Moiré fractals in twisted graphene layers
Deepanshu Aggarwal, Rohit Narula, and Sankalpa Ghosh
Department of Physics, Indian Institute of Technology Delhi, Hauz Khas,New Delhi 110016
(Dated: February 19, 2024)
Twisted bilayer graphene (TBLG) subject to a sequence of commensurate external periodic po-
tentials reveals the formation of moiré fractals (MF) that share striking similarities with the cen-
tral place theory (CPT) of economic geography, thus uncovering a remarkable connection between
twistronics and the geometry of economic zones. MFs arise from the self-similarity of the emergent
hierarchy of Brillouin zones (BZ), forming a nested subband structure within the bandwidth of
the original moiré bands. We derive the fractal generators (FG) for TBLG under these external
potentials and explore their impact on the hierarchy of the BZ edges and the wavefunctions at the
Dirac point. By examining realistic super-moiré structures (SMS) and demonstrating their equiva-
lence to MFs with periodic perturbations under specific conditions, we establish MFs as a general
description for such systems. Furthermore, we uncover parallels between the modification of the BZ
hierarchy and magnetic BZ formation in Hofstadter’s butterfly (HB), allowing us to construct an
incommensurability measure for MFs vs. twist angle. The resulting bandstructure hierarchy bolsters
correlation effects, pushing more bands within the same energy window for both commensurate and
incommensurate TBLG.
I.
INTRODUCTION
Fractals are fascinating structures that are found in
both natural and abstract forms, from the intricate pat-
terns of Romanesco broccoli to the complex geometry
of the Mandelbrot set [1, 2]. Iterated function systems
(IFS) are a powerful tool for generating fractals, with
many unusual geometries emerging as attractors [3], e.g.,
Koch curve [4]. Iterated fractals (IF) involve applying a
generator recursively to a starting shape –an initiator.
Particularly interesting are IFs generated by,
x2 + βx −
 LN −β2
/3 = 0,
(1)
where for β ∈N if it has a discriminant D ∈Z, then LN ∈
N generate a triangular lattice with integral coordinates
[5] (FIG.1(a)).
In this work we show that Eq.(1) describes an emergent
fractality when commensurate or incommensurate moiré
patterns in TBLG [7–43] are subjected to a sequence of
superlattice periodic potentials (SOPP) (FIG.1(b)), hav-
ing the same moiré periodicity as structures on which
they are applied, but twisted by an angle restoring com-
mensuration. The sequence of iterated edges of the first
Brillouin zone (FBZ) forms a fractal (FIG.2, 4) with di-
mensions determined by LN (the number of unit cells in
a newly formed BZ fitting a unit cell of the preceding BZ
at each iteration).
The emergent fractality of Eq.(1), dubbed as the moiré
fractal (MF) resembles the hierarchy and fractality of
economic geography’s CPT as pioneered by Christaller
[44–46] and Lösch [47], which terms LN as Löschian
numbers [48].
This connection emerges when densely-
packed hexagonal trade areas centered on settlements are
multiply-stacked with trade areas representing smaller
settlements [46, 47, 49–54] (FIG.1(c)).
The MF frac-
tal dimension (Df) provides quantitative information
about the bandstructure of realistic SMS e.g., multi-
ple graphene or hexagonal boron nitride (hBN)-graphene
layers [55–60]. Further, we establish an analogy with HB
[61, 62] explaining the topological quantization of Hall
conductivity [63, 64], thus formulating an incommensu-
ration measure for moiré structures.
(a)
(b)
(c)
Hamlet
Village
Town
City
Higher order settlements
K = 3 Central place hierarchy
j = 0, , ,
1 2 3
j = 0
j = 0,1
j = 0, ,
1 2
FIG. 1.
(a) LN, β and x over triangular coordinates (the
axes are 60◦w.r.t.
each other).
The intersection of β
(magenta)- and x (green)-lines identify the intersection points
(x, y = x+β) at which LN exists. (b) The real-space creation
of each iteration: TBLG is created by stacking graphene lay-
ers with the top layer being the zeroth iteration j = 0 and the
bottom layer being the next j = 1. The iterations j = 2, 3, . . .
are created by applying the z-independent external periodic
potentials identically to both the graphene layers. (c) The
K = 3 CPT hierarchy (since a hexagon of each layer encloses
three hexagons of an adjacent layer) where each layer corre-
sponds to a particular order of settlement [6]. LN is analogous
to K.
arXiv:2306.04580v3  [cond-mat.mes-hall]  16 Feb 2024

2
(a1)
Brillouin zone (BZ)
Edge of the BZ 
(a2)
(a3)
(b1)
(b2)
(b3)
E (eV)
E (eV)
DOS
(a4)
(b4)
ρK(r)
min
max
WS cell encloses
maxima and minima
y/a(2)
x/a(2)
y/a(2)
2
2
2
j = 1
j = 2
1
7
4
6
5
3
2
7
8
9
11
12
10
13
5
6
1
3
4
2
FIG. 2. (a1), (b1) The BZ for j = 1 (black), 2 (magenta) 3 (green), where
n
b(1)
1 , b(1)
2
o
are the reciprocal lattice primitive
vectors for j = 1. (a2), (b2) The fractal structures at the BZ edges. The dashed red hexagon represents the initiator, i.e.,
the BZ of SLG rotated by θ/2. The solid blue lines outline the fractal structures’ outer boundary. The generators (inside)
attach alternately to the initiator, forming the fractal structure. The copies of the BZ at each iteration are added such that
they overlap with the BZ of SLG (red solid line in (a1, b1)) and these overlaps lead to IFs. The insets between (a1), (a2) and
(b1), (b2) display reciprocal and real-space lattice vectors for q = 3, p = 1 and q = 2, p = 1, respectively. The shift between
the Dirac points is equal to the hexagon’s side length for q = 3, p = 1, and twice its side for q = 2, p = 1. (a3) and (b3) The
bandstructures for V0 = 1.2 meV and the DOS (with a Gaussian smearing of 0.002 meV). (a4) and (b4) show ρnk(r) of the
lowest conduction band (dashed-dotted line) at the Dirac point (see text, Appendix-H).
II.
THE HAMILTONIAN AND EMERGENT
FRACTALITY IN MOIRÉ FRACTALS
The
rapid
progress
in
the
fabrication
of
two-
dimensional (2D) layered materials e.g., TBLG has stim-
ulated interest in the effect of substrates [65–68]. Such
materials experience an external potential with a moiré-
like periodicity when placed on substrates with matching
[69] or mismatched layers [70, 71] which may be modeled
as perturbations to the Hamiltonian of TBLG (HTBLG)
via external periodic potentials [71–73].
HTBLG sub-
jected to SOPP at j-th iteration is
Hj =
ˆhk(θ/2) + Pj
i=1 Vi(r)
T(r)
T †(r)
ˆhk(−θ/2) + Pj
i=1 Vi(r)

