#  Linear response theory for cavity QED materials

Juan Román-Roche  Instituto de Nanociencia y Materiales de Aragón (INMA),
CSIC-Universidad de Zaragoza, Zaragoza 50009, Spain  Departamento de Física de
la Materia Condensada, Universidad de Zaragoza, Zaragoza 50009, Spain  Álvaro
Gómez-León  Institute of Fundamental Physics IFF-CSIC, Calle Serrano 113b,
28006 Madrid, Spain  Fernando Luis  Instituto de Nanociencia y Materiales de
Aragón (INMA), CSIC-Universidad de Zaragoza, Zaragoza 50009, Spain
Departamento de Física de la Materia Condensada, Universidad de Zaragoza,
Zaragoza 50009, Spain  David Zueco  Instituto de Nanociencia y Materiales de
Aragón (INMA), CSIC-Universidad de Zaragoza, Zaragoza 50009, Spain
Departamento de Física de la Materia Condensada, Universidad de Zaragoza,
Zaragoza 50009, Spain

###### Abstract

We present a rigorous framework for linear response theory in cavity QED
materials. Our approach leverages the collective coupling between light and
matter, drawing parallels with large-N theories in quantum field theory. We
derive closed formulas for various responses of both the cavity and the
matter. Our theory is validated by recovering established results for the
Dicke model and the Quantum Hall Effect. Additionally, we discover novel
excitations in quantum magnets, where the cavity binds magnon pairs into
localized states.

The control of quantum matter with quantum light is a common pursuit in
quantum optics. Initially, the focus was on minimalistic matter such as single
atoms and molecules. Due to the weak light-matter coupling, it was realized
that photons need to be confined in cavities, giving rise to cavity quantum
electrodynamics (cQED) [1, 2]. This field has since evolved to consider more
complex forms of matter as well. First, to extract information about matter,
and more recently, to push the boundaries of light-matter interaction to
explore whether quantum light, either a few photons or vacuum states, can
alter the properties of matter [3, 4, 5]. Seminal experimental demonstrations
modifying and controlling conductivity [6, 7], magnetism [8] and the metal-to-
insulator transition [9] led to envisioning novel phenomenology emerging from
the hybridization of light and matter, such as modifications of chemical
reactions [10], changes in the critical temperature in superconductivity [11,
12], or alterations in magnetism [13, 14], topology [15, 16, 17, 18, 19, 20],
ferroelectricity [21, 22, 23] and transport in disordered systems [24, 25,
26].

The physics behind these phenomena can be understood through various
approximations, starting from the non-relativistic limit of the Dirac
equation, which yields the Pauli Hamiltonian [27, 28, 29]. In its simplest
forms, this Hamiltonian simplifies to well-known models such as the Jaynes-
Cummings and quantum Rabi models, where a single-mode cavity interacts with a
two-level system through a dipolar interaction term [30, 31, 32]. Both models
are exactly solvable [33] and their importance in quantum optics and
information cannot be overstated. Their generalizations, respectively the
Tavis-Cummings and Dicke models, describe a set of $N$ independent two-level
systems (TLSs) coupled to a single-mode cavity [34, 35]. Remarkably, the
partition function of the Dicke model is exactly solvable in the thermodynamic
limit, $N\to\infty$ [36, 37, 38]. This solution for the partition function of
the Dicke model has been exploited to obtain effective Hamiltonians for
interacting materials coupled to a cavity [29].

