11institutetext: Anton Pannekoek Institute for Astronomy, University of
Amsterdam, Postbus 94249, 1090 GE Amsterdam, The Netherlands
11email<EMAIL_ADDRESS>
# Discovery of a probable very fast extragalactic nova in a symbiotic
binary††thanks: A reproduction package for this paper is available at
https://doi.org/10.5281/zenodo.6817684.

David Modiano and Rudy Wijnands

Very fast novae are novae which evolve exceptionally quickly (on timescales of
only days). Due to their rapid evolution, very fast novae are challenging to
detect and study, especially at early times. Here we report the discovery,
which was made as part of our Transient UV Objects project, of a probable very
fast nova in the nearby spiral galaxy NGC 300. We detected the rise to the
peak (which is rarely observed for very fast novae) in the near-ultraviolet
(NUV), with the first detection just $\sim$ 2 hours after the eruption
started. The peak and early stages of the decay were also observed in UV and
optical bands. The source rapidly decayed (2 NUV magnitudes within 3.5 days),
making it one of the fastest novae known. In addition, a likely quiescent
counterpart was found in archival near-infrared Spitzer and VIRCAM images but
not in any deep optical and UV observations, indicating a very red spectral
shape in quiescence. The outburst and quiescence properties suggest that the
system is likely a symbiotic binary. We discuss this new transient in the
context of very fast novae in general and specifically as a promising
supernova Type Ia progenitor candidate, due to its very high inferred WD mass
($\sim$ 1.35 M⊙; determined by comparing this source to other very fast
novae).

###### Key Words.:

methods: data analysis – methods: observational – ultraviolet: stars –
binaries: symbiotic – novae

##  1 Introduction

White dwarfs (WDs) in binary systems can accrete matter from their companions.
If the companion is a main-sequence (MS) star (although in rare cases it can
be another WD), then mass transfer occurs via Roche-Lobe overflow and the
system is usually known as a cataclysmic variable (CV; for an extended review
see Warner 1995). If the donor is an evolved giant, the orbital separation is
large (at least hundreds of days) and the system is known as a symbiotic
binary (or a symbiotic; see Mikołajewska 2012; Munari 2019 for reviews). In
this case, the WD accretes mass from the strong stellar wind of the companion,
although for symbiotics with short orbital periods ($\lesssim$600 days) Roche-
Lobe overflow may also occur (and could be the primary method of accretion;
Mikolajewska 2010) .

When Roche-Lobe overflow is the dominant mode of accretion (i.e. in CVs and
short-period symbiotics), then the system can undergo explosive thermonuclear
outbursts known as novae (Gallagher & Starrfield 1978; Chomiuk et al. 2021).
These eruptions arise because a hydrogen-rich layer of accreted material
accumulates on the WD surface, causing the temperature and density to increase
and eventually reach the critical conditions for unstable nuclear burning
(Starrfield et al. 1972; Bode & Evans 2008). This triggers a thermonuclear
runaway event (TNR), which causes the accreted envelope to rapidly expand, and
consequently the system to brighten in the ultraviolet (UV) and optical by up
to a factor of 1 million (8–15 visual magnitudes) on timescales of days to
weeks (for reviews see Bode & Evans 2008; Starrfield et al. 2016; Chomiuk et
al. 2021).

Novae are often classified according to their light curve properties, in
particular the timescale of their decay. The ‘speed class’ is commonly
quantified by the parameters $t_{2}$ and $t_{3}$, which indicate the time
taken for the brightness of the nova to decline from its peak by 2 and 3
visual magnitudes, respectively. The classes range from ‘very fast’ ($t_{2}$ ¡
10 d) to ‘very slow’ ($t_{2}$ ¿ 150 d; Payne-Gaposchkin 1964). Despite some
caveats (see Sect. 4), this parameter is important because the speed of decay
is linked to the WD mass, with the fastest-evolving novae likely to harbour
the most massive WDs ($\sim$1.4 M⊙; Hachisu & Kato 2016; Shara et al. 2018;
Hachisu & Kato 2018; Darnley 2021). Thus, very fast novae are of interest
because of their extreme nature both in terms of light curve properties (and
the underlying physical processes) and WD mass.

So far, a few tens of very fast novae have been discovered, but with only
seven originating in symbiotics111Here is it worth making a brief note on
terminology with regards to novae. The term ‘symbiotic novae’ is usually
reserved for very slow (years to decades) and low-amplitude (2–6 magnitudes)
outbursts that occur in typical, wide-orbit (orbital periods $\gtrsim$800
days) symbiotics (Allen 1980). Due to the physical properties of these slow
novae (low accretion rates, low WD masses, and wind-only accretion) and their
observational signatures (very slow and very faint), it is important to
distinguish them from all other novae. Thus, novae originating in symbiotics
are referred to as such, and not as ‘symbiotic novae’. In this paper, we do
not include the latter when discussing novae. (Schaefer 2010; Darnley et al.
2016; Jencson et al. 2021 and references therein). There are still many
uncertainties regarding the mechanisms of these remarkable outbursts, such as
the temperature evolution (e.g. Darnley et al. 2016) and the geometrical
structure of the expanding envelope (e.g, Cao et al. 2012). In addition, many
very fast novae exhibit plateaus in their light curve (e.g. Strope et al.
2010) which are also not well understood (possibly they arise from the
irradiation of the accretion disc by emission from the WD; Hachisu et al.
2000).

