# Halperin States of Particles and Holes in Ideal Time Reversal Invariant
Pairs of Chern Bands and The Fractional Quantum Spin Hall Effect in Moiré
MoTe2

Inti Sodemann Villadiego  Institut für Theoretische Physik, Universität
Leipzig, Brüderstraße 16, 04103, Leipzig, Germany

(August 28, 2024)

###### Abstract

An experiment in moiré MoTe2 bilayers reported the first observation of a
topologically ordered state with zero Hall conductivity and half of the edge
conductance of a standard time-reversal invariant quantum spin Hall insulator.
This state is believed to emerge at total filling one of a pair of bands with
Chern numbers $C=\pm 1$ related by time reversal symmetry. By viewing these
bands as a pair of Landau levels with opposite magnetic fields, and starting
from a parent magnet with one filled band, we demonstrate that a class of
Halperin states constructed by adding particles to the empty Chern band and
holes to the occupied Chern band have all the properties observed in MoTe2.
Remarkably, these states break time-reversal symmetry but have exactly zero
Hall conductivity and helical edge conductance of $e^{2}/2h$. These states
also feature a spinless composite fermion with the same charge as the electron
but split equally between both valleys. In a standard Halperin 331 state, this
particle would be a neutral Bogoliubov composite fermion. However, in our
context this composite fermion is charged but remains itinerant because it is
split into the two valleys that effectively experience opposite magnetic
fields. The existence of such charged itinerant particles is a key difference
between Landau levels with opposite magnetic fields and standard multi-
components Landau levels, where all the itinerant particles are charge
neutral, such as the magneto-roton of the Laughlin state or the Bogoliubov
composite fermion of the Moore-Read state. When the electron density changes
away from the ideal filling and these itinerant charged particles are added to
the parent state, the disorder potential is less efficient at localizing them
as compared to standard Lanadau levels. This can explain why the state in
MoTe2 did not display a robust Hall plateau upon changing the electron
density.

Introduction. The dream to realize fractional quantum Hall states without
magnetic fields has recently come true with the observations of fractional
quantum anomalous Hall effect in twisted MoTe2 bilayers (tMoTe2) Cai  _et al._
(2023); Zeng  _et al._ (2023); Park  _et al._ (2023); Xu  _et al._ (2023) and
in pentalayer graphene on hBN Lu  _et al._ (2024). These striking discoveries
follow a line of earlier observations in moiré twisted bilayer graphene of
anomalous Sharpe  _et al._ (2019); Tseng  _et al._ (2022), anomalous integer-
quantized Serlin  _et al._ (2020); Stepanov  _et al._ (2021), and fractional
Chern insulators at relatively small fields Xie  _et al._ (2021), as well as
integer-quantized anomalous Hall effect MoTe2/WTe2 bilayers Li  _et al._
(2021a); Zhao  _et al._ (2024) and in tMoTe2 Anderson  _et al._ (2023). These
fractional anomalous quantum Hall states have been conceptualized as the
result of spontaneous magnetism leading to particles polarizing into a flat
Chern band where analogues of fractional quantum Hall states can become
favorable Ledwith  _et al._ (2020); Repellin and Senthil (2020); Abouelkomsan
_et al._ (2020); Wilhelm  _et al._ (2021); Liu  _et al._ (2021); Li  _et al._
(2021b); Sheffer and Stern (2021); Crépel and Fu (2023); Dong  _et al._
(2023a); Goldman  _et al._ (2023); Morales-Durán  _et al._ (2023); Reddy and
Fu (2023); Song  _et al._ (2023); Guo  _et al._ (2023); Dong  _et al._
(2023b); Zhou  _et al._ (2023); Dong  _et al._ (2023c); Kwan  _et al._ (2023);
Wang  _et al._ (2024); Yu  _et al._ (2024); Xu  _et al._ (2024).

Figure 1: (a) Two flat moiré bands from $K$ and $K^{\prime}$ valleys and
opposite spins and Chern numbers $C=\pm$ are simplified as a pair of Landau
levels with opposite magnetic fields. (b) Profile of the PH-Halperin 331 state
where holes of the $K^{\prime}$ valley with spin $\downarrow$ and electrons of
the $K$ valley with spin $\uparrow$ are added to a reference Integer Chern
ferromagnet occupying valley $K^{\prime}$. This state has a spinless itinerant
composite fermion-like particle ($f_{e}$) with the same electric charge of the
electron but split equally into each valley (see Fig.2).

However, a recent remarkable experiment in tMoTe2 Kang  _et al._ (2024a) has
reported the first evidence of a fractionalized state that does not fit into
the standard paradigm of a fractional quantum Hall states resulting from
partially filling a Chern band. This experiment reported the striking
observation of an insulating state with vanishing Hall conductivity,
$\sigma_{xy}=0$, but with a quantized fractional edge conductance of
$e^{2}/2h$, namely, the system behaves as if having an edge that is half of
the standard time-reversal-invariant quantum spin Hall insulator Kane and Mele
(2005a, b); Bernevig and Zhang (2006). The moiré bands of interest for this
setting originate from the valence bands at $K$ and $K^{\prime}$ valleys of
MoTe2 which are spin-split due to a large uniaxial spin-orbit field and
related by time-reversal symmetry. Upon twisting two layers of MoTe2 by a few
degrees a moiré pattern with a skyrmion texture of interlayer tunneling
appears and gives rise to time-reveral pairs of flat bands with opposite Chern
number originating from $K$ and $K^{\prime}$ valleys Wu  _et al._ (2019);
Devakul  _et al._ (2021); Wang  _et al._ (2020). Related models also apply to
bilayers of other transition metal dichalcogenides, such as WTe2, where
recently the double spin quantum Hall effect has been reported Kang  _et al._
(2024b).

The presumptive fractional quantum spin Hall effect in tMoTe2 is observed at a
filling of $\nu=-3$ Kang  _et al._ (2024a) (three holes per moire unit cell).
At filling $\nu=-1$ the system behaves as an anomalous integer quantum Hall
state with $\sigma_{xy}=e^{2}/h$ Kang  _et al._ (2024a), indicating that holes
polarize onto a single moiré Chern band with unit Chern number. At filling
$\nu=-2$ the system behaves as a time reversal invariant quantum anomalous
spin Hall state with $\sigma_{xy}=0$ and conductance of $e^{2}/h$ per edge,
indicating that holes fill equally a time-reversal invariant pair of bands
with opposite Chern numbers $C=\pm 1$. At filling $\nu=-3$ the system displays
an insulating state with vanishing Hall conductivity $\sigma_{xy}=0$ and edge
conductance of $3e^{2}/2h$ per edge. We interpret this observation, as
resulting from a non-trivial incompressible state constructed at half-filling
of a pair of time-reversal invariant flat Chern bands residing on top of the
simpler time-reversal-invariant vacuum of fully filled pair of Chern bands.
Therefore, from here on we will focus on a single time-reversal invariant pair
of flat Chern bands at an effective electron filling $\nu=1$ (see Fig.1(a)).

Reference Kang  _et al._ (2024a) and a few subsequent theoretical proposals
Zhang (2024); Shi  _et al._ (2024); May-Mann  _et al._ (2024); Jian and Xu
(2024), interpreted the observation of vanishing Hall effect, $\sigma_{xy}=0$,
as indicative that the ground state has time-reversal-invariant symmetry. We
will, however, deviate from this point of view and propose instead a fully
gapped incompressible state that spontaneously breaks time reversal symmetry
but which has all the key properties observed in experiment, including the
vanishing Hall conductivity and the fractional $e^{2}/2h$ helical conductance
per edge. Our proposal is that tMoTe2 realizes a correlated Halperin state
Halperin (1983) with an equal number and electrons and holes added to a parent
Ising Chern insulator at $\nu=1$, which we call a “PH-Halperin” state. Our
states are distinct from the Halperin states constructed in the standard
quantum Hall setting Halperin (1983) because the electrons reside on Chern
bands with opposite Chern number. They are also distinct from Halperin-like
states of electrons discussed in the fractional quantum spin-Hall setting
Bernevig and Zhang (2006) because they generically break time-reversal
symmetry, and have unequal occupation of both valleys (see Fig.1(b)).

