of 2
Supplementary Materials for ”Entropy-Enhanced Fractional Quantum Anomalous Hall Effect”
Gal Shavit
1, 2
1
Department of Physics and Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, California 91125, USA
2
Walter Burke Institute of Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA
OBTAINING THE TRANSITION TEMPERATURE
Beginning with Eq. (1) in the main text, we equate the free energies of the
F
and
I
phases, plugging in the edge entropy,
U
I
P
π
2
k
2
B
T
2
3
h
X
j
I
1
v
j
=
U
F
P
π
2
k
2
B
T
2
3
h
X
i
F
1
v
i
,
(S1)
where
P
is the perimeter of the device, and
i
and
j
sum over the edge modes in the
F
and
I
phases, respectively. Isolating the
transition temperature
̃
T
,
k
B
̃
T
=
v
u
u
u
t
U
F
U
I
π
2
3
P
X
i
F
1
v
i
X
j
I
1
v
j
1
,
(S2)
We may recover Eq. (4) in the main text via the definitions
U
F
U
I
= Ω
δu
,
v
=
h
P
i
F
1
v
i
P
j
I
1
v
j
i
1
, and
3Ω
/
π
2
P

α
geo
.
L
.
RHOMBOHEDRAL PENTALAYER GRAPHENE BAND STRUCTURE
We employ a similar calculation scheme as the one used in Ref. [1], which adapted tight-binding parameters from a model
of rhombohedral trilayer graphene [2], as well as density functional theory parameters from Ref. [3]. Assuming a fully-isospin-
polarized parent phase (i.e., a “quarter metal”) as the parent state, we perform the calculation for only one valley. The tight
binding Hamiltonian in momentum
k
space is given by
H
penta
=
2
u
d
v
4
π
v
3
π
0
1
2
γ
2
0
0
0
0
2
u
d
+
δ γ
1
v
4
π
0
0
0
0
0
0
v
4
π γ
1
u
d
+
u
a
v
4
π
v
3
π
0
1
2
γ
2
0
0
v
3
π
v
4
π vπ u
d
+
u
a
γ
1
v
4
π
0
0
0
0
0
0
v
4
π
γ
1
u
a
v
4
π
v
3
π
0
1
2
γ
2
1
2
γ
2
0
v
3
π
v
4
π vπ u
a
γ
1
v
4
π
0
0
0
0
0
0
v
4
π γ
1
u
d
+
u
a
v
4
π
v
3
π
0
0
1
2
γ
2
0
v
3
π
v
4
π vπ
u
d
+
u
a
γ
1
v
4
π
0
0
0
0
0
0
v
4
π
γ
1
2
u
d
+
δ vπ
0
0
0
0
1
2
γ
2
0
v
3
π
v
4
π
2
u
d
.
(S3)
The Hamiltonian is written in the layer-sublattice basis,
(
ψ
1
A
k
1
B
k
2
A
k
2
B
k
3
A
k
3
B
k
4
A
k
4
B
k
5
A
k
5
B
k
)
,
where
ψ
ℓσ
k
annihilates an electron in layer
, sublattice
σ
, at momentum
k
relative to the
K
valley point. In our notation,
π
=
k
x
+
ik
y
, and
v
i
=
3
2
i
,
with
a
= 0
.
246
nm. The rest of the parameters are:
γ
= 2600
meV,
γ
1
= 356
.
1
meV,
γ
2
=
15
meV,
γ
3
=
293
meV,
γ
4
=
144
meV,
δ
= 12
.
2
meV,
u
a
= 16
.
4
meV.
FUBINI-STUDY METRIC
After diagonalization the band structure, we focus on the lowest-lying conduction band, spanned by the Bloch wavefunctions
|
u
k
. Employing similar conventions as the ones in Ref. [4], we define the quantum geometrical tensor
η
αβ
(
k
) =
α
u
k
|
β
u
k
u
k
|
u
k
α
u
k
|
u
k
⟩⟨
u
k
|
β
u
k
u
k
|
u
k
2
,
(S4)
2
with the shorthand
α
=
∂k
α
. The Berry curvature
and Fubini-Study metric
g
are then
Ω = 2Im
η
yx
, g
αβ
= Re
η
αβ
.
(S5)
In the main text, we plot the quantity
tr
g
=
g
xx
+
g
yy
. The relative change in the trace of the quantum metric with two
different parameters, e.g., different interlayer potential strength is simply
g
= 2
tr
g
(1)
tr
g
(2)
tr
g
(1)
+ tr
g
(2)
.
(S6)
Here the subscripts
(
i
)
indicate different value of parameters for which the band structure was calculated.
[1] Z. Dong, A. S. Patri, and T. Senthil, Theory of fractional quantum anomalous hall phases in pentalayer rhombohedral graphene moir
́
e
structures (2023), arXiv:2311.03445 [cond-mat.str-el].
[2] F. Zhang, B. Sahu, H. Min, and A. H. MacDonald, Phys. Rev. B
82
, 035409 (2010).
[3] Y. Park, Y. Kim, B. L. Chittari, and J. Jung, Phys. Rev. B
108
, 155406 (2023).
[4] P. J. Ledwith, G. Tarnopolsky, E. Khalaf, and A. Vishwanath, Phys. Rev. Res.
2
, 023237 (2020).