# Cascade of multi-electron bubble phases in monolayer graphene at high Landau
level filling

Fangyuan Yang  Department of Physics, University of California at Santa
Barbara, Santa Barbara CA 93106, USA  Ruiheng Bai  Department of Physics,
University of California at Santa Barbara, Santa Barbara CA 93106, USA
Alexander A. Zibrov  Department of Physics, University of California at Santa
Barbara, Santa Barbara CA 93106, USA  Sandeep Joy  Department of Physics, Ohio
State University, Columbus, Ohio 43210, USA  Takashi Taniguchi  National
Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Kenji
Watanabe  National Institute for Materials Science, 1-1 Namiki, Tsukuba
305-0044, Japan  Brian Skinner  Department of Physics, Ohio State University,
Columbus, Ohio 43210, USA  Mark O. Goerbig  Laboratoire de Physique des
Solides, CNRS UMR 8502, Université Paris-Saclay, 91405 Orsay Cedex, France
Andrea F. Young<EMAIL_ADDRESS>Department of Physics, University of
California at Santa Barbara, Santa Barbara CA 93106, USA

###### Abstract

The phase diagram of an interacting two-dimensional electron system in a high
magnetic field is enriched by the varying form of the effective Coulomb
interaction, which depends strongly on the Landau level index. While the
fractional quantum Hall states that dominate in the lower energy Landau levels
have been explored experimentally in a variety of two-dimensional systems,
much less work has been done to explore electron solids owing to their subtle
transport signatures and extreme sensitivity to disorder. Here we use chemical
potential measurements to map the phase diagram of electron solid states in
$N=2$, $N=3$, and $N=4$ Landau levels in monolayer graphene. Direct comparison
between our data and theoretical calculations reveals a cascade of density-
tuned phase transitions between electron bubble phases up to two, three or
four electrons per bubble in the N=2, 3 and 4 Landau levels respectively.
Finite temperature measurements are consistent with melting of the solids for
T$\approx$1K.

In an electron solid, spatial translation symmetry is spontaneously broken so
that the ground state charge density forms a periodic structure incommensurate
with the underlying crystal lattice. One known example is obtained in high
Landau levels (LLs) in two-dimensional (2D) electron systems. Theoretically,
the phase diagram is expected to host a rich interplay of competing
phasesKoulakov  _et al._ (1996); Fogler  _et al._ (1996); Moessner and Chalker
(1996); Haldane  _et al._ (2000); Shibata and Yoshioka (2001); Fogler (2002);
Côté  _et al._ (2003); Goerbig  _et al._ (2004a). A unique feature of electron
solids in higher LLs is that a variable number of electrons may cluster on
each site of the emergent crystal. The formation of the phases—known as
“electron bubbles”—is driven by the structure of the electronic form factors
in the LLs. Electron bubble phases were first identified in the GaAs 2D
electron gas by the observation of re-entrant integer quantum Hall effect
(RIQHE) in transport measurementLilly  _et al._ (1999); Du  _et al._ (1999),
in which the crystallized electrons freeze and no longer contribute to the
Hall conductivity. Similar phases are also expected in graphenePapic  _et al._
(2011); Knoester  _et al._ (2016); Zhang and Joglekar (2007), and recent
measurements have confirmed their existenceChen  _et al._ (2019); Zeng  _et
al._ (2019). While the existence of electron solids is straightforward to
confirm using transport measurements, distinguishing them from each other to
construct a comprehensive phase diagram is not. To this end, other
experimental methods, such as microwave spectroscopyLewis  _et al._ (2002),
surface acoustic wave transmissionMsall and Dietsche (2015); Friess  _et al._
(2017), and tunnelling spectroscopy Jang  _et al._ (2016) have been developed
to study vibrating modes related to the lattice structure of electron solids.
More recently, temperature dependent transport has shown that the same RIQH
state may host more than one bubble phase, distinguished by different melting
temperaturesRo  _et al._ (2019); Fu  _et al._ (2019); Ro  _et al._ (2020).
However, a detailed phase diagram of the electron bubble phases across
different LLs, long been predicted by theory, has not been conclusively
established.

Measuring thermodynamic properties provides a probe of quantities directly
related to the ground state energy, offering a chance to map out a complete
phase diagram independent of the detailed transport phenomenology of the
ground state. In this Letter, we use chemical potential measurements Yang  _et
al._ (2021) to construct just such a phase diagram for partially filled LLs in
monolayer graphene. Our data demonstrate the existence of multiple distinct
electron bubble phases characterized by different bubble sizes. By directly
comparing our data with mean-field-theory calculations, we establish a one-to-
one correlation between the filling factor and the electron bubble morphology.

Our measurement is performed in a graphene/hBN heterostructure assembled using
standard dry pickup techniquesWang  _et al._ (2013). Two graphene monolayers
are separated by an hBN dielectric layer of 40nm thickness, with additional
hBN dielectric and graphite gates forming a four-plate capacitor geometry. The
top graphene serves as a charge detector, which combined with a feedback loop
allows us to determine changes in chemical potential of the bottom ‘sample’
graphene accuratelyYang  _et al._ (2021).

