# Theory of fractional quantum Hall liquids coupled to quantum light and
emergent graviton-polaritons

Zeno Bacciconi  SISSA — International School for Advances Studies, via Bonomea
265, 34136 Trieste, Italy  ICTP — The Abdus Salam International Centre for
Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy  Hernan B.
Xavier  SISSA — International School for Advances Studies, via Bonomea 265,
34136 Trieste, Italy  ICTP — The Abdus Salam International Centre for
Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy  Iacopo
Carusotto  INO-CNR Pitaevskii BEC Center and Dipartimento di Fisica,
Università di Trento, 38123 Povo, Italy  Titas Chanda  Department of Physics,
Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552,
India  Marcello Dalmonte  ICTP — The Abdus Salam International Centre for
Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy

###### Abstract

Recent breakthrough experiments have demonstrated how it is now possible to
explore the dynamics of quantum Hall states interacting with quantum
electromagnetic cavity fields. While the impact of strongly coupled non-local
cavity modes on integer quantum Hall physics has been recently addressed, the
effects on fractional quantum Hall (FQH) liquids – and, more generally,
fractionalized states of matter – remain largely unexplored. In this work, we
develop a theoretical framework for the understanding of FQH states coupled to
quantum light. In particular, combining analytical arguments with tensor
network simulations, we study the dynamics of a $\nu=1/3$ Laughlin state in a
single-mode cavity with finite electric field gradients. We find that the
topological signatures of the FQH state remain robust against the non-local
cavity vacuum fluctuations, as indicated by the endurance of the quantized
Hall resistivity. The entanglement spectra, however, carry direct fingerprints
of light-matter entanglement and topology, revealing peculiar polaritonic
replicas of the $U(1)$ counting. As a further response to cavity fluctuations,
we also find a squeezed FQH geometry, encoded in long-wavelength correlations.
We additionally observe that moving to strong cavity field gradients leads to
an instability towards a sliding Tomonaga-Luttinger liquid phase, featuring a
strong density modulation in the gradient direction. Finally, by exploring the
low-energy excited spectrum inside the FQH phase, we identify a new
quasiparticle, the graviton-polariton, arising from the hybridization between
quadrupolar FQH collective excitations (known as gravitons) and light. We
discuss the experimental implications of our findings and possible extension
of our results to more complex scenarios.

##  I Introduction

The possibility of controlling quantum matter properties via cavity embedding
has sparked a lot of interest in recent years [1, 2, 3, 4]. Vacuum
fluctuations of strongly confined electromagnetic modes have been proposed as
handles on various phenomena [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], and
pioneering experiments have demonstrated the non-trivial role of cavity
quantum electrodynamics (QED) set-ups in shaping matter properties [16, 17,
18, 19, 20]. A particularly intriguing framework is that of topological
phases, whose traditional many-body understanding faces fundamentally new
questions due to the non-local nature of the cavity degree of freedom.

On this point, a recent breakthrough experiment [16] has shown that the Hall
conductivity of the integer quantum Hall (IQH) effect can be affected by a
split-ring cavity in the ultra-strong coupling regime, even in the absence of
any driving. This effect has been proposed to arise from cavity mediated
hoppings [21] involving cyclotron transitions to higher Landau levels (LLs)
assisted by vacuum photons or, more recently, as a consequence of cavity
losses [22]. Although a rather good understanding of IQH states coupled to
quantum light has been developed, its effect on the much richer physics of
fractional quantum Hall (FQH) matter [23, 24] remains largely unexplored.

The FQH effect [23] is a fundamentally different state of matter which,
differently from its integer counterpart, genuinely arise from many-body
correlations. In FQH phases, new collective degrees of freedom emerge, such as
anyonic quasi-particles [25], one of the smoking gun of topological order, or
magnetoroton modes [26, 27] also dubbed at long wavelength as gravitons [28]
given the link to a more recent geometric description of FQH correlations [29,
30, 31, 32]. Another unique feature of topologically ordered phases is their
peculiar many-body entanglement structure [33, 34], which naturally connects
to spectral properties and has been demonstrated an useful tool for the
classification of quantum phases of matter. What is the fate of such rich and
profound phenomena under the action of quantum light? This is the question we
address in this work.

We put forward a theory describing FQH liquids coupled to quantum light. The
starting point of our analysis is a careful modeling of electrons coupled to
light, that preserves a restricted gauge invariance exactly. We introduce a
lowest Landau level (LLL) projected minimal coupling, following Ref. [35],
which neglects cyclotron transitions and focuses on intra-LL physics. This
highlights the role of cavity field gradients which, in agreement with Kohn’s
theorem [36], are essential to non-trivially couple to electronic correlations
within the LLL.111We note that the electric quadrupole moment of quantum Hall
liquids [37], visible only by field gradients, has been proposed as
fundamental in their understanding.

As a proof of principle, we investigate a simplified scenario given by the
$\nu=1/3$ Laughlin state coupled to a cavity with a constant gradient, and no
losses. The setting is schematically depicted in Fig. 1. The full quantum
dynamics of the system (light and matter) is studied numerically with a novel
hybrid tensor network ansatz that combines the success in representing FQH
states [38, 39] using matrix product states (MPS) [40, 41, 42], with cavity-
matter correlations in 1D [43, 44, 45, 46]. This allows us to investigate in
an unbiased manner both ground and excited states properties, using algorithms
based on the density-matrix-renormalization group (DMRG) [47, 48] and on the
time-dependent variational principle (TDVP) [49, 50].

We show that the topological properties of the Laughlin state remain stable
against the introduction of the non-local mode, as signaled by the quantized
Hall resistivity, which we calculate by adapting the original flux insertion
argument [51]. However, entanglement properties are drastically affected: new
entanglement spectrum features arise in form of copies of the typical $U(1)$
counting of FQH states [33], which appear separated by a polariton
entanglement gap. This reveals a collective coupling with matter excitations
at zero transferred momenta which we also detect in spectral properties. These
matter excitations are revealed to be gravitons, i.e., the long-wavelength
part of the magnetoroton dispersion, and, once they hybridize with cavity
photons, become graviton-polaritons. They give rise to a typical polariton
doublet, which can be used in spectroscopy experiments as a smoking gun for
the strong-coupling regime. Through a simple effective model that almost
matches quantitatively with the finite size numerical simulations, we also
provide analytical predictions for the collective Rabi frequency in terms of
the known graviton quadrupole moment.

Another major consequence of the cavity mode on the ground state is the
squeezing of the FQH metric [29], that is revealed by a striking change in
long-wavelength correlations and by the spatially anisotropic profile of
electron correlation holes. Inspecting field gradients which are strong at the
single particle level, we also find a cavity mediated instability accompanied
by the softening of the full magnetoroton dispersion. The strong coupling
phase has strong density modulations organized as stripes which can be
understood as a sliding Tomonaga-Luttinger liquid phase, reminiscent of other
transitions mediated by different anisotropy sources [52].

Figure 1: Illustration of the FQH bar placed inside the cavity. The electrons
are confined to move in the 2D plane under the action of a perpendicular
magnetic field $\bm{B}_{\mathrm{ext}}$. The in-plane component of the cavity
electric field pierces the principal edges of the strip perpendicularly.
Periodic boundary conditions on $y$ are depicted as dotted edges, while open
boundary conditions on $x$ are depicted as a sharp cut. The inset shows the
mode profile with a uniform gradient used in most of the discussion.

##### Summary of results.—

Before describing the structure of the work, we summarize here our main
findings:

  * •

A microscopic QED theory describing the coupling of quantum light to electrons
in the LLL, pointing out that constant fields have no effects on the latter;

  * •

Resilience of quantized Hall resistivity upon the introduction of the non-
local cavity-mode degree of freedom, with the latter imprinting an anisotropic
FQH geometry in the ground state;

  * •

A new entanglement structure in hybrid quantum Hall states, where the role of
quantum light is to introduce a “band” of chiral Luttinger liquids multiplets,
each with an approximately quantized photon number;

  * •

The prediction of graviton-polariton modes, describing the hybridization of
the typical magnetoroton mode with quantum light in the setup we consider;

  * •

The prediction of an instability of Laughlin states to sliding Luttinger
liquids at a very strong light-matter coupling for cavities describing
constant electric field gradients.

The paper is organized as follows.

In Section II we describe the system under study, starting from a brief recap
on LLs and the microscopic derivation of the light-matter coupling by means of
a minimal substitution restricted to the LLL, ensuring gauge invariance and a
proper treatment of the strong coupling regime. We discuss the role of cavity
field gradients, and detail the numerical methods we use to solve the full
cavity-matter problem.

In Section III we present a detailed study of the nature of the FQH
topological order in the presence of a strongly coupled non-local cavity
degree of freedom. We do so by looking at two key markers, the entanglement
spectrum structure and the transverse Hall resistivity. We also provide
numerical evidence that cavity vacuum fluctuations lead to a squeezed FQH
geometry.

In Section IV, we discuss the phase diagram and the instability to a sliding
Tomonaga-Luttinger liquid phase in the regime of strong field-gradients at the
single electron level. We first present finite size numerical results, and
then motivate the existence of an instability on the basis of a photon mean-
field decoupling. We present a qualitative field theory describing the sliding
Tomonaga-Luttinger liquids, where light-matter correlations can be
reintroduced to some extent.

In Section V we investigate bulk spectral properties of the hybrid light-
matter FQH state. We start by reviewing the magnetoroton spectrum, the low
energy gapped neutral excitations on top of FQH states. Then we present
numerical evidence for the effect of the cavity degree of freedom on the full
magnetoroton dispersion, from the formation of the hybrid graviton-polariton
to the lowering of the magnetoroton minimum as precursor of the instability.
We then construct an effective model which builds on top of the Girvin-
MacDonald-Platzman (GMP) treatment of magnetorotons, and is able to capture
analytically the salient features of the graviton-polaritons that we observe
numerically.

In Section VI, we discuss the connection of our findings to realistic
experimental scenarios taking as a reference a split-ring resonator [16].
While energy scales match, we highlight how carefully designed resonators with
strong field gradients are needed to reach strong coupling to FQH physics. We
also comment on the possible role of cavity screening of Coulomb interactions
neglected in our treatment.

To conclude in Section VII, we summarize our results and draw a more general
picture on how confined electromagnetic modes can be used to both probe and
control correlations in the LLL.

##  II The model

In this section we start by reviewing LL physics (Sec. II.1) and the split-
ring cavity set-up (Sec. II.2). Then, we present the first result of this
work, namely the derivation of a QED Hamiltonian for LLL electrons coupled to
a cavity (Sec. II.3). After discussing some of the physical implications of
the obtained model (Sec. II.4), we spell out the full Hamiltonian used in the
rest of the paper (Sec. II.5) and describe the MPS ansatz used to solve it
(Sec. II.6).

###  II.1 Landau Levels

We consider a collection of $N_{e}$ electrons with mass $m_{e}$ and elementary
charge $q=|e|$ living on the $(x,y)$ plane under a strong magnetic field
$\bm{B}_{\mathrm{ext}}=(0,0,B)$ in the $z$-direction (Fig. 1), represented in
the so called Landau gauge by an external vector potential
$\bm{A}_{\mathrm{ext}}=(0,xB,0)$. We choose to work with periodic boundary
conditions along $y$, implying a cylinder geometry with a finite circumference
$L_{y}$. The single-particle eigenstates can be written as [53]:

$\psi_{n,k}(x,y)=\frac{1}{\sqrt{L_{y}}}e^{iky}\phi_{n}(x-X_{k}),\qquad
k=\frac{2\pi}{L_{y}}j_{k},$ (1)

with $n=0,1,2,...,$ being the LL index and $j_{k}\in\mathbb{Z}$ labeling the
momentum along $y$. Here $\phi_{n}$ are eigenfunctions of the harmonic
oscillator with characteristic length equal to the magnetic length
$l_{B}=\sqrt{\frac{\hbar c}{eB}}$, frequency equal to the cyclotron frequency
$\omega_{B}=\frac{eB}{m_{e}}$, and centered around $X_{k}=kl_{B}^{2}$. In
presence of an external potential $W$, which can account for both disorder and
confining potential, the single-particle Hamiltonian for the LLs takes the
following general form:

$\hat{\mathcal{H}}_{0}=\sum_{n,k}n\omega_{B}\,\hat{n}_{k,n}+\sum_{k,k^{\prime},n}W_{k,k^{\prime}}^{n}\hat{c}^{\dagger}_{k,n}\hat{c}_{k^{\prime},n}$
(2)

where $\hat{c}_{k,n}$ is the annihilation operator for the orbital $(k,n)$,
$\hat{n}_{k,n}=\hat{c}^{\dagger}_{k,n}\hat{c}_{k,n}$, and
$W_{k,k^{\prime}}^{n}$ are the matrix element of the external potential,
assumed diagonal in the LL index for simplicity. Importantly, we focus on the
LLL ($n=0$) where the single-particle Hamiltonian then reads:

$\hat{H}_{0}=\mathcal{P}\hat{\mathcal{H}}_{0}\mathcal{P}=\sum_{k,k^{\prime}}W^{0}_{k,k^{\prime}}\hat{c}^{\dagger}_{k}\hat{c}_{k^{\prime}},$
(3)

where $\mathcal{P}$ is a projector onto the LLL, and we dropped the LL index
on the fermionic operator. In the following we are going to focus on $W=0$,
except when explicitly stated.

The other important ingredient are two-body interactions represented by a
central potential $V(|\bm{r}_{1}-\bm{r}_{2}|)$. Neglecting LL mixing and
projecting onto the LLL, the interaction term can be written as:

$\hat{H}_{\mathrm{int}}=\sum_{k_{1},k_{2},k_{3},k_{4}}V_{k_{1},k_{2},k_{3},k_{4}}\hat{c}^{\dagger}_{k_{1}}\hat{c}^{\dagger}_{k_{2}}\hat{c}_{k_{3}}\hat{c}_{k_{4}}$
(4)

where $V_{k_{1},k_{2},k_{3},k_{4}}$ are the matrix elements of $V(r)$ in the
LLL. In order to keep the analysis simple, we adopt the first Haldane
pseudopotential [54], i.e., the shortest range fermionic interaction, for
which the Laughlin wavefunction is an exact zero-energy ground-state. Its
matrix element on the cylinder are [54]:

$\displaystyle V_{k_{1},k_{2},k_{3},k_{4}}$
$\displaystyle=V_{0}\delta_{k_{1}+k_{2}}^{k_{3}+k_{4}}\frac{\sqrt{2\pi}}{L_{y}}\left[(k_{1}-k_{3})^{2}-(k_{2}-k_{3})^{2}\right]$
$\displaystyle\times
e^{-\frac{1}{2}(k_{1}-k_{3})^{2}}e^{-\frac{1}{2}(k_{2}-k_{3})^{2}},$ (5)

where the energy scale of the interaction is set to $V_{0}=1$, and the
$1/L_{y}$ factor guarantees the interaction term to be extensive. In absence
of disorder, both total number of particles $\hat{N}=\sum_{k}\hat{n}_{k}$ and
total momentum along the $y$ direction $\hat{K}_{y}=\sum_{k}k\,\hat{n}_{k}$
are conserved and can be fixed to $N_{e}$ and $K_{y}$. We will consider a
finite cylinder in the $x$ direction by truncating in the orbital space, e.g.,
$j_{k}=-(M-1)/2,\dots,(M-1)/2$ with $M$ being the total number of orbitals. In
particular, we fix $N_{e}$ with the condition $M=3N_{e}-2$ such that the
interaction Hamiltonian has a unique ground state at $K_{y}=0$, which is the
$\nu=1/3$ Laughlin state.

