# Tuning Fermi liquids with sub-wavelength cavities

Riccardo Riolo  Dipartimento di Fisica dell’Università di Pisa, Largo Bruno
Pontecorvo 3, I-56127 Pisa, Italy  Andrea Tomadin  Dipartimento di Fisica
dell’Università di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy
Giacomo Mazza  Dipartimento di Fisica dell’Università di Pisa, Largo Bruno
Pontecorvo 3, I-56127 Pisa, Italy  Reza Asgari  Department of Physics,
Zhejiang Normal University, Jinhua, Zhejiang 321004, China  School of Physics,
Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5531, Iran
Allan H. MacDonald  Department of Physics, The University of Texas at Austin,
Austin, TX 78712, USA  Marco Polini  Dipartimento di Fisica dell’Università di
Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy  ICFO-Institut de Ciències
Fotòniques, The Barcelona Institute of Science and Technology, Av. Carl
Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain

###### Abstract

The question of whether or not passive sub-wavelength cavities can alter the
properties of quantum materials is currently attracting a great deal of
attention. In this Letter we show that the Fermi liquid parameters of a two-
dimensional metal are modified by cavity polariton modes, and that these
changes can be monitored by measuring a paradigmatic magneto-transport
phenomenon, Shubnikov–de Haas oscillations in a weak perpendicular magnetic
field. This effect is intrinsic, and totally unrelated to disorder. As an
illustrative example, we carry out explicit calculations of the Fermi liquid
parameters of graphene in a planar van der Waals cavity formed by natural
hyperbolic crystals and metal gates.

## Introduction.

Magnetic oscillation phenomena Shoenberg ; Landau09  like the de Haas–van
Alphen effect, oscillations of the magnetization of a metal, are a powerful
probe of the physics of itinerant electron systems. The frequencies of de
Haas–van Alphen oscillations, for example, are an historically important probe
of the Fermi surfaces of metals Shoenberg . In metallic two-dimensional
electron systems (2DESs) strong magnetic oscillations appear in the
longitudinal resistivity $\rho_{xx}$ Ando_RMP_1982  (Shubnikov–de Haas (SdH)
oscillations) at magnetic fields weaker than those at which the quantum Hall
effect occurs. Importantly, these can be used to access the Landau Fermi
liquid parameters Rice_AnnPhys_1965 ; Pines_and_Nozieres ; Mahan_2000 ;
Giuliani_and_Vignale of 2DESs such as the quasiparticle effective mass
$m^{\ast}$ (or, equivalently, quasiparticle velocity $v^{\ast}_{\rm F}$) and
the interaction enhanced $g$-factor $g^{\ast}$. Measurements in ultra-clean
systems, including 2DESs confined to silicon inversion layers
Fang_PhysRev_1968 ; Smith_PRL_1972 ; Pudalov_PRL_2002 , AlAs/AlGaAs quantum
wells Vakili_PRL_2004 ; Padmanabhan_PRL_2008 ; Gokmen_PRL_2008 ;
Gokmen_PRB_2009 , GaAs/AlGaAs quantum wells Tan_PRL_2005 ; Tan_PhysicaE_2006 ;
Chiu_PRB_2011 , ${\rm LaAlO}_{3}/{\rm SrTiO}_{3}$ interfaces Caviglia_PRL_2010
; BenShalom_PRL_2010 , SiGe/Si/SiGe quantum wells Melnikov_SciRep_2023 , and
atomically-thin materials Novoselov_Nature_2005 ; Zhang_Nature_2005 ;
Elias_NatPhys_2011 ; Zou_PRB_2011 ; Fallahazad_PRL_2016 , report large
deviations of $m^{\ast}$ and $g^{\ast}$ from the non-interacting values
$m_{\rm e}$ and $g$.

The aim of this Letter is to investigate whether SdH oscillations are affected
by the fluctuations of an electromagnetic field confined to a small volume of
space by a sub-wavelength Novotny_2012 ; Basov_Nanophotonics_2021 ;
Plantey_ACS_2021 optical cavity. The possibility of tailoring the ground state
and transport properties of an electron system solely by altering the
fluctuations of a passive cavity (i.e. a cavity in the absence of optical
pumping) is clearly potentially exciting and has recently captured a great
deal of attention Bloch_Nature_2022 ; Schlawin_APR_2022 ;
GarciaVidal_Science_2021 ; Genet_PT_2021 ; Rubio_NatureMater_2021 .

In 2DESs, transport is a particularly convenient probe to monitor passive
cavity tuning of many-electron properties. The pioneering theoretical work on
cavity-induced modifications of magneto-transport has been performed by Ciuti
and co-workers Bartolo_PRB_2018 ; Ciuti_PRB_2021 ; Arwas_PRB_2023 . Recent
experiments Paravicini_Bagliani_NaturePhys_2019 ; scalari_science_2012 ;
muravev_prb_2013 ; maissen_prb_2014 ; smolka_science_2014 ;
Keller_nanolett_2017 ; Appugliese_science_2022 have shown that the magneto-
transport properties of a 2DES in a GaAs/AlGaAs quantum well can be modified
by coupling the electronic degrees of freedom to sub-wavelegth cavities in the
Terahertz (THz) spectral range. In the case of split-ring THz resonators, for
example, Ref. Paravicini_Bagliani_NaturePhys_2019 reports a suppression of the
amplitude of SdH traces, but no changes in the frequency $B_{\rm F}$, in the
passive regime comment_Ciuti .

