# Quantum feedback at the solid-liquid interface:
flow-induced electronic current and  its negative contribution to friction

Baptiste Coquinot1,2 Lydéric Bocquet1 Nikita Kavokine2,3
<EMAIL_ADDRESS>1 Laboratoire de Physique de l’École
Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université
Paris Cité, 24 rue Lhomond, 75005 Paris, France
2 Center for Computational Quantum Physics, Flatiron Institute, 162 5th
Avenue, New York, NY 10010, USA
3  Department of Molecular Spectroscopy, Max Planck Institute for Polymer
Research, Ackermannweg 10, 55128 Mainz, Germany

###### Abstract

An electronic current driven through a conductor can induce a current in
another conductor through the famous Coulomb drag effect. Similar phenomena
have been reported at the interface between a moving fluid and a conductor,
but their interpretation has remained elusive. Here, we develop a quantum-
mechanical theory of the intertwined fluid and electronic flows, taking
advantage of the non-equilibrium Keldysh framework. We predict that a globally
neutral liquid can generate an electronic current in the solid wall along
which it flows. This hydrodynamic Coulomb drag originates from both the
Coulomb interactions between the liquid’s charge fluctuations and the solid’s
charge carriers, and the liquid-electron interaction mediated by the solid’s
phonons. We derive explicitly the Coulomb drag current in terms of the solid’s
electronic and phononic properties, as well as the liquid’s dielectric
response, a result which quantitatively agrees with recent experiments at the
liquid-graphene interface. Furthermore, we show that the current generation
counteracts momentum transfer from the liquid to the solid, leading to a
reduction of the hydrodynamic friction coefficient through a quantum feedback
mechanism. Our results provide a roadmap for controlling nanoscale liquid
flows at the quantum level, and suggest strategies for designing materials
with low hydrodynamic friction.

##  I Introduction

Figure 1: Mechanisms of hydrodynamic Coulomb drag. (a) Schematic
representation of the phenomenon under study: a liquid flow induces an
electronic current in a solid along which it flows. (b)-(c) Schematic of the
two limiting mechanisms of current generation. In (b), electrons (full lines)
are driven directly by Coulomb interactions (dashed line) with liquid charge
fluctuations (hydrons, wavy line). In (c) electrons are driven by phonons
(wavy line), that are excited through hydrodynamic friction with the liquid.

New functionalities in nanoscale fluid transport have been achieved by
exploiting analogies with condensed matter phenomena. The analogy between
surface charge in a nanochannel and doping in a semiconductor Bocquet and
Charlaix (2010) has led to the development of nanofluidic diodes Vlassiouk and
Siwy (2007); Picallo  _et al._ (2013), and transistors Karnik  _et al._
(2006); the similarity between ionic and electronic Coulomb interactions
Kavokine  _et al._ (2021) allows for ionic Coulomb blockade Kavokine  _et al._
(2019); Tanaka  _et al._ (2017); Feng  _et al._ (2016); Chen  _et al._ (2021).
More recently, it has been suggested that – beyond mere analogies –,
nanofluidic transport can directly couple to electronic effects within the
channel wall, as the solid-liquid interface can host fluctuation-induced
electromagnetic phenomena Volokitin and Persson (2007, 2017) : energy and
momentum transfer mediated by interfacial charge fluctuations. For instance,
it has been predicted that a quantum contribution to hydrodynamic friction
results from the interaction of charge fluctuations in the liquid with
electronic excitations in the solid Kavokine  _et al._ (2022).

A more straightforward example of such a liquid-electron coupling is
apparently provided by the numerous observations of a liquid-flow-induced
current (or voltage drop) within a solid wall Ghosh  _et al._ (2003); Newaz
_et al._ (2012); Ho Lee  _et al._ (2013); Yin  _et al._ (2014); Yang  _et al._
(2018); Rabinowitz  _et al._ (2020); Marcotte  _et al._ . The mechanisms
proposed to explain the current generation include the buildup of a streaming
potential Cohen (2003); Newaz  _et al._ (2012), charging/discharging of a
pseudo capacitance Yin  _et al._ (2014); Yang  _et al._ (2018); Park  _et al._
(2017), or adsorbed/desorbed ion hopping Persson  _et al._ (2004); Dhiman  _et
al._ (2011). In all these cases, the liquid is in fact simply acting on the
solid as an average external potential. Yet, some of the most recent
experimental results Ho Lee  _et al._ (2013); Marcotte  _et al._ cannot be
explained by the above-mentioned mechanisms.  In ref. Ho Lee  _et al._ (2013),
the generation of an open-circuit voltage across a millimeter-sized graphene
sample due to the flow of various liquids was reported – external potential
effects could not account for these results as the liquids were ion-free. In
ref. Marcotte  _et al._ , we have carried out analogous experiments with a
thousand times smaller sample dimensions, which exclude any mesoscale charge
inhomogeneities. Our observation of a liquid-flow-induced electronic current
thus suggests to examine the possibility of "intrinsic" current generation in
the framework of fluctuation-induced electromagnetic phenomena, which would be
analogous to the Coulomb drag effect in condensed matter physics (Fig. 1a).

In Coulomb drag, an electric current driven through a conductor induces a
current in a nearby – yet electrically insulated – conductor. This is due to
charge fluctuations in the driven conductor creating particle-hole excitations
with non-zero momentum in the passive conductor, which results in a current,
provided that the conductors are not particle-hole symmetric Narozhny and
Levchenko (2016). A similar process could occur at the interface between a
solid and a flowing liquid: excitations in the solid would then be generated
through Coulomb interactions with (collective) charge fluctuations in the
liquid, that we shall in the following call _hydrons_ (Fig. 1b). Pioneering
attempts at describing such a mechanism have been made by Volokitin and
Persson. They applied the general theory of momentum transfer between two
media through evanescent electromagnetic waves to determine the trans-
resistivity of two closely-spaced solids Volokitin and Persson (2001). Later,
they expressed the electric field induced in a solid by a flowing ionic
solution Volokitin and Persson (2008), and studied the effect of substrate
optical phonons on the electric current in a graphene sample Volokitin and
Persson (2011). Their approach, however, remains macroscopic, in the sense
that the interacting media are described at the level of their dielectric
functions. It is thus unable to reproduce the most general theoretical result
that has been established for solid-solid Coulomb drag Narozhny and Levchenko
(2016), and a more microscopic theory is thus also required to rigorously
describe the solid-liquid analogue.

An alternative Coulomb drag mechanism, where the liquid-electron interaction
is mediated by the solid’s acoustic phonons (in short, _phonon drag_ , see
Fig. 1c), has been first proposed by Kràl and Shapiro, and formalized in a
Boltzmann equation framework Král and Shapiro (2001). Phonon drag has been
invoked, for instance, to account for the experimental results of ref. Ho Lee
_et al._ (2013). It has also been suggested as a mechanism of momentum
transfer between two fluids separated by a solid wall Andreev and Meierovich
(1971).

In this paper, we develop a microscopic theory of electronic current
generation at a solid-liquid interface, that includes both types of solid-
liquid interactions.  Accounting for the physics at play required us to adopt
a new theoretical strategy, at odds with existing approaches to comparable
problems. In particular, a mesoscopic description of the solid at the level of
its dielectric function Volokitin and Persson (2001, 2008, 2011) was
insufficient, since it is imprecise with regard to the mechanisms by which
electrons relax their momentum. Conversely, descriptions based on the
Boltzmann equation for the electrons Král and Shapiro (2001); Narozhny and
Levchenko (2016) accurately capture the electron relaxation mechanisms, but
fail to systematically include electron-electron interactions, thus missing
the effect of the solid’s plasmon modes, which can play a key role in solid-
liquid systems Yu  _et al._ .

To overcome these limitations, we made use of the non-equilibrium Keldysh
framework of many-body quantum theory Kavokine  _et al._ (2022). Our
diagrammatic description allows for the inclusion of all interactions in a
systematic way, possibly in the framework of numerical methods such as
diagrammatic Monte Carlo Prokof’ev and Svistunov (1998); Gull  _et al._
(2011); Bertrand  _et al._ (2019). Proceeding with controlled approximations,
we derived an explicit expression for the electronic current generated by
liquid flow. Our description of electron relaxation being fully microscopic,
we could compare the relative importance of phonon-mediated and direct Coulomb
solid-liquid interactions and unveil their interplay. Strikingly, we found
that the current generation triggers a _quantum feedback_ mechanism at the
solid-liquid interface, that reduces the total hydrodynamic friction.  Our
results account qualitatively for the flow-induced voltage reported in ref. Ho
Lee  _et al._ (2013) and agree quantitatively with our own flow-induced
current measurements Marcotte  _et al._ .

The paper is organized as follows. In Sec. II we present our model and state
the main results. In Sec. III, the formal derivation is carried out. The
reader interested only in the physical outcomes may skip directly to Sec. IV,
where we evaluate explicitly the Coulomb drag current  and compare it to the
experimental results of ref. Marcotte  _et al._ . In Sec. V, we derive the
hydrodynamic friction renormalization resulting from the current generation;
finally Sec. VI establishes our conclusions.

#### Units and conventions.

We set the Boltzmann constant $k_{\rm B}=1$ (that is, we express the
temperature in energy units), but otherwise use SI units throughout the text.
Matrices are denoted with bold capital letters. We use the following
convention for the $d$-dimensional Fourier transform:

$\displaystyle\hat{F}(\textbf{q},\omega)$ $\displaystyle=$
$\displaystyle\int\textnormal{d}^{d}\textbf{r}\textnormal{d}t\,F(\textbf{r},t)e^{-i\textbf{q}\cdot\textbf{r}+i\omega
t},$ $\displaystyle F(\textbf{r},t)$ $\displaystyle=$
$\displaystyle\int\frac{\textnormal{d}^{d}\textbf{q}\textnormal{d}\omega}{(2\pi)^{d+1}}\,\hat{F}(\textbf{q},\omega)e^{i\textbf{q}\cdot\textbf{r}-i\omega
t}.$

##  II Model and main results

We consider a two-dimensional solid occupying the plane $z=0$, in contact with
a semi-infinite liquid occupying the half-space $z>0$, and flowing along the
$x$ direction with a velocity $\textbf{v}_{\rm\ell}$, as depicted in Fig. 1a.
The system is at temperature $T=300~{}\rm K$ (or 26 meV). The flow field
$\textbf{v}_{\rm\ell}$ is assumed uniform in the interfacial liquid layer
Kavokine  _et al._ (2022). The liquid interacts with the solid through Coulomb
forces. The corresponding _electron-hydron_ Hamiltonian is

$\mathcal{H}_{\rm
h/e}(t)=\int\textnormal{d}\textbf{r}\textnormal{d}\textbf{r}_{\rm
e}\,n_{\ell}(\textbf{r}-\textbf{v}_{\ell}t,t)V^{\rm
C}(\textbf{r}-\textbf{r}_{\rm e})n_{\rm e}(\textbf{r}_{\rm e},t),$ (1)

where $n_{\ell}$ and $n_{\rm e}$ are the liquid and solid charge density,
respectively, and $V^{\rm C}(\mathbf{r})=e^{2}/(4\pi\epsilon_{0}r)$ is the
Coulomb potential. Following Kavokine  _et al._ (2022), we treat $n_{\ell}$ as
a free bosonic field, whose correlation functions are related to the liquid’s
dielectric response. An additional liquid-solid interaction originates from
short-range repulsion forces which result in "classical" hydrodynamic friction
Kavokine  _et al._ (2022); Bocquet and Barrat (2007). We will assume that this
hydrodynamic friction transfers momentum to the solid’s acoustic phonons Král
and Shapiro (2001): those phonons then acquire a non-zero average momentum.
This effect can be modeled by adding a shift $\mathbf{v}_{\rm ph}$ to the
phonon (sound) velocity. The electron-phonon interaction Hamiltonian is then
of the form

$\mathcal{H}_{\rm
ph/e}(t)=\int\textnormal{d}\textbf{r}\textnormal{d}\textbf{r}_{\rm
e}\,\phi_{\rm ph}(\mathbf{r}-\mathbf{r}_{\rm e}-\mathbf{v}_{\rm ph}t)n_{\rm
e}(\textbf{r}_{\rm e},t).$ (2)

Here, $\phi_{\rm ph}$ is proportional to the local lattice displacement; its
exact expression depends on the particular solid under consideration. The
"phonon wind" velocity $\mathbf{v}_{\rm ph}$ is not known _a priori_ and it
will be determined self-consistently by establishing the system’s momentum
balance (see Sec. IV.3).

The treatment of this model within non-equilibrium perturbation theory allows
us to obtain two key analytical results. First, we obtain an explicit
expression for the electronic current density $\mathbf{j}$ induced in the
solid by the liquid flow for each electronic band:

$\langle\textbf{j}\rangle=\frac{e\hbar}{T}\int_{-\infty}^{+\infty}\frac{\mathrm{d}\omega}{2\pi}\int_{0}^{+\infty}\frac{\mathrm{d}q}{2\pi}\,\frac{q^{2}\left(\nabla_{q}\xi_{q}\right)\mathbf{v}_{\rm
e}(q)}{\textnormal{cosh}^{2}\left[\frac{\hbar(\omega+\xi_{q})}{2T}\right]}\frac{\tau_{q}}{1+\tau_{q}^{2}\omega^{2}}.$
(3)

Here $\tau_{q}$ is the lifetime of the quasiparticle at energy
$\hbar\xi_{\mathbf{q}}=u_{\mathbf{q}}-\mu$ where $u_{\mathbf{q}}$ is the band
dispersion and $\mu$ is the chemical potential. $\mathbf{v}_{e}$ is the
electron "wind velocity", which is a linear combination of the phonon wind
velocity $\mathbf{v}_{\rm ph}$ and the hydron wind velocity (or simply, liquid
flow velocity) $\mathbf{v}_{\ell}$:

$\mathbf{v}_{\rm e}(q)=\frac{\tau_{\mathbf{q}}}{\tau^{\rm
ph/e}_{\mathbf{q}}}\textbf{v}_{\rm ph}+\frac{\tau_{\mathbf{q}}}{\tau^{\rm
h/e}_{\mathbf{q}}}\textbf{v}_{\rm\ell}.$ (4)

$1/\tau^{\rm ph/e}_{\mathbf{q}}$ and $1/\tau^{\rm h/e}_{\mathbf{q}}$ are the
phonon and hydron contributions to the total quasiparticle scattering rate
$1/\tau_{\mathbf{q}}$. This result is valid under a few reasonable assumptions
on the electronic self-energy (see Sec. III), and as long as the electronic
structure has no band crossings close to the Fermi level.

Second, we predict a reduction of the hydrodynamic friction $\lambda$
coefficient due to the current generation. We recall that the solid-liquid
friction force is given by $\mathbf{F}=-\lambda\mathcal{A}\mathbf{v}_{\ell}$,
where $\cal A$ is the surface area. Accounting for the current generation,
$\lambda$ is modified from its "bare" value $\lambda_{0}$ according to

$\lambda=\frac{\lambda_{0}}{1+(\lambda_{\rm h/ph}+\frac{\tau}{\tau^{\rm
ph/e}}\lambda_{\rm h/e})/\lambda_{\rm um}}.$ (5)

Here, $\lambda_{\rm h/ph}$ and $\lambda_{\rm h/e}$ are the phononic and
electronic contributions to the fluctuation-induced solid-liquid friction
Kavokine  _et al._ (2022); $\lambda_{\rm
um}=3\zeta(3)T^{3}/(2\pi\hbar^{2}c^{4}\tau_{\rm um})$ has the dimension of a
friction coefficient, and is expressed in terms of the sound velocity $c$ in
the solid, and the typical phonon lifetime $\tau_{\rm um}$. We demonstrate
that this friction reduction is a quantum effect, that takes its roots in the
solid’s electronic excitations.

