# Charge transport in chiral solids as a possible tool in search of dark
matter signals

Marek Rogatko<EMAIL_ADDRESS>Karol I. Wysokinski
<EMAIL_ADDRESS>Institute of Physics, Maria Curie-Sklodowska
University, pl. Marii Curie-Sklodowskiej 1, 20-031 Lublin, Poland

###### Abstract

As the existing techniques to directly detect dark matter particles in
laboratory experiments continue to produce negative signals, the research
community considers novel approaches including those relying on condensed
matter systems with non-trivial quantum properties. For example, finding
signatures of dark matter in transport characteristics of solids would be an
important step on the road to detect this illusive component of the mass of
our Universe. As a first step in this direction we have recently derived the
modified kinetic equation taking into account two coupled $U(1)$-gauge fields,
one being the standard Maxwell electromagnetic field and other corresponding
to the dark sector. The resulting Boltzmann kinetic equation is modified by
the Berry curvature which couples to both visible and dark sector gauge
fields. The linear and nonlinear longitudinal and/or Hall currents in
topological matter may signal the existence of dark particle induced
electromagnetic fields. The usage of appropriately designed resonant cavity to
enhance the electric field may help to reach the goal. The conservative
estimates show that the technique should have the sensitivity of the order of
$10^{-9}$ or better and may help to partially fill the gap between XENON1T and
haloscope experiments.

##  I Introduction

The only evidences of a dark matter bertone2018  come from the gravitational
observations. The signals include galaxy rotation curves, gravitational
lensing and the gravitational structures at cosmological scales frenk2012 .
The search for this elusive, albeit dominant, component of the matter in our
Universe by various techniques continues gaskins2016  to give negative
effects. The direct unique signals of dark matter in the laboratory
experiments would provide a breakthrough comparable to the first observation
of the gravitational waves, which also resulted in the proposal that
primordial black holes may in fact constitute the dark matter bird2016 . The
absence of evidence for dark-matter particles despite extensive search results
in a “sense of ‘crisis’ in the dark-matter particle community” bertone2018a .
Observing this state of the affairs the authors of bertone2018a  propose: “We
argue that diversifying the experimental effort and incorporating astronomical
surveys and gravitational-wave observations is our best hope of making
progress on the dark-matter problem.” On the other hand, this situation
provokes the appearance of novel detection techniques and theoretical models
of dark matter (DM).

The mass of DM particle is unknown and may span many orders of magnitude
battaglieri2017 . Some novel proposals consider dark matter candidates with
huge masses as, e.g., the mentioned bird2016  primary black holes or the
recently suggested fractionally charged gravitino with the mass at Planck
scale meissner2019 . Many of other recent proposals concentrate on the
opposite end and consider relatively low masses in the $eV$ range or even much
below. The detection schemes of small mass candidates are necessarily very
different from the existing techniques.

Many of the new proposals base on observations of condensed matter phenomena.
It is condensed matter in which elementary excitations are observed at low
energies and may be used as detectors of small mass dark matter particles.
Various novel methods have been recently discussed kim2023 ; dixit2021 ;
liang2018 ; rogatko2023 ; hong2022 ; hochberg2022 ; hochberg2016 ; knapen2021
; liang2022 ; chen2022 ; bloch2022 ; hochberg2018 ; budnik2018 ; knapen2017 ;
knapen2018 ; bar18  proposing to use both, standard condensed matter systems
and quantum materials with topologically non-trivial band structure. The
latter include inter alia detecting DM with help of the superconducting
materials rog16 ; rog15a ; rog16a ; rog17 ; kic21  or superconducting qubits
dixit2021 , graphene-based Josephson junctions kim2023  or by observing chiral
magnetic effect in topologically non-trivial quantum matter hong2022 ;
rogatko2023 .

Other detection techniques encompass the search for bosonic dark matter via
absorption in superconductors hochberg2016 , using superfluid helium
knapen2017 , optical phonons in polar materials knapen2018 , observations of
colour centres production in crystals budnik2018 , the usage of bulk three-
dimensional Dirac semimetals hochberg2018  or multilayered optical devices
bar18 . The future will tell which of them turns out to be successful. The
usage of various condensed matter systems and detection techniques involving
charge carriers, phonons, light and quasiparticles in superconductors to
detect light dark matter has been recently reviewed kahn2022 ; mitridate2023 .

The popular candidates for ultralight DM bosonic particles are axions (more
generally axion like particles) and dark photon. They manifest as dark
electric (or magnetic) field oscillating with frequency
$\tilde{\omega}=m_{DP}c^{2}/\hbar$, where $m_{DP}$ is the dark particle mass,
$c$ denotes speed of light and $\hbar$ is reduced Planck constant. In this
paper we concentrate on the massive dark photon as an important candidate.

Our starting point will be $(3+1)$-dimensional action describing visible
sector comprising Maxwell $U(1)$-gauge field and hidden one with massive
$U(1)$-gauge auxiliary field $\tilde{A}_{\mu}$ (dark photon), interacting with
the ordinary Maxwell field by the so-called kinetic mixing term. The action is
provided by

$S_{M-dark~{}photon}=\int
d^{4}x\Big{(}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}-\frac{\alpha}{2}F_{\mu\nu}\tilde{F}^{\mu\nu}-\frac{m_{DP}^{2}}{2}\tilde{A}_{\mu}\tilde{A}^{\mu}\Big{)},$
(1)

where $\alpha$ is a coupling constant between two sectors, standing in front
of the kinetic mixing term. $m_{DP}$ denotes here the mass of dark photon.
$F_{\mu,\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, and
$\tilde{F}_{\mu,\nu}=\partial_{\mu}\tilde{A}_{\nu}-\partial_{\nu}\tilde{A}_{\mu}$
denote the field strength tensors of the standard and dark photon,
respectively. The coupling $\alpha$ is expected to be $\ll 1$.

The idea that dark photon can be a candidate for dark matter has been widely
exploited on various backgrounds, both theoretically hol86 -abe08a  and
experimentally til15 -an23 . The model in question possesses also some
possible astrophysical confirmations jea03 -bod15 . We have cited only some
illustrative examples due to the vast amount of work authorising this
blossoming field of researches.

Dark photons, the ultralight candidates for dark matter, interact with the
Standard Model particles via kinetic mixing term. After appropriate rotation
and redefinitions, one can eliminate kinetic mixing term and arrive at the
Lagrangian for dark photon, Maxwell field and electromagnetic current dphbook
. Thus, on the solid state physics background we can say that dark photon
interacts with electrons in solids. More generally, the scattering of dark
particle like axion (in the presence of external magnetic field) or dark
photon produces small electric field oscillating with frequency
$\tilde{\omega}$ at the surface of a metal or superconductor iwazaki2020 ;
iwazaki2021 ; kishimoto2022  and this can possibly be detected.

