting open questions involving spatially inhomogeneous
systems can be studied using our lattice approach. The control of the
electronic structure of the chiral modes by the shape of the domain wall, a
study of networks of domain walls and the effect of impurities are some
examples. We hope to study these and other questions in the near future.

Note Added: While our manuscript was being finalized for submission, an
experimental work using STM on domain walls in twisted monolayer-bilayer
graphene has appeared, in which the QAHE edge states have been imaged for the
first time [57]. Although closely related to the system under study here, it
is different in detail. The main idea of our theory can be adapted
straightforwardly to this work.

Acknowledgments: This work was supported in part by NSF DMR-2026947 (AK, RKK).
GM is grateful to the US-Israel BSF for partial support under grant no.
2016130. GM and RKK acknowledge the Aspen Center for Physics, NSF PHY-1607611
(GM, RKK) for its hospitality. GM is also grateful to the International Center
for Theoretical Sciences, Bangalore for its hospitality while this work was
being completed. The authors are grateful to the University of Kentucky Center
for Computational Sciences and Information Technology Services Research
Computing for their support and use of the Lipscomb Compute Cluster and
associated research computing resources.

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