Figure 1:  FQH and electron solid states in graphene monolayer probed by
chemical potential measurements. (a) Chemical potential change as a function
of effective filling factor, $\nu^{*}\equiv\nu-\lfloor\nu\rfloor$, in the
$N=0$, (b) $N=1$, (c) $N=2$, (d) $N=3$, and (e) $N=4$ LLs. In the $N=0$ and
$N=1$ LLs, FQH states are observed as jumps in $\nu$ at $\nu^{*}=p/(2p\pm 1)$
and $\nu^{*}=p/(4p\pm 1)$ ($p=1,2,3,...$), a selection of which are labeled.
For $N\geq 2$, broad oscillatory features dominate, which we associated with
electron solids. The $N=2$ LL is a marginal case where fractional quantum Hall
states and electron bubbles compete within a narrow range of filling factors.
All data measured at $B=13T$ and $T=15mK$.

Fig. 1 presents the chemical potential $\mu$ measured across individual LLs
with orbital quantum numbers $N=0$, 1, 2, 3, and 4. The qualitative behavior
of $\mu$ depends strongly on $N$. For $N=0$ and $N=1$ (Figs. 1a-b) fractional
quantum Hall states are favored, with incompressible states (manifesting here
as nearly discontinuous jumps in $\mu$) observed at filling factors associated
with two-flux and four-flux composite fermion sequencesJain (1989). For
$\nu^{*}>-1/5$ (or $\nu^{*}<-4/5$) within the $N=0$ and $N=1$ LL, $\mu$
changes smoothly, showing a large negative inverse compressibility
$d\mu/d\nu$Bello  _et al._ (1981). This behavior has been identified with the
formation of Wigner crystal states in previous experiments in both
GaAsEisenstein  _et al._ (1992, 1994) and grapheneZhou  _et al._ (2019); Yang
_et al._ (2021).

For $N\geq 2$ (Figs. 1c-e), a qualitatively different behavior is observed,
with $\mu$ dominated by much weaker oscillatory features that are not
associated with any particular fractional $\nu$. As we elaborate upon below,
these features are signatures of multi-electron bubble states. Bubble states
are generically expected in higher LLs due to the nature of the single-
particle wave functions, which feature multiple nodes. This form factor
considerably modifies the Coulomb repulsion at short distances, favoring
charge-density-wave-type states instead of incompressible fractional quantum
Hall states. In the $N=2$ LL, our measurement reveals a competition between
the FQH states observed at $\nu^{*}=-1/5$ and $-4/5$ and electron bubble
states, as reported previouslyChen  _et al._ (2019). In the $N=3$ and $N=4$
LLs, the electron bubble phases are favored over the entire range of filling
factors, manifesting as a slow modulation of $\mu$ and $d\mu/d\nu$, as shown
in 2a-b. The number of oscillatory features increases with $N$. In the $N=3$
and $N=4$ LL we observed three and four pairs of features, related by particle
hole symmetry about $\nu^{*}=1/2$, respectively.

Figure 2:  Electronic compressibility and temperature dependence of electron
bubble phases.  (a) $d\mu/d\nu$ in the $N=3$ and (b) $N=4$ LLs. The data is
obtained via numerical differentiation of $\mu$ measured at $13T$ and $15mK$.
Within each LL, the four symmetry breaking levels are plotted by blue, red,
orange and purple curves with increasing $|\nu|$. The curves are offset as
indicated by the gray dashed lines. Stars indicate the center of the regions
identified with electron bubble states. (c) Temperature dependence of electron
bubble states in $N=3$ and (d) $N=4$ LLs, measured at $B=13T$.

The panels of Fig. 2a-b show $d\mu/d\nu$ measured over a range spanning
several LLs each, grouped by their orbital quantum number. For the $N=3$
orbital (Fig. 2a), the four curves depicted are acquired in filling factor
ranges corresponding to each of the four symmetry broken levels spanning
$-10<\nu<-6$. Due to limitations on the range of the electrostatic gates, for
the $N=4$ LL (Fig. 2b) only $-12<\nu<-10$ is shown. Remarkably, the repetition
of the pattern of $\mu$ oscillations across different symmetry-broken levels
indicates that this physics is independent of the spin and valley order. We
may conclude that the formation of the bubble phases is governed only by
single-component LL physics; as a consequence, the bubbles are not expected to
be accompanied by complex spin or valley textures as have been shown to play a
role in lower LLsZhou  _et al._ (2019); Liu  _et al._ (2022).

The energy scale characterizing the bubble phases may be directly accessed via
the temperature dependence, shown in Fig. 2c-d. Signatures of the bubble
phases disappear rapidly for $T\approx 1-2K$ in the $N=3$ LL, and below 1K in
the $N=4$. This is consistent with the general scale of the chemical potential
changes associated with these phases, which are on the order of a few hundred
$\mu eV$, as well as previously reported transport dataChen  _et al._ (2019).
The order of magnitude of this scale is consistent with simplified Lindemann
criterionLindemann (1910) for crystal melting, according to which the thermal
position fluctuations need to be roughly $15\%$ of the lattice spacing to make
the crystal melt. Within the harmonic approximation for the crystal, one
obtains critical temperatures in the $\sim 1$ K range (see supplementary
material). Notably, the energy scale of the bubble phases is considerably
smaller than that of the fractional quantum Hall physics in the lower LLs,
where gaps (at comparable magnetic fields) typically are in the $>10K$ range.