Figure 1: A side view of the planar sub-wavelength cavity analyzed in this
work. A 2D crystal which hosts a Fermi liquid (thick line) of areal density
$n_{0}$ located at $z=0$ is encapsulated between two hyperbolic dielectrics
(light grey) described by frequency-dependent uniaxial permittivity tensors
$\hat{\bm{\epsilon}}(\omega)={\rm
diag}(\epsilon_{x}(\omega),\epsilon_{x}(\omega),\epsilon_{z}(\omega))$. The
bottom (top) slab has thickness $d$ ($d^{\prime}$). Metal gates (hatched grey)
fill the two half-spaces $z>d^{\prime}$ and $z<-d$. We assume that the
dielectrics are identical.

\begin{overpic}[percent]{fig_2_a.pdf}
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\end{overpic}\begin{overpic}[percent]{fig_2_b.pdf}
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Figure 2: (Color online) Dynamical structure factor $S(q,\omega)$ (in units of
$k_{\text{F}}/v_{\text{F}}$) as a function of the wave vector $q$ and
frequency $\omega$. Results for graphene encapsulated by hBN—Panel (a)—and
$\text{Bi}_{2}\text{Se}_{3}$—Panel (b). In both panels,
$n_{0}=$3\text{\times}{10}^{12}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$ and
$d=d^{\prime}=$50\text{\,}\mathrm{n}\mathrm{m}$$ (parameters described in Fig.
1). The white solid line is the 2DES plasmon frequency calculated by
artificially substituting $F(q,0)$ for $F(q,\omega)$ in Eq. 5, thereby
neglecting cavity dynamics. Notice that the plasmon dispersion is linear at
small $q$ because of the presence of metal gates (see Fig. 1).

Here we focus on the temperature ($T$) dependence of the SdH oscillations.
According to the Lifshitz–Kosevich (LK) formula Lifshitz_1955 ;
Isihara_JPhysC_1986 ; Sharapov_PRB_2004 ; Kopelevich_PRL_2004 ;
Kuppersbusch_PRB_2017 , the oscillatory component of the magneto-resistance is
given by LK_comment

$\frac{\delta\rho_{xx}(B)}{\rho_{0}}=R(B,T)\cos\left(2\pi\left(\frac{B_{\rm
F}}{B}+\frac{1}{2}+\gamma\right)\right),$ (1)

where $\rho_{0}$ is the resistance at $B=0$, $B_{\rm F}=\phi_{0}n_{0}/N_{\rm
f}$ is the frequency of the oscillations in $1/B$, $\phi_{0}=hc/e$ is the
magnetic flux quantum ($c$ is the speed of light in vacuum and $e$ the
elementary charge), $n_{0}$ is the carrier density, $N_{\rm f}$ is a
degeneracy factor, and $\gamma\in[0,1)$ is a Berry phase Mikitik_PRL_1999 .
The amplitude $R(B,T)$ of the SdH oscillations is proportional to (and fully
controlled by) the dimensionless quantity

$A(B,T)=\frac{2\pi^{2}k_{\rm B}T/\hbar\omega_{\rm c}}{\sinh(2\pi^{2}k_{\rm
B}T/\hbar\omega_{\rm c})}e^{-\pi/\omega_{\rm c}\tau}.$ (2)

Here, $\tau$ is the momentum relaxation time, $m_{\rm c}$ the cyclotron mass,
and $\omega_{\rm c}\equiv eB/m_{\rm c}c$ is the cyclotron frequency. The LK
formula is derived under the assumptions of weak fields, i.e.
$\hbar\omega_{\rm c}\ll\mu$, where $\mu$ is the chemical potential. The
$\tau$-dependent Dingle factor Shoenberg  captures the reduction of the
amplitude of the SdH oscillations due to electron-impurity scattering. Note
that inelastic processes (such as electron-electron collisions) affect only
$R(B,T)$ and not the SdH oscillation frequency $B_{\rm F}$ Shoenberg . Eq. (2)
is routinely used to fit the measured $T$-dependence of the amplitude of the
SdH oscillations at a fixed value of $B$ to extract an experimental value of
$m_{\rm c}$.

---
Figure 3: Feynman diagrams considered in this work. Top: the irreducible self-
energy $\Sigma_{\lambda}(\bm{k},\mathrm{i}\omega_{m})$ in the $G0W$
approximation, Eq. (8). Bottom: the dynamically screened potential $W$ in the
random-phase approximation, as in Eq. (9). Here, $V$ represents the Coulomb
interaction dressed by the cavity environment, and $\chi_{0}$ the polarization
function of the 2DES.

We now explain why and how SdH oscillations are modified by a sub-wavelength
cavity. According to the LK formula (1), a passive cavity can act via both
extrinsic and intrinsic microscopic mechanisms. The extrinsic mechanism is
active if the cavity environment alters the momentum relaxation time $\tau$
with respect to the case where no cavity is present, and is clearly irrelevant
in clean 2DES (i.e. for $\omega_{\rm c}\tau\gg 1$). The intrinsic mechanism,
which is the focus of this Letter, is a cavity-induced change in the
interaction-induced modification of the cyclotron mass $m_{\rm c}$, which we
evaluate as $m_{\rm c}=\hbar k_{\rm F}/v^{\ast}_{\rm F}$, where $k_{\rm F}$ is
the Fermi wave number and $v^{\ast}_{\rm F}$ is the quasiparticle velocity
Giuliani_and_Vignale . We show by explicit calculation that a sub-wavelength
cavity—such as the van der Waals cavity like in Fig. 1—changes $m_{\rm c}$ and
all Landau parameters of the normal Fermi liquid relative to their values in
vacuum. The reason is as follows: in the absence of the cavity, the spectrum
or dynamical structure factor $S(q,\omega)$ of a clean 2DES displays a sharp
plasmon mode, which carries a large fraction of the total f-sum rule spectral
weight Pines_and_Nozieres ; Giuliani_and_Vignale ; Mahan_2000 , and a
continuum of particle-hole excitations. When coupling between the 2DES and the
cavity is allowed, spectral weight flows from the electronic degrees of
freedom to the polariton modes of the cavity, Fig. 2. This leads ultimately to
changes in the Landau parameters of the 2DES.