###  II.2 Cavity set-up

We consider a single-mode cavity model, inspired by the split-ring resonator
used in Ref. [16]. The split-ring mode can be understood in terms of an LC
resonance [55, 56] at frequency $\omega_{c}=1/\sqrt{LC}$ with $C$ being an
effective capacitance and $L$ being an effective inductance. By shrinking the
capacitor region, one can reach impressive enhancement of the vacuum electric
field fluctuations and in some cases enter the strong coupling regimes even at
the single electron level [57] . The free quantized Hamiltonian for the LC
resonator can be written in terms of the electric and magnetic field energy
density as:

$\hat{H}_{c}=\int\differential^{3}r\bigg{[}\frac{\epsilon_{0}}{2}\hat{\bm{E}}_{c}^{2}+\frac{1}{2\mu_{0}}(\bm{\nabla}\times\hat{\bm{A}_{c}})^{2}\bigg{]}=\omega_{c}\hat{a}^{\dagger}\hat{a},$
(6)

where $\hat{\bm{E}}_{c}$ is the electric field and $\hat{\bm{A}}_{c}$ is the
vector potential. These are expanded on a single quantized bosonic mode as:

$\displaystyle\hat{\bm{A}}_{c}$
$\displaystyle=A_{c}\bm{f}_{c}(\hat{a}+\hat{a}^{\dagger}),$ (7)
$\displaystyle\hat{\bm{E}}_{c}$
$\displaystyle=i\omega_{c}A_{c}\bm{f}_{c}(\hat{a}-\hat{a}^{\dagger}),$ (8)

with $\bm{f}_{c}$ the mode function, and
$A_{c}=\sqrt{\hbar/(2\omega_{c}\epsilon_{0}V_{\mathrm{mode}})}$ the intensity
of cavity vacuum fluctuations expressed as a function of the effective mode
volume $V_{\mathrm{mode}}$. With this choice we have
$\int\differential^{3}r\bm{f}_{c}\cdot\bm{f}_{c}=V_{\mathrm{mode}}$. In the
region of interest, i.e., where the Hall bar is placed, we will assume to have
negligible cavity magnetic field component
$\hat{\bm{B}}_{c}=\bm{\nabla}\times\hat{\bm{A}}_{c}\simeq 0$.

We focus on the specific case of vanishing electric field component in the $y$
direction and generic field along $x$. The in-plane part of the mode function
can be written as:

$\bm{f}_{c}^{\parallel}(\bm{r})=f_{c}(x)\bm{u}_{x}$ (9)

with $f_{c}(x)$ being a generic function and $\bm{u}_{x}$ being the unit
vector in the $x$ direction.

In the ideal case of infinite parallel mirror plates [55] living in the $y-z$
plane, we have $f_{c}(x)=f_{c}$ independent of $x$. However, relevant field
gradients are expected [16] and can be controlled to some extent. Note that
the gradient of the cavity mode function in 3D is constrained by Gauss’s law
such that, in the case of uniform dielectric within the capacitor plates, we
have $\bm{\nabla}\cdot\bm{f}_{c}=0$. This does not constrain the in-plane
gradients which can indeed be finite. The split-ring set-up naturally
implements an electric field perpendicular to the edges of the Hall bar [16],
modeled here as the open edges of the cylinder in our configuration (Fig. 1).
We remark that this particular choice of cavity configuration differs from
that considered in Ref. [58] where the electric field is exactly parallel to
edge of the system.

###  II.3 How to couple the LLL to quantum light

Naive truncations of light-matter interactions in strong coupling regimes have
been shown to be problematic [59, 60], often leading to photon condensation
transitions that contravene well established no-go theorems [61, 62]. A lot of
attention has been dedicated to devise controlled and effective low-energy
models for truncated electronic systems strongly coupled to quantized
electromagnetic modes [35, 63]. Here we perform a minimal substitution as
discussed in Ref. [35], which enforces a restricted gauge invariance on the
model [62], avoiding the emergence of a false photon condensation transition.

The key idea is performing the minimal substitution only after the degrees of
freedom are truncated. In the present case, it means after the LLL projection
and the single-mode approximation for the cavity. This can be implemented by a
projected unitary transformation
$\hat{U}=\hat{\mathcal{P}}\hat{\mathcal{U}}\hat{\mathcal{P}}$ applied to the
electronic projected Hamiltonian. The unitary transformation
$\hat{\mathcal{U}}$ is the one applying the “standard” minimal substitution
which, in the continuum, shifts the electronic momenta as
$\hat{\bm{p}}\rightarrow\hat{\bm{p}}-\frac{q}{c}\hat{\bm{A}}$.

In the following, we derive the effective cavity QED Hamiltonian for LLL
electrons following the aforementioned procedure, both in the Coulomb gauge
and in the Dipole gauge [35].

##### Coulomb gauge.—

We start by introducing the cavity pseudopotential $\chi(\bm{r})$ defined, up
to a constant, as:

$\chi(\bm{r})=\frac{e}{c}A_{c}\int_{\bm{r}_{0}}^{\bm{r}}\differential\bm{r}^{\prime}\cdot\bm{f}_{c}(\bm{r}^{\prime}).$
(10)

This uniquely defines the cavity vector potential
$\hat{\bm{A}}_{c}=\frac{c}{e}(\hat{a}+\hat{a}^{\dagger})\bm{\nabla}\chi$ in
absence of magnetic fields. The unitary transformation $\hat{U}$ implementing
the coupling to electrons in the LLL then reads:

$\displaystyle\hat{U}=\exp\left[i(\hat{a}+\hat{a}^{\dagger})\sum_{k,k^{\prime}}\chi_{k,k^{\prime}}^{0,0}\hat{c}^{\dagger}_{k,0}\hat{c}_{k^{\prime},0}\right],$
(11)

where we have kept the LLL index $n=0$ and the matrix elements are in general
defined as:

$\chi^{n,n^{\prime}}_{k,k^{\prime}}=\int\differential
r\,\psi_{k,n}^{*}(\bm{r})\,\chi(\bm{r})\,\psi_{k^{\prime},n^{\prime}}(\bm{r}).$
(12)

For our specific choice of cavity mode, and neglecting second order
derivatives of the cavity field, we can write a more explicit expression for
the matrix element:

$\displaystyle\chi_{k,k^{\prime}}^{n,n^{\prime}}$
$\displaystyle=el_{B}A_{c}f_{c}(X_{k})\sqrt{n}(\delta_{n,n^{\prime}+1}+\delta_{n,n^{\prime}-1})\delta_{k,k^{\prime}}$
$\displaystyle+\chi(X_{k})\delta_{n,n^{\prime}}\delta_{k,k^{\prime}}.$ (13)

Here $l_{B}$ is the magnetic length, $X_{k}$ the center of the orbital with
momentum $k$. There are two rather different terms, while the first one drives
inter-LL cyclotron transition, the second one dresses intra-LL physics. We
remark that in Eq. (11) the cyclotron transitions are completely neglected.
From now on we are going to shorten the notation of the matrix element of
interest as $\chi^{0,0}_{k,k^{\prime}}=\chi_{k}\delta_{k,k^{\prime}}$.

The QED Hamiltonian in the Coulomb gauge is then obtained by applying the
restricted unitary to the LLL Hamiltonian:

$\hat{H}^{\mathrm{coul}}=\hat{H}_{c}+\hat{U}(\hat{H}_{\mathrm{int}}+\hat{H}_{0})\hat{U}^{\dagger}.$
(14)

In order to understand the effect of the unitary $\hat{U}$, we note that it
corresponds to a simple dressing of the fermionic operators:

$\hat{c}_{k}\rightarrow e^{i(\hat{a}+\hat{a}^{\dagger})\chi_{k}}\hat{c}_{k}.$
(15)

Interestingly, the case of electric fields which are purely polarized on $x$
correspond to the so-called Peierls substitution.

##### Dipole gauge.—

An equivalent formulation of the problem can be carried out in the Dipole
gauge. This can be obtained from the Coulomb gauge by applying a projected
Power-Zienau-Wolley (PZW) transformation, e.g., the reverse unitary
transformation $\hat{U}^{\dagger}$:

$\displaystyle\hat{H}^{\mathrm{dip}}=\hat{U}^{\dagger}\hat{H}^{\mathrm{coul}}\hat{U}=\hat{H}_{\mathrm{int}}+\hat{H}_{0}+\hat{U}^{\dagger}\hat{H}_{c}\hat{U}.$
(16)

The unitary now acts on the cavity part of the Hamiltonian, effectively
shifting the cavity operator as:

$\hat{a}\rightarrow\hat{a}-i\sum_{k}\chi_{k}\hat{n}_{k}.$ (17)

This give rise to a very simple structure to the Dipole gauge Hamiltonian:

$\displaystyle\hat{H}^{\mathrm{dip}}=\hat{H}_{\mathrm{int}}+\hat{H}_{0}+\hat{H}_{c}+\hat{H}_{DP}+\hat{H}_{PP},$
(18)

with:

$\displaystyle\hat{H}_{DP}$
$\displaystyle=i\omega_{c}(\hat{a}-\hat{a}^{\dagger})\sum_{k}\chi_{k}\hat{n}_{k},$
(19) $\displaystyle\hat{H}_{PP}$
$\displaystyle=\omega_{c}\left(\sum_{k}\chi_{k}\hat{n}_{k}\right)^{2}.$ (20)

The subscripts $DP$ and $PP$ comes from the fact that we can express the
light-matter interaction terms via the dimensionless scalar displacement
$\hat{D}$ and polarization $\hat{P}$ operators:

$\displaystyle\hat{D}=\frac{i}{\sqrt{2}}(\hat{a}-\hat{a}^{\dagger}),\qquad\hat{P}=\sqrt{2}\sum_{k}\chi_{k}\hat{n}_{k},$
(21)

which summed up give the scalar electric field:

$\displaystyle\hat{E}=\frac{i}{\sqrt{2}}(\hat{a}-\hat{a}^{\dagger})+\sqrt{2}\sum_{k}\chi_{k}\hat{n}_{k}$
(22)

The field nature of the electric field can be recovered by reintroducing the
mode function.

We now want to remark that gauge transformations change the meaning of both
cavity and matter operators. Hence one should focus on physical observables
which are instead gauge invariant. For example, the electric field of the
cavity is expressed differently in the two gauges:

$\displaystyle\text{Coulomb}:\;\;\hat{\bm{E}}_{c}=i\omega_{c}A_{c}(\hat{a}-\hat{a}^{\dagger})\bm{f}_{c},$
(23)
$\displaystyle\text{Dipole}:\;\;\hat{\bm{E}}_{c}=i\omega_{c}A_{c}\left[(\hat{a}-\hat{a}^{\dagger})-2\sum_{k}\chi_{k}\hat{n}_{k}\right]\bm{f}_{c}.$
(24)

In this sense the no-go theorems [61, 62], which forbid a macroscopic coherent
occupation of the cavity in the Coulomb gauge, constrain the ground state
coherent occupation in the Dipole gauge as:

$\langle\hat{a}\rangle_{C}=0\;\;\Longrightarrow\;\;\langle\hat{a}\rangle_{D}=i\sum_{k}\chi_{k}\langle\hat{n}_{k}\rangle_{D},$
(25)

where $\langle\dots\rangle_{C(D)}$ denotes the expectation value in the
Coulomb (Dipole) gauge. We also remark that there is an extra freedom in the
choice of an overall constant in $\chi_{k}$, e.g., the origin of our system,
which guarantees us that we can always find a basis where
$\langle\hat{a}\rangle_{D}=0$ also in the Dipole gauge.

###  II.4 Role of gradients

In a clean system, gradients are fundamental to couple the cavity field to
electrons within the LLL. This is a consequence of the celebrated Kohn’s
theorem [36], whose corollary is that a uniform field can only couple to the
cyclotron mode [22], generating transitions among different Landau levels. In
view of that we now consider a further simplified cavity mode by considering a
linear expansion $f_{c}(x)=C_{0}+C_{1}x$ and leave the discussion of more
complicated modes for Sec. VI. For this particular shape of the mode function,
the LLL matrix elements are:

$\chi_{k}=eA_{c}C_{0}(kl_{B}^{2}-x_{0})+\frac{1}{2}eA_{c}C_{1}(kl_{B}^{2}-x_{0})^{2},$
(26)

with $x_{0}$ the origin for the integration in Eq. (10). In the Coulomb gauge,
the matrix elements of the coupling can be readily understood by the dressing
of $\hat{H}_{\mathrm{int}}$ and $\hat{H}_{0}$. For the interaction we have
that the four-body terms
$\hat{c}^{\dagger}_{k_{1}}\hat{c}^{\dagger}_{k_{2}}\hat{c}_{k_{3}}\hat{c}_{k_{4}}$
get dressed with the following phase factor:

$\displaystyle\exp\left[i(\hat{a}+\hat{a}^{\dagger})(\chi_{k_{1}}+\chi_{k_{2}}-\chi_{k_{3}}-\chi_{k_{4}})\right].$
(27)

However, from momentum conservation along $y$, we have that
$k_{1}+k_{2}=k_{3}+k_{4}$, and for the explicit expression of $\chi_{k}$ the
dressing phase can be written as:

$\displaystyle\exp\left[ieA_{c}C_{1}l_{B}^{4}(\hat{a}+\hat{a}^{\dagger})(k_{1}-k_{4})(k_{4}-k_{2})\right].$
(28)

At this point, we highlight two important properties:

  * •

First, as anticipated from the Kohn’s theorem [36], the constant part of the
electric field ($C_{0}$) is completely decoupled;

  * •

Second, an uniform gradient of the electric field generates a simple, uniform
coupling.

The latter follows by the fact that only differences in momenta, hence
relative distance on $x$, appear. Alternatively, sticking to the dipole gauge
formulation, one can see the constant part of the field ($C_{0}$) only couples
to conserved quantities, such as the number of electrons and momentum along
$y$. In contrast, the gradient ($C_{1}$) couples to the $xx$-component of the
quadrupole moment operator,
$\hat{Q}^{xx}\propto\sum_{kk^{\prime}}\int\differential^{2}r\,x^{2}\psi_{k}^{*}(\bm{r})\psi_{k^{\prime}}(\bm{r})\hat{c}^{\dagger}_{k}\hat{c}_{k^{\prime}}$,
associated to electrons in the LLL [37].

Another way to avoid Kohn’s restriction and actually couple the LLL to a
constant electric field is via the presence of an external potential for the
electrons. Either a confining potential or disorder will do the job, realizing
however quite different scenarios. A confining potential on $x$ will have a
dominant effect at the edges where its variation are stronger. More in
details, the origin of the coupling is an effective shift of the single-
particle orbitals $X_{k}\simeq
k\,l_{B}^{2}+\frac{l_{B}^{2}}{\omega_{B}}\partial_{x}W(x)$ that makes Eq. (27)
non-trivial even for a constant electric field. In the case of bulk disordered
systems, the single-particle potential depends on both coordinates, $W(x,y)$,
and the coupling arises directly from the dressing of $\hat{H}_{0}$ since
momentum along $y$ is no longer conserved. Although both are promising
options, we reserve their study to future works.

###  II.5 Hamiltonian

We now focus on the effect of uniform electric field gradients by choosing
$f_{c}(x)=C_{1}x$ and setting to zero the external potential $W=0$. Moreover,
we choose to work in the Dipole gauge where the Hamiltonian reads:

$\displaystyle\hat{H}$
$\displaystyle=\hat{H}_{\mathrm{int}}+\omega_{c}\hat{a}^{\dagger}\hat{a}+i\omega_{c}g(\hat{a}-\hat{a}^{\dagger})\sum_{k}\Big{(}\frac{k^{2}}{2}-\kappa_{0}\Big{)}\hat{n}_{k}$
$\displaystyle+\omega_{c}g^{2}\bigg{[}\sum_{k}\Big{(}\frac{k^{2}}{2}-\kappa_{0}\Big{)}\hat{n}_{k}\bigg{]}^{2},$
(29)

with $\hat{H}_{\mathrm{int}}$ the interaction Hamiltonian for the first
Haldane pseudopotentials (Eq. (4)), $\kappa_{0}$ the overall constant in the
definition of $\chi_{k}$ related to the choice of origin and
$g=eC_{1}A_{c}l_{B}^{2}$ a dimensionless coupling constant proportional to the
field gradient $C_{1}$. The energy scale of the interaction, the magnetic
length and the cavity frequency are all set to unity
$V_{0}=\omega_{c}=l_{B}=1$, unless specified otherwise.

