serving as a toy model of 4D gravity, it also reveals its own subtleties, the
most striking of which is a field-dependent cocycle in the charge algebra.
Whether such a field-dependent cocycle also occurs in 4D Einstein–Maxwell
theory, say when using the Koszul bracket, is an open question which we keep
for future work.

Having a complete control over the 3D setup, several interesting prospects are
now in sight:

  * •

3D flat holography with radiation. A natural follow-up of our construction of
the Einstein–Maxwell solution space and its symmetries is to seek a candidate
holographic description. The field-dependent cocycle (5.15) is expected to
play a key role in that context, since it affects the structure of the
symmetry group, which in turn affects coadjoint orbits and unitary
representations. In particular, it will be interesting to investigate how
geometric actions for BMS3 [70, 73, 69, 76, 78, 77] are modified by the
coupling to electrodynamics. An alternative approach would be a 3D version of
celestial holography, built by recasting the scattering amplitudes of 3D
Maxwell theory in terms of conformal correlators on the celestial circle.
Finally, it is natural to wonder how results on flat-space entanglement [129,
130] generalize in the presence of matter fields.

  * •

3D Maxwell/scalar duality. Since the 3D Maxwell field is dual to a scalar, a
natural question is how this duality is implemented at the level of asymptotic
charges. Indeed, such a puzzle also appears for 4D scalars whose soft theorem
[131, 132] a priori has no asymptotic symmetry counterpart (scalars have no
gauge symmetries). The puzzle is resolved by noting that a 4D scalar is dual
to a 2-form gauge theory with non-trivial asymptotic symmetries [133, 134,
135]. In 3D Einstein–Maxwell theory, one faces the opposite situation, where
the asymptotic structure of the Maxwell field is known but that of its dual is
not. An approach that settles the issue in one stroke would be to study the
Einstein–Maxwell-dilaton system. Relatedly, 3D Maxwell theory admits a gauge-
invariant Chern–Simons mass term, presumably leading to rich asymptotic
symmetries. This could also be a setup to analyse the peculiar properties of
3D quantum electrodynamics (QED3), including the effects of its one-loop
Chern–Simons mass term [136, 137, 138, 139] on soft photons and asymptotic
symmetries.

  * •

3D soft photons and memories. Since 3D (Einstein–)Maxwell theory possesses a
propagating massless photon, it admits a soft photon theorem [140] and, in
principle, (sub)n-leading soft photon theorems as well. Although such 3D soft
photons have been mentioned in the literature [141], their detailed
investigation has been superficial so far. What makes the issue especially
attractive is that 3D Maxwell theory, being super-renormalizable, is badly
infrared divergent as a result [142]. In that sense, the infrared problem in
QED3 is more violent than in 4D. This has partly been addressed using
Faddeev–Kulish dressing [143], but a full picture of the infrared triangle
[17] is still missing. We briefly mentioned this in sections 2.3 and 4.2 when
pointing out that 3D electromagnetic memory fails to match with the leading
U$(1)$ charge. A natural continuation of our work is thus to ask how the soft
photon theorem relates to memory and asymptotic symmetries.

  * •

4D Einstein–Maxwell. Perhaps the most natural follow-up of the present work is
to study the asymptotic structure of 4D Einstein–Maxwell theory. This has
partly been explored in [88, 89], although the emphasis was on WZ charges in
the first reference, and on dimensional reduction from 5D in the second one.
The analysis can be vastly extended by studying the detailed structure of the
solution space, including flux-balance laws encoding the soft graviton and
soft photon theorems (or alternatively their memory counterpart). In fact,
such a setup is ideal to investigate the interplay between soft photons and
soft gravitons, which to our knowledge is an open question. In addition, one
should compute the charge algebra using the Koszul bracket in order to see
whether it gives rise to a field-dependent cocycle as in the present 3D model.

  * •

4D Einstein–Rosen waves. As stressed throughout this work, the analogy between
3D Einstein–Maxwell and 4D pure gravity can be understood through the prism of
dimensional reduction [59]. Interestingly, this dimensional reduction also
reveals that the 3D theory actually describes a highly symmetric 4D system,
namely cylindrical gravitational waves, or so-called Einstein–Rosen waves
[86]. Since we have worked out the detailed asymptotic structure of 3D
Einstein–Maxwell theory, it would now be interesting to reinterpret these
results from the 4D vacuum point of view, where they could serve to
investigate the radiative properties of Einstein–Rosen waves. This is
potentially a setup where 4D holography in the presence of radiation is more
approachable than in the case of arbitrary asymptotically flat spacetimes.

  * •

Hamiltonian approach. A general question that stems from our work is whether
ambiguities of surface charges can be fixed by avoiding the covariant
formalism and using instead a Hamiltonian approach. The latter has indeed been
recently successful at either re-deriving known 4D asymptotic symmetries [144,
145, 146, 147, 148, 149] or unveiling new asymptotic structures altogether
[150, 151, 152, 153]. In the present case, one might attempt to recover the
field-dependent central extension (5.15) from a canonical analysis, where
integrability issues would be absent by design.

## Acknowledgements

We thank Glenn Barnich, Adrien Fiorucci and Romain Ruzziconi for key
discussions on asymptotic symmetries with fluxes, as well as Mathieu
Beauvillain, Steve Carlip, Daniel Grumiller, Ali Seraj, Shahin Sheikh-Jabbari
and Simone Speziale for fruitful interactions. The work of J.B. was partially
supported by the Swiss National Science Foundation through the NCCR SwissMAP.
The work of S.M. is supported by the LABEX Lyon Institute of Origins
(ANR-10-LABX-0066) within the Plan France 2030 of the French government
operated by the National Research Agency (ANR). The work of B.O. is supported
by the F.R.S.–FNRS and by the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No.
846244.

