# Propagation of shear stress in strongly interacting metallic Fermi liquids
enhances transmission of terahertz radiation

D. Valentinis1,2 J. Zaanen3 D. van der Marel1,∗ 1Department of Quantum Matter
Physics, University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva 4,
Switzerland  2Institute for Theoretical Condensed Matter physics, Karlsruhe
Institute of Technology, Wolfgang-Gaede Straße 1, 76131 Karlsruhe, Germany
3Institute-Lorentz for Theoretical Physics, Leiden University, PO Box 9506,
NL-2300 RA Leiden, The Netherlands<EMAIL_ADDRESS>
(March 29, 2021)

###### Abstract

A highlight of Fermi-liquid phenomenology, as explored in neutral 3He, is the
observation that in the collisionless regime shear stress propagates as if one
is dealing with the transverse phonon of a solid. The existence of this
$``$transverse zero sound$"$ requires that the quasiparticle mass enhancement
exceeds a critical value. Could such a propagating shear stress also exist in
strongly correlated electron systems? Despite some noticeable differences with
the neutral case in the Galilean continuum, we arrive at the verdict that
transverse zero sound should be generic. We present an experimental setup that
should be exquisitely sensitive in this regard: the transmission of terahertz
radiation through a thin slab of heavy-fermion material will be strongly
enhanced at low temperature and accompanied by giant oscillations, which
reflect the interference between light itself and the $``$material photon$"$
being the actual manifestation of transverse zero sound in the charged Fermi
liquid.

##  I Introduction

The elucidation of the Fermi liquid as a unique state of matter is a highlight
of twentieth century physics [1]. It has a precise identity only at strictly
zero temperature. At times large compared to $\hbar/(k_{B}T)$ it can be
adiabatically continued to the high-temperature limit and it is therefore
indistinguishable from a classical fluid - the $``$collision-full regime$"$ .
However, at energies $\hbar\omega>k_{B}T$ (the $``$collisionless regime$"$)
the unique nature of the zero-temperature state can be discerned, being
different from either the non-interacting Fermi gas or a thermal fluid.
Instead, the closest analogy is with ordered matter characterized by
spontaneous symmetry breaking: the Fermi surface takes the role of order
parameter, acting however very differently from its $``$bosonic$"$ analogues.
A case in point is zero sound. The $F_{0}^{S}$ Landau parameter encapsulates
the effects of interactions on the compressibility. When this is finite, a
propagating collective mode splits off from the Lindhard continuum of
particle-hole excitations. The Fermi surface $``$hardens$"$ and zero sound is
the coherent s-wave breathing motion of this $``$membrane$"$ corresponding to
an oscillation of fermion density in space and time. Historically the neutral
Fermi liquid realized in 3He has formed the experimental theater to test these
general notions. The observation of the sound attenuation maximum at
$\hbar\omega\approx k_{B}T$ separating the hydrodynamic protection of first
(thermal) sound and the $``$Fermi-surface$"$ protected zero sound has been
seminal in this regard [2].

The strongly interacting Fermi liquid has yet another, less well known zero-
temperature property. This revolves around the response to external shear
stress. Shear rigidity is supposed to be uniquely associated with the breaking
of translations, with the ramification that a solid exhibits a propagating
shear mode (the transverse acoustic phonon, TA) and a liquid a relaxational
response instead. The spatial angular momentum $``$spin-1$"$ Landau parameter
regulates the mass enhancement of quasiparticles, $m^{\ast}/m=1+F_{1}^{S}/3$.
Abrikosov and Khalatnikov [3, 4] predicted that for $F_{1}^{S}>6$ a new
propagating mode forms - see Fig. 1a. This mode propagates in fact shear
stress ($``$transverse zero sound$"$), entailing a coherent shear deformation
of the Fermi sphere [4]. This may be confusing at first sight: the Fermi
liquid is a liquid also at zero temperature. The resolution is that the static
shear rigidity of a solid is associated with the spin-2 channel involving two
space directions (e.g. Refs. 5, 6). Propagating shear requires only one space
direction: it is spin-1 and can therefore be reconciled with a system that is
hydrostatically a liquid. Hence, shear-stress propagation in a Fermi liquid is
unique in the regard that it is an inherently dynamical phenomenon. As for
longitudinal zero sound, temperature dictates the parameter space in which
transverse zero sound propagates. The Fermi liquid exhibits a
$``$viscoelastic-like$"$ behavior [7], in the sense that at short times
(collisional regime) shear is relaxing while it propagates in the long-time
collisionless regime. This prediction was confirmed in the 1970’s by
transverse ultrasound measurements in 3He [8].

Figure 1:  Transverse collective mode spectrum of neutral and charged Fermi
liquids.  (a) Sketch of the real part of the dispersion relation of transverse
sound in a neutral Fermi liquid, corresponding to a pole in the transverse
susceptibility. For $F_{1}^{S}>6$ (solid line) transverse sound propagates at
frequencies above the Lindhard continuum (purple-shaded area); for
$F_{1}^{S}<6$ transverse sound is Landau-damped (dashed line), and the pole
disappears deep inside the continuum at an $F_{1}^{S}$-dependent frequency
marked by the red dot. (b) Sketch of the real part of the dispersion relation
of shear collective modes (polaritons) in a charged Fermi liquid, in the long-
wavelength propagating-shear regime with $F_{1}^{S}>6$, obtained from the
Fermi-liquid transverse dielectric functin in linear response. The interaction
of the photon root (dashed golden line) with the transverse sound of the
neutral case (panel a) generates the usual plasmon-polariton (solid orange
line), which propagates above the plasma frequency $\omega_{p}$, and the
shear-polariton (red solid line), which bears the signatures of Fermi-surface
shear rigidity in the charged case. The shear-polariton is repelled by the
interaction with the nearby photon root at low frequency (red-shaded area), so
that it bends with quadratic dispersion and submerges into the continuum at
$q\propto\omega_{p}/c$.

Electrons in solids also form Fermi liquids. We address here the following
question: could it be that propagating shear is ubiquitous in the large
variety of heavy Fermi liquid metals [9]? We will present here the case that
it appears to be natural for such collective modes to exist in these systems -
see Fig. 1b. Propagating shear in solids has been overlooked up to now for the
simple reason that it is quite difficult to measure. The difficulty is that
one can only exert shear forces on electrons via electromagnetic (EM) fields.
To this end, gradients have to be applied and special experimental conditions
are required to overcome the kinematic mismatch between the electron and light
velocities.

Figure 2:  Simulations of optical transmission experiments at normal incidence
on a charged Fermi liquid slab.  (a) Schematics of the transmission
experiments on a charged Fermi-liquid slab of thickness $d$. The incident
electric field $E(z,\omega)$ produces the plasmon-polariton and the shear-
polariton in the slab, which correspond to wave vectors $q_{+}(\omega)$ and
$q_{-}(\omega)$ for each frequency. Each mode is the sum of two
counterpropagating rays from Fabry-Perot internal reflections, which lead to
the transmitted field $E_{T}(z,\omega)$. (b) Schematic depiction of the Fermi-
surface shear deformation induced by transverse sound. (c) Slab transmission
modulus $|t(\omega)|=|E_{T}(z,\omega)/E(z,\omega)|$, as a function of
frequency $\omega/\omega_{p}$ for a Fermi liquid in the Galilean continuum at
temperatures $T=300$ K (red line) and $T=1$ K (blue line). We assume a slab
thickness $d=10c/\omega_{p}$ ($\sim 0.1\mu$m for a density $n=10^{23}$ cm-3),
Landau parameters $F_{0}^{S}=1$ and $F_{1}^{S}=20$, and the slip length
corresponding to specular interface scattering with $hh^{\prime}k_{F}^{2}=3.8$
(cf. sec. IV). The shear mode submerges into the continuum below the dashed
green line. (d) Same data as in panel c on logarithmic scale. The gray dashed
line shows the result for a standard Ohmic conductor. (e) Slab transmission
modulus $|t(\omega)|$ for a charged Fermi liquid including momentum relaxation
due to Umklapp scattering with $\alpha_{U}=0.5$, and acoustic phonons with
Debye temperature $\hbar\omega_{D}/k_{B}=$500 K and electron-phonon coupling
constant $\lambda=0.1$. All other parameters are as in panels c-d. (f) Same
data as in panel e on logarithmic scale. The gray dashed line is the result
for an Ohmic conductor.