Beyond equilibrium, linear response is a key quantity that directly links to
experiments. In particular, linear response theory (LRT) explains the
formation of polaritons, a topic extensively explored in the literature.
Various models, mostly bilinear models of coupled quantum oscillators, have
been employed to interpret experimental results [6, 9]. This context led us to
investigate whether the exact solutions and effective theories for the
partition function of Dicke-like models in the thermodynamic limit can be
generalized to LRT, thereby providing a rigorous framework for cavity QED
materials. In this letter, we demonstrate that in the thermodynamic limit a
LRT can be derived that relies solely on the bare responses of light and
matter. Our approach leverages the fact that, in the thermodynamic limit, a
mean-field-like treatment of light and matter interactions becomes exact [36,
39], akin to the large-N theories in quantum field theory (QFT) where the
saddle-point approximation is exact. We emphasize that our approach does not
depend on linearization or bosonization. To validate our theory, we recover
previous findings for the Dicke model [40] and the Quantum Hall Effect [41,
42]. Additionally, we discuss the coupling of an Ising magnet to a cavity,
announcing novel polaritons hybridizing magnon pairs and photons, and the
potential emergence of _bound polariton states_ where the cavity acts as an
impurity and the material as a continuum of excitations.

## Model and its exact LRT theory.-

We consider a material collectively coupled to a single-mode cavity. The
Hamiltonian reads ($\hbar=1$)

$H=H_{\rm m}+\Omega
a^{\dagger}a+g\left(a+a^{\dagger}\right)C_{x}+\zeta\frac{g^{2}}{\Omega}C_{x}^{2}\,.$
(1)

Here, $\Omega$ is the cavity frequency and $[a,a^{\dagger}]=1$ are bosonic
operators. $H_{m}$ is the Hamiltonian of the bare material and remains
unspecified, as does $C_{x}$, which is the matter coupling operator. Although
our theory does not require a single-mode simplification, we employ it here
for clarity. Our approach relies on the collective nature of the matter
coupling operator, i.e. $C_{x}\sim N$, with $N$ the number of matter degrees
of freedom. The light matter coupling constant $g=\lambda/\sqrt{N}$ represents
the coupling per particle, with $\lambda$ the collective coupling. The last
term is an optional $P^{2}$ term, toggled by $\zeta\in\\{0,1\\}$. It enables
handling different scenarios of light-matter coupling. For $\zeta=1$, the
Hamiltonian describes electric dipoles, typically studied in the dipole gauge,
where the $P^{2}$ term ensures gauge-invariance. For $\zeta=0$, the
Hamiltonian may describe a gas of charged particles in the Coulomb gauge or a
system of magnetic dipoles.

Our theory utilizes a coherent-path-integral formulation of the partition
function. We use it to obtain two-point thermal Green functions of the form
$G^{\rm
t}_{A,B}(\tau-\tau^{\prime})=-\langle\mathcal{T}_{\tau}A(\tau)B(\tau^{\prime})\rangle$,
with $\tau\in[0,\beta]$ the imaginary time and $\beta$ the inverse
temperature. When expressed in terms of the matsubara frequencies,
$\omega_{m}$, these can be related to retarded response functions by analytic
continuation $i\omega_{m}\to\omega+i0^{+}$ [43][Chap. 7]. We do not aim to
provide here an exhaustive derivation of the theory, but rather an overview of
the key steps leading to our main result. For a full derivation and an
alternative approach based on the equations of motion for the Green functions,
we refer the reader to the longer paper accompanying this letter [44].

The first step is a partial integration over the cavity fields, resulting in a
matter-only action with an induced retarded collective interaction of the form
$S_{\rm ind}=\frac{1}{2}\int_{\tau,\tau^{\prime}}C_{x}(\tau)\frac{1}{N}V_{\rm
ind}(\tau-\tau^{\prime})C_{x}(\tau^{\prime})$ [45, 46] with

$V_{\rm
ind}(\omega_{m})=2\lambda^{2}\frac{\Omega^{2}(\zeta-1)+\zeta\omega_{m}^{2}}{\Omega(\omega_{m}^{2}+\Omega^{2})}\,.$
(2)

Note that $V_{\rm ind}$ combines the cavity-mediated interaction and the
$P^{2}$ term. More generally it may combine the cavity-mediated interaction
with intrinsic collective interactions. Following Refs. [47, 48], this induced
interaction is eliminated with a Hubbard-Stratonovich transformation. The
transformation decouples the matter operators, $C_{x}$, while introducing an
auxiliary field $\varphi$. Then, a partial integration over the matter degrees
of freedom results in an action for the auxiliary field of the form