We also introduce some relevant observables that we are going to use
throughout the paper. The expressions we are going to give are valid for the
Dipole gauge, where cavity operators are dressed rather than matter ones.
First the real-space charge density:

$\hat{n}(x)=\frac{1}{L_{y}}\sum_{k}\hat{n}_{k}|\phi_{0}(x-X_{k})|^{2},$ (30)

which is independent on position $y$ because of the translational invariance.
In order to get information about correlations in the $y$ direction we will
use the density-density correlations from the $G^{(2)}$ defined as:

$G^{(2)}(\bm{r}_{1},\bm{r}_{2})=\langle\hat{\psi}^{\dagger}(\bm{r}_{1})\hat{\psi}^{\dagger}(\bm{r}_{2})\hat{\psi}(\bm{r}_{2})\hat{\psi}(\bm{r}_{1})\rangle.$
(31)

Another important quantity is the guiding center density operator [26, 64]
which in second quantization reads:

$\hat{\bar{\rho}}(\bm{q})=\frac{1}{M}e^{-iq_{x}q_{y}/2}\sum_{k}e^{-iq_{x}k}\hat{c}^{\dagger}_{k}\hat{c}_{k+q_{y}}.$
(32)

From this we can define the connected guiding center static structure factor:

$S(\bm{q})=\bra{\mathrm{GS}}\delta\hat{\bar{\rho}}(-\bm{q})\delta\hat{\bar{\rho}}(\bm{q})\ket{\mathrm{GS}},$
(33)

with
$\delta\hat{\bar{\rho}}=\hat{\bar{\rho}}-\bra{\mathrm{GS}}\hat{\bar{\rho}}\ket{\mathrm{GS}}$
and its dynamical counterpart:

$\displaystyle S(\omega,\bm{q})=\frac{1}{M}$
$\displaystyle\sum_{n}|\bra{n}\delta\hat{\bar{\rho}}(\bm{q})\ket{\mathrm{GS}}|^{2}\delta_{\eta}(\omega-
E_{n}+E_{\mathrm{GS}})$ $\displaystyle-$
$\displaystyle\\{\omega\to-\omega\\},$ (34)

with $\ket{n}$ being a many-body eigenstate with energy $E_{n}$ and $\eta$
being a broadening parameter that should be sent to zero. Regarding the cavity
we will use its density of states as a way to probe polaritons at finite
frequency:

$\displaystyle D_{c}(\omega)=$
$\displaystyle\sum_{n}|\bra{n}\hat{a}^{\dagger}\ket{\mathrm{GS}}|^{2}\delta_{\eta}(\omega-
E_{n}+E_{\mathrm{GS}})$ $\displaystyle-$
$\displaystyle\sum_{n}|\bra{n}\hat{a}\ket{\mathrm{GS}}|^{2}\delta_{\eta}(\omega+E_{n}-E_{\mathrm{GS}}),$
(35)

which can be obtained from the retarded cavity Green’s function as
$D_{c}(\omega)=-\frac{1}{\pi}\mathrm{Im}\,G^{R}_{c}(\omega)$ with
$G^{R}_{c}(t)=i\theta(t)\langle[\hat{a}(t),\hat{a}^{\dagger}(0)]\rangle$. We
remark that in the ultra-strong coupling regime a precise calculation for the
outcome of transmission/reflection experiments should also take into account
anomalous correlations [65, 66] and use the gauge invariant electric field
rather than gauge dependent cavity operator $\hat{a}$ [67]. We leave these
refinements for a future work.

###  II.6 Numerical methods

In order to study the strongly-coupled light-matter system in Eq. (II.5), we
perform DMRG simulations for the ground state and a combination of TDVP and
exact diagonalization (ED) for spectral functions. DMRG methods have been
extensively used in the context of FQH systems to find in an unbiased way the
ground state of microscopic Hamiltonians [38, 68, 69]. The cylinder geometry
in particular allow for a very direct mapping of the LLL spanned by a single
quantum number $k$ to a quasi-1D chain with long range interactions. In the
case of the first Haldane pseudopotential the range is finite and depends on
the circumference of the cylinder $O(L_{y})$. The price to pay in order to use
a 1D MPS ansatz is twofold: (i) The MPO representation of the interaction
Hamiltonian is going to require a large bond dimension; and (ii) the MPS bond
dimension will need to grow exponentially with $L_{y}$.

Figure 2: Cartoon for the hybrid MPS ansatz. The cavity photon is placed at
the first site of the MPS, while the rest of the MPS sites represent the LLL
wavefunctions $\psi_{k}$.

####  II.6.1 Hybrid MPS ansatz

Here we introduce a hybrid cavity-matter finite MPS ansatz where the photon is
placed at the beginning of the MPS (Figure 2), also used in Refs [45, 44, 43,
46]. Empirically we observed that this MPS ansatz is still efficient enough in
representing the non-local correlations of the cavity mode, provided that the
bond dimension $m$ is large enough. The light-matter interactions are also
easy to represent as an MPO due to their infinite range nature [70]. Here the
choice of the Dipole gauge is actually important to avoid the dressing of the
4-body operators in the only matter part which can become quite expensive,
differently from the dressing of 2-body operators [43]. Moreover we make an
explicit use of the freedom in the choice the overall constant in the light-
matter matrix elements $\kappa_{0}$ (Eq. (II.5)) such that the constraint
given in Eq. (25) gives zero coherent occupation of the photon
$\langle\hat{a}\rangle_{D}=0$. This guarantees a good convergence with the
truncation of the cavity Hilbert space at $d_{\mathrm{ph}}=64$. Note that the
total number of electrons and total momentum along $y$ are still good quantum
numbers that we conserve in our simulations. We limit the bond dimension of
the MPS up to $m=1600$ which allows us to keep the truncation error always
below $10^{-6}$.

Apart from ground state properties via DMRG, we also focus on excited states
and dynamical properties with different methods. Regarding the excited states,
it is possible to directly get good results from the local effective
Hamiltonians constructed during DMRG runs, as discussed in Ref. [71]. In
particular local targeting of the excited states has been found to be quite
accurate for critical system in 1D [71], owing the success to the delocalized
nature of the low lying spectrum. As discussed more in depth in Sec. V.2 we
find that this method gives good qualitative results and is even able to
capture mixed light-matter polariton states. There we further study dynamical
properties via time evolution with TDVP (two site updates) or directly using
Lanczos methods in ED [72]. For these TDVP runs we limit the bond dimension of
the MPS to $m=200$ which still guarantees good convergence for small
circumferences at a reduced computational cost.

Interestingly, the TDVP algorithm can also be used to implement change of
gauges via the unitary $\hat{U}$. In particular we divide $\hat{U}$ in many
steps and apply them sequentially as commonly done in TDVP evolutions. The
“Hamiltonian” of this gauge change is:

$\hat{h}_{\mathrm{gauge}}=(\hat{a}+\hat{a}^{\dagger})\sum_{k}\chi_{k}\hat{n}_{k},$
(36)

so that:

$\hat{U}=\exp(-i\hat{h}_{\mathrm{gauge}}),$ (37)

and, in this units, the evolution time is $\tau=1$. The fact that the MPS
representation of the many-body ground state in a different gauge remains
efficiently compressible (i.e., small enough bond dimension) is not guaranteed
a priori and has to be checked. We find this to be case in the FQH phase.
During the change of gauge we keep the bond dimension of the MPS constant.

##  III Topological order in cavity

\begin{overpic}[width=130.08731pt]{flux} \put(0.0,60.0){(a)} \end{overpic}
\begin{overpic}[width=130.08731pt]{sigma_16_30.pdf} \put(25.0,20.0){(b)}
\end{overpic} \begin{overpic}[width=130.08731pt]{sigmaexp_16_30.pdf}
\put(25.0,75.0){(c)} \end{overpic}

Figure 3: (a) Sketch of the flux insertion; as a unit of time $\tau$ passes a
unit of flux $\Phi_{0}$ is inserted resulting in relative shift in position of
the cylinder along $x$. (b) Inverse resistivity in the bulk $x_{0}\in[-2,2]$
as measured by the density (Eq. (44)) with a shaded green area and quantized
value $\rho_{xy}^{-1}=1/3$ as a dashed red line. At strong cavity field
gradients $g$ the quantization is lost. (c) Exponential decay from the edge of
density modulations $\hat{n}(x)$ around the bulk quantized value. $x=0$ is the
center of the cylinder while $x=20$ is already outside of it. Obtained via
DMRG at system size $L_{y}=16$ and $N_{e}=30$.

The presence of a genuine non-local degree of freedom makes the present
setting not immediately classifiable within the framework of topological
phases of matter. In the context of cavity mediated topology, a lot of
attention has been dedicated to regimes where the cavity degree of freedom can
be integrated out, both in the quantum materials context [11, 10, 21, 73] and
in cold atom set-ups [74, 75, 76, 77]. Integrating out the non-local mode
generally produces effective long-range interactions which simplify the
picture of a mixed cavity-matter system and require the inspection of a
matter-only model. This neglects by construction light-matter entanglement and
give a direct interpretation of cavity mediated topology in terms of
“standard” topology.

Here, we are interested in the opposite situation, where light-matter
entanglement cannot be neglected - and, as we show below, does carry key
signatures of topological order.

Few works [44, 78, 79, 80] have been recently investigating the questions
above in the context of Symmetry Protected Topological (SPT) phases. In
particular in Ref. [44] pointed out that, in the case of Majorana fermions,
respecting the symmetry which protects the topology is fundamental. The FQH
effect instead belongs to a fundamentally different set of topological states
which display topological order - that is, where order is intrinsically
related to emergent gauge theories and entanglement. Addressing the interplay
of topological order and quantum light is thus a completely distinct challenge
with respect to the above mentioned SPTs.

As a paradigm of FQH, we are going to study the effect of the non-local cavity
degree of freedom on the topological properties of a $\nu=1/3$ Laughlin state.
In particular we focus on two markers, first on the Hall resistivity (Sec.
III.1) and second on the entanglement properties (Sec. III.2). They represent
two key aspects of topological order: quantization of transport properties,
and fingerprints in the entanglement structure of a state. Once the stability
of the FQH phase is understood, we further discuss (Sec. III.3) the effect of
cavity fluctuations on the emergent geometry of the FQH state.

###  III.1 Hall resistivity via flux insertion

A key feature of the FQH effect is the fractionally quantized transverse
resistivity $\rho_{xy}$. An easy way to probe this quantity numerically on the
cylinder geometry is via the so-called adiabatic flux insertion [51], sketched
in Fig. 3(a).

Let us now consider the adiabatic insertion of a single magnetic flux quanta
$\Phi_{0}=2\pi\hbar/e$ in the cylinder over a time $\tau$. This process can be
described by a uniform time-dependent vector potential
$\bm{A}_{p}(t)=\frac{\Phi_{0}}{L_{y}}\frac{t}{\tau}\bm{u}_{y}$, so that
$\Phi(t)=\Phi_{0}t/\tau$. The vector potential $\bm{A}_{p}(t)$ in turns
generate an electric field
$\bm{E}_{p}=-\partial_{t}\bm{A}_{p}=\frac{\Phi_{0}}{\tau L_{y}}\bm{u}_{y}$,
which is constant in time and directed towards the $y$ direction. Hence, we
can find the Hall resistivity by calculating the current
$\hat{J}_{x}=\hat{\bm{J}}\cdot\bm{u}_{x}$ flowing transversely to the electric
field $\bm{E}_{p}$. Since our system is homogeneous in $y$ but in general not
along $x$ the result is a spatial dependent resistivity:

$\rho_{xy}(x)=\frac{E_{y}}{e\langle\hat{J}_{x}(x)\rangle}.$ (38)

For the FQHE the bulk value is expected to be quantized as
$\rho_{xy}=\frac{1}{\nu}\frac{h}{e^{2}}$ with $\nu=1/3$ in this work.

In the adiabatic limit $\tau\rightarrow\infty$ and assuming the existence of a
many-body gap, we can focus on the ground state of the instantaneous total
Hamiltonian $\hat{H}(\Phi)$ which in principle depends on $\Phi$. Now we know
that a constant vector potential $\bm{A}_{p}$ does not couple to the LLL.
Indeed, the effect of a flux $\Phi$ is to change the $y$-momentum quantization
of the single particle orbitals $\psi_{n,k}$ (Sec. II.1) and consequently
their position:

$k_{\Phi}=\frac{2\pi}{L_{y}}m_{k}+\frac{2\pi}{L_{y}}\frac{\Phi}{\Phi_{0}},\qquad
X_{k_{\Phi}}=k_{\Phi}\,l_{B}^{2}$ (39)

with $m_{k}\in\mathbb{Z}$. This means that the projection $\mathcal{P}_{\Phi}$
to the LLL depends on $\Phi$. This is not an issue as long as we focus on
adiabatic processes. Given Eq. (39), we can now inspect how the many-body
light-matter Hamiltonian $\hat{H}(\Phi)$ looks like. The electron-electron
interaction clearly remains unchanged as it only depends on difference of
momenta. The light-matter coupling, for this purpose, is more conveniently
formulated within the Coulomb gauge formulation (Sec. II.3). Here the
interaction is only controlled by difference in momenta
$k_{\Phi}-k_{\Phi}^{\prime}$ and hence the full Hamiltonian $\hat{H}(\Phi)$
remains unchanged. As a key consequence we have that the ground state
wavefunction in second quantization $\ket{\mathrm{GS}_{\Phi}}$, hence fixing
an orbital basis, will be the same:

$\hat{H}(\Phi)\equiv\hat{H}(\Phi=0)\;\;\Rightarrow\;\;\ket{\mathrm{GS}_{\Phi}}\equiv\ket{\mathrm{GS}_{\Phi=0}}.$
(40)

It is important to stress that in Eq. (40), we have not used the equality
sign. The reasons is that the Hilbert spaces in which the two sides of the
equations are defined are not, strictly speaking, the same: they refer to
different LLL projection $\hat{\mathcal{P}}_{\Phi}$. In order to perform a
meaningful comparison, we can consider relevant physical observables such as
the charge density $\hat{n}(x)$, which is expressed as:

$\displaystyle\hat{n}(x)=\frac{1}{L_{y}}\sum_{k_{\Phi}}\hat{n}_{k_{\Phi}}|\phi_{0}(x-X_{k_{\Phi}})|^{2},$
(41)

and can be evaluated at different $\Phi$ by just using the $\Phi=0$ solution
with a change in the single particle orbitals $\phi_{0}(x-X_{k_{\Phi}})$.

In order to calculate the current $\langle\hat{J}_{x}\rangle$ we use the
continuity equation:

$\bm{\nabla}\cdot\langle\hat{\bm{J}}\rangle=\partial_{t}\langle\hat{n}\rangle,$
(42)

where $\hat{\bm{J}}$ is the current operator. Given our finite cylinder
geometry we can integrate both sides of the equation above in a region
$\mathcal{V}=\\{(x,y)\,\mathrm{s.t.}\ x<x_{0}\\}$ to get:

$\langle\hat{J}_{x}(x_{0})\rangle=\partial_{t}\int_{-\infty}^{x_{0}}\differential
x\langle\hat{n}(x)\rangle_{\Phi(t)}$ (43)

with $\hat{J}_{x}(x_{0})$ the current density along $x$ at position $x_{0}$.
Because of translational invariance on $y$ the current and the density do not
depend on $y$ making the integration over $y$ trivial. Using Eq. (41) and the
ramp protocol $\Phi(t)\propto t$ we can express the local transverse
resistivity in terms of the static density:

$\rho_{xy}(x_{0})=\frac{1}{2\pi\langle\hat{n}(x_{0})\rangle}\frac{h}{e^{2}},$
(44)

with $h/e^{2}$ being the Von Klitzing constant and the factor $2\pi$ is the
area occupied by a single-particle state (in units of $l_{B}^{2}$). The above
equation directly links the bulk density to the fractional Hall response of
the system. In particular for a topologically ordered state in the class of
the Laughlin $1/3$ one expects the bulk density to be constant
$2\pi\langle\hat{n}(x_{0})\rangle=1/3$.

In Fig. 3(b) we show the bulk resistivity ($x_{0}\in[-2,2]$) as a function of
cavity field gradients $g$ for a specific system size ($L_{y}=16$,
$N_{e}=30$). The resistivity is quantized up to exponential corrections even
at relatively large values of the cavity field gradient $g\simeq 2$. The
nature of the ground state in the regime of non-quantized Hall response will
be better characterized in Sec IV. In Fig. 3(c) we depict the deviations of
the density from its bulk value when approaching an edge of the cylinder
($x=0$ is the center of the system while $x=20$ is outside of it). For all
depicted couplings in the FQH phase, the corrections decay exponentially in
the bulk with a correlation length $l_{\mathrm{edge}}$ which increase with
$g$.

It has been recently argued [22] that the finite lifetime of the cavity mode
can give a correction to the quantized transverse conductivity at temperature
$T=0$ in the IQH regime. We note that at the many-body level the flux
insertion argument can be adapted to the case of weak cavity losses (see
Appendix A). The key physical insight is that the steady state of the whole
system (cavity+matter) is at thermal equilibrium with the bath [81], hence the
ground state results regarding the Hall resistivity are expected to hold
provided that the photonic bath is at a small enough temperature. It would be
interesting to check, in the spirit of Ref. [22], what finite temperature
corrections are. We leave this to future work.