##  II Results

We will explain an experimental setup where propagating shear should give rise
to spectacular, counter-intuitive signals. The configuration is conceptually
straightforward: it involves the transmittance of light through a thin
metallic layer (see Fig. 2a) as a function of radiation frequency and for a
fixed layer thickness. Dealing with a weakly interacting Fermi-liquid metal,
the incident light is heavily attenuated inside the layer, which becomes
essentially nontransparent: the EM field strength is damped exponentially on a
length scale set by the skin depth. This behavior continues up to the plasma
frequency $\omega_{p}$. However, upon increasing the quasiparticle mass, a
critical point is reached for $m^{\ast}\geq 3m$: above this threshold a
collective mode of coherent shear deformation of the Fermi sphere propagates
inside the material (see Fig. 2b), and the transmittance starts to severely
oscillate as function of frequency (see Fig. 2c-f). In the ideal case of
extremely clean samples and very low temperatures, the transmittance can
become so large that at the maximum of the oscillation the metallic layer may
become quasi-transparent, at a thickness where the weakly interacting metal
would block all the incident light (see Fig. 2c-d)! Recently Khoo et al.
studied shear sound of interacting electrons in a different setting, namely in
two-dimensional systems [10] and in narrow strips [11].

The plan for the remainder of this paper is as follows. On the basis of the
semiclassical Boltzmann kinetic theory applied to the Fermi Liquid (the Landau
kinetic equation [3]) we will present here a quantitative phenomenology for
the transverse optical response of heavy Fermi-liquid metals in the Galilean
continuum. We will firsts re-derive the Abrikosov-Khalatnikov results for the
neutral Fermi liquid, which form a useful template for the comparison with the
charged Fermi liquid. We will then turn to the charged Fermi liquid, focusing
on the implications of shear-stress propagation for the optical properties
associated with the transverse response of heavy-mass quasiparticles. We will
subsequently compute quantitative predictions for the transmission experiment.
We will finish with a discussion of how such computations generalize to heavy-
mass Fermi liquids realized in solids. We will suggest an experimental
protocol to observe propagating shear modes, concluding with the potential to
use this as a new form of high precision $``$Fermi-surface spectroscopy$"$ .
The latter may be used to shed a new light on the mysteries of strongly
correlated electron systems.

Let us start out reproducing the Abrikosov and Khalatnikov results [3, 4] for
the neutral Fermi liquid. The Landau kinetic equation contains all the
information required to analyze the linear response to shear perturbations of
given momentum $\bm{q}$ and frequency $\omega$. This information is contained
in the transverse susceptibility $\chi_{T}(\bm{q},\omega)$, while the
quasiparticle interactions are parametrized in terms of the momentum-
conserving collision time $\tau_{c}\approx\hbar E_{F}/[(\hbar\omega)^{2}+(\pi
k_{B}T)^{2}]$ in the Galilean continuum 111For all calculations in this work
we employed the momentum-conserving collision time $\tau_{c}$ (a single-
particle quantity) in accordance with Eq. (24) in sec. IV. This is different
from the sound-attenuation or optical scattering rates (two-particle
quantities), which for a Fermi liquid are $\propto\hbar
E_{F}/[(\hbar\omega)^{2}+(2\pi k_{B}T)^{2}]$ [44]. (cf. sec. IV).

Figure 3:  Diagram of the shear response in neutral and charged Fermi liquids.
(a) Shear-response diagram of the zero-temperature neutral Fermi liquid as a
function of excitation energy $\hbar\omega/E_{F}$ and $F_{1}^{S}$, calculated
from the transverse-sound velocity $v_{s}(\omega)=\omega(q)/q$, and using the
Fermi-liquid collision time $\tau_{c}$. Notice that the zero-temperature phase
diagram is valid at any electron density $n$, since at $T=0$ the collision
time $\tau_{c}$ becomes a universal function of $\hbar\omega/E_{F}$. The
threshold $F_{1}^{S}=6$ separates the Lindhard regime, where the Lindhard
continuum dominates the response and the transverse susceptibility has no
pole, and the propagating shear regime, where transverse sound propagates. (b)
Neutral shear-response diagram at $T=3$ K and $E_{F}=7.6$ eV (electron density
$n=10^{23}$ cm-3). For $F_{1}^{S}<6$, the red solid line marks the crossover
between the viscous regime of hydrodynamic damped transverse sound and the
Lindhard regime. For $F_{1}^{S}>6$, the condition $\omega=v_{F}^{\ast}$Re$q$
(dashed golden line) identifies the crossover between viscous and propagating
shear regimes. (c) Long-wavelength shear-response diagram of the charged Fermi
liquid at $T=0$ and $E_{F}=7.6$ eV, calculated from the dispersion relation of
the shear-polariton and the Fermi-liquid optical conductivity in Kubo
formalism. The shear-polariton emerges from the continuum at the solid blue
line. Deep inside the continuum, the response is dominated by the Lindhard
continuum and yields the phenomenology of anomalous skin effect. (d) Charged
shear-response diagram at $T=3$ K and $E_{F}=7.6$ eV. The red line shows the
condition $\delta_{0}=l_{c}$, with $\delta_{0}$ the EM skin depth in anomalous
regime and $l_{c}$ the collisional mean free path, which indicates the
crossover between the anomalous skin effect and viscous regimes, with the
latter implying hydrodynamic response.

##  III Discussion

Transverse zero sound appears as a pole of $\chi_{T}(\bm{q},\omega)$ above the
critical $F_{1}^{S}$, representing the oscillating shear deformation of the
Fermi surface (see Fig. 2b). This reproduces the main Abrikosov-Khalatnikov
result as indicated in Fig. 1a: the shear mode appears as an anti-bound state
formed from the Lindhard continuum. In the collisionless limit
$\tau_{c}\rightarrow+\infty$ and at large $F_{1}^{S}$, this mode has a
velocity $v_{s}=v_{F}\sqrt{(1+F_{1}^{S}/3)/5}$ [13], which is enhanced with
respect to the velocity $v_{F}^{\ast}$ of Fermi-surface quasiparticles. The
quasiparticle Fermi velocity is renormalized by interactions according to
$v^{\ast}_{F}=\hbar k_{F}/m^{\ast}=v_{F}/(1+F_{1}^{S}/3)$ where $v_{F}=\hbar
k_{F}/m$ is the bare Fermi velocity and $k_{F}=(3\pi^{2}n)^{1/3}$ is the Fermi
wave vector in a three-dimensional Fermi liquid, with $n$ electron density.
Therefore, for small $q$ and $F_{1}^{S}>6$ transverse sound is protected
against the Landau damping associated with entering the Lindhard electron-hole
continuum [3]. We notice that for $F_{1}^{S}<6$ a remnant of the shear mode
still exists at small momenta, but it is critically damped due to the decay in
the Lindhard continuum (dashed line in Fig. 1a), and it ceases to exist
altogether deep inside the continuum (dot in Fig. 1a) [4].

For future purposes it is convenient to sketch a $``$phase diagram$"$ of the
shear response in the neutral Fermi liquid as a function of $\omega$ and
$F_{1}^{S}$, both for zero- (a) and finite temperature (b): Fig. 3. This will
later serve as a basis to understand the transverse response of quasiparticles
endowed with electric charge, where the latter allows for the interaction of
Fermi-surface shear modes with photons.

