# Intertwined fractional quantum anomalous Hall states and charge density
waves

Xue-Yang Song  Department of Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139  Chao-Ming Jian  Department of Physics, Cornell
University, Ithaca, New York 14853  Liang Fu  Department of Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139  Cenke Xu
Department of Physics, University of California, Santa Barbara, CA 93106

###### Abstract

Motivated by the recent experimental breakthrough on the observation of the
fractional quantum anomalous Hall (FQAH) effects in semiconductor and graphene
moiré materials, we explore the rich physics associated with the coexistence
of FQAH effect and the charge density wave (CDW) order that spontaneously
breaks the translation symmetry. We refer to a state with both properties as
“FQAH-crystal”. We show that the interplay between FQAH effect and CDW can
lead to a rich phase diagram including multiple topological phases and
topological quantum phase transitions at the same moiré filling. In
particular, we demonstrate the possibility of direct quantum phase transitions
from a FQAH-crystal with Hall conductivity $\sigma_{H}=-2/3$ to a trivial CDW
insulator with $\sigma_{H}=0$, and more interestingly, to a QAH-crystal with
$\sigma_{H}=-1$.

##  I Introduction

The recent advance in the fabrication and control of two-dimensional (2D) van
der Waals heterostructures has enabled the development of moiré superlattices
that feature tunable mini-bands. Topologically nontrivial mini-bands in moiré
materials provide an ideal avenue to search for topological states of matter.
As an example, evidence of fractional quantum Hall effect in fractionally
filled moiré band was observed in twisted bilayer graphene under a magnetic
field above $\sim 5$T or higherXie  _et al._ (2021); Spanton  _et al._ (2018).
More recently, thermodynamic and transport measurements revealed the
fractional quantum anomalous Hall (FQAH) effect at zero magnetic field in
twisted TMD homobilayersCai  _et al._ (2023); Zeng  _et al._ (2023); Park  _et
al._ (2023); Xu  _et al._ (2023) and rhombohedral pentalayer graphene/hBN
superlatticeLu  _et al._ (2023).

The discovery of FQAH effect points towards a fertile ground for studying
strong interaction effect in topological moiré bands of 2D materials. Twisted
TMD homobilayers feature spin-valley-locked moiré bands with opposite Chern
numbers in the two valleys Wu  _et al._ (2019); Yu  _et al._ (2020); Devakul
_et al._ (2021). At finite carrier density, Coulomb interaction drives
spontaneous valley polarization, and FQAH effect is anticipated at fractional
fillings of the valley-polarized Chern band Devakul  _et al._ (2021); Li  _et
al._ (2021); Crépel and Fu (2023). Interaction induced FQAH states in Chern
bands are also known as (zero-field) fractional Chern insulators in the
literature Sun  _et al._ (2011); Tang  _et al._ (2011); Neupert  _et al._
(2011); Sheng  _et al._ (2011); Regnault and Bernevig (2011). The highly
tunable nature of moiré systems, with abundant tuning knobs such as twist
angle, displacement field, electrostatic doping and gate screening, offers a
large parameter space to explore FQAH states and proximate phasesMorales-Durán
_et al._ (2023); Wang  _et al._ (2023); Reddy  _et al._ (2023); Qiu  _et al._
(2023); Crépel  _et al._ (2023); Goldman  _et al._ (2023); Dong  _et al._
(2023a); Liu  _et al._ (2023); Xu  _et al._ (2023); Reddy and Fu (2023); Song
_et al._ (2023); Yu  _et al._ (2023); Abouelkomsan  _et al._ (2023).

In this work, we explore the rich physics of a state with coexisting FQAH and
CDW (a state that we refer to as FQAH-crystal), and its proximate phases that
occur at the same moiré band filling under zero magnetic field. Our study is
motivated by the observed phase transition at hole filling $\nu=-2/3$ in
twisted bilayer MoTe2 under a displacement field from an FQAH state with
quantized Hall conductance $\sigma_{H}=-2/3$ (in the unit of $e^{2}/h$) to an
insulating state Park  _et al._ (2023); Xu  _et al._ (2023). We consider a
scenario in which the FQAH state spontaneously breaks the translational
symmetry of the moiré lattice. We call this state a FQAH-crystal, analogous to
the notion of “Hall crystal” introduced in Ref. Tešanović  _et al._ , 1989.
Our consideration of FQAH-crystal is partly motivated by recent numerical
finding of a softened magneto-roton gap in $\sigma_{H}=-2/3$ FQAH states Reddy
_et al._ (2023), which suggests incipient CDW order with tripled
$\sqrt{3}\times\sqrt{3}$ moiré unit cells (as illustrated in Fig.1) at
experimentally relevant twist angles.

We show by field theory analysis that a variety of strongly correlated phases
can be found in the vicinity of FQAH-crystal. These include a trivial CDW
insulator with $\sigma_{H}=0$ and a $\sigma_{H}=-1$ QAH state with CDW order,
which we call QAH-crystal. While the QAH-crystal phase has been proposed in
moiré systems under the name of topological charge density wave Polshyn  _et
al._ (2022); Pan  _et al._ (2022); Xu  _et al._ (2023), its connection to FQAH
physics Kol and Read (1993); Kourtis and Daghofer (2014); Dong  _et al._
(2023b) and phase transitions have received little attention before. All the
phases considered in this work possess the same type of CDW order, but are
distinguished by their different topological properties.

We further show that direct and (potentially) continuous phase transitions
between these topologically distinct phases are theoretically allowed.
Interestingly, these phase transitions can be described by (2+1)D quantum
electrodynamics (QED) with a Chern-Simons (CS) term of the $\mathrm{U}(1)$
gauge field coupled to either fermionic or bosonic charges. In our theory, the
transition from the FQAH-crystal with $\sigma_{H}=-2/3$ to the trivial CDW
insulator is described by a fermionic QED with two flavors Dirac fermions at
low energy coupled with a U(1) gauge field with a CS term at level-$1/2$. On
the other hand, its transition to a QAH-crystal with $\sigma_{H}=-1$ is
described by either bosonic or fermionic QEDs. These two descriptions are dual
to each other based on the boson-fermion duality web that was actively
discussed in recent years Chen  _et al._ (1993); Seiberg  _et al._ (2016);
Wang  _et al._ (2017); Song and Zhang (2022).

We note that direct transitions between a standard FQAH state (without any
spontaneous symmetry breaking) and exotic CDW states, including with
topological order, were studied in Ref. Song  _et al._ , 2023, whereas our
work starts from a FQAH-crystal (with $\sigma_{H}=-2/3$ and CDW order), and
obtains different proximate phases. In particular, we highlight the
possibility of a QAH-crystal (with $\sigma_{H}=-1$ and CDW order) as a
proximate phase and a direct phase transition between FQAH-crystal and QAH-
crystal at $\nu=-2/3$.

##  II Phases at $\nu=-2/3$

The most prominent state observed experimentally in the homobilayer TMD moiré
system is the $\sigma_{H}=\pm 2/3$ FQAH state at hole filling $n_{h}=2/3$,
i.e., at hole density of 2/3 per moiré unit cell. Since this state is shown to
be fully spin/valley polarized by magnetic circular dichroism measurements, in
the following we consider a spinless electron system at charge density
$\nu=-2/3$.

Throughout the discussion below we postulate the presence of a charge density
wave (CDW) which triples the unit cell (examples shown in Fig 1), such that
the holes are at integer filling with respect to the enlarged moiré unit cell.
We will show that under tripling of the unit cell, it is natural to construct
phases with Hall conductivity $\sigma_{H}=-2/3$, $-1$, $0$ in a unified
formalism. Furthermore, there can be direct and (potentially) continuous
quantum phase transitions between any of the two states mentioned above,
though a direct transition between the $\sigma_{H}=-2/3$ state and the trivial
insulator with $\sigma_{H}=0$ requires certain discrete space-time symmetries.

For the purpose of constructing these phases and describing their properties,
it is convenient to use the standard parton construction. One can formally
write the hole operator as $c=\Phi f$, where the bosonic parton $\Phi$ carries
the physical electric charge, and the charge-neutral parton $f$ is a fermion.
The electric charge can actually be assigned arbitrarily between $\Phi$ and
$f$, which should not change the final physics. The parton construction
formally enlarges the Hilbert space of the holes, which can be remedied by
coupling $\Phi$ and $f$ both to an internal dynamical $\mathrm{U}(1)$ gauge
field $a$, with charge $\pm 1$ respectively. The dynamical $\mathrm{U}(1)$
gauge field enforces a local constraint which equates the local density of $f$
to that of $\Phi$. The physical state of holes is obtained by enforcing the
relation of hole density to that of the partons, i.e.,
$\nu_{h}=\nu_{\Phi}=\nu_{f}=2/3$ with respect to the original moiré unit cell.
Importantly, in the presence of a CDW order that triples the unit cell, both
the holes and partons are at integer fillings with respect to the enlarged
unit cell.

Figure 1:  (Left) The moiré Brillouin zone formed when twisting the original
Brillouin zone (two large hexagons) of each TMD layer. When charge density
wave forms that triples the unit cell, schematically shown on the right on the
moiré superlattices, the Brillouin zone is further folded to the center
smallest hexagon on the left.

Figure 2:  (Left) The zoomed in moiré Brillouin zone with the symmetry
$C_{2y}\mathcal{T}$ shown that protects degeneracy of $2$ generic Dirac
cones(black dots) aligned along vertical axis. The arrows indicate the action
of $C_{2y}$ which flips the horizontal axis. When combined with time reversal,
$C_{2y}\mathcal{T}$ flips the vertical axis of the Brillouin zone and hence
protects the degeneracy of Dirac cone as plotted. When charge density wave
forms a stripe pattern schematically shown on the right, the symmetry is
preserved and a direct transition from $\sigma_{H}=-2/3\rightarrow 0$ could be
realized.