###  III.2 Entanglement spectrum

\begin{overpic}[width=127.9171pt]{entspectrum_20_34_L.pdf}
\put(20.0,65.0){(a)}
\end{overpic}\begin{overpic}[width=127.9171pt]{entspectrum_20_34.pdf}
\put(20.0,20.0){(b)}
\end{overpic}\begin{overpic}[width=155.66676pt]{entspectrumC_20_34.pdf}
\put(17.0,17.0){(c)} \end{overpic}

Figure 4: Entanglement spectra obtained via DMRG for the asymmetric
bipartition of Eq. (47) on system size $L_{y}=20$ and $N_{e}=34$. The charge
sector of the shown spectra is $N_{e}/2=17$ while the momentum along $y$ gives
the x-axis. In panel (a) the state is pure Laughlin ($g=0$) while in panel
(b,c) we have a finite light-matter coupling $g=0.2$ in Dipole and Coulomb
gauge respectively. The color represent the number of photons in each Schmidt
state and the colorbar is the same for all three panels

The entanglement spectrum at bipartition of a topologically ordered state can
be used to detect topological order [33, 82, 38, 83]. In particular, one
expect to find information about the edge theory of the topological state
under consideration. This procedure, dubbed entanglement spectroscopy, is
widely used as a theoretical tool and has also been also proposed as an
experimental protocol to detect topology in cold atom systems [83]. The deep
roots of this topology-entanglement connection lie in a very general results
in relativistic quantum field theory, the Bisognano-Wichmann theorem [84, 85],
which dictates closed functional form expressions for the entanglement (or
modular) Hamiltonian, and explain the Li-Haldane result [86]. A natural
question to ask is, what is left about this topology footprint on entanglement
in the presence of quantum light.

Before discussing our hybrid cavity-matter setting, we review some general
concepts. Let us define the pure state $\ket{\Psi}$ of the full system and the
density matrix $\hat{\rho}_{A}$ of a subsystem $A$ as
$\hat{\rho}_{A}=\Tr_{B}\left[\ket{\Psi}\bra{\Psi}\right]$ with $B$ being the
rest of the system. In general, we can write:

$\hat{\rho}_{A}=\exp\left(-\hat{H}_{ee}\right)=\exp\left(-\sum_{{q},i}\xi_{\\{q\\},i}\ket{\phi_{\\{q\\},i}}\bra{\phi_{\\{q\\},i}}\right),$
(45)

where $\hat{H}_{ee}$ is the entanglement Hamiltonian, $\xi_{\\{q\\},i}$ the
entanglement energies, $\ket{\phi_{\\{q\\},i}}$ the Schmidt vectors
corresponding to the bipartition, and $(\\{q\\},i)$ an index tuple labeling
the quantum numbers $\\{q\\}$ and the Schmidt state $i$. Following the Li-
Haldane conjecture [33, 34], the entanglement spectrum for a bipartition in
the bulk must follow, at low energies, the Hamiltonian of the edge. For
Laughlin states, the edge theory is a chiral Luttinger liquid ($\chi$LL) [87,
88] which, once the $U(1)$ charge sector is fixed, gives a specific
fingerprint in terms of degeneracies at each total momentum:

$(d_{0},d_{1},d_{2},d_{3},d_{4},d_{5},\dots)=(1,1,2,3,5,7,\dots),$ (46)

with $d_{k}$ being the degeneracy at momentum quantum number $k$. At finite
sizes, the degeneracies are usually broken but a gap still separate a
universal low energy part from a non-universal part of the entanglement
spectrum [34]. This remark that this also happens for energy spectra at
physical edges [89] and not only for entanglement spectra in bulk
bipartitions.

In the case of a cavity embedded systems there is no clear notion of pure
state bipartition as the non-local bosonic mode cannot be “divided” in two. As
already done in [44, 45], one needs to define asymmetric bipartitions where
the cavity mode resides on one side. A possible choice is the following:

$\hat{\rho}_{X_{k}>0}=\Tr_{x<0;c}\left[\ket{\Psi}\bra{\Psi}\right]$ (47)

where $\ket{\Psi}$ is the many-body ground state, $\hat{\rho}_{X_{k}>0}$ is
the reduced density matrix for electrons in orbitals $k$ with $X_{k}>0$ and
$\Tr_{X_{k}<0;c}$ is the trace over electrons at $X_{k}<0$ and the cavity mode
$c$. One can, however, always recover the notion of an only matter bipartition
by considering the electronic density matrix
$\hat{\rho}_{el}=\Tr_{c}[\ket{\Psi}\bra{\Psi}]$. Taking its bipartition will
give as a result the same density matrix of the asymmetric bipartition with
the cavity in Eq. (47).

In light of the discussion about different possible representations of the
cavity-matter system (Sec. II.3), we want to stress that (not unexpectedly)
the entanglement spectrum is not a gauge invariant quantity. Indeed, it can
change under global unitary transformations $\hat{U}$ which implement the
change of gauge, since a global change of basis can change the reduced density
matrix of subsystems. However, as discussed in Sec. II.6, we can easily change
the gauge in which a state $\ket{\Psi}$ is represented by applying the unitary
$\hat{U}$ and just check whether the entanglement spectrum displays features
that are stable against the change of gauge.

####  III.2.1 Entanglement spectrum bands and polariton entanglement gap

In Fig. 4 we show the DMRG results for the entanglement spectrum of the
asymmetric bipartition in Eq. (47) at $g=0$ (a) and at finite $g=0.2$ in the
Dipole (b) and Coulomb (c) gauges. In particular, we fix the number of
particles of the bipartition to be $N_{e}/2$ and look at momentum quantum
number $K_{y}$. At finite light-matter coupling $g$ the entanglement spectrum
still show the $\chi$LL counting, but the higher energy part clearly changes.
In order to understand the difference between entanglement eigenvectors, we
color the dots based on the number of photons in the respective Schmidt state:

$n^{ph}_{\\{q\\},i}=\bra{\phi_{\\{q\\},i}}\hat{a}^{\dagger}\hat{a}\ket{\phi_{\\{q\\},i}},$
(48)

where the Schmidt states are readily available from the total MPS. This
highlights a very informative pattern. The $\chi$LL counting is repeated for
number of photons roughly equals to integers. We empirically find for each of
these branches:

$n^{ph}_{k,j}\simeq j,\qquad\xi_{k,j}\simeq\epsilon_{k}+j\Delta_{pol},$ (49)

with $\epsilon_{k}$ being a size dependent non-universal dispersion and
$j=0,1,2,\dots$ being an integer. The quantity $\Delta_{pol}$, which we call
polariton entanglement gap, controls the separation between different sectors
of the $\chi$LL with different number of photons and needs to be finite to
preserve the $\chi$LL structure.

Figure 5: Polariton entanglement gap extrated from DMRG entanglement spectra
in the middle of the cylinder for different number of particles $N_{e}$ in the
Dipole gauge (circles) and Coulomb gauge (stars) as a function of the
collective coupling $g\sqrt{N_{e}}$. $L_{y}=20$

In Fig. 5 we study the polariton entanglement gap dependence with system size.
We show its dependence as a function of a re-scaled collective coupling
$g\sqrt{N_{e}}$ for different number of particles $N_{e}$. The perfect
collapse highlights the collective nature of polariton excitations, i.e., they
are controlled by the collective coupling $g\sqrt{N_{e}}$. By showing the
entanglement gap for both gauges it is evident that this is a gauge dependent
quantity, nonetheless both gauges reveal the same collective behavior.
Moreover, we find that the value of $\Delta_{pol}$ does not depend on $L_{y}$
(not shown).

The nature of this gapped polaritonic excitation will be clarified in Sec. V,
where we show that a strong hybridization between a collective emergent
electronic mode, the $q\simeq 0$ part of the magnetoroton spectrum, and the
cavity is taking place. An important consequence of the collective coupling is
that taking thermodynamic limit $N_{e}\rightarrow\infty$ at fixed $g$ breaks
the $\chi$LL counting and the topological order of the state. The stability of
the FQH phase at finite $g$ then needs to be understood in a mesoscopic sense
– another important scale $g\sqrt{N_{e}}$ is controlling the many-body gap of
the FQH phase and hence its topological order.

###  III.3 Cavity control of FQH geometry

Another important property of a FQH state which have received a lot of
interest in recent years is its intrinsic geometry or metric [29, 30, 31, 90,
91]. This controls many of its ground state correlations and its long-
wavelength gapped excitations quanta, in analogy with gravitational theories
that have been dubbed as “gravitons” [30, 31, 32]. Here we focus on ground
state leaving for Sec. V the discussion about excitations.

A key ground state footprint of non-trivial geometry can be found in guiding-
center correlations [92, 91], in particular via the guiding-center structure
factor $S(\bm{q})$ \- see Eq. (33) - at small momenta. For a gapped FQH state
it is possible to show [92, 91] that:

$S(\bm{q})\simeq
S_{4}^{xx}q_{x}^{4}+2S_{4}^{xy}q_{x}^{2}q_{y}^{2}+S_{4}^{yy}q_{y}^{4}.$ (50)

In the case of unbroken rotational invariance, we further have
$S_{4}^{xx}=S_{4}^{xy}=S_{4}^{yy}=s_{4}$, with $s_{4}$ satisfying the Haldane
bound $s_{4}\geq(1-\nu)/24\nu$ saturated by the Laughlin state [92, 93].
Anisotropic interactions and/or anisotropic LLs [29] induce an intrinsic
anisotropic geometry on the FQH liquid.

\begin{overpic}[width=433.62pt]{S4xx.pdf} \end{overpic} Figure 6: Long
wavelength static structure factor parameter $S^{xx}_{4}$ as a function of the
collective coupling $g\sqrt{N_{e}}$ for different number of electrons $N_{e}$
at fixed cylinder circumference $L_{y}=20$. The black dashed line represent
the Haldane bound $s_{4}=(1-\nu)/24\nu$ valid for isotropic interactions and
saturated by the Laughlin at $g=0$ in the thermodinamic limit. Inset: Static
structure factor as a function of momenta along $y$ for $g=0$ (red) and $g=1$
(blue) showing enhancement of the small wavevector part. The dashed red line
represent the thermodinamic limit behaviour $S(q_{y})=s_{4}\;q_{y}^{4}$ of the
Laughlin state.

In Fig. 6 we show the effect of cavity fluctuations on the long-wavelength
properties of $S(\bm{q})$. The behaviour for momenta along $x$ is still
quartic and the proportionality factor $S_{4}^{xx}$ is plotted against the
collective coupling $g\sqrt{N_{e}}$. To extract it we directly look at the
definition of $S(q_{x})$ and expand for small $q_{x}$:

$\displaystyle S_{4}^{xx}=\lim_{q_{x}\rightarrow
0}\frac{1}{4!}\partial_{q_{x}}^{4}S(q_{x})=\frac{1}{4M}\sum_{k,k^{\prime}}k^{2}k^{\prime
2}\langle\delta\hat{n}_{k}\delta\hat{n}_{k^{\prime}}\rangle,$ (51)

with $\delta\hat{n}_{k}=\hat{n}_{k}-\langle\hat{n}_{k}\rangle$. Finite size
effect with both $N_{e}$ and $L_{y}$ are expected and are signaled by the
discrepancy with the known value $s_{4}=(1-\nu)/24\nu$ at $g=0$ shown as a
dashed black line. Nonetheless the finite $g$ reduction is consistent and
likely hold in the thermodynamic limit. The extraction of all three parameters
in Eq. (50) is hindered by strong finite size effects along $y$. This is
highlighted in the inset of Fig. 6 where we show the full dependence of
$S(q_{y})$ which cannot capture the $q_{y}^{4}$ dependence. Qualitatively we
still see that correlations along $y$ are enhanced at finite $g$, as opposed
to those on $x$.

\begin{overpic}[width=132.25092pt]{density_16_30.pdf} \put(20.0,75.0){(a)}
\end{overpic}\begin{overpic}[width=147.4292pt]{g2_fqh_20.pdf}
\put(20.0,70.0){(c)}
\end{overpic}\begin{overpic}[width=132.25092pt]{Sord_20.pdf}
\put(20.0,75.0){(e)}
\end{overpic}\begin{overpic}[width=132.25092pt]{density_20_30.pdf}
\put(20.0,75.0){(b)}
\end{overpic}\begin{overpic}[width=147.4292pt]{g2_fqh_80.pdf}
\put(20.0,70.0){(d)}
\end{overpic}\begin{overpic}[width=132.25092pt]{cavityord_20.pdf}
\put(20.0,75.0){(f)} \par\end{overpic}

Figure 7: Left panels (a,b) show the orbital occupancies from Laughlin state
at $g=0$ (red) to stripe phase at $g=3$ (blue) for two circumferences of the
cylinder $L_{y}=16$ (a) and $L_{y}=20$ (b) at fixed $N_{e}=30$. Note that at
fixed number of particles the size of the system along $x$ reduces. Center
panels (c,d) show the densisty-density correlations $G^{(2)}(0,\bm{r})$ in the
two phases, at $g=1$ for the FQH phase in panel (c) and at $g=4$ for the
stripe phase in panel (d). In the FQH phase (panel (c)) we also compare the
shape of the correlation hole between $g=1$ (white line) and the Laughlin
$g=0$ (blue line) by tracking the local maximum of the $G^{(2)}$. For both
panels (c,d) $L_{y}=20$ and $N_{e}=34$. Right panels (e,f) show the behaviour
of two quantities across the transition for different $N_{e}$ and fixed
$L_{y}=20$. Panel (e) show the peak height of the static structure factor as
defined in the main text and serve as an order parameter. Panel (f) show the
electric field fluctuations (full lines) and the contribution coming from the
polarization (dashed lines).

Another key signature of a distorted metric is the shape of the correlation
hole of the $G_{2}$, shown in Fig. 7(c) for $g=1$ (white line) and compared
with the circular one of the $g=0$ Laughlin case (blue line). As already
noticed for other anysotropic FQH model wavefunctions [94], we also find that
the short distance behaviour of the $G_{2}(r)$ changes from $r^{6}$ to $r^{2}$
going from isotropic ($g=0$) to anisotropic cases ($g>0$). We also note that
the effect at long wavelength shown in Fig. 6 is greater in magnitude with
respect to the reshaping of the correlation hole. As a rule of thumb, we
expect the latter to be more sensible to short wavelength properties
(electron-electron interactions) while the former to long-wavelength (cavity-
mediated interactions). We suggest this to be related to the cavity induced
modifications of the magnetoroton dispersion (Sec. V) which are collectively
enhanced ($g\sqrt{N_{e}}$) only around the graviton part $q\to 0$ while at
finite $q$ seems only to see the single-particle coupling ($g$). This is
intuitively related to the fact that a uniform field gradient couples
collectively to a uniform quadrupolar excitation, the graviton ($q\to 0$), and
not to quadrupolar excitations at finite momenta, the rest of the magnetoroton
dispersion ($q$ finite).

##  IV Cavity-induced sliding Tomonaga-Luttinger liquid

In this section, we analyze the nature of the large coupling regime $g\gtrsim
1$. In particular we first provide numerical evidence for a cavity mediated
instability to a stripe phase with strong density modulation along the $x$
direction (Sec. IV.1). Then we argue on the basis of a mean-field argument how
the squeezed geometry analyzed in Sec III.3 is stabilizing the FQH phase,
pushing the transition to higher values of the coupling $g$ (Sec. IV.2).
Finally, we complete the understanding of the strong coupling phase as a
sliding Tomonaga-Luttinger liquid and discuss the effect of fluctuations (Sec.
IV.3).

###  IV.1 Numerical results

In Fig. 7 we present a sample of the numerical results obtained via DMRG
simulations as discussed in Sec. II.6. In order to get immediate insight on
the state of the system we can look at the orbital density $\hat{n}_{k}$ in
the left panels (a) and (b). The Laughlin state ($g=0$ red line) shows a
quantized bulk density $\hat{n}_{k}$ with modulations only close to the edge
due to the open boundaries. A strong bulk modulation instead appears around
$g\simeq 2-3$ for both system sizes $L_{y}=16$ and $L_{y}=20$. The modulation
in orbital space follows a different pattern for the two system sizes. In
particular, we have that the number of electron per peak are $n_{S}=3$ and
$n_{S}=4$ for system system sizes $L_{y}=16$ and $L_{y}=20$ respectively. In
order to understand this we recall that the distance between two neighboring
orbitals decreases as $\Delta X_{k}=2\pi/L_{y}$. Hence to keep the distance
between the peaks $\lambda_{S}$ independent of $L_{y}$, the number of particle
per peak $n_{S}$ must increase. This constrains the generic behavior of
$\lambda_{S}$ to be

$\lambda_{S}=\frac{2\pi}{L_{y}}3n_{s}(L_{y}),$ (52)

where we expect $n_{s}(L_{y})\propto L_{y}$ in the thermodynamic limit.