$Z=\oint_{\varphi}e^{-Nf[\varphi]}\,,$ (3)

with

$f[\varphi]=\frac{1}{2}\int_{\tau,\tau^{\prime}}\varphi(\tau)V_{\mathrm{ind}}^{-1}(\tau-\tau^{\prime})\varphi(\tau^{\prime})+\mathcal{G}_{\rm
m}^{0}[\varphi]\,.$ (4)

Here $\mathcal{G}_{\rm m}^{0}[\xi]=-N^{-1}\log Z_{\rm m}[\xi]$ is the
generating functional for connected correlation functions of the coupling
operator, $C_{x}$, of the bare matter, with

$Z_{\rm m}[\xi]=\oint_{c}e^{-(S_{\rm m}+i\int_{\tau}\xi(\tau)C_{x}(\tau))}\,.$
(5)

It is crucial to note that we have chosen to make $\varphi$ not scale with
$N$. Thus, the action’s extensivity becomes explicit with the prefactor $N$
above, with $f$ being an intensive quantity. As a consequence, the partition
function lends itself to be computed with a saddle-point approximation for
large $N$. The saddle point approximation consists on a series expansion of
$f$ around is minimum, $\varphi_{\rm{sp}}$, where the number of terms required
to obtain a good approximation decreases with $N$. In fact, in the
thermodynamic limit, $N\to\infty$, the saddle-point approximation becomes
exact and it is warranted to truncate the expansion at second order, which
reveals the propagator of the auxiliary field, $W$, to be

$(NW)^{-1}=V_{\rm ind}^{-1}+\tilde{\chi}_{xx,0}\,,$ (6)

with

$\tilde{\chi}_{xx,0}(\tau-\tau^{\prime})=\frac{1}{N}\langle
C_{x}(\tau)C_{x}(\tau^{\prime})\rangle_{\varphi_{\rm{sp}}}^{\rm c}\,.$ (7)

Here $\langle\cdot\rangle_{\varphi_{\rm{sp}}}$ indicates the expectation value
for the bare matter subjected to a field $i\varphi_{\rm sp}$. Relations
between the photonic propagator, the matter response functions and the
auxiliary field propagator can be derived from functional derivatives of the
corresponding generators [44], allowing one to obtain the cavity response,
$D(\tau)=G^{\rm t}_{a,a^{\dagger}}(\tau,0)$,

$D(\omega_{m})=D_{0}(\omega_{m})-\lambda^{2}D_{0}(\omega_{m})\chi_{xx}(\omega_{m})D_{0}(\omega_{m})\,.$
(8)

and the matter responses, $\chi_{ab}(\tau)=-N^{-1}G^{\rm
r}_{C_{a},C_{b}}(\tau,0)$,

$\displaystyle\chi_{xa}(\omega_{m})=\frac{\tilde{\chi}_{xa,0}(\omega_{m})}{1+V_{\rm
ind}(\omega_{m})\tilde{\chi}_{xx,0}(\omega_{m})}\,,$ (9a)
$\displaystyle\chi_{ax}(\omega_{m})=\frac{\tilde{\chi}_{ax,0}(\omega_{m})}{1+V_{\rm
ind}(\omega_{m})\tilde{\chi}_{xx,0}(\omega_{m})}\,,$ (9b)
$\displaystyle\begin{split}&\left.\chi_{ab}\right|_{a,b\neq
x}(\omega_{m})=\tilde{\chi}_{ab,0}(\omega_{m})\\\
&\qquad\qquad\qquad\quad\;-\frac{\tilde{\chi}_{ax,0}(\omega_{m})V_{\rm
ind}(\omega_{m})\tilde{\chi}_{xb,0}(\omega_{m})}{1+V_{\rm
ind}(\omega_{m})\tilde{\chi}_{xx,0}(\omega_{m})}\,,\end{split}$ (9c)

in terms of $V_{\rm{ind}}$ and the bare cavity and matter responses, $D_{0}$
and $\tilde{\chi}_{ab}$ respectively. Here, some notation has been introduced:
since we are not always interested in the response of the matter through the
operator that couples to the cavity, $C_{x}$, we also consider responses
including $C_{x}$ and another operator and responses including two alternative
operators.