In the recent paper rogatko2023  we considered the chiral magnetic effect
induced by dark photon in the Dirac semimetal. Chiral magnetic effect denotes
the current flowing along the applied magnetic field in conducting material
with a Berry curvature; in the mentioned paper the current flows along dark
magnetic field $\tilde{\textbf{B}}$. In that scenario the very existence of
the signal relies on the kinetic coupling, which is a source of the dark
magnetic field. The estimates show that the expected dark magnetic field is
small and thus detection of the non-dissipative current will be difficult, but
not impossible. In contrast, the work hong2022  also proposing measurement of
the chiral magnetic effect, assumes application of the external magnetic field
and detection of axion-like particles. In that work the current flows along
the direction of the external magnetic field B.

In the present paper, we extend our previous work and consider detection of
dark matter via measurement of anomalous transport in Dirac or Weyl semimetals
armitage2018 ; lv2021 , with broken time reversal or inversion symmetry. Dirac
or Weyl semimetals are three or two-dimensional condensed matter systems with
topologically non-trivial energy spectrum, which is characterized by the
existence of Dirac nodes and zero, or very small energy gap between conduction
and valence bands. The oscillating electric field induced by dark particles
like axions or dark photons in a two-dimensional Weyl or Dirac semimetals,
could be identified by observing in a such systems two kinds of electric
currents flowing in mutually perpendicular directions, i.e., the longitudinal
and anomalous Hall current. The latter appears at linear electric field order
if the time reversal symmetry is broken. On the other hand, at higher order in
electric field strength, the anomalous Hall effect appears even when the
Hamiltonian is time reversal invariant, but other symmetries like inversion
are broken sodemann2015 . Two dimensional transition metal dichalcogenides
wang2018 ; chang2023  or three-dimensional Weyl semimetals armitage2018  could
serve as potential candidates to observe the effect and thus as detectors of
dark matter fields.

The organization of the rest of the paper is following. In Sec. II we briefly
recall the Boltzmann kinetic theory application to chiral systems, like Weyl
or Dirac semimetals in the Maxwell E and dark sector $\tilde{\textbf{E}}$
electric fields. We solve the Boltzmann equation in Sec. III. The currents in
the system with two electric fields are calculated in Sec. IV. The estimation
of the dark particle induced electric field is a subject of Sec. V, while its
possible enhancement in a resonant cavity is discussed in Sec. VI. The
estimates of the sensitivity of the proposed measurements are presented in
Sec. VII. We end up with the discussion of a few scenarios to identify signal
of dark matter in Sec. VIII.

##  II Boltzmann kinetic equation for systems with Berry curvature and dark
matter

Standard Boltzmann kinetic theory applied to Weyl and Dirac systems requires a
modification which takes into account the existence of the Berry curvature
$\bm{\Omega}(\mathbf{k})$, in such materials. The existence of a non-trivial
electron spectrum and resulting Berry curvature is known to modify the
equations of motion of the electron velocity $\dot{\mathbf{r}}$ and momentum
$\dot{\mathbf{k}}$, as well as, a phase space volume $d\mathbf{r}d\mathbf{k}$
xiao2010 . Strictly speaking, the Boltzmann kinetic equation has its standard
form. However, the velocity of a particle is modified by the presence of Berry
curvature and this induces the modification of the particles’ dynamics.

On the other hand, Boltzmann equation allows one to find the local
distribution function mahan  $f(r,\textbf{k},t)$ in the presence of external
perturbations. In Weyl or Dirac semimetals with nodes of different chirality,
the distribution may depend on the node in question. Thus, in general we
denote the distribution around the node $c$ by
$f_{c}(\mathbf{r},\mathbf{k},t)$. It changes in time according to the relation

$\frac{\partial f_{c}(\mathbf{r},\mathbf{k},t)}{\partial
t}+\mathbf{\dot{r}}_{c}\cdot\bm{\nabla}_{\mathbf{r}}f_{c}(\mathbf{r},\mathbf{k},t)+\mathbf{\dot{k}_{c}}\cdot\bm{\nabla}_{\mathbf{k}}f_{c}(\mathbf{r},\mathbf{k},t)=I_{coll}[f_{c}(\mathbf{r},\mathbf{k},t)].$
(2)

Here $I_{coll}[f_{c}(\mathbf{r},\mathbf{k},t)]$ is the collision integral.

In systems with Berry phase the time dependence of r and k near the node $c$
is found xiao2010 ; son2013  and is provided by

$\displaystyle\mathbf{\dot{r}}_{c}=\frac{1}{D_{c}(\mathbf{k})}\left[\mathbf{v}_{c}+\frac{e}{\hbar}(\mathbf{v}_{c}\cdot\bm{\Omega}_{c})\mathbf{B}+\frac{e}{\hbar}(\mathbf{E}\times\bm{\Omega}_{c})\right]$
(3)
$\displaystyle\hbar\mathbf{\dot{k}}_{c}=\frac{1}{D_{c}(\mathbf{k})}\left[-e\mathbf{E}-e\mathbf{v}_{c}\times\mathbf{B}-\frac{e^{2}}{\hbar}(\mathbf{E}\cdot\mathbf{B})\bm{\Omega}_{c}\right].$
(4)

In the above equation $\bm{\Omega}_{c}$ denotes the wave vector dependent
Berry curvature $\bm{\Omega}_{c}(\mathbf{k})$ at node $c$ and
$D_{c}(\mathbf{k})=1+\frac{e}{\hbar}(\mathbf{B}\cdot\bm{\Omega}_{c})$ is the
phase space correction factor xiao2010 .

Our interest is focused on the solution of Boltzmann equation which simplifies
considerably in homogeneous systems and in the absence of external magnetic
field sodemann2015 ; dantas2021 ; zhang2021 . With such an assumptions of
system homogeneity and absence of magnetic field, the distribution function
$f(r,\textbf{k},t)=f(\textbf{k},t)$ is independent on r. We suppose also the
validity of relaxation time approximation. The Boltzmann equation reduces to

$\frac{\partial f(\textbf{k},t)}{\partial t}+\dot{k}^{a}\frac{\partial
f(\textbf{k},t)}{\partial
k_{a}}=-\frac{f(\textbf{k},t)-f_{0}(\textbf{k},0)}{\tau}.$ (5)

For simplicity, we neglect here the explicit dependence on $c$ and assume the
relaxation time $\tau$ is isotropic and constant.