##  Appendix A Constructing the solution space

This appendix provides details on the construction of the Einstein–Maxwell
solution space in sections 3.2 and 6.1. The construction follows the Bondi
hierarchy [6], so that equations of motion are split into (i) hypersurfacce
constraints, (ii) genuine evolution equations, and (iii) trivial equations
that are automatically satisfied.

###  A.1 Hypersurface equations

Within the Einstein–Maxwell equations of motion (3.2), consider those that are
purely spatial and contain at least one radial index. We solve each of them in
turn, using as initial inputs the Maxwell fall-offs (2.9), the stress tensor
(2.16) and the Bondi metric ansatz (3.6).

#### Einstein equation $\boldsymbol{(rr)}$.

Begin by considering the field equation

$\displaystyle E_{rr}=\frac{2}{r^{2}}\big{(}r\partial_{r}\beta-
F_{r\phi}^{2}\big{)}=0.$ (A.1)

In radial gauge with $A_{r}=0$, this determines $\beta$ in terms $A_{\phi}$
(and no other Maxwell component), with $A_{\phi}$ given by (2.9b). The
equation is thus solved to order $E_{rr}=\mathcal{O}(r^{-5})$ by the expansion

$\displaystyle\beta=\beta_{0}+\frac{\beta_{1,0}}{r}+\frac{\beta_{3/2,0}}{r^{3/2}}+\frac{1}{r^{2}}\big{(}\beta_{2,0}+(\ln
r)\beta_{2,1}\big{)}+\frac{1}{r^{5/2}}\big{(}\beta_{5/2,0}+(\ln
r)\beta_{5/2,1}\big{)}+\mathcal{O}(r^{-3}),$ (A.2)

where $\beta_{0}(u,\phi)$ is an integration constant that we henceforth set to
zero, while the subleading terms are

$\displaystyle\beta_{1,0}$ $\displaystyle=-\frac{1}{4}C^{2},$ (A.3a)
$\displaystyle\beta_{3/2,0}$ $\displaystyle=-\frac{2}{3}CA_{\phi}^{\ell},$
(A.3b) $\displaystyle\beta_{2,1}$
$\displaystyle=\frac{1}{4}CA_{\phi}^{1/2,1},$ (A.3c)
$\displaystyle\beta_{2,0}$
$\displaystyle=-\frac{1}{2}\big{(}A_{\phi}^{\ell}\big{)}^{2}+\frac{1}{8}C\big{(}2A_{\phi}^{1/2,0}-3A_{\phi}^{1/2,1}\big{)},$
(A.3d) $\displaystyle\beta_{5/2,1}$
$\displaystyle=\frac{2}{5}\big{(}A_{\phi}^{\ell}A_{\phi}^{1/2,1}+CA_{\phi}^{1,1}\big{)},$
(A.3e) $\displaystyle\beta_{5/2,0}$
$\displaystyle=\frac{2}{25}\Big{(}A_{\phi}^{\ell}\big{(}5A_{\phi}^{1/2,0}-8A_{\phi}^{1/2,1}\big{)}+C\big{(}5A_{\phi}^{1,0}-3A_{\phi}^{1,1}\big{)}\Big{)}.$
(A.3f)

Note that (A.3a) is reminiscent of the analogous link between shear and the
function $\beta$ in 4D gravity in Bondi gauge.

#### Einstein equation $\boldsymbol{(r\phi)}$.

Still using the stress tensor (2.16) and the Bondi metric (3.6), consider the
Einstein equation

$\displaystyle
E_{r\phi}=\frac{\beta^{\prime}}{r}-\partial_{r}\beta^{\prime}+\frac{r}{2}e^{-2\beta}\Big{(}(3-2r\partial_{r}\beta)\partial_{r}U+r\partial_{r}^{2}U\Big{)}-2e^{-2\beta}F_{r\phi}(UF_{r\phi}+F_{ru})=0.$
(A.4)

This is solved to order $E_{r\phi}=\mathcal{O}(r^{-3})$ by

$\displaystyle
U=U_{0}+\frac{U_{3/2,0}}{r^{3/2}}+\frac{1}{r^{2}}\Big{(}U_{2,0}+(\ln
r)U_{2,1}+(\ln r)^{2}U_{2,2}\Big{)}+\frac{1}{r^{5/2}}\Big{(}U_{5/2,0}+(\ln
r)U_{5/2,1}\Big{)}+\mathcal{O}(r^{-3}),$ (A.5)

where $U_{0}(u,\phi)$ is undetermined while the subleading terms are

$\displaystyle U_{3/2,0}$ $\displaystyle=-\frac{8}{3}CE,$ (A.6a)
$\displaystyle U_{2,2}$ $\displaystyle=\frac{1}{4}CA_{u}^{1/2,1},$ (A.6b)
$\displaystyle U_{2,1}$
$\displaystyle=-2A_{\phi}^{\ell}E+\frac{1}{2}C\left(A_{u}^{1/2,0}-\frac{3}{2}A_{u}^{1/2,1}-2C^{\prime}\right),$
(A.6c) $\displaystyle U_{2,0}$ $\displaystyle=-L,$ (A.6d) $\displaystyle
U_{5/2,1}$
$\displaystyle=-\frac{8}{5}\big{(}EA_{\phi}^{1/2,1}+A_{\phi}^{\ell}A_{u}^{1/2,1}+CA_{u}^{1,1}\big{)},$
(A.6e) $\displaystyle U_{5/2,0}$ $\displaystyle=\text{(lengthy)}.$ (A.6f)

Note that (A.4) is second-order in radial derivatives of $U$, so one gets two
integration constants here. The first is $L(u,\phi)$ in (A.6d), eventually
identified with the bare angular momentum aspect. The second is the
aforementioned $U_{0}(u,\phi)$ in (A.5), which is analogous to $\beta_{0}$ in
(A.2) since it specifies the induced boundary metric on $\mathscr{I}^{+}$; we
always set $U_{0}=0$ from now on. Also note that we omit writing the component
(A.6f) because its form is long without being particularly illuminating; it is
included here merely for consistency, because the Einstein equation at order
$E_{r\phi}=\mathcal{O}(r^{-3})$ does fix the component $U_{5/2,0}$.