These results are derived in detail in the following sections. In Sec. IV, we
evaluate the flow-induced current for different material systems, and
successfully compare our predictions with experimental data Marcotte  _et al._
. In Sec. V, we show that the correction to hydrodynamic friction in Eq. (5)
can be non-negligible, and leads to significant hydrodynamic slippage in
systems where it would not typically be expected.

##  III Non-equilibrium perturbation theory

###  III.1 Description in the Keldysh framework

We describe the system’s dynamics in terms of three types of real-time Green’s
functions: the Retarded, Advanced and Keldysh Green’s functions, defined, for
both bosons and fermions, according to

$\displaystyle\left\\{\begin{array}[]{l}G^{\rm
R}(\textbf{r},t,\textbf{r}^{\prime},t^{\prime})=-i\theta(t-t^{\prime})\langle[\psi(\textbf{r},t),\psi^{\dagger}(\textbf{r}^{\prime},t^{\prime})]_{s}\rangle,\\\
G^{\rm
A}(\textbf{r},t,\textbf{r}^{\prime},t^{\prime})=i\theta(t^{\prime}-t)\langle[\psi(\textbf{r},t),\psi^{\dagger}(\textbf{r}^{\prime},t^{\prime})]_{s}\rangle,\\\
G^{\rm
K}(\textbf{r},t,\textbf{r}^{\prime},t^{\prime})=-i\langle[\psi(\textbf{r},t),\psi^{\dagger}(\textbf{r}^{\prime},t^{\prime})]_{-s}\rangle,\end{array}\right.$
(9)

where $\psi^{\dagger}$ and $\psi$ are the particles’ creation and annihilation
operators, and $[A,B]_{\pm}=AB\pm BA$, $s$ being $+$ for fermions and $-$ for
bosons. The Retarded and Advanced Green’s functions contain information on the
system’s elementary excitations. For non-interacting electronic quasiparticles
in a translationally-invariant system at equilibrium, the Fourier-transformed
Green’s functions are

$G^{\rm R,A}_{0}(\textbf{q},\omega)=\frac{1}{\omega-\xi_{\textbf{q}}\pm
i0^{+}}$ (10)

where $\hbar\xi_{\textbf{q}}=u_{\textbf{q}}-\mu$ is the quasiparticle energy:
$u_{\textbf{q}}$ is the band dispersion and $\mu$ is the chemical potential.
The Keldysh Green’s function contains information on the quasiparticle
distribution. At equilibrium, it satisfies the fluctuation-dissipation
theorem:

$G^{\rm K}(\textbf{q},\omega)=\frac{2i}{f(\omega)}\textnormal{Im}[G^{\rm
R}(\textbf{q},\omega)]$ (11)

where $f(\omega)=\textnormal{cotanh}\left(\frac{\hbar\omega}{2T}\right)$.
Given its importance for the subsequent discussion, we recall the derivation
of this result in Appendix A.1. For non-interacting electrons, Eqs. (10) and
(11) yield

$G^{\rm K}(\textbf{q},\omega)=(2n_{\rm F}(\omega)-1)\times
2i\pi\delta(\omega-\xi_{\mathbf{q}}),$ (12)

where we recover indeed the Fermi-Dirac distribution $n_{\rm
F}(\omega)=1/(e^{\hbar\omega/T}+1)$.

We use the letter $D$ to denote bosonic Green’s functions. For free bosons
(such as phonons) with dispersion $\omega_{\mathbf{q}}$ at equilibrium,

$D^{\rm R,A}=\frac{\omega_{\mathbf{q}}}{(\omega\pm
i0^{+})^{2}-\omega_{\mathbf{q}}^{2}}.$ (13)

The bosonic fluctuation-dissipation theorem reads (see Appendix A.1)

$D^{\rm K}(\textbf{q},\omega)=2if(\omega)\textnormal{Im}[D^{\rm
R}(\textbf{q},\omega)].$ (14)

so that, at equilibrium

$D^{\rm K}=(2n_{\rm B}(\omega)+1)\times
i\pi\left[\delta(\omega-\omega_{\mathbf{q}})-\delta(\omega+\omega_{\mathbf{q}})\right],$
(15)

where we recover the Bose-Einstein distribution $n_{\rm
B}(\omega)=1/(e^{\hbar\omega/T}-1)$.

The Keldysh Green’s functions are therefore the analogues of the occupation
distribution functions in the approximate Boltzmann formalism. They will be
key in determining the non-equilibrium state of the system. Indeed, as shown
in Appendix A.2, the current density (within one electronic band) is given by

$\langle\mathbf{j}\rangle=2ie\int\frac{\mathrm{d}\mathbf{q}\mathrm{d}\omega}{(2\pi)^{3}}(\nabla_{\mathbf{q}}\xi_{\mathbf{q}})G^{\rm
K}(\mathbf{q},\omega).$ (16)

In addition, the non-equilibrium density-density response function, which will
be required for obtaining the correction to the hydrodynamic friction
coefficient, can be computed starting from the non-equilibrium Green’s
functions (see Appendix C.1).

Figure 2: Non-equilibrium diagrammatic expansion. (a) Dyson equation for the
matrix Green’s function G (thick line). The thin line is the bare Green’s
function $\textbf{G}^{0}$. (b) First order self-energy diagram, which is
computed for each of the electron-boson interactions. (c) Dyson equation for
the random phase approximation (RPA) screening of the boson propagator (dashed
line). (d) Dyson equation for the boson-electron vertex, within RPA. (e)
Diagrammatic expansion for the electric current, which is related to the
Keldysh Green’s function at equal point in space and time. The square
represents a gradient operator (multiplication by $\mathbf{q}$ in momentum
space). (f) Notation for the electron and boson Green’s functions. We have
absorbed the RPA renormalisation of the vertices into the definition of the
bosonic propagator.

###  III.2 Dyson equation

Our task is now to compute the non-equilibrium Green’s functions in the
presence of the perturbations $\mathcal{H}_{\rm h/e}(t)$ and $\mathcal{H}_{\rm
ph/e}(t)$. In the Keldysh formalism, we consider the matrix Green’s function

$\displaystyle\textbf{G}=\left(\begin{array}[]{cc}G^{\rm R}&G^{\rm K}\\\
0&G^{\rm A}\end{array}\right).$ (19)

The perturbation series may be partially resummed by introducing a (matrix)
self-energy $\bf\Sigma$. The Green’s function then satisfies the non-
equilibrium Dyson equation

$\bf G=G_{0}+G_{0}\otimes\bm{\Sigma}\otimes G,$ (20)

which is represented diagrammatically in Fig. 2a. Here, $\otimes$ represents
convolution in space and time, as well as matrix multiplication. We assume
that the system is translationally invariant parallel to the interface, and
that it has reached a steady state: we may then Fourier-transform Eq. (20).
With the convolutions becoming products in Fourier space, and using that
$G_{0}^{\rm A}(\mathbf{q},\omega)=G_{0}^{\rm R}(\mathbf{q},\omega)^{*}$ and
$\Sigma^{\rm A}(\mathbf{q},\omega)=\Sigma^{\rm R}(\mathbf{q},\omega)^{*}$, we
obtain

$\displaystyle G^{\rm R,A}(\mathbf{q},\omega)$ $\displaystyle=$
$\displaystyle\dfrac{G_{0}^{\rm R,A}-|G_{0}^{\rm R}|^{2}\Sigma^{\rm
A,R}}{\left|1-G_{0}^{\rm R}\Sigma^{\rm R}\right|^{2}},$ (21) $\displaystyle
G^{\rm K}(\mathbf{q},\omega)$ $\displaystyle=$ $\displaystyle\dfrac{G_{0}^{\rm
K}+|G_{0}^{\rm R}|^{2}\Sigma^{\rm K}}{\left|1-G_{0}^{\rm R}\Sigma^{\rm
R}\right|^{2}}.$ (22)

Using Eqs. (10) and (11) for the equilibrium Green’s functions, we find that
the first term in Eq. (22) vanishes if the self-energy is non-zero. Then,
recalling Eq. (16), we obtain a first very general expression for the flow-
induced electric current (within a given electronic band):

$\langle\mathbf{j}\rangle=2ie\int\frac{\mathrm{d}\mathbf{q}\mathrm{d}\omega}{(2\pi)^{3}}(\nabla_{\mathbf{q}}\xi_{\mathbf{q}})\dfrac{|G_{0}^{\rm
R}|^{2}\Sigma^{\rm K}}{\left|1-G_{0}^{\rm R}\Sigma^{\rm R}\right|^{2}}.$ (23)

We note that this expression is valid far from equilibrium, and that it allows
for systematic inclusion of electron-electron interactions.

###  III.3 Non-equilibrium self-energy

In order to proceed, we need to evaluate the non-equilibrium self-energy
$\bm{\Sigma}$. It contains contributions from both the electron-phonon and
electron-hydron interaction. We do not consider here any contribution of
electron-electron interactions to the self-energy. We expect this to be
reasonable as long as electron-phonon and electron-hydron scattering dominate
electron-electron scattering, which is typically the case at room temperature
Polini and Geim (2020). Neglecting diagrams where the phonon and hydron
propagators cross, they may be computed separately:
$\bm{\Sigma}=\bm{\Sigma}_{\rm h/e}+\bm{\Sigma}_{\rm ph/e}$. Furthermore, the
two contributions are in fact formally identical, since both the phonons and
the hydrons are free bosons coupled to the electrons. Therefore, we only need
to compute a generic electron-boson self-energy, resulting from a perturbation
of the form given in Eq. (1).

We will consider a single diagram for this self-energy, as shown in Fig. 2b;
we verify that higher order diagrams are indeed negligible under most
conditions (see Appendix B.5). We account for the electronic screening of the
bosonic propagator and boson-electron interaction vertices within the random
phase approximation (Fig. 2c-d). We will absorb the screened vertices into the
definition of the full bosonic propagator, that we denote as D. The electric
current, which is expressed in terms of the Keldysh Green’s function evaluated
at equal points in space and time, can then be represented as a sum of "ice
cone" diagrams (Fig. 2e). These diagrams are reminiscent of the Aslamazov-
Larkin diagrams Aslamasov and Larkin (1968) that typically represent Coulomb
drag in condensed matter systems Kamenev and Oreg (1995); Narozhny  _et al._
(2012); Kamenev and Levchenko (2009). Their evaluation typically involves the
computation of a non-linear susceptibility (triangle diagram). In our case,
this complication can be avoided, as the non-equilibrium self-energy can be
readily evaluated.

Using the Keldysh formalism Feynman rules for the boson-fermion interaction
Rammer (2007) (see Appendix B) we obtain the components of the self-energy
diagram in Fig. 2b as:

$\displaystyle\Sigma^{\rm R,A}(\mathbf{q},\omega)$ $\displaystyle=$
$\displaystyle-\int\frac{\mathrm{d}\mathbf{q}^{\prime}\mathrm{d}\omega^{\prime}}{(2\pi)^{3}}\mathcal{M}(\mathbf{q}-\mathbf{q}^{\prime},\mathbf{q})\left(f(\omega^{\prime}-\mathbf{q}^{\prime}\\!\cdot\\!\mathbf{v}_{\rm
b})+\frac{1}{f(\xi_{\mathbf{q}-\mathbf{q}^{\prime}})}\right)\frac{\mathrm{Im}\,[D^{R}(\mathbf{q}^{\prime},\omega^{\prime}-\mathbf{q}^{\prime}\\!\cdot\\!\mathbf{v}_{\rm
b})]}{\omega-\omega^{\prime}-\xi_{\mathbf{q}-\mathbf{q}^{\prime}}\pm i0^{+}},$
(24) $\displaystyle\Sigma^{\rm K}(\mathbf{q},\omega)$ $\displaystyle=$
$\displaystyle
2i\pi\int\frac{\mathrm{d}\mathbf{q}^{\prime}}{(2\pi)^{3}}\mathcal{M}(\mathbf{q}-\mathbf{q}^{\prime},\mathbf{q})\left(1+\frac{f(\omega-\xi_{\mathbf{q}-\mathbf{q}^{\prime}}-\mathbf{q}^{\prime}\\!\cdot\\!\mathbf{v}_{\rm
b})}{f(\xi_{\mathbf{q}-\mathbf{q}^{\prime}})}\right)\mathrm{Im}\,[D^{R}(\mathbf{q}^{\prime},\omega-\xi_{\mathbf{q}-\mathbf{q}^{\prime}}-\mathbf{q}^{\prime}\\!\cdot\\!\mathbf{v}_{\rm
b})],$ (25)

and using $\mathrm{Im}\,[1/(\epsilon+i0^{+})]=-\pi\delta(\epsilon)$ yields

$\mathrm{Im}\left[\Sigma^{\rm
R}(\mathbf{q},\omega)\right]=\pi\int\frac{\mathrm{d}\mathbf{q}^{\prime}}{(2\pi)^{3}}\mathcal{M}(\mathbf{q}-\mathbf{q}^{\prime},\mathbf{q})\left(f(\omega-\xi_{\mathbf{q}-\mathbf{q}^{\prime}}-\mathbf{q}^{\prime}\\!\cdot\\!\mathbf{v}_{\rm
b})+\frac{1}{f(\xi_{\mathbf{q}-\mathbf{q}^{\prime}})}\right)\mathrm{Im}\,[D^{R}(\mathbf{q}^{\prime},\omega-\xi_{\mathbf{q}-\mathbf{q}^{\prime}}-\mathbf{q}^{\prime}\\!\cdot\\!\mathbf{v}_{\rm
b})].$ (26)

Here $\mathbf{v}_{\rm b}$ is the boson wind velocity, and
$\mathcal{M}(\mathbf{q}-\mathbf{q}^{\prime},\mathbf{q})\equiv|\langle\mathbf{q}-\mathbf{q}^{\prime}|e^{-i\mathbf{q}^{\prime}\mathbf{r}}|\mathbf{q}\rangle|^{2}$
are matrix elements computed between the electronic states
$|\mathbf{q}\rangle$. The fluctuation-dissipation theorem in Eq. (11) is
therefore not satisfied for the non-equilibrium self-energy. Nevertheless, we
may always express the Keldysh component in the form

$\Sigma^{\rm
K}(\textbf{q},\omega)=\frac{2i}{f(\omega-\mathbf{q}\\!\cdot\\!\mathbf{v}_{\rm
eff}(\mathbf{q},\omega))}\textnormal{Im}[\Sigma^{\rm R}(\textbf{q},\omega)],$
(27)

which defines $\mathbf{v}_{\rm eff}$ as a frequency and momentum dependent
_effective wind velocity_.

Figure 3: Non-equilibrium self energy. (a) Imaginary part of the electron-
boson self-energy (at the Fermi level) for the four electron-boson couples
discussed in the text, as a function of the charge carrier (electron or hole)
density. Electron-phonon and electron-hydron scattering occur at a similar
rate in the 2DEG model, whereas in graphene electron-hydron scattering is much
more efficient than electron-phonon scattering. (b). Same as in (a), but
plotted as a function of frequency: $\omega=0$ corresponds to the Fermi-level.
The relatively weak variations within the thermal window (grey rectangle)
justify the use of the impurity approximation. (c) Deviation of the effective
electronic wind velocity (defined in the text) from the bosonic wind velocity,
as a function of the latter’s magnitude.