In essence, the response diagram of Fig. 3a-b outlines the conditions on
$\omega$ and $F_{1}^{S}$ for transverse sound to develop in the neutral
system, as testified by the emergence of a pole in the susceptibility
$\chi_{T}(\bm{q},\omega)$ (cf. sec. IV) in the presence of the Fermi-liquid
collision time $\tau_{c}$ [4, 8, 14]. An analysis of the mode dispersion
relation $\omega(q)$ for $F_{1}^{S}>6$ confirms that the real (i.e. reactive)
part of $\omega(q)$ prevails over its imaginary (i.e. dissipative) part for
Re$\omega(q)\geq v^{\ast}_{F}q$: indeed transverse sound propagates nearly
undamped outside of the Lindhard continuum, consistently with the sketch in
Fig. 1a. Such propagation occurs in the regime $\omega\tau_{c}\gg 1$, and in
the large-$F_{1}^{S}$ collisionless limit the mode velocity reaches
$v_{s}^{\infty}$.

It is instructive to translate the qualitative criterion
Re$\omega(q)=v^{\ast}_{F}q$, which characterizes the crossover from overdamped
to propagating shear at the edge of the continuum, into a condition
Re$\omega(q)\geq v^{\ast}_{F}q$ on the mode velocity
$v_{s}(\omega,F_{1}^{S})=\omega(q)/q$ as commonly defined in the 3He
literature [4, 8]. We determine such condition numerically, as marked by the
dashed orange line in Fig. 3a-b. Conversely, for $F_{1}^{S}<6$ the pole in
$\chi_{T}(\bm{q},\omega)$ is heavily Landau-damped at low frequency, in the
hydrodynamical/collisional regime $\omega\tau_{c}\ll 1$, and it disappears at
an $F_{1}^{S}$-dependent frequency evaluated numerically and marked by the red
solid line in Fig. 3a-b.

At zero temperature (Fig. 3a) the physical meaning of the above analysis is
very simple: above the critical $F_{1}^{S}$ the response is governed by the
propagating shear mode, and otherwise one recovers the incoherent response of
the Fermi gas, devoid of collective modes. In the finite-temperature case
(Fig. 3b) the crossover from the high-frequency zero sound to the low-
frequency thermal regime occurs at a frequency $\hbar\omega\ll k_{B}T$ , as
known in the 3He community: such frequency is so low because we need many
collisions during an excitation cycle, i.e. $\omega\tau_{c}\ll 1$, to
establish local equilibrium among quasiparticles and enable hydrodynamic
behaviour. In thermal regime there is no longer a critical change associated
with $F_{1}^{S}$: we are dealing with the Galilean-invariant continuum, which
is always governed by Navier-Stokes hydrodynamics characterized by the
viscosity of the finite-temperature Fermi liquid $\nu(0)=v_{F}^{2}\tau_{c}$,
as follows from the kinetic theory [3, 15] (cf. sec. IV). With regard to
hydrodynamics, it will become clear that it is largely irrelevant for the
electron systems: extremely clean systems such as graphene are required to
observe hydrodynamic flow [16, 17, 18, 19, 20, 21], while the latter is
actually very difficult to detect by radiative means.

Let us now turn to the less well-charted charged Fermi liquid - to the best of
our knowledge the transverse response for large $F_{1}^{S}$ was only addressed
in the book by Nozieres and Pines [1] in the form of an exercise. A first
ramification of the Coulomb interactions is a renormalization of the Landau
parameters [22, 23]. More importantly, one has now to address the linear
response to EM sources, which can be handled by time-dependent mean-field
(RPA) using the Kubo formalism (cf. sec. IV). The way this works in the
longitudinal channel is well known. The zero (longitudinal) sound mode gets
$``$dressed$"$ by the Coulomb interaction, with the effect that it is promoted
to a plasmon with dispersion large $\sim\sqrt{\omega_{p}^{2}+v_{F}^{2}q^{2}}$
where $\omega_{p}=\sqrt{ne^{2}/\epsilon_{0}m}$ is the electron plasma
frequency.

However, light is transversally polarized. This means that, in the presence of
a propagating transverse mode in the material, the latter blends with light
through radiation-matter interaction, which can be rationalized in terms of
simple linear mode coupling [24]. As for the plasmon, this is a universal
affair and one may as well first consider a Wigner crystal formed by
electrical charges [5]. The shear modes of such ensemble are the electronic
analogues of transverse acoustic phonons in ionic lattices, and such
$``$phonons$"$ couple to photons possessing a much larger velocity (of light).
A first product of this coupling is the usual $``$light-like$"$ transversally
polarized plasmon-polariton, which is universal: indeed, at $q=0$ both
transverse and longitudinal $``$plasmons$"$ precisely match. However, its
partner is the phonon-like $``$shear-polariton$"$: this mode continues to be
massless, but it acquires a quadratic dispersion $\omega\sim q^{2}$ at small
momenta [5, 6] (a similar mechanism occurs in a particular viscoelastic-like
holographic strange metal, see reference 25.) The shear mode of the heavy-mass
Fermi liquid shares an analogous fate, see Fig. 1b. The only qualitative
difference with the neutral case of Fig. 1a is that, for very small momenta
being less than $q\approx\omega_{p}/c$ (cf. sec. IV), the shear mode dips
below the upper bound of the Lindhard continuum at $\omega\approx v_{F}q$,
thus falling prey to Landau damping. Actually, according to the kinetic
theory, Landau damping further modifies the mode dispersion, which becomes
$\omega\propto q^{4}$ at vanishing frequencies [13].

Once again, the only way to exert forces at finite frequency on the charged
Fermi liquid is by electromagnetic radiation. This in turn implies
restrictions on which intrinsic properties of this system can actually be
measured: a primary consequence is that the response to the EM field is
inherently a boundary phenomenon. The reason is that an oscillating EM field
is screened by the metal, with the ramification that the transverse response
corresponding to the screening currents decreases exponentially away from the
boundary, the classical skin effect. In an Ohmic conductor the skin depth is
$\delta_{s}=\sqrt{2\rho/\mu\omega}$ where $\rho$ and $\mu$ are the DC
resistivity and permeability of the conductor, respectively. This in turn
assumes the Drude model, departing from a free Fermi gas, characterized by an
overall scattering time $\tau_{K}$ due to momentum relaxation. In the Galilean
continuum this is absent, and one has to reconsider the skin effect mechanism.
The answer is well known at low temperature and small $F_{1}^{S}$ in the form
of the anomalous skin effect, elucidated a long time ago by Reuter and
Sondheimer [26]. Upon lowering temperature, the mean free path of the
electrons $l_{mf}=v_{F}\tau$ (either $\tau_{c}$, $\tau_{K}$ or both) exceeds
the length over which the transverse electric field penetrates the metal. The
screening is now entirely due to the Lindhard continuum, with the outcome that
the skin depth saturates as a function of temperature at a value
$\delta_{0}=\\{(4c^{2}v_{F})/(3\pi\omega\omega_{p}^{2})\\}^{1/3}$.

This implies in turn that in the Galilean continuum the finite-temperature
hydrodynamical fluid (e.g. Fig. 3b) is actually hidden from observation in a
large frequency regime! Hydrodynamics sets in at a scale larger than the
collision length $l_{c}=v_{F}^{\ast}\tau_{c}$, but we just learned that the
screening is complete at the length scale $\delta_{0}<l_{c}$. In this regime
the hydrodynamical fluid responds therefore like a free Fermi gas. Upon
raising temperature $l_{c}$ decreases, and when $\delta_{0}\approx l_{c}$ a
crossover occurs to a regime where hydrodynamical currents take over the
screening (cf. sec. IV). In an earlier paper we analyzed the ramifications of
such screening [15]. These observations may be of relevance in the context of
graphene-like systems, but we leave a detailed analysis to a future
publication.

As for the neutral case, we can sketch an electromagnetic response diagram:
Fig. 3c-d. In the presence of electric charge, the role of
$\chi_{T}(\bm{q},\omega)$ is taken by the transverse dielectric function
$\epsilon_{T}(\bm{q},\omega)$ , with the two quantities linked by the Kubo
formula (cf. sec. IV). The fact that the Fermi liquid is interrogated by
photons, with dispersion relation $\omega$ =c q (see Fig. 1b), fixes momentum
so that the only independent parameters are $\omega$ and $F_{1}^{S}$.
Polaritons are self-consistent solutions of Maxwell’s equations in the Fermi
liquid, i.e. $q^{2}c^{2}/\omega^{2}=\epsilon_{T}(\bm{q},\omega)$: in other
words, we look at the photon self-energy to deduce the response of the matter.