###  II.1 Phases tuned by fermionic parton $f$

In the following we will construct a series of states by making $\Phi$ a
bosonic fractional quantum Hall state with Hall conductivity $-1/2$. Each
state can also be equivalently constructed employing the composite fermion
picture through vortex attachment. As is well known in the context of Landau
level systems, composite fermions experience a modified residual magnetic
field, and prominent fractional quantum Hall states are formed at integer
filling of composite fermion Landau levels Jain (1989). As we show later, in
(moiré) lattice systems, the mean-field state of composite fermions allows
much richer possibilities, leading to a series of new statesSong  _et al._
(2023).

— $\sigma_{H}=-2/3$ State:

The $\sigma_{H}=-2/3$ state at filling $\nu=-2/3$ is the most prominent state
observed experimentally in the homobilayer TMD moiré system. This state can be
constructed naturally using the parton formalism, where $\Phi$ and $f$ each
forms its own “mean field state”. With the assumption of the existence of a
background charge density wave that triples the unit cell, the original Chern
band in the moiré Brillouin zone (BZ) would split into three bands in the
folded moiré BZ, and the fermionic parton $f$ would fill two out of the three
bands due to its $2/3$ filling. One natural way to construct the $2/3$ state,
is for the bosonic parton $\Phi$ to form a $\nu=-1/2$ Laughlin state, and at
the mean field level the fermionic parton $f$ fills two low energy bands in
the folded BZ with Chern numbers $+1,+1$. As we mentioned previously, the
$\sigma_{H}=-2/3$ state constructed here has FQAH effect as well as
spontaneous translation symmetry breaking, which we refer to as FQAH-crystal.
Later, we will demonstrate with a composite fermion construction that, the
existence of the $\sigma_{H}=-2/3$ state itself does not have to break the
translation symmetry. However, all of the nearby states within our formalism
must break the translation symmetry. This observation motivates us to focus on
the scenario where the translation symmetry is already broken in the
$\sigma_{H}=-2/3$ state.

Here we would like to demonstrate that the parton construction given above is
a natural state for holes at filling $2/3$ of the moiré unit cell. We note
that the flux $\phi_{\Phi}$ per moiré unit cell felt by the parton $\Phi$ is
not necessarily equal to the physical flux seen by holes $\phi_{h}$, due to
the internal gauge field $a$ coupled to both $\Phi$ and $f$. In the continuum,
the total fluxes seen by the hole and the partons should in general obey the
relation $\phi_{h}=\phi_{\Phi}+\phi_{f}$. In twisted semiconductor bilayersWu
_et al._ (2019); Yu  _et al._ (2020); Morales-Durán  _et al._ (2023) and other
continuum systemsPaul  _et al._ (2023) where holes fill a valley polarized
Chern band, despite being at zero magnetic field, the holes experience an
effective flux $\phi_{h}=-1$ per moiré unit cell produced by the periodic
skyrmion spin (or layer pseudospin) texture in real space111we are not
discussing a tight-binding model on the moiré scale, hence the magnetic flux
cannot be removed through large gauge transformation, and we could further set
$\phi_{\Phi}=-4/3$ to allow $\Phi$ to form a Laughlin $\nu=-1/2$ state. This
leaves $\phi_{f}=1/3$ and $\nu_{f}=2/3$. The fermionic partons hence are
naturally allowed to fill two Landau levels, equivalent to filling two bands
with Chern number +1. In fact, with the CDW order that triples the unit cell
that we postulate, the $f$ feel $\tilde{\phi}_{f}=1$ with a filling
$\tilde{\nu}_{f}=2$ per enlarged unit cell, hence $f$ could naturally form an
insulator with total Chern number $C=2$.

In terms of the Chern-Simons theory, this state corresponds to the following
Lagrangian

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle-\frac{2}{4\pi}b\wedge
db+\frac{1}{2\pi}b\wedge da-\frac{1}{4\pi}a\wedge(a_{1}+a_{2})$ (1)
$\displaystyle+$ $\displaystyle\sum_{i=1,2}\frac{1}{4\pi}a_{i}\wedge da_{i}.$
(3)

The gauge field $b$ is the “dual” of the current of the bosonic parton $\Phi$;
$a_{1}$ and $a_{2}$ are the dual of the fermionic parton $f$ that fills the
two Chern bands with Chern number $(+1,+1)$; $a$ is the gauge field that
couples to both $\Phi$ and $f$. The CS Lagrangian Eq. 3 can also be written in
a more compact form using the $K-$matrix Wen and Zee (1992):

$\displaystyle{\cal L}=\frac{1}{4\pi}K_{2/3,IJ}a^{I}\wedge da^{J},$ (4)

where

$\displaystyle K_{2/3}=\begin{pmatrix}-2&0&0&1\\\ 0&1&0&-1\\\ 0&0&1&-1\\\
1&-1&-1&0\end{pmatrix},$ (5)

and $a^{I}=(b,a_{1},a_{2},a)$. We will hereafter abbreviate $a\wedge db$ as
$adb$ without loss of clarity. The topological ground state degeneracy is
given by the determinant of $K_{2/3}$, which in this case is $3$. To derive
the Hall conductivity of this state, we need to couple $a^{I}$ to the external
electromagnetic field $A$, through a “charge vector” Wen and Zee (1992). In
the current construction, the charge vector is $v=(1,0,0,0)$, meaning that
only the bosonic parton $\Phi$ carries electric charge +1. By integrating out
all the dynamical gauge field $a^{I}$, one can show that the total Hall
conductivity of the state is $\sigma_{H}=-2/3$, i.e.
$\sigma_{H}=vK^{-1}v^{T}=-2/3$.

An alternative picture is that, the $2/3$ state can be viewed as holes at
filling $1$, forming a $\nu=-1$ integer quantum hall state, together with
electrons at filling $1/3$, forming a $\nu=1/3$ Laughlin state. The $K-$matrix
for this construction is

$\displaystyle K^{\prime}_{2/3}=\begin{pmatrix}-1&0\\\ 0&3\end{pmatrix}$ (6)

with the charge vector $v=(1,-1)$. The first diagonal element of
$K^{\prime}_{2/3}$ describes the $\nu=-1$ quantum hall state and the second
diagonal element describes $\nu=1/3$ Laughlin state. The $K-$matrix in Eq. 5
is related to the $K^{\prime}-$matrix in Eq. 6 (up to $2$ extra fields that
describe a trivial, neutral sector) by a similarity transformation in
$SL(4,Z)$:

$\displaystyle W^{T}K_{2/3}W=\begin{pmatrix}-1&0&0&0\\\ 0&3&0&0\\\ 0&0&0&-1\\\
0&0&-1&0\end{pmatrix},$ $\displaystyle W=\begin{pmatrix}-1&1&1&0\\\
-1&2&0&0\\\ 0&-1&0&0\\\ -2&2&1&-1\end{pmatrix}.$ (7)

The third picture of constructing the $2/3$ state is through the composite
fermion (CF) and flux attachment. As we mentioned before, when the holes fill
a valley polarized Chern band, the physics is topologically equivalent to
integer quantum Hall state where a hole sees a $\phi_{h}=-1$ magnetic flux
quantum through each moiré unit cell. Then a composite fermion is constructed
by binding the hole with $2$-flux quanta of a gauge field $a$, i.e. the
composite fermions will see total gauge flux $\phi_{cf}=\phi_{h}+2\rho_{cf}$.
When the hole density is 2/3 per moiré unit cell, the density of $\phi_{cf}$
would be 1/3 flux quantum per moiré unit cell. Hence the composite fermions
would naturally fill two Landau Levels of $\phi_{cf}$, and form an integer
quantum Hall state with composite fermion Hall conductivity $\sigma_{cf}=2$.
When inserting an extra flux density $\delta\phi_{cf}$ into the system,
composite fermion density $\delta\rho_{cf}=\sigma_{cf}\delta\phi_{cf}$ will be
accumulated; and the total Hall conductivity is the ratio between the extra
density of composite fermions and the density of extra magnetic flux:
$\sigma_{H}=\delta\rho_{cf}/\delta\phi_{h}=\sigma_{cf}/(1-2\sigma_{cf})=-2/3$.
This composite fermion construction for the $2/3$ state does not break
translation. The composite fermion picture for understanding the observed FQAH
states in twisted bilayer MoTe2 is strongly supported by recent numerical
studies Goldman  _et al._ (2023); Dong  _et al._ (2023a); Reddy and Fu (2023).

To formally implement this flux attachmentBarkeshli and McGreevy (2012), we
introduce a noncompact gauge field $b$ whose charge is the flux of $a$. We
will demonstrate that a $\mathrm{U}(1)_{-2}$ CS term for $b$ attaches $2$
units of fluxes of $a$ to the composite fermion, which also carries charge
under gauge field combination $a+A$. The Lagrangian of all the field mentioned
above reads:

$\displaystyle\mathcal{L}=-\frac{2}{4\pi}bdb+\frac{1}{2\pi}bda+\mathcal{L}_{CF}[\psi,a+A],$
(8)

where the mutual CS term between $b,a$ implies that the flux of $a$ is charged
under $b$, and the last term of Eq. 8 is the CF Lagrangian capturing the
physics that the CF ($\psi$) is coupled to $a+A$.