Finally, let us return to the issue of finding the saddle-point,
$\varphi_{\rm{sp}}$. The condition that $\delta f[\varphi]/\delta\varphi=0$
yields a self-consistent equation that defines the saddle-point as

$\varphi_{\rm sp}(\omega_{m})=-\frac{i}{N}V_{\rm ind}(\omega_{m})\langle
C_{x}(\omega_{m})\rangle_{\varphi_{\rm sp}}\,.$ (10)

We assume that $\varphi_{\rm{sp}}$ be constant, meaning $\varphi_{\rm
sp}(\tau)=\varphi_{\rm sp}=\varphi_{\rm sp}(\omega_{m}=0)$ (all other
frequency components being zero). Thus, Eq. (10) simplifies to

$\varphi_{\rm sp}=-\frac{i}{N}V_{\rm ind}(\omega_{m}=0)\langle
C_{x}\rangle_{\varphi_{\rm sp}}\,.$ (11)

This equation indicates that $\varphi_{\rm sp}$ is self-consistently
proportional to the expectation value of $C_{x}$ for the bare matter subjected
to a field $i\varphi_{\rm sp}=\frac{1}{N}V_{\rm ind}(\omega_{m}=0)\langle
C_{x}\rangle$. This can reversely be viewed as a self-consistent condition for
$\langle C_{x}\rangle$, precisely the one that arises when computing $\langle
C_{x}\rangle$ using the mean-field Hamiltonian

$H_{\rm eff}^{\rm MF}=H_{\rm m}+\frac{2\lambda^{2}(\zeta-1)}{N\Omega}\langle
C_{x}\rangle C_{x}-\frac{\lambda^{2}(\zeta-1)}{N\Omega}\langle
C_{x}\rangle^{2}\,.$ (12)

This is the mean-field theory of the effective Hamiltonian that arises from
taking the static limit in $V_{\rm ind}$ in the effective action [29],
justifying (11). Therefore, $\tilde{\chi}_{ab,0}$ are the response functions
of the mean field matter Hamiltonian of Eq. (12). These response functions
will depend on $\langle C_{x}\rangle$ which can be computed by solving $H_{\rm
eff}^{\rm MF}$ variationally with respect to $\langle C_{x}\rangle$. It is
important to note that in the absence of symmetry breaking, that is, when
$\langle C_{x}\rangle=0$, or if $\zeta=1$, the effective matter Hamiltonian is
just the bare matter Hamiltonian, making $\tilde{\chi}_{ab,0}=\chi_{ab,0}$
equivalent to the bare matter response. Equations (9b), (9b) and (9c) have a
structure analogous to the response functions typically obtained with a random
phase approximation, like those derived in [47]. We emphasize that they are
valid across the entire range of parameters and for both normal and
superradiant phases.

