# Effective theory for matter in non-perturbative cavity QED

Juan Román-Roche  Instituto de Nanociencia y Materiales de Aragón (INMA),
CSIC-Universidad de Zaragoza, Zaragoza 50009, Spain  David Zueco  Instituto de
Nanociencia y Materiales de Aragón (INMA), CSIC-Universidad de Zaragoza,
Zaragoza 50009, Spain

(August 27, 2024)

###### Abstract

Starting from a general material system of $N$ particles coupled to a cavity,
we use a coherent-state path integral formulation to produce a _non-
perturbative_ effective theory for the material degrees of freedom. We tackle
the effects of image charges, the $A^{2}$ term and a multimode arbitrary-
geometry cavity. The resulting (non-local) action has the photonic degrees of
freedom replaced by an effective position-dependent interaction between the
particles. In the large-$N$ limit, we discuss how the theory can be cast into
an effective Hamiltonian where the cavity induced interactions are made
explicit. The theory is applicable, beyond cavity QED, to any system where
bulk material is linearly coupled to a diagonalizable bosonic bath. We
highlight the differences of the theory with other well-known methods and
numerically study its finite-size scaling on the Dicke model. Finally, we
showcase its descriptive power with three examples: photon condensation, the
2D free electron gas in a cavity and the modification of magnetic interactions
between molecular spins; recovering, condensing and extending some recent
results in the literature.

###### Contents

  1. I Introduction
  2. II Summary of main results
  3. III Model
    1. III.1 A gas of charged particles
    2. III.2 A system of localized emitters
  4. IV Deriving the effective matter model
    1. IV.1 Effective action: Euclidean path integral formulation
    2. IV.2 Obtaining an effective matter Hamiltonian in the large-$N$ limit
    3. IV.3 Relation between matter and light observables
  5. V Testing the validity of the effective theory
    1. V.1 Comparison with other effective theories
    2. V.2 Finite $N$ and approach to the thermodynamic limit
  6. VI Applications
    1. VI.1 Equilibrium superradiance (photon condensation) with electric coupling
      1. VI.1.1 Uniform cavity fields
      2. VI.1.2 Non-uniform cavity fields
    2. VI.2 The 2D free electron gas in a cavity
    3. VI.3 Magnetic cavity QED
  7. VII Conclusion and outlook
  8. A Light-matter Hamiltonian for a system of localized emitters
  9. B Entirely Hamiltonian formulation

##  I Introduction

The development of cavity QED is marked by the achievement of strong light-
matter coupling [1]. This experimental milestone opened the possibility of
entangling light and matter. Since then, experiments have advanced to be able
to control and manipulate a few atoms and photons in a fully quantum way.
Nowadays, there are many situations where strong coupling is achieved. Natural
atoms or molecules as well as artificial systems such as quantum dots or
superconducting circuits [2]. Some of these systems allow us to push light-
matter coupling to new regimes. One of them is ultrastrong coupling, where the
interaction strength is comparable to the bare energies of cavity photons and
atomic transitions [3, 4]. Here, correlations arise such that the ground state
is already an entangled state with both virtual matter and photon excitations
[5, 6]. In the same spirit as the original motivation of cavity QED -to
control atoms with quantum, rather than classical, light- a new field of
research is emerging. It is known as cavity QED materials [7, 8, 9]. Here,
instead of a few atoms, the modification of bulk material properties due to
nonperturbative light-matter correlations is investigated. Thus, the focus is
on the physics of a macroscopic number of particles coupled to the quantized
electromagnetic field. Quantum light fluctuations have recently been shown to
modify chemical reactivity [10, 11], excitonic transport [12, 13, 14],
superconductivity [15, 16, 17, 18] the ferroelectric phase in quantum
paramagnetic materials [19, 20, 21] and ferromagnetism [22, 23].

At the same time, the cavity-QED community has gained awareness of the
critical role that certain ubiquitous approximations can have in the predicted
behaviour of a system, often leading to wrong theoretical predictions that
later fail to achieve experimental confirmation. This is especially important
in the non-perturbative regime, which is precisely the regime where
modifications of matter occur. Perhaps the most discussed example in the
recent literature is equilibrium superradiance (photon condensation). This
quantum phase transition was originally predicted for electric dipoles in a
uniform electric field (Dicke model) [24, 25, 26], but it has since been shown
to be just a consequence of the breaking of gauge invariance in theories
missing the infamous $A^{2}$ term [27, 28]. Equilibrium superradiance can be
achieved, nevertheless, in a system of electric charges with spatially-varying
electromagnetic (EM) fields [29, 30, 31] or in systems with magnetic dipoles
[22], even when gauge-invariance is restored, but it serves as an example of
the profound consequences that an incomplete description of cavity QED can
have. Several works have also pointed out that the two level approximation is
a source of mistakes if performed, without additional care, in the Coulomb
gauge [32, 33]. Finally, image charges constitute a potentially relevant
electrostatic contribution to the effective interactions between charges in a
cavity [34, 35]. Thus, a complete formulation of cavity QED must include them
along the dynamical transverse fields of the cavity in the Coulomb gauge. This
is crucial to later resolve the contributions of the two to the modification
of matter properties.

Another important caveat in the study of hybrid systems in cavity QED is the
fact that, when studying the interplay of light and matter in a strong
coupling scenario, the separation between “light” and “matter” blurs. This
occurs not only because of the well-understood creation of hybrid polaritonic
excitations but, more fundamentally, because the definition of “light” and
“matter” subsystems is gauge-dependent [36]. This observation is perfectly
compatible with the gauge-invariance of physical observables but it raises
questions regarding our labelling of the underlying cause of certain phenomena
[32, 37]. When we say that an effective interaction is _light-mediated_ , is
this a universal statement?, or can a change of gauge make it seem as though
the interaction is _matter-mediated_? Here, we avoid descending into this
philosophical rabbit hole by defining the bare matter first. We start by
establishing the Hamiltonian that describes the behaviour of a material system
in the absence of a confined electromagnetic field, i.e. without a cavity.
This is motivated by the fact that most material properties are well captured
by such a description, such as superconductivity, ferromagnetism, electric and
thermal conductivity, etc.. In this sense, there exist experimentally
accessible observables that can be measured for the bare material system.
Then, we study what happens when such a material is placed within a cavity.
For that, we couple it to the quantized electromagnetic field in the Coulomb
gauge for an arbitrary cavity geometry. The _experimentally relevant question_
is then: if we measure the same experimentally accessible observables now, to
what extent are they altered? If there is a change, we naturally attribute it
to the effect of the cavity, even if in a microscopic description of the
hybrid light-matter system it is impossible to universally distinguish between
the two subsystems.

In this work we develop the theoretical analogue to this experimental
procedure by deriving an effective theory of matter inside a cavity.
Generally, an effective theory is found when a subsystem is systematically
eliminated from the dynamical description of a larger system. Perturbative
effective theories have widespread use in cavity QED. Notable examples are
adiabatic elimination [38] or the dispersive theory [39, 40, 41]. However, for
our purpose of studying strong light-matter correlations, it is necessary to
consider non-perturbative techniques. A seminal example is the Feynman-Vernon
influence functional within the path integral formalism, developed for quantum
systems coupled to a continuum (open systems) [42, 43, 44, 45]. Recently,
coherent state path integrals have been used to define effective models in
cavity QED [46, 47, 48]. Here, we build upon these works: we use a coherent-
state path integral to define an _exact effective theory_ for a general
material gauge-invariantly coupled to a general cavity. This exact treatment
yields a non-local effective action that becomes local in the large-$N$ limit.
In this limit, the theory can be cast into an effective matter Hamiltonian
where the effect of light vacuum fluctuations takes the explicit form of
induced matter-matter interactions. The range and form of these interactions
depends on the light-matter Hamiltonian and the geometry of the cavity. The
theory is compared against those effective models commonly employed in the
field. Unlike theories based on the adiabatic elimination of fast variables,
our theory is valid in the full light-matter coupling regime. We also test its
finite size scaling numerically via exact diagonalisation. In doing that, we
show that our theory matches the exact results already for $N\sim 10^{2}$.
Finally, we test the power of our formalism by deriving recent results from
the literature.

The remainder of the manuscript is organised as follows. In Section II we
provide a graphical and non-technical summary of the main results of the
paper. Section III is devoted to constructing a comprehensive model of matter
inside an electromagnetic cavity. Then, Section IV constitutes the main result
of the paper: we trace out the cavity to derive the effective model. In Sec. V
we compare the theory against other effective models and test its finite size
scaling. In Sec. VI we study different material systems within a cavity to
showcase the descriptive power of our effective theory. We close the paper
with some conclusions. Technical details and concepts of minor importance are
sent to the appendices.

Figure 1: Schematic summary of the derivation of the effective theory. We
begin by defining a general matter model in a). We accompany the illustration
with an example Hamiltonian, but the theory extends to other models as well.
The matter is then minimally coupled to an electromagnetic cavity in b). The
corresponding Hamiltonian describes a broad range of models, which are encoded
in the coupling constants $\\{c_{\bm{k}}\\}$ and operators
$\\{\mathcal{\hat{C}}_{\kappa}\\}$, and in $\mathcal{\hat{H}}_{M}^{\prime}$.
In c), the light degrees of freedom are non-perturbatively eliminated with a
coherent state path integral, yielding a non-local action with effective
matter-matter couplings. The non-local contributions vanish in the large-$N$
limit and an exact effective Hamiltonian is obtained in d).

##  II Summary of main results

The main focus of this work is the theoretical description of a broad range of
materials when these are coupled to an electromagnetic cavity. To avoid
subtleties, we fix our starting hypothesis. We begin by asserting that a
material is an extensive collection of typically a macroscopic number $N$ of
particles that can be isolated (to a good approximation) and their behaviour
described by a Hamiltonian $\mathcal{\hat{H}}_{M}$. This is illustrated in
Fig. 1(a). We then couple this material to a quantized electromagnetic cavity
and our formalism begins. For the cavity, we focus on a complete description
including the $A^{2}$ term, the Zeeman coupling and image charges from the
cavity boundaries. We consider a multi-mode and non-uniform electromagnetic
field and we do not specify any particular geometry, leaving the spatial
dependence encoded in the mode functions of the EM field. We must disclaim
that, in order to diagonalize the photonic Hamiltonian in the model of a gas
of charged particles, we simplify certain mode-mode interactions arising from
the $A^{2}$ term. This is a common occurrence in the literature and its
adequacy is discussed in the text. We show that due to the fact that the
light-matter Hamiltonian is quadratic in bosonic (light) operators, it can be
brought to the linear form shown in Fig. 1(b), which corresponds to Eq. (17).
This is a general Hamiltonian that can be used to describe other systems
besides the ones that we treat in this paper. It describes the linear coupling
of an unspecified extensive subsystem of interest to a diagonal(ized) bosonic
ensemble. In our case, the role of the bosons is played by cavity photons, but
the Hamiltonian could also be used to describe, e.g., phononic or magnonic
ensembles.

Then, we use a coherent-state path integral formulation of the partition
function of the system to trace out the photonic degrees of freedom (DOFs). In
an _exact_ treatment, we obtain an effective action (33) in which the light-
matter coupling has been replaced by an cavity-field-mediated interaction
between the constituents of the material subsystem. This interaction is
generally non-local in imaginary time, as illustrated in Fig. 1(c). Then, we
show that in the large-$N$ limit the action becomes local, moving from an
effective action to an effective Hamiltonian (41) as shown in Fig. 1(d). In
this effective Hamiltonian, the role of the cavity is made explicit in two
ways: first, the bare coulombic interactions are modified by the image
charges; second, an interpretable additional term introduces effective
interactions mediated by the transverse fields of the cavity. In the process,
we recover relations between photonic and material observables, such that
information about the cavity is accessible even when the latter is traced out.
The theory is tested on the Dicke model against other effective techniques,
showing that it is not equivalent to perturbative techniques . Its finite size
scaling is also numerically tested, showing that our theory becomes exact
already for $N\sim 10^{2}$. To illustrate the applicability of our formalism,
we finish the manuscript by applying it to three different material systems in
a cavity. First, we revisit the problem of photon condensation. Within our
effective model, we recover the no-go theorem for superradiance in the single-
uniform-mode limit and we show how neglecting the $A^{2}$ term leads to a
wrong prediction of the phase transition. Our result rules out phase
transitions of any order in a single formulation and is valid at finite
temperatures. This is a generalization of [28, 49]. Then, we obtain critical
conditions for photon condensation with spatially-varying electromagnetic
fields, recovering [31]. Then, we solve the 2D free electron gas in a cavity,
recovering the results in [50]. The last system that we solve is an ensemble
of molecular nanomagnets. These couple to the cavity solely by the Zeeman
term, avoiding the $A^{2}$ problem, and have been shown to be a suitable
candidate to showcase cavity-modified ferromagnetism as well as photon
condensation [22]. We obtain the effective spin-spin interactions for an
arbitrary magnetic field geometry.

##  III Model

We begin by fixing the model under consideration. First, we consider a general
class of material subsystems, composed of $N$ identical particles. Depending
on the degree to which the particles are bounded to a particular position in
space we can distinguish two different families of systems. A gas, where the
particles are free to occupy any position of space, like an electron gas in a
metallic lattice; and a system of emitters where each particle or group of
particles is bounded to a non-dynamical core, such as a system of neutral
atoms or molecules where the motion of the nucleus is disregarded and the
focus is placed on the electron(s) motion. We also discuss the limit case of
completely fixed emitters such as magnetic spins. The two families differ
slightly in the way they couple to EM fields. Nevertheless, the degree of
parallelism in their analytical description is high, so we will only discuss
the more complex case of the gas in the main text, and comment briefly on the
derivation for a system of localized emitters, highlighting the differences
with the former and reserving the detailed derivation for Appendix A.