Very fast novae are additionally noteworthy because they have been proposed as
progenitors of Type Ia supernovae (SNe; Maoz et al. 2014; Hillman et al. 2016;
Hachisu & Kato 2018). Although there is general agreement in that Type Ia SNe
result from the thermonuclear detonation of a WD after its mass has increased
to the Chandrasekhar limit ($\sim$1.4 M⊙; Nomoto 1982), it is not yet clear
how this is achieved. Very fast novae may provide an efficient pathway to
sufficiently increasing the mass of a WD, due to their high WD masses and the
fact that they can grow (Hillman et al. 2016). A particularly likely channel
may be the very fast novae which originate in symbiotics (e.g. Patat et al.
2011). However, although not precisely known, the rates of very fast novae
appear to be too low to account for all Type Ia SNe (although this is true of
all the likely progenitor channels, suggesting that there are multiple
channels; see e.g. Schaefer 2010).

Recently, we have initiated the Transient UV Objects (TUVO) Project, through
which we search for and study transients in the relatively underutilised UV
wavelength range. Within this project, we focus on the UV data obtained by the
Ultraviolet and Optical Telescope (UVOT, Roming et al. 2005; Breeveld et al.
2010) aboard the Neil Gehrels Swift Observatory (Swift, Gehrels et al. 2004).
Through our specialised pipeline used to analyse these UVOT data (Modiano et
al. 2022), we regularly detect serendipitous new and archival transients.
Since novae exhibit strong UV outbursts, this method is well-suited to the
discovery and study of these systems. In this letter, we present the TUVO
discovery and a photometric study of TUVO-22albb, a new UV transient we
detected in the nearby spiral galaxy NGC 300 (1.88 Mpc; Gieren et al. 2005).
We determine it to be most likely one of the rare, very fast novae originating
in symbiotics, and find that it is one of the fastest-evolving novae known.

##  2 Observations

###  2.1 Discovery

Figure 1: Top panel: UVOT $UVW1$ exposure of NGC 300 in which TUVO-22albb was
discovered (Observation ID 00030848003, exposure 15). Bottom left panel:
zoomed-in view of the top panel image, but centred on TUVO-22albb. The
transient is labelled. Bottom right panel: $UVW1$ reference image during
quiescence (Observation ID 00031210007, exposure 1). North is up and east is
left.

Using our pipeline, we detected a transient in an archival222Archival Swift
data are hosted at the High Energy Astrophysics Science Archive Research
Centre (HEASARC); at https://heasarc.gsfc.nasa.gov/cgi-bin/W3Browse/swift.pl.
UVOT exposure of NGC 300, taken using the $UVW1$ filter (central wavelength
$2600$ $\AA$; Poole et al. 2008). The source was denoted as TUVO-22albb. We
report the position at right ascension=00:54:34.26 and
declination=-37:42:50.7, with an error circle of
$0.5\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ (due to the UVOT astrometric
solution; see Poole et al. 2008). In Fig. 1 we show the discovery image, which
was taken on Dec. 31, 2006, and cutouts of the discovery and reference images
zoomed in on the transient, clearly showing the transient only visible in the
discovery image.

A preliminary light curve was automatically created by our pipeline using all
archival UVOT data covering the position of the source (shown in Appendix A).
This data set includes 164 observations, and as per the UVOT data
structure333https://www.swift.ac.uk/analysis/uvot/files.php, each observation
consists of one or multiple individual exposures per filter. The full data set
spanned $\sim$16 years, was of highly varying cadence (between hours and
years), and contained data in all six UVOT
filters444https://www.swift.ac.uk/analysis/uvot/filters.php. This long-term
light curve indicated that the source was only detected during a $\sim$ 10 day
period, so we selected these exposures with which to carry out a more
customised analysis (see Appendix B for details of the observations used). We
note that the late stages of the outburst may have continued for several
weeks, but there were no UVOT observations in this period. The gap between the
last detection and the subsequent non-detection is $\sim$20 days.

###  2.2 Outburst photometry

To characterise the outburst brightness evolution of the transient, we used
all the selected outburst exposures to construct a light curve, which we
display in Fig. 2. Before performing the photometry, we inspected all the
exposures during the outburst to check for secure detections of the source. We
noted that during the later stages of the outburst ($\gtrsim$ 6 days after
peak), the source had dimmed such that it was not clearly detectable in every
individual exposure. Therefore, for these data we co-added some exposures (see
Appendix B for details).

We then carried out photometry on all the individual or resulting co-added
exposures. All fluxes were obtained using the standard UVOT photometry tool
uvotsource555https://heasarc.gsfc.nasa.gov/lheasoft/ftools/headas/uvotsource.html.
The methods used for photometry, including aperture selection and dealing with
diffuse background emission, are described in Appendix C. The fluxes were
corrected for extinction using the Galactic dust and extinction maps available
at NASA/IPAC Infrared Science
Archive666https://irsa.ipac.caltech.edu/applications/DUST/. Extinction in the
line of sight of TUVO-22albb is small ($E_{B-V}=0.01$), so this had only a
small effect (typically $\sim$5 %) on the light curve. We note that the $U$,
$UVW1$, and $UVW2$ UVOT filter transmission is contaminated by a red leak.
However, several studies (e.g. Brown et al. 2010; Siegel et al. 2012, 2014)
have found that this does not have a significant effect for hot
($\gtrsim$5000–10 000 K) sources such as novae at early times (see Sect. 4),
so we did not correct for it. For easier comparison with other similar sources
for which light curves are often provided in magnitudes, the final corrected
fluxes were also converted back into (AB) magnitudes; these are provided in
Appendix C (Table 3) for all observations in which a detection was made.