## Testing our theory.-

We demonstrate the theory with the Dicke model [34, 49]. The matter
Hamiltonian in Eq. (1) is simply $H_{\rm
m}=\frac{\omega_{z}}{2}\sum_{j}\sigma_{j}^{z}$ and the coupling operator is
$C_{x}=\sum_{j}\sigma_{j}^{x}$. Due to the absence of intrinsic interactions,
the mean-field effective Hamiltonian describes independent spins and the
corresponding bare matter response functions can be easily obtained [44].
Figure 1 shows the cavity response for this model with the exact polaritons
overlayed, with and without the $P^{2}$ term. The exact polaritons were
obtained with a two-oscillator solution of the Dicke model in the
thermodynamic limit [40, 44]. The agreement between our theory and the exact
polaritons indicates that the theory can accurately describe the model in the
normal and superradiant phases. The normal phase is characterized by the usual
polaritonic physics that can be captured with a model of two coupled
oscillators, one corresponding to the cavity and another to the bare matter.
The superradiant phase transition is signaled by a complete softening of the
lower polariton, which is prevented by the $P^{2}$ term. In the deep
superradiant regime, the lower polariton becomes purely photonic, while the
upper polariton corresponds to a material described by the effective matter
Hamiltonian (12). It is crucial to note that while one might be tempted to
think that $H_{\rm{eff}}^{\rm{MF}}$ contains purely material excitations and,
therefore, is decoupled from the cavity, this interpretation cannot be
correct, since the matter response is resolved through the cavity. The point
is that the effective interactions present in $H_{\rm{eff}}^{\rm{MF}}$ stem
from virtual photons.

Figure 1: Cavity response, $D$, of the Dicke model as a function of the
collective coupling, $\lambda$, without the $P^{2}$ term, $\zeta=0$, (left)
and with the $P^{2}$ term, $\zeta=1$, (right). The dashed lines correspond to
the polaritons computed with a two-oscillator solution [40]. The spins and
cavity are resonant, $\omega_{z}=\Omega$.

We have also tested the theory on another exactly solvable model, a 2D
electron gas homogeneously coupled to a cavity and subjected to a
perpendicular classical magnetic field. This model has been used to explain
the breakdown of topological protection that underlies the observed
modifications of the quantum Hall effect by coupling to a cavity [24, 7, 41,
42]. Our theory precisely recovers the optical and DC conductivities of the
model as well as the formation of Landau polaritons [42, 41]. This second
benchmark demonstrates the applicability of the theory to electronic light-
matter systems described in the Coulomb gauge, where the emergence of a no-go
theorem for the modification of the equilibrium properties of the system,
stemming from the $A^{2}$ term, is much more subtle than the clear no-go
afforded by the $P^{2}$ term in the dipole gauge [Cf. Eqs. (2) and (12) and
Ref. [44] for a detailed comparison].

## Cavity QED with magnetic materials.-

Finally, we consider a toy model of a magnetic material, the Ising chain in
transverse field, with transverse coupling to the cavity. Alternatively, the
linear response theory of the Ising chain with longitudinal coupling to the
cavity has been studied in Ref. [50]. The matter Hamiltonian reads

$H_{\rm
m}=\frac{\omega_{x}}{2}\sum_{j}\sigma_{j}^{x}-J\sum_{j}\sigma_{j}^{z}\sigma_{j+1}^{z}$
(13)

and the coupling operator in (1) is $C_{x}=\sum_{j}\sigma_{j}^{x}$. The full
light-matter model is termed the Dicke-Ising model [51, 52, 53, 54]. For
vanishing transverse field the model has a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry. The first symmetry corresponds
to a spin-flip, $\sigma_{j}^{z}\to-\sigma_{j}^{z}$, and in the bare Ising
model is spontaneously broken in a second-order phase transition from a
paramagnetic to a ferromagnetic phase. The second symmetry corresponds to a a
simultaneous cavity-field and spin flip, $a\to-a$ and
$\sigma_{j}^{x}\to-\sigma_{j}^{x}$, and in the bare Dicke model is
spontaneously broken in a second-order phase transition from a normal to a
superradiant phase. Their combination gives rise to a first-order phase
transition in the Dicke-Ising model between two symmetry-broken phases: a
paramagnetic superradiant phase for large $g^{2}/\omega_{c}$ and a
ferromagnetic normal phase for large $J$. A non-zero classical transverse
field breaks the Dicke symmetry but the model still features a first-order
phase transition between a superradiant phase where the cavity field direction
is fixed by the classical field to a symmetry-broken ferromagnetic phase.