Below, we first introduce a general theoretical framework, summarized in Fig.
3, that captures this physics and then present detailed calculations for the
case of graphene embedded in the van der Waals cavity sketched in Fig. 1
containing metal gates and natural hyperbolic crystals Sun_ACS_2014 ;
Basov_Science_2016 ; Low_NatMater_2017 ; Zhang_Nature_2021 , such as hexagonal
Boron Nitride (hBN) Dai_science_2014 ; Caldwell_naturecommun_2014 ;
Li_NatCommun_2015 ; Dai_NatCommun_2015 and ${\rm Bi}_{2}{\rm S}_{3}$
Eva_NatCommun_2021 ; footnote_topologicalinsulators . A specific illustration
of the above mentioned spectral weight transfer is illustrated in Fig. 2,
where we plot $S(q,\omega)$ Pines_and_Nozieres ; Giuliani_and_Vignale ;
Mahan_2000 for graphene in the cavity shown in Fig. 1.

## Polariton-mediated effective electron-electron interactions.

We consider an interacting many-electron system embedded in a passive cavity,
containing (hyperbolic) dielectric media and metal gates. Because condensed
matter is only weakly relativistic, the coupling between matter and
electromagnetic degrees of freedom can be separated Cohen_1997 into non-
radiative and radiative contributions. Non-radiative photons mediate density-
density electron-electron interactions, while radiative ones mediate current-
current interactions andolina_arxiv_2022 that are weaker by $(v_{\rm
F}/c)^{2}$, where $v_{\rm F}$ is the bare Fermi velocity. For DC transport
properties, the effects of photon-mediated density-density interactions is
dominant with respect to the one of current-current interactions. (Current-
current interactions can yield large modifications of response functions at
finite frequency Flick_ACS_2019 ; Amelio_PRB_2021 , but have negligible
effects on static properties. In the static limit, current-current
interactions reduce to classical magneto-statics nataf_prl_2019 ;
andolina_prb_2020 ; guerci_prl_2020 ; mazza_prb_2023 .)

After integrating out cavity degrees of freedom, the effective density-density
interaction between electrons in a sub-wavelength cavity is given by
$V({\bm{r}},{\bm{r}}^{\prime},\omega)=e^{2}g^{({\rm
s})}({\bm{r}},{\bm{r}}^{\prime},\omega)$, where $g^{({\rm
s})}({\bm{r}},{\bm{r}}^{\prime},\omega)$ andolina_arxiv_2022 ; GF_comment is
the scalar Green’s function for the frequency-dependent Poisson equation
Landau09 :

$-\partial_{\alpha}[\epsilon_{\alpha\beta}(\bm{r},\omega)\partial_{\beta}\phi(\bm{r},\omega)]=4\pi\rho^{\rm
ext}(\bm{r},\omega).$ (3)

Here, $\phi(\bm{r},\omega)$ is the electrical potential, $\rho^{\rm
ext}(\bm{r},\omega)$ is the external charge density, and
$\epsilon_{\alpha\beta}(\bm{r},\omega)$ is the frequency-dependent complex
permittivity tensor of the dielectric environment, which is assumed to be
local in space footnote_spatial_nonlocality . Losses enter the problem via the
imaginary part of the permittivity tensor. Metal gates can be included by
imposing Jackson that, at metal-dielectric interfaces, the tangential
component of the electric field vanishes and the normal component of the
displacement field is $4\pi\sigma$, where $\sigma$ is the surface charge
density. Eq. (3) can be rigorously derived andolina_arxiv_2022 in the
generalized Coulomb gauge Glauber_PRA_1991 .

$V({\bm{r}},{\bm{r}}^{\prime},\omega)$ describes the interaction between two
charges at positions ${\bm{r}}$ and ${\bm{r}}^{\prime}$. In free space, it
reduces to the ordinary, instantaneous Coulomb interaction Jackson , i.e.
$v(|{\bm{r}}-{\bm{r}}^{\prime}|)=e^{2}/|{\bm{r}}-{\bm{r}}^{\prime}|$. In a
sub-wavelength cavity, however, it can be vastly different from
$v(|{\bm{r}}-{\bm{r}}^{\prime}|)$. In particular, phonon polaritons in the
metallo-dielectric cavity environment yield a retarded frequency-dependent
interaction. In the $\omega=0$ limit, $V({\bm{r}},{\bm{r}}^{\prime},0)$
physically describes static screening due to polarization charges in the
metallo-dielectric environment. At finite $\omega$,
$V({\bm{r}},{\bm{r}}^{\prime},\omega)$ encodes the polariton modes of the
latter (including losses), which appear as poles of
$V({\bm{r}},{\bm{r}}^{\prime},z)$ viewed as a function of the complex
frequency $z$.

Given $V({\bm{r}},{\bm{r}}^{\prime},\omega)$, one can understand how the
properties of the interacting many-electron system change due to the presence
of the passive cavity by simply replacing the bare Coulomb interaction
$v(|{\bm{r}}-{\bm{r}}^{\prime}|)$ with the retarded interaction
$V({\bm{r}},{\bm{r}}^{\prime},\omega)$ in the framework of diagrammatic many-
body perturbation theory Hedin_1969 ; Mahan_2000 ; Giuliani_and_Vignale . This
leads to generalized Hedin equations, from which one can calculate, for
example, the one-body electron Green’s function
$G({\bm{r}},{\bm{r}}^{\prime},\omega)$, the irreducible self-energy
$\Sigma({\bm{r}},{\bm{r}}^{\prime},\omega)$ Hedin_1969 ; Mahan_2000 ;
Giuliani_and_Vignale , and the density-density response function
$\chi_{nn}({\bm{r}},{\bm{r}}^{\prime},\omega)$ Mahan_2000 ;
Giuliani_and_Vignale . Poles in the latter quantity occur at the collective
excitation energies of the 2DES/cavity hybrid system.