In the following, we describe the EM phenomenology through the analytical
solutions for polaritons obtained deep in the low- and high-frequency regimes
(cf. sec. IV), which are sufficient to understand the essential physics. We
leave the full numerical computation of polaritons near crossovers between
different regimes for a subsequent work, but here we provide robust
qualitative estimates on where such crossovers occur. In the high-frequency,
long-wavelength regime $\omega\gg v_{F}^{\ast}$Re$q$, at leading order we find
two polariton branches (cf. sec. IV): one is the usual plasmon-polariton root,
which propagates above the plasma frequency. The other, lower-frequency root
is the $``$shear-polariton$"$ , a genuine product of Fermi-surface shear
interacting with light. As such, the shear-polariton is a collective mode
rooted in time-dependent mean field, being in this regard in the same category
as the plasmon or even a TA phonon. Via continuity, the shear-polariton is
associated with a coherent shear current, which takes over the screening from
the Lindhard excitations as soon as it springs into existence.

Fig. 3c-d shows that, compared to the neutral case, charged propagating shear
takes over above a critical, finite frequency. This expresses the fact that at
large wavelengths the shear mode becomes overdamped because of its dispersion
bending from linear to quadratic due to EM forces at $q\leq\omega_{p}/c$ (see
Fig. 1b), which causes the mode to dive into the Lindhard continuum [1]. The
qualitative criterion for the shear-polariton to emerge from the continuum is
that its dispersion obeys $\omega\gg v_{F}^{\ast}$Re$q$. We estimate the
latter condition analytically (cf. sec. IV), as shown by the blue line in Fig.
3c-d. Notice that the shear-polariton never propagates for $F_{1}^{S}<6$: this
is because light needs to couple to propagating transverse sound in the
material to generate the propagating shear-polariton, and transverse sound
propagation necessitates $F_{1}^{S}>6$ as previously explained.

On the other hand, in the low-frequency, short-wavelength regime $\omega\ll
v_{F}^{\ast}$Re$q$ and at finite $\tau_{c}$, the leading-order expansion of
the Fermi-liquid dielectric function yields anomalous skin effect (cf. sec.
IV) due to the incoherent response of the continuum to EM fields. A final
difference with the neutral case is that the apparent onset of the
hydrodynamical $``$viscous$"$ regime is at a much lower frequency than in the
neutral case. But as we explained, this is in a way a deception: although in
principle the bulk may well already be in the thermal regime, the boundary
layer where screening currents are running is still too thin for the fluid to
reach local thermal equilibrium, given that the collision length is large
compared to the skin depth, and the ensuing Fermi-gas response (anomalous skin
effect) is insensitive to the difference between the thermal- and zero-
temperature liquid. We qualitatively estimate the crossover between Lindhard
and hydrodynamical regimes by solving $\delta_{0}=l_{c}$ numerically with the
Fermi-liquid collision time $\tau_{c}$: this gives the red curve in Fig. 3d.

Having established where to look for the propagating shear in the Fermi-liquid
metal, the next question is how to find it. In this regard we wish to propose
an experiment yielding a unique, qualitative signal that will unambiguously
confirm or falsify the existence of the shear mode. This revolves around
shining light through a thin, micrometer-sized slab of metal. In a normal
metal, the field strength of the radiation decays on the length scale set by
the (anomalous) skin depth, which is small compared to the width of the slab.
In these conditions, the slab is opaque. However, the propagating shear mode
is in essence a photon $``$made from matter$"$, being only different in the
regard that its velocity is much smaller than the light velocity. The coupled
light-matter system is therefore characterized by two propagating modes: the
light-like plasmon- and matter-like shear-polariton, characterized by wave
vectors $q_{+}(\omega)$ and $q_{-}(\omega)$ -see Fig. 2a. These modes
propagate with different velocities, while both are in part reflected back at
each interface. Given the difference in the real part of their momenta, their
phases evolve differently along the path through the material. Accordingly –
pending the frequency of the incoming light – the two polaritons interfere
constructively or destructively at the slab boundaries, giving rise to a
transmitted field that exhibits oscillations as function of frequency (see
Fig. 2b). Given the propagating nature of both polaritons, one already
anticipates that these oscillations in the transmittance of the slab may be
very large, as confirmed by our computations: Fig. 2c-d.

We addressed a similar set up some time ago, dealing instead with a Navier-
Stokes liquid characterized by its relaxational shear mode [15]. This
computation is trivially adapted to the present propagating shear – it is just
an exercise involving the transmission and reflection of two linearly coupled
evanescent waves in the slab (cf. sec. IV). In the hydrodynamical case the
shear mode is overdamped, with a complex momentum dominated by its imaginary
part while it has still a finite real ($``$propagating$"$ ) part. We found out
that this sufficed to form similar oscillations as function of frequency,
being however very small while the transmission as a whole is strongly
attenuated [15]. Dealing with the propagating shear mode, its imaginary
momentum is small compared to its real one, which entails that the slab
becomes quite transparent and characterized by large oscillations in the
transmittance as a function of frequency.

Using the results of the kinetic theory as an input, we can realistically
simulate the transmission of slabs containing Galilean-invariant Fermi liquids
(cf. sec. IV). For the results in Fig. 2 we assumed a slab thickness
$d=10c/\omega_{p}$ ($\cong$ 0.1 $\mu$m for a density $n=10^{23}$ cm-3), Landau
parameters $F_{0}^{S}=1$, and $F_{1}^{S}=20$, and the slip length
corresponding to specular interface scattering with $hh^{\prime}k_{F}^{2}=3.8$
allowing for a maximal transmission of $|t(\omega)|\approx$0.06 in the
frequency range between $10^{-3}\omega_{p}$ and 0.1$\omega_{p}$. As we just
discussed, at small frequencies up to the green dashed line the shear mode is
overdamped, while spectacular transmittance oscillations set in upon entering
the propagation regime (see blue line in Fig. 2d). One infers that the overall
transmission is much larger than the standard result for an ohmic conductor
(dashed gray line in Fig. 2d). Up to this point we ignored the lifetime of the
shear mode. This is accounted for by the kinetic theory, parametrized by the
frequency dependence of the collision time, $\tau_{c}\propto\omega^{-2}$. This
attenuates the shear mode at high frequencies, with the consequence that the
enhanced transmission decreases rapidly as illustrated in the log-log plot
(Fig. 2). Raising temperature has a very similar effect since
$\tau_{c}(\omega=0,T)\propto T^{-2}$. (red line in Fig. 2d).

The conclusion is that, in order to detect the propagating shear oscillations
in a heavy-mass Fermi liquid, temperature should be small compared to $E_{F}$,
while small frequencies (of order of $10^{-3}$ of the plasma frequency) are
required, just above the point where the propagating shear escapes the long-
wavelength Landau damping sketched in Fig. 1b.

As illustrated in the above, the Landau kinetic theory has remarkable powers
in its capacity to deliver a quantitative phenomenology. However, it requires
Galilean invariance, which is badly broken in the experimentally available
charged Fermi-liquids: the metallic state of electrons formed in solids. Such
breaking of Galilean invariance has the fundamental ramification that total
momentum is no longer conserved. Considering the longitudinal response of the
Fermi-liquid, total momentum conservation entails that the sound mode (or the
plasmon) is infinitely long-lived both in the zero- and finite-temperature
cases in the long-wavelength limit. The same principle applies to transverse
zero sound, although it needs in addition the sharp zero-temperature Fermi
surface.

