# Giant intrinsic anomalous terahertz Faraday rotation in the magnetic Weyl
semimetal Co2MnGa at room temperature

Xingyue Han  Department of Physics and Astronomy, University of Pennsylvania,
Philadelphia, Pennsylvania 19104, USA  Anastasios Markou  Max-Planck-Institut
fur Chemische Physik fester Stoffe, 01187 Dresden, Germany  Jonathan Stensberg
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia,
Pennsylvania 19104, USA  Yan Sun  Max-Planck-Institut fur Chemische Physik
fester Stoffe, 01187 Dresden, Germany  Claudia Felser  Max-Planck-Institut fur
Chemische Physik fester Stoffe, 01187 Dresden, Germany  Liang Wu
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Pennsylvania, Philadelphia, Pennsylvania 19104, USA

###### Abstract

We report measurement of terahertz anomalous Hall conductivity and Faraday
rotation in the magnetic Weyl semimetal Co2MnGa thin films as a function of
the magnetic field, temperature and thickness, using time-domain terahertz
spectroscopy. The terahertz conductivity shows a thickness-independent
anomalous Hall conductivity of around 600 $\Omega^{-1}\cdot cm^{-1}$ at room
temperature, and it is also frequency-independent from 0.2-1.5 THz. The
magnitude of the longitudinal and Hall conductivities, the weak spin-orbit
coupling, and the very close positioning of Weyl points to the chemical
potential all satisfy the criteria for intrinsic anomalous Hall conductivity.
First-principle calculation also supports the frequency-independent intrinsic
anomalous Hall conductivity at low frequency. We also find a thickness-
independent Faraday rotation of 59 ($\pm 6$) mrad at room temperature, which
comes from the intrinsic Berry curvature contribution. In the thinnest 20 nm
sample, the Faraday rotation divided by the sample thickness reaches around 3
mrad/nm due to Berry curvature, and is the largest reported at room
temperature. The giant Verdet constant of the order of 10 6 rad m -1 T -1 at
room temperature and the large Hall angle around 8.5 $\%$ from 0.2-1.5 THz
indicates that Co2MnGa is promising for THz spintronics at room temperature.

Introduction

The last decade witnessed an explosion of research on topological states of
matter characterized by the topological properties of the bulk wave-
functionsMoore (2010); Hasan and Kane (2010); Qi and Zhang (2011). Topological
insulators are robust to adiabatic perturbation as long as the bulk gap is not
closedHasan and Kane (2010); Qi and Zhang (2011). Weyl semimetals are newly
discovered topological states of matter without a bulk gap and with open Fermi
surface arcs when either time-reversal or inversion symmetry is brokenWan  _et
al._ (2011); Burkov and Balents (2011); Xu  _et al._ (2011); Weng  _et al._
(2015); Huang  _et al._ (2015); Xu  _et al._ (2015); Lv  _et al._ (2015); Yang
_et al._ (2015). These materials have accidental band touching at pairs of
points with different chirality in the momentum space. Near these points, the
quasi-particles (low-energy excitations) can be described by Weyl equations
first proposed by Hermann Weyl in 1929Weyl (1929). As a result, these touching
points in the band structure are called Weyl points, and the quasi-particles
near them resemble Weyl fermions. The bulk wave functions in Weyl semimetals
acquire a Berry phase as they move around the Weyl point because each Weyl
node behaves like a fictional magnetic field known as Berry curvature. These
Weyl points (monopoles in k space) are also topologically protected because
translation-invariant perturbations are identical to moving Weyl points in the
momentum space, unless they meet in the zone boundary and annihilate with each
other.

Weyl semimetals host many exotic phenomena including interesting temperature
and frequency dependence in optical conductivityHosur  _et al._ (2012); Xu
_et al._ (2020); Ni  _et al._ (2020), novel quantum oscillations related with
Fermi arcsPotter  _et al._ (2014), giant second harmonic generationWu  _et
al._ (2017a); Patankar  _et al._ (2018), as well as a chiral magnetic effect
and intrinsic anomalous Hall effectBurkov (2015). Experimental signatures of
Weyl semi-metals are still far behind the advance of various theoretical
proposals. Thus far, the widely experimentally studied Weyl semi-metal
materials are non-magnetic, but with broken inversion symmetryXu  _et al._
(2015); Lv  _et al._ (2015); Yang  _et al._ (2015); Wu  _et al._ (2017b).
Fermi arcs in magnetic Co3Sn2S2 and Co2MnGa were identified recently
Belopolski  _et al._ (2019); Morali  _et al._ (2019), which established direct
evidence for the magnetic Weyl semimetals. Among them, Co2MnGa is particularly
interesting as it is a room temperature ferromagnetic with a large anomalous
Hall effect and a high curie temperature at $T_{C}$=690 K Manna  _et al._
(2018); Markou  _et al._ (2019). Bulk Co2MnGa is a Heusler compound, which has
a cubic face-centered structure with space group $Fm\bar{3}m$ (No. 225). Large
anomalous Nernst effects are also observed in both bulk and thin filmsSakai
_et al._ (2018); Reichlova  _et al._ (2018); Park  _et al._ (2020).

The anomalous Hall effect (AHE) has been studied extensively in conventional
ferromagnetic materials Nagaosa  _et al._ (2010); Lee  _et al._ (2004). The
intrinsic AHE is a scattering-independent process first proposed by Karplus
and Luttinger Karplus and Luttinger (1954), which depends only on the
topological band structure with the contribution of the Berry curvature in the
momentum space Nagaosa  _et al._ (2010); Xiao  _et al._ (2010); Armitage  _et
al._ (2018). The AHE can be greatly enhanced when Fermi energy is close to
band (anti-)crossings such as the Weyl points Fang  _et al._ (2003); Yao  _et
al._ (2004); Kübler and Felser (2016). Due to its exotic band structure,
magnetic Weyl semimetals (WSMs) such as Co2MnGa are a good platform to study
the intrinsic AHE Yan and Felser (2017); Hosur and Qi (2013). When the
chemical potential is close to the Weyl points, the nonzero net Berry
curvature effect is dominating, and a large intrinsic anomalous Hall effect
can be calculated by the Kubo formula and compared with experiments Nagaosa
_et al._ (2010).