(2)
where ˆhk(θ) = vF σθ ·

ˆk −Kθ
[74] describes single-
layer graphene (SLG) rotated by θ, vF
[75] is the
Fermi velocity, and Kθ is the rotated, right-valley
Dirac point.
The transformed Pauli matrices σθ =
e−iσzθ/2(σx, σy)eiσzθ/2 provide the rotation. The expres-
sions for the interlayer hopping matrices T(r) [11, 76–78]
are given in appendix-F. The external potential Vj(r)
exhibits a periodicity: Vj(r + n1 t(j)
1
+ n2 t(j)
2 ) = Vj(r),
where the primitive vectors (PV) t(j)
1,2 = [R(θ)] a(j−1)
1,2
.
θ = θr is the twist between the moiré pattern and the
moiré external potential (mEP), leading to commensu-
ration between mEP at each j-th iteration and TBLG
with all potentials up to the (j −1)-th iteration. R(θr)
denotes a 2D rotation matrix at these commensurate an-
gles. The condition for commensuration under such rota-
tion maps an integer pair n = {n1, n2} to m = {m1, m2}
(Appendix-D). The integral solutions:
m1, m2, n1 and
n2 satisfy the necessary and sufficient condition when
the matrix elements assume only rational values [12, 79],
leading to a set of Diophantine equations (Appendix-D)
whose solutions provide the PV of the commensurate su-
percell, i.e.,
n
a(2)
1 , a(2)
2
o
in terms of the PV of the pre-
ceding structure.
For the SOPP of FIG.1, V1(r) = 0 at j = 1, H1 in (2)
becomes HTBLG, while the potentials for j ≥2 are non-
zero. For j = 2, V2(r) is periodic with t(2)
1,2 = [R(θr)] a(1))
1,2
where a(1)
1,2 are the PV of commensurate TBLG. On re-
peating this procedure, the commensuration of Vj(r)
and the structure upto the (j −1)th-level is spanned by
n
a(j)
1 , a(j)
2
o
. We demonstrate this via a cosine potential
V0 with only six Fourier components (Appendix-F). The
intrinsic Coulomb interactions can be modeled using such
an onsite mEP having the same periodicity as the moiré

3
pattern[80–83] as a starting ansatz for a self-consistent
calculation. For the first iteration of V2(r) (Appendix-
G-FIGs.11-15) various SMS e.g., trilayer graphene [55–
57, 84–86], four-layer graphene [58, 59, 87, 88], and tri-
layer hBN-G-hBN [60, 89] are modeled by (2) represent-
ing an MF and a weak periodic perturbation whose de-
tails depend on the system considered. V0 can be con-
trolled by [90] the interlayer separation (d) and the in-
terlayer bias (VSTM), essentially the bias applied to a
scanning tunneling microscope (STM) tip, i.e., |VSTM| ≈
20 −500 meV [91, 92].
Typically, V0 ∼1.2 meV for
|VSTM| = 45 meV given d ∼1 nm.
Each commensuration of either TBLG or TBLG plus
the mEPs gives (Appendix-D)
A(j−1)
FBZ /A(j)
FBZ = LN = p2
1 + p2
2 + p1 p2
(3)
where A(j)
FBZ =
b(j)
1
× b(j)
2
 represents the area of the first
BZ at j-th iteration and a(j)
i
· b(j)
k
= 2πδik ∀i, k = 1, 2
and p1(p, q), p2(p, q) ∈Z+ (Appendix-D-FIG.8), with p, q
being co-prime numbers. For a hexagonal lattice, p2 =
p1 + β such that LN with β = 0 lie on the line X = Y
while the numbers with β > 0 lie on lines parallel to X =
Y (FIG.1(a)), converting Eq.(3) into Eq.(1), therefore
making each LN lie at an intersection of the x- and β-
rays (FIG.1(a)).
For each LN and β, the corresponding FG are gen-
erated by applying IFS to one side of the initiator A0
[3], i.e., the rotated BZ of SLG (a constituting layer
in TBLG) (details in appendix-A). Applying this FG to
each arm of A0 generates the edges of successive BZs,
continuing the recursive process to produce a sequence
of BZs.
The hierarchical construct in CPT i.e., LN exhibits
successively smaller regions within a trade area at each
stage. Number theory [93] identifies conditions for LN
fulfilling Eq.(1). This connection between the CPT lat-
tice partition and Eq.(1) facilitates the systematic deter-
mination of lattice coordinates for economic zones and
the corresponding FG responsible for CPT associated
with LN. For TBLG in the presence of specific mEPs,
we begin with the transformation mappings of the FGs
for q = 3,p = 1 and q = 2, p = 1 corresponding to
θ ∼21.79◦[94] and θ ∼32.20◦. Df = log(nc)/ log(s) for
the attractor A [1] where s = √LN is the contractivity
factor (Appendix-A-FIG.5).
III.
IMPLICATIONS OF THE ABOVE
CONSTRUCTION FOR THE BAND STRUCTURE
FIG.2(a1),(b1) display the superimposed BZs for j =
1, 2, 3 corresponding to the hierarchical mEP applied to
two commensurate structures at θ = θr ∼21.79◦and
θ = θr ∼32.20◦.
The FG shape is shown in the
next column.
For both, it is applied alternately out-
side and inside the edges of the initiators (red dashed
lines, FIGs. 2(a2),(b2)), exhibiting emergent fractality
0.002
FIG. 3. ∆A/A(j−1)
FBZ
vs. θ: the blue rectangles represent com-
mensurate/closest angles for odd q and p = 1. The first BZ
area for incommensurate angles is determined using moiré
vectors[96]. The inset confirms the absence of the apparent
multivaluedness in the lowest values of the V-regions between
the dashed lines, with a separation of ∼0.002◦.
with Df = 1.129 and Df = 1.255. The real-space fractals
corresponding to q = 3,p = 1 are shown in appendix-B-
FIG.7. The RLV magnitudes for i = 1, 2 form a Cauchy
sequence [95] as limj→∞
 b±θ/2
i

s(j−1)

→0 whose conver-
gence rates depend on q, p (Solid blue and red lines in
the insets of FIG.2).
The commuting translation operators (TOs) at the j-
th iteration with the Hamiltonian (2) for the rotated
mEP with PV a(j)
1
= p1a(j−1)
1
+ p2a(j−1)
2
and a(j)
2
=
−p2a(j−1)
1
+(p1 + p2) a(j−1)
2
differ from the PV at the (j−
1)-th iteration. They satisfy ˆTa(j)
1
ˆTa(j)
2
= ˆTa(j)
2
ˆTa(j)
1 , lead-
ing to a large real-space supercell with a more squeezed
BZ scaled by s (FIG.2(a1),(b1)), conventionally termed
as a mini zone (MZ) [68, 97]. This has similarities with
the HB problem [61, 98, 99] where the magnetic TOs do
not commute. Consequently for a TR-symmetric, com-
mensurate TBLG in a rotated mEP we define a dimen-
sionless incommensurability measure [100] ,
∆A(θ)
A(j−1)
FBZ
=
 
1 −⌊A(j−1)
FBZ /A(j)
FBZ⌋
A(j−1)
FBZ /A(j)
FBZ
!
(4)
where ⌊. . . ⌋denotes the greatest integer function.
Following [11, 96, 101] for any θ between TBLG and
mEP (FIG.2(b1)), b(j)
1,2 of the mBZ are obtained from
∆K = Kθ/2 −K−θ/2.
These definitions coincide for
p = 1 and odd integer q with the lattice vectors giv-
ing the hexagonal BZ side length as ℓ=

2b(1)
1
+b(1)
2
3

(FIG.2(a1),(b1)), such that A(j−1)
FBZ /A(j)
FBZ = LN ⇒∆A =
0, corresponding to the blue points in FIG.3.

