# Disorder-driven phase transitions in bosonic fractional quantum Hall liquids

Chao Han and Zhao Liu  Zhejiang Institute of Modern Physics, Zhejiang
University, Hangzhou 310027, China

###### Abstract

We investigate the disorder-driven phase transitions in bosonic fractional
quantum Hall liquids at filling factors $f=1/2$ and $f=1$ in the lowest Landau
level. We use the evolution of ground-state entanglement entropy, fidelity
susceptibility, and Hall conductance with increasing disorder strength to
identify the underlying phase transitions. The critical disorder strengths
obtained from these different quantities are consistent with each other,
validating the reliability of our numerical calculations based on exact
diagonalization. At $f=1/2$, we observe a clear transition from the bosonic
Laughlin state to a trivial insulating phase. At $f=1$, we identify a direct
phase transition from the non-Abelian bosonic Moore-Read state to a trivial
insulating phase, although some signs of a disorder-induced intermediate
fractional quantum Hall phase were recently reported for the $f=5/2$ fermionic
cousin.

##  I Introduction

The phase transition is one of the most classical and long-standing problem in
condensed matter physics. While Landau’s symmetry-breaking theory gains great
success in describing phase transitions, the discovery of integer Klitzing
_et al._ (1980) and the more complex, interaction-induced fractional quantum
Hall (FQH) effects Tsui  _et al._ (1982) in two-dimensional electron gases
(2DEGs) penetrated by strong magnetic fields clearly indicates the existence
of novel topological phases WEN (1990) beyond the symmetry-breaking paradigm.
Since the 1980s, various FQH states, their competing phases, and the
transitions between them have been extensively studied Laughlin (1983);
Haldane (1983); Jain (1989); Moore and Read (1991); Morf (1998); Stormer
(1999); McDonald and Haldane (1996); Ardonne and Schoutens (1999); Pan  _et
al._ (2003); Gervais  _et al._ (2004); Jolicoeur (2007); Papić  _et al._
(2010); Archer and Jain (2013); Geraedts  _et al._ (2015); Liu  _et al._
(2015); Peterson  _et al._ (2015); Pakrouski  _et al._ (2015); Thiebaut  _et
al._ (2015); Gils  _et al._ (2009); Samkharadze  _et al._ (2016).

Disorder is a ubiquitous ingredient that may drive phase transitions in FQH
systems. Although weak disorder is responsible for the characteristic plateaus
of Hall conductance when the FQH effect occurs, sufficiently strong disorder
can destroy FQH phases by closing the spectral and mobility gaps.
Unfortunately, due to the exponentially large Hilbert space of the underlying
many-body system as well as the breaking of spatial symmetry, it is
challenging to study in the microscopic level how disorder affects FQH phases.
In the past decades, only a few numerical studies tackle this problem for
fermionic FQH systems Sheng  _et al._ (2003); Wan  _et al._ (2005); Friedman
_et al._ (2011); Liu and Bhatt (2016, 2017); Zhu and Sheng (2019). By tracking
the evolution of the ground-state energy gap, Hall conductance, and
entanglement entropy as a function of disorder strength, disorder-driven
transitions from Abelian and non-Abelian FQH phases to trivial phases were
identified in microscopic models. In particular, for electrons at filling
$f=5/2$ where numerical simulations support either the non-Abelian fermionic
Moore-Read (MR) state or its particle-hole (PH) conjugate anti-MR state as the
ground state in the zero-disorder limit Moore and Read (1991); Morf (1998);
Rezayi and Haldane (2000); Levin  _et al._ (2007); Lee  _et al._ (2007);
Peterson  _et al._ (2008a); Pakrouski  _et al._ (2015); Rezayi (2017), it was
found that enhancing disorder with a finite correlation length might first
drive the system into an intermediate FQH phase before completely ruining the
topological order Zhu and Sheng (2019). This observation may provide deep
insight into the nature of the mysterious $f=5/2$ FQH effect which is still
under debate Son (2015); Wan and Yang (2016); Zucker and Feldman (2016);
Banerjee  _et al._ (2018); Mishmash  _et al._ (2018); Mross  _et al._ (2018);
Wang  _et al._ (2018).

Most studies of FQH physics focus on electrons because of the direct relevance
to 2DEG experiments. However, it has been proposed that bosons in rapidly
rotating atomic gases or optical lattices can also form FQH states, where an
effective magnetic field is created by rotation or laser beams Cooper  _et
al._ (2001); Regnault and Jolicoeur (2003); Rezayi  _et al._ (2005); Sørensen
_et al._ (2005); Regnault and Jolicoeur (2007); Cooper (2008); Fetter (2009);
Möller and Cooper (2009); Cooper and Dalibard (2013). From the viewpoint of
the wave function, there is a correspondence between the fermionic and bosonic
Read-Rezayi series Read and Rezayi (1999); Cooper  _et al._ (2001), including
the Laughlin Laughlin (1983) and MR states, in which one can obtain a bosonic
state from a fermionic one by simply removing a proper Jastrow factor to
ensure the correct statistics. This fact immediately raises several
interesting questions. How does disorder affect bosonic FQH states? Is the
disorder-driven physics for bosonic FQH states similar to the corresponding
fermionic cases? In particular, is there a disorder-induced intermediate FQH
phase between the bosonic MR state and the trivial strong-disorder limit?

To investigate these problems, in this paper we study the effect of disorder
on bosonic FQH liquids at filling fractions $f=1/2$ and $f=1$ in the lowest
Landau level (LLL). For contact-interacting bosons at $f=1/2$, we observe a
direct phase transition from the bosonic Laughlin state to a trivial
insulator, which is identified by the special behavior of the ground-state
entanglement entropy, fidelity susceptibility, and Hall conductance at the
critical disorder strength. This transition is similar to the one from the
$f=1/3$ fermionic Laughlin state to an insulating phase reported in Refs.
Sheng  _et al._ (2003); Wan  _et al._ (2005); Liu and Bhatt (2016). However,
such a similarity between bosons and fermions seems to be absent at the MR
filling. As reported in Ref. Zhu and Sheng (2019), a disorder-induced
intermediate FQH phase for $f=5/2$ Coulomb-interacting fermions might exist.
By contrast, we have checked $f=1$ bosons interacting via either the contact
repulsion or the Coulomb potential, but do not find any clear signature of an
intermediate phase. Instead, we identify only a single transition from the
bosonic MR state to a trivial insulating phase. Promising candidates for the
disorder-induced intermediate phase of $f=5/2$ fermions include a PH symmetric
puddle structure composed of fermionic MR and anti-MR domains Mross  _et al._
(2018); Wang  _et al._ (2018); Lian and Wang (2018); Zhu and Sheng (2019) and
a PH symmetric state Jolicoeur (2007); Zucker and Feldman (2016); Balram  _et
al._ (2018); Rezayi  _et al._ (2021). As the usual PH transformation is only
defined for fermions, the absence of an intermediate phase for $f=1$ bosons in
our models does not contradict with the possible existence of an intermediate
PH symmetric phase for $f=5/2$ fermions.

The remainder of this paper is organized as follows. In Sec. II, we introduce
our model and method in detail, including the many-body Hamiltonian, the
disorder model, and the definitions of ground-state entanglement entropy,
fidelity susceptibility, and Hall conductance which are used to identify the
phase transitions. In Sec. III, we discuss the results at $f=1/2$, which will
also benchmark the validity of our method. Then we will move to $f=1$ in Sec.
IV. Finally, our conclusions, discussions and outlook are presented in Sec. V.