## Illustrative example: a planar vdW cavity.

In order to illustrate the power of the approach we just described— while
keeping the treatment of the cavity at a fully analytical level—we focus in
what follows on a planar sub-wavelength cavity based on a vdW heterostructure
Geim_Nature_2013 , which is sketched in Fig. 1. The active material, where one
measures magneto-transport, is a 2D crystalline material located at $z=0$.
This is encapsulated between two homogeneous and uniaxial dielectric slabs of
different thicknesses, the slab on top (bottom) having thickness $d^{\prime}$
($d$). The electrical permittivity tensor of these dielectrics is
$\hat{\bm{\epsilon}}(\omega)={\rm
diag}(\epsilon_{x}(\omega),\epsilon_{x}(\omega),\epsilon_{z}(\omega))$. Note,
finally, the presence of top and bottom metal gates. Without uniaxial
dielectrics we would be dealing with an ordinary Fabry-Pérot cavity
andolina_arxiv_2022 .

Setups like the one in Fig. 1 are routinely fabricated in laboratories
throughout the world. The most studied case Geim_Nature_2013 is the one in
which the 2D material is single-layer or Bernal-stacked bilayer graphene and
the dielectric material is hBN Dai_science_2014 ; Caldwell_naturecommun_2014 ;
Li_NatCommun_2015 ; Dai_NatCommun_2015 . Infinite possibilities are clearly
possible: many other 2D conducting materials can be chosen (including twisted
2D materials) as well as low-loss dielectrics. For illustration purposes,
below we will focus on single-layer graphene (SLG) encapsulated between two
natural hyperbolic dielectrics Sun_ACS_2014 ; Basov_Science_2016 ;
Low_NatMater_2017 ; Zhang_Nature_2021 .

In order to find $V({\bm{r}},{\bm{r}}^{\prime},\omega)$ analytically, we
follow an approach first proposed by Keldysh Keldysh_1979 and recently used in
Ref. Tomadin_prl_2015 in the context of vdW heterostructures. Since the setup
in Fig. 1 is translationally invariant in the $\hat{\bm{x}}$–$\hat{\bm{y}}$
plane, the scalar Green’s function $g^{({\rm
s})}(z,z^{\prime},|{\bm{r}}_{\|}-{\bm{r}}^{\prime}_{\|}|,\omega)$ depends only
on $|{\bm{r}}_{\|}-{\bm{r}}^{\prime}_{\|}|$. Here, ${\bm{r}}_{\|}$ and
${\bm{r}}^{\prime}_{\|}$ denote 2D positions in the
$\hat{\bm{x}}$–$\hat{\bm{y}}$ plane. The effective electron-electron
interaction (EEEI) that is relevant for the many-body physics of the 2DES
located at $z=0$ is
$V(0,0,|{\bm{r}}_{\|}-{\bm{r}}^{\prime}_{\|}|,\omega)=e^{2}g^{({\rm
s})}(0,0,|{\bm{r}}_{\|}-{\bm{r}}^{\prime}_{\|}|,\omega)$. We therefore
introduce its 2D Fourier transform Giuliani_and_Vignale with respect to
$r=|{\bm{r}}_{\|}-{\bm{r}}^{\prime}_{\|}|$:

$V(q,\omega)\equiv 2\pi e^{2}\int_{0}^{+\infty}rg^{({\rm
s})}(0,0,r,\omega)J_{0}(qr)\,\mathrm{d}r,$ (4)

where $J_{0}(x)$ is a Bessel function of the first kind. Once again, the
frequency dependence of $V(q,\omega)$ encodes all the information about
polaritons while its $\omega\to 0$ limit andolina_arxiv_2022 encodes the
physics of static screening due to the presence of metal gates and
polarization charges.

The polariton-mediated propagator of the planar vdW cavity sketched in Fig. 1
is given by the following expression: $V(q,\omega)=v_{q}F(q,\omega)$, where
$v_{q}\equiv 2\pi e^{2}/q$ is the non-retarded 2D Coulomb propagator
Giuliani_and_Vignale and

$F(q,\omega)\equiv\frac{2\sinh{\left[q\sqrt{\frac{\epsilon_{x}(\omega)}{\epsilon_{z}(\omega)}}d\right]}\sinh{\left[q\sqrt{\frac{\epsilon_{x}(\omega)}{\epsilon_{z}(\omega)}}d^{\prime}\right]}}{\epsilon_{z}(\omega)\sqrt{\frac{\epsilon_{x}(\omega)}{\epsilon_{z}(\omega)}}\sinh{\left[q\sqrt{\frac{\epsilon_{x}(\omega)}{\epsilon_{z}(\omega)}}(d+d^{\prime})\right]}}$
(5)

is a form factor Tomadin_prl_2015 ; Woessner_naturemater_2015 ;
Lundeberg_Science_2017 ; Alonso_NatureNano_2017 ; Kim_NatureCommun_2020
entirely due to the presence of the dielectrics. Optical phonons in the latter
are responsible for the frequency dependence of $F(q,\omega)$. The components
of the uniaxial dielectric tensor are usually parametrized via a Lorentz
oscillator model. In the case of a single mode, one has