Interestingly, we will see that PH-Halperin states with $\sigma_{xy}=0$
feature an emergent spin-less gapped composite-fermion-like particle that
carries a total charge identical to the usual electron but equally split into
halves in each spin-valley resolved Chern band (see Fig.1(b)). Because this
particle is split into occupying equally opposite Chern bands it behaves like
the itinerant neutral dipolar particles of the standard quantum Hall setting
(i.e. the magneto-roton of Laughlin states, or the Bogoliubov composite
fermion of the Moore-Read state), even though it is charged. The existence of
these itinerant charged particles is a unique feature of states constructed in
pairs of Landau levels with opposite magnetic fields not present in ordinary
Landau levels. These particles are harder to localize by the disorder
potential as a result of an uncertainty relation between the average position
and the relative distance of the two components residing in opposite Chern
bands. The disorder potential tries to localize their average position, while
interactions try to localize their relative distance. As a result there is a
competition, and when interactions dominate, disorder will be less effective
at pinning and localizing these itinerant charged particles and the
corresponding analogue of the Hall plateau of these PH-Halperin states is
expected to be more fragile relative standard quantum Hall states. This can
explain why Ref.Kang  _et al._ (2024a) did not observe a robust plateau, but
instead a smooth deviation of the Hall conductivity away from zero as the
electron density was changed away from the ideal filling where the
incompressible state was observed.

Ideal model and symmetries. We consider a highly idealized model in which the
Chern bands arising from both valleys of tMoTe2 (labeled $\uparrow$ and
$\downarrow$) are viewed as the $n=0$ Landau levels of a Hamiltonian for two
species of particles experiencing opposite magnetic fields:

$H_{\uparrow}=\frac{(\mathbf{p}-\mathbf{A}_{0}-\delta\mathbf{A}^{\uparrow})^{2}}{2m},\
H_{\downarrow}=\frac{(\mathbf{p}+\mathbf{A}_{0}-\delta\mathbf{A}^{\downarrow})^{2}}{2m},$
(1)

Here $\mathbf{A}_{0}$ is a spatially uniform magnetic field
$\nabla\times\mathbf{A}_{0}=B_{0}\mathbf{\hat{z}}$ of equal magnitude and
opposite sign on the two valleys. The area of the moiré unit cell can be
intepreted as the area of one flux quantum $a_{UC}=2\pi l_{B}^{2}=2\pi/B_{0}$
111We use units of electron charge $q_{e}=1$ and $\hbar=1$, but will restore
explicit units at the end in a few selected formulas.. One crucial symmetry of
the Hamiltonian and the states that we will consider is the
$U_{\uparrow}(1)\times U_{\downarrow}(1)$ associated with separate particle
number of each valley. This symmetry allows us to couple the system to two
probe gauge fields, with vector potentials denoted by
$\delta\mathbf{A}^{\uparrow,\downarrow}$, which we view as weak and slowly
varying in space and time. Thus, the net magnetic and electric fields
experienced by the particles in the two valleys can be different:

$\displaystyle\begin{aligned}
&\nabla\times(\mathbf{A}_{0}+\delta\mathbf{A}^{\uparrow})\equiv
B_{e}^{\uparrow}\mathbf{\hat{z}},\ {\bf
E}_{e}^{\uparrow}\equiv-\partial_{t}\delta\mathbf{A}^{\uparrow},\\\
&\nabla\times(-\mathbf{A}_{0}+\delta\mathbf{A}^{\downarrow})\equiv
B_{e}^{\downarrow}\mathbf{\hat{z}},\ {\bf
E}_{e}^{\downarrow}\equiv-\partial_{t}\delta\mathbf{A}^{\downarrow}.\end{aligned}$
(2)

Another important symmetry of our Hamiltonian is time reversal,
$T=i\sigma^{y}K$ ($T^{2}=-1$), which exchanges the two valleys (present for
$\delta\mathbf{A}^{\uparrow,\downarrow}=0$). However, the states of our
interest will spontaneously break this symmetry. For concreteness it is useful
to imagine that the system has an interacting Hamiltonian where particles in
the same and in opposite valleys interact with different potentials
$V_{\uparrow\uparrow}(\mathbf{r}_{i}-\mathbf{r}_{j})$ and
$V_{\uparrow\downarrow}(\mathbf{r}_{i}-\mathbf{r}_{j})$ Stefanidis and
Sodemann (2020). We will imagine that the relevant physics emerges from
projecting this Hamiltonian onto the $n=0$ pair of Landau levels from Eq.(1),
but we will not make explicit detailed use of the specific microscopic form of
the interactions in our discussion.

Halperin states of particles and holes. We start from an Ising Chern magnet as
a reference vacuum in which particles fully occupy the valley with spin
$\downarrow$ with corresponding Chern number $C=-1$. A common trick in this
setting that allows to map the problem into a usual quantum Hall setting of
two two-components with the same magnetic field, is to perform a particle-hole
conjugation on the $\downarrow$ particles (see e.g. Stefanidis and Sodemann
(2020); Kwan  _et al._ (2021, 2022)). Thus, effectively, the $\uparrow$
particles and the $\downarrow$ holes behave like two species of particles in
the same field $B_{e}^{\uparrow}=B_{h}^{\downarrow}=B_{0}$ in an $n=0$ Landau
level with a total number of flux quanta $N_{\phi}=A/a_{UC}=B_{0}A/2\pi$ (for
$\delta\mathbf{A}^{\uparrow,\downarrow}=0$). This is a setting well studied in
multi-layer and multi-component quantum Hall systems, but one important
difference is that the sign of inter-flavor interaction,
$V_{\uparrow\downarrow}$, would be flipped upon such partial particle-hole
conjugation.

In order to construct states with the same total electron density as the Ising
Chern magnet reference vacuum, we add as many holes to the $C=-1$ band as
electrons to the $C=1$ band,
$N_{e}^{\uparrow}=N_{h}^{\downarrow}=N_{\phi}-N_{e}^{\downarrow}$. We will
assume that these electrons and holes are forming a correlated Halperin $mmn$
state Halperin (1983) with wave-function:

$\Psi^{PH}_{mmn}=\prod_{i<j}^{N_{e}^{\uparrow}}(z_{i}-z_{j})^{m}\prod_{i<j}^{N_{h}^{\downarrow}}(w_{i}-w_{j})^{m}\prod_{i,j}^{N_{e}^{\uparrow},N_{h}^{\downarrow}}(z_{i}-w_{j})^{n},$
(3)

where $m$ is an odd integer, $n$ is a non-negative integer, and we have
omitted the standard exponential factors of $n=0$ Landau level wave-functions.
Notice that for $n>0$ in the above states electrons of $\uparrow$ component
have repulsive correlations with the holes of $\downarrow$ component,
suggesting that in order to stabilize these states there needs to be an
effective attraction between electrons of $\uparrow$ and $\downarrow$
components, or at least a reduction of the repulsion $V_{\uparrow\downarrow}$
relative to $V_{\uparrow\uparrow}$ so that electrons gain correlation energy
relative to those they have in the Ising Chern magnet vacuum. Investigating
the mechanism behind these energetics is an important problem, but in this
study our goal will be instead to understand the properties of these PH
Halperin states under the assumption that they are indeed the stable
incompressible ground states of the system.