4
For generic (q, p) the two definitions do not coincide
and ∆A/A(j−1)
FBZ
shifts upward from the ∆A = 0 line
by differing amounts (FIG.3).
The angles correspond-
ing to the minima of the V-regions are commensurate
angles where the ratio becomes rational (Appendix-E).
FIG.3(inset) shows that these minima are not vertically
collinear: the lateral shift is ∼0.002◦.
FIG.2(a3),(b3) display the bandstructures for j = 1
(HTBLG) and j = 2 in the presence of mEP V2(r),
and the DOS. Precisely 2
 2e2 ln(nc)/Df −1

(Appendix-
C) bands, per Eq.3, populate the bandgap of 6.38 eV for
θr ∼21.79◦and 5.04 eV for θr ∼32.20◦between the two
lowest bands at Γ-point, without the mEP. Additional
bands at K and K′ occupy an even narrower range within
the bandgap vs. the Γ-point, with the two lowest bands
meeting at the Dirac point. FIG.2(a4),(b4) provide the
probability density ρnk = |ψnk|2 of the Bloch wavefunc-
tions obtained by diagonalizing (2) at the Dirac point.
The number of maxima and minima within the Wigner-
Seitz cell for the lowest conduction band is 2e2 ln(nc)/Df
(Appendix-H). The MFs in (2) therefore provide precise
control over the number of in-gap states given Df by
changing θr, which is in contrast to other such methods
for small-angle TBLG [102–104] and SMS [105, 106].
For magic-angle TBLG (MATBLG) [8, 11, 107, 108],
the flat bands facilitate various correlated phases [22,
23, 109–116].
The FGs apply alternately outside and
inside the edges of their initiators, which are the mBZ
of MATBLG (red dashed lines in FIGs. 4(a),(b)), also
showing the superimposed BZs for j = 1, 2, 3 at the first
magic angle θ ∼1.05◦).
For FIG.4(a): θr ∼13.17◦
and Df = 1.093, while for FIG.4(b): θr ∼21.79◦and
Df = 1.129. The change in Df and LN alters the num-
ber of bands pushed towards the Fermi level EF at the
Γ-point within a bandgap of ∼13.76 meV for MATBLG,
preserving the emergent fractality similar to commensu-
rate structures. An additional 2(e2 ln(nc)/Df −1) inner
bands again exhibit significantly reduced curvature com-
pared to the original flat bands.
FIG.10 verifies that
TBLG’s renormalized vF remains indifferent to the pres-
ence of hierarchical mEP, despite the shift in EF . The
Hamiltonian (2) ignores lattice relaxation effects, namely
the variations in the interlayer hopping amplitudes in the
AA-BB- and AB-BA-rich regions. Their [24, 28, 101, 117]
inclusion doesn’t alter this emergent fractality linked to
the bandstructures (Appendix-F-FIG.9).
Much like how the recursive stacking of nets in CPT
leads to a proliferation of smaller settlements within a
designated area, the recursive mEP in TBLG produce an
expanding count of electronic bands within a defined en-
ergy range, thereby amplifying correlation effects, with
both scenarios characterized by LN. The impact of this
band engineering and emergent fractality on electronic
correlation can be understood through the Hubbard pa-
rameter ratio between the interaction energy U and the
band-width tW (the difference between the band max-
ima and minima). Since in (2), HTBLG and the rotated
MEP have different translational symmetry, limV0→0
U
tW
doesn’t yield the same ratio for pristine TBLG [23]. How-
ever this ratio depends on mEP V0 when EF lies within
the flat band,
U
tW ≫1.
Under an mEP, U →
U
s ,for
U = e2θ/4πκϵ0a without any mEP (Appendix-I), where
e is electronic charge and κ = 4. However,the closest flat
band near EF experiences a significantly larger reduction
in bandwidth vs. s. E.g., in FIG.4(a), the bandwidth de-
creases from ∼6meV to ∼0.23meV (FIG.16), leading to
∼9 times increase in
U
tW vs. MATBLG. The effective
mass m∗scales as √n/vF at EF , increasing with the su-
perlattice density n scaled by LN; so does DOS.
IV.
CONCLUSION
In summary, we have introduced MFs in TBLG sub-
jected to a sequence of commensurate, rotated external
potentials, thus establishing a framework for studying
their properties and drawing parallels between emergent
fractality in sp2 carbons and the agglomeration of CPT
trade zones. The bandstructure of several SMS can be
understood via MFs and a weak perturbation enabling
insertion of a controlled number of in-gap bands deter-
mined by Df. We analyzed the restructuring of the moiré
unit cell and established an incommensurability measure,
linking it to correlation effects and Df.
The MFs re-
mained robust despite corrugation effects.
Amidst the emerging domain of SMS, including tri-
layer [57, 84–86], tetralayer [87, 118–120], and pentalayer
graphene [88, 121], scenarios where slight rotations of
SLG interact with thin graphitic crystals [122, 123], dis-
(a)
(b)
j = 1
j = 2
y/a(2)
y/a(2)
min
max
WS cell encloses
maxima and minima
1
2
3
4
5
6 7
8
9
10
11
12
13
14
15
16 17
18
19
1
2
3
4
5
6
7
x/a(2)
FIG. 4.
The bandstructures at θr ∼21.79◦in (a) and
θr ∼13.17◦in (b) along K′ −M −Γ −K. The twist be-
tween the layers is the first magic angle θ ∼1.05◦. The in-
terlayer hopping parameters: tAA/BB = tAB/BA = 110 meV
(Appendix-F) and V0 = 1.2 meV. (left to right) The emer-
gent fractal structure in reciprocal space, 2D-bandstructure:
orange (blue) bands without (with) mEP, the DOS, and the
spatial profiles of ρK(r) of the lowest conduction band with
j = 2. Changing θr from (b) to (a) modifies Df, shifting more
bands towards EF .

5
similar layers like encapsulated SLG and bilayer graphene
between hBN layers [60, 89, 124, 125], and moiré lattices
in photonic crystals [126] and ultra-cold atom systems
[127–129], we’ve provided a general framework allowing
such systems to be understood as MFs under weak per-
turbations.
The increase in the number of MF bands near EF can
be detected via differential tunneling conductance mea-
surements for the corresponding SMS under suitable con-
ditions (Appendix-J) as was done for quasicrystals with a
Penrose tiling [130]. Real-space scanning probes like the
quantum twisting microscope [94] offer another approach
by gating the vdW device on a rotating platform, thereby
enabling a rotated moiré effect. The optical conductiv-
ity, absorption coefficient and the photocurrent which de-
pend on the interband transitions [43, 131, 132], and ex-
periments analogous to cryogenic nanoscale photovoltage
measurements [133] (Appendix-J) on hBN-encapsulated
TBLG may also exhibit the MF signatures. Future re-
search will investigate MF symmetry [134], implications
for strong-correlation physics, extension to external po-
tentials without common moiré periodicity [135–137],
any nontrivial topological properties embedded in our
incommensuration measure, and the possibility of frac-
tality in moiré quasicrystals [57, 138–141].
ACKNOWLEDGMENTS
SG is supported by MTR/2021/000513 funded by
SERB, DST, Govt. of India. DA is supported by a UGC
fellowship.
Appendix A: The properties of the fractal
generators (FG)
Here we provide details of the transformation map-
pings for FGs corresponding to q = 3, p = 1 and for
q = 2, p = 1, respectively, whose effect on the band struc-
ture was shown in FIG. (2) of the main text. More exam-
ples with additional discussion are provided in Table-I.
The first case corresponds to the commensurate angle
θ ∼21.79◦which has recently been explored experimen-
tally for pristine TBLG [94]. The number of contraction
mappings, that is, an associated cardinal number nc of
IFS W = {wn : n = 1, 2, . . . , nc} [3] is obtained for each
LN and β.
In this case LN is found to be 7 and lie
at the intersection of x = 1 and β = 1 lines as shown
in FIG. 1(a) of the main text. Correspondingly, nc of
the IFS comes out to be 3, giving W = {w1, w2, w3},
where w1 = R(−ϕ1) I
s , w2 = R(ϕ2) I
s + w1, w3 =
w1 + w2 with s = √LN =
b(j−1)
1/2
/
b(j)
1/2
 being the
contractivity factor and I is a 2 × 2 identity matrix.
For {p1, p2, s} = {1, 2,
√
7}, ϕ1 = cos−1  2p1+p2
2s