Another advantage of Co2MnGa is that thin films, which are ideal for device
applications, can be fabricated by sputtering. The samples studied in this
work are grown by sputtering on MgO substrates and capped with 3 nm Al, the
growth details of which were the subject of previous studyMarkou  _et al._
(2019). At room temperature, transport measurement on thin films show a large
anomalous Hall conductivity (AHC) of 814 $\Omega^{-1}\cdot cm^{-1}$ and a
large anomalous Hall angle (AHA) of 10.5 %Markou  _et al._ (2019). However,
the spectrum of the anomalous Hall conductivity, $\sigma_{xy}(\omega)$, and
Hall angle, $\theta_{H}(\omega)$, at terahertz frequency remain missing.
Future high-speed spintronic devices will crucially rely on such AHE phenomena
at terahertz (THz) frequencies. Current DC AHE transport measurements
significantly lag behind other interests of information carriers such as
electrons in field-effect transistors featuring cut-off frequencies of around
1 THz Del Alamo (2011). Therefore, a proper understanding of the thin-film
properties at THz frequencies is required. Another intriguing aspect is the
possibly giant magneto-optical Faraday effect, i.e. the rotation of the light
polarization of the transmitted light for the magnetic media, resulting from
the enhanced Hall conductivity on the transitions near the Weyl point. The
magnitude of the Faraday rotation is proportional to the terahertz Hall
conductivity. This novel optical property inherent to the magnetic WSM remains
to be explored.

In our work, we study the THz AHC, AHA and Faraday rotation spectra of Co2MnGa
thin films using magneto-optical methods. We employ time-domain terahertz
spectroscopy (TDTS) to measure optical longitudinal conductivity and complex
Faraday rotation at 0.2-1.5 THz. Giant THz AHC and AHA at RT around 600
$\Omega^{-1}\cdot cm^{-1}$ and 8.5 % are observed. AHE remains operative from
DC up to 1.5 THz with a flat frequency response in thin films. The optical
Faraday rotation and AHC both show a thickness-independent behavior at RT. One
of the samples exhibits a 2.5 mrad/nm Faraday rotation divided by thickness,
which is the largest value reported at RT. Using first-principle calculation,
we show that the AHC and Faraday rotation are dominating from the intrinsic
contribution.

Figure 1:  Longitudinal conductivity spectra $\sigma_{xx}(\omega)$ of a 40 nm
Co2MnGa thin film from 2 K to 280 K. The solid lines are the real part. The
dashed lines are the imaginary part.

Results and Discussion

Figure 2:  A) A sketch of the experimental geometry of the Faraday rotation
measurement. B) Field dependent Faraday angle spectra of a 40 nm Co2MnGa
sample at 290 K. C) Field dependence of the average Faraday angle over the
frequency range of 0.2-1.5 THz of the 40 nm Co2MnGa sample. D) Thickness
independent Faraday angle under 2 T at 290 K. E) Temperature dependence of
Faraday angle of the 40 nm Co2MnGa under 2.5 T. F) Thickness-normalized
Faraday rotation and anomalous Hall conductivity for Co2MnGa and several
magnetic thin films. The values of Co2MnGa, Co3Sn2S2Okamura  _et al._ (2020),
Mn3SnMatsuda  _et al._ (2020); Khadka  _et al._ (2020), and SrRuO3Shimano  _et
al._ (2011) are measured by THz measurement. The values of GdFeSeifert  _et
al._ (2021) and DyCo5Seifert  _et al._ (2021) are estimated from THz
measurement. The values of FeMiyasato  _et al._ (2007), CoMiyasato  _et al._
(2007), NiMiyasato  _et al._ (2007), GdMiyasato  _et al._ (2007), FePtHe  _et
al._ (2012), Co2MnAlSakuraba  _et al._ (2020), Co2MnSiSakuraba  _et al._
(2020), Mn5Ge3Zeng  _et al._ (2006) and Mn4NShen  _et al._ (2014) are
estimated from transport measurement.

In our TDTS, a laser beam from a fiber laser (center wavelength 780nm,
duration 82 fs) is split by a beam splitter into pump and probe beams. On the
pump side, a photoconductive antenna with 34 V bias voltage and 1.8 kHz
modulation frequency is used to generate THz pulses. Four off-axis parabolic
mirrors (OAPs) are placed in a confocal geometry to collect and focus the THz
beam. Thin film sample is put on the focus of the second and third OAP. On the
probe side, the time delayed probe beam hits the detector antenna to detect
the transmitted THz electric field using optical sampling method. The entire
THz path is enclosed in a purge box with relative humidity <3% to avoid water
absorption. We present the zero-field terahertz conductivity on these films
first. Utilizing TDTS, we obtain the complex longitudinal conductivity
spectra, $\sigma_{xx}(\omega)$, as we measure both the amplitude and the phase
of the transmission without using Kramers–Kronig transformation Wu  _et al._
(2013, 2015). In the thin-film approximation, the longitudinal conductivity
can be extracted from the following relation:

$T_{xx}(\omega)=\dfrac{n+1}{n+1+d\sigma_{xx}(\omega)Z_{0}}exp(i\dfrac{\omega}{c}\Delta
L(n-1)).$ (1)

Here $T_{xx}$ is the transmission coefficient of the thin film, $n$ is the
refractive index of the substrate, $d$ is the thickness of the film, $Z_{0}$
is the impedance of free space, $\Delta L$ is the thickness mismatch between
the sample and the bare substrate. Fig. 1 shows the real and imaginary part of
$\sigma_{xx}(\omega)$ of 40 nm Co2MnGa from 2 K to 280 K. The temperature
dependence is quite weak as the the real conductivity changes only by 8 $\%$
from 2 K to 280 K. The results follow the Drude model,
$\sigma_{xx}(\omega)=\sigma_{0}/(1-i\omega\tau)-i\epsilon_{0}(\epsilon_{\infty}-1)\omega$,
where $\sigma_{0}$ is the DC conductivity, $\tau$ is the scattering time,
$\epsilon_{0}$ is the vacuum permittivity, and $\epsilon_{\infty}$ is a
constant that describes lattice polarizability at high frequency. The flat
spectra in the real part (solid lines) and the small linear spectra of Im
$\sigma_{xx}(\omega)$ (dashed lines) both indicates a short $\tau$. Fitting
shows $1/\tau$ is around 3-5 THz in the whole temperature range. The magnitude
of the real part of the longitudinal conductivity falls into one of the
criteria for intrinsic AHE Miyasato  _et al._ (2007); Onoda  _et al._ (2006)
as discussed below.