$\displaystyle\epsilon_{\ell}(\omega)$ $\displaystyle=$
$\displaystyle\epsilon_{\ell}(\infty)+\frac{\epsilon_{\ell}(0)-\epsilon_{\ell}(\infty)}{1-(\omega/\omega_{\ell}^{\rm
T})^{2}-\mathrm{i}\gamma_{\ell}\omega/(\omega_{\ell}^{\rm T})^{2}},$ (6)

with $\ell=x$ or $z$. Here, $\epsilon_{\ell}(0)$ and $\epsilon_{\ell}(\infty)$
are the static and high-frequency dielectric constants, respectively, while
$\omega^{\rm T}_{\ell}$ is the transverse optical phonon frequency in the
direction $\ell$. The longitudinal optical phonon frequency $\omega^{\rm
L}_{\ell}$ satisfies the Lyddane–Sachs–Teller relation $\omega^{\rm
L}_{\ell}=\omega_{\ell}^{\rm
T}\sqrt{\epsilon_{\ell}(0)/\epsilon_{\ell}(\infty)}$. The quantity
$\gamma_{\ell}$ parametrizes losses. Eq. (6) describes hBN accurately and
realistic values of the parameters can be found e.g. in Ref.
Woessner_naturemater_2015 . Multi-mode parametrizations of
$\epsilon_{\ell}(\omega)$ for ${\rm Bi}_{2}{\rm S}_{3}$ can be found in Ref.
Eva_NatCommun_2021 .

An inspection of Eq. (6) in the limit $\gamma_{\ell}\to 0$ shows that a
dielectric described by this frequency-dependent permittivity tensor is
hyperbolic: in the lower (upper) reststrahlen band, which is defined by the
inequality $\omega^{\rm T}_{z}<\omega<\omega^{\rm L}_{z}$ ($\omega^{\rm
T}_{x}<\omega<\omega^{\rm L}_{x}$), the quantity
$\epsilon_{x}(\omega)/\epsilon_{z}(\omega)$ takes negative values. Inside the
reststrahlen bands, the dressed propagator $V(q,\omega)$ therefore displays
poles, which physically correspond to standing hyperbolic phonon polaritons
(SHPPs). In the case of negligible losses (i.e. for $\gamma_{\ell}=0$), they
can be found by looking at the zeroes of the denominator in Eq. (5):

$q_{n}(\omega)=n\frac{\pi}{(d+d^{\prime})\sqrt{|\epsilon_{x}(\omega)/\epsilon_{z}(\omega)|}},$
(7)

with $n=1,2,\dots$ commensurability . Note that the polariton wavelength
$\lambda_{\rm p}=\pi/q_{n}(\omega)\ll\lambda_{0}$, where
$\lambda_{0}=\pi(c/\omega)$ is the free-space wavelength. This is a
distinctive feature of sub-wavelength cavities. We now proceed to quantify how
the Landau parameters of single-layer graphene (SLG) are modified by these
modes.

## Fermi liquid theory in a graphene cQED setup.

The Landau parameters of a normal Fermi liquid are controlled by the one-body
electron Green’s function Hedin_1969 ; Mahan_2000 ; Giuliani_and_Vignale . In
a (massless Dirac fermion) continuum-model of SLG Kotov_RMP_2012 , the
physical (i.e. retarded) electron Green’s function $G^{\rm
ret}_{\lambda}({\bm{k}},\omega)$ is a function of a conserved wave vector
${\bm{k}}$ and depends on a band index $\lambda=\pm$. It satisfies the Dyson
equation (setting $\hbar=1$), $G^{\rm
ret}_{\lambda}({\bm{k}},\omega)=[\omega-\xi_{{\bm{k}},\lambda}-\Sigma^{\rm
ret}_{\lambda}({\bm{k}},\omega)]^{-1}$, where $\xi_{{\bm{k}},\lambda}=\lambda
v_{\rm F}k-\mu$ are single-particle band energies measured from the chemical
potential $\mu$ and $\Sigma^{\rm ret}_{\lambda}({\bm{k}},\omega)$ is the
retarded self-energy Mahan_2000 ; Giuliani_and_Vignale . The latter quantity
needs to be approximated. In a normal Fermi liquid, a good approximation is
the so-called $G0W$ approximation Mahan_2000 ; Giuliani_and_Vignale ;
Hedin_1969 ; Quinn_Ferrell ; Hedin_physrev_1965 ; Hybertsen_PRB_1986 in which
the electron self-energy is expanded to first order in the dynamically
screened Coulomb interaction $W(q,\mathrm{i}\Omega)$. The latter is
approximated at the level of the random phase approximation (RPA) Mahan_2000 ;
Giuliani_and_Vignale . The corresponding Feynman diagrams are displayed in
Fig. 3. (The same approximation can be used when translational invariance is
not at play Hedin_1969 and $G$, $\Sigma$, and $W$ depend on ${\bm{r}}$,
${\bm{r}}^{\prime}$, and $\omega$.)

In the case of SLG, the continuum-model $G0W$ self-energy Polini_ssc_2007 ;
Polini_prb_2008

$\Sigma_{\lambda}({\bm{k}},\mathrm{i}\omega_{n})=-k_{\rm
B}T\sum_{\lambda^{\prime}}\int\frac{\mathrm{d}^{2}{\bm{q}}}{(2\pi)^{2}}\sum_{m=-\infty}^{+\infty}W(q,\mathrm{i}\Omega_{m})\\\
\times
F_{\lambda\lambda^{\prime}}(\theta_{{\bm{k}},{\bm{k}}-{\bm{q}}})G^{(0)}_{\lambda^{\prime}}({\bm{k}}-{\bm{q}},\mathrm{i}\omega_{n}-\mathrm{i}\Omega_{m}),$
(8)

where $\omega_{n}=(2n+1)\pi k_{\rm B}T$ is a fermionic Matsubara frequency,
the sum runs over all the bosonic Matsubara frequencies $\Omega_{m}=2m\pi
k_{\rm B}T$,
$F_{\lambda\lambda^{\prime}}(\theta_{{\bm{k}},{\bm{k}}-{\bm{q}}})=[1+\lambda\lambda^{\prime}\cos{(\theta_{{\bm{k}},{\bm{k}}-{\bm{q}}})}]/2$
is the SLG chirality factor, $\theta_{{\bm{k}},{\bm{k}}-{\bm{q}}}$ is the
angle between ${\bm{k}}$ and ${\bm{k}}-{\bm{q}}$,
$G^{(0)}_{\lambda}({\bm{k}},\mathrm{i}\omega_{n})=1/(\mathrm{i}\omega_{n}-\xi_{{\bm{k}},\lambda})$
is the non-interacting temperature Green’s function, and

$W(q,\mathrm{i}\Omega)=\frac{V(q,\mathrm{i}\Omega)}{1-V(q,\mathrm{i}\Omega)\chi_{0}(q,\mathrm{i}\Omega)}$
(9)

is the dynamically screened interaction on the imaginary frequency axis. Here,
$\chi_{0}(q,\mathrm{i}\Omega)$ is the SLG non-interacting density-density
response function, evaluated on the imaginary-frequency axis Barlas_PRL_2007 .