Bulk properties of PH-Halperin states. We begin by establishing a matrix
generalization of the Streda relation between valley resolved magnetic fields
and particle densities. To do so we hypothetically allow the net effective
magnetic fields experienced by the $\uparrow$ electrons and $\downarrow$
holes, to be slightly different from the background field $B_{0}$ generated by
the moiré potentials, by considering non-zero
$\delta\mathbf{A}^{\uparrow,\downarrow}$ in Eqs.(2). The field experienced by
the $\downarrow$ holes is $B_{h}^{\downarrow}=-B_{e}^{\downarrow}$. We also
allow the number of particles to be different ($N_{e}^{\uparrow}\neq
N_{h}^{\downarrow}$) relative to the ideal ground state. Let us imagine that
the PH-Halperin state is placed on a manifold of area $A$ without boundaries
such as the torus or the sphere. Matching the areas of the droplets of both
spin components from Eq.(3) so that no fractional quasiparticles are added and
the system remains in its deformed incompressible ground state under these
sligthly modified conditions, leads to the following relations in the
thermodynamic limit:

$\left(\begin{array}[]{c}B_{e}^{\uparrow}\\\ B_{h}^{\downarrow}\\\
\end{array}\right)=2\pi\left(\begin{array}[]{cc}m&n\\\ n&m\\\
\end{array}\right)\left(\begin{array}[]{c}n_{e}^{\uparrow}\\\
n_{h}^{\downarrow}\\\ \end{array}\right),$ (4)

with $n_{e}^{\uparrow}=N_{e}^{\uparrow}/A$,
$n_{h}^{\downarrow}=N_{h}^{\downarrow}/A$. By converting back to electron
variables, $B_{e}^{\downarrow}=-B_{h}^{\downarrow}$,
$n_{e}^{\downarrow}=B_{h}^{\downarrow}/2\pi-n_{h}^{\downarrow}$, we obtain the
following layer-resolved Streda relations:

$\left(\begin{array}[]{c}n_{e}^{\uparrow}\\\ n_{e}^{\downarrow}\\\
\end{array}\right)=\frac{1}{2\pi}\left(\begin{array}[]{cc}\frac{m}{m^{2}-n^{2}}&\frac{n}{m^{2}-n^{2}}\\\
\frac{n}{m^{2}-n^{2}}&\frac{m}{m^{2}-n^{2}}-1\\\
\end{array}\right)\left(\begin{array}[]{c}B_{e}^{\uparrow}\\\
B_{e}^{\downarrow}\\\ \end{array}\right).$ (5)

As we will see, a wealth of physical properties of the PH Halperin states can
be derived from the above relations using a few reasonable physical
assumptions. For example, the electron filling of the moiré bands for each
spin component can be obtained by restoring the magnetic fields to the value
generated by the moiré potential in the absence of perturbations
($B_{e}^{\uparrow}=-B_{e}^{\downarrow}=B_{0}$), and are given by:

$\nu_{e}^{\uparrow}\equiv\frac{2\pi n_{e}^{\uparrow}}{B_{0}}=\frac{1}{m+n},\
\nu_{e}^{\downarrow}\equiv\frac{2\pi
n_{e}^{\downarrow}}{B_{0}}=1-\frac{1}{m+n}.$ (6)

Moreover the Eqs.(5) also govern the valley-resolved current density response
to local electric fields in the bulk. This can be derived by imagining that
the PH Halperin states are placed in a geometry without boundaries so that
there is a full gap to all excitations and that the probe fields
$\delta\mathbf{A}^{\uparrow,\downarrow}$ are varied weakly and slowly in time
and space. Assuming that the ground state evolves adiabatically under such
perturbations and that Eqs.(5) holds locally in space and time, one can derive
a relation between local currents and electric fields. This follows from
combining the continuity equations for the particle densities and recasting
Faraday’s laws as 2D continuity equations as follows:

$\partial_{t}n_{e}^{\uparrow,\downarrow}=-\nabla\cdot{\bf
j}_{e}^{\uparrow,\downarrow},\
\partial_{t}B_{e}^{\uparrow,\downarrow}=-\nabla\cdot({\bf
E}_{e}^{\uparrow,\downarrow}\times\hat{{\bf z}}).$ (7)

By combining the above with Eqs.(5) and assuming that currents are locally
orthogonal to electric fields so that the latter are dissipationless and do
not perform work, one obtains the following valley resolved Hall conductivity
matrix:

$\left(\begin{array}[]{c}{\bf j}_{e}^{\uparrow}\\\ {\bf j}_{e}^{\downarrow}\\\
\end{array}\right)=\frac{e^{2}}{h}\left(\begin{array}[]{cc}\frac{m}{m^{2}-n^{2}}&\frac{n}{m^{2}-n^{2}}\\\
\frac{n}{m^{2}-n^{2}}&\frac{m}{m^{2}-n^{2}}-1\\\
\end{array}\right)\left(\begin{array}[]{c}{\bf
E}_{e}^{\uparrow}\times\hat{{\bf z}}\\\ {\bf
E}_{e}^{\downarrow}\times\hat{{\bf z}}\\\ \end{array}\right),$ (8)

where we have restored the explicit units of electrical conductivity for
convenience. The above formula demonstrates that the PH Halperin states
feature a non-trivial quantized Hall-drag response, whereby an electrical
field driving only one valley can induce a quantized Hall current in the other
valley, analogous to that of standard Halperin states in Landau levels Yang
(1998). From the above we can get the valley-resolved electric currents in
response to a net physical electrical field that acts identically on both
valleys to be:

${\bf j}_{e}^{\uparrow}=\frac{e^{2}}{h}\left(\frac{1}{m-n}\right){\bf
E}\times\hat{{\bf z}},\ {\bf
j}_{e}^{\downarrow}=\frac{e^{2}}{h}\left(\frac{1}{m-n}-1\right){\bf
E}\times\hat{{\bf z}}.$ (9)

and the net bulk electrical conductivity is therefore the sum of the above
coefficients and given by:

$\sigma_{xy}=\frac{e^{2}}{h}\left(\frac{2}{m-n}-1\right).$ (10)

Interestingly, we see that the Hall conductivity of the PH-Halperin states is
not simply the sum or the difference of the filling factors of the two valleys
from Eq.(6). And even more remarkably, the above implies that the subset of PH
Halperin states satisfying $m=n+2$ have exactly zero Hall conductivity
$\sigma_{xy}=0$, and fractional-$1/2$ spin-resolved Hall conductivities of
equal magnitude and opposite sign:

${\bf j}_{e}^{\uparrow}=-{\bf j}_{e}^{\downarrow}=\frac{e^{2}}{2h}{\bf
E}\times\hat{{\bf z}}.$ (11)

In other words, their bulk electrical response would be exactly half of a
standard time-reversal invariant quantum spin-Hall insulator state Kane and
Mele (2005a, b); Bernevig and Zhang (2006) with valley number conservation in
which the two valley resolved bands with opposite Chern would be fully
occupied. However, notice that such PH Halperin states are not time-reversal
invariant. This can be seen from Eq.(6) by noting that generically the
fillings of the two components are different
$\nu_{e}^{\uparrow}\neq\nu_{e}^{\downarrow}$, for example the PH 331 state the
fillings are $\nu_{e}^{\uparrow}=1/4$ and $\nu_{e}^{\downarrow}=3/4$ (see
Fig.1(b)).

Edge properties of PH Halperin states. Let us now analyze the properties of
the PH-Halperin states at the edge. From analogy to fractional quantum Hall
states of correlated holes, such as the $\nu=2/3$ particle-hole conjugate to
the Laughlin state Kane  _et al._ (1994); Wen (1991a, 1990, b); MacDonald
(1990), one expects a non-trivial edge profile in which the ideal bulk PH-
Halperin state might be surrounded by a strip with fillings
$(\nu_{e}^{\downarrow},\nu_{e}^{\uparrow})=(1,0)$, since the PH-Halperin
droplet is carved out from a parent Ising-Chern magnet vacuum of the
$\downarrow$ valley (see Fig.1(a)), although more complex variants are
certainly possible (see e.g.Wang  _et al._ (2013)). Given the complexities of
the edge even in standard quantum Hall settings and considering the additional
uncertainties about physical ingredients that are special to moiré materials,
we will not attempt here to develop a detailed microscopic description of the
edge. Instead we will follow the spirit of thermodynamic considerations such
as those discussed in Ref.Cooper  _et al._ (1997), and appeal to the
assumption that in each edge there is good local thermodynamic equilibration
between the particles in each of the valleys, but no inter-valley scattering
processes that violate the conservation of the number particles in each
valley, and thus we will assume that different chemical potentials for each
valley at the edge are well defined.

