# Stability of even-denominator fractional quantum Hall states in systems with
strong Landau-level mixing

Wenchen Luo  School of Physics and Electronics, Central South University,
Changsha, China 410083  Shenglin Peng  School of Information Science and
Technology, Northwest University, Xi’an, China 710127  School of Physics and
Electronics, Central South University, Changsha, China 410083  Hao Wang
Shenzhen Institute for Quantum Science and Engineering, Southern University of
Science and Technology, Shenzhen, China 518055  Yu Zhou<EMAIL_ADDRESS>Department of Physics, Jiangsu University of Science and Technology, Jiangsu,
China 212003  Tapash Chakraborty<EMAIL_ADDRESS>Department
of Physics, Brock University, St. Catharines, ON, Canada L2S 3A1  Department
of Physics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2

###### Abstract

Mixing of Landau levels has been understood to be essential in governing the
nature of the ground state for the even-denominator fractional quantum Hall
effect. The incompressibility of the ground state at filling factor $5/2$ in
the strong Landau-level-mixed systems, such as the ZnO quantum well, is not
always stable. Here we present an approach to generally deal with this kind of
systems and satisfactorily explain the recent experiments [Falson et al. Sci.
Adv. 4, eaat8742 (2018)] by implementing the screening plus the thickness
effect. Further, the phase diagrams of the incompressibility of the ground
state indicate that the phase transitions can be explicitly extracted by
observing the lowest gap of the collective modes when the magnetic field and
the width of the quantum well are tuned. We also predict the incompressibility
of the two-dimensional electron gas in higher Landau levels in another strong
Landau-level-mixed system, viz., the black phosphorene, by considering the
screening effect where the relevant even-denominator fractional quantum Hall
effects can possibly be observed.

For the past four decades, the quantum Hall effects have been the epitome of
elegant phenomena in condensed matter physics review . Ever since the
observation of the even-denominator fractional quantum Hall effect (FQHE)
odfqhe , it has been recognized that some difficulties remain in explaining
this effect. Experimentally, the spin polarization of the $5/2$ FQHE state is
still debated argue  and very high-quality samples are an absolute requirement
to resolve the issue. A couple of trial wave functions have been proposed thus
far for this system mr ; trialwf . However, the numerical results are somewhat
size dependent so that the trial wave function candidates are not so
satisfactory and show relatively poor performance when compared with the
numerical wave functions. It is also possible that both the long-range and the
short-range interactions are important but the Haldane’s pseudopotentials are
limited to a spherical geometry. In a toroidal geometry the numerical
strategy, although not entirely trouble free, can be a powerful alternative
approach.

In practice, the even-denominator FQHE is proposed to be useful in topological
quantum computation nonab  since the non-Abelian excitations in the even-
denominator FQHE are robust against many environment noises due to the
topological protection. Determining the nature of the even-denominator FQHE
states is therefore important for development of quantum computation. The
Moore-Read pfaffian trial wave function (or its particle-hole conjugate anti-
pfaffian state) is the most plausible candidate for the ground state, albeit
the overlap between this wave function and the numerical results is not being
high. A particle-hole symmetric pfaffian state for the ground state is
proposed son  that is believed to be stable when the Landau level (LL) mixing
and the influence of disorders in the two-dimensional electron gas (2DEG) are
included feldman . Further studies have indicated the existence of a phase
transition when the LL mixing increases luo3 . The incompressibility of the
2DEG depends on the strength of the LL mixing which can be effectively
considered by screening the Coulomb potential. In the framework of screening,
the surprise missing of the 5/2 plateau in ZnO quantum well has been explained
falson ; luo1 . In a more recent experiment performed in 2018, however, Falson
et al reported that the $5/2$ FQHE is recovered in one sample falson2 , which
could not be explained by our previous theoretical approach. In order to
understand the strong LL mixed 2DEG in general, and why the $5/2$ FQHE is
survived in one sample but is still missing in another sample of the 2018
experiment, we need to consider the screening effect together with the
thickness effect associated with the quantum well in our analysis below.