##  3 Analysis and results

Figure 2: Outburst light curve of TUVO-22albb. Day zero is defined as
MJD=54099.9, which is the time of the exposure during the peak of the
outburst. Data points with error bars on the time indicate exposures that were
co-added, and open circles indicate exposures in which the source was not
detected (see Appendices B and C for details).

###  3.1 Outburst: light curve properties

The rise and peak of the outburst were observed only using the $UVW1$ filter;
the initial rapid decline was observed only using the $U$ filter; and the
slower subsequent decay was observed using all six UVOT filters.
Unfortunately, observations between the filters do not overlap in time (except
the late $B$ and $V$ observations), so we cannot precisely discern the profile
of the entire light curve. However, at maximum brightness the spectral energy
distribution (SED) of this source is likely strongest in the $U$ or near-UV
(NUV; e.g. Jencson et al. 2021), so a reasonable assumption is that the SED
between the $U$ and $UVW1$ bands (which are only separated by $\sim$ 800
$\AA$) at the peak was fairly flat. Since these were the two filters used for
the exposures during and immediately after the light curve peak, the early
light curve profile is reasonably well approximated by the combination of the
early $UVW1$ and $U$ data. Overall, the outburst light curve (Fig. 2) of the
transient can be broadly characterised by a very rapid (hours) rise, followed
by a initial steep decay until t $\sim$ 0.6 d and then a subsequent plateau
and slower decay phase (where we have defined t = 0 as MJD=54099.9, which is
the time of the peak of the light curve in the $UVW1$ filter).

Figure 3: Fits to the $UVW1$ (left panel) and $U$ (right panel) light curves.
Detections are shown as solid circles and non-detections are shown as open
circles. The rise in $UVW1$ is fit with a straight line using the first three
detections. The decays for both $U$ and $UVW1$ are fit with exponential
functions. For $U$, the data shown with squares were not included in the fit
(see Sect. 3.1 for details). For $UVW1$, all detections (starting from the
peak) were used in the fit. The inferred $t_{2}$ from the fit for $UVW1$ is
3.5 days, just before the cluster of data.

After $\sim$1 day, a drop in flux by a factor of 2 occurs between the $U$ and
$UVW2$ band data but this happens in less than 2 hours. This suggests that it
does not represent a real decrease in overall luminosity but that the source
was fainter in $UVW2$ than in $U$, indicating an SED peaking towards the NUV
or $U$. Therefore, the assumption of a flat SED between the $UVW1$ and $U$
filters during the first day of the outburst is not valid for the entire SED
(and indeed is only a first-order approximation). However, for the remainder
of the light curve (i.e. during the slow decline phase starting from day
$\sim$2) the very similar fluxes and proximity (in time) of observations
between the different filters suggest that the SED rapidly switched to
exhibiting a fairly flat shape across the full UV-optical range.

The start time of the eruption can be very well constrained by the non-
detections in the data obtained using the $UVW1$ filter just a few hours
before the first detection (using the same filter). We approximated the rise
with a straight-line fit to the earliest three $UVW1$ detections (Fig. 3, left
panel). This gave a flux of zero at 5.3 hours (0.22 days) before the peak,
which gives a good indication of the rise time.

The decay stage is more complex than a simple exponential decay. Nonetheless,
exponential fits can still be illustrative of the source behaviour, and allow
us to estimate $t_{2}$. Therefore, we fitted an exponential decay function to
the $U$ rapid-decline data (up to $\sim$0.6 days) together with the late-time
detections at ¿8 days. The fit is shown in Fig. 3 (right panel). We found the
best-fit $\tau_{U}$ = -1.15 $\pm$ 0.04 d-1. Due to the decline break, we
cannot fit the decay in $U$ before and after 0.6 days with the same
exponential function, so we cannot constrain $t_{2,U}$ well. However, in order
to quantitatively compare the fast initial and slower later declines, we also
fitted an exponential function to the data omitting the steep initial decline,
obtaining a much slower $\tau_{U}$ = -0.48 $\pm$ 0.04 d-1. Assuming the $U$
peak brightness was similar to that for $UVW1$, an upper limit for $t_{2,U}$
can be estimated as 4.3 days, which is the time between the peak and the first
$U$ exposure for which the brightness had declined by more than 2 magnitudes.

We also fitted an exponential decay function to the $UVW1$ data, displayed in
Fig. 3 (left panel), with the best fit giving $\tau_{UVW1}$ = -0.66 $\pm$ 0.04
d-1. This indicates slower decay than that inferred from the t $\leq$ 0.6 day
$U$ data, which reflects a combination of the faster early decay before 0.6
days and shallower subsequent decay. The $t_{2,UVW1}$ value we infer from the
exponential fit is 3.5 days777Since the peak is clearly observed in $UVW1$, a
robust upper limit for $t_{2}$ independent of the fitting can be determined by
calculating the time between the peak and the $UVW1$ exposure during which the
brightness had decreased by 2.2 magnitudes. This gives $t_{2}$ ¡ 3.55 days,
which is very close the the value of 3.5 days inferred from the exponential
fit.. There is not sufficient sampling in the $UVW1$ band for it to be useful
to fit the light curve with two different exponential functions. Due to the
brightness of the background emission at the source location, the photometry
never exhibits a 3 magnitude decrease from the peak, so we are unable to
determine $t_{3}$. However, at t = 8.3 days it has declined by $\sim$ 2.9
magnitudes, so we take this as a lower limit for $t_{3,UVW1}$.