One can now evaluate explicitly the self-energy. The calculations steps are
detailed in Appendix B. In a few words, we assume for simplicityÂ that the
liquid is water, whose interfacial charge fluctuations are described in terms
of a _surface response function_. As demonstrated by the extensive molecular
dynamics simulations of ref. Kavokine  _et al._ (2022), this response function
for waterÂ can be modelled as a sum of two Debye peaks, so that

$D_{\rm w}^{\rm
R}(\mathbf{q},\omega)=-\frac{1}{\hbar}\frac{V_{q}^{C}}{\epsilon(q)}\sum_{k=1,2}\frac{f_{k}}{1-i\epsilon(q)\omega/\omega_{\mathrm{D},k}},$
(28)

where $V_{q}^{C}=e^{2}/(2\epsilon_{0}q)$ is the Fourier-transformed Coulomb
potential, $\omega_{\rm D,1}\approx 1.5~{}\rm meV$ and $\omega_{\rm
D,2}\approx 20~{}\rm meV$ are the Debye frequencies and $f_{1,2}$ are the
corresponding oscillator strengths; $\epsilon(q)$ is the RPA dielectric
function of the electronic system, that accounts for the screening of the
interaction vertices. These results can be extended to any other liquid using
a relevant description of their response function. The acoustic phonon
propagator can be written as Bruus and Flensberg (2004)

$D_{\rm ph}^{\rm R}(\mathbf{q},\omega)=\frac{1}{\hbar}V_{q}^{\rm
ph/e}\frac{\omega_{\mathbf{q}}^{2}}{(\omega+i0^{+})^{2}-\omega_{\mathbf{q}}^{2}},$
(29)

with $\omega_{\mathbf{q}}=cq$, $c$ being the phonon velocity, and
$V_{\mathbf{q}}^{\rm ph/e}$ is the material-dependent screened electron-phonon
interaction. For the description of the electronic system, we consider two
different models: a two-dimensional electron gas (2DEG) with an effective mass
$m$; and graphene, treated within the Dirac cone approximation, characterized
by the constant Fermi velocity $v_{\rm F}$. We refer to Appendix B for the
details on the associated electronic structure and dielectric properties. Both
models are assumed invariant by translation. The crystallographic structure of
solid then appears only in the electronic propagator. More realistic models of
solid may also be implemented in the theory, at the cost of higher
technicality in the calculations.

The results of these calculations are summarized in Fig. 3. Fig. 3a shows the
imaginary part of the retarded self energy at the Fermi level,
$\mathrm{Im}\left[\Sigma^{\rm R}(q=k_{\rm F},\omega=0)\right]$, computed by
numerical integration according to Eq. (26), where we have separated the
hydron and phonon contributions. This quantity represents the scattering rate
of the low-energy electronic quasiparticles. The electron-hydron interaction,
as well as the electron-phonon interaction in the 2DEG are essentially
screened Coulomb interactions and they yield a similar order of magnitude for
the associated scattering rate. On the other hand, the electron-phonon
interaction in graphene has a peculiar form (see Appendix B.3), so that the
corresponding scattering rate is 2 to 3 orders of magnitudes lower. This will
have an importance for the global momentum balance discussed in Sec. IV.3.

For the purpose of computing the Coulomb drag current, we need to determine
how the electron-boson scattering affects the electronic distribution
function, that varies typically over a scale $T$ around the Fermi level. If
the boson energy is much smaller than $T$, we may approximate it as 0: the
electrons then see a random static impurity potential. Within this _impurity
approximation_ , we further assume

$\left\\{\begin{array}[]{ll}&\mathrm{Im}\,[\Sigma^{\rm
R}(\mathbf{q},\omega)]\approx\mathrm{Im}\,[\Sigma^{\rm
R}(\mathbf{q},\xi_{\mathbf{q}})]\equiv-1/\tau_{\mathbf{q}},\\\
&\mathrm{Re}\,[\Sigma^{\rm R}(\mathbf{q},\omega)]\approx 0.\end{array}\right.$
(30)

Then, as detailed in Appendix B.3, the electron-boson scattering rate can be
computed as

$\frac{1}{\tau_{\mathbf{q}}}=\pi\frac{T}{\hbar^{2}}\int\frac{\mathrm{d}\mathbf{q}^{\prime}}{(2\pi)^{2}}V_{\mathbf{q}-\mathbf{q}^{\prime}}\delta(\xi_{\mathbf{q}}-\xi_{\mathbf{q}^{\prime}}),$
(31)

where $V$ is the electron-boson interaction. Fig. 3b shows the frequency
dependence of the scattering rate (imaginary part of the self-energy) at
$q=k_{\rm F}$, at a fixed chemical potential $\mu=100~{}\rm meV$ for both 2DEG
and graphene. One observes that the variation of the self-energy in a window
of width $2T$ around zero frequency (grey rectangle in Fig. 3b) is relatively
weak, justifying the use of the impurity approximation (dashed lines in Fig.
3b) in the following computations.

If one neglects the angular dependence the integrand in Eq. (31), the
scattering rate assumes an intuitive Fermi golden rule form:

$\frac{1}{\tau_{\mathbf{q}}}\approx\pi\frac{T}{\hbar}V_{\mathbf{q}}N(u_{\mathbf{q}}),$
(32)

where $N(u_{\mathbf{q}})$ is the density of states at energy $u_{\mathbf{q}}$.
Here, the quantity $\hbar\omega_{\rm b}V_{\mathbf{q}}/\mathcal{A}$ plays the
role of the squared matrix element, and $T/\hbar\omega_{\rm b}$ is the number
of bosonic modes on which the electrons can scatter, $\hbar\omega_{\rm b}$
being the typical bosonic energy (see Appendix B.4 for a detailed derivation).

The simplified expression in Eq. (32) allows us to understand the scalings
observed in Figs. 3 a-b. The frequency dependence of the self-energy (Fig. 3b)
is roughly consistent with it being proportional to the density of states,
which is independent of energy in the 2DEG ($N(u)=\theta(u)\times
m/\hbar^{2}$) and proportional to the energy in graphene
($N(u)=\theta(u)\times 2u/(\pi\hbar)$, for the upper Dirac cone). In a 2DEG
with reasonable electronic density, the screening length is much shorter than
the Fermi wavelength, so that for $q\sim k_{\rm F}$, $V_{\mathbf{q}}\approx
1/(2N(u_{\mathbf{q}}))$: we then obtain a "Planckian" scattering time
$\tau_{\mathbf{q}}\approx 2\hbar/\pi T$ Zaanen (2004). This result could be
expected on dimensional grounds. Indeed, once the boson energy has been
neglected, the temperature is the only energy scale in the problem. The
electron-boson scattering rate evaluated at the Fermi momentum is then
expected to be nearly independent of electronic density, as observed in Fig.
3a. The situation is different in graphene, where the screened Coulomb
potential scales like $1/q$ for $q\sim k_{\rm F}$ and the density of states
$N(u_{\mathbf{q}})\propto q$ since $u_{\mathbf{q}}=\hbar v_{\rm F}q$.
Therefore, we expect again the self-energy to weakly depend on the momentum
(hence, the electronic density) for the electron-hydron scattering. For the
electron-phonon scattering, where the effective potential does not depend on
$q$, we expect $\tau_{\mathbf{q}}\propto 1/q$, so that $\tau_{k_{\rm
F}}\propto n^{-1/2}$, consistently with Fig. 3a.

We now come back to the Keldysh component of the self-energy, which we
evaluate numerically according to Eq. (25). We then compute the effective
electronic velocity $\mathbf{v}_{\rm eff}$ defined in Eq. (27), at $q=k_{\rm
F}$ and $\omega=0$. The deviation of $v_{\rm eff}$ from the boson wind
velocityÂ $v_{\rm b}$ is plotted in Fig. 3c as a function of $v_{\rm b}$, for
the different electron-boson couples. While the liquid flow velocity can be as
low as $1~{}\rm\mu m/s$, we will find that phonon wind velocities can reach
thousands of m/s. For this whole range of boson velocities $v_{\rm b}$ we find
that $v_{\rm eff}$ remains within 5% of $v_{\rm b}$, with a stronger deviation
appearing only for $\mathbf{v}_{\rm b}$ in excess of $1~{}\rm km\cdot s^{-1}$.
We may therefore safely assume $\mathbf{v}_{\rm eff}\approx\mathbf{v}_{\rm
b}$, and evaluate $\Sigma^{\rm K}$ according to a _quasi-equilibrium_
fluctuation-dissipation theorem:

$\Sigma^{\rm
K}(\textbf{q},\omega)=\frac{2i}{f(\omega-\mathbf{q}\\!\cdot\\!\mathbf{v}_{\rm
b})}\textnormal{Im}[\Sigma^{\rm R}(\textbf{q},\omega)],$ (33)

which differs from the equilibrium version only by the frequency shift
$\mathbf{q}\\!\cdot\\!\mathbf{v}_{\rm b}$. We note here the power of the
Keldysh framework, which allows us to control the approximation leading to Eq.
(33), and potentially explore conditions where it no longer holds.

###  III.4 Quasi-equilibrium state

When Eq. (33) is satisfied for all electron-boson self-energies $\Sigma_{j}$,
we will say that the system is in a _quasi-equilibrium state_. Within the
impurity approximation, we denote
$\textnormal{Im}[\Sigma_{j}]=-1/\tau^{j}_{\mathbf{q}}$; then,
$\tau_{\mathbf{q}}^{-1}=\sum_{j}(\tau^{j}_{\mathbf{q}})^{-1}$ is the total
electron scattering rate at the energy $\xi_{\mathbf{q}}$. Eqs. (21) and (22)
for the non-equilibrium Green’s functions now become:

$\displaystyle G^{\rm R}(\mathbf{q},\omega)$ $\displaystyle=$
$\displaystyle\dfrac{\sum_{j}\textnormal{Re}[\Sigma_{j}]+i\tau^{-1}_{\mathbf{q}}}{(\omega-\xi_{\textbf{q}})^{2}+\tau^{-2}_{\mathbf{q}}}$
(34) $\displaystyle G^{\rm K}(\mathbf{q},\omega)$ $\displaystyle=$
$\displaystyle\dfrac{2i}{f(\omega-\textbf{q}\\!\cdot\\!\textbf{v}_{\rm
e}(q))}\dfrac{\tau^{-1}_{\mathbf{q}}}{(\omega-\xi_{\textbf{q}})^{2}+\tau^{-2}_{\mathbf{q}}}$
(35)

where the _electron wind velocity_ defined as,

$\mathbf{v}_{\rm
e}(q)=\sum_{j}\frac{\tau_{\mathbf{q}}}{\tau^{j}_{\mathbf{q}}}\textbf{v}_{j},$
(36)

is a convex combination of the different boson wind velocities
$\textbf{v}_{j}$; Eq. (35) is valid as long as these are small compared to the
Fermi velocity. We note that we may include static impurities as an additional
scatterer with zero velocity. Physically, each bosonic wind blows on the
electrons through its electron-boson interaction. Each bosonic velocity
contributes to the total wind velocity with a weight that is given by the
corresponding electron-boson scattering rate.

We therefore find that the non-equilibrium Green’s function satisfies the same
quasi-equilibrium fluctuation-dissipation theorem as the individual self-
energies:

$G^{\rm
K}(\textbf{q},\omega)=\frac{2i}{f(\omega-\textbf{q}\\!\cdot\\!\textbf{v}_{\rm
e}(q))}\textnormal{Im}[G^{\rm R}(\textbf{q},\omega)]$ (37)

Eqs. (34) and (37) provide a complete picture of the non-equilibrium
electronic state. Due to the scattering on the different bosons, the spectral
function is broadened, and acquires a width $\tau_{\mathbf{q}}^{-1}$. In
addition, the occupation of the broadened states undergoes a Doppler shift
$\mathbf{q}\\!\cdot\\!\mathbf{v}_{e}$ with respect to the equilibrium Fermi-
Dirac occupation. This becomes apparent if one evaluates the actual electronic
density in energy-momentum space (see Appendix A.2):

$n_{\rm e}(\textbf{q},\omega)=\frac{G^{\rm
K}(\mathbf{q},\omega)}{2i}+\,\textnormal{Im}[G^{\rm R}(\textbf{q},\omega)],$
(38)

which reduces to

$n_{\rm e}(\textbf{q},\omega)=n_{F}(\omega-\textbf{q}\cdot\textbf{v}_{\rm
e}(q))\frac{2\tau}{1+\tau^{2}(\omega-\xi_{\textbf{q}})^{2}}$ (39)

where $n_{F}(\omega)=1/(e^{\hbar\omega/T}+1)$ is the Fermi-Dirac distribution.
The electrons appear to acquire an average velocity equal to the electron’s
wind velocity, which corresponds precisely to the electric current that we
evaluate in the next section.

We make one last remark concerning the electron density-density response
function. Starting from its expression in terms of the Green’s functions we
demonstrate in Appendix C.1 that it satisfies a quasi-equilibrium fluctuation-
dissipation theorem as long as the Green’s functions satisfy one:

$\chi_{\rm e}^{\rm
K}(\textbf{q},\omega)=2if(\omega-\textbf{q}\\!\cdot\\!\textbf{v}_{\rm
e}(q))\textnormal{Im}[\chi_{\rm e}^{\rm R}(\textbf{q},\omega)].$ (40)

This result will be important for the evaluation of fluctuation-induced
friction forces in the next section.

##  IV Flow-induced electric current

###  IV.1 General expression

Starting from the formal expression in Eq. (16), and using the quasi-
equilibrium Green’s function in Eq. (35), we immediately obtain, after angular
integration and to first order in the wind velocity, an explicit expression
for the flow-induced electronic current:

$\langle\textbf{j}\rangle=\frac{e\hbar}{T}\int_{-\infty}^{+\infty}\frac{\mathrm{d}\omega}{2\pi}\int_{0}^{+\infty}\frac{\mathrm{d}q}{2\pi}\,\frac{q^{2}\left(\nabla_{q}\xi_{q}\right)\mathbf{v}_{\rm
e}(q)}{\textnormal{cosh}^{2}\left[\frac{\hbar(\omega+\xi_{q})}{2T}\right]}\frac{\tau_{q}}{1+\tau_{q}^{2}\omega^{2}}.$
(41)

The wind velocity $\mathbf{v}_{e}(q)$ is given by Eq. (36). In the presence of
electron-hydron, electron-phonon and electron-impurity scattering, it
explicitly writes

$\mathbf{v}_{\rm e}(q)=\frac{\tau_{\mathbf{q}}}{\tau^{\rm
ph/e}_{\mathbf{q}}}\textbf{v}_{\rm ph}+\frac{\tau_{\mathbf{q}}}{\tau^{\rm
h/e}_{\mathbf{q}}}\textbf{v}_{\rm\ell}$ (42)

where $1/\tau_{\mathbf{q}}=1/\tau^{\rm ph/e}_{\mathbf{q}}+1/\tau^{\rm
h/e}_{\mathbf{q}}+1/\tau^{\rm im}_{\mathbf{q}}$, the last term being the
impurity scattering rate.

We thus obtain our first main result, as anticipated in Sec. II: we predict
current generation in the solid due to the flow of a neutral liquid.

