# Magnetic Weyl semimetals as a source of circularly polarized THz radiation

Jeremy Hansen<EMAIL_ADDRESS>Department of Physics and Astronomy, Iowa
State University, Ames, Iowa, 50011, USA  Kazuki Ikeda
<EMAIL_ADDRESS>Co-design Center for Quantum Advantage, Stony
Brook University, Stony Brook, New York 11794-3800, USA  Department of Physics
and Astronomy, Stony Brook University, New York 11794-3800, USA  Dmitri E.
Kharzeev<EMAIL_ADDRESS>Co-design Center for Quantum
Advantage, Stony Brook University, Stony Brook, New York 11794-3800, USA
Department of Physics and Astronomy, Stony Brook University, New York
11794-3800, USA  Department of Physics and RIKEN-BNL Research Center,
Brookhaven National Laboratory, Upton, New York 11973, USA  Qiang Li
<EMAIL_ADDRESS>Department of Physics and Astronomy, Stony Brook University,
New York 11794-3800, USA  Condensed Matter Physics and Materials Science
Department, Brookhaven National Laboratory, Upton, NY 11973, USA  Kirill
Tuchin<EMAIL_ADDRESS>Department of Physics and Astronomy, Iowa State
University, Ames, Iowa, 50011, USA

###### Abstract

We propose to use the electromagnetic radiation induced by a few MeV electron
beam in magnetic Weyl semimetals as a source of the circularly polarized
photons in the THz frequency range.

_Introduction_. — A fast charged particle traversing a chiral medium emits, by
means of the chiral anomaly, a peculiar form of circularly polarized
electromagnetic radiation known as the Chiral Cherenkov radiation
Tuchin:2018sqe ; Hansen:2020irw ; Tuchin:2018mte . It appears as a result of
anomalous response of the chiral medium to the electromagnetic field of a fast
charge. Therefore, it should appear in a variety of systems ranging from quark
gluon plasma to Weyl and Dirac semimetals Huang:2018hgk . In this letter we
argue that Chiral Cherenkov radiation in Weyl semimetals is a useful source of
polarized electromagnetic radiation in the practically important THz frequency
range.

_Photon dispersion in Weyl semimetals_.— An external electromagnetic field in
Weyl semimetals induces the anomalous Hall current

$\displaystyle\bm{j}_{\text{AH}}=\bm{b}\times\bm{E}\,,$ (1)

where $\bm{b}=(\alpha/\pi)\bm{\Delta}$, with $\bm{\Delta}$ being the
separation of Weyl nodes in momentum space and $\alpha$ the fine structure
constant. A typical Weyl semimetal has more than one pair of nodes in which
case $\bm{\Delta}$ stands for an effective displacement. It was suggested in
Zubkov:2023dpe  that the total displacement is the vector sum of all
displacements: $\bm{\Delta}=\sum_{i}\bm{\Delta}_{i}$. In this case
$\bm{\Delta}$ is finite only in magnetic Weyl semimetals but vanishes in non-
magnetic ones. For example, in Co3Sn2S2 $b=3.8$ eV Zubkov:2023dpe ;
wang2018large .

In the presence of the current (1) the dielectric tensor takes form
$\varepsilon_{ij}=\varepsilon\delta_{ij}-i\epsilon_{ijk}b_{k}/\omega$, where
$\varepsilon\approx 1/(1+\omega_{p}^{2}/\omega^{2})$ for
$\omega\gg\omega_{p}$. The photon dispersion relation is a solution of the
Fresnel equation

$\displaystyle\left|k_{i}k_{j}-k^{2}\delta_{ij}+\omega^{2}\varepsilon_{ij}\right|=0\,$
(2)

in units where $\hbar=c=1$, which we assume henceforth. The dispersion
relation has the general form $\omega^{2}-k^{2}=\mu^{2}(\bm{k},\lambda)$ where
$\bm{k}$ is the wave vector and, setting $\omega_{p}=0$ for the sake of
presentation simplicity, Qiu:2016hzd

