# Drag resistance mediated by quantum spin liquids

Raffaele Mazzilli  Max-Planck-Institut für Festkörperforschung, 70569
Stuttgart, Germany  Alex Levchenko  Department of Physics, University of
Wisconsin-Madison, Madison, Wisconsin 53706, USA  Elio J. König  Max-Planck-
Institut für Festkörperforschung, 70569 Stuttgart, Germany

###### Abstract

Recent advances in material synthesis made it possible to realize two-
dimensional monolayers of candidate materials for a quantum spin liquid (QSL)
such as $\alpha$-RuCl3,1T-TaSe2 and 1T-TaS2. In this work, we propose an
experimental setup that exploits nonlocal electrical probes to gain
information on the transport properties of a gapless QSL. The proposed setup
is a spinon-mediated drag experiment: a current is injected in one of the two
layers and a voltage is measured on the second metallic film. The overall
momentum transfer mechanism is a two-step process mediated by Kondo
interaction between the local moments in the quantum spin liquid and the spins
of the electrons. In the limit of negligible momentum relaxed within the QSL
layer, we calculate the drag relaxation rate for Kitaev, $\mathbb{Z}_{2}$, and
U(1) QSLs using Aslamazov-Larkin diagrams. We find, however, that the case of
dominant momentum relaxation within the QSL layer is far more relevant and
thus develop a model based on the Boltzmann kinetic equation to describe the
proposed setup. Within this framework we calculate the low temperature scaling
behavior of the drag resistivity, both for U(1) and $\mathbb{Z}_{2}$ QSLs with
Fermi surfaces. In some regimes we find a crossover in the temperature scaling
that is different between the $\mathbb{Z}_{2}$ and U(1) QSL both because of
the non-Fermi liquid nature of the U(1) QSL, and because of the qualitatively
different momentum relaxation mechanism within the QSL layer. Our findings
suggest that parameters of the system can be tuned to make the spinon-mediated
drag a significant fraction of the total transresistance.

##  I Introduction

Quantum Spin Liquids (QSLs) Savary and Balents (2016); Zhou  _et al._ (2017);
Knolle and Moessner (2019) have enjoyed increased interest in the past years.
Theoretically, one major driving force is the fascination for topological
quantum states of matter Wen (2019) with strong entanglement, which are
mathematically equivalent to certain topological quantum error correction
codes proposed in quantum information theory Kitaev (2003, 2006). What is
equally important, are the advances in materials science and synthesis.
Amongst the QSL candidate materials which were most actively discussed in
recent years, particular attention is devoted to exfoliable materials such as
the Kitaev candidate Kasahara  _et al._ (2018); Yokoi  _et al._ (2021); Czajka
_et al._ (2021); Bruin  _et al._ (2022) $\alpha$-RuCl3 as well as the
transition metal dichalcogenides Kratochvilova  _et al._ (2017); Law and Lee
(2017); Murayama  _et al._ (2020); Ribak  _et al._ (2017) 1T-TaS2 and
1T-TaSe2.

The study of monolayers and few layers of these van-der-Waals QSL candidate
materials benefits from a variety of advantages: first, in the monolayer
limit, interlayer valence bonds can not form, thereby precluding the
stabilization of topologically trivial interlayer valence bond solids; second,
the principle of van-der-Waals-LEGO Geim and Grigorieva (2013) allows to
combine the functionality of 2D QSL sheets with superconductors,
semiconductors, and metals in a single device and, as an example,
$\alpha$-RuCl3 on graphene devices were reported in the literature Mashhadi
_et al._ (2019). Moreover, the stacking of van-der-Waals materials also allows
to artificially create 2D QSL materials, for example in twisted transition-
metal-dichalcogenide heterostructures Zare and Mosadeq (2021); Kiese  _et al._
(2022). Third, the functionality of van-der-Waals heterostructures also allows
for unprecedented experimental techniques to probe QSL behavior. These include
tunneling spectroscopy, as studied theoretically Mross and Senthil (2011);
Morampudi  _et al._ (2017); Carrega  _et al._ (2020); Chen and Lado (2020);
Feldmeier  _et al._ (2020); König  _et al._ (2020); Jia  _et al._ (2021) and
experimentally Ruan  _et al._ (2021); Chen  _et al._ (2022) in different
setups. Moreover, anyonic interference experiments to detect the topological
order in the QSL state were proposed by interfacing them with quantum Hall and
superconducting states Aasen  _et al._ (2020); Kishony and Berg (2021). It
appears plausible that this wide variety of advantages may thus counteract the
main drawback of thin films, namely that standard bulk probes, such as neutron
scattering Kratochvilova  _et al._ (2017); Banerjee  _et al._ (2016, 2017)
experiments are inefficient in the mono-layer or few-layer limit.

Figure 1:  a) Experimental setup under consideration: a QSL sheet is
interposed between two metallic layers, a current is applied in the upper
layer and a voltage is measured in the lower layer. b) Schematic
representation of momentum transfer between the QSL layers and the metallic
leads: momentum transfer is a two-step process. c) Summary of main results
[see Eqs. (16a), (17a), (31a), (32a), (34), (36) for details and discussion]
i.e. the temperature dependence of drag in different regimes of experimentally
tunable parameters, temperature $T$, and density $n$ in the metallic leads
depicted for the three different types of QSLs under consideration.

Motivated by the need for probing 2D QSL materials, in this paper we propose a
spinon induced drag experiment in which instead of a dielectric separating two
metallic films, as in a Coulomb drag Narozhny and Levchenko (2016) experiment,
we interpose a QSL material to mediate the interaction, see Fig. 1a. In the
limit when no momentum is deposited in the QSL layer, we calculate the
temperature dependence of the nonlocal drag resistivity using standard
Aslamazov-Larkin diagrams for three examples of gapless QSL states, including
Kitaev QSLs and $\mathbb{Z}_{2}$ as well as U(1) QSLs with a spinon Fermi
surface. Additionally, in the presence of a spinon Fermi surface, we also
employ a Boltzmann-like framework, in which momentum relaxation inside the QSL
layer is accounted for. Our study is thus applicable to the quantum states
proposed for $\alpha$-RuCl3 (Kitaev-like $\mathbb{Z}_{2}$ QSL with Dirac
nodes) and the 1T-TaS2 family (for which there are experimental signatures of
U(1) QSL with spinon Fermi surface Ruan  _et al._ (2021); Chen  _et al._
(2022)).

The standard mechanism of momentum transfer between the layers mediated by
Coulomb interaction results in a temperature $T$ dependent drag rate (defined
in Sec. II.1) Narozhny and Levchenko (2016):
$\tau^{-1}_{D}=\mathcal{C}(T^{2}/E_{F})\mathrm{min}(1,{T_{d}}/{T})$, where
$T_{d}=v_{F}/d$ and $E_{F}$, $v_{F}$ and $d$ are the Fermi energy, Fermi
velocity and distance between the conducting layers (here and in what follows
and we set $\hbar=1=k_{B}$). The dimensionless interaction parameter
$\mathcal{C}\sim r_{s}^{-2}(k_{F}d)^{-4}$ (expressed in terms of $r_{s}\sim
e^{2}/v_{F}\epsilon_{r}$ and Fermi wave-vector $k_{F}$) controls the magnitude
of the effect. It is sensitive to the external screening and dielectric
environment encoded in $\epsilon_{r}$. Drag due to exchange of bosons other
than photons were considered in the literature before, including phonon-
mediated mediated drag Bønsager  _et al._ (2000); Khveshchenko (2000). The
most extensively studied example of drag resistance in the case of non-Fermi
liquids (nFLs) corresponds to the mutual friction between two copies of
quantum Hall liquids at half filling. In this case Chern-Simons field theories
of fermions coupled to the gauge field and also complimentary hydrodynamic
ideas find their powerful applications Ussishkin and Stern (1997); Sakhi
(1997); Kim and Millis (1999); Patel  _et al._ (2017). Spinon contributions to
the Coulomb drag of nFLs Pereira and Sela (2010); Zou  _et al._ (2010) and
within moiré bilayer systems Zhang and Vishwanath (2020) were also considered,
as well as a variety of (nonlocal) transport probes of spinons Chen  _et al._
(2013); Chatterjee and Sachdev (2015); Okamoto (2016); Werman  _et al._
(2018); Aftergood and Takei (2020); Zhuang and Marston (2021). However, to the
best of our knowledge, a spinon-mediated drag experiment as a probe of QSL
physics has not been proposed before.

To conclude this introduction, we recall that in contrast to $\mathbb{Z}_{2}$
QSLs, the gauge field in U(1) QSLs is always strongly coupled to the spinons
and large-$N$ methods are commonly used to access this non-trivial physics
Altshuler  _et al._ (1994); Kim  _et al._ (1994); Hermele  _et al._ (2004);
Lee (2009); Mross  _et al._ (2010). In the case of a spinon Fermi surface, a
nFL phase arises which displays similar behavior as electrons at a nematic
quantum critical point Oganesyan  _et al._ (2001); Dell’Anna and Metzner
(2007); Metlitski and Sachdev (2010); Maslov  _et al._ (2011); Holder and
Metzner (2015), in particular a single particle lifetime
$\tau\sim{\epsilon}^{-2/3}$ and a momentum relaxation time $\tau_{2}\sim
T^{-4/3}$, where $\epsilon$ is the energy of the excitation and $T$ the
temperature. At the same time, the non-trivial power-laws induced by gauge
field fluctuations cancel in the polarization operator (i.e. the structure
factor) in the low-energy long-wavelength limit Kim  _et al._ (1994);
Altshuler  _et al._ (1994); Lee (2009); Mross  _et al._ (2010); Metlitski and
Sachdev (2010) such that we find little difference between the
$\mathbb{Z}_{2}$ and U(1) QSL in the regime when no momentum is deposited into
the QSL layer.

In the opposite limit, where momentum is lost in the QSL layer
$\mathbb{Z}_{2}$ and $U(1)$ QSLs differ. In order to employ a Boltzmann-like
approach to the drag resistivity across the U(1) QSL we follow a technique
originally due to Prange and Kadanoff Prange and Kadanoff (1964); Kim  _et
al._ (1995); Nave and Lee (2007); Kim and Pépin (2009); Hackl and Thomale
(2011) (see also Appendix B.1), noting that the retarded spinon self energy
$\Sigma^{R}(\textbf{k},{\epsilon})$ is independent of the modulus of the
momentum k. Then the spectral weight $A(\textbf{k},{\epsilon})$ becomes a
peaked function of $|\textbf{k}|$ and can be used to define a pseudo-
distribution function $f(\hat{k},{\epsilon})$. Within Keldysh technique, we
derive the coupling of the Boltzmann-like equation describing the insulating
QSL to the conventional Boltzmann equations describing the conducting layers.

This paper is organized as follows. In Sec. II we describe the experimental
setup of a spinon induced drag experiment, offer a phenomenological
perspective and introduce the model for microscopic calculations. In Sec. III
we use known results from Aslamazov-Larkin diagrammatics to calculate the drag
relaxation rate for the Kitaev QSL, and for $\mathbb{Z}_{2}$ as well as U(1)
QSL with Fermi surface. Sec. IV is devoted to the limit of momentum being
relaxed inside the QSL and we present the derivation and solution of the
system of coupled Boltzmann equations. We conclude with a discussion and
outlook and relegate important technical details to two appendices.

##  II Model and Experimental Setup

###  II.1 Spinon mediated drag

The proposed setup is designed to measure spinon-mediated drag between two
conducting two-dimensional electron systems. A layer of a QSL material is
interposed between two metallic layers and a current $\textbf{j}_{3}$ is
passed through the active metallic layer (layer 3). The spins of the free
electrons Kondo couple with the local moments constituting the QSL material
(layer 2), which then couple to the spins of the free electrons in the second
metallic layer, also referred to as “passive” layer 1, thus inducing an
electric field $\textbf{E}_{1}$. The proportionality constant between the
field and current is the definition of drag conductivity

$\textbf{j}_{3}=\sigma_{D}\textbf{E}_{1},$ (1)

and thereby to the drag rate

$\frac{1}{\tau_{D}}\simeq\frac{n_{3}e^{2}}{m_{1}}\frac{\sigma_{D}}{\sigma_{1}\sigma_{3}}.$
(2)

Here $n_{i}$ is the carrier density in layer $i=1,3$, $m_{1}$ is the
electron’s effective mass in layer 1, $e$ is the electric charge, and
$\sigma_{1,3}$ the conductivities of the metallic layers. While our
calculations are performed for metallic layers given by 2D electron gases
(2DEG) with parabolic dispersion, the final temperature dependence of our
results should also hold for doped graphene.