###  IV.2 Comparison with literature results

To our knowledge, our result in Eq. (41) is not found in the literature. Its
closest analogue is the general expression for solid-solid Coulomb drag, Eq.
(15) in ref. Narozhny and Levchenko (2016), since it is derived at the same
level of theory. However, the solid-solid result is not directly applicable to
the solid-liquid case, since it involves a non-linear current-voltage response
function, which is not defined for a liquid that is assumed insulating within
our model. The result in Eq. (36) is in fact simpler, because, as compared to
the solid-solid case, one of the interacting fermionic systems is replaced
with a free bosonic field. We note that, recently, a theory of Coulomb drag
has been developed for a system of two graphene sheets where the electrons are
in the hydrodynamic regime Levchenko  _et al._ (2022). However, the analogy
with our system is mostly semantic, as the fluctuations of an electron liquid
are very different from hydron modes, which are similar to strongly damped
optical phonons.

In the framework of fluctuation-induced electromagnetic phenomena, Volokitin
and Persson have proposed an expression for the electric field $E$ induced in
a 2D electron gas (2DEG) by a liquid flowing along its surface Volokitin and
Persson (2008). They considered only the contribution of direct Coulomb
interactions, and did not include any phonon effects. Under the assumption
that this electric field equilibrates the solid-liquid quantum friction force,
they obtain $neE=\lambda_{\rm h/e}v_{\rm\ell}$, where $n$ is the electron
density in the 2DEG and $\lambda_{\rm h/e}$ is the quantum friction
coefficient (see Table 1). In order to convert this to a current density, one
has to assume a Drude-like conductivity for the 2DEG, $\sigma=ne^{2}\tau/m$,
where $\tau$ is a momentum-independent relaxation time and $m$ is the
effective mass. Then, using Ohm’s law $j=\sigma E$, one obtains

$j=\frac{e\tau}{m}\lambda_{\rm e/h}v_{\rm\ell}.$ (43)

Even under the assumption of momentum-independent relaxation times and a
parabolic band structure, we find that the direct Coulomb contribution in Eq.
(41) does not reduce to Eq. (43). Indeed, if we further assume a quasiparticle
scattering rate that is small compared to the thermal energy $\hbar/\tau\ll
T$, the Coulomb contribution in Eq. (36) becomes

$\langle j\rangle=\frac{e\tau}{m}\times\int\frac{d^{2}q}{(2\pi)^{2}}\hbar
q\left[\frac{\pi
q}{\textnormal{cosh}^{2}\left[\frac{\hbar\xi_{q}}{2T}\right]}\right]\frac{1}{\tau^{\rm
e/h}}v_{\ell}\equiv\frac{e\tau}{m}\lambda_{\rm e/h}^{*}v_{\ell}.$ (44)

Comparing to Eq. (43), we may identify an effective friction coefficient
$\lambda_{\rm e/h}^{*}$, whose expression differs from the usual $\lambda_{\rm
e/h}$ as obtained, for example, in Kavokine  _et al._ (2022) (see also Eq.
(58)). In particular, the photon tunneling rate is given by the liquid’s
contribution to the electronic self-energy, $1/\tau_{\rm
e/h}\equiv\mathrm{Im}\,\Sigma_{\rm h}$, rather than by the overlap of surface
excitation spectra; the two expressions would in fact be equivalent only if
the electrons were non-interacting.

We conclude that the microscopic Keldysh-formalism approach was instrumental
in obtaining a rigorous description of electronic current generation by liquid
flow. We would like to stress the generality of the approach, since it
formally allows for any interactions to be taken into account to any desired
level of precision. Our most general result is in fact given by Eq. (23),
where the electric current is expressed in terms of the electronic self-
energy. This self-energy may be computed within various numerical schemes (in
particular, diagrammatic Monte Carlo Prokof’ev and Svistunov (1998); Gull  _et
al._ (2011); Bertrand  _et al._ (2019)), and thus the solid-liquid Coulomb
drag maybe studied within regimes where our impurity or quasi-equilibrium
approximations no longer hold.

###  IV.3 Wind velocity : global momentum balance

Figure 4: Momentum balance. (a) Diagram representing the momentum fluxes in the solid-liquid system, separated into four subsystems, represented by the rectangular boxes. The liquid (flowing at velocity $\mathbf{v}_{\ell}$) is a momentum source, and the "crystal" is a momentum sink. The momentum fluxes in and out of the phonon and electron subsystems need to be balanced in the steady state. The phonons (electrons) accumulate a momentum $\Delta\textbf{P}_{\rm ph}$ ($\Delta\textbf{P}_{\rm e}$), corresponding to a Doppler shift $\hbar qv_{\rm ph}$ ($\hbar qv_{e}$) of their momentum distribution. $\tau_{\rm um}$ is the phonon umklapp scattering time and $\tau_{\rm im}$ is the electron impurity scattering time. (b) Phonon and electron wind velocities (normalized by the liquid velocity) as a function of the electronic density, for $\lambda_{\rm um}\approx 2~{}\rm N\cdot s\cdot m^{-3}$ and $\lambda_{0}=2\times 10^{6}~{}\rm N\cdot s\cdot m^{-3}$. Interaction |  Momentum transfer |  Force | Scattering time
---|---|---|---
Classical friction |  Liquid | $\rightarrow$ | Phonons | $\textbf{F}_{0}$ | = | $\lambda_{0}\mathcal{A}\textbf{v}_{\ell}$ |
Phonon-hydron interaction |  Liquid | $\leftrightarrow$ | Phonons | $\textbf{F}_{\rm h/ph}$ | = | $\lambda_{\rm h/ph}\mathcal{A}(\textbf{v}_{\ell}-\textbf{v}_{\rm ph})$ |
Electron-hydron interaction |  Electrons | $\leftrightarrow$ | Liquid | $\textbf{F}_{\rm h/e}$ | = | $\lambda_{\rm h/e}\mathcal{A}(\textbf{v}_{\ell}-\textbf{v}_{\rm e})$ | $\tau_{\rm h/e}$
Electron-phonon interaction |  Phonons | $\leftrightarrow$ | Electrons | $\textbf{F}_{\rm ph/e}$ | = | $\lambda_{\rm ph/e}\mathcal{A}(\textbf{v}_{\rm e}-\textbf{v}_{\rm ph})$ | $\tau_{\rm ph/e}$
Umklapp processes |  Phonons | $\rightarrow$ | Crystal | $\textbf{F}_{\rm um}$ | = | $\Delta\textbf{P}_{\rm ph}/\tau_{\rm um}=\lambda_{\rm um}\mathcal{A}\textbf{v}_{\rm ph}$ | $\tau_{\rm um}$
Impurities |  Electrons | $\rightarrow$ | Crystal | $\textbf{F}_{\rm im}$ | = | $\Delta\textbf{P}_{\rm e}/\tau_{\rm im}$ | $\tau_{\rm im}$
Total friction |  Liquid | $\rightarrow$ | Phonons+Electrons | F | = | $\lambda\mathcal{A}\textbf{v}_{\ell}=(\lambda_{0}+\delta\lambda)\mathcal{A}\textbf{v}_{\ell}$ |
Table 1: Forces in the solid-liquid system. List of interactions that may
transfer momentum between the different components of the solid-liquid system,
and notations for the associated forces, friction coefficients and scattering
times.

In order to evaluate the current in Eq. (41), we require one last ingredient,
which is the velocity $\mathbf{v}_{\rm ph}$ of the phonon wind. As mentioned
in Sec. II, we evaluate it self-consistently, by enforcing momentum
conservation in the solid-liquid system.

In the stationary state, the phonons accumulate a momentum that we denote
$\Delta\textbf{P}_{\rm ph}$. We model this momentum accumulation by giving all
the phonons an average velocity $\mathbf{v}_{\rm ph}$: this means shifting the
phonon distribution according to

$n_{\rm ph}(\textbf{q},\omega)=n_{\rm B}(\omega-\textbf{q}\cdot\textbf{v}_{\rm
ph})\times 2\pi\delta(\omega-cq)$ (45)

where $n_{\rm B}(\omega)=1/(e^{\hbar\omega/T}-1)$ is the Bose-Einstein
distribution and $c$ is the sound velocity. Then,

$\frac{\Delta\textbf{P}_{\rm
ph}}{\mathcal{A}}=\int\frac{\textnormal{d}^{2}\textbf{q}}{(2\pi)^{2}}\hbar\textbf{q}\,n_{\rm
B}\left(qc-\textbf{q}\cdot\textbf{v}_{\rm ph}\right),$ (46)

which becomes, to first order in $\mathbf{v}_{\rm ph}$,

$\frac{\Delta\textbf{P}_{\rm
ph}}{\mathcal{A}}=\frac{3\zeta(3)}{2\pi}\frac{T^{3}}{\hbar^{2}c^{4}}\textbf{v}_{\rm
ph},$ (47)

$\zeta$ being the Riemann function. The contributions to the momentum flux in
and out of the phonon system are summarized in Fig. 4a, and the associated
notations are explicited in Table 1. The phonons receive momentum from the
flowing liquid through the classical, roughness-induced contribution to the
hydrodynamic friction force,
$\textbf{F}_{0}=\lambda_{0}\mathcal{A}\mathbf{v}_{\ell}$, and through the
phononic contribution $\textbf{F}_{\rm h/ph}$ to fluctuation-induced friction
Kavokine  _et al._ (2022). In Appendix C.2, we extend the framework of ref.
Kavokine  _et al._ (2022) to account for the non-equilibrium state of the
solid, and show that as long as the quasi-equilibrium fluctuation-dissipation
theorem (Eq. (40)) holds for the system’s density response functions, the
fluctuation-induced friction force is proportional to the differential
velocity:

$\textbf{F}_{\rm h/ph}=\lambda_{\rm
h/ph}\mathcal{A}(\mathbf{v}_{\ell}-\mathbf{v}_{\rm ph}).$ (48)

We note that, conversely, $\textbf{F}_{0}$ – the classical, roughness induced
friction – does not depend on the phonon velocity: indeed, it originates
largely in defects on the solid’s surface, which do not move even if the
phonons accumulate momentum.

The phonons lose momentum mainly through umklapp processes; we denote
$\tau_{\rm um}$ the corresponding relaxation time. Formally, in an umklapp
process, the interference of multiple phonons converts their momentum into a
global translation of the crystal lattice. In practice, however, the 2D
material layer remains fixed, and the momentum is transferred to the
underlying substrate, which we do not describe explicitly. The momentum lost
by the phonons per unit time and unit area through umklapp processes is
therefore

$\frac{\textbf{F}_{\rm um}}{\mathcal{A}}=\frac{\Delta\textbf{P}_{\rm
ph}}{\tau_{\rm
um}\mathcal{A}}=\frac{3\zeta(3)}{2\pi}\frac{T^{3}}{\hbar^{2}c^{4}\tau_{\rm
um}}\textbf{v}_{\rm ph}\equiv\lambda_{\rm um}\mathbf{v}_{\rm ph},$ (49)

where we have defined the umklapp friction coefficient $\lambda_{\rm um}$. In
graphene, $\tau_{\rm um}\approx 10~{}\rm ps$ Klemens and Pedraza (1994) and
$c\approx 2\cdot 10^{4}$ m/s Nika  _et al._ (2009); Ochoa  _et al._ (2011);
Cong  _et al._ (2019), so that $\lambda_{\rm um}\approx 2~{}\rm N\cdot s\cdot
m^{-3}$. In addition, the phonons loose momentum through quantum friction with
the conduction electrons, which is analogous to the water-electron quantum
friction studied in ref. Kavokine  _et al._ (2022), with the Coulomb
interaction being replaced by the electron-phonon interaction. This type of
friction is also known as a _current-induced force_ in the context of
nanoscale electron transport Di Ventra (2008). The corresponding momentum flux
is $\textbf{F}_{\rm ph/e}=\lambda_{\rm ph/e}\mathcal{A}(\mathbf{v}_{\rm
ph}-\mathbf{v}_{\rm e})$. In a steady state, the incoming and outgoing
momentum fluxes (or forces) must compensate:

$\textbf{F}_{0}+\textbf{F}_{\rm h/ph}=\textbf{F}_{\rm um}+\textbf{F}_{\rm
ph/e}.$ (50)

Using Eq. (42) for the electronic velocity $\mathbf{v}_{\rm e}$, we then
obtain explicitly the phonon wind velocity as

$\textbf{v}_{\rm ph}=\frac{\lambda_{0}+\lambda_{\rm
h/ph}+\frac{\tau}{\tau_{\rm h/e}}\lambda_{\rm ph/e}}{\lambda_{\rm
um}+\lambda_{\rm h/ph}+\left(1-\frac{\tau}{\tau_{\rm ph/e}}\right)\lambda_{\rm
ph/e}}\mathbf{v}_{\rm\ell}.$ (51)

This formula is consistent with the roughness-induced friction (with
coefficient $\lambda_{0}$) being a momentum source, and the umklapp processes
(with equivalent friction coefficient $\lambda_{\rm um}$) being a momentum
sink. We do not have a practical way of evaluating the acoustic phonon
contribution $\lambda_{\rm h/ph}$ to the fluctuation-induced friction.
Nevertheless, guided by the qualitative ideas of ref. Kavokine  _et al._
(2022), we expect it to be very small, since it is associated with in-plane
lattice displacements at wavelengths much larger than the atomic spacing, that
have short-range contact interactions with the liquid. Henceforth, we will
assume it to be negligible compared to $\lambda_{\rm um}$ and $\lambda_{0}$.
The electron-phonon friction coefficient, on the other hand, is evaluated
explicitly in Sec. V, and we find that $\lambda_{\rm ph/e}\ll\lambda_{0}$
under all practical conditions; it is, however, comparable to $\lambda_{\rm
um}$ at large electronic density. Therefore, in the absence of impurities, the
phonon wind velocity may be simplified, after some rearrangements, to

$\mathbf{v}_{\rm ph}\approx\frac{\lambda_{0}}{\lambda_{\rm um}+\lambda_{\rm
ph/e}\tau/\tau_{\rm h/e}}\mathbf{v}_{\ell}.$ (52)

Typically, on a molecularly rough surface, the roughness-induced friction
coefficient is of orderÂ $\lambda_{0}\sim 10^{5}-10^{6}~{}\rm N\cdot s\cdot
m^{-3}$ Bocquet and Charlaix (2010), so that $\mathbf{v}_{\rm
ph}\gg\mathbf{v}_{\ell}$: the phonon velocity is orders of magnitude larger
than the flow velocity. Fig. 4b shows the phonon and electron wind velocities
(normalized by $\mathbf{v}_{\ell}$) as a function of the electronic density.
Under all practical conditions the phonon wind is faster than the electron
wind, whose velocity is reduced by the electron-hydron interaction. This
results in the electron-hydron interaction actually making a _negative_
contribution to the Coulomb drag, as explained in the next section.

One may draw an analogy between current generation by hydrodynamic Coulomb
drag and current generation through the photoelectric effect. The former
benefits from strong electron-phonon interactions, as those help transfer
momentum from the liquid to the electrons. Conversely, the latter is
suppressed by electron-phonon scattering, as phonon drag slows down the photo-
generated charge carriers: reducing phonon drag is a key challenge in the
engineering of Perovskite materials, which are the state-of-the-art for
photoelectric panels Yang  _et al._ (2015); Kentsch  _et al._ (2018); Zhang
_et al._ (2021). Such a complementarity calls for further studies of the
potential interplay between these two phenomena.