Besides the intrinsic contribution to AHE, there are two other kinds of
extrinsic contributions: skew scattering and side jump Nagaosa  _et al._
(2010). In ferromagnets, the combination of the total anomalous Hall and
longitudinal conductivity show a crossover behavior in three regions according
to the magnitude of longitudinal conductivityMiyasato  _et al._ (2007); Onoda
_et al._ (2006): dirty regime ($\sigma_{xx}<10^{4}$ $\Omega^{-1}\cdot
cm^{-1}$), intermediate regime ($\sigma_{xx}=10^{4}-10^{6}$ $\Omega^{-1}\cdot
cm^{-1}$), and extreme conducting regime ($\sigma_{xx}>10^{6}$
$\Omega^{-1}\cdot cm^{-1})$. The intrinsic contribution always dominates in
the intermediate regime. The magnitude of the real part of the longitudinal
conductivity in Co2MnGa falls at the lower boundary in the intermediate
regime. Often, to separate the three contributions, the scaling relation
between longitudinal conductivity and anomalous Hall conductivity is used. The
contrition of the skew scattering to the AHC is proportional to the square of
longitudinal conductivity $\sigma_{xy}^{A-skew}\propto\sigma_{xx}^{2}$, while
the intrinsic and side-jump contributions are both independent of the
longitudinal conductivity. Previous DC transport on Co2MnGa bulk crystal used
the scaling relation and revealed that the intrinsic AHC is around $10^{3}$
$\Omega^{-1}\cdot cm^{-1}$ Manna  _et al._ (2018); Belopolski  _et al._
(2019).

Nevertheless, it is very difficult to separate the intrinsic contribution and
side jump in DC transport Nagaosa  _et al._ (2010). Interestingly, Co2MnGa
also satisfies the other two criteria that favors intrinsic contribution Onoda
_et al._ (2006). 1) The AHC is on the order of 1000
$\Omega^{-1}$$\cdot$$cm^{-1}$. The intrinsic AHC value in the thin film and
bulk crystal was reported to be 1138 $\Omega^{-1}$$\cdot$$cm^{-1}$Markou  _et
al._ (2019) and 1164 $\Omega^{-1}$$\cdot$$cm^{-1}$Manna  _et al._ (2018)
respectively. 2) The anticrossing point (the Weyl point) is only around 80 meV
above the chemical potentialBelopolski  _et al._ (2019). Because of these
three criteria, one can perform first-principles calculations to reliably
predict the frequency-dependent intrinsic contribution and compare it with
experiments.

With a set of freestanding wire-grid polarizers as shown in Fig. 2A, our TDTS
can resolve the polarization state of the THz signal and measure the frequency
dependent AHC. Three THz wire-grid polarizers (P1,P2,P3, extinction ratio>2000
at 1 THz) are used to measure the Faraday angle. P1 aligns the incident
polarization vertically. P2 is mounted on a rotation stage (not shown in the
figure) to selectively pass the vertical electric field ($E_{x}$) or
horizontal electric field ($E_{y}$). P3 is fixed at 45∘ so that $E_{x}$ and
$E_{y}$ have the same response at the detector. The polarization change before
and after the Co2MnGa film is the Faraday rotation
$\theta_{F}(\omega)=\dfrac{E_{y}(\omega)}{E_{x}(\omega)}$. These measurements
were performed under an out-of-plane magnetic field up to 7 T. To exclude the
nonmagnetic effects such as birefringence from the windows, we apply $\pm$B to
get a symmetrized Faraday angle spectra
$\theta_{F}(\omega,H)=[\theta_{F}^{meas}(\omega,H)-\theta_{F}^{meas}(\omega,-H)]/2$.
Fig. 2B shows the Faraday angle $\theta_{F}(\omega)$ of the 40 nm sample at
290 K. The rotation has a weak dependence on the frequency. The field
dependence is similar to the known magnetization curve with a saturation field
$H_{s}\approx 1.5$ TMarkou  _et al._ (2019) as shown in Fig. 2C. Note that
usually the Faraday rotation is proportional to the thickness and
characterized by the Verdet constant Meyers  _et al._ (1987). The large
$\theta_{F}$ around 60 mrad at room temperature is nearly thickness
independent as in Fig. 2D, which mainly contribute from the terahertz
anomalous Hall effect. $\theta_{F}$ increases monotonically and reaches 80
mrad as the temperature cools down to 2K (see Fig. 2E). Below 1 Tesla, the
Verdet constant of the 40 nm sample reaches 10 6 $rad$ $m^{-1}T^{-1}$, which
is of similar size of the giant magneto-optical effect in topological
insulators HgTe Shuvaev  _et al._ (2011) and Bi2Se3 Valdés Aguilar  _et al._
(2012); Wu  _et al._ (2015) at low temperature.

In Fig. 2F, we show the normalized Faraday rotation by thickness of a few
magnetic materials at room temperature and low temperature. Co2MnGa has the
largest room temperature value of 3 mrad/nm to our best knowledge. It is 2-3
orders of magnitude larger than the typical transition metals Fe, Co, Ni, and
GdMiyasato  _et al._ (2007). It is 15-30 times larger than the value in the
Weyl antiferromagnet candidate Mn3Sn Matsuda  _et al._ (2020); Khadka  _et
al._ (2020) and the ferromagnet Co-based Heusler compound Co2MnSi Sakuraba
_et al._ (2020). The value in Co2MnGa is twice larger than another half
metallic ferromagnet, Co2MnAl Sakuraba  _et al._ (2020), which is known to
exhibit similar AHE from bulk crystal with Co2MnGaLi  _et al._ (2020). Even if
we make the comparison at low temperature, 3 mrad/nm is similar to the
magnetic Weyl semimetal Co3Sn2S2Okamura  _et al._ (2020), and larger than any
other materials (see Fig.2F). The realization of the giant Faraday rotation at
room temperature overcomes the cryogenic temperature limit, which makes it
promising in spintronics applications. A recent work measures the broadband
THz anomalous Hall conductivity in the ferrimagnets DyCo5 and GdFe Seifert
_et al._ (2021), and we compute the Faraday rotation by using the reported
transverse and longitudinal conductivity. The small anomalous Hall
conductivity in DyCo5 probably indicates an extrinsic origin from the side
jump or skewing scattering. GdFe is interesting as it also shows both large
anomalous Hall conductivity and Faraday rotation, but whether the origin of
both are intrinsic needs further investigation. Note that the GdFe Seifert
_et al._ (2021) samples studied previously were amorphous. Therefore the Berry
curvature from the Bloch wave function is ill-defined, and it is not straight-
forward to study the intrinsic effect in GdFe Seifert  _et al._ (2021). As
discussed below, we will show that the large anomalous Hall conductivity and
Faraday rotation have an intrinsic orgin in Co2MnGa.