$\displaystyle\mu^{2}(\bm{k},\lambda)=\frac{b^{2}}{2}-\lambda\,\text{sgn}(\bm{k}\cdot\bm{b})\sqrt{(\bm{k}\cdot\bm{b})^{2}+b^{2}/4}\,.$
(3)

Here $\lambda=\pm 1$ is the photon circular polarization. Let $\beta$ be the
angle between $\bm{b}$ and $\bm{k}$, see Fig. 2. There are two branches of the
dispersion relation: (i) the gapped branch: $\lambda\cos\beta>0$, $\mu^{2}>0$
and (ii) the gapless branch $\lambda\cos\beta<0$, $\mu^{2}<0$. These are
depicted in Fig. 1. Photon radiation is associated with the latter.

Figure 1: Chiral branches of the photon dispersion relation. Arbitrary units.

_Photon radiation_.— The initial and final states of electron are solutions to
the Dirac equation with energies $E$, $E^{\prime}$ and momenta $\bm{p}$,
$\bm{p}^{\prime}$. The wave function of the radiated photon with energy and
momentum $\omega$, $\bm{k}$ and polarization $\lambda$ reads
alekseev1966quantization

$\displaystyle\bm{A}=$
$\displaystyle\bm{e}\left(\frac{k\,v}{2\omega^{2}\varepsilon_{ij}e^{*}_{i}e_{j}}\right)^{1/2}e^{i\bm{k}\cdot\bm{x}-i\omega
t}$ (4)

where $\bm{e}$ is the polarization vector for a given $\lambda$ and
$v=\partial\omega/\partial k$ is the group velocity. The explicit expression
for $\bm{e}$ can be found in Tuchin:2018mte . Computing the transition
amplitude $e^{-}\to e^{-}\gamma$ and using the standard procedure to obtain
the transition rate we obtain for the number of photons emitted per unit time
in unit solid angle in the frequency interval $d\omega$ Tuchin:2018mte :

$\displaystyle\frac{dW}{d\Omega d\omega}=$
$\displaystyle\frac{\alpha}{16\pi}\sum_{\lambda}\delta(\omega+E^{\prime}-E)\frac{k^{3}}{EE^{\prime}\omega^{2}\varepsilon_{ij}e_{i}^{*}e_{j}}$
$\displaystyle
4e^{*}_{i}e_{j}\left[p_{i}p_{j}^{\prime}+p_{j}p_{i}^{\prime}+\delta_{ij}(EE^{\prime}-\bm{p}\cdot\bm{p}^{\prime}-m^{2})\right]\,,$
(5)

where $\bm{p}^{\prime}=\bm{p}-\bm{k}$. The energy conservation expressed by
the delta-function is satisfied only for the gapless branch $\mu^{2}<0$.
Similarly to the conventional Cherenkov radiation, the Chiral Cherenkov one is
radiated even by a particle moving with constant velocity, which corresponds
to the limit $\omega\ll E$ when electron’s recoil is small. In this case, the
radiating electron moves in phase with the wave. As the result both types of
Cherenkov radiation have infinite coherence length in contrast to the
bremsstrahlung.

Thus far we discussed the Chiral Cherenkov radiation by a single electron. A
typical electric current pulse consists of $N=10^{11}$ electrons (equivalent
to the electron bunch charge 16 nC). The corresponding power $dI=NdP$.

The explicit form of the rate is rather bulky. We therefore consider two
special cases.