Let us begin by describing the edge when the system is globally in
thermodynamic equilibrium (all edges included). There are two global chemical
potentials $\mu_{e}^{\uparrow},\mu_{e}^{\downarrow}$ for electrons in each
valley. Particles in each valley are confined to some area $A$ by
electrostatic potentials with an associated force ${\bf
E}_{0}^{\uparrow,\downarrow}=-\hat{{\bf n}}E_{0}^{\uparrow,\downarrow}$, where
$E_{0}^{\uparrow,\downarrow}>0$ and $\hat{{\bf n}}$ is the local unit vector
normal to the edge and pointing outwards from the sample (see Fig.2). From
Eq.(11) we expect that there are net currents in equilibrium at the edge,
which are flowing in opposite directions for the two spin components, and
which for our conventions would be counterclockwise for the $\uparrow$
particles and clockwise for the $\downarrow$ particles (see Fig.2). Following
standard principles of equilibrium thermodynamics, we can define a grand-
canonical free energy,
$G=E-\mu_{e}^{\uparrow}N_{e}^{\uparrow}-\mu_{e}^{\downarrow}N_{e}^{\downarrow}$,
where $E$ is the total energy of the system, whose differential is:

$dG=TdS-
N_{e}^{\uparrow}d\mu_{e}^{\uparrow}-N_{e}^{\downarrow}d\mu_{e}^{\downarrow}-AM_{e}^{\uparrow}dB_{e}^{\uparrow}-AM_{e}^{\downarrow}dB_{e}^{\downarrow}+\cdots$
(12)

where $M_{e}^{\uparrow,\downarrow}$ are the particle number magnetization
densities for the two valleys, and the $\cdots$ include the work differentials
of variations of the Hamiltonian with respect to parameters other than
$B_{e}^{\uparrow,\downarrow}$. From the above we obtain the following Maxwell
relations:

$\frac{\partial M_{e}^{j}}{\partial\mu_{e}^{i}}=\frac{\partial
n_{e}^{i}}{\partial B_{e}^{j}},\ i,j\in\\{\uparrow,\downarrow\\}.$ (13)

Figure 2: (a) Depiction of the half-quantized helical curent transport at the
edge of the PH-Halperin states with $m=n+2$. (b) Quasiparticle lattice of the
331 PH-Halperin state. The most elementary anyons are denoted by
$v_{e\uparrow},v_{e\uparrow},v_{h\uparrow},v_{h\uparrow}$. Their bound states
include charged spinless composite-fermion like particles $f_{e}$ and $f_{h}$,
and the neutral spin-carrying semions $s_{\uparrow}$ and $s_{\downarrow}$. $c$
is the local inter-valley cooper-pair, $e_{\downarrow}$ is the local electron
in valley $\downarrow$ and $m_{\uparrow}$ is a local charge-neutral spin-1
inter-valley magnon. Spinless particles along the pure charge axis are
itinerant charged particles, which is impossible in usual Landau levels where
charged particles are also drifting particles which experience Lorentz force.

Now from the assumption of incompressibility of the bulk, one concludes that
any changes of the magnetizations, $\delta M_{e}^{\uparrow,\downarrow}$ in
response to small variations of the chemical potentials
$\delta\mu_{e}^{\uparrow,\downarrow}$ which remain within the bulk gap, must
arise entirely from variations of the currents localized at the edges of the
system. By integrating the relation between magnetization and current
densities ${\bf j}_{e}^{\uparrow,\downarrow}=\nabla\times(\hat{{\bf
z}}M_{e}^{\uparrow,\downarrow})$, one gets the relation between edge currents
and magnetization ${\bf
I}_{e}^{\uparrow,\downarrow}=M_{e}^{\uparrow,\downarrow}\hat{{\bf
z}}\times\hat{{\bf n}}$ Cooper  _et al._ (1997). Then by computing the right-
hand-side of Eq.(13) from Eq.(5), one obtains the relation between variations
of edge currents and global chemical potentials:

$\left(\begin{array}[]{c}\delta{\bf I}_{e}^{\uparrow}\\\ \delta{\bf
I}_{e}^{\downarrow}\\\ \end{array}\right)=(\hat{{\bf z}}\times\hat{{\bf
n}})\frac{e^{2}}{h}\left(\begin{array}[]{cc}\frac{m}{m^{2}-n^{2}}&\frac{n}{m^{2}-n^{2}}\\\
\frac{n}{m^{2}-n^{2}}&\frac{m}{m^{2}-n^{2}}-1\\\
\end{array}\right)\left(\begin{array}[]{c}\delta\mu_{e}^{\uparrow}\\\
\delta\mu_{e}^{\downarrow}\\\ \end{array}\right),$ (14)

Now, following the spirit of Landauer-Büttiker approach Landauer (1970);
Büttiker (1986), in a transport experiment one expects that different edges
connecting various voltage leads generally have different chemical potentials
for the two spins components. Since the net direction of the
$\uparrow,\downarrow$ edge currents is opposite, we assume that the chemical
potential of these modes is in equilibrium with the voltage lead from which
each of them emanates, in analogy to the standard Landauer-Büttiker picture
for the quantum-spin-Hall effect Kane and Mele (2005a). In particular we see
from the above that the total electric current at the edge, $\delta{\bf
I}_{e}=\delta{\bf I}_{e}^{\uparrow}+\delta{\bf I}_{e}^{\downarrow}$, is:

$\delta{\bf I}_{e}=(\hat{{\bf z}}\times\hat{{\bf
n}})\frac{e^{2}}{h}\left(\frac{1}{m-n}\right)\delta\mu_{e}^{\uparrow}-(\hat{{\bf
z}}\times\hat{{\bf
n}})\frac{e^{2}}{h}\left(1-\frac{1}{m-n}\right)\delta\mu_{e}^{\downarrow},$
(15)

and, therefore, we see that for the special PH Halperin states of our interest
with $m=n+2$ each edge behaves as a helical edge conductor with opposite
current directions for each spin and fractional conductance $e^{2}/2h$, namely
half of the usual quantum spin Hall effect, in agreement with experiment Kang
_et al._ (2024a):

$\delta{\bf I}_{e}=(\hat{{\bf z}}\times\hat{{\bf
n}})\frac{e^{2}}{2h}(\delta\mu_{e}^{\uparrow}-\delta\mu_{e}^{\downarrow}).$
(16)

Bulk quasiparticles of PH Halperin states. There is a key distinction between
standard multicomponent systems in Landau levels with the same magnetic field,
and our current setting of pairs of Landau levels with opposite magnetic
fields which becomes evident when we analyze their quasi-particles. In
standard Landau levels only fully neutral particles are itinerant and can move
in straight trajectories in the presence of magnetic field. Such itinerant
neutral particles include the excitons in integer quantum Hall ferromagnets,
the magneto-roton in the Laughlin state, and the Bogoliubov composite fermion
in paired states such as the Moore-Read or the standard 331 Halperin state. In
the current setting, however, it is possible to have charged quasi-particles
that are itinerant, when the charge of the particle is equally split between
the two valleys that experience opposite magnetic fields. The converse of this
is also possible, namely in the current setting there are neutral particles
that behave as the ordinary charged particles in standard Landau levels, as
demonstrated for the inter-valley excitons of the Ising Chern magnet which
have exactly opposite charges in the two Landau levels with opposite magnetic
field, as discussed in Refs.Stefanidis and Sodemann (2020); Kwan  _et al._
(2021, 2022). Therefore, we would like to introduce a notion to distinguish
particles not only as charged and neutral, but also as itinerant or drifting
quasiparticles. Quasi-particles carry definite valley charges, which we denote
by a vector $(\delta N_{e}^{\uparrow},\delta N_{e}^{\downarrow})$ indicating
their total electric charge in each of the two valleys (in units where the
electron charge is 1). Thus the total quasi-particle electric charge is
$\delta N_{e}^{\uparrow}+\delta N_{e}^{\downarrow}$. Now, in the context of a
pair of Landau levels where valleys have opposite magnetic fields, we will say
that a quasi-particle is itinerant if their valley polarization is zero,
namely if:

$\delta N_{e}^{\uparrow}=\delta N_{e}^{\downarrow},\ {\rm(itinerant\
quasiparticle),}$ (17)

otherwise we will call the quasiparticle a drifting quasiparticle. For
example, when we add quasiparticles to the trivial vacuum of fully empty
valleys, the electrons would have numbers $(\delta N_{e}^{\uparrow},\delta
N_{e}^{\downarrow})=(1,0)$ or $(\delta N_{e}^{\uparrow},\delta
N_{e}^{\downarrow})=(0,1)$, and thus they would be charged drifting particles,
whereas a Cooper-pair-like bound state of two electrons with $(\delta
N_{e}^{\uparrow},\delta N_{e}^{\downarrow})=(1,1)$ would be a charged
itinerant particle. Conversely, an ordinary exciton with $(\delta
N_{e}^{\uparrow},\delta N_{e}^{\downarrow})=(1,-1)$, which can be added to the
Ising Chern magnet vacuum, would be a neutral drifting particle.