In the model with standard electric field $E^{a}(t)=Re~{}(E^{a}_{0}e^{i\omega
t})$ and dark matter electric field
$\tilde{E}^{a}(t)=Re~{}(\tilde{E}_{0}^{a}e^{i\tilde{\omega}t})$, the electron
charge (-$e$) changes its momentum as a result of the electric force
rogatko2023

$\hbar\dot{k}^{a}=-e\Big{[}E^{a}(t)+\frac{\alpha}{2}\tilde{E}^{a}(t)\Big{]}=-e~{}Re~{}\Big{(}E^{a}_{0}e^{i\omega
t}+\frac{\alpha}{2}{\tilde{E}_{0}^{a}}e^{i\tilde{\omega}t}\Big{)},$ (6)

where $\alpha$ is the coupling constant in the kinetic mixing between two
$U(1)$-gauge fields in equation (1). It is a source of the change of the total
electric field acting upon an electron in the background of dark matter
sector.

##  III Solution of the Boltzmann equation in the presence of Maxwell and dark
matter fields

This section is devoted to the solution of the equation (5). We shall find
corrections to the distribution function to an arbitrary order in the electric
fields $\textbf{E}(t)$ and $\tilde{\textbf{E}}(t)$. For a standard Maxwell
electric field only, the solutions can be found in Refs. sodemann2015 ;
dantas2021 ; zhang2021 . Even though it is possible to generalize the previous
work dantas2021  and obtain the non-perturbative solution of the equation (5)
we limit the discussion to quadratic order. To this end, in this section we
follow Zhang et al. zhang2021 .

To commence with, one writes the total distribution function as a sum given by

$f(\textbf{k},t)=Re\Big{(}f_{0}+f_{1}+f_{2}+\cdots f_{n}+\cdots\Big{)},$ (7)

where $f_{n}$ is assumed to vanish as a power $E_{0}^{n},\tilde{E}_{0}^{n}$.
Having in mind the relations (5)- (6) and counting the powers of electric
fields, we derive the set of equations

$\displaystyle\tau\partial_{t}f_{0}(\textbf{k},t)$ $\displaystyle=$
$\displaystyle 0,$ (8) $\displaystyle\tau\partial_{t}f_{1}+f_{1}$
$\displaystyle=$ $\displaystyle\frac{e\tau}{2\hbar}\left(E_{0}^{a}e^{i\omega
t}+E_{0}^{*a}e^{-i\omega
t}+\frac{\alpha}{2}\tilde{E}_{0}^{a}e^{i\tilde{\omega}t}+\frac{\alpha}{2}\tilde{E}_{0}^{*a}e^{-i\tilde{\omega}t}\right)\partial_{a}f_{0},$
(9) $\displaystyle\tau\partial_{t}f_{2}+f_{2}$ $\displaystyle=$
$\displaystyle\frac{e\tau}{2\hbar}\left(E_{0}^{a}e^{i\omega
t}+E_{0}^{*a}e^{-i\omega
t}+\frac{\alpha}{2}\tilde{E}_{0}^{a}e^{i\tilde{\omega}t}+\frac{\alpha}{2}\tilde{E}_{0}^{*a}e^{-i\tilde{\omega}t}\right)\partial_{a}f_{1},$
(10) $\displaystyle\cdots,$ $\displaystyle\tau\partial_{t}f_{n}+f_{n}$
$\displaystyle=$ $\displaystyle\frac{e\tau}{2\hbar}\left(E_{0}^{a}e^{i\omega
t}+E_{0}^{*a}e^{-i\omega
t}+\frac{\alpha}{2}\tilde{E}_{0}^{a}e^{i\tilde{\omega}t}+\frac{\alpha}{2}\tilde{E}_{0}^{*a}e^{-i\tilde{\omega}t}\right)\partial_{a}f_{n-1},$
$\displaystyle etc.$

We have used here the shorthand notations
$\partial_{t}=\frac{\partial}{\partial t}$ and
$\partial_{a}=\frac{\partial}{\partial k_{a}}$. The above equations can be
solved order by order.

Equation (8) comprises a statement that the equilibrium distribution function
does not depend on time $f_{0}(\textbf{k},t)=f_{0}(\textbf{k})$. Further, one
assumes that the initial distribution is given by the Fermi-Dirac function
corresponding to temperature $T$ and chemical potential $\mu$. Namely, we have

$f_{0}(\textbf{k})=\frac{1}{e^{\frac{\varepsilon(\textbf{k})-\mu}{k_{B}T}}+1}.$
(12)

The solution of equation (9) is simple and and can be written as

$f_{1}(\textbf{k},t)=Re~{}\Big{(}f_{1}^{\omega}e^{i\omega
t}+f_{1}^{\tilde{\omega}}e^{i\tilde{\omega}t}\Big{)},$ (13)

where we have denoted

$\displaystyle f_{1}^{\omega}$ $\displaystyle=$
$\displaystyle\frac{e\tau}{\hbar}\frac{E_{0}^{a}~{}\partial_{a}f_{0}}{(1+i\omega\tau)},$
(14) $\displaystyle f_{1}^{\tilde{\omega}}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2}\frac{e\tau}{\hbar}\frac{\tilde{E}_{0}^{a}~{}\partial_{a}f_{0}}{(1+i\tilde{\omega}\tau)}.$
(15)

The second order term $f_{2}(\textbf{k},t)$, for two oscillating fields,
introduces various harmonic functions with frequencies
$2\omega,2\tilde{\omega},\omega\pm\tilde{\omega}$. Interestingly, the
frequency independent term also contributes at the second order. After some
algebra one arrives at the relation provided by

$f_{2}(\textbf{k},t)=Re~{}\Big{(}f_{2}^{(0)}+f_{2}^{(2\omega)}e^{2i\omega
t}+f_{2}^{(2\tilde{\omega})}e^{2i\tilde{\omega}t}+f_{2}^{(\omega+\tilde{\omega})}e^{i(\omega+\tilde{\omega})t}+f_{2}^{(\omega-\tilde{\omega})}e^{i(\omega-\tilde{\omega})t}\Big{)},$
(16)