Figure 3:  A) Field dependent Hall conductivity spectra of the 40 nm Co2MnGa
at 290 K. B) Field dependence of the average Hall conductivity over frequenncy
range of 0.2-1.5 THz of the 40nm Co2MnGa at 3 K and 290 K. The intercepts of
dashed lines on the vertical axis are the anomalous Hall conductivity
$\sigma_{xy}^{A}$. C) Thickness independent Hall conductivity $\sigma_{xy}$
under 2 T at 290 K. D,E) First principle calculation of RT Hall conductivity
with $E_{F}$=-80 meV and 0 meV. We define the Weyl point position as 0 meV.
F)Hall angle spectra of the 40 nm Co2MnGa at 6 K and 290 K

.

The terahertz Hall conductivity spectra $\sigma_{xy}(\omega)$ can be extracted
from the Faraday rotation by the relation

$\sigma_{xy}(\omega)=\theta_{F}(\sigma_{xx}+\dfrac{n+1}{dZ_{0}})$ (2)

where $n$ is the refractive index of the substrate, $Z_{o}$ is the vacuum
impedance and $d$ is the thickness of the film. FIG. 3A shows the terahertz
Hall conductivity $\sigma_{xy}(\omega)$ of 40 nm Co2MnGa. Consistent with the
flat spectra of longitudinal conductivity and Faraday rotation, it is also
independent of frequency. The mean value between 0.2 and 1.5 THz under each
field is plotted in FIG. 3B, again scaling with the magnetizationMarkou  _et
al._ (2019). According to the Hall conductivity formula:

$\sigma_{xy}=R_{0}H+\sigma_{xy}^{A}$ (3)

The total Hall conductivity $\sigma_{xy}(\omega)$ consists of the ordinary
Hall effect, $R_{0}H$, where $R_{0}$ is the ordinary Hall coefficient, and the
anomalous Hall effect, $\sigma_{xy}^{A}$. The ordinary Hall conductivity
scales with the magnetic field. The anomalous term can be obtained by
extrapolating the high field data to zero field (FIG. 3B intersection). The 40
nm Co2MnGa shows a giant value of $|\sigma_{xy}^{A}|=930$ $\Omega^{-1}\cdot
cm^{-1}$ at 2 K, and $|\sigma_{xy}^{A}|=590$ $\Omega^{-1}\cdot cm^{-1}$ at 290
K. These results are consistent with the DC transport values on the 40 nm
Co2MnGa Markou  _et al._ (2019). Similar to RT Faraday rotation, the RT Hall
effect is also thickness independent, with the Hall conductivity value around
600 $\Omega^{-1}\cdot cm^{-1}$ as shown in FIG. 3C. This can be qualitatively
analysed from Eq.(2). As the sample thickness increases, the first term,
$\sigma_{xx}$, slightly increases, but the second term $\frac{n+1}{dZ_{0}}$
decreases slightly as well. The changes in the two terms tend to cancel each
other, resulting in a thickness independence behavior in the 20 to 100 nm
range. However, the main reason for the thickness-independent Faraday rotation
is due to the thickness-independent intrinsic anomalous Hall conductivity as
discussed below. We would like to point out that substrates from different
vendors and different capping could lead to a change of 10$\%$ of AHC at room
temperature. Note that the experiment value $|\sigma_{xy}^{exp}|\sim 600$
$\Omega^{-1}\cdot cm^{-1}$ is very close to the theoretical prediction of
intrinsic Hall effect in the order of $|\sigma_{xy}^{int}|\sim e^{2}/ha\sim
670$ $\Omega^{-1}\cdot cm^{-1}$ for Co2MnGa Miyasato  _et al._ (2007), where
$a$ is the lattice constant. This indicates that the intrinsic contribution
(the Berry curvature) is the leading effect.

The large scattering rate in Co2MnGa makes it difficult to separate the
intrinsic and side jump contributions via the scaling of $\sigma_{xy}$ versus
$\sigma_{xx}^{2}$ at terahertz frequency, as the measured THz frequency range
is comparable to the scattering rate Nagaosa  _et al._ (2010). Nevertheless,
as we discussed above, because Co2MnGa satisfies the three criteria for
intrinsic anomalous Hall conductivity in terms of longitudinal conductivity,
Hall conductivity, and chemical potential, it is quite accurate to use density
functional theory (DFT) to calculate and identify the intrinsic contribution.
We calculate the electronic band structure based on DFT by employing the full-
potential local-orbital code (FPLO) with localized atomic basis Koepernik and
Eschrig (1999). The exchange and correlation energies were considered in the
generalized gradient approximation (GGA) level following the
Perdew–Burke–Ernzerhof parametrization schemePerdew  _et al._ (1996).
Following experimental results, we set a ferromagnetic structure with magnetic
moment along $z$ directionManna  _et al._ (2018). We projected the Bloch
wavefunction into high symmetric atomic-orbital-like Wannier functions
Koepernik  _et al._ (2021) and constructed the tight-binding model Hamiltonian
by the Wannier function overlap. Based on the tight-binding model Hamiltonian,
the DC anomalous Hall conductivity and THz conductivity were computed by
following the Kubo formula approach in linear response approximation and clean
limit Kubo (1957); Greenwood (1958); Xiao  _et al._ (2010); Nagaosa, Naoto and
Sinova, Jairo and Onoda, Shigeki and MacDonald, Allan H and Ong, Nai Phuan
(2010), with AHC

$\displaystyle\sigma_{xy}^{DC}(E_{F})=2e^{2}\hbar\int_{BZ}\frac{d^{3}k}{(2\pi)^{3}}\underset{E_{n}(k)\leq
E_{F}}{\sum}f_{n}(k)$ (4) $\displaystyle Im\underset{m\neq
n}{\sum}\frac{<u_{n}(k)|\hat{v}_{x}|u_{m}(k)><u_{m}(k)|\hat{v}_{y}|u_{n}(k)>}{(E_{m}(k)-E_{n}(k))^{2}+\eta^{2}}$

and terahertz conductivity

$\displaystyle\sigma_{xy}(\hbar\omega)=ie^{2}\hbar\int_{BZ}\frac{d^{3}k}{(2\pi)^{3}}\underset{m\neq
n}{\sum}\frac{f_{m}(k)-f_{n}(k)}{E_{m}(k)-E_{n}(k)}$ (5)
$\displaystyle\frac{<u_{n}(k)|\hat{v}_{x}|u_{m}(k)><u_{m}(k)|\hat{v}_{y}|u_{n}(k)>}{E_{m}(k)-E_{n}(k)-(\hbar\omega+i\eta)}$

where $f_{n}(k)$ is the Fermi-Dirac distribution, $E_{n}(k)$ is the eigenvalue
of $n$th band with eigenstate $|u_{n}(k)>$,
$\hat{v}_{x}(k)=\frac{1}{\hbar}\frac{\partial\hat{H}(k)}{\partial k_{x}}$ is
the velocity operator, $\hbar\omega$ is the transition energy, and $\eta$ is a
smearing parameter to avoid numerical divergence. (Here we set $\eta$=0.1
meV.) We used a dense $k$-grid of $240^{3}$ for the numerical integration. As
shown in FIG. 3D, when the Weyl points are around 80 meV above the chemical
potential, it matches the anomalous THz Hall conductivity, which also agrees
with previous transport and ARPES studies Markou  _et al._ (2019); Belopolski
_et al._ (2019). FIG. 3D shows the intrinsic anomalous THz Hall conductivity
over a larger frequency range with resonant features associated with Berry
curvature contribution, which we hope future experiments could explore.