##  II Model and method

###  II.1 Model

We consider $N$ bosons in a two-dimensional (2D) random potential $U(\bf{r})$
on an $L_{1}\times L_{2}$ rectangular torus penetrated by a uniform
perpendicular magnetic field. Bosons interact with each other via a
translationally invariant two-body interaction $V({\bf r})$. After using the
magnetic length $\ell_{B}$ as the length unit and choosing the Landau gauge,
we can write the LLL single-particle orbitals as

$\displaystyle\psi_{m}({\bf
r})=\left(\frac{1}{\sqrt{\pi}L_{2}}\right)^{\frac{1}{2}}\sum_{n=-\infty}^{+\infty}$
$\displaystyle e^{i\frac{2\pi}{L_{2}}\left(m+nN_{\phi}\right)y}$ (1)
$\displaystyle\times
e^{-\frac{1}{2}\left[x-\frac{2\pi}{L_{2}}\left(m+nN_{\phi}\right)\right]^{2}},$

where ${\bf r}=(x,y)$ is the real-space coordinate, $N_{\phi}$ is the number
of magnetic flux quanta penetrating the torus, and $m=0,\dots,N_{\phi}-1$ is
the orbital index. As required by the magnetic translational invariance on the
torus, we have $L_{1}L_{2}=2\pi N_{\phi}$ Haldane (1985). To approach the 2D
limit and preclude other competing phases such as the stripe phase Fogler ;
Cooper  _et al._ (2005); Seki and Ino (2008), we choose the square torus with
$L_{1}=L_{2}=\sqrt{2\pi N_{\phi}}\equiv L$ throughout this paper.

We focus on the strong-field situation such that both the interaction and
disorder are small compared with the Landau level spacing. In this case, the
system’s Hamiltonian is $\sum_{i<j}^{N}V({\bf r}_{i}-{\bf
r}_{j})+\sum_{i=1}^{N}U({\bf r}_{i})$ projected to the LLL, where ${\bf
r}_{i}$ is the coordinate of the $i$th boson. In the Fock space spanned by the
single-particle orbitals [Eq. (1)], the LLL-projected Hamiltonian after second
quantization takes the form of

$\displaystyle H=$
$\displaystyle\sum_{m_{1},m_{2},m_{3},m_{4}=0}^{N_{\phi}-1}V_{m_{1},m_{2},m_{3},m_{4}}c_{m_{1}}^{\dagger}c_{m_{2}}^{\dagger}c_{m_{3}}c_{m_{4}}$
(2)
$\displaystyle+\sum_{m_{1},m_{2}=0}^{N_{\phi}-1}U_{m_{1},m_{2}}c_{m_{1}}^{\dagger}c_{m_{2}},$

where $c^{{\dagger}}_{m}$ ($c_{m}$) creates (annihilates) a boson in the LLL
orbital $m$, and the interaction and disorder matrix elements are

$\displaystyle V_{\left\\{m_{i}\right\\}}=$
$\displaystyle\frac{1}{2}\delta_{m_{1}+m_{2},m_{3}+m_{4}}^{\bmod
N_{\phi}}\sum_{s,t=-\infty}^{+\infty}\delta_{t,m_{1}-m_{4}}^{\bmod
N_{\phi}}V_{\mathbf{q}}$ (3) $\displaystyle\times
e^{-\frac{1}{2}|\mathbf{q}|^{2}}e^{i\frac{2\pi
s}{N_{\phi}}\left(m_{1}-m_{3}\right)}$

and

$U_{\left\\{m_{i}\right\\}}=\sum_{s,t=-\infty}^{+\infty}\delta_{t,m_{1}-m_{2}}^{\bmod
N_{\phi}}U_{\mathbf{q}}e^{-\frac{1}{4}|\mathbf{q}|^{2}}e^{\mathrm{i}\frac{\pi
s}{N_{\phi}}\left(2m_{1}-t\right)},$ (4)

respectively. In Eqs. (3) and (4), $\delta_{i,j}^{\bmod N_{\phi}}$ is the
periodic Kronecker delta function with period $N_{\phi}$, ${\bf
q}=(q_{x},q_{y})=(2\pi s/L_{1},2\pi t/L_{2})$ with $|{\bf
q}|^{2}=q^{2}_{x}+q^{2}_{y}$, and $V_{\bf q}=\frac{1}{2\pi N_{\phi}}\int
V({\bf r})e^{-{\rm i}{\bf q}\cdot{\bf r}}d{\bf r}$ and
$U_{\bf{q}}=\frac{1}{2\pi N_{\phi}}\int U({\bf r})e^{-{\rm i}{\bf q}\cdot{\bf
r}}d{\bf r}$ are the Fourier transforms of $V({\bf r})$ and $U({\bf r})$,
respectively. To study the effects of correlated disorder, we use the Gaussian
correlated random potential for $U({\bf r})$, which satisfies $\left\langle
U(\mathbf{r})U\left(\mathbf{r}^{\prime}\right)\right\rangle=\frac{W^{2}}{2\pi\xi^{2}}e^{-\frac{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{2}}{2\xi^{2}}}$
and $\left\langle
U_{\mathbf{q}}U_{\mathbf{q^{\prime}}}\right\rangle=\frac{W^{2}}{2\pi
N_{\phi}}\delta_{\mathbf{q},-\mathbf{q}^{\prime}}e^{-q^{2}\xi^{2}/2}$, where
$\xi$ is the characteristic correlation length, $W$ is the disorder strength,
and $\left\langle\cdots\right\rangle$ represents the sample average. If
$\xi=0$, we return to the Gaussian white noise. When averaging over $N_{s}$
samples, we estimate the error bar of quantity $A$ by
$\sqrt{\left(\left\langle A^{2}\right\rangle-\langle
A\rangle^{2}\right)/\left(N_{s}-1\right)}$.

###  II.2 Entanglement entropy

Topologically ordered phases are characterized by the underlying pattern of
quantum entanglement Eisert  _et al._ (2010); Kitaev and Preskill (2006);
Levin and Wen (2006). Previous works have found that the ground-state
entanglement entropy and its derivative with respect to the disorder strength
provide sharp signatures of the disorder-driven phase transitions in FQH
systems Liu and Bhatt (2016, 2017); Zhu and Sheng (2019). Here, we will apply
this diagnosis to bosonic systems. For all cases that we consider, the system
at zero and weak disorder is in a topological FQH phase (the bosonic Laughlin
state at $f=1/2$ and the bosonic MR state at $f=1$) with a well-defined
ground-state manifold consisting of $D$ approximately degenerate states, which
are separated by a finite energy gap from other highly excited states. This
ground-state degeneracy is a character of the underlying FQH topological order
Wen and Niu (1990). Therefore, it is natural to extract the entanglement
entropy from this ground-state manifold at weak disorder. For consistency, we
will always choose the lowest $D$ eigenstates $|\Psi_{i=1,\ldots,D}\rangle$ of
the many-body Hamiltonian equation (2) as the ground-state manifold and
calculate its entanglement entropy at all disorder strengths, even if the
quasidegeneracy among states in this manifold disappears after the FQH phase
is destroyed by strong disorder.

Having defined the ground-state manifold, we divide the whole system into two
subsystems $A$ and $B$ to calculate the entanglement entropy between them. We
consider two kinds of half-half bipartition: (i) the orbital cut Haque  _et
al._ (2007), for which $A$ consists of orbitals $m=0,\dots,\left\lceil
N_{\phi}/2\right\rfloor-1$ while $B$ consists of the remaining orbitals,
respectively, where $\left\lceil a\right\rfloor$ is the integer part of $a$;
and (ii) the real-space cut Sterdyniak  _et al._ (2012); Dubail  _et al._
(2012), for which $A$ is the region with $x\in[0,L/2],y\in[0,L]$ and $B$ is
the complement of $A$. The entanglement entropy between $A$ and $B$ can be
measured by the von Neumann entropy
$S(\rho)=-\operatorname{Tr}\rho_{A}\ln\rho_{A}$, where $\rho$ is the density
matrix of the ground-state manifold and $\rho_{A}=\operatorname{Tr}_{B}\rho$
is the reduced density matrix of part $A$. Here, $\rho$ can be chosen as
either the average over all states in the ground-state manifold, i.e.,
$\bar{\rho}=\frac{1}{D}\sum_{i=1}^{D}\left|\Psi_{i}\right\rangle\left\langle\Psi_{i}\right|$,
or a single state
$\rho_{i}=\left|\Psi_{i}\right\rangle\left\langle\Psi_{i}\right|$. These two
choices correspond to the entanglement entropy $S(\bar{\rho})$ and
$\bar{S}=\frac{1}{D}\sum_{i=1}^{D}S(\rho_{i})$, respectively. Both
$S(\bar{\rho})$ and $\bar{S}$ include the contributions of all
$|\Psi_{i}\rangle$’s, thus minimizing the finite-size effect. We have checked
that they give similar results, but $\bar{S}$ suffers from larger finite-size
effects. Therefore our discussion will be based on the results of
$S(\bar{\rho})$ in what follows.