The role of momentum (non-)conservation is thereby identical regardless of
whether one is dealing with the longitudinal or transverse electromagnetic
response, and we know precisely how this works for the former. In the Fermi
liquid, the large Umklapp momentum scattering associated with a perfectly
periodic lattice is irrelevant at long wavelength, and what remains is elastic
scattering against impurities. Furthermore, at finite temperature (and energy)
momentum relaxation also acts through electron-electron interactions and
electron-phonon coupling. Good metals are $``$nearly hydrodynamical$"$ , in
the sense that the width of the Drude peak, which represents the momentum
relaxation rate, is small. Such scale governs the decay of transverse sound as
well. In a Fermi liquid, the main consequence of a finite-momentum lifetime is
that the hydrodynamical flow of the continuum is in any but the most perfect
solids (like graphene) immediately destroyed and replaced by the Drude (Ohmic)
regime, where momentum is damped in the lattice by the individual
quasiparticles [27]. Consequently, the hydrodynamic regime in Fig. 3d is
replaced by the Ohmic skin effect.

However, the propagating shear is only affected $``$perturbatively$"$ : it
will decay faster compared to the continuum case due to momentum relaxation.
To offer some intuition regarding these numbers, we show in Fig. 2e-f a
realistic example (cf. sec. IV). Compared to Fig. 2c-d we introduce two
additional scattering channels: the temperature- and frequency-dependent
momentum relaxation time $\tau_{K}(T,\omega)$ which parametrizes the $T^{2}$
contribution rooted in a realistic $``$Umklapp efficiency$"$ $\alpha_{U}=0.5$
[15, 13], and the coupling to phonons, which is particularly efficient in
relaxing momentum. Transverse zero sound will couple to transverse lattice
vibrations, and we consider such acoustic phonons with a Debye temperature
$\hbar\Omega_{D}/k_{B}=$ 500 K and an electron-phonon coupling constant
$\lambda=0.1$. The outcome is that phonons and Umklapp effectively conceal any
signal of shear propagation at high temperatures, but at low temperatures the
oscillations are still clearly discernible. However, even for a relatively
weak electron-phonon coupling, the phonons represent an efficient decay
channel for transverse zero sound: as the figure suggests, one better looks
for transverse shear at frequencies below the typical phonon frequency scale
in a given material.

In addition to slab transmission, surface impedance measurements provide in
principle simpler access to the EM skin depth which discerns the different
regimes in Fig. 3d [15, 13]. However, explicit computation [28] shows that the
differences between the signals in propagating-shear, Ohmic and anomalous skin
effect regimes are minimal, once momentum relaxation is taken into account.
Hence surface impedance experiments on bulk heavy-mass Fermi-liquids are only
subsidiary with respect to thin-film optical spectroscopy in identifying
Fermi-surface propagating shear.

A real complication is that invariably strong periodic background potentials
are required to endow the Fermi surface of electron systems with a large
quasiparticle mass. In turn, these strongly renormalized Fermi liquids are
rooted in a strongly interacting microscopic physics, which is typically
described by Kondo lattice models, by the proximity to quantum phase
transitions, and so forth [29]. As a matter of fact, it is not at all
understood why a heavy Fermi liquid emerges in the deep infrared, but this is
the domain where the experimental study of transverse zero sound may have
considerable potential as a spectroscopic technique, due to its exquisite
sensitivity to the characteristics of the Fermi surface.

There is leeway for theoretical guidance in the form of the tradition on
determining the renormalized band structure governing the low-temperature
Fermi-surface topology. This is typically a rather complex affair for real
solids, with a variety of sheets characterized by different effective masses
[30]. In such case, the shape of the Fermi surface, as well as the
quasiparticle mass and transverse sound, are associated with projections based
on the point-group symmetries of the crystal. This implies that, for instance,
the velocity and the damping of transverse sound become anisotropic, but this
can be dealt with in principle, as demonstrated in the theory of elasticity
and quite recently also in hydrodynamics [31, 32].

Can transverse zero sound be measured in the laboratory, to be exploited as a
spectroscopic tool to study the mysteries of electron systems with heavy
quasiparticle mass? From the above we have established the required
conditions. First, one needs good metals characterized by a small resistivity
at low temperatures, to assure that transverse sound does not decay too
rapidly through momentum relaxation. To avoid complications associated with
extreme anisotropy, the ideal candidate materials for transverse-shear optical
spectroscopy should be preferably rather isotropic and three-dimensional, with
a band structure as simple as possible. Most importantly, they should be
characterized as Fermi liquids with large quasiparticle mass enhancement:
since a large $F_{1}^{S}$ protects transverse sound from Landau damping, the
higher the mass the better. Examples of candidate materials fulfilling these
criteria are $\alpha$-cerium [33, 34], MnSi [35], and UPd2Al3 [36]. These
should be tailored into submicron-width thin layers and then cooled to low
temperature, while subsequently the transmission should be measured as
function of frequency in the terahertz range: the sweet-spot frequency window
to look for lies in between the low-frequency regime, where the traditional
skin effect takes over, and the regime where phonons damp transverse sound. It
is a beneficial circumstance that the technology to realize these conditions
in the laboratory has just become available [37].

###### Acknowledgements.

We acknowledge stimulating discussions with A. Morpurgo, C. Berthod, J.
Schmalian, A. B. Kuzmenko, G. A. Inkof, M. Baggioli and F. Pientka. This work
was supported by the Swiss National Science Foundation through grants no.
200020$-$ 179157 (D.v.d.M.), and the Early Postdoc Mobility grant P2GEP2
181450 (D.V.).
Author contributions
DV, JZ and DvdM conceived the project and discussed the model, DV performed
calculations, DV, JZ and DvdM wrote the paper.
Competing interests
None.
Materials & Correspondence
Not applicable.

##  IV Methods

In this section we summarize the main features of the model used to obtain all
results discussed in the main text. Additional details of the formalism are
described in Ref. 13.

###  IV.1 Landau kinetic equation

We start from the Landau kinetic equation, which determines the first-order
deviation (or “displacement") $\epsilon_{\bm{k}}(\bm{q},\omega)$ of the
quasiparticle distribution function, with momentum $\bm{k}$ on the Fermi
surface, with respect to global thermodynamic equilibrium, driven by external
perturbations of momentum $\bm{q}$ and frequency $\omega$. Perturbation
sources include quasiparticle interactions, collisions, momentum-relaxing
scattering and external driving forces like electromagnetic fields. In
general, the kinetic equation reads [1, 3, 13]

$\left(\bm{q}\cdot\bm{v}_{\bm{k},\sigma}-\omega\right)\epsilon_{\bm{k}}(\bm{q},\omega)+\bm{q}\cdot\bm{v}_{\bm{k},\sigma}\delta\left[E\left(\bm{k},\sigma\right)-\mu\right]\\\
\cdot\left\\{\sum_{\bm{k}^{\prime},\sigma^{\prime}}\textsl{f}_{\bm{k},\sigma,\bm{k}^{\prime},\sigma^{\prime}}\epsilon_{\bm{k}^{\prime}}(\bm{q},\omega)\delta\left[E_{n}(\bm{k}^{\prime},\sigma^{\prime})-\mu\right]\right\\}\\\
={\mathscr{I}}_{coll}(\bm{q},\omega)+{\mathscr{I}}_{r}(\bm{q},\omega)\\\
-\frac{e}{mv_{F}^{\ast}}\left(\bm{q}\cdot\bm{v}_{\bm{k},\sigma}\right)\delta\left[E\left(\bm{k},\sigma\right)-\mu\right]\bm{k}\cdot\bm{A}(\bm{q},\omega),$
(1)

where $\bm{v}_{\bm{k},\sigma}$ is the quasiparticle velocity for momentum
$\bm{k}$ and spin
$\sigma=\left\\{\left|\uparrow\right\rangle,\left|\downarrow\right\rangle\right\\}$,
$\textsl{f}_{\bm{k},\sigma,\bm{k}^{\prime},\sigma^{\prime}}$ encode short-
ranged interactions between quasiparticles,
${\mathscr{I}}_{coll}(\bm{q},\omega)$ is the momentum-conserving collision
integral, ${\mathscr{I}}_{r}(\bm{q},\omega)$ is the integral for momentum-
relaxing scattering, $\bm{A}(\bm{q},\omega)$ is the applied vector potential,
and the delta functions $\delta\left[E_{n}(\bm{k},\sigma)-\mu\right]$ select
quasiparticle states at the chemical potential $\mu$. Considering a three-
dimensional system, we standardly expand the interaction term in angular
harmonics using Legendre polynomials $\wp_{l}(\cos\theta^{{}^{\prime}})$ [3,
7, 38, 13], and we focus on the spin-symmetric channel for density
excitations, characterized by the Landau parameters $F_{l}^{s}$. The result is