Theoretically, the ground state of the interacting electron system in a
partially filled high-N Landau level is expected to evolve through a series of
multi-electron bubble phases, as illustrated in Fig. 3a for the case of N=4.
These crystalline phases can be described within a mean-field approach as
presented in detail in the Supplementary Material. Fig. 3a shows the cohesive
energy per particle for the bubble crystals with $M$ electrons per lattice
site as a function of the effective filling factor $\nu^{*}$. The cohesive
energy is the energy per particle, from which we have already subtracted the
Hartree-Fock energy of a featureless electronic liquidGoerbig  _et al._
(2004a) as well as the charging energy of the parallel plate capacitor in
which the sample is embedded. For a fixed value of $M$, the energy of the
triangular bubble crystals depends on the spacing $\Lambda_{B}=\sqrt{4\pi
M/\sqrt{3}\nu^{*}}l_{B}$ between the bubbles, which in turn depends on the
effective filling $\nu^{*}$. Here, $l_{B}=\sqrt{\hbar/eB}$ is the magnetic
length.

One obtains a family of curves, with minima at positions described
approximately by $\nu^{*}\sim M/N$. The $M$-bubble phase is realized whenever
it is lowest in energy within a a certain filling-factor range. Within a given
Landau level, the maximum stabilized value of $M$ equals $N$. Theoretically,
one may even stabilize a bubble phase with $M=N+1$ in the vicinity of a half-
filled, singly-degenerate Landau level ($\nu^{*}\sim 1/2$)Côté  _et al._
(2003). However, this phase is thought to compete energetically with a stripe
phase; we find no evidence for it in the experimental data.

Notably, the family of minimum energy curves shown in Fig. 3b are not convex
upon variation of $\nu^{*}$, a signature of thermodynamic instability to the
formation of mixed phases in which parts of the sample area are occupied by
crystals with differing number of electrons per bubble. However, we note that
for our experimental geometry, the variations in internal energy caused by the
bubble phases are dwarfed by the electrostatic energy of the electron gas.
Taking this into account, mixed phases are only found in a range
$\delta\nu\approx 2\times 10^{-3}$ (see supplementary information) in the
vicinity of the level crossings visible in Fig. 3b. In this picture, then, we
expect a succession of pure bubble phases, separated by sharp phase
transitions.

To facilitate comparison between experiment and theory, in Fig. 4a-c, we plot
the experimentally measured $\mu$ scaled by the Coulomb energy,
$E_{c}=e^{2}/(\epsilon\ell_{B})$. Each panel presents $\mu$ measured at
different values of the magnetic field $B$ for the same LL fillings, with an
offset of $0.01E_{C}$ between curves introduced for clarity. The $\mu$
modulations observed in the curves are almost identical in these units, as
expected given the Coulomb-driven nature of the electron bubble phases. Fig.
4d-f presents the calculated chemical potential of electron bubble phases in
the $N=2$, $N=3$, and $N=4$ LLs in the absence of disorder. The solid curves
are obtained from the calculated energy per particle $E$ of the $M$-bubble
phases via $\mu=\partial\left(\nu E\right)/\partial\nu$Goerbig  _et al._
(2004a). Note that in these calculations, we restore the contribution of the
featureless background charge omitted above in the calculation of the cohesive
energy. Our calculations account for screening caused by both the dielectric
environment as well as inter-Landau level excitations in the grapheneShizuya
(2007); Roldán  _et al._ (2010). As in the N=0 and N=1 Landau levelsYang  _et
al._ (2021), accurately accounting for screening is required for quantitative
agreement between experiment and theory in graphene.

Figure 3:  Cohesive energy for electron bubble states. (a) Schematic depiction
of electron bubble phases in the $N=4$ LL. (b) Calculated cohesive energy for
the $N=4$ LL (see supplementary information for details). The ground state is
obtained by tracing the lowest energy state at each filling factor, which is
highlighted by colored lines. The color codes here match those in panel (a).
Figure 4:  Quantitative comparison with theoretical model of electron bubble
cascade. (a) $\mu(\nu)$ at several magnetic fields in the $N=2$, (b) $N=3$,
and (c) $N=4$ Landau level. The data at $B=31.5T$ and $20T$ were measured at
$300mK$, while the data at $13T$, $7T$, and $5T$ were measured at $15mK$. The
chemical potential change is presented in units of the Coulomb energy
$E_{c}=\frac{e^{2}}{\epsilon l_{B}}\approx
12.5meV\cdot\sqrt{B/\mathrm{Tesla}}$. The red, orange, and purple curves are
offset by $-0.01E_{c}$, $-0.02E_{c}$, and $-0.03E_{c}$ from the blue curve,
respectively. (d) Chemical potential calculated by mean field-theory (solid
lines, see supplementary materials) in the $N=2$, (e) $N=3$, and (f) $N=4$
Landau level. The dashed lines in these panels are chemical potential taking
disorder broadening into account. The pink, blue, purple, and green color bars
represent the domain of stability for the $M=1$, $M=2$, $M=3$, and $M=4$
electron bubble phases within the disorder broadened model, respectively. The
gray regions represent broadened phase transitions where neighboring pure
electron bubble phases coexist. Panel (a)-(c) use the same color codes to
label the corresponding regions identified by experiments from the sign of the
compressibility.

Despite the comparative simplicity of our model, it agrees quantitatively with
the data in the overall scale of the chemical potential modulation across the
Landau level, as well as in the locations of the various bubble phases, which
we identify with positive compressibility regions for the $M>=2$. However, in
contrast to the theoretical model, where the phase transitions are sharp, in
the experimental data the phase transitions are marked by broad regions of
negative compressibility typically rather than sharp jumps. It is natural to
associate these regions with a mixed phase arising from disorder potentials.
To capture this physics, we convolve the disorder-free curves with a Gaussian
‘inhomogenous broadening’ of width $\Delta\nu=0.015$ at $13T$. Given the
negligible quantum capacitance in the bubble regime, this is equivalent to an
energy broadening $\Delta E=7.5meV$. The dashed curves in Fig. 4d-f show the
results of this model. We use the same color code to label the regions
associated with pure and mixed electron bubble phases in both experimental and
simulation data in the figure; the disordered model quantitatively reproduces
the key missing feature of the experimental data, replacing the cusps of the
disorder-free model with negative compressibility regimes as observed
experimentally.