###  II.3 Fidelity susceptibility

We will also use the ground-state fidelity and fidelity susceptibility to
identify the disorder-driven phase transitions. For a parameter-dependent
Hamiltonian, fidelity is originally the overlap between two ground states at
different values of parameters Zanardi and Paunković (2006). In our case, it
can be generalized to the total overlap between two ground-state manifolds at
disorder strength $W$ and $W+\Delta W$, i.e.,

$F(W,\Delta
W)=\left(\frac{1}{D}\sum_{i,j=1}^{D}|\left\langle\Psi_{i}(W)|\Psi_{j}(W+\Delta
W)\right\rangle|^{2}\right)^{1/2}.$ (5)

Since the fidelity measures the similarity between states, the change in the
ground-state manifold at the phase transition should be reflected by a
dramatic drop in the fidelity across the critical point. A closely related
quantity is the fidelity susceptibility, defined as

$\chi(W)=\lim_{\Delta W\rightarrow 0}\frac{-2\ln F(W,\Delta W)}{\Delta
W^{2}},$ (6)

which should show a sharp peak diverging with the increasing system size at
the phase transition and can describe the universality class of the transition
You  _et al._ (2007); Gu  _et al._ (2008).

###  II.4 Hall conductance

The Hall conductance $\sigma_{H}$ can unambiguously distinguish FQH phases
with nonzero quantized $\sigma_{H}$ from insulating phases with $\sigma_{H}=0$
that could exist at strong disorder Sheng  _et al._ (2003); Wan  _et al._
(2005). $\sigma_{H}$ is simply proportional to the many-body Chern number of
the ground state Niu  _et al._ (1985). After imposing the twisted boundary
condition $\mathcal{T}_{\mu}\psi_{m}({\bf r})=e^{i\theta_{\mu}}\psi_{m}({\bf
r})$ on the single-particle-orbital equation (1), where $\mathcal{T}_{\mu}$ is
the magnetic translation operator Haldane (1985) in the $\mu=x,y$ direction
and $\theta_{\mu}$ is the boundary phase, we can calculate the many-body Chern
number

$C_{j}=\frac{i}{2\pi}\iint_{0}^{2\pi}d\theta_{x}d\theta_{y}\left(\left\langle\frac{\partial\Psi_{j}}{\partial\theta_{x}}\bigg{|}\frac{\partial\Psi_{j}}{\partial\theta_{y}}\right\rangle-\left\langle\frac{\partial\Psi_{j}}{\partial\theta_{y}}\bigg{|}\frac{\partial\Psi_{j}}{\partial\theta_{x}}\right\rangle\right)$
(7)

for the state $|\Psi_{j}\rangle$ in the ground-state manifold. Numerically, we
separate the $\theta_{x}$-$\theta_{y}$ space into dense meshes and calculate
the sum of the accumulated Berry phase (divided by $2\pi$) in
$|\Psi_{j}\rangle$ along the edges of each mesh to get $C_{j}$. Then we
compute the total many-body Chern number $C=\sum_{j=1}^{D}C_{j}$ of the
ground-state manifold, which gives the average ground-state Hall conductance
$\sigma_{H}=(C/D)q^{2}/h$ with $q$ being the boson’s charge. For an FQH phase
at filling $f$, we expect $C=fD$ and $\sigma_{H}=fq^{2}/h$. For an insulating
phase, we expect $C=0$ and $\sigma_{H}=0$.

##  III $f=1/2$ bosonic Laughlin filling

Let us first study the effect of disorder on the $f=1/2$ bosonic Laughlin
state to examine the validity of the diagnoses proposed in Secs. II.2–II.4. In
this section we assume that the bosons interact via the two-body contact
repulsion with constant $V_{\bf q}=2/N_{\phi}$ in Eq. (3), i.e., the zeroth
Haldane’s pseudopotential Haldane (1983). In this case, the ground state in
the absence of disorder is the model bosonic Laughlin state which is exactly
two-fold degenerate on the torus. The corresponding wave function on an
infinite plane Laughlin (1983) takes the simple form

$\Psi^{f=1/2}_{\text{Laughlin}}=\prod_{j<k}\left(z_{j}-z_{k}\right)^{2}\exp\left[-\frac{1}{4}\sum_{i}\left|z_{i}\right|^{2}\right],$
(8)

where $z_{j}=x_{j}-iy_{j}$ is the complex coordinate of the $j$th boson on the
plane. For the disorder term in Eq. (4), we first consider the Gaussian white
noise. One can imagine that the ground state stays in the bosonic Laughlin
phase when the disorder is weak but enters an insulating phase at strong
disorder with all bosons pinned at the minimum of the disorder potential.
Therefore at least one phase transition must occur at an intermediate disorder
strength.

Figure 1: The orbital-cut entanglement entropy $S^{o}(\bar{\rho})$ for
$N=4$-$9$ contact-interacting bosons at $f=1/2$. Disorder is modeled by the
Gaussian white noise. (a) $\langle S^{o}(\bar{\rho})\rangle$ vs $W$. (b)
$\langle dS^{o}(\bar{\rho})/dW\rangle$ vs $W$. For the finite-size scaling
analysis of $\langle S^{o}(W)\rangle$ and $\langle dS^{o}/dW\rangle$, we plot
$[\langle S^{o}(W)\rangle-\langle S^{o}(W_{c})\rangle]/L$ and $\langle
dS^{o}/dW\rangle/L^{1+1/\nu}$ vs $L^{1/\nu}(W-W_{c})$ in (c) and (d),
respectively, where we use $W_{c}=0.7$ and $\nu=0.6$ and neglect the smallest
system size $N=4$. We average over 10 000 samples for $N=4$-$7$, 3500 samples
for $N=8$, and 700 samples for $N=9$. Markers in (a)–(d) with the same color
refer to the same system size.

To identify the phase transition, we measure the orbital-cut entanglement
entropy (OEE), $S^{o}(\bar{\rho})$, as a function of disorder strength. At
each $W$, we consider different disorder configurations and calculate the
sample-averaged OEE $\left\langle S^{o}(\bar{\rho})\right\rangle$. Unlike for
the $f=1/3$ fermionic Laughlin state, where $\left\langle
S^{o}(\bar{\rho})\right\rangle$ monotonically decreases with increasing $W$
Liu and Bhatt (2016, 2017), here we find that $\left\langle
S^{o}(\bar{\rho})\right\rangle$ first grows and then decays when disorder
strength grows [Fig. 1(a)]. A peak in $\left\langle
S^{o}(\bar{\rho})\right\rangle$ develops near $W=0.7$ for all system sizes,
which seems to indicate a phase transition at $W\approx 0.7$.