$\left(qv_{\bm{k},\sigma}\cos{\theta}-\omega\right)\epsilon_{\bm{k}}(\bm{q},\omega)\\\
+qv_{\bm{k},\sigma}\cos{\theta}\int\frac{d\Omega^{{}^{\prime}}}{4\pi}\sum_{l=0}^{+\infty}F_{l}^{s}\wp_{l}(\cos\theta^{{}^{\prime}})\epsilon_{\bm{k}^{{}^{\prime}}}(\bm{q},\omega)\\\
={\mathscr{I}}_{coll}(\bm{q},\omega)+{\mathscr{I}}_{r}(\bm{q},\omega)-\frac{e}{mv_{F}^{\ast}}qv_{\bm{k},\sigma}\cos{\theta}\left[\bm{k}\cdot\bm{A}(\bm{q},\omega)\right],$
(2)

where $\theta$ is the angle between the wave vector $\bm{q}$ and the
quasiparticle velocity $v_{\bm{k},\sigma}$, and $d\Omega$ is the solid angle
element in three dimensions. We further expand the Fermi-surface displacement
function $\epsilon_{\bm{k}}(\bm{q},\omega)$ in the spin-symmetric channel in
terms of spherical harmonics $\mathscr{Y}_{l}^{m}(\theta,\phi)$ [13]:

$\epsilon_{\bm{k}}(\bm{q},\omega)=\sum_{l=0}^{+\infty}\sum_{m=-l}^{+l}\epsilon_{l,m}^{s}\mathscr{Y}_{l}^{m}(\theta,\phi).$
(3)

In the following we consider the first transverse mode with $m=1$, generating
shear currents in the first spin-symmetric interaction channel. Therefore we
truncate the sums over $l$ in Eqs. (2) and (3) to $l=1$, so that
$\sum_{l=0}^{+\infty}F_{l}^{s}\wp_{l}(\cos\alpha)\equiv
F_{0}^{S}+F_{1}^{S}\cos{\alpha}$, and
$\epsilon_{\bm{k}}(\bm{q},\omega)=\sum_{l=0}^{+\infty}\epsilon_{l,1}^{s}\mathscr{Y}_{l}^{1}(\theta,\phi)\equiv\epsilon^{s}(\theta)e^{i\phi}$,
where $\epsilon^{s}(\theta)$ collects the $\theta$-dependent part of the
displacement. Integration over the angles $\theta^{\prime}$ and
$\phi^{\prime}$ in Eq. (2) with the displacement
$\epsilon^{s}(\theta)e^{i\phi}$ gives null result for terms involving
$F_{0}^{S}$ [13], and we are left with

$\left(\cos{\theta}-s\right)\epsilon^{s}(\theta)e^{i\phi}\\\
+\cos{\theta}\int_{0}^{\pi}\frac{\sin{\theta^{\prime}}d\theta^{\prime}}{4\pi}F_{1}^{S}\sin\theta\sin{\theta^{\prime}}\pi
e^{i\phi}\epsilon^{s}(\theta^{\prime})\\\
=\frac{1}{qv_{F}^{\ast}}{\mathscr{I}}_{coll}(\bm{q},\omega)+\frac{1}{qv_{F}^{\ast}}{\mathscr{I}}_{r}(\bm{q},\omega)+\frac{e}{mv_{F}^{\ast}}\cos{\theta}\left[\bm{k}\cdot\bm{A}(\bm{q},\omega)\right],$
(4)

where

$s=\frac{\omega}{qv_{F}^{\ast}},$ (5)

and $v_{F}^{\ast}=v_{F}/(1+F_{1}^{S}/3)$ is the velocity of quasiparticles on
the Fermi surface – i.e. at the Fermi wave vector $\bm{k}\equiv\bm{k}_{F}$ –
renormalized with respect to $v_{F}=\hbar k_{F}/m$ and $k_{F}$ by
quasiparticle interactions.
Eq. (4) is the starting point of our analysis: it describes coherent shear
vibrations of the Fermi surface with the dispersion relation (5) in response
to collisions, interactions, momentum-relaxing scattering and applied vector
potentials. Notice that Eq. (4) does not depend on $F_{0}^{S}$, since shear
deformations generate transverse currents with no net density flow.

In the following sections we solve Eq. (4) for the transverse susceptibility
in the Galilean continuum by explicitating the form of the collision integral
as prescribed by Fermi-liquid theory [4]. Then, we analyze the effect of
momentum-relaxing scattering in relaxation-time approximation [7] considering
impurities, Umklapp processes and acoustic phonons as independent scattering
sources.

###  IV.2 Fermi-liquid transverse susceptibility

Starting from the kinetic equation (4), we can obtain the transverse (current)
susceptibility of the Fermi liquid in closed form thanks to the assumption of
only two harmonic components of short-range interactions. We first neglect
both the collision and the momentum-relaxing integrals, i.e.
${\mathscr{I}}_{coll}(\bm{q},\omega)={\mathscr{I}}_{r}(\bm{q},\omega)=0$, and
we assume no interactions. In this case, from the expression of the
paramagnetic current density one derives the noninteracting transverse
paramagnetic susceptibility [13]:

$\chi_{T}^{0}(\bm{q},\omega)=3\frac{nm^{\ast}}{m^{2}}{\cal I}(s),$ (6)

where $m^{*}=m\left(1+\frac{F_{1}^{S}}{3}\right)$ is the effective mass of
Landau quasiparticles, and

${\cal
I}(s)=-\frac{1}{3}+\frac{1}{2}s^{2}+\frac{s}{4}\left(1-s^{2}\right)\ln\left(\frac{s+1}{s-1}\right).$
(7)

We now allow for momentum-conserving collisions in the Galilean continuum,
i.e. with no momentum relaxation. The conservation of particle number, energy
and momentum in collisions imposes constraints on the form of the collision
integral ${\mathscr{I}}_{coll}(\bm{q},\omega)$ [3]. The latter is written is
terms of a single collision time $\tau_{c}=\tau_{c}(\omega,T)$, which may
depend on frequency $\omega$ and temperature $T$, as [4, 13]

$\frac{1}{qv_{F}^{\ast}}{\mathscr{I}}_{coll}(\bm{q},\omega)={\mathscr{I}}_{coll}\left[\epsilon^{s}(\theta)e^{i\phi}\right]\\\
=-\frac{\epsilon^{s}(\theta)-\left[\epsilon^{s}(\theta)\right]_{\rm{av}}-3\left[\epsilon^{s}(\theta)\sin\theta\right]_{\rm{av}}\sin\theta}{i\omega\tau_{c}}se^{i\phi}$
(8)

where the notation
$\left[\cdot\right]_{\rm{av}}=\int_{0}^{\pi}\frac{(\sin\theta)^{2}d\theta}{4}\cdot$
denotes the angular average with respect to $\theta$. Notice that for purely
shear stresses $\left[\epsilon^{s}(\theta)\right]_{\rm{av}}=0$ [4]. Again the
definition of the paramagnetic current density allows one to derive the
interacting transverse paramagnetic susceptibility [13]. We have

$\chi_{T}(\xi)=\frac{X_{T}^{0}(\xi)}{1-3\left(\frac{F_{1}^{S}}{3}-\beta\right)\frac{m^{2}}{3nm^{\ast}}X_{T}^{0}(\xi)+\beta}\\\
=\frac{n}{m}\frac{3\left(1+\frac{F_{1}^{S}}{3}\right){\cal
I}(\xi)}{1-3\left(\frac{F1^{s}}{3}-\beta\right){\cal I}(\xi)+\beta}.$ (9)

where

$\xi=s\left(1+\frac{i}{\omega\tau_{c}}\right),$ (10a)
$\beta=\frac{1}{i\omega\tau_{c}-1}.$ (10b)

