# Tunable even- and odd-denominator fractional quantum Hall states
in trilayer graphene

Yiwei Chen1† Yan Huang1† Qingxin Li1† Bingbing Tong2,3 Guangli Kuang4
Chuanying Xi4 Kenji Watanabe5 Takashi Taniguchi6 Guangtong Liu2,3,7∗ Zheng
Zhu8 Li Lu2,3,7 Fu-Chun Zhang8,9,10 Ying-Hai Wu11∗ Lei Wang1,10∗ 1National
Laboratory of Solid-State Microstructures, School of Physics, Nanjing
University, Nanjing, 210093, China  2Beijing National Laboratory for Condensed
Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  3Hefei National Laboratory, Hefei 230088, China.  4Anhui
Province Key Laboratory of Condensed Matter Physics at Extreme Conditions,
High Magnetic Field Laboratory of the Chinese Academy of Science, Hefei
230031, China.  5Research Center for Electronic and Optical Materials,
National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan
6Research Center for Materials Nanoarchitectonics, National Institute for
Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  7Songshan Lake
Materials Laboratory, Dongguan 523808, China  8Kavli Institute of Theoretical
Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China
9CAS Center for Excellence in Topological Quantum Computation, University of
Chinese Academy of Sciences, Beijing, 100049, China  10Collaborative
Innovation Center of Advanced Microstructures, Nanjing University, Nanjing,
210093, China  11School of Physics and Wuhan National High Magnetic Field
Center, Huazhong University of Science and Technology, Wuhan 430074, China
†These authors contributed equally to this work.  ∗Corresponding authors,
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
The fractional quantum Hall (FQH) states are exotic quantum many-body phases
whose elementary charged excitations are neither bosons nor fermions but
anyons, obeying fractional braiding statistics. While most FQH states are
believed to have Abelian anyons, the Moore-Read type states with even
denominators, appearing at half filling of a Landau level (LL), are predicted
to possess non-Abelian excitations with appealing potentials in topological
quantum computation. These states, however, depend sensitively on the orbital
contents of the single-particle LL wavefunction and the mixing between
different LLs. Although they have been observed in a few materials, their non-
Abelian statistics still awaits experimental confirmation. Here we show
magnetotransport measurements on Bernal-stacked trilayer graphene (TLG), whose
unique multiband structure facilitates the interlaced LL mixing, which can be
controlled by external magnetic and displacement fields. We observe a series
of robust FQH states including even-denominator ones at filling factors
$\nu=-9/2$, $-3/2$, $3/2$ and $9/2$. In addition, we are able to finetune the
LL mixing and crossings to drive quantum phase transitions of these half-
filling states and their neighboring odd-denominator ones, exhibiting a
related emerging and waning behavior. Our results establish TLG as a
controllable system for tuning the weights of LL orbitals and mixing strength,
and a fresh platform to seek for non-Abelian quasi-particles.

Electrons confined in a two-dimensional system under a perpendicular magnetic
field develop quantized energy levels. And fully filling such Landau levels
(LLs) one by one gives rise to the integer quantum Hall states [1]. Within a
LL, the strong Coulomb interaction dominates over kinetic energy in the highly
degenerated LL flatband, further conduces to the emergence of FQH states at
certain fractional fillings $\nu$ [2]. In most cases, the denominators of
$\nu$ are odd integers, which finds an explanation in that FQH states can be
understood effectively as integer quantum Hall states of composite fermions
[3]. An important exception to the odd-denominator rule is the 5/2 FQH state
found in the second LL of GaAs [4]. Based on extensive experimental and
theoretical investigations [5, 6, 7, 8, 9], the most probable explanation of
this state is the Moore-Read theory [10] and its extensions, which provide us
the Pfaffian, anti-Pfaffian, and particle-hole symmetric Pfaffian wave
functions as candidates [11, 12, 13, 14]. The elementary charged excitations
of these states obey non-Abelian braiding statistics and may be utilized to
perform fault-tolerant quantum computation that are topologically protected at
the fundamental level [15]. In recent years, even-denominator FQH states have
been observed in several other systems and some of them are also believed to
host non-Abelian anyons [16, 17, 18, 19, 20, 21, 22].

Fig. 1:  Band structure of TLG and its quantum Hall states. a, Schematic of
the TLG device with graphite top-gate and bottom-gate isolated by hBN. b, The
crystal structure of Bernal stacked TLG and the parameters in its tight-
binding model. c, The low-energy band structure of TLG without the
displacement field $D$ in the vicinity of the $\mathbf{K}_{+}$ valley. d, The
color map of the longitudinal conductance $\sigma_{xx}$ plotted versus carrier
density $n$ and magnetic field $B$ at 1.5 K and $D=0$ mV/nm. The filling
factors defined at $B$ = 14 T is given below the bottom axis. The diamond
pattern at $B\approx 5$ T is attributed to level crossings. A plethora of FQH
states are observed above 10 T. e, $\sigma_{xx}$ as a function of $\nu$ for
different temperatures at $B$ = 14 T and $D$ = 0 mV/nm. The filling factors
$\nu=2$, $\pm 6$ and $\pm 10$ are marked by green, yellow, and blue shaded
regions, respectively. In the range $-6{\leqslant}\nu{\leqslant}-6$, there are
two MLG levels and four BLG levels for each spin component (see Fig. 3f).
Fig. 2:  Even- and odd-denominator FQH states in the filling factor range
$\boldsymbol{-6{\leqslant}\nu{\leqslant}6}$. a, Top panel: the color map of
the longitudinal conductance $\sigma_{xx}$ plotted versus $\nu$ and the
displacement field $D$. The magnetic field is $B=14$ T and the base
temperature is 15 mK. FQH states with both even and odd denominators are
observed at various filling factors $\nu=p/q$ ($p$ and $q$ are integers) as
signified by minima of $\sigma_{xx}$. The FQH states exhibit intricate
evolution with $D$. Bottom panel: sketch map of the top panel in which the FQH
states are marked for clarity. Grey shaded regions label the integer quantum
Hall states. Red, blue, green lines label the FQH states with $q=2$, $3$, and
$5$, respectively. b-c, The line plots of $\sigma_{xx}$ and the Hall
conductance $\sigma_{xy}$ for $1{\leqslant}\nu{\leqslant}2$ with $B=16$ T and
$D=0$ mV/nm (b) and $4{\leqslant}\nu{\leqslant}5$ with $B=14$ T, $D=-153$
mV/nm (c). A few FQH states are indicated by the dashed lines and the
associated values of $\nu$ are given below. The short horizontal lines mark
the quantized plateau of $\sigma_{xy}$.

We focus on Bernal stacked trilayer graphene (TLG) here and report even-
denominator FQH states in this system for the first time as well as a plethora
of odd-denominator ones. In recent years, extensive investigations of
monolayer and bilayer graphene (MLG, BLG) [17, 18, 23, 19, 20], and
experimental attempts have also been made on TLG. However, FQH states in TLG
remained elusive with only tenuous traces speculated [24]. Compared to MLG and
BLG, TLG possesses richer and more delicate band structure tunability [25].
Under zero displacement field, the band structure of TLG can be decomposed to
a combination of MLG and BLG (with hopping parameters that are different from
the actual monolayer and bilayer systems), but this fact by no means implies
that the physics of TLG is a trivial repetition of MLG and BLG. In the
presence of a magnetic field, the LLs originate from the MLG and BLG parts are
not separated in energy but intersect with each other. If a vertical
displacement field is introduced, the decomposition is no longer valid as the
MLG and BLG parts hybridize. Each single-particle eigenstate is a
superposition of the solutions in the non-relativistic (NR) Landau problem and
its weights in different NR levels vary with external fields. The separations
between LLs can be tuned to generate many different orderings. For GaAs, the
NR second LL that hosts the $5/2$ state is sandwiched between the lowest and
third LLs. In contrast, one level in TLG that is similar to the NR second LL
may be surrounded from above and blow by other levels that have various
different orbital contents. The LL mixing between them is very sophisticated
and may results in intricate competition between strongly correlated states.

The structure of our TLG devices is depicted in Fig. 1a, where two graphite
gates are separated from TLG by insulating hBN (see Extended Data Fig. 1 for
optical images of our device). By applying voltages $V_{tg}$ on the top gate
and $V_{bg}$ on the bottom gate, the carrier density $n$ and the displacement
field $D$ can be tuned independently as: $n=(C_{b}V_{bg}+C_{t}V_{tg})/e$ and
$D=(C_{b}V_{bg}-C_{t}V_{tg})/2$, where $C_{b}$, $C_{t}$ are average geometric
capacitances for the bottom and top gates. The lattice structure of TLG is
shown in Fig. 1b together with the Slonczewski–Weiss–McClure (SWMc) parameters
in its tight-binding description [26]. The potential difference between the
top and bottom layers caused by $D$ is denoted as $2\Delta_{1}$. An additional
variable $\Delta_{2}$ was proposed to characterize the intrinsic charge
imbalance between the outer and middle layers [25]. The low-energy band of TLG
for $D=0$ is shown in Fig. 1c, where MLG-like linear and BLG-like quadratic
components can be discerned. When a perpendicular magnetic field $B$ is
applied, the linear and quadratic bands give rise to two sets of LLs that
scale as $\sqrt{B}$ and $B$, respectively. Some levels from these two sets may
intersect with each other when $B$ is varied. This can be seen in Fig. 1d that
presents the Landau fan diagram measured at $1.5$ K. A few level crossings are
observed around $B$ = 5 T for $2{\leqslant}\nu{\leqslant}6$, as was also
reported in previous works [27, 24, 28]. When the magnetic field increase to
11 T, minimal longitudinal conductance $\sigma_{xx}$ was observed at all
integer filling factors $-10{\leqslant}\nu{\leqslant}10$, which signifies a
complete lifting of the spin and valley degeneracies of the LLs by electron
correlation. Meanwhile, a variety of well-developed FQH states emerge. The
temperature dependence of $\sigma_{xx}$ at 14 T are displayed in Fig. 1e for
many FQH states with denominator $3$, whose thermal activation behaviour
clearly demonstrates their incompressible nature. An interesting feature is
the different onset magnetic field for some states in the range
$6{\leqslant}\nu{\leqslant}10$. FQH states at $\nu=19/3,23/3,25/3,29/3$ were
observed even before integer states appear at $\nu=7,9$ (see Extended Data
Fig. 2 for $\sigma_{xx}$ in the $\nu-B$ plane). In contrast, FQH states begin
to develop at $\nu=20/3,22/3,26/3,28/3$ near 10 T, at which point LL
degeneracies at $\nu=7,9$ has already been lifted. This phenomenon suggests
that some FQH states are intimately connected with the lifting of spin-valley
degeneracy at $\nu=7,9$.