###  IV.4 Quantitative estimates: Coulomb drag vs. phonon drag

Figure 5: Coulomb drag current. Electro-fluidic conductivity $\sigma_{\rm
el}=\langle\textbf{j}\rangle/\mathbf{v}_{\rm\ell}$ as computed with Eq. (41),
for different models of the solid’s band structure. (a) Two-dimensional
electron gas, with different values of the effective mass $m$ (expressed in
units of electron mass $m_{\rm e}$). (b) Semi-conductor with electron and hole
masses $m=0.1m_{\rm e}$, and band gap $2u_{\rm g}$. (c) Graphene, with
constant Fermi velocity in the Dirac cone approximation. In (a)-(c) we assumed
$\frac{\lambda_{0}}{\lambda_{\rm um}}=10^{6}$. The dashed lines correspond to
the prediction of Eq. (54), with no chemical potential dependence of the
electronic wind velocity.

We are now in position to evaluate the electronic current according to eqs.
(41) and (42). In order to make quantitatives estimates, we will use
$\lambda_{\rm um}\approx 2~{}\rm N\cdot s\cdot m^{-3}$ as computed for
graphene in the previous section, and $\lambda_{0}=2\times 10^{6}~{}\rm N\cdot
s\cdot m^{-3}$, which is reasonable for water on a large area exhibiting someÂ
ripples and defects. Furthermore, we make use of the values for $\lambda_{\rm
ph/e}$ computed in Sec. V B. Then, at not too large electronic density,
$\mathbf{v}_{\rm ph}\approx(\lambda_{0}/\lambda_{\rm um})\mathbf{v}_{\ell}\sim
10^{6}\mathbf{v}_{\ell}$. As in Sec. III, we will consider two models for the
electronic structure and electron-phonon interaction: a two-dimensional
electron gas (2DEG) with an effective mass $m$, and graphene, treated within
the Dirac cone approximation, characterized by the constant Fermi velocity
$v_{\rm F}$. In addition we will consider a model of a direct band gap
semiconductor as the combination of an electron gas and a hole gas. For the
two-band systems we will evaluate the total current as the sum of the currents
in the two bands, which amounts to neglecting interband scattering. Fig. 5
shows schematics of the three band structures under consideration

We define the _electro-fluidic conductivity_ as

$\sigma_{\rm ef}\equiv{\langle j\rangle\over v_{\ell}}.$ (53)

We plot in Fig. 5a-c the electro-fluidic conductivity as a function of the
chemical potential $\mu$. One first notes the difference in scaling of
$\sigma_{\rm ef}$ with $\mu$ between the three model systems. These scalings
are most conveniently understood in the limit of weak interactions and low
temperature $\hbar/\tau_{k_{\rm F}}\ll T\ll\mu$. The first inequality means
that the broadening of the electronic distribution due to electron-boson
scattering is negligible compared to the thermal broadening. The second
inequality means that the Fermi-Dirac distribution is well approximated by a
step function. Eq. (41) accordingly simplifies to

$\langle\textbf{j}\rangle\approx 2ev_{\rm F}N(\mu)\times\hbar k_{\rm
F}\mathbf{v}_{\rm e}(k_{\rm F}),$ (54)

where $N(\mu)$ is the density of states at the Fermi level. The effect of the
temperature appears to cancel out, leaving us with a transparentÂ expression
that is intuitive in a zero-temperature picture of Coulomb drag. The current
is the electronic charge times the electronic velocity (which is the Fermi
velocity), times the charge carrier density contributing to the current. The
latter is the density of states at the Fermi level, times the energy range
around the Fermi level in which the charge carriers can contribute to the
current: this is given by the "Doppler shift" $\hbar k_{F}v_{e}$.

In the 2DEG, the density of states $N(\mu)=m/\hbar^{2}$ is independent of
chemical potential, and $v_{\rm F}k_{\rm F}\propto\mu$: one expects a linear
scaling of $\sigma_{\rm ef}$ with $\mu$ when $\mu\gg T$. A correction to this
scaling comes from the chemical potential dependence of the electron-phonon
friction coefficient: $\lambda_{\rm ph/e}\propto\mu$ (see Appendix C.3), which
contributes to reducing the phonon wind velocity at high $\mu$ (see Fig. 4b).
As a consequence, the current is expected to saturate at high chemical
potential. Our model for the semi-conductor is a combination of two-
dimensional electron and hole gases, therefore a linear scaling of
$\sigma_{\rm ef}$ is obtained for both positive and negative chemical
potential (Fig 5b). The electro-fluidic conductivity is suppressed when the
chemical potential is within the band gap because of the lack of charge
carriers, similarly to the 2DEG at negative chemical potential. In graphene,
$N(\mu)\propto\mu$, $k_{\rm F}\propto\mu$ and $v_{\rm F}$ is independent of
$\mu$: we thus expect $\sigma_{\rm ef}\propto\mu|\mu|$ (the sign of
$\sigma_{\rm ef}$ reflects the nature – electron or hole – of the charge
carriers). Fig. 5c shows a slight deviation from this quadratic scaling, which
is due to the dependence on chemical potential of the electron-phonon
scattering time that contributes to the wind velocity $\mathbf{v}_{e}$ (see
Fig. 3c and Fig. 4b). We note that we consider here a simplified model of
graphene that neglects interband scattering, or any effect of charge
inhomogeneities that could cause a non-vanishing Coulomb drag current at
charge neutrality Gorbachev  _et al._ (2012); Song and Levitov (2012).

In the electron gas model, $\sigma_{\rm ef}$ is tunable by the charge carrier
effective mass. As the effective mass increases, the Fermi velocity is
reduced, but the density of states and the Fermi momentum increase: the latter
dominate, and overall, at low enough chemical potential ($\mu\lesssim
0.2~{}\rm eV$) $\sigma_{\rm ef}\propto m$. However, at larger chemical
potential, the electron-phonon friction coefficient, that scales as $m^{2}$
(see Appendix C.3), reduces the phonon wind velocity (see Fig. 4b), so that
$\sigma_{\rm ef}\propto 1/m$. These scalings, illustrated in Fig. 5b, suggests
that flat-band materials are likely to exhibit a significant hydrodynamic
Coulomb drag effect.

At similar chemical potential, the electro-fluidic conductivity is found to be
about two orders of magnitude larger in the 2DEG than in graphene. For
instance at $\mu\sim 100$ meV, $\sigma_{\rm ef}\approx 20~{}\rm nA\cdot
s\cdot\mu m^{-2}$ for the 2DEG and $\sigma_{\rm ef}\approx 0.1~{}\rm nA\cdot
s\cdot\mu m^{-2}$ for graphene. This difference is mainly due to the wind
velocity $\mathbf{v}_{e}$, which is determined according to Eq. (42) from the
phonon wind velocity $\mathbf{v}_{\rm ph}$ and the flow velocity
$\mathbf{v}_{\ell}$, with $\mathbf{v}_{\rm ph}\gg\mathbf{v}_{\ell}$ (see Fig.
4b). In the 2DEG the electron-phonon and electron-hydron scattering times are
similar, so that $\mathbf{v}_{e}\approx\mathbf{v}_{\rm ph}/2$. In graphene,
the electron-hydron scattering is much faster than the electron-phonon
scattering, so that $\mathbf{v}_{e}\sim 10^{-2}\mathbf{v}_{\rm
ph}\gg\mathbf{v}_{\ell}$. Thus, despite the different orders of magnitude, the
phonon drag is the main driving force for the electronic current in both model
systems. However, this does not imply that the electron-hydron interactions
are negligible: in fact, they reduce the "bare" phonon drag current by a
factor of 2 in the 2DEG and by a factor of $10^{2}$ in graphene, by providing
a supplementary momentum relaxation pathway for the electrons. In other words,
the electron-hydron interaction makes a negative contribution to the Coulomb
drag. This effect is at the root of the quantum feedback phenomenon discussed
in Sec. V.

###  IV.5 Comparison with experiment

Figure 6: Comparaison with the experiment of Marcotte et al. Marcotte  _et
al._ . (a) Schematic of the experimental setup. Thanks to an atomic force
microscope (AFM), a liquid droplet is deposited onto the surface of a
multilayer graphene sample connected to two metallic electrodes. Actuation of
a piezoelectric scanner results in motion of the drop on the carbon surface at
a velocity $v_{\rm drop}$. (b) Quantitative comparison between the
experimental data for the electro-fluidic conductivity $\sigma_{\rm ef}$ from
three different devices and theoretical predictions, parametrized by the
wrinkling number $W$ that account for the wrinkle density on the sample
surface.

Lastly, we compare the quantitative estimates obtained from our model with the
results of the companion experimental paper (ref. Marcotte  _et al._ ). In
ref. Marcotte  _et al._ , an atomic force microscope is used to deposit a
liquid droplet on the surface of a strongly wrinkled multilayer graphene
sample, connected to two metallic electrodes. When the droplet is set in
horizontal motion (at a velocity $\mathbf{v}_{\rm drop}$), an electric current
is generated in the sample (see Fig. 6a). The roughness-induced friction force
(per unit area) can be estimated as $\mathbf{F}_{\rm
drop}=\lambda_{0}\mathbf{v}_{\rm drop}$, with $\lambda_{\rm drop}\approx
3\cdot 10^{6}$ N.s.m${}^{-3}\times W$ Marcotte  _et al._ , where $W\approx 10$
is a dimensionless parameter (dubbed wrinkling number) accounting for the
wrinkle density. We note that $\mathbf{v}_{\rm drop}$ may be different from
the interfacial velocity $\mathbf{v}_{\ell}$ used as an input parameter in our
theory, depending on the hydrodynamic flow profile within the drop. However,
we do not need to explicitly determine $\mathbf{v}_{\ell}$ since the wind
velocity $\mathbf{v}_{\rm e}$ is dominated by the phonon wind $\mathbf{v}_{\rm
ph}$. Because of the strong wrinkling, we choose to model the multilayer
graphene sample as a zero-gap semiconductor with effective mass
$m^{*}=0.1m_{e}$, $m_{e}$ being the electron mass. Then, since the
experimental electronic density remains low, $\mathbf{v}_{\rm
e}\approx\mathbf{v}_{\rm ph}/2\approx\textbf{F}_{\rm drop}/(2\lambda_{\rm
um})$. At a chemical potential $\mu=20$ meV, the experimentally measured
electro-fluidic conductivities are in the range 2-8 nA.s.$\mu$m-2, which is
reproduced by our theoretical prediction (eq. (41)), for the wrinkling number
$W$ in the range 5 - 20 (see Fig. 6b). Experimentally, the chemical potential
is set by the bias $\Delta V$ applied between a grounded electrode and a
working electrode: $\mu\approx\Delta V/2$. Thus, the zero-gap semi-conductor
model accounts for the experimentally observed linear scaling of $\sigma_{\rm
ef}$ with the DC bias voltage.

Overall, our theory is quantitatively consistent with the experiments of ref.
Marcotte  _et al._ and is able to account for the particularly strong Coulomb
drag currents (in the 10 nA range) that were generated by the motion of a
micrometer-sized droplet at a few $\mu\rm m/s$. The key factor that is
responsible for a strong current is the hydrodynamic friction force, which is
large in the experiments of ref. Marcotte  _et al._ due to the high viscosity
of the liquids used and the wrinkling of the sample surface: this results in
an efficient transfer of a significant amount of momentum to the sample’s
phonon modes. The crucial role of phonons is further supported by the fact
that the effect could be observed with a non-ionic silicon oil, where liquid-
electron Coulomb interactions are negligible.

##  V Quantum feedback and current-induced negative friction

Figure 7: Quantum feedback and  current-induced negative friction. (a)
Schematic representation of the quantum feedback phenomenon: a fraction of the
momentum transferred to the solid by classical friction is returned to the
fluid by the solid’s internal excitations. (b) Electron-boson friction
coefficients as a function of the electron density for a 2DEG with
$m=0.1~{}\rm m_{\rm e}$ and graphene, evaluated according to (58). The
calculations has been carried out using the electronic susceptibilities in the
zero-temperature limit and in the RPA approximation. (c) Hydrodynamic friction
coefficient $\lambda$ (in units of the classical contribution $\lambda_{0}$)
for a 2DEG with $m=0.1m_{\rm e}$ and graphene.

###  V.1 Derivation

Our analysis in Sec. IV revealed that the hydrodynamic Coulomb drag current is
determined by a subtle combination of electron-phonon and electron-hydron
interactions. Indeed, the electron-hydron scattering provides a supplementary
momentum relaxation pathway for the electrons, which prevents them from
aligning to the phonon wind velocity. This immediately implies that the
electrons actually transfer momentum _to_ the flowing liquid: they make a
_negative_ contribution to the hydrodynamic friction force, as shown
schematically in Fig. 7. In this section, we explicitly evaluate this negative
contribution and assess its practical consequences.

The liquid interacts with the solid through the classical roughness-induced
friction force $\textbf{F}_{0}$, and through the phononic and electronic
contributions to the fluctuation-induced (quantum) friction $\textbf{F}_{\rm
h/ph}$ and $\textbf{F}_{\rm h/e}$, respectively. The total hydrodynamic
friction coefficient $\lambda$ is then defined according to

$\lambda\mathcal{A}\mathbf{v}_{\rm\ell}=\textbf{F}_{0}+\textbf{F}_{\rm
h/ph}+\textbf{F}_{\rm h/e}.$ (55)

Introducing the individual friction coefficients as in Sec. IV B (see Table 1
for notations),

$\lambda\mathbf{v}_{\rm\ell}=\lambda_{0}\mathbf{v}_{\ell}+\lambda_{\rm
h/ph}(\mathbf{v}_{\ell}-\mathbf{v}_{\rm ph})+\lambda_{\rm
h/e}(\mathbf{v}_{\ell}-\mathbf{v}_{e}).$ (56)

In the limit where $\lambda_{0}$ dominates all other friction coefficients,
and using eqs. (42) and (52) for the velocities $\mathbf{v}_{\rm e}$ and
$\mathbf{v}_{\rm ph}$, a rearrangement yields:

$\lambda=\lambda_{0}-\delta\lambda\quad\textnormal{with}\quad\delta\lambda=\frac{\lambda_{0}}{1+\frac{\lambda_{\rm
um}+\frac{\tau}{\tau_{\rm im}}\lambda_{\rm ph/e}}{\lambda_{\rm
h/ph}+\frac{\tau}{\tau_{\rm ph/e}}\lambda_{\rm h/e}}},$ (57)

where we have included the possibility for the electrons to lose momentum
through impurity scattering (at a rate $\tau_{\rm im}^{-1}$). This is our
second key result, anticipated in Sec. II. Strikingly, $\delta\lambda$
represents a negative contribution to hydrodynamic friction. It is, however,
always smaller than $\lambda_{0}$, so that the total friction coefficient
remains positive, and there is no violation of the laws of thermodynamics. But
a key observation is that it becomes equal to $\lambda_{0}$ in the absence of
impurity or umklapp scattering. This amounts to formally considering a solid
that is unable to relax momentum: then, even if the solid has a rough surface,
all the momentum the liquid loses through the classical friction
$\textbf{F}_{0}$ is sent back by the solid’s electronic and phononic
fluctuations, so that the total friction vanishes. In practice, however, there
is always some amount of momentum relaxation that keeps the friction from
vanishing. The  net reduction in hydrodynamic friction is ultimately obtained
by balancing the relaxation processes (umklapp and impurity scattering), and
the processes that allow the solid to return momentum to the liquid: the
phonon-hydron and electron-hydron interactions, the latter corresponding to
quantum friction Kavokine  _et al._ (2022).