To include damping of transverse currents due to momentum relaxation, we have
to utilize the full kinetic equation (4) with the integral
${\mathscr{I}}_{r}(\bm{q},\omega)$. A convenient way to relate the momentum-
relaxing transverse susceptibility to the result (9) without relaxation is
provided by reference 7: we define the momentum-relaxing integral in single
relaxation time approximation as

$\frac{1}{qv_{F}^{\ast}}{\mathscr{I}}_{r}(\bm{q},\omega)=-\frac{\epsilon^{s}(\theta)-\epsilon^{r}(\theta)}{i\omega\tau_{K}}se^{i\phi}.$
(11)

According to Eq. (11), momentum relaxation tends to restore a “locally
relaxed" equilibrium distribution function, characterized by the displacement
$\epsilon^{r}(\theta)$ [7], which would be in equilibrium without relaxation
but in the presence of an effective vector potential
$\bm{A}_{r}(\bm{q},\omega)$ modified by relaxation. The scattering processes
described by the time $\tau_{K}$ conserve particle number but not current [7],
and may be $\omega$\- and $T$-dependent. This way, we obtain the momentum-
relaxing transverse susceptibility [13]:

$\chi_{T}(\bm{q},\omega)=\frac{n}{m}\frac{\omega}{\omega+i/\tau_{K}}\frac{\left(1+\tilde{\beta}\right)\left[1+3\cal{I}(\zeta)\right]}{1-3\left(\frac{F_{1}^{S}}{3}-\tilde{\beta}\right)\cal{I}(\zeta)+\tilde{\beta}}.$
(12)

where

$\zeta=\xi+s\frac{i}{\omega\tau_{K}}=s\left(1+\frac{i}{\omega\tau_{c}}+\frac{i}{\omega\tau_{K}}\right),$
(13)

and $\tilde{\beta}=\beta\xi/\zeta$. We recognize a pole at the denominator of
Eqs. (12) and (9), in the presence or absence of relaxation respectively: this
collective mode is transverse sound, studied and observed in the electrically
neutral Fermi liquid realized in 3He [4, 8]. The dispersion relation of
transverse sound stems from the condition that the susceptibility diverges for
a given real frequency $\omega$ and complex momentum $\bm{q}$:

$\chi_{T}(\bm{q},\omega)^{-1}=0$ (14)

Eq. (14) determines the velocity of the transverse collective mode with first
Landau parameter $F_{1}^{S}$ and collisions encoded by
$\tau_{c}=\tau(\omega,T,F_{1}^{S})$. This mode has been labeled $``$transverse
sound$"$ in the liquid-helium literature, although physically it amounts to a
shear oscillation of the Fermi surface. The numerical solutions of Eqs. (9)
and (14) are shown schematically in Fig. 1a.

For $F_{1}^{S}<6$, there is an $F_{1}^{S}$-dependent value of $\omega_{th}$
above which Eq. (14) does not have solutions. Physically, this means that no
Fermi-surface collective modes are allowed for $\omega>\omega_{th}$, and the
only possible excitations are incoherent electron-hole pairs. Knowing the
dependence $\tau_{c}(\omega,T,F_{1}^{S})$, in the Fermi-liquid model – i.e.
Eq. (24) – one finds the numerical solution $\omega(T,F_{1}^{S})$ which is
displayed with the red solid curves in Fig. 3a,b. For $F_{1}^{S}>6$ Eq. (14)
has solutions for all frequencies, but above a certain threshold frequency
$\omega_{\rm{th}}$ the solutions are outside the particle-hole continuum and
thus free from Landau damping. This threshold is set by solving Eq. (14)
together with the condition

${\omega_{\rm{th}}}{\mathop{\rm Re}\nolimits}{q^{-1}}=v_{F}^{\ast}$ (15)

The orange dashed curves in Fig. 3a,b represent the numerical solution of
$\omega_{th}$ as a function of $F_{1}^{S}$.

###  IV.3 Dielectric function and polaritons

The effect of the long range Coulomb interaction is described by the
transverse dielectric function, which can be obtained from the paramagnetic
transverse susceptibility (12) through the Kubo formula. We have

$\varepsilon_{T}(\bm{q},\omega)=1-\frac{e^{2}}{\varepsilon_{0}\omega^{2}}{\chi_{T}}(\bm{q},\omega)\\\
=1-\frac{(\omega_{p})^{2}}{\omega\left(\omega+i/\tau_{K}\right)}\frac{\left(1+\tilde{\beta}\right)\left[1+3\mathscr{I}(\zeta)\right]}{1-3\left(\frac{F_{1}^{S}}{3}-\tilde{\beta}\right)\mathscr{I}(\zeta)+\tilde{\beta}}.$
(16)

where $\omega_{p}=\sqrt{ne^{2}/(m\varepsilon_{0})}$ is the electron plasma
frequency. Notice that the poles of the dielectric function (16) correspond to
the transverse sound dispersion relation (14), i.e. the shear resonance of the
Fermi surface. Due to the charge of the electrons, these resonances couple to
radiation through the shear stresses exerted by photons. The radiation-matter
interaction gives rise to new collective modes, consisting of the combined
shear response of photons and Fermi-surface quasiparticles: the polaritons.
Mathematically, polaritons satisfy the self-consistent solutions of Maxwell
equations inside the Fermi liquid [1]:

$\frac{q^{2}c^{2}}{\omega^{2}}=\varepsilon_{T}(\bm{q},\omega)$ (17)

Analytical solutions of Eqs. (16) and (17) are available in the regimes
$q\rightarrow 0$ and $q\rightarrow\infty$: the leading-term solution in the
$q\rightarrow 0$ limit reproduces the phenomenology of viscous charged fluids
[15], while in the $q\rightarrow\infty$ regime we retrieve the phenomenology
of anomalous skin effect [26, 39, 13]. In the regime $\omega\gg v_{F}^{\ast}$
Re$q$, the leading-order expansion of Eq. (17) for
$\omega/(v_{F}^{\ast}q)\rightarrow\infty$ gives [13]

${\varepsilon_{T}}(\bm{q},\omega)=1-\frac{(\omega_{p})^{2}}{\omega^{2}+i\omega/\tau_{K}+i\omega\tilde{\nu}(\omega)q^{2}},$
(18)

where

$\tilde{\nu}(\omega)=\left(1+\frac{F_{1}^{S}}{3}\right)\frac{(v_{F}^{*})^{2}\tau_{c}}{5\left(1+\tau_{c}/\tau_{K}-i\omega\tau_{c}\right)}.$
(19)

In the limit $\tau_{c}\ll\tau_{K}$, that is when momentum-conserving
collisions are much more frequent than momentum-relaxing scattering, Eq. (18)
becomes [13]

$\epsilon_{T}(\bm{q},\omega)=1-\frac{(\omega_{p})^{2}}{\omega^{2}+i\omega/\tau_{K}+i\left(\omega+i/\tau_{K}\right)\nu(\omega)q^{2}}\\\
\approx
1-\frac{(\omega_{p})^{2}}{\omega^{2}+i\omega/\tau_{K}+i\omega\nu(\omega)q^{2}},$
(20)

where we have defined the generalized shear modulus of the isotropic Fermi
liquid [7, 40, 41]

$\nu\left(\omega\right)=\frac{{1+F_{1}^{S}/3}}{5}\frac{(v_{F}^{\ast})^{2}}{{1/{\tau_{c}}-i\omega}}.$
(21)

Eqs. (20) and (21) can be derived from the combination of the linearized
Navier-Stokes equation for transverse currents and of Maxwell’s equations
under the assumption that $\tau_{c}\ll\tau_{K}$. In the propagating shear
regime Eqs. (17) and (18) give rise to two degenerate polariton branches for
each frequency, characterized by the complex wave vectors