_High energy approximation_.— Assume that $E\gg m$ and $\omega\gg|\mu|$. The
latter condition is equivalent to $\omega\gg b,\omega_{p}$. The Chiral
Cherenkov radiation rate (Magnetic Weyl semimetals as a source of circularly
polarized THz radiation) can then be written in a compact form Tuchin:2018sqe
; Tuchin:2018mte ; Hansen:2020irw :

$\displaystyle\frac{dW}{d\omega d\Omega}=$
$\displaystyle\frac{\alpha\omega}{2\pi
E}\delta(\omega^{2}\vartheta^{2}+\kappa)$
$\displaystyle\times\left\\{fE\left(\frac{\omega^{2}}{2E^{2}}-\frac{\omega}{E}+1\right)-m^{2}\frac{\omega}{E}\right\\}\theta(-\kappa)\,,$
(6)

where $f=\lambda b\cos\beta$, and $\beta$ is now the angle between the
incident electron direction and the vector $\bm{b}$ since in this
approximation the angle $\vartheta$ between $\bm{k}$ and $\bm{p}$ is small,
see Fig. 2.

Figure 2: The geometry of the photon radiation. The three vectors are
generally not in the same plane, however the azimuthal angle is not shown. At
high energy $\vartheta\ll\beta$, i.e. $\beta$ is approximately the angle
between $\bm{p}$ and $\bm{b}$ (angle of incidence).

In the right-hand-side $\theta$ is the step function and we defined

$\displaystyle\kappa=-\left(1-\frac{\omega}{E}\right)\omega
f+\frac{m^{2}\omega^{2}}{E^{2}}\,.$ (7)

The kinematic condition $\kappa<0$ can be satisfied only if $f>0$. Since
$\mu^{2}\approx-f\omega$ it is equivalent to $\mu^{2}<0$. Additionally, it
imposes the upper limit on the photon spectrum:

$\displaystyle\omega<E(1+m^{2}/fE)^{-1}\,.$ (8)

The lower limit is proportional to $\omega_{p}$, but in the high-energy
approximation it is neglected. The differential spectrum of the radiation
power is given by

$\displaystyle\frac{dP}{d\omega d\Omega}=\omega\frac{dW}{d\omega d\Omega}$ (9)

solid lines in Figs. 3,4,5 display the high-energy approximation of the
radiated photon spectrum, its angular distribution and dependence on angle of
incidence (viz. angle between the electron momentum $\bm{p}$ and the vector
$\bm{b}$).

Figure 3: Intensity spectrum of Chiral Cherenkov radiation by a pulse of
$N=10^{11}$ electrons of energy $E=3$ MeV, $\beta=0$ in a Weyl semimetal with
$b=3.8$ eV (green), $b=0.19$ eV (orange), and $b=0.01$ eV (blue). Solid line:
Eq. (Magnetic Weyl semimetals as a source of circularly polarized THz
radiation), dashed line: Eq. (Magnetic Weyl semimetals as a source of
circularly polarized THz radiation) with $\omega_{p}=0.6$ meV, dotted line:
bremsstrahlung spectrum in Co3Sn2S2 computed according to Workman:2022ynf .
1/cm $\approx$ 0.03 THz.  Figure 4: Angular distribution of Chiral Cherenkov
radiation. $N=10^{11}$, $E=3$ MeV, $\beta=0$, $b=3.8$ eV. Eq. (Magnetic Weyl
semimetals as a source of circularly polarized THz radiation) and Eq.
(Magnetic Weyl semimetals as a source of circularly polarized THz radiation)
are indistinguishable. Figure 5: Variation of Chiral Cherenkov radiation
intensity with electron angle of incidence (the polar angle). The two loops
have opposite polarization. $N=10^{11}$, $E=3$ MeV, $b=3.8$ eV.

The photon emission rate Eq. (Magnetic Weyl semimetals as a source of
circularly polarized THz radiation) can be further simplified by noting that
in semimetals $m^{2}\gg fE$:

$\displaystyle\frac{dW}{d\omega d\Omega}\approx\frac{\alpha
f}{2\pi\omega}\delta\left(\vartheta^{2}-\frac{f}{\omega}+\frac{m^{2}}{E^{2}}\right)\theta\left(\frac{E^{2}f}{m^{2}}-\omega\right)\,,$
(10)

where the step-function follows from (8). Multiplying by $\omega$ and
integrating over $d\Omega=\pi d\vartheta^{2}$ we obtain the power spectrum