Since the PH Halperin states are abelian topologically ordered states, their
topological properties can be captured by an abelian Chern-Simons theory with
a K-matrix and layer resolved charge vectors Wen (2004) given by:

$K=\left(\begin{array}[]{ccc}m&n&0\\\ n&m&0\\\ 0&0&-1\\\ \end{array}\right),\
q_{\uparrow}=\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\ \end{array}\right),\
q_{\downarrow}=\left(\begin{array}[]{c}0\\\ -1\\\ 1\\\ \end{array}\right).$
(18)

All the bulk quasiparticles are expected to be fully gapped and can be labeled
by vectors of integers ${\it{\bf l}}^{T}=(l_{1},l_{2},l_{3})$. Their self-
exchange statistical angle (topological spin) and layer resolved charges are
given by $\theta_{\bf l}=\pi{\bf l}^{T}K^{-1}{\bf l}$, $\delta
N_{e}^{\uparrow/\downarrow}=q_{\uparrow/\downarrow}^{T}K^{-1}{\bf l}$, while
their braiding statistics (statistical phase of a particle after full loop
around another particle) is $\theta_{{\bf l},{\bf l^{\prime}}}=2\pi{\bf
l}^{T}K^{-1}{\bf l^{\prime}}$ Wen (2004). The most elementary anyons are:

$v_{e\uparrow}=\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\
\end{array}\right),v_{h\uparrow}=\left(\begin{array}[]{c}-1\\\ 0\\\ 0\\\
\end{array}\right),v_{e\downarrow}=\left(\begin{array}[]{c}0\\\ -1\\\ 0\\\
\end{array}\right),v_{h\downarrow}=\left(\begin{array}[]{c}0\\\ 1\\\ 0\\\
\end{array}\right).$ (19)

where $v_{e\uparrow}$,$v_{h\uparrow}$ denote the Laughlin-type quasi-electron
and quasi-hole particles in the $\uparrow$ component, and similarly,
$v_{e\downarrow}$,$v_{h\downarrow}$ the corresponding quasi-particles for
$\downarrow$ component. All of these quasi-particles are anyons with
fractional self-exchange statistical angle $\theta_{v}=\pi m/(m^{2}-n^{2})$.
The quasi-electrons have a bump of electron density relative to the background
PH Halperin ground state, and the quasi-holes a depletion, so that the layer-
resolved charges of these quasi-particles are:

$v_{e\uparrow}:(\delta N_{e}^{\uparrow},\delta
N_{e}^{\downarrow})=\left(\frac{m}{m^{2}-n^{2}},\frac{n}{m^{2}-n^{2}}\right).$
(20)

The corresponding layer resolved charges of the quasi-electron
$v_{e\downarrow}$ are obtained by swapping $\uparrow\leftrightarrow\downarrow$
in the above formula, and those of the quasi-holes
$v_{h\uparrow},v_{h\downarrow}$ are minus those of the corresponding quasi-
electrons. From the above, we see that the $v$-type anyons are drifting quasi-
particles because they have un-equal occupation of the two valleys whenever
$m\neq n$, such as in the PH $331$ state. Interestingly, we see that the total
charge of these quasi-particles is:

$v_{e\uparrow}:\ \delta N_{e}=\frac{1}{m-n},$ (21)

And therefore, for the PH-Halpering states with $m=n+2$, these quasi-particles
carry 1/2 of the electron charge which is unequally split between both valleys
according to Eq.(20). This contrasts with the standard Halperin states in
multicomponent Landau levels (LLs), whose smallest quasiparticles have charge
$(\delta N_{e})_{\rm LLs}=1/(m+n)$ which is split as $(\delta
N_{e}^{\uparrow},\delta N_{e}^{\downarrow})_{\rm
LLs}=(m/(m^{2}-n^{2}),-n/(m^{2}-n^{2})))$. For example the standard, 331
Halperin state has smallest quasiparticles with total charge $1/4$ which is
split as $(3/8,-1/8)$, whereas our 331 PH Halperin state has a smallest
quasiparticles with total charge $1/2$ which is split as $(3/8,1/8)$. This
difference can be understood as a result of the particle-hole conjugation of
the $\downarrow$ component, which basically maps the layer pseudospin
polarization of the quasiparticle in the usual Landau levels onto the total
quasiparticle charge in the current setting: $(\delta N_{e}^{\uparrow}-\delta
N_{e}^{\downarrow})_{\rm LLs}\rightarrow(\delta N_{e}^{\uparrow}+\delta
N_{e}^{\downarrow})$.

Let us now consider some of the quasi-particles that are obtained as bound
states of pairs of the above elementary anyons. The ${\bf l}$ vectors of these
bound states are obtained by adding those of the elementary anyons, and we
label them as:

$f_{e}=v_{e\uparrow}\times\ v_{e\downarrow}=\left(\begin{array}[]{c}1\\\ -1\\\
0\\\ \end{array}\right),\ f_{h}=v_{h\uparrow}\times\
v_{h\downarrow}=\left(\begin{array}[]{c}-1\\\ 1\\\ 0\\\ \end{array}\right),$
(22)

$s_{\uparrow}=v_{e\uparrow}\times\
v_{h\downarrow}=\left(\begin{array}[]{c}1\\\ 1\\\ 0\\\ \end{array}\right),\
s_{\downarrow}=v_{h\uparrow}\times\
v_{e\downarrow}=\left(\begin{array}[]{c}-1\\\ -1\\\ 0\\\ \end{array}\right).$
(23)

The layer-resolved charges and the self-statistical exchange phase of these
quasi-particles are:

$f_{e}:(\delta N_{e}^{\uparrow},\delta
N_{e}^{\downarrow})=\left(\frac{1}{m-n},\frac{1}{m-n}\right),\
\theta_{f}=\frac{2\pi}{m-n},$ (24)

$s_{\uparrow}:(\delta N_{e}^{\uparrow},\delta
N_{e}^{\downarrow})=\left(\frac{1}{m+n},-\frac{1}{m+n}\right),\
\theta_{s}=\frac{2\pi}{m+n}.$ (25)

The particle $f_{h}$ has the same statistics and opposite layer-resolved
charges of $f_{e}$, while $s_{\downarrow}$ has the same statistics and
opposite charges of $s_{\uparrow}$. Therefore, we see that the $f_{e},f_{h}$
are spinless particles with equal occupations of both layers and therefore are
also charged itinerant particles, while the $s_{\uparrow},s_{\downarrow}$ are
neutral spinful particles with opposite occupations of layers, and thus are
drifting particles.

Remarkably, for the special class of PH-Halperin with $m=n+2$ that are
consistent with the experiments of Ref.Kang  _et al._ (2024a), the
$f_{e},f_{h}$ particles are spinless fermions ($\theta_{f}=\pi$) with the same
charge of an ordinary electron and hole respectively, but fractionalized onto
halves residing on each spin component $(\delta N_{e}^{\uparrow},\delta
N_{e}^{\downarrow})=\pm(1/2,1/2)$. Despite being charged these particles are
therefore intinerant quasiparticles. Moreover their fermionic statistics imply
that if a finite density of these particles is added to the PH-Halperin state
vacuum, they can naturally form a compressible metallic itinerant state. In
contrast for these special states with $m=n+2$, the
$s_{\uparrow},s_{\downarrow}$ have in general fractional self-statistics
($\theta_{s}=\pi/(n+1)$) and are drifting non-intinerant particles.

The spinless fermions $f_{e},f_{h}$ are non-local emergent particles and
should not be confused with the microscopic electron or hole, even though they
have the same statistics and total charge. This can be seen for example by
noting that the statistical phase for moving $f_{e}$ around a loop encircling
the anyon $v_{e\uparrow}$ is $\theta_{f_{e},v_{e\uparrow}}=2\pi/(m-n)$. For
the special states with $m=n+2$ this is $\theta_{f_{e},v_{e\uparrow}}=\pi$,
which implies that this fermion $f_{e}$ sees the anyon $v_{e\uparrow}$ as an
effective $\pi$-flux Abrikosov-like vortex. Such resemblance to the quasi-
particles of a superconductor is not a coincidence; it is known that in
ordinary two-component Landau levels the Halperin states with $m=n+2$ can be
thought of as an inter-layer paired state of composite fermions Read and Green
(2000); Haldane and Rezayi (1988); Greiter  _et al._ (1992); Ho (1995); Dimov
_et al._ (2008); Sharma  _et al._ (2024); Read and Rezayi (1996). In ordinary
Landau levels the $f$ particles would have been the charge neutral Bogoliubov
fermions descending from the composite fermion upon pairing, and would have
been an itinerant particle with opposite charges in the two layers. However,
as we discussed before, our setting of valleys with opposite magnetic fields
maps into the traditional Landau level setting after particle-hole conjugating
one of the valleys. Under such transformation, the Bogoliubov composite
fermion becomes a charged particle but retains its itinerant character.