We further locate the critical point $W_{c}$ and extract the critical exponent
$\nu$ of the correlation length via a scaling analysis of the OEE. As the OEE
evolves smoothly with the disorder strength, we expect a continuous phase
transition at $W_{c}$, with the correlation length

$\lambda\propto\left|W-W_{c}\right|^{-\nu}$ (9)

near the critical point, where $\nu$ is the critical exponent. Considering
that the entanglement entropy obeys the area law, we argue the scaling form

$S^{o}(W)-S^{o}(W_{c})\propto Lg[L^{\frac{1}{\nu}}(W-W_{c})]$ (10)

for the OEE, where $g$ is a universal function. By plotting the linear density
of the OEE $[S^{o}(W)-S^{o}(W_{c})]/L$ versus $L^{\frac{1}{\nu}}(W-W_{c})$, we
find that all data points in Fig. 1(a) (except those of the smallest system
size $N=4$) collapse onto a single curve when we choose $W_{c}\approx 0.7\pm
0.05$ and $\nu\approx 0.6$ [Fig. 1(c)]. Such a scaling analysis provides a
convincing indication of a phase transition at $W_{c}\approx 0.7\pm 0.05$.
Note that the critical exponent $\nu$ obtained here is close to the value for
$f=1/3$ fermions Liu and Bhatt (2016).

Compared with the entanglement entropy itself, its derivative with respect to
the disorder strength may provide a sharper fingerprint of the underlying
phase transition Liu and Bhatt (2016, 2017). We approximate
$dS^{o}(\bar{\rho})/dW$ in each sample as $[S^{o}(\bar{\rho})|_{W+\Delta
W}-S^{o}(\bar{\rho})|_{W}]/\Delta W$ with $\Delta W=0.001W$, where
$S^{o}(\bar{\rho})|_{W+\Delta W}$ is evaluated by only changing the magnitude
of $W$ by a small percentage but keeping the disorder configuration fixed. The
sample-averaged result is shown in Fig. 1(b). We observe one peak and one
valley in $\langle dS^{o}(\bar{\rho})/dW\rangle$ when $\langle
S^{o}(\bar{\rho})\rangle$ grows and decays with increasing $W$, respectively.
The magnitudes of both the peak and valley grow with increasing system size.
While it is difficult to confirm by our exact-diagonalization calculations in
relatively small systems, we may expect that the peak and valley both move to
the critical $W_{c}$ and their magnitudes diverge in the thermodynamic limit.
In this case, $\langle dS^{o}(\bar{\rho})/dW\rangle$ goes to positive
(negative) infinity when $W\rightarrow W_{c}$ from the left (right) side,
which is a sharp signature of the phase transition. For finite systems, such
divergence to positive and negative infinity is replaced by the peak and
valley, respectively. Naively we can estimate $W_{c}$ as the position where
$\langle dS^{o}(\bar{\rho})/dW\rangle\approx 0$, which again gives
$W_{c}\approx 0.7$. Moreover, according to Eq. (10), we suggest the scaling
form

$dS^{o}(W)/dW\propto L^{1+1/\nu}g^{\prime}[L^{\frac{1}{\nu}}(W-W_{c})]$ (11)

to the entropy derivative, where $g^{\prime}$ means the derivative of the
function $g$. In Fig. 1(d), we plot $\langle dS^{o}(W)/dW\rangle/L^{1+1/\nu}$
as a function of $L^{\frac{1}{\nu}}(W-W_{c})$ for $W_{c}=0.7$ and $\nu=0.6$.
Indeed, all data points in Fig. 1(b) (except those of the smallest system size
$N=4$) collapse to a single curve near the critical point.

At first glance, the non-monotonic behavior of the orbital-cut entanglement
entropy in Fig. 1(a) seems counterintuitive. With increasing disorder
strength, bosons are more likely trapped at the minima of the disorder
potential, and thus we would expect weaker real-space entanglement between
different regions of the system. Since the OEE is often argued to be a
reasonable, approximated measure of the real-space entanglement between two
subregions of the system due to the Gaussian localization of LLL orbitals
Haque  _et al._ (2007); Li and Haldane (2008), one may expect a monotonically
decaying OEE when disorder becomes stronger. However, this is contrary to our
numerical observation that the OEE evolves non-monotonically with the disorder
strength. We attribute this discrepancy to the imprecise capture of the real-
space entanglement by the orbital cut: All LLL orbitals have nonzero support
across the whole torus, so the orbital cut cannot precisely match the true
real-space cut Sterdyniak  _et al._ (2012); Dubail  _et al._ (2012). Compared
with fermions, this mismatch between orbital-cut and real-space entanglement
may be more obvious for bosons due to their multiple occupation in each LLL
orbital. Indeed, we did not observe the non-monotonic behavior of the OEE for
fermions in previous works Liu and Bhatt (2016, 2017).

Figure 2: (a) The sample-averaged real-space-cut entanglement entropy $\langle
S^{r}(\bar{\rho})\rangle$ and (b) the entropy derivative $\langle
dS^{r}(\bar{\rho})/dW\rangle$ vs $W$ for $N=4$-$6$ contact-interacting bosons
at $f=1/2$. Disorder is modeled by the Gaussian white noise. We average over
10 000 samples for $N=4$-$5$ and 1500 samples for $N=6$. Markers in (a) and
(b) with the same color refer to the same system size.

In Fig. 2(a), we show the real-space-cut entanglement entropy (REE)
$S^{r}(\bar{\rho})$ obtained by a true real-space bipartition of the system
instead of an orbital cut. Unlike $\langle S^{o}(\bar{\rho})\rangle$, $\langle
S^{r}(\bar{\rho})\rangle$ indeed monotonically decays with increasing disorder
strength, which is consistent with the picture that bosons are more likely
trapped at minima of the disorder potential such that the REE becomes smaller.
The derivative of $\langle S^{r}(\bar{\rho})\rangle$ with respect to the
disorder strength exhibits a single minimum which gets deeper and flows to
smaller $W$ for larger system sizes [Fig. 2 (b)]. Although we can only
calculate the REE for at most six bosons because its computational cost is
larger than that of the OEE Sterdyniak  _et al._ (2012); Dubail  _et al._
(2012), we still try a linear fitting of the minimum position of $\langle
dS^{r}(\bar{\rho})/dW\rangle$ versus $1/N$ and obtain $W_{c}\approx 0.78$ in
the thermodynamic limit, which is consistent with the critical point indicated
by the OEE.

Figure 3: The sample-averaged fidelity susceptibility $\langle\chi(W)\rangle$
vs $W$ for $N=4$-$9$ contact-interacting bosons at $f=1/2$. Disorder is
modeled by the Gaussian white noise. We average over 10 000 samples for
$N=4$-$7$, 3500 samples for $N=8$, and 700 samples for $N=9$. The vertical
dashed line indicates $W=0.7$, which is the critical disorder strength
extracted from the entanglement quantities in Fig. 1.

The diagnoses above based on entanglement measures suggest a single phase
transition with increasing $W$. To confirm this, we look for a clue of the
phase transition directly at the wave-function level. At each $W$, we fix the
disorder configuration and change $W$ by $\Delta W=0.001W$ to evaluate the
fidelity $F(W,\Delta W)$ in Eq. (5). Then the fidelity susceptibility
$\chi(W)$ is obtained from Eq. (6), which is further averaged over different
disorder configurations. We find a single pronounced maximum in
$\langle\chi(W)\rangle$ which grows for larger system sizes and probably
diverges in the thermodynamic limit (Fig. 3). Such a single peak indicates a
single abrupt change in the ground-state manifold, i.e., a single phase
transition. The location of the peak for the largest three system sizes gives
a rough estimate of $W_{c}\approx 0.75$, agreeing with the prediction of the
entanglement analysis.