To gain insight into the problem, we briefly reiterate a simple Drude-like
treatment based on the coupled equations of motion

$\displaystyle\dot{\textbf{v}}_{1}$
$\displaystyle=-\frac{\textbf{v}_{1}}{\tau_{1}}-\frac{\textbf{v}_{1}-\textbf{v}_{{2}}}{\tau_{T}}-\frac{e}{m_{1}}\textbf{E}_{1},$
(3) $\displaystyle\dot{\textbf{v}}_{2}$
$\displaystyle=-\frac{\textbf{v}_{{2}}}{\tau_{2}}-\frac{\textbf{v}_{{2}}-\textbf{v}_{1}}{\tau_{T}}-\frac{\textbf{v}_{{2}}-\textbf{v}_{3}}{\tau_{T}},$
(4) $\displaystyle\dot{\textbf{v}}_{3}$
$\displaystyle=-\frac{\textbf{v}_{3}}{\tau_{3}}-\frac{\textbf{v}_{3}-\textbf{v}_{{2}}}{\tau_{T}}.$
(5)

Here, the momentum relaxation rates within each layer are denoted
$\tau_{1,2,3}$ and the rate of momentum transfer is $\tau_{T}$. For simplicity
we assume parabolic bands in all layers. Here, and in most of the main text,
we consider a symmetric setup with equivalent layers $1,3$, while details on
non-symmetric setups are relegated to the appendices. The drag rate in the
limit $\tau_{1}=\tau_{3}\ll\tau_{T}$ is thus

$\frac{1}{\tau_{D}}=\frac{\tau_{2}}{\tau_{T}(\tau_{T}+2\tau_{2})}.$ (6)

While this formula is phenomenologically valuable in the following discussion
of limits, the remainder of the paper is devoted to the microscopic
calculation of $1/\tau_{D}$. The competition between the interlayer scattering
and the intralayer scattering gives rise to two regimes. (i) For
$\tau_{T}{\ll}\tau_{2}$ the momentum is quasi-instantaneously transferred from
metallic layers to the QSL and from the latter to the second metal. The final
result for $1/\tau_{D}$ is independent of $1/\tau_{2}$, since intralayer
relaxation is unimportant there is no need for the description that makes use
of the Boltzmann equation approach in the QSL layer. The microscopic theory
for this regime is presented in Sec. III. (ii) For $\tau_{T}\gg\tau_{2}$, the
momentum transferred to the spinons relaxes before being transferred to the
second lead. To account for these relaxation effects in the presence of a
spinon Fermi surface, we employ a quantum Boltzmann equation, see Sec. IV. We
highlight that regime (i) and (ii) correspond to the limits
$\tau_{D}\ll\tau_{2}$ and $\tau_{D}\gg\tau_{2}$, respectively.

###  II.2 Models of QSLs

In this section, we briefly summarize the models for the QSL state which we
use. First, we consider QSL states with spinon Fermi surface.

As mentioned above a QSL comes in with the emergence of gauge fields.
Theoretically the most prominent example of a QSL with spinon Fermi surface is
a U(1) QSL, where the gauge degrees of freedom have their own Maxwellian
dynamics and couple to the spin degrees of freedom

$\hat{S}^{a}=\psi^{\dagger}_{\sigma}\sigma^{a}_{\sigma\sigma^{\prime}}\psi_{\sigma^{\prime}}.$
(7)

where $\psi_{\sigma}$ are fermionic spinons and the $\sigma^{a}$s describe the
spin and we use Einstein summation convention. As we will shortly explain, we
consider $\sigma^{a}$ to be SU(N) generators. The interacting action for such
system is Lee and Lee (2005); Nave and Lee (2007)

$S_{f}=\sum_{\sigma=1}^{N}\int_{x}\bar{\psi}_{\sigma}(x)\left[i\partial_{t}+\frac{a_{0}}{\sqrt{N}}+\frac{1}{2m_{f}}\left(\bm{\nabla}+i\frac{\textbf{a}}{\sqrt{N}}\right)^{2}\right]\psi_{\sigma}(x),$
(8)

Given the absence of an expansion control parameter as $\alpha=\frac{1}{137}$
in QED, we use a large $N$ expansion to deal with Eq. (8). Strictly speaking,
a large $N$ expansion yields reliable results only when taken along with a
small ${\varepsilon}$ expansion Lee (2009); Mross  _et al._ (2010), where
${\varepsilon}$ appears in the free action of the gauge boson

$S_{\rm MW}=\int
d\textbf{k}\,d\omega\,\frac{|\textbf{k}|^{\varepsilon+1}}{e^{2}}|a(\textbf{k},\omega)|^{2},$
(9)

with the limits $\\{\frac{1}{N},\varepsilon\\}\rightarrow 0$ in a way that the
product $N{\varepsilon}$ is constant. Despite this, following a common wisdom
Lee (2021) that the dynamics of the spinons is not affected by the double
expansion, we use the large $N$ limit directly for ${\varepsilon}=0$.

As mentioned in the introduction, the U(1) QSL with Fermi surface forms a nFL.
It is important to point out that the polarization operator and thus the spin
susceptibility for the U(1) QSL for small $q$ and $\omega$ retains its low
energy behavior due to mutual cancellation between vertex and self-energy
corrections Kim  _et al._ (1994); Altshuler  _et al._ (1994); Lee (2009);
Mross  _et al._ (2010); Metlitski and Sachdev (2010).

We compare the nFL of the U(1) QSL with Fermi surface to the case of
$\mathbb{Z}_{2}$ QSLs with Fermi surface. In this case gauge fields are gapped
and can be disregarded at lowest energies. The spinons form a Fermi liquid,
and for simplicity of comparison to the U(1) case, we stick to the case with
parabolic spinon dispersion and a spin representation as in Eq. (7). While the
case of $\mathbb{Z}_{2}$ QSLs with Fermi surface is discussed much less in the
literature, it does appear in certain exactly soluble models Baskaran  _et
al._ (2009); Yao  _et al._ (2009).

Finally, we also calculate the drag relaxation time in a Kitaev spin liquid
with no Fermi surface. Despite the low-energy Majorana spinon excitations, the
low-energy spin response is gapped Knolle  _et al._ (2014) in the integrable
Kitaev limit. However, taking into account integrability breaking Heisenberg
and symmetric superexchange interactions the retarded spin-spin correlation
function is gapless and given by Song  _et al._ (2016)

$C^{R}(\textbf{q},\omega)=C_{K}\sqrt{(v_{{F2}}q)^{2}-(\omega+i0^{+})^{2}},$
(10)

where $C_{K}$ is a constant of dimension
$1/(\text{energy}\times\text{length})^{2}$.

###  II.3 Microscopic QSL-metal coupling

The interlayer momentum and energy transfer is microscopically encoded in the
Kondo coupling between the local moments of the quantum magnet and the spins
of the free electrons

$H=\frac{\mathcal{J}_{K}}{2N}\sum_{\textbf{x}}\hat{S}^{a}(\textbf{x})\cdot(c^{\dagger}_{\sigma_{3}}(\textbf{x}){\sigma}^{a}_{\sigma_{3},\sigma_{4}}c_{\sigma_{4}}(\textbf{x})),$
(11)

where $c$ and $c^{\dagger}$ are the annihilation and creation operators for
the electrons in the metal and $\mathcal{J}_{K}$ is the Kondo coupling
constant.

In the case of QSLs with a Fermi surface the effective coupling of spinons and
electrons for the continuum model is

$\begin{split}H=\frac{J_{K}}{2N}\int_{\textbf{x}}(\psi^{\dagger}_{\sigma_{1}}(\textbf{x})&{\sigma}^{a}_{\sigma_{1},\sigma_{2}}\psi_{\sigma_{2}}(\textbf{x}))(c^{\dagger}_{\sigma_{3}}(\textbf{x}){\sigma}^{a}_{\sigma_{3},\sigma_{4}}c_{\sigma_{4}}(\textbf{x})).\end{split}$
(12)

Here, $J_{K}\sim\mathcal{J}_{K}a^{2}$, with $a$ being the lattice spacing and
the subscripts $\sigma_{i}$ are spin indices and we use the large $N$
expansion for our calculations, so ${\sigma_{i}}=1,\dots,N$ and the generators
of SU(N) have the property $\text{tr}[\sigma^{a}\sigma^{b}]=2\delta_{ab}$.

###  II.4 Momentum relaxation in QSLs with Fermi surface

Before continuing with the microscopic calculation of the drag current, we
summarize below the expected low-temperature behavior of the momentum
relaxation rate $1/\tau_{2}$ in QSLs with spinon Fermi surface.

For the $\mathbb{Z}_{2}$ QSL with a Fermi surface, for small $\omega$ and $q$
the situation is similar to that of a Fermi liquid Pal  _et al._ (2012). In a
clean, interacting Fermi liquid on a lattice, Umklapp scattering is the
leading momentum relaxation channel. In the presence of impurities, the low
temperature relaxation time is temperature independent, therefore:

$\frac{1}{\tau_{2}}\propto\begin{cases}T^{0},&\text{Impurity-dominated
scattering;}\\\ T^{2},&\text{Umklapp-dominated scattering.}\end{cases}$ (13)

In contrast, for the nFL U(1) QSL with Fermi surface, the spinons interact
with gapless gauge bosons. If the gauge bosons are assumed to form a thermal
bath Nave and Lee (2007), which can serve as a sink of energy and momentum,
then the relaxation happens via the momentum transferred from the spinons to
the gauge bosons. Clearly, this requires that the gauge bosons are an open
quantum system, which thermalizes faster than the rate of returning the energy
and momentum to the spinons. If the overall system of spinons and gauge bosons
is approximately closed, and thus momentum conserving, spinon-gauge boson
interactions have to be supplemented by Umklapp and impurity scattering. It
was shown Lee (2021) that if Umklapp scattering is dominant, and there is weak
impurity scattering, the momentum relaxation rate is qualitatively different
with respect to the case of scattering with thermal gauge bosons. We summarize
the two different situations described above, and additionally quote the
result for impurity scattering:

$\frac{1}{\tau_{2}}\propto\begin{cases}T^{0}&\text{Impurity scattering;}\\\
T^{4/3}&\text{Gauge boson scattering;}\\\ T&\text{Umklapp scattering.}\\\
\end{cases}$ (14)

##  III Theory without momentum relaxation in the QSL layer

In the case of $\tau_{D}^{-1}\gg\tau_{2}^{-1}$ with no momentum relaxation
within the QSL layer the drag rate is calculated with the formula Jauho and
Smith (1993); Narozhny and Levchenko (2016)

$\displaystyle\frac{1}{\tau_{\rm{D}}}$
$\displaystyle=\frac{1}{{p^{2}_{F1}}m_{{1}}}\int_{0}^{\infty}\frac{d\omega/T}{\sinh^{2}(\omega/2T)}$
$\displaystyle\times\int_{0}^{\infty}dq\,q^{3}|C(\textbf{q},\omega)|^{2}{\rm{Im}}[\Pi_{{1}}(\textbf{q},\omega)]^{2}.$
(15)

For simplicity, we concentrate on the case of metallic layers with equal
scattering rate, mass and Fermi energy, and polarization operator
$\Pi_{1}(\textbf{q},\omega)=\Pi_{3}(\textbf{q},\omega)$.

For the standard Coulomb drag, $C(\textbf{q},\omega)$ can be approximated by
the electrostatic potential between a point charge on the active layer and a
point charge of the passive layer. In the present case it is proportional to
the spin-spin correlation function with additional factor $J_{K}^{2}/N$.

The support for the integration in Eq. (15) is given by domain of the
imaginary parts of the polarization operator of metallic layers. As we explain
in more details below (cf. Figs. 2, 4, 5) there are combinations of the
parameters $\\{v_{Fi},p_{Fi}\\}$ for $i=1,2,3$, and of the soft frequency cut-
off that create regimes in which temperature scaling of the drag rate is
qualitatively different. In the next paragraphs we calculate the drag
relaxation rate separately for a $\mathbb{Z}_{2}$ Kitaev QSL and both for a
$\mathbb{Z}_{2}$ and a U(1) QSL together.

###  III.1 $\mathbb{Z}_{2}$ QSL with Dirac excitations (Kitaev QSL)

Figure 2: Temperature dependence of the relaxation rate in the regime of fast
spinons (panel a) and slow spinons (panel b) in a $\mathbb{Z}_{2}$ Kitaev QSL.
In the case of fast spinons there is no crossover temperature below the Fermi
temperature because in panel-b the line $\omega=2v_{F2}q$ never crosses the
shaded area of particle-hole continuum. In contrast, for slow spinons, such a
crossover takes place.

In the case of Kitaev QSLs we apply the formula from Eq. (15) with the spin-
correlator as in Eq. (10). As illustrated in Fig. 2 there are two
qualitatively different regimes, namely $v_{F2}>v_{F1}$ (fast spinons) and
$v_{F2}<v_{F1}$ (slow spinons). The support of
$\text{Im}[C^{R}(\textbf{q},\omega)]$ at $\omega>v_{F2}q$, which is indicative
of the continuum of particle-hole excitations about the Dirac point, overlaps
with the particle-hole continuum $\omega<v_{F1}q$ of the Fermi-liquid like
metallic bands only in the second case. This situation allows for a crossover
temperature $T_{C}{\simeq 2v_{F2}p_{F1}}$, above which the relative
contribution of real processes dominates over virtual quantum processes. As
shown in the Appendix A.2, in the limit of fast spinons we find

$\frac{1}{\tau_{D}}\simeq\frac{1}{\tau_{0}^{NC,K}}\left(\frac{T}{E_{F1}}\right)^{2}\log{\left(\frac{E_{F1}}{T}\right)},$
(16a) with
$\frac{1}{\tau_{0}^{NC,K}}=\frac{4}{3}\left(\frac{J_{K}^{2}}{N}\right)^{2}(C_{K}\nu_{{F1}})^{2}T_{C}^{2}E_{F1}.$
(16b)

The result of Eq. (16a) is shown in panel a) of Fig. 2. In the limit of slow
spinons we find instead

$\frac{1}{\tau_{D}}\simeq\frac{1}{\tau_{0}^{C,K}}\left(\frac{T}{T_{C}}\right)^{2}\mathrm{max}\left[1,\left(\frac{T}{T_{C}}\right)^{2}\right]\log{\left(\frac{E_{F1}}{T}\right)}$
(17a) and
$\frac{1}{\tau_{0}^{C,K}}=\left(\frac{T_{C}}{E_{F1}}\right)^{2}\frac{1}{\tau_{0}^{NC,K}}.$
(17b)

The result of Eq. (17a) is shown in panel c) of Fig. 2.

###  III.2 $\mathbb{Z}_{2}$ and U(1) QSLs with Fermi surface

We now turn to the situation of QSLs with Fermi surface. In the case of
spinons coupled to a $\mathbb{Z}_{2}$ gapped gauge field, the dynamical spin-
spin correlation function retains the Fermi liquid like behavior of the
Lindhard function. It is a more nontrivial fact that even for U(1) spin-
liquids the dynamical spin-spin correlation function is Fermi liquid-like for
small $\omega$ and $q$. As such, using the Aslamazov-Larkin diagrammatic
approach given by Eq. (15), U(1) and $\mathbb{Z}_{2}$ QSL can mostly be
treated at equal footing.