To close this section we would like to connect the above fractionalized quasi-
particles to the standard local microscopic particles. The standard electrons
and holes with valley-spins $\uparrow$ and $\downarrow$ can be obtained as:

$e_{\uparrow}=(v_{e\uparrow})^{m}\times\
(v_{h\downarrow})^{n}=\left(\begin{array}[]{c}m\\\ n\\\ 0\\\
\end{array}\right),\ e_{\downarrow}=(v_{h\uparrow})^{n}\times\
(v_{e\downarrow})^{m}=\left(\begin{array}[]{c}-n\\\ -m\\\ 0\\\
\end{array}\right),$ (26)

$h_{\uparrow}=(v_{h\uparrow})^{m}\times\
(v_{e\downarrow})^{m}=\left(\begin{array}[]{c}-m\\\ -n\\\ 0\\\
\end{array}\right),\ h_{\downarrow}=(v_{e\uparrow})^{n}\times\
(v_{h\downarrow})^{m}=\left(\begin{array}[]{c}n\\\ m\\\ 0\\\
\end{array}\right),$ (27)

The above are local fermions, with charges $(\delta N_{e}^{\uparrow},\delta
N_{e}^{\downarrow})$ respectively given by $\\{(1,0),(0,1),(-1,0),(0,-1)\\}$.
Interestingly, the inter-valley Cooper-pair-like boson obtained as the bound
state of two ordinary electrons in the two valleys can be alternatively also
obtained as the bound state of an $(m-n)$ multiple of $f_{e}$ composite-
fermion-like particles:

$c=e_{\uparrow}\times\
e_{\downarrow}=(f_{e})^{m-n}=\left(\begin{array}[]{c}m-n\\\ n-m\\\ 0\\\
\end{array}\right),$ (28)

The above can be recognized as such inter-valley Cooper-pair because it is a
self-boson which is local (i.e. has trivial full braiding with any other
quasi-particle modulo $2\pi$) and carries charges $(\delta
N_{e}^{\uparrow},\delta N_{e}^{\downarrow})=(1,1)$. For the PH-Halperin states
with $m=n+2$ that are consistent with experiments, Eq.(30) can be interpreted
as saying that a bound state of a pair $f_{e}$ composite-fermions forms an
ordinary Cooper-pair, or alternatively, that the $f_{e}$ composite-fermion is
one half of an ordinary Cooper-pair. We see that the Cooper pair is therefore
the simplest fully local itinerant charged particle in these states.

Conversely, the inter-valley magnon-exciton can be obtained either as a bound
state of an $\uparrow$ electron and a $\downarrow$ hole or alternatively as an
$(m+n)$ multiplet of the $s$ anyons from Eq.(23), as follows:

$m_{\uparrow}=e_{\uparrow}\times\
h_{\downarrow}=(s_{\uparrow})^{m+n}=\left(\begin{array}[]{c}m+n\\\ m+n\\\ 0\\\
\end{array}\right),$ (29)

and, analogously, we also have a valley-spin raising particle:

$m_{\downarrow}=e_{\uparrow}\times\
h_{\downarrow}=(s_{\downarrow})^{m+n}=\left(\begin{array}[]{c}-m-n\\\ -m-n\\\
0\\\ \end{array}\right).$ (30)

these local magnon-like particles carry charges $(\delta
N_{e}^{\uparrow},\delta N_{e}^{\downarrow})$ given respectively by $(1,-1)$
and $(-1,1)$, and thus are neutral spin-1 bosons, but are drifting (not
intinerant) particles. We see, therefore, that the PH-Halperin states realize
a beautiful and colorful pattern of spin-charge separation.

Localization of itinerant vs drifting particles and the robustness of the Hall
plateau. We believe that the existence of itinerant charged particles can have
important consequences on the robustness of the Hall plateau when the system
is doped away from the precise filling of the ideal incompressible state. To
argue for this, we begin by reviewing the mechanism behind the existence of a
Hall plateau for a standard Laughlin or Halperin states in a usual Landau
levels in the same magnetic field. The ideal fractional quantum Hall fluids
exist at some precise fractional proportion between the electron density and
the magnetic field. However, when the magnetic field or electron density are
slightly changed away from this exact proportion, an experimental Hallmark of
the fractional quantum Hall effect is that $\sigma_{xy}$ remains precisely
quantized at the value associated with the ideal fluid as if this proportion
had not changed. This occurs because the excess particles away from the ideal
filling are accommodated in the form of Laughin-like quasi-particles added on
top of the ideal vacuum. In standard Landau levels, these quasi-particles are
charged and therefore drifting (non-itinerant) and thus easily localized and
pinned by the disorder potential. For sufficiently small densities of these
quasi-particles they remain pinned at disconnected locations and surrounded by
the ideal Laughlin-like fluid with its chiral edges intact and not connected
through the bulk of the sample. Therefore experiments probing these edges
still observe the same quantized behavior as if the extra pinned quasi-
particles were not there.

Therefore, we see that the robustness of the Hall plateau requires several key
ingredients beyond the mere existence of a parent ideal incompressible fluid,
and a particularly crucial ingredient is that the disorder potential is
effective at pinning the charged quasiparticles added to the parent ideal
fluid. As we have mentioned, in the traditional Laughlin and Halperin states
in standard Landau levels, all charged quasiparticles are drifting particles
(such as the Laughlin anyon or the electon) whereas the itinerant particles
are all neutral particles (such as the magneto-roton or the Bogoliubov-
composite-fermion of the Moore-Read state). Because charged particles are non-
itinerant they can be easily pinned by the disorder potential, and Hall
plateaus tend to be robust.

However, in the current context of time-reversal invariant pairs of Landau
levels with opposite magnetic fields, charged quasiparticles can be itinerant
(such as the charged $f_{e},f_{h}$ composite fermions of PH Halperin states
discussed in the previous section). Therefore, when a net excess charge is
added to the ideal incompressible PH Halperin state so that the filling
deviates slightly from the ideal total filling $\nu=1$, the system might be
doped with a finite density of itinerant charged quasi-particles which cannot
be efficiently pinned by the disorder potential. In the special case of the
$f_{e},f_{h}$ particles these are moreover charged fermions with can naturally
form a fermi-fluid-like metallic state at finite density. These intinerant
charged particles will easily move across the sample and electrically connect
the initially disconnected edges through the bulk. The disorder potential will
scatter them but it will be much less efficient at pinning and localizing
them, as compared to the drifting quasi-particles. This will lead to smooth
deviations of the Hall and longitudinal conductivities away from those of the
parent ideal PH Halperin state and in particular degrade the quantization of
its edge conductance. This is consistent with the experimental observations of
Ref.Kang  _et al._ (2024a), which observed as smooth variation of the
longitudinal and Hall resistances away from the pressumed ideal filling of
$\nu=-3$ and no clear sign of a robust Hall plateau.

Notice that this mechanism might not be as efficient at destroying the Hall
plateu of an simple Ising Chern number in which particles occupy a single
valley Chern band, such as the state believed to be realized at $\nu=-1$ in
Ref.Kang  _et al._ (2024a). This is because in this case we expect that the
simplest itinerant charged quasiparticle is the cooper-pair-like bound state
of two electrons in separate valleys. Notice that such Cooper would have to be
made from two electrons on the two valleys, and since one of the valley-
resolved Chern bands would be fully occupied in the Ising Chern magnet at
$\nu=-1$, then this Cooper-pair would have put an electron on one of the empty
and gapped moire bands that is not related by time reversal to the occupied
band. Thus such Cooper-pair-like bound states might have a larger gap that
single electrons and holes (which are not itinerant), or perhaps no good
cooper-pair like bound state of these electrons which is well separated from
the continuum might appear at low energies. Thus it might not be energetically
favorable to add it to the Ising Chern magnet vacuum when the charge density
deviates away from the ideal filling but instead to add usual electrons or
holes which are drifting (not itinerant) and thus more easily pinned by the
disorder. This is consistent with the observation of the Hall plateau for the
state at $\nu=-1$ in Ref.Kang  _et al._ (2024a).