Next we investigate the states with $-6{\leqslant}\nu{\leqslant}6$ at lower
temperatures in detail. The color map of $\sigma_{xx}$ versus $\nu$ and $D$ is
plotted in Fig. 2a with the left (right) panel of the top row showing the hole
(electron) side. The FQH states are sketched in the bottom row for clarity
(see Extended Data Fig. 3 for the complete map). For various filling factors
of the form $\widetilde{\nu}=\nu-[\nu]=s/(2s+1)$ and $1-s/(2s+1)$ ($[\nu]$ is
the greatest integer less than or equal to $\nu$ and $s=1,2$), the observed
minima of $\sigma_{xx}$ suggest the existence of FQH states. These states are
illustrated as blue and green lines in the sketch panel. Besides these odd-
denominator states, even-denominator FQH states were found at
$\nu=-9/2,-3/2,3/2,9/2$ and highlighted by red lines. The $\nu=-9/2$ and $3/2$
states can be realized at $D=0$ and remain stable over a wide range of $D$,
but the $\nu=-3/2$ and $9/2$ states only appears when a finite $D$ is applied.
In fact, the $\nu=9/2$ state can only be observed in a very narrow range of
$D$. We plot $\sigma_{xx}$ together with the Hall conductance $\sigma_{xy}$
around $\nu=3/2$ in Fig. 2b and the same quantities around $\nu=9/2$ in Fig.
2c. The concomitant appearance of exponentially suppressed $\sigma_{xx}$ and
quantized $\sigma_{xy}$ clearly demonstrate that FQH states are realized at
many different $\nu$. Interestingly, a weak minima can be seen at $\nu=20/13$,
which may be a composite fermion state or a Pfaffian daughter state [29]. As
shown in Extended Data Fig. 4, similar features are observed in the vicinity
of $\nu=-9/2,-3/2$. In contrast, there is no signature of daughter states
associated with $\nu=9/2$ but FQH states are observed clearly at
$\widetilde{\nu}=2/5$ and 3/5.

Fig. 3:  Evolution of FQH states with magnetic field. a-c, The color maps of
the longitudinal conductance $\sigma_{xx}$ plotted versus $B$ and $\nu$ at 15
mK. The displacement field $D$ is zero in a and c and 217 mV/nm in b. d,
$\sigma_{xx}$ as a function of $\nu$ at $B=16$ T. The vertical lines and the
numbers mark some FQH states. e, The energy gaps of the FQH states marked in d
deduced from thermal activation measurements. The data points are drawn as
blue rectangles with heights proportional to their fitting uncertainty. The
gaps for three odd-denominator states are fitted linearly using the red dash
line $\Delta=\hbar eB_{\rm eff}/m_{\rm CF}$ ($B_{\rm
eff}=(1-2\widetilde{\nu})B$ is the effective magnetic field for composite
fermions, and $m_{\rm CF}$ is composite fermion mass). Its inset shows the gap
at $\nu=-9/2$ for different $B$. f, The theoretically computed LLs with
$\Delta_{1}=\Delta_{2}=0$. The $\mathbf{K}_{+}$ ($\mathbf{K}_{-}$) valley is
represented using solid (dashed) lines. The scheme of six levels is shown on
the right of this panel. The MLG and BLG levels are marked by the subscripts
${\rm M/B}$ and their orbital contents are given by the numbers $0/1$. The
splitting between the two BLG levels in the same valley is denoted as
$\epsilon^{\pm}_{10}$. g, Schematic of the chiral graviton spectral functions
$I_{\pm}(\omega)$. Left panel: Chiral gravitons are excited when the relative
angular momentum of electron pairs $m$ is changed by $2$.
$I_{+}(\omega)(I_{-}(\omega))$ is the spectral function of the operator that
increases (decreases) $m$. Right panel: Particle-hole symmetry within one LL
ensures that $I_{+}(\omega)=I_{-}(\omega)$, but LL mixing causes asymmetry
between them. $\epsilon_{10}$ denotes the gap between the two LLs under
consideration.

For a fixed displacement field, the dependence of $\sigma_{xx}$ on $B$ around
$\nu=-9/2,-3/2,3/2$ is presented in Fig. 3a,b,c. The existence of odd-
denominator four-flux FQH states ($\nu=-32/7,-41/9,-12/7,-9/7,12/7$)
underscores the high quality of our sample. The three even-denominator FQH
states are robust in a considerable range of magnetic field. The line plot of
$\sigma_{xx}$ at $B=16$ T with $-5{\leqslant}\nu{\leqslant}-4$ is shown in
Fig. 3d on which a few FQH states are indicated. The energy gaps of some
states in Fig. 3d are deduced from their thermal activated behaviour and
presented in Fig. 3e (see Extended Data Fig. 5 for the fitting). For the odd-
denominator ones at $\widetilde{\nu}=2/5,3/7,4/9$, the data can be understood
using the composite fermion theory. A remarkable prediction of this theory is
that the energy gap decreases to zero linearly as the filling factor
$\widetilde{\nu}$ approaches 1/2 [30]. The gap values are fitted using
$\Delta=\hbar eB_{\rm eff}/m_{\rm CF}$ in Fig. 3e, where $B_{\rm
eff}=(1-2\widetilde{\nu})B$ is the effective magnetic field for composite
fermions and $m_{\rm CF}=1.99m_{e}$ is the composite fermion mass. It is
obvious that this rule is violated by the $\nu=-9/2$ FQH state whose energy
gap is well above zero. The inset of Fig. 3e shows the evolution of the gap at
$\nu=-9/2$ with $B$. It gets larger when $B$ increases as one would expect for
a state driven by interaction.

Fig. 4:  Displacement and magnetic fields tuning of FQH states in the filling
factor range $\boldsymbol{4{\leqslant}\nu{\leqslant}5}$. a-b, The color map of
longitudinal conductance $\sigma_{xx}$ plotted versus $D$ (with $B=14$ T fixed
in a) or $B$ (with $D=-153$ mV/nm fixed in b) and $\nu$ at 15 mK. Phase
transitions of FQH states are observed at $\nu=9/2,32/7,23/5$. c, The
temperature dependence of $\sigma_{xx}$ versus $\nu$ for $D=-153$ mV/nm and
$B=14$ T. Its inset shows the Arrhenius fitting of the thermal activation gap
at $\nu=9/2$.

To understand the even-denominator states, we first inspect the LLs of TLG
presented in Fig. 3f (spin is suppressed for simplicity). The parameter
$\Delta_{2}$ is fixed at zero in most parts of our discussion, but a small
value would not be detrimental either. In the absence of $D$, the LLs can be
labeled as MLG and BLG ones. For $B\;{\gtrsim}\;8$ T and
$-6{\leqslant}\nu{\leqslant}6$, there are three levels in each of the
$\mathbf{K}_{\pm}$ valley and they are labeled using $N_{\rm M/B}=0,1$. The
subscript ${\rm M/B}$ refers to their MLG or BLG origin and the numbers $0$
and $1$ characterize the orbital contents of the single-particle states. When
the MLG and BLG parts are hybridized by a nonzero $D$, the single-particle
eigenstates are six-dimensional vectors consist of NR Landau orbits. For one
eigenstate of the TLG LL, the largest weight may reside in the the NR lowest
LL, then it would be denoted as NR0. If there is a substantial weight in the
NR second LL, it would be denoted as NR1. The $\mathbf{K}_{\pm},N_{\rm B}=0$
levels are of the NR0 type and the $\mathbf{K}_{\pm},N_{\rm B}=1$ levels are
of the NR1 type. It is well-known that the NR second LL is favorable for
realizing FQH state at half filling, as exemplified by the 5/2 state in GaAs
[4]. Using this information, we can provide a simple picture for the
$\nu=-9/2$ and $3/2$ states at $D=0$. As indicated in Fig. 3f, electrons fill
all the LLs below $\mathbf{K}_{-},N_{\rm B}=0$ at $\nu=-6$. An extra $3/2$
filling of electrons are added to arrive at $\nu=-9/2$. The Zeemann splitting
is $E_{\rm Z}=1.62$ meV at 14 T whereas the separation $\epsilon^{-}_{10}$
between $\mathbf{K}_{-},N_{\rm B}=1$ and $\mathbf{K}_{-},N_{\rm B}=0$ is
$4.59$ meV. If there is no interaction, electrons would populate the
$\mathbf{K}_{-},N_{\rm B}=0$ level with both spin projections. However,
Coulomb interaction is likely to favor maximal spin polarization, so the
electrons may instead occupy the $\mathbf{K}_{-},N_{\rm B}=0$ and
$\mathbf{K}_{-},N_{\rm B}=1$ levels with spin down. This leads to a half-
filled NR1 level in which the Moore-Read type states may be realized.

This picture for $\nu=-9/2$ is corroborated using exact diagonalization
results on the torus. There are six quasi-degenerate ground states, which is
consistent with the prediction for the Moore-Read type states as well as
previous results in BLG [31, 32, 33, 34]. This result alone cannot tell us if
the state is of the Pfaffian, anti-Pfaffian, or particle-hole symmetric
Pfaffian type. To this end, we have computed the chiral graviton spectral
functions [35, 36]. As illustrated in Fig. 3g, these quantities are designed
to reveal the relative angular momentum of electron pairs. It has been shown
that the dominant chirality is negative (positive) for the Pfaffian (anti-
Pfaffian) wave function [36]. If we only keep the $\mathbf{K}_{-},N_{\rm B}=1$
in our calculation, particle-hole symmetry ensures that the two chiralities
are the same. After incorporating LL mixing with the $\mathbf{K}_{-},N_{\rm
B}=0$ level (and excluding all other levels), the Pfaffian state becomes the
favored one. This is consistent with the possible existence of a daughter
state at $\nu=-5+7/13$. More details are given in the Supplementary
Information. In general, when a NR0/NR1 doublet has $3/2$ filling, one may
expect to see an even-denominator FQH state. For the $\nu=3/2$ state, the
$\mathbf{K}_{+},N_{\rm B}=0$ and $\mathbf{K}_{+},N_{\rm B}=1$ levels are
partially occupied. It should also be of the Pfaffian type in view of the weak
minima at $\nu=20/13=1+7/13$. For the range of $B$ that have been studied, it
may be sufficient to keep only the two BLG levels, but we should keep in mind
that the MLG levels are not too far away in energy. If $B$ increases to about
29 T, the $\mathbf{K}_{+},N_{\rm B}=1$ level would be equidistant from
$\mathbf{K}_{+},N_{\rm B}=0$ and $\mathbf{K}_{+},N_{\rm M}=0$. This results in
exotic LL mixing that has not been explored in previous works and the fate of
the $3/2$ state would be very interesting. A higher magnetic field is require
to do the same thing for the $\mathbf{K}_{-}$ valley.