where the frequency dependent coefficients imply

$\displaystyle f_{2}^{(0)}$ $\displaystyle=$
$\displaystyle\left(\frac{e\tau}{\hbar}\right)^{2}\left[\frac{E_{0}^{*b}E_{0}^{a}~{}\partial_{ab}f_{0}}{2(1+i\omega\tau)}+\left(\frac{\alpha}{2}\right)^{2}\frac{\tilde{E}_{0}^{*b}\tilde{E}_{0}^{a}~{}\partial_{ab}f_{0}}{2(1+i\tilde{\omega}\tau)}\right],$
(17) $\displaystyle f_{2}^{(2\omega)}$ $\displaystyle=$
$\displaystyle\left(\frac{e\tau}{\hbar}\right)^{2}\frac{E_{0}^{b}E_{0}^{a}~{}\partial_{ab}f_{0}}{2(1+i\omega\tau)(1+2i\omega\tau)},$
(18) $\displaystyle f_{2}^{(2\tilde{\omega})}$ $\displaystyle=$
$\displaystyle\alpha^{2}\left(\frac{e\tau}{2\hbar}\right)^{2}\frac{\tilde{E}_{0}^{b}\tilde{E}_{0}^{a}~{}\partial_{ab}f_{0}}{2(1+i\tilde{\omega}\tau)(1+2i\tilde{\omega}\tau)},$
(19) $\displaystyle f_{2}^{(\omega+\tilde{\omega})}$ $\displaystyle=$
$\displaystyle\alpha\left(\frac{e\tau}{2\hbar}\right)^{2}\left[\frac{E_{0}^{b}\tilde{E}_{0}^{a}~{}\partial_{ab}f_{0}}{(1+i\tilde{\omega}\tau)}+\frac{\tilde{E}_{0}^{b}{E}_{0}^{a}~{}\partial_{ab}f_{0}}{(1+i{\omega}\tau)}\right]\frac{1}{[1+i(\omega+\tilde{\omega})\tau]},$
(20) $\displaystyle f_{2}^{(\omega-\tilde{\omega})}$ $\displaystyle=$
$\displaystyle\alpha\left(\frac{e\tau}{2\hbar}\right)^{2}\left[\frac{E_{0}^{b}\tilde{E}_{0}^{*a}~{}\partial_{ab}f_{0}}{(1-i\tilde{\omega}\tau)}+\frac{\tilde{E}_{0}^{*b}{E}_{0}^{a}~{}\partial_{ab}f_{0}}{(1+i{\omega}\tau)}\right]\frac{1}{[1+i(\omega-\tilde{\omega})\tau]}.$
(21)

The higher order corrections can be obtained in a similar way, if needed. In
the next section we use the above formulae to calculate the ac currents in
chiral system.

##  IV Currents in chiral systems affected by the dark field

We are mainly interested in detecting dark matter fields which may arise in
the rare scattering processes of dark particles with ordinary matter
particles. For this purpose, we analyse the currents flowing in general
systems, i.e., those with and without Berry curvature and only later we
discuss possible detection scenarios. For chiral systems the velocity, the
distribution function and the Berry curvature, depend on the node index $c$.
However, in the following formulae we neglect this dependence and restrict our
attention to the discussion of the currents to the second order in the field
strength. To find the total current one has to take all nodes existing in a
given material.

In the kinetic equation approach the current flowing in d-dimensional system
is calculated from the relation

$j^{a}(t)=-e\int\frac{d^{d}k}{(2\pi)^{d}}\dot{r}^{a}f(\textbf{k},t),$ (22)

and can be found to an arbitrary order in electric field. However, in a chiral
system influenced by dark matter sector, the velocity $\dot{\mathbf{r}}$ is
modified by the Berry curvature and dark electric field rogatko2023

$\dot{r}^{a}=v^{a}(\textbf{k})-\frac{e}{\hbar}\epsilon^{abc}\Omega_{b}(\textbf{k})\Big{(}E_{c}(t)+\frac{\alpha}{2}\tilde{E}_{c}(t)\Big{)}.$
(23)

In the linear (in electric fields) order, the longitudinal currents flow
whatever the symmetry of the crystal, however the existence of Hall effect
requires breaking the time reversal symmetry. In the non-linear regime the
Hall current can be measured even in time reversal symmetric systems. It
appears at second sodemann2015  or higher zhang2021 ; zhuang2023 ; zhang2023 ;
xiang2023  orders in the electric field.

We calculate the current from the relation (22), with the usage of the
equation (23), first expanding the distribution function with respect to
powers of electric fields, as given in (7) and (6). To the lowest order, i.e.,
using $f(\textbf{k},t)=Re(f_{0})$, one obtains

$j_{0}^{a}(t)=-e\int\frac{d^{d}k}{(2\pi)^{d}}\left[v^{a}(\textbf{k})-\frac{e}{\hbar}\epsilon^{abc}\Omega_{b}~{}Re\Big{(}E_{c}(t)+\frac{\alpha}{2}\tilde{E}_{c}(t)\Big{)}\right]f_{0}(\textbf{k}).$
(24)

This can be rewritten as follows:

$j_{0}^{a}(t)=Re~{}\Big{(}j_{0}^{a}+j_{0}^{a(\omega)}e^{i\omega
t}+j_{0}^{a(\tilde{\omega})}e^{i\tilde{\omega}t}\Big{)},$ (25)

where we have denoted

$\displaystyle j_{0}^{a(0)}$ $\displaystyle=$
$\displaystyle-e\int\frac{d^{d}k}{(2\pi)^{d}}v^{a}(\textbf{k})f_{0}(\textbf{k}),$
(26) $\displaystyle j_{0}^{a(\omega)}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{\hbar}\int\frac{d^{d}k}{(2\pi)^{d}}\epsilon^{abc}\Omega_{b}E_{(0)c}f_{0}(\textbf{k}),$
(27) $\displaystyle j_{0}^{a(\tilde{\omega})}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2}\frac{e^{2}}{\hbar}\int\frac{d^{d}k}{(2\pi)^{d}}\epsilon^{abc}\Omega_{b}\tilde{E}_{(0)c}f_{0}(\textbf{k}).$
(28)

The total current related to the unperturbed distribution function $f_{0}$
consists of three terms. The first of them is proportional to the velocity
$v^{\alpha}(\textbf{k})$ and is independent of the electric field. This
rectified current $j_{0}^{a(0)}$ is purely classical. Its existence does not
require Berry curvature. However, it vanishes due to the fact that velocity is
an odd function of momentum $v^{a}(\textbf{k})=-v^{a}(-\textbf{k})$. This is
correct, as there can not be a current flow in the equilibrium.

On the other hand, the anomalous velocity induced by the Berry curvature
(second term in Eq. (23)) also gives the currents proportional to the
unperturbed distribution function and to electric field. To the first order in
the field, the current possesses two harmonic contributions bounded with the
standard and dark field of frequency $\omega$ and $\tilde{\omega}$,
respectively. It is this second contribution, that appearance signals the dark
sector effect. This is true, even in the experimental setup with $\omega=0$,
i.e., for the static standard electric field $E^{a}(t)=E_{0}^{a}$.