###  3.2 Quiescence

The long-term UVOT light curve (Fig. 5) did not reveal any detections outside
of the outburst period, indicating that only one outburst was observed by
UVOT. To constrain the source brightness during quiescence, we co-added all
pre- and post-outburst UVOT exposures per filter, which ranged in time from
2007 to 2022 (see Appendix A). This resulted in data sets with total exposure
times of $\sim$17 ks, $\sim$23 ks, $\sim$66 ks, $\sim$91 ks, $\sim$91 ks, and
$\sim$98 ks in the filters $V$, $B$, $U$, $UVW1$, $UVM2$, and $UVW2$,
respectively. The source was not detected in any of the co-added exposures.
While the detection limits of these exposures overall reached 23-24
magnitudes, the diffuse emission due to NGC 300 at the source position
resulted in weaker upper limits for the quiescent counterpart. To determine
these, we ran uvotsource at the source position (when no source is detected
the tool calculates a 3-$\sigma$ upper limit). We obtained upper limits of
18.6, 18.7, 19.5, 20.5, 20.7, and 20.5 magnitudes in the filters $V$, $B$,
$U$, $UVW1$, $UVM2$, and $UVW2$, respectively (see Appendix C). Comparing the
$U$ and $UVW1$ quiescence upper limits to their corresponding peak magnitudes
of $\sim$17.9, this suggests outburst amplitudes of at least 1.6 and 2.6
magnitudes, respectively. However, during quiescence the source was likely
much fainter than the upper limits (see Sect. 4) so these are very weak lower
limits to the amplitudes.

We searched various publicly available astronomical databases for archival
observations covering the position of the source (see Table A.1 in Modiano et
al. 2022 for details of all the catalogues we probe automatically in the TUVO
project). We found no previous classifications of this source in any of these
databases, and no detections in X-ray, UV, or optical catalogues. We looked at
archival optical ($g$ band; 380–550nm) images from OmegaCam (Kuijken 2011)
mounted on the ESO-VLT Survey Telescope (Shanks et al. 2015) and did not
detect the source, with an upper limit of $\sim$ 23.4 magnitudes (5-$\sigma$;
the provided detection limit). This significantly improves the optical upper
limits on the quiescent counterpart determined from UVOT.

Potential matches to our source, which we define as any catalogued source
within $1\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ of the position of
TUVO-22albb, were found in archival near-infrared (NIR) images obtained by the
Spitzer (Werner et al. 2004) Infrared Array Camera (IRAC; Fazio et al. 2004)
and by
VIRCAM888https://www.eso.org/sci/facilities/paranal/instruments/vircam/inst.html
mounted on the VISTA telescope (Sutherland et al. 2015; source IDs SSTSL2
J005434.20-374251.3 and 473505681893, respectively). The Spitzer-IRAC
catalogued photometry measurements for this source in Channel 1 (3.1–3.9
$\mu$m) and Channel 2 (3.9–5.1 $\mu$m) were 17.9$\pm$0.1 and 18.2$\pm$0.1
magnitudes, respectively. The VISTA-VIRCAM $J$ (1.15–1.4 $\mu$m) and $K_{s}$
(1.95–2.35 $\mu$m) band photometry measurements were 19.1$\pm$0.1 and
18.2$\pm$0.2 magnitudes, respectively.

###  3.3 X-ray data

We also examined the data taken with the X-ray telescope (XRT; Burrows et al.
2005) aboard Swift, which operates simultaneously with the UVOT. Combining
both quiescence and outburst observations did not result in a detection at the
source position (see Appendix D for details). However, the observation
subsequent to the last detection in the light curve, which was taken $\sim$30
days after the peak, possibly revealed a very faint X-ray source (albeit with
SNR $\sim$ 1.9) coincident with the position of TUVO-22albb. Using the
ximage999https://www.swift.ac.uk/analysis/xrt/xrtcentroid.php tool, we found a
count rate of 0.004$\pm$0.002 c s-1, and we used the
Webpimms101010https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3pimms/w3pimms.pl
tool available from HEASARC to calculate a flux in the 0.3-10 keV band of
1.9$\times$10-13 erg cm-2 s-1 (for details see Appendix D). Assuming the
distance to NGC 300 of 1.88 Mpc, this corresponds to an X-ray luminosity of
8.1$\times$1037 erg s-1.

##  4 Discussion

Using our method to search for UV transients in Swift UVOT data (Modiano et
al. 2022), we discovered an archival transient in the nearby spiral galaxy NGC
300. The UV and optical light curve was exceptionally well sampled, including
detections during the rise, peak, and early (¡ 10 days) decay stages. We used
the light curve to characterise the evolution of the outburst. We further
investigated the quiescent counterpart using both Swift data and additional
publicly available optical and NIR images.