###  III.1 A gas of charged particles

A gas of charged particles is generally described by a Hamiltonian of the form

$\mathcal{\hat{H}}_{M}=\sum_{j}\frac{\hat{\bm{p}}_{j}^{2}}{2m}+\mathcal{\hat{H}}_{qq}+\mathcal{\hat{H}}_{qQ}+\sum_{j}\hat{v}(\hat{\bm{r}}_{j})+\sum_{i\neq
j}\hat{V}(\hat{\bm{r}}_{i},\hat{\bm{r}}_{j})\,.$ (1)

Here, $\hat{\bm{r}}_{j}$ and $\hat{\bm{p}}_{j}$ are the canonical position and
momentum operators of the $j$-th particle, such that
$[\hat{r}_{i,\alpha},\hat{p}_{j,\beta}]=i\delta_{ij}\delta_{\alpha\beta}$.
Note that $\hbar=1$ throughout the text. We also include four potential terms.
Coulombic interactions between charges, including self-interactions, are
encapsulated in $\mathcal{\hat{H}}_{qq}$. $\mathcal{\hat{H}}_{qQ}$ contains
possible interactions with non-dynamical charges, e.g. with an ionic lattice.
The potentials $\hat{v}(\hat{\bm{r}}_{j})$ and
$\hat{V}(\hat{\bm{r}}_{i},\hat{\bm{r}}_{j})$ represent additional unspecified
single-particle and two-particle potentials, respectively. This matter
Hamiltonian can be summarized as
$\mathcal{\hat{H}}_{M}=\mathcal{\hat{T}}+\mathcal{\hat{H}}_{qq}+\mathcal{\hat{H}}_{qQ}+\mathcal{\hat{V}}$.
We are interested in the effects of coupling this material subsystem to the
quantized electromagnetic field in a confined region of space, i.e. a cavity.
To be consistent with the inclusion of Coulombic interactions, we will
describe the coupling of the matter and light subsystems in the Coulomb gauge.
This yields a Hamiltonian in minimal coupling format for the complete system

$\mathcal{\hat{H}}=\sum_{j}\frac{(\hat{\bm{p}}_{j}-q\hat{\bm{A}}(\hat{\bm{r}}_{j}))^{2}}{2m}+\mathcal{\hat{H}}^{\prime}_{qq}+\mathcal{\hat{H}}^{\prime}_{qQ}+\mathcal{\hat{V}}+\mathcal{\hat{H}}_{\rm
EM}-\frac{g\mu_{B}}{2}\sum_{j}\hat{\bm{\sigma}}_{j}\hat{\bm{B}}(\hat{\bm{r}}_{j})\,.$
(2)

The effect of the coupling is fourfold. Most trivially, we must account for
the energy of the electromagnetic field, hence $\mathcal{\hat{H}}_{\rm EM}$.
Secondly, the canonical momentum now differs from the mechanical momentum,
this is apparent in the kinetic term, which now includes the vector potential
$\hat{\bm{A}}(\hat{\bm{r}}_{j})$ at each particles’ position. The third and
perhaps more subtle effect is the modification of the Coulombic interactions
between charges $\mathcal{\hat{H}}_{qq}\to\mathcal{\hat{H}}^{\prime}_{qq}$,
$\mathcal{\hat{H}}_{qQ}\to\mathcal{\hat{H}}^{\prime}_{qQ}$, which now include
direct as well as image-charge mediated interactions [35, 34]. The latter
arise as an electrostatic effect of imposing the boundary conditions of a
cavity on the Coulomb potential. Finally, we include the Zeeman coupling
between the particles’ spin and the magnetic field
$\hat{\bm{B}}(\hat{\bm{r}}_{j})$. Here
$\hat{\bm{\sigma}}_{j}=(\hat{\sigma}_{j}^{x},\hat{\sigma}_{j}^{y},\hat{\sigma}_{j}^{z})$
is the Pauli vector, whose components are the Pauli matrices,
$\mu_{B}=\frac{e}{2m}$ is the Bohr magneton and $g$ the Landé factor. W.l.o.g.
we are discussing spin $1/2$ particles. In fact, Eq. (2) is the many-particle
generalization of the Pauli-Fiertz Hamiltonian for non-relativistic spin $1/2$
particles, bar the spin-orbit coupling.

In the Coulomb gauge, the redundancy in the description of the dynamical
variables is eliminated by constraining the vector potential to be transverse,
i.e. $\divergence{A}=0$. The quantized vector potential then reads

$\hat{\bm{A}}(\hat{\bm{r}}_{j})=\sum_{\kappa}A_{\bm{k}}\left(\hat{\bm{u}}_{\kappa}(\hat{\bm{r}}_{j})\hat{a}_{\kappa}+\hat{\bm{u}}^{*}_{\kappa}(\hat{\bm{r}}_{j})\hat{a}_{\kappa}^{\dagger}\right)\,,$
(3)

where $\kappa$ is a four-vector containing the polarization index $\sigma=1,2$
and the wavevector $\bm{k}$: $\kappa\equiv\\{\bm{k},\sigma\\}$. Note that
$A_{\bm{k}}=A_{-\bm{k}}$. We also introduce the bosonic annihilation and
creation operators of the $\kappa$-th mode: $a_{\kappa}$,
$a_{\kappa}^{\dagger}$, which obey the canonical commutation relations
$[a_{\kappa},\hat{a}_{\kappa^{\prime}}^{\dagger}]=\delta_{\kappa\kappa^{\prime}}$.
To maintain as much generality as possible, we have refrained from using a
particular spatial dependence for the vector potential. For a specific model,
the geometry of the cavity will determine the spatial quantization of the
wavevector $\bm{k}$ and in turn, the functional form of the mode functions
$u_{\kappa}(\bm{r})$ [51]. In any case, the following properties hold for the
mode functions: $u_{-\bm{k},\sigma}(\bm{r})=u_{\bm{k},\sigma}^{*}(\bm{r})$,
$u_{\bm{k},\sigma}(\bm{r})\dotproduct\bm{k}=0$,
$\int_{V}dVu^{*}_{\kappa}(\bm{r})u_{\kappa^{\prime}}(\bm{r})=\delta_{\kappa\kappa^{\prime}}$.
Note that in Eq. (3) the mode functions have been promoted to mode operators,
since they depend on the position operator of each particle (Cf. App. A). By
definition, $\hat{\bm{B}}=\curl{\hat{\bm{A}}}$, so

$\hat{\bm{B}}(\hat{\bm{r}}_{j})=\sum_{\kappa}B_{\bm{k}}\left(\hat{\bm{u}}_{\perp,\kappa}(\hat{\bm{r}}_{j})\hat{a}_{\kappa}+\hat{\bm{u}}_{\perp,\kappa}^{*}(\hat{\bm{r}}_{j})\hat{a}_{\kappa}^{\dagger}\right)\,,$
(4)

where we have defined the transverse mode functions
$\bm{u}_{\perp,\kappa}(\bm{r}_{j})=\frac{1}{|\bm{k}|}\curl{\bm{u}_{\kappa}(\bm{r}_{j})}$
and $B_{\bm{k}}=A_{\bm{k}}|\bm{k}|=A_{\bm{k}}\omega_{\bm{k}}/c$. Finally, we
can express $\mathcal{\hat{H}}_{\rm EM}$ as

$\mathcal{\hat{H}}_{\rm
EM}=\sum_{\kappa}\omega_{\bm{k}}\hat{a}_{\kappa}^{\dagger}\hat{a}_{\kappa}\,.$
(5)

To facilitate the analytical treatment of the Hamiltonian, it is convenient to
expand the kinetic term and substitute in Eqs. (3) and (4).

$\begin{split}\mathcal{\hat{H}}&=\mathcal{\hat{H}}_{M}^{\prime}+\sum_{\kappa}\omega_{\bm{k}}\hat{a}_{\kappa}^{\dagger}\hat{a}_{\kappa}-\sum_{j}\sum_{\kappa}\frac{q}{m}\hat{\bm{p}}_{j}A_{\bm{k}}\left(\hat{\bm{u}}_{\kappa}(\hat{\bm{r}}_{j})\hat{a}_{\kappa}+h.c.\right)\\\
&+\sum_{j}\frac{q^{2}}{2m}\sum_{\kappa,\kappa^{\prime}}A_{\bm{k}}A_{\bm{k}^{\prime}}\left(\hat{\bm{u}}_{\kappa}(\hat{\bm{r}}_{j})\hat{a}_{\kappa}+h.c.\right)\left(\hat{\bm{u}}_{\kappa^{\prime}}(\hat{\bm{r}}_{j})\hat{a}_{\kappa^{\prime}}+h.c.\right)-\frac{g\mu_{B}}{2}\sum_{j}\sum_{k}\hat{\bm{\sigma}}_{j}B_{\bm{k}}\left(\hat{\bm{u}}_{\perp,\kappa}(\hat{\bm{r}}_{j})\hat{a}_{\kappa}+h.c.\right)\,.\end{split}$
(6)

Here, $\mathcal{\hat{H}}_{M}^{\prime}$ is just $\mathcal{\hat{H}}_{M}$ but
with $\mathcal{\hat{H}}_{qq}$ ($\mathcal{\hat{H}}_{qQ}$) replaced by
$\mathcal{\hat{H}}_{qq}^{\prime}$ ($\mathcal{\hat{H}}_{qQ}^{\prime}$) To
condense the notation, we define the operators

$\displaystyle\hat{U}_{\kappa,\kappa^{\prime}}=N^{-1}\sum_{j}\hat{\bm{u}}_{\kappa}(\hat{\bm{r}}_{j})\hat{\bm{u}}_{\kappa^{\prime}}(\hat{\bm{r}}_{j})\,,$
(7)
$\displaystyle\hat{\bar{U}}_{\kappa,\kappa^{\prime}}=N^{-1}\sum_{j}\hat{\bm{u}}_{\kappa}(\hat{\bm{r}}_{j})\hat{\bm{u}}_{\kappa^{\prime}}^{\dagger}(\hat{\bm{r}}_{j})\,,$
(8)

and the constants

$\displaystyle\Delta_{\bm{k},\bm{k}^{\prime}}=\frac{Nq^{2}A_{\bm{k}}A_{\bm{k}^{\prime}}}{2m}\,,$
(9) $\displaystyle
c_{\bm{k}}^{e}=qA_{\bm{k}}\sqrt{\frac{\omega_{\bm{k}}}{m}}\,,$ (10)
$\displaystyle c_{\bm{k}}^{m}=\frac{g\mu_{B}B_{\bm{k}}}{2}\,.$ (11)

With these, we write the terms that depend quadratically on the bosonic
operators as

$\begin{split}\mathcal{\hat{H}_{\rm
ph}}=&\sum_{\kappa}\omega_{\bm{k}}\hat{a}_{\kappa}^{\dagger}\hat{a}_{\kappa}+\sum_{\kappa,\kappa^{\prime}}\Delta_{\bm{k},\bm{k}^{\prime}}\left(\hat{U}_{\kappa,\kappa^{\prime}}\hat{a}_{\kappa}\hat{a}_{\kappa^{\prime}}+\hat{\bar{U}}_{\kappa,\kappa^{\prime}}\hat{a}_{\kappa}\hat{a}_{\kappa^{\prime}}^{\dagger}+h.c.\right)\,.\end{split}$
(12)

Analytical tractability demands that we neglect mode-mode interactions (Cf.
App. A) and assume
$\hat{U}_{\bm{k},\sigma,-\bm{k},\sigma}=\hat{\bar{U}}_{\bm{k},\sigma,\bm{k},\sigma}=\mathbb{I}$,
leaving only the momentum-conserving terms, and without spatial dependence

$\begin{split}\mathcal{\hat{H}_{\rm
ph}}\approx&\sum_{\bm{k},\sigma}\omega_{\bm{k}}\hat{a}_{\bm{k},\sigma}^{\dagger}\hat{a}_{\bm{k},\sigma}+\sum_{\kappa}\Delta_{\bm{k}}\left(\hat{a}_{\bm{k},\sigma}\hat{a}_{-\bm{k},\sigma}+\hat{a}_{\bm{k},\sigma}\hat{a}_{\bm{k},\sigma}^{\dagger}+h.c.\right)\,,\end{split}$
(13)

where $\Delta_{\bm{k}}\equiv\Delta_{\bm{k},\bm{k}}$. The approximation is
exact for a homogeneous gas. Furthermore, for gasses that have a second order
phase transition to a non-homogeneous phase, the approximation is valid in the
vicinity of the transition [31]. This suggests that the resulting model can be
used to predict and locate transitions to non-homogeneous gas phases. It is
expected that the effect of the approximation will be quantitative and will
skew the calculation of observables in the non-homogeneous phase.
$\mathcal{\hat{H}}_{\rm ph}$ can now be diagonalized with a Bogoliubov
transformation

$\hat{a}_{\bm{k},\sigma}^{\dagger}=\cosh(\theta_{\bm{k}})\hat{b}_{\bm{k},\sigma}^{\dagger}-\sinh(\theta_{\bm{k}})\hat{b}_{-\bm{k},\sigma}\,.$
(14)

One finds that
$\cosh\left(\theta_{\bm{k}}\right)=\left(\lambda_{\bm{k}}+1\right)/(2\sqrt{\lambda_{\bm{k}}})$
and
$\sinh\left(\theta_{\bm{k}}\right)=\left(\lambda_{\bm{k}}-1\right)/(2\sqrt{\lambda_{\bm{k}}})$
and $\lambda_{\bm{k}}=\sqrt{1+4\Delta_{\bm{k}}/{\omega_{\bm{k}}}}$. As a
result, the complete Hamiltonian can be written as

$\begin{split}\mathcal{\hat{H}}=&\mathcal{\hat{H}}_{M}^{\prime}+\sum_{\kappa}\omega_{\bm{k}}\lambda_{\bm{k}}\hat{b}_{\kappa}^{\dagger}\hat{b}_{\kappa}-\sum_{j}\sum_{\kappa}c_{\bm{k}}^{e}\lambda_{\bm{k}}^{-1/2}\sqrt{\frac{1}{m\omega_{\bm{k}}}}\hat{\bm{p}}_{j}\left(\hat{\bm{u}}_{\kappa}(\hat{\bm{r}}_{j})\hat{b}_{\kappa}+h.c.\right)-\sum_{j}\sum_{k}c_{\bm{k}}^{m}\lambda_{\bm{k}}^{-1/2}\hat{\bm{\sigma}}_{j}\left(\hat{\bm{u}}_{\perp,\kappa}(\hat{\bm{r}}_{j})\hat{b}_{\kappa}+h.c.\right)\,.\end{split}$
(15)

Defining the coupling operator and coupling constant

$c_{\bm{k}}\mathcal{\hat{C}}_{\kappa}=\sum_{j}\lambda_{\bm{k}}^{-1/2}\left(c_{\bm{k}}^{e}\sqrt{\frac{1}{m\omega_{\bm{k}}}}\hat{\bm{p}}_{j}\hat{\bm{u}}_{\kappa}(\hat{\bm{r}}_{j})+c_{\bm{k}}^{m}\hat{\bm{\sigma}}_{j}\hat{\bm{u}}_{\perp,\kappa}(\hat{\bm{r}}_{j})\right)\,,$
(16)

we can finally write the Hamiltonian as

$\mathcal{\hat{H}}=\mathcal{\hat{H}}_{M}^{\prime}+\sum_{\kappa}\tilde{\omega}_{\bm{k}}\hat{b}_{\kappa}^{\dagger}\hat{b}_{\kappa}-\sum_{\kappa}c_{\bm{k}}\left(\mathcal{\hat{C}}_{\kappa}\hat{b}_{\kappa}+h.c.\right)\,,$
(17)

where $\tilde{\omega}_{\bm{k}}=\omega_{\bm{k}}\lambda_{\bm{k}}$.