When the displacement field is turned on, the FQH states evolves in different
manners. The disappearance of the $\nu=-9/2$ and $3/2$ states can be explained
qualitatively as shown in Extended Data Fig. 6b. The $\mathbf{K}_{-},N_{\rm
B}=0$ level goes down when $D$ increases and eventually crosses with two NR2
type levels. In this regime, the $\nu=-9/2$ state would not correspond to a
half-filled NR1 type level, so no even-denominator FQH state is expected. The
experimental data in Fig. 2a indeed suggests that a level crossing occurs at
$D\approx 80$ mV/nm. The $\mathbf{K}_{+},N_{\rm B}=0$ and
$\mathbf{K}_{+},N_{\rm B}=1$ levels changes slowly with $D$, but they do cross
with each other at quite large $D$. Another important change is that the
weight of $\mathbf{K}_{+},N_{\rm B}=1$ in the NR second LL decreases. It is
thus plausible that the $3/2$ state gradually weakens as $\epsilon^{+}_{10}$
decreases, which is consistent with the experimental data in Fig. 2a. In
contrast to these two filling factors, a state emerges at $\nu=-3/2$ when $D$
becomes sufficiently large. For the case with $D=0$, the
$\mathbf{K}_{+},N_{\rm B}=0$ level with spin down is half filled and no FQH
state is expected. As $D$ increases, the $\mathbf{K}_{-},N_{\rm B}=1$ level
goes up slowly and eventually crosses with $\mathbf{K}_{+},N_{\rm B}=0$ at
large $D$. This crossing occurs at smaller $D$ if a small positive
$\Delta_{2}$ is invoked (see Extended Data Fig. 6c). After the crossing, the
electrons fully occupy the $\mathbf{K}_{\pm},N_{\rm B}=0$ levels with both
spin components at $\nu=-2$. The $-3/2$ state would correspond to a half
filled NR1 level and it is likely of the Pfaffian type given the weak feature
observed at $\nu=-2+7/13$ (see Extended Data Fig. 4).

Finally, we study the $\nu=9/2$ FQH state that only appears in a very small
window of $D$ and $B$. The color map of $\sigma_{xx}$ around $\nu=9/2$ is
presented in Fig. 4a,b (with one parameter fixed and the other varied). As the
$\nu=9/2$ state emerges, the gap at $\nu=23/5$ closes and then reopens. This
implies that a phase transition has occurred between two different $\nu=23/5$
states. The $\nu=32/7$ state simply disappears when the $\nu=9/2$ state is
observed. On the contrary, the states at $\nu=22/5$ and $31/7$ remain stable
and no transition is found. By fitting the thermal activation data in Fig. 4c,
the gap at $\nu=9/2$ is found to be $\sim 0.51$ K. For our tight-binding model
at $\Delta_{2}=0$, the active levels at $\nu=9/2$ are of the NR0 type, which
is unfavorable for realizing even-denominator states. If we change
$\Delta_{2}$ to -10 meV, the Pfaffian state would be realized at $\nu=9/2$.
This analysis is not very satisfactory because such a value of $\Delta_{2}$ is
not quite reasonable and it is difficult to explain why this state only appear
in a narrow range of $D$. To this end, we may consider multi-component FQH
states whose spin and/or valley indices are not polarized. The Halperin 331
state is a well-known two-component state with $\nu=1/2$ [37]. Another example
is the Jain state constructed from the parton theory [38, 39]. For a suitable
range of $D$, two NR0 type levels are almost degenerate. At $\nu=9/2$, two NR0
type levels with the same valley index are half filled. The interaction
between different valleys may be altered by valley anisotropic terms [40] such
that the Jain state is stabilized [41]. This mechanism was proposed to explain
the $1/2$ state observed in MLG [23], and a more detailed analysis is needed
to check if it also works for TLG.

In summary, our results unveil the rich odd- and even- denominator FQH states
in TLG and underscore the extraordinary tunability due to intricate interplay
of spin, valley, and orbital degrees of freedom. While the odd-denominator
states are most acceptably described by the composite fermion theory, other
candidates such as the Read-Rezayi states with non-Abelian Fibonacci anyons
[42] are also possible and deserve further investigations. On the other hand,
it would be fruitful to further explore the consequences brought by the
evolution of LL eigenstates and their mixing and crossings. By varying
external fields, we may switch between multiple Abelian and non-Abelian FQH
states as well as non-FQH states, and continuous quantum phase transitions may
be feasible in some of these process. Since FQH states are not described by
the Landau paradigm based on symmetry breaking, their transitions are most
likely not captured by the standard Landau-Ginzburg-Wilson theory [43]. The
low-energy effective theory of FQH states generally involve Chern-Simons gauge
fields [44], so there could be many exotic transitions described by strongly
coupled quantum field theory.

## References

  * Klitzing  _et al._ [1980] K. v. Klitzing, G. Dorda, and M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance, Phys. Rev. Lett. 45, 494 (1980).
  * Tsui  _et al._ [1982] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two-dimensional magnetotransport in the extreme quantum limit, Phys. Rev. Lett. 48, 1559 (1982).
  * Jain [1989a] J. K. Jain, Composite-fermion approach for the fractional quantum hall effect, Phys. Rev. Lett. 63, 199 (1989a).
  * Willett  _et al._ [1987] R. Willett, J. P. Eisenstein, H. L. Störmer, D. C. Tsui, A. C. Gossard, and J. H. English, Observation of an even-denominator quantum number in the fractional quantum hall effect, Phys. Rev. Lett. 59, 1776 (1987).
  * Halperin and Jain [2020] B. I. Halperin and J. K. Jain, eds.,  _Fractional Quantum Hall Effects: New Developments_ (World Scientific Publishing, 2020).
  * Radu  _et al._ [2008] I. P. Radu, J. B. Miller, C. M. Marcus, M. A. Kastner, L. N. Pfeiffer, and K. W. West, Quasi-particle properties from tunneling in the $\nu=5/2$ fractional quantum hall state, Science 320, 899 (2008).
  * Venkatachalam  _et al._ [2011] V. Venkatachalam, A. Yacoby, L. Pfeiffer, and K. West, Local charge of the $\nu=5/2$ fractional quantum hall state, Nature 469, 185 (2011).
  * Banerjee  _et al._ [2018] M. Banerjee, M. Heiblum, V. Umansky, D. E. Feldman, Y. Oreg, and A. Stern, Observation of half-integer thermal hall conductance, Nature 559, 205 (2018).
  * Dutta  _et al._ [2022] B. Dutta, W. Yang, R. Melcer, H. K. Kundu, M. Heiblum, V. Umansky, Y. Oreg, A. Stern, and D. Mross, Distinguishing between non-abelian topological orders in a quantum hall system, Science 375, 193 (2022).
  * Moore and Read [1991] G. Moore and N. Read, Nonabelions in the fractional quantum hall effect, Nucl. Phys. B 360, 362 (1991).
  * Lee  _et al._ [2007] S.-S. Lee, S. Ryu, C. Nayak, and M. P. A. Fisher, Particle-hole symmetry and the $\nu=5/2$ quantum hall state, Phys. Rev. Lett. 99, 236807 (2007).
  * Levin  _et al._ [2007] M. Levin, B. I. Halperin, and B. Rosenow, Particle-hole symmetry and the pfaffian state, Phys. Rev. Lett. 99, 236806 (2007).
  * Son [2015] D. T. Son, Is the composite fermion a dirac particle?, Phys. Rev. X 5, 031027 (2015).
  * Zucker and Feldman [2016] P. T. Zucker and D. E. Feldman, Stabilization of the particle-hole pfaffian order by landau-level mixing and impurities that break particle-hole symmetry, Phys. Rev. Lett. 117, 096802 (2016).
  * Nayak  _et al._ [2008] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, Non-abelian anyons and topological quantum computation, Rev. Mod. Phys. 80, 1083 (2008).
  * Falson  _et al._ [2015] J. Falson, D. Maryenko, B. Friess, D. Zhang, Y. Kozuka, A. Tsukazaki, J. H. Smet, and M. Kawasaki, Even-denominator fractional quantum hall physics in zno, Nat. Phys. 41, 347 (2015).
  * Zibrov  _et al._ [2017] A. A. Zibrov, C. R. Kometter, H. Zhou, E. M. Spanton, T. Taniguchi, K. Watanabe, M. P. Zaletel, and A. F. Young, Tunable interacting composite fermion phases in a half-filled bilayer-graphene landau level, Nature 549, 360 (2017).
  * Li  _et al._ [2017] J. I. A. Li, C. Tan, S. Chen, Y. Zeng, T. Taniguchi, K. Watanabe, J. Hone, and C. R. Dean, Even denominator fractional quantum hall state in bilayer graphene, Science 358, 648 (2017).
  * Kim  _et al._ [2018] Y. Kim, A. C. Balram, T. Taniguchi, K. Watanabe, J. K. Jain, and J. H. Smet, Even denominator fractional quantum hall states in higher landau levels of graphene, Nature Physics 15, 154 (2018).
  * Huang  _et al._ [2022] K. Huang, H. Fu, D. R. Hickey, N. Alem, X. Lin, K. Watanabe, T. Taniguchi, and J. Zhu, Valley isospin controlled fractional quantum hall states in bilayer graphene, Phys. Rev. X 12, 031019 (2022).
  * Shi  _et al._ [2020] Q. Shi, E.-M. Shih, M. V. Gustafsson, D. A. Rhodes, B. Kim, K. Watanabe, T. Taniguchi, Z. Papić, J. Hone, and C. R. Dean, Odd- and even-denominator fractional quantum hall states in monolayer wse2, Nat. Nanotechnol. 15, 569 (2020).
  * Hossain  _et al._ [2023] M. S. Hossain, M. K. Ma, Y. J. Chung, S. K. Singh, A. Gupta, K. W. West, K. W. Baldwin, L. N. Pfeiffer, R. Winkler, and M. Shayegan, Valley-tunable even-denominator fractional quantum hall state in the lowest landau level of an anisotropic system, Phys. Rev. Lett. 130, 126301 (2023).
  * Zibrov  _et al._ [2018] A. A. Zibrov, E. M. Spanton, H. Zhou, C. Kometter, T. Taniguchi, K. Watanabe, and A. F. Young, Even-denominator fractional quantum hall states at an isospin transition in monolayer graphene, Nat. Phys. 14, 930 (2018).
  * Stepanov  _et al._ [2016] P. Stepanov, Y. Barlas, T. Espiritu, S. Che, K. Watanabe, T. Taniguchi, D. Smirnov, and C. N. Lau, Tunable symmetries of integer and fractional quantum hall phases in heterostructures with multiple dirac bands, Phys. Rev. Lett. 117, 076807 (2016).
  * Serbyn and Abanin [2013] M. Serbyn and D. A. Abanin, New dirac points and multiple landau level crossings in biased trilayer graphene, Phys. Rev. B 87, 115422 (2013).
  * Koshino and McCann [2011] M. Koshino and E. McCann, Landau level spectra and the quantum hall effect of multilayer graphene, Phys. Rev. B 83, 165443 (2011).
  * Taychatanapat  _et al._ [2011] T. Taychatanapat, K. Watanabe, and P. Taniguchi, Takashiand Jarillo-Herrero, Quantum hall effect and landau-level crossing of dirac fermions in trilayer graphene, Nat. Phys. 7, 621 (2011).
  * Datta  _et al._ [2017] B. Datta, S. Dey, A. Samanta, H. Agarwal, A. Borah, K. Watanabe, T. Taniguchi, R. Sensarma, and M. M. Deshmukh, Strong electronic interaction and multiple quantum hall ferromagnetic phases in trilayer graphene, Nat. Comm. 8, 14518 (2017).
  * Levin and Halperin [2009] M. Levin and B. I. Halperin, Collective states of non-abelian quasiparticles in a magnetic field, Phys. Rev. B 79, 205301 (2009).
  * Du  _et al._ [1993] R. R. Du, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, and K. W. West, Experimental evidence for new particles in the fractional quantum hall effect, Phys. Rev. Lett. 70, 2944 (1993).
  * Apalkov and Chakraborty [2011] V. M. Apalkov and T. Chakraborty, Stable pfaffian state in bilayer graphene, Phys. Rev. Lett. 107, 186803 (2011).
  * Papić  _et al._ [2011] Z. Papić, R. Thomale, and D. A. Abanin, Tunable electron interactions and fractional quantum hall states in graphene, Phys. Rev. Lett. 107, 176602 (2011).
  * Snizhko  _et al._ [2012] K. Snizhko, V. Cheianov, and S. H. Simon, Importance of interband transitions for the fractional quantum hall effect in bilayer graphene, Phys. Rev. B 85, 201415 (2012).
  * Zhu  _et al._ [2020] Z. Zhu, D. N. Sheng, and I. Sodemann, Widely tunable quantum phase transition from moore-read to composite fermi liquid in bilayer graphene, Phys. Rev. Lett. 124, 097604 (2020).
  * Liou  _et al._ [2019] S.-F. Liou, F. D. M. Haldane, K. Yang, and E. H. Rezayi, Chiral gravitons in fractional quantum hall liquids, Phys. Rev. Lett. 123, 146801 (2019).
  * Haldane  _et al._ [2021] F. D. M. Haldane, E. H. Rezayi, and K. Yang, Graviton chirality and topological order in the half-filled landau level, Phys. Rev. B 104, L121106 (2021).
  * Halperin [1983] B. I. Halperin, Theory of the quantized hall conductance, Helv. Phys. Acta 56, 75 (1983).
  * Jain [1989b] J. K. Jain, Incompressible quantum hall states, Phys. Rev. B 40, 8079 (1989b).
  * Moran  _et al._ [2012] N. Moran, A. Sterdyniak, I. Vidanović, N. Regnault, and M. V. Milovanović, Topological $d$-wave pairing structures in jain states, Phys. Rev. B 85, 245307 (2012).
  * Kharitonov [2012] M. Kharitonov, Phase diagram for the $\nu=0$ quantum hall state in monolayer graphene, Phys. Rev. B 85, 155439 (2012).
  * Wu [2022] Y.-H. Wu, Two-component parton fractional quantum hall state in graphene, Phys. Rev. B 106, 155132 (2022).
  * Read and Rezayi [1999] N. Read and E. Rezayi, Beyond paired quantum hall states: Parafermions and incompressible states in the first excited landau level, Phys. Rev. B 59, 8084 (1999).
  * Sachdev [2011] S. Sachdev,  _Quantum Phase Transitions_ (Cambridge University Press, Cambridge, 2011).
  * Wen [1995] X.-G. Wen, Topological orders and edge excitations in fractional quantum hall states, Advances in Physics 44, 405 (1995).
  * Wang  _et al._ [2013] L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, One-dimensional electrical contact to a two-dimensional material, Science 342, 614 (2013).
  * Yang  _et al._ [2017] B. Yang, Z.-X. Hu, C. H. Lee, and Z. Papić, Generalized pseudopotentials for the anisotropic fractional quantum hall effect, Phys. Rev. Lett. 118, 146403 (2017).
  * Gagliano and Balseiro [1987] E. R. Gagliano and C. A. Balseiro, Dynamical properties of quantum many-body systems at zero temperature, Phys. Rev. Lett. 59, 2999 (1987).