The data presented suggest that TUVO-22albb is most likely a very fast nova
originating in a symbiotic. The arguments in favour of this classification
are:

  * •

The spatial coincidence with NGC 300. This suggests an association with this
galaxy and that it is not a fainter, foreground Milky Way transient such as a
stellar flare or a dwarf nova (DN; a 2–6 mag outburst caused by accretion disc
instabilities in a CV; see Lasota 2001). We note that flares are further ruled
out by the transient duration: flares last between a seconds and $\sim$ 1 day
(e.g. Shibayama et al. 2013; Günther et al. 2020), while this transient was
active for at least 9 days. If the association with NGC 300 is correct, this
implies a distance of 1.88 Mpc (Gieren et al. 2005) and consequently peak
absolute (AB) magnitudes of $M_{UVW1}$ = $-$8.3 and $M_{U}$ $\leq$ $-$8.4.
This is consistent with typical nova luminosities (see e.g. Schaefer 2010;
Strope et al. 2010).

  * •

The shape and speed of evolution of the light curve. The rapid rise, occurring
in just $\sim$5 hours, and the fast decay in the subsequent days, closely
resemble that of other very fast novae (e.g. Darnley et al. 2016; Jencson et
al. 2021). A plot displaying the light curves of some very fast novae
originating in symbiotics together with that of TUVO-22albb is displayed in
Fig. 4 (see below for a more extended discussion comparing TUVO-22albb to
similar sources). The $t_{2}$ of $\sim$3.5 days places TUVO-22albb in the
speed class of very fast novae and it is one of the fastest known novae.

  * •

The evolution of the SED. The data are consistent with the SED peaking in the
U or NUV at early times (t ¡ 1 day) and then flattening. For very fast novae,
it is expected that the material ejected expands and cools, therefore shifting
the SED from a hot, blue continuum to a flatter SED (e.g. Darnley 2021; see
also the extended discussion below).

  * •

The brightness and colour of the likely quiescent counterpart we found in NIR
images. For this source $J$-$K_{S}$ = 0.9 and $M_{Ks}$ = $-$8.2 (at the
distance of NGC 300). Using the colour-magnitude diagrams of evolved giants
from Cioni et al. (2006) and Whitelock et al. (2009), we find that this source
is brighter than the tip of the red giant (RG) branch ($M_{Ks}\sim-6$), and
consistent with asymptotic giant branch (AGB) stars. The likely presence of an
evolved giant indicates that the system is a symbiotic, and specifically we
suggest that the companion in TUVO-22albb is an AGB star.

  * •

The lack of detections in archival optical and UV images down to $\sim$23.4
magnitudes. This further rules out the source being a nearby (Milky Way) DN,
because its quiescent optical brightness should have been ¡ 23 mags (moreover,
the quiescent SED in DNe is expected to be flat, not very red). Similarly,
stellar flares do not cause brightenings of more than a few magnitudes (e.g.
Günther et al. 2020). Unfortunately, the Hubble Space Telescope did not image
the exact position of our source (this would have provided a very good
constraint on the quiescence magnitude and therefore the outburst amplitude).

  * •

The possible X-ray source coincident with the position of TUVO-22albb in an
XRT image taken $\sim$30 days after the outburst peak. The luminosity of the
source ($\sim$8$\times$1037 erg s-1; assuming it was a real detection) is
similar to that of other X-ray detections of novae days to weeks after the
peak. For example, for the very fast nova U Scorpii soft X-rays were detected
and peak in brightness $\sim$30 days after the optical maximum (Mason et al.
2012). Similar X-ray detections were made for very fast novae in symbiotics
(peaking at varying times after the optical nova peak), such as V745 Sco (Page
et al. 2015), V3890 Sgr (Page et al. 2020), and RS Oph (Page et al. 2022).
X-ray detections are likely influenced by the orbital phase during the
observations (e.g. Mason et al. 2012).

Although the $t_{2}$ parameter does not encapsulate the many complications and
diversities exhibited by nova light curves, it is still a good indication of
the speed class and can be used to infer properties of the system. The
$t_{2,NUV}$ of 3.5 days makes TUVO-22albb one the fastest-evolving novae
known. This source then belongs to the class of very fast novae originating in
symbiotics. To our knowledge, until now only eight of these sources were
known: the famous RS Ophiuci (e.g. Munari & Valisa 2022), along with T CrB
(Iłkiewicz et al. 2016), V745 Sco (Sekiguchi et al. 1990), V1534 Sco (Joshi et
al. 2015), V3890 Sgr (Page et al. 2020), V407 Cyg (Munari et al. 2011), M31N
2008-12a (Darnley et al. 2016), and the recently discovered AT 2019qyl
(Jencson et al. 2021). Among these, TUVO-22albb together with AT 2019qyl
exhibit the third fastest $t_{2}$ (3.5 days; although the fastest, V745 Sco,
was not observed at peak, so its $t_{2}$ of 2.0 days is uncertain). Fig. 4
shows the light curves of TUVO-22albb along with four other very fast novae
originating in symbiotics for which data are available.

Figure 4: Light curves of TUVO-22albb and four well-known very fast novae (for
which data are available), all of which likely originate in symbiotics. The
TUVO-22albb light curves are shown with thicker lines. All light curves have
been shifted such that the time of the peaks is at day zero and the magnitude
of the peaks is set to zero. The data for M31N 2008-12a were obtained from the
UVOT data archive and processed using our pipeline (see Sect. 2.1 and Modiano
et al. 2022); for the other sources their data were obtained from the American
Association of Variable Star Observers (https://www.aavso.org/).