We also measure the Hall angle at THz frequency,
$\theta_{H}=\dfrac{\sigma_{xy}(\omega)}{\sigma_{xx}(\omega)}$, as it will be
useful for field-effect transistors around 1 THz Del Alamo (2011). A large
Hall angle $\sim 8.5\%$ is observed at 290 K, as shown in FIG. 3F. It
increases as temperature decreases and reaches a maximum with 10 % at 2 K.
Co2MnGa is one of few materials that exhibits large AHE and large Hall angle
at the same time. Since AHE and spin Hall effect are generated by the same
mechanisms, the large AHE in Co2MnGa guarantees a large spin Hall effect,
which was reported recentlyLeiva  _et al._ (2021). Looking forward, we believe
that our observation of large THz anomalous Hall conductivity, Faraday
rotation, and Hall angle from the intrinsic contribution at room temperature
will be critical to use Co2MnGa for future topological spintronics at the THz
frequencies.

X.H., J.S. and L.W. acknowledge the support for this project from the ARO
under the Grant W911NF1910342. The acquisition of the cryostat is supported by
the ARO under the Grant W911NF2020166. X.H. and J.S. are also partially
supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative, Grant
GBMF9212 to L.W. The acquisition of the laser for the THz system is support
from a seed grant at NSF supported University of Pennsylvania Materials
Research Science and Engineering Center (MRSEC)(DMR-1720530). L.W.
acknowledges partial summer support from the NSF EAGER grant (DMR-2132591).