For $f(\textbf{k},t)$ linear in both fields ($f_{1}(\textbf{k},t)$), one finds
the current, which contains terms linear and quadratic in electric fields. The
linear terms result from the velocity v, while quadratic ones are due to
$\textbf{E}\times\mathbf{\Omega}$ term in relation (23). Thus, we obtain the
total time dependent current consisting of various components, each
oscillating at a specific frequency

$\displaystyle j_{1}^{a}(t)$ $\displaystyle=$ $\displaystyle
Re~{}\Big{(}j_{1}^{a(0)}+j_{1}^{a(\omega)}e^{i\omega
t}+j_{1}^{a(\tilde{\omega})}e^{i\tilde{\omega}t}+j_{1}^{a(2\omega)}e^{2i\omega
t}$ $\displaystyle+$ $\displaystyle
j_{1}^{a(2\tilde{\omega})}e^{2i\tilde{\omega}t}+j_{1}^{a(\omega+\tilde{\omega})}e^{i(\omega+\tilde{\omega})t}+j_{1}^{a(\omega-\tilde{\omega})}e^{i(\omega-\tilde{\omega})t}\Big{)},$

where the various coefficients are given by the following formulae:

$\displaystyle j_{1}^{a(0)}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{2\hbar}\int\frac{d^{d}k}{(2\pi)^{d}}\epsilon^{abc}\Omega_{b}\left[E_{(0)c}f_{1}^{*(\omega)}+\frac{\alpha}{2}\tilde{E}_{(0)c}f_{1}^{*(\tilde{\omega})}\right],$
(30) $\displaystyle j_{1}^{a(\omega)}$ $\displaystyle=$
$\displaystyle-e\int\frac{d^{d}k}{(2\pi)^{d}}v^{a}(\textbf{k})f_{1}^{(\omega)},$
(31) $\displaystyle j_{1}^{a(\tilde{\omega})}$ $\displaystyle=$
$\displaystyle-e\int\frac{d^{d}k}{(2\pi)^{d}}v^{a}(\textbf{k})f_{1}^{(\tilde{\omega})},$
(32) $\displaystyle j_{1}^{a(2\omega)}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{2\hbar}\int\frac{d^{d}k}{(2\pi)^{d}}\epsilon^{abc}\Omega_{b}E_{(0)c}f_{1}^{(\omega)},$
(33) $\displaystyle j_{1}^{a(2\tilde{\omega})}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2}\frac{e^{2}}{2\hbar}\int\frac{d^{d}k}{(2\pi)^{d}}\epsilon^{abc}\Omega_{b}\tilde{E}_{(0)c}f_{1}^{(\tilde{\omega})},$
(34) $\displaystyle j_{1}^{a(\omega+\tilde{\omega})}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{2\hbar}\int\frac{d^{d}k}{(2\pi)^{d}}\epsilon^{abc}\Omega_{b}\left[E_{(0)c}f_{1}^{(\tilde{\omega})}+\frac{\alpha}{2}\tilde{E}_{(0)c}f_{1}^{(\omega)}\right],$
(35) $\displaystyle j_{1}^{a(\omega-\tilde{\omega})}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{2\hbar}\int\frac{d^{d}k}{(2\pi)^{d}}\epsilon^{abc}\Omega_{b}\left[E_{(0)c}f_{1}^{*(\tilde{\omega})}+\frac{\alpha}{2}\tilde{E}_{(0)c}^{*}f_{1}^{(\omega)}\right],$
(36)

with $f_{1}^{(\omega)}$ and $f_{1}^{(\tilde{\omega})}$ given by the relations
(14) and (15). Note, that the latter function contains the factor
$\frac{\alpha}{2}$, which makes both coefficients (35) and (36) to be linear
in the coupling constant $\alpha$.

##  V Estimation of the dark photon induced electric field

The very recent measurements of the local dark matter densities conducted by
LAMOST DR5 and Gaia DR2 experiments guo20 ; loe22 , reveal that our Galactic
disc is immersed in dark matter halo with a characteristic mass density
$\rho_{DM}=0.5GeV/cm^{3}$. In addition, the latest observations of the
gravitational lensing seem to indicate that the model of wave-like dark matter
better fits the observed lensing features (’brightnesses and positions of
multiply-lensed images’), then the model of particle-like dark matter
amruth2023 .

The dark matter density is expected to be bounded with the amplitude of the
dark photon, and time and space components of its vector potentials
$\tilde{{A^{0}}}$ and $\tilde{\textbf{A}}$, which are subject to time and
space dependences kishimoto2022

$\displaystyle\tilde{A}^{0}(\textbf{r},t)$ $\displaystyle=$
$\displaystyle-\tilde{\textbf{v}}\cdot\tilde{\textbf{A}}_{c}\cos(m_{DP}t-m_{DP}\textbf{v}\cdot\textbf{r}),$
(37) $\displaystyle\tilde{\textbf{A}}(\textbf{r},t)$ $\displaystyle=$
$\displaystyle\tilde{\textbf{A}}_{c}\cos(m_{DP}t-m_{DP}\textbf{v}\cdot\textbf{r}).$
(38)

Furthermore, the time component of the dark photon field is suppressed by the
velocity of dark sector around the Earth $|\textbf{v}|\approx 10^{-3}c$. The
field oscillates with frequency $m_{DP}$ and thus provides a window for its
detection by frequency dependent measurements. The amplitude
$\tilde{\textbf{A}}_{c}$ is related to the density of the dark matter
$\rho_{DM}$, depending on dark photon mass beringer2012 ; iwazaki2020 ;
iwazaki2021 ; kishimoto2022

$\rho_{DM}=\frac{1}{2}m^{2}_{DP}\tilde{\textbf{A}}^{2}_{c}.$ (39)

The dark photon induced electric field E at the surface of the metal with
conductivity $\sigma$ implies kishimoto2022

${\textbf{
E}}=\alpha\sqrt{\frac{m_{DP}}{2\sigma}}\tilde{\textbf{A}}_{c}[\cos(m_{DP}t)-\sin(m_{DP}t)]=\alpha\sqrt{\frac{m_{DP}}{\sigma}}\tilde{\textbf{A}}_{c}\cos(m_{DP}t+\frac{\pi}{4}).$
(40)

The dark photon induced electric field oscillates with frequency
$\tilde{\omega}=\frac{m_{DP}c^{2}}{\hbar}$ and is phase shifted by $\pi/4$, in
comparison to the behaviour assumed in Sec. II.

Combining (39) and (40) one finds the magnitude of the dark matter induced
electric field amplitude

$\tilde{\textbf{E}}_{0}=c\sqrt{2\mu_{0}\rho_{DM}},$ (41)

in standard units. Using dark matter density 0.5GeV$/cm^{3}$ we get
$\tilde{\textbf{E}}_{0}\approx 130\frac{V}{m}$. This field oscillates at
frequency in GHz regime, induces in vacuum the electric field, oscillating
with the same frequency and small magnitude. Namely one has

$\textbf{E}_{0}=\alpha\tilde{\textbf{E}}_{0},$ (42)

as discussed earlier.

##  VI Enhancing the induced field

The induced electric field is scaled down by the kinetic mixing term $\propto$
coupling. Due to the fact that $\alpha\ll 1$ it would be of great importance
to amplify its value. One possibility is to implement the cavity resonators.
In such devices the electric field is reflected from two parallel mirrors
distance $L$ apart from each other. The wave moving towards one the mirrors is
reflected back changing the sign of the wave-vector and the sign of the
amplitude. The appropriate boundary conditions require the vanishing of the
field component parallel to the mirror. Depending on the details of the setup
the electric field inside the resonator may be further increased.