The equations of motion with respect to $b$ and $a$ lead to the following
relations:

$\displaystyle\frac{\delta\mathcal{L}}{\delta b_{0}}=0\ \rightarrow\
\frac{da}{2\pi}=\frac{2db}{2\pi},$ (9) (10)
$\displaystyle\frac{\delta\mathcal{L}}{\delta a_{0}}=0\ \rightarrow\
\rho_{cf}=\frac{db}{2\pi}.$ (11)

Combining the equations we obtain the relation $2\rho_{cf}=\frac{da}{2\pi}$,
which corresponds to the picture of flux attachment: each CF is bound with two
flux quanta of gauge field $a$.

When the CF fermion $\psi$ fills Chern bands with total Chern number $C_{cf}$
(an integer), we need to introduce $|C_{cf}|$ copies of gauge fields $a_{i}$,
which are dual to the current of the CFs:

$\displaystyle\mathcal{L}_{CF}[\psi,a+A]=\textrm{sgn}(C_{cf})\sum_{i=1}^{C_{cf}}\left(\frac{1}{4\pi}a_{i}da_{i}+\frac{1}{2\pi}a_{i}d(a+A)\right),$
(12)

where each self-CS term of $a_{i}$ describes the CF filling a complete Landau
Level (or equivalently Chern band with Chern number $1$). The flux current of
$a_{i}$, which is the dual of the CF current, couple to $a+A$.

For $C_{cf}=2$, after combining Eq. 8 and Eq. 12 we eventually arrive at
exactly the same $K-$matrix as that in Eq. 5, albeit the charge vector now is
$(0,1,1,0)$, which corresponds to shifting the electric charge from the
bosonic parton $\Phi$ to the fermionic parton $f$, and it still leads to
$\sigma_{H}=-2/3$. The charge vector could be transformed to $(1,0,0,0)$, by
relabeling $a\rightarrow a-A$. Then the two formalisms based on parton and CF
yield exactly identical $K-$matrices and Hall conductivity.

— $\sigma_{H}=0$ and $\sigma_{H}=-1$ states:

To construct a trivial insulator phase with $\sigma_{H}=0$, we can still fix
the bosonic parton at a $\nu=-1/2$ Laughlin state, and let the fermionic
parton $f$ fill two bands with Chern numbers $+1,-1$ respectively. The $K$
matrix of this state is similar to Eq. 5, with the diagonal component $K_{33}$
changed to $-1$. This change will lead to the Hall conductivity
$\sigma_{H}=0$, without any topological degeneracy.

An integer QAH state with $\sigma_{H}=-1$ and coexisting CDW order (referred
to as the QAH-crystal state) can be constructed by removing the row and column
of the $K$ matrix that involves the second band of the fermionic parton,
meaning that the fermionic parton $f$ now fills bands with total Chern numbers
$+1$.

In the composite fermion picture, when the CF forms a $\nu=+1$ quantum hall
state, inserting a $+1$ $\phi_{h}$-flux quantum is accompanied by $-2$ units
of fluxes of $a$, and accumulating $-1$ charge of CF since there is a total
$-1$ unit of extra flux $\phi_{cf}$, i.e. the CFs form a $\nu=-1$ state with
respect to $\phi_{h}$. This state eventually corresponds to the
$\sigma_{H}=-1$ state. The $K-$matrix can be similarly deduced and is
equivalent to that deduced from the parton construction.

The trivial insulator with $\sigma_{H}=0$ corresponds to the CF forming a
trivial insulator, which leads to trivial electromagnetic response. The
$K-$matrices of the three states constructed in this subsection are summarized
in table 1.

– Translation breaking enforced by filling: Importantly, when the translation
symmetry of the moiré superlattice is preserved, it is not possible (at least
within CF mean-field theory) to have a direct transition from a state that
fills a CF band with Chern number $C_{cf}=2$ and correspondingly Hall
conductivity $\sigma_{H}=-2/3$, to states with $C_{cf}=1,0$ and
$\sigma_{H}=-1,0$. Since the effective field seen by the CFs is
$\phi_{cf}=1/3$ flux quanta through each moiré plaquette, the CFs obey the
magnetic translation symmetry:
$T_{1}T_{2}=T_{1}^{-1}T_{2}^{-1}e^{\mathrm{i}2\pi/3}$ for the two elementary
translations $T_{1,2}$ that enclose a moiré unit cell. The mean-field spectrum
of the CFs will be $3$ fold degenerate as the magnetic translation admits
representations with minimal dimensions of $3$. Burkov and Balents (2005)

Therefore the translation symmetry guarantees that the change of the Chern
number $\Delta C_{cf}$ across a transition must have $\Delta C_{cf}$ equal to
multiples of $3$, as it is given by the number of gapless Dirac cones in the
spectrum. Hence if the transitions $\sigma_{H}=-2/3\rightarrow-1$ and
$\sigma_{H}=-2/3\rightarrow 0$ are described by changing the band Chern number
of the composite fermions, they can only occur when the translation symmetry
of the moire lattice is broken, i.e. they can only occur when there is a
background charge density wave order. In particular, the simplest scenario of
CDW is to triple the unit cell, rendering the magnetic translation trivial and
permitting a direct transition with $\Delta C_{cf}=-1,-2$ etc. In the next
section, we shall demonstrate explicitly direct transition from
$\sigma_{H}=-2/3$ FQAH-crystal to either $\sigma_{H}=0$ CDW or $\sigma_{H}=-1$
QAH-crytsal.

Phase | CF | Parton $f$ ($\Phi$ in Laughlin $-\frac{1}{2}$ state) | Electron+holes($\nu=-1$ IQH) | K matrix
---|---|---|---|---
$\sigma_{H}=-\frac{2}{3}$ | $C_{cf}=2$ | $C_{f}=2$ | Electron in $\frac{1}{3}$ Laughlin | $\begin{pmatrix}-2&0&0&1\\\ 0&1&0&-1\\\ 0&0&1&-1\\\ 1&-1&-1&0\end{pmatrix}\simeq\begin{pmatrix}-1&0&0&0\\\ 0&3&0&0\\\ 0&0&0&-1\\\ 0&0&-1&0\end{pmatrix}$
$\sigma_{H}=0$ | $C_{cf}=0$ | $C_{f}=0$ | Electron in $\nu=1$ IQH | $\begin{pmatrix}-2&0&0&1\\\ 0&1&0&-1\\\ 0&0&-1&-1\\\ 1&-1&-1&0\end{pmatrix}\simeq\begin{pmatrix}-1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&-1\\\ 0&0&-1&0\end{pmatrix}$
$\sigma_{H}=-1$ | $C_{cf}=1$ | $C_{f}=1$ | Electron in trivial insulator | $\begin{pmatrix}-2&0&1\\\ 0&1&-1\\\ 1&-1&0\end{pmatrix}\simeq\begin{pmatrix}-1&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix}$
Table 1: Summary of the $3$ formalisms that describes the three states with
$\sigma_{H}=-2/3,-1,0$ respectively. When the composite fermions (CF) fill
Chern bands with total Chern number $C_{cf}$, the physical Hall conductivity
is $\sigma_{H}=\frac{C_{cf}}{1-2C_{cf}}$.

###  II.2 Phases tuned by bosonic parton $\Phi$

In all the three states constructed in the last subsection, the bosonic parton
$\Phi$ always form a $\nu=-1/2$ bosonic Laughlin state. Starting with the
FQAH-crystal state with $\sigma_{H}=-2/3$, two more states can be constructed
by changing the physics of the bosonic parton $\Phi$. These states/phases are
summarized in the global phase diagram Fig. 3.

One such state is an insulator without any Hall conductivity, but it has a
neutral topological order and neutral chiral edge states, leading to quantized
thermal Hall effect. This quantum thermal Hall insulator can be obtained from
the FQAH-crystal, by driving the bosonic parton $\Phi$ into a trivial
insulator. When $\Phi$ is in a trivial insulator, there is no nontrivial
response to the external electromagnetic field as $\Phi$ is the parton that
carries the electric charge. However, this state must still have a nontrivial
topological order, as the $K$ matrix of this state corresponds to Eq. 5 after
removing the components that involve gauge field $b$. The determinant of the
remaining $3\times 3$ $K$ matrix is $2$, and it is equivalent to a simple
semion topological order. The semion topological order can also be revealed by
integrating out $a_{1}$ and $a_{2}$, which yields a level-2 CS term for the
gauge field $a$. This semion topological order with zero Hall conductivity is
one of the states discussed in Ref. Song  _et al._ , 2023.

The other state is an QAH-crystal state with Hall conductivity
$\sigma_{H}=+2$. This state can be constructed from FQAH-crystal by driving
the bosonic parton $\Phi$ into a “superfluid” state. In the condensate of
$\Phi$, the hole operator $c$ is identified with $f$, and since $f$ fills two
bands with Chern number $+1$, this leads to a QAH-crystal state with Hall
conductivity $\sigma_{H}=2$. An QAH state without topological order is
possible as we assumed a background CDW that triples the unit cell.

Figure 3: A schematic global phase diagram in terms of the parton
construction, tuned by both physics of bosonic parton $\Phi$ (vertical
direction) and fermionic parton $f$ (horizontal directions) starting from the
FQAH-crystal. We discuss interesting critical theories among the phases shown
in Sec. III.