As in the Kitaev QSL there are two regimes, i.e. the limit of fast spinons
($v_{F2}>v_{F1}$) and the limit of slow spinons ($v_{F2}<v_{F1}$). In each of
these two cases, one may distinguish the case of small ($p_{F2}<p_{F1}$) or
large ($p_{F2}>p_{F1}$) spinon Fermi surface. The kinematic constraints for
these cases are also illustrated in Fig. 4 below.

For $\mathbb{Z}_{2}$ and U(1) QSLs a crossover temperature scale, $T_{C}\simeq
2v_{\rm min}p_{\rm min}\ll E_{F1,2}$, appears due to kinematic constraints in
the cases of slow spinons with large Fermi surface, see Fig. 4. We found that
in the limit of fast spinons

$\frac{1}{\tau_{D}}\simeq\left(\frac{J_{K}^{2}}{N}\right)^{2}\frac{(2\nu_{F2}\nu_{F1})^{2}}{E_{F1}}T^{2}\log{\biggl{(}\frac{E_{F1}}{T}\biggr{)}},$
(18)

while in the limit of slow spinons

$\begin{split}\frac{1}{\tau_{D}}\simeq&\frac{1}{4}\left(\frac{J_{K}^{2}}{N}\right)^{2}\frac{(\nu_{F1}\nu_{F2})^{2}}{E_{F1}}T^{2}\log{\left(\frac{E_{F1}}{T}\right)}\\\
\end{split}$ (19)

for $T<T_{C}$, and

$\frac{1}{\tau_{D}}\simeq\left(\frac{J_{K}^{2}}{N}\right)^{2}\frac{(2\nu_{F2}\nu_{F1})^{2}}{E_{F1}}T^{2}\log{\left(\frac{E_{F1}}{T}\right)}$
(20)

for $T>T_{C}$. Several comments are in order in relation to results presented
in this section. First, the term $\log{\left(\frac{E_{F1}}{T}\right)}$ appears
as a finite temperature regularization of the divergence at the edge of the
integration domain. Generically it appears in the lifetime of 2D Fermi liquids
and also for phonon-mediated drag Bønsager  _et al._ (2000); Khveshchenko
(2000). Second, the application of the above results to the case of U(1) QSLs
is valid only for spinons with large Fermi surface. In the opposite case, the
momentum integral extends beyond the limit of $q\ll p_{F2}$ for which the
spin-correlation function is known to take a Fermi liquid form. Third, for
both U(1) and $\mathbb{Z}_{2}$ QSL we find that the $1/\tau_{D}\ll 1/\tau_{2}$
under realistic conditions. A posteriori it is thus evident that the
physically relevant drag rate should be obtained in the limit of fast momentum
relaxation, which is treated using Boltzmann transport theory in the next
section.

##  IV Theory with momentum relaxation in the QSL layer

###  IV.1 General formalism

In this section, we present the effective Boltzmann treatment to account for
intra-layer momentum relaxation in QSLs with Fermi surface. Within this
formalism there are three coupled equations that describe how momentum is
relaxed in a two-step process from the passive layer to the active layer, see
Fig. 1 b). We are interested in the stationary state in which the applied
electric field in layer 1 generates a current that dissipates momentum within
this layer due to temperature independent impurity scattering (assumed to be
dominant) and transfers momentum to the QSL, that relaxes momentum within
itself and transfers momentum to the second metallic sheet. We assume the
linear response regime to be valid (small fields applied, small deviations
from equilibrium). This setup is described by the system of equations

$\displaystyle-e\textbf{E}_{1}\cdot\frac{\partial
f_{1}^{eq}}{\partial\textbf{p}_{1}}$ $\displaystyle=-\frac{\delta
f_{1}}{\tau_{1}}{+I^{\rm{coll}}_{2{\rightarrow}1}};$ (21a) $\displaystyle 0$
$\displaystyle=-\frac{\delta
f_{2}}{\tau_{2}}+I^{\rm{coll}}_{1{\rightarrow}2}+I^{\rm{coll}}_{3{\rightarrow}2};$
(21b) $\displaystyle 0$ $\displaystyle=-\frac{\delta
f_{3}}{\tau_{3}}+I^{\rm{coll}}_{2{\rightarrow}3}.$ (21c)

Here, $f_{1,3}=f_{1,3}(\textbf{p})$ are the distribution functions in the
metallic layers, and $f_{2}=f_{2}(\textbf{p})$ ($f_{2}=f_{2}(\hat{p},\omega)$)
is the distribution function (pseudo-distribution function) in the QSL in the
case of a $\mathbb{Z}_{2}$ ($U(1)$) QSL with spinon Fermi surface. We treat
intralayer momentum relaxation in the relaxation time approximation, where
$\delta f_{i}=f_{i}-f_{i}^{\rm eq}$ with $i=1,2,3$. In our treatment the
momentum relaxation rate within each layer including the QSL is an external
phenomenological parameter that allows for the description in the variety of
physical situations described in Eqs. (13)–(14). The additional collision
integrals $I^{\rm{coll}}_{j\rightarrow i}(\textbf{p}_{i})$ account for the
transfer of momentum between the layers and give rise to the drag resistivity.
They are treated beyond relaxation time approximation and are derived in the
next section. Linearizing the distribution function we can write the drag
current as

$\textbf{j}_{3}=-\frac{e}{m_{3}}\int\frac{d^{2}p_{3}}{(2\pi)^{2}}\delta
f_{3}(\textbf{p}_{3})\textbf{p}_{3}.$ (22)

Thus, the solution of Eqs. (21) and (22) lead to the calculation of
$1/\tau_{D}$.

###  IV.2 Collision integral of the momentum transfer

Figure 3: Self-energy diagram associated with the Kondo interaction between
layers (dashed lines). Using the standard derivation of the Boltzmann equation
from Keldysh technique, this diagram leads to Eq. (23) describing the transfer
of energy and momentum between layers.

In this section, we summarize the calculation of the collision integral
describing the momentum transfer between adjacent layers, leaving details to
Appendix B.1. Using the Kondo coupling, Eq. (12), the Dyson equation in the
Schwinger-Keldysh formalism, and a gradient expansion, the leading self-energy
contribution due to Kondo coupling, Fig. 3, leads to

$\displaystyle I^{\rm{coll}}_{j\rightarrow i}=$
$\displaystyle\frac{J_{K}^{2}}{N}\int(d{p}_{i^{\prime}})(d{p}_{j})(d{p}_{j^{\prime}}){\delta({P}_{i}+{P}_{j}-{P}_{i^{\prime}}-{P}_{j^{\prime}})}$
$\displaystyle\times$
$\displaystyle\left\\{f^{\rm{eq}}_{i}f^{\rm{eq}}_{j}[1-f^{\rm{eq}}_{i^{\prime}}][1-f^{\rm{eq}}_{j^{\prime}}](\psi_{i}+\psi_{j}-\psi_{i^{\prime}}-\psi_{j^{\prime}})\right\\}.$
(23)

In this expression we have absorbed a constant of order unity into $J_{K}$,
and employed the linearization of the Boltzmann equation using

$\delta f_{i}=f_{i}^{\rm eq}(1-f_{i}^{\rm eq})\psi_{i}.$ (24)

The meaning of the $P_{i}$ entering the three dimensional delta function
enforcing energy and momentum conservation is explained below. For the non-
linearized expression, see Eq. (77) of the Appendix.

It is important to stress several points concerning the notation and the
fundamental distinction between a Fermi liquid and a nFL. For the metal layer
and the $\mathbb{Z}_{2}$ QSL

$\int(d{p})(...)=\int\frac{d^{2}p}{(2\pi)^{2}}(...),\;f^{\rm
eq}_{{i}}=f^{eq}(\epsilon_{\textbf{p}_{i}}),\;{\psi_{i}=\psi(\textbf{p}_{i})}.$
(25)

Consequently, the energy-momentum three-vector is
$P_{i}=(\textbf{p}_{i},\epsilon_{\textbf{p}_{i}})$. In contrast, for the U(1)
QSL layer, because of the compromised definition of quasi particles,

$\int(d{p})(...)=\nu\int
d\epsilon\langle...\rangle_{{\hat{p}}},\;f^{eq}_{i}=f(\epsilon_{i}),\;{\psi_{i}=\psi(\hat{p}_{i},\epsilon_{i})},$
(26)

where $\nu=m_{2}/2\pi$ is the density of states per spin and
$\textbf{p}_{i}=p_{F2}\hat{p}_{i}$, is implied everywhere, so that
$P_{i}=(p_{F2}\hat{p}_{i},\epsilon_{i})$, in this case.

###  IV.3 General solution for the drag rate

The system of Eqs. (21)–(22) can be manipulated to derive a general formula
for the drag conductivity (see Appendix B.2 for details):

$\displaystyle\frac{1}{\tau_{D}}$
$\displaystyle=-\frac{(2\pi)^{5}}{n_{1}n_{3}}\frac{\tau_{2}}{2T}\left(\frac{J_{K}^{2}}{N}\right)^{2}\int
dQdQ^{\prime}(\textbf{q}\cdot\hat{E}_{1})(\textbf{q}^{\prime}\cdot\hat{E}_{1})$
$\displaystyle\times\frac{\mathrm{Im}[\Pi_{3}^{\mathrm{R}}(Q)]\mathrm{Im}[\Pi_{1}^{\mathrm{R}}(Q^{\prime})]I_{2}(Q,Q^{\prime})}{\sinh{\bigl{(}\frac{\omega}{2T}\bigr{)}}\sinh{\bigl{(}\frac{\omega^{\prime}}{2T}\bigr{)}}\sinh{\bigl{(}\frac{\omega+\omega^{\prime}}{2T}\bigr{)}}},$
(27)

with $Q=(\textbf{q},\omega)$ and
$dQ=\frac{d^{2}q}{(2\pi)^{2}}\frac{d\omega}{2\pi}$, and we have defined

$\begin{split}I_{2}(Q,Q^{\prime})&=\int(d{p}_{2})(d{p}_{2^{\prime}})(d{p}_{2^{\prime\prime}})\delta^{(3)}({P_{2^{\prime\prime}}-P_{2^{\prime}}+Q})\times\\\
&\times\delta^{(3)}({P_{2^{\prime\prime}}-P_{2}-Q^{\prime}})(f_{2}-f_{2^{\prime}}).\end{split}$
(28)

Apart from the behavior of $\tau_{2}$, the integral $I_{2}(Q,Q^{\prime})$
encodes the main difference between the the $\mathbb{Z}_{2}$ and the U(1) QSL.

###  IV.4 $\mathbb{Z}_{2}$ QSL with Fermi surface

Figure 4: Temperature dependence of the spinon-mediated drag in
$\mathbb{Z}_{2}$ QSLs, Eqs. (31a)–(32a), and regions of kinematic constraints
for the momentum transfer. The combination of Fermi velocities and momenta of
spinons and electrons leads to four regimes (particle-hole continua in the
metallic lead (QSL) are shaded light blue (dark purple)). The panels b) and h)
are symmetric under the exchange $1\leftrightarrow 2$ and differ from each
other only for the value of $\tau_{0}^{{C,\mathbb{Z}_{2}}}$, whose definition
is generally given by Eq. (32b) and $T_{C}=2v_{\mathrm{min}}p_{\mathrm{min}}$.
Finally, panels b) and c) are symmetric (for $1\leftrightarrow 2$) and only
differ for the value of $\tau_{0}^{{NC,\mathbb{Z}_{2}}}$ defined in Eq. (31b).

In the case of the $\mathbb{Z}_{2}$ QSL with the Fermi surface we calculate
the object $I_{2}(Q,Q^{\prime})$ to be

$\begin{split}&{\langle
I_{2}(Q,Q^{\prime})({\textbf{q}\cdot\hat{E}_{1}})({\textbf{q}^{\prime}\cdot\hat{E}_{1}})\rangle_{\hat{q},\hat{q}^{\prime}}}={-}\frac{4T\sinh{\bigl{(}\frac{\omega+\omega^{\prime}}{2T}\bigr{)}}}{(2\pi)^{6}\omega\omega^{\prime}E_{F2}}\\\
&\times\biggl{(}\omega+\frac{q^{2}}{2m_{2}}\biggr{)}\biggl{(}\omega^{\prime}+\frac{q^{\prime
2}}{2m_{2}}\biggr{)}{\rm{Im}}\left[\Pi_{2}^{R}(\textbf{q},\omega)\right]{\rm{Im}}\left[\Pi_{2}^{R}(\textbf{q}^{\prime},\omega^{\prime})\right],\\\
\end{split}$ (29)

which holds in the regime

$\frac{\omega}{v_{\rm{F}}}\left(1+\frac{T}{2E_{F}}\right)<q<\left(2p_{\rm{F}}-\frac{\omega}{v_{\rm{F}}}\right)\left(1-\frac{T}{2E_{F}}\right)$
(30)

that defines the dominant contribution to the integrals entering the drag
rate.

The insertion of Eq. (29) into the general formula for the drag rate, Eq.
(27), implies that the domain of integration over $Q,Q^{\prime}$ is bounded by
the overlap of particle-hole continua in both metallic and QSL Fermi liquids,
see Fig. 4. Physically, this means that only real scattering processes between
particle-hole pairs of adjacent layers allow for momentum transfer. As a
consequence, the combinations of the parameters $p_{Fi}$ and $v_{Fi}$ give
rise to four regimes that we indicate formally by the Cartesian product
$(v_{F1}>v_{F2},v_{F1}<v_{F2})\times(p_{F1}>p_{F2},p_{F1}<p_{F2})$. This is
analogous to the regimes introduced in Sec. III.2, but in contrast to the
present discussion, we considered there the limit of absent momentum
relaxation in the QSL and thereby also included virtual particle-hole
excitations in the QSL.