We emphasize that the $G0W$ theoretical framework presented in Eqs. (8, 9) is
valid also in the presence of losses in the cavity. In this case, however,
care needs to be exercised in defining the quantities that enter Eq. (9).
Indeed, any thermal Green’s function $O(\mathrm{i}\Omega)$ satisfies Landau09

$O(\mathrm{i}\Omega)=\begin{cases}O^{\rm ret}(\mathrm{i}\Omega),&\text{for
$\Omega>0$,}\\\ O^{\rm adv}(\mathrm{i}\Omega),&\text{for $\Omega<0$,}\\\
\end{cases}$ (10)

where $O^{\rm ret}(\omega)$ and $O^{\rm adv}(\omega)$ are the corresponding
retarded and advanced Green’s functions, respectively. Note that, $O^{\rm
adv}(z)=[O^{\rm ret}(z^{*})]^{*}$.

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Figure 4: Plot of the quasiparticle Fermi velocity
$v^{*}_{\text{F}}/v_{\text{F}}$ defined in Eqs. (14, 17a, 17b, 17c) as a
function of the electron density $n_{0}$ for (a) hBN and (b)
$\text{Bi}_{2}\text{Se}_{3}$. In both cases, $d^{\prime}=d$ and the values of
$d$ have been indicated in the legend.

We also introduce the retarded density-density response function in the RPA
approximation

$\chi_{nn}(q,\omega)=\frac{\chi_{0}(q,\omega)}{1-V(q,\omega)\chi_{0}(q,\omega)},$
(11)

where $\chi_{0}(q,\omega)$ is the non-interacting (retarded) polarization
function on the real-frequency axis Shung_PRB_1986 ; Wunsch_NJP_2016 ;
Hwang_PRB_2006 and the polariton-mediated EEEI $V(q,\omega)$ has been
introduced above. The dynamical structure factor $S(q,\omega)$ plotted in Fig.
2 is related to $\chi_{nn}(q,\omega)$ by the fluctuation-dissipation theorem
Pines_and_Nozieres ; Mahan_2000 ; Giuliani_and_Vignale , i.e.
$S(q,\omega)=-(\hbar/\pi)\Theta(\omega)\operatorname{Im}\chi_{nn}(q,\omega)$,
where $\Theta(\omega)$ is the Heaviside step function.

Note that we can express $W(q,\mathrm{i}\Omega)$ in Eq. (9) as:

$W(q,\mathrm{i}\Omega)=V(q,\mathrm{i}\Omega)+V^{2}(q,\mathrm{i}\Omega)\chi_{nn}(q,\mathrm{i}\Omega).$
(12)

The first term in the previous equation is responsible for the exchange
interaction between a quasiparticle at the Fermi energy and the occupied Fermi
sea. The second term, instead, represents the interaction with particle-hole
and collective virtual fluctuations. Both terms are dressed by SHPPs.

Landau Fermi liquid parameters are controlled by the physical (i.e. retarded)
self-energy $\Sigma^{\rm ret}_{+}({\bm{k}},\omega)$, which can be obtained
from the analytical continuation
$\mathrm{i}\omega_{n}\to\omega+\mathrm{i}0^{+}$ from imaginary to real
frequencies. For definiteness, we consider the case of electron-doped SLG with
positive chemical potential $\mu$ (results for $\mu<0$ are identical because
of particle-hole symmetry). The quasiparticle weight factor $Z$ and the
renormalized Fermi velocity $v^{\ast}_{\rm F}$ (in units of the bare value
$v_{\rm F}$) can be expressed in terms of the wave-vector and frequency
derivatives of the retarted self-energy $\Sigma^{\rm
ret}_{+}({\bm{k}},\omega)$ evaluated at the Fermi surface ($k=k_{\rm F}$) and
at the quasiparticle pole $\omega=\xi_{+}({\bm{k}})$ Mahan_2000 ;
Giuliani_and_Vignale :

$\displaystyle
Z=\frac{1}{1-\left.\partial_{\omega}\operatorname{Re}\Sigma^{\rm
ret}_{+}({\bm{k}},\omega)\right|_{k=k_{\rm F},\omega=0}},$ (13)
$\displaystyle\frac{v^{\ast}_{\rm F}}{v_{\rm F}}=\frac{\displaystyle 1+(v_{\rm
F})^{-1}\left.\partial_{k}\operatorname{Re}\Sigma^{\rm
ret}_{+}({\bm{k}},\omega)\right|_{k=k_{\rm
F},\omega=0}}{1-\left.\partial_{\omega}\operatorname{Re}\Sigma^{\rm
ret}_{+}({\bm{k}},\omega)\right|_{k=k_{\rm F},\omega=0}}.$ (14)