We note in closing several open questions raised by our work. First, while
electron solids evidently dominate the ground states for $N>2$, it is likely
that they appear in the lower LLs as well, but are difficult to detect with
bulk methods where their subtle thermodynamic or transport phenomenology may
be overwhelmed by the incompressibility of the fractional quantum Hall states.
Second, it is unclear whether the particular orbital wavefunctions of single-
and multi-layer graphene may lead to any particularities in the electron solid
ground states as compared to semiconductor systems. Finally, our disorder
model is likely to be gross oversimplification. In particular, the lack of
observed magnetic field dependence in the sharpness of the phase transitions
is at odds with a model of quenched disorder where the effective broadening
$\Delta E$ would be expected to be magnet field independent. These and other
questions might be directly resolved via scanning tunneling microscopy
measurements of the real space structure of these phasesCoissard  _et al._
(2022); Liu  _et al._ (2022), as well as more detailed theoretical modeling
that accounts for the interplay of disorder, finite temperature, and
mesoscopic phase separation.

###### Acknowledgements.

The authors acknowledge discussions with M. Zaletel. This work was primarily
supported by Office of Naval Research under award N00014-23-1-2066. A.F.Y.
acknowledges the additional support of the Gordon and Betty Moore Foundation
EPIQS program under award GBMF9471. A portion of this work was performed at
the National High Magnetic Field Laboratory, which is supported by National
Science Foundation Cooperative Agreement No. DMR-1644779 and the State of
Florida. This work made use of shared facilities supported by the National
Science Foundation through Enabling Quantum Leap: Convergent Accelerated
Discovery Foundries for Quantum Materials Science, Engineering and Information
(Q-AMASE-i) award number DMR-1906325. K.W. and T.T. acknowledge support from
JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233). S.J. and B.S.
were supported by the NSF under Grant No. DMR-2045742.

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Supplementary Information

##  I Model of electron bubble states in N$>=$2 LL

###  S1 Energy of the bubble phases

The energy of the bubble crystals with $M$ electrons per bubble at a partial
filling $\nu^{*}$ is readily calculated with the help of the formulaGoerbig
_et al._ (2004a)

$E^{M}_{coh}(\nu^{*})=\frac{\nu^{*}}{2{\pi}l^{2}_{B}M}\sum_{\mu_{1},\mu_{2}}u^{HF}_{N}(G_{\mu_{1},\mu_{2}})\frac{J^{2}_{1}(\sqrt{2M}|G_{\mu_{1},\mu_{2}}|)}{|G_{\mu_{1},\mu_{2}}|^{2}}$
(S1)

where $l_{B}$ is the magnetic length, and
$G_{\mu_{1},\mu_{2}}={\mu_{1}}\mathbf{g_{1}}+{\mu_{2}}\mathbf{g_{2}}$ are
reciprocal lattice vectors, with
$\mathbf{g_{1}}=2{\pi}(\mathbf{e}_{x}-\mathbf{e}_{y}/\sqrt{3})/{\Lambda}_{B}$
and $\mathbf{g_{2}}=4{\pi}\mathbf{e}_{y}/\sqrt{3})/{\Lambda}_{B}$, where

${\Lambda}_{B}=\sqrt{\frac{4{\pi}M}{\sqrt{3}\nu^{*}}}l_{B}$ (S2)

is the lattice spacing. Furthermore,

$u^{HF}_{N}(\mathbf{q})=v_{N}(\mathbf{q})-\frac{2{\pi}l^{2}_{B}}{A}\sum_{\mathbf{p}}v_{N}(\mathbf{p})e^{-i(p_{x}q_{y}-p_{y}q_{x})l^{2}_{B}}$
(S3)

is the Hartree-Fock potential in the N-th LL, with

$v_{N}(\mathbf{q})=\frac{2{\pi}e^{2}}{{\epsilon}q}|F_{N}(ql_{B})|^{2}$ (S4)

being the effective interaction in this level, in terms of the LL form factor
($N\geq 1$)

$F_{N}(\mathbf{q})=\frac{1}{2}\left[L_{N}\left(\frac{x}{2}\right)+L_{N-1}\left(\frac{x}{2}\right)\right]e^{-x/4}$
(S5)

which takes into account the overlap between the LL wave functions, in terms
of Laguerre polynomials $L_{N}(x)$. Furthermore, we have used the measure
$\sum_{\mathbf{q}}=[A/(2\pi)^{2}]{\int}d^{2}q$, where $A$ is the total area.
Notice that the cohesive energy is an energy per particle, where the reference
energy is that of the uniform electron gas, where we do not only take into
account its exchange energy

$\begin{split}E_{ref}(\nu^{*})&=-\frac{\nu^{*}}{4{\pi}l^{2}_{B}}u^{HF}_{N}(\mathbf{q}=0)=-\frac{\nu^{*}}{2A}\sum_{\mathbf{p}}v_{N}(\mathbf{p})\\\
&=-\frac{\nu^{*}}{2}\left(\frac{e^{2}}{{\epsilon}l_{B}}\right){\int}^{\infty}_{0}dx[F_{N}(x)]^{2}\end{split}$
(S6)

where $x=ql_{B}$. In order to calculate the chemical potential of the system,
one needs to take into account also this contribution,