Figure 4: The sample-averaged many-body Chern number $\langle C\rangle$ vs $W$
for $N=4$-$7$ contact-interacting bosons at $f=1/2$. Disorder is modeled by
the Gaussian white noise. We average over 10 000 samples for $N=4$-$5$, 3000
samples for $N=6$, and 1500 samples for $N=7$. The vertical dashed line
indicates $W=0.7$, which is the critical disorder strength extracted from the
entanglement quantities in Fig. 1. Figure 5: The orbital-cut entanglement
entropy $S^{o}(\bar{\rho})$ for even $N=6$-$12$ Coulomb-interacting bosons at
$f=1$. Disorder is modeled by the Gaussian white noise. (a) $\langle
S^{o}(\bar{\rho})\rangle$ vs $W$. (b) $\langle dS^{o}(\bar{\rho})/dW\rangle$
vs $W$. For the finite-size scaling analysis of $\langle S^{o}(W)\rangle$ and
$\langle dS^{o}/dW\rangle$, we plot $[\langle S^{o}(W)\rangle-\langle
S^{o}(W_{c})\rangle]/L$ and $\langle dS^{o}/dW\rangle/L^{1+1/\nu}$ vs
$L^{1/\nu}(W-W_{c})$ in (c) and (d), respectively, where we use $W_{c}=0.16$
and $\nu=0.5$. We average 10 000 samples for $N=6$ and $8$, 2000 samples for
$N=10$, and 500 samples for $N=12$. Markers in (a)–(d) with the same color
refer to the same system size.

In the end, we track the evolution of the many-body Chern number $C$ of the
ground-state manifold as a function of disorder strength, which can
unambiguously distinguish trivial insulators from FQH states. In the Laughlin
phase, we should have $C=1$, while we expect $C=0$ in the insulating phase.
For each disorder configuration, we twist the boundary conditions and evaluate
$C$ by Eq. (7). Then we average $C$ over various disorder configurations.
Indeed, we observe $\langle C\rangle=1$ for weak disorder strengths (Fig. 4).
A decaying of $\langle C\rangle$ starts at $W\approx 0.7$, signaling the
collapse of the Laughlin phase. The drop in $\langle C\rangle$ becomes steeper
for larger system sizes, tending towards a step function in the thermodynamic
limit. Note that this critical disorder strength obtained from the many-body
Chern number is consistent with the values extracted from the entanglement
analysis and fidelity susceptibility, thus validating the reliability of all
these diagnoses.

Beside the Gaussian white noise, we have also checked the Gaussian correlated
random potential with nonzero correlation length $\xi$ for $f=1/2$ contact-
interacting bosons and obtain similar results (not shown here to avoid
repetition).

##  IV $f=1$ bosonic Moore-Read filling

Having diagnosed the disorder-driven collapse of the $f=1/2$ bosonic Laughlin
state, we now study the fate of the $f=1$ bosonic MR state in the presence of
disorder. This bosonic MR state can be chosen as the model MR state, whose
wavefunction on an infinite plane is

$\Psi^{f=1}_{\text{MR}}=\operatorname{Pf}\left(\frac{1}{z_{i}-z_{j}}\right)\prod_{i<j}\left(z_{i}-z_{j}\right)\exp\left[-\frac{1}{4}\sum_{k}\left|z_{k}\right|^{2}\right],$
(12)

with “Pf” standing for Pfaffian. Alternatively, we can also choose the ground
state of a two-body interaction which is in the same topological phase as the
model MR state. Previously, the Coulomb interaction was used to study the
disorder-driven phase transitions and the disorder-induced intermediate FQH
phase for fermions at the MR filling $f=5/2$ Zhu and Sheng (2019). Therefore,
for a better comparison with fermions, in this section we also use a two-body
interaction and focus on the disorder effect on its ground state in the $f=1$
bosonic MR phase. This choice also reduces the computational cost, because
obtaining the model bosonic MR state requires the numerical diagonalization of
the three-body contact repulsion for an even number of bosons Greiter  _et
al._ (1991).

On the torus geometry, the model bosonic MR state is three-fold degenerate in
the $\mathbf{K}=(0,0)$, $\mathbf{K}=(0,N/2)$, and $\mathbf{K}=(N/2,0)$
momentum sectors Repellin  _et al._ (2015), respectively. We hence need to
seek a two-body interaction whose lowest three eigenstates are (i) in the same
momentum sectors as the model MR states, (ii) approximately three-fold
degenerate, (iii) separated from higher-energy levels by a robust energy gap,
and (iv) close to the model MR states. One natural candidate for such a two-
body interaction is the contact repulsion used in Sec. III, which is realistic
for bosons in cold-atom setups. However, we observe a strong finite-size
effect for this type of interaction. For relatively small systems of
$N=6,8,10,12$ bosons, although the ground states of the two-body contact
repulsion are in the same momentum sectors as the model bosonic MR states and
the overlaps with the model states are decent, the energy gap protecting the
ground-state manifold strongly fluctuates with the system size and can even
become smaller than the splitting between the three ground states (see Table
1). Considering that we can only efficiently deal with at most $N=12$ bosons
at $f=1$ once disorder is turned on, the strong finite-size effect of the two-
body contact repulsion in small clean systems will lead to messy results at
nonzero disorder. Indeed, in this case we find it very difficult to estimate
$W_{c}$ and $\nu$, as shown in Appendix A.

Table 1: The ground-state energy gap $\Delta$ and the ground-state splitting $\delta$ at zero disorder for $f=1$ bosons. We consider both the two-body contact repulsion and the Coulomb interaction corrected by a finite layer thickness $w=4\ell_{B}$. We set $q^{2}/(4\pi\epsilon\ell_{B})=1$ for the Coulomb interaction, where $\epsilon$ is the dielectric constant. After sorting all energy levels in ascending order, $\Delta$ is defined as the energy difference between the fourth level and the third level, and $\delta$ is the energy difference between the third level and the lowest level. For both interactions, the lowest three eigenstates are always in the same momentum sectors as the model bosonic MR state, forming the ground-state manifold. The total overlaps $\mathcal{O}=\sum_{i=1}^{3}|\langle\Psi_{i}|\Psi_{i}^{\mathrm{MR}}\rangle|^{2}/3$ between the ground-state manifold $\\{|\Psi_{i}\rangle\\}$ and the model bosonic MR states $\\{|\Psi_{i}^{\mathrm{MR}}\rangle\\}$ are also given. For $N=8$ and $N=10$ contact-interacting bosons, although the ground-state overlap with the model MR state is decent, $\Delta$ drops a lot and becomes smaller than $\delta$, implying that the ground-state manifold of these two systems is not in a robust MR phase. | Contact | Coulomb with $w=4\ell_{B}$
---|---|---
| $\Delta$ | $\delta$ | $\mathcal{O}$ | $\Delta$ | $\delta$ | $\mathcal{O}$
$N=6$ | 0.2531 | 0.0997 | 0.9582 | 0.0758 | 0.0356 | 0.9648
$N=8$ | 0.0864 | 0.2028 | 0.8939 | 0.0443 | 0.0212 | 0.9676
$N=10$ | 0.0928 | 0.1374 | 0.8968 | 0.0663 | 0.0212 | 0.9412
$N=12$ | 0.3328 | 0.0879 | 0.8080 | 0.1190 | 0.0114 | 0.9109
$N=14$ | 0.2771 | 0.1252 | 0.8152 | 0.1010 | 0.0206 | 0.9018
$N=16$ | 0.3074 | 0.0420 | 0.7215 | 0.1251 | 0.0026 | 0.8749

Because the two-body contact repulsion is not ideal for our purpose, we
examine long-range interactions. Motivated by the fact that the $f=5/2$
fermionic MR phase can be significantly enhanced by the Coulomb interaction
with a finite layer thickness $w\approx 4\ell_{B}$ caused by the quasi-2D
nature of the system Peterson  _et al._ (2008b), we check the energy spectra
and ground-state wave functions of this interaction with $w=4\ell_{B}$ for
$f=1$ bosons. Although the Coulomb interaction is not as pertinent as the
contact repulsion for bosons in cold-atom setups, we think it is a reasonable
choice for theoretical interest, as it can provide higher-order Haldane’s
pseudopotentials Haldane (1983). In this case, with the $V_{\bf q}$ given in
Ref. Peterson  _et al._ (2008b), we find that there are indeed three
approximately ground states in the correct momentum sectors. Remarkably,
compared with the two-body contact repulsion, the Coulomb ground states have
higher overlaps with the model bosonic MR state. Moreover, the energy gap
suffers from a much weaker finite-size effect and is always larger than the
ground-state splitting (Table 1). We hence proceed to study the disorder
effect on the $f=1$ bosonic MR state stabilized by the the Coulomb interaction
with $w=4\ell_{B}$. We emphasize that the trick of using the Coulomb
interaction is only for suppressing the finite-size effect in numerically
tractable systems. The results of both contact and Coulomb interactions are
qualitatively similar, so the underlying physics should not depend on the
choice of the interaction.