We first discuss the limit
$({v_{F1}}>{v_{F2}}\land{p_{F1}}>{p_{F2}})\lor({v_{F1}}<{v_{F2}}\land{p_{F1}}<{p_{F2}})$,
corresponding to panels d) and f) of Fig. 4. As we assume the low-temperature
degenerate Fermi gas limit of either the metallic plates or the QSL, the
minimal Fermi energy $E_{\text{min}}=\text{min}\left(E_{F1},E_{F2}\right)$
bounds the temperature limiting the energy integration. Hence, the drag rate
in the symmetric case is

$\frac{1}{\tau_{D}}=\frac{1}{\tau_{0}^{NC,\mathbb{Z}_{2}}}\left(\frac{T}{E_{\text{min}}}\right)^{2},$
(31a) with
$\frac{1}{\tau_{0}^{NC,\mathbb{Z}_{2}}}=\frac{1}{4\pi^{7}}\left(\frac{J_{K}^{2}}{N}\right)^{2}\left(\frac{E^{2}_{\text{min}}}{E_{F1}E_{F2}}\right)\frac{\nu_{2}p_{\text{min}}^{4}}{v_{F1}^{2}},$
(31b)

as shown in panels c) and e) of Fig. 4.

We discuss next the regimes $(v_{F1}>v_{F2}\land
p_{F2}<p_{F1})\lor(v_{F1}<v_{F2}\land p_{F1}>p_{F2})$. As shown in panels b)
and h) Fig. 4 the crossover temperature $T_{C}\simeq
2v_{\text{min}}p_{\text{min}}$ [with
$v_{\text{min}}=\min\left(v_{F1},v_{F2}\right)$ and
$p_{\text{min}}=\min\left(p_{F1},p_{F2}\right)$] emerges. For $T<T_{C}$, the
frequency integration is bounded by $T$, while the overlap of particle-hole
continua limits the $\omega$ integral at $T_{C}$ for $T_{C}<T$. Importantly,
the crossover temperature scale is below $E_{\rm min}$ and thus relevant for
the regime of applicability of our theory. The drag rate in these regimes is
thus

$\frac{1}{\tau_{D}}=\frac{1}{\tau_{0}^{C,\mathbb{Z}_{2}}}\left(\frac{T}{T_{C}}\right)^{2}\mathrm{min}\left[1,\left(\frac{T_{C}}{T}\right)^{2}\right],$
(32a) with
$\frac{1}{\tau^{C,\mathbb{Z}_{2}}_{0}}=\pi\left(\frac{E_{\text{min}}}{T_{C}}\right)^{2}\frac{1}{\tau^{NC,\mathbb{Z}_{2}}_{0}}.$
(32b)

These results are plotted in panels a) and g) in Fig. 4.

###  IV.5 U(1) QSL with Fermi surface

Figure 5: Temperature dependence of the spinon-mediated drag in $U(1)$ QSLs,
per Eqs. (34)–(36), and kinematic constraints for the momentum transfer.
Panels a) and c) illustrate that $\tau_{2}^{-1}\gg\tau_{D}^{-1}$, meaning that
momentum relaxation in the QSL is faster than interlayer momentum transfer.
The latter underpins applicability of the semiclassical Boltzmann approach.

In the case of the U(1) QSL we find

$\begin{split}&{\bigl{\langle}I_{2}(Q,Q^{\prime})(\textbf{q}\cdot\hat{E_{1}})(\textbf{q}^{\prime}\cdot\hat{E}_{1})\bigr{\rangle}_{\hat{q},\hat{q}}}=\\\
&=-\frac{\nu^{3}_{2}}{\pi
p_{F2}^{2}}{({\omega+\omega^{\prime}})}\frac{\theta(2p_{F2}-q)q}{\sqrt{(2p_{F2})^{2}-q^{2}}}\frac{\theta(2p_{F2}-q^{\prime})q^{\prime}}{\sqrt{(2p_{F2})^{2}-q^{\prime
2}}},\end{split}$ (33)

Insertion of Eq. (33) into Eq. (27) leads to the general equation for the drag
rate in the case of U(1) QSL. As opposed to the $\mathbb{Z}_{2}$ case, Eq.
(29), Eq. (33) only contains a sharp momentum cutoff $\theta(2p_{F2}-q)$ that
is a consequence of the fact that for this pseudo quasiparticles the momentum
is pinned on the Fermi surface but no bound in the frequency integrals over
$\omega$, $\omega^{\prime}$. This is a consequence of the diffuse nature of
the fermionic spectral weight in a nFL with self-energy
$\Sigma\sim\epsilon^{2/3}$, which allows for real scattering processes even
when energy-conservation would be violated in a Fermi liquid with sharp quasi-
particles.

The altered kinematic constraints, see Fig. 5, for the nFL lead to
qualitatively different regimes and behaviors of $\tau_{D}^{-1}(T)$, because
in this case the velocity $v_{F2}$ does not play any role. We can therefore
distinguish two limits: (i) $p_{F2}>p_{F1}$ in which case, as shown in Fig. 5
b), the $\omega$-integration is bounded by $T<E_{F1}$, and there is no other
relevant energy scale; (ii) the limit $p_{F2}<p_{F1}$, in which case the
integration can be bounded by the domain integration.

In the symmetric regime for $p_{F2}\gg p_{F1}$ the drag rate is found in the
form

$\frac{1}{\tau_{D}}=\frac{1}{\tau_{0}^{NC,U(1)}}\left(\frac{T}{E_{F1}}\right)^{\frac{8}{3}},$
(34)

with

$\frac{1}{\tau_{0}^{NC,U(1)}}=\frac{128}{\pi^{2}}\left(\frac{J_{K}^{2}}{N}\right)^{2}\frac{\nu_{1}\nu_{3}\nu_{2}^{3}}{p_{F2}^{2}}\frac{E_{F1}^{4}}{T_{C}^{2}}\left(\frac{E_{F2}}{E_{F1}}\right)^{1/3}.$
(35)

This result is plotted in panel a) of Fig. 5. In the limit of $p_{F2}\ll
p_{F1}$ the temperature dependence depends on a crossover temperature
$T_{C}=2v_{F1}p_{F,}$ and the relaxation rate is given by

$\frac{1}{\tau_{D}}=\frac{1}{\tau_{0}^{C,U(1)}}\left(\frac{T}{T_{C}}\right)^{\frac{8}{3}}\mathrm{min}\left(1,\frac{T_{C}}{T}\right)^{2},$
(36)

where the prefactor is given by

$\frac{1}{\tau_{0}^{C,U(1)}}=\frac{1}{\tau_{0}^{NC,U(1)}}\left(\frac{1}{4}\right)\left(\frac{p_{F1}}{p_{F2}}\right)^{2}\left(\frac{T_{C}}{E_{F1}}\right)^{\frac{14}{3}}.$
(37)

##  V Summary, Discussion, and outlook

In summary, we have derived the spinon-mediated induced drag rate for
heterostructures as displayed in Fig. 1 a) for a three types of QSLs: Kitaev
QSL beyond the integrable limit; $\mathbb{Z}_{2}$ QSL with Fermi surface; U(1)
QSL with Fermi surface. We have distinguished the situation of the slow
momentum relaxation within the QSL layer (as compared to the drag rate) from
the situation of fast momentum relaxation. In the first case, we have derived
the drag rate using Aslamazov-Larkin diagrammatics, while we employed a
Boltzmann-like description for the second case. We conclude that the latter
limit appears experimentally more realistic (see dashed curves in Figs. 4,5).
The semiclassical Boltzmann approach assumes the Fermi wavelength to be short
as compared to the mean free path, which is directly applicable only to QSLs
with a Fermi surface in a proper range of parameters. In contrast to earlier
studies Chen  _et al._ (2013); Chatterjee and Sachdev (2015); Okamoto (2016);
Werman  _et al._ (2018); Aftergood and Takei (2020); Zhuang and Marston
(2021), the transport phenomena discussed here are purely electrical, and do
not rely on the spin injection through spin-orbit interaction in the leads.

We now summarize and discuss main results as displayed in Figs. 1 c) (for
graphene) and 6 (for 2DEGs with parabolic dispersion) as a function of
experimentally tunable parameters, namely temperature and charge carriers
density $n\sim p_{F1}^{2}$ in the metallic leads. The difference between the
two figures is due to the independence (square-root dependence) of the Fermi
velocity on the carrier density in graphene (in a 2DEG with parabolic
dispersion). Note that in all our Boltzmann calculations we assumed a
quadratic dispersion both for the metallic layers and for the QSL layer.
However, as previously discussed, the results most crucially depend on the
Fermi-liquid like particle-hole continua of the metallic leads. At small
momenta excitations, these are the same for graphene or 2DEGs with parabolic
dispersion. Therefore, our calculations are expected to qualitatively describe
metallic layers made of doped graphene, as well (see also Sec. II.1).

Figure 6: The figure shows the scaling behavior of $\tau_{D}^{-1}$ as a
function of temperature and charge carriers in the metallic layers and in the
symmetric regime, in the case of the metallic plates having a parabolic
dispersion. Only the results for the $\mathbb{Z}_{2}$ and U(1) QSL with Fermi
surface are reported because the notion of $m_{2}$ does not exist in the
context of a Kitaev QSL.

Except for the Kitaev QSL, our approach assumes a degenerate Fermi gas $T\ll
E_{F}$ both for the metallic leads and the spinons. The purple region in all
the panels in Figs. 1 c) and 6 is beyond this assumption. The remaining areas
of the parameter space contain characteristic crossover temperature scales,
above which the power exponent of the temperature dependence of the drag rate
changes. This is illustrated by means of the color coding in Figs. 1 c) and 6
and a consequence of the kinematic constraints of energy and momentum
transfer.

Specifically, in the case of the Kitaev QSL, a crossover in temperature occurs
$T_{C}=2v_{F2}p_{F1}$ below the red region in the case of slow spinons
($v_{F2}<v_{F1}$). In a $\mathbb{Z}_{2}$ QSL with Fermi surface there is a
crossover temperature $T_{C}=2v_{\text{min}}p_{\text{min}}$ in both the cases
of fast and slow spinons. Note that for graphene leads, Fig. 1 c), in the case
of fast spinons $T_{C}=2v_{F1}p_{F2}$, so that the characteristic crossover
temperature is not a function of density $n$. Finally, for the U(1) QSL with
Fermi surface the drag relaxation time has a crossover temperature
$T_{C}=2v_{F1}p_{F2}$ only in the limit of fast spinons. We stress that this
is a characteristic difference as compared to the case of the $\mathbb{Z}_{2}$
QSL and ultimately a consequence of the smeared fermionic spectral weight due
to the non-Fermi-liquid behavior of the U(1) QSL. Another key distinction of
the U(1) QSL are the fractional powers in the temperature dependence, which
reflect the momentum relaxation rate $1/\tau_{2}\sim T^{4/3}$ inside the QSL.

We conclude with a discussion of experimental consequences of this study.
First, as in all 2DEG double-layers, Coulomb interaction provides the leading
mechanism for drag current generation. A way to experimentally suppress the
contribution of Coulomb interaction without affecting the spinon drag
contribution is to embed the heterostructure in the environment of a strong
dielectric, such as SrTiO3, which are often employed experimentally for gating
purposes. Second, the spinon mediated drag is a fourth order effect in Kondo
coupling $J_{K}$ and thus naively small in amplitude. However, the reference
energy scale determining the drag rate is given by a combination of $E_{F1}$
and the microscopic energy scale of the QSL $J$. While the former can be
reduced by gating whereby the spinon induced drag increases, the ratio
$J_{K}/J$ determines whether the heterostructure is in a topologically
ordered, fractionalized Fermi liquid Senthil  _et al._ (2003) (FL∗) with small
Fermi surface (small $J_{K}/J$) or a Kondo-screened phase of a topologically
trivial Fermi liquid (FL) with large Fermi-surface (large $J_{K}/J$).
Importantly, also the second regime appears experimentally realistic Mashhadi
_et al._ (2019).

We conclude with a speculation about the behavior of the drag rate as a
function of $J_{K}/J$ as the FL∗ to FL transition is approached and
hypothesize that the $1/\tau_{D}\sim J_{K}^{4}$ increase levels off at the
transition and gives way to a regime of perfect drag, akin to the situation of
perfect drag in the presence of an exciton condensate Nandi  _et al._ (2012).
Indeed, the mean field description of the Kondo-screened phase implies that
only the symmetrized wavefunction of electrons of layers 1 and 3 form the FL
with large Fermi surface, while the antisymmetric wave function of electrons
constitutes an independent band with small Fermi surface. The homogeneous dc
field $\textbf{E}_{1}$ does not allow for interband coupling and has support
in both bands, the same is true for the current operator $\textbf{j}_{3}$. By
consequence, electrons in both bands are accelerated and equally contribute to
drag and transport current, ultimately leading to the perfect drag phenomenon
(same local and non-local conductances). It will be an interesting problem for
the future to study this effect in more detail, as well as the impact of
magnetic field on the spinon mediated drag.

###### Acknowledgements.

We gratefully acknowledge useful discussions with Inti Sodemann, Ciaran
Hickey, Marko Burghard. A.L. is grateful to the Max Planck Institute for Solid
State Research for hospitality, where this work was initiated, and to the
Alexander von Humboldt Foundation for the financial support.

##  Appendix A Limit of absent momentum relaxation within QSL layer:
Aslamazov-Larkin-Diagram

In this appendix, we present details complementing discussion in Sec. III. As
argued in the main text, when the condition
$\tau_{2}^{-1}\ll\tau_{\rm{D}}^{-1}$ is satisfied, $\tau_{D}^{-1}$ is
analogous to the Coulomb drag one Narozhny and Levchenko (2016), as determined
by the Aslamazov-Larkin diagram (see Eq. (15)).