$\begin{split}{\mu}_{\nu^{*}}&=\frac{\partial[\nu^{*}(E^{M}_{coh}+E_{ref})]}{{\partial}{\nu^{*}}}\\\
&=E^{M}_{coh}+\nu^{*}\frac{{\partial}E^{M}_{coh}}{{\partial}{\nu^{*}}}-\nu^{*}\left(\frac{e^{2}}{{\epsilon}l_{B}}\right){\int}^{\infty}_{0}dx[F_{N}(x)]^{2}\end{split}$
(S7)

The integral in the last term is given by

${\int}^{\infty}_{0}dx[F_{N}(x)]^{2}=\frac{515}{1024}\sqrt{\frac{\pi}{2}}=0.630$
(S8)

Notice that the last term in Eq. (S7) can be viewed, similarly to the
reference energy (S6), as a reference chemical potential $\mu_{ref}(\nu^{*})$
that is due to the underlying structure-less electron liquid, and one has

$\mu_{ref}(\nu^{*})=2E_{ref}(\nu^{*})$ (S9)

Figure S1: Theoretical calculation of the cohesive energy for electron bubble
phases in the $N=3$ Landau level. The blue lines represent the cohesive energy
of M-electron bubble phases.

###  S2 Energy and chemical potential of the bubble phases for a screened
interaction potential in the $N=3$ Landau level

The above calculations do not taken into account the role of (non-local)
screening, but only the dielectric environment in the form of the overall
dielectric constant $\epsilon=4.5$ (for BN-encapsulated graphene). In
principle the screening of the Coulomb energy due to the metallic top and
bottom gates at distance $d$ from the graphene also needs to be taken into
account. If we have a top gate at distance $d_{1}$ and a bottom gate at
distance $d_{2}$, the $2{\pi}e^{2}/{\epsilon}q$ Coulomb potential needs to be
multiplied by a ”form factor”

$f_{d_{1},d_{2}}(q)=2\frac{\tanh(d_{1}q)\tanh(d_{2}q)}{\tanh(d_{1}q)+\tanh(d_{2}q)}$
(S10)

which reduces to $f_{d,d}(q)=\tanh(dq)$ in the case of a symmetric arrangement
of the gates, with $d=d_{1}=d_{2}$. However, in the present experiment, we can
safely neglect the effect of the gate screening, which are at distances much
larger compared with the magnetic length (and the cyclotron radius), so that
$dq\sim d/l_{B}\gg 1$, in which limits $f_{d_{1},d_{2}}\simeq 1$.

Figure S2: Static dielectric function for $N=3$ as a function of the wave
vector, calculated from Eq.(S14).

In contrast to the gate screening, one needs to take into account the
screening due to virtual excitations to Landau levels adjacent to $N=3$. This
is captured in the (static) dielectric function, which one obtaines from an
RPA calculation

$v^{scr}_{N}(\mathbf{q})=\frac{2{\pi}e^{2}}{\epsilon\epsilon_{N}(q)q}|F_{N}(nl)B)|^{2}$
(S11)

in terms of the polarizability $\Pi^{0}_{n}(q,\omega)$ for non-interacting
electrons in a magnetic fieldShizuya (2007); Roldán  _et al._ (2010). The
polarizability consists of a sum over all inter-Landau-level transitions
allowed by the Pauli principle and has a particular form due to the presence
of inter-band excitations from the valence to the conduction band. In contrast
to the polarizability of the usual 2D electron gas, it is not cut off above
twice the Fermi wave vactor $k_{F}\simeq\sqrt{2n}/l_{B}$, but continues to
grow linearly. This linearity provides us with a contribution to the
dielectric constant in the large wave-vector limit

$\epsilon_{n}(q\rightarrow\infty)=1+\pi\alpha_{G}/2+\alpha_{G}k_{F}/q$ (S12)

where $\alpha_{G}=e^{2}/{\hbar}v_{F}\epsilon=2.2/\epsilon\simeq 0.49$ is the
graphene fine-structure constantRoldán  _et al._ (2010). Furthermore, the
asymptotic behavior in the large-$q$ limit shows that one retrieves a simple
Thomas-Fermi screening on top of this constant background, as indicated by the
last term in the above equation. In th opposite small wave-vector limit, the
static dielectric function behaves as

$\epsilon_{n}(q\rightarrow 0)=1+2\pi\alpha_{G}N^{3/2}ql_{B}$ (S13)

while one has a maximum that scales as $\epsilon_{n}(q\sim
1/R_{C})\sim\alpha_{G}n$, where $R_{C}=\sqrt{2N}l_{B}$ is the graphene
cyclotron radius. A convenient interpolation formula that reproduces well the
static dielectric function is given by

$\begin{split}\epsilon_{n}(q)=1+2\pi\alpha_{G}N\left[\frac{qR_{C}}{\sqrt{2}}e^{-q^{2}R_{C}^{2}/2}+\left(q^{2}l_{B}^{2}+\frac{2\sqrt{2}N}{\pi}ql_{B}\right)\left(1-e^{-1/(2q^{2}R_{C}^{2})}\right)\right]\end{split}$
(S14)

The form of the static dielectric function for the Landau level $N=3$ is shown
in Fig. S2.