and
ϕ2 = π
3 −ϕ1. Df of the attractor A is obtained using
LN as Df = log(nc)/ log(s) [1].
TABLE I. Each commensuration is characterized uniquely by
a pair of coprime-integers (q, p). The Löschian number LN
and the FG are obtained by identifying the number of sides
nc in the FG. The fractal dimension (Df) corresponding to the
various q, p are calculated using LN and nc [142]. The outer
boundary of the mini-zones superposed over the FBZ of SLG
is generated by attaching the FG on a hexagonal initiator.
θ
q p LN β nc Generator
Df
21.79◦3 1
7
1 3
1.129
32.20◦2 1 13 2 5
1.255
13.17◦5 1 19 1 5
1.093
38.21◦5 3
7
1 3
1.129
17.90◦11 3 31 4 9
1.280
...
To understand wn’s action on one of the sides of A0
(BZ of SLG), rotated clockwise with the angle θ/2 ∼
10.89◦, and represented by reciprocal lattice vector u =
1
3

2b−θ/2
1
+ b−θ/2
2

(see FIG. 5), where b−θ/2
1
and b−θ/2
2
are the two reciprocal lattice vectors of the rotated
graphene layers, we note that w1 shortens |u| by s =
√
7
times and rotates u clockwise by an angle cos−1 2/
√
7

,
providing the first side of the FG, namely u1.
Simi-
larly, the successive mappings are u2,3 = w2,3u.
The
second case mentioned in the main text, corresponds to
the commensuration θ ∼32.20◦with q = 2 and p = 1,
for which LN = 13 and nc = 5. Considering the side
of the initiator to be identical to u, the mappings be-
come w1 = R (ϕ1) I
s ;
w2 = R(−ϕ2) I
s + w1;
w3 =
w1 + w2;
w4 = w2 + w3;
w5 = R(−( π
3 + ϕ2)) I
s + w4
where p1 = 1, p2 = 3 and s =
√
13. The application of
the contraction mappings wn’s for the first two entries in
Table-(I) are shown in FIG. 5.
The other FGs in Table-(I), such as for q=5 and p=3
lead to the same value LN = 7 and β = 1 as for q = 3
and p = 1, but the shift between the Dirac points for
commensurate TBLG is ∆K = bc
2.
However, the FG
remains identical to the one for q = 3 and p = 1. In
fact, the mappings for the FGs corresponding to the other
values of q being an odd number with p = 1 remain the
same with an increasing cardinal number nc. Therefore,
the class of commensurate structures with q being an
(a)
(b)
w1
w2
w3
w1
w2
w3
w4
w5
FIG. 5. (a) The stepwise creation of the FG for LN = 7 with
q = 3 and p = 1, and (b) LN = 13 with q = 2 and p = 1. The
red dashed line is the side u of A0 as defined in the Sec. A

6
j = 1, 2
(a)
q = 11, p = 3
Df = 1.28
(b) j = 3
FIG. 6.
(a) The fractal corresponding to one more entry in
the Table-I where q = 11 and p = 3 having LN = 31 with
β = 4 at an angle of θr = 17.89◦for j = 1, 2. (b) the fractal
corresponding to the iteration j = 3.
odd number and p = 1 associate with the FGs having
the same shape but with a different number of sides. If
one draws a line that is perpendicular to the red dashed
line in FIG. 5(a) which bisects the FG, it exactly cuts
it into two pieces with one becomeing the other with a
rotation of π in the plane containing the FG. In the case
of q = 11 and p = 3, the shift ∆K = bc
2 is the same as
for q = 5 and p = 3 but the FG is asymmetric about the
perpendicular bisector. The corresponding FG is shown
in FIG. (6). It shows the j = 1, 2-level iterations in (a)
and the one for j = 3 in (b). Similarly, for q = 4 and
p = 1, the shift in the Dirac points ∆K = 2 (2bc
1 + bc
1) /3
is very different from the previous cases but the FG has
a similar shape as for q = 2 and p = 1 where ∆K is
identical.
Appendix B: The real-space construction of the
moiré fractals
While the contraction mappings {w1, w2, w3, . . . } for
a moiré fractal in real-space corresponding to a given
value of q, p remain identical, the initiator however is
different. In reciprocal-space, the FG is applied to the
side of a hexagon of a rotated SLG in the commensu-
rate case while it’s applied to an arm of the moiré BZ for
the incommensurate case to obtain the iterated fractal.
However, the initiator in the real-space is the arm of a
hexagon which corresponds to the highest-level of itera-
tion of the chosen j-values. E.g., FIG. 7(b) shows the
moiré fractal corresponding to q = 3 and p = 1. Since
the j-values considered are 0, 1, 2, 3, the initiator is the
side of the hexagon corresponding to the commensurate
cell of j = 3.
(a)
(b)
(c)
w1
w2
w3
21.79
2
q = 3, p = 1
Df = 1.129
FIG. 7.
(a) The real-space structure of commensurate TBLG
at θ ∼21.79◦and two levels of periodic potentials for j = 2, 3.
The direct-lattice primitive vectors are also shown for each
level. (b) The corresponding moiré fractal along the edges of
the Wigner-Seitz unit-cell at j = 3. (c) The initiator, the arm
of the hexagon corresponding to j = 3, and the FG.
Appendix C: The relation of the number of bands
with the fractal dimension (Df)
As given in the main text, the number of bands for
θ = θr within the bandgap of the two lowest bands at
the Γ-point is 4e2 ln(nc)/Df −2 (see FIG. 2(a3) & (b3) of
the main text), while for θ ̸= θr, the number of bands
is 2e2 ln(nc)/Df −2 (see FIG. 4 in the main text). The
dependence of the number of bands on the cardinal num-
ber nc and the fractal dimension Df is obtained from LN
which is related to nc and Df as
Df =
ln(nc)
ln(√LN)
(C1)
After some rearrangement, the crucial quantity LN can
be written as
LN = e2 ln(nc)/Df
(C2)
The number of bands depends upon LN and therefore, it
depends upon nc and Df through (C2).
Appendix D: Derivation of Eq. (3) of the main text
For a general 2D-Bravais lattice case, the direct lattice
primitive vectors of the constituting layers a1 and a2 are
not necessarily orthogonal, i.e., a1 · a2 = a1 a2 cos(ϕ) ̸=
0 and also |a1| ̸= |a2|.
Therefore, the general square
matrix that maps an integer pair n = {n1, n2} to m =
{m1, m2} is given by
m =

cos θr −ϵ(σz cos(ϕ)a1 + iσya2)
a1 sin(ϕ)
sin θr

n
(D1)