###  III.2 A system of localized emitters

The main difference between a gas and a system of localized emitters relies on
the fact that in the former case the dynamical particles are free to move on
the entire volume of the system, whereas in the latter their movement is
bounded to a non-dynamical core. A limit case is given by systems in which no
movement occurs, and instead the dynamics affect some other degree of freedom,
this is the case of spins of fixed positions, such as the ones that compose a
magnetic crystal. If the length scale of the bounded motion is much smaller
than the wavelength of the EM field, one can consider that the EM field is
constant and equal at any point in the trajectory of an individual particle
$\hat{\bm{A}}(\hat{\bm{r}}_{j}+\bm{R}_{j})\approx\hat{\bm{A}}(\bm{R}_{j})$:
where $\bm{R}_{j}$ is the position of the corresponding non-dynamical core.
This is the long-wavelength approximation. Assuming that we can work within
the long-wavelength limit, the EM field operators $\hat{\bm{A}}$,
$\hat{\bm{B}}$ act only on the Hilbert space of the photons, i.e. they do not
contain position operators acting on the Hilbert space of the matter. Instead,
they are parametrized by the positions of the non-dynamical cores. This subtle
difference implies that, unlike for the gas, the terms in the Hamiltonian
quadratic in bosonic operators can be exactly diagonalized by a Bogoliubov
transformation. The final Hamiltonian (17) (See Fig. 1(b)) has the same
functional form in both cases, the differences are encoded in the coupling
constants $\\{c_{\bm{k}}\\}$ and coupling operators
$\\{\mathcal{\hat{C}}_{\kappa}\\}$, and of course in $\mathcal{\hat{H}}_{M}$.
For a detailed derivation on the case of localized emitters see App. A.

Before advancing, we want to emphasize that Hamiltonian (17) or the equivalent
(99) represents a general light-matter Hamiltonian. In this Section (Appendix
A), we arrived to it by considering the particular case of a gauge-invariant
description of a gas of charged particles (an ensemble of localized emitters)
coupled to the multi-mode electromagnetic field of a cavity, but this approach
is not unique. For instance, the Zeeman coupling to the magnetic field is
often neglected and the description of the EM field is simplified by making
the single-mode approximation. Neglecting the $A^{2}$ term is also a
ubiquitous approximation. The important fact is that all these variations
arise as particularizations of Hamiltonian (17). More over, systems of
magnetic molecules where the orbital degrees of freedom can be disregarded,
which have received attention for the possibility of hosting the superradiant
phase transition [22], can also be described by this Hamiltonian. In this
case, the matter Hamiltonian $\mathcal{\hat{H}}_{M}^{\prime}$ and the coupling
operator $\mathcal{\hat{C}}_{\bm{k}}$ would exclusively contain spin degrees
of freedom. In these examples, we simplify the picture by considering only
cavity fields, but the matter DOFs can be coupled to classical external fields
as well. For instance, it is common in systems of magnetic molecules to induce
Zeeman level splitting with an external magnetic field perpendicular to the
microwave magnetic field of the cavity. The inclusion of a classical field is
trivial and indeed will be considered in Sec. VI when we cover magnetic cavity
QED.

We have discussed some relevant examples for cavity QED and quantum optics,
but in general, any matter system which is linearly coupled to light and is
quadratic in the photonic operators can have its Hamiltonian brought to Eq.
(17). In essence, the formalism presented hereinafter is just a recipe to
systematically eliminate bosonic degrees of freedom in a hybrid system.
Furthermore, we show that in the large-$N$ limit, which is typically the
relevant one in cavity QED materials, the effective action becomes local and
the effective description for the matter is exactly Hamiltonian. Thus
eliminating the need for mean field or perturbative approximations.

##  IV Deriving the effective matter model

Our goal in this section is to take a Hamiltonian of the form of Eq. (17) and
trace out the photonic degrees of freedom. To do so, we will use a coherent
path integral approach to define an effective matter action. One can foresee
that the light-matter coupling present in (17) will translate into an
effective matter-matter coupling that will appear in the effective action.
Then, we obtain an effective Hamiltonian from the effective action in the,
$N\to\infty$, thermodynamic limit.

###  IV.1 Effective action: Euclidean path integral formulation

Let us recall that the partition function of a single-mode bosonic system
described by a Hamiltonian $\mathcal{\hat{H}}$ can be written in Euclidean
path integral form as

$Z=\int\mathcal{D}(z,\bar{z})e^{-S}\,,$ (18)

with Euclidean action

$S=\int_{0}^{\beta}\left(\bar{z}\partial_{u}z+H(z,\bar{z})\right)du\,,$ (19)

where $z$, $\bar{z}$ are complex-valued continous fields satisfying the
boundary condition $\bar{z}(0)=\bar{z}(\beta)$, $z(0)=z(\beta)$ and
$\mathcal{D}(z,\bar{z})=\lim_{N\to\infty}\prod_{n=1}^{N}d(z^{n},\bar{z}^{n})$,
with $d(z^{n},\bar{z}^{n})=d\real z^{n}d\imaginary z^{n}$ [52, chapter 4].
Here $H(z,\bar{z})$ is obtained by replacing $a$ by $z$ and $a^{\dagger}$ by
$\bar{z}$ in the normal-ordered form of $\mathcal{\hat{H}}$, where
$a$($a^{\dagger}$) are bosonic annihilation (creation) operators. This result
is readily generalized to the multimode case

$Z=\prod_{\kappa}\int\mathcal{D}(z_{\kappa},\bar{z}_{\kappa})e^{-S_{\kappa}}$
(20)

with

$S_{\kappa}=\int_{0}^{\beta}\left(\bar{z}_{\kappa}\partial_{u}z_{\kappa}+H_{\kappa}(z_{\kappa},\bar{z}_{\kappa})\right)du\,.$
(21)

We now focus our attention on a hybrid light-matter system such as the one
described by Hamiltonian (17). With
$\ket{\bm{z}}=\bigotimes_{\kappa}\ket{z_{\kappa}}$, its partition function is
given by

$Z=\Tr_{M}\left(\int\prod_{\kappa}\frac{d^{2}z_{\kappa}}{\pi}\bra{\bm{z}}e^{-\beta\mathcal{\hat{H}}}\ket{\bm{z}}\right)\,.$
(22)

We assume that one could formulate a coherent-path-integral partition function
for the full system, so we assign an action to the matter Hamiltonian
$\mathcal{\hat{H}}_{M}^{\prime}\to S_{M}$ and a field to the coupling operator
$\mathcal{\hat{C}}_{k}\to C_{k}$,
$\mathcal{\hat{C}}_{k}^{\dagger}\to\bar{C}_{k}$. Knowledge of the precise
dependence of $S_{M}$, $C_{k}$ and $\bar{C}_{k}$ on material coherent fields
is not required for what follows. With these definitions, the total action can
be written as $S=S_{M}+\sum_{\kappa}S_{\kappa}$, with

$S_{\kappa}=\int_{0}^{\beta}\left(\bar{z}_{\kappa}\partial_{u}z_{\kappa}+\tilde{\omega}_{\bm{k}}\bar{z}_{\kappa}z_{\kappa}-c_{\bm{k}}\left(C_{\kappa}z_{\kappa}+\bar{C}_{\kappa}\bar{z}_{\kappa}\right)\right)du\,,$
(23)

For such a system, we can define an effective matter action by tracing only
over the light degrees of freedom

$Z=\Tr_{M}\left(e^{-S_{M}}\prod_{\kappa}\int\mathcal{D}(z_{\kappa},\bar{z}_{\kappa})e^{-S_{\kappa}}\right)=:Z_{0}\Tr_{M}\left(e^{-S_{\rm
eff}}\right)\,.$ (24)

At this point, $Z_{0}$ is just a constant prefactor. It should be noted that
$S_{\rm eff}$ is useful because it makes explicit the influence of the cavity
on the matter DOFs. In particular, the type of interactions that emerge. In
addition, as we show in Sec. IV.3, no information is lost because light
observables can be written in terms of matter observables, which can be
calculated from $S_{\rm eff}$.

We are thus interested in computing

$\prod_{\kappa}\int\mathcal{D}(z_{\kappa},\bar{z}_{\kappa})e^{-S_{\kappa}}\,.$
(25)

Since the trajectories are periodic $z(0)=z(\beta)$,
$\bar{z}(0)=\bar{z}(\beta)$, we can expand the fields as a Fourier series

$\displaystyle
z_{\kappa}(u)=\frac{1}{\sqrt{\beta}}\sum_{n}z_{\kappa}(\omega_{n})e^{i\omega_{n}u}\,,$
(26)
$\displaystyle\bar{z}_{\kappa}(u)=\frac{1}{\sqrt{\beta}}\sum_{n}\bar{z}_{\kappa}(\omega_{n})e^{-i\omega_{n}u}\,.$
(27)

with the Matsubara frequencies $\omega_{n}=\frac{2\pi n}{\beta}$. Inserting
Eqs. (26), (27) in the action (23) yields

$S_{\kappa}=\sum_{n}\left(\left(i\omega_{n}+\tilde{\omega}_{\bm{k}}\right)\bar{z}_{\kappa}(\omega_{n})z_{\kappa}(\omega_{n})-c_{\bm{k}}z_{\kappa}(\omega_{n})C_{\kappa}(\omega_{n})-c_{\bm{k}}\bar{z}_{\kappa}(\omega_{n})\bar{C}_{\kappa}(\omega_{n})\right)\,,$
(28)

where

$\displaystyle
C_{\kappa}(\omega_{n})=\frac{1}{\sqrt{\beta}}\int_{0}^{\beta}duC_{\kappa}(u)e^{i\omega_{n}u}\,,$
(29)
$\displaystyle\bar{C}_{\kappa}(\omega_{n})=\frac{1}{\sqrt{\beta}}\int_{0}^{\beta}du\bar{C}_{\kappa}(u)e^{-i\omega_{n}u}\,.$
(30)

The Jacobian of the Fourier transformation is unity, so the functional
differential over temperature trajectories in Eq. (25) is simply replaced with
the differential over frequency trajectories. The result is simply a
collection of $n$-dimensional Gaussian integrals for each mode $\kappa$,
yielding

$\prod_{\kappa}\int\mathcal{D}(z_{\kappa},\bar{z}_{\kappa})e^{-S_{\kappa}}=\prod_{\kappa}Z_{\kappa}\exp\left[\frac{c_{\bm{k}}^{2}}{\beta\tilde{\omega}_{\bm{k}}}\iint_{0}^{\beta}dudu^{\prime}C_{\kappa}(u)\bar{C}_{\kappa}(u^{\prime})K_{\kappa}(u-u^{\prime})\right]\,,$
(31)

with the kernel

$K_{\kappa}(u-u^{\prime})=\sum_{n}\frac{\tilde{\omega}_{\bm{k}}}{i\omega_{n}+\tilde{\omega}_{\bm{k}}}e^{i\omega_{n}(u-u^{\prime})}\,.$
(32)

At this point, the prefactor $Z_{0}$ defined in Eq. (24) acquires meaning:
$Z_{0}=\prod_{\kappa}Z_{\kappa}$, where
$Z_{\kappa}=\left[1-\exp(-\beta\tilde{\omega}_{\bm{k}})\right]^{-1}$ is the
partition function of a Harmonic oscillator of frequency
$\tilde{\omega}_{\bm{k}}$. We can now put together the effective action as
defined in Eq. (24)

$S_{\rm
eff}=S_{M}-\sum_{\kappa}\frac{c_{\bm{k}}^{2}}{\beta\tilde{\omega}_{\bm{k}}}\iint_{0}^{\beta}dudu^{\prime}C_{\kappa}(u)\bar{C}_{\kappa}(u^{\prime})K_{\kappa}(u-u^{\prime})\,.$
(33)

After tracing out the light degrees of freedom, the light-matter interaction
appears translated into a temperature-non-local interaction between matter
degrees of freedom. The elimination of the cavity allows us to study the
modification of material properties from the point of view of the matter
subsystem. However, the non locality of the effective interaction implies that
Eq. 33 does not have straightforward analytical applicability. In particular,
we cannot recover an equivalent Hamiltonian. That is, we cannot undo our
agnostic substitutions $\mathcal{\hat{H}}_{M}^{\prime}\to S_{M}$,
$\mathcal{\hat{C}}_{k}\to C_{k}$ and
$\mathcal{\hat{C}}_{k}^{\dagger}\to\bar{C}_{k}$, to revert the description of
the matter subsystem from an action-based one to a Hamiltonian-based one. This
is because not all fields depend on the same temperature in this non-local
action. This should not come as a surprise, since in classical statistical
mechanics when integrating a subpart of the whole system the magnitudes that
involve the remaining DOFs can be calculated from a temperature-dependent
effective Hamiltonian [53]. Equivalently, in bipartite quantum systems, the
(temperature dependent) Hamiltonian of mean force describes the equilibrium
properties of a partition [54, 55]. On the other hand, (33) is formally
analogous to the effective action after tracing out the environment DOFs in
the path integral formulation of open quantum systems [44].

What is less intuitive is what we show in the next subsection, namely that the
kernel becomes local in the thermodynamic limit $N\to\infty$, recovering a
time-local effective interaction with a clear Hamiltonian equivalent.

###  IV.2 Obtaining an effective matter Hamiltonian in the large-$N$ limit

The effective action (33) only depends on $N$ through the scaling of the
involved fields, which they inherit from the corresponding operators. We
assume that the light-matter Hamiltonian (17) is not ill-defined in the
thermodynamic limit, i.e. that all its terms scale as $N$. Otherwise some of
them would become negligible for $N\to\infty$, rendering the system exactly
solvable and, thus, of little analytical interest. It is also important to
note that in order to have a macroscopically well-defined thermodynamic limit
the density of particles in the material $\rho=N/V$ must be finite, and thus
the volume $V$ must scale like the number of particles $N$, i.e. $V\to\infty$
in the thermodynamic limit. This implies that $b$, $b^{\dagger}$ scale as
$\sqrt{N}$ and $c_{\bm{k}}$ scales as $1/\sqrt{N}$, with both
$\mathcal{\hat{H}}_{M}^{\prime}$ and $\mathcal{\hat{C}}_{\kappa}$ scaling as
$N$.