In order to go beyond a qualitative discussion, we evaluate the electron-
hydron and electron-phonon friction coefficients in the framework of ref.
Kavokine  _et al._ (2022). A general electron-boson friction coefficient is
given by

$\lambda_{\rm
b/e}=\frac{\hbar^{2}}{8\pi^{2}T}\int_{0}^{\infty}\frac{\textnormal{d}\omega\textnormal{d}q}{\textnormal{sinh}^{2}\left(\frac{\hbar\omega}{2T}\right)}\,q^{3}\,\frac{\textnormal{Im}[\chi_{\rm
e}^{\rm R}]\textnormal{Im}[D^{\rm R}]}{|1-\chi_{\rm e}^{\rm R}D^{\rm
R}|^{2}},$ (58)

where $D^{\rm R}$ is the retarded bosonic propagator in which the electron-
boson interaction has been absorbed (see Sec. III). We evaluate Eq. (58) by
numerical integration for both the graphene and 2DEG model systems, using zero
temperature RPA expressions for the electronic susceptibility Wunsch  _et al._
(2006); Mihaila . The results are plotted in Fig. 7b as a function of the
electronic density. The relative values of the friction coefficients are
consistent with the respective scattering times: indeed, one roughly expects
$\lambda_{\rm b/e}\propto\tau^{-1}_{\rm b/e}$ (see Appendix C3). In
particular, in the graphene case $\lambda_{\rm ph/e}\ll\lambda_{\rm um}\approx
2~{}\rm N\cdot s\cdot m^{-3}$ for all reasonable electronic densities, while
in the 2DEG model this is true only for low enough electronic densities.

In any case the term proportional to $\lambda_{\rm ph/e}$ in Eq. (57) is
negligible in the limit of low impurity scattering: $1/\tau_{\rm im}\to 0$. If
we further assume $\lambda_{\rm h/ph}\ll\frac{\tau}{\tau_{\rm
ph/e}}\lambda_{\rm h/e}$ following the argument of Sec. IV B, Eq. (57)
simplifies to

$\lambda=\lambda_{0}-\delta\lambda\quad\textnormal{with}\quad\delta\lambda=\frac{\lambda_{0}}{1+\frac{\tau_{\rm
ph/e}}{\tau}\frac{\lambda_{\rm um}}{\lambda_{\rm h/e}}}.$ (59)

The resulting estimate for $\delta\lambda$ is plotted in Fig. 7c. In doped
graphene, with an electronic density $10^{14}~{}\rm cm^{-2}$, we find that the
negative quantum contribution reduces the hydrodynamic friction by about 12%.
In a 2DEG with parabolic dispersion, the quantum contribution to friction is
stronger, so that at the same electronic density (and taking the graphene
value for $\lambda_{\rm um}$) the friction reduction is nearly 80%.

###  V.2 Discussion

According to Eq. (59), the negative contribution to friction represents a
_quantum feedback_ of the solid on the liquid flow: it originates from
electronic excitations returning momentum that the liquid has lost due to the
surface roughness. It is significant if the quantum friction coefficient
$\lambda_{\rm h/e}$ is non-negligible compared to the umklapp friction
coefficient. This is not a very stringent requirement: it is in fact satisfied
for our two model systems, at least in part of the considered electron density
range.

The negative quantum friction effect is thus predicted to be quantitatively
important. It complements the picture of quantum hydrodynamic friction
developed in ref. Kavokine  _et al._ (2022), where the solid was assumed to
remain in an equilibrium state. This was justified in particular for a truly
semi-infinite solid, where the surface can quickly transmit momentum to the
bulk. In a 2D material that is weakly coupled to the underlying substrate, we
find that momentum accumulation and the resulting Coulomb drag current are
non-negligible. As electrons receive momentum from the liquid through the
phonon-mediated interaction, they begin to move faster than the liquid, thus
helping it flow along the surface and reducing the total hydrodynamic friction
force. In this way, quantum friction can play a significant role even on non-
atomically-flat surfaces.

Negative quantum friction may provide a clue as to why the experimentally
measured water friction on graphene Xie  _et al._ (2018) appears to be much
lower than what is predicted by essentially all molecular simulations Kannam
_et al._ (2013); Tocci  _et al._ (2014, 2020); Thiemann  _et al._ . Even in ab
initio simulations at the Born-Oppenheimer level, it is not possible to
account for Coulomb drag or fluctuation-induced quantum friction.
Nevertheless, our quantitative estimates cannot account for the full extent of
the discrepancy, and the specific case of water on pristine graphene will be
the subject of further investigation.

Ultimately, negative quantum friction provides a previously unexplored pathway
for designing surfaces with low hydrodynamic friction. Friction reduction is
first achieved by minimizing momentum relaxation in the solid: the best case
scenario is a suspended 2D material, or a 2D material weakly coupled to its
substrate. Then, the water-solid quantum friction coefficient needs to be
maximized, so as to allow for efficient momentum transfer back to the liquid.
Conditions for high quantum friction have been detailed in ref. Kavokine  _et
al._ (2022); typically, quantum friction benefits from high electronic
densities and large effective masses. As outlined above, for the momentum
transfer to be efficient, the quantum friction coefficient needs to be large
compared to $\lambda_{\rm um}$, which is several orders of magnitude smaller
than a typical hydrodynamic friction coefficient: many materials may satisfy
this requirement. Potential candidates include magic angle twisted bilayer
graphene Stauber and Kohler (2016), and metallic transition metal
dichalcogenides such as $\rm VS_{2}$ and $\rm TaS_{2}$ da Jornada  _et al._
(2020).

##  VI Conclusion: the importance of Quantum Interfacial Effects

In this work, we introduce a novel perspective on hydrodynamics at solid-
liquid interfaces. The usual description of such interfaces relies on the
continuum Navier-Stokes equation, or – going down to the molecular scale – on
the interactions between fluid molecules and the surface corrugation. This
approach becomes insufficient to account for more advanced quantum couplings
that arise between the liquid and the solid.

Specifically, we start here by addressing the question of electronic current
generation by interfacial liquid flows. Using the non-equilibrium Keldysh
framework, we systematically investigate the mechanisms that couple fluid
motion to the electronic degrees of freedom within the solid material, which
include Coulomb and phonon-mediated interactions. The theory provides
quantitative estimates for the electro-fluidic conductivity – defined as the
electronic current response to the fluid motion –, which are fully
corroborated by experimental reports.

Going further, our theoretical framework reveals a quantum feedback mechanism,
that provides a negative contribution to hydrodynamic friction: in a very
counter-intuitive way, quantum effects may reduce hydrodynamic friction at the
solid-liquid interface. The Coulomb drag current can in fact be "faster" than
the liquid flow, and return momentum to the liquid through electron-hydron
scattering. The resulting _negative quantum friction_ provides a unique
opportunity to tune hydrodynamic friction by choosing specific electronic
properties of the confining wall. This broadens the scope of solid-liquid
quantum friction beyond the water-carbon interfaces discussed in ref. Kavokine
_et al._ (2022), as we find that it can play a role even for materials with
non-negligible surface roughness.

More generally, our results provide a new way of thinking about the
interaction of liquids with solids, and in particular water-solid interfaces.
By bridging fluid dynamics and condensed matter theory, we picture the
interface dynamics in terms of the collective excitations of both the liquid
and the solid, instead of real-space molecular interactions. Water charge
fluctuations – which we dub _hydrons_ – couple to the quantum excitations
inside the confining solid as first proposed in ref. Kavokine  _et al._
(2022). Here, this approach bears fruit by accounting for existing
experimental results, and predicting a novel quantum feedback effect. It opens
the way to quantum engineering of fluid transport: quantum effects can become
valuable assets for future water technologies.

The emerging interface between hydrodynamics, electrodynamics, condensed
matter physics and quantum mechanics, is an uncharted territory that begs for
further exploration.

## Acknowledgements

The authors acknowledge fruitful discussions with A. Marcotte, M. Lizee and A.
Siria. The Flatiron Institute is a division of the Simons Foundation. B.C.
acknowledges funding from a J.-P. Aguilar grant of the CFM Foundation, and
thanks Olivier Parcollet for hosting him at the Flatiron Institute.  N.K.
acknowledges support from a Humboldt fellowship. L.B. acknowledges funding
from the EU H2020 Framework Programme/ERC Advanced Grant agreement number
785911-Shadoks.

##  Appendix A Useful results in the Keldysh formalism

A reader interested in an extensive description of the formalism may consult
Rammer (2007); Kamenev (2011); Di Ventra (2008).

###  A.1 Fluctuation-dissipation theorem

We consider particles (bosons or fermions) with creation and annihilation
operators $\psi^{\dagger}(\textbf{r},t)$ and $\psi(\textbf{r},t)$,
respectively. We define the real-time Green’s functions

$\left\\{\begin{array}[]{l}G^{<}(\textbf{r},t,\textbf{r}^{\prime},t^{\prime})=\mp
i\langle\psi^{\dagger}(\textbf{r}^{\prime},t^{\prime})\psi(\textbf{r},t)\rangle\\\
G^{>}(\textbf{r},t,\textbf{r}^{\prime},t^{\prime})=-i\langle\psi(\textbf{r},t)\psi^{\dagger}(\textbf{r}^{\prime},t^{\prime})\rangle\end{array}\right.$
(60)

where the upper sign is for the bosons. At equilibrium, the mean value of an
operator $A$ is defined according to

$\langle
A\rangle=\frac{1}{\textnormal{Tr}\left(e^{-\frac{\mathcal{H}}{T}}\right)}\textnormal{Tr}\left(e^{-\frac{\mathcal{H}}{T}}A\right),$
(61)

where $\mathcal{H}$ is the total Hamiltonian.

Using the cyclicity of the the trace we deduce that at equilibrium

$G^{<}\left(\textbf{r},t+\frac{i\hbar}{T},\textbf{r}^{\prime},t^{\prime}\right)=\pm
G^{>}(\textbf{r},t,\textbf{r}^{\prime},t^{\prime}).$ (62)

Upon Fourier transformation,

$G^{<}(\textbf{q},\omega)=\pm e^{-\hbar\omega/T}G^{>}(\textbf{q},\omega).$
(63)

In terms of the R, A, K components used throughout the main text,

$\displaystyle\left\\{\begin{array}[]{l}G^{\rm R}-G^{\rm A}=G^{>}-G^{<}\\\
G^{\rm K}=G^{>}+G^{<}\end{array}\right.$ (66)

From Eqs. (63) and (66) we deduce the fluctuation-dissipation theorem:

$G^{\rm K}(\textbf{q},\omega)=2i\frac{1\pm e^{-\hbar\omega/T}}{1\mp
e^{-\hbar\omega/T}}\textnormal{Im}\left[G^{\rm R}(\textbf{q},\omega)\right].$
(67)

This proves Eqs. (11) and (14) of the main text.

###  A.2 Density and current

The particle density is given by
$n(\textbf{r},t)=\langle\psi^{\dagger}(\textbf{r},t)\psi(\textbf{r},t)\rangle$,
that is, $n(\textbf{r},t)=\pm iG^{<}(\textbf{r},t,\textbf{r},t)$. Using Eq.
(66), the density can be expressed as

$n=\pm\frac{i}{2}\left(G^{\rm K}-G^{\rm R}+G^{\rm A}\right)$ (68)

which yields Eq. (38). Let us notice that in real space and at equal times
$G^{\rm R}(\textbf{r},t,\textbf{r}^{\prime},t)-G^{\rm
A}(\textbf{r},t,\textbf{r}^{\prime},t)$ reduces to a constant.

Let us now derive an expression for the average electric current, forgetting
for simplicity the spin degree of freedom. In the Heisenberg picture, the one-
particle Hamiltonian is of the form

$\mathcal{H}(t)=\int\textnormal{d}\textbf{r}\,\psi^{\dagger}(\textbf{r},t)H_{\textbf{r}}\psi(\textbf{r},t)$
(69)

where $H_{\textbf{r}}$ is a differential operator acting in real space. Going
to momentum space, this operator becomes a function of the quasi-momentum
$\mathbf{q}$ of the electron. Within a given band,

$\mathcal{H}(t)=\int\textnormal{d}\textbf{q}\,\psi^{\dagger}_{\textbf{q}}(t)\hbar\xi_{\textbf{q}}\psi_{\textbf{q}}(t).$
(70)

The Hamiltonian determines the dynamics of the density operator:

$\displaystyle\partial_{t}n_{\rm e}(\textbf{r},t)$ $\displaystyle=$
$\displaystyle\frac{1}{i\hbar}[\psi^{\dagger}(\textbf{r},t)\psi(\textbf{r},t),\mathcal{H}(t)]$
(71) $\displaystyle=$
$\displaystyle\frac{1}{2\hbar}\left(H_{\textbf{r}}-H_{\textbf{r}^{\prime}}\right)G^{\rm
K}(\textbf{r},t,\textbf{r}^{\prime},t)|_{\textbf{r}=\textbf{r}^{\prime}}.$
(72)

On the other hand the electronic density satisfies the conservation equation

$e\partial_{t}n_{\rm
e}(\textbf{r},t)+\nabla\cdot\langle\textbf{j}(\textbf{r},t)\rangle=0.$ (73)

from which we deduce

$\displaystyle\langle\textbf{j}(\textbf{r},t)\rangle=\frac{e}{2\hbar}\left([H_{\textbf{r}},\textbf{r}]-[H_{\textbf{r}^{\prime}},\textbf{r}^{\prime}]\right)G^{\rm
K}(\textbf{r},t,\textbf{r}^{\prime},t)|_{\textbf{r}=\textbf{r}^{\prime}}.~{}~{}~{}~{}~{}~{}~{}~{}$
(74)

Going to Fourier space and assuming translational invariance in time and
space, we obtain

$\displaystyle\langle\textbf{j}\rangle=ie\int\frac{\textnormal{d}\textbf{q}\textnormal{d}\omega}{(2\pi)^{3}}\,\left(\nabla_{\mathbf{q}}\xi_{\mathbf{q}}\right)\,G^{\rm
K}(\textbf{q},\omega).$ (75)

Multiplying by 2 to account for the spin degeneracy we recover Eq. (16) of the
main text. An additional factor can be included to account for a valley
degeneracy.