$\displaystyle\frac{dP}{d\omega}=\frac{\alpha
f}{2}\theta\left(\frac{E^{2}f}{m^{2}}-\omega\right)\,,$ (11)

and the the total power

$\displaystyle P=\frac{\alpha f^{2}E^{2}}{2m^{2}}=\frac{\alpha
b^{2}\cos^{2}\beta E^{2}}{2m^{2}}.$ (12)

Integrating (10) over $\omega$ furnishes the angular distribution of the
radiation

$\displaystyle\frac{dW}{d\Omega}=\frac{\alpha
f}{2\pi}\frac{1}{\vartheta^{2}+m^{2}/E^{2}}\,.$ (13)

Evidently, the total radiation power increases as $b^{2}$, while the UV cutoff
of the spectrum as $b$. It is therefore advantageous to have as high $b$ as
possible to obtain the highest power of the Chiral Cherenkov radiation. It
also guarantees, as per Figs. 3,4, that the radiation power of the polarized
photons by means of the Chiral Cherenkov mechanism is larger than that of the
non-polarized photons through the conventional sources.

_Electron incident along $\bm{b}$_.— According to (12) the radiation power
vanishes when the incident electron moves perpendicularly to the vector
$\bm{b}$. Therefore, in practice one would like to position the Weyl semimetal
sample such that $\bm{p}\times\bm{b}=0$. In this case (Magnetic Weyl
semimetals as a source of circularly polarized THz radiation) significantly
simplifies. Perfoming the angular integral yields the rate:

$\displaystyle\frac{dW}{d\omega}=$
$\displaystyle\frac{(\omega^{2}-\mu^{2})(EE^{\prime}-m^{2})-(2\omega
E-\mu^{2})(2\omega
E^{\prime}+\mu^{2})}{4(\omega^{2}-\mu^{2})\omega^{2}E|\bm{p}|}$
$\displaystyle\times\frac{\alpha}{2}\left(1+\omega_{p}^{2}/\omega^{2}\right)\theta(\omega_{+}-\omega),$
(14)

where $\mu^{2}=\omega_{p}^{2}-\lambda b\omega$, $\bm{p}^{2}=E^{2}-m^{2}$ and

$\displaystyle\omega_{+}=$ $\displaystyle\bigg{[}\lambda
b(2\bm{p}^{2}+\omega_{p}^{2})+2E\omega_{p}^{2}+$
$\displaystyle+2|\bm{p}|\sqrt{\left(\lambda
bE-\omega_{p}^{2}\right)^{2}-m^{2}\left(b^{2}+4\omega_{p}^{2}\right)}\bigg{]}$
$\displaystyle\times$ $\displaystyle\left[\lambda b(4E+\lambda
b)+4m^{2}\right]^{-1}.$ (15)

In the high energy limit $\omega\gg b$ and $E\gg m$ these equations reduce to
those displayed in the previous section. Dashed lines in Figs. 3,4 represent a
calculation of intensity with Eq. (Magnetic Weyl semimetals as a source of
circularly polarized THz radiation). Evidently, the high-energy approximation
is quite accurate all the way down to the IR region $\omega\sim\omega_{p}$
where our approximation for the dielectric constant $\varepsilon$ breaks down
anyway.

_Dirac semimetals_.—In Dirac semimetals $\bm{b}=0$. However, by applying
external parallel magnetic $B$ and electric $\mathcal{E}$ fields one can
generate the chiral charge. The corresponding axial chemical potential reads
Fukushima:2008xe :

$\mu_{5}=\frac{3}{4}\frac{v^{3}}{\pi^{2}}e^{2}\frac{\mathcal{E}B}{T^{2}}\tau_{V},$
(16)

where $\tau_{V}$ is the chirality relaxation time, and $v$ is the Fermi
velocity of Dirac quasiparticles. The chiral anomaly manifests itself as the
Chiral Magnetic Effect which is induction of electric current in the direction
of the applied magnetic field:

$\displaystyle\bm{j}_{\text{CM}}=\sigma_{\chi}\bm{B}\,,$ (17)

where the chiral magnetic conductivity is
$\sigma_{\chi}=(2/\pi)\alpha\mu_{5}$.