To understand how a large $N$ affects the effective action it is best to
exploit the scaling properties of the photonic operators before they are
traced out. This is done following the reasoning of Wang and Hioe [25].
Because the photonic operators scale as $\sqrt{N}$, one can define rescaled
operators $b/\sqrt{N}$, $b^{\dagger}/\sqrt{N}$ whose $N\to\infty$ limit is
well-defined and whose commutator scales as $1/N$, and thus vanishes in the
large-$N$ limit. This allows one to write

$\int\prod_{\kappa}\frac{d^{2}z_{\kappa}}{\pi}\bra{\bm{z}}e^{-\beta\mathcal{\hat{H}}}\ket{\bm{z}}=\int\prod_{\kappa}\frac{d^{2}z_{\kappa}}{\pi}e^{-\beta\expectationvalue{\mathcal{\hat{H}}}{\bm{z}}}\,.$
(34)

Defining

$\tilde{Z}=\Tr_{M}\left(\int\prod_{\kappa}\frac{d^{2}z_{\kappa}}{\pi}e^{-\beta\expectationvalue{\mathcal{\hat{H}}}{\bm{z}}}\right)\,,$
(35)

with $\ket{\bm{z}}=\bigotimes_{\kappa}\ket{z_{\kappa}}$ and

$\expectationvalue{\mathcal{\hat{H}}}{\bm{z}}=\mathcal{\hat{H}}_{M}^{\prime}+\sum_{\kappa}\tilde{\omega}_{\bm{k}}|z_{\kappa}|^{2}+\sum_{\kappa}c_{\bm{k}}\left(\mathcal{\hat{C}}_{\kappa}z_{\kappa}+h.c.\right)\,,$
(36)

we note that taking the trace over matter of the lhs and rhs of Eq. (34)
yields the equality $Z=\tilde{Z}$ (Cf. (22)). The same equality can be
obtained from the bounds derived by Hepp and Lieb in their original study of
equilibrium superradiance [26]

$\tilde{Z}\leq Z\leq e^{\beta\sum_{\kappa}\omega_{\bm{k}}}\tilde{Z}\,,$ (37)

In the thermodynamic limit $N\to\infty$ and provided the number of modes does
not scale with $N$, the correction term
$\exp\left[\beta\sum_{\kappa}\omega_{\bm{k}}\right]$ in the upper bound
becomes negligible and the partition function is given exactly by
$Z=\tilde{Z}$. With this equality at hand, we backtrack to Eqs. (23) and (24)
and note that if one assumes constant photon fields, the path integral becomes
a Gaussian integral over the initial states and Eq. (24) becomes precisely
$\tilde{Z}$. Thus, in the path integral formulation, the large-$N$ limit
implies constant paths for the photons. This resembles the classical limit in
the coherent state path integral formulation of fields [56, chapter 2]. We may
enforce constant fields at any point in the derivation of the effective
theory. In particular, we may fulfill our initial desire of finding the
large-$N$ limit of the final effective action (33). To do so, it is important
to realize that the constant field condition equates to truncating the
Matsubara summation to the $n=0$ mode or, equivalently, to setting
$\omega_{n}=0$ for all $n$. This realization will allow us to enforce the
consequences of the large-$N$ limit on the non-local kernel (32). Before doing
so, it is convenient to do some manipulations

$K_{\kappa}(\tau)=\sum_{n}\frac{\tilde{\omega}_{\bm{k}}^{2}e^{i\omega_{n}\tau}}{\omega_{n}^{2}+\tilde{\omega}_{\bm{k}}^{2}}-\sum_{n}\frac{i\omega_{n}\tilde{\omega}_{\bm{k}}e^{i\omega_{n}\tau}}{\omega_{n}^{2}+\tilde{\omega}_{\bm{k}}^{2}}=\bar{K}_{\kappa}(\tau)-\frac{1}{\tilde{\omega}_{\bm{k}}}\partial_{\tau}\bar{K}_{\kappa}(\tau)\,.$
(38)

The first term can be split into a collection of delta functions and another
kernel

$\bar{K}_{\kappa}(\tau)=\sum_{n}e^{i\omega_{n}\tau}-\sum_{n}\frac{\omega_{n}^{2}e^{i\omega_{n}\tau}}{\omega_{n}^{2}+\tilde{\omega}_{\bm{k}}^{2}}=\beta\sum_{n}\delta(\tau-\beta
n)-\bar{\bar{K}}_{\kappa}(\tau)\,.$ (39)

The delta functions provide the local action that we seek when integrating
over the interval $[0,\beta]$. Crucially, $\bar{\bar{K}}_{\kappa}(\tau)=0$ in
the large-$N$ limit. To see this, recall that the large-$N$ limit equates to
restricting to constant photonic fields, which in turn equates to setting
$\omega_{n}=0$ for all $n$. Returning to the second term in (38), because it
is odd in $\tau$, it can be seen that it does not contribute to the action
when the matter coupling operators are Hermitian and thus
$C_{\kappa}=\bar{C}_{\kappa}$. In a more general case, it can contribute.
Regardless, in the large-$N$ limit, it vanishes for the same reason as
$\bar{\bar{K}}_{\kappa}$ and the action is purely local.

So in the thermodynamic limit the effective action becomes local in time

$S_{\rm
eff}=S_{M}-\sum_{\kappa}N\frac{c_{\bm{k}}^{2}}{\tilde{\omega}_{\bm{k}}}\int_{0}^{\frac{\beta}{N}}dvC_{\kappa}(v)\bar{C}_{\kappa}(v)=S_{M}-\sum_{\kappa}\frac{c_{\bm{k}}^{2}}{\tilde{\omega}_{\bm{k}}}\int_{0}^{\beta}duC_{\kappa}(u)\bar{C}_{\kappa}(u)\,.$
(40)

Accordingly, we can recover the corresponding effective Hamiltonian

$\mathcal{\hat{H}}_{\rm
eff}=\mathcal{\hat{H}}_{M}^{\prime}-\sum_{\kappa}\frac{c_{\bm{k}}^{2}}{\tilde{\omega}_{\bm{k}}}\mathcal{\hat{C}}_{k}\mathcal{\hat{C}}_{k}^{\dagger}\,,$
(41)

such that $Z=Z_{0}\Tr_{M}\left[\exp(-\beta\mathcal{\hat{H}}_{\rm
eff})\right]$. An alternative derivation of the effective Hamiltonian can be
found in App. B. Note also that what we show here is just a particular way of
deriving the effective Hamiltonian, but constant photon paths can be enforced
at any point along the derivation of the effective action, e.g. at the level
of the kernel but before extracting the Dirac deltas. In those cases –the
calculation may involve the use of the Hubbard–Stratonovich transformation–
one always ends up with a Gaussian integral over photonic DOFs akin to the one
discussed in App. B which also leads to the same effective Hamiltonian.

Equation (41) shows that in the large-$N$ limit the effect of the cavity can
be described via an effective Hamiltonian in which the induced interactions
are explicit (second term on the r.h.s. of Eq. (41)). Notice that sometimes
similar effective interactions are argued by invoking approximations, such as
perturbation or mean field theory. Here we show that they are exact in the
thermodynamic limit for the matter. Besides, it is a rather general result. We
emphasize that the cavity-matter coupling has been discussed under general
grounds in Section III, and Eq. (41) follows from it.

###  IV.3 Relation between matter and light observables

Our effective theory puts emphasis on the behaviour of the matter subsystem,
to the point where the cavity is eliminated from the Hamiltonian. However, for
a complete characterization of the full system one must be able to compute
photonic fields, $\expectationvalue{b_{\kappa}}$,
$\expectationvalue{b_{\kappa}^{\dagger}}$, and the photon number which
determine the intensity of the electromagnetic fields in the cavity. We
reconcile this neccesity with our formalism by noting that in the process of
tracing out the photonic degress of freedom we can extract a relationship
between the expectation values of matter and light observables. To illustrate
the idea, let us work out the case of computing $\langle
b_{\kappa^{\prime}}+b_{\kappa^{\prime}}^{\dagger}\rangle$. We extend our
Hamiltonian to the form
$\mathcal{\hat{H}}(\lambda)=\mathcal{\hat{H}}-\lambda(b_{\kappa^{\prime}}+b_{\kappa^{\prime}}^{\dagger})$,
such that we can write, as is customary in statistical mechanics,
$\beta\langle
b_{\kappa^{\prime}}+b_{\kappa^{\prime}}^{\dagger}\rangle=\lim_{\lambda\to
0}\partial_{\lambda}\ln Z(\lambda)$, with
$Z(\lambda)=\Tr(\exp[-\beta\mathcal{\hat{H}}(\lambda)])$. At the same time, we
can rewrite $\mathcal{\hat{H}}(\lambda)$ as

$\mathcal{\hat{H}}(\lambda)=\mathcal{\hat{H}}_{M}^{\prime}+\sum_{\kappa}\tilde{\omega}_{\bm{k}}\hat{b}_{\kappa}^{\dagger}\hat{b}_{\kappa}-\sum_{\kappa}c_{\bm{k}}\left(\mathcal{\hat{C}}_{\kappa}(\lambda)\hat{b}_{\kappa}+h.c.\right)\,,$
(42)

where
$\mathcal{\hat{C}}_{\kappa}(\lambda)=\mathcal{\hat{C}}_{\kappa}-\delta_{\kappa,\kappa^{\prime}}\lambda/c_{\bm{k}}$.
Following our theory, the corresponding effective Hamiltonian is given by

$\mathcal{\hat{H}}_{\rm
eff}(\lambda)=\mathcal{\hat{H}}_{M}^{\prime}-\sum_{\kappa}\frac{c_{\bm{k}}^{2}}{\tilde{\omega}_{\bm{k}}}\mathcal{\hat{C}}_{\kappa}(\lambda)\mathcal{\hat{C}}_{\kappa}^{\dagger}(\lambda)\,.$
(43)

Then, following our definition of the effective action (41), we can express
the partition function as
$Z(\lambda)=Z_{0}\Tr_{M}\left(\exp[-\beta\mathcal{\hat{H}}_{\rm
eff}(\lambda)]\right)$, such that

$\expectationvalue{b_{\kappa^{\prime}}+b_{\kappa^{\prime}}^{\dagger}}=\lim_{\lambda\to
0}Z(\lambda)^{-1}Z_{0}\Tr_{M}\left[-\frac{c_{\bm{k}^{\prime}}}{\tilde{\omega}_{\bm{k}^{\prime}}}\left(\mathcal{\hat{C}}_{\kappa^{\prime}}(\lambda)+\mathcal{\hat{C}}_{\kappa^{\prime}}^{\dagger}(\lambda)\right)e^{-\beta\mathcal{\hat{H}}_{\rm
eff}(\lambda)}\right]=-\frac{c_{\bm{k}^{\prime}}}{\tilde{\omega}_{\bm{k}^{\prime}}}\expectationvalue{\mathcal{\hat{C}}_{\kappa^{\prime}}+\mathcal{\hat{C}}_{\kappa^{\prime}}^{\dagger}}_{M}\,.$
(44)

Proceeding in similar fashion, we can compute other observables such as

$\expectationvalue{b_{\kappa^{\prime}}-b_{\kappa^{\prime}}^{\dagger}}=\frac{c_{\bm{k}^{\prime}}}{\tilde{\omega}_{\bm{k}^{\prime}}}\expectationvalue{\mathcal{\hat{C}}_{\kappa^{\prime}}-\mathcal{\hat{C}}_{\kappa^{\prime}}^{\dagger}}_{M}\,.$
(45)

Another observable of interest is the photon number of the $\kappa$-th mode

$\expectationvalue{b_{\kappa^{\prime}}^{\dagger}b_{\kappa^{\prime}}}=-\frac{1}{\beta}\partial_{\tilde{\omega}_{\bm{k}^{\prime}}}\ln
Z=n_{B}(\beta,\tilde{\omega}_{\bm{k}^{\prime}})+\left(\frac{c_{\bm{k}^{\prime}}}{\tilde{\omega}_{\bm{k}^{\prime}}}\right)^{2}\expectationvalue{\mathcal{\hat{C}}_{\kappa^{\prime}}\mathcal{\hat{C}}_{\kappa^{\prime}}^{\dagger}}_{M}\,,$
(46)

where $n_{B}$ is the bosonic occupation number of a free Harmonic oscillator

$n_{B}(\beta,\tilde{\omega}_{\bm{k}^{\prime}})=\frac{1}{e^{\beta\tilde{\omega}_{\bm{k}^{\prime}}}-1}\,.$
(47)

We find that the photon number has two sources at finite temperature, one
corresponding to the thermal fluctuations of the collection of free Harmonic
oscillators that constitute the cavity, and another arising from the coupling
to matter.

These relations may be important for computing measurable outputs of a cavity
QED material. For example, if we probe the cavity-matter system through the
transmissivity of the former, the transmitted signal is proportional to the
dynamical response of the system. This is in turn related to two point
correlators of bosonic fields that can be directly computed from relation (46)
or analogues.

