PHYSICAL REVIEW B
110
, L201406 (2024)
Letter
Entropy-enhanced fractional quantum anomalous Hall effect
Gal Shavit
Department of Physics and Institute for Quantum Information and Matter,
California Institute of Technology
,
Pasadena, California 91125, USA
and Walter Burke Institute of Theoretical Physics,
California Institute of Technology
, Pasadena, California 91125, USA
(Received 9 September 2024; revised 22 October 2024; accepted 5 November 2024; published 15 November 2024)
Strongly interacting electrons in a topologically nontrivial band may form exotic phases of matter. An
especially intriguing example of which is the fractional quantum anomalous Hall phase, recently discovered
in twisted transition metal dichalcogenides and in moiré graphene multilayers. However, it has been shown to
be destabilized in certain filling factors at sub-100 mK temperatures in pentalayer graphene, in favor of a novel
integer quantum anomalous Hall phase [Z. Lu, Extended quantum anomalous Hall states in graphene/hBN moiré
superlattices,
arXiv:2408.10203
]. We propose that the culprit stabilizing the fractional phase at higher temper-
atures is its rich edge state structure. Possessing a multiplicity of chiral modes on its edge, the fractional phase
has lower free energy at higher temperatures due to the excess edge mode entropy. We make distinct predictions
under this scenario, including the system-size dependency of the fractional phase entropic enhancement, and
how the phase boundaries change as a function of temperature.
DOI:
10.1103/PhysRevB.110.L201406
Introduction.
The fractional quantum anomalous Hall ef-
fect [
1
,
2
] (FQAH) is the zero magnetic field analog of the
celebrated fractional quantum Hall effect [
3
,
4
]. Recent ad-
vances in moiré materials have recently enabled experimental
observation of this topological strongly correlated phase of
matter in twisted transition metal dichalcogenides [
5
–
8
], and
crystalline graphene/hexagonal boron nitride (hBN) moiré su-
perlattices [
9
–
11
].
A recent experiment [
12
] revealed an unexpected surprise:
Throughout much of the phase diagram in Ref. [
9
], the ob-
served FQAH phases at various filling factors are
not
the
actual ground state of the system. When the electronic temper-
ature was lowered, a different phase superseded the FQAH.
This phase is characterized by vanishing longitudinal resis-
tivity and a quantized anomalous Hall response
R
xy
≈
h
/
e
2
.
Moreover, it extends to a finite range of densities, with little
regards to commensuration with the underlying moiré struc-
ture. It has thus been dubbed the extended integer quantum
anomalous Hall phase (EIQAH).
To understand the root cause of this phenomenon, a better
understanding of both the FQAH phases and the EIQAH is
required. A recent proposal [
13
] has associated the EIQAH
with a Wigner-crystal-like phase formed on top of an integer
quantum anomalous Hall state. An attractive feature of this
theory is that it accounts for some of the nonlinear trans-
port signatures in Ref. [
12
]. However, the temperature-driven
transition into the FQAH remains unresolved. Reference
[
14
] posited that the “leftover” electrons in the EIQAH
Anderson-localize due to disorder. Heating the EIQAH delo-
calizes these electrons, therefore allowing interaction effects
to take root, such that the system crosses over to a FQAH
phase.
In this Letter, we propose a different theoretical expla-
nation for the observed phenomenon, relying only on the
topological distinction between the two competing phases,
and on the mesoscopic character of the experimental devices.
Namely, excess gapless edge modes in the FQAH phase en-
dow it with a higher entropy. This naturally results in a phase
boundary which favors the FQAH over the EIQAH as temper-
ature increases (see Fig.
1
).
It is rather suggestive that the temperature-stabilized
FQAHs appear at filling factors where the analogous frac-
tional quantum Hall phases have a rich nonuniversal edge
content [
15
–
18
]. Whereas one naturally expects the EIQAH
to host one chiral edge state, at a FQAH filling of, e.g.,
∼
2
/
3,
one may encounter as many as four downstream
/
upstream
modes [
19
,
20
] which carry heat and thus contribute to the
overall edge entropy [
21
].
We provide key predictions which may confirm or inval-
idate the proposed scenario. Namely, under our assumptions
the entropic effect is necessarily device-size dependent, and
should be virtually undetectable for a large enough sample.