In the presence of the current (17) the photon dispersion relation acquires a
branch with $\mu^{2}<0$ in a way very similar to Weyl semimetals. In fact all
formulas derived in the high-energy limit, apply at finite $\mu_{5}$ as well
upon the substitution $f=\lambda\sigma_{\chi}$ Tuchin:2018sqe ; Hansen:2020irw
.

In practice, however, the values of $\sigma_{\chi}$ are much smaller than $b$
as can be seen in Fig. 6. The separation of the Weyl nodes in a chiral
semimetal is of the order $\Delta\sim 10^{-2}(\pi/a)$. For example, in TaAs it
is $\Delta=80$ eV Xu:2015cga ; Lv:2015pya . The corresponding $b$-parameter in
(1) is of magnitude $b=0.19$ eV. In contrast, in a magnetic Weyl semimetal
Co3Sn2S2 $b=3.8$ eV Zubkov:2023dpe ; wang2018large .

Figure 6: The temperature and magnetic field dependence of $\mu_{5}$ at
$\mathcal{E}=1~{}mV/m$. The chiral magnetic conductivity is
$\sigma_{\chi}=4.6\cdot 10^{-3}\mu_{5}$.

_Summary._ — The Chiral Cherenkov radiation appears due to anomalous response
of chiral media to the electromagnetic field of a fast charge. In magnetic
Weyl semimetals, for an intense electron beam of a few MeV energy, the
intensity of the Chiral Chernkov radiation can exceed the intensity of usual
bremsstrahlung in the THz frequency range. Moreover, the THz CC radiation is
circularly polarized, which can be very useful for practical applications.

## Acknowledgements

We thank Sebastian Grieninger, Fangcheng He and Andrea Palermo for a useful
discussion. This work was supported by the U.S. Department of Energy, Office
of Science, National Quantum Information Science Research Centers, Co-design
Center for Quantum Advantage (C2QA) under Contract No.DE-SC0012704 (KI, DK),
and the U.S. Department of Energy under Grants DE-FG88ER41450 (DK) and DE-
SC0012704 (DK), This work of JH and KT was supported in part by the U.S.
Department of Energy Grant No. DE-SC0023692. The work of QL was supported by
the U.S. Department of Energy, Office of Basic Energy Sciences, Division of
Materials Sciences and Engineering, under contract No. DE-SC0012704.

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$r$-process material, including $\approx\\!0.01-0.04\,M_{\odot}$ of heavy
($A>136$) $r$-process material and $\approx\\!0.03-0.65\,M_{\odot}$ of light
($A<136$) $r$-process material, and $\approx\\!0.14-0.26\,M_{\odot}$ of 56Ni.

In comparison to Siegel et al. (2019), the updated model presented here tends
to predict moderately less heavy $r$-process material, more light $r$-process
material, and more 56Ni. This is the result of i) a more detailed treatment of
the disk accretion rate onto the BH, ii) an additional nucleosynthesis regime
of light $r$-process material only at high accretion rates
$>\\!\dot{M}_{\nu,\text{r}-\text{p}}$ (cf. Eq. (26)), and iii) a detailed
evolution of the nucleosynthesis regimes throughout the accretion process as a
result of BH growth. Overall, however, the mass ranges of all nucleosynthesis
products found here broadly agree with Siegel et al. (2019). In particular,
our refined analysis still predicts a sizable amount of lanthanide-bearing
$r$-process ejecta of $\approx\\!0.04-0.7\,M_{\odot}$ across various
progenitor models of Heger et al. (2000). These results remain consistent with
Miller et al. (2020), insofar that the mass accretion and generation of disk
winds occur over a wide range of accretion rates, which drift through
nucleosynthesis regimes characterized by varying degrees of neutrino
irradiation (cf. Eq. (26)). Interestingly, our new models result in disk-wind
56Ni yields approaching the values required to explain the light curves of
observed GRB SNe (e.g., Cano et al. 2016), without a prompt shock-heated
explosion (e.g., Barnes et al. 2018) or explosive nucleosynthesis at larger
radii in the disk (Zenati et al., 2020).