7
80
p1
p2
q
0
20
40
60
0
25
50
75
100
40
50
p1
p2
20
30
40
50
0
10
20
30
FIG. 8. p1 vs p2 for two maximum values of q. Evidently,
the points arrange themselves in hexagons and each point is
associated with a Löschian number LN.
Here ϵ = sgn [(a1 × a2)z], a1/2 =
a1/2
, and ϕ represents
the angle between a1 and a2. For the present hexagonal
case , |a1| = |a2| and ϕ = 60◦, the necessary and suffi-
cient condition for the integer solutions m1, m2, n1 and
n2 demands the matrix elements to assume only rational
values [79]. The commensurate angle θr then becomes
θr(q, p) = 2 tan−1(p/
√
3q)
(D2)
where q > p > 0. As p/q →0 gives θr →0◦and p/q →1
gives θr →60◦. The commensurate structures are dis-
tinguished on the basis of δ = gcd(p, 3) and the direct
lattice primitive vectors are

ac
1
ac
2

=

p1
p2
−p2 p1 + p2
 
a1
a2

(D3)
where p1 = (3q −p) /γ and p2 = 2p/γ for δ = 3 and
p1 = (q −p) /γ and p2 = (q −p) /γ for δ = 1 and the
quantity γ = gcd [3q + p, 3q −p]. For both the cases δ =
1 and δ = 3, the two elements in first row p1 and p2
are positive integers Z+.
Corresponding to the direct
space primitive vectors ac
1 and ac
2, the reciprocal space
primitive vectors bc
1 and bc
2 are defined such that ac
i ·bc
j =
2πδij ∀i, j = 1, 2.
Then, the number LN of the BZ
hexagons of commensurate cell enclosed within the BZ
of SLG formed by {b1, b2} are,
LN = |(b1 × b2) · ˆz|
|bc
1 × bc
2 · ˆz| = p2
1 + p2
2 + p1 p2
(D4)
1.
Fractal dimension (Df) in the metric space of
p1, p2
An irregular hexagon is formed for higher magnitudes
of p1, p2 along with the smaller hexagons for lower magni-
tudes. This self-similar, small irregular hexagons lead to
the fractality in the metric space (Z2, Euclidean). The
fractality in the arrangement of the coefficients p1, p2
in this space is shown in FIG. (8).
Here we calcu-
late the Hausdorff dimension Df in the metric space
(Z2, Euclidean) of coefficients p1, p2 with the Euclidean
distance. We calculate the Df using the box-counting
theorem [1, 4]
Df = lim
k→∞
log(nk)
log(2k)
(D5)
where nk is number of smaller polygons that completely
fit inside a bigger polygon at the kth-iteration.
There
are 3k small irregular hexagons that fit inside the bigger
irregular hexagon at the kth-iteration and therefore the
fractal dimension is then given by
Df = ln(3)
ln(2) ∼1.585
Note that the fractality in this metric space of p1, p2 is
different from the fractality that we observed in moiré
fractals where Df is a function of q, p that characterizes a
given commensuration whereas Df ∼1.585 in the metric
space of p1, p2 is constant.
Appendix E: More on the incommensuration
measure
It can be shown that for a generic (q, p):
∆K =
( 2p
3γ

2b(j)
1
+ b(j)
2

if gcd(p, 3) = 1
2p
3γ b(j)
2
if gcd(p, 3) = 3
(E1)
where γ = gcd (3q −p, 3q + p) [143]. For q = 2n+1 with
n = 1, 2, 3, . . . and p = 1, the shift ∆K always equals
the side-length of the hexagon (ℓ) as shown in FIG. 2(a1)
of the main text [12]. Further, the moiré lattice vectors
coincide with the lattice vectors of the commensurate cell.
In this case, we get,
A(j−1)
FBZ
A(j)
FBZ
= LN =⇒
∆A
A(j−1)
FBZ
= 0
(E2)
Similarly, if q = 2n with n = 1, 2, 3, . . . and p = 1, the
shift |∆K| = 2ℓas shown in FIG. 2(b1) of main text, and
the ratio
A(j−1)
FBZ
A(j)
FBZ
= 4χ(q) + 1
4
=⇒
∆A
A(j−1)
FBZ
=
1
4χ(q) + 1
(E3)
where χ(q) is a positive integer dependent on q.
Appendix F: The details of the potential, interlayer
tunnelling matrices and the band structures with
corrugation effects
The cosine potential that we considered in the main
text, has a cosine profile, namely
Vj(r) = V0
3
X
i=1
cos

G(j)
i
· r

(F1)

8
50
25
0
25
50
FIG. 9. The inclusion of corrugation effects leads to different
interlayer hopping parameters, namely, tAA,BB = 79.7 meV
and tAB,BA = 97.5 meV [96].
This does not change the
2e2 ln(nc)/Df −2 bands inserted within the band-gap of the
two lowest bands at the Γ-point (shown by the two arrows)
in the absence of mEP and hence the corrugated TBLG also
does not affect the emergence of fractality.
(a)
(b)
EF (meV)
FIG. 10.
(a) The renormalized Fermi velocity v∗
F /vF as a
function of α2 and hence the twist angle θ. (b) The Fermi
energy (EF ) as a function of α2.
Similar to the case of a
pristine TBLG [11] or an unrotated mEP to TBLG [83], the
renormalized Fermi velocity in the presence of a rotated mEP
remains unaffected both in the absence and the presence of
the corrugation effect.
where G(j)
1 , G(j)
2
and G(j)
3
= −G(j)
1
−G(j)
2
are the recip-
rocal lattice vectors satisfying t(j)
i
· G(j)
k
= 2πδik ∀i, k =
1, 2.
V0 is the strength of the potential and the t(j)
i -
vectors are defined in the main text.
The spatially-dependent interlayer-tunneling T(r) in
the Hamiltonian in Eq. 2 of the main text is given by
[11, 77, 78]
T(r) =
3
X
i=1
Tie−iqi·r =
3
X
i=1
{σ0tAA,BB + [σx cos(i −1)ϕ + σy sin(i −1)ϕ] tAB,BA} e−iqi·r
(F2)
where tAA,BB and tAB,BA are the interlayer hopping
parameters in the local AA/BB-regions and AB/BA-
regions, respectively. σ0 is a second order identity matrix
and (σx, σy) are the Pauli matrices.
The band struc-
tures in FIG. 2 and FIG. 4 of the main text are ob-
tained by considering the hopping parameters tAA,BB =
tAB,BA = 110 meV [11] where we ignore the variations in
the hopping parameters which occur due to corrugation
or atomic relaxations [24] in different regions. The inclu-
sion of this effect leads to different interlayer hopping pa-
rameters in both the regions, namely, tAA,BB = 79.7 meV
and tAB,BA = 97.5 meV [96]. The band structures with
these values of hopping parameters are shown in FIG. 9.
The emergent fractality remains unaffected and when
the fractal dimension is changed from Df = 1.129 to
Df = 1.093 in going from left to right, an indentical
number 2e2 ln(nc)/Df −2 of bands are inserted within the
band-gap of ∼13 meV at the Γ-point.
Appendix G: Some realistic systems to realize the
model Hamiltonian (2) of the main text at the first
iteration of the potential
To supplement our assertion in the main text that the
moiré fractals introduced through the model Hamiltonian
H2(r) in Eq.(2) of the main text can provide a general
description of a number of realistic super-moiré systems,
we shall explicitly provide the modelling for three repre-
sentative super-moiré systems in terms of moiré fractals.
For the first iteration j = 2 of the mEP, the super-moiré
Hamiltonian H(r) can be written as a combination of
the hamiltonian of the moiré-fractal and a weak periodic
perturbation as,
H(r) = H2(r) + Hpert(r)
(G1)
where the periodic perturbation Hpert(r) satisfies
Hpert(r + n1a(2)
1
+ n2a(2)
2 ) = Hpert(r)
(G2)
where a(2)
i
for i = 1, 2 are the primitive lattice vectors
of the super-moiré cell as defined in the main text. To
do this let us note that it has already been established
[69, 70] that for moiré system such as graphene/graphene