In the supplementary material we discuss a toy model for bound states of two
particles in the case of opposite and also the same magnetic fields. This
model illustrates than that when the bound state is a charged but itinerant
particle, there is a competition between the interaction that tries to
localize the relative distance between the two charges that experience
opposite magnetic fields while delocalizing the average position of the center
of the bound state. Thus when the interaction scale associated with binding
energy of the two components is larger than the disorder potential that tries
to localize its average position, the disorder becomes inefficient at pinning
and ultimately localizing the itinerant particle. This picture suggests that
disorder is less efficient at localizing bound states of particles whose
constituents reside in bands with opposite Chern numbers, even in comparison
to completely trivial bands that have no Berry phase geometry.

Summary, discussion and outlook. We have demonstrated that a subset of
Halperin states where particles are added to an empty band with Chern number
$C=1$ and holes are added to a filled band with opposite Chern number $C=-1$,
have exactly zero Hall conductivity in spite of spontaneously breaking time-
reversal symmetry. These states are those for which the intra-flavor ($m$) and
inter-flavor ($n$) exponents are related as $m=n+2$. Moreover, in the presence
of separate particle number conservation for the two flavors, they feature a
fractional helical edge conductance of $e^{2}/2h$ per edge. Therefore these
states behave from the charge transport point of view as half of the standard
quantum spin Hall states Kane and Mele (2005a, b); Bernevig and Zhang (2006),
and are, therefore, consistent with those observed in moiré tMoTe2 in Ref.Kang
_et al._ (2024a).

We have also emphasized, using these PH-Halperin states as examples, a crucial
difference between the standard setting of multi-component systems in Landau
levels with common magnetic fields, and our setting of pairs of Landau levels
with opposite magnetic fields. Namely, in standard Landau levels all itinerant
quasi-particles are neutral (e.g. excitons in quantum Hall ferromagnets,
magneto-roton of Laughlin state, Bogoliubov composite-fermion of Moore-Read
state), and charged particles are drifting particles which experience Lorentz
force. However, in the current setting of pairs of Landau levels with opposite
magnetic fields it is possible to have charged particles that are itinerant
when their charge is equally split between the valleys experiencing
effectively opposite magnetic fields. In particular, we have seen that the
Bogoliubov composite fermion of the 331 PH-Halperin state is a charged
itinerant spinless fermion in the current setting. The disorder potential can
scatter these particles, but it is much less efficient at pinning and
localizing them, because they are itinerant. As a result one expects that if
these particles are added to the parent ideal PH-Halperin state for example by
changing the electron density slightly away from the precise filling, the
disorder potential is less efficient at localizing them and they can make a
conducting metallic fluid that will degrade the quantization of the
conductance. This is consistent with the experimental observation of no robust
Hall plateau surrounding the ideal filling fraction where the fractional
quantum spin Hall effect is observed, but instead a smooth variation of the
Hall resistance away from zero as function of filling in Ref.Kang  _et al._
(2024a).

There are several important open questions for future studies which we would
like to mention. One set of questions pertains to investigating experimental
probes which could distinguish the PH-Halperin states from other possible
states. In this regard, we would like to highlight that the PH-Halperin states
we have discussed have quasi-particles with minimal charge equal to $1/2$ (see
Fig.2). This is in contrast with typical paired states realized in half-filled
Landau levels (including the standard Halperin 331 and the Moore-Read states)
which have quasiparticles with minimal charge $1/4$. For example, measuring
this charge would distinguish our state from a state in the same universality
class of a time-reversal invariant pair of Moore-Read states constructed in
the $C=\pm 1$ bands. Measuring the fractional charge of quasi-particles is
difficult but there are important precedents from current-noise experiments in
quantum Hall settings Saminadayar  _et al._ (1997); De-Picciotto  _et al._
(1998). Anyon interferometers could also be used to measure the quasi-particle
charge and statistics Nakamura  _et al._ (2020). Other interesting observables
to further consider include the spin/valley Hall-drag, which is expected to be
quantized according to Eqs.(8), and could be probed by inter-facing the PH-
Halperin state with another magnetic state that would act as spin battery
inducing a non-zero spin-Hall-voltage. Other possibilities include also
measuring the thermal Hall conductance Banerjee  _et al._ (2017, 2018) which
is expected to ideally be non-zero and the same as an integer quantum Hall
effect, since the K-matrix from Eq.(18) has two negative and one positive
eigen-value. This would distinguish the PH-Halperin states from those with
time-reversal symmetry, which are ideally expected to have vanishing thermal
Hall conductance.

Another important open problem is to understand which microscopic models and
interactions stabilize the PH-Halperin states in tMoTe2. Clearly there are
many non-trivial realistic aspects of moiré tMoTe2 that our current discussion
is missing. However we would like to highlight that even understanding the
competition of the PH-Halperin states and other states in ideal pairs of
Landau levels with opposite magnetic fields remains relatively unexplored. We
would like to comment in this regard that Ref.Stefanidis and Sodemann (2020)
demonstrated that for simple toy models of short-ranged gaussian interactions
with a range comparable to the moiré unit cell, the Ising Chern magnet becomes
unstable when the intra-flavor repulsions $V_{\uparrow,\uparrow}$ become about
$30\%$ stronger than inter-flavor repulsion $V_{\uparrow,\downarrow}$.
References Stefanidis and Sodemann (2020); Kwan  _et al._ (2022) proposed that
one possible set of states emerging in this setting might be Laughlin states
of excitons. But it would be important to examine these possibilities in
detailed many-body numerical studies and search for PH-Halperin and other
possible states.

Our PH-Halperin states clearly illustrate that for a pair of Chern bands with
valley Chern numbers $C=\pm 1$, having a gapped state with zero Hall
conductivity does not imply absence of valley polarization or time-reversal
symmetry. Conversely having zero valley polarization in this same setting does
not imply a zero Hall conductivity, as exemplified by the excitonic Laughlin
states constructed in Refs.Stefanidis and Sodemann (2020); Kwan  _et al._
(2022), which can be valley unpolarized but have a quantized integer Hall
conductivity $\sigma_{xy}=e^{2}/h$. These examples highlight that in the
setting of pairs of valleys with opposite Chern numbers it is important to
exercise some caution as the observation of integer quantized Hall
conductivities could be secretly disguising non-trivial fractionalized states.

Finally, we would like to close by mentioning that there are other interesting
PH-Halperin states that we have not focused on because they do not fit the
experimental observations of Ref.Kang  _et al._ (2024a), but which might
appear in other experimental settings and are also interesting theoretically.
For example from Eq.(10) we see that another interesting subset of PH Halperin
states with integer Hall conductivity, $\sigma_{xy}=e^{2}/h$, are those with
$m=n+1$, such as the PH 332 state, which has fractional valley fillings
$\nu_{e}^{\uparrow}=1/5$ and $\nu_{e}^{\downarrow}=4/5$. Another interesting
states are those with $m=n-1$, such as PH 112, which would have Hall
conductivity $\sigma_{xy}=-3e^{2}/h$, and valley fillings
$\nu_{e}^{\uparrow}=1/3$ and $\nu_{e}^{\downarrow}=2/3$, and those with
$m=n-2$, such as PH 113, which would have Hall conductivity
$\sigma_{xy}=-2e^{2}/h$, and valley fillings $\nu_{e}^{\uparrow}=1/4$ and
$\nu_{e}^{\downarrow}=3/4$. These states further highlight the aforementioned
point that integer quantization of the Hall conductivity might hide behind
more interesting states in disguise.