Figure 6: The sample-averaged fidelity susceptibility $\langle\chi(W)\rangle$
for even $N=6$-$12$ Coulomb-interacting bosons at $f=1$. Disorder is modeled
by the Gaussian white noise. We average 10 000 samples for $N=6$ and $8$, 2000
samples for $N=10$, and 500 samples for $N=12$. The vertical dashed line
indicates $W=0.16$, which is the critical disorder strength extracted from the
entanglement quantities in Fig. 5. Figure 7: The sample-averaged orbital-cut
entanglement entropy $\langle S^{o}(\bar{\rho})\rangle$, entropy derivative
$\langle dS^{o}(\bar{\rho})/dW\rangle$, fidelity susceptibility
$\langle\chi(W)\rangle$, and many-body Chern number $\langle C\rangle$ for
even $N=6$-$12$ Coulomb-interacting bosons at $f=1$. Disorder is modeled by
the Gaussian correlated random potential with $\xi=0.5\ell_{B}$ [(a)–(d)] and
$\xi=\ell_{B}$ [(e)–(h)]. In (a)–(c) and (e)–(g), we average 10 000 samples
for $N=6$ and $8$, 2000 samples for $N=10$, and 500 samples for $N=12$. Due to
the larger computational cost of the many-body Chern number, which requires
integrating over boundary conditions, we average 10 000 samples for $N=6$,
5000 samples for $N=8$, and 1000 samples for $N=10$ in (d) and (h). Markers in
(a)–(h) with the same color refer to the same system size.

Now we switch on disorder. One can imagine that the system must enter an
insulating phase at sufficiently strong disorder as all bosons are pinned at
the minimum of the disorder potential. Therefore we expect at least one phase
transition from the bosonic MR state to an insulating phase. Note that the
$f=5/2$ fermionic MR or anti-MR state is ultimately replaced by a composite-
fermion liquid rather than an insulator Zhu and Sheng (2019). This difference
is due to the different statistics between bosons and fermions. In the
presence of disorder, each Landau level is broadened into a Landau band, and
all single-particle states except those at the band center are localized.
While in the strong-disorder limit all bosons can pile in one localized
single-particle state with the lowest energy, $f=5/2$ fermions must occupy
half of all single-particle states in the second Landau band including the
delocalized states at the band center, as two fermions cannot occupy the same
single-particle state.

To proceed, we first model disorder by the Gaussian white noise. Similar to
the Laughlin case, the OEE $\left\langle S^{o}(\bar{\rho})\right\rangle$ first
increases and then decreases with increasing $W$ for all system sizes [Fig.
5(a)], leading to a peak and a valley in the entropy derivative $\left\langle
dS^{o}(\bar{\rho})/dW\right\rangle$ [Fig. 5(b)]. By contrast, we find that the
REE decays monotonically (not shown here). We believe that the critical point
$W_{c}$ can be identified by maximum of the OEE, and the OEE derivative
diverges to positive (negative) infinity on the left (right) side of the
critical point in the thermodynamic limit. Because the OEE data in Fig. 5,
especially the entropy derivative, suffer from a stronger finite-size effect
than the $f=1/2$ Laughlin case, we use the data of the largest two system
sizes, $N=10$ and $N=12$, to do the scaling analysis to extract $W_{c}$. We
apply the scaling forms in Eqs. (10) and (11) to $\left\langle
S^{o}(\bar{\rho})\right\rangle$ and $\left\langle
dS^{o}(\bar{\rho})/dW\right\rangle$, respectively. With the rescaled
variables, we find that all data points almost locate on a single curve with
$W_{c}\approx 0.16$ and $\nu\approx 0.5$ [Figs. 5(c) and 5(d)].

To examine the critical $W$ extracted from the entanglement quantities, we
calculate the fidelity susceptibility $\langle\chi(W)\rangle$ (Fig. 6).
Similar to the Laughlin case, all $\langle\chi(W)\rangle$ curves exhibit a
pronounced maximum which gets higher for larger system size. While the
position of the peak varies with the system size, it is located at $W\approx
0.16$ for the largest system, $N=12$, which matches the $W_{c}$ extracted from
the OEE.

The similar behavior of the OEE to the $f=1/2$ Laughlin case and the single
peak in the fidelity susceptibility suggest a direct phase transition from the
bosonic MR state to an insulating phase when the disorder is modeled by the
Gaussian white noise. Motivated by the intriguing observation that $f=5/2$
fermions subjected to disorder with a finite correlation length $\xi$ might
evolve to an intermediate FQH phase Zhu and Sheng (2019), we repeat the
studies in Figs. 5 and 6 by using the Gaussian correlated random potential
with nonzero $\xi$. We consider $\xi/\ell_{B}=0.5$-$2$, for which a disorder-
induced intermediate FQH phase seems to exist for $f=5/2$ fermions Zhu and
Sheng (2019). However, for $f=1$ bosons, we cannot see any clear signature of
an intermediate phase within this range of $\xi$. The results for
$\xi=0.5\ell_{B}$ and $\xi=\ell_{B}$ are shown in Fig. 7. In these cases, the
OEE and the fidelity susceptibility look qualitatively similar to the $\xi=0$
case (i.e., the Gaussian white noise). While the curves at $\xi>0$ are
stretched compared with the $\xi=0$ case such that the critical $W$ moves to
larger values [Figs. 7(a)–7(c), 7(e)–7(g)], we can only observe the signature
of a single phase transition, as reflected by the single pronounced peak in
the fidelity susceptibility [Figs. 7(c) and 7(g)]. This is also true for
larger $\xi$ despite stronger finite-size effects. We also find that the many-
body Chern number stays at $\langle C\rangle=3$ at weak disorder and decays at
sufficiently large $W$ [Figs. 7(d) and 7(h)], which confirms the transition
from the bosonic MR state to an insulating phase.

Figure 8: The sample-averaged orbital-cut entanglement entropy $\langle
S^{o}(\bar{\rho})\rangle$, entropy derivative $\langle
dS^{o}(\bar{\rho})/dW\rangle$, and fidelity susceptibility
$\langle\chi(W)\rangle$ for even $N=6$-$12$ contact-interacting bosons at
$f=1$. Disorder is modeled by the Gaussian correlated random potential with
$\xi=0$ [(a)–(c)], $\xi=0.5\ell_{B}$ [(d)–(f)], and $\xi=\ell_{B}$ [(g)–(i)].
We average 10 000 samples for $N=6$ and $8$, 2000 samples for $N=10$, and 500
samples for $N=12$. Markers in (a)–(i) with the same color refer to the same
system size.