9
d = 3d0
d0
3θr/2
θr/2
-θr/2
-θr/2
d = 3d0
d = 3d0
d0
d0
θr/2
3θr/2
θr/2
(a)
(b)
(c)
d0
1
2
1'
1
1
2
3
2
2'
1'
Axis of rotation
FIG. 11. A (a) trilayer-, (b) a four-layer-twisted graphene, and (c) a trilayer boron nitride-Graphene-boron nitride system. In
the above
denote carbon,
boron and
nitrogen atoms, respectively. The angle θr is one of the commensurate angles and
d0 ∼0.335 nm is the interlayer separation in TBLG. The number besides each layer represents the layer index used in the text
in appendix- H.
System
Hpert(r)
Trilayer system

ℏ2v2
F Meff
11′ (r)σz
0
0
ℏ2v2
F Meff
21′ (r)σz+I2V eff
21′ (r)−I2V eff
11′ (r)

Tetralayer system

ℏ2v2
F Meff
11′ (r)σz+I2V eff
12′
0
0
I2V eff
21′ (r)+ℏ2v2
F Meff
21′ (r)σz+I2V eff
22′ (r)+ℏ2v2
F Meff
22′ (r)σz−I2V eff
11′ (r)

TABLE II. Explicit form of Hpert for trilayer and tetralayer graphene systems after ignoring the vector potential term. The
quantities I2 is the second order identity matrix and σz is z component of the Pauli matrix. V S(r) in (G3) consists of various
moiré potentials V eff
ij
between the layers. The integer subscripts ij in different terms appeared above, refer to the layer indices
as indicated in FIG. 11. The detailed expressions for various terms in Hpert each super-moiré system. are given in the text.
or graphene-hBN, the full Hamiltonian can be written as
an effective one-layer Hamiltonian experiencing a moiré-
periodic potential as,
Vmoiré = V S(r)I2 + ℏ2v2
F M eff(r)σz + ℏvF eAeff(r) · σ
(G3)
y/a(1)
(a)
(b)
x/a(1)
FIG. 12. (a) The spatial variation of the inversion symmetry
preserving part V eff(r), where the colorbar is in the units of
t2
0/VSTM, and (b) the spatial variation of the mass-dependent
term M eff
11′(r) that breaks the inversion symmetry. Here the
colorbar is in the units of
√
3t2
0/VSTM.
In the left hand side (LHS) of Eq. G3 the strength
of each term is of the same order, and is directly pro-
portional to the square of the interlayer hopping param-
eter and can be controlled by the interlayer bias VSTM
(its inversely proportional) which is typically of the or-
der of the bias applied to the tip of the scanning tun-
nelling microscope (STM) that ranges from 20−500 meV
[91, 92]. V S(r) preserves the inversion symmetry, the ef-
fective mass term M eff(r) breaks the inversion symmetry
while the effective vector potential Aeff(r) represents a
pseudo magnetic field. In the rest of the calculation, we
ignore the effective vector potential (G3) since it does
not change the proposed insertion of in-gap bands. Thus
the first two terms of the LHS of Eq. G3 form a Hpert
whose details for both the tri- and tetra-layer super-moiré
graphene systems are given in Table-II. In the following
we provide the specifics for each super-moiré system.

10
1.
A graphene trilayer system
We consider an AAA-stacked trilayer system [55–57]
such that the top graphene layer is at a distance d =
3 d0 ∼1 nm where d0 is the interlayer perpendicular dis-
tance between the remaining two layers. The configura-
tion is shown in FIG. 11(a), where the top layer is rotated
to an angle 3θr/2, the middle layer is rotated to θr/2
while the bottom layer is rotated at an angle −θr/2 such
that the relative misorientation between any two layers
is θr. Due to the relatively large distance (d/d0 > 1), the
top layer couples only weakly with the remaining two
layers. The details of the various terms that appeared in
the corresponding Hamiltonian H as had appeared in G1
are as follows:
H2(r) =
h1(θr/2) + V eff
11′(r)
T12(r)
T †
12(r)
h2(−θr/2) + V eff
11′(r)

(G4)
The integer subscripts ij that appeared in different terms
in the Hamiltonian (G4), again refer to the layer indices
as indicated in FIG. 11.
These terms, as well as the
terms that appeared in Hpert are shown in Table-II have
the detailed expression as follows:
V eff
l1′ (r) = 6t2
l1′
VSTM
+
t2
l1′
VSTM
3
X
j=1
cos

Gl1′
j
· r

(G5a)
ℏ2v2
F M eff
l1′(r) =
√
3t2
l1′
VSTM
3
X
j=1
sin

Gl1′
j
· r

(G5b)
For the moiré reciprocal lattice vector, the superscript lj
indicates the layers involved and it’s numbered according
to FIG. 11, with l = 1, 2. The subscript numbers such
reciprocal lattice vectors between the two surfaces and
Gl1′
3
= −Gl1′
1
−Gl1′
2 . t11′ = tAA/BB(d) = tAB/BA(d) ∼
7.31 meV, t21′ = tAA/BB(d + d0) = tAB/BA(d + d0) ∼
1.41 meV. As a representative value, for VSTM = 40 meV,
the strength of the potential becomes t2
11′/VSTM
∼
1.334 meV and t2
21′/VSTM ∼0.05 meV.
To proceed further with the calculation we note that
for a commensurate angle θr corresponding to q, p in
(D2), the two coprime integers for 2θr are q′, p′ that can
be obtained from q, p as p′ = 6pq/gcd(6qp, 3q2 −p2) and
q′ = (3q2 −p2)/gcd(6qp, 3q2 −p2). After doing straight-
forward algebra, one can obtain the relation between the
reciprocal lattice vectors of two interfaces as

G11′
1
G11′
2

=

Z1(q, p) Z2(q, p)
Z3(q, p) Z4(q, p)
 