In the case of QSL with Fermi surface and for small $\omega$ and $q$, the spin
correlation function is proportional to the free electron polarization
operator

$\Pi_{{i}}^{\rm{R}}(\textbf{q},\omega)=\int(d{k})\,\frac{f^{eq}(\epsilon_{\textbf{k}})-f^{eq}(\epsilon_{\textbf{k}+\textbf{q}})}{\epsilon_{\textbf{k}}-\epsilon_{\textbf{k}+\textbf{q}}-\omega+i0^{+}},$
(38)

with $\epsilon_{\textbf{k}}=|\textbf{k}|^{2}/2m-E_{Fi}$, $E_{Fi}$ being the
Fermi energy, with $f^{eq}$ being the Fermi-Dirac equilibrium distribution
function.

###  A.1 Properties of the polarization operator

Despite the strong coupling nature and the nFL behavior of the U(1) QSL, it is
still possible to show that for small $\omega$ and $q$ the polarization
operator for the $\mathbb{Z}_{2}$ and the U(1) QSLs are essentially the same
as the free electron polarization operator Altshuler  _et al._ (1994).
Furthermore, if $\Pi^{R}(\textbf{q},\omega)_{T=0,E_{F}=E}$ is a smooth
function in an interval with a width of $\delta E\simeq{T}$ around $E=E_{F}$,
we can use the zero temperature expression of the polarization operator also
in the finite temperature situation. At zero temperature ${T}=0$ the
polarization operator is Giuliani and Vignale (2005)

$\begin{split}&\text{Im}\left[\Pi^{R}(\textbf{q},\omega)\right]=-\nu\frac{m}{q^{2}}\\{\theta((v_{F}q)^{2}-\omega_{-}^{2})\sqrt{(v_{F}q)^{2}-\omega_{-}^{2}}\\\
&-\theta((v_{F}q)^{2}-\omega_{+}^{2})\sqrt{(v_{F}q)^{2}-\omega_{+}^{2}}\\};\\\
&\text{Re}\left[\Pi^{R}(\textbf{q},\omega)\right]=-\nu\biggl{\\{}1+\nu\frac{m}{q^{2}}\\{\text{sign}(\omega_{-})\theta(\omega_{-}^{2}-(v_{F}q)^{2})\times\\\
&\times\sqrt{\omega_{-}^{2}-(v_{F}q)^{2}}-\text{sign}(\omega_{+})\theta(\omega_{+}^{2}-(v_{F}q)^{2})\sqrt{\omega_{+}^{2}-(v_{F}q)^{2}}\\}\biggr{\\}},\end{split}$
(39)

where $\omega_{\pm}=\left(\omega\pm\frac{q^{2}}{2m}\right)$ and $\nu$ is the
density of states at the Fermi surface. To define quantitatively the
applicability condition of $T\to 0$ expressions one turns to the imaginary
part of the finite temperature polarization operator that can be written as
follows

$\displaystyle{\rm{Im}}[\Pi^{R}(\mathbf{q},\omega)]$
$\displaystyle=-\nu\frac{m}{q^{2}}\int_{-\infty}^{+\infty}dE\left(\frac{\partial
f^{eq}}{\partial E}\right)\times$
$\displaystyle\times\biggl{\\{}\sqrt{(v_{F}q)^{2}-\omega_{-}^{2}+\frac{2q^{2}}{m}(E-E_{F})}$
$\displaystyle-\sqrt{(v_{\rm{F}}q)^{2}-\omega_{+}^{2}+\frac{2q^{2}}{m}(E-E_{F})}\biggr{\\}}.$
(40)

Here, it is assumed that the argument of the square roots is positive
(otherwise the expression vanishes per definition).

We now comment on the role of temperature in determining the boundaries of the
integral support. We note that the term $(v_{F}q)^{2}-\omega_{\pm}^{2}$ would
vanish in proximity of the boundary of the support of the imaginary part of
the polarization operator. The only term that keeps the square root smooth is
$\frac{2q^{2}}{m}(E-E_{F})$ which is bounded by $T$, so that Eq. (39) will
always be valid provided that the condition

$(v_{F}q)^{2}-\omega_{\pm}^{2}>\frac{2q^{2}}{m}T$ (41)

is satisfied. This leads to a restriction in the integration over $q$ given by
Eq. (30). This integration domain is represented in blue in Fig. 7. It should
be pointed out that in this regime the expansion of
$\Pi_{T=0}(\mathbf{q},\omega)$ coincides with $\Pi_{T\neq
0}(\mathbf{q},\omega)$. Also $\Pi_{T=0}^{R}(\mathbf{q},\omega)$ can be
expanded if $\omega_{\pm}^{2}<<(v_{F}q)^{2}\longrightarrow|\omega_{\pm}|\ll
v_{F}q$. This condition is less restrictive than (41), so since we limit the
integration in this domain, we can expand
${\mathrm{Im}}[\Pi^{R}(\mathbf{q},\omega)]$ to obtain

$\mathrm{Im}[\Pi^{R}(\mathbf{q},\omega)]\simeq\frac{\nu\omega}{\sqrt{(v_{F}q)^{2}-\bigl{(}\omega^{2}+\bigl{(}\frac{q^{2}}{2m}\bigr{)}^{2}\bigr{)}}}.$
(42)

where we have dropped to subscript $T=0$ because from now on we will deal only
with the $T=0$ polarization operator. The expression is valid for the whole
integration regime and especially for the QSL we will make large use of the
approximation $q\ll p_{F}$ that leads to

$\Pi^{R}(\mathbf{q},\omega)\simeq\nu\left(1+i\frac{\omega}{\sqrt{(v_{F}q)^{2}-\omega^{2}}}\right).$
(43)

Figure 7: Integration domain used for integrals in Eq. (15) (light blue
shading). Solid lines are given by Eq. (30). The areas in purple are
discarded, their contributions are higher orders of $\frac{T}{E_{F}}$. Panels
a) and b): $\mathbb{Z}_{2}$ QSL with Fermi surface in the case of fast and
slow spinons respectively. As shown in panel b), in the case of slow spinons
there is a crossover in the behavior of $|C^{R}(\textbf{q},\omega)|$. The same
considerations old true for the Kitaev QSL (panels c) and d)) with a
difference for the crossover of the spin correlation function over the line
$\omega=v_{F2}q$.

###  A.2 $\mathbb{Z}_{2}$ QSL with Dirac excitations (Kitaev QSL)

We rewrite Eq. (15) as follows

$\frac{1}{\tau_{D}}={\left(\frac{J_{K}^{2}}{N}\right)^{2}}\frac{1}{p_{F1}^{2}mT}\int_{0}^{\infty}\frac{d\omega}{\sinh^{2}{\bigl{(}\frac{\omega}{2T}\bigr{)}}}I_{q}(\omega)$
(44)

and consider separately limits of slow and fast spinons as discussed in the
main text.

####  A.2.1 Limit of fast spinons

We start with the case $v_{F2}>v_{F1}$ in which
$I_{q}(\omega)=I^{{NC}}_{q}(\omega)$,

$\begin{split}&I^{{NC}}_{q}(\omega)=\int
dqq^{3}\left(\frac{(\nu_{1}\omega)^{2}}{(v_{F1}q)^{2}-(\frac{q^{2}}{2m})^{2}-\omega^{2}}\right)\times\\\
&\times
C_{K}^{2}((v_{F2}q)^{2}-\omega^{2})=(C_{K}\nu_{1})^{2}(\omega^{2}v_{F2}^{2}I_{1}-\omega^{4}I_{2}),\end{split}$
(45)

here the NC stands for non-crossover, indicating that the crossover line is
outside of the integration domain, and

$I_{1}=\int_{\frac{\omega}{v_{F1}}\left(1+\frac{T}{2E_{F1}}\right)}^{-\frac{\omega}{v_{F1}}\left(1-\frac{T}{2E_{F1}}\right)+2p_{F1}\left(1-\frac{T}{2E_{F1}}\right)}\frac{dqq^{5}}{(v_{F1}q)^{2}-(\frac{q^{2}}{2m})^{2}-\omega^{2}},$
(46)

$I_{2}=\int_{\frac{\omega}{v_{F1}}\left(1+\frac{T}{2E_{F1}}\right)}^{-\frac{\omega}{v_{F1}}\left(1-\frac{T}{2E_{F1}}\right)+2p_{F1}\left(1-\frac{T}{2E_{F1}}\right)}\frac{dqq^{3}}{(v_{F1}q)^{2}-(\frac{q^{2}}{2m})^{2}-\omega^{2}}.$
(47)

The integral $I_{1}$ can be evaluated to be

$I_{1}=8p_{F1}^{2}m^{2}\left[\log{\left(\frac{E_{F1}}{T}\right)-1-\log{\left(1+\frac{\omega}{2T}\right)}}\right],$
(48)

while the integral $I_{2}$ can be calculated to be

$I_{2}=2m^{2}\left[\log{\left(\frac{E_{F1}}{T}\right)-\log{\left(1+\frac{\omega}{2T}\right)}}\right].$
(49)

Integration over $\omega$ of Eq. (45) yields the results presented in Eq.
(16a) and (16b) and shown in panel a) of Fig. 2.

####  A.2.2 Limit of slow spinons

We turn now to the regime in which $v_{F2}<v_{F1}$ which contains a typical
crossover temperature $T_{C}=2v_{F2}p_{F2}$ and frequency

$\omega_{C}=T_{C}\left(1-\frac{T}{2E_{F1}}\right),$ (50)

see Fig. 7 d). The $\omega$ integral appears exactly in the same form as (44),
but this time we write

$\begin{split}&I^{{C}}_{q}=C_{K}^{2}[\omega^{4}I_{3}+\omega^{2}I_{4}],\end{split}$
(51)

with

$\displaystyle I_{3}$
$\displaystyle=\int_{\frac{\omega}{v_{F1}}\left(1+\frac{T}{2E_{F1}}\right)}^{\frac{\omega}{v_{{2}}}}\frac{dq\,q^{3}}{(v_{F1}q)^{2}-(\frac{q^{2}}{2m})^{2}-\omega^{2}}$
$\displaystyle-\int_{\frac{\omega}{v_{{2}}}}^{\left(2p_{F1}-\frac{\omega}{v_{F1}}\right)\left(1-\frac{T}{2E_{F1}}\right)}\frac{dq\,q^{3}}{(v_{F1}q)^{2}-(\frac{q^{2}}{2m})^{2}-\omega^{2}}$
$\displaystyle=2m^{2}\biggl{\\{}\log{\left(\frac{T}{E_{F1}}\right)}-2\log{\left(1-\left(\frac{\omega}{T_{C}}\right)^{2}\right)}$
$\displaystyle+\log{\biggl{(}1+\frac{\omega}{2T}\biggr{)}}\biggr{\\}};$ (52)

and

$\displaystyle I_{4}$
$\displaystyle=\int_{\frac{\omega}{v_{{2}}}}^{\left(2p_{F1}-\frac{\omega}{v_{F1}}\right)\left(1-\frac{T}{2E_{F1}}\right)}\frac{dq\,q^{5}{v_{F2}^{2}}}{(v_{F1}q)^{2}-(\frac{q^{2}}{2m})^{2}-\omega^{2}}$
$\displaystyle-\int_{\frac{\omega}{v_{F1}}\left(1+\frac{T}{2E_{F1}}\right)}^{\frac{\omega}{v_{{2}}}}\frac{dq\,q^{5}{v_{F2}^{2}}}{(v_{F1}q)^{2}-(\frac{q^{2}}{2m})^{2}-\omega^{2}}$
$\displaystyle=-2m^{2}T_{C}^{2}\biggl{\\{}\left(1-\left(\frac{\omega}{T_{C}}\right)^{2}\right)-2\log{\biggl{(}1-\left(\frac{\omega}{T_{C}}\right)^{2}}\biggr{)}$
$\displaystyle+\log{\biggl{(}1-\frac{\omega}{2T}}\biggr{)}+\log{\left(\frac{T}{E_{F1}}\right)}\biggr{\\}}.$
(53)

We separately study the case $T<T_{C}$ and the case $T>T_{C}$. In the first
case the integration is bounded by $T$ we find

$\frac{1}{\tau_{D}}\simeq
4{\left(\frac{J_{K}^{2}}{N}\right)^{2}}\frac{(C_{K}\nu_{1})^{2}}{E_{F1}}T_{C}^{2}T^{2}\log{\left(\frac{E_{F1}}{T}\right)}$
(54)

In the opposite case $T>T_{C}$, the $\omega$ integral splits into two

$\frac{1}{\tau_{D}}=\frac{1}{\tau_{D}^{(1)}}+\frac{1}{\tau_{D}^{(2)}}$ (55)

where we can define

$\frac{1}{\tau_{D}^{(1)}}={\left(\frac{J_{K}^{2}}{N}\right)^{2}}\int_{0}^{T_{C}\left(1-\frac{T}{2E_{F1}}\right)}\frac{d\omega}{\sinh^{2}{\bigl{(}\frac{\omega}{2T}\bigr{)}}}I_{q}^{C}(\omega),$
(56)

where $I_{q}^{C}(\omega)$ defined as in (51), while

$\frac{1}{\tau_{D}^{(2)}}={\left(\frac{J_{K}^{2}}{N}\right)^{2}}\int_{T_{C}}^{T}\frac{d\omega}{\sinh^{2}{\bigl{(}\frac{\omega}{2T}\bigr{)}}}I_{q}^{NC}(\omega),$
(57)

with $I_{q}^{NC}(\omega)$ defined as in (45) but with a minus sign in front.
We find that

$\frac{1}{\tau_{D}^{(1)}}\simeq{\left(\frac{J_{K}^{2}}{N}\right)^{2}}\frac{16}{3}\frac{(C_{K}\nu_{1})^{2}}{E_{F1}}(T_{C})^{3}T\log{\left(\frac{E_{F1}}{T}\right)},$
(58)

and

$\frac{1}{\tau_{D}^{(2)}}\simeq
4{\left(\frac{J_{K}^{2}}{N}\right)^{2}}\frac{(C_{K}\nu_{1})^{2}}{E_{F1}}T^{4}\log{\left(\frac{E_{F1}}{T}\right)}.$
(59)

In the $T/T_{C}\gg 1$ limit
$\frac{1}{\tau_{D}^{(1)}}{\ll}\frac{1}{\tau_{D}^{(2)}}$, therefore
$\frac{1}{\tau_{D}}\simeq{\frac{1}{\tau_{D}^{(2)}}}.$ Our results are
summarized in Eq. (17a) and (17b) and shown in panels c) and d) of Fig. 2.