Moreover, the temperature dependence of the phase bound-
ary between the phases should follow
T
∝
√
δ
u
, with
δ
u
the
ground state energy difference between the phases. Finally,
the appreciable evolution of the EIQAH-FQAH boundary
as a function of electric displacement field and temperature
suggests that this
δ
u
should be relatively insensitive to dis-
placement field changes.
Theory.
We aim to provide a phenomenological charac-
terization of the competition between the extended quan-
tum anomalous Hall phase (
I
) and the fractional quantum
anomalous Hall phase (
F
), via a description of the free
energy associated with these phases. Assuming a tempera-
ture scale far below the respective transition temperatures of
these phases to a metallic phase, we approximate the free
energies as
G
μ
(
x
,
T
)
=
U
μ
(
x
)
−
TS
edge
μ
,
(1)
with the index
μ
=
I
,
F
for the appropriate phase. Here,
x
is
a parameter tuning a phase transition between the integer and
2469-9950/2024/110(20)/L201406(5)
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©2024 American Physical Society
GAL SHAVIT
PHYSICAL REVIEW B
110
, L201406 (2024)
FQAH
EIQAH
Metal
FIG. 1. Schematic description of the phase diagram emerging
from our model. The ground state of the system is tuned by some pa-
rameter, e.g., displacement field
D
, determining the energy difference
between the fractional quantum anomalous Hall phase (FQAH) on
the left side, and the extended integer quantum anomalous Hall phase
(EIQAH) on the right. The left and right insets illustrate a possible
edge picture of the respective phases. The integer EIQAH phase has
a single chiral edge mode, whereas the fractional phase may host a
rich plethora of edge modes, each contributing to the overall entropy.
At higher temperatures, the phase boundary (dark blue solid line)
shifts as
T
∝
√
δ
u
[see Eq. (
4
)], favoring the phase with higher edge
state entropy. As the linear dimension of the system
L
increases,
the shift of the boundary becomes more subtle (red dotted line) and
the entropy-driven transition may be preempted by a transition to a
metallic phase.
fractional phases. In the experiment, this parameter is, e.g., an
electric displacement field.
Let us denote by
and
P
the area and perimeter of the
device. We will define the potential energy density differ-
ence between the two phases
δ
u
≡
(
U
F
−
U
I
)
/
, which is a
phenomenological parameter of our theory. It is determined
by details of the underlying microscopic theory of both the
fractional and integer phases, which are beyond the scope of
this Letter. The entropy density of the phase
μ
originating in
its linearly dispersing edge states is
s
edge
μ
=
S
edge
μ
/
P
=
π
2
k
2
B
T
3 ̄
h
∑
i
∈
μ
1
v
i
,
(2)
where
v
i
are the edge mode velocities in the phase
μ
, and
k
B
is
the Boltzmann constant. It is then useful to define the velocity
scale
v
∗
=
[
∑
i
∈
F
1
v
i
−
∑
j
∈
I
1
v
j
]
−
1
,
(3)
where we assume
v
∗
>
0, implying the fractional phase edge
states carry excess entropy as compared to the integer phase. If
all velocities are equal to ̄
v
,
v
∗
=
̄
v
/
N
δ
, with
N
δ
the difference
in the number of edge states between
F
and
I
. Notice that
v
∗
is dominated by the lowest velocities in the system (and hence
by modes carrying the highest density of states). Furthermore,
in the presence of multiple edge modes (as hypothesized for
the quantum Hall analogs for the relevant filling factors in the
experiment), intermode interactions tend to strongly renor-
malize the velocities [
22
,
23
]. Generically, one or several of
the renormalized velocities will be significantly lower than the
bare ones.
We note it has been well established that nonlinearities
in the edge spectrum arise for FQH phases, at energy scales
which are a fraction of the FQH gap [
24
,
25
]. These non-
linearities soften the edge mode dispersion, give rise to an
enhancement of the edge state density of states, and conse-
quently lead to a higher entropy than the simplified description
in Eq. (
2
). Assuming similar physics persist for the FQAH
edge states, this will make the described entropy-driven effect
even more pronounced.
The transition between the two phases, which is of first
order by construction [we consider the competition be-
tween the
G
μ
described in Eq. (
1
)], is then given by the
condition [
26
]
k
B
̃
T
=
√
α
geo
L
̄
h
v
∗
×
√
δ
u
(
̃
x
)
,
(4)
where we have conveniently defined the geometrical constant
3
/
(
π
2
P
)
≡
α
geo
L
, relating the surface-to-perimeter ratio of
the device to its linear extent
L
[for, e.g., a square,
α
geo
=
3
/
(4
π
2
)].