The reflection process is always related to some losses of the signal.
However, for high quality factor $Q$ of the resonator the reflection
coefficient $r$ is close to unity. This is because gue2023

$r=1-\frac{\pi}{2Q}.$ (43)

In typical resonators the standing waves form and this results in a very
inhomogeneous electromagnetic fields. The field inhomogeneity makes standard
resonators not applicable for the purpose of transport measurements and more
complicated systems are needed. This can be achieved by supplementing the
resonators with tuners (metallic reflectors) which change the boundary
conditions and may give more homogeneous fields, at the expense that not all
frequencies can be effectively used. With large quality factors one may get a
resonator of arbitrary size and working frequency. The details depend on the
size, shape and construction of tuners. Smaller resonators allow for larger
usable frequencies. Such devices are usually called electromagnetic
reverberation chambers hill2009 . It is a complicated task to calculate the
electric field enhancement in a resonator of arbitrary shape.

However, the order of magnitude estimation may be obtained by considering the
simplest possible device consisting of two parallel metallic mirrors similar
to that considered in gue2023 . It has been shown that both the external
electric field E and the dark matter introduced field $\tilde{\textbf{E}}$ get
enhanced. The idea is as follows. Consider a resonator in a form of
parallelepiped with two mirrors characterized by the reflection coefficient
$r$. The electric field propagating with a wave vector k and frequency
$\omega$ as a plane wave
$\textbf{E}=Re[\textbf{E}_{0}e^{i(\textbf{k}\cdot\textbf{r}-\omega t)}]$, with
the amplitude $\textbf{E}_{0}$, is reflected back on the mirror with
probability $r$ changing the direction and wave vector to opposite one. During
its travel from one mirror to another it also gets additional phase
$e^{i{kL}}$, where $k$ is component of k along the direction between the
mirrors. After many such reflections, the resulting total field
$\textbf{E}_{tot}(t)$ at $x=0$, which is taken to be in the middle of the
cavity, is found to be gue2023

$\textbf{E}_{tot}(t)=Re\left[\textbf{E}_{0}\frac{e^{i\frac{kL}{2}}}{1+re^{ikL}}e^{-i\omega
t}\right].$ (44)

Comparing this relation with our definition of the field amplitude given
before equation (6), one notices that the factor

$\textbf{E}_{0tot}=\textbf{E}_{0}\frac{e^{i\frac{kL}{2}}}{1+re^{ikL}},$ (45)

plays a role of $\textbf{E}^{*}_{0}$, used previously. Thus for an experiment
utilising cavity one replaces everywhere in the previous formulae
$\textbf{E}_{0}$ by $\textbf{E}_{0tot}^{*}$.

As far as the dark matter field is concerned, we have to notice that according
to the observations, it exists everywhere in the space around the Earth,
moving with relative velocity $v\approx 10^{-3}c$ and is considered as a
coherent over typical experimental distances. This leads to the conclusion
that it produces the electric field $\alpha\tilde{\textbf{E}}$ in both
mirrors. However, only the component perpendicular to the mirrors, denoted
$\tilde{\textbf{E}}_{\perp}$, propagates and in effect only this component is
enhanced. Adding the constant in space component one finds that the total dark
matter field inside the cavity at $x=0$, is given by gue2023

$\tilde{\textbf{E}}_{tot}(t)=\left[\tilde{\textbf{E}}_{0}+2\tilde{\textbf{E}}_{0\perp}\frac{e^{i\frac{kL}{2}}}{1+re^{ikL}}\right]e^{-i\tilde{\omega}t}.$
(46)

Once again we notice that the role of a complex amplitude of the dark matter
field $\tilde{\textbf{E}}^{*}_{0}$, used previously, is played by the
following relation:

$\tilde{\textbf{E}}_{0tot}=\left[\tilde{\textbf{E}}_{0}+2\tilde{\textbf{E}}_{0\perp}\frac{e^{i\frac{\tilde{k}L}{2}}}{1+re^{i\tilde{k}L}}\right],$
(47)

where the frequency and the wave vector of dark matter massive field in the
laboratory frame is calculated (for arbitrary boost direction) from

$\displaystyle\frac{\tilde{\omega}{{}^{\prime}}}{c}=\gamma(\frac{\omega}{c}-\textbf{v}\cdot\textbf{k})\approx\frac{\omega}{c},$
(48)
$\displaystyle\tilde{\textbf{k}}{{}^{\prime}}=\tilde{\textbf{k}}+\frac{1}{v^{2}}(\gamma-1)(\tilde{\textbf{k}}\cdot\textbf{v})\textbf{v}-\gamma\frac{\omega\textbf{v}}{c^{2}},$
(49)

where $\gamma=1/\sqrt{1-\frac{v^{2}}{c^{2}}}$. The approximate equality in the
first equation is motivated by the fact that $\gamma\approx 1$. Similarly,
this shows that
$\tilde{\textbf{k}}{{}^{\prime}}\approx-\frac{\omega\textbf{v}}{c^{2}}$, for
the dark photon mode of interest, $\tilde{\textbf{k}}=0$. The wave vector
$\tilde{\textbf{k}}^{\prime}$ can be neglected for macroscopic resonators,
when $\tilde{k}^{\prime}L\ll 2\pi$. The massless photon moves with
$k=\frac{\omega}{c}$.

To proceed, we rewrite (45) and (47) in terms of photon frequencies as

$\displaystyle\textbf{E}_{0tot}=\textbf{E}_{0}\frac{e^{i\frac{\omega
L}{2c}}}{1+re^{i\frac{\omega L}{c}}}=\textbf{E}_{0}B(\omega),$ (50)
$\displaystyle\tilde{\textbf{E}}_{0tot}=\tilde{\textbf{E}}_{0}\left[1+2\beta\frac{e^{i\frac{\tilde{\omega}L}{2c}}}{1+re^{i\frac{\tilde{\omega}L}{c}}}\right]=\tilde{\textbf{E}}_{0}A(\tilde{\omega}),$
(51)

and the parameter $\beta=1/\sqrt{3}$ characterises the average value of the
projection of dark photon field on the unit vector $\hat{\mathbf{n}}$
perpendicular to the mirror gue2023 . The frequency dependent factors
$A(\tilde{\omega})$ and $B(\omega)$ provide the appropriate complex
amplification amplitudes.

Figure 1: (color online) The dependence of the modulus of amplification factor
$W(\tilde{\omega},\omega)=A(\tilde{\omega})B^{*}(\omega)$ of the
$\textbf{j}_{1}^{\omega-\tilde{\omega}}$ current defined in Eq. (36). We
measure frequencies in unit of $c/L$ and assume the modest value of the
reflection coefficient $r=0.95$.