The screened interaction potential is now obtained by dividing Eq. S4 by
$\epsilon_{n}(q)$

$v^{scr}_{N}(\mathbf{q})=\frac{2{\pi}e^{2}}{{\epsilon}{\epsilon}_{N}(q)q}|F_{N}(ql_{B})|^{2}$
(S15)

Naturally, this affects also the exchange as well as the Hatree-Fock potential
Eq. S3. Finally, one notices that the reference energy S6 now reads

$E^{scr}_{ref}(\nu^{*})=-\frac{\nu^{*}}{2}\left(\frac{e^{2}}{{\epsilon}l_{B}}\right){\int}^{\infty}_{0}dx\frac{[F_{N}(x)]^{2}}{\epsilon_{N}(x)}$
(S16)

The integral yields the numerical value

${\int}^{\infty}_{0}dx\frac{[F_{N}(x)]^{2}}{\epsilon_{N}(x)}=0.143$ (S17)

and one notices here already a reduction by a factor of 4.5. One thus obtains
the reference energy

$E^{scr}_{ref}(\nu^{*})=-0.0715\nu^{*}\left(\frac{e^{2}}{{\epsilon}l_{B}}\right)$
(S18)

and the reference chemical potential of the uncorrelated liquid state.

$\mu^{scr}_{ref}(\nu^{*})=-0.143\nu^{*}\left(\frac{e^{2}}{{\epsilon}l_{B}}\right)$
(S19)

The cohesive energies are calculated by numerical integration since there is
no compact analytical formula for the exchange potential in the case of the
screened interaction potential. The energy of the cohesive energy for the
$N=3$ LL are given by the fitted formulas (to order three in the filling
factor measured from the energy minimum; Fig. S3b.)

$\frac{E^{scr,M=1}_{coh}(\nu^{*})}{e^{2}/{\epsilon}l_{B}}=-0.0311+0.952(\nu^{*}-0.132)^{2}+5.14(\nu^{*}-0.132)^{3}$
(S20)

$\frac{E^{scr,M=2}_{coh}(\nu^{*})}{e^{2}/{\epsilon}l_{B}}=-0.0312+0.412(\nu^{*}-0.21)^{2}+0.812(\nu^{*}-0.21)^{3}$
(S21)

and

$\frac{E^{scr,M=3}_{coh}(\nu^{*})}{e^{2}/{\epsilon}l_{B}}=-0.0291+0.287(\nu^{*}-0.265)^{2}+0.014(\nu^{*}-0.265)^{3}$
(S22)

Figure S3: Theoretical calculation of energy per electrons for electron bubble
phases in the $N=2$(a), $N=3$(b), and $N=4$(c) LLs. The blue lines represent
the energy of M-electron bubble phases. In the $N=2$ LL, we also plot the
energy of the Laughlin states for partial filling factor $\nu^{*}=1/3$ and
$\nu^{*}=1/5$. While at $\nu^{*}=1/3$ the electron bubble phase has lower
energy, at $\nu^{*}=1/5$ the FQH state is more stable, which indeed agree with
the experimental observation. Around half filling, previous calculation in
GaAs shows the unidirectional CDW phase dominatesGoerbig  _et al._ (2004b),
which we didn’t consider here.

###  S3 Energy and chemical potential of the $N=2$ bubble phases for a
screened interaction potential

In the $N=2$ LL, the FQH states are in competition with the electron bubble
states. Here we show the calculation of the energy of the Laughlin states at
$\nu^{*}=1/3$ and $\nu^{*}=1/5$. Their cohesive energy can be calculated by
the Haldane’s pseudo-potential

$\frac{V^{N}_{2m+1}}{e^{2}/{\epsilon}l_{B}}=2{\pi}{\int}^{\infty}_{0}dxxv^{scr}_{N}(x)L_{2m+1}(x^{2})e^{-x^{2}/2}$
(S23)

and the cohesive energy

$E^{L}_{coh}(\nu^{*}=1/(2s+1))=\frac{\nu}{\pi}\sum^{\infty}_{m=0}c^{s}_{2m+1}V^{N}_{2m+1}$
(S24)

The coefficients $c^{s}_{2m+1}$ characterize the Laughlin state at
$\nu^{*}=1/(2s+1)$ and can be calculated, within a good approximation, with
the help of sum rules from the plasma analogyGoerbig  _et al._ (2004a). With
this method and taking into account the reference energy

$E^{scr}_{ref}(\nu^{*})=-0.1\nu^{*}\left(\frac{e^{2}}{{\epsilon}l_{B}}\right)$
(S25)

One obtains

$E^{L}(\nu^{*}=1/3)=-0.0646\left(\frac{e^{2}}{{\epsilon}l_{B}}\right)$ (S26)

and

$E^{L}(\nu^{*}=1/5)=-0.0708\left(\frac{e^{2}}{{\epsilon}l_{B}}\right)$ (S27)

for the energy per particle of the Laughlin states. We notice that only the
quantum liquid at $\nu^{*}=1/5$ is stable, while the $1/3$-state has a higher
energy than the $M=2$ bubble phase at the same filling (Fig. S3a, black
crosses).

The cohesive energies of the pure bubble crystals now read

$\frac{E^{scr,M=1}_{coh}(\nu^{*})}{e^{2}/{\epsilon}l_{B}}=-0.0478+0.478(\nu^{*}-0.16)^{2}+2.53(\nu^{*}-0.16)^{3}$
(S28)

$\frac{E^{scr,M=2}_{coh}(\nu^{*})}{e^{2}/{\epsilon}l_{B}}=-0.0439+0.319(\nu^{*}-0.245)^{2}-0.0217(\nu^{*}-0.245)^{3}$
(S29)

and

$\frac{E^{scr,M=3}_{coh}(\nu^{*})}{e^{2}/{\epsilon}l_{B}}=-0.038+0.347(\nu^{*}-0.305)^{2}-0.539(\nu^{*}-0.305)^{3}$
(S30)

The $M=3$ bubble crystal in the vicinity of $\nu^{*}=1/2$ is in competition
with a stripe phase or a highly anisotropic Wigner crystal, and therefore
might not exist as the ground state.