Our collapsar model is also in good agreement with properties of observed
GRBs. We check for consistency of our collapsar model with observed GRBs in
terms of GRB timescales and energies. We assume that the accreted mass onto
the BH is proportional to the radiated $\gamma$-ray energy, that is,
$L_{\gamma}\propto\eta\dot{m}_{\rm acc}c^{2}$, where $L_{\gamma}$ is the
observed gamma-ray luminosity and $\eta$ is an efficiency parameter.

A necessary requirement for collapsar accretion to explain observed GRBs is
that the evolution time of the accretion rate be smaller or equal to the
typical time required to generate a GRB in the engine frame, i.e.,
$\tau_{\dot{m}_{\mathrm{acc}}}\lesssim\tau_{\mathrm{GRB}}$. The timescale
$\tau_{\dot{m}_{\mathrm{acc}}}$ increases with time, typically expected as a
power-law $\tau_{\dot{m}_{\mathrm{acc}}}\propto t^{\alpha}$, where
$\alpha\simeq 1$. Following Siegel et al. (2019), we define

$\tau_{\dot{m}_{\mathrm{acc}}}\equiv\left(\frac{\mathrm{d}\ln\dot{m}_{\mathrm{acc}}}{\mathrm{d}t}\right)^{-1}.$
(D1)

Furthermore, let $t_{\mathrm{GRB}}$ denote the time relative to the onset of
disk accretion at which the equality
$\tau_{\dot{m}_{\mathrm{acc}}}=\tau_{\mathrm{GRB}}$ is reached. Consistency
then requires

$t_{\mathrm{GRB}}\sim\tau_{\mathrm{GRB}}.$ (D2)

The GRB duration in the engine frame is determined by (Bromberg et al., 2012;
Sobacchi et al., 2017)

$\tau_{\mathrm{GRB}}=\tau_{\gamma}+\tau_{\mathrm{b}},$ (D3)

where $\tau_{\gamma}$ is the observed duration of a GRB in the engine rest
frame and $\tau_{\mathrm{b}}$ is the time required for the jet to drill
through the stellar envelope. Assuming a typical value of
$\tau_{\mathrm{b}}=57^{+13}_{-10}\,\mathrm{s}$ (Sobacchi et al., 2017), and a
typical observed GRB duration of $\tau_{\gamma}=T_{90}/(1+z)=9\,\mathrm{s}$,
with a characteristic $T_{90}\simeq 27\,\mathrm{s}$ and redshift $z\simeq 2$
(Narayana Bhat et al., 2016), one finds $\tau_{\mathrm{GRB}}\approx
66\,\mathrm{s}$. The top panel of Fig. 17 shows that $t_{\rm GRB}\sim\tau_{\rm
GRB}$ essentially throughout the parameter space. Similar results are found
for the other models of Heger et al. (2000). Hence, consistency with observed
GRB durations according to Eq. (D2) holds.

Consistency with typical observed GRB energies requires that

$\displaystyle E_{\gamma,{\rm iso}}f_{\rm b}\frac{\tau_{\rm
GRB}}{\tau_{\gamma}}$ $\displaystyle\ll$ $\displaystyle\int_{t_{\rm
GRB}}L_{\rm jet}\,{\rm d}t$ $\displaystyle<$ $\displaystyle\int_{t_{\rm
GRB}}\dot{m}_{\rm acc}c^{2}\,{\rm d}t\equiv M_{\rm acc,GRB}c^{2},$

where $E_{\gamma,{\rm iso}}\sim 1\times 10^{53}$ erg is the typical isotropic-
equivalent gamma-ray energy of observed GRBs and $f_{\rm b}\simeq 0.006$ is
the beaming fraction (Goldstein et al., 2016), $L_{\rm jet}$ is the luminosity
of the accretion-powered jet, and $M_{\rm acc,GRB}$ is the accreted mass onto
the BH through the disk during the GRB timescale $t_{\rm GRB}$. With these
values, the condition translates into