##  Appendix B Evaluation of the self-energy

###  B.1 General expression

In this appendix we evaluate the electron-boson self-energy diagram in Fig.
2b. We compute the self-energy in a given electronic band with dispersion
$\xi_{\mathbf{q}}$ and eigenstates $|\mathbf{q}\rangle$, and we neglect
interband electron scattering: this is reasonable as long as the boson energy
is small compared to the Fermi level. Applying the Keldysh formalism Feynman
rules in real space, we find Rammer (2007):

$\left\\{\begin{array}[]{l}\Sigma^{\rm R,A}=\dfrac{i}{2}\left(D^{\rm
R,A}G^{\rm K}_{0}+D^{\rm K}G^{\rm R,A}_{0}\right)\\\ \\\ \Sigma^{\rm
K}=\dfrac{i}{2}\left(G^{\rm R}_{0}-G^{\rm A}_{0}\right)\left(D^{\rm R}-D^{\rm
A}\right)+\dfrac{i}{2}D^{\rm K}G^{\rm K}_{0}\end{array}\right.$ (76)

Starting with the retarded component, the products become convolutions in
Fourier space:

$\Sigma^{\rm
R}(\mathbf{q},\omega)=\frac{i}{2}\int\frac{\mathrm{d}\mathbf{q}^{\prime}\mathrm{d}\omega^{\prime}}{(2\pi)^{3}}\mathcal{M}(\mathbf{q}-\mathbf{q}^{\prime},\mathbf{q})\left[D^{\rm
R}(\mathbf{q}^{\prime},\omega^{\prime}-\mathbf{q}^{\prime}\\!\cdot\\!\mathbf{v}_{\rm
b})G_{0}^{\rm K}(\mathbf{q}-\mathbf{q}^{\prime},\omega-\omega^{\prime})+D^{\rm
K}(\mathbf{q}^{\prime},\omega^{\prime}-\mathbf{q}^{\prime}\\!\cdot\\!\mathbf{v}_{\rm
b})G_{0}^{\rm
R}(\mathbf{q}-\mathbf{q}^{\prime},\omega-\omega^{\prime})\right],$ (77)

where
$\mathcal{M}(\mathbf{q}-\mathbf{q}^{\prime},\mathbf{q})\equiv|\langle\mathbf{q}-\mathbf{q}^{\prime}|e^{-i\mathbf{q}^{\prime}\mathbf{r}}|\mathbf{q}\rangle|^{2}$.
Using Eqs. (10) and (11) for the equilibrium Green’s functions, as well as the
bosonic fluctuation-dissipation theorem in Eq. (14), we obtain

$\Sigma^{\rm
R}(\mathbf{q},\omega)=\frac{i}{2}\int\frac{\mathrm{d}\mathbf{q}^{\prime}}{(2\pi)^{3}}\mathcal{M}(\mathbf{q}-\mathbf{q}^{\prime},\mathbf{q})\left[\int\mathrm{d}\omega^{\prime}2if(\omega^{\prime}-\mathbf{q}^{\prime}\\!\cdot\\!\mathbf{v}_{\rm
b})\frac{\mathrm{Im}\,[D^{\rm
R}(\mathbf{q}^{\prime},\omega^{\prime}-\mathbf{q}^{\prime}\\!\cdot\\!\mathbf{v}_{\rm
b})]}{\omega-\omega^{\prime}-\xi_{\mathbf{q}-\mathbf{q}^{\prime}}+i0^{+}}-\frac{2i\pi}{f(\xi_{\mathbf{q}-\mathbf{q}^{\prime}})}D^{\rm
R}(\mathbf{q}^{\prime},\omega-\xi_{\mathbf{q}-\mathbf{q}^{\prime}}-\mathbf{q}^{\prime}\\!\cdot\\!\mathbf{v}_{\rm
b})\right].$ (78)

Introducing the Lehmann representation for the bosonic propagator,

$D^{\rm
R}(\mathbf{q},\omega)=-\frac{1}{\pi}\int_{-\infty}^{+\infty}\mathrm{d}\omega^{\prime}\frac{\mathrm{Im}\,[D^{\rm
R}(\mathbf{q},\omega^{\prime})]}{\omega-\omega^{\prime}+i0^{+}},$ (79)

Eq. (78) becomes

$\Sigma^{\rm
R}(\mathbf{q},\omega)=-\frac{1}{\hbar}\int\frac{\mathrm{d}\mathbf{q}^{\prime}\mathrm{d}\omega^{\prime}}{(2\pi)^{3}}\mathcal{M}(\mathbf{q}-\mathbf{q}^{\prime},\mathbf{q})\frac{\mathrm{Im}\,[D^{\rm
R}(\mathbf{q}^{\prime},\omega^{\prime}-\mathbf{q}^{\prime}\\!\cdot\\!\mathbf{v}_{\rm
b})]}{\omega-\omega^{\prime}-\xi_{\mathbf{q}-\mathbf{q}^{\prime}}+i0^{+}}\left(f(\omega^{\prime}-\mathbf{q}^{\prime}\\!\cdot\\!\mathbf{v}_{\rm
b})+\frac{1}{f(\xi_{\mathbf{q}-\mathbf{q}^{\prime}})}\right)$ (80)

which is Eq. (24) of the main text. Following these exact same steps, we may
check that $\Sigma^{\rm A}(\mathbf{q},\omega)=\Sigma^{\rm
R}(\mathbf{q},\omega)^{*}$. Making use again of fluctuation-dissipation
relations, we find for the Keldysh component of the self-energy

$\Sigma^{\rm
K}(\mathbf{q},\omega)=-2i\int\frac{\mathrm{d}\mathbf{q}^{\prime}\mathrm{d}\omega^{\prime}}{(2\pi)^{3}}\mathcal{M}(\mathbf{q}-\mathbf{q}^{\prime},\mathbf{q})\left(1+\frac{f(\omega^{\prime}-\mathbf{q}^{\prime}\\!\cdot\\!\mathbf{v}_{\rm
b})}{f(\omega-\omega^{\prime})}\right)\mathrm{Im}\,[D^{\rm
R}(\mathbf{q}^{\prime},\omega^{\prime}-\mathbf{q}^{\prime}\mathbf{v})]\mathrm{Im}\,[G^{\rm
R}(\mathbf{q}-\mathbf{q}^{\prime},\omega-\omega^{\prime})].$ (81)

Using that $\mathrm{Im}\,[G^{\rm
R}(\mathbf{q},\omega)]=-\pi\delta(\omega-\xi_{\mathbf{q}})$, we obtain

$\Sigma^{\rm
K}(\mathbf{q},\omega)=i\int\frac{\mathrm{d}\mathbf{q}^{\prime}}{(2\pi)^{2}}\mathcal{M}(\mathbf{q}-\mathbf{q}^{\prime},\mathbf{q})\left(1+\frac{f(\omega-\xi_{\mathbf{q}-\mathbf{q}^{\prime}}-\mathbf{q}^{\prime}\\!\cdot\\!\mathbf{v}_{\rm
b})}{f(\xi_{\mathbf{q}-\mathbf{q}^{\prime}})}\right)\mathrm{Im}\,[D^{\rm
R}(\mathbf{q}^{\prime},\omega-\xi_{\mathbf{q}-\mathbf{q}^{\prime}}-\mathbf{q}^{\prime}\\!\cdot\\!\mathbf{v}_{\rm
b})],$ (82)

which is Eq. (25) of the main text.

###  B.2 Bosonic propagators

The expressions for the boson (hydron and phonon) propagators that are
relevant for our model systems are given in the main text (Eqs. (28) - (29)).
Here, we provide a few additional details, in particular concerning the
derivation of the hydron propagator. Starting from the electron-hydron
Hamiltonian in Eq. (1), the bare hydron propagator is defined as

$D^{\rm R}_{\rm w,0}(\mathbf{q},t)=-i\theta(t)(V^{\rm
C}_{\mathbf{q}})^{2}\langle[n_{\rm s}(\mathbf{q},t),n_{\rm
s}(-\mathbf{q},0)]\rangle,$ (83)

with

$n_{\rm
s}(\mathbf{q},t)=\int\mathrm{d}\bm{\rho}\int_{0}^{+\infty}\mathrm{d}ze^{-i\mathbf{q}\bm{\rho}}e^{-qz}n_{\rm\ell}(\mathbf{r},t).$
(84)

We have isolated here the component of the position $\mathbf{r}$ that is
perpendicular to the interface: $\mathbf{r}=\bm{\rho}+z\mathbf{e}_{z}$.
Identifying the water density-density response function $\chi^{\rm R}_{\rm
w}(\mathbf{q},z,z^{\prime},\omega)$, we obtain

$\begin{split}D^{\rm R}_{\rm w,0}(\mathbf{q},\omega)&=(V_{\mathbf{q}}^{\rm
C})^{2}\int_{0}^{+\infty}\mathrm{d}z\mathrm{d}z^{\prime}e^{q(z+z^{\prime})}\chi^{\rm
R}_{\rm w}(\mathbf{q},z,z^{\prime},\omega)\\\ &\equiv-V_{\mathbf{q}}^{\rm
C}g^{\rm R}_{\rm w}(\mathbf{q},\omega),\end{split}$ (85)

recovering the water surface response function $g^{\rm R}_{\rm
w}(\mathbf{q},\omega)$ that was studied extensively in ref. Kavokine  _et al._
(2022). It was found to be well described by the sum of two Debye peaks:

$g^{\rm R}_{\rm
w}(\mathbf{q},\omega)=\sum_{k=1,2}\frac{f_{k}}{1-i\omega/\omega_{\mathrm{D},k}},$
(86)

with $\omega_{\rm D,1}\sim 1~{}\rm meV$ and $\omega_{\rm D,2}\sim 100~{}\rm
meV$, and $f_{1}\approx f_{2}\sim 0.5$. Accounting for the RPA screening as
per the diagrams in Fig. 2c-d, the full hydron propagator becomes

$D_{\rm w}^{\rm R}(\mathbf{q},\omega)=\frac{D^{\rm R}_{\rm
w,0}(\mathbf{q},\omega)(1+V_{\mathbf{q}}^{\rm C}\chi^{\rm R}_{\rm
e}(\mathbf{q},\omega))^{2}}{1-D^{\rm R}_{\rm w,0}(\mathbf{q},\omega)\chi^{\rm
R}_{\rm e}(\mathbf{q},\omega)},$ (87)

where $\chi_{\rm e}^{\rm R}$ is the electronic density-density response
function. Using the definition of the dielectric function
$\epsilon(\mathbf{q},\omega)=1/(1-V_{\mathbf{q}}^{\rm C}\chi_{\rm e}^{\rm
R}(\mathbf{q},\omega))$, and neglecting its frequency dependence on the scale
of the Debye frequencies, we recover Eq. (28) of the main text.

For acoustic phonons in the framework of a jellium model, the fully screened
propagator is Bruus and Flensberg (2004):

$D_{\rm ph}^{\rm R}(\mathbf{q},\omega)=\frac{1}{\hbar}\frac{V_{q}^{\rm
C}}{\epsilon(q)}\frac{\omega_{\mathbf{q}}^{2}}{(\omega+i0^{+})^{2}-\omega_{\mathbf{q}}^{2}},$
(88)

with $\omega_{\mathbf{q}}=cq$, $c$ being the sound velocity. The electron-
phonon interaction is essentially a screened Coulomb interaction. Conversely,
in graphene, the electron-phonon interaction has a peculiar form Ochoa  _et
al._ (2011), so that

$D_{\rm ph}^{\rm R}(\mathbf{q},\omega)=\frac{1}{\hbar}V^{\rm
ph/e}\frac{\omega_{\mathbf{q}}^{2}}{(\omega+i0^{+})^{2}-\omega_{\mathbf{q}}^{2}},$
(89)

with $V^{\rm ph/e}=g^{2}/(2\rho c^{2})$; here, $g\approx 3$ eV, is the
electron-phonon coupling and $\rho\approx 7.6\cdot 10^{-7}$ kg/m2 is the mass
per unit area.

In the 2DEG, the matrix elements $\mathcal{M}$ are unity. We use the Thomas-
Fermi approximation for the dielectric function: $\epsilon(q)=1+q_{\rm TF}/q.$
Since the density of states is independent of energy, the Thomas-Fermi
wavevector $q_{\rm TF}$ does not depend on the Fermi level: $q_{\rm
TF}=2/a_{0}$, with $a_{0}=4\pi\epsilon_{0}\hbar^{2}/(me^{2})$ the Bohr radius
at the effective mass. In graphene,
$\mathcal{M}(\mathbf{q}-\mathbf{q}^{\prime},\mathbf{q})=\frac{1}{2}(1+\cos\theta_{\mathbf{q}-\mathbf{q}^{\prime},\mathbf{q}})$,
where $\theta_{\mathbf{q}-\mathbf{q}^{\prime},\mathbf{q}}$ is the angle
between $\mathbf{q}-\mathbf{q}^{\prime}$ and $\mathbf{q}$ Hwang and Das Sarma
(2007). For the RPA dielectric function, a full analytical expression can be
found in the literature Hwang and Das Sarma (2007); Wunsch  _et al._ (2006).

###  B.3 Impurity approximation

The self-energies can be simplified within the impurity approximation, which
amounts to taking the limit of vanishing bosonic frequency. This is expected
to be reasonable as long as the bosonic frequencies are much lower than
$T/\hbar$. For the hydron propagator, as $\omega_{\rm D,1,2}\to 0$,

$f(\omega)\mathrm{Im}\,[D^{\rm
R}_{w}(\mathbf{q},\omega)]\to-2\pi\frac{T}{\hbar^{2}}\frac{V_{\mathbf{q}}^{C}}{\epsilon(\mathbf{q})}\delta(\omega).$
(90)

For the acoustic phonons, as $\omega_{\mathbf{q}}\to 0$,

$f(\omega)\mathrm{Im}\,[D^{\rm R}_{\rm
ph}(\mathbf{q},\omega)]\to-2\pi\frac{T}{\hbar^{2}}\frac{V_{\mathbf{q}}^{\rm
ph/e}}{\epsilon(\mathbf{q})}\delta(\omega).$ (91)

Thus, within the impurity approximation, the hydron and phonon propagators
become formally identical.

As the bosonic frequencies are taken to 0, the terms that are not proportional
to $f(\omega)$ in Eq. (80) become negligible. We then obtain

$\Sigma^{\rm
R}(\mathbf{q},\omega)=\frac{T}{\hbar^{2}}\int\frac{\mathrm{d}\mathbf{q}^{\prime}}{(2\pi)^{2}}\frac{\mathcal{M}(\mathbf{q}^{\prime},\mathbf{q})V_{\mathbf{q}-\mathbf{q}^{\prime}}}{\omega-(\mathbf{q}-\mathbf{q}^{\prime})\\!\cdot\\!\mathbf{v}_{\rm
b}-\xi_{\mathbf{q}^{\prime}}+i0^{+}},$ (92)

with $V=V^{\rm C}_{\mathbf{q}}/\epsilon(q)$ or $V^{\rm ph/e}$. In the usual
treatment of impurity scattering Bruus and Flensberg (2004), one further
neglects the real part of the self-energy, and the frequency dependence of the
imaginary part: $\mathrm{Im}\,[\Sigma^{\rm
R}(\mathbf{q},\omega)]\approx-1/\tau_{\mathbf{q}}\equiv\mathrm{Im}\,[\Sigma^{\rm
R}(\mathbf{q},\xi_{\mathbf{q}})]$. The quasiparticle scattering rate at
wavevector $\mathbf{q}$ is given by

$\frac{1}{\tau_{\mathbf{q}}}=\pi\frac{T}{\hbar^{2}}\int\frac{\mathrm{d}\mathbf{q}^{\prime}}{(2\pi)^{2}}\mathcal{M}(\mathbf{q},\mathbf{q}^{\prime})V_{\mathbf{q}-\mathbf{q}^{\prime}}\delta(\xi_{\mathbf{q}}-\xi_{\mathbf{q}^{\prime}})$
(93)

which is Eq. (31) of the main text. An explicit estimate can be obtained under
the assumption $V_{\mathbf{q}-\mathbf{q}^{\prime}}\approx V_{\mathbf{q}}$ (and
$\mathcal{M}(\mathbf{q},\mathbf{q}^{\prime})=1$). Then, by changing variables
$\textnormal{d}\mathbf{q}^{\prime}=(2\pi)^{2}N(u)\textnormal{d}u$, where
$N(u)$ is the density of states at energy $u$, we deduce

$\frac{1}{\tau_{\mathbf{q}}}\approx\pi\frac{T}{\hbar}V_{\mathbf{q}}N(u_{\mathbf{q}})$
(94)

which is Eq. (32) of the main text.