## Methods

The devices were fabricated using our “pick-up method” [45] to achieve a
multi-layer heterostructure with the TLG encapsulated by two flakes of
hexagonal boron nitride (hBN) and thin graphite flakes as the top and bottom
gates. The stacks were annealed under a high vacuum at 350 ∘C for 2 hours.
Electron-beam lithography was used to write a etch mask to define the Hall-bar
geometry and the electrodes. Redundant regions were etched away by CHF3/O2
plasma [45]. Finally the TLG and gates were edge-contacted [45] by e-beam
evaporating thin metal layers consisting of Cr/Pd/Au (1 nm/15 nm/100 nm).

The transport measurements were performed in two systems, a dilution fridge
with a base temperature of 15 mK and a VTI fridge down to 1.5 K, and both are
with superconducting magnets. All data were taken using the standard four-
terminal configuration with lock-in amplifier techniques by sourcing an AC
current $I$ between 10 and 100 nA at a frequency of 17.777 Hz. The data of
longitudinal conductance $\sigma_{xx}$ and Hall conductance $\sigma_{xy}$ are
obtained from the measured resistances by
$\sigma_{xx}=\rho_{xx}/(\rho_{xx}^{2}+R_{xy}^{2})$ and
$\sigma_{xy}=R_{xy}/(\rho_{xx}^{2}+R_{xy}^{2})$.

## Data availability

The data that support the findings of this study are available from the
corresponding authors upon request.

## Code availability

The code that support the findings of this study are available from the
corresponding authors upon request.

## Acknowledgments

L.W. acknowledges support from the National Key Projects for Research and
Development of China (Grant Nos. 2021YFA1400400, 2022YFA1204700), National
Natural Science Foundation of China (Grant No. 12074173) and Natural Science
Foundation of Jiangsu Province (Grant No. BK20220066). Y.H.W. acknowledges
support from the National Natural Science Foundation of China (Grant No.
12174130). Z.Z. acknowledges the National Natural Science Foundation of China
(Grant No.12074375), the Strategic Priority Research Program of CAS (Grant
No.XDB33000000). K.W. and T.T. acknowledge support from the JSPS KAKENHI
(Grant Numbers 21H05233 and 23H02052) and World Premier International Research
Center Initiative (WPI), MEXT, Japan. F.C.Z. acknowledges partial support from
China Ministry of Sci and Tech (grant 2022YFA1403902), Priority Program of CAS
grant No XDB28000000, NSFC grant No 11674278, and Chinese Academy of Sciences
under contract No. JZHKYPT-2021-08. G.L. acknowledges the support by the NSFC
of China (grant No. 9206520), the National Basic Research Program of China
from the MOST (grant No. 2022YFA1602803), and the Strategic Priority Research
Program of the Chinese Academy of Sciences (grant No. XDB33010300). This work
is supported by the Synergic Extreme Condition User Facility and by the
Innovation Program for Quantum Science and Technology (Grant No.
2021ZD03026001).

## Author Contributions

L.W. conceived and designed the experiment. Y.C., Y.H., and Q.L. fabricated
the samples. Y.C., Y.H., Q.L., B.T., G.L., and L.L. performed the transport
measurements. Y.C., L.W., Y.H.W., F.C.Z., and Z.Z analyzed the data. Y.H.W.
conducted theoretical analysis. K.W. and T.T supplied the hBN crystals. Y.C.,
Y.H.W. and L.W. wrote the manuscript with input from all co-authors.

## Competing interests

The authors declare no competing interest.

Extended Data Fig. 1:  Optical image of one typical device. a, The optical
image of the released stack after transferring. The blue (black) line denotes
the boundary of top (bottom) graphite and the red line denotes the boundary of
trilayer graphene (TLG). The scalebar corresponds to 10 $\mu$m. b, The
completed device with Hall bar geometry that avoids the region with bubbles,
after the deposition of Au/Pd/Cr contacts. The scalebar corresponds to 5
$\mu$m.

Extended Data Fig. 2:  Quantum Hall states in the range
$\boldsymbol{0{\leqslant}\nu{\leqslant}10}$. The color map of $\sigma_{xx}$
measured at at $D=0$ mV/nm and 1.5 K plotted versus $\nu$ and $B$. The
$\nu=19/3,23/3$ states appear at a lower $B$ compared to the integer quantum
Hall state $\nu=7$, while the $\nu=20/3,22/3$ states appear after it.

Extended Data Fig. 3:  FQH states in the range
$\boldsymbol{-6{\leqslant}\nu{\leqslant}6}$. a, The hole side. b, The electron
side. Both panels are measured at $B=14$ T and 15 mK.