G21′
1
G21′
2

(G6)
where Z1, . . . , Z4 are integers and are functions of q, p.
As an example, for q = 3 and p = 1, the two coprime
integers are obtained as q′ = 13 and p′ = 9 yielding
Z1 = 1, Z2 = −1, Z3 = 1 and Z4 = 2, in (G6. With
these, we calculate the first order correction to the energy
eigenvalues of H2(r) sue to Hpert(r). The unperturbed
eigenvalues of H2(r) and the corrected eigenvalues to the
first order in perturbation Hpert are shown in FIG. 13 for
q = 3 and p = 1 for different values of VSTM. From the
band structure, we see that the first order corrections
to the energy eigenvalues of H2(r) become smaller as
the interlayer bias VSTM increases.
Therefore, we can
conclude that the number of bands 4e2 ln(nc)/Df −2 do
not change under the effect of the perturbation.
2.
A graphene tetralayer system [87, 88]
The second example of the super-moiré structure that
we consider is where TBLG is sandwiched between two
graphene layers, each of which lie at a distance d = 3d0
from it as shown in FIG. 11(b).
For this system too
the unperturbed Hamiltonian can be presented as H2(r)
in (G4). For the expression of Hpert we only retain the
interlayer coupling between the nearest-neighbor layers
which are dominant over the strengths of the interlayer
potential in the next-nearest neighbor coupling. There-
fore, we ignore V eff
12′(r), V eff
12′(r) and ℏ2v2
F M eff
21′(r) in the
Hpert in Table-II, and Hpert finally becomes
E (eV)
(a)
(b)
FIG. 13. The band structure corresponding to the trilayer
graphene system with increasing values of the interlayer bias
voltage (a) VSTM = 10 meV, and (b) VSTM = 40 meV. The
solid orange lines show the band structure in the absence
of the potential while the blue lines represent the eigenval-
ues of H2(r) without the perturbation, while the cyan lines
show the band structure of H2(r) + Hpert(r) calculated upto
the first order in perturbation. The dotted cyan lines show
4e2 ln(nc)/Df −2 bands within the bandgap of lowest two bands
at the Γ-point.
Hpert(r) =

ℏ2v2
F M eff
11′(r)σz
0
0
V eff
22′(r) + ℏ2v2
F M eff
22′(r)σz −V eff
11′(r)

(G7)

11
E (eV)
(a)
(b)
q = 3, p = 1
q = 2, p = 1
FIG. 14. The band structure of the tetralayer graphene sys-
tem as shown in FIG. 11(b) for VSTM = 20 meV. The solid
orange lines show the band structure in the absence of the po-
tential while the blue lines represent the eigenvalues of H2(r)
without the perturbation, while the cyan lines show the band
structure of H2(r) + Hpert(r) calculated up to the first order
in perturbation. The dotted cyan lines show 4e2 ln(nc)/Df −2
bands within the bandgap of lowest two bands at the Γ-point.
Explicit form of different terms in H2(r) and Hpert(r)
can be written as
V eff
ll′ (r) = 6t2
ll′
VSTM
+
t2
ll′
VSTM
3
X
j=1
cos

Gll′
j · r

(G8a)
ℏ2v2
F M eff
ll′ (r) =
√
3t2
ll′
VSTM
3
X
j=1
sin

Gll′
j · r

(G8b)
The different superscripts and subscripts used have the
same meaning as in the previous case. With the Hpert
given in (G7) we calculate the first order correction to the
energy eigenvalues. The unperturbed and the perturbed
band structure to the first order in perturbation theory
are shown in FIG. 13 for q = 3 and p = 1 for comparison.
Evidently, the number of bands 4e2 ln(nc)/Df −2 do not
change under the effect of the perturbation up to the
leading order correction.
3.
A trilayer system of dissimilar layers
To show that the modelling of a super-moiré struc-
ture with the Hamiltonian H2(r) of moiré fractal and
a periodic perturbation holds beyond merely multi-layer
graphene systems, we consider a system of dissimilar lay-
ers [60] such that a graphene layer is sandwiched between
two hexagonal boron nitride (hBN) layers which are a dis-
tance d0 apart as shown in FIG. 11(c). This system can
be modelled as a moiré fractal at the first iteration of the
potential without any perturbing potential, but with a
modified H2(r). Namely,
H2(r) →H′
2(r) = −iℏvF σθr/2 · ∇+ V21(r) + V23(r)
(G9)
where V2l(r) is the effective periodic potential due to the
hBN layer-l = 1, 3 on the graphene layer-2. This is in
contrast to the examples 1 and 2 above since the Hamil-
tonian H′
2(r) itself describes the moiré-fractal at the first
iteration and the perturbation is zero. To progress fur-
ther, we first find the relative misorientation between the
hBN and graphene-layer using
θ = sin−1[(1 + δ) sin(ϕ)] −ϕ
(G10)
such that they make a commensurate angles θr between
them i.e., we want the difference |ϕ1 −ϕ2| to be one of
the commensurate angle θr(q, p) where ϕ is the orienta-
tion of the moiré-pattern with the graphene layer. Each
potential consists of three terms [70], where the different
spatially-dependent terms are given as
V eff
2l (r) = −3t2
0
 1
VN
+ 1
VB

−t2
0e−iψ
 1
VN
+ ω 1
VB

3
X
l=1
cos α2l
l (r)
(G11a)
ℏ2v2
F meff
2l (r) = −
√
3t2
0e−iψ
 1
VN
+ ω 1
VB

3
X
l=1
sin α2l
l (r)
(G11b)
ℏevF Aeff
2l (r) = −2t2
0e−iψ
 1
VN
+ ω 1
VB

3
X
l=1
{cos[ϕ(l + 1)]ˆx + sin[ϕ(l + 1)]ˆy} cos α2l
l (r)
(G11c)
where t0 = 0.152 eV, ω = ei 2π/3, ψ ∼−0.29 rad, VN =
3.34 eV and VB = −1.4 eV for boron and nitride atoms
[73], and
α2l
l = G2l
l · r + ψ + 2π/3
(G12)
For three different values of q, p and hence the θr, the
band structures along the high symmetry path X-Y-K-
X are shown in Fig 15.
It may be noted that in the
band structure plot we have chosen the path through
the high-symmetry points differently as compared to the
one used in the preceding two examples of super-moiré

12
q = 3, p = 1
q = 5, p = 1
E (meV)
GM
1
R(θr)GM
2
R(θr)GM
1
GM
2
b(2)
1
b(2)
2
(a)
(b)
FIG. 15. (a) The two blue and dashed blue hexagons show
the moiré BZ of the top and bottom moiré interfaces for the
trilayer hBN-G-hBN shown in FIG. 11(c).
The inner red
hexagon is the super-moiré BZ. (b) The band structures for
two different values of q, p along the high-symmetry path X-
Y -K-X [73] as shown in (b). The solid blue lines show the
band structures of the graphene-hBN system with V2(r) = 0
while the solid red lines represent graphene sandwiched be-
tween two hBN layers as in FIG.11(c). The two dark-gray
arrows show the bandgap between two lowest bands at the
Y -point where the 2e2 ln(nc)/Df −1 bands are inserted.
structures consisting only out of graphene layers. This
is in accordance with the convention used in [73]. The
high-symmetry point Y encloses 2e2 ln nc/Df −1 in-gap
bands within the bandgap of the lowest two bands shown
by the double-headed arrow. This example of dissimilar
layers also exhibits the robustness of the insertion of a
controlled number of bands determined by the fractal di-
mension Df of the moiré fractal. Therefore, the moiré
fractal can also explain the band structure of such a sys-
tem.
Appendix H: Discussion on the probability-density
plot for of the moiré-fractal wavefunctions given in
FIG. 2 and FIG. 4 of the main text
The probability density corresponding to the wave
functions of the MF at the first iteration j = 2 for the
Hamiltonian (2) of the main text were plotted in FIG. 2
(a4) and (b4) and also in the third column of FIG. 4 in the
main text. To that purpose we have calculated the spatial
variation of the probability density ρnk(r) = |ψnk(r)|2
of the Bloch states, where n is the band index and k is
the Bloch wave vector. Specifically, we calculated ρnk(r)
corresponding to the conduction band at the Dirac point
over the area in real-space covering the first super-moiré-
cell. θ = θr(3, 1) ∼21.79◦and θr(2, 1) ∼32.20◦, the
ρK(r) is shown in the FIGs. 2(a4) & (b4) of the main
text and θ ∼1.05◦and θr(3, 1) ∼21.79◦is shown in
FIGs. 4(a)and for θ ∼1.05◦and θr(5, 1) ∼13.17◦in
FIG. 4(b).
For both the plots the WS unit cell is shown with solid
black lines (see FIGs. 2 & 4) that encloses 2e2 ln(nc)/Df
local maxima or minima of ρc(v)K(r). We numerically
verified that this happens for both the high-symmetry K
and M-points, while the number of maxima or minima
enclosed by the WS cell at the Γ-point is different.
Appendix I: Calculation of the Hubbard parameters
The Hubbard interaction UH can be written as [145]
UR′m′,Rm =
X
XX′
Z
dr′ dr
ϕX′
m′(r′, R′)