###  B.4 Fermi’s golden rule

Eq. (94) can be obtained by writing down a simplified Fermi golden rule for an
electron-phonon interaction of the form

$\mathcal{H}_{\rm
ph/e}(t)=g\mathcal{A}^{2}\int\frac{\textnormal{d}\textbf{q}\textnormal{d}\textbf{k}}{(2\pi)^{4}}\,\psi^{\dagger}_{\mathbf{q}}(t)b^{\dagger}_{\textbf{k}}(t)\psi_{\mathbf{q}+\textbf{k}}(t)+\textnormal{h.c.}$
(95)

where $g$ is the coupling constant and $b^{\dagger}$ the creation operator of
the phonon (h.c. stands for hermitian conjugate). Fermi’s golden rule
predicts:

$1/\tau_{\mathbf{q}}\approx\frac{2\pi}{\hbar}g^{2}\mathcal{A}N(u_{\mathbf{q}})n_{\rm
B}(\omega_{\mathbf{q}}).$ (96)

Here, the Bose distribution $n_{\rm B}$ counts the number of modes on which
the electrons can scatter. Assuming $\hbar\omega_{\mathbf{q}}\ll T$,

$1/\tau_{\mathbf{q}}\approx\frac{2\pi}{\hbar}g^{2}\mathcal{A}N(u_{\mathbf{q}})T/\hbar\omega_{\mathbf{q}}.$
(97)

With our usual notations the coupling constant $g$ is absorbed inside the
effective potential $V_{\mathbf{q}}$, according to
$V_{\mathbf{q}}=g^{2}\mathcal{A}/\epsilon(q)\omega_{\mathbf{q}}$. Therefore,
we recover the estimate provided in Eqs. (32) and (94) from the Fermi’s golden
rule.

For the electron-hydron interaction the computation is more involved since we
would in principle have to consider the superposition of many hydron modes.
Qualitatively, reproducing the above reasoning for each of the modes yields
exactly the same result.

###  B.5 Higher order diagrams: extension of the Migdal theorem

Figure 8: Diagrammatic expansions. (a) First and second order diagrams for the
electron-boson self-energy. (b) Diagrammatic representation of the electronic
susceptibility in terms of the renormalized Green’s functions (thick lines).

Up to now, we have only considered the first order diagram for the self-
energy. Here, we provide an estimate for higher order diagrams and show that
they can reasonably be neglected. The $n$-th order self-energy diagram contain
$n$ bosonic propagators and $2n-1$ electronic propagators. Such a diagram
includes $n$ loops, therefore there are $n$ momentum integrals and $n$
frequency integrals. The first few self-energy diagrams are displayed in Fig.
8a.

In the impurity approximation, the bosonic propagators take the form

$\displaystyle\left\\{\begin{array}[]{l}\textnormal{Im}\left[D^{\rm
R}(\mathbf{q},\omega)\right]\approx-\pi\dfrac{\omega}{\hbar}V_{\mathbf{q}}\delta(\omega)=0\\\
\\\ D^{\rm
K}(\mathbf{q},\omega)\approx-4i\pi\dfrac{T}{\hbar^{2}}V_{\mathbf{q}}\delta(\omega)\end{array}\right.$
(101)

Therefore, in the expression of the self energy the only non-vanishing terms
are those that only include the Keldysh component of the bosonic propagator.
In the Keldysh diagrammatic rules each vertex must be summed over the dynamic
indices. However, when restricting to the Keldysh propagator for the bosons,
the dynamic index of the electronic Green function is preserved at each vertex
Rammer (2007). We shall focus on the retarded component of the self energy.
When imposing these external dynamic indices, the electronic propagators can
only be retarded or advanced Green functions thanks to the trigonal structure.
To obtain an estimate of the self energy we will neglect the real part of the
retarded/advanced electronic Green function, which then reduces to its
imaginary part:

$G^{\rm R,A}(\mathbf{q},\omega)\approx\mp
i\pi\delta(\omega-\xi_{\mathbf{q}}).$ (102)

Putting everything together, we have $n$ integrations over internal 2D
momenta, $n$ integrations over internal frequencies, $n$ bosonic Keldysh Green
functions, each of them introducing a Dirac distribution that fixes a
frequency, and $2n-1$ electronic propagators, each of them introducing a Dirac
distribution that fixes a momentum, and a factor
$1/(\nabla_{q}\xi_{\mathbf{q}})$. After integration of all the Dirac
distributions there only remains an angular integration. The structure of the
diagram determines the numerical factor coming from the angular integration,
and the momentum at which the $n$ interactions $V_{\mathbf{q}}$ and the $2n-1$
velocities $\nabla_{q}\xi_{\mathbf{q}}$ are evaluated. To provide an order of
magnitude, we evaluate them at the Fermi wavevector and take the angular
factor equal to 1. We then obtain the following order of magnitude for the
$n$-th order self-energy diagram:

$\Sigma^{(n)}\sim\left(\frac{TV_{k_{\rm F}}}{\hbar^{2}}\right)^{n}\frac{k_{\rm
F}}{v_{\rm F}^{2n-1}}.$ (103)

For $n=1$ we recover the estimate for the first order self energy calculated
above (Eq. (94)). Comparing the $n$-th and first order self-energies,

$\frac{\Sigma^{(n)}}{\Sigma^{(1)}}\sim\left(\frac{TV_{k_{\rm
F}}}{\hbar^{2}v_{\rm F}^{2}}\right)^{n-1}\equiv\alpha^{n-1}.$ (104)

Let us estimate the coefficient $\alpha$ for the different models.

For the 2DEG, $V_{k_{\rm F}}$ is the screened the Coulomb potential. For
reasonable electronic densities ($k_{\rm F}\ll 1/a_{0}$), the dielectric
constant is $\epsilon(k_{\rm F})\approx\frac{2}{a_{0}k_{\rm F}}$ where $a_{0}$
is the Bohr radius computed at the effective mass. Thus, $V_{k_{\rm
F}}\approx\hbar^{2}/m$ and $\alpha\sim T/\mu$. Therefore, neglecting higher
order diagrams is valid as long as the chemical potential is large compared to
the temperature.

For graphene, $\epsilon(k_{\rm F})\approx\frac{e^{2}}{\epsilon_{0}\hbar v_{\rm
F}}\sim 1$. For the electron-hydron interaction, we obtain again $\alpha\sim
T/\mu$. For the electron-phonon interaction, we find $\alpha\sim\frac{TV^{\rm
eff}}{\hbar v_{\rm F}^{2}}\sim 10^{-4}$: higher order self-energy diagrams can
always be neglected.

##  Appendix C Fluctuation-induced friction forces

###  C.1 Susceptibility renormalisation

In this section, we compute the non-equilibrium electronic susceptibility
(density-density response function) $\bm{\chi}_{\rm e}$ starting from the
electronic Green, according to the diagrammatic definition given in Fig. 8b.
We neglect in particular vertex corrections due to the electron-boson
interaction. This diagram is formally equivalent to the diagram for the first
order self energy Kamenev (2011). Thus, the calculation is similar, if one
replaces the bosonic propagator D by the _backward_ electronic Green’s
function $G_{\rm
B}^{ij}(\textbf{r}_{1},t_{1},\textbf{r}_{2},t_{2})=G^{ji}(\textbf{r}_{2},t_{2},\textbf{r}_{1},t_{1})$,
where $i,j$ are the Keldysh dynamical indices. For convenience we denote in
the following $\check{F}(x)=F(-x)$. Using that in real space $\check{G}^{\rm
R,A}=(G^{\rm A,R})^{*}$ and $\check{G}^{K}=-(G^{K})^{*}$ by construction, the
backward propagators become in the Keldysh trigonal representation and in
Fourier space:

$\displaystyle\left\\{\begin{array}[]{l}G_{\rm B}^{\rm
R,A}(\mathbf{q},\omega)=(G^{\rm A,R})^{*}(-\mathbf{q},-\omega)\\\ G_{\rm
B}^{K}(\mathbf{q},\omega)=-(G^{K})^{*}(-\mathbf{q},-\omega)\end{array}\right.$
(107)

Let us notice that the backward electronic Green function satisfies the same
fluctuation-dissipation theorem as the forward electronic Green’s function G.
Using Eq. (76) and the fluctuation-dissipation theorem in Eq. (37), we obtain

$\left\\{\begin{array}[]{l}\chi^{R}_{\rm e}=-\left(G_{\rm
B}^{R}\ast\frac{\textnormal{Im}[G^{\rm R}]}{f}+\frac{\textnormal{Im}[G^{\rm
R}_{\rm B}]}{f}\ast G^{R}\right)\\\ \\\ \chi^{\rm K}_{\rm
e}=-2i\left(\textnormal{Im}[G^{\rm R}_{\rm B}]\ast\textnormal{Im}[G^{\rm
R}]+\frac{\textnormal{Im}[G^{\rm R}_{\rm
B}]}{f}\ast\frac{\textnormal{Im}[G^{\rm R}]}{f}\right)\end{array}\right.$
(108)

where at frequency $\omega$ and momentum $\mathbf{q}$, $f$ stands for
$f(\omega-\mathbf{q}\cdot\mathbf{v}_{\rm e}(q))$. As long as
$\mathbf{q}\cdot\mathbf{v}_{\rm e}(q)\ll\omega$, we may wompare these formulas
point by point, and using the trigonometric identities
$f^{-1}_{\omega^{\prime}}+f^{-1}_{\omega-\omega^{\prime}}=f^{-1}_{\omega}\frac{1-f^{-1}_{\omega^{\prime}}}{1-f^{-1}_{\omega}f^{-1}_{\omega^{\prime}}}$
and
$1+f^{-1}_{\omega^{\prime}}f^{-1}_{\omega-\omega^{\prime}}=\frac{1-f^{-1}_{\omega^{\prime}}}{1-f^{-1}_{\omega}f^{-1}_{\omega^{\prime}}}$,
we deduce a the quasi-equilibrium fluctuation-dissipation theorem for the
susceptibility:

$\chi_{\rm e}^{\rm
K}(\textbf{q},\omega)=2if(\omega-\textbf{q}\\!\cdot\\!\textbf{v}_{\rm
e}(q))\textnormal{Im}[\chi_{\rm e}^{\rm R}(\textbf{q},\omega)]$ (109)

which is Eq. (40) of the main text.

###  C.2 Electron-boson friction force

We now generalize the result of ref. Kavokine  _et al._ (2022) for
fluctuation-induced quantum friction to account for the non-equilibrium state
of the solid. The electron-boson force per unit surface can be expressed as
Kavokine  _et al._ (2022):

$\frac{\langle\textbf{F}_{\rm
b/e}\rangle}{\mathcal{A}}=\frac{\hbar}{4\pi}\int\frac{\textnormal{d}\omega\textnormal{d}\textbf{q}}{(2\pi)^{3}}\,\textbf{q}\,\chi_{\rm
b/e}^{\rm K}(\textbf{q},\omega).$ (110)

where the Keldysh cross correlation $\chi_{\rm b/e}^{\rm K}$ is given by

$\chi^{\rm K}_{\rm eb}=-\frac{\chi_{\rm e}^{\rm A}D^{\rm K}+\chi_{\rm e}^{\rm
K}D^{\rm R}}{|1-\chi_{\rm e}^{\rm R}D^{\rm R}|^{2}},$ (111)

with all the correlation function being computed in the non-equilibrium state.
We now use the _quasi-equilibrium_ fluctuation-dissipation theorems in Eqs.
(14) and Eq. (109), to obtain

$\chi^{\rm K}_{\rm eb}=-2i\Delta f\frac{\textnormal{Im}[\chi_{\rm e}^{\rm
R}]\textnormal{Im}[D^{\rm R}]}{|1-\chi_{\rm e}^{\rm R}D^{\rm R}|^{2}}.$ (112)

where $\Delta f=f(\omega-\textbf{q}\cdot\mathbf{v}_{\rm
e})-f(\omega-\textbf{q}\cdot\textbf{v}_{\rm b})$. Therefore, we find that as
long as the quasi-equilibrium condition holds, the electron-boson friction
coefficient is computed as if the electrons were at equilibrium:

$\lambda_{\rm
b/e}=\frac{\hbar^{2}}{8\pi^{2}T}\int_{0}^{\infty}\frac{\textnormal{d}\omega\textnormal{d}q}{\textnormal{sinh}^{2}\left(\frac{\hbar\omega}{2T}\right)}\,q^{3}\,\frac{\textnormal{Im}[\chi_{\rm
e}^{\rm R}]\textnormal{Im}[D^{\rm R}]}{|1-\chi_{\rm e}^{\rm R}D^{\rm R}|^{2}}$
(113)

which is Eq. (58) of the main text. However, the friction force is now
proportional to the differential velocity: $\langle\textbf{F}_{\rm
b/e}\rangle/\mathcal{A}=\lambda_{\rm b/e}(\mathbf{v}_{\rm b}-\textbf{v}_{\rm
e})$.

###  C.3 Scaling of the electron-boson friction coefficient

In this section, we provide a qualitative approach to electron-boson friction,
that is able to predict the scaling of the friction coefficient with
electronic density and effective mass. In the reference frame where the bosons
do not move, the electrons are subject to a wind velocity $\mathbf{v}_{\rm
e}-\mathbf{v}_{\rm b}$. They relax by scattering on the bosons at a rate
$\tau_{\rm b/e}^{-1}$. The force (or momentum flux) per unit surface is then
given by

$-\frac{\textbf{F}_{\rm b/e}}{\mathcal{A}}\sim\frac{\hbar k_{\rm F}\times
N(\mu)\hbar k_{\rm F}(\mathbf{v}_{\rm e}-\mathbf{v}_{\rm b})}{\tau_{\rm
b/e}}.$ (114)

This is the momentum of an electron (at the Fermi level), times the number of
electrons that are able to scatter (in a zero temperature picture), times the
scattering rate. The scaling of the friction force with the inverse of the
electron-boson scattering time is consistent with the relation between
resistivity and electronic friction coefficient proposed by Persson Persson
(1991, 2000).

Therefore, using Eq. (94), we find that the electron-boson friction
coefficient scales as

$\lambda_{\rm b/e}\sim\pi\hbar k_{\rm F}^{2}N(\mu)^{2}TV_{k_{\rm F}}.$ (115)

Let us notice that in this approximation the electron-boson friction
coefficient does not depend on the dynamics of the bosons but only on the
electronic structure and the interaction potential: this is in fact the
analogue of the impurity approximation for the friction coefficient. For a
2DEG with reasonable electronic density, the screened Coulomb potential is
roughly independent of the Fermi level and the effective mass. Since the
density of states $N(\mu)=m/\hbar^{2}$ is constant and $k_{\rm
F}\sim\sqrt{m\mu}$ we expect $\lambda_{\rm b/e}\propto m^{2}\mu\propto mn$
where $n$ is the charge carriers density, for both the electron-phonon and
electron-hydron interactions. In graphene, the screened Coulomb potential
scales as $1/k_{\rm F}\propto 1/\mu$, the density of states
$N(\mu)=2\mu/(\pi\hbar^{2}v_{\rm F}^{2})$ scales as $\mu$ and $k_{\rm
F}=\mu/v_{\rm F}$. Therefore, we expect $\lambda_{\rm
b/e}\propto\mu^{3}\propto n^{3/2}$ for the electron-hydron interaction. On the
other hand, using that the effective potential $V_{\mathbf{q}}$ does not
depend on $q$ for the electron-phonon interaction in graphene, we expect
$\lambda_{\rm b/e}\propto\mu^{4}\propto n^{2}$. We thus recover the scalings
of the full numerical results displayed in Fig. 7b.

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