##  V Conclusions and Discussions

In this paper, we present a systematic exact-diagonalization study of
disorder-driven quantum phase transitions for the $f=1/2$ and $f=1$ bosonic
FQH systems. We identify the phase transition by tracking the entanglement
entropy, fidelity susceptibility, and many-body Chern number (Hall
conductance) of the ground-state manifold as a function of disorder strength.
The critical disorder strengths $W_{c}$ obtained from different quantities are
consistent with each other, validating the reliability of our numerical
results. At both $f=1/2$ and $f=1$, we identify a single phase transition from
the FQH state (bosonic Laughlin state for $f=1/2$ and bosonic MR state for
$f=1$) to an insulating phase, no matter whether the correlation length in the
disorder potential is zero or not. While the location of the transition
appears to be robust, the critical exponent $\nu$ of the transition extracted
from the finite-size scaling analysis of the OEE data ($\nu\approx 0.6$ at
$f=1/2$ and $\nu\approx 0.5$ at $f=1$) violates the expected Harris-Chayes
inequality $\nu\geq 2/d$ for $d$-dimensional disordered systems Harris (1974);
Chayes  _et al._ (1986). Such violation was also observed in fermionic FQH
systems Liu and Bhatt (2016, 2017) and many-body localization transitions
Kjäll  _et al._ (2014); Luitz  _et al._ (2015). Our results may motivate
further studies of the critical exponent of disorder-driven phase transitions
in FQH systems by more powerful numerical techniques and experiments.

The fate of the $f=1$ bosonic MR state in the presence of disorder looks
different from that of the $f=5/2$ fermionic MR or anti-MR state. At least for
the interactions and the range of the disorder correlation length considered
by us ($\xi/\ell_{B}=0.5$-$2$), we observe a direct transition to an
insulating phase for the former, while for the latter the system might first
enter an intermediate FQH phase before finally becoming a composite-fermion
liquid Zhu and Sheng (2019). The suspected intermediate phase in the fermionic
case was argued to be a PH symmetric phase described by either the fermionic-
MR–anti-MR puddle structure Mross  _et al._ (2018); Wang  _et al._ (2018);
Lian and Wang (2018); Zhu and Sheng (2019) or a PH symmetric wavefunction
Jolicoeur (2007); Zucker and Feldman (2016); Balram  _et al._ (2018); Rezayi
_et al._ (2021). Since the usual PH transformation is only well defined for
fermions, it is indeed impossible for bosons to form a PH symmetric
intermediate phase induced by disorder. Therefore our results do not
contradict with the existence of a PH symmetric intermediate phase of $f=5/2$
fermions. In Ref. Wang and Senthil (2016), a generalized PH transformation is
defined for bosons, by which one can construct a bosonic anti-MR state at
$f=1$. However, unlike in the fermionic case where the MR and anti-MR states
have the same energies for PH symmetric Hamiltonians, this bosonic anti-MR
state could have a much higher energy than the bosonic MR state. Indeed, the
absence of an intermediate phase in our numerical results seems to suggest
that the bosonic anti-MR state is not energetically favored by the Hamiltonian
we use.

A future direction following our work could be exploring the disorder effect
on dipolar interacting bosons, for which the FQH states compete with stripe
and bubble phases Cooper  _et al._ (2005); Seki and Ino (2008). Alternatively,
one can also consider disorder-driven phase transitions when the MR state
competes with the composite-fermion liquid for bosons Wu and Jain (2015). It
would be interesting to check whether an intermediate phase can appear in
these cases. Another future direction could be to study the disorder-driven
phase transitions for bosonic FQH states in optical lattice setups, i.e., the
bosonic fractional Chern insulators Parameswaran  _et al._ (2013); Bergholtz
and Liu (2013); Neupert  _et al._ (2015). In that case, the notable
differences between a Chern band and the LLL, such as the multiband structure
and high Chern number, may host new interesting phenomena.

###### Acknowledgements.

We thank Bo Yang, Wei Zhu, and Xin Wan for fruitful discussions. This work is
supported by the National Natural Science Foundation of China through Grant
No. 11974014.

##  Appendix A $f=1$ with the contact interaction

Here, we study the disorder-driven phase transitions for contact-interacting
bosons at $f=1$. In Fig. 8, we present the orbital-cut entanglement entropy,
the entropy derivative, and the fidelity susceptibility for different
correlation lengths of disorder. Compared with Figs. 5–7, the results with the
contact interaction are qualitatively similar to the Coulomb-interacting case.
Overall, there is no sign of an intermediate FQH phase. However, the data in
Fig. 8 are much messier than those in Figs. 5–7. As we argue in the main text,
this is due to the strong finite-size effect of the energy gap in small clean
systems of $N=6,8,10,12$. For these system sizes, once we turn on disorder,
the critical strength of disorder required to destroy the MR state strongly
fluctuates with the system size. Even with the largest two systems, $N=10$ and
$N=12$, we find it difficult to make a reliable scaling analysis for the
entanglement entropy and its derivative to extract $W_{c}$ and $\nu$.
Moreover, with increasing system size, the peak position of the fidelity
susceptibility changes a lot, and the peak height does not monotonically grow,
making it difficult to estimate $W_{c}$. All these results suggest that much
larger system sizes are needed to study the disorder-driven phase transitions
for contact-interacting bosons at $f=1$. Alternatively, one can choose the
Coulomb interaction to suppress the finite-size effect, as done in the main
text.

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Q(u,v)=Q(g,g)+\alpha^{2}\int(a\phi^{2}_{x}+2b\phi_{x}\phi_{y}+c\phi^{2}_{y})g^{2}d\mu+\alpha(T\phi,g^{2})_{L^{2}(d\mu)}.$

$L$ is uniformly parabolic so we can bound $||Q(g,g)||$ by
$C||g||^{2}_{L^{2}(d\mu)}$ for some constant $C>0$. Using the fact that
$|\nabla\phi|\leq 1$, the second term and the third term are bounded by
$\alpha^{2}||g||^{2}_{L^{2}(d\mu)}$, $C|\alpha|\cdot||g||^{2}_{L^{2}(d\mu)}$
for some constant $C>0$, respectively. So we obtained (129).

##### Case II:

In a neighborhood of the tangent Kimura edge, say suppose in a neighborhood of
a corner which is the intersection of two Kimura edges, $L$ is degenerate.
Under local coordinates (127),

$\displaystyle
L=x\partial_{xx}+y\partial_{yy}+xyb(x,y)\partial_{xy}+d_{1}(x,y)\partial_{x}+d_{2}(x,y)\partial_{y}.$

We refer to [14, §3.1] to see that

$\displaystyle
Q_{\text{sym}}(u,v)=-\int\left(xu_{x}v_{x}+yu_{y}v_{y}+\frac{1}{2}xyb(u_{x}v_{y}+u_{y}v_{x})\right)d\mu$

and $T$ is a tangent vector field with logarithmic singularities:

$\displaystyle
T=x\left(\alpha_{1}+\beta_{1}\text{ln}x+\gamma_{1}y\text{ln}y\right)\partial_{x}+y\left(\alpha_{2}+\beta_{2}\text{ln}x+\gamma_{2}\text{ln}y\right)\partial_{y}$

where the functions $\alpha_{i},\beta_{i},\gamma_{i}$ are smooth. Because
$y=0$ is a tangent edge, the coefficient of $\text{ln}y\partial_{x}$ vanishes
at the edge $y=0$.

Take $u=e^{\alpha\phi}g,v=e^{-\alpha\phi}g$. Then,

$\displaystyle(Lu,v)=Q(g,g)+\alpha^{2}\int
g^{2}(x\phi^{2}_{x}+y\phi^{2}_{y}+axy\phi_{x}\phi_{y})d\mu+\alpha(T\phi,g^{2})_{L^{2}(d\mu)}.$

By [14, Lemma 3.1], $||Q(g,g)||\leq C||g||^{2}_{L^{2}(d\mu)}$ for some
constant $C>0$. Using the fact that $|\nabla\phi|\leq 1$, the second term is
bounded by $\alpha^{2}||g||^{2}_{L^{2}(d\mu)}$. $T$ has logarithmic
singularities near $x=0$ and we choose $\phi$ so that
$|\phi_{y}\text{ln}x|\leq x$. Then the third term is also bounded by
$C|\alpha|\cdot||g||^{2}_{L^{2}(d\mu)}$ for some constant $C>0$. We thus
obtained (129).