The corresponding mean-field effective Hamiltonian is that of a Ising chain in
transverse field

$H_{\rm{eff}}^{\rm{MF}}=\frac{\tilde{\omega}_{x}}{2}\sum_{j}\sigma_{j}^{x}-J\sum_{j}\sigma_{j}^{z}\sigma_{j+1}^{z}+\frac{N\lambda^{2}}{\Omega}m_{x}^{2}\,,$
(14)

with $\tilde{\omega}_{x}=\omega_{x}-4\lambda^{2}/\Omega m_{x}$ and
$m_{x}=N^{-1}\sum_{j}\langle\sigma_{j}^{x}\rangle$. The transverse field is a
combination of the external field and the mean-field cavity-induced
interaction. In the thermodynamic limit, $N\to\infty$, the ground-state energy
per spin is given by

$e_{0}(m_{x})=\frac{\lambda^{2}}{\Omega}m_{x}^{2}-\frac{1}{2}\int_{-\pi}^{\pi}\frac{dk}{2\pi}\epsilon_{k}\,,$
(15)

with

$\epsilon_{k}=\sqrt{(2J)^{2}+\tilde{\omega}_{x}^{2}-4J\tilde{\omega}_{x}\cos
k}\,.$ (16)

Solving variationaly allows us to compute the equilibrium value of $m_{x}$
numerically and subsequently the response function $\tilde{\chi}_{xx,0}$,
which depends on the value of $m_{x}$. At zero temperature and in the
continuum limit, $\tilde{\chi}_{xx,0}$ is given by

$\tilde{\chi}_{xx,0}(\omega)=-16J^{2}\int_{-\pi}^{\pi}\frac{dk}{\pi}\frac{\sin^{2}k}{\epsilon_{k}(\omega_{+}^{2}-4\epsilon_{k}^{2})}\,.$
(17)

We find that $\tilde{\chi}_{xx,0}$ has poles at $2\epsilon_{k}$. This stems
from the fact that the coupling operator, $C_{x}$, creates and destroys
excitations in pairs of opposite momentum. Thus $\chi_{xx}$ and $D$ will
reflect how the zero-momentum sector of the two-excitation band (a double-
energy replica of the single-excitation band) of the Ising model hybridizes
with the cavity photon. The fact that the excitations are created in pairs of
opposite momenta allows one to probe the full band, despite the collective
nature of $C_{x}$.

Figure 2: Cavity response, $D$, of the Dicke model as a function of the
collective coupling, $\lambda$, for different values of the classical
transverse field, $\omega_{x}$. The yellow dashed lines correspond to a fit of
the polaritons with a two-oscillator model. The top right insets show the
magnetization. The dotted lines mark the edges of the band of the mean-field
effective Hamiltonian (14). The Ising interaction is set to $4J=\Omega$.

This feature brings novel phenomenology that we summarize in Fig. 2, where the
cavity response (8) is plotted for different scenarios. In all panels, we set
$4J=\Omega$, such that the two-excitation band is in resonance with the cavity
frequency. Figure 2(a) shows the case of vanishing classical field,
$\omega_{x}=0$. In this case, the model is non-dispersive in the normal phase,
as the only source of transverse field is the effective mean field. Instead of
a band, the model has a collection of degenerate excitations with energy $2J$:
domain walls. Accordingly, the zero-momentum sector of the two-excitation band
is a degenerate collection of the domain-wall pairs that correspond to single-
spin flips. This is a typical situation where the cavity is coupled to a
collective mode hybridizing with the cavity photon, forming polaritons whose
energy can be fitted by a model of two coupled oscillators of frequencies
$\Omega$ and $4J$:

$2\Omega_{\pm}^{2}=4J^{2}+\Omega^{2}\pm\sqrt{\left(4J^{2}-\Omega^{2}\right)^{2}+32\lambda^{2}J\Omega}\,.$
(18)

At the first-order phase transition, the effective mean field acquires a non-
zero value, opening the band. The lower polariton hardens to become the cavity
photon in the deep superradiant regime.