According to the previous discussion the transport experiments aimed at the
discovery of dark matter will concentrate on the currents related to dark
photon induced electric field. These include the anomalous Hall current (28),
which existence does not require external field. Its value is enhanced by the
amplitude $A(\tilde{\omega})$. Similarly, the longitudinal current (32), which
also does not require external field is amplified by the same factor. The
remaining two currents which are of linear order in $\alpha$, namely relations
(35) and (36) can be rewritten in terms of bar amplitudes and enhancement
factors. One finds that the current depending on the sum of frequencies is
enhanced by $A^{*}(\tilde{\omega})B^{*}(\omega)$, while that corresponding to
the difference of frequencies is enhanced by the complex amplitude given by
$A(\tilde{\omega})B^{*}(\omega)$. These enhancement factors are complex and
their maximal values depend on the reflection coefficient $r$. For cavities
with large $Q$ values $r$ is very close to 1 and the amplification factor may,
for particular frequencies, attain very large values.

For the illustration purposes we use $r=0.95$ and calculate the enhancement as
a function of frequency $\omega$ and $\tilde{\omega}$, in units of $c/L$. In
the figure we plot the absolute value of the amplification factor
$W(\tilde{\omega},\omega)=A(\tilde{\omega})B^{*}(\omega)$ which enters the
current $\textbf{j}_{1}^{\omega-\tilde{\omega}}$ defined by the relation (36).
The amplification factor is large and gets larger with the increase of the $r$
value. For $r=0.999$ it reaches the value as large as a few thousands.

##  VII Sensitivity analysis

As discussed in previous sections, at the linear order in the coupling
$\alpha$, the signals from dark photon of frequency $\tilde{\omega}$ which we
expect to detect by using external high frequency $\omega$ radiation may
appear at the set
$\\{\tilde{\omega},\omega+\tilde{\omega},\omega-\tilde{\omega}\\}$ of
frequencies. Signal at frequency $\tilde{\omega}$ appears at the linear order
of the field $\mathbf{\tilde{E}}$, while the other at the second order of
fields. As visible from the equations (33,35,36) the signals are proportional
to $\mathbf{\tilde{E}}$ in the linear case and the combinations of the fields
$\mathbf{\tilde{E}}\mathbf{E}$ enter the high frequency anomalous non-linear
Hall signals. It has to be mentioned that high frequency transport
measurements of the longitudinal and Hall currents can utilise various
techniques. The analysis of the projected experimental sensitivity may depend
on the particular measurement technique. Here we concentrate on one source of
experimental uncertainty which is the fluctuations in the amplitude of the
field. The estimate of the limit on the coupling $\alpha$ due to this source
are analogous to those in Ref. gue2023  and we shall follow their analysis.

The anomalous Hall currents we are interested in are of linear order in
$\alpha$ and proportional to the products of two fields. For the measurements
in the cavity one has to note that the cavity enhances both the field and its
fluctuations. The latter are related to the power fed into the cavity and to
the power fluctuations. Modelling the injected field amplitude by stochastic
component means that the field reads
$\mathbf{E}(\mathbf{r},t)=Re\left[\mathbf{E}_{0}e^{i(\mathbf{k}\mathbf{r}-\omega
t)}+\frac{\Delta\mathbf{E}_{0}}{2}(e^{i(\mathbf{(k-k_{0})}\mathbf{r}-(\omega-\omega_{0})t)}+e^{i(\mathbf{(k+k_{0})}\mathbf{r}-(\omega+\omega_{0})t)})\right]$.
Here, $k_{0},\omega_{0}$ is the wave vector and frequency of the amplitude
modulations. In principle, all involved phases can be shifted by arbitrary
constants, but we neglect here such phase shifts for simplicity. In analogy to
the calculations in Section (VI) we propagate all components of the field with
fluctuating amplitude and get at $\mathbf{r}=0$

$\mathbf{E}_{tot}(t)=Re\left[\mathbf{E}_{0}B(\omega)e^{-i(\omega
t+\phi)}+\frac{\Delta\mathbf{E}_{0}}{2}B(\omega_{-})e^{-i(\omega_{-}t+\phi_{-})}+\frac{\Delta\mathbf{E}_{0}}{2}B(\omega_{+})e^{-i(\omega_{+}t+\phi_{+})}\right].$
(52)

We reintroduced arbitrary phases $\phi$ and $\phi_{0}$ and denoted
$\phi_{\pm}=\phi\pm\phi_{0}$, $\omega_{\pm}=\omega\pm\omega_{0}$. The function
$B(\omega)=\frac{e^{i\frac{\omega L}{2c}}}{1+re^{i\frac{\omega L}{c}}}$ is the
same enhancement factor as in the relation (50).

To be on the save side one should take into account the enhanced fields at the
minimum and enhanced noise at the maximal values. This leads to the
conservative estimation. The order of the magnitude calculation of the
accuracy, using the parameters relevant for our proposal of the same value as
in gue2023 , leads to a rough sensitivity estimation of the order of
$\alpha\approx 10^{-10}\div 10^{-9}$. Importantly, even with the conservative
estimate the technique may be of importance, as it fills the gap between the
existing measurements by XENON1T and haloscope experiments as visible from the
Fig. (2). In the figure the thick yellow horizontal line shows the limit of
the kinetic mixing corresponding to our sensitivity estimate $=10^{-9}$. The
expected increase of sensitivity to $10^{-10}$ due to correlation between two
signals at two frequencies further closes the existing gap. This is
represented by the thin yellow line in the figure.

Figure 2: (color online) The expected constrained of the discussed technique
on the kinetic mixing $\alpha$ is marked on the standard diagram sensitivity
by the yellow solid lines. Thick line is for the conservative and thin one for
the optimistic sensitivity estimate as discussed in the text. In both cases
the proposed technique allows for partial filling of the gap between
haloscopes’ and XENON1T’s measurements.

We now provide a discussion of the experimental setup and the mass range which
can be accessed by the present technique. The measurements of ac longitudinal
and anomalous Hall currents are among standard tools in the condensed matter
field. The limitation on the upper limit of the dark photon mass is related to
its expected de Broglie wave-length $\lambda_{DP}=\frac{h}{m_{DP}v_{DP}}$,
which should be greater than the typical size of the measured sample $L_{s}$.
Using $L_{s}\approx 1mm\div 1cm$, $v_{DP}\approx 10^{-3}c$ one gets the upper
mass limit $\approx 6\div 60meV$. Knowing the dark matter density $\rho$ and
assuming that it is composed of dark photons of the above mass only we
estimate that there exist about $10^{11}\div 10^{10}$ of them in each cubic
$cm$ of space around Earth.

The limit on the lower mass range depends on the particular detection scheme.
We discuss here the most promising scheme requiring simultaneous measurements
of non-linear anomalous Hall currents at two frequencies
$\omega\pm\tilde{\omega}$, distance $2\tilde{\omega}$ apart. If the frequency
$\tilde{\omega}$ is too small the total frequency shift $2\tilde{\omega}$ may
not exceed the width of the experimental signal peak making the frequency
shift unobservable. Even though the precise numbers do depend on experimental
setup we expect that other unexplored mass range, namely $m_{DP}<10^{-14}$ eV
($c.f.$ wide unexplored range of masses in Fig. (2)), is not accessible in the
current scheme.