$M_{\rm acc,GRB}\gg 2.5\times 10^{-3}\,M_{\odot},$ (D5)

which we find is satisfied throughout the parameter space where the peak
accretion rate reaches $\dot{m}_{\rm acc}>10^{-4}\,M_{\odot}\,\text{s}^{-1}$
(cf. Fig. 17, bottom panel), which we take as a threshold to postulate a
successful jet (Sec. 2.2). Similar findings apply to other models of Heger et
al. (2000). We therefore find that our ordinary collapsar models are
consistent both with typical GRB duration times and energies, including drill
time.

Figure 18 shows a parameter-space scan for model 250.25, a typical mass gap
collapsar model. The GRB properties are in good agreement with observational
constraints. While the GRB durations are typically similar to ordinary
collapsars, the accreted mass during the GRB phase may be up to a factor
$\sim\\!10$ higher. One may thus speculate that these models give rise to GRBs
that may be a factor $\sim\\!10$ more luminous or energetic, if the gamma-ray
luminosity tracks accreted mass.

##  Appendix E Gravitational-wave emission

Figure 19: Mininum (top) and maximum (bottom) frequency of the predicted
$l=m=2$ gravitational wave emission during the gravitationally unstable phase
of collapsar accretion, for the same model shown in Fig. 9. Final BH mass
contours are drawn as in previous figures.

We calculate the gravitational-wave strain of emitted gravitational waves by
approximating ‘the lump’ of the unstable disk (assumed to correspond to an
over-density of $\delta\rho/\rho\gtrsim 0.1$; Shibata et al. 2021; Wessel et
al. 2021) and the central BH as two orbiting point masses. The frequencies of
gravitational-wave emission can thus be directly predicted from the evolution
of the disk angular velocity according to the collapsar model in Sec. 2.2. The
maximum and minimum gravitational-wave frequencies vary considerably across
the $\\{f_{\rm K},r_{\rm b}\\}$ parameter space (see Fig. 19 for our fiducial
model).

According to the quadrupole formula, assuming that the orbital radius only
slowly changes with respect to the orbital frequency, the plus ($h_{+}$) and
cross ($h_{\times}$) polarizations of the gravitational waves at distance $r$
and inclination $\iota$ of the disk with respect to the observer can be
written as

$\displaystyle\mskip-20.0muh_{+}(t)$ $\displaystyle=$
$\displaystyle\frac{4G}{rc^{4}}\mu r_{\rm disk}^{2}\Omega_{\rm
K,disk}^{2}\frac{1+\cos^{2}\iota}{2}\cos[\Phi(t)],$ (E1)
$\displaystyle\mskip-20.0muh_{\times}(t)$ $\displaystyle=$
$\displaystyle\frac{4G}{rc^{4}}\mu r_{\rm disk}^{2}\Omega_{\rm
K,disk}^{2}\cos\iota\sin[\Phi(t)],$ (E2)

where $\Phi(t)\equiv\int_{t_{0}}^{t}2\Omega_{\rm
K,disk}(t^{\prime})\,\rm{d}t^{\prime}$, with $t=t_{0}$ denoting the start time
of the gravitational instability. These expressions apply to the $l=m=2$ mode,
while $\Omega_{\rm K,disk}$ is replaced by $0.5\Omega_{\rm K,disk}$ for the
$m=1$ mode. Furthermore, $\mu=M_{\rm lump}M_{\bullet}/(M_{\rm
lump}+M_{\bullet})$ is the reduced mass of the lump–BH system, and we set
$M_{\rm lump}=f_{\rm lump}M_{\rm disk}$ with $f_{\rm lump}=0.2$. Uncertainties
in the value of $f_{\rm lump}$ can be absorbed into uncertainties of the scale
height of the disk and the threshold mass fraction $f_{\rm disk,thr}\equiv
M_{\bullet}/M_{\rm disk}$ at which gravitational instabilities set in (cf. Eq.
(25)). We neglect corrections $\propto\dot{\mu}$ to Eqs. (E1) and (E2) due to
a time-dependent reduced mass $\mu$ as a result of accretion onto the black
hole, assuming that $M_{\bullet}$ changes only weakly over the course of
gravitational-wave emission.