#### Maxwell equation $\boldsymbol{(r)}$.

Now turn to the radial Maxwell equation $\nabla^{\mu}F_{\mu r}=0$, which only
depends on the metric through $\beta$ and $U$. Owing to the Bondi metric
(3.6), its explicit form is

$\displaystyle r^{2}\nabla^{\mu}F_{\mu r}=F_{\phi
r}^{\prime}+re^{-2\beta}\Big{(}\partial_{r}\big{(}r(UF_{r\phi}+F_{ru})\big{)}-2r\partial_{r}\beta(UF_{r\phi}+F_{ru})\Big{)}=0.$
(A.7)

This equation determines the subleading terms $A_{u}^{m>0,n}$ of the
electrostatic potential in terms of $A_{\phi}$ and the Coulombic data $E$.
More precisely, solving recursively in $1/r$, one gets the leading-order
behaviour

$\displaystyle\nabla^{\mu}F_{\mu r}=\mathcal{O}(r^{-3})\quad$
$\displaystyle\Rightarrow\quad\begin{cases}A_{u}^{1/2,1}=0,\\\
A_{u}^{1/2,0}=2C^{\prime}\end{cases}$ (A.8)

which coincides with the Minkowskian Maxwell equations (2.10). As for the
subleading components, they satisfy non-linear relations

$\displaystyle\nabla^{\mu}F_{\mu r}=\mathcal{O}(r^{-4})\quad$
$\displaystyle\Rightarrow\quad\begin{cases}A_{u}^{1,1}=0,\\\
A_{u}^{1,0}=\displaystyle\big{(}A_{\phi}^{\ell}\big{)}^{\prime}-\frac{5}{6}C^{2}E,\\\
A_{u}^{3/2,2}=0,\\\
A_{u}^{3/2,1}=\displaystyle-\frac{2}{9}\Big{(}3CEA_{\phi}^{\ell}+\big{(}A_{\phi}^{1/2,1}\big{)}^{\prime}\Big{)},\\\
A_{u}^{3/2,0}=\text{(lengthy)}.\end{cases}$ (A.9)

Note, as in (A.6f), that we omit to write the long and obscure expression of
$A_{u}^{3/2,0}$; it is merely mentioned here for consistency when solving
Maxwell’s equation $\nabla^{\mu}F_{\mu r}=0$ at order $r^{-4}$.

The identifications (A.9) should be contrasted with their pure electromagnetic
analogue: with the fall-offs (2.9), Maxwell’s equations without sources would
yield the linear relations

$\displaystyle\nabla^{\mu}F_{\mu r}=\mathcal{O}(r^{-4})\quad$
$\displaystyle\Rightarrow\quad\begin{cases}A_{u}^{1,1}=0,\\\
A_{u}^{1,0}=\displaystyle\big{(}A_{\phi}^{\ell}\big{)}^{\prime},\\\
A_{u}^{3/2,2}=0,\\\
A_{u}^{3/2,1}=\displaystyle-\frac{2}{9}\big{(}A_{\phi}^{1/2,1}\big{)}^{\prime},\vspace{0.12cm}\\\
A_{u}^{3/2,0}=\displaystyle-\frac{2}{9}\big{(}A_{\phi}^{1/2,0}\big{)}^{\prime}+\frac{4}{27}\big{(}A_{\phi}^{1/2,1}\big{)}^{\prime}.\end{cases}$
(A.10)

The non-linear contributions in (A.9) are due to gravitational backreaction on
the Maxwell field.

#### Einstein equation $\boldsymbol{(ru)}$.

We finally turn to the last hypersurface equation:

$\displaystyle E_{ru}$
$\displaystyle=\frac{1}{r^{2}}\left(VF_{r\phi}^{2}-rV\partial_{r}\beta+\frac{r}{2}\partial_{r}V+rU(r\partial_{r}\beta-\beta)^{\prime}+\frac{1}{2}\partial_{r}(r^{2}U^{\prime})-e^{2\beta}\big{(}\beta^{\prime
2}+\beta^{\prime\prime}\big{)}\right)$ (A.11) $\displaystyle\phantom{\
=}+e^{-2\beta}\big{(}U^{2}F_{r\phi}^{2}-F_{ur}^{2}\big{)}+\frac{r}{4}e^{-2\beta}\Big{(}(2r\partial_{r}\beta-3)\partial_{r}U^{2}-2rU\partial_{r}^{2}U-r(\partial_{r}U)^{2}\Big{)}=0.$
(A.12)

The solution is given to order $E_{ru}=\mathcal{O}(r^{-3})$ by

$\displaystyle V=(\ln
r)V_{0,1}+V_{0,0}+\frac{V_{1/2,0}}{r^{1/2}}+\mathcal{O}(r^{-1}),$ (A.13)

with

$\displaystyle V_{0,1}$ $\displaystyle=2E^{2},$ (A.14a) $\displaystyle
V_{0,0}$ $\displaystyle=2M,$ (A.14b) $\displaystyle V_{1/2,0}$
$\displaystyle=\frac{8}{3}\big{(}2EC^{\prime}-CE^{\prime}\big{)}.$ (A.14c)

The equation of motion (A.11) is first-order in radial derivatives of $V$, so
it contains a single integration constant $M(u,\phi)$ in (A.14b), which is
eventually identified with the bare mass aspect.