Extended Data Fig. 4:  Evidence of the Levin-Halperin daughter states. a-b,
$\sigma_{xx}$ and $\sigma_{xy}$ versus $\nu$ in the range
$-5{\leqslant}\nu{\leqslant}-4$ ($B=14.9$ T and $D=0$ mV/nm in a) and
$-2{\leqslant}\nu{\leqslant}-1$ ($B=14.6$ T and $D=217$ mV/nm in b). Both
panels exhibit a small dip in $\sigma_{xx}$ at $\nu-[\nu]=7/13$ that may be
interpreted as one of the Levin-Halperin daughter states.  Extended Data Fig.
5:  Arrhenius fitting for the gaps of FQH states in the range
$\boldsymbol{-5{\leqslant}\nu{\leqslant}-4}$. a, The linecuts of $R_{xx}$
versus $\nu$ at different temperatures $T$. b, The Arrhenius fittings for
$\nu=-5+2/5,3/7,4/9,1/2,2/3$ using the formula $R_{xx}\sim\exp(-\Delta/2T)$.
Extended Data Fig. 6:  LL diagrams obtained by numerical calculations. a, The
same diagram as in Fig. 3f of the main text but plotted in a different way
here for comparison. The solid (dashed) lines represent the $\mathbf{K}_{+}$
($\mathbf{K}_{-}$) valley. The orbital content of a level is indicated as
NR$x$ if it has a substantial weight in the non-relativistic LL with index
$x$. b, The evolution of LLs with $\Delta_{1}$ when $B=14$ T and
$\Delta_{2}$=0. The MLG and BLG levels in panel a are hybridized by
$\Delta_{1}$. The green $\mathbf{K}_{\pm},N_{\rm M}=0$ levels cross with each
other at $\Delta_{1}\approx 17$ meV. The blue $\mathbf{K}_{+},N_{\rm B}=0$
level and the red $\mathbf{K}_{+},N_{\rm B}=1$ level cross with each other at
$\Delta_{1}\approx 40$ meV. The blue $\mathbf{K}_{-},N_{\rm B}=0$ level
crosses with two NR2 levels at $\Delta_{1}\approx 40$ meV. c, The evolution of
LLs with $\Delta_{2}$ when $B=14$ T and $\Delta_{1}=17.31$ meV. One NR1 level
at $\Delta_{2}=-10$ meV that may host the $\nu=9/2$ state is indicated. One
NR1 level in the $\mathbf{K}_{-}$ valley crosses with one NR0 level in the
$\mathbf{K}_{+}$ valley at $\Delta_{2}\approx 3.5$ meV. More details about
these calculations are given in the Supplementary Information.

Supplementary Information for

Tunable even- and odd-denominator fractional quantum Hall states

in trilayer graphene

Yiwei Chen et al.

This Supplementary Information provides more details about the theoretical
analysis of fractional quantum Hall states in trilayer graphene.

##  I Models

To begin with, we define the single-particle tight-binding model for trilayer
graphene (TLG). There are two sublattice sites $A$ and $B$ in each layer of
TLG and their layer indices are appended as subscripts $1,2,3$. The distance
between Carbon atoms is $a=0.142$ nm and the lattice constant is
$\widetilde{a}=\sqrt{3}a$. As illustrated in Fig. 1 of the main text, six
hopping constants $\gamma_{i}$ ($i=0,1,\cdots,5$) are considered in the
Slonczewski-Weiss-McClure (SWMc) parametrization. The distance between layers
along the vertical direction is $d=0.335$ nm. An additional parameter $\delta$
is used to account for the onsite potential of $B_{1}$, $A_{2}$, and $B_{3}$
lattice sites. If a displacement field is present, the potential difference
between the top and bottom layer is quantified by $\Delta_{1}$. Based on
symmetry consideration, it was proposed another parameter $\Delta_{2}$ should
be introduced to describe the deviation of potential on the middle layer from
the mean of the potentials on top and bottom layers. In principle,
$\Delta_{2}$ could be nonzero even when $\Delta_{1}$ is zero. Let us choose
the basis to be $A_{1},B_{1},A_{2},B_{2},A_{3},B_{3}$. The tight-binding
Hamiltonian is

$\displaystyle\mathcal{H}=\left[\begin{matrix}\Delta_{1}+\Delta_{2}&\gamma_{0}t^{*}(\mathbf{k})&\gamma_{4}t^{*}(\mathbf{k})&\gamma_{3}t(\mathbf{k})&\gamma_{2}/2&0\\\
\gamma_{0}t(\mathbf{k})&\delta+\Delta_{1}+\Delta_{2}&\gamma_{1}&\gamma_{4}t^{*}(\mathbf{k})&0&\gamma_{5}/2\\\
\gamma_{4}t(\mathbf{k})&\gamma_{1}&\delta-2\Delta_{2}&\gamma_{0}t^{*}(\mathbf{k})&\gamma_{4}t(\mathbf{k})&\gamma_{1}\\\
\gamma_{3}t^{*}(\mathbf{k})&\gamma_{4}t(\mathbf{k})&\gamma_{0}t(\mathbf{k})&-2\Delta_{2}&\gamma_{3}t^{*}(\mathbf{k})&\gamma_{4}t(\mathbf{k})\\\
\gamma_{2}/2&0&\gamma_{4}t^{*}(\mathbf{k})&\gamma_{3}t(\mathbf{k})&-\Delta_{1}+\Delta_{2}&\gamma_{0}t^{*}(\mathbf{k})\\\
0&\gamma_{5}/2&\gamma_{1}&\gamma_{4}t^{*}(\mathbf{k})&\gamma_{0}t(\mathbf{k})&\delta-\Delta_{1}+\Delta_{2}\end{matrix}\right]$
(S1)

where
$t(\mathbf{k})=-1-2\cos(k_{x}\widetilde{a}/2)\exp(i\sqrt{3}k_{y}\widetilde{a}/2)$
is a summation over nearest neighbors. We adopt the parameters
$\gamma_{0}=3.1$, $\gamma_{1}=0.39$, $\gamma_{2}=-0.028$, $\gamma_{3}=0.315$,
$\gamma_{4}=0.041$, $\gamma_{5}=0.05$, and $\delta=0.046$ (all in units of eV)
for our calculations [25]. The band structure with $\Delta_{1}=\Delta_{2}=0$
is presented in Fig. 1 of the main text.

Next we turn to the Landau levels (LLs) of TLG. The energy bands have two
valleys in the hexagonal Brillouin zone with momentum
$\mathbf{K}_{\pm}=({\pm}4\pi/3,0)$. In the vicinity of these valleys, the
Hamiltonian can be expanded to yield

$\displaystyle\mathcal{H}_{\mathbf{K}_{+}}=$
$\displaystyle\left[\begin{matrix}\Delta_{1}+\Delta_{2}&v_{0}\pi^{-}&v_{4}\pi^{-}&v_{3}\pi^{+}&\gamma_{2}/2&0\\\
v_{0}\pi^{+}&\delta+\Delta_{1}+\Delta_{2}&\gamma_{1}&v_{4}\pi^{-}&0&\gamma_{5}/2\\\
v_{4}\pi^{+}&\gamma_{1}&\delta-2\Delta_{2}&v_{0}\pi^{-}&v_{4}\pi^{+}&\gamma_{1}\\\
v_{3}\pi^{-}&v_{4}\pi^{+}&v_{0}\pi^{+}&-2\Delta_{2}&v_{3}\pi^{-}&v_{4}\pi^{+}\\\
\gamma_{2}/2&0&v_{4}\pi^{-}&v_{3}\pi^{+}&-\Delta_{1}+\Delta_{2}&v_{0}\pi^{-}\\\
0&\gamma_{5}/2&\gamma_{1}&v_{4}\pi^{-}&v_{0}\pi^{+}&\delta-\Delta_{1}+\Delta_{2}\end{matrix}\right]$
(S2)

and

$\displaystyle\mathcal{H}_{\mathbf{K}_{-}}=$
$\displaystyle\left[\begin{matrix}\Delta_{1}+\Delta_{2}&-v_{0}\pi^{+}&-v_{4}\pi^{+}&-v_{3}\pi^{-}&\gamma_{2}/2&0\\\
-v_{0}\pi^{-}&\delta+\Delta_{1}+\Delta_{2}&\gamma_{1}&-v_{4}\pi^{+}&0&\gamma_{5}/2\\\
-v_{4}\pi^{-}&\gamma_{1}&\delta-2\Delta_{2}&-v_{0}\pi^{+}&-v_{4}\pi^{-}&\gamma_{1}\\\
-v_{3}\pi^{+}&-v_{4}\pi^{-}&-v_{0}\pi^{-}&-2\Delta_{2}&-v_{3}\pi^{+}&-v_{4}\pi^{-}\\\
\gamma_{2}/2&0&-v_{4}\pi^{+}&-v_{3}\pi^{-}&-\Delta_{1}+\Delta_{2}&-v_{0}\pi^{+}\\\
0&\gamma_{5}/2&\gamma_{1}&-v_{4}\pi^{+}&-v_{0}\pi^{-}&\delta-\Delta_{1}+\Delta_{2}\end{matrix}\right]$
(S3)

where ${\hbar}v_{i}=3a\gamma_{i}/2$, $\pi^{-}=\hbar(k_{x}-ik_{y})$, and
$\pi^{+}=\hbar(k_{x}+ik_{y})$. For our purpose, it is convenient to change the
basis to

$\displaystyle\frac{A_{1}-A_{3}}{\sqrt{2}},\frac{B_{1}-B_{3}}{\sqrt{2}},\frac{A_{1}+A_{3}}{\sqrt{2}},\frac{B_{1}+B_{3}}{\sqrt{2}},A_{2},B_{2}$
(S4)

such that $\mathcal{H}_{\mathbf{K}_{\pm}}$ has the block form

$\displaystyle\left[\begin{matrix}\mathcal{H}_{\rm MLG}&\mathcal{H}_{\rm
mix}\\\ \mathcal{H}^{{\dagger}}_{\rm mix}&\mathcal{H}_{\rm BLG}\\\
\end{matrix}\right]$ (S5)

with a two-dimensional monolayer graphene (MLG) part $\mathcal{H}_{\rm MLG}$,
a four-dimensional bilayer graphene (BLG) part $\mathcal{H}_{\rm BLG}$, and a
mixing part

$\displaystyle\mathcal{H}_{\rm mix}=\left[\begin{matrix}\Delta_{1}&0&0&0\\\
0&\Delta_{1}&0&0\\\ \end{matrix}\right]$ (S6)

This representation clearly shows that the TLG Hamiltonian can be decomposed
to a combination of MLG and BLG parts in the absence of $\Delta_{1}$. In the
$\mathbf{K}_{+}$ valley, we have

$\displaystyle\mathcal{H}^{+}_{\rm
MLG}=\left[\begin{matrix}-\frac{\gamma_{2}}{2}+\Delta_{2}&v_{0}\pi^{-}\\\
v_{0}\pi^{+}&-\frac{\gamma_{5}}{2}+\delta+\Delta_{2}\end{matrix}\right]\qquad\mathcal{H}^{+}_{\rm
BLG}=\left[\begin{matrix}\frac{\gamma_{2}}{2}+\Delta_{2}&v_{0}\pi^{-}&-\sqrt{2}v_{4}\pi^{-}&\sqrt{2}v_{3}\pi^{+}\\\
v_{0}\pi^{+}&\frac{\gamma_{5}}{2}+\delta+\Delta_{2}&\sqrt{2}\gamma_{1}&-\sqrt{2}v_{4}\pi^{-}\\\
-\sqrt{2}v_{4}\pi^{+}&\sqrt{2}\gamma_{1}&\delta-2\Delta_{2}&v_{0}\pi^{-}\\\
\sqrt{2}v_{3}\pi^{-}&-\sqrt{2}v_{4}\pi^{+}&v_{0}\pi^{+}&-2\Delta_{2}\end{matrix}\right].$
(S7)