Figure 20: Maximum strain amplitude of the characteristic strain $h_{c}$ of
gravitational waves from non-axisymmetric instabilities in self-gravitating
collapsar disks across the parameter space of $\\{r_{\rm b},f_{\rm K}\\}$ for
our fiducial model shown in Fig. 2 with $p=4.5$, $f_{\rm K}=0.3$ and $r_{\rm
b}=1.5\times 10^{9}$ cm. Characteristic strains range from $\sim
10^{-24}-10^{-22}$ depending on the rotation profile of the progenitor. Final
BH mass contours are drawn as in previous figures.

Figure 21: Detection horizons for gravitational waves from our fiducial model
shown in Fig. 2 with $p=4.5$, $f_{\rm K}=0.3$ and $r_{\rm b}=1.5\times 10^{9}$
cm for advanced LIGO at design sensitivity (top) and Big Bang Observer
(bottom), assuming optimal matched filtering and a signal-to-noise ratio of 8.
While for aLIGO the detection horizon is typically limited to $\lesssim 100$
Mpc, BBO will be able to detect such sources up to typically several hundred
Mpc to several Gpc, with particular sensitivity for progenitors with low-
angular momentum ‘cores’ (medium to large values of $r_{\rm b}$). Contours
delineate final BH masses as in previous figures.

We characterize gravitational-wave emission in the frequency domain (positive
frequencies $f$ only) by computing the characteristic strain, defined as

$h_{c}=2f|\tilde{h}^{\rm res}(f)|.$ (E3)

For an estimate of the horizon distance we assume that the detector receives a
signal from a directly overhead source and hence the optimal strain response
at the detector can be written as

$\tilde{h}^{\rm res}=\sqrt{|\tilde{h}_{+}|^{2}+|\tilde{h}_{\times}|^{2}},$
(E4)

where $\tilde{h}_{+}$ and $\tilde{h}_{\times}$ are the Fourier transforms of
the respective polarization strain amplitudes that we compute employing a
Tukey window function limited to the physical frequencies between the maximum
and minimum frequency expected from disk evolution (Fig. 19). We compare
gravitational wave signals to detector sensitivity in terms of the amplitude
spectral density $\sqrt{S_{h}}$ (Moore et al., 2015),

$\sqrt{S_{h}}=2\sqrt{f}|\tilde{h}^{\rm res}|,$ (E5)

where $S_{h}$ denotes the power spectral density, and calculate the signal-to-
noise ratio (SNR) using an optimal filter (Moore et al., 2015),

$\mathrm{SNR}=\left(\int df\frac{h_{c}^{2}}{S_{n}f^{2}}\right)^{\frac{1}{2}}.$
(E6)

Characteristic strains of superKN collapsars range from $\sim
10^{-24}-10^{-22}$ depending on the rotation profile of the progenitor. A
representative example is shown in Fig. 20. Detection horizons for advanced
LIGO and BBO assuming $\text{SNR}=8$ are shown in Fig. 21, while those for CE,
ET, and DECIGO are presented in Fig. 13.

## Data Availability

The pre-collapse stellar models from Renzo et al. (2020b) and Heger et al.
(2000), are available at https://zenodo.org/record/3406357 and
https://2sn.org/stellarevolution/rotation/, respectively. Data of our model
runs together with visualization scripts will be made available at
https://doi.org/10.5281/zenodo.5639697.

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