Acknowledgements. I am especially indebted to my colleague Bernd Rosenow for
many important and stimulating discussions on this subject, and, in
particular, for teaching me about various aspects of the physics of quantum
Hall edges which were crucial for me to gain deep appreciation of the
experimental findings of Ref.Kang  _et al._ (2024a). I am also thankful to the
organizers and participants of three conferences which were crucial for
gaining early insights for developing these ideas: the “Workshop Fractional
Quantum Anomalous Hall Effect and Fractional Chern Insulators” at the Max-
Planck-Institute for the Physics of Complex Systems in Dresden, the “FTPI
March Meeting” at the Physics Department of the University of Minnesota
supported by the Simons Foundation, and the 2024 March Meeting in Minneapolis
of the American Physical Society. In these conferences I enjoyed stimulating
discussions in particular with Yang Zhang, Ady Stern, Liang Fu, Allan
MacDonald, Cenke Xu, T. Senthil, and Andrei Bernevig. I am also thankful for
the support from the Deutsche Forschungsgemeinschaft (DFG) through research
grant project number 518372354.

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Supplementary Material: Halperin States of Particles and Holes in Time
Reversal Invariant Pairs of Landau Levels and Their Possible Presence in Moiré
MoTe2

##  I A toy model of competition of interactions and localization of charged
itinerant particles in pairs of time-reversal invariant Landau levels.

To illustrate this mechanism more concretely, let us consider a toy-model of
two particles with different charges $q_{1},q_{2}$ and experiencing different
magnetic fields $B_{1},B_{2}$. The operator that generates translations within
a Landau level is the magnetic momentum, which for each of these particles
takes the form:

${\bf Q}_{a}^{1}=\boldsymbol{\pi}_{a}^{1}+B_{1}q_{1}\epsilon_{ab}{\bf
r}_{b}^{1},\ {\bf
Q}_{a}^{2}=\boldsymbol{\pi}_{a}^{2}+B_{2}q_{2}\epsilon_{ab}{\bf r}_{b}^{2}.$
(S1)

These particles have non-commutative magnetic momenta:

$[{\bf Q}_{a}^{i},{\bf Q}_{b}^{j}]=i\hbar\epsilon_{ab}\delta^{ij}q_{i}B_{i}.$
(S2)

Now the operator that generates the translations of these two particles
together while preserving their relative positions is the total magnetic
momentum, which has commutation relations:

${\bf Q}_{a}\equiv{\bf Q}_{a}^{1}+{\bf Q}_{a}^{2},[{\bf Q}_{a},{\bf
Q}_{b}]=i\hbar\epsilon_{ab}\delta^{ij}(q_{1}B_{1}+q_{2}B_{2}).$ (S3)

From the above it follows that a particle made as a bound state of these two
particles would be itinerant if the above momenta commute and otherwise it
would be drifting. For the standard Landau level setting where each particle
experiences equal magnetic fields $B_{1}=B_{2}$, we see that only charge-
neutral bound states ($q_{1}=-q_{2}$) are itinerant while charged bound states
are drifting. On the other hand for the case of pairs of time-reversal
invariant Landau levels with opposite fields $B_{1}=-B_{2}$, we have the
converse situation where charge neutral ($q_{1}=-q_{2}$) bound states are
drifting, whereas a charged bound state with equal charges on the two
components ($q_{1}=q_{2}$) is itinerant.

Despite having a commutative total momentum, these itinerant bound states in
time-reversal-invariant pairs of Landau levels have non-trivial differences
with respect to trivial intinerant particles and their bound states in a
trivial band without any Berry phases whatsoever. To see this, let us consider
introduce the guiding center position (or projected position) operators for
each particle defined as follows:

${\bf R}_{a}^{i}=-\frac{\epsilon_{ab}}{q_{i}B_{i}}{\bf Q}_{b}^{i}={\bf
r}_{a}^{i}-\frac{\epsilon_{ab}}{q_{i}B_{i}}\boldsymbol{\pi}_{b}^{i}.$ (S4)

We can then define a relative distance ${\bf d}$ and average position ${\bf
R}$ of the pair of particles as follows:

${\bf d}\equiv{\bf R}^{1}-{\bf R}^{2},{\bf R}\equiv\frac{{\bf R}^{1}+{\bf
R}^{2}}{2},$ (S5)

These operators obey the following algebra:

$\left[{\bf d}_{a},{\bf
d}_{b}\right]=\left(\frac{1}{q_{1}B_{1}}+\frac{1}{q_{2}B_{2}}\right)i\hbar\epsilon_{ab},$
(S6)

$\left[{\bf R}_{a},{\bf
R}_{b}\right]=\frac{1}{4}\left(\frac{1}{q_{1}B_{1}}+\frac{1}{q_{2}B_{2}}\right)i\hbar\epsilon_{ab},$
(S7)

$\left[{\bf R}_{a},{\bf
d}_{b}\right]=\frac{1}{2}\left(\frac{1}{q_{1}B_{1}}-\frac{1}{q_{2}B_{2}}\right)i\hbar\epsilon_{ab},$
(S8)

We see therefore that in Landau levels with $B_{1}=B_{2}$ the algebras of
relative distance and average position commute for bound states of particles
with equal charge $q_{1}=q_{2}$. However, for time-reversal pairs of Landau
levels with $B_{1}=-B_{2}$ and bound states of particles with equal charge
$q_{1}=q_{2}$, the relative distance operator does not commute with the
average position operator. The interaction between the particles depends on
their relative distance and thus would like to fix this operator, whereas the
disorder potential wants to localize the particles and pin their average
positions. But the Heisenberg uncertainty prevents optimizing both of these
properties simultaneously. There is, therefore, a competition between disorder
and interactions. If the interaction is stronger, the system will prefer
making more certain the relative distance ${\bf d}$ leading to strong quantum
fluctuations on the average position, ${\bf R}$, thus reducing the ability of
disorder to localize and pin these particles. This is a delocalization
mechanism that would not be present in trivial bands with fully trivial Berry
phases where projected position operators commute.

To illustrate this let us consider a toy harmonic Hamiltonian of the form:

$H=\sum_{a}\frac{U}{l^{2}_{B}}{\bf d}_{a}{\bf
d}_{a}+\frac{V}{l^{2}_{B}}\left({\bf R}_{a}^{1}{\bf R}_{a}^{1}+{\bf
R}_{a}^{2}{\bf R}_{a}^{2}\right)$ (S9)

the term proportional to $U$ is a cartoon for an attractive interaction
potential that tries to minimize the distance between the particles and $V$ is
a cartoon for a disorder potential trying to pin the particles at the origin.
Let us assume that the two particles have the same charge $q_{1}=q_{2}=q$, but
let us contrast two cases: (a) when $B_{1}=B_{2}=B$ and (b) $B_{1}=-B_{2}=B$.
We have taken a unit length $l_{B}=(\hbar/|qB|)^{1/2}$. In case (a) one finds
two decoupled Harmonic oscillators one for the ${\bf d}$ and one ${\bf R}$
variables with frequencies and mean fluctuations in the ground state given by:

$\hbar\omega_{\bf d}=4U+V,\left\langle{\bf d}_{x}^{2}+{\bf
d}_{y}^{2}\right\rangle=2l_{B}^{2},$ (S10)

$\hbar\omega_{\bf R}=V,\left\langle{\bf R}_{x}^{2}+{\bf
R}_{y}^{2}\right\rangle=\frac{l_{B}^{2}}{2},$ (S11)

We see therefore that there is no competition between interactions and
disorder in this case, and the relative and average position can be
simultaneously localized on a similar length scale comparable to the magnetic
length $l_{B}$ irrespective of the values of $U$ and $V$. Now in case (b) we
find two degenerate oscillators, one generated by the pair of operators ${\bf
d}_{x},{\bf R}_{y}$, and another generated by the pair ${\bf d}_{y},{\bf
R}_{x}$. The frequency of these oscillators and fluctuations in the ground
state are:

$\hbar\omega=\sqrt{V\left(4U+V\right)},$ (S12)

$\left\langle{\bf R}_{x}^{2}+{\bf
R}_{y}^{2}\right\rangle=\frac{l_{B}^{2}}{2}\sqrt{\frac{4U+V}{V}},\left\langle{\bf
d}_{x}^{2}+{\bf d}_{y}^{2}\right\rangle=2l_{B}^{2}\sqrt{\frac{V}{4U+V}}.$
(S13)

The above clearly illustrates the competing tendencies of interactions and
disorder in case (b). When the interaction dominates the disorder ($U\gg V$)
the system optimizes the localization of the relative distance ${\bf d}$ at
the expense of delocalizing the average position ${\bf R}$.