The 2DEG discussed here has a unique property that the LL gap is very small
compared to the Coulomb energy gap. The LL mixing is therefore strong and
influences the properties of the ground state, and its incompressibility may
be absent. A dimensionless quantity is defined to characterize the LL mixing
strength: the ratio of the Coulomb interaction strength $E_{\rm
C}=e^{2}/{\epsilon\ell}$ to the LL gap $E_{\rm cyc}=\hbar\omega_{c}$,
$\kappa=E_{\rm C}/E_{\rm cyc}$, where $\epsilon$ is the dielectric constant of
the material, $\ell=25.656/\sqrt{B}$ nm is the magnetic length, and
$\omega_{c}$ is the cyclotron frequency in the magnetic field $B$. This ratio
is typically small in a conventional GaAs quantum well in a strong magnetic
field, and is a constant $0.5\sim 0.8$ in graphene graphene . In contrast, it
is very large in ZnO, $\kappa>10$ (the same order as in black phosphorene).
The perturbation theory approach, including the 3-body interaction which is
based on the expansion of $\kappa$ perturbation  is not appropriate for these
systems. To overcome this difficulty, we make use of the screened Coulomb
potential in the relevant LL, which is obtained by integrating out all the
other LLs, to effectively describe the LL mixing.

As has been done in previous works luo1 , we also consider the electron-
electron interaction described by the screened Coulomb potential in our
present approach. However, if we just consider the pure 2DEG, we only have
unstable $5/2$ FQHE in both systems considered in the experiment falson2 ;
joe_more , which clearly conflicts with the experimental observations. It has
been suggested that we need to consider the third dimension, the thickness, of
the quantum well in order to get a better understanding of the present
situation. For simplicity, the confinement of the quantum well is
approximately described either by a parabolic potential parabolic ; luo2  or
an infinite square potential peterson . We will compare these two
approximations, of which the results are supposed to be coincident.

The many-body Hamiltonian is given by

$\displaystyle
H=\sum_{n,i,\sigma}E_{n,\sigma}c_{n,i,\sigma}^{{\dagger}}c_{n,i,\sigma}+\frac{1}{2}\sum_{\sigma,\sigma^{\prime}}\sum_{\begin{subarray}{c}n_{1},\ldots,n_{4}\\\
i_{1},\ldots,i_{4}\end{subarray}}V_{\left(s\right),i_{1},i_{2},i_{3},i_{4}}^{n_{1},n_{2},n_{3},n_{4}}$
$\displaystyle\times\,c_{n_{1},i_{1},\sigma}^{{\dagger}}c_{n_{2},i_{2},\sigma^{\prime}}^{{\dagger}}c_{n_{3},i_{3},\sigma^{\prime}}c_{n_{4},i_{4},\sigma},$
(1)

where $c$ is the electron operator, $E_{n,\sigma}$ is the kinetic energy of
the LL $n$ with spin $\sigma$, $n_{i}$ is the LL index and $i_{1,\ldots,4}$
are the guiding center indices. The Coulomb interaction matrix element
$V_{\left(s\right),i_{1},i_{2},i_{3},i_{4}}^{n_{1},n_{2},n_{3},n_{4}}$ depends
on the confinement potential, which will be explicitly given in the following
cases. We perform the exact diagonalization scheme to numerically solve the
Hamiltonian with the translational symmetry in the toroidal geometry haldane ;
book ; exactdiag . The collective modes of the system at zero temperature are
obtained, and the incompressibility or instability of the FQHE state is then
determined.

The virtual process between these LLs in the density response function should
be excluded to avoid double counting of the correlations if more LLs are
involved in the Hamiltonian. In the following, we need to analyze the screened
dielectric functions in detail for different confinement potentials.

The non-interacting Hamiltonian can be exactly solved in a parabolic potential
in a tilted (or perpendicular) magnetic field, and the screened Coulomb
potential has been studied luo2 . We will numerically evaluate the collective
modes in different magnetic fields and different widths of the quantum well
based on the formulations of Ref. luo2 .

For an infinite square potential, the non-interacting Hamiltonian can also be
exactly solved when the magnetic field is perpendicular to the $xy$ plane, in
which the $z$ component of the wave function can be separated,
$\Psi_{m,n,X}\left(\mathbf{r},z\right)=\sqrt{2/L_{z}}\sin\left(m\pi
z/L_{z}\right)\psi_{n,X}\left(\mathbf{r}\right)$, where $L_{z}$ is the width
of the quantum well, $m$ is the Landau band index, $n$ is the LL index, $X$ is
the guiding center index, and $\psi_{n,X}\left(\mathbf{r}\right)$ is the wave
function of a conventional 2DEG in a magnetic field in the Landau gauge.