###  A.3 QSL with Fermi surface ($\mathbb{Z}_{2}$ and U(1))

As in the case of the Kitaev QSL, we distinguish the situation of slow and
fast spinons. For $v_{F2}>v_{F1}$ both the real and the imaginary parts of the
QSL polarization operator contribute, while for $v_{F2}<v_{F1}$ there is a
region $0<q<\frac{\omega}{v_{F2}}\left(1+\frac{T}{2E_{F1}}\right)$, in which
only the real part contributes, and
$\frac{\omega}{v_{F2}}\left(1+\frac{T}{2E_{F1}}\right)<q<-\frac{\omega}{v_{F1}}\left(1-\frac{T}{2E_{F1}}\right)+2p_{F1}\left(1-\frac{T}{2E_{F1}}\right)$,
in which both real and imaginary parts contribute to the QSL polarization
operator, see Fig. 7.

####  A.3.1 Limit of fast spinons

For $v_{F2}>v_{F1}$, Eq. (15) becomes

$\begin{split}\frac{1}{\tau_{D}}=&{\left(\frac{J_{K}^{2}}{N}\right)^{2}}\frac{1}{p_{F}^{2}\,mT}\int_{0}^{\infty}d\omega\frac{\omega^{2}}{\sinh^{2}{\bigl{(}\frac{\omega}{2T}\bigr{)}}}\times\\\
&\times\int{dq}q^{3}\frac{(\nu_{2}{v_{F2}}q)^{2}}{(v_{F2}q)^{2}-\omega^{2}}\frac{\nu^{2}_{1}}{(v_{F1}q)^{2}-\bigl{(}\omega^{2}+\bigl{(}\frac{q^{2}}{2m}\bigr{)}^{2}\bigr{)}},\\\
\end{split}$ (60)

which is only valid if $p_{F2}\gg p_{F1}$. Paying attention to the integration
limits explicitly expressed in (30), and approximating
$\omega^{2}\sinh^{-2}{\bigl{(}\frac{\omega}{2T}\bigr{)}}\simeq(2T)^{2}\theta(T-\omega)$,
this leads to the relaxation time reported in the main text in Eq. (18).

####  A.3.2 Limit of slow spinons

Now we turn to the case $v_{F2}<v_{F1}$ for which Eq. (15) can then be written
as

$\frac{1}{\tau_{D}}={\left(\frac{J_{K}^{2}}{N}\right)^{2}}\frac{(2T)^{2}}{p_{F1}^{2}mT}\int_{0}^{T}d\omega\\{I_{q}^{(A)}+{I_{q}^{(B)}}\\},$
(61)

with

$I_{q}^{(A)}=\int_{\frac{\omega}{v_{F1}}\left(1+\frac{T}{2E_{F1}}\right)}^{\frac{\omega}{v_{F2}}\left(1+\frac{T}{2E_{F1}}\right)}\frac{dqq^{3}(\nu_{1}\nu_{2})^{2}}{(v_{F1}q)^{2}-(\omega^{2}+\bigl{(}\frac{q^{2}}{2m}\bigr{)}^{2})};$
(62)
$\begin{split}I_{q}^{(B)}=&\int_{\frac{\omega}{v_{F2}}\left(1+\frac{T}{2E_{F1}}\right)}^{{\left(2p_{F1}-\frac{\omega}{v_{F1}}\right)}\left(1-\frac{T}{2E_{F1}}\right)}\frac{dqq^{3}(\nu_{2}{v_{F2}}q)^{2}}{(v_{F2}q)^{2}-\omega^{2}}\\\
&\times\frac{\nu^{2}_{1}}{(v_{F1}q)^{2}-\bigl{(}\omega^{2}+\bigl{(}\frac{q^{2}}{2m}\bigr{)}^{2}\bigr{)}}.\end{split}$
(63)

In the presence of a Fermi surface, the crossover temperature is
$T_{C}=2v_{F2}p_{F1}$. We find that

$\begin{split}&I_{q}^{(A)}+I_{q}^{(B)}=2(\nu_{2}\nu_{1}m)^{2}\biggl{\\{}\log{\left(1-\left(\frac{\omega}{T_{C}}\right)^{2}\right)}+\\\
&+\frac{1}{16}\frac{1}{1-\bigl{(}\frac{\omega}{T_{C}}\bigr{)}^{2}}\biggl{[}\log{\biggl{(}\frac{E_{F1}}{T}\biggr{)}}-\log{\biggl{(}1+\frac{\omega}{2T}\biggr{)}}+\\\
&+\log{\biggl{(}1-\biggl{(}\frac{\omega}{T_{C}}\biggr{)}^{2}\biggr{)}}\biggr{]}\biggr{\\}}.\end{split}$
(64)

Therefore we found that the integral in (61) has the form of Eq. (19) for
$T<\omega_{C}$. We can argue that this is consistent with a smooth crossover
from $v_{F2}>v_{F1}$ to $v_{F2}<v_{F1}$. In the case of $T>T_{C}$ the omega
integral again splits into two, allowing the definition of the scattering rate
as

$\frac{1}{\tau_{D}}=\frac{1}{\tau_{D}^{(1)}}+\frac{1}{\tau_{D}^{(2)}},$ (65)

where we can define

$\frac{1}{\tau_{D}^{(1)}}={\left(\frac{J_{K}^{2}}{N}\right)^{2}\frac{1}{p_{F1}^{2}mT}}\int_{0}^{T_{C}}\frac{d\omega\,{\omega^{2}}}{\sinh^{2}{\bigl{(}\frac{\omega}{2T}\bigr{)}}}I_{q}^{C}(\omega),$
(66)

where $I_{q}^{C}(\omega)=I_{q}^{(A)}+I_{q}^{(B)}$, and

$\frac{1}{\tau_{D}^{(2)}}={\left(\frac{J_{K}^{2}}{N}\right)^{2}\frac{1}{p_{F1}^{2}mT}}\int_{T_{C}}^{T}\frac{d\omega\,{\omega^{2}}}{\sinh^{2}{\bigl{(}\frac{\omega}{2T}\bigr{)}}}I_{q}^{NC}(\omega),$
(67)

with

$I_{q}^{NC}=\int_{\frac{\omega}{v_{F1}}\left(1+\frac{T}{2E_{F1}}\right)}^{-\frac{\omega}{v_{F1}}+2p_{F1}\left(1-\frac{T}{2E_{F1}}\right)}dq\frac{(\nu_{1}\nu_{2})^{2}q^{3}}{(v_{F1}q)^{2}-(\frac{q^{2}}{2m})^{2}-\omega^{2}}.$
(68)

We thus find

$\frac{1}{\tau_{D}^{(1)}}\simeq{\left(\frac{J_{K}^{2}}{N}\right)^{2}}\frac{(\nu_{2}\nu_{1})^{2}}{E_{F1}}\frac{T_{C}}{T}T^{2}\,\biggl{(}6\log{\left(\frac{E_{F1}}{T}\right)-340\biggr{)}},$
(69)

and

$\frac{1}{\tau_{D}^{(2)}}\simeq\frac{(2\nu_{2}\nu_{1})^{2}}{E_{F1}}T^{2}\log{\left(\frac{E_{F1}}{T}\right)}.$
(70)

Because of the suppressing term $(T_{C}/T)$, we conclude that
$\frac{1}{\tau_{D}}\simeq\frac{1}{\tau_{D}^{(2)}}$, as declared in Eq. (20).

##  Appendix B Theory with momentum relaxation within QSL layer

In this Appendix we present details on the calculation of the drag resistivity
in the case when momentum is relaxed within the QSL layer. We focus on the
situation of a spinon Fermi surface.

###  B.1 Collision integral from Metal-QSL coupling

In this section, we present details on the derivation of the collision
integral accounting for the interlayer momentum transfer, see Sec. IV.2 of the
main text. This calculation is based on the Kondo interaction Hamiltonian
describing the coupling between spinons and the electrons in the metal, Eq.
(11). This Hamiltonian gives rise to a term in the self energy of the spinon
$\hat{\Sigma}(p)$ ($p=(\textbf{p},\epsilon)$) as displayed in Fig. 3 and
enters the Keldysh component of the Dyson equation leading to Kamenev (2011)

$-i\left(\big{[}G^{R}_{i}\big{]}^{-1}\circ
F_{i}-F_{i}\circ\big{[}G^{A}_{i}\big{]}^{-1}\right)=\mathbb{I}^{coll}_{i}$
(71)

with ‘$\circ$’ denoting concatenation of operators and ”off-shell” collision
integral

$\mathbb{I}^{coll}_{{i}}(p)=i\left[\Sigma^{K}_{{i}(p)}-(\Sigma^{R}_{{i}}(p)-\Sigma^{A}_{{i}}(p))F_{K{i}}(p)\right].$
(72)

Here the subscript ${{i}}$ denotes that this collision integral will enter the
Boltzmann equation of layer $i$ ($i=2$ for spinons), the superscripts $K$,
$R$, and $A$ stand for the Keldysh, retarded, and advanced components of the
spinon self energy, respectively, and $F_{i}(\textbf{p},\omega)$ is related to
the (quasi-) distribution function (see Eq. (75) below).

The self energy components are

$\displaystyle\Sigma^{K}_{i}(p)$
$\displaystyle=\left(\frac{J_{K}^{2}}{2N}\right)\int
dq\,dk\,dk^{\prime}G_{i}^{R-A}(q)G^{R-A}_{j}(k)G^{R-A}_{j}(k^{\prime})$
$\displaystyle\times\\{F_{i}(q)(F_{j}(k)F_{j}(k^{\prime})-1)-(F_{j}(k)-F_{j}(k^{\prime}))$
$\displaystyle\times\delta(p+k^{\prime}-q-k),$ (73)

where $G^{R-A}=G^{R}-G^{A}$ is the difference of retarded and advanced Green’s
functions and

$\begin{split}&(\Sigma^{R-A})_{i}(p)F_{i}(p)=\left(\frac{J_{K}^{2}}{2N}\right)\int
dqdkdk^{\prime}G^{R-A}_{i}(q)G^{R-A}_{j}(k)\times\\\ &\times
G^{R-A}_{j}(k^{\prime})\\{F_{i}(p)(F_{j}(k)F_{j}(k^{\prime})-1)+\\\
&+F_{i}(p)F_{j}(q)(F_{j}(k^{\prime})-F_{j}(k))\\}\times\delta(p+k^{\prime}-q-k).\end{split}$
(74)

Here, ${G_{i}}(\textbf{k},\omega)$ is the dressed Green’s function in layer
$i$. The integration is defined as $\int dp={\frac{1}{2\pi}}\int(dp)d\omega$.

The on-shell distribution function is defined as

$2f_{i}-1=\begin{cases}\int\frac{d{\xi}}{2\pi}({G}^{R}_{i}-{G}^{A}_{i})(p)F_{i}(p),&{i=2\text{
and U(1) QSL},}\\\
{\int\frac{d\epsilon}{2\pi}({G}^{R}_{i}-{G}^{A}_{i})(p)F_{i}(p),}&{\text{else.}}\end{cases}$
(75)

Here, $f_{2}=f_{2}(\epsilon,\hat{p})$ is implied for layer 2 in the case of
the U(1) QSL and $f_{i}=f_{i}(\textbf{p})$ in all other cases. To derive the
”on-shell” collision integral ($I^{coll}_{i}$), we need to multiply by
$\frac{G^{R-A}_{i}}{{2\pi}}$ and perform an additional external integration
over $\xi$ (in the case $i=2$ and a U(1) QSL) or $\epsilon$ (in all other
cases):

$I^{coll}_{i\rightarrow
j}=\begin{cases}\int\frac{d\xi}{{2\pi}}(G^{R}_{i}-G^{A}_{i})(p)\mathbb{I}^{coll}_{i}(p),&{i=2\text{and
U(1) QSL},}\\\
\int\frac{d\epsilon}{{2\pi}}(G^{R}_{i}-G^{A}_{i})(p)\mathbb{I}^{coll}_{i}(p),&\text{{else}.}\end{cases}$
(76)

Finally, using Eqs. (72), (B.1) to (76) we obtain the collision integral

$\begin{split}I^{\rm{coll}}_{j\rightarrow
i}(\textbf{p}_{i})=&(2\pi)^{3}\frac{J_{K}^{2}}{N}\int(d{p}_{i^{\prime}})(d{p}_{j})(d{p}_{j^{\prime}})\delta(\textbf{p}_{i}+\textbf{p}_{j}-\textbf{p}_{i^{\prime}}-\textbf{p}_{j^{\prime}})\delta(\epsilon_{i}+\epsilon_{j}-\epsilon_{i^{\prime}}-\epsilon_{j^{\prime}})\times\\\
\times&\left\\{f(\textbf{p}_{i^{\prime}})f(\textbf{p}_{j^{\prime}})[1-f(\textbf{p}_{i})][1-f(\textbf{p}_{j})]-f(\textbf{p}_{i})f(\textbf{p}_{j})[1-f(\textbf{p}_{i^{\prime}})][1-f(\textbf{p}_{j^{\prime}})]\right\\}.\end{split}$
(77)

This collision integral describes the momentum and energy transferred from
layer $j$ to $i$ due to Kondo coupling. We stress that it is valid both for a
$\mathbb{Z}_{2}$ and a U(1) QSL assuming that the measure of integration is
interpreted accordingly, see Eqs. (25)–(26). Using the linear expansion about
the equilibrium distribution function, Eq. (24), this concludes the derivation
of Eq. (23) of the main text. A factor of $(2\pi)^{3}$ is reabsorbed into
$J_{K}$ in the rest of the paper concerning the Boltzmann approach.