In order to check the 2D stripe nature of the phase we show the density-
density correlator $G^{(2)}(\bm{r}_{1},\bm{r}_{2})$ in the center panels (c)
and (d) as defined in Eq. (31). In the FQH phase ($g=1$ panel (c)) we find the
characteristic correlation hole at short distances and a constant value at
long distances as one should expect from a liquid state. Note that, as
discussed in Sec. III.3, the anisotropic nature of the cavity mode reflects
into an anisotropic correlation hole. To highlight this, two lines are shown
which tracks the maximum of the $G^{(2)}$, one for the Laughlin at $g=0$ (blue
line) and one for the $g=1$ case (white line) whose $G^{(2)}$ is actually
plotted. In the strong coupling regime instead ($g=4$ panel (d)) there is
clear ordering on $x$ and absence of ordering on $y$, confirming the
interpretation of the phase as stripe phase. Weak density modulations on $y$
are present within the stripe, likely due to finite size effects and not to a
true crystalline order in 2D.

In the right panels (e,f) we show two quantities related to the matter and
cavity components. For the electrons we consider the maximum of the
disconnected static structure factor $S_{uc}(\bm{q})$ at finite momenta where
it is expected to scale with $N_{e}$ in a charge density wave phase and not in
a liquid phase. $S_{uc}(\bm{q})$ is defined as in $S(\bm{q})$ (Eq. (33)) but
with the full guiding center density operator $\hat{\bar{\rho}}(\bm{q})$
replacing fluctuations $\delta\hat{\bar{\rho}}(\bm{q})$ in the definition.
Panel (e) shows the normalized finite-momenta peak height of the disconnected
static structure factor $\max S_{uc}(\bm{q})/N_{e}$ which is found around
$\bm{q}\simeq(1.75,0)$. At fixed $L_{y}$ the transition point seems to shift
towards higher values of $g$ with increasing $N_{e}$. However, a more careful
investigation is hindered by commensurability effects on $y$ which likely
renormalize the energy cost of the stripes for finite $L_{y}$. We note that
considering the thermodynamic limit in the $x$ direction with an infinite MPS
is impossible with the present cavity-matter structure.

For the cavity mode, we consider the scalar electric field $\hat{E}$ (Eq.
(22)). Since its expectation value vanishes identically because of Eq. (25),
we look at its fluctuations in panel (f) as a function of $g$ for different
system sizes (full lines). There we also show how, in the Dipole gauge, most
of the fluctuations come from the $\hat{P}^{2}$ term (dashed lines). The terms
$\hat{D}\hat{P}$ and $\hat{D}^{2}$ are still macroscopic even though they
almost cancel each other. Fluctuations peak in the vicinity of the value of
$g$ where strong stripe order forms and decrease afterwards. This indicate a
crucial role of vacuum electric field gradient fluctuations in driving the
instability.

###  IV.2 Photon mean-field argument

In order to get an insight of the effect of the field gradients we perform a
mean-field decoupling of the photon and matter degrees of freedom in the
Dipole gauge. Due to the linear nature of the coupling term, the mean-field
state generally reads:

$\ket{\Psi}=\ket{\alpha}\ket{\psi},$ (53)

with $\ket{\psi}$ being an electronic wavefunction defined on the LLL and
$\ket{\alpha}$ being a coherent state. Minimizing the mean-field state energy
over the coherent state value gives:

$\alpha=i\sum_{k}\chi_{k}\bra{\psi}\hat{n}_{k}\ket{\psi}.$ (54)

In order to find the best mean-field ansatz, one should then take the photon
mean-field Hamiltonian for the electrons:

$\hat{H}_{\mathrm{PMF}}=\hat{H}_{\mathrm{int}}+\omega_{c}\left[\sum_{k}\chi_{k}\big{(}\hat{n}_{k}-\bra{\psi}\hat{n}_{k}\ket{\psi}\big{)}\right]^{2},$
(55)

and find the ground state $\ket{\psi}$ with a self-consistent procedure [43].

This minimization still requires the solution of a complicated many-body
Hamiltonian. However we can now clearly see two different terms competing, the
first is the electron-electron interaction and the second is due to effective
cavity mediated interactions. The latter has a very specific form: e.g., when
calculated on the ground state $\ket{\psi}$, it is the variance of the
polarization operator $\hat{P}$. This has two important consequences. First it
is a positive number and second it is, in general, an extensive quantity,
being proportional to $N_{e}$. These are not specific to the FQH set-up, but a
more general property of Dipole gauge Hamiltonians where the so-called self
polarization term is present. For other models where only the linear coupling
is present, e.g., the Dicke model, the all-to-all nature of the mode leads to
collective enhanced cavity-mediated interactions which contribute
superextensively, and grow as $g^{2}N_{e}^{2}$. For this reason one can
usually work with a collective coupling $g\sqrt{N_{e}}$. Here instead we
clearly see that the cavity mediated interactions are controlled by the so-
called single particle coupling constant $g$.

We can now discuss two limiting cases: $g=0$ and $g\rightarrow\infty$. In the
first case, we recover a pure Laughlin state, eigenstate of the first part of
the Hamiltonian. In the second case instead, there will be a massive
degeneracy of states with no fluctuations of the polarization operator
$\hat{P}$ lifted only by the interactions $\hat{H}_{\mathrm{int}}$. All
product states in the orbital basis will, for example, have no orbital density
fluctuations and hence give identically zero contributions to the second term.
Among these states we can write down stripe states with index $n$:

$\ket{S_{n}}=\ket{\\{0_{n}1_{n}0_{n}\\}},$ (56)

where $0_{n}$ ($1_{n}$) indicates the repetition of zero (one) electron in $n$
consecutive orbitals and the curly bracket indicate the repetition of such
unit cell. Note that these states satisfy the bulk filling factor $1/3$ by
construction. The electron-electron interaction energy contribution of these
states is also relatively simple to calculate. The mean-field energy of the
stripe state $\ket{S_{n}}$ is:

$\displaystyle
E_{S_{n}}=\bra{S_{n}}\hat{H}_{\mathrm{int}}\ket{S_{n}}=\sum_{k_{1},k_{2}\in
K_{\mathrm{occ}}}2V_{k_{1},k_{2},k_{2},k_{1}},$ (57)

where $K_{\mathrm{occ}}$ is the set of orbitals which are occupied in
$\ket{S_{n}}$, and $V_{k_{1},k_{2},k_{2},k_{1}}$ are matrix elements given by
Eq. (II.1). By changing the circumference of the cylinder $L_{y}$ the energy
per particle will also change, as shown in Fig. 8. For each number of particle
per stripe $n$ there is an $L_{y}$ that minimizes the mean-field energy and
this increases with $n$.

Figure 8: Interaction energy of stripe state $\ket{S_{n}}$ in the photon mean-
field approximation with $n$ going from $n=1$ (dark blue) to $n=6$ (yellow) as
a function of cylinder circumference $L_{y}$ size. Dashed vertical lines show
that for $L_{y}=20$ and $L_{y}=16$ used in Fig. 7 the states with $n=4$ and
$n=3$ are favoured. Inset: approximate behavior of $E_{S_{n}}$ as a function
of $1/k_{\mathrm{F}}\simeq L_{y}/3n$ in the thermodynamic limit. The two main
contributions to the interaction energy, same stripe (dotted) and neighboring
stripes (dashed), sum up to $E_{S_{n}}$ (red full line). The minimum is around
$k_{\mathrm{F}}^{-1}\simeq 1.6$ ($\lambda_{S}\simeq 5$).

This behavior can be well understood by working in the thermodynamic limit
$L_{y}\to\infty$. In this setting, the stripe state can be essentially viewed
as an array of filled Fermi seas, each with Fermi momentum $k_{\mathrm{F}}=\pi
n/L_{y}$, and whose centers are separated in momentum by $\Delta
k=6k_{\mathrm{F}}$. The circumference $L_{y}$ controls the range of the
electron-electron interactions $V_{k_{1},k_{2},k_{2},k_{1}}$ on the momentum
basis via an exponential $\propto e^{-(k_{1}-k_{2})^{2}/2}$. Thus, if we start
from small $L_{y}$, i.e., large $k_{\mathrm{F}}$ for some $n$ fixed, the most
prominent contribution to $E_{S_{n}}$ must come from interactions between
electrons living on the same stripe. Then, by increasing $L_{y}$, interactions
among electrons on different stripes are expected to become stronger. We thus
expand the energy $E_{S_{n}}$ as the sum

$E_{S_{n}}=E_{S_{n}}^{(0)}+E_{S_{n}}^{(1)}+\cdots,$ (58)

where $E_{S_{n}}^{(0)}$ and $E_{S_{n}}^{(1)}$ respectively represent the
energy contributions of electrons on the same and nearest neighbor stripes.
Taking the continuum limit $\sum_{k}\to\frac{L_{y}}{2\pi}\int\differential k$,
we estimate these two energy contributions (details on appendix B) to be

$E_{S_{n}}^{(0)}=2N_{e}\Big{[}\erf(\sqrt{2}k_{\mathrm{F}})+\frac{2e^{-2k_{\mathrm{F}}^{2}}-2}{\sqrt{2\pi}k_{\mathrm{F}}}\Big{]},$
(59)

and

$\displaystyle E_{S_{n}}^{(1)}$
$\displaystyle=4N_{e}\Big{[}\erf(\sqrt{8}k_{\mathrm{F}})-3\erf(\sqrt{18}k_{\mathrm{F}})+2\erf(\sqrt{32}k_{\mathrm{F}})$
$\displaystyle+\frac{e^{-8k_{\mathrm{F}}^{2}}-2e^{-18k_{\mathrm{F}}^{2}}+e^{-32k_{\mathrm{F}}^{2}}}{\sqrt{2\pi}k_{\mathrm{F}}}\Big{]},$
(60)

where $\erf(x)$ is the Gauss error function. In Fig. 8(b) we plot the
approximate form of $E_{S_{n}}$ as a function of $1/k_{\mathrm{F}}$. Within
this approximation, we find the minimum lies at $k_{\mathrm{F,{min}}}\simeq
0.62$, with a corresponding energy gap $E_{S_{n}}/N_{e}\simeq 0.25$. This
means the number of electrons per stripe $n_{S}\approx L_{y}/5$ is only
dictated by $L_{y}$ in the thermodynamic limit as discussed above, see Eq.
(52).

On the other hand we have that the Laughlin state $\ket{\Psi_{L}}$ is an
eigenstate of $\hat{H}_{\mathrm{int}}$ with zero energy. This means the
variational energy of the Laughlin state depends only on the light-induced
interaction:

$E_{L}=\omega_{c}\sum_{kk^{\prime}}\chi_{k}\chi_{k^{\prime}}\bra{\Psi_{L}}\delta\hat{n}_{k}\delta\hat{n}_{k^{\prime}}\ket{\Psi_{L}},$
(61)

where we have defined
$\delta\hat{n}_{k}=\hat{n}_{k}-\bra{\Psi_{L}}\hat{n}_{k}\ket{\Psi_{L}}$. Given
that $\chi_{k}\propto k^{2}$, it is possible to compute this energy
contribution via the long-wavelength behaviour of the static structure factor
$S(\bm{q})$. From Eq. (51), we observe that $E_{L}$ is directly tied to the
$S_{4}^{xx}$ component, so when we send $N_{e}\to\infty$, the mean-field
energy scales as

$E_{L}/N_{e}\approx S_{4}^{xx}\omega_{c}g^{2}/\nu,$ (62)

where $\nu=1/3$.

The outcome is that a transition to the stripe phase must take place once the
energy of the Laughlin state $E_{L}$ becomes equal or larger than the energy
$E_{S_{n}}$. Replacing the value for the unperturbed Laughlin state,
$S_{4}^{xx}=(1-\nu)/24\nu$, we find the magnitude of the critical point
$g=g_{\star}$ goes as $\omega_{c}g_{\star}^{2}\approx 1$. This is far below
the observed values in the numerics, which happen close to $g\simeq 3$. One
key reason behind this difference is the renormalization of $S_{4}^{xx}$ as a
function of the light-matter interaction as shown in Fig. 6. Given
$S_{4}^{xx}$ becomes smaller, the energy of the real ground state scales
slower than $g^{2}$, and the transition is pushed towards a higher value of
$g$. In particular, by using the numerical data for $S_{4}^{xx}$ as a function
of $g$, we have checked the true ground state energy agrees well with the
formula in Eq. (62). The strong resilience of the FQH liquid to cavity
fluctuations is then to be attributed to the change in its geometry,
interpreted as a hidden variational parameter [29].

###  IV.3 Field theory for the sliding Tomonaga-Luttinger liquid phase

To further clarify the nature of the sliding Tomonaga-Luttinger liquid phase
[95, 96, 97], we build a set of low-energy excitations on top of our mean-
field ground state. In the large light-matter coupling limit, we assume the
energy penalty to introduce $\delta n_{k}$ fluctuations is so large we can
only spare to build excitation with definite density occupation $n_{k}=0,1$.
This is a great simplification that essentially allow us to treat the LLL
electrons as an array of Tomonaga-Luttinger liquids. This state is naturally
gapped along $x$, but remain gapless along the $y$ direction, and can be
viewed as an archetype of a smectic metal state [95], a state with zero shear
modulus [98]. The gapless modes take the form of particle-hole excitations
near the Fermi points of each stripe $k_{\sigma}=\pm
k_{\mathrm{F}}+6k_{\mathrm{F}}\sigma$, where $\sigma=0,\pm 1,\pm 2,\dots$
denotes the stripe index. To verify this, we first consider a particle
excitation with momentum $k=k_{\mathrm{F}}+q$, assuming $q\ll k_{\mathrm{F}}$
so the excitation lies close to the right Fermi point of the stripe
$\sigma=0$. In this limit, similar to the analysis carried out in Appendix B,
we approximate the energy cost of adding one electron by the integral:

$\displaystyle\varepsilon(k_{\mathrm{F}}+q)$
$\displaystyle\simeq\sqrt{\frac{2}{\pi}}\int_{-k_{\mathrm{F}}}^{k_{\mathrm{F}}}\differential
k(k_{\mathrm{F}}-k+q)^{2}e^{-\tfrac{1}{2}(k_{\mathrm{F}}-k+q)^{2}},$ (63)

where we integrate over the nearest stripe only, throwing out energy
contributions from distant stripes that are suppressed by the exponential
factor. Evaluating the integral in the limit $q\to 0$, we find the linear
dispersion relation
$\varepsilon(k_{\mathrm{F}}+q)\approx\varepsilon(k_{\mathrm{F}})+v_{\mathrm{F}}q+\cdots$,
where
$v_{\mathrm{F}}=\sqrt{\frac{2}{\pi}}4k_{\mathrm{F}}^{2}e^{-2k_{\mathrm{F}}^{2}}$
plays the role of the Fermi velocity.

It thus follows that the particle-hole excitation
$c_{k+q}^{\dagger}c_{k}\ket{S_{n}}$ in the vicinity of the Fermi level has a
dispersion relation of the form

$\omega(q)\approx v_{\mathrm{F}}|q|+\cdots.$ (64)

This behavior is compatible with a sliding Tomonaga-Luttinger liquid (sTLL)
model description of the stripe phase, characterized by the following low-
energy Hamiltonian:

$H_{\mathrm{sTLL}}=\sum_{\sigma}\frac{v}{2}\int\differential
y\Big{[}K(\partial_{y}\theta_{\sigma})^{2}+\frac{1}{K}(\partial_{y}\phi_{\sigma})^{2}\Big{]},$
(65)

where the boson fields $\phi_{\sigma}$ represent phase fluctuations of the
charge density along each stripe $\sigma$, denoted as
$\delta\hat{n}_{\sigma}=-\partial_{y}\phi_{\sigma}/\sqrt{\pi}$, and the
Luttinger parameter $K$ embodies generic intra-stripe interactions. The
effective model has a large (scaling with $L_{x}$) number of additional $U(1)$
symmetries, corresponding to separate number conservation on each stripe. This
is reflected in the invariance of the sTLL Hamiltonian under discrete sliding
transformations [98] of the form

$\phi_{\sigma}(y)\to\phi_{\sigma}(y)+f_{\sigma},$ (66)

where $f_{\sigma}$ is an arbitrary function over discrete values $\sigma$.