Following some standard manipulations Giuliani_and_Vignale ; Quinn_Ferrell ,
the retarded self-energy can be expressed in a form convenient for numerical
evaluation, as the sum $\Sigma^{\rm
ret}_{\lambda}({\bm{k}},\omega)=\Sigma^{\rm
res}_{\lambda}({\bm{k}},\omega)+\Sigma^{\rm line}_{\lambda}({\bm{k}},\omega)$
of a contribution from the interaction of quasiparticles at the Fermi energy,
the residue contribution $\Sigma^{\rm res}_{\lambda}({\bm{k}},\omega)$, and a
contribution from interactions with quasiparticles far from the Fermi energy
and via both exchange and virtual fluctuations, the line contribution
$\Sigma^{\rm line}_{\lambda}({\bm{k}},\omega)$. In the zero-temperature limit

$\Sigma^{\rm
res}_{\lambda}({\bm{k}},\omega)=\sum_{\lambda^{\prime}=\pm}\int\frac{\mathrm{d}^{2}{\bm{q}}}{(2\pi)^{2}}W(q,\omega-\xi_{\lambda^{\prime}}({\bm{k}}-{\bm{q}}))\\\
\times F_{\lambda\lambda^{\prime}}(\theta_{{\bm{k}},{\bm{k}}-{\bm{q}}})\\\
\times\left[\Theta(\xi_{\lambda^{\prime}}({\bm{k}}-{\bm{q}}))-\Theta(\xi_{\lambda^{\prime}}({\bm{k}}-{\bm{q}})-\omega)\right]$
(15)

and

$\Sigma^{\rm
line}_{\lambda}({\bm{k}},\omega)=-\sum_{\lambda^{\prime}=\pm}\int\frac{\mathrm{d}^{2}{\bm{q}}}{(2\pi)^{2}}F_{\lambda\lambda^{\prime}}(\theta_{{\bm{k}},{\bm{k}}-{\bm{q}}})\\\
\times\int_{-\infty}^{+\infty}\frac{\mathrm{d}\Omega}{2\pi}\frac{W({\bm{q}},\mathrm{i}\Omega)}{\omega-\mathrm{i}\Omega-\xi_{\lambda^{\prime}}({\bm{k}}-{\bm{q}})}.$
(16)

In what follows we will calculate the following three quantities:

$\displaystyle\frac{v^{\ast}_{\text{F}}}{v_{\text{F}}}\biggr{|}_{\text{free\,space}}\equiv\frac{v^{\ast}_{\text{F}}}{v_{\text{F}}}\biggr{|}_{F(q,\omega)\to
1},$ (17a)
$\displaystyle\frac{v^{\ast}_{\text{F}}}{v_{\text{F}}}\biggr{|}_{\text{non-
hyperbolic}}\equiv\frac{v^{\ast}_{\text{F}}}{v_{\text{F}}}\biggr{|}_{F(q,\omega)\to
F(q,0)},$ (17b)
$\displaystyle\frac{v^{\ast}_{\text{F}}}{v_{\text{F}}}\biggr{|}_{\text{hyperbolic}}\equiv\frac{v^{\ast}_{\text{F}}}{v_{\text{F}}}\biggr{|}_{F(q,\omega)}.$
(17c)

The quantity in Eq. (17a) physically represents the quasiparticle Fermi
velocity (measured in units of the bare Fermi velocity $v_{\rm F}$) in
graphene, in the total absence of screening stemming from nearby dielectrics
and metal gates. It is mathematically calculated by forcing the full dynamical
propagator $V(q,\omega)$ to coincide with the instantaneous Coulomb free-space
propagator, i.e. $V(q,\omega)\to v_{q}$. The quantity $v^{\ast}_{\rm
F}|_{\text{free space}}$ is clearly measurable Elias_NatPhys_2011 .

The quantity in Eq. (17b) physically represents the quasiparticle Fermi
velocity (in units of $v_{\rm F}$) in a graphene sheet embedded in the cavity
sketched in Fig. 1. However, it is calculated by forcing $V(q,\omega)$ to
attain its static value, i.e. $V(q,\omega)\to V(q,0)=v_{q}F(q,0)$. The
quantity in Eq. (17b) therefore includes static screening from the dielectrics
and top/bottom metal gates. Notice that: i) in the $V(q,\omega)\to V(q,0)$
limit, top and bottom dielectrics are not hyperbolic and ii) $v^{\ast}_{\rm
F}|_{\text{non-hyperbolic}}$ is not measurable but it is an important
theoretical construct, as emphasized below.

Finally, the quantity in Eq. (17c) is the quasiparticle Fermi velocity (in
units of $v_{\rm F}$) in a graphene sheet embedded in the cavity sketched in
Fig. 1 and calculated with the full retarded propagator $V(q,\omega)$. This
last quantity includes the role of SHPPs.

\begin{overpic}[percent]{fig_5_a.pdf}
\put(0.0,80.0){\makebox(0.0,0.0)[tl]{(a)}}
\end{overpic}\begin{overpic}[percent]{fig_5_b.pdf}
\put(0.0,80.0){\makebox(0.0,0.0)[tl]{(b)}} \end{overpic}

Figure 5: (Color online) Cavity QED effect on the quasiparticle Fermi velocity
$v^{*}_{\rm F}$. Color plots of $\Delta_{\rm QED}$—as defined in Eq. (18)—as a
function of the electron density $n_{0}$ and dielectric thickness
$d=d^{\prime}$. Panel (a): the case of hBN. Panel (b): the case of
$\text{Bi}_{2}\text{Se}_{3}$.