##  V Testing the validity of the effective theory

Effective theories are a common tool in theoretical physics. To contextualize
ours, we compare it, analytically and numerically, against other common ones
in the fields of condensed matter and quantum optics. Importantly, we also
perform a numerical analysis of its finite-size scaling. Because other
effective theories are model dependent and to facilitate the numerical
analysis, our testbed here will be the Dicke model,

$\mathcal{\hat{H}}=\frac{\omega_{z}}{2}\sum_{j}\hat{\sigma}_{j}^{z}+\omega_{c}\hat{a}^{\dagger}\hat{a}+\frac{\lambda}{\sqrt{N}}\sum_{j}\hat{\sigma}_{j}^{x}\left(\hat{a}+\hat{a}^{\dagger}\right)\,.$
(48)

In the thermodynamic limit, it is exactly solvable, with the free energy per
site given by [57]

$\begin{dcases}-\beta
f=\ln\left[2\cosh(\beta\frac{\omega_{z}}{2})\right]&\text{if}\quad\lambda<\lambda_{c}\quad\text{or}\quad\beta>\beta_{c}\,,\\\
-\beta
f=\ln\left[2\cosh\left(\beta\frac{4\lambda^{2}}{\omega_{c}}\sigma\right)\right]-\beta\frac{4\lambda^{2}}{\omega_{c}}\sigma^{2}+\beta\frac{\omega_{c}\omega_{z}^{2}}{16\lambda^{2}}&\text{if}\quad\lambda>\lambda_{c}\quad\text{and}\quad\beta<\beta_{c}\,,\end{dcases}$
(49)

where $\lambda_{c}$ is given by $4\lambda_{c}^{2}=\omega_{c}\omega_{z}$,
$\beta_{c}$ is given by
$\omega_{c}\omega_{z}=4\lambda^{2}\tanh(\beta_{c}\omega_{z}/2)$ provided
$\lambda>\lambda_{c}$ and $\sigma$ is found by solving
$2\sigma=\tanh(4\beta\lambda^{2}\sigma/\omega_{c})\neq 0$. The Dicke model,
besides being a well-known toy model, has the added benefit of being
numerically amenable, by virtue of conserving the total spin
$\hat{\bm{S}}^{2}=\hat{S}_{x}^{2}+\hat{S}_{y}^{2}+\hat{S}_{z}^{2}$. Where
$\hat{S}_{\nu}=\frac{1}{2}\sum_{j}\hat{\sigma}_{j}^{\nu}$. The numerical
diagonalization can be done on each spin $S$ sector separately, with the total
partition function given by [24]

$Z=\sum_{S=s_{0}}^{N/2}\Omega(S,N)Z_{S}\,,$ (50)

where $s_{0}=0$ ($1/2$) if $N$ is even (odd) and

$\Omega(S,N)=\frac{N!(2S+1)}{\left(N/2-S\right)!\left(N/2+S+1\right)!}\,.$
(51)

Using total spin operators, the Hamiltonian can be rewritten as

$\mathcal{\hat{H}}=\omega_{z}\hat{S}_{z}+\omega_{c}\hat{a}^{\dagger}a+2g\hat{S}_{x}\left(\hat{a}+\hat{a}^{\dagger}\right)\,,$
(52)

With $g=\lambda/\sqrt{N}$. The corresponding effective Hamiltonian, according
to our theory, reads

$\mathcal{\hat{H}}_{\rm
eff}=\omega_{z}\hat{S}_{z}-\frac{4g^{2}}{\omega_{c}}\hat{S}_{x}^{2}\,.$ (53)

A useful tool to facilitate the numerical diagonalization of the full Dicke
model is the Polaron transformation. The Polaron transformation is given by
$\hat{U}_{\rm P}=\exp(-\hat{\alpha}\hat{S}_{x})$ with
$\hat{\alpha}=2\zeta(\hat{a}^{\dagger}-\hat{a})$ and $\zeta=g/\omega_{c}$. The
resulting effective Hamiltonian $\mathcal{\hat{H}}_{\rm P}=\hat{U}_{\rm
P}^{\dagger}\mathcal{\hat{H}}\hat{U}_{\rm P}$ reads

$\mathcal{\hat{H}}_{\rm
P}=\omega_{z}\left(\hat{S}_{z}\cosh\hat{\alpha}-i\hat{S}_{y}\sinh\hat{\alpha}\right)+\omega_{c}\hat{a}^{\dagger}a-\frac{4g^{2}}{\omega_{c}}\hat{S}_{x}^{2}\,.$
(54)

The Polaron frame is particularly well-suited for exact diagonalization
because the displacement of the bosons by $\hat{S}_{x}$ cancels the finite
expectation value of $\hat{a}$, $\hat{a}^{\dagger}$ in the ordered phase. As a
result, the number of bosons in the Polaron frame is only dependent on the
temperature and the bosonic state-space can be truncated to a much smaller
excitation cutoff than would otherwise be required [19, 58]. This, paired with
the conservation of the total spin, allows one to reach system sizes of $N\sim
100$. As we show below, this is effectively "close" to the thermodynamic
limit. For this reason, we use the Hamiltonian in the Polaron frame (54) to do
exact diagonalization of the full Dicke model.

###  V.1 Comparison with other effective theories

First, let us discuss the taxonomy of effective theories. We can distinguish
two families: Hamiltonian theories, with unitary dynamics; and descriptions
based on master equations and non-Hermitian Hamiltonians, with dissipative
dynamics [59]. Because our theory belongs to the former, we will only consider
other Hamiltonian theories in the comparison. We can also distinguish between
perturbative and non-perturbative theories. In this regard, it must be noted
that our effective Hamiltonian was derived in the $N\to\infty$ limit. As such,
it can be regarded as a zeroth order perturbation theory with $1/N$ as the
perturbative parameter. The extension to a full-fledged perturbation theory
that can be taken to any order in $1/N$ is a work in progress. However, in
this context we will use the term perturbative to refer only to parameters
that affect the light-matter coupling regime, such as the light-matter
coupling strength or the detuning between the cavity frequency and the
characteristic energy scale of the matter subsytem. It is in this sense that
our theory is non-perturbative. The simplest example of a perturbative theory
would be standard perturbation theory on the light-matter coupling. To yield
an effective Hamiltonian with perturbation theory, a separation of energy
scales (large detuning) is required, with the more energetic sector of the
spectrum mediating effective interactions on a low-energy subspace [60, 61]. A
less cumbersome way to obtain the same effective description is by means of a
Schrieffer–Wolff unitary transformation [62]. We will thus use the latter in
our comparison. Regarding non-perturbative alternatives to our effective
theory, it is interesting to consider the Polaron transformation itself [19].
For the Dicke model, $\expectationvalue{\hat{a}^{\dagger}-\hat{a}}$ vanishes
in the thermodynamic limit, so $\hat{\alpha}$ in Eq. (54) vanishes as well.
This means that, in the $N\to\infty$ limit, light and matter decouple in the
Polaron Hamiltonian (54) and it coincides with our effective Hamiltonian (53)
.

The aforementioned methods, namely the Schrieffer–Wolff and Polaron
transformations, have the downside of being model-dependent. This is because
the application of the unitary tranformations onto the original Hamiltoninan
requires knowledge about the commutation relation between the matter part of
the Hamiltonian and the matter part of the transformations. The matter
subsystem must be specified a priori because knowledge of its algebra is
required to obtain the effective theory. In contrast, our effective theory was
obtained in a model-independent fashion. In all fairness, standard
perturbation theory in the fast-cavity limit can also yield an effective
matter model without knowledge of the matter subsystem. But in general,
perturbation theory is model dependent [61].

For the Dicke model, the Schrieffer-Wolf transformation is given by
$\hat{U}_{\rm
SW}=\exp(-\xi_{-}\mathcal{\hat{X}}_{-}-\xi_{+}\mathcal{\hat{Y}}_{-})$, with
$\xi_{\pm}=g/(\omega_{c}\pm\omega_{z})$ and

$\displaystyle\mathcal{\hat{X}}_{\pm}=\hat{S}_{-}\hat{a}^{\dagger}\pm\hat{S}_{+}\hat{a}\,,$
(55)
$\displaystyle\mathcal{\hat{Y}}_{\pm}=\hat{S}_{+}\hat{a}^{\dagger}\pm\hat{S}_{-}\hat{a}\,.$
(56)

The resulting effective Hamiltonian $\mathcal{\hat{H}}_{\rm SW}=\hat{U}_{\rm
SW}^{\dagger}\mathcal{\hat{H}}\hat{U}_{\rm SW}$ can be expanded perturbatively
and truncated at first order in $\xi_{\pm}$, yielding

$\mathcal{\hat{H}}_{\rm
SW}\approx\omega_{z}\hat{S}_{z}+\omega_{c}\hat{a}^{\dagger}a-\frac{2g^{2}\omega_{z}}{\omega_{c}^{2}-\omega_{z}^{2}}\hat{S}_{z}\left(\hat{a}+\hat{a}^{\dagger}\right)^{2}-\frac{4g^{2}\omega_{c}}{\omega_{c}^{2}-\omega_{z}^{2}}\hat{S}_{x}^{2}\,.$
(57)

Consequently, the parameters $\xi_{\pm}$ must be small, which requires a large
detuning $|\omega_{c}-\omega_{z}|\gg g$.

In the fast-cavity limit $\omega_{c}\gg\omega_{z}$, the Hamiltonian simplifies
to

$\mathcal{\hat{H}}_{\rm
SW}\approx\omega_{z}\hat{S}_{z}+\omega_{c}\hat{a}^{\dagger}\hat{a}-\frac{4g^{2}}{\omega_{c}}\hat{S}_{x}^{2}\,.$
(58)

The spin and the cavity decouple and the matter part coincides with
$\mathcal{\hat{H}}_{\rm eff}$. Taking the route of standard perturbation
theory in this regime of detuning, one arrives at $\mathcal{\hat{H}}_{\rm
eff}$ as well.

The slow-cavity regime is rather nuanced. For starters, there is no light-
matter decoupling in this limit

$\mathcal{\hat{H}}_{\rm
SW}\approx\omega_{z}\hat{S}_{z}+\omega_{c}\hat{a}^{\dagger}a+\frac{2g^{2}}{\omega_{z}}\hat{S}_{z}\left(\hat{a}+\hat{a}^{\dagger}\right)^{2}\,.$
(59)

In addition, the resulting model is ill-defined. The Hamiltonian commutes with
$\hat{S}_{z}$, so one can fix the quantum number $m_{z}$, defined as
$\hat{S}_{z}\ket{S,m_{z}}=m_{z}\ket{S,m_{z}}$. The Hamiltonian for an $m_{z}$
sector is quadratic in bosonic operators and can be diagonalized with a
Bogoliubov transformation, $\mathcal{\hat{H}}_{\rm
SW}(m_{z})=\epsilon(m_{z})\hat{b}^{\dagger}\hat{b}$. The resulting boson has a
frequency

$\epsilon(m_{z})=\sqrt{\omega_{c}^{2}+\frac{8g^{2}\omega_{c}}{\omega_{z}}m_{z}}\,,$
(60)

that vanishes for $m_{z}=-S$ at the critical coupling of the Dicke model
($4\lambda_{c}^{2}=\omega_{c}\omega_{z}$) and is imaginary thereafter. The
vanishing is a proper signature of the phase transition but becoming imaginary
after is a symptom of the Hamiltonian being ill-defined. This is prevented by
keeping the $\hat{S}_{x}^{2}$ term of $\mathcal{\hat{H}}_{\rm SW}$ that was
neglected in passing from Eq. (57) to Eq. (59). Thus, in the slow-cavity limit
the Schrieffer–Wolff Hamiltonian does not admit simplifications nor does it
coincide with our effective Hamiltonian. This is a manifestation of the
underlying perturbation theory, which requires a separation of energy scales
that is unattainable when the cavity is taken as the less energetic subsystem,
because its number of excitations (its energy) its unbounded. Additionally,
the regime of resonant cavity and spins is not accessible to the
Schrieffer–Wolff theory. These differences are numerically confirmed in Fig. 2
for the free energy per site of the Dicke model at high and low temperatures.
We observe that the two theories agree in the fast-cavity regime, as expected,
and deviate in the slow-cavity regime. Importantly, it is the Schrieffer–Wolff
theory that deviates significantly from the full Dicke model past the critical
point, whereas our effective theory shows the correct asymptotic behaviour.
Let us emphasize again that the Schrieffer–Wolff theory is being used as a
convenient proxy for standard perturbation theory.

The Schrieffer–Wolff Hamiltonian (57) requires a larger cutoff than the
Polaron Hamiltonian, specially in the slow-cavity regime. For this reason, the
system size has been kept at a conservative $N=30$ for the comparison in Fig.
2 and the bosonic cutoff has been adjusted to reach convergence of the free
energy. In practice, this meant a cutoff of $N_{\rm ph}=100$ in the worst
case.

###  V.2 Finite $N$ and approach to the thermodynamic limit

For an effective Hamiltonian that is only exact in the thermodynamic limit, it
is interesting to test its finite-size scaling. We continue the numerical
analysis on the Dicke model and in Fig. 3 we focus solely on our effective
theory, which allows us to lower the cutoff to $N_{\rm ph}=10$ and explore
much larger system sizes. We distinguish three regimes of fast cavity,
resonance and slow cavity and find that each presents a characteristic finite-
size scaling. In the fast cavity limit, the effective theory quickly converges
to the full Dicke model, already for $N=30$, and then they both slowly
converge to the $N\to\infty$ analytical solution later, with relative
differences of $\sim 2\%$ at the critical point for $N=150$. As one moves
towards the slow-cavity regime, the convergence of the effective theory to the
full Dicke model slows down, whereas the convergence of the full Dicke model
to the $N\to\infty$ analytical solution accelerates. Here we have focused in
the free energy at low temperature, which for $\beta\omega_{c}=5$ is
practically equivalent to the ground-state energy, because it is the regime in
which the effective theory is most challenged to meet the full Dicke model. In
contrast, at high temperature convergence is obtained much faster, akin to the
fast-cavity regime at low temperature, as can be seen in the top row of Fig. 2
already at $N=30$. In summary, we find that the effective theory scales well
with $N$, with relative differences with the full Dicke model $<1\%$ at the
critical point for $N\sim 100$ in every regime. In the high temperature and/or
fast-cavity regime, the convergence is even faster, ocurring at $N\sim 30$.

These numerical results imply an important advantage of our effective theory
in regards to its applicability to any light-matter coupling regime, hence the
surname non-perturbative. They also prove that it is not equivalent to
standard perturbation theory. As a caveat, we acknowledge that the theory has
only been numerically compared and tested in the Dicke model, although we
expect the theory to be equally applicable to other systems. As a matter of
fact, in the next section we employ the theory to rederive and generalize
results in other models.

Figure 2: Comparison of the free energy per site computed by exact
diagonalization of our effective Hamiltonian, exact diagonalization of the
Schrieffer–Wolff Hamiltonian, exact diagonalization of the Dicke model in the
Polaron frame and the exact analytical solution in the thermodynamic limit.
The system size is $N=30$ and the cutoff is $N_{\rm ph}=100$ throughout.
Figure 3: Finite-size scaling of the free energy per site computed with our
effective theory. Top: Maximum relative difference between the free energies
computed by exact diagonalization of our effective Hamiltonian, exact
diagonalization of the Dicke model in the Polaron frame and the exact
analytical solution in the thermodynamic limit. Bottom: Plots of the free
energy at small and large system sizes. a) In the fast-cavity regime
$\omega_{z}/\omega_{c}=1/7$. b) At resonance $\omega_{z}/\omega_{c}=1$. c) In
the slow-cavity regime $\omega_{z}/\omega_{c}=2$. For reasons of numerical
intractability, values for $N=150$ were unattainable in the slow-cavity
regime. The temperature is $\beta\omega_{c}=5$ and the cutoff is $N_{\rm
ph}=10$ throughout.

##  VI Applications

Here, we showcase the effectiveness of the formalism developed in Section IV
in the study of several models that have received recent attention in the
literature, namely: equilibrium superradiance (photon condensation) with
electric coupling, the 2D free electron gas in a cavity, and the modification
of interactions in magnetic materials coupled to a cavity (magnetic cavity
QED).