Let us now reintroduce light-matter interactions. We proceed
phenomenologically, adding the cavity term $\delta
H_{c}=\omega_{c}\hat{a}^{\dagger}\hat{a}$, and then performing the dipole-
gauge shift:
$\hat{a}\to\hat{\mathfrak{a}}=\hat{a}-i\sum_{\sigma}\chi_{\sigma}\int\differential
y\,\delta\hat{n}_{\sigma}(y)$, where $\chi_{\sigma}$ takes into account field
variations along the $x$ direction. Thus all cavity-related couplings are
summarized into $\delta
H_{c}=\omega_{c}\hat{\mathfrak{a}}^{\dagger}\hat{\mathfrak{a}}$, which we add
to the sTLL Hamiltonian Eq. (65). However, as it couples individually to the
zero modes of the stripes, $\delta H_{c}$ commutes with $H_{\mathrm{sTLL}}$
and this sort of light-matter interactions are harmless to the stripe phase.

The seemingly innocuous $\delta H_{c}$ interaction actually plays a role when
we try to recover the FQH phase from electron-electron interactions. As in the
traditional coupled-wire construction of FQH states [99], we observe that the
transition to the $\nu=1/m$ FQH state is driven by the strong-coupling limit
of the inter-wire interaction $\delta
H_{\mathrm{IWC}}=-\kappa\sum_{\sigma}\int\differential
y\,\cos\\{\sqrt{\pi}[m(\phi_{\sigma}+\phi_{\sigma+1})-\theta_{\sigma}+\theta_{\sigma+1}]\\}$.
This term describes a correlated electron-hopping mechanism among neighboring
stripes that preserves both total number of electrons and the total momentum
along the $y$ direction. The introduction of $\delta H_{\mathrm{IWC}}$ thus
breaks the enlarged global symmetry of the $H_{\mathrm{sTLL}}$, allowing the
number of electrons in each stripe to fluctuate. We hence note that $\delta
H_{\mathrm{IWC}}$ competes with $\delta H_{c}$ in the presence of field
gradients, meaning that the positive-definite cavity term $\delta H_{c}$
penalizes changes in the number of electrons in the stripes provoked by
$\delta H_{\mathrm{IWC}}$.

To complete the analysis of symmetry allowed perturbations, we comment on the
effect of interactions that preserve the number of electrons inside each
stripe. These sort of interactions are not quenched by the coupling
$\chi_{\sigma}$, and take the form of forward scattering interactions $\delta
H_{\mathrm{FWS}}$ and CDW interactions $\delta H_{\mathrm{CDW}}$ among
different stripes. For the short-ranged, first Haldane pseudopotential these
corrections are expected to be small, but they are likely to play a greater
role for long-ranged potentials, such as the Coulomb interaction. Forward
scattering interactions can be generically described as

$\delta H_{\mathrm{FWS}}=\sum_{\sigma_{1}\sigma_{2}}\int\differential
y\,\partial_{y}\bm{\varphi}_{\sigma_{1}}^{T}\cdot\bm{V}_{\sigma_{1}\sigma_{2}}\cdot\partial_{y}\bm{\varphi}_{\sigma_{2}},$
(67)

where $\bm{\varphi}_{\sigma}=(\phi_{\sigma},\theta_{\sigma})^{T}$, and the
$2\times 2$ matrices
$\bm{V}_{\sigma_{1}\sigma_{2}}=\bm{V}_{|\sigma_{1}-\sigma_{2}|}$ control the
strength of the interactions. As discussed in Refs. [97, 96], the addition of
$\delta H_{\mathrm{FWS}}$ does not spoil the sliding symmetry of the decoupled
theory, causing a Gaussian deformation of the model $H_{\mathrm{sTLL}}$ [96].
In contrast, CDW interactions, such as $\delta
H_{\mathrm{CDW}}=\eta_{\mathrm{CDW}}\sum_{\sigma}\int\differential
y\,\cos[\sqrt{4\pi}(\phi_{\sigma}-\phi_{\sigma+1})]$, favor phase locking
between different stripes. In particular, when $\eta_{\mathrm{CDW}}$ is
relevant, the sTLL state flows to an anisotropic Wigner crystal, exhibiting
$2k_{\mathrm{F}}$ density modulations along the stripes [97]. We note further
corrections to the light-matter coupling $\chi_{\sigma}$ disfavor this
scenario, by introducing a coupling that penalizes changing the relative
numbers of left and right fermions in each stripe. That is, if we assume field
variations are appreciable across distances $\Delta x\approx
2k_{\mathrm{F}}l_{B}^{2}$, the light-matter model should include the
additional coupling
$\hat{\mathfrak{a}}\to\hat{\mathfrak{a}}^{\prime}=\hat{\mathfrak{a}}-i\sum_{\sigma}\tilde{\chi}_{\sigma}\int\differential
y\,\hat{\jmath}_{\sigma}(y)$, with
$\hat{\jmath}_{\sigma}=-\partial_{y}\theta_{\sigma}/\sqrt{\pi}$ and
$\tilde{\chi}_{\sigma}$ describing the field variation along each stripe. Due
to the short-range nature of the Haldane pseudopotential, even a moderately
large $\tilde{\chi}_{\sigma}$ may seem enough to prevent the stripe to
crystallize. We stress however the detailed interplay among these competing
operators lies beyond our current mean-field ansatz.

##  V Spectral properties

In this section we discuss the effect of the cavity mode on the bulk spectral
properties of the FQH liquid. We first give an introduction to the neutral
excitations in absence of the cavity mode (Sec. V.1), e.g., the magnetoroton
spectrum, using the so-called single-mode approximation [26] for the
magnetorotons due to Girvin, MacDonald and Platzman. In Sec. V.2 we present
numerical results which map out the phenomenology of the low-lying excited
state spectrum in presence of the cavity mode. Finally, we provide a simple
effective polariton model that captures the essential physics of hybridized
cavity-matter excitations, the graviton-polaritons, and test its predictions
against numerics (Sec. V.3).

###  V.1 Magnetoroton spectrum

\begin{overpic}[width=130.08731pt]{mrmode2} \put(10.0,73.0){(a)}
\end{overpic}\begin{overpic}[width=130.08731pt]{Sqdyn_10_10_0.pdf}
\put(10.0,73.0){(b)}
\end{overpic}\begin{overpic}[width=130.08731pt]{Sqdyn_10_10_2.pdf}
\put(10.0,73.0){(c)}
\end{overpic}\begin{overpic}[width=142.65869pt]{gaps_16_21_nol.pdf}
\put(15.0,67.0){(d)}
\end{overpic}\begin{overpic}[width=142.65869pt]{gaps_16_30_nol.pdf}
\put(15.0,67.0){(e)}
\end{overpic}\begin{overpic}[width=126.62025pt]{eff_spectrum_16_30.pdf}
\put(15.0,76.0){(f)} \end{overpic}

Figure 9: (a) Magnetoroton dispersion as calculated from the SMA (red) with a
sketch of the two-magnetoroton continuum. The SMA magnetoroton gap
$\Delta_{\mathrm{min}}\simeq 0.6$ occurs at a finite wavevector
$q_{\mathrm{min}}\simeq 1.7$ and a gap is also present at small wavevectors
$\Delta_{0}\simeq 1.5$ (b,c) Dynamical structure factor as a function of
momentum along $x$ at $g=0$ (b) and $g=0.8$ (c). The red dashed line is a
guide to the eye that signals the energy of the first excited state at $g=0$.
$L_{y}=10$ and $N_{e}=10$ in (b) and (c). (d),(e) Low energy spectrum in the
$K_{y}=0$ sector as a function of $g$ calculated via DMRG for different number
of particles $N_{e}=21$ (d) and $N_{e}=30$ (e) at the same $L_{y}=16$. The
colorbar represent the number of photons in each state and the vertical grey
dashed lines are a guide to the eye for different regimes. The orange dashed
line represent the lower polariton frequency obtained from the bare graviton-
polariton model Eq. (V.3). (f) Comparison between the exact DMRG results and
the eigenfrequencies of the effective model in Eq. (V.3) with coupling
parameter $\gamma_{q}$ shown in the inset. The orange dashed line represent
again the lower polariton frequency from the bare graviton polariton model Eq.
(V.3). $L_{y}=16$ in (d),(e),(f). The cavity frequency is always
$\omega_{c}=1.$

Magnetorotons are the lowest-energy excitations above the gapped FQH ground
state. They manifest as charge density modulations within the LLL that arise
from the bound state of a quasi-electron and a quasi-hole [100, 101]. Crucial
to the stability of the FQH state, the magnetoroton spectrum has been subject
of intense study [26, 101, 64, 52, 102]. Their dispersion relation exhibits a
pronounced minima at finite wavevector $q\sim 1/l_{B}$, which descends below
the two-particle continuum, see Fig. 9(a). The minima is known as the
magnetoroton gap, and is the precursor of ordered phases such as the Wigner
crystal and stripes [52].

More recently the $q\rightarrow 0$ magnetorotons have been associated to the
fluctuations of the quantum Hall geometry tensor [29, 37]. These emergent
spin-2 gravitons (gapped and non-relativistic) have a strong quadrupolar
component with a vanishing dipole moment and have a preferred chirality, exact
for model wavefunctions. We remark that chiral FQH gravitons have also been
recently detected in inelastic scattering experiments with circularly
polarized light [28].

We now briefly review a simple physical picture of the magnetoroton mode,
provided by Girvin, MacDonald and Platzman (GMP) with the single-mode
approximation (SMA) [26]. Note that here single mode does not refer to our
cavity model. The SMA is a variational ansatz that describes the magnetoroton
excitations as long wavelength charge modulations on top of the liquid ground
state of uniform density. With this picture in mind, GMP built a set of
excited states as:

$\ket{\bm{q}}=\hat{\bar{\rho}}_{\bm{q}}\ket{\Psi_{L}},$ (68)

where $\hat{\bar{\rho}}_{\bm{q}}$ is the guiding center density operator in
the LLL defined in Eq. (32). Using the static structure factor $S(\bm{q})$ to
normalize the variational state, the excitation energy then becomes fixed by
the ratio:

$\Delta_{\mathrm{SMA}}(\bm{q})=\frac{F(\bm{q})}{S(\bm{q})},\qquad
F(\bm{q})=\bra{\bm{q}}\hat{H}_{\mathrm{int}}-E_{0}\ket{\bm{q}},$ (69)

where $F(\bm{q})$ is the oscillator strength. The LLL density operator
$\hat{\rho}_{\bm{q}}$ obeys the Lie algebra:
$\commutator{\hat{\rho}_{\bm{q}}}{\hat{\rho}_{\bm{q}}^{\prime}}=2i\sin(\frac{1}{2}\bm{q}\times\bm{q}^{\prime})\hat{\rho}_{\bm{q}+\bm{q}^{\prime}}$,
named after GMP [26, 64], which allows to express the oscillator strength
$F(\bm{q})$ as a sole function of $S(\bm{q})$ and the interaction potential
[26].

As shown in Fig. 9(a), the GMP ansatz captures the essential features of the
magnetoroton mode, reproducing a fully gapped mode with a mininum at finite
momenta. It is worth noting that the SMA overestimates the magnetoroton mode
energy gap as the wave vector is increased. This is a well-known shortcoming
of the ansatz [26], given that the density operator also couples to high-
energy states containing a greater number of quasi-electron and quasi-hole
pairs [64].

In our notations, the SMA predicts a gap minimum of
$\Delta_{\mathrm{min}}\simeq 0.6$ close to the momenta $q_{\mathrm{min}}\simeq
1.4$. This is to be contrasted with the ED calculation of the dynamical
structure factor shown in Fig. 9(b). We observe the actual gap is smaller,
around $\Delta_{\mathrm{min}}\simeq 0.4$ with a corresponding wave vector
$q_{\mathrm{min}}\simeq 1.7$. Long-wavelength magnetorotons hide inside the
two-quasi-particle continuum as the SMA predicts $\Delta_{0}\simeq 1.5$. While
the SMA fails to predict a quantitatively correct magnetorton gap, it is
believed to capture the graviton energy $\Delta_{0}$. It is less clear however
up to what extent the two particle continuum damps the pristine magnetoroton
dispersion found in the SMA.

###  V.2 Numerical results

We now investigate numerically the effect of the cavity mode on the low-lying
FQH bulk spectrum. Note that because of the choice of “hard” boundary the edge
states are gapped and only bulk excitations remain. In order to track the
magnetoroton dispersion we look at the dynamical structure factor
$S(\bm{q},\omega)$ (Eq. (II.5)). The dynamical structure factor probes the
response of the system to density excitation at a certain frequency $\omega$.
In the following we will focus on the response to modulation along $x$ so on
$S(q_{x},\omega)$. In Fig. 9 we show this quantity for $g=0$ (b) and for
$g=0.6$ (c) at a small system size ($L_{y}=10$ and $N_{e}=10$) accessible via
ED. Note that due to open boundary condition on $x$ the momenta is not a good
quantum number and excitations are expected to spread over a finite region of
$q_{x}$. The cavity is clearly lowering the magnetoroton gap with no
particular modification of the wavevector at which the minimum is found
$q_{\mathrm{min}}\simeq 1.75$. Note that this quantity is close to the
ordering wavevector of the stripe phase in the mean-field limit
$2\pi/\lambda_{S}\simeq 1.86$ thus indicating the finite momentum magnetoroton
minimum as a precurson of the instability.

In order to access larger system sizes we look directly at the excited state
spectrum by targeting excited states via local effective Hamiltonians
constructed during DMRG calculations [71] as explained in Section II.6.
Moreover by looking at the excited states we can gain more information also on
the cavity degree of freedom. It is important to remark that this method
allows us to probe excitations in the $K_{y}=0$ sector only. In figure 9(d,e)
we show the low-lying excited states as a function of $g$ for number of
particles: $N_{e}=21$ (d) and $N_{e}=30$ (e) at $L_{y}=16$. The colors of the
lines represent the strength of the matrix element
$|\bra{n}\hat{a}^{\dagger}\ket{0}|$ which enters in the cavity density of
states $D_{c}(\omega)$ (see Eq. (II.5)), helping us to spot the polaritonic
character of the states. The orange dashed line is an analytical prediction
for the lower polariton that will be discussed in Sec. V.3. We can distinguish
roughly 4 different regimes, delineated with vertical grey dashed lines in
Fig. 9. These are the following:

  1. 1.

Near $g=0$, the low lying states are part of the magnetoroton dispersion
around the $q_{\mathrm{min}}$ and they start at around
$\Delta_{\mathrm{min}}\simeq 0.4$ which is the bulk neutral gap of the
Laughlin state. All these states remain with approximately zero photons above
the ground state and hence have a strong electronic component.

  2. 2.

At a value of the coupling $g_{P}$, a polariton state (i.e., a state with
strong photon component) comes down in energy from the higher bare cavity
frequency $\omega_{c}=1$ at $g=0.$ and becomes the first excited state. This
happens sooner for larger systems, for system sizes shown in Figure 9 we find
$g_{P}=0.5$ vs $g_{P}=0.65$ ($N_{e}=30$ vs $N_{e}=21$).

  3. 3.

close to the instability region, a magnetoroton state with strong electronic
character becomes again the lowest energy state and almost closes the gap. The
gap reduces with bigger $N_{e}$ as well as the overall region shifts to higher
$g$.

  4. 4.

In the stripe phase a small gap, increasing with $g$, is present. Since we are
probing the $K_{y}=0$ sector of the sTLL phase, as discussed in Sec. IV.3, the
excitations will be gapped and hybridized with the cavity photons. Higher
excited states also carry a strong hybridization and correspond to multi-
photon transitions (not shown).

Note that having the polariton state lower in energy with respect to the
magnetoroton implies that the gap protecting the topological order from finite
temperatures will be the polariton gap, as pointed out in Ref. [22] for the
IQH case. We then want to remark that there seems to be a difference between
the dependence with system size of the polariton state and of the magnetoroton
dispersion. While the latter is empirically controlled by $g$, the polariton
state sees the collective enhancement thus it’s controlled by $g\sqrt{N_{e}}$.
In the next subsection we propose an effective model to explain this feature
and the nature of the polariton state.