The screened dielectric function is given by

$\epsilon_{s}\left(q\mathbf{,}q_{z}\right)=1-\frac{4\pi
e^{2}}{\left(q^{2}+q_{z}^{2}\right)\epsilon}\chi_{nn}^{0}\left(q\mathbf{,}q_{z}\right),$
(2)

where the three-dimensional momentum contains the in-plane momentum $q$ and
the $z$-component momentum $q_{z}$.
$\chi_{nn}^{0}\left(q\mathbf{,}q_{z}\right)$ is the noninteracting retarded
density-density response function computed in the random phase approximation
in the static limit sm . If we consider that the relevant LLs are all in the
lowest band, i.e., $m_{i}=1$, then the screened Coulomb interaction matrix
element is

$\displaystyle
V_{\left(s\right),i_{1},i_{2},i_{3},i_{4}}^{n_{1},n_{2},n_{3},n_{4}}=\frac{e^{2}}{\epsilon\ell}\frac{2}{\pi
N_{s}}\overline{\sum_{\mathbf{q}}}V_{i_{1},i_{2},i_{3},i_{4}}^{n_{1},n_{2},n_{3},n_{4}}\left(\mathbf{q}\right)$
(3)
$\displaystyle\times\,\int_{0}^{\infty}\frac{dq_{z}\ell}{\epsilon_{s}\left(q,q_{z}\right)\left(q^{2}+q_{z}^{2}\right)\ell^{2}}\frac{32\pi^{4}\left[1-\cos\left(q_{z}L_{z}\right)\right]}{\left(4\pi^{2}q_{z}L_{z}-q_{z}^{3}L_{z}^{3}\right)^{2}},$

where $\overline{\sum}$ means that the term of $q=0$ is excluded in the
summation,
$V_{i_{1},i_{2},i_{3},i_{4}}^{n_{1},n_{2},n_{3},n_{4}}\left(\mathbf{q}\right)$
is the Coulomb interaction matrix element for a 2DEG in a conventional zero-
width quantum well without screening sm . The integral includes the width and
screening corrections.

Figure 1: The lowest energy gaps in the collective modes for different
magnetic fields and widths of the wave functions in the ZnO quantum wells.
Other parameters are extracted from Ref. falson , $m^{*}=0.44m_{e}$ and
$\epsilon=8.5$, and $N_{e}=11$. The confinement of the 2DEG in the $z$
direction is (a) parabolic potential and (b) infinite square well. When the
gap is below $10^{-4}\,{e^{2}}/{\epsilon\ell}$ (could be negative), which is
shown in the purple region, the 2DEG is not incompressible. The crosses mark
the sample in Ref. falson , where $5/2$ FQHE is missing.

Let us check the theory to see whether it agrees with the experiment falson
by using the parameters: half width of the wave function ($W=L_{z}/1.5$) about
$5-6$ nm, dielectric constant $\epsilon=8.5$, effective mass of the electron
$m^{*}=0.44m_{e}$ with the electron mass $m_{e}$ . We first assume the
incompressible phase as the ground state, and evaluate the lowest excitation
gap in the collective modes separating the incompressible ground state from
the excitations. If the gap is larger than $10^{-4}\,e^{2}/{\epsilon\ell}$,
then the FQHE is supposed to be stable. The reason for this particular choice
of $10^{-4}$ is because this energy corresponds to about $20$ mK, which
guarantees that the thermal fluctuation can not overwhelm the
incompressibility of the system, as otherwise the FQHE can not be observed in
the current experimental condition. It is clearly shown in Fig. 1 that the
collective modes (at the crosses) are softened at $B=3.75$ T and the
translational invariant liquid phase no longer has the lowest energy. The 2DEG
is compressible no matter what kind of potential is chosen, which coincides
with the experimental observation and the previous theoretic work luo1 .
Figure 1 also shows the phase diagram of the ground state at $\nu=5/2$ for
different values of magnetic fields and widths of the wave function, since the
qualitative variation of the gap must represent the changes in the ground
states. The phase diagrams for the two different potentials are qualitatively
similar, i.e., the 5/2 FQHE is only stable when the quantum well is thick and
the LL mixing is weak (small $\kappa$). Note that the LLs are crossing and the
noninteracting ground states are changed in the white regions in Fig. 1.

Figure 2: Lowest gaps in the collective modes for different magnetic fields
and the widths of the wave functions in the ZnO quantum well with
$m^{*}=0.3m_{e}$ and $\epsilon=8.5$ represent the phase diagram in the strong
LL mixed system. The confinement of the 2DEG in the $z$ direction is (a) a
parabolic potential and (b) an infinite square well. The left cross and the
right dot mark the phase of samples b and a in Ref. falson2 , respectively.
The first three principle components of the wave functions, which are
extracted from the lines shown in (a) and (b), respectively, are shown in (c)
and (d).