###  VI.1 Equilibrium superradiance (photon condensation) with electric
coupling

Despite its long history, the problem of the existence (or lack thereof) of a
superradiant phase transition when an ensemble of atoms are immersed in a
cavity has attracted renewed attention. After 50 years from its original
theoretical description, the transition has not been experimentally measured
[63] and there is vivid debate to elucidate the reason. So far, the
understanding is as follows. If one adopts the correct gauge-invariant
description of a single uniform cavity mode, the phase transition, defined as
the appearance of a finite population of transverse photons, is forbidden.
This has been proven true even without the dipole approximation [28, 37, 49,
32]. This transition is often referred to as _equilibrium superradiance_ or
_photon condensation_ , although some ambiguity still surrounds these terms
[36]. Moreover, critical conditions have been obtained in the case of non-
uniform cavity fields [29, 30, 31]. Subtleties arise from the fact that the
separation between light and matter subsystems is manifestly gauge-dependent.
For that reason, we prefer to characterize the phase transition in terms of
gauge invariant observables such as the transverse photon number. However,
although theoretically valid, this choice suffers from the fact that not all
observables, in particular the transverse photon number, are experimentally
measurable, a handicap that challenges their validity as signatures of a phase
transition. To avoid this pitfall, we propose to look at this issue from a
different perspective: instead of arguing whether the phase transition has a
“light” or a “matter” signature, we will simply compare the behaviour of the
material when it is placed outside a cavity, i.e. in a region of space with an
unconfined electromagnetic field, against its behaviour when it is placed
inside a cavity, i.e. in a region of space where the electromagnetic field is
confined. Note that this is _the experimentally relevant question_. In fact,
alterations of matter and condensation of photons are two sides of the same
coin.

Our formalism is particularly well suited to answer this question because it
eliminates the cavity, reformulating the model in terms of the original (in
free space) matter Hamiltonian plus an effective interaction mediated by the
cavity. Whether or not a material behaves differently after placing it inside
a cavity will be answered by studying the impact, or lack thereof, that the
effective coupling has in its behaviour.

We center on the study on a matter model such as the one in Eq. 17, where the
Zeeman coupling to the electron spins is neglected. Accordingly, we have
$c_{\bm{k}}=\lambda_{\bm{k}}^{-1/2}c_{\bm{k}}^{e}$ and

$\mathcal{\hat{C}}_{\kappa}=\sum_{j}\sqrt{\frac{1}{m\omega_{\bm{k}}}}\hat{\bm{p}}_{j}\hat{\bm{u}}_{\kappa}(\hat{\bm{r}}_{j})\,.$
(61)

Then, in the thermodynamic limit, the effective matter Hamiltonian, according
to Eq. 41, is

$\begin{split}\mathcal{\hat{H}}_{\rm
eff}=\mathcal{\hat{H}}_{M}^{\prime}-\sum_{\kappa}\frac{(c_{\bm{k}}^{e})^{2}}{\omega_{\bm{k}}+4\Delta_{\bm{k}}}\sum_{ij}\frac{1}{m\omega_{\bm{k}}}(\hat{\bm{p}}_{j}\hat{\bm{u}}_{\kappa}(\hat{\bm{r}}_{j}))(\hat{\bm{u}}_{\kappa}^{\dagger}(\hat{\bm{r}}_{i})\hat{\bm{p}}_{i})\end{split}$
(62)

We have an effective all-to-all coupling between the electrons’ momenta. The
strength of the coupling between a pair of electrons depends on their
positions. We treat here the case of the electron gas, but the results that
follow extend to the case of localized atoms as they are independent of
whether the EM fields depend on the position of the electrons dynamically or
parametrically.

####  VI.1.1 Uniform cavity fields

If we make the approximation to a setup with a single uniform mode $\bm{k}\to
0$, $\sigma=1$, we end up with

$\mathcal{\hat{H}}_{\rm
eff}=\mathcal{\hat{H}}_{M}^{\prime}-\frac{(c_{0}^{e})^{2}}{\omega_{0}+4\Delta_{0}}\sum_{ij}\frac{1}{m\omega_{0}}(\bm{e}_{1}\hat{\bm{p}}_{j})(\bm{e}_{1}\hat{\bm{p}}_{i})\,,$
(63)

where $\bm{e}_{1}$ is the polarization vector of the electromagnetic mode. Now
that the spatial dependence of the coupling is no longer present, the coupling
is uniform and all-to-all, a situation that can be treated exactly with a mean
field theory in which
$\hat{\bm{p}}_{j}=\expectationvalue{\hat{\bm{p}}}+\delta\hat{\bm{p}}_{j}$.

$\begin{split}\mathcal{\hat{H}}_{\rm
eff}&=\mathcal{\hat{H}}_{M}^{\prime}-2\frac{4\Delta_{0}}{\omega_{0}+4\Delta_{0}}\sum_{j}\frac{(\bm{e}_{1}\hat{\bm{p}}_{j})\expectationvalue{\bm{e}_{1}\hat{\bm{p}}}}{2m}+N\frac{4\Delta_{0}}{\omega_{0}+4\Delta_{0}}\frac{\expectationvalue{\bm{e}_{1}\hat{\bm{p}}}^{2}}{2m}\\\
&=\sum_{j}\frac{1}{2m}\left(\hat{\bm{p}}_{j}-\frac{4\Delta_{0}}{\omega_{0}+4\Delta_{0}}\bm{e}_{1}\expectationvalue{\bm{e}_{1}\hat{\bm{p}}}\right)^{2}+\mathcal{\hat{H}}_{qq}^{\prime}+\mathcal{\hat{V}}+N\frac{4\Delta_{0}\omega_{0}}{(\omega_{0}+4\Delta_{0})^{2}}\frac{\expectationvalue{\bm{e}_{1}\hat{\bm{p}}}^{2}}{2m}\,.\end{split}$
(64)

For convenience, we have expressed the electric coupling $c_{0}^{e}$ in terms
of $\Delta_{0}$. Now we observe that

$\hat{\bm{p}}_{j}-\frac{4\Delta_{0}}{\omega_{0}+4\Delta_{0}}\bm{e}_{1}\expectationvalue{\bm{e}_{1}\hat{\bm{p}}}=[\hat{\bm{r}}_{j},\mathcal{\hat{H}}_{\rm
eff}]\,,$ (65)

and since $\langle[\hat{\bm{r}}_{j},\mathcal{\hat{H}}_{\rm eff}]\rangle=0$, we
have

$\expectationvalue{e_{1}\hat{\bm{p}}}-\frac{4\Delta_{0}}{\omega_{0}+4\Delta_{0}}\expectationvalue{\bm{e}_{1}\hat{\bm{p}}}=0\,.$
(66)

Note that this condition is only satisfied if
$\expectationvalue{e_{1}\hat{\bm{p}}}=0$. This is because

$\frac{4\Delta_{0}}{\omega_{0}+4\Delta_{0}}\leq 1\;.$ (67)

In fact, one could express the critical condition as
$\omega_{0}+4\Delta_{0}=4\Delta_{0}$, which is clearly never satisfied.
Consequently, we find that $\mathcal{\hat{H}}_{\rm
eff}=\mathcal{\hat{H}}_{M}^{\prime}$, which serves as proof that the effect of
the transverse cavity fields on the matter system is non-existent. If there is
a phase transition, its origin lies within $\mathcal{\hat{H}}_{M}^{\prime}$.
Meaning that it is either caused by intrinsic interactions in the material
$\mathcal{\hat{H}}_{M}$ or induced by electrostatic effects arising from the
interaction between real charges and image charges.

To reinforce this result, let us compare it with the case in which the
diamagnetic term is omitted in the light-matter interaction. This equates to
setting $\Delta_{0}=0$. Of course, one must then be careful not no express the
electric coupling $c_{0}^{e}$ in terms of $\Delta_{0}$ as we have done before,
otherwise we cannot zero $\Delta_{0}$ independently. Then, the critical
condition becomes

$\frac{N(c_{0}^{e})^{2}}{\omega_{0}^{2}}=1\,,$ (68)

which can be satisfied if the coupling $c_{0}^{e}$ is large enough. We have
thus not only arrived to the well-known no-go theorem for single mode
cavities, but we are also able to explain the presence of critical behaviour
when the diamagnetic term is ignored. We must also emphasize that the results
obtained here are valid at any temperature and that no assumptions have been
made in the derivation regarding the order of the phase transition. So the
result is a no-go theorem for both continuous and discontinuous phase
transitions. This generalizes the works of Refs. [28, 49] that discuss the
zero temperature case of second and first order quantum phase transitions.

####  VI.1.2 Non-uniform cavity fields

Let us return now to the multimode case described by Eq. (62). For finite, but
low-enough temperatures, minimizing the free energy is equivalent to
minimizing the energy (See SM from [22]). In that case, a phase transition
will occur if the condition $\Tr_{M}(\mathcal{\hat{H}}_{\rm
eff}\rho)\leq\Tr_{M}(\mathcal{\hat{H}}_{\rm eff}\rho_{0})$ is met, where
$\rho_{0}$ is the density matrix describing the thermal ensemble given by
$\mathcal{\hat{H}}_{M}$ and $\rho$ is a tentative density matrix describing
the thermal ensemble given by $\mathcal{\hat{H}}_{\rm eff}$. The critical
condition can be written as

$\Tr_{M}(\mathcal{\hat{H}}_{M}\rho)-\Tr_{M}(\mathcal{\hat{H}}_{M}\rho_{0})\leq\sum_{\kappa}\frac{c_{\bm{k}}^{2}}{\tilde{\omega}_{\bm{k}}}\Tr_{M}(\mathcal{\hat{C}}_{k}\mathcal{\hat{C}}_{k}^{\dagger}\rho)-\sum_{\kappa}\frac{c_{\bm{k}}^{2}}{\tilde{\omega}_{\bm{k}}}\Tr_{M}(\mathcal{\hat{C}}_{k}\mathcal{\hat{C}}_{k}^{\dagger}\rho_{0})\,.$
(69)

Making use of the relation between light and matter observables (46) this can
be reexpressed as

$\Tr_{M}(\mathcal{\hat{H}}_{M}\rho)-\Tr_{M}(\mathcal{\hat{H}}_{M}\rho_{0})\leq\sum_{\kappa}\tilde{\omega}_{\bm{k}}\expectationvalue{b_{\kappa}^{\dagger}b_{\kappa}}\,.$
(70)

Close to the phase transition, the left term can be expressed in terms of
$\langle b_{\kappa}^{\dagger}b_{\kappa}\rangle$ using the stiffness theorem
[64]. In particular, we can consider the zero temperature case, where the
critical condition is written in terms of $\ket{\psi}$ the tentative matter
ground state, and $\ket{\psi_{0}}$ the matter ground state in the absence of
light-matter coupling

$\expectationvalue{\mathcal{\hat{H}}_{M}}{\psi}-\expectationvalue{\mathcal{\hat{H}}_{M}}{\psi_{0}}\leq\sum_{\kappa}\tilde{\omega}_{\bm{k}}\expectationvalue{b_{\kappa}^{\dagger}b_{\kappa}}\,.$
(71)

Moreover, we are not restricted to considering only electric coupling as in
Eq. (61), the Zeeman coupling can be included in $\mathcal{\hat{C}}_{\kappa}$
as well, either on its own or combined with the electric coupling. With these
extension, we are able to recover the full results of Ref. [31], demonstrating
that our formalism is able to provide a complete description of the phenomenon
of photon condensation.

###  VI.2 The 2D free electron gas in a cavity

The free electron gas is a paradigmatic model in condensed matter physics. It
was originally formulated by Sommerfeld [65] for the study of the thermal and
electronic properties of materials, and it later became the building block of
the Fermi liquid theory, developed by Landau [66, 64]. Its interacting
generalization, known as the homogeneous electron gas or jellium model, is the
basis for advanced computational techniques such as density functional theory
(DFT) and the local density approximation (LDA) [67, 68]. Lowering the
dimensionality, 2D electron gases can be found in semiconductor devices in
solid state physics [69]. They also describe the integer quantum Hall effect
when subjected to strong magnetic fields and low temperature [70, 71]. Having
established that the electron gas is an interesting model exhibiting rich
phenomenology, it seems suggestive to explore the extent to which its
behaviour can be altered by coupling it to a cavity. The first steps in this
direction have already experimentally confirmed the breakdown of topological
protection by cavity vacuum fields [72], and predicted the existence of a
polaritonic Hofstadter butterfly and modified integer Hall conductance [73]
and cavity-mediated hopping [74], in the integer quantum Hall effect.

Ref. [50] is dedicated to the exact analytical solution of the 2D free
electron gas in a cavity, making it the perfect testbed for our effective
theory. In this subsection we recover their results straightforwardly with our
formalism. To adhere to the same assumptions as the authors, we will consider
a multimode description of the cavity fields, but with the peculiarity that
the wavelength of every mode be large enough that the electrons see the mode
as uniform. In that case we have
$\hat{\bm{u}}_{\kappa}(\hat{\bm{r}}_{j})\equiv\bm{e}_{\sigma}$ where
$\bm{e}_{\sigma}$ is the polarization vector of the mode, but we retain the
summation in $\kappa$ when summing over modes. In addition, the mode-mode
interactions arising from the diamagnetic term are completely disregarded. On
the matter side, free electrons are simply described by the Hamiltonian

$\mathcal{\hat{H}}_{M}=\sum_{j}\frac{\hat{\bm{p}}_{j}^{2}}{2m}\,,$ (72)

where the 2D nature of the system is reflected in the fact that the momenta
are confined to a plane: $\hat{\bm{p}}_{j}=(\hat{p}_{j,x},\hat{p}_{j,y})$. So
the complete Hamiltonian for the system is given by

$\begin{split}\mathcal{\hat{H}}=&\mathcal{\hat{H}}_{M}+\sum_{\kappa}\omega_{\bm{k}}\hat{a}_{\kappa}^{\dagger}\hat{a}_{\kappa}+\sum_{j}\sum_{\kappa}\frac{q}{m}(\hat{\bm{p}}_{j}\bm{e}_{\sigma})A_{\bm{k}}\left(\hat{a}_{\kappa}+a_{\kappa}^{\dagger}\right)+\sum_{j}\frac{q^{2}}{2m}\sum_{\kappa}A_{\bm{k}}^{2}\left(\hat{a}_{\kappa}+\hat{a}_{\kappa}^{\dagger}\right)^{2}\,.\end{split}$
(73)

Note here that image charges from the cavity boundaries are not included in
the description. The terms quadratic in photon operators can be brought to
diagonal form by a Bogoliubov transformation

$\hat{a}_{\bm{k},\sigma}^{\dagger}=\cosh(\theta_{\bm{k}})\hat{b}_{\bm{k},\sigma}^{\dagger}-\sinh(\theta_{\bm{k}})\hat{b}_{\bm{k},\sigma}\,,$
(74)

where
$\cosh\left(\theta_{\bm{k}}\right)=\left(\lambda_{\bm{k}}+1\right)/(2\sqrt{\lambda_{\bm{k}}})$,
$\sinh\left(\theta_{\bm{k}}\right)=\left(\lambda_{\bm{k}}-1\right)/(2\sqrt{\lambda_{\bm{k}}})$
and $\lambda_{\bm{k}}=\sqrt{1+4\Delta_{\bm{k}}/{\omega_{\bm{k}}}}$. The
resulting Hamiltonian is