###  V.3 Effective model

\begin{overpic}[width=135.72458pt]{Ac_eff.pdf} \put(0.0,60.0){(a)}
\end{overpic}\begin{overpic}[width=135.72458pt]{Ac_ed.pdf} \put(0.0,60.0){(b)}
\end{overpic}\begin{overpic}[width=120.98299pt]{Ac_tdvp.pdf}
\put(-2.0,65.0){(c)} \end{overpic}

Figure 10: Cavity density of states $D_{c}(\omega)$ in the Dipole gauge with
different methods. The ED results (b) and effective model (a) show the avoided
crossing when changing the cavity frequency $\omega_{c}$, signalling the
strong coupling between the graviton mode and the cavity mode. Dashed red
lines mark the frequency range between the two-particle continuum and the
magnetoroton gap. In the effective model there is a residual small coupling to
the finite $q$ part of the magnetoroton while the ED results only highlight a
coupling to the two-particle continuum. In panel (c) we show TDVP results with
fixed cavity frequency $\omega_{c}=1.5$, almost resonant with the graviton,
and we change $N_{e}$ and $L_{y}$. By fitting with two lorenzians (dashed
lines) we extract the Rabi splittings between the two polariton resonances
(Table 1). System size $(N_{e},L_{y})$ in (a) is $(10,10)$, in (b) is
$(30,16)$, in (c) are described in the legend. The couplings for $(a,b,c)$ are
respectively $g=0.1,0.1/\sqrt{3},0.1$ . Broadening parameters $\eta=0.02$ in
(a,b) and $\eta=0.05$ in (c).

We introduce an effective model to describe the coupling between magnetorotons
and the cavity field. Treating the magnetorotons as free bosons, we express
the matter Hamiltonian as a collection of independent oscillators:
$H_{\mathrm{MR}}\approx\sum_{\bm{q}}\Delta(\bm{q})\hat{b}^{\dagger}_{\bm{q}}\hat{b}_{\bm{q}}$,
where $\hat{b}_{\bm{q}}$ represents the bosonic excitation of the magnetoroton
at momentum $\bm{q}$, and $\Delta(\bm{q})$ denotes the energy dispersion. For
the sake of simplicity we neglect interactions. In the original model the
coupling to the cavity is performed via the polarization operator
$\hat{P}=\frac{1}{2}\sum_{k}k^{2}\hat{n}_{k}$. For the effective model we
replace it with
$\hat{P}\to\sum_{q}\gamma_{q}(\hat{b}_{q}+\hat{b}_{q}^{\dagger})$, where
$\gamma_{q}$ represents the effective coupling governing the transition, and
the momenta $\bm{q}=(q,0)$ is restricted to the $x$ direction since the
electric field does not depend on $y$. The effective model Hamiltonian is then
expressed as

$\displaystyle\hat{H}_{\mathrm{eff}}$
$\displaystyle=\omega_{c}\hat{a}^{\dagger}\hat{a}+\sum_{q}\Delta(q)\hat{b}^{\dagger}_{q}\hat{b}_{q}+\omega_{c}g^{2}\Big{[}\sum_{q}\gamma_{q}(\hat{b}_{q}+\hat{b}^{\dagger}_{q})\Big{]}^{2}$
$\displaystyle+i\omega_{c}g(\hat{a}-\hat{a}^{\dagger})\sum_{q}\gamma_{q}(\hat{b}_{q}+\hat{b}^{\dagger}_{q}),$
(70)

where we factor out the $g$ coupling in order to facilitate the power
counting. The sum on $q$ should be sensitive to boundary conditions: in the
present case of an finite cylinder of width $L_{x}$ we will consider modes
with momentum $q=\pm(2j_{q}+1)\pi/L_{x}$ ($j_{q}=1,\dots,M/2$), whose
symmetric and antisymmetric combination form even and odd standing waves with
proper boundary conditions. Then, being Eq. (V.3) a quadratic bosonic
Hamiltonian, it can be easily solved via Bogoliubov-Hopfield transformations.

To draw a comparison with the numerical simulations of the actual FQH plus
cavity setup, we fix the effective parameters of our model by using the SMA.
The energy dispersion follows from the variational ansatz in Eq. (69) and is
sketched in Fig. 9(a). The light-matter interaction parameters $\gamma_{q}$
are obtained from the matrix elements of the polarization operator
$\hat{P}=\frac{1}{2}\sum_{k}k^{2}\hat{n}_{k}$ with respect to the Laughlin
ground state and the SMA excited states:

$\displaystyle\gamma_{q}=\frac{1}{\sqrt{MS(q)}}\sum_{k,k^{\prime}}\frac{1}{2}k^{2}e^{-ik^{\prime}q}\bra{\Psi_{L}}\delta\hat{n}_{k}\delta\hat{n}_{k^{\prime}}\ket{\Psi_{L}}.$
(71)

The matrix element $\gamma_{q}$ is shown in the inset of Fig. 9(f). We observe
it displays a prominent peak as $q\to 0$ and some smaller oscillations for
finite wave vectors. These have a period of roughly $2\pi/L_{x}$ and are
caused by the open cylinder geometry. The behaviour at $q\to 0$ can be
obtained from the long-wavelength expansion of the structure factor
$S(q)\simeq S_{4}^{xx}q^{4}$, which in the thermodynamic limit yields

$\lim_{q\to 0}\gamma_{q}=-\sqrt{N_{e}S_{4}^{xx}/\nu}\propto\sqrt{N_{e}}.$ (72)

The collective enhancement factor $\sqrt{N_{e}}$ is signalling that the
graviton is expected to be a good collective excitation able to couple to a
uniform field gradient.

We remark that in Eq. (72) there is no explicit dependence on $L_{y}$, just on
the parameter $S_{4}^{xx}$ which controls the long-wavelength correlations of
the FQH liquid. The behaviour at finite $q$ is a boundary effect and is also
the regime where the SMA should be taken with a grain of salt. The collective
enhancement of the $q\to 0$ mode only suggests that the effective model can be
further simplified to bare graviton-polariton model with a single collective
matter mode $\hat{b}_{q=0}$ plus the cavity mode $\hat{a}$:

$\displaystyle\hat{H}_{\text{eff},0}$
$\displaystyle=\omega_{c}\hat{a}^{\dagger}\hat{a}+\Delta_{0}\hat{b}_{0}^{\dagger}\hat{b}_{0}+i\omega_{c}g\gamma_{0}(\hat{a}-\hat{a}^{\dagger})(\hat{b}_{0}+\hat{b}_{0}^{\dagger})$
$\displaystyle+\omega_{c}g^{2}\gamma_{0}^{2}(\hat{b}_{0}+\hat{b}_{0}^{\dagger})^{2}$
(73)

with $\Delta_{0}$ the graviton energy and $\gamma_{0}$ taken from Eq. (72). By
means of a Bogoliubov-Hopfield transformation we can directly get the two
polariton energies resulting from Eq. (V.3):

$\omega_{P\pm}^{2}=\frac{1}{2}\left(\omega_{c}^{2}+\Delta_{0}^{2}+\Omega^{2}\pm\sqrt{(\omega_{c}+\Delta_{0}+\Omega)^{2}-4\omega_{c}^{2}\Delta_{0}^{2}}\right)$
(74)

where we have introduced the Rabi frequency:

$\Omega=2g\gamma_{0}\sqrt{\omega_{c}\Delta_{0}}.$ (75)

In Fig. 9(f) we compare the predictions of the effective models (orange lines)
with the low-lying energy spectrum obtained from DMRG simulations (blue
lines). We observe that effective model $\hat{H}_{\mathrm{eff}}$ successfully
captures the emergence of the polariton mode that comes down in energy as a
function of $g$. This energy closely follows the lower polariton energy
$\omega_{P_{-}}$ where only an effective $q=0$ mode is taken into account.
Small deviations could be explained by a finite $g$ renormalization of the
$\gamma_{0}$ parameter. In stark contrast, the effective model misses the gap
softening of other magnetoroton states which live at finite $q$. The gap
renormalization of the magnetoroton mode seems indeed very important at strong
coupling, where it goes even below the polariton mode and becomes the smallest
gap along the instability towards the sliding Tomonaga-Luttinger liquid phase,
as seen Fig. 9(d,e). In this respect we remark that even at $g=0$ the
effective model, being based on the SMA, does not capture the correct gap.
Interactions, both matter-matter and cavity-mediated, play a key role in the
renormalization of the gap and capturing them requires the treatment of the
full many-body problem beyond the effective model.

####  V.3.1 Spectroscopy of the graviton-polariton

Given the hybrid nature of the polariton excitations [103], it is possible to
have distinct simple spectroscopic signature of both modes signalling the
strong-coupling regime. To this end we show (Fig. 10) the cavity density of
states $D_{c}(\omega)$ as defined in Eq. (II.5). Here we compare the
prediction of the effective model (b) with ED (a) and TDVP (c) results. The ED
and effective model result indicates the dominant hybridization at small $g$
occurs at the energy scales of the graviton $\Delta_{0}$. In ED we also spot
subdominant couplings to other states inside the two-particle continuum which
are not captured in the effective model. The latter instead predicts a weight
on the finite $q$ part of the magnetoroton that in ED is not visible (region
between the red dashed lines). The predicted collective enhanchment of the
graviton hybridization is confirmed by the TDVP results. By fitting the
spectral function with two lorenzians we can extract the Rabi splitting
between the two polariton resonances which are reported in Table 1.

$(N_{e},L_{y})$ | $(15,10)$ | $(15,14)$ | $(30,14)$ | Effective model
---|---|---|---|---
$\Omega$ | 0.53 | 0.53 | 0.77 | $1.0\times\omega_{c}g\sqrt{N_{e}}$
$\Omega/\omega_{c}g\sqrt{N_{e}}$ | 0.91 | 0.91 | 0.94 | 1.0
Table 1: Rabi splittings between graviton-polaritons resonances obtained by
fitting TDVP results (Fig. 10(c)) compared with the effective model prediction
in the thermodynamic limit and assuming $\Delta_{0}=\omega_{c}=1.5$. Small
corrections due to small detunings are expected, as well as finite size
corrections with both $N_{e}$ and $L_{y}$ on $\Delta_{0}$ and $\Omega$.

According to the graviton-polariton model (Eq. (74)) and assuming a resonance
condition $\Delta_{0}=\omega_{c}=1.5$ the Rabi splitting at small enough
$g\sqrt{N_{e}}$ coincide with the Rabi frequency, and hence should increase
with $\sqrt{N_{e}}$. Indeed we find that $\Omega$ increases by a factor
$1.45\simeq\sqrt{2}$ when the number of particles $N_{e}$ is doubled and it
does not change with $L_{y}$, confirming $\gamma_{0}$ to be independent of it.
The precise value is also quite close to the graviton-polariton effective
model and the discrepancy gets smaller as the circumference $L_{y}$ is
increased.

##  VI Experimental discussion

In this section we discuss more in detail the implication of our theoretical
and numerical findings for relevant experimental conditions in solid state
systems.

##### Non-uniform gradients.—

The choice of studying uniform gradients has simplified our analysis so far as
it enabled to characterize a simpler uniform effect of the cavity mode.
However in realistic scenarios [16] this is not in general the case. In Fig.
11 we argue how most of our discussion remains qualitatively relevant upon the
introduction of non-uniform field gradients. We compare two of our main
results, the strong density modulations and the graviton-polaritons, in two
scenarios for the mode function $f_{c}(x)$: (i) a straight line (blue) and
(ii) a parabola (orange). The latter mimics the presence of stronger field
gradients near the plate of the LC resonator as detailed in Ref. [16]. Note
that we choose to compare two cases such that the “average” gradient is the
same, namely in the case (ii) the gradient is $0$ in the bulk and twice the
uniform value of case (i) at the edge.

Fig. 11(a) highlights that only the local value of the field gradient matters,
i.e. for strong gradients only at the edge only the latter will show this
feature. Instead in Fig. 11(b) we give evidence for the formation of the
graviton-polariton doublet also for the non-uniform case. Being a collective
excitation, the local values of the gradient are averaged and thus still show
the same collective enhancement. The bigger Rabi splitting observed in the
non-uniform case can be rationalized by noting that the effective coupling
constant is likely averaged as $g_{\mathrm{eff}}\simeq\sqrt{\int\differential
x\,g^{2}(x)}$ thus giving two slightly different Rabi splittings couplings in
the two cases.

\begin{overpic}[width=433.62pt]{nonuniform.pdf} \put(20.0,100.0){(a)}
\put(20.0,40.0){(b)} \end{overpic} Figure 11: Sample of numerical results
comparing uniform and non-uniform cavity electric field gradients. In panel
(a) we show DMRG results for the orbital occupation in a uniform gradient
(blue) and a non-uniform gradient (orange). The corresponding mode profiles
$f_{c}(x)$ are shown in the inset which correspond to a straight line
$f_{c}=gx$ (uniform) and a parabola $f_{c}=2gx^{2}/L_{x}$ (non-uniform). Note
that with this choice the non-uniform case has field gradients going form zero
(bulk) to twice the value of the uniform case (edge). In (a)
$N_{e}=30$,$L_{y}=16$ and $g=2.5$ so that the uniform case is unstable towards
a uniform modulation but the non-uniform case clearly show two different
behaviours in the bulk and at the edge. In panel (b) we show the cavity
density of states in the two cases obtained via TDVP which still show the
formation of graviton-polaritons with fitted rabi splittings $\Omega=0.53$ and
$\Omega=0.59$ for uniform and non-uniform case respecively. In (b) $N_{e}=15$,
$L_{y}=10$ and $g=0.1$.

We further note that the assumption of a fixed mode function $f_{c}$ is also
an approximation which implicitly derive from the single-mode restriction of
QED. Setting a precise limit of validity for this approximations is an open
problem for most of cavity QED set-ups which try to achieve non-perturbative
couplings and is beyond the scope of this work.

##### Energy scales.—

We now verify that typical energy scales of the electronic component
(interaction $V$) and the cavity (frequency $\omega_{c}$) can match. The
strength of interactions are controlled by the Coulomb energy
$E_{C}=e^{2}/4\pi\epsilon_{0}\epsilon_{r}l_{B}\simeq
56\,\mathrm{meV}\sqrt{B[\mathrm{T}]}/\epsilon_{r}$ which for a typical GaAs
quantum well ($n_{e}=10^{11}\,\mathrm{cm}^{-2}$ and $\epsilon_{r}=13$) at
filling $\nu=1/3$ ($l_{B}=\sqrt{\nu/2\pi n_{e}}\simeq 7\,\mathrm{nm}$ and
$B=\hbar c/el_{B}^{2}\simeq 12\,\mathrm{T}$) give $E_{C}\simeq
14\,\mathrm{meV}$. The energy of the collective modes is then a fraction of
the Coulomb energy, i.e., in Ref. [28] the graviton energy has been measured
as $\Delta_{0}\simeq 0.05E_{C}\simeq 0.65\,\mathrm{meV}$ for $\nu=1/3$. The
precise value will however depend on other experimental parameters such as the
well thickness and disorder strength [104]. On the other side, typical split-
ring resonator have similar frequencies $\omega_{c}\simeq 2\pi\times
0.1-1\mathrm{Thz}\simeq 0.4-4\,\mathrm{meV}$. Although the frequency regime
match we remark that, once the filling $\nu$ is fixed, there is very little
tunability of the ratio $\Delta_{0}/\omega_{c}$ needed for example to measure
the polariton anticrossing (Fig. 10). Changing $E_{C}$ via a change in
magnetic field $\delta B$ imply also a change in the filling $\delta\nu$ which
is expected destabilize many-body collective excitations [27, 28]. We suggest
that using graphene samples could resolve this issue by implemeting a gate-
tunable density $n_{e}$ [105] and thus achieving a tunable interaction energy
$E_{C}$ at constant filling $\nu$.

