of 5
Correlation-driven topological phases in
magic-angle twisted bilayer graphene
In the format provided by the
authors and unedited
Nature | www.nature.com/nature
Supplementary information
https://doi.org/10.1038/s41586-020-03159-7
Supplementary Information: Correlation-driven Topologi-
cal Phases in Magic-Angle Twisted Bilayer Graphene
Youngjoon Choi, Hyunjin Kim, Yang Peng, Alex Thomson, Cyprian Lewandowski, Robert Polski,
Yiran Zhang, Harpreet Singh Arora, Kenji Watanabe, Takashi Taniguchi, Jason Alicea, Stevan
Nadj-Perge
Theoretical Modeling
Hofstadter spectrum from Continuum model
We will follow the approach used in Ref. 1,2. The
continuum model Hamiltonian for TBG at
K
-valley (in monolayer graphene BZ) can be written in
the following matrix form
H
=
(
h
(
θ/
2)
T
(
r
)
T
(
r
)
h
+
(+
θ/
2)
)
(1)
where the intralayer Hamiltonian is given by
h
(
θ/
2) =
~
v
F
(
ˆ
p
+
q
h
)
·
σ
θ/
2
,
(2)
and the interlayer coupling is written as
T
(
r
) =
w
2
j
=0
T
j
exp(
i
q
j
·
r
)
. Here,
q
0
=
k
θ
(0
,
1)
,
q
1
=
k
θ
(
3
,
1)
/
2
,
q
2
=
k
θ
(
3
,
1)
/
2
,
q
h
=
k
θ
(
3
/
2
,
0)
,
k
θ
= 8
π
sin(
θ/
2)
/
(3
a
)
. The matrices
are defined as
T
0
=
ησ
0
+
σ
1
,
T
1
=
ησ
0
+
ζσ
+
+
̄
ζσ
,
T
2
=
ησ
0
+
ζσ
+
̄
ζσ
+
, with
ζ
= exp(
i
2
π/
3)
,
where the Pauli matrices
σ
1
,
2
,
3
,
σ
±
= (
σ
1
±
2
)
/
2
correspond to the sublattice degree of freedom.
In the following, we choose the Fermi velocity
v
F
= 9
.
38
×
10
6
m/s, the interlayer coupling
strength
w
= 110
meV, and
η
= 0
.
4
which takes into account the effects of lattice relaxation.
In the presence of perpendicular magnetic field
B
=
B
ˆ
z
, we substitute
ˆ
p
ˆ
p
+
e
A
. We
further take the Landau gauge
A
=
B
(
y,
0)
.
We can rewrite the intralayer Hamiltonian as
h
(
θ/
2) =
~
ω
c
(
0
a
p
x
e
±
iθ/
2
a
p
x
e
iθ/
2
0
)
(3)
where
ω
c
=
2
v
F
/l
b
,
l
b
=
~
/eB
, and
a
p
x
=
1
2
[
(( ˆ
p
x
+
3
2
k
θ
)
l
b
y/l
b
)
i
ˆ
p
y
l
b
]
(4)
a
p
x
=
1
2
[
(( ˆ
p
x
+
3
2
k
θ
)
l
b
y/l
b
) +
i
ˆ
p
y
l
b
]
(5)
1
Numerically, we can diagonalize the Hamiltonian using the basis
{|
L,σ,n
;
k
=
|
k
〉 ⊗
|
L,σ,n,k
〉}
(truncated up to some index
n
), where
L
=
is the layer index,
σ
is the sub-
lattice index, and
ˆ
p
x
|
k
=
k
|
k
, and
n
labels the
n
th harmonic oscillator eigenstates such that
a
k
|
n,k
=
n
|
n
1
,k
,
a
k
|
n,k
=
n
+ 1
|
n
+ 1
,k
. Moreover, we have
a
k
= exp(
i
̃
p
y
kl
2
b
)
a
exp(
i
̃
p
y
kl
2
b
)
,
(6)
with
a
a
0
, and
exp(
i
ˆ
p
y
kl
2
b
)
|
n,
0
=
|
n,k
.
Let us compute the matrix elements of the Hamiltonian in the above basis.
Intralayer Hamiltonian.
In this basis, the annihilation operator has the following matrix elements
L
,n
;
k
|
a
k
|
L,σ,n
;
k
=
n
1
δ
kk
δ
LL
δ
σσ
δ
n
,n
1
(7)
Interlayer Hamiltonian.