###  A.2 Evolution equations

We now consider the dynamical part of the equations of motion (3.2),
containing angular or time components, but no radial component. Again, we
consider each such equation in turn.

#### Maxwell equation $\boldsymbol{(\phi)}$.

We start with the Maxwell equation $\nabla^{\mu}F_{\mu\phi}=0$, which
determines the time evolution of the coefficients of $A_{\phi}$. Solving
recursively in $1/r$, we find the leading-order evolution equations

$\displaystyle\nabla^{\mu}F_{\mu\phi}=\mathcal{O}(r^{-2})\quad$
$\displaystyle\Rightarrow\quad\begin{cases}\dot{A}_{\phi}^{\ell}=E^{\prime},\\\
\dot{A}_{\phi}^{0}=E^{\prime}+G^{\prime},\\\
\dot{A}_{\phi}^{1/2,1}=\displaystyle-\frac{3}{4}CE^{2},\vspace{0.12cm}\\\
\dot{A}_{\phi}^{1/2,0}=\displaystyle\frac{1}{4}\Big{(}7CE^{2}-3CM+6C^{\prime\prime}\Big{)},\end{cases}$
(A.15)

while at subleading order one has

$\displaystyle\\!\\!\nabla^{\mu}F_{\mu\phi}=\mathcal{O}(r^{-{5/2}})\quad$
$\displaystyle\Rightarrow\quad\begin{cases}\dot{A}_{\phi}^{1,1}=0,\vspace{0.12cm}\\\
\dot{A}_{\phi}^{1,0}=\displaystyle\frac{1}{3}\Big{(}C^{2}E^{\prime}-4MA_{\phi}^{\ell}+2E\big{(}L-7CC^{\prime}\big{)}+2\big{(}A_{\phi}^{\ell}\big{)}^{\prime\prime}\Big{)}.\end{cases}$
(A.16)

#### Maxwell equation $\boldsymbol{(u)}$.

Turning to the remaining Maxwell equation, we find

$\displaystyle\lim_{r\to\infty}(r\nabla^{\mu}F_{\mu u})=0\quad$
$\displaystyle\Rightarrow\quad\dot{A}_{u}^{\ell}=0.$ (A.17)

This is in fact the only contribution in the temporal Maxwell equation.
Indeed, using the fact that $\nabla^{\mu}F_{\mu r}=0=\nabla^{\mu}F_{\mu\phi}$
have already been solved, along with the form of the metric in Bondi gauge,
one deduces from $\nabla^{\mu}\nabla^{\nu}F_{\mu\nu}=0$ that
$\partial_{r}(r\nabla^{\mu}F_{\mu u})=0$, which means that there is a single
term in the radial expansion.

#### Einstein equations $\boldsymbol{(u\phi)}$ and $\boldsymbol{(uu)}$.

Finally, consider the Einstein evolution equations $E_{u\phi}$ and $E_{uu}$.
For the former, one finds

$\displaystyle\lim_{r\to\infty}(rE_{u\phi})=0\quad$
$\displaystyle\Rightarrow\quad\dot{L}=M^{\prime}+EE^{\prime}+\frac{1}{2}\big{(}CN^{\prime}-3NC^{\prime}\big{)},$
(A.18)

which yields the evolution equation of angular momentum in (3.10). Similarly,
one has

$\displaystyle\lim_{r\to\infty}(rE_{uu})=0\quad$
$\displaystyle\Rightarrow\quad\dot{M}=-2N^{2},$ (A.19)

which is the Bondi mass loss equation in (3.10).

The point of the Bondi hierarchy is the following: it turns out that all
subleading equations in $E_{u\phi}$ and $E_{uu}$ are automatically satisfied
on-shell by virtue of the previous equations. We now show this using Bianchi
identities.

###  A.3 Trivial equations

Using the Bianchi identity for the Einstein tensor, one can write

$\displaystyle
0=2\sqrt{-g}\,\nabla_{\mu}G^{\mu}_{\nu}=2\sqrt{-g}\,\nabla_{\mu}(E^{\mu}_{\nu}+T^{\mu}_{\nu})=2\partial_{\mu}(\sqrt{-g}\,E^{\mu}_{\nu})+\sqrt{-g}\,E_{\rho\sigma}\partial_{\nu}g^{\rho\sigma}+2\sqrt{-g}\,\nabla_{\mu}T^{\mu}_{\nu}.$
(A.20)

Since all three Maxwell equations have been solved at this stage, the stress
tensor is conserved, and the last term above drops. Since $E_{r\mu}=0$ has
already been solved as well, putting $\nu=r$ in (A.20) and using
$g^{uu}=0=g^{u\phi}$ yields $E_{\phi\phi}\partial_{r}g^{\phi\phi}=0$. Thus the
Bianchi identity automatically implies $E_{\phi\phi}=0$ when $E_{r\mu}=0$ and
when Maxwell’s equations are satisfied.