The shape and speed of evolution of the light curve of TUVO-22albb clearly
resembles those of other very fast novae in symbiotics. The decline is
slightly slower than those of M31N 2008-12a and V745 Sco, slightly faster than
that of V1534 Sco, and very similar to that of AT 2019qyl (as also suggested
by the same $t_{2}$ of 3.5 days). The shape of the decay is also similar. We
note that due to the inhomogenous sampling for each particular filter for
TUVO-22albb, the light curve shape suggested by the connected lines is
slightly misleading; most likely the true decline is exponential-like (see
Sect. 3) and therefore matches the behaviour of AT 2019qyl even more clearly.
We note that this comparison is based on different filters (due to data
availability). However, the general behaviour of the decay in very fast novae
light curves is typically very similar between filters in the optical to NUV;
therefore comparisons of the decay shapes still give insightful information
about similarities between different systems.

In Table 1 we compile basic properties (the absolute magnitude at maximum
brightness $M_{peak}$, $t_{2}$, $t_{3}$, and the most likely companion type)
of all the known very fast novae which are thought to originate in symbiotics
(to our knowledge). By $t_{2}$ and $t_{3}$, TUVO-22albb ($t_{2}$ = 3.5 d and
$t_{3}$ ¿ 8.3 d) is slightly faster than the averages of $\sim$4.4 d and
$\sim$10.3 d, respectively (where the averages include TUVO-22albb). The
absolute magnitude of TUVO-22albb at its peak shows that it is on the brighter
side of similar sources, though our observations at the peak were in the NUV
($U$ and $UVW1$) while other sources typically have only $V$-band observations
at the peak. The table further highlights the remarkable similarity of this
source with AT 2019qyl, in terms of peak brightness, decay properties, and
most likely companion stars.

Table 1: Comparison table of properties of all the known (to our knowledge)
very fast novae thought to originate in symbiotics.

Object |  $M_{peak}$ (filter) | $t_{2}$ | $t_{3}$ | Donor
---|---|---|---|---
RS Oph |  $-$8.0 ($V$) | 7 | 14 | RG
V745 Sco |  $-$7.9 ($V$) | 2 | 4 | RG
V1534 Sco |  $-$5.8 ($V$) | 5.6 | 9.2 | RG
T CrB |  $-$6.9 ($V$) | 4 | 6 | RG
V407 Cyg |  $-$9.0 ($V$) | 5.9 | 24 | Mira
M31N 2008-12a |  $-$6.0 ($V$) | 1.65 | 2.47 | RG
V3890 Sgr |  $-$5.2 ($V$) | 6 | 14 | RG
AT2019qyl |  $-$9.2 ($V$), -9.8 ($U$) | 3.5 | 10.3 | AGB
TUVO-22albb |  $-$9.2 ($U$) | 3.5 | ¿8.3 | AGB

Notes. The data for M31N 2008-12a are taken from Darnley et al. (2016); for
V3890 from Strope et al. (2010); for AT2019qyl from Jencson et al. (2021); for
TUVO-22albb from this paper; for all others from Hachisu & Kato (2018) and
references therein. As novae magnitudes are usually provided in the Vega
system, all magnitudes in this table (including for TUVO-22albb) are given in
the Vega system (this is in contrast with the AB system which we use for other
magnitudes quoted this paper).

Here we highlight a limitation in comparisons using $t_{2}$. A striking
feature of the light curve of TUVO-22albb is the sharp break in the decline at
$\sim$0.6 days, which marks the start of a plateau phase. Such plateaus have
been observed in some novae, mostly fast novae with short recurrence times.
Indeed, this feature has been suggested as a telltale signature for such
systems (Schaefer 2010; Darnley et al. 2016) and thus strengthens our
interpretation of TUVO-22albb as a member of this class of object. The
tendency is that the faster the nova, the earlier the plateau phase begins
(e.g. Strope et al. 2010); however, to our knowledge no nova exhibits a
plateau beginning as early as 0.6 days. For instance, the plateau phase for
M31N 2008-12a (one of the fastest novae known, with a similar light curve to
TUVO-22albb but with $t_{2}$ ¡ 2 days) begins at $\sim$4 days. This highlights
the caveats associated with using the $t_{2}$ parameter: for most very fast
novae, the plateau phase (if present) occurs after $t_{2}$ has passed.
Therefore, when comparing the early decay speed of TUVO-22albb to that of
other very fast novae, our reported $t_{2}$ is artificially high (i.e. slow).
Extrapolating the early rapid decay observed in $U$ (i.e. ignoring the plateau
data; see Fig. 3, right panel) results in a faster $t_{2}$ of 2.2 days.

Plateaus in nova light curves are thought to be caused by the irradiation of
the disc by the radiation emitted from the nuclear burning on the WD surface
(which becomes visible as the envelope recedes), combined with the decreasing
flux from photons emitted in the ejecta (Schaefer 2010; Hachisu & Kato 2000).
Hachisu et al. (2000) modelled fast nova light curves and find that several
properties of the system, including the accretion disc radius, the thickness
of the disc, and the irradiation efficiency might affect the plateau. However,
it is not clear what would cause the very early plateau phase of TUVO-22albb.