$q^{2}=\frac{\omega^{2}}{2c^{2}}+\frac{i\omega}{2\nu(\omega)}\left\\{1\pm\frac{\sqrt{\tau_{K}\left\\{\omega\left[c^{2}+i\nu(\omega)\omega\right]^{2}-4ic^{2}\nu(\omega)(\omega_{p})^{2}\right\\}+i\left[c^{2}+i\nu(\omega)\omega\right]^{2}}}{c^{2}\sqrt{\omega\tau_{K}+i}}\right\\},$
(22)

One solution of Eq. (22) is the usual plasmon-polariton which propagates above
$\omega_{p}$, as also found in the standard Drude model, and the other slower-
dispersing root is the shear-polariton. The two roots are schematically shown
in Fig. 1b. The shear-polariton root of Eq. (22) emerges from the electron-
hole continuum, thus ceasing to be Landau-damped, for $\omega>\omega_{th}$. We
can obtain a quantitative estimation of the latter condition by reverting to
Eq. (15) and noticing that, for $F_{1}^{S}>6$, the shear-polariton emerges
from the continuum in the regime $\omega\tau_{c}\gg 1$, Re$q\gg$Im$q$. This
allows us to find the analytical solution

${\omega_{th}}=\frac{{{\omega_{p}}}}{c}\frac{{3v_{F}^{\ast}}}{{\sqrt{\left({F_{1}^{S}-6}\right)\left[{1-{{\left({v_{F}^{\ast}/c}\right)}^{2}}}\right]}}}$
(23)

Eq. (23) is shown by the blue solid curves in Fig. 3c,d. It confirms the early
order-of-magnitude estimate by Nozières and Pines [1], that is
$\mathrm{Re}q=\omega/v_{F}^{\ast}\approx\omega_{p}/c$ at the boundary with the
electron-hole continuum. In the region between the blue solid lines and the
red dashed lines in Fig. 3c,d, the shear-polariton propagates but it cannot be
described by the viscous-liquid phenomenology of Eq. (20). The dispersion of
the shear-polariton is robust against momentum relaxation, as it is negligibly
affected by the value of $\tau_{K}$ in the propagating regime.

###  IV.4 Collision and momentum relaxation rates

For the electron-electron momentum-conserving collision time
$\tau_{c}=\tau_{c}(\omega,T)$ we employ the Fermi-liquid result [42, 14, 29]

$\frac{\hbar}{\tau_{c}}=\frac{\pi}{12}\left[\frac{7}{24}(A_{1}^{S})^{2}+\frac{5}{8}(A_{0}^{S})^{2}-\frac{5}{12}A_{0}^{S}A_{1}^{S}\right]\times\\\
\frac{(\hbar\omega)^{2}+(\pi k_{B}T)^{2}}{E_{F}^{\ast}},$ (24)

where $A_{l}^{j}=F_{l}^{j}/\left[1+F_{l}^{j}/\left({2l+1}\right)\right]$. Eq.
(24) holds for an interaction comprising the Landau parameters $F_{0}^{S}$ and
$F_{1}^{S}$, consistently with the assumptions made in obtaining the kinetic
equation (4). The momentum transfer rate, $1/\tau_{K}$, results from impurity
scattering, $1/\tau_{i}$, electron-phonon scattering, $1/\tau_{e-ph}$, and
Umklapp scattering. The Umklapp term is given by a fraction
$\alpha_{U}\in\left(0,1\right)$ of the electron-electron collisions. Typical
values in transition-metal compounds are $\alpha_{U}\approx 0.5$ [42].
Consequently

$\frac{1}{{{\tau_{K}}}}=\frac{{{\alpha_{U}}}}{{{\tau_{c}}}}+\frac{1}{{{\tau_{e-ph}}}}+\frac{1}{{{\tau_{i}}}}$
(25)

We calculate the electron-phonon relaxation rate from the many-body self-
energy associated with scattering between electrons and an Einstein phonon
branch [13]. Explicitly,

$\frac{\hbar}{\tau_{e-ph}(\omega,T)}=\frac{2}{1-\frac{\lambda\hbar\omega_{D}}{\pi
k_{B}T}\mathscr{Q}\left(\frac{k_{B}T}{\hbar\omega_{D}}\right)}\pi\lambda\hbar\omega_{D}\mathscr{S}\left(\frac{\omega}{\hbar\omega_{D}},\frac{k_{B}T}{\hbar\omega_{D}}\right),$
(26a)
$\mathscr{Q}(t)=\int_{0}^{1}duu^{2}\mathrm{Im}\left\\{\psi^{\prime}\left(\frac{1}{2}+i\frac{u}{2\pi
t}\right)\right\\}$ (26b)
$\mathscr{S}\left(x,t\right)=t^{3}\int_{0}^{\frac{1}{t}}duu^{2}\left(\frac{1}{e^{u-\frac{x}{t}}+1}+\frac{1}{e^{u+\frac{x}{t}}+1}+\frac{2}{e^{u}-1}\right).$
(26c)

where $\lambda$ is the electron-phonon coupling constant, $\hbar\omega_{D}$ is
the Debye energy, and $\psi^{\prime}(z)$ is the first derivative of the
Digamma function. For the impurity scattering rate, we assume a positive
constant $\tau_{i}$ consistently with first Born approximation.

###  IV.5 Thin-film optical transmission in propagating-shear regime

We now turn to the computational method for the optical transmission of a
Fermi-liquid thin film of thickness $d$ in the propagating shear regime. The
transmission coefficients at the vacuum/material interfaces follow from the
condition that the electromagnetic wave in the vacuum has to match the
frequency-degenerate polariton modes, Eq. (22), inside the film. To calculate
these coefficients, we exploit the analogy between Fermi-liquid
electrodynamics for $\omega\gg v_{F}^{\ast}$Re$q$ and the electromagnetic
response of viscous charged fluids [15]: appropriate additional constitutive
relations at the slab boundaries can be formulated in terms of the slip length
$\lambda$, in accordance with Ref. 15, which closes the system of equations
for the reflection and transmission coefficients of each polariton mode at the
slab boundaries. A total of six linear equations stem from the continuity of
the electric field and of its derivative, and the constitutive relation from
the linearized Navier-Stokes equation. The solution of such linear equations
gives the transmission coefficient of the film $t_{\rm film}=t_{\rm
film}(\omega)$ relative to the transmission in vacuum $t_{v}$:

$t_{\rm film}\left(\omega\right)/{t_{v}}={e^{-i\omega
d/c}}\left[{t_{+}}{e^{i{q_{+}}d}}+{\theta_{+}}{e^{-i{q_{+}}d}}+{t_{-}}{e^{i{q_{-}}d}}\right.\\\
\left.+{\theta_{-}}{e^{-i{q_{-}}d}}\right],$ (27)

where the indices refer to the $\pm$ branches of Eq. (22), $t_{v}$ and the
amplitudes $t_{\pm}$, $\theta_{\pm}$ are determined numerically through Eq.
(9) of Ref. 15.
We microscopically determine the slip length following Ref. 43:

$\lambda^{s}=\frac{1}{h^{2}(h^{\prime})^{2}(k_{F})^{4}}\frac{45\pi^{2}}{(k_{F})^{4}\hbar}nm\nu(0)=\frac{3}{h^{2}(h^{\prime})^{2}(k_{F})^{4}}v_{F}^{*}\tau_{c},$
(28a)
$\lambda^{d}=\frac{8\pi^{2}}{(k_{F})^{4}\hbar}nm\nu(0)=\frac{8}{15}v_{F}^{*}\tau_{c},$
(28b)

where $h$ and $h^{\prime}$ are amplitude and correlation length of the
interface roughness, $k_{F}$ is the Fermi wave vector, and $\lambda^{s}$
($\lambda^{d}$) refer to specular (diffuse) scattering. In Fig. 2c-f we
compare the results for $t(\omega)=t_{\rm film}\left(\omega\right)/{t_{v}}$
from the propagating-shear model, Eq. (27), with the standard Drude model
(Ohmic conductor) for thin-film transmission.

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