Finally, for $\nu=(\phi,u)$, the Bianchi identity (A.20) gives
$\partial_{\mu}(\sqrt{-g}\,E^{\mu}_{\phi})=0=\partial_{\mu}(\sqrt{-g}\,E^{\mu}_{u})$.
Since by now $E_{r\phi}=0=E_{\phi\phi}$, the first of these identities reduces
to $\partial_{r}(rE_{u\phi})=0$. This means that there is a single non-trivial
term at order $r^{-1}$ in the Einstein equation $E_{u\phi}$; it turns out to
be the evolution equation (A.18) for angular momentum. Finally, we turn to the
second Bianchi identity above. Since now $E_{ru}=0=E_{u\phi}$, the identity
reduces to $\partial_{r}(rE_{uu})=0$. There is thus a single non-trivial term
in $E_{uu}$, namely the evolution equation (A.19) for the mass aspect.

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# Coulomb drag between two strange metals

E. Mauri<EMAIL_ADDRESS>H.T.C. Stoof<EMAIL_ADDRESS>Institute for
Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,
Utrecht University, Princentonplein 5, 3584 CC Utrecht, The Netherlands

###### Abstract

We study the Coulomb drag between two strange-metal layers using the Einstein-
Maxwell-Dilaton model from holography. We show that the low-temperature
dependence of the drag resistivity is $\rho_{D}\propto T^{4}$, which strongly
deviates from the quadratic dependence of Fermi liquids. We also present
numerical results at room temperature, using typical parameters of the
cuprates, to provide an estimate of the magnitude of this effect for future
experiments. We find that the drag resistivity is enhanced by the plasmons
characteristic of the two-layer system.

††preprint: APS/123-QED

Introduction. — The cuprates have been of interest to both the experimental
and theoretical physics community for the past three decades due to their
peculiar metallic properties that cannot be explained within the standard
Fermi-liquid framework [1, 2]. In this class of materials, that have in common
a layered structure of $\text{CuO}_{2}$ planes, strong Coulomb interactions
between the electrons give rise to unusual and still unclear quantum behavior,
of which the high-temperature superconducting phase is of most technological
importance. Above the maximum critical temperature of the superconducting
phase, we find the so-called strange metal, whose anomalous linear-in-$T$
resistivity [3, 4, 5, 6, 7, 8] characterizes its non-Fermi liquid behavior,
even if the superconductivity is suppressed by a magnetic field [9]. Given the
importance of understanding the physics at play in the strange-metal phase in
order to also get a better grasp on the instability towards high-temperature
superconductivity, there have been a variety of attempts to model the strongly
interacting cuprates [10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. One technique
in particular is the application of the AdS/CFT correspondence, or holographic
duality [20], conjectured and first developed by the string-theory community,
but that has proven to be a powerful tool to qualitatively study low-energy
properties of strongly interacting quantum matter as well [21, 22, 23].

In this Letter we use a holographic Einstein-Maxwell-Dilaton model of the
strange metal [24] to study a two-layer system where the interaction among the
different layers is only mediated by the long-range Coulomb force. The
importance of such a setting is motivated by the experimental relevance of
multi-layered systems. For simplicity most of the holographic studies of the
cuprates focus on describing the behavior of a single-layer, however, there
has been recent experimental interests in probing also the bulk response of
the cuprates [25, 26]. Experimental results in both hole- and electron-doped
cuprates show that the density response is governed by the inter-layer Coulomb
interaction, giving rise to low-energy acoustic plasmon modes. Plasmons are
ubiquitous in condensed-matter materials due to the electric charge of the
electron, but only recently a way to capture the charged nature of the system
has been proposed in holography [27, 28, 29, 30, 31], opening the door to the
possibility of studying ‘holographic’ plasmons and providing a simplified
framework to test experimental results on acoustic plasmons in multi-layered
cuprates [30].

Figure 1:  Experimental set-up for a Coulomb drag measurement. A current $I$
flows through the active layer and the voltage drop $\Delta V=-R_{D}I$ induced
in the passive layer is observed to measure the drag resistance. In the
holographic description under study we have two types of interactions, i.e.,
strong in-plane interactions leading to strange-metal behavior (blue) and the
long-range Coulomb interactions (red).

In view of these developments, our main aim is here to present a link between
holographic predictions and experimental results by studying the density
response of two holographic strange-metal layers. In particular, by
considering the Coulomb drag resistivity, we are able to obtain also
quantitative estimates of the drag effect in the hope to provide a verifiable
prediction to either validate the holographic model or to show its
shortcomings. Here Coulomb drag refers to a transport phenomenon of a system
of closely spaced but electrically isolated layers, where a current driven
through the ‘active’ layer gives rise to a current in the nearby ‘passive’
layer due to inter-layer scattering of the charge carriers mediated by the
long-range Coulomb interaction that couples density fluctuations among the
different layers [32]. A typical experimental set-up is shown in Fig. 1, where
a known current is driven through one layer and the drag effect, parametrized
by the drag resistivity $\rho_{D}$, is studied by measuring the voltage drop
in the passive layer.

In the holographic description we are using, the system is characterized by
two kinds of interactions, a strong one that is of short range and therefore
does not contribute to the inter-layer scattering, and the long-range Coulomb
interaction that instead also gives rise to the coupling between the separate
layers. The success of the holographic duality relies on the fact that it
naturally describes a strongly interacting system, such as a strange metal
[33], and we thus can take two copies of such a model to describe the two
layers. On the other hand, it is important to realize that holographic models
present always a ‘neutral’ response, where there are no photons in the system.
However, by performing a so-called double-trace deformation [34], we have
recently shown how to add within holography also the long-range Coulomb
screening effects to multi-layered models on top of the strong neutral
interactions [30]. We use this procedure here to couple the holographic
strange-metal layers in order to study the Coulomb drag resistivity, as we
explain in detail in the following.