The SED evolution of TUVO-22albb is difficult to discern because of the lack
of simultaneous data in multiple filters. Nonetheless, as suggested in Sect.
3.1, at the peak of the outburst the SED is consistent with having its maximum
in the $U$ and NUV followed by a rapid (a few days) evolution towards a flat
spectrum across the UV-optical during the decay of the outburst. This is
similar to the SED evolution for both AT 2019qyl and M31N 2008-12a, where the
SEDs are observed to peak in the NUV at maximum brightness followed by a
flattening of the SED over the course of the first $\sim$ 10 days (see Fig. 5
in Jencson et al. 2021 for AT 2019qyl and Fig. 16 in Darnley et al. 2016 for
M31N 2008-12a). Flat UV-optical SEDs are expected throughout the outburst for
very fast novae in symbiotics but not those in CVs (Darnley et al. 2016;
Hachisu & Kato 2016). These similarities therefore further suggest a symbiotic
nature for TUVO-22albb. Rapid SED evolution is indicative of a fast evolution
of the temperature, which is achievable with a relatively low mass ejecta
shell (e.g. Jencson et al. 2021), so this is consistent with the low $t_{2}$
of TUVO-22albb.

As discussed Sect. 1, very fast nova light curves are indicative of massive
WDs. While this general connection is well-established (e.g. Hachisu & Kato
2016; Shara et al. 2018; Hachisu & Kato 2018; Darnley 2021), the variety of
different parameters of a nova that affect the light curve111111For example,
the chemical composition of the WD and of the accreted matter, internal shocks
in the ejecta, dust formation in the ejecta, and interactions of the ejecta
with the giant’s wind in the case of a symbiotic. See (Chomiuk et al. 2021)
and references therein for a discussion. imply that obtaining precise
estimates of the WD mass based solely on the speed class is difficult.
Nonetheless, useful first approximations can be inferred. Very fast novae
should harbour massive WDs of at least $1.25$ M⊙ (Yaron et al. 2005; Hillman
et al. 2016; Shara et al. 2018), which gives a likely lower limit for the WD
of TUVO-22albb. By comparing $t_{2}$ for TUVO-22albb to those of other very
fast novae in symbiotics (e.g. T CrB with $t_{2}$=4 d and MWD=1.38 M⊙; V1534
with $t_{2}$=5.6 d and MWD=1.37 M⊙; V745 Sco with $t_{2}$=2 d and MWD=1.38 M⊙;
Hachisu & Kato 2018), we suspect that the mass of the WD is in fact at least
1.35 M⊙, which is near the maximum possible mass for a WD.

The progenitors of Type Ia SNe, which are still not known, must have WD masses
close to the Chandrasekhar mass limit of $\sim$1.4 M⊙ (Nomoto 1982). Very fast
novae already have WDs with masses close to this ($>$1.25 M⊙), but to be
viable Type Ia SN progenitor candidates, their WDs must be able to grow
further. Hillman et al. (2016) show that WDs in novae can grow in mass to the
Chandrasekhar limit for a large range of accretion rates and initial WD
masses. The very fast novae which originate in symbiotics may represent an
even more likely channel to Type Ia SNe than those in CVs (Munari & Renzini
1992; Patat et al. 2011; Darnley et al. 2011). This may be because the
presence of the evolved giant in symbiotics results in a higher accretion rate
(Darnley et al. 2011), which allows the WD to grow more quickly (assuming not
all the accreted material is ejected; e.g. Hillman et al. 2016). The very fast
nature of TUVO-22albb and consequently its high inferred WD mass, together
with its likely origin in a symbiotic, overall make it an optimal candidate
for the Type Ia SN progenitor channel.

###### Acknowledgements.

DM is partly supported by the Netherlands Research School for Astronomy
(NOVA).
We acknowledge the use of public data from the Swift data archive.
This research has made use of the VizieR catalogue access tool, CDS,
Strasbourg, France (DOI : 10.26093/cds/vizier), and NASA’s Astrophysics Data
System Bibliographic Services.
We thank the referee greatly for the very helpful and insightful suggestions
to improve this paper.

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##  Appendix A Long-term UVOT light curve

In Fig. 5 we provide the long-term UVOT light curve of TUVO-22albb. Details of
how this light curve is created are described in (Modiano et al. 2022; Sect.
2.6), and we refer to the reproduction package of this paper for further
details. The light curve only shows one outburst detected from the source.
This outburst is extensively discussed in the main text of the paper.

##  Appendix B Co-addition of exposures

Here we provide details for the observations used to create the light curve
shown in Figs. 2 and 3. Table 2 summarises the observations that were used to
study the outburst of the source.

As discussed in Sect. 2.2, some exposures in which there was no clear
detection of the source were co-added. We chose the intervals over which to
co-add exposures such that each co-added exposure resulted in a clear
detection. The co-addition of exposures and the intervals that were co-added
are detailed in Table 2. Co-addition was always performed independently for
each filter, using the HEASoft (v6.29) tool
uvotimsum121212https://heasarc.gsfc.nasa.gov/lheasoft/ftools/headas/uvotimsum.html.

For the two pre-outburst observations (ObsIDs 00030848002 and 00030848003), we
co-added all exposures to check for a detection. Since no detection was made
in either filter in the resulting co-added exposures, in Figs. 2 and 3 and in
Table 2 we only include the individual exposures for those observations.

For all co-added exposures, the time stamps used for the light curve shown in
Fig. 2 are the average of the time stamps of their constituent exposures,
weighted by exposure time. The standard deviation on the time stamps (also
weighted by exposure time) was used as the error on the time. For the
exposures that were not co-added, we define the error bars on the time as the
exposure times. However, these were not included in Fig. 2 because they were
much smaller than the size of the markers.