Let us now move on to the new experiments falson2  where the effective mass is
$m^{*}=0.3m_{e}$ at magnetic fields $B=9.6$ T (sample a) and $7.2$ T (sample
b), corresponding to $\kappa=5.4$ and $6.2$, respectively. The perturbation
theory is still not very useful. Again, we compute the phase diagrams for this
effective mass in parabolic and infinite square potentials, shown in Fig. 2.
It is clear that for the infinite square well, we can quantitatively find that
the 5/2 FQHE is stable in sample a and unstable in sample b (in Ref. falson2
), while the incompressibility of the 2DEG can be obtained qualitatively in
the parabolic potential. It implies that the infinite square well may be a
better approximation to describe the quantum well. Moreover, the second
incompressible states found in tilted magnetic field in Ref. falson2  can be
explained by the phase transition induced by the LLs crossing when the Zeeman
coupling is in excess of the LL gap luo2 .

The phase diagrams are not qualitatively different from those of
$m^{*}=0.44m_{e}$. Some important features can be found here. First, there is
an arc-like region isolated around the compressible region where the gaps
become smaller or even negative for both types of potentials. It is similar to
the case discussed in Ref. luo3 ; dopedgaas  where a topological phase
transition was suggested. Here we also observe that the LL mixing strength
causes the phase transition. Besides, the width of the quantum well is even
involved to induce more than one phase transitions, which is not expected
earlier. Roughly, when $B\cdot W<30$ T$\cdot$nm where $W$ is the wave function
width, the 2DEG is compressible. Second, the energy gaps are somewhat size-
dependent. The gap becomes smaller when the electron number is increased.
However, dealing with a larger number of electrons is not currently feasible,
and we expect that for larger systems the FQHE in sample a still survives. We
examine its stability in a 3-LL model, i.e., we consider the Hamiltonian with
three LLs $N=0,1,2$ with $N_{e}=15$ and the virtual process in these three LLs
are excluded in the screening. The collective mode clearly supports that the
ground state is incompressible.

To understand more about the incompressible phase, we perform the principle
component (PC) analyze pca  to the ground state at zero momentum, as shown in
Figs. 2(c) and (d). The wave functions are extracted along the black lines in
Figs. 2(a) and (b). The first PC is almost the same in the arc region, which
means that the ground states in the arc region share the same phase.

We next report on our study of black phosphorene, another strong LL mixed
system. Here we need to consider its bilayer structure in which the inter-
layer Coulomb potential is different from the intra-layer Coulomb potential.
We are required to extend the formula of screening to a bilayer system, i.e.,
the Coulomb potential and the density response function should be replaced by
matrices,

$V^{s}(q)=\left(\begin{array}[]{cc}V^{s}_{11}(q)&V^{s}_{12}(q)\\\
V^{s}_{21}(q)&V^{s}_{22}(q)\end{array}\right),$ (4)

where $V^{s}_{kl}(q)$ is the screened Coulomb potential between layers $k$ and
$l$ sm .

Black phosphorene has a rectangular crystal lattice. There are four atoms
$A,B,C,D$ in a unit cell in which $A,B$ are in layer 1 and $C,D$ are in layer
2. It can be simplified to a two-band model when we only work on the bands
near the Fermi surface bptwobands . The positive and negative filling factors
correspond to the conduction and the valance bands, respectively.

The single-particle wave function is written as

$\psi_{n,X}^{bp}=\frac{1}{\sqrt{2}}\sum_{m}\left(\begin{array}[]{c}u_{n,m}\psi_{m,X}\left(\left|B\right\rangle+\left|C\right\rangle\right)\\\
v_{n,m}\psi_{m,X}\left(\left|A\right\rangle+\left|D\right\rangle\right)\end{array}\right),$
(5)

where $u_{n,m}$ and $v_{n,m}$ can be obtained by diagonalizing the non-
interacting Hamiltonian which is given in Refs. bptwobands ; bpfqhe . The
Coulomb interaction matrix element reads

$\displaystyle
V_{\left(s\right),i_{1},i_{2},i_{3},i_{4}}^{n_{1},n_{2},n_{3},n_{4}}=\frac{e^{2}}{\epsilon_{bp}\ell}\overline{\sum_{\mathbf{q}}}\frac{V_{11}^{s}+V_{12}^{s}}{2N_{s}}\sum_{j_{1},\ldots,j_{4}}V_{i_{1},i_{2},i_{3},i_{4}}^{j_{1},j_{2},j_{3},j_{4}}\left(\mathbf{q}\right)$
$\displaystyle\left(u_{n_{1},j_{1}}^{\ast}u_{n_{4},j_{4}}+v_{n_{1},j_{1}}^{\ast}v_{n_{4},j_{4}}\right)\left(u_{n_{2},j_{2}}^{\ast}u_{n_{3},j_{3}}+v_{n_{2},j_{2}}^{\ast}v_{n_{3},j_{3}}\right).$

Note that the inter-layer distance $d$ may not be negligible luo4 , although
$d$ is as small as $2.13$ Å.