Taking into account realistic parameters from the exper-
iment in Ref. [
12
], one may arrive at a rough estimate of
δ
u
given Eq. (
4
) characterizes the transition. Considering an
electronic density of
n
e
≈
0
.
5
×
10
12
cm
−
2
, and parameters
v
∗
∼
10
3
m
/
s,
̃
T
∼
0
.
1K,
L
∼
1 μm, the difference in en-
ergies between the fractional and extended integer phase
δ
u
/
(
k
B
n
e
) is on the order of
O
(5 mK/electron).
Let us take a different perspective, considering specifically
the out-of-plane electric displacement field as controlling the
transition,
x
=
D
. We may formulate the evolution of the
phase boundary in the
D
-
T
plane as a Clausius-Clapeyron
relation,
dD
dT
=
S
edge
F
−
S
edge
I
δμ
D
,
(5)
where
δμ
D
/
=
(
∂δ
u
)
/
(
∂
D
) is the difference of the out-
of-plane electric dipole moment between the fractional and
integer phases. Equation (
5
) then implies that the effect
we describe here, stabilization of the phase with a richer
edge content by the edge entropy, is greatly enhanced
if
the energy difference between the phases is insensitive to
changes in the displacement field
. As the relevant experimen-
tal regime lies in the high-field region with
D
≈
1V
/
nm,
this is actually quite likely—the displacement field effects on
the single-particle properties of the relevant band are nearly
saturated.
An in-depth study of the effect of
D
on the stability of the
various FQAH states, and on its energetic competition with
the EIQAH phase (whose nature is yet unresolved) is beyond
the scope of this Letter. Such a study requires of course,
among other things, consideration of the electron-electron
interactions necessary to facilitating a fractional phase. Let
us, however, examine a single-particle property which has
been shown to play a key role in stabilizing the fractional
quantum anomalous Hall phase in a given band, namely the
trace of the Fubini-Study metric tr
g
[
27
–
36
]. For simplic-
ity, we omit the effect of the moiré-inducing hBN layer,
since electrons are mostly polarized to the graphene layer far
away from it in the relevant regime. A change of 5% to the
potential difference between the outermost graphene layers
L201406-2
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(a)
(b)
(c)
FIG. 2. Insensitivity of the quantum geometry to displacement
field. (a) Trace of the Fubini-Study metric around the valley
K
point for an interlayer potential difference of 40 meV. The white
contour marks the Fermi surface corresponding to the density
n
e
=
6
×
10
11
cm
−
2
. (b) Same as (a), with an interlayer potential difference
of 38 meV. (c) The relative difference between (a) and (b), which is
significantly less than the relative potential change. Dashed and solid
lines correspond to the Fermi surfaces in (a) and (b), respectively.
Details of the calculations for this figure appear in the Supplemental
Material [
26
].
hardly has any effect on this quantum geometric property,
as seen by comparing Figs.
2(a)
and
2(b)
.InFig.
2(c)
we
plot the relative change of this metric, which remains much
smaller than the relative change in
D
throughout the majority
of the enclosed Fermi volume at the relevant densities. This
simple calculation demonstrates that at the experimentally
relevant values of displacement field, a decisive parameter
controlling the FQAH physics remains quite insensitive to
changes in
D
. Such an insensitivity is consistent with a very
small
δμ
D
.
In the thermodynamic limit,
L
→∞
, and the entropy-
driven transition temperature diverges. Referring again to
Fig.
1
, the phase boundary between the FQAH and EIQAH
phases becomes nearly vertical below the transition tempera-
ture to some gapless phase. Practically, one expects to “lose”
the transition once a device exceeds a certain size
̃
L
,for
which the transition temperature exceeds the temperature at
which the fractional phase forms,
̃
T
>
T
c
. This length scale is
approximately
̃
L
≈
(
k
B
T
c
)
2
α
geo
̄
h
v
∗
δ
u
.
(6)
Using the estimate for
δ
u
we obtained above, given
T
c
which
is of order of a few hundred mK, one may estimate
̃
L
∼
O
(10 μm). If the excess edge density of states (
∝
1
/
v
∗
) is large
enough, or the potential energy difference
δ
u
is rather small,
the EIQAH-to-FQAH “melting” can be observed for realistic
device sizes.