##  III Quantum Phase Transitions

So far, we have constructed five different states centered around the “2/3”
state, in a phase diagram tuned by mean field physics of the bosonic parton
$\Phi$ and the fermionic parton $f$. These states can also be equally well
constructed through other formalisms, including the composite fermions and
flux attachment. In this section, we discuss the quantum phase transitions
between these states. Here we stress that the FQAH state we start with is in
fact a FQAH-crystal, while the starting point of Ref. Song  _et al._ , 2023
was a FQAH state without spontaneous translation symmetry breaking. Hence
different proximate phases and phase transitions are obtained in these two
papers. For example, in our current case the most natural $\sigma_{H}=0$
insulator state next to the $\sigma_{H}=-2/3$ state is a trivial insulator
with CDW, while in Ref. Song  _et al._ , 2023 the $\sigma_{H}=0$ state has a
topological order that would lead to nontrivial thermal Hall signal.

###  III.1 $\sigma_{H}=-2/3\rightarrow 0$ transition

To drive a transition between the FQAH-crystal state with $\sigma_{H}=-2/3$
and a trivial CDW insulator with $\sigma_{H}=0$, we can keep the bosonic
parton $\Phi$ in the $\nu=-1/2$ state unchanged, and only change the total
Chern number of the fermionic parton bands from $C=2$ to $C=0$, which can be
realized by changing one of the occupied bands from $C=+1$ to $C=-1$. If there
is a direct transition between these two states, it must involve two Dirac
fermions at low energy. The complete critical theory reads:

$\displaystyle\mathcal{L}_{1-2}$ $\displaystyle=$
$\displaystyle\sum_{i=1,2}\bar{\psi}_{i}\gamma\cdot(\mathrm{i}\partial-a)\psi_{i}+m\bar{\psi}\psi+\frac{1}{2\pi}ad(\alpha-b)$
(13) $\displaystyle+$ $\displaystyle\frac{1}{4\pi}\alpha
d\alpha-\frac{2}{4\pi}bdb+\frac{1}{2\pi}Adb,$ (14)

where $\alpha,b$ are the dual fields associated with the filled $C=1$ band of
$f$ (hence the $U(1)_{1}$ for $\alpha$) and the boson $\Phi$ currents,
respectively. Two Dirac fermions must exist at low energy at the transition
for one $f$-band to change from $C=1$ ($m>0$) to $C=-1$ ($m<0$).

The theory Eq. 14 can be simplified at the cost of losing the proper
quantization of the CS terms. Integrating out $\alpha$ generates
$-1/(4\pi)ada$.and integrating out $b$ generates $1/(8\pi)(A-a)d(A-a)$. The
simplified theory then reads:

$\displaystyle\mathcal{L}_{2;-\frac{1}{2}}$ $\displaystyle=$
$\displaystyle\sum_{i=1,2}\bar{\psi}_{i}\gamma\cdot(\mathrm{i}\partial-a)\psi_{i}+m\bar{\psi}\psi-\frac{1}{8\pi}ada$
(15) $\displaystyle-$ $\displaystyle\frac{1}{4\pi}adA+\frac{1}{8\pi}AdA.$

We use the notation $\mathcal{L}_{N_{f};k}$ to label the QED Lagrangian with
$N_{f}$ flavors of Dirac fermions and a Chern-Simons term at level $k$.

The degeneracy of the two Dirac fermions can be guaranteed by extra discrete
space-time symmetries. In the absence of displacement field, the entire
homobilayer twisted TMD moiré system has a $C_{2y}$ symmetry, a two-fold
rotation along the vertical axis in Fig. 1, as well as a time-reversal
symmetry $\mathcal{T}$. Both symmetries exchange the two valleys. Hence, each
valley of the system holds a composite symmetry of $C_{2y}\mathcal{T}$. This
composite symmetry sends $(k_{x},k_{y})\rightarrow(k_{x},-k_{y})$. The
degeneracy of the two Dirac fermions that is needed for a direct
“$\sigma_{H}=-2/3\rightarrow\sigma_{H}=0$” transition in our set-up depends on
the type of CDW order. For example, if the CDW is a stripe order along the $y$
direction with modulation along the $x$ direction as shown in Fig. 2, the two
Dirac points could still be located at $(k_{x},-k_{y})$ and $(k_{x},-k_{y})$
points of the BZ, and their degeneracy is still protected by the
$C_{2y}\mathcal{T}$ symmetry. In contrast, if there is a
$\sqrt{3}\times\sqrt{3}$ CDW order with $C_{3}$ symmetry shown in fig 1, there
is no scenario where $C_{3},C_{2y}\mathcal{T}$ together protect two and only
two degenerate Dirac cones. Here we note that an out-of-plane displacement
field in principle breaks the $C_{2y}$ symmetry as it exchanges the two
layers, hence under a displacement field the transition may split into two.

Although the microscopic symmetry $C_{2y}$ of the system may be broken by a
displacement field, extra effective symmetries may still exist in the physics
of the moire minibands. For example, there is an extra discrete symmetry of
the Hamiltonian that describes one valley of the system Wu  _et al._ (2019),
which is a composite of $R_{x}:y\rightarrow-y$, and a “time-reversal” that
acts on this one-valley Hamiltonian. This symmetry still exists with the
presence of the displacement field. If this symmetry is a good approximate
symmetry of the moire miniband, it can also protect the degeneracy of two
Dirac points and a direct transition of changing Chern number by $2$, as was
observed in model studies in Ref. Wu  _et al._ , 2019; Reddy  _et al._ , 2023.

An alternative description for the same transition from the $\sigma_{H}=-2/3$
state to the $\sigma_{H}=0$ state can be constructed using the “electron-hole
picture”: The $\sigma_{H}=-2/3$ state can be viewed as a composition of holes
at the $\nu=-1$ IQH state, and electrons in the Laughlin $\nu=1/3$ state. To
drive a transition to the $\sigma_{H}=0$ state we need a transition of the
electrons from the $\nu=1/3$ state to the $\nu=1$ IQH state. The critical
theory reads:

$\displaystyle\mathcal{L}_{1-2;eh}$ $\displaystyle=$
$\displaystyle\sum_{i=1,2}\bar{\psi}\gamma\cdot(\mathrm{i}\partial-a)\psi+m\bar{\psi}\psi-\frac{1}{2\pi}ad\beta$
(16) $\displaystyle+$ $\displaystyle\frac{2}{4\pi}\beta
d\beta+\frac{1}{2\pi}Ad\beta+\frac{1}{4\pi}AdA,$ (17)

The last term $+\frac{1}{4\pi}AdA$ accounts for the hole $\nu=-1$ state that
stays unchanged throughout the transition. The rest of the Lagrangian
describes the transition of the electrons from the $\nu=1/3$ state to a
$\nu=1$ IQH state. When the $2$ Dirac cones are gapped out by a mass,
integrating out the $\psi$’s give

$\displaystyle\mathcal{L}_{m}$ $\displaystyle=$ $\displaystyle\frac{-{\rm
sgn}(m)}{4\pi}ada-\frac{1}{2\pi}ad\beta+\frac{2}{4\pi}\beta d\beta$ (18)
$\displaystyle+$ $\displaystyle\frac{1}{2\pi}Ad\beta+\frac{1}{4\pi}AdA.$ (19)

It is straightforward to verify that for $m>0$ ($m<0$), the theory describes
states with Hall responses $\sigma_{H}=-2/3$ ($\sigma_{H}=0$) respectively.
Starting with Eq. 17, the theory simplifies again to
$\mathcal{L}_{2;-\frac{1}{2}}$ after integrating out $\beta$.

###  III.2 $\sigma_{H}=-2/3\rightarrow-1$ transition

From the parton picture, this transition involves changing a fermion $f$’s
state from integer quantum Hall state with $\nu=+1$ to $\nu=0$. This
transition can be described by QED with one Dirac fermion in the infrared, the
critical theory reads:

$\displaystyle\mathcal{L}_{1-3}$ $\displaystyle=$
$\displaystyle\bar{\psi}\gamma\cdot(\mathrm{i}\partial-a)\psi+m\bar{\psi}\psi-\frac{1}{8\pi}ada+\frac{1}{2\pi}ad(\alpha-b)$
(20) $\displaystyle+$ $\displaystyle\frac{1}{4\pi}\alpha
d\alpha-\frac{2}{4\pi}bdb+\frac{1}{2\pi}Adb,$ (21)

We have added $-\frac{1}{8\pi}ada$ to properly regularize a single Dirac cone,
which arises from another massive Dirac fermion which must exist in the same
band as $\psi$. Here, the single Dirac fermion is written in the convention
that, by changing the sign of $m$, $\psi$ would generate a level $\pm 1/2$ CS
term for $a$. Integrating out $b,\alpha$, we obtain a simplified theory (again
at the cost of not properly quantizing the CS term)

$\displaystyle\mathcal{L}_{1;-1}$ $\displaystyle=$
$\displaystyle\bar{\psi}\gamma\cdot(\mathrm{i}\partial-a)\psi+m\bar{\psi}\psi-\frac{1}{4\pi}ada$
(22) $\displaystyle+$ $\displaystyle\frac{1}{8\pi}AdA-\frac{1}{4\pi}adA.$

Similarly, when the Dirac cone is gapped out by a positive mass term
$m\bar{\psi}\psi$, integrating out the $\psi$’s give

$\displaystyle\mathcal{L}_{m>0}=-\frac{3}{8\pi}ada+\frac{1}{8\pi}AdA-\frac{1}{4\pi}adA,$
(23)

which generates Hall conductivity $\sigma_{H}=-2/3$. While $m<0$ one has

$\displaystyle\mathcal{L}_{m<0}=-\frac{1}{8\pi}ada+\frac{1}{8\pi}AdA-\frac{1}{4\pi}adA,$
(24)

and integrates out $a$ leaves $\frac{1}{4\pi}AdA$, describing a state with
$\sigma_{H}=-1$.