Coulomb drag resistivity. — Our starting point is the drag resistivity within
the random-phase approximation that is given by the expression [35, 36, 37,
38]

$\displaystyle\rho_{D}=\frac{\hbar^{2}}{16\pi^{3}e^{2}n_{1}n_{2}k_{B}T}\int_{-\infty}^{+\infty}\frac{\mathrm{d}\omega}{\sinh^{2}\left(\hbar\omega/2k_{B}T\right)}\int\mathrm{d}\bm{q}~{}\bm{q}^{2}\frac{\text{Im}[\Pi_{1}(\omega,\bm{q})]}{\lvert\Pi_{1}(\omega,\bm{q})\rvert^{2}}\lvert\chi_{12}(\omega,\bm{q})\rvert^{2}\frac{\text{Im}[\Pi_{2}(\omega,\bm{q})]}{\lvert\Pi_{2}(\omega,\bm{q})\rvert^{2}}\text{
,}$ (1)

where $n_{1}$ and $n_{2}$ are the densities of the active and passive layers,
respectively, $\Pi_{i}(\omega,\bm{q})$ are the corresponding intra-layer
density-density response functions, and the dynamically screened inter-layer
density response is given by

$\displaystyle\chi_{12}=-\frac{\Pi_{1}\Pi_{2}^{*}}{\Pi_{1}\Pi_{2}\frac{e^{2}}{\epsilon
q}\sinh{(q\ell)}+\left[\frac{2q\epsilon}{e^{2}}-\Pi_{1}-\Pi_{2}\right]e^{q\ell}}\text{
,}$ (2)

with $q=\lvert\bm{q}\rvert$, $\ell$ the separation between the two layers, and
$\epsilon$ the dielectric constant of the insulating material surrounding the
layers. This approximation ignores the effect of vertex corrections. However,
it has been observed experimentally that the electron self-energy in the
cuprates depends mostly on frequency. This feature is also captured in our
holographic model where the low-energy theory is described by a ‘semi-local’
quantum critical phase, where the electron self-energy
$\hbar\Sigma(\omega,\bm{k})$ scales like $\omega^{2\nu_{k}}$, with the
momentum dependence only entering in the power $\nu_{k}$ [24, 39]. We
therefore expect Eq. (1) to be a valid approximation for the Coulomb drag
resisitvity for a cuprate two-layer system in the strange-metal regime.

From Eqs. (1) and (2) we see that the only input we need to compute the drag
resistivity are the single-layer response functions $\Pi_{i}(\omega,\bm{q})$.
The power of our holographic approach now lies in the fact that it allows us
to compute this response in a strongly interacting system. Since the behavior
of the drag resistivity depends solely on the nature of the inter-layer
interactions, and not on disorder, it may reveal interesting features of the
cuprates, and possibly verify if the proposed holographic model of the strange
metal correctly describes this phenomenon.

Einstein-Maxwell-Dilaton theory. — Let us briefly review how we obtain the
strange-metal response function from holography. The holographic duality
states the equivalence between a strongly interacting quantum theory and a
classical gravitational theory with one additional spatial dimension, where
the coordinate $r$ of this additional dimension parametrizes the energy scale
of the quantum field theory. In particular low-energy physics is described by
the geometry in the deep interior ($r\to 0$) of the spacetime, while the near-
boundary geometry ($r\to\infty$) characterize the ultra-violet details of the
dual field theory. The thermodynamics of the holographic strange-metal model
proposed by Gubser et al. [24], is described by a solution to the coupled set
of Einstein equations for the metric $g_{\mu\nu}$, Maxwell equations for the
$U(1)$ gauge field $A_{\mu}$, and the Klein-Gordon equation for the real
scalar field $\phi$, known as the dilaton [40, 41]:

$\displaystyle R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=T_{\mu\nu}\text{
,~{}~{}}\nabla_{\mu}(e^{\phi/\sqrt{3}}F^{\mu\nu})=0\text{ ,}$
$\displaystyle\nabla_{\mu}\nabla^{\mu}\phi=\frac{e^{\phi/\sqrt{3}}}{4\sqrt{3}}F^{2}-\frac{2\sqrt{3}}{L^{2}}\sinh(\phi/\sqrt{3})\text{
,}$ (3)

where $R_{\mu\nu}$ and $R$ are the Ricci tensor and scalar, respectively, and
the energy-momentum tensor equals

$\displaystyle T_{\mu\nu}=$
$\displaystyle\frac{1}{2}\partial_{\mu}\phi\partial_{\nu}\phi+\frac{e^{\phi/\sqrt{3}}}{2}F_{\mu}^{~{}\sigma}F_{\nu\sigma}$
(4) $\displaystyle-
g_{\mu\nu}\left(\frac{e^{\phi/\sqrt{3}}}{8}F^{2}+\frac{(\partial_{\sigma}\phi)^{2}}{4}+\frac{3}{L^{2}}\cosh(\phi/\sqrt{3})\right)\text{
.}$

Finally, $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field
strength and $L$ is the AdS radius. In particular, as we are interested in a
regime where the underling lattice structure of the cuprates does not play a
role in the drag resistivity, the fields in Eq. (3) are a function of the
radial coordinate $r$ only. This set of equations for
$(g_{tt}=-1/g_{rr},g_{xx}=g_{yy},A_{t},\phi)$ then support a fully analytical
black-hole solution [24], with a non-zero temperature and entropy, and with
$A_{t}(r)$ setting a non-zero density in the dual boundary theory through the
radially conserved quantity
$n\equiv\sqrt{-g(r)}~{}e^{\phi(r)/\sqrt{3}}F^{rt}(r)$, $g(r)$ being the
determinant of the metric.