We note that in the presence of disorder, specifically of the
kind that couples differently to the
F
and
I
phases, the impact
of the edge state entropy may be further
enhanced
. Domain
walls separating the topologically distinct phases contain pro-
tected edge modes which carry entropy at finite temperature.
A transition from a uniform
I
phase at low temperatures to
a domain-patterned phase at higher temperature, analogous
to the so-called Chern mosaic [
37
], is expected in such a
scenario. In such a mosaic phase,
α
geo
effectively becomes
much smaller, since one replaces the perimeter of the device
P
,
with the overall length of domain walls in the system, roughly
proportional to
P
times the number of domains. The interplay
between disorder and domain wall entropy-driven transitions
has been thoroughly explored in Ref. [
38
]. The signature this
sort of transition would have on
transport
is not universal,
would depend on the position of the domains relative to the
contacts, and would thus be hard to reproduce.
Discussion.
We have presented a scenario accounting for
the emergence of FQAH upon heating of an EIQAH ground
state. The culprit in this scenario is the excess entropy of
the FQAH edge states. If the ground state energy difference
between the phases is sufficiently small, the system may gain
energy by “partially melting” in its edges. This is enabled
by transitioning into the intermediate FQAH, which allows
additional gapless excitations on its edge.
Clearly, the free-energy gain on the edge is expected to
be eclipsed by the bulk
δ
u
>
0 contribution for large enough
systems, and thus the behavior of the phase boundary will be
strongly system-size dependent. This distinct prediction of our
model indicates that
themovementofthephaseboundaryinthe
D-T plane should be inversely proportional to L
, as inferred
by Eq. (
5
).
As the transition is dominated by edge contributions, our
theory further predicts this effect should be enhanced (i.e.,
the FQAH more stable against the EIQAH) at filling fractions
which host a larger abundance of edge states. We carefully
speculate that this effect may explain why certain FQAHs
disappear completely at low temperatures, whereas others sur-
vive for a finite range of displacement fields. An exhaustive
mapping of the hierarchy of the stabilized FQAHs at inter-
mediate temperatures is necessary in order to make this point
more concrete.
In addition to experimental predictions, our proposal raises
issues to be addressed by the theory underlying the FQAH-
EIQAH competition. Namely, in the picture described above
the energetic difference between the phase
δ
u
is rather small.
Moreover,
δ
u
should be relatively insensitive to changes of the
displacement field around the experimentally relevant range,
D
≈
1V
/
nm. These points may provide a valuable clue to-
wards disentangling the two phases, as well as a testable
hypothesis in their numerical simulations.
The predictive power of Eq. (
4
), suggesting the phase
boundary takes the shape
̃
T
∝
√
δ
u
( ̃
x
), is somewhat dimin-
ished by the lack of knowledge of the functional dependence
of
δ
u
(
x
). Improved theoretical understanding of the phase
competition will help resolve this issue and provide an ideal
setting to test Eq. (
4
). Alternative explanations resulting
in the same square-root behavior are possible, as the na-
ture of collective excitations in FQAH and EIQAH is far
from understood. These include the anomalous magneto-roton
[
39
,
40
], which is gapped, and presumably contributes very
little entropy at low temperatures. Phonon excitations due to
spontaneous crystallization in the EIQAH are expected to be
gapped due to the weak moiré potential [
13
], and even if they
are not (or the gap is very small), the phonons should give
an entropic advantage to the EIQAH and not the FQAH. We
note that the observed FQAH itself may harbor an additional
hidden translation-symmetry-breaking order [
41
], yet the re-
sulting phonons are also expected to be gapped for the same
reason—the underlying moiré lattice.
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In conclusion, the rich landscape of FQAH, EIQAH,
anomalous Hall crystals, and their interplay in moiré flat-
band systems is far from being understood. The anomalous
appearance of the exotic FQAH phase at higher tempera-
tures is one of many observations calling to question our
understanding of these materials. Drawing inspiration from
the presumably similar physics of the fractional quantum Hall
effect, we illustrate how this phenomenon can be related to
the
topological difference
between the FQAH and the EIQAH
ground states, regardless of their intricate collective excitation
spectra.
Acknowledgments.
G.S. acknowledges enlightening dis-
cussions with Gil Refael, as well as support from the Walter
Burke Institute for Theoretical Physics at Caltech, and from
the Yad Hanadiv Foundation through the Rothschild fellow-
ship.
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