$\begin{split}\mathcal{\hat{H}}=&\mathcal{\hat{H}}_{M}+\sum_{\kappa}\omega_{\bm{k}}\lambda_{\bm{k}}\hat{a}_{\kappa}^{\dagger}\hat{a}_{\kappa}+\sum_{j}\sum_{\kappa}c_{\bm{k}}^{e}\lambda_{\bm{k}}^{-1/2}\sqrt{\frac{1}{m\omega_{\bm{k}}}}(\hat{\bm{p}}_{j}\bm{e}_{\sigma})\left(\hat{b}_{\kappa}+\hat{b}_{\kappa}^{\dagger}\right)\,.\end{split}$
(75)

At this point, the paralelism with Eq. 17 is complete by simply setting
$c_{\bm{k}}=c_{\bm{k}}^{e}\lambda_{\bm{k}}^{-1/2}$ and
$\mathcal{\hat{C}}_{\kappa}=\mathcal{\hat{C}}_{\kappa}^{\dagger}=\sqrt{\frac{1}{m\omega_{\bm{k}}}}\bm{e}_{\sigma}\hat{\bm{P}}$,
with $\hat{\bm{P}}=\sum_{j}\hat{\bm{p}}_{j}$. Consequently, the effective
Hamiltonian is given by

$\mathcal{\hat{H}}_{\rm
eff}=\sum_{j}\frac{\hat{\bm{p}}_{j}^{2}}{2m}-\sum_{\kappa}\frac{(c_{\bm{k}}^{e})^{2}}{\tilde{\omega}_{\bm{k}}\lambda_{\bm{k}}}\frac{1}{m\omega_{\bm{k}}}\left(e_{\sigma}\hat{\bm{P}}\right)^{2}=\frac{1}{2m}\left[\sum_{j}\frac{\hat{\bm{p}}_{j}^{2}}{2m}-\left(\sum_{\bm{k}}\frac{4\Delta_{\bm{k}}}{\omega_{\bm{k}}+4\Delta_{\bm{k}}}\right)\frac{1}{N}\sum_{\sigma}\left(e_{\sigma}\hat{\bm{P}}\right)^{2}\right]\,.$
(76)

The reader should note that this Hamiltonian corresponds precisely with the
ground-state energy obtained in [50]. We find the added benefit that the
Hamiltonian offers more flexibility in the amount of calculations that can be
done from here. For instance finite temperature calculations, in contrast to
the ground-state energy, which is limited in that regard.

###  VI.3 Magnetic cavity QED

As a final example we explore how to modify the magnetic properties of
materials. Ferromagnetic enhancement by collective strong coupling to a cavity
has already been demonstrated experimentally with oxide nanoparticles [23], in
connection with the modification of superconducting properties [18]. Here,
instead, we consider molecular nanomagnets. The vast majority of them are
neutral and present a near zero electric dipole. Their behaviour is then
accurately described by an effective “giant”-spin Hamiltonian
$\mathcal{\hat{H}}_{S}$, that includes the effect of magnetic anisotropy and
the Zeeman coupling ${\mathcal{\hat{H}}}_{\rm
Z}=-g\mu_{B}\hat{\bm{S}}\cdot\hat{\bm{B}}$ to a magnetic field. With spin
operators $[S_{i},S_{j}]=i\epsilon_{ijk}S_{k}$. These molecules can be coupled
to superconducting cavities [75] and have been recently studied as a good
candidate to achieve equilibrium condensation, and beyond, to achieve the
modification of the ferromagnetism of materials from the coupling to
electromagnetic cavities [22]. To study the situation where an ensemble of
such molecular spins is placed within a cavity, we will consider that
$\mathcal{\hat{H}}_{S}$ only contains the coupling to external classical
fields, and treat separately the coupling of the spins to the quantized
magnetic field of an electromagnetic cavity. The resulting full Hamiltonian
reads [76]

$\mathcal{\hat{H}}=\mathcal{\hat{H}}_{S}^{\prime}+\mathcal{\hat{H}}_{\rm
EM}-g\mu_{B}\sum_{j}\hat{\bm{S}}_{j}\hat{\bm{B}}(\bm{r}_{j})\,.$ (77)

We write $\mathcal{\hat{H}}_{S}^{\prime}$ to indicate that the effect of image
spins from the cavity boundaries can be taken into account, modifying the
behaviour of the spin ensemble beyond the Zeeman coupling. Owing to the fact
that the molecules are completely described by their spin degrees of freedom
$\hat{\bm{S}}_{j}$, their positions are assumed fixed at positions
$\bm{r}_{j}$. The intensity of the cavity field at each position will
determine the strength of the individual couplings. In contrast to previous
examples, the positions $\bm{r}_{j}$ act simply as parameters of the model and
not as operators in a Hilbert space, they do not constitute a dynamical degree
of freedom. Introducing the explicit forms of the electromagnetic Hamiltonian
and magnetic field, the Hamiltonian reads

$\mathcal{\hat{H}}=\mathcal{\hat{H}}_{S}+\sum_{\kappa}\omega_{\bm{k}}\hat{a}_{\kappa}^{\dagger}\hat{a}_{\kappa}-g\mu_{B}\sum_{j}\sum_{\kappa}\hat{\bm{S}}_{j}B_{\bm{k}}\left(\bm{u}_{\perp,\kappa}(\bm{r}_{j})\hat{a}_{\kappa}+h.c.\right)\,.$
(78)

Identifying the couplings $c_{\bm{k}}=g\mu_{B}B_{\bm{k}}$ and the coupling
operators

$\mathcal{\hat{C}}_{\kappa}=\sum_{j}\hat{\bm{S}}_{j}\bm{u}_{\perp,\kappa}(\bm{r}_{j})\,,$
(79)

we arrive to the effective Hamiltonian

$\mathcal{\hat{H}}_{\rm
eff}=\mathcal{\hat{H}}_{S}-\sum_{\kappa}\frac{c_{\bm{k}}^{2}}{\omega_{\bm{k}}}\sum_{ij}\left(\hat{\bm{S}}_{i}\bm{u}_{\perp,\kappa}(\bm{r}_{i})\right)\left(\hat{\bm{S}}_{j}\bm{u}_{\perp,\kappa}^{*}(\bm{r}_{j})\right)\,.$
(80)

This is a generalization to an arbitrary cavity geometry of the effective
Hamiltonian presented in [22]. We can see that the degree to which two distant
spins couple depends on the amplitude of the transverse mode functions at each
position, but also on the degree to which each spin aligns with the local
magnetic field. This is a trait inherited from the original Hamiltonian (77),
where we can see that the energy is lowered when all the spins are parallel to
the local magnetic field. The extent to which this local interaction with the
magnetic field will translate to an effective ferromagnetic interaction among
the spins is dictated by the spatial uniformity of the magnetic field. Only if
we consider the simplest case of a uniform single-mode field $\bm{k}\to 0$,
$\sigma=1$ will we have an effective ferromagnetic all-to-all interaction
between spins. This situation is only an ideal limit-case of the more general
picture in which some degree of non-uniformity is unavoidable. This caveat is
particularly important because, as explained in [22], in order to achieve
significant modifications of the behaviour of the spin system, the filling
factor -a proxy for the degree to which the magnetic molecules occupy the
region of space where the field has a significant intensity- must be
maximized. If we consider a molecular crystal, there is a clear trade-off
between increasing the crystal size to cover a larger portion of the cavity
volume and keeping the crystal within a small-enough region of space that the
experienced field is uniform.

##  VII Conclusion and outlook

We have established an operational way to understand how matter is modified
when it is immersed in a cavity. We have explicitly isolated the effect of
quantum fluctuations of light. Taking advantage of the fact that the EM field
is a free bosonic field, we have traced it our exactly with the coherent path
integral formalism to obtain an effective theory of matter. Our theory comes
in the form of an effective action containing only matter degrees of freedom,
where the effect of light is given by a non-local kernel. In the large-$N$
limit, this kernel becomes local and the theory can be written as a matter
Hamiltonian containing effective matter-matter interactions. Our starting
point is the minimal coupling of light and matter in an arbitrary geometry, so
our results are quite general. Besides, our theory can be adapted to describe
the coupling to other bosonic excitations, such as phonons, plasmons or
magnons.

This effective Hamiltonian has been tested on the Dicke model. We have shown
that it is more versatile than perturbative effective theories. Its ability to
cover the full light-matter coupling regime merits its denomination as non-
perturbative. This general theory has been in three examples of current
relevance. First, photon condensation was understood from our effective
theory. In particular, its non-existence when uniform cavity fields are
considered and gauge invariance is respected, and the condition for its
existence in the case of non-uniform cavity fields. Then, we recovered the
exact solution for the 2D free electron gas in a cavity within our effective
formalism. Finally, we studied a system of molecular spins in a cavity, laying
the foundations of magnetic cavity QED. In the limit case of uniform fields,
the effective interactions are ferromagnetic and all-to-all. Conversely, for
general non-uniform fields, the effective spin-spin interactions are position
dependent. This could be leveraged to engineer complex interactions between
distant spins, benefiting from the different directions and intensities of the
mode functions at different locations within the cavity. The appearance of
all-to-all position-dependent interactions is not exclusive to magnetic
systems, but instead intrinsic to the formalism, so the engineering of
interactions, with applications in sight, can also be explored in other
systems. Unlike other coupling mechanisms, the intensity of the interaction
between two constituents of a material is not inversely determined by their
distance but instead depends on the intensity of the cavity field at the
particles positions. If they both lie at antinodes of the cavity field, the
coupling will be maximal irrespective of their distance, hinting at the
possibility of inducing long-range interactions within materials. These have
been studied to generate non-geometrical frustration, to induce quantum spin
liquid phases [77]. Importantly enough, our treatment is exact and does not
rely on perturbative expansions on the coupling, detuning, etc. Therefore, the
effective Hamiltonian is as valid as the underlying modelization of the light-
matter coupling. Thus, the tailored interactions can be used for, _e.g._
quantum simulators in the whole range of matter-matter interaction strengths
[78, 79, 80]. On the other hand, the explicitness of the effective Hamiltonian
provides a direct way to classify the obtained models, in terms of symmetries,
topology, integrability, etc.

From a theoretical perspective, the theory may find several extensions. For
instance by considering the coupling of matter to non-bosonic, e.g. fermionic
or spin, degrees of freedom. Alternatively, staying within the realm of cavity
QED, it might be illuminating to develop the dynamical counterpart of this
equilibrium effective theory, using real time path integrals and linear
response theory. Finally, a systematic treatment of the non-local kernel, in
relation to bridging the gap from finite to infinite $N$, remains to be found.

###### Acknowledgements.

The authors acknowledge clarifying discussions with Peter Rabl, Adam Stokes,
Ahsan Nazir, Pierre Nataf, Thierry Champel and Denis Basko. Also funding from
the EU (QUANTERA SUMO and FET-OPEN Grant 862893 FATMOLS), the Spanish
Government Grant PID2020-115221GB-C41/AEI/10.13039/501100011033, the Gobierno
de Aragón (Grant E09-17R Q-MAD) and the CSIC Quantum Technologies Platform
PTI-001.

##  Appendix A Light-matter Hamiltonian for a system of localized emitters

Let us now adapt the formulation of the gas model presented in Sec. III to the
case of localized emitters. In the case of fixed emitters, such as atoms or
molecules, the charged particles are electrons with $q=-e$, where $e$ is the
electron’s charge, the Hamiltonian is given by

$\mathcal{\hat{H}}_{M}=\sum_{j}\frac{\hat{\bm{p}}_{j}^{2}}{2m}+\mathcal{\hat{H}}_{qq}+\mathcal{\hat{H}}_{qQ}+\sum_{j}\hat{v}(\hat{\bm{r}}_{j}+\bm{R}_{j})+\sum_{i\neq
j}\hat{V}(\hat{\bm{r}}_{i}+\bm{R}_{i},\hat{\bm{r}}_{j}+\bm{R}_{j})\,.$ (81)

Here, $\hat{\bm{r}}_{j}$ and $\hat{\bm{p}}_{j}$ are the relative position and
momentum operators of the $j$-th electron with respect to its bounding core,
$\bm{R}_{j}$ is the position of the core of the $j$-th electron. Note that
several electrons can share the same core. The set of $\\{\bm{R}_{j}\\}$
constitute fixed parameters of each model. In this case
$\mathcal{\hat{H}}_{qQ}$ constitutes the Coulombic interaction that bounds
each electron to its respective core. Once again this matter Hamiltonian can
be summarized as
$\mathcal{\hat{H}}_{M}=\mathcal{\hat{T}}+\mathcal{\hat{H}}_{qq}+\mathcal{\hat{H}}_{qQ}+\mathcal{\hat{V}}$.
After coupling this material subsystem to the quantized electromagnetic field
in a cavity, the Hamiltonian for the complete system in the Coulomb gauge
reads

$\mathcal{\hat{H}}=\sum_{j}\frac{(\hat{\bm{p}}_{j}-q\hat{\bm{A}}(\bm{R}_{j}))^{2}}{2m}+\mathcal{\hat{H}}^{\prime}_{qq}+\mathcal{\hat{H}}^{\prime}_{qQ}+\mathcal{\hat{V}}+\mathcal{\hat{H}}_{\rm
EM}-\frac{g\mu_{B}}{2}\sum_{j}\hat{\bm{\sigma}}_{j}\hat{\bm{B}}(\bm{R}_{j})\,.$
(82)

Note that in contrast to the gas, the fact that the dynamic charges are
bounded allows us to take the long-wavelength approximation, assuming that the
electromagnetic field is constant at each particle’s location, such that
$\hat{\bm{A}}(\hat{\bm{r}}_{j}+\bm{R}_{j})\approx\hat{\bm{A}}(\bm{R}_{j})$,
$\hat{\bm{B}}(\hat{\bm{r}}_{j}+\bm{R}_{j})\approx\hat{\bm{B}}(\bm{R}_{j})$.
The main difference with the description of the gas is that now the
electromagnetic fields are operators acting solely on the Hilbert space of the
light. The mode functions that describe their spatial dependence are not
promoted to operators since they depend on the fixed positions
$\\{\bm{R}_{j}\\}$. After expanding the kinetic term and substituting the
explicit forms of the vector potential and the magnetic field, we obtain