## References

  * Moore (2010) J. E. Moore, Nature 464, 194 (2010).
  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
  * Wan  _et al._ (2011) X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Physical Review B 83, 205101 (2011).
  * Burkov and Balents (2011) A. Burkov and L. Balents, Physical Review Letters 107, 127205 (2011).
  * Xu  _et al._ (2011) G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Physical Review Letters 107, 186806 (2011).
  * Weng  _et al._ (2015) H. Weng, C. Fang, Z. Fang, B. A. Bernevig, and X. Dai, Physical Review X 5, 011029 (2015).
  * Huang  _et al._ (2015) S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, B. Wang, N. Alidoust, G. Bian, M. Neupane, C. Zhang, _et al._ , Nature Communications 6 (2015).
  * Xu  _et al._ (2015) S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, _et al._ , Science 349, 613 (2015).
  * Lv  _et al._ (2015) B. Lv, H. Weng, B. Fu, X. Wang, H. Miao, J. Ma, P. Richard, X. Huang, L. Zhao, G. Chen, _et al._ , Physical Review X 5, 031013 (2015).
  * Yang  _et al._ (2015) L. Yang, Z. Liu, Y. Sun, H. Peng, H. Yang, T. Zhang, B. Zhou, Y. Zhang, Y. Guo, M. Rahn, _et al._ , Nature Physics 11, 728 (2015).
  * Weyl (1929) H. Weyl, Zeitschrift für Physik A Hadrons and Nuclei 56, 330 (1929).
  * Hosur  _et al._ (2012) P. Hosur, S. Parameswaran, and A. Vishwanath, Physical review letters 108, 046602 (2012).
  * Xu  _et al._ (2020) B. Xu, Z. Fang, M.-Á. Sánchez-Martínez, J. W. Venderbos, Z. Ni, T. Qiu, K. Manna, K. Wang, J. Paglione, C. Bernhard, _et al._ , Proceedings of the National Academy of Sciences 117, 27104 (2020).
  * Ni  _et al._ (2020) Z. Ni, B. Xu, M.-Á. Sánchez-Martínez, Y. Zhang, K. Manna, C. Bernhard, J. Venderbos, F. de Juan, C. Felser, A. G. Grushin, _et al._ , npj Quantum Materials 5, 1 (2020).
  * Potter  _et al._ (2014) A. C. Potter, I. Kimchi, and A. Vishwanath, Nature communications 5 (2014).
  * Wu  _et al._ (2017a) L. Wu, S. Patankar, T. Morimoto, N. L. Nair, E. Thewalt, A. Little, J. G. Analytis, J. E. Moore, and J. Orenstein, Nature Physics 13, 350 (2017a).
  * Patankar  _et al._ (2018) S. Patankar, L. Wu, B. Lu, M. Rai, J. D. Tran, T. Morimoto, D. E. Parker, A. G. Grushin, N. Nair, J. Analytis, _et al._ , Physical Review B 98, 165113 (2018).
  * Burkov (2015) A. Burkov, Journal of Physics: Condensed Matter 27, 113201 (2015).
  * Wu  _et al._ (2017b) L. Wu, S. Patankar, T. Morimoto, N. L. Nair, E. Thewalt, A. Little, J. G. Analytis, J. E. Moore, and J. Orenstein, Nature Physics 13, 350 (2017b).
  * Belopolski  _et al._ (2019) I. Belopolski, K. Manna, D. S. Sanchez, G. Chang, B. Ernst, J. Yin, S. S. Zhang, T. Cochran, N. Shumiya, H. Zheng, _et al._ , Science 365, 1278 (2019).
  * Morali  _et al._ (2019) N. Morali, R. Batabyal, P. K. Nag, E. Liu, Q. Xu, Y. Sun, B. Yan, C. Felser, N. Avraham, and H. Beidenkopf, Science 365, 1286 (2019).
  * Manna  _et al._ (2018) K. Manna, L. Muechler, T.-H. Kao, R. Stinshoff, Y. Zhang, J. Gooth, N. Kumar, G. Kreiner, K. Koepernik, R. Car, J. Kübler, G. H. Fecher, C. Shekhar, Y. Sun, and C. Felser, Phys. Rev. X 8, 041045 (2018).
  * Markou  _et al._ (2019) A. Markou, D. Kriegner, J. Gayles, L. Zhang, Y.-C. Chen, B. Ernst, Y.-H. Lai, W. Schnelle, Y.-H. Chu, Y. Sun, and C. Felser, Phys. Rev. B 100, 054422 (2019).
  * Sakai  _et al._ (2018) A. Sakai, Y. P. Mizuta, A. A. Nugroho, R. Sihombing, T. Koretsune, M.-T. Suzuki, N. Takemori, R. Ishii, D. Nishio-Hamane, R. Arita, _et al._ , Nature Physics 14, 1119 (2018).
  * Reichlova  _et al._ (2018) H. Reichlova, R. Schlitz, S. Beckert, P. Swekis, A. Markou, Y.-C. Chen, D. Kriegner, S. Fabretti, G. Hyeon Park, A. Niemann, _et al._ , Applied Physics Letters 113, 212405 (2018).
  * Park  _et al._ (2020) G.-H. Park, H. Reichlova, R. Schlitz, M. Lammel, A. Markou, P. Swekis, P. Ritzinger, D. Kriegner, J. Noky, J. Gayles, _et al._ , Physical Review B 101, 060406 (2020).
  * Nagaosa  _et al._ (2010) N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).
  * Lee  _et al._ (2004) W.-L. Lee, S. Watauchi, V. Miller, R. Cava, and N. Ong, Science 303, 1647 (2004).
  * Karplus and Luttinger (1954) R. Karplus and J. Luttinger, Physical Review 95, 1154 (1954).
  * Xiao  _et al._ (2010) D. Xiao, M.-C. Chang, and Q. Niu, Reviews of modern physics 82, 1959 (2010).
  * Armitage  _et al._ (2018) N. Armitage, E. Mele, and A. Vishwanath, Reviews of Modern Physics 90, 015001 (2018).
  * Fang  _et al._ (2003) Z. Fang, N. Nagaosa, K. S. Takahashi, A. Asamitsu, R. Mathieu, T. Ogasawara, H. Yamada, M. Kawasaki, Y. Tokura, and K. Terakura, Science 302, 92 (2003).
  * Yao  _et al._ (2004) Y. Yao, L. Kleinman, A. MacDonald, J. Sinova, T. Jungwirth, D.-s. Wang, E. Wang, and Q. Niu, Physical review letters 92, 037204 (2004).
  * Kübler and Felser (2016) J. Kübler and C. Felser, EPL (Europhysics Letters) 114, 47005 (2016).
  * Yan and Felser (2017) B. Yan and C. Felser, Annual Review of Condensed Matter Physics 8, 337 (2017).
  * Hosur and Qi (2013) P. Hosur and X. Qi, Comptes Rendus Physique 14, 857 (2013).
  * Del Alamo (2011) J. A. Del Alamo, Nature 479, 317 (2011).
  * Okamura  _et al._ (2020) Y. Okamura, S. Minami, Y. Kato, Y. Fujishiro, Y. Kaneko, J. Ikeda, J. Muramoto, R. Kaneko, K. Ueda, V. Kocsis, _et al._ , Nature communications 11, 1 (2020).
  * Matsuda  _et al._ (2020) T. Matsuda, N. Kanda, T. Higo, N. Armitage, S. Nakatsuji, and R. Matsunaga, Nature communications 11, 1 (2020).
  * Khadka  _et al._ (2020) D. Khadka, T. R. Thapaliya, S. H. Parra, X. Han, J. Wen, R. F. Need, P. Khanal, W. Wang, J. Zang, J. M. Kikkawa, L. Wu, and S. X. Huang, Science Advances 6, eabc1977 (2020).
  * Shimano  _et al._ (2011) R. Shimano, Y. Ikebe, K. Takahashi, M. Kawasaki, N. Nagaosa, and Y. Tokura, EPL (Europhysics Letters) 95, 17002 (2011).
  * Seifert  _et al._ (2021) T. S. Seifert, U. Martens, F. Radu, M. Ribow, M. Berritta, L. Nádvorník, R. Starke, T. Jungwirth, M. Wolf, I. Radu, _et al._ , Advanced Materials 33, 2007398 (2021).
  * Miyasato  _et al._ (2007) T. Miyasato, N. Abe, T. Fujii, A. Asamitsu, S. Onoda, Y. Onose, N. Nagaosa, and Y. Tokura, Physical review letters 99, 086602 (2007).
  * He  _et al._ (2012) P. He, L. Ma, Z. Shi, G. Guo, J.-G. Zheng, Y. Xin, and S. Zhou, Physical Review Letters 109, 066402 (2012).
  * Sakuraba  _et al._ (2020) Y. Sakuraba, K. Hyodo, A. Sakuma, and S. Mitani, Physical Review B 101, 134407 (2020).
  * Zeng  _et al._ (2006) C. Zeng, Y. Yao, Q. Niu, and H. H. Weitering, Physical Review Letters 96, 037204 (2006).
  * Shen  _et al._ (2014) X. Shen, A. Chikamatsu, K. Shigematsu, Y. Hirose, T. Fukumura, and T. Hasegawa, Applied Physics Letters 105, 072410 (2014).
  * Wu  _et al._ (2013) L. Wu, M. Brahlek, R. V. Aguilar, A. V. Stier, C. M. Morris, Y. Lubashevsky, L. S. Bilbro, N. Bansal, S. Oh, and N. P. Armitage, Nature Physics 9, 410 (2013).
  * Wu  _et al._ (2015) L. Wu, W.-K. Tse, M. Brahlek, C. M. Morris, R. V. Aguilar, N. Koirala, S. Oh, and N. P. Armitage, Physical Review Letters 115, 217602 (2015).
  * Onoda  _et al._ (2006) S. Onoda, N. Sugimoto, and N. Nagaosa, Physical review letters 97, 126602 (2006).
  * Meyers  _et al._ (1987) R. A. Meyers _et al._ ,  _Encyclopedia of physical science and technology_ (Academic Press, 1987).
  * Shuvaev  _et al._ (2011) A. Shuvaev, G. Astakhov, A. Pimenov, C. Brüne, H. Buhmann, and L. Molenkamp, Physical review letters 106, 107404 (2011).
  * Valdés Aguilar  _et al._ (2012) R. Valdés Aguilar, A. V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, and N. P. Armitage, Phys. Rev. Lett. 108, 087403 (2012).
  * Li  _et al._ (2020) P. Li, J. Koo, W. Ning, J. Li, L. Miao, L. Min, Y. Zhu, Y. Wang, N. Alem, C.-X. Liu, _et al._ , Nature communications 11, 1 (2020).
  * Koepernik and Eschrig (1999) K. Koepernik and H. Eschrig, Phys. Rev. B 59, 1743 (1999).
  * Perdew  _et al._ (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Koepernik  _et al._ (2021) K. Koepernik, O. Janson, Y. Sun, and J. Brink, arXiv preprint arXiv:2111.09652 (2021).
  * Kubo (1957) R. Kubo, Journal of the Physical Society of Japan 12, 570 (1957).
  * Greenwood (1958) D. Greenwood, Proceedings of the Physical Society (1958-1967) 71, 585 (1958).
  * Nagaosa, Naoto and Sinova, Jairo and Onoda, Shigeki and MacDonald, Allan H and Ong, Nai Phuan (2010) Nagaosa, Naoto and Sinova, Jairo and Onoda, Shigeki and MacDonald, Allan H and Ong, Nai Phuan, Reviews of modern physics 82, 1539 (2010).
  * Leiva  _et al._ (2021) L. Leiva, S. Granville, Y. Zhang, S. Dushenko, E. Shigematsu, T. Shinjo, R. Ohshima, Y. Ando, and M. Shiraishi, Physical Review B 103, L041114 (2021).