Figures 2(b), (c) and (d) show the case of non-zero classical field,
$\omega_{x}\neq 0$. Then, the Ising band (16) is already open in the normal
phase and we can understand the model as an impurity model, where the impurity
role is played by the cavity. The original collective coupling of the spins to
the cavity translates into a momentum-dependent coupling of the Bogoliubov
fermions that diagonalize the effective Ising model to the cavity
$C_{x}\propto\sum_{k}\eta_{k}(\gamma_{k}^{\dagger}\gamma_{-k}^{\dagger}-\rm{h.c.})$
[55][Chap. 10]. The coupling, $\eta_{k}=2J\sin k/\epsilon_{k}$, has an
exponentially localized profile in real space. Thus, the analogy with the
impurity is complete. Moreover, the equation for the poles of $D(\omega)$ can
be shown to be

$F(\omega)=\omega_{+}^{2}-\Omega^{2}-4\lambda^{2}\Omega\int_{-\pi}^{\pi}\frac{dk}{2\pi}\eta_{k}^{2}\frac{4\epsilon_{k}}{\omega_{+}^{2}-4\epsilon_{k}^{2}}=0\,,$
(19)

which can be compared with the equation for the eigenvalues of a discrete
system coupled to a continuum with a finite bandwidth, in our case the band of
the Ising model given by (16). The possibility of bound states appearing is
well known [56, 57, 58]. These states belong to the discrete spectrum and
solve $F(\omega)=0$ for $\omega$ outside the band. Much is known about bound
states, particularly that their existence is essentially conditioned by the
bandwidth, the coupling, and the position of the impurity [59, 60], in our
case, the cavity. In Fig. 2, we see how tuning the bandwidth with $\omega_{x}$
results in the appearance of one bound state for $\omega_{x}=0.1\Omega$, two
for $\omega_{x}=0.05\Omega$, or none for $\omega_{x}=0.2\Omega$. Furthermore,
we understand their increased visibility (with respect to the band) when they
appear, as it is well known that the contribution of the impurity (the cavity)
is finite in the bound states. Additionally, the localized nature of bound
states explains why the polariton formula (18) accounts well for their energy.
We are not aware of this physics being discussed before, so we have decided to
call these new states _bound polariton states_ (BPS). They are, in some way,
complementary to the usual bound states in quantum optics where matter
localizes photons around the impurity. In this case, it is the cavity that
localizes spin excitations, marking a new phenomenology in cavity QED
materials. This occurs already in the normal phase, and we attribute its
exotic nature to the interplay between matter and light correlations, which in
the Ising-Dicke model can be studied exactly. It would be interesting to
consider how generalizable this phenomenology is.

## Conclusion.-

In this letter, we have presented a LRT for cavity QED materials, discussing
exact formulas that go beyond perturbative theories of light-matter coupling.
We have demonstrated that the broad applicability of our theory does not
compromise its simplicity, and we have provided a guide on how to use it. In
addition to the formal development, we have discovered novel excitations
called bound polariton states (BPS), where the cavity binds magnon pairs into
localized states, adding to the existing polariton panorama [61]. Our work
extends beyond light-matter systems, such as in the coupling to phonon-
polaritons [62] or magnons, where recent experiments with perovskites have
demonstrated the superradiant quantum phase transition [63].

We acknowledge discussions with Katharina Lenk, Martin Eckstein, Andreas Wipf,
Yuto Ashida, Luis Martín-Moreno, Fernando Falceto and José G. Esteve. Despite
the enormous impediments in spending the budget, the authors must acknowledge
funding from the grant TED2021-131447B-C21 funded by
MCIN/AEI/10.13039/501100011033 and the EU ‘NextGenerationEU’/PRTR. We also
acknoledge the Gobierno de Aragón (Grant E09-17R Q-MAD), Quantum Spain and the
CSIC Quantum Technologies Platform PTI-001. J. R-R acknowledges support from
the Ministry of Universities of the Spanish Government through the grant
FPU2020-07231.

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