##### Proof of (125)

Fix $p\in P$ that is not on a quadratic edge or tangent Kimura edge, $q\in
H\times[A,1],r_{1},r_{2}>0$ so that $B_{r_{1}}(p),B_{r_{2}}(q)\subset P$.
Notice that if $q$ is an edge point on a Kimura edge, $B_{r_{2}}(q)$ is taken
to be the semi-ball that lies in $P$.

Let $\chi_{1},\chi_{2}$ be the indicatrix functions of
$B_{r_{1}}(p),B_{r_{2}}(q)$. Instead of deriving the mean value inequalities
from local Sobolev inequality in [34, Theorem 5.2.9], we directly import two
mean value inequalities. If $q\in\mathring{P}$, $L$ is elliptic in
$B_{r_{2}}(q)$ and we can apply the sup norm inequality with $m=1$ in [27,
Theorem 6.17]. If $q$ is on a transverse Kimura edge, we can apply [17, Lemma
A.6] and the proof in Corollary 4.3. Apart from these two modifications,
following the proof in [34], we show that for
$t\geq\max(r^{2}_{1},r_{2}^{2})$,

$\displaystyle
p_{t}(p,q)\leq\frac{C}{\sqrt{\mu(B_{1})\mu(B_{2})}}\text{exp}\left[\alpha^{2}t+\alpha(\phi(p)-\phi(q))+C_{0}|\alpha|t+|\alpha|(r_{1}+r_{2})+C_{1}t\right].$

Taking $\alpha=-\frac{\phi(p)-\phi(q)}{2t}$, then for
$t\geq\epsilon^{-2}\max(r^{2}_{1},r^{2}_{2},\epsilon^{4})$,

$\displaystyle
p_{t}(p,q)\leq\frac{C}{\sqrt{\mu(B_{1})\mu(B_{2})}}\text{exp}\left[-\frac{(\phi(p)-\phi(q))^{2}}{4t}+\left(\frac{C_{0}}{2}+1\right)|\phi(p)-\phi(q)|+C_{1}t\right].$

##### Proof of (126)

Having obtained the pointwise upper bound of $p_{t}(p,q)$, we can estimate
$v(t,x)$. We choose $\phi(q)=\phi(p)-C_{2}x$ with some constant $C$ so that
$|\nabla\phi|\leq 1$. Since $h|_{H\times[0,A]}\equiv 1$, by definition of
$v(t,x)$ (92), we only need to integrate $y$ in $[A,1]$. We use the pointwise
upper bounded above to obtain the following estimate of $v(t,x)$:

$\displaystyle v(t,x)\leq
C\cdot\text{exp}\left[-\frac{x^{2}}{4C_{2}t}+C_{0}|x|+C_{1}t\right],$ (130)

for some constant $C_{0},C_{1},C_{2}>0$. Next we estimate
$\int\psi^{\prime}(f)v(t,x)dx$. We use Hölder’s inequality with
$\frac{1}{r}+\frac{1}{s}=1$ to obtain

$\displaystyle\left|\int\psi^{\prime}(f)v(t,x)dx\right|\leq||\psi^{\prime}(f)\mu^{1-\frac{1}{p}}||_{\frac{p}{p-1}}\cdot||\mu^{\frac{1}{p}-1}v^{1-\frac{1}{ps}}||_{pr}\cdot||v^{\frac{1}{ps}}||_{ps}.$
(131)

First it is easy to see

$\displaystyle||v^{\frac{1}{ps}}||_{ps}=O(e^{\frac{\lambda_{0}}{ps}t})$ (132)

has exponential decay with rate $\frac{\lambda_{0}}{ps}$. We next estimate the
first term. Since $\psi(x)\sim x^{p}$ as $x\to\infty$ and taking into account
that $\psi$ is negative only on a finite interval, we have

$\displaystyle||\psi^{\prime}(f)\mu^{1-\frac{1}{p}}||^{\frac{p}{p-1}}_{\frac{p}{p-1}}\leq
M_{1}\int|\psi|\mu\leq M_{1}\int\psi\mu+M_{2}.$ (133)

For the second term, notice that $\mu^{-1}$ has asymptotic behavior
$O(e^{a|x|})$ for $a>0$, and $v(t,x)$ decays quadratically in $|x|$ by (130).
So the second term is intergable while it may have exponential growth with
respect to $t$. Indeed after a direct calculation

$\displaystyle||\mu^{\frac{1}{p}}v^{1-\frac{1}{ps}}||_{pr}$
$\displaystyle=O\left(\sqrt{t}\cdot\text{exp}\left[4\left(C_{0}+\frac{a(p-1)}{p-\frac{1}{s}}\right)^{2}(1-\frac{1}{ps})\right]\right)$
$\displaystyle=O\left(\sqrt{t}\cdot\text{exp}\left[4\left(C_{0}+a\right)^{2}(1-\frac{1}{ps})\right]\right)$

since $\frac{1}{s}<1,p<2$. Regarding (132), for $0<\epsilon<1$, we take
$\frac{1}{ps}=1-\epsilon\frac{\lambda_{0}}{4(C_{0}+a)^{2}+\lambda_{0}}$ so
that

$\displaystyle||\mu^{\frac{1}{p}-1}v^{1-\frac{1}{ps}}||_{pr}\cdot||v^{\frac{1}{ps}}||_{ps}=O(e^{(1-\epsilon)\lambda_{0})}).$
(134)

Taken $\eqref{98},\eqref{99},\eqref{100}$ and an interpolation of arithmetic-
geometric mean inequality together, we obtain

$\displaystyle\left|\int\psi^{\prime}(f)v(t,x)dx\right|\leq\epsilon\int\psi\mu+\sqrt{t}e^{(1-\epsilon)\lambda_{0}t}.$
(135)

∎

######  Lemma A.2.

There exists a stratification as described in (71).

###### Proof.

We first take a finite covering of $H$ by coordinate neighborhoods of the form
$R$ based at points of $H$. Let $k_{1}$ be the minimum of the height (normal
direction in the local coordinates) of these rectangles. Then by taking the
closure of these rectangles and shrinking the height, $P_{[k_{0},k_{1}]}$ is
covered by closed rectangles that satisfy the condition 72. We can do the
similar job at each level of the corners. Specifically, for a corner $p$, the
level $L_{\rho(p)}:=\rho=\rho(p)$ is the disjoint union

$L_{\rho(p)}=L_{\rho(p),\max}\bigcup L_{\rho(p),i}$

where $L_{\rho(p),\max}$ are the collection of points that attain the local
maximum. $L_{\rho(p),\max}$, $L_{\rho(p),i}$ are both compact and a disjoint
union of finite points and the edges out of condition 5.2.c. We cover
$L_{\rho(p),\max},L_{\rho(p),i}$ by finite open sets of the form $R$ or $T$.
Again, by taking the closure and shrinking the height of these open sets,
there are $\epsilon_{1},\epsilon_{2}\geq 0$ so that
$P_{[\rho(p)-\epsilon_{1},\rho(p)+\epsilon_{2}]}$ is covered by the closed
sets that satisfy the condition 72. Up to now, the remaining parts of $P$ we
have not covered are a disjoint unions of level intervals. Suppose $P_{[a,b]}$
is one of level interval, we first cover $P_{[a,b]}$ by finite open rectangles
with two sides lie on two $\rho$-levels, then we can cut these rectangles so
that $P_{[a,b]}$ is covered by closed rectangles that satisfy the condition
72. Thus we can cover these level intervals by the desired stratification. ∎

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