From standard boson-fermion duality Chen  _et al._ (1993); Seiberg  _et al._
(2016) , the critical theory $\mathcal{L}_{1;-1}$ is dual to

$\displaystyle\mathcal{L}_{1;-1}$ $\displaystyle\leftrightarrow$
$\displaystyle|(\partial-\mathrm{i}\beta)\phi|^{2}+\frac{1}{4\pi}\beta
d\beta+\frac{1}{2\pi}ad\beta-\frac{1}{8\pi}ada$ (25) $\displaystyle+$
$\displaystyle\frac{1}{8\pi}AdA-\frac{1}{4\pi}adA.$

Here $\beta$ is another gauge field that couples to the dual bosonic field
$\phi$. One could verify that the massive and condensed phase of $\phi$
corresponds to $\mathcal{L}_{m>(<)0}$ respectively, yielding
$\sigma_{H}=-2/3,-1$. One can also directly perform the duality transformation
from Eq. 21.

Integrating $a$ in the dual bosonic theory leaves

$\displaystyle\mathcal{L}_{1;-1}\leftrightarrow|(\partial-\mathrm{i}\beta)\phi|^{2}+\frac{3}{4\pi}\beta
d\beta-\frac{1}{2\pi}\beta dA+\frac{1}{4\pi}AdA.$ (26)

This Chern-Simons-matter theory with $\phi$ coupled to a $\mathrm{U}(1)$ gauge
field with a CS term with level-3 is the standard theory that describes a
transition between a trivial insulator and a fractional quantum Hall state
with three-fold topological degeneracy Wen and Wu (1993). Combined with the
last term $\frac{1}{4\pi}AdA$ which corresponds to an extra $\nu=-1$ IQH
layer, the theory describes a transition between states with $\sigma_{H}=-2/3$
and $\sigma_{H}=-1$. This FQAH-crystal to QAH-crystal transition also admits
another description in terms of bosonic partons, which is a modified version
of the FQAH to QAH+CDW transition discussed in Ref.Song  _et al._ , 2023
driven by the condensation of $3$ ‘vortex’ fields coupled to a
$\mathrm{U}(1)_{3}$ Chern-Simons term. It is worth noting that this modified
vortex condensation theory takes the same form as Eq. 26. 222The multiple
vortex fields are required as the $2/3$ state there in Ref.Song  _et al._ ,
2023 preserved translation, hence vortex degeneracy enforced by fractional
filling. When CDW orders are assumed here and integer filling is reached for
enlarged unit cell, the transition is modified to a single vortex $\phi$
condensation transition discussed here.

###  III.3 $\sigma_{H}=-2/3\rightarrow+2$ transition

Another potentially direct transition is between state 1 and 5, i.e. a
transition from a FQAH-crystal state with $\sigma_{H}=-2/3$ to an QAH-crystal
state with $\sigma_{H}=+2$. In the parton construction, this requires changing
the state of $\Phi$ from a $\nu=-1/2$ Laughlin state to a “superfluid” state.
This transition of $\Phi$ was discussed in Ref. Barkeshli and McGreevy, 2014,
2012, and it is described by a QED with two flavors of Dirac fermions and a CS
term at level $-1$. In our notation, the critical theory of $\Phi$ is
described by a Lagrangian ${\cal L}_{2;-1}$, and the Dirac fermions are
charges of the gauge field $b$, i.e. the dual of the current of $\Phi$. To
describe the transition between states 1 and 5, we need to couple $b$ to
several other gauge fields $a$, $a_{i}$ as in Eq. (3). After integrating out
the $a$ and $a_{i}$, we arrive at the critical theory between state 1 and 5:

$\displaystyle{\cal L}_{1-5}=\tilde{\cal L}_{2;-1/2}=$ (27) (28)
$\displaystyle\sum_{i=1,2}\bar{\chi}_{i}\gamma\cdot(\mathrm{i}\partial-b)\chi_{i}+m\bar{\chi}\chi-\frac{1}{8\pi}bdb+\frac{1}{2\pi}Adb.$
(29)

We note that here the Dirac fermion $\chi_{i}$ is different from the fermions
in the previous sections, as it is charged under $b$(hence the critical theory
$\tilde{\cal L}_{2,-1/2}$ is different from $\mathcal{L}_{2;-1/2}$ in Eq. (15)
with different charge assignment). The degeneracy of two Dirac cones is again
protected by $C_{2y}\mathcal{T}$ in a stripe order.

Integrating out the fermion $\chi_{i}$, we obtain the following action

$\displaystyle{\cal L}_{m}=-\frac{{\rm
sgn}(m)}{4\pi}bdb-\frac{1}{8\pi}bdb+\frac{1}{2\pi}Adb.$ (30)

It is straightforward to verify that, for $m>0$ ($m<0$), the final Hall
conductivity is $\sigma_{H}=-2/3$ ($\sigma_{H}=+2$).

###  III.4 Transitions involving quantum thermal Hall insulator

Ref. Song  _et al._ , 2023 proposed a proximate insulating phase of the
$\sigma_{H}=-2/3$ FQAH state to be one with vanishing Hall response and a
neutral topological order (TO) described by the $\mathrm{U}(1)_{2}$ CS terms.
In the current framework, this state could be obtained by putting the bosonic
parton $\Phi$ in a Mott insulator state. Formally, this amounts to eliminating
the dual $b$ of the boson current, i.e. setting $b=0$, in the construction of
the $2/3$ state in Eq. (3). Physically, it means that the bosonic sectors are
trivially gapped in the low energy. Integrating out $a$ then sets
$a_{1}=a_{2}$, and one gets a $\mathrm{U}(1)_{2}$ CS coupling of an internal
gauge field, describing the neutral TO, signified by a quantized thermal hall
response nevertheless.

The transition from quantum thermal Hall insulator to $\sigma_{H}=-2/3$ FQAH-
crystal or $\sigma_{H}=+2$ QAH-crystal, can hence be obtained by tuning the
$\Phi$ out of the Mott phase to a Laughlin $-1/2$ state (for state 1), or a
superfluid (for state 5), respectively.

The Mott to Laughlin $-1/2$ transition is realized by condensing the
“vortices” of the bosonic partonsin the Laughlin $-1/2$ stateSong  _et al._
(2023). The critical theory describing the transition from state 1 to state 4
hence reads

$\displaystyle\mathcal{L}_{1-4}=|(\partial-\mathrm{i}b)\Phi_{v}|^{2}-\frac{2}{4\pi}bdb+\frac{1}{2\pi}Adb$
$\displaystyle\frac{1}{2\pi}ad(b-a_{1}-a_{2})+\sum_{i=1,2}\frac{1}{4\pi}a_{i}da_{i}.$
(31)

The first term describes the condensation of the vortices $\Phi_{v}$, which is
indicated by its coupling to $b$, whose flux equals the boson density. As
$\Phi_{v}$ condenses, the field $b$ acquires a gap and could be ignored.
Hence, one arrives at the quantum thermal Hall insulator with $\sigma_{H}=0$.
An insulator phase of $\Phi_{v}$ just leaves the Lagrangian for the $2/3$
state Eq. (3). A similar transition was discussed in ref Song  _et al._ ,
2023, albeit with $3$ vortex fields enforced by fractional filling $2/3$.

The transition from quantum thermal Hall insulator to QAH-crystal with
$\sigma_{H}=+2$ is given by the standard boson Mott-superfluid transition,

$\displaystyle\mathcal{L}_{4-5}=|(\partial-\mathrm{i}a-\mathrm{i}A)\Phi|^{2}-\frac{1}{2\pi}ad(a_{1}+a_{2})+\sum_{i=1,2}\frac{1}{4\pi}a_{i}da_{i}.$
(32)

When $\Phi$ is gapped, the remaining last $2$ terms describes the neutral
topological order. The condensation of $\Phi$ sets $a=-A$, and the CS terms of
$a_{i}$’s then give a Hall conductivity $\sigma_{H}=2$.

##  IV Summary

In this work we discussed the phase diagram centered around a FQAH-crystal
state with $\sigma_{H}=-2/3$ state at filling $-2/3$, motivated by recent
experiments.Various phases and phase transitions can be obtained by tuning the
physics of the bosonic and fermionic partons, including a direct transition
between the $\sigma_{H}=-2/3$ state and a trivial insulating state with
$\sigma_{H}=0$ observed in recent experiments. Interestingly, we also find a
direct transition between the $\sigma_{H}=-2/3$ FQAH state and a
$\sigma_{H}=-1$ QAH state.

Our formalism and conclusions can easily be generalized to other FQAH states.
For example, if the bosonic parton $\Phi$ forms a $\nu=-1/p$ Laughlin state
(with even integer $p$), and the fermionic parton (or composite fermion) $f$
fills Chern bands with total Chern number $C_{cf}$, we would end up with an
FQAH state with Hall conductivity

$\displaystyle\sigma_{H}=\frac{C_{cf}}{1-pC_{cf}}.$ (33)

In particular, when $p=-2,C_{cf}=1$, i.e. the composite fermions fill a Chern
band with $C_{cf}=1$, one constructs a Laughlin state with $\sigma_{H}=1/3$.
When the CFs go through a transition from $C_{cf}=1\rightarrow 0$, the
electronic state transitions from $\sigma_{H}=1/3\rightarrow 0$. The critical
theory is similar to that of $\sigma_{H}=-2/3\rightarrow-1$ with the
Lagrangian $\mathcal{L}_{1;-1}$ in Eq. (22), with an only difference of an
integer quantum hall layer described by $-1/(4\pi)AdA$. The critical theory
for the transition hence reads

$\displaystyle\mathcal{L}_{1/3\rightarrow
0}=\mathcal{L}_{1;-1}-\frac{1}{4\pi}AdA.$ (34)

More states and phase transitions can be constructed by changing the “mean
field states” of $\Phi$ and $f$. This general construction will be useful to
understand the growing number of FQAH states Lu  _et al._ (2023) observed in
this rapidly developing field.