The dilaton field is such that the emergent low-energy theory is described by
a semi-local quantum critical phase with both the dynamical critical exponent
$z=\infty$ and the hyperscaling violation exponent $\theta=-\infty$ divergent
but with a fixed ratio equal to $-1$. The dynamical exponent $z=\infty$
implies that at energies well below the Fermi energy, quantum critical
correlations can only depend on momentum through the power of the frequency
dependence, while $\theta/z=-1$ ensures that the entropy goes linearly to zero
at zero temperature [21]. Ultimately, this scaling allows for the description
of a quantum critical phase with a linear-in-$T$ resistivity [39] and an
electron self-energy that is dominated by its frequency dependence as desired
for a strange metal [42].

In order to also compute the response of the holographic strange metal to
small external perturbations, we have to linearize the gravitational equations
in Eq. (3) around the black-hole solution. We obtain in this way a set of
coupled equations for the fluctuations $(\delta g_{tt},\delta g_{tx},\delta
g_{xx},\delta g_{yy},\delta A_{t},\delta A_{x},\delta\phi)$ that can only be
solved numerically. According to the holographic dictionary [33, 21, 23, 22],
finding such a solution with infalling-wave boundary conditions at the black-
hole horizon, allows us to extract all the retarded Green’s function of the
system, and hence the density-density response function $\Pi(\omega,\bm{q})$,
by studying the near-boundary behavior of the field fluctuations.

Results. — We work in each layer with electrons at the same fixed equilibrium
density $n$, and we study the holographic density response function, that is
anticipated to describe the physics of the strange metal near the Fermi
momentum in the regime where the dispersion of the electrons can be
linearized. Therefore we replace from now on the speed of light in the
gravitational theory with the Fermi velocity, i.e., $c\rightarrow v_{F}$. In
the low-temperature regime and at energies and momenta much smaller than the
Fermi energy and Fermi momentum, respectively, we find that the response
function is well approximated by a hydrodynamic model of the form [43]

$\displaystyle\Pi(\omega,q)=\frac{q^{2}\left(\omega{\mathcal{D}}+iv_{s}^{2}D_{d}\chi
q^{2}\right)}{\omega^{3}+i\omega^{2}q^{2}(2D_{s}+D_{d})-\omega
q^{2}v_{s}^{2}-iv_{s}^{2}D_{d}q^{4}}$ (5)

that contains a linear sound mode at $\omega=\pm
v_{s}q-iD_{s}q^{2}+\mathcal{O}(q^{4})$, and a diffusive mode at
$\omega=-iD_{d}q^{2}+\mathcal{O}(q^{4})$, with $\mathcal{D}$ the Drude weight,
and $\chi=-\lim_{q\to 0,\,\omega\to 0}\Pi(\omega,\bm{q})$ the (hydrodynamic)
compressibility. This form of $\Pi$ implies that the integrand in Eq. (1)
contains, for our symmetric case $\Pi_{i}\equiv\Pi$ (it is straightforward to
generalize the results to layers with different densities and three different
surrounding dielectric constants [44]), in addition to a diffusive mode
$\omega=-iD_{d}q^{2}$, the out-of-phase acoustic plasmon mode and the in-phase
optical plasmon characteristic of a two-layer system [45, 46, 47, 48, 44]. The
latter frequencies are

$\displaystyle\omega=$
$\displaystyle\pm\sqrt{v_{s}^{2}+\frac{e^{2}\mathcal{D}\ell}{2\epsilon}}q+\mathcal{O}\left(q^{2}\right)$
(6) $\displaystyle\omega=$
$\displaystyle\pm\sqrt{\frac{e^{2}\mathcal{D}q}{\epsilon}}\left(1+\frac{v_{s}^{2}-e^{2}\mathcal{D}\ell/2\epsilon}{2e^{2}\mathcal{D}/\epsilon}q\right)+\mathcal{O}\left(q^{2}\right)\text{
.}$ (7)

All three modes are clearly visible in the numerical result for the integrand
in Fig. 2.

Figure 2:  Typical integrand in Eq. (1), normalized to peak height, showing
the in-phase and out-of phase plasmon modes, and the low-energy diffusive
mode.

These modes thus determine the low-temperature behavior of the drag
resistivity. We emphasize here that compared to a purely hydrodynamic approach
as in Refs. [49, 50], the holographic model gives also a prediction for the
coefficients in Eq. (5) and for their temperature dependence, allowing us to
fully determine the dominant low-temperature behavior of $\rho_{D}$. In
particular, we numerically explored the pole structure of the retarded density
correlators in the limit of $T\rightarrow 0$ to find that in this low-
temperature hydrodynamics [51, 52, 53], the sound diffusion coefficient equals
$D_{s}=(1/6\sqrt{3})k_{B}T/\hbar n$, whereas the other diffusion constant
obeys [21] $D_{d}=(4\pi/\sqrt{3})k_{B}T/\hbar n$. Both are fully determined by
background (thermodynamic) quantities, as is the zero-temperature Drude weight
$\mathcal{D}=(1/3^{1/4})v_{F}\sqrt{n}/\hbar$. We further find a speed of
sound, that differs from the $v_{F}/\sqrt{2}$ of a conformal field theory as
conformal invariance is broken in the low-energy limit by the dilaton field.
We obtain $v_{s}\approx 0.76v_{F}$, that is slightly higher than expected from
the thermodynamic potential $\Omega$ via
$v_{s}=(-\partial\Omega/\partial\epsilon)^{1/2}=\sqrt{17/31}v_{F}$.
Accordingly, $\chi=\mathcal{D}/v_{s}^{2}\approx 1.32\sqrt{n}/\hbar v_{F}$ so
that the height of the diffusive peak is determined by the subleading behavior
$(\mathcal{D}/v_{s}^{2}-\chi)/2\approx 2.56T^{2}/\sqrt{n}(\hbar v_{F})^{3}$,
with the proportionality constant obtained numerically.