###  B.2 General solution for drag rate

We proceed to solve the system of equations (21) to calculate the drag current
defined in Eq. (22). We remark that to the leading order in $J_{K}$

$\psi_{1}\sim O(J_{K}^{0})\longrightarrow\psi_{2}\sim
O(J_{K}^{2})\longrightarrow\psi_{3}\sim O(J_{K}^{4}).$ (78)

As the first step, we plug Eq. (21c) into (22) and obtain

$\textbf{j}_{3}=-\frac{e\,\tau_{3}}{m_{3}}\int(d{p}_{3})I^{\rm{coll}}_{2\rightarrow
3}\,\textbf{p}_{3}.$ (79)

Using Eq. (23) and the solution of Eq. (21a),

$\psi_{1}=-e\frac{1}{m_{1}T}\tau_{1}\textbf{E}_{1}\cdot\textbf{p}_{1},$ (80)

the drag current can be written as

$\displaystyle\textbf{j}_{3}$
$\displaystyle=-\frac{e\,\tau_{3}}{2m_{3}}\frac{J_{K}^{2}}{N}\int(d{p}_{2})(d{p}_{2^{\prime}})(d{p}_{3})(d{p}_{3^{\prime}})(\textbf{p}_{3}-\textbf{p}_{3^{\prime}})$
$\displaystyle\times\delta^{(3)}(P_{2}+P_{3}-P_{2^{\prime}}-P_{3^{\prime}})f_{2}f_{3}\tilde{f}_{2^{\prime}}\tilde{f}_{3^{\prime}}(\psi_{2}-\psi_{2^{\prime}})$
(81)

where we have used the symmetry properties of the collision integral to write
the term $\textbf{p}_{3}-\textbf{p}_{3^{\prime}}$.

At the next wtep, we determine $\psi_{2}$ from Eq. (21b). The sum of the two
collision integrals can be written as

$\begin{split}&f_{2}\tilde{f}_{2}\psi_{2}=\tau_{2}(I^{\rm{coll}}_{1{\rightarrow}2}+I^{\rm{coll}}_{3\rightarrow
2})\simeq\\\
&\simeq\frac{J_{K}^{2}\tau_{2}}{N}\int(d{p}_{2^{\prime\prime}})(d{p}_{1})(d{p}_{1^{\prime}})f_{2}f_{1}\tilde{f}_{2^{\prime\prime}}\tilde{f}_{1^{\prime}}(\psi_{1}-\psi_{1^{\prime}})\\\
&\times\delta^{(3)}(P_{2}+P_{1}-P_{2^{\prime\prime}}-P_{1^{\prime}}),\end{split}$
(82)

in which we have neglected the higher orders in $J_{K}$. We can use the
symmetry properties of $\textbf{j}_{3}$ and write

$\begin{split}&\textbf{j}_{3}=-\frac{e\,\tau_{3}}{m_{3}}\frac{J_{K}^{2}}{N}\int(d{p}_{2})(d{p}_{2^{\prime}})(d{p}_{3})(d{p}_{3^{\prime}})(\textbf{p}_{3}-\textbf{p}_{3^{\prime}})\times\\\
&\times\delta^{(3)}(P_{2}+P_{3}-P_{2^{\prime}}-P_{3^{\prime}})f_{2}f_{3}\tilde{f}_{2^{\prime}}\tilde{f}_{3^{\prime}}\psi_{2}.\end{split}$
(83)

Subsequently we use Eq. (82) and the relations

$\begin{split}&\delta^{(3)}(P_{2}+P_{3}-P_{2^{\prime}}-P_{3^{\prime}})=\int
dQ(2\pi)^{3}\delta^{(3)}({P_{3}-P_{3^{\prime}}-Q}),\times\\\
&\times\delta^{(3)}({P_{2}-P_{2^{\prime}}+Q})\\\
&\delta^{(3)}(P_{2}+P_{1}-P_{2^{\prime\prime}}-P_{1^{\prime}})=\int
dQ^{\prime}(2\pi)^{3}\delta^{(3)}({P_{2}-P_{2^{\prime\prime}}-Q^{\prime}})\times\\\
&\times\delta^{(3)}({P_{1}-P_{1^{\prime}}+Q^{\prime}}),\\\ \end{split}$ (84)

to rewrite the current in the following way:

$\begin{split}&\textbf{j}_{3}=\lambda\int dQ\,dQ^{\prime}\int
d\\{({p})\\}(\textbf{p}_{3}-\textbf{p}_{3^{\prime}})(\textbf{p}_{1}-\textbf{p}_{1^{\prime}})\cdot\textbf{E}_{1}\times\\\
&\times\delta^{(3)}({P_{3}-P_{3^{\prime}}-Q})\delta^{(3)}({P_{2^{\prime\prime}}-P_{2^{\prime}}+Q})\times\\\
&\times\delta^{(3)}({P_{2^{\prime\prime}}-P_{2}-Q^{\prime}})\delta^{(3)}({P_{1^{\prime}}-P_{1}+Q^{\prime}})\times\\\
&(f_{3}\tilde{f}_{3^{\prime}})(f_{1}\tilde{f}_{1^{\prime}})(f_{2}\tilde{f}_{2^{\prime}}),\\\
\end{split}$ (85)

where the coefficient is defined as
$\lambda=-e^{2}\frac{\tau_{1}\tau_{2}\tau_{3}}{m_{1}m_{3}T}(2\pi)^{10}\left(\frac{J_{K}^{2}}{N}\right)^{2}$
and we used the shorthand notation for the integration measure $d\\{({p})\\}$.
We make use of the identity

$f_{p}\tilde{f}_{p^{\prime}}=\frac{f_{p}-f_{p^{\prime}}}{1-e^{(\epsilon_{p}-\epsilon_{p^{\prime}})/T}}$
(86)

and the energy conserving delta functions to get

$\begin{split}&\textbf{j}_{3}=-e^{2}\frac{\tau_{1}\tau_{2}\tau_{3}}{2m_{1}m_{3}T}\left(\frac{J_{K}^{2}}{N}\right)^{2}(2\pi)^{4}\times\\\
&\times\int
dQdQ^{\prime}\frac{{\mathrm{Im}}[\Pi_{3}^{R}(Q)]{{\mathrm{Im}}}[\Pi_{1}^{R}(Q^{\prime})]I_{2}(Q,Q^{\prime})}{\sinh{\bigl{(}\frac{\omega}{2T}\bigr{)}}\sinh{\bigl{(}\frac{\omega^{\prime}}{2T}\bigr{)}}\sinh{\bigl{(}\frac{\omega+\omega^{\prime}}{2T}\bigr{)}}}\textbf{q}(\textbf{q}^{\prime}\cdot\textbf{E}_{1}),\\\
\end{split}$ (87)

where we have used the definition Eq. (28) of the main text. Expression in Eq.
(87) is valid both for the $\mathbb{Z}_{2}$ and the U(1) QSL and the origin of
Eq. (27) of the main text.

###  B.3 $\mathbb{Z}_{2}$ QSL: Fermi liquid behavior

Here we calculate $I_{2}(Q,Q^{\prime})$ for the $\mathbb{Z}_{2}$ QSL. We take
into account also the integration over $\hat{q}$ and $\hat{q}^{\prime}$, so
evaluating the object

$\begin{split}&\langle
I_{2}(Q,Q^{\prime})(\vec{q}\cdot\hat{E}_{1})(\vec{q}^{\prime}\cdot\hat{E}_{1})\rangle_{\hat{q},\hat{q}^{\prime}}=\int\frac{d^{2}p_{2}^{\prime\prime}}{(2\pi)^{2}}\\\
&\bigl{\langle}(\vec{q}\cdot\hat{E}_{1})\delta(\omega_{q}^{\pm}-\textbf{v}_{2}^{{}^{\prime\prime}}\cdot\textbf{q})\rangle_{\hat{q}}\,\,\langle(\vec{q}\cdot\hat{E}_{1})\delta(\omega^{\prime-}_{q^{\prime}}-\textbf{v}_{2}^{{}^{\prime\prime}}\cdot\textbf{q}^{\prime})\rangle_{q^{\prime}}\times\\\
&\times(f(\epsilon_{p_{2}^{\prime\prime}}-\omega^{\prime})-f(\epsilon_{p_{2}^{\prime\prime}}+\omega))\bigr{\rangle}_{\hat{q},\hat{q}^{\prime}},\end{split}$
(88)

where we have defined $\omega_{q}^{\pm}=\omega\pm\frac{q^{2}}{2m_{2}}$. We
calculate the object

$\langle(\textbf{q}\cdot\hat{E}_{1})\delta(\textbf{v}_{2}^{\prime\prime}\cdot\textbf{q}-\omega_{q}^{\pm})\rangle_{\hat{q}}=\frac{\hat{p}_{2}^{\prime\prime}\cdot\hat{E}_{1}}{\pi
v_{{2}}^{\prime\prime}}\frac{\omega_{q}^{\pm}}{\sqrt{(v_{{2}}^{\prime\prime}q)^{2}-\omega_{q}^{\pm
2}}},$ (89)

and use the approximation

$f(\epsilon_{p_{2}^{\prime\prime}}-\omega^{\prime})-f(\epsilon_{p_{2}^{\prime\prime}}+\omega)\\!\\!\underbrace{\simeq}_{\omega,\omega^{\prime}\ll
T}\\!\\!2T\delta\left(\epsilon+\frac{\omega-\omega^{\prime}}{2}\right)\sinh\left(\frac{\omega+\omega^{\prime}}{2T}\right).$
(90)

Recalling the operative expression for the polarization operator (42) and that
$\omega\ll E_{F1}$ we can write

$\begin{split}&\langle
I_{2}(Q,Q^{\prime}(\vec{q}\cdot\hat{E}_{1})(\vec{q}^{\prime}\cdot\hat{E}_{1}))\rangle_{\hat{q},\hat{q}^{\prime}}={-}\frac{4T\sinh{\bigl{(}\frac{\omega+\omega^{\prime}}{2T}\bigr{)}}}{(2\pi)^{6}\omega\omega^{\prime}E_{F2}}\\\
&\times\omega^{+}_{q}\omega^{+}_{q^{\prime}}\mathrm{Im}\Pi_{2}^{R}(\textbf{q},\omega)\mathrm{Im}\Pi_{2}^{R}(\textbf{q}^{\prime},\omega^{\prime}).\\\
\end{split}$ (91)

The drag conductivity can be the concisely expressed as

$\sigma_{D}=e^{2}\left(\frac{J_{K}^{2}}{N}\right)^{2}\frac{\tau_{1}\tau_{2}\tau_{3}}{2\pi
m_{1}m_{2}m_{3}p_{F2}^{2}}I_{12}I_{32}$ (92)

where we have defined

$I_{i3}=\int
d\textbf{q}d\omega\frac{q^{2}}{\omega\sinh{\bigl{(}\frac{\omega}{2T}\bigr{)}}}\mathrm{Im}\Pi_{i}^{R}(\textbf{q},\omega)\mathrm{Im}\Pi_{2}^{R}(\textbf{q},\omega).$
(93)

We recognize the existence of four regimes for $I_{i2}$, $i=1,3$):
$(\frac{v_{F2}}{v_{Fi}}>1,\frac{v_{F2}}{v_{Fi}}<1)\times(\frac{p_{F2}}{p_{F{i}}}>1,\frac{p_{F2}}{p_{Fi}}<1)$.
We start with the limit
$(\frac{v_{Fi}}{v_{F2}}>1\land\frac{p_{Fi}}{p_{F2}}>1)\lor(\frac{v_{Fi}}{v_{F2}}<1\land\frac{p_{Fi}}{p_{F2}}<1)$.
In this limit the Fermi energy of either the metallic plates or the QSL bounds
the energy integration as show in panel d) and f) of Fig. 4. We find that

$I_{i2}=\left(\frac{T}{\pi}\right)^{2}\frac{\nu_{i}\nu_{2}}{v_{Fi}v_{F2}}4p^{2}_{\text{min}},$
(94)

and, using Eq. (93) we find

$\frac{\sigma_{D}}{e^{2}}=\left(\frac{J_{K}^{2}}{N}\right)^{2}\frac{2\tau_{1}\tau_{2}\tau_{3}p^{4}_{\text{min}}T^{4}}{\pi^{5}m_{1}m_{2}m_{3}p_{F2}^{2}}\left(\frac{\nu_{1}\nu_{2}}{v_{F1}v_{F2}}\right)^{2}.$
(95)

The results for the drag rate in the symmetric case are presented in Eq. (31a)
and Eq. (31b), and shown in panels c) and e) of Fig. 4.

We now explore the regime in which
$(\frac{v_{Fi}}{v_{F2}}>1\land\frac{p_{Fi}}{p_{F2}}<1)\lor(\frac{v_{Fi}}{v_{F2}}<1\land\frac{p_{Fi}}{p_{F2}}>1)$.
As shown in panels b) and h) of Fig. 4, we can define a crossover temperature
$T_{C}=2v_{\text{min}}p_{\text{min}}$. We find that

$I_{i2}=\frac{4\nu_{i}\nu_{2}p^{2}_{\text{min}}T^{2}}{\pi^{2}v_{Fi}v_{F2}}\mathrm{min}\left(1,\frac{T_{C}}{T}\right).$
(96)

Squaring the term $I_{i2}$, where we now take $v_{F1}=v_{F3},p_{F1}=p_{F3}$,
and using Eq. (93), we obtain

$\frac{\sigma_{D}}{e^{2}}=\left(\frac{J_{K}^{2}}{N}\right)^{2}\frac{\tau_{1}\tau_{2}\tau_{3}}{2\pi^{8}E_{F2}}\frac{T^{4}p_{\text{min}}^{4}}{v_{F1}^{2}v_{F2}^{2}}\mathrm{min}\left(1,\frac{T_{C}}{T}\right)^{2},$
(97)

hence in the symmetric case we obtain the results presented in Eq. (32a) and
(31b), and plotted in panels a) and g) in Fig. 4.