In the $\mathbf{K}_{-}$ valley, we have

$\displaystyle\mathcal{H}^{-}_{\rm
MLG}=\left[\begin{matrix}-\frac{\gamma_{2}}{2}+\Delta_{2}&-v_{0}\pi^{+}\\\
-v_{0}\pi^{-}&-\frac{\gamma_{5}}{2}+\delta+\Delta_{2}\end{matrix}\right]\qquad\mathcal{H}^{-}_{\rm
BLG}=\left[\begin{matrix}\frac{\gamma_{2}}{2}+\Delta_{2}&-v_{0}\pi^{+}&\sqrt{2}v_{4}\pi^{+}&-\sqrt{2}v_{3}\pi^{-}\\\
-v_{0}\pi^{-}&\frac{\gamma_{5}}{2}+\delta+\Delta_{2}&\sqrt{2}\gamma_{1}&\sqrt{2}v_{4}\pi^{+}\\\
\sqrt{2}v_{4}\pi^{-}&\sqrt{2}\gamma_{1}&\delta-2\Delta_{2}&-v_{0}\pi^{+}\\\
-\sqrt{2}v_{3}\pi^{+}&\sqrt{2}v_{4}\pi^{-}&-v_{0}\pi^{-}&-2\Delta_{2}\end{matrix}\right].$
(S8)

If we turn on a perpendicular magnetic field generated by the vector potential
$\mathbf{A}$, miniaml coupling is achieved by the substitution

$\displaystyle\pi^{-}\rightarrow\Pi^{-}=\pi^{-}-e(A_{x}-iA_{y}),\quad\pi^{+}\rightarrow\Pi^{+}=\pi^{+}-e(A_{x}+iA_{y}).$
(S9)

In the non-relativistic Landau problem, interlevel ladder operators are
defined as

$\displaystyle\widehat{\mathsf{a}}=\frac{\ell_{B}}{\sqrt{2}\hbar}\Pi^{-}\qquad\widehat{\mathsf{a}}^{{\dagger}}=\frac{\ell_{B}}{\sqrt{2}\hbar}\Pi^{+}$
(S10)

where $\ell_{B}=\sqrt{\hbar/(eB)}$ is the magnetic length. For our TLG
Hamiltonian $\mathcal{H}_{\mathbf{K}^{\pm}}$, this amounts to the replacement

$\displaystyle\gamma_{i}\Pi^{-}\rightarrow\frac{3\gamma_{i}a}{\sqrt{2}\ell_{B}}\mathsf{a},\quad\gamma_{i}\Pi^{+}\rightarrow\frac{3\gamma_{i}a}{\sqrt{2}\ell_{B}}\mathsf{a}^{{\dagger}}.$
(S11)

The number operator $\widehat{\mathsf{a}}^{{\dagger}}\widehat{\mathsf{a}}$ can
be defined as usual, and its eigenstates $|n\rangle$ satisfy

$\displaystyle\widehat{\mathsf{a}}|n\rangle=\sqrt{n}|n-1\rangle,\quad\widehat{\mathsf{a}}^{{\dagger}}|n\rangle=\sqrt{n+1}|n+1\rangle$
(S12)

The eigenstates of $\mathcal{H}_{\mathbf{K}_{\pm}}$ can be expressed as

$\displaystyle\begin{bmatrix}f_{00}|0\rangle+f_{01}|1\rangle+f_{02}|2\rangle+\ldots\\\
f_{10}|0\rangle+f_{11}|1\rangle+f_{12}|2\rangle+\ldots\\\
f_{20}|0\rangle+f_{21}|1\rangle+f_{22}|2\rangle+\ldots\\\
f_{30}|0\rangle+f_{31}|1\rangle+f_{32}|2\rangle+\ldots\\\
f_{40}|0\rangle+f_{41}|1\rangle+f_{42}|2\rangle+\ldots\\\
f_{50}|0\rangle+f_{51}|1\rangle+f_{52}|2\rangle+\ldots\\\ \end{bmatrix},$
(S13)

where each row of this spinor is an infinite summation and cannot be computed
exactly. In our calculation, a proper truncation is imposed such that the
summation terminates at a finite $n$. If we are only interested in the
eigenvalues close to zero, a moderate $n$ would be sufficient. For example,
the Landau level diagrams are obtained for $n=19$. On the other hand, the
many-body problem uses $n=7$.

It should be emphasized that some aspects of the energy spectrum depend
sensitively on the system parameters. A precise determination of the
parameters is difficult, but a certain degree of consensus has emerged in the
past decade. In the absence of $\Delta_{1}$, the LLs can be labeled as MLG and
BLG ones as we have done in Fig. 3f of the main text. More generally, when the
MLG and BLG parts are hybridized by nonzero $\Delta_{1}$, it is more useful to
inspect the orbital content of each level. If $|0\rangle$ has the largest
weight in Eq. S13, it would be denoted as non-relativistic 0 (NR0). If
$|1\rangle$ has a substantial weight in Eq. S13, it would be denoted as non-
relativistic 1 (NR1). Naturally, the $\mathbf{K}_{\pm},N_{\rm B}=0$ levels are
of the NR0 type and the $\mathbf{K}_{\pm},N_{\rm B}=1$ levels are of the NR1
type. The effects of changing the parameters shall be discussed when we study
the many-body problem. The Zeemann coupling is $E_{\rm
Z}=g\mu_{B}B=0.058g\;B[\text{Tesla}]\;\text{meV}$ with $g=2$. It is $1.62$ meV
at $B=14$ T and is smaller than the energy separation between the NR0 and NR1
levels. The hexagonal boron-nitride has dielectric constants
$\varepsilon^{\perp}_{\rm BN}=3$ in the perpendicular direction and
$\varepsilon^{\parallel}_{\rm BN}=6.6$ within the two-dimensional plane. The
interaction between electrons is the screened Coulomb potential

$\displaystyle V_{\rm
SC}(\mathbf{q})=\frac{e^{2}}{4\pi\varepsilon_{0}\varepsilon^{\parallel}_{\rm
BN}\ell_{B}}\frac{2\pi}{q}\tanh(qd)=\frac{2\pi}{q}\tanh(qd)\;\frac{56.2}{\varepsilon^{\parallel}_{\rm
BN}}\;\sqrt{B[\text{Tesla}]}\;\text{meV},$ (S14)

when $d$ is the distance between top and bottom graphite gates.

##  II Methods

In our numerical calculation, electrons are placed on a rectangular torus
whose two sides are described by the vectors

$\displaystyle\mathbf{L}_{1}=L_{1}\widehat{e}_{v}\qquad\mathbf{L}_{2}=L_{2}\widehat{e}_{y}$
(S15)

The flux magnetic flux $N_{\phi}$ through the torus is quantized to satsify
$L_{1}L_{2}=2\pi\ell^{2}_{B}N_{\phi}$. Let us consider non-relativistic
electrons with anisotropic mass as described by the single-particle
Hamiltonian

$\displaystyle\mathcal{H}_{\rm NR}(\Lambda)=\frac{1}{2M}\left[\Lambda
p^{2}_{x}+\frac{1}{\Lambda}\left(p_{y}-eBx\right)^{2}\right].$ (S16)

The single-particle wave functions in the LLs are

$\displaystyle\phi^{\alpha}_{m}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\frac{1}{\left(2^{\alpha}\alpha!\sqrt{\pi}\ell_{B}L_{2}\sqrt{\Lambda}\right)^{1/2}}\sum^{\mathbb{Z}}_{k}\exp\left\\{-\frac{1}{2\Lambda}\left[\frac{x}{\ell_{B}}-\frac{2{\pi}\ell_{B}}{L_{2}}\left(m+kN_{\phi}\right)\right]^{2}+i\frac{2{\pi}y}{L_{2}}\left(m+kN_{\phi}\right)\right\\}$
(S17) $\displaystyle\times$ $\displaystyle
H_{\alpha}\left\\{\frac{1}{\sqrt{\Lambda}}\left[\frac{x}{\ell_{B}}-\frac{2{\pi}\ell_{B}}{L_{2}}\left(m+kN_{\phi}\right)\right]\right\\},$

where $\alpha\in\mathbb{N}$ is the Landau level index and
$m\in[0,\ldots,N_{\phi}-1]$. For a generic two-body interaction potential
$V(\mathbf{r}_{1}-\mathbf{r}_{2})$, the many-body Hamiltonian can be written
as

$\displaystyle H_{\rm
MB}(\Lambda)=\frac{1}{2L_{1}L_{2}}\sum_{\mathbf{q}}V(\mathbf{q}):\rho(\mathbf{q})\rho(-\mathbf{q}):$
(S18)

with

$\displaystyle
V(\mathbf{r}_{1}-\mathbf{r}_{2})=\frac{1}{L_{1}L_{2}}\sum_{\mathbf{q}}V(\mathbf{q})\;\exp\left[-i\mathbf{q}\cdot(\mathbf{r}_{1}-\mathbf{r}_{2})\right]\qquad\mathbf{q}=\frac{2\pi}{L_{1}}q_{1}\widehat{e}_{x}+\frac{2\pi}{L_{2}}q_{2}\widehat{e}_{y}$
(S19)

Based on previous experience, it is useful to define

$\displaystyle|\mathbf{q}(\Lambda)|^{2}=\Lambda\left(\frac{2{\pi}\ell_{B}}{L_{1}\sin\theta}q_{1}-\frac{2{\pi}\ell_{B}\cot\theta}{L_{2}}q_{2}\right)^{2}+\frac{1}{\Lambda}\left(\frac{2{\pi}\ell_{B}}{L_{2}}q_{2}\right)^{2}$
(S20)

and the form factor

$\displaystyle
F_{\alpha\beta}(q_{1},q_{2},\Lambda)=\frac{1}{\sqrt{\alpha!\beta!2^{\alpha+\beta}}}\sum^{{\rm
min}[\alpha,\beta]}_{k}2^{k}k!\;\binom{\alpha}{k}\;\binom{\beta}{k}\left[i\sqrt{\Lambda}\frac{2{\pi}\ell_{B}}{L_{1}}q_{1}-\frac{2\pi\ell_{B}}{\sqrt{\Lambda}L_{2}}q_{2}\right]^{\alpha-k}\left[i\sqrt{\Lambda}\frac{2{\pi}\ell_{B}}{L_{1}}q_{1}+\frac{2\pi\ell_{B}}{\sqrt{\Lambda}L_{2}}q_{2}\right]^{\beta-k}$
(S21)