The matrix elements of the interlayer Hamiltonian can be evaluated using
〈−
,n
;
k
|
T
j
e
i
q
j
·
r
|
+
,σ,n
;
k
= (
T
j
)
σ
σ
δ
k
,k
q
jx
e
iq
jy
kl
2
b
e
iq
jy
3
k
θ
l
2
b
/
2
e
iq
jx
q
jy
l
2
b
/
2
F
n
,n
(
z
j
)
(8)
where
z
j
=
q
jx
+
iq
jy
2
l
b
.
The function
F
n
,n
(
z
)
is
F
n
,n
(
z
) =
n
!
n
!
(
̄
z
)
n
n
exp(
−|
z
|
2
/
2)
L
(
n
n
)
n
(
|
z
|
2
)
n
n
n
!
n
!
z
n
n
exp(
−|
z
|
2
/
2)
L
(
n
n
)
n
(
|
z
|
2
)
n
< n.
(9)
Hofstadter spectrum from the ten-band model
Here, we briefly describe the ten-band model
for the TBG at magic angle, first introduced in Ref. 3.
The ten-band model is defined on a triangular lattice with basis vectors
a
1
= (
3
/
2
,
1
/
2)
and
a
2
= (0
,
1)
. We write the Bravais lattice sites as
r
=
r
1
a
1
+
r
2
a
2
or simply as
r
= (
r
1
,r
2
)
,
where
r
1
,
2
Z
. Within each unit cell, there are ten orbitals which are distributed on three different
sites, as indicated by the different colors in Fig. 1. Explicitly, there are three orbitals,
p
z
,
p
+
, and
p
, on every triangular lattice site (orange).
Each of the three kagome sites (black) within a unit cell hosts an
s
orbital. Finally, both A
and B sublattices of the honeycomb sites (green) have
p
+
and
p
orbitals. Throughout this work,
these ten orbitals are ordered as
c
r
= (
τ
z,
r
+
,
r
,
r
1
,
r
2
,
r
2
,
r
A
+
,
r
A
,
r
B
+
,
r
B
,
r
)
T
,
where
τ
,
κ
, and
η
denote operators on the triangular, kagome, and honeycomb sites respectively.
2
a
1
a
2
(1)
(2)
(3)
(A)
(B)
Supplementary Information Figure 1
|
Lattice and orbitals for the ten-band model. The orange
solid circles denote the triangular sites with
p
z
,
p
+
, and
p
orbitals. The black solid circles cor-
respond to the three types of kagome sites, labeled as (1), (2), and (3). On each of these kagome
sites, there is an
s
orbital. The green empty circles indicate the honeycomb sites, either type A or
type B, with
p
+
and
p
orbitals on each of them.
When an external perpendicular magnetic field is applied, we insert a Peierls phase
exp(
i
j
)
with
θ
i
j
=
e
~
r
j
r
i
A
·
d
l
(10)
into the hopping amplitude from
r
i
to
r
j
.
When the flux per unit cell is a rational number
p/q
, with
p,q
Z
, namely
Φ =
p
q
φ
0
, φ
0
=
hc
e
,
(11)
we can take the periodic Landau gauge
4
A
=
0
2
πq
[
(
ξ
1
−b
ξ
1
c
)
b
2
ξ
2
n
=
−∞
δ
(
ξ
1
n
+ 0
+
)
b
1
]
,
(12)
where
b
1
and
b
2
are reciprocal lattice vectors. The advantage of the periodic Landau gauge is the
system is periodic in both
a
1
and
a
2
directions, with periods
a
1
and
q
a
2
, respectively.
3
References:
1. Bistritzer,
R.
&
MacDonald,
A.
H.
Moir
́
e
butterflies
in
twisted
bilayer
graphene.
Phys. Rev. B
84
,
035440
(2011).
URL
https://link.aps.org/doi/10.1103/PhysRevB.84.035440
.
2. Hejazi,
K.,
Liu,
C.
&
Balents,
L.
Landau
levels
in
twisted
bilayer
graphene and semiclassical orbits.
Phys. Rev. B
100
, 035115 (2019).
URL
https://link.aps.org/doi/10.1103/PhysRevB.100.035115
.
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ile topology of magic-angle bilayer graphene.
Phys. Rev. B
99
, 195455 (2019).
URL
https://link.aps.org/doi/10.1103/PhysRevB.99.195455
.
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Periodic landau gauge and quantum hall ef-
fect in twisted bilayer graphene.
Phys. Rev. B
88
,
125426 (2013).
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https://link.aps.org/doi/10.1103/PhysRevB.88.125426
.
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