###  S4 Energy and chemical potential of the $N=4$ bubble phases for a
screened interaction potential

Finally we show the calculated energy per particle of the bubble phases and
the associated chemical potentials in the $N=4$ LL (Fig. S3c). As expected
from the scaling $M_{max}=N$ of the bubble crystal with the maximum number of
particles per bubble in the $N$-th LL, we need to consider a bubble crystal
with four electrons per bubble now. (Notice that the originally proposed
$M_{max}=N+1$ bubble crystal is usually covered by the stripe or the
anisotropic Wigner crystal phase around half filling). In the $N=4$ LL, the
reference energy is given by

$E^{scr}_{ref}(\nu^{*})=-0.0559\nu^{*}\left(\frac{e^{2}}{{\epsilon}l_{B}}\right)=\frac{\mu^{scr}_{ref}(\nu^{*})}{2}$
(S31)

while the cohesive energies of the pure bubble crystals read

$\frac{E^{scr,M=1}_{coh}(\nu^{*})}{e^{2}/{\epsilon}l_{B}}=-0.0221+1.409(\nu^{*}-0.11)^{2}+7.77(\nu^{*}-0.11)^{3}$
(S32)

$\frac{E^{scr,M=2}_{coh}(\nu^{*})}{e^{2}/{\epsilon}l_{B}}=-0.023+0.625(\nu^{*}-0.182)^{2}+1.69(\nu^{*}-0.182)^{3}$
(S33)

$\frac{E^{scr,M=3}_{coh}(\nu^{*})}{e^{2}/{\epsilon}l_{B}}=-0.0229+0.454(\nu^{*}-0.235)^{2}-0.635(\nu^{*}-0.235)^{3}$
(S34)

and

$\frac{E^{scr,M=4}_{coh}(\nu^{*})}{e^{2}/{\epsilon}l_{B}}=-0.0216+0.342(\nu^{*}-0.275)^{2}-0.823(\nu^{*}-0.275)^{3}$
(S35)

##  II Theoretical estimate for the range of phase coexistence

Here we consider the general criteria for phase coexistence between two
distinct electronic phases at zero temperature and in the absence of disorder,
and we estimate the maximum range of filling factor $\delta\nu$ that such a
coexistence regime can occupy.

Consider a phase transition between two distinct phases of the electron
system, which we label as 1 and 2. If $E_{i}(n)$ denotes the energy per
electron as a function of the electron density $n$ in phase $i=1,2$, then at
zero temperature the free energy per unit area in phase $i$ is
$f_{i}(n)=nE_{i}(n)$. In the usual Maxwell construction for phase coexistence,
one considers that the total free energy for a fixed number of electrons is
minimized with respect to the area fraction $x$ that is occupied by one of the
two phases (say, phase 2). That is, if $n_{1},n_{2}$ denote the electron
densities in spatial regions occupied by phases $1$ and $2$ respectively, then
the region of phase coexistence can be found by minimizing the total free
energy $(1-x)f_{1}(n_{1})+xf_{2}(n_{2})$ with respect to $x$, $n_{1}$, and
$n_{2}$ for a fixed average density $\bar{n}=(1-x)n_{1}+xn_{2}$. From this
procedure one can easily show that, within the regime of phase coexistence,
the two phases have the same chemical potential $\mu_{i}=df_{i}(n)/dn$.

In 2D electron systems, however, there is an additional Coulombic energy cost
associated with the difference between the local electron density ($n_{1}$ or
$n_{2}$) and the charge density of the positive background (which for large
gate separation can be treated as spatially uniform, with charge density
$+e\bar{n}$), which leaves one phase with a net positive charge and the other
with a net negative charge. This Coulombic energy suppresses phase
coexistence, and generally prohibits the formation of macroscopic phase
domains (since a region of size $L$ and charge density $\eta$ has a Coulombic
self-energy per unit area that grows extensively with the region size,
$\sim\eta^{2}L$). Thus, phase coexistence, where present, can only take the
form of “microemulsion”-type phases Spivak and Kivelson (2004); Spivak  _et
al._ (2010), in which stripes or droplets of the less abundant phase are
interspersed periodically among the more abundant phase.

To derive the criteria for phase coexistence including this Coulomb energy
cost, one should minimize the total free energy per unit area

$f=\left(1-x\right)f_{1}\left(n_{1}\right)+xf_{2}\left(n_{2}\right)+e_{m},$
(S36)

where $e_{m}$ is the energy density of mixing, which contains both
electrostatic and surface energy terms. The energy $e_{m}$ has been considered
in detail in Ref. Ortix  _et al._ (2006) for both the droplet and the stripe
geometry. For the case of droplet configurations, $e_{m}$ is well-approximated
by

$e_{m}=\frac{8}{\sqrt{3}}|n_{2}-n_{1}|\sqrt{\frac{e^{2}\gamma}{\epsilon}}x(1-x),$
(S37)

where $\gamma$ is the surface tension. (In stripe configurations there is an
additional logarithmic factor Ortix  _et al._ (2006).) The corresponding
optimal droplet size $R_{d}$ is

$R_{d}=\frac{\sqrt{3}}{2\sqrt{x\left(1-x\right)}}\frac{\sqrt{e^{2}\gamma/\epsilon}}{\left(e^{2}/\epsilon\right)|n_{2}-n_{1}|}.$
(S38)

Thus, both the droplet size and the energy of mixing depend on the parameter
$\beta\equiv\sqrt{e^{2}\gamma/\epsilon}$, which has units of energy. While in
the usual Maxwell construction the two phases have the same chemical
potential, the inclusion of the term $e_{m}$ implies that two coexisting
phases have a difference in chemical potential $\approx 8\beta/\sqrt{3}$.