## Numerical results and discussion.

We now present our main numerical results. In Fig. 4 we show the dependence of
the quasiparticle Fermi velocity $v^{\ast}_{\rm F}$ (in units of $v_{\rm
F}=10^{6}~{}{\rm m}/{\rm s}$) on carrier density $n_{0}$, for
$d=d^{\prime}=50~{}{\rm nm}$. Panel (a) in this figure refers to SLG
encapsulated in hBN. In this case and at high electron densities, the value of
$v^{*}_{\rm F}$ calculated with the retarded EEEI $V(q,\omega)$ differs little
from the result $v^{*}_{\rm F}|_{\text{free space}}$ obtained with the
instantaneous Coulomb interaction $v_{q}$. At low electron densities, however,
we notice two things: i) $v^{*}_{\rm F}|_{\text{hyperbolic}}>v^{*}_{\rm
F}|_{\text{free space}}$, which is surprising since one would expect the
opposite inequality, due to the role played by screening from metal gates and
polarization charges in hBN and ii) the density dependence of $v^{*}_{\rm
F}|_{\text{hyperbolic}}$ is different from that of $v^{*}_{\rm F}|_{\text{free
space}}$. While the latter grows logarithmically near the charge neutrality
point Kotov_RMP_2012 ; Elias_NatPhys_2011 , the former saturates to a constant
value. This is a qualitative effect, which may be verified experimentally. In
Fig. 4 we also present results for the asymptotic regime
$d=d^{\prime}\to\infty$ (distance between the metal gates is sent to
infinity). The fact that the effect of SHPPs persists also in this limit is
peculiar to the case of density-density interactions. In the case of current-
current interactions mediated by radiative photons Ashida_PRL_2023 ,
fingerprints of SHPPs disappear in the limit $d=d^{\prime}\to\infty$ (see Fig.
3 of Ref. Ashida_PRL_2023 ).

An important figure of merit to quantify the role of virtual polaritonic
excitations (i.e. the role of genuine QED effects) on the quasiparticle Fermi
velocity is the following:

$\Delta_{\rm
QED}=\frac{v^{*}_{\text{F}}\rvert_{\text{hyperbolic}}-v^{*}_{\text{F}}\rvert_{\text{non-
hyperbolic}}}{v^{*}_{\text{F}}\rvert_{\text{non-hyperbolic}}}~{}.$ (18)

The larger $\Delta_{\rm QED}$ the larger the role of virtual SHPPs. The
smaller $\Delta_{\rm QED}$ the larger is the (trivial) role of static
screening from polarization charges in the dielectric and metal gates. In
other words, large values of $\Delta_{\rm QED}$ certify that changes in the
quasiparticle Fermi velocity in a given cavity are not a trivial consequence
of static screening.

The quantity $\Delta_{\rm QED}$ is plotted in Fig. 5(a) for the case of SLG
encapsulated in hBN. We clearly see that $\Delta_{\rm QED}$ is small, on the
order of $1.5\%$ at best, and weakly dependent on $n_{0}$ and dielectric
thickness $d$. We believe that the reason is that hBN reststrahlen bands occur
at very high energies, the lower one starting at $\omega=\omega^{\rm
T}_{z}\simeq 97~{}{\rm meV}$. On the contrary, the SLG plasmon carries a large
spectral weight (in fact, the total spectral weight) at small values of $q$
and $\omega>v_{\rm F}q$. This “mismatch” in the $(q,\omega)$ plane between the
SLG plasmon and the hBN (lower) reststrahlen band suppresses spectral flow
from the matter degrees of freedom to the phonon polariton ones.

To show that this is indeed the case, we have also calculated the
quasiparticle Fermi velocity $v^{*}_{\rm F}$ for SLG encapsulated in ${\rm
Bi}_{2}{\rm Se}_{3}$. Results for this case are reported in Fig. 4(b). We
immediately see that the cavity plays a much stronger role, both at the static
and polaritonic levels. The fact that $v^{*}_{\rm F}|_{\text{non-hyperbolic}}$
is much smaller than $v^{*}_{\rm F}|_{\text{free space}}$ stems from the fact
that ${\rm Bi}_{2}{\rm Se}_{3}$ has a much smaller band gap than hBN, and
therefore much larger values of the static in-plane and out-of-plane
permittivities. The fact that $v^{*}_{\rm F}|_{\text{hyperbolic}}$ is larger
than $v^{*}_{\rm F}|_{\text{non-hyperbolic}}$ is due to the fact that ${\rm
Bi}_{2}{\rm Se}_{3}$ has reststrahlen bands at much lower energies, the lower
one occurring at energies on the order of $8~{}{\rm meV}$. Fig. 5 shows the
quantity $\Delta_{\rm QED}$ as a function of system parameters. Without fine
tuning, the QED effect on the quasiparticle Fermi velocity is much larger in
this cavity rather than in the case of hBN.

A systematic quest of hyperbolic dielectrics with reststrahlen bands at ultra-
low energies in order to maximize $\Delta_{\rm QED}$ is beyond the scope of
the present Letter. Nevertheless, our work shows that measurable QED effects
emerge in DC magneto-transport in planar van der Waals cavities. We expect
that these effects can be greatly enhanced in cavities where SHPPs and matter
degrees of freedom are subject to lateral (and not only vertical) confinement,
as in the case of Ref. Hanan_arXiv_2022 . From a theoretical point of view, it
will be very interesting to extend the theory presented in this work to (a)
real and moiré crystals and (b) strongly correlated materials Loon_npj_2023 .

## Acknowledgements.

M. P. wishes to thank G. M. Andolina, M. Ceccanti, F. H. L. Koppens, and A. R.
Hamilton for many useful discussions. R. R. and A. T. acknowledge the
“National Centre for HPC, Big Data and Quantum Computing”, under the National
Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.4
funded from the European Union - NextGenerationEU. G. M. was supported by the
Rita-Levi Montalcini program of the MUR. A. H. M. was supported by a grant
from the Keck foundation. M. P. was supported by the European Union’s Horizon
2020 research and innovation programme under the Marie Sklodowska-Curie grant
agreement No. 873028 - Hydrotronics and by the MUR - Italian Minister of
University and Research under the “Research projects of relevant national
interest - PRIN 2020” - Project No. 2020JLZ52N, title “Light-matter
interactions and the collective behavior of quantum 2D materials (q-LIMA)”.

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