Acknowledgement We thank insightful discussions with Xiaodong Xu. LF thanks
related collaborations with Aidan Reddy, Hart Goldman, Nisarga Paul, Ahmed
Abouelkomsan and Emil Bergholtz. XYS thanks collaborations and discussions
with T. Senthil and Y-H Zhang. XYS is supported by the Gordon and Betty Moore
Foundation EPiQS Initiative through Grant No. GBMF8684 at the Massachusetts
Institute of Technology. CMJ is supported by a faculty startup grant at
Cornell University. LF is supported by the Air Force Office of Scientific
Research (AFOSR) under Award No. FA9550-22-1-0432. CX is supported by the
Simons foundation through the Simons Investigator program.

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# Quantifying Influencer Effects on Affective Polarization
††thanks: The research was sponsored by the Army Research Office and was
accomplished under Grant Number W911NF-22-1-0035. The views and conclusions
contained in this document are those of the authors and should not be
interpreted as representing the official policies, either expressed or
implied, of the Army Research Office or the U.S. Government. The U.S.
Government is authorized to reproduce and distribute reprints for Government
purposes notwithstanding any copyright notation herein.

Rezaur Rashid  Department of Computer Science
UNC Charlotte
Charlotte, USA
<EMAIL_ADDRESS>Joshua Melton  Department of Computer Science
UNC Charlotte
Charlotte, USA
<EMAIL_ADDRESS>Ouldouz Ghorbani  Department of Computer Science
UNC Charlotte
Charlotte, USA
<EMAIL_ADDRESS>Siddharth Krishnan  Department of Computer Science
UNC Charlotte
Charlotte, USA
<EMAIL_ADDRESS>Shannon Reid  Department of Criminal Justice and
Criminology
UNC Charlotte
Charlotte, USA
<EMAIL_ADDRESS>Gabriel Terejanu  Department of Computer Science
UNC Charlotte
Charlotte, USA
<EMAIL_ADDRESS>
###### Abstract

In an era where digital platforms increasingly mediate public discourse,
grasping the complexities and nuances in affective polarization—especially as
influenced by key figures on social media—has never been more vital. This
study delves into the intricate web of interactions on Twitter, now rebranded
as ’X’, to unravel how influencer-led conversations catalyze shifts in public
sentiment, laying bare the complex dynamics that underpin online polarization.
Employing a novel methodological framework that includes counterfactual
analysis, we analyze scenarios with and without specific influencer-led
conversations. Our findings illuminate the significant role influencers play
in shaping public discourse, offering insights into the mechanisms of online
polarization and suggesting pathways for future research to mitigate
divisiveness and explore new methods for quantifying affective polarization.
This research contributes to the broader understanding of digital
communication’s impact on societal polarization, underscoring the importance
of detailed analysis in developing strategies to foster a more cohesive
digital public sphere.

###### Index Terms:

social networks, twitter, stance labeling, gun control, climate change

##  I Introduction

The transformative impact of social media on political communication cannot be
overstated. As a platform for instantaneous information exchange, social media
has redefined the way individuals engage with political content, debate
policy, and form community bonds [1, 2]. However, this digital revolution has
also given rise to a less auspicious phenomenon: affective polarization [3, 4,
5]. This term encapsulates the growing emotional divide between individuals
with opposing political stances, characterized not only by differences in
policy opinion but also by deep-seated animosity and distrust [6, 7].

Affective polarization on social media is particularly pernicious due to its
potential to erode the foundations of democratic discourse, replacing reasoned
debate with hostile confrontation. It transforms disagreement into contempt,
making compromise and consensus-building increasingly elusive [8, 9]. The
implications of this trend extend beyond the digital realm, influencing real-
world political engagement and the broader societal fabric [10, 11].

At the heart of this polarization are influential users—individuals and
entities with significant followings and the ability to shape public
discourse. These influencers can act as catalysts, amplifying existing divides
or bridging gaps in understanding through their engagement with contentious
topics [12, 13]. In the current digital age, the reach and impact of such
figures are unprecedented, necessitating a closer examination of their role in
the affective polarization process [14].

Our study aims to fill a critical gap in the literature by quantifying the
effects of conversations initiated by influential users on the polarization
observed in Twitter discourses. Leveraging computational methods, we dissect
the emotional tone and polarization of social media conversations, tracing the
shifts in sentiment that occur when influencer-led conversations are
introduced or removed from the discourse. By doing so, we seek to provide a
granular understanding of how influencers sway public sentiment and contribute
to the emotional polarization of online communities.

Our paper makes the following contributions:

  * •

This research introduces a counterfactual analysis framework to quantitatively
assess how conversations led by influencers affect polarization on Twitter.

  * •

We implement a novel methodology that compares polarization scores with and
without the presence of influential conversations, shedding light on the
direct impact of these discourses.

  * •

We offer a detailed analysis of how influential users shape emotional dynamics
within contentious topics such as climate change and gun control, providing
quantifiable measures of their influence on affective polarization.

This study not only advances our understanding of affective polarization but
also has practical implications for the design and governance of social media
platforms. By identifying the influential figures and the factors that
exacerbate or mitigate polarization, platform designers and policymakers can
better navigate the challenges posed by this phenomenon and work towards a
more informed and less divided public discourse.

##  II Literature Review

The phenomenon of affective polarization has been extensively documented in
political psychology, with recent studies revealing its escalation on social
media platforms [15, 16]. Affective polarization extends beyond ideological
disagreements, encapsulating emotional responses that manifest as mutual
dislike and distrust among those with opposing political allegiances [17].

The literature on this subject reveals a complex ecosystem where media,
political figures, and ideological silos interact to deepen divisions. Social
media platforms, with their algorithmically curated content, have been
implicated as amplifiers of this division, fostering environments where
inflammatory rhetoric and moral outrage thrive [18, 19, 20]. Echo chambers, a
topic of much debate within the literature, are said to contribute to
polarization by reinforcing ideological conformity and limiting exposure to
diverse viewpoints [21]. While some studies suggest that these echo chambers
might insulate users from contrary opinions, thus exacerbating polarization,
others argue against this insularity, suggesting that exposure to opposing
views does not necessarily diminish polarization and may, in some cases,
intensify it [22, 23].

Advancements in quantifying affective polarization have led to the development
of metrics that capture the emotional content and toxicity in social media
interactions. Studies now classify users by political partisanship and analyze
the emotions and language used in their communications, providing a subtle
understanding of the affective component of polarization [6, 24]. This line of
research highlights that negative emotions and toxicity are not randomly
distributed but correlate with the network distance in social media
interactions, suggesting structural properties of these networks influence the
emotional tone of online discourse [25].

Kramer et al.[26] assert that emotional states are contagious so that whether
expressed by other people or appeared on Newsfeed have a direct impact on our
emotions. Another finding of their study is that as opposed to prevalent
perception, non-verbal language and inter-personal interactions are not
required for contagion of the emotions. The study by Cha et al.[27] explores
influence patterns on social media and identifies the most important role
playing factors. Betts and Bliuc [28] shows that an influencer with extreme
opinions will invariably accelerate the pace of polarization, and this impact
grows in proportion to their influence and level of engagement. The impact of
a neutral influencer, however, depends on the society’s openness to differing
viewpoints.

Our research builds upon these findings by specifically examining how
influential figures impact the affective landscape of online discourses. By
systematically evaluating the presence and absence of high-profile
conversations, we offer a unique perspective on the role influencers play in
either mitigating or exacerbating affective polarization. Through this lens,
we contribute to a deeper understanding of the dynamics at play in social
media’s political discourse to provide insights for digital communication
strategies aimed at reducing polarization.

##  III Methodology

The goal of this study is to methodically quantify the impact of influencers
on affective polarization on social media platforms. To achieve this, we
implement a counterfactual analysis framework, evaluating scenarios with and
without specific influencer-led conversations, as illustrated in Figure 1.
This approach allows us to isolate the effect of these conversations on
polarization dynamics. The following subsections will delve into the specifics
of our data collection strategies, the construction of interaction networks,
and the application of sentiment analysis and quantifying polarization
metrics, providing a comprehensive understanding of our methodological
approach and the analytical techniques employed to evaluate the influencer
impact on online social interactions.

Figure 1: Illustrative example of an interaction network with and without a
conversation, highlighting the structural differences.

###  III-A Data

####  III-A1 Data Collection

Our research utilized a comprehensive dataset extracted from Twitter,
encompassing a wide array of tweets related to two prominent and contentious
political issues–climate change and gun control. We used Hashtags to scrape
the initial set of relevant tweets related to each topic. To fully capture
conversation cascades and user interactions, we then recursively collect all
referenced tweets from each tweet in the initially matched set. This ensures
that all available tweets from relevant conversations are included in our
analysis, even if particular tweets do not explicitly use any of the hashtags.
We also retrieved user metadata, including the number of likes, replies,
retweets, etc. to identify influential users, as shown in Table I.