##### Coupling strength.—

The other important discussion is on the magnitude of the dimensionless
coupling constant $g$ used in this work, in particular its relation with the
neglected cavity losses. To do so we remind the coupling definition:

$g=eA_{c}l_{B}^{2}\partial_{x}f_{c}(x)$ (76)

where $f_{c}$ is the mode function and
$A_{c}=\sqrt{\hbar/(2\omega_{c}\epsilon_{0}V_{\mathrm{mode}})}$ the strength
of vacuum fluctuations which account for the mode confinement. We take as a
reference for our estimate the split-ring resonator used in Ref. [16]. We
extract an average electric field gradient of roughly
$\partial_{x}E^{x}_{c}=\omega_{c}A_{c}\partial_{x}f_{c}(x)\simeq
10^{5}\,\mathrm{V/m^{2}}$ in relevant sample area $S\simeq 40\mathrm{\mu
m}\times 200\mathrm{\mu m}$ and a cavity frequency $\omega_{c}=2\pi\times
140\,\mathrm{GHz}$. Considering again $n_{e}=10^{11}\,\mathrm{cm}^{-2}$ and
$\nu=1/3$ which give $l_{B}=7\,\mathrm{nm}$, we get:

$g\simeq\frac{10^{5}\,\mathrm{eV/m^{2}}}{2\pi\times
140\,\mathrm{GHz}}(7\,\mathrm{nm})^{2}\simeq 10^{-9}.$ (77)

The number of particle in the estimated sample area $S$ is $N_{e}=n_{e}S\simeq
8\times 10^{6}$, giving a rather small collective coupling
$g\sqrt{N_{e}}\simeq 3\times 10^{-6}$. Using Eq. (75) for the Rabi frequency
we would get:

$\frac{\Omega}{\omega_{c}}=2g\gamma_{0}\sqrt{N_{e}}\simeq 6\times 10^{-6}.$
(78)

Since quality factors of split ring resonators are rather small
$Q=\omega_{c}/\Gamma\simeq 10$ we find that current designs of split-ring
resonators fall in the weak coupling regime $\Omega/\omega_{c}<Q^{-1}$. In
particular the empirically observed resilience of FQH features in Ref. [16]
can be understood from the above estimate of the relevant Rabi frequency
related to gradients. Considering other type of resonators could be beneficial
as well as a more ad-hoc design for the resonators which enhances field
gradients. In this respect we remark that a strictly uniform gradient is not
necessary and also slowly varying gradients couple to the $q\simeq 0$
collective mode. Then we also mention that slightly greater couplings can be
also be achieved by looking at more dilute FQH liquids. For example reducing
the density of the quantum well gives $g\sqrt{N_{e}}\propto
l_{B}^{2}\sqrt{n_{e}}\propto 1/\sqrt{n_{e}}$ or focusing on higher fractions
like $\nu=n+\bar{\nu}$ (with $n$ integer and $\bar{\nu}<1$) gives
$g\sqrt{N_{e}^{\text{eff}}}\propto\nu\sqrt{N_{e}\bar{\nu}/\nu}$ with
$N_{e}^{\text{eff}}$ the electrons available in the last partially filled LL.

##### Cavity-mediated interactions.—

As a last remark we discuss experimental relevance of our findings regarding
ground state properties (Sec. IV). A realistic resonator will display non-
uniform gradients, usually stronger close to the metallic plates, and as such
drive instabilities only close to the edges. The single particle coupling $g$
which governs the stripe instability is estimated to be small when accounting
for only a single mode. The single mode model however likely underestimates
these effects and one would also need to take into consideration the effect of
the cavity on the rest of the electromagnetic continuum [56]. Without entering
into the details, we point out two different contributions that are neglected
in our modelling. The first is the presence of higher frequency LC resonances
or other plasmonic modes which and are expected to give contributions similar
to what has been discussed so far. The second important one is the screening
of Coulomb interactions [56] which we have neglected by considering a model
interaction, the first Haldane pseudopotential, independent from the cavity
setting. Changing the precise form of the only-matter interaction potential
could provide a nice handle on the isotropy of the quantum metric in the
underlying FQH liquid [29] as well on the magnetoroton dispersion or even
drive itself instabilities in higher LL [52]. In parallel, it would be also
interesting to study the effect of cavity-screened interactions on transport
in the Integer Quantum Hall regime where a loss of quantized conductivity has
been observed [16] and attributed to renormalized single-particle properties
[21]. A more careful analysis on these effects is left for future
investigation.

##  VII Conclusions and outlook

In this work we have developed a theoretical framework, readily generalizable
to experiments, in which it is possible to understand key effects of confined
electromagnetic modes on lowest Landau level physics. Importantly, together
with the proposed QED model, we have discussed a tensor network architecture
which allow for an in depth study of ground state and spectral properties for
large number of particles.

This has allowed us to uncover many aspects of the interplay between quantum
light and Fractional Quantum Hall physics. Here we summarize and discuss the
main points:

  * •

Constant cavity fields are completely decoupled from intra-Landau level
correlations [36]. The coupling strength of the quantum light-matter
interaction is directly proportional to field gradients and quadrupole moment
of the electron fluid. Ref. [16] reported a higher resilience of FQH physics
to cavity vacuum fluctuations with respect to the IQH effect. This can be
traced back to the smaller value of the effective Rabi frequency (Eq. (78))
coming from relatively weak field gradients.

  * •

The quantized Hall response of the FQH liquid is stable against non-local
cavity fluctuations. This theoretical results for FQH states is parallel to
recent precise measurements of quantized Hall resistivity in the integer
quantum Hall regime [19].

  * •

A new entanglement structure in hybrid quantum Hall states is found. The role
of quantum light is to introduce a “band” of chiral Luttinger liquids
multiplets, each with an approximately quantized photon number and separated
by a finite entanglement polariton gap.

  * •

We predict the formation of graviton-polariton modes, describing the
hybridization of the long-wavelength magnetorotons with cavity photons. The
Rabi frequency is collectively enhanced and is directly proportional to the
quadrupole matrix element of the gravitons. We confirm numerically the
presence of a polariton doublet in the cavity density of states, smoking gun
of the strong coupling regime.

  * •

Cavity vacuum fluctuations can squeeze the FQH geometry [29]. The latter can
be thought as a hidden variational parameter for the FQH liquid that thus is
able to adapt to the long-range anisotropic interactions induced by the
cavity.

  * •

At electric field gradients strong at the single particle level we find the
FQH phase to be unstable towards sliding Luttinger liquids with strong density
modulations in the gradient direction. The instability crucially depend on the
local value of the electric field gradient, thus non-uniform gradients give
rise to non-uniform states.

The simple scenario we discussed here gives a proof of principle on how cavity
vacuum fields can be used to probe and control intra-Landau level
correlations. This hopefully lays the ground for more detailed and complex
many body investigations as well as serve as a motivation for a more careful
analysis on which experimental cavity set-up best adapts to the need of strong
field gradients. Here we give few possible interesting future directions.

As a first example, we remark that the peculiar chiral nature of the graviton
excitation [28] has not entered our discussion, mainly because of the choice
of the cavity mode. However, a set-up with a pair of chiral cavity modes have
been recently demonstrated to be in a chiral-selective strong coupling regime
with cyclotron transition [106]. We expect this set-up to be sensible to the
chirality of the gravitons. This will also likely play a role in the FQH
regime of a recently demonstrated optical pumping experiment [107], paving the
way for a coherent control of FQH excitations. Tuning the cavity-modes can
also be interesting to explore periodic spatial modulation with periodicity on
the scale of few magnetic lengths $\simeq 10-100\mathrm{nm}$, achievable with
subwavelength set-ups. This will results in coupling to finite momentum
magnetoroton excitations. Another effect, likely important away from perfect
fillings, is the interplay between the cavity electric field and disorder
which allows for a non-trivial coupling to the constant part of the field.

From a more theoretical perspective, it would be interesting to extend our
treatment to more complex FQH phases and the effect of a non-local mode on
non-abelian topological oredered states, such as the Moore-Read state. We can
anticipate for example that the presence of multiple magnetoroton branches
will enrich the entaglement structure on one side and the landscape of
polaritons on the other, possibly giving distinct signatures to the nature of
the phases. Moving away from gapped liquids; it is also tempting to imagine
scenarios where, starting from gapless states such as the $\nu=1/2$ composite
Fermi sea, the cavity can be used as a handle to induce instabilities towards
desired states. It would also be intriguing to connect some of our findings -
which are inherently related to bulk properties - to recent works focusing on
edge dynamics [58].

From an experimental perspective, one immediate follow up is extending our
findings to different setups, including atomic systems in cavities and other
solid state platforms. In those contexts, it will be interesting to exploit
the strong matter-light correlations we report as a mean to probe so far
unexplored features of quantum Hall states, such as their entanglement
structure, and eventually understand whether such correlations can be utilized
to access and control the functioning of topological quantum memories.

## Acknowledgments

We thank G.M. Andolina, A. Asenjo-Garcia, D. Chang, G. Chiriacó, O. Dmytruck,
D. Fausti, R. Fazio, A. Nardin, M. Schirò, and O. Zilberberg for discussions.
M.D. was partly supported by the QUANTERA DYNAMITE PCI2022-132919, by the EU-
Flagship programme Pasquans2, by the PNRR MUR project PE0000023-NQSTI, and by
the PRIN programme (project CoQuS). H.B.X. was supported by MIUR Programme
FARE (MEPH). Z.B. and M.D. would like to thank the Institut Henri Poincaré
(UAR 839 CNRS-Sorbonne Université) for hospitality, and the group of C. Ciuti
for inspiring discussions during the stay. T.C. acknowledges the support of
PL-GRID infrastructure for the computational resource for a part of the
numerical simulations. MPS-based numerical calculations have been performed
with the help of ITensors.jl [108] and TenNetLib.jl [109] libraries.

##  Appendix A Cavity losses

We consider the effect of losses by introducing a weakly coupled photonic bath
with Hamiltonian $\hat{H}_{B}$ and temperature $T_{B}$. In the weak system-
bath coupling regime it is possible to derive a Lindblad master equation for
the evolution of the system density matrix [81], where by system here we mean
both cavity and electrons together. Under standard approximations in this
derivation (Born, Markov and secular) it is possible to show that the steady
state of the system is going to be a thermal state [81]:

$\hat{\Theta}=\frac{1}{\Tr\left[\exp\left(-\frac{1}{T_{B}}\hat{H}\right)\right]}\exp\left(-\frac{1}{T_{B}}\hat{H}\right)$
(79)

where $\hat{\Theta}$ is the steady state density matrix of the electron-cavity
system and $\hat{H}$ its Hamiltonian. The relevant information of equation 79
is that for a weakly coupled bath, i.e. weak cavity losses, it is sufficient
to look at the cavity plus electron system, independently on the strength of
the cavity coupling. Moreover we remark that there is no conceptual difference
with the coupling to an external electronic bath, the steady state for weakly
coupled thermal environment is indeed a thermal state. In the
$T_{B}\rightarrow 0$ limit we then have that the state of the system is up to
exponential corrections the ground state of the electron-cavity Hamiltonian
$\hat{H}$.

The deep consequence then is that the topological properties of the system
will not be affected by cavity losses when the temperature of the photon bath
is low. For example we can adapt the adiabatic flux insertion procedure of
Section III.1 to the case of an environment, assuming adiabaticity can still
be reached in the open system scenario. The reason behind this robustness it
is that the quantized resistivity is a ground state property and no photon
losses can occur from the ground state when $T_{B}=0$. At finite photon bath
temperature $T_{B}>0$ corrections to the quantization of transport are
expected by the fact that we are giving a finite weight to excited states.
Non-trivial effects may arise when the bath temperature for the electrons
$T_{B}^{el}$ is different from the bath temperature of the cavity $T_{B}^{c}$
[17, 110], giving rise to a competition whose study is beyond the scope of
this work.

##  Appendix B Stripe energy estimation

In this appendix we explicit our derivation for the approximate form of the
mean-field energy $E_{S_{n}}$ associated to the stripe state
$\ket{S_{n}}=\ket{\\{0_{n}1_{n}0_{n}\\}}$. From the photon mean-field
argument, the energy $E_{S_{n}}$ is given by the expectation value
$E_{S_{n}}=\bra{S_{n}}\hat{H}_{\mathrm{int}}\ket{S_{n}}$ as shown in Eq. (57).
Writing out explicitly the matrix elements of $\hat{H}_{\mathrm{int}}$, we
have

$E_{S_{n}}=\frac{\sqrt{8\pi}}{L_{y}}\sum_{k_{1},k_{2}\in
K_{\mathrm{occ}}}(k_{1}-k_{2})^{2}e^{-(k_{1}-k_{2})^{2}/2},$ (80)

where we recall $K_{\mathrm{occ}}$ denotes the set of orbitals that are
occupied in the stripe state. Note the sums over $k_{1}$ and $k_{2}$ run
independently. Let us then proceed to the thermodynamic limit by sending
$L_{y},n\to\infty$, while keeping their ratio $n/L_{y}$ fixed. From the ratio
$n/L_{y}$ we define the Fermi momentum $k_{\mathrm{F}}=\pi n/L_{y}$ of the
stripes. With this, we may replace the original restricted sum over
$K_{\mathrm{occ}}$ by $\sum_{k\in
K_{\mathrm{occ}}}\to\sum_{\sigma}\frac{L_{y}}{2\pi}\int_{-k_{\mathrm{F}}}^{k_{\mathrm{F}}}\differential
q$, where we sum over stripes labeled by the integer index $\sigma$, and
integrate over occupied states within each stripe $|q|<k_{\mathrm{F}}$. The
electronic filling $\nu=1/m$, with $m=3$, fixes the relative distance among
neighboring stripes, so we further replace the original momenta $k$ in the
summand by $k\to q+2mk_{\mathrm{F}}\sigma$ within the integral, where $q$ is
the momenta measured from the stripe center. We are then led to

$\displaystyle E_{S_{n}}$
$\displaystyle=\frac{L_{y}/\pi}{\sqrt{2\pi}}\sum_{\sigma_{1}\sigma_{2}}\int_{-k_{\mathrm{F}}}^{k_{\mathrm{F}}}\differential
q_{1}\differential q_{2}(q_{12}+6k_{\mathrm{F}}\sigma_{12})^{2}$
$\displaystyle\times e^{-\frac{1}{2}(q_{12}+6k_{\mathrm{F}}\sigma_{12})^{2}},$
(81)

where the integrand only depends on the relative variables
$q_{12}=q_{1}-q_{2}$ and $\sigma_{12}=\sigma_{1}-\sigma_{2}$.

To extract a series, which only depends on the relative stripe coordinate
$|\sigma_{12}|$, we assume translation invariance along $x$, so that we may
simplify the formula in Eq. (B) to

$\displaystyle E_{S_{n}}$
$\displaystyle=\frac{N_{e}}{\sqrt{2\pi}k_{\mathrm{F}}}\sum_{\sigma_{12}}\int_{-k_{\mathrm{F}}}^{k_{\mathrm{F}}}\differential
q_{1}\differential q_{2}(q_{12}+6k_{\mathrm{F}}\sigma_{12})^{2}$
$\displaystyle\times e^{-\frac{1}{2}(q_{12}+6k_{\mathrm{F}}\sigma_{12})^{2}},$
(82)

where we use that the total number of stripes must be fixed so that
$N_{\mathrm{stripes}}=N_{e}/n$. In this form we see the energy can be
naturally expanded as $E_{S_{n}}=\sum_{r}E_{S_{n}}^{(r)}$ for
$r=|\sigma_{12}|$, where thanks to the exponential factor, these contributions
decrease quite quickly as a function of $r$.

Let us then evaluate the first two contributions, corresponding to
$\sigma_{12}=0$ and $\sigma_{12}=\pm 1$. When $\sigma_{12}=0$, both $q_{1}$
and $q_{2}$ belong to the same stripe. The integral is elementary and gives
the energy contribution

$E_{S_{n}}^{(0)}=2N_{e}\big{[}\erf(\sqrt{2}k_{\mathrm{F}})+\frac{2e^{-2k_{\mathrm{F}}^{2}}-2}{\sqrt{2\pi}k_{\mathrm{F}}}\big{]},$
(83)

where $\erf(x)$ is the error function, defined as
$\erf(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}\differential t\,e^{-t^{2}}$.
Likewise, we evaluate the contribution for $\sigma_{12}=\pm 1$, which comes
from the interaction of neighboring stripes. The integral yields the same
result for both $\sigma_{12}=+1$ and $\sigma_{12}=-1$, of course, and the
energy contribution $E_{S_{n}}^{(1)}$ is found to be

$\displaystyle E_{S_{n}}^{(1)}$
$\displaystyle=4N_{e}\Big{[}\erf(\sqrt{8}k_{\mathrm{F}})-3\erf(\sqrt{18}k_{\mathrm{F}})+2\erf(\sqrt{32}k_{\mathrm{F}})$
$\displaystyle+\frac{e^{-8k_{\mathrm{F}}^{2}}-2e^{-8k_{\mathrm{F}}^{2}}+e^{-32k_{\mathrm{F}}^{2}}}{\sqrt{2\pi}k_{\mathrm{F}}}\Big{]}.$
(84)

These two functions are plotted in the inset of Fig. 8. When added together
they provide an estimate for the minimum of $E_{S_{n}}$, which sets the
minimum energy gap to the stripe state as $E_{S_{n}}/N_{e}\simeq 0.25$ and
fixes the Fermi momentum of that stripe state to $1/k_{\mathrm{F,min}}\simeq
1.62$.

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