###  B.4 U(1) QSL: non-Fermi liquid behavior

Analogously to the $\mathbb{Z}_{2}$ case, we need to solve Eq. (87) for a U(1)
QSL, so we first need to calculate the quantity $I_{2}(Q,Q^{\prime})$. Using
the definitions (26) we can write

$\begin{split}&I_{2}(Q,Q^{\prime})=\nu_{2}^{3}\int
d\hat{p}_{2}d\hat{p}_{2^{\prime}}d\hat{p}_{2^{\prime\prime}}d\epsilon_{2}d\epsilon_{2^{\prime}}d\epsilon_{2^{\prime\prime}}\\\
&\delta^{(2)}({p_{F2}\hat{p}_{2^{\prime\prime}}-p_{F2}\hat{p}_{2^{\prime}}+\textbf{q}})\delta^{(2)}({p_{F2}\hat{p}_{2^{\prime\prime}}-p_{F2}\hat{p}_{2}-\textbf{q}^{\prime}})\times\\\
&\times\delta(\epsilon_{2^{\prime\prime}}-\epsilon_{2^{\prime}}+\omega)\delta(\epsilon_{2^{\prime\prime}}-\epsilon_{2}-\omega^{\prime})(f(\epsilon_{2})-f(\epsilon_{2^{\prime}})).\\\
\end{split}$ (98)

Integrating out
$\delta(\epsilon_{2^{\prime\prime}}-\epsilon_{2}-\omega^{\prime})\delta(\epsilon_{2^{\prime\prime}}-\epsilon_{2^{\prime}}+\omega)$
and integrating in $d\epsilon_{2}$ we find

$\begin{split}&I_{2}(Q,Q^{\prime})=\nu_{2}^{3}{(\omega+\omega^{\prime})}\int
d\hat{p}_{2}d\hat{p}_{2^{\prime}}d\hat{p}_{2^{\prime\prime}}\delta^{{(2)}}(p_{F2}\hat{p}_{2^{\prime\prime}}+\\\
&-p_{F2}\hat{p}_{2^{\prime}}+\textbf{q})\times\delta^{{(2)}}({p_{F2}\hat{p}_{2^{\prime\prime}}-p_{F2}\hat{p}_{2}-\textbf{q}^{\prime}}).\end{split}$
(99)

We use the relation $\int d\hat{p}=\frac{2\pi}{p_{F}}\int
d\textbf{p}\delta(|\textbf{p}|-p_{F})$ for the integrations over
$d\hat{p}_{2^{\prime}}$ and $d\hat{p}_{2^{\prime\prime}}$ to rewrite Eq. (99)
as

$\begin{split}&I_{2}(Q,Q^{\prime})=\nu_{2}^{3}\left(\frac{2\pi}{p_{F2}}\right)^{2}{(\omega+\omega^{\prime})}\\\
&\times\int
dp_{2}^{"}\delta(|p_{F2}\hat{p}_{2"}+\textbf{q}|-p_{F2})\delta(|p_{F2}\hat{p}_{2"}-\textbf{q}^{\prime}|-p_{F2}).\\\
\end{split}$ (100)

It can be shown that

$\begin{split}&{\bigl{\langle}I_{2}(Q,Q^{\prime})(\textbf{q}\cdot\hat{E_{1}})(\textbf{q}^{\prime}\cdot\hat{E}_{1})\bigr{\rangle}_{\hat{q},\hat{q}}}=\\\
&=-\frac{\nu_{2}^{3}}{\pi
p_{F2}^{2}}{({\omega+\omega^{\prime}})}\frac{\theta(2p_{F2}-q)q}{\sqrt{(2p_{F2})^{2}-q^{2}}}\frac{\theta(2p_{F2}-q^{\prime})q^{\prime}}{\sqrt{(2p_{F2})^{2}-q^{\prime
2}}},\end{split}$ (101)

so that the expression used to calculate the drag conductivity is

$\displaystyle\sigma_{D}=\varsigma\int\frac{(\omega+\omega^{\prime})dqdq^{\prime}d\omega
d\omega^{\prime}}{\sinh{\bigl{(}\frac{\omega}{2T}\bigr{)}}\sinh{\bigl{(}\frac{\omega^{\prime}}{2T}\bigr{)}}\sinh{\bigl{(}\frac{\omega+\omega^{\prime}}{2T}\bigr{)}}}$
$\displaystyle\times\frac{q^{2}\,{{\rm{Im}}}[\Pi_{3}^{R}(q,\omega)]}{\sqrt{(2p_{F2})^{2}-q^{2}}}\frac{q^{\prime
2}\,{\rm{Im}}[\Pi_{1}^{R}(q^{\prime},\omega^{\prime})]}{\sqrt{(2p_{F2})^{2}-q^{\prime
2}}},$ (102)

where the integrations over the modulus of the exchanged momenta is restricted
to $0<q<2p_{F2}$ and the integration in the frequencies takes has the domain
$-\infty<\omega<\infty$. The prefactor is

$\varsigma=\frac{e^{2}}{8\pi^{3}}\left(\frac{J_{K}^{2}}{N}\right)^{2}\frac{\nu_{2}^{3}}{T}\frac{\tau_{1}\tau_{2}\tau_{3}}{m_{1}m_{3}p_{F2}^{2}}$
(103)

####  B.4.1 Limit of small spinon Fermi momentum

For $\\{p_{F1,F3}\\}\gg p_{F2}$ we use Eq. (43) for the polarization operator.
We write $\sigma_{D}$ as

$\sigma_{D}=\varsigma C(T,p_{F1},p_{F2},p_{F3})$ (104)

and we only focus on the $C(T,p_{F1},p_{F2},p_{F3})$ term. To extract a first
analytical estimate we approximate the function of $\omega$ and
$\omega^{\prime}$ like

$\begin{split}&\frac{\omega+\omega^{\prime}}{\sinh{\bigl{(}\frac{\omega}{2T}\bigr{)}}\sinh{\bigl{(}\frac{\omega^{\prime}}{2T}\bigr{)}}\sinh{\bigl{(}\frac{\omega+\omega^{\prime}}{2T}\bigr{)}}}\simeq\frac{{8T}}{\bar{\omega}\bar{\omega}^{\prime}}\\\
&\times\theta(T-\omega)\theta(T-\omega^{\prime})\theta(T-(\omega+\omega^{\prime})),\end{split}$
(105)

where $\bar{\omega}=\frac{\omega}{T}$. From Eqs. (102), (105) we obtain

$\begin{split}&C(T,p_{F1},p_{F2},p_{F3})=32T^{3}\int_{0}^{1}d\bar{\omega}\int_{0}^{1-\bar{\omega}}d\bar{\omega}^{\prime}\frac{1}{\bar{\omega}\bar{\omega}^{\prime}}\\\
&\times\iint\limits_{0}^{2p_{F2}}dqdq^{\prime}\frac{q^{2}\,{\rm{Im}}[\Pi_{3}^{R}(q,\omega)]}{\sqrt{(2p_{F2})^{2}-q^{2}}}\frac{q^{\prime
2}\,{\rm{Im}}[\Pi_{1}^{R}(q^{\prime},\omega^{\prime})]}{\sqrt{(2p_{F2})^{2}-q^{\prime
2}}}.\\\ \end{split}$ (106)

With little manipulation one can write

$\begin{split}&C(T,T_{C1},T_{C3})=32T^{5}\nu_{1}\nu_{3}(2p_{F2})^{4}\int_{0}^{1}d\bar{\omega}\int_{0}^{1-\bar{\omega}}d\bar{\omega}^{\prime}\\\
&\int_{0}^{1}d\bar{q}\int_{0}^{1}d\bar{q}^{\prime}\frac{\bar{q}^{2}}{\sqrt{1-\bar{q}^{2}}}\frac{\bar{q}^{\prime
2}}{\sqrt{1-{q}^{\prime 2}}}\\\
&\times\frac{\theta(\frac{T_{C3}}{T}\bar{q}-\bar{\omega})}{\sqrt{(\frac{T_{C3}}{T})^{2}\bar{q}^{2}-\bar{\omega}^{2}}}\frac{\theta(\frac{T_{C1}}{T}\bar{q}-\bar{\omega}^{\prime})}{\sqrt{(\frac{T_{C1}}{T})^{2}\bar{q}^{\prime
2}-\bar{\omega}^{\prime 2}}},\end{split}$ (107)

with $T_{Ci}=2v_{F2}p_{i}$. Integrating in $q$ and $q^{\prime}$ we find

$\begin{split}&C(T,T_{C1},T_{C3})=\frac{32T^{5}}{T_{C1}T_{C3}}(2p_{F2})^{4}\nu_{1}\nu_{3}\int_{0}^{1}d\bar{\omega}\int_{0}^{1-\bar{\omega}}d\bar{\omega}^{\prime}\\\
&\times\theta\left[1-\bar{\omega}\left(\frac{T}{T_{C3}}\right)\right]\theta\left[1-\bar{\omega}^{\prime}\left(\frac{T}{T_{C1}}\right)\right]\times\\\
&\times
E\left[1-\omega\left(\frac{T}{T_{C3}}\right)^{2}\right]E\left[1-\omega^{\prime}\left(\frac{T}{T_{C1}}\right)^{2}\right],\\\
\end{split}$ (108)

with E(x) being the elliptic integral of the second kind. There are two
regimes that need to be discussed, $T\gg T_{C1,3}$ and $T\ll T_{C1,3}$,
distinguished by the fact that for $T>T_{C1,3}$ the energy integral is bounded
by $T_{C}$ and not $T$. The integration domain in the $\omega$,
$\omega^{\prime}$ is $(0,T_{C3})\times(0,T_{C1})$ and the
$\omega$,$\omega^{\prime}$ dependence is factorized yielding

$\begin{split}&C(T,T_{C1},T_{C3})=\frac{32T^{5}}{T_{C1}T_{C3}}(2p_{F2})^{4}\nu_{1}\nu_{3}\\\
&\times\mathrm{min}\left(1,\frac{T_{C1}}{T}\right)\mathrm{min}\left(1,\frac{T_{C3}}{T}\right).\end{split}$
(109)

So we find that the drag relaxation rate in the non symmetric regime is

$\begin{split}&\frac{1}{\tau_{D}^{(U(1),C)}}=\frac{8}{\pi^{4}}\frac{(\nu_{2})^{3}}{E_{F1}v_{F1}v_{F3}}\left(\frac{J_{K}^{2}}{N}\right)^{2}\tau_{2}T^{4}\\\
&\times\mathrm{min}\left(1,\frac{T_{C1}}{T}\right)\mathrm{min}\left(1,\frac{T_{C3}}{T}\right).\end{split}$
(110)

In the symmetric regime the results are the ones presented in Eqs. (36), (37)
of the main text, and plotted in panel c) of Fig. 5.

####  B.4.2 Limit of large spinon Fermi momentum

For $p_{F2}>p_{F1}$ we write the term $C(T,p_{F1},p_{F2},p_{F3})$ defined in
Eq. (104) as:

$\begin{split}&C(T,p_{F1},p_{F2},p_{F3})=32T^{5}\nu_{1}\nu_{3}\int_{0}^{1}d\bar{\omega}\int_{0}^{1-\bar{\omega}}d\bar{\omega^{\prime}}\\\
&\biggl{(}\int_{\frac{{\omega}}{2v_{3}}\left(1+\frac{T}{E_{F3}}\right)}^{\frac{{\omega}}{2v_{3}}\left(1-\frac{T}{E_{F3}}\right)+2p_{F3}\left(1-\frac{T}{E_{F3}}\right)}dqI(q,v_{F3},m_{3})\biggr{)}\times\\\
&\times\biggl{(}\int_{\frac{{\omega^{\prime}}}{2v_{F1}}\left(1+\frac{T}{E_{F3}}\right)}^{\frac{{\omega^{\prime}}}{2v_{F1}}\left(1-\frac{T}{E_{F3}}\right)+2p_{F1}\left(1-\frac{T}{E_{F3}}\right)}dqI(q,v_{F1},m_{1})\biggr{)},\end{split}$
(111)

where the integrand of the two momenta integrals is defined as

$I(q,v_{Fi},m_{i})=\frac{q^{2}}{\sqrt{(2p_{F2})^{2}-q^{2}}\sqrt{(v_{Fi}q)^{2}-{\omega}^{2}-\left(\frac{q^{2}}{2m_{i}}\right)^{2}}}.$
(112)

We calculated $C(T,p_{F1},p_{F2},p_{F3})$ to be

$C(T,p_{F1},p_{F2},p_{F3})=(32)^{2}\nu_{1}\nu_{3}m_{1}m_{3}\frac{E_{F3}}{T_{C3}}\frac{E_{F1}}{T_{C1}}T^{5}.$
(113)

The drag relaxation rate in the non-symmetric case yields

$\begin{split}&\frac{1}{\tau_{D}}=\frac{128}{\pi^{2}}\frac{\nu_{2}^{3}\nu_{1}\nu_{3}}{p_{F2}^{2}}\left(\frac{J_{K}^{2}}{N}\right)^{2}\frac{E_{F3}}{T_{C1}T_{C3}}T^{4}\tau_{2}.\end{split}$
(114)

In the symmetric regime Eq. (114) becomes Eq. (34) of the main text, as it is
plotted in panel a) of Fig. 5.

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