TABLE I: Dataset details

Dataset Timeline Total Tweets (million) Total Conversations (million) Total
Users (million) #likes (billion) #replies (million) #retweets (billion) Gun
Control 2022.01.01 - 2022.12.31 2.65 2.26 0.94 1.46 120.69 7.49 Climate Change
2021.06.01 - 2022.05.31 7.24 6.46 2.04 30.39 149.05 2.64

TABLE II: Example Subset of Influential Users

Dataset Twitter Account Tweet Count Total Likes (million) Total Retweets
(million) Total Replies (million) Stance Gun Control User 1 1,241 272.40 25.55
19.69 anti User 2 1,665 66.79 10.35 10.77 pro User 3 1,750 37.74 4.75 2.75 pro
User 4 1,760 31.75 5.07 4.15 pro User 5 746 22.81 5.25 1.15 pro Climate Change
User 1 1,766 190.47 16.64 11.84 believe User 2 76 185.99 48.82 9.99 believe
User 3 1,945 57.79 8.05 6.79 believe User 4 33 26.43 7.53 0.65 believe User 5
55 24.12 5.61 1.14 believe

####  III-A2 Data Exploration

A preliminary assessment revealed that despite the high volume of
conversations captured within our datasets, a notable portion of these
conversations are limited in engagement as depicted in Figure 2. This
observation indicates that a substantial quantity of conversations remains
solitary, devoid of further interaction that could potentially influence the
polarization of social discourse. In light of this, our analysis is confined
to threads manifesting a minimum engagement criteria of 20 tweets and 10
distinct users.

(a) Gun Control Dataset

(b) Climate Change Dataset

Figure 2: Bar chart distributions of conversation characteristics: (a) Gun
Control and (b) Climate Change topics, displaying the frequency of
conversations by number of tweets (left) and number of users (right).

####  III-A3 Influential Users

The study incorporates a selection of the top influential users from each
dataset, identified by their capacity to attract substantial engagement, as
evidenced by metrics such as accumulated likes (e.g. Table II presents a
representative subset). This methodology is predicated on the premise that
these opinion leaders wield a considerable impact on the discourse permeating
online platforms. Despite the extensive user base encapsulated within our
datasets, the chosen influencers serve as a microcosm for wider interaction
trends, with their content demonstrating a significant resonance amongst a
broader audience. From these users adhering to our engagement criteria, we
honed in on 5,500 gun control and 8,421 climate change conversations for a
detailed examination which resulted in 2,103 influential users for the gun
control dataset and 3,701 for the climate change dataset. This analysis of
influencer-initiated conversations over an annual span enables us to discern
the dynamics of influencer-driven polarization and to extrapolate these
findings to reflect overarching tendencies on the platform.

###  III-B Quantifying Polarization with E/I Index

In our study, we employ a robust multi-model sentiment analysis approach to
assess the emotional content of social media interactions. Utilizing sentiment
analysis models—VADER, BERTweet, and RoBERTa—we analyze the textual content of
tweets to capture a broad spectrum of affective expressions. This method
provides a sentiment profile for each tweet integral to our subsequent
polarization analysis.

To quantify affective polarization, we adopt the E/I Index methodology, which
measures the ratio of external (between-group) to internal (within-group)
interactions within Twitter discourse networks. This approach, directly
adapted from the work by Tyagi et al. [10], allows us to quantitatively assess
the degree of in-group solidarity and out-group engagement—key indicators of
affective polarization. We construct interaction networks based on retweet and
reply data to evaluate:

  * •

In-group Solidarity: The frequency and sentiment of interactions within a
group, indicate cohesion.

  * •

Out-group Engagement: The nature of cross-group interactions, often marked by
conflict or disagreement.

These interactions are enriched with emotional valences through our sentiment
analysis framework, enhancing our understanding of the dynamics at play. For
instance, in the gun control context, we observe polarization patterns such as
Pro-Anti and Anti-Pro, where advocates from either side express negative
sentiments toward their opposites, signaling a shift from mere support or
opposition to active advocacy or contention. These categorizations not only
demonstrate the affective landscape within these debates but also illuminate
how specific interactions contribute to broader societal polarization.

###  III-C User Stance Labeling using Graph Neural Networks (GNNs)

Our approach to user stance labeling employs a two-stage pipeline that
integrates textual and social interaction data, as established in [29].

##### Initial Label Generation

Initially, we deploy a reciprocal label propagation algorithm within a user-
hashtag bipartite graph. This method effectively generates labels by
associating users with probable stance categories based on their interactions
with specific hashtags. This labeling serves as a flexible precursor to more
definitive stance categorization, accommodating the preliminary subtleties and
complexities of user stances. Notably, this approach yields stance labels for
a limited subset of users, compared to a significantly larger population, and
does not take into account any social interactions between users.

##### Expanding Stance Labeling with GNNs

We expand stance labeling by constructing a comprehensive user-user
interaction network, where nodes represent users and edges signify connections
between authors and targets of their interactions. We make use of BERTweet to
embed textual content into node features. We then train a graph neural network
classifier on this interaction graph to predict node labels, using the seed
users labeled by the hashtag method as the training set. This approach is
grounded in the principles of linguistic and network homophily [30, 31, 32],
which provide a theoretical foundation for understanding how users’ language
and interactions reflect their ideological alignments. Moreover, the GNN
predicts node labels by generating a probability distribution over possible
stances for each user.

##### Optimization of Classification Thresholds

To find the users in a stance group, we employed a dual-threshold approach,
utilizing Threshold_1 and Threshold_2 to categorize users based on their
stance probability. Through a systematic grid search, we optimized the
threshold combination to minimize misclassification near the decision
boundary, maximizing the F1 score. This involved comparing predicted labels
against known heuristic stances from the hashtag method and a subset of
manually labeled users with domain expertise. The optimal thresholds were
established as Threshold_1 ($\leq 0.40$) for ’pro’ or ’believers’ stances and
Threshold_2 ($\geq 0.60$) for ’anti’ or ’disbelievers’, with intermediate
probabilities classified as ’undecided’. Consequently, a significant number of
users and their initiated conversations were excluded from both datasets due
to their stance being labeled as ’undecided’.

###  III-D Approach to Quantifying Influencer Impact on Affective Polarization

Our study employs a multifaceted approach to quantify the influence of key
Twitter influencers on affective polarization, integrating comparative
analyses with strategic subgraph constructions. We center our methodology
around the concept of a counterfactual scenario, where polarization scores are
calculated in scenarios both with and without specific influencer-led
conversations, as illustrated in Figure 1. This comparative method is crucial
for isolating the shifts in polarization attributable to these conversation
networks, providing insights into how individual conversations can sway public
sentiment. However, it’s important to note the observational nature of our
approach; while it reveals significant insights into the dynamics of
influencer conversations, it does not assert direct causation.

To enhance our analysis, we focus on constructing subgraphs around specific
influencers and their follower networks, recognizing that the comprehensive
interaction graphs on platforms like Twitter, which involve millions of daily
interactions, can obscure the effects of smaller-scale conversation networks.
For instance, in our detailed case study of ’User X’s Twitter interactions, we
selectively identified a subset of approximately 1,000 followers from a larger
pool of 15,000 active users. We constructed a subgraph that included these
followers and their adjacent connections, encompassing around 2,000 users in
total.

This subgraph methodology provides a focused lens through which we can observe
and analyze the polarization dynamics more clearly. Initially, we compute the
polarization within this subgraph by including ’User X’s conversation.
Subsequently, to distinctly understand the conversation’s specific impact, we
recalibrated the polarization after removing the 400 users directly involved
in the conversation. This technique of contrasting polarization scores with
and without the conversation offers a transparent view of how specific
conversations influence the network’s polarization dynamics.

By focusing on influencer-centric subgraphs, we uncover the subtle yet
significant impacts of specific conversations on affective polarization,
avoiding the dilution of findings by the broader network’s noise.

##  IV Results

Our analysis reveals a temporal sensitivity in polarization, underscored by
fluctuations that correspond closely with real-world events and influencer-led
conversations. Specifically, our temporal examination—illustrated in Figures 3
and 4—highlights notable shifts in polarization within the contexts of gun
control and climate change discourse.

For instance, in Figure 3(a) during the summer of 2022, the gun control
dataset exhibited significant spikes in the daily Twitter conversation,
correlating with high-profile shooting incidents in the United States. This
period, notably marked by the tragic Texas Robb Elementary School shooting,
saw intensified influencer activity leading conversations that either
amplified or attempted to bridge divides in public sentiment. Similarly, in
the climate change discourse (Figure 3(b)), pivotal events such as the release
of the Intergovernmental Panel on Climate Change (IPCC) report and the United
Nations Climate Change Conference were mirrored by peaks, pointing to the
reactive nature of online discourse to global climate events.

Following these events, Figure 4 illustrates how influential-led conversation
changes the dynamic of polarization. For both, gun-control and climate-change
datasets, we observe that when a certain event occurs, the polarization score
increases compared to the situation it was before the event happened. These
findings not only demonstrate the temporal alignment of polarization with
external events but also highlight the role of influencers in steering the
conversation, thereby affecting the polarization landscape.

(a) Gun Control Dataset

(b) Climate Change Dataset

Figure 3: Frequency distributions of daily conversation counts over the
specified time frame, stratified by topic: (a) Gun Control and (b) Climate
Change.