In subsequent calculations, we will need their derivatives

$\displaystyle\left.\frac{d|\mathbf{q}(\Lambda)|^{2}}{d\Lambda}\right|_{\Lambda=1}=\left(\frac{2{\pi}\ell_{B}}{L_{1}}q_{1}\right)^{2}-\left(\frac{2{\pi}\ell_{B}}{L_{2}}q_{2}\right)^{2}$
(S22)

and

$\displaystyle\left.\frac{dF_{\alpha\beta}(q_{1},q_{2})}{d\Lambda}\right|_{\Lambda=1}=\frac{1}{\sqrt{\alpha!\beta!2^{\alpha+\beta}}}\sum^{{\rm
min}[\alpha,\beta]}_{k}2^{k}k!\;\binom{\alpha}{k}\;\binom{\beta}{k}$ (S23)
$\displaystyle\times$
$\displaystyle\left\\{\frac{1}{2}(\alpha-k)\left(i\frac{2{\pi}\ell_{B}}{L_{1}}q_{1}-\frac{2\pi\ell_{B}}{L_{2}}q_{2}\right)^{\alpha-k-1}\left(i\frac{2{\pi}\ell_{B}}{L_{1}}q_{1}+\frac{2\pi\ell_{B}}{L_{2}}q_{2}\right)^{\beta-k+1}\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{2}(\beta-k)\left(i\frac{2{\pi}\ell_{B}}{L_{1}}q_{1}-\frac{2\pi\ell_{B}}{L_{2}}q_{2}\right)^{\alpha-k+1}\left(i\frac{2{\pi}\ell_{B}}{L_{1}}q_{1}+\frac{2\pi\ell_{B}}{L_{2}}q_{2}\right)^{\beta-k-1}\right\\}$

If $\alpha=\beta$, the second derivative can be simplified to

$\displaystyle\left.\frac{dF_{\alpha\alpha}(q_{1},q_{2})}{d\Lambda}\right|_{\Lambda=1}=-\left[\left(\frac{2{\pi}\ell_{B}}{L_{1}}q_{1}\right)^{2}-\left(\frac{2\pi\ell_{B}}{L_{2}}q_{2}\right)^{2}\right]$
(S24) $\displaystyle\times$
$\displaystyle\frac{1}{\sqrt{\alpha!\alpha!2^{2\alpha}}}\sum^{\alpha-1}_{k}2^{k}k!\;\binom{\alpha}{k}\;\binom{\alpha}{k}(\alpha-k)\left[i\frac{2{\pi}\ell_{B}}{L_{1}}q_{1}-\frac{2\pi\ell_{B}}{L_{2}}q_{2}\right]^{\alpha-k-1}\left[i\frac{2{\pi}\ell_{B}}{L_{1}}q_{1}+\frac{2\pi\ell_{B}}{L_{2}}q_{2}\right]^{\alpha-k-1}.$

To study the FQH states, we keep two LLs with single-particle energy
$\epsilon^{(\alpha)}$ and single-particle wave functions

$\displaystyle|\widetilde{\phi}^{(\alpha)}\rangle=\begin{bmatrix}\sum^{7}_{n=0}f^{(\alpha)}_{0n}|n\rangle\\\
\sum^{7}_{n=0}f^{(\alpha)}_{1n}|n\rangle\\\
\sum^{7}_{n=0}f^{(\alpha)}_{2n}|n\rangle\\\
\sum^{7}_{n=0}f^{(\alpha)}_{3n}|n\rangle\\\
\sum^{7}_{n=0}f^{(\alpha)}_{4n}|n\rangle\\\
\sum^{7}_{n=0}f^{(\alpha)}_{5n}|n\rangle\\\ \end{bmatrix}.$ (S25)

The superscript $\alpha=0,1$ labels the two levels, the first subscript
$0,1,\ldots,5$ labels the components of the spinor, and the second subscript
$n=0,1,\cdots,7$ denotes the non-relativistic LL indices. These two levels are
called active and other levels are neglected. The creation (annihilation)
operator associated with $|\widetilde{\phi}^{(\alpha)}\rangle$ is denoted as
$C^{{\dagger}}_{{\alpha}m}$ ($C_{{\alpha}m}$). The many-body Hamiltonian is

$\displaystyle H_{\rm MB}(\Lambda)$ $\displaystyle=$
$\displaystyle\sum_{\alpha,m}\epsilon^{(\alpha)}C^{{\dagger}}_{{\alpha}m}C_{{\alpha}m}+\frac{1}{2L_{1}L_{2}}\sum_{\\{\alpha_{i}\\}}\sum_{\\{m_{i}\\}}\sum_{q_{1},q_{2}}V_{\rm
SC}(\mathbf{q})\exp\left[-\frac{1}{2}|\mathbf{q}(\Lambda)|^{2}\ell^{2}_{B}-i\frac{2{\pi}q_{1}}{N_{\phi}}\left(m_{1}-m_{4}\right)\right]$
(S26) $\displaystyle\times$
$\displaystyle\widetilde{F}_{\alpha_{1}\alpha_{3}}(-q_{1},-q_{2},\Lambda)\widetilde{F}_{\alpha_{2}\alpha_{4}}(q_{1},q_{2},\Lambda)\;\widetilde{\delta}_{m_{1},m_{3}-q_{2}}\widetilde{\delta}_{m_{2},m_{4}+q_{2}}\;C^{{\dagger}}_{\alpha_{1}m_{1}}C^{{\dagger}}_{\alpha_{2}m_{2}}C_{\alpha_{4}m_{4}}C_{\alpha_{3}m_{3}}$

where $\widetilde{\delta}$ is a generalized Kroneker function defined as

$\displaystyle\widetilde{\delta}_{s,t+q_{2}}=1\qquad\text{if and only
if}\qquad s\;\text{mod}\;N_{\phi}=(t+q_{2})\;\text{mod}\;N_{\phi}$ (S27)

and

$\displaystyle\widetilde{F}_{\alpha_{1}\alpha_{3}}(-q_{1},-q_{2},\Lambda)$
$\displaystyle=$
$\displaystyle\sum^{5}_{i=0}\sum^{7}_{n_{1},n_{2}}f^{(\alpha_{1})*}_{in_{1}}f^{(\alpha_{3})}_{in_{2}}F_{n_{1}n_{2}}(-q_{1},-q_{2},\Lambda)$
$\displaystyle\widetilde{F}_{\alpha_{2}\alpha_{4}}(q_{1},q_{2},\Lambda)$
$\displaystyle=$
$\displaystyle\sum^{5}_{i=0}\sum^{7}_{n_{1},n_{2}}f^{(\alpha_{2})*}_{in_{1}}f^{(\alpha_{4})}_{in_{2}}F_{n_{1}n_{2}}(q_{1},q_{2},\Lambda).$
(S28)

It is obvious that the total momentum $Y=\sum_{i}m_{i}$ is conserved. The low-
energy eigenstates of $\widehat{H}_{\rm MB}(1)$ are computed by exact
diagonalization. Due to the exponential growth of Hilbert space dimension, the
maximal number of electrons is severely constrained. This problem can be
partially mitigated if we impose a constraint on the number of electrons in
the higher LL. If the total filling factor of the two levels is $3/2$ and
there is no interaction, then the lower level is fully occupied and the higher
level is half filled. As interaction is turned on, it is possible for all
electrons to move to the higher level, but this configuration is not
energetically favored. In our calculation, we only retain the configurations
in which the higher level does not deviate too much from half filling.

The purpose of studying general $\Lambda$ is to define chiral graviton
operators and compute their spectral functions [35]. If we choose
$\Lambda=1+\xi$ and treat $\xi$ as a small parameter, the Hamiltonian can be
expanded as

$\displaystyle H_{\rm MB}(\Lambda)=H_{\rm MB}(1)+\xi\mathcal{O}$ (S29)

with graviton operator

$\displaystyle\mathcal{O}$ $\displaystyle=$
$\displaystyle\frac{1}{2L_{1}L_{2}}\sum_{\\{\alpha_{i}\\}}\sum_{\\{m_{i}\\}}\sum_{q_{1},q_{2}}\;V(\mathbf{q})\exp\left[-\frac{1}{2}|\mathbf{q}|^{2}\ell^{2}_{B}-i\frac{2{\pi}q_{1}}{N_{\phi}}(m_{1}-m_{4})\right]$
(S30) $\displaystyle\times$
$\displaystyle\widetilde{\delta}_{m_{1},m_{3}-q_{2}}\widetilde{\delta}_{m_{2},m_{4}+q_{2}}\;C^{{\dagger}}_{\alpha_{1}m_{1}}C^{{\dagger}}_{\alpha_{2}m_{2}}C_{\alpha_{4}m_{4}}C_{\alpha_{3}m_{3}}$
$\displaystyle\times$
$\displaystyle\left\\{-\frac{1}{2}\left[\left(\frac{2{\pi}\ell_{B}}{L_{1}}q_{1}\right)^{2}-\left(\frac{2{\pi}\ell_{B}}{L_{2}}q_{2}\right)^{2}\right]\widetilde{F}_{\alpha_{1}\alpha_{3}}(-q_{1},-q_{2},1)\widetilde{F}_{\alpha_{2}\alpha_{4}}(q_{1},q_{2},1)\right.$
$\displaystyle\left.+\left.\frac{d\widetilde{F}_{\alpha_{1}\alpha_{3}}(-q_{1},-q_{2},\Lambda)}{d\Lambda}\right|_{\Lambda=1}\widetilde{F}_{\alpha_{2}\alpha_{4}}(q_{1},q_{2},1)+\widetilde{F}_{\alpha_{1}\alpha_{3}}(-q_{1},-q_{2},1)\left.\frac{d\widetilde{F}_{\alpha_{2}\alpha_{4}}(q_{1},q_{2},\Lambda)}{d\Lambda}\right|_{\Lambda=1}\right\\}.$

It is more complicated than previously used ones because we have two LLs here
[35, 36]. While the physics of gravitons in two LLs may be very interesting,
it is more convenient to simplify the problem by going back to one level. If
the total filling factor of the two LLs is $3/2$, the eigenstates obtained by
exact diagonalization will be projected to the higher level. The summation
over $\alpha$ in $\mathcal{O}$ will be limited to the higher level. The
operator thus obtained is still not good enough because it cannot reveal the
graviton chirality. To this end, we change the prefactors in Eqs. (S22) and
(S24) as

$\displaystyle\left[\left(\frac{2{\pi}\ell_{B}}{L_{1}}q_{1}\right)^{2}-\left(\frac{2\pi\ell_{B}}{L_{2}}q_{2}\right)^{2}\right]\quad\Rightarrow\quad\left(\frac{2{\pi}\ell_{B}}{L_{1}}q_{1}\mp\frac{2\pi\ell_{B}}{L_{2}}q_{2}\right)^{2}$
(S31)

to define chiral graviton operators $\mathcal{O}_{\pm}$. This replacement is
motivated by the connection between chiral graviton operators and anisotropic
Haldane pseudopotentials [46].