This approach gives two primary criteria for the existence of a mixed phase,
which ultimately place a constraint on the range of average density $\bar{n}$
that such a mixed phase may occupy. First, the difference in chemical
potential between the two phases must be equal to $8\beta/\sqrt{3}$. The
chemical potential difference is maximal when the two phases have a density
nearly equal to the value $n_{c}$ at which the two phases have the same
energy, $E_{1}(n_{c})=E_{2}(n_{c})$. Correspondingly phase coexistence is only
permitted when

$\beta<\frac{\sqrt{3}}{8}\left|\mu_{2}(n_{c})-\mu_{1}(n_{c})\right|.$ (S39)

So phase coexistence requires the parameter $\beta$ to be small enough.

On the other hand, if $\beta$ is too small, then the length scale associated
with microemulsion droplets becomes unrealistically small [see Eq. (S38)].
Specifically, phase coexistence in the sense of mesoscopic domains of two
distinct phases is not possible unless the corresponding droplet size is much
larger than the inter-electron spacing: $R_{d}\gg n_{1,2}^{-1/2}$. This
condition sets a _lower_ bound on the parameter $\beta$:

$\beta\gg\frac{e^{2}|n_{2}-n_{1}|}{\sqrt{3}\epsilon n_{1,2}^{1/2}}.$ (S40)

Combining Eqs. (S39) and (S40) produces an inequality for the maximum
difference in density between the two phases:

$|n_{2}-n_{1}|\ll\frac{3}{8}\frac{\left|\mu_{2}(n_{c})-\mu_{1}(n_{c})\right|n_{c}^{1/2}}{e^{2}/\epsilon}.$
(S41)

Inserting units of the magnetic length, we arrive at

$\delta\nu\ll\frac{3\sqrt{2\pi}}{8}\frac{\left|\delta\mu(\nu_{c})\right|}{e^{2}/\epsilon
l_{B}}\nu_{c}^{1/2},$ (S42)

where $\delta\nu$ is the window of filling factor that corresponds to phase
coexistence, $\nu_{c}$ is the filling factor at which the two phases have the
same energy, and $\delta\mu(\nu_{c})$ is the difference in chemical potential
between the phases at $\nu=\nu_{c}$.

For the case depicted in Fig. 3 of the main text, which illustrates three
phase transitions between distinct bubble phases, Eq. (S42) gives
$\delta\nu<1.8\times 10^{-3}$, $\delta\nu<2.5\times 10^{-3}$, and
$\delta\nu<2.8\times 10^{-3}$ for the respective windows of phase coexistence.

##  III Lindemann criterion for the melting of the bubble phase

In order to understand the orders of magnitude for the melting of the bubble
crystal, one may appeal to the phenomenological Lindemann criterion Lindemann
(1910) according to which a crystal melts if the temperature-induced position
fluctuations $\Delta x$ are fraction of the lattice spacing $\Lambda_{B}$. The
thumb rule is given by Lindemann (1910)

$\Delta x=C\Lambda_{B},\qquad\text{with}\qquad C\simeq 0.15.$ (S43)

Within the harmonic approximation, the potential-energy scale is given by

$E_{\text{pot}}\simeq\frac{e^{2}}{\epsilon\Lambda_{B}}\left(\frac{\Delta
x}{\Lambda_{B}}\right)^{2},$ (S44)

which needs to be identified with the thermal energy $k_{B}T$ so that one
obtains for the critical temperature

$k_{B}T^{*}\simeq C^{2}\frac{e^{2}}{\epsilon\Lambda_{B}}\simeq
C^{2}\sqrt{\frac{\sqrt{3}\nu^{*}}{4\pi M}}\left(\frac{e^{2}}{\epsilon
l_{B}}\right),$ (S45)

where we have made use of the expression (S2) for the spacing in the bubble
crystal. If we consider a Coulomb energy scale of

$\frac{e^{2}}{k_{B}\epsilon l_{B}}\simeq
110\times\sqrt{B[\text{T}]}\,\text{K},$ (S46)

which is appropriate for graphene encapsulated in h-BN, this yields roughly
400 K at a magnetic field of $B=13$ T, as in our experiments (see Fig. 2 of
the main text). For a filling factor $\nu^{*}=0.2$, where we expect an $M=2$
bubble phase in $N=4$, this yields a critical temperature $T^{*}\simeq 1$ K,
in good agreement with our experimental findings.

Notice, however, that the above arguments remain valid only on the level of
orders of magnitude. Indeed, since the crystals do not consist of pointlike
lattice sites but rather of complex bubbles with an internal structure, the
melting mechanism in the bubble phase may be more complicated than it is
suggested by the simple Lindemann criterion. Furthermore, the interaction is
not given by a simple Coulomb potential, but it has strong corrections at
short distances that are expected to affect the electron dynamics inside each
bubble.