$\begin{split}\mathcal{\hat{H}}=&\mathcal{\hat{H}}_{M}^{\prime}+\sum_{\kappa}\omega_{\bm{k}}\hat{a}_{\kappa}^{\dagger}\hat{a}_{\kappa}-\sum_{j}\sum_{\kappa}\frac{q}{m}\hat{\bm{p}}_{j}A_{\bm{k}}\left(\bm{u}_{\kappa}(\bm{R}_{j})\hat{a}_{\kappa}+h.c.\right)\\\
+&\sum_{j}\frac{q^{2}}{2m}\sum_{\kappa,\kappa^{\prime}}A_{\bm{k}}A_{\bm{k}^{\prime}}\left(\bm{u}_{\kappa}(\bm{R}_{j})\hat{a}_{\kappa}+h.c.\right)\left(\bm{u}_{\kappa^{\prime}}(\bm{R}_{j})\hat{a}_{\kappa^{\prime}}+h.c.\right)-\frac{g\mu_{B}}{2}\sum_{j}\sum_{k}\hat{\bm{\sigma}}_{j}B_{\bm{k}}\left(\bm{u}_{\perp,\kappa}(\bm{R}_{j})\hat{a}_{\kappa}+h.c.\right)\,.\end{split}$
(83)

To condense the notation, we define the quantities

$\displaystyle
U_{\kappa,\kappa^{\prime}}=N^{-1}\sum_{j}\bm{u}_{\kappa}(\bm{R}_{j})\bm{u}_{\kappa^{\prime}}(\bm{R}_{j})\,,$
(84)
$\displaystyle{\bar{U}}_{\kappa,\kappa^{\prime}}=N^{-1}\sum_{j}\bm{u}_{\kappa}(\bm{R}_{j})\bm{u}_{\kappa^{\prime}}^{*}(\bm{R}_{j})\,.$
(85)

With these, we write the terms that depend quadratically on the bosonic
operators as

$\begin{split}\mathcal{\hat{H}_{\rm
ph}}=\sum_{\kappa}\omega_{\bm{k}}\hat{a}_{\kappa}^{\dagger}\hat{a}_{\kappa}+\sum_{\kappa,\kappa^{\prime}}\Delta_{\bm{k},\bm{k}^{\prime}}\left(U_{\kappa,\kappa^{\prime}}\hat{a}_{\kappa}\hat{a}_{\kappa^{\prime}}+{\bar{U}}_{\kappa,\kappa^{\prime}}\hat{a}_{\kappa}\hat{a}_{\kappa^{\prime}}^{\dagger}+h.c.\right)\,.\end{split}$
(86)

Whereas in the case of a gas the photonic Hamiltonian could not be
diagonalized in the general case, the fact that the coefficients of the
quadratic terms are now c-numbers allows us to exactly diagonalize the
photonic terms by means of a Bogoliubov transform. First, we define
$\Psi^{t}=(\hat{a}_{\kappa_{1}}^{\dagger},\ldots,\hat{a}_{\kappa_{M}}^{\dagger},\hat{a}_{\kappa_{1}},\ldots,\hat{a}_{\kappa_{M}})$,
which allows us to write the Hamiltonian as

$\mathcal{\hat{H}}_{\rm
ph}=\frac{1}{2}\Psi^{\dagger}\mathbb{H}\Psi\,,\qquad\text{with}\qquad\mathbb{M}=\begin{pmatrix}\mathbb{H}_{1}&\mathbb{H}_{2}\\\
\mathbb{H}_{2}^{\dagger}&\mathbb{H}_{1}^{*}\end{pmatrix}\,,$ (87)

where the $M\times M$ blocks $\mathbb{H}_{1}$ and $\mathbb{H}_{2}$ are defined
in terms of their matrix elements

$\displaystyle(\mathbb{H}_{1})_{\kappa,\kappa^{\prime}}=2\Delta_{\kappa,\kappa^{\prime}}\bar{U}_{\kappa,\kappa^{\prime}}+\delta_{\kappa\kappa^{\prime}}\omega_{\bm{k}}\,,$
(88)
$\displaystyle(\mathbb{H}_{2})_{\kappa,\kappa^{\prime}}=2\Delta_{\kappa,\kappa^{\prime}}U_{\kappa,\kappa^{\prime}}\,.$
(89)

The Bogoliubov transform is determined by a transformation matrix
$\mathbb{T}$, obeying the pseudounitarity condition
$\mathbb{T}^{\dagger}\mathbb{I}_{-}\mathbb{T}=\mathbb{I}_{-}$, which defines a
new set of bosonic operators
$\Phi=\mathbb{T}^{-1}\Psi=(\hat{b}_{\kappa_{1}}^{\dagger},\ldots,\hat{b}_{\kappa_{M}}^{\dagger},\hat{b}_{\kappa_{1}},\ldots,\hat{b}_{\kappa_{M}})$
and a new Hamiltonian matrix
$\mathbb{\tilde{H}}=\mathbb{T}^{\dagger}\mathbb{H}\mathbb{T}$ such that
$\mathbb{\tilde{H}}$ is diagonal [52]. The resulting Hamiltonian can be
written as

$\mathcal{\hat{H}_{\rm
ph}}=\sum_{\kappa}\tilde{\omega}_{\bm{k}}b_{\kappa}^{\dagger}b_{\kappa}\,.$
(90)

The pseudounitarity of $\mathbb{T}$ ensures that the new bosonic operators
obey the canonical commutation relations. The $\mathbb{I}_{-}$ block matrix is
defined as

$\mathbb{I}_{-}=\begin{pmatrix}\mathbb{I}_{M}&0\\\
0&-\mathbb{I}_{M}\end{pmatrix}\,,$ (91)

with $\mathbb{I}_{M}$ the identity matrix of dimension $M$. The old bosonic
operators can be written in terms of the new ones as

$\hat{a}_{\kappa}=\sum_{\kappa^{\prime}}(\alpha_{\kappa,\kappa^{\prime}}\hat{b}_{\kappa^{\prime}}+\beta_{\kappa,\kappa^{\prime}}\hat{b}_{\kappa^{\prime}}^{\dagger})\,.$
(92)

With this, and defining

$\displaystyle\tilde{A}_{\bm{k}}\tilde{\bm{u}}_{\kappa}(\bm{R}_{j})=\sum_{\kappa^{\prime}}A_{\bm{k}^{\prime}}\left(\bm{u}_{\kappa^{\prime}}(\bm{R}_{j})\alpha_{\kappa,\kappa^{\prime}}+\bm{u}_{\kappa^{\prime}}^{*}(\bm{R}_{j})\beta_{\kappa,\kappa^{\prime}}^{*}\right)\,,$
(93)
$\displaystyle\tilde{B}_{\bm{k}}\tilde{\bm{u}}_{\perp,\kappa}(\bm{R}_{j})=\sum_{\kappa^{\prime}}B_{\bm{k}^{\prime}}\left(\bm{u}_{\perp,\kappa^{\prime}}(\bm{R}_{j})\alpha_{\kappa,\kappa^{\prime}}+\bm{u}_{\perp,\kappa^{\prime}}^{*}(\bm{R}_{j})\beta_{\kappa,\kappa^{\prime}}^{*}\right)\,,$
(94)

and the constants

$\displaystyle\tilde{c}_{\bm{k}}^{e}=q\tilde{A}_{\bm{k}}\sqrt{\frac{\omega_{\bm{k}}}{m}}\,,$
(95)
$\displaystyle\tilde{c}_{\bm{k}}^{m}=\frac{g\mu_{B}\tilde{B}_{\bm{k}}}{2}\,,$
(96)

we can conveniently write the total Hamiltonian as

$\begin{split}\mathcal{\hat{H}}=\mathcal{\hat{H}}_{M}^{\prime}+\sum_{\kappa}\omega_{\bm{k}}\lambda_{\bm{k}}\hat{b}_{\kappa}^{\dagger}\hat{b}_{\kappa}-\sum_{j}\sum_{\kappa}\tilde{c}_{\bm{k}}^{e}\sqrt{\frac{1}{m\omega_{\bm{k}}}}\hat{\bm{p}}_{j}\left(\tilde{\bm{u}}_{\kappa}(\bm{R}_{j})\hat{b}_{\kappa}+h.c.\right)-\sum_{j}\sum_{k}\tilde{c}_{\bm{k}}^{m}\hat{\bm{\sigma}}_{j}\left(\tilde{\bm{u}}_{\perp,\kappa}(\bm{R}_{j})\hat{b}_{\kappa}+h.c.\right)\,.\end{split}$
(97)

Defining the coupling operator

$c_{\bm{k}}\mathcal{\hat{C}}_{\kappa}=\sum_{j}\left(\tilde{c}_{\bm{k}}^{e}\sqrt{\frac{1}{m\omega_{\bm{k}}}}\hat{\bm{p}}_{j}\tilde{\bm{u}}_{\kappa}(\hat{\bm{R}}_{j})+\tilde{c}_{\bm{k}}^{m}\hat{\bm{\sigma}}_{j}\tilde{\bm{u}}_{\perp,\kappa}(\hat{\bm{R}}_{j})\right)\,,$
(98)

we can finally write the Hamiltonian as

$\mathcal{\hat{H}}=\mathcal{\hat{H}}_{M}^{\prime}+\sum_{\kappa}\tilde{\omega}_{\bm{k}}\hat{b}_{\kappa}^{\dagger}\hat{b}_{\kappa}-\sum_{\kappa}c_{\bm{k}}\left(\mathcal{\hat{C}}_{\kappa}\hat{b}_{\kappa}+h.c.\right)\,.$
(99)

Which is formally identical to Hamiltonian (17), the only difference lies in
how $\tilde{\omega}_{\bm{k}}$, $c_{\bm{k}}$, $\mathcal{\hat{C}}_{\kappa}$ and
$\hat{b}_{\kappa}$ are defined in this case.

##  Appendix B Entirely Hamiltonian formulation

The effective Hamiltonian (41) derived in the previous subsection can also be
deduced directly with an alternative derivation that does not rely on the path
integral approach and the definition of an effective action. Instead, we
borrow from Hepp and Lieb’s original study of equilibrium superradiance [26]
the following bounds for the partition function

$\tilde{Z}\leq Z\leq e^{\beta\sum_{\kappa}\omega_{\bm{k}}}\tilde{Z}\,,$ (100)

where $Z$, the proper partition function, and $\tilde{Z}$, its lower bound,
are defined as

$\displaystyle Z$
$\displaystyle=\Tr_{M}\left(\int\prod_{\kappa}\frac{d^{2}z_{\kappa}}{\pi}\bra{\bm{z}}e^{-\beta\mathcal{\hat{H}}}\ket{\bm{z}}\right)\,,$
(101) $\displaystyle\tilde{Z}$
$\displaystyle=\Tr_{M}\left(\int\prod_{\kappa}\frac{d^{2}z_{\kappa}}{\pi}e^{-\beta\expectationvalue{\mathcal{\hat{H}}}{\bm{z}}}\right)\,,$
(102)

with $\ket{\bm{z}}=\bigotimes_{\kappa}\ket{z_{\kappa}}$ and

$\expectationvalue{\mathcal{\hat{H}}}{\bm{z}}=\mathcal{\hat{H}}_{M}^{\prime}+\sum_{\kappa}\tilde{\omega}_{\bm{k}}|z_{\kappa}|^{2}+\sum_{\kappa}c_{\bm{k}}\left(\mathcal{\hat{C}}_{\kappa}z_{\kappa}+h.c.\right)\,.$
(103)

In the thermodynamic limit $N\to\infty$ and provided the number of modes does
not scale with $N$, the correction term
$\exp\left[\beta\sum_{\kappa}\omega_{\bm{k}}\right]$ in the upper bound
becomes negligible and the partition function is given exactly by
$\tilde{Z}=Z$. Then, we can define an effective Hamiltonian as

$\tilde{Z}=\Tr_{M}\left(\int\prod_{\kappa}\frac{d^{2}z_{\kappa}}{\pi}e^{-\beta\expectationvalue{\mathcal{\hat{H}}}{\bm{z}}}\right)=\tilde{Z}_{0}\Tr_{M}\left(e^{-\beta\mathcal{\hat{H}}_{\rm
eff}}\right)\,.$ (104)

Note that in this context the $\\{z_{\kappa}\\}$ are complex numbers. We are
thus interested in computing

$\int\prod_{\kappa}\frac{d^{2}z_{\kappa}}{\pi}e^{-\beta\expectationvalue{\mathcal{\hat{H}}}{\bm{z}}}\,,$
(105)

which is just a collection of multidimensional complex Gaussian integrals. The
resulting Hamiltonian is simply

$\mathcal{\hat{H}}_{\rm
eff}=\mathcal{\hat{H}}_{M}^{\prime}-\sum_{\kappa}\frac{c_{\bm{k}}^{2}}{\tilde{\omega}_{\bm{k}}}\mathcal{\hat{C}}_{k}\mathcal{\hat{C}}_{k}^{\dagger}\,,$
(106)

just like Eq. 41. Note that in order to carry out the Gaussian integral, one
has to be able to write

$e^{-\beta\left(\mathcal{\hat{H}}_{M}^{\prime}+\sum_{\kappa}\tilde{\omega}_{\bm{k}}|z_{\kappa}|^{2}+\sum_{\kappa}c_{\bm{k}}\left(\mathcal{\hat{C}}_{\kappa}z_{\kappa}+h.c.\right)\right)}=e^{-\beta\mathcal{\hat{H}}_{M}}e^{-\beta\left(\sum_{\kappa}\tilde{\omega}_{\bm{k}}|z_{\kappa}|^{2}+\sum_{\kappa}c_{\bm{k}}\left(\mathcal{\hat{C}}_{\kappa}z_{\kappa}+h.c.\right)\right)}$
(107)

which holds because the matter Hamiltonian and the coupling operator commute
$[\mathcal{\hat{H}}_{M}/N,\mathcal{\hat{C}}_{\kappa}/N]=0$ in the
thermodynamic limit $N\to\infty$. To see this, note that both
$\mathcal{\hat{H}}_{M}$ and $\mathcal{\hat{C}}_{\kappa}$ are extensive and
both can be written as a series in powers of the position and momentum
operators, the commutation of position and momentum yields a dirac delta for
the $i$th and $j$th particle, eliminating one of the two sums over the N
particles. The result of the commutator is still extensive, so its division by
$N^{2}$ will scale as $1/N$. The Gaussian integrals also yield
$\tilde{Z}_{0}=\prod_{\kappa}\tilde{Z}_{\kappa}$, where
$\tilde{Z}_{\kappa}=1/\beta\tilde{\omega}_{\bm{k}}$ is the high-temperature
limit of the partition funtion of a Harmonic oscillator of frequency
$\tilde{\omega}_{\bm{k}}$, i.e.
$\tilde{Z}_{\kappa}=\lim_{T\to\infty}Z_{\kappa}$.

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