The eigenstates of $\widehat{H}_{\rm MB}(1)$ are denoted as $|\Psi_{n}\rangle$
and the associated eigenvalues are $E_{n}$. The total weights of the chiral
graviton operators are

$\displaystyle
W_{\pm}=\langle\Psi_{0}|\mathcal{O}^{{\dagger}}_{\pm}\mathcal{O}_{\pm}|\Psi_{0}\rangle$
(S32)

and the normalized spectral functions are

$\displaystyle
I_{\pm}(\omega)=\sum_{n}\frac{\left|\langle\Psi_{n}|\mathcal{O}_{\pm}|\Psi_{0}\rangle\right|^{2}}{W_{\pm}}\delta(\omega-
E_{n}+E_{0}).$ (S33)

In finite-size systems with discrete energy levels, $I_{\pm}(\omega)$ can be
computed using a method based on Lanczos tridiagonalizatio and continued
fraction [47]. We introduce a rescaled operator
$\mathcal{F}=\mathcal{O}_{\pm}/\sqrt{W_{\pm}}$ and define its Green’s function

$\displaystyle G(t)=\langle\Psi_{0}|\exp(iH_{\rm
MB}t)\mathcal{F}^{{\dagger}}\exp(-iH_{\rm MB}t)\mathcal{F}|\Psi_{0}\rangle.$
(S34)

Fourier transform yields

$\displaystyle G(\omega)=\int
dt\;\exp(i{\omega}t)G(t)=\sum_{n}\frac{\left|\langle\Psi_{n}|\mathcal{F}|\Psi_{0}\rangle\right|^{2}}{\omega-
E_{n}+E_{0}}=\sum_{n}\frac{w^{2}_{n}}{\omega-p_{n}}$ (S35)

where $p_{n}$ are the poles and $w^{2}_{n}$ are the weights. Using
$|F_{0}\rangle=\mathcal{F}|\Psi_{0}\rangle$ as the initial vector, the Lanczos
algorithm generates a tridiagonal matrix. $E_{0}+p_{n}$ is the $n$-th
eigenvalue of this matrix and $\omega_{n}$ is the first element of the $n$-th
eigenstate.

##  III Results

Let us consider the many-body state at $\nu=-9/2$ that occurs without the
displacement field. The magnetic field is chosen to be $14$ T so we have
$\ell_{B}=6.842$ nm, $d=8.697\ell_{B}$ and
$\frac{e^{2}}{4\pi\varepsilon_{0}\varepsilon^{\parallel}_{\rm
BN}\ell_{B}}=31.86$ meV. The LL diagram in Extended Data Fig. 6a is computed
using the parameters listed above with $\Delta_{2}=0$ meV. It is helpful to
begin our analysis with the $\nu=-6$ state. If the interaction is turned off,
electrons fill all the LLs below the $\mathbf{K}_{-},N_{\rm B}=0$ level in
Extended Data Fig. 6a. The presence of interaction is not expected to
fundamentally change the $\nu=-6$ state since the single-particle gap below
$\mathbf{K}_{-},N_{\rm B}=0$ is about $30$ meV. The $\nu=-9/2$ state has an
additional $3/2$ filling of electrons compared to the $\nu=-6$ state. The most
likely configuration is that electrons are spin-valley polarized and fill the
$\mathbf{K}_{-},N_{\rm B}=0$ and $\mathbf{K}_{-},N_{\rm B}=1$ levels. It is
essential to keep both levels given their separation is only $4.59$ meV. The
energy spectra obtained by exact diagonalization for the system with $36$
electrons and $N_{\phi}=24$ are presented in Fig. S1. Here we only show the
low-energy eigenvalues with total momentum $0{\leq}Y{\leq}11$ because the
other half is obtained trivially by magnetic translation. In Fig. S1a, there
are exactly 24 (12) electrons in the $\mathbf{K}_{-},N_{\rm B}=0$
($\mathbf{K}_{-},N_{\rm B}=1$) level, hence no mixing is present. In contrast,
one or two electrons are allowed to escape from the $\mathbf{K}_{-},N_{\rm
B}=0$ level in Fig. S1b,c. For all cases, there are six quasi-degenerate
ground states, which is consistent with the prediction for the Moore-Read type
states. This information cannot distinguish the Pfaffian, anti-Pfaffian, or
particle-hole symmetric Pfaffian states. To this end, we turn to the chiral
graviton spectral functions shown in Fig. S2. For the ground states in Fig.
S1a, particle-hole symmetry in the $\mathbf{K}_{-},N_{\rm B}=1$ level ensures
that the two chiralities have the same the total weights. As soon as LL mixing
is turned on, $W_{-}$ becomes much larger than $W_{+}$, which suggests that
the ground states in Fig. S1b,c are of the Pfaffian type. In addition, we have
also studied the cases with $33$ and $39$ electrons. While ground states
cannot be defined for them, their chiral graviton spectral functions also
reveal that the negative chirality dominates in the presence of LL mixing. On
the experimental side, the weak feature observed at $\nu=-5+7/13$ can be
ascribed to the Levin-Halperin daughter state of the Pfaffian state.

Fig. S1: Low-energy eigenvalues of the $\boldsymbol{\nu=-9/2}$ state. There
are 36 electrons in total and the number of electrons that is allowed to
escape from $\mathbf{K}_{-},N_{\rm B}=0$ is denoted as $M_{c}$. Fig. S2:
Chiral graviton spectral functions for the $\boldsymbol{\nu=-9/2}$ state. The
three columns correspond to the states in Fig. S1. The top row is for negative
chirality and the bottom row is for positive chirality. In each panel,
different symbols are used to represent the three ground states as shown in
panel a. For total momentum $Y=6$, the lowest energy state is called $a$ and
the second lowest state is called $b$. The three numbers in each panel are the
total weights of the spectral functions.

Naively, one may expect to obtain an even-denominator FQH state when a NR0/NR1
doublet has $3/2$ filling. This could happen at
$\nu=-6+3/2,-4+3/2,-2+3/2,0+3/2$, but only two of them are actually observed.
The absence of FQH states at $\nu=-4+3/2$ and $-2+3/2$ is not easy to explain.
It may simply because that the sample quality is still insufficient. Another
possibility is the competition between spin-valley polarized states and other
states that are not fully polarized. The existence of a nonzero $\Delta_{2}$
may reduce some single-particle gaps and/or lead to level crossings. We hope
that more in-depth experimental and theoretical studies in the future can shed
some light on this issue. While the $\nu=3/2$ state was indeed observed, its
nature is more subtle. The active levels are expected to be
$\mathbf{K}_{+},N_{\rm B}=0$ and $\mathbf{K}_{+},N_{\rm B}=1$, but the
$\mathbf{K}_{+},N_{\rm M}=0$ level is not too far in energy. For our choice of
SWMc parameters, the two single-particle gaps between these three levels are
$5.22$ meV and $14.8$ meV. The two gaps can be brought to comparable values by
increasing the magnetic field and/or changing the SWMc parameters. The fate of
the Pfaffian state in this process is unclear but very interesting.

The FQH states at $\nu=-3/2$ and $9/2$ are more difficult to pin down
quantitatively. The application of a displacement field results in many LL
crossings that is further complicated by a nonzero $\Delta_{2}$. If the
electric field is chosen to be $D=155$ mV/nm, we would have $\Delta_{1}=17.31$
meV. The evolution of LLs with $\Delta_{2}$ is shown in Extended Data Fig. 6c.
For $\Delta_{2}=0$ meV, the active levels at $\nu=-3/2$ and $9/2$ are of the
NR0 type. If $\Delta_{2}=0$ meV is fixed but $D$ increases to a very large
value or $D$ is fixed but $\Delta_{2}$ increases to $\sim 5$ meV, one NR1
level in the $\mathbf{K}_{-}$ goes up and crosses with one NR0 level in
$\mathbf{K}_{+}$. This enables a Pfaffian type state at $\nu=-3/2$ but the
$\nu=9/2$ state is still mysterious. If $\Delta_{2}$ is changed to -10 meV,
the top most level (in the $-6{\leq}\nu{\leq}6$ range) would be of the NR1
type. In this case, the $\nu=9/2$ state corresponds to half filling of one
spin component of this level, and its mixing with the NR0 type level below it
would select the Pfaffian state, as confirmed by the results in Fig. S3 and
Fig. S4. If we go beyond the one-component spin-valley polarized states, other
even-denominator FQH states could be possible. Two examples are the Halperin
331 state [37] and the Jain state constructed from parton theory [38]. In this
scenario, there is no need to invoke $\Delta_{2}$, but our analysis is still
applicable for a small nonzero $\Delta_{2}$. As shown in Extended Data Fig.
6b, two NR0 type levels evolve with $\Delta_{1}$ and become quasi-degenerate
in a suitable range. After incorporating the spin degree of freedom, there are
four levels whose total filling factor is $5/2$. For the integer part $2$,
electrons could populate the same spin but different valleys or the same
valley but different spins. The fractional part $1/2$ corresponds to half
filling of two NR0 type levels with the same spin or valley index. To realize
the Halperin or Jain state, the interlevel interaction should be weaker
compared to the intralevel interaction. If the two levels belong to the same
valley but have opposite spins, it is difficult to envisage spin-dependent
that would favor the Halperin or Jain state. In contrast, the interaction
between different valleys may be altered by valley anisotropic terms that
arise from lattice scale corrections to the Coulomb potential [40]. It has
been proposed that the Jain state could be realized using suitable valley
anisotropic terms [41]. To check if this mechanism also works for the TLG, a
detailed analysis of valley anisotropic terms would be necessary. This is an
interesting topic that is left for future works.

Fig. S3: Low-energy eigenvalues of the $\boldsymbol{\nu=9/2}$ state. There are
36 electrons in total and the number of electrons that is allowed to escape
from the NR0 level is denoted as $M_{c}$. Fig. S4: Chiral graviton spectral
functions for the $\boldsymbol{\nu=9/2}$ state. The three columns correspond
to the states in Fig. S3. The symbols are the same as in Fig. S2.