The hydrodynamic model in Eq. (5), together with the above results, can be
integrated to compute the drag resistivity according to Eq. (1). Doing so
shows that in the low-temperature limit and $e^{2}\chi\ell/\epsilon\gg 1$ the
behavior is dominated by the diffusive mode, that implies $\rho_{D}(T)\propto
T^{4}$, strikingly different from the Fermi liquid result $\rho_{D}(T)\propto
T^{2}$ due to thermal broadening of the Fermi surface [54, 35, 36]. By
considering the diffusive mode alone in Eq. (5), using that in the limits of
interest $\Pi\approx\chi$, the drag resistivity can be determined analytically
to yield

$\displaystyle\rho_{D}=\frac{3\zeta(3)}{32\pi^{2}}\frac{\hbar}{e^{2}}\frac{(\mathcal{D}/v_{s}^{2}-\chi)^{2}}{\chi^{2}n\ell^{2}(e^{2}\chi\ell/\epsilon)^{2}}\propto\frac{T^{4}}{\ell^{4}}\text{
.}$ (8)

For the cuprates ($e^{2}\chi\ell/\epsilon\approx 2.6$) the full hydrodynamic
result is considerable larger than expected from diffusion alone but has the
same temperature dependence as shown in the inset of Fig. 3.

Figure 3:  Coulomb drag at high temperature (blue), where it is determined by
the plasmons, together with the fit $\rho_{D}\propto e^{-\Delta/T}$ and
$\Delta\approx 2760$ K (dashed red). In the inset the low-temperature results
obtained from hydrodynamics (black) and diffusion only (dashed grey).

It is important to notice that the temperature dependence of the dissipative
coefficients $D_{d}$ and $D_{s}$ is characteristic of the particular
holographic model used. However, the linear dependence on $T$ of these
coefficients comes from the linear-in-$T$ resistivity, that is a fundamental
requirement of any strange-metal model. We would also like to highlight that,
due to the introduction of the Coulomb interactions into the holographic
model, we are able to obtain also the magnitude of the drag effect. In a
bottom-up approach to holography we are only able to compute response
functions up to an unknown factor related to (dimensionless) Newton’s constant
$16\pi G\hbar/c^{3}L^{2}$. The introduction of the Coulomb interaction, and
hence the plasma frequency, allows us to match this unknown constant with the
experimentally observed plasma frequency and uniquely fix the magnitude of the
density response.

For the curves given in Fig. 3, we use the density $n=7.1\times
10^{14}\,\text{cm}^{-2}$, the Fermi velocity obeying $\hbar
v_{F}=3\,\text{eV\AA}$ as typical in the cuprates, and the inter-layer spacing
$\ell=7.5\,\text{\AA}$ that is of the order of the effective inter-layer
distance of LCCO planes used in the experiment on plasmons by Hepting et al.
[25]. In addition, a value of the above-mentioned overall constant equal to
one and also $e^{2}/\epsilon\hbar v_{F}=1$ turns out to give a plasmon
dispersion of the same magnitude as observed in this experiment. For this
choice of parameters, the low-temperature hydrodynamic behavior gives a very
small resistivity. However, we numerically considered also temperatures at and
above room temperature. Here we are in a regime that lies outside the region
of validity of the hydrodynamic model in Eq. (5), and the dominant
contribution to the drag resistivity comes from higher energies where the
acoustic plasmon, and to a lesser extent the optical plasmon, play a major
role in the magnitude of the effect. In this regime the temperature behavior
is mostly due to the thermal factor $1/\sinh^{2}{(\hbar\omega/2k_{B}T)}$ in
Eq. (1) and hence it gives an exponential growth, leading to a resistivity of
similar magnitude as in drag experiments with two-dimensional electron gases.

Conclusion and outlook. — We have here studied by holography the Coulomb drag
between two strange metals and have shown how to determine the drag
resistivity in such a strongly interacting system. We performed the
computation within a particular model for which an analytical background
solution exists and showed the important role of the linear acoustic plasmon
mode in the drag resistivity at easily accessible temperatures.

The holographic model used captures various qualitative features of the low-
energy properties of the strange metal. However, we also have to keep in mind
that it describes a system with a very different ultra-violet behavior
compared to a laboratory cuprate material as the electronic dispersion is
implicitly linearized and no lattice bandstructure is involved in the
calculation. This may affect the quantitative estimate of the drag resistivity
somewhat. A important matter of future studies is the thermodynamics of the
strange metal, as an analytical understanding of the observed values for the
speed of sound and susceptibility is still lacking. We, nevertheless, hope
that the theory presented here may stimulate further experiments on drag
transport to test holographic quantum matter in general and strange metals in
particular.

Acknowledgments. — We are grateful to Alberto Cortijo for helpful suggestions
and discussions. This work is supported by the Stichting voor Fundamenteel
Onderzoek der Materie (FOM) and is part of the D-ITP consortium, a program of
the Netherlands Organization for Scientific Research (NWO) that is funded by
the Dutch Ministry of Education, Culture and Science (OCW).

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