Black phosphorene is also a large $\kappa$ system ($\kappa>5.6$ when $B<10$ T)
due to its heavy effective mass. The mobility of this system is sufficiently
high so that the $-4/3$ FQHE has already been observed bpfqheexp . We check
the stability of this odd denominator FQHE for verification of our present
approach. With screening, the $\nu=\pm 1/3,\pm 4/3$ FQHEs are stable sm ,
which is not surprising and is compatible with the experiment results. We then
explore whether the even denominator FQHE is observable in this material.
Interestingly, the even denominator states $\pm 5/2$ and $\pm 7/2$ FQHEs are
not stable, since the eigen wave function is composed of different
$\psi_{m,i}$. The weights of $\psi_{m\neq 1,i}$ play important roles in
destabilizing these FQHEs.

Figure 3:  The lowest gaps in the collective modes of the LL $N=2$ black
phosphorene states in different magnetic fields, (a) without screening, (b)
$|\nu|=9/2$ with screening, and (c) $|\nu|=11/2$ with screening.

For the higher LL $N=2$ ($|\nu|=9/2,11/2$), the FQHEs survive due to the LL
screening. As shown in Fig. 3, the 2DEG may not be incompressible until $B>40$
T if the screening is not considered, since the FQHE is not stable for
$N_{e}=5$. However, at $\nu=11/2$ the FQHE can be stabilized for a much lower
magnetic field ($B>26$ T in the conduction band) for $N_{e}=5.$ Besides, the
gap does not change too much when $N_{e}\geq 7$. For $\nu=9/2$, although the
gap decreases a bit when $N_{e}$ is increased from 7 to 11, this gap is still
of the order of $10^{-3}{e^{2}}/{\epsilon_{bp}\ell}$ which is one order larger
than that in ZnO. We thus expect that $11/2$ FQHE can be observed, but the
mobility of this material may need to be further increased. We also assume
that the $9/2$ FQHE state is observable (provided the smaller system of
$N_{e}=5$ can be ignored). That prediction requires experimental verification.

Summary and remarks: From the phase diagrams of the ground state at $\nu=5/2$
in the strong LL mixing region, we find that the two stable FQHE regions (with
large energy gap from the incompressible ground state) for both the parabolic
and infinite square well potentials roughly coincide: the arc-like region
surrounded by low (or negative) energy gap regions. This means that the phase
transitions between the incompressible phase and the compressible phase occur
more than once when either $\kappa$ or width is varied. Moreover, even in the
incompressible phase region, the topological phase transition between the
(anti-)pfaffian state and the particle-hole symmetric state which is suggested
to be stabilized by the LL mixing feldman , could be also possible by tuning
the magnetic field luo3  or the width of the quantum well. The topological
features of the ground state should be determined in a thermal transport
experiment thermal . We believe that the particle-hole symmetry Rezayi  can be
conserved by the extremely strong LL mixing, albeit the topological order of
the $5/2$ FQHE is still puzzling (especially when $\kappa\sim 1$), since the
numerical evidence has confirmed that the $5/2$ FQHE can be stabilized at
shift $-1$ on a sphere when $\kappa$ is large enough luo3 . This argument is
also compatible with the screening theory employed here, in which the two-body
interaction with renormalized Coulomb interaction does not break the particle-
hole symmetry.

By combining the screening and finite width corrections, we have successfully
explained the latest experiments in ZnO. In the strong LL mixing limit, this
strategy effectively takes the related correlations of other LLs into account
and obeys the conservation laws. Moreover, we predict the stability of the
$5/2$ FQHE and present the phase diagrams at this filling factor for different
magnetic fields and quantum well widths. The phase diagram should be amenable
to verification by the experiments. Another kind of strong LL mixed system,
the black phosphorene, is also considered and we have shown that the even
denominator FQHEs are stabilized in higher LLs by the screening. This is also
expected to be observed experimentally. The screening theory is expected to be
applicable for study of the FQHE in black arsenic (in the same nitrogen family
as phosphorus) with spin-orbit coupling blackarsenic  which also has strong LL
mixing, and is probably helpful to understand the even denominator FQHE in
monolayer WSe2 WSe2 . Indeed, our present approach has laid the foundation for
future studies of strongly LL mixed systems.

This work has been supported by the NSF-China under Grant No. 11804396. We are
grateful to the High Performance Computing Center of Central South University
for partial support of this work. W.L. thanks Xinyu He for programming
assistance.

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