By definition the observed currents do not depend on the mass of the sample or
the acquisition time. Moreover, the sensitivity estimate does not depend on
the mass/frequency range and is thus valid from the lowest (in actual
experimental practice of order of hundred Hz) up to THz frequencies.

Additionally to the above estimates on the sensitivity we propose another
piece of information related to the issue and a particular technique of
measuring the anomalous Hall current. One way to learn about the high
frequency anomalous Hall conductance of materials is to measure the change of
the angle between the beams of linearly polarised light incident and reflected
from the surface of material displaying the Hall effect shimano2002 . This
technique known as Kerr effect has been widely utilised in measuring Hall
effect in superconductors with time reversal breaking order parameter
kapitulnik2009  without external magnetic field. To measure the tiny effects
in Sr2RuO4 kiw2012  and other superconductors special instrument was built
xia2006  reaching the accuracy of Kerr angle measurements as small as a few
$nrad$ at energy of light quanta $=0.8eV$. Combining this with recent
calculations sonowal2019  which show that in Weyl semimetals the Kerr angle is
of order of $0.1rad$ one gets the realistic estimate of sensitivity for the
existing apparatus and typical materials of the order of $10^{-8}$. We propose
to measure three different signals, so the statistical correlations among them
can be utilised to identify the events even if the individual measurements are
less precise.

##  VIII Discussion of detection scenarios and conclusions

We consider massive dark photon as a source of the electric field which
couples to the standard one by kinetic mixing characterised by the coupling
$\alpha$. With both, Maxwell and dark photon fields of respective frequencies
$\omega$ and $\tilde{\omega}$ taken into account, we have solved the Boltzmann
equation up to the first order in the fields, although, the proposed scheme is
valid to all orders. Moreover, the closed non-perturbative solution (valid to
arbitrary order of the electric fields) can also be obtained by
straightforward generalization of the recent paper dantas2021 . The
alternating longitudinal and anomalous Hall currents induced by the sum of the
external electric field and the field due to the dark photon are calculated up
to the second order in the fields and up to the linear order in the coupling
$\alpha$. There are three contributions of order of $\alpha^{1}$. The
conservative estimations show the sensitivity which sets the limit about
$10^{-9}\div 10^{-10}$. The proposed experiments may fill the gap between the
XENON1T and haloscope measurements, as visible from the Fig. (2). The existing
instrumentation should potentially allow detection of the effect in Weyl
semimetals.

Various detecting scenarios can be thought about in the context of the present
proposal. Nevertheless, one has to recall that the $\alpha$-coupling constant
in the kinetic mixing term is expected to be small, $\alpha\ll 1$, so the
terms of order $\alpha^{2}$ can be probably neglected as unmeasurable, even
with application of resonant electromagnetic cavities. This is true for the
second term of the rectified current (30) and for the current (34) which is
also proportional to $\alpha^{2}\tilde{\textbf{E}}^{2}_{(0)}$.

It is important to have a possibility to measure all currents which are of the
first order in $\alpha$. These are of the longitudinal and the anomalous Hall
character and show up at various field orders and frequencies as discussed
earlier. Thus the material to be used should have a band structure allowing
for the Berry curvature $\bm{\Omega(\textbf{k})}$ at some wave vector(s) k.
Only in such systems with two (or more) wave vectors not related by time
reversal symmetry there is a possibility to observe anomalous Hall effect in
the linear order, i.e., $\propto\tilde{\textbf{E}}$. The appearance of the
nonlinear anomalous Hall effect does not require breaking of time reversal
symmetry.

The expectation is that some of the two dimensional transition metal
dichalcogenides or other materials pronin2021  with non-vanishing Berry
curvature could serve the purpose. These materials are relatively clean and
their two dimensional character make the high frequency transport measurements
easier than their three dimensional counterparts. The expectation that the two
dimensional system may in some cases be better probes comes from the
observation that, e.g., in graphene a very large non-linear Kerr effect
zhang2012  has been measured. However, not the dimensionality itself, but the
distance $\Delta k_{W}$ between the Weyl nodes seems to be the most important
parameter, as the recent calculations suggest sonowal2019 , because the Kerr
signal is directly proportional to it.

We mention here the most obvious scenarios, leaving more detailed analysis to
the future work. One can imagine (i) measuring the current (28), which is the
ac analogue of the anomalous Hall haldane2004  current in the topological
material with broken time reversal symmetry. In a similar manner (ii) the
longitudinal current given by (32) is linear in $\tilde{\textbf{E}}_{(0)}$.
This current can also be spotted in a standard metallic system without Berry
curvature. Both these currents appear when the dark particle occasionally
collides with the studied material and produces oscillating electric field
$\propto\alpha\tilde{\textbf{E}}(t)$ in the absence of external standard
electric field $\textbf{E}=0$.

Other scenario (iii) relies on the application of a strong ac standard
electric field E of frequency $\omega$ and studying longitudinal and Hall
currents at frequencies $\omega\pm\tilde{\omega}$. This is potentially very
precise measurement which, however, requires the knowledge of frequency
$\tilde{\mathbf{\omega}}$ zong2023 . Unfortunately, the dark field frequency
is unknown and thus the apparatus and measurement details have to be tuned for
each expected mass, in a way very much similar as in other existing or
proposed experiments. However, the value of the applied frequency $\omega$ may
allow to entangle the signals as resulting from the matter as they appear
symmetrically with respect to applied frequency $\omega\pm\tilde{\omega}$.

Surely, the simultaneous observation of all three currents is potentially the
most desired scenario. The rough estimations suggest that the technique
sensitivity could be better than $10^{-10}$, especially if the temporal
correlations among three different signals will be utilised.

Our proposal supplements other techniques based on the condensed matter
systems. The important aspect of the present proposal is the possibility to
measure essentially simultaneously a few currents using topological quantum
materials (with Berry curvature). One expects the existence of three signals
with one longitudinal and two ac anomalous Hall currents. They appear at a few
frequencies, e.g., $\tilde{\omega},\omega\pm\tilde{\omega}$, what provides
additional control on the effect and an additional experimental knob, which
can be effectively used to identify the mass of the dark photon.

In summary, we have proposed to search for the dark matter induced electric
fields by means of the transport currents in chiral Weyl or Dirac semimetals.
The discussed here detection scheme of the dark matter is based on the
observations of ac longitudinal and Hall currents.

###### Acknowledgements.

K.I. W. and M. R. were partially supported by Grant No. 2022/45/B/ST2/00013 of
the National Science Center, Poland.

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