$\mathbf{q}=(l_{1},...,l_{g})$ $(l_{1}=d$, $l_{g}=l$), where $E$ is in Eq.
(7), (8), or (9).

By Theorems 5 and 8 in (Bartlett and Maass, 2003), we know that ${\rm
VCdim}(\mathcal{H}_{\mathbf{q},E}^{\sigma,\lambda})<+\infty$. Additionally,
Proposition 4 implies that Assumption 1 holds and
$\sup_{{h}\in\mathcal{H}_{\mathbf{q}}^{\sigma}}|\\{\mathbf{x}\in\mathcal{X}:{h}(\mathbf{x})\in\mathcal{Y}\\}|=+\infty$,
when $|\mathcal{X}|=+\infty$. Hence, Theorem 5 implies that OOD detection is
not learnable in $\mathscr{D}^{s}_{XY}$ for
$\mathcal{H}_{\mathbf{q}}^{\sigma}$, when $|\mathcal{X}|=+\infty$.

Fifth, we prove that if $|\mathcal{X}|<+\infty$, there exists a sequence
$\mathbf{q}=(l_{1},...,l_{g})$ $(l_{1}=d$ and $l_{g}=l)$ such that OOD
detection is learnable in $\mathscr{D}^{s}_{XY}$ for for $\mathcal{H}^{\rm
in}\bullet\mathcal{H}^{\rm b}$, where $\mathcal{H}^{\rm
b}=\mathcal{H}_{\mathbf{q},E}^{\sigma,\lambda}$ for any sequence
$\mathbf{q}=(l_{1},...,l_{g})$ $(l_{1}=d$, $l_{g}=l$), where $E$ is in Eq.
(7), (8), or Eq. (9).

Based on Lemma 22, we only need to show that
$\\{\mathbf{v}\in\mathbb{R}^{l}:E(\mathbf{v})\geq\lambda\\}$ and
$\\{\mathbf{v}\in\mathbb{R}^{l}:E(\mathbf{v})<\lambda\\}$ both contain
nonempty open sets for different score functions $E$.

Since
$\max_{k\in\\{1,...,l\\}}\frac{\exp{(v^{k})}}{\sum_{c=1}^{l}\exp{(v^{c})}}$,
$\max_{k\in\\{1,...,l\\}}\frac{\exp{(v^{k}/T)}}{\sum_{c=1}^{K}\exp{(v^{c}/T)}}$
and $T\log\sum_{c=1}^{l}\exp{(v^{c}/T)}$ are continuous functions, whose
ranges contain $(\frac{1}{l},1)$, $(\frac{1}{l},1)$, $(0,+\infty)$ and
$(0,+\infty)$, respectively.

Based on the property of continuous function ($E^{-1}(A)$ is an open set, if
$A$ is an open set), we obtain that
$\\{\mathbf{v}\in\mathbb{R}^{l}:E(\mathbf{v})\geq\lambda\\}$ and
$\\{\mathbf{v}\in\mathbb{R}^{l}:E(\mathbf{v})<\lambda\\}$ both contain
nonempty open sets. Using Lemma 22, we complete the fifth step.

Sixth, we prove that if $|\mathcal{X}|<+\infty$, then there exists a sequence
$\mathbf{q}=(l_{1},...,l_{g})$ $(l_{1}=d$ and $l_{g}=l)$ such that for any
sequence $\mathbf{q}^{\prime}=(l_{1}^{\prime},...,l_{g^{\prime}}^{\prime})$
satisfying that $\mathbf{q}\lesssim\mathbf{q}^{\prime}$, OOD detection is
learnable in $\mathscr{D}^{s}_{XY}$ for $\mathcal{H}^{\rm
in}\bullet\mathcal{H}^{\rm b}$, where $\mathcal{H}^{\rm
b}=\mathcal{H}_{\mathbf{q}^{\prime},E}^{\sigma,\lambda}$, where $E$ is in Eq.
(7), (8), or Eq. (9).

In the fifth step, we have proven that Eqs. (7), (8), and (9) meet the
condition in Lemma 22. Therefore, Lemma 22 implies this result. We complete
the proof when the hypothesis space $\mathcal{H}$ is score-based.

##  Appendix P Proof of Theorem 15

See 15

Proof [Proof of Theorem 15] Using a similar strategy of Theorem 14, we can
prove this theorem by Theorem 7 and Lemma 13.

##  Appendix Q Proof of Theorem 16

See 16

Proof [Proof of Theorem 16]

1) By Lemma 3, we conclude that Learnability in $\mathscr{D}_{XY}^{\mu,b}$ for
$\mathcal{H}\Rightarrow$ Condition 1.

2) By Proposition 3 and Proposition 4, we know that when $K=1$, there exist
$h_{1},h_{2}\in\mathcal{H}$, where $h_{1}=1$ and $h_{2}=2$, here $1$
represents ID, and $2$ represent OOD. Therefore, we know that when $K=1$,
$\inf_{h\in\mathcal{H}}R_{D}^{\rm in}(h)=0$ and
$\inf_{h\in\mathcal{H}}R_{D}^{\rm out}(h)=0$, for any
$D_{XY}\in\mathscr{D}_{XY}^{\mu,b}$.