2
UC(r′, r)
ϕX
m(r, R)
2
(I1)
where UC(r′, r) is the screened Coulomb interaction, e
is the electronic charge and ϕX
m(r, R) is the Wannier or-
bital of the sublattice X and band index m that is cen-
tered at the R-th lattice site. For pristine TBLG, the
localized Wannier orbitals can be constructed from the
Bloch states of the Hamiltonian H1 in Eq. (2) of the main
text corresponding to the two flat bands near the Fermi
level and these orbitals are centered at the local AB/BA-
regions of the moiré pattern [144, 146, 147]. Following
this prescription [147], for the onsite Hubbard interac-
tion U0 one can write
U0 ∝e2
a(1)
(I2)
where e is the electronic charge and the moiré wavelength
a(1) provides the cut-off for the screening. In the moiré
fractal model due to the presence of the mEP in Hj for
j ≥2 in Eq.(2) of the main text, the Bloch states ψ(j)
nk(r)
corresponding to the jth iteration of the potential have
the Bloch periodicity corresponding to the moiré super-
cell, that can be expressed as
ψ(j)
nk(r + a(j)) = eik·a(j)ψ(j)
nk(r)
(I3)
The translational-invariant Wannier functions, made out
of superposing these Bloch states, will be centered at the
lattice sites given by a(j). Accordingly, the onsite Hub-
bard interaction U0 that has the lattice constant as a
cut-off length, will be scaled. However it may be men-
tioned that the Wannier orbitals are constructed through
self-consistent ab-initio calculations and depend upon
the number of chosen bands [146–148]. In a moiré frac-
tal, 2e2 ln(nc/Df ) −2 inner bands can be considered since
they are well-separated from the other higher bands (see
FIG. 4 in main text and FIG. 5 here). A more detailed
calculation may lead to a more precise quantitative esti-
mate of U0 in the MF, but this is beyond the scope of the
current manuscript. Nevertheless, following the above ar-
gument we can estimate the onsite Hubbard interaction
for the jth iteration of the potential as
U (j)
0
∝
e2
ϵ a(j)
(I4)
If U0 is the onsite Hubbard interaction for pristine TBLG,
then for j = 2, it becomes U0 →U0/s where s is the

13
tw/tw0
V0 (meV)
V0 (meV)
(a)
(b)
tAA,BB = tAB,BA = 110 meV
tAA,BB = 79.7 meV
tAB,BA = 97.5 meV
FIG. 16. The bandwidth of the conduction and valence band in the presence of mEP for q = 3, p = 1 for (a) tAA,BB = tAB,BA =
110 meV [11], and (b) tAA,BB = 79.7 meV and tAB,BA = 97.5 meV [24, 144]. The bandwidth tw is in the unit of tw0 which is
the bandwidth of pristine TBLG at θ ∼1.05◦.
contractivity factor as defined in the main text.
Par-
ticularly, for q = 3 and p = 1, the contractivity factor
s =
√
7.
For the bandwidth tW , however we do not
have any such simple scaling argument. The bandwidth
tW = max(En)−min(En), where En is the energy of the
nth band, can be determined as a function of the strength
of the potential V0. Hence we determine this numerically
and present the results in FIG. 16(a) and (b). It can be
seen that that the band width depends on both the V0 as
well as the hopping parameters. Even though the band-
width gets significantly reduced as compared to pristine
TBLG, the full behavior is not amenable to simple ex-
planation. As an example, tw/tw0 ∼0.04 for the con-
duction band at V0 = 1.2 meV as shown in FIG. 16(a)
for tAA/BB = tAB/BA = 110 meV [11]. Therefore, the
Hubbard ratio U/tw becomes 1/(
√
7 × 0.04) ∼9.4 of the
ratio of pristine TBLG.
Appendix J: More on the experimental signatures of
moiré fractals
In the presence of the first iteration of an external
potential V2(r), the in-gap bands near the Fermi sur-
face, situated within the energy window of the lowest two
bands, contribute to a greater number of dips and peaks
in the density of states (refer to FIG. 2 and 4 of the
main text). This effect arises from the curvature of these
additional bands, resulting in changes to the density of
states within that energy range.
Since the differential
conductance (dI / dV ) in the various real-space probes is
proportional to the DOS of the sample at a particular
bias volatge Vbias [149],
dI / dV ∝ρsample(−e Vbias)
(J1)
where e is the fundamental electron charge.
The in-
creased DOS of the sample leads to an increased con-
ductance within a given energy window. In case of the
quantum twisting microscope [94], which offers better
real-space probing due to the local interference at the tip,
the increased number of states at a particular location r
may lead to an enhanced coupling among these states
and therefore alter the transport properties in contrast
to the case when there is no such external potential.
The calculation of experimentally measurable optical
properties involves the calculation of the optical matrix
elements between the Bloch states of different (same)
band indices, i.e., the interband (intraband) transitions
[43, 131, 132]. Due to the greater number of states avail-
able within a given energy window, there may be non-
vanishing optical matrix elements between the induced
in-gap states that can further tune those properties and
therefore optical measurements also provide a technique
for characterizing MFs.
A recent article [133] reported cryogenic near-field op-
toelectronic measurements of hBN-encapsulated MAT-
BLG where the photovoltage measurements revealed a
supermoiré pattern whose periodicity was embedded in
the photovoltage response.
In their experiment, the
top graphene layer was aligned with the top encapsu-
lating hBN layer, while the bottom graphene layer was
twisted by an angle θhBN wrt the hBN layer, and the two
graphene layers were relatively twisted by θTBLG. Since
the moiré-wavelength for the graphene-hBN interface is
limited to ∼14 nm due to the lattice mismatch of 1.8 %,
there is a finite mismatch in the moiré-wavelengths of
both the graphene-hBN and the graphene-graphene in-
terfaces at smaller angles. For MFs in hBN-encapsulated
graphene (FIG.15), both the hBN layers with respect
to the graphene layer are rotated to the same angle,
thereby enabling the same moiré wavelengths in both the
graphene-hBN interfaces. Thus for the two cases with
(q, p) = (3, 1) and (q, p) = (5, 1) the supermoiré wave-
lengths become λSM ∼36.13 nm and λSM ∼60.24 nm,
respectively. A similar photovoltage response measure-
ment in hBN-encapsulated graphene may also show su-

14
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