(a) Texas Shooting

(b) Illinois Shooting

(c) Colorado Shooting

(d) Hurricane Ida

(e) Heat Wave

(f) IPCC Annual Report

Figure 4: Temporal dynamics of polarization scores in response to events. The
top trio of subfigures (a-c) illustrates changes in polarization preceding and
following events related to gun control, with ’(P $\rightarrow$ A)’ denoting
shifts from pro-to-anti gun control stance and ’(A $\rightarrow$ P)’ denoting
shifts from anti-to-pro stance. The bottom trio (d-f) depicts similar changes
for climate change-related events, with ’(B $\rightarrow$ D)’ representing
shifts from believe-to-disbelieve in climate change and ’(D $\rightarrow$ B)’
for shifts from disbelieve-to-believe stance direction. TABLE III: Climate
Change Dataset: The effect of a single conversation by influential users on
daily polarization score.

User User’s Stance Number of Tweets in the Conversation Number of Followers in
the Conversation (#disbelieve / #believe) Number of Followers Interacting on
That Day (#disbelieve / #believe) Stance Direction Polar. Score Without Conv.
Polar. Score With Conv. User 1 believe 104 41 / 50 904 / 1,332 disbelieve
$\rightarrow$ believe -0.091 -0.135 believe $\rightarrow$ disbelieve 0.463
0.398 User 2 believe 187 131 / 24 457 / 333 disbelieve $\rightarrow$ believe
0.115 0.185 believe $\rightarrow$ disbelieve 0.378 0.402 User 3 disbelieve 157
68 / 14 302 / 100 disbelieve $\rightarrow$ believe -0.137 -0.136 believe
$\rightarrow$ disbelieve 0.633 0.471 User 4 believe 134 51 / 37 356 / 356
disbelieve $\rightarrow$ believe 0.149 0.181 believe $\rightarrow$ disbelieve
-0.228 -0.247 User 5 disbelieve 66 43 / 2 395 / 86 disbelieve $\rightarrow$
believe 0.144 0.152 believe $\rightarrow$ disbelieve 0.248 0.237

TABLE IV: Gun Control: The effect of a single conversation by influential
users on daily polarization score.

User User’s Stance Number of Tweets in the Conversation Number of Followers in
the Conversation (#anti / #pro) Number of Followers Interacting on That Day
(#anti / #pro) Stance Direction Polar. Score Without Conv. Polar. Score With
Conv. User 1 pro 746 334 / 325 1,211 / 1,040 anti $\rightarrow$ pro 0.147
0.252 pro $\rightarrow$ anti 0.247 -0.200 User 2 anti 1,333 497 / 391 856 /
707 anti $\rightarrow$ pro 0.079 0.229 pro $\rightarrow$ anti 0.220 -0.087
User 3 pro 57 20 / 22 312 / 356 anti $\rightarrow$ pro 0.085 0.099 pro
$\rightarrow$ anti -0.022 0.121 User 4 anti 164 37 / 100 176 / 286 anti
$\rightarrow$ pro 0.247 0.121 pro $\rightarrow$ anti 0.204 0.241 User 5 pro
129 16 / 50 867 / 1,879 anti $\rightarrow$ pro 0.043 0.041 pro $\rightarrow$
anti 0.069 0.085

TABLE V: Gun Control Dataset: Impact of conversation initialized by
influential authors (Twitter users) on polarization scores. The ’% increase’
or ’% decrease’ denotes the percentage of cases, (how) polar score changes
upon adding a conversation compared to the total number of conversations.

Twitter User User Type Stance (Gun Control) Number of Conversations
Polarization: Change (pro $\rightarrow$ anti) Polarization: Change (anti
$\rightarrow$ pro) User 1 Media Outlet anti 5 100.0 % - decrease 100.0 % -
increase User 2 Media Outlet pro 6 66.67 % - decrease 100.0 % - decrease User
3 Commentator pro 9 88.89 % - decrease 77.78% - decrease User 4 Journalist
anti 18 88.89 % - decrease 61.11% - increase User 5 Politician pro 7 85.71 % -
increase 85.71% - decrease User 6 Politician pro 14 64.29 % - decrease 78.57%
- decrease User 7 Legal Analyst pro 38 76.32 % - decrease 71.05% - increase
User 8 Media Outlet anti 12 66.67 % - increase 75.00% - increase User 9
Attorney anti 7 71.43 % - increase 71.43% - increase User 10 Politician pro 26
53.85 % - decrease 61.54% - decrease

TABLE VI: Climate Change Dataset: Impact of conversation initialized by
influential authors (Twitter users) on polarization scores. The ’% increase’
or ’% decrease’ denotes the percentage of cases, (how) polar score changes
upon adding a conversation compared to the total number of conversations.

Twitter User User Type Stance (Climate Change) Number of Conversations
Polarization: Change (believe $\rightarrow$ disbelieve) Polarization: Change
(disbelieve $\rightarrow$ believe) User 1 Scientific Org. believe 5 60.00% -
decrease 100.0 % - increase User 2 Public Figure disbelieve 5 80.00% -
decrease 100.0 % - increase User 3 Media Personality disbelieve 8 50.00% -
increase 87.5 % - decrease User 4 Politician believe 7 85.71% - decrease 57.14
% - decrease User 5 Media Outlet believe 41 51.22% - decrease 85.37 % -
increase User 6 Actor believe 6 83.33% - decrease 50.0 % - increase User 7
Climate Activist believe 9 77.78% - increase 55.56 % - decrease User 8 Media
Personality disbelieve 8 75.00% - increase 62.5 % - increase User 9 Climate
Advocate believe 7 57.14% - decrease 71.43 % - decrease User 10 Politician
believe 19 57.89% - decrease 63.16 % - increase

TABLE VII: Details of conversations’ effects based on stances for both gun
control and climate change. The ’% increase’ or ’% decrease’ denotes the
percentage of cases, (how) polar score changes upon adding a conversation
compared to the total number of conversations.

Dataset Stance Number of Conversations Polarization Score: Change Gun Control
(pro $\rightarrow$ anti) (anti $\rightarrow$ pro) pro 1,940 63.40% - decrease
52.99% - increase anti 1,704 55.52% - decrease 53.28% - decrease Climate
Change (believe $\rightarrow$ disbelieve) (disbelieve $\rightarrow$ believe)
believe 1,561 50.99% - decrease 64.25% - increase disbelieve 596 50.67% -
decrease 58.22% - decrease

###  IV-A Effect of an Influential-led Conversation

The shifts in polarization are quantified by examining the changes in
polarization scores when specific influencer-led conversation is either
included or excluded from the analysis. This approach helps isolate the direct
impact of a single conversation on the overall polarization score that day.

For example, as illustrated in Table III, the removal of a conversation by
’User 3’, a prominent disbeliever in climate change, resulted in a notable
shift in the polarization scores. Specifically, the polarization score for the
believer-to-disbeliever direction increased from 0.471 to 0.633, indicating a
more pronounced divide when the disbeliever’s influence was absent from the
conversation. This change, representing a magnitude of 0.162 in the
polarization score, underscores the significant role that influencer-led
interactions play in shaping public sentiment and polarization.

Similarly, in the context of gun control as detailed in Table IV, the
exclusion of a conversation led by an anti-gun control user (i.e. ’User 2’)
resulted in a decrease in the anti-pro polarization score. This demonstrates
how conversations led by users opposed to gun control amplify polarization
towards anti-gun stances when present, and reduce it when removed.

###  IV-B Insight of Influential Users

In this section, we examine the collective influence of conversations
initiated by influential users on shaping public sentiment and polarization
across various themes. We employ a systematic approach to quantify the impact
of these users by assessing the frequency and effect of their conversations on
polarization shifts.

Tables V and VI present data on influential users who actively engage in
discussions and initiate several conversations related to gun control and
climate change, respectively. These tables reveal the percentage of
conversations that increase or decrease polarization, given a specified number
of conversations within the context of these topics, thereby illuminating the
effects of influential users on social media dynamics.

###  IV-C Insight of Conversations Within Stance-Group

Table VII provides a more comprehensive view by analysing all the
conversations initiated by users within a certain stance. Interestingly, we
observe that conversations led by users with anti or disbeliever stances often
result in decreased polarization compared to their opposing stance group.
Overall, the insights gleaned from these influencer-led conversations across
different thematic areas demonstrate the critical role of influential users in
modulating the affective landscape of online communities.

##  V Conclusion

This study provides a detailed analysis of the mechanics of affective
polarization on social media, particularly highlighting the crucial role of
influential users in modulating public sentiment on platforms like Twitter. By
implementing a counterfactual framework, which evaluates scenarios with and
without specific influencer-led conversations, we offer a unique
methodological contribution that allows us to isolate and quantify the
influence of such conversations on the dynamics of polarization. Our findings
demonstrate that influencers have the potential to amplify or mitigate
divisiveness across polarizing topics like gun control and climate change. We
reveal how subtle shifts in influencer-driven dialogues can significantly
affect public discourse by employing a comparative analysis coupled with
cultivated computational techniques, such as subgraph construction. This
approach deepens our understanding of how specific conversations impact
polarization that are critical to social media dynamics.

Despite these advancements, our study faced methodological challenges,
particularly the limitations inherent in sentiment analysis based on the
presence of positive or negative words in tweets and the sporadic nature of
meaningful social media interactions. The small scale of interactions within
specific conversations compared to the vast daily interactions across the
platform posed significant challenges in achieving statistical significance.
These challenges underscore the complexity of capturing the sophisticated
landscape of online discourse and point to the need for developing more
refined metrics and methodologies.

Moreover, this research advances scholarly understanding of online
polarization by pioneering the use of counterfactual scenarios, significantly
illuminating the complex dynamics of social media influence.

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