By Condition 1, we obtain that $\inf_{h\in\mathcal{H}}R_{D}(h)=0$, for any
$D_{XY}\in\mathscr{D}_{XY}^{\mu,b}$. Because each domain $D_{XY}$ in
$\mathscr{D}_{XY}^{\mu,b}$ is attainable, we conclude that Risk-based
Realizability Assumption holds.

We have proven that Condition 1$\Rightarrow$ Risk-based Realizability
Assumption.

3) By Theorems 5 and 8 in (Bartlett and Maass, 2003), we know that ${\rm
VCdim}(\phi\circ\mathcal{H}_{\mathbf{q}}^{\sigma})<+\infty$ and ${\rm
VCdim}(\mathcal{H}_{\mathbf{q},E}^{\sigma,\lambda})<+\infty$. Then, using
Theorem 12, we conclude that Risk-based Realizability Assumption$\Rightarrow$
Learnability in $\mathscr{D}_{XY}^{\mu,b}$ for $\mathcal{H}$.

4) According to the results in 1), 2) and 3), we have proven that

Learnability in $\mathscr{D}_{XY}^{\mu,b}$ for
$\mathcal{H}\Leftrightarrow$Condition 1$\Leftrightarrow$ Risk-based
Realizability Assumption.

5) By Lemma 4, we conclude that Condition 4$\Rightarrow$Condition 1.

6) Here we prove that Learnability in $\mathscr{D}_{XY}^{\mu,b}$ for
$\mathcal{H}\Rightarrow$Condition 4. Since $\mathscr{D}^{\mu,b}_{XY}$ is the
prior-unknown space, by Theorem 1, there exist an algorithm
$\mathbf{A}:\cup_{n=1}^{+\infty}(\mathcal{X}\times\mathcal{Y})^{n}\rightarrow\mathcal{H}$
and a monotonically decreasing sequence $\epsilon_{\rm cons}(n)$, such that
$\epsilon_{\rm cons}(n)\rightarrow 0$, as $n\rightarrow+\infty$, and for any
$D_{XY}\in\mathscr{D}_{XY}^{\mu,b}$,

$\begin{split}&\mathbb{E}_{S\sim D^{n}_{X_{\rm I}Y_{\rm I}}}\big{[}R^{\rm
in}_{D}(\mathbf{A}(S))-\inf_{h\in\mathcal{H}}R^{\rm
in}_{D}(h)\big{]}\leq\epsilon_{\rm cons}(n),\\\ &\mathbb{E}_{S\sim
D^{n}_{X_{\rm I}Y_{\rm I}}}\big{[}R^{\rm
out}_{D}(\mathbf{A}(S))-\inf_{h\in\mathcal{H}}R^{\rm
out}_{D}(h)\big{]}\leq\epsilon_{\rm cons}(n).\end{split}$

Then, for any $\epsilon>0$, we can find $n_{\epsilon}$ such that
$\epsilon\geq\epsilon_{\rm cons}(n_{\epsilon})$, therefore, if
$n={n_{\epsilon}}$, we have

$\begin{split}&\mathbb{E}_{S\sim D^{n_{\epsilon}}_{X_{\rm I}Y_{\rm
I}}}\big{[}R^{\rm in}_{D}(\mathbf{A}(S))-\inf_{h\in\mathcal{H}}R^{\rm
in}_{D}(h)\big{]}\leq\epsilon,\\\ &\mathbb{E}_{S\sim D^{n_{\epsilon}}_{X_{\rm
I}Y_{\rm I}}}\big{[}R^{\rm
out}_{D}(\mathbf{A}(S))-\inf_{h\in\mathcal{H}}R^{\rm
out}_{D}(h)\big{]}\leq\epsilon,\end{split}$

which implies that there exists $S_{\epsilon}\sim D^{n_{\epsilon}}_{X_{\rm
I}Y_{\rm I}}$ such that

$\begin{split}&R^{\rm
in}_{D}(\mathbf{A}(S_{\epsilon}))-\inf_{h\in\mathcal{H}}R^{\rm
in}_{D}(h)\leq\epsilon,\\\ &R^{\rm
out}_{D}(\mathbf{A}(S_{\epsilon}))-\inf_{h\in\mathcal{H}}R^{\rm
out}_{D}(h)\leq\epsilon.\end{split}$

Therefore, for any equivalence class $[D_{XY}^{\prime}]$ with respect to
$\mathscr{D}_{XY}^{\mu,b}$ and any $\epsilon>0$, there exists a hypothesis
function $\mathbf{A}(S_{\epsilon})\in\mathcal{H}$ such that for any domain
$D_{XY}\in[D_{XY}^{\prime}]$,

$\mathbf{A}(S_{\epsilon})\in\\{h^{\prime}\in\mathcal{H}:R_{D}^{\rm
out}(h^{\prime})\leq\inf_{h\in\mathcal{H}}R_{D}^{\rm
out}(h)+\epsilon\\}\cap\\{h^{\prime}\in\mathcal{H}:R_{D}^{\rm
in}(h^{\prime})\leq\inf_{h\in\mathcal{H}}R_{D}^{\rm in}(h)+\epsilon\\},$

which implies that Condition 4 holds. Therefore, Learnability in
$\mathscr{D}_{XY}^{\mu,b}$ for $\mathcal{H}\Rightarrow$Condition 4.

7) Note that in 4), 5) and 6), we have proven that

Learnability in $\mathscr{D}_{XY}^{\mu,b}$ for
$\mathcal{H}\Rightarrow$Condition 4$\Rightarrow$Condition 1, and Learnability
in $\mathscr{D}_{XY}^{\mu,b}$ for $\mathcal{H}\Leftrightarrow$Condition 1,
thus, we conclude that Learnability in $\mathscr{D}_{XY}^{\mu,b}$ for
$\mathcal{H}\Leftrightarrow$Condition 4$\Leftrightarrow$Condition 1.

8) Combining 4) and 7), we have completed the proof.

##  Appendix R Proof of Theorem 17

See 17

Proof [Proof of Theorem 17] The result can be obtained by Theorems 13 and 3.

##  Appendix S Proof of Theorem 18

See 18

Proof [Proof of Theorem 18] Using Proposition 3 and Proposition 4, we obtain
that $\inf_{h\in\mathcal{H}}R_{D}^{\rm in}(h)=0$ and
$\inf_{h\in\mathcal{H}}R_{D}^{\rm out}(h)=0$. Then, Theorem 1 implies this
result.

Note that if we replace the activation function $\sigma$ (ReLU function) in
Theorem 18 with any other activation functions, Theorem 18 still hold.

##  Appendix T Proof of Theorem 19

See 19

Proof [Proof of Theorem 19] This is a conclusion of Lemma 2.