##  Appendix C Photometry

Here we provide details about how we optimised the uvotsource photometry in
order to obtain the most accurate results. We refer to the reproduction
package of this paper for all the files and details needed to perform the
photometry as described. We additionally provide a table of all the magnitudes
measured from the UVOT data for all images in which a detection was made
(Table 3). The magnitudes given were converted from fluxes after the
corrections described in Sect. 2.2.

The aperture used for photometry was selected as a circular region of radius
$3.5\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ centred at the position of the
transient. The aperture was checked by eye for each exposure, and the centre
was manually adjusted if needed to account for small-scale misalignments
between exposures (of $\sim$0.5$\arcsec$; Poole et al. 2008). The radius of
the aperture was chosen so as to optimise the signal-to-noise (SNR) on the
data during the outburst peak (and then not modified for all other
observations for consistency). Specifically, this was done in order to keep
contamination from sources close to the transient position minimal. We note
however that a smaller aperture radius of 1.5$\arcsec$ was tested and had
almost no effect on the photometry, because uvotsource corrects for aperture
size. However, the measured fluxes still suffer from contamination from the
high background emission at the source position (due to unresolved stars and
diffuse emission objects in NGC 300; see Fig. 1). To take this effect into
account, we used the same method described in Modiano et al. (2020), who
encountered a similar issue with diffuse emission in the globular cluster 47
Tuc. The method consists of determining the flux at the source position by
carrying out photometry using the same source aperture, but during quiescence
(i.e. when the source was not detected; see Sect. 3.2). This ‘contamination
flux’ is equal to the flux from other sources in NGC 300 at the source
position. For each measured flux of the transient during the outburst, we
subtracted this contamination flux, independently for each filter.

The instrumental background is accounted for automatically by uvotsource using
a user-provided background region. We selected a background region near the
edge of the exposures (far from the centre of NGC 300), to avoid over-
correcting (because the galaxy background is already accounted for in our
method).

For the pre-outburst $UVM2$ and $UVW1$ exposures, no source was detected in
any of the exposures nor in the co-added exposures. We therefore reported the
photometry on these data (in Fig. 2) for individual exposures. The photometry
for these exposures was performed in the same exact way as for the detections,
including the removal of the diffuse emission. Therefore they are all exhibit
roughly zero flux (minor departures from zero, including to negative fluxes,
are likely due to small-scale variations in the diffuse emission at the source
position). To represent these as non-detections we, display them as open
circles in Fig. 2.

To determine upper limits from the co-added quiescence exposures built from
the entire archival UVOT data set of NGC 300 (except the outburst
observations), we used a smaller aperture size of
$1.2\text{\,}\mathrm{\SIUnitSymbolArcsecond}$, to avoid contamination from
bright objects close to the source position.

##  Appendix D XRT data

Here we provide details regarding the XRT data analysed. We used the online
XRT analysis tool131313https://www.swift.ac.uk/user_objects/ developed by
Evans et al. (2009) to build combined X-ray images of the field of TUVO-22albb
in the energy ranges 0.3–10 keV, 0.3–1.5 keV, and 1.5–10 keV. We combined the
outburst observations (ObsIDs 00030848001–00030848012; all in photon counting,
PC, mode), resulting in a total of $\sim$84 ks of exposure time. We separately
combined all quiescence observations (all other ObsIDs; also all in PC mode),
which accounted for $\sim$500 ks of exposure time. No detections were made at
the source position in either the combined outburst images or the combined
quiescent images (in any of the energy ranges). We estimated upper limits
using the Webpimms141414https://heasarc.gsfc.nasa.gov/cgi-
bin/Tools/w3pimms/w3pimms.pl tool available from HEASARC, by inputting the
predicted column density NH (at the source position) obtained using the NH
tool151515https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3nh/w3nh.pl, an
expected power law index of 1.8 (similar to other quiescent symbiotics; see
e.g. Bozzo et al. 2022), and the 3-$\sigma$ confidence interval prescriptions
from Gehrels (1986). We report unabsorbed X-ray 0.3–10 keV flux upper limits
of 3.7$\times$10-15 erg cm-2 s-1 and 6.2$\times$10-16 erg cm-2 s-1 for
outburst and quiescence, respectively. Using the distance to NGC 300 of 1.88
Mpc, we report upper limits on the outburst and quiescence X-ray luminosity of
1.6$\times$1036 erg s-1 and 2.6$\times$1035 erg s-1, respectively.

We inspected all individual XRT images during outburst by eye, and found no
detections. We also checked the observation carried out $\sim$30 days after
the outburst peak (ObsID 00035866004), and found a possible, faint detection
(SNR ¡ 2), shown in Fig. 6. For the fluxes derived from this tentative
detection, we used the same inputs as above (power-law index and NH), together
with the measured count rate, to determine the flux (see Sect. 3.3). We did
not check any subsequent XRT data by eye (all further observations were
obtained over a year later).

Figure 5: The long-term UVOT light curve of TUVO-22albb in six filters as
automatically generated by our pipeline (for details about the pipeline see
Modiano et al. 2022). Triangles indicate upper limits. The outburst which is
the focus of this study is labelled; no additional outbursts were detected.
Table 2: UVOT observations used to construct the light curve shown in Fig. 2.

Filter | ObsID | No. of exposures | Total exp. time (s) | Co-added exposures? | Notes
---|---|---|---|---|---
$UVM2$ | 00030848002 | 10 | 9982 | No |  Source never detected∗
$UVW1$ | 00030848003 | 15 | 9695 | No |  Source never detected∗
$U$ | 00030848004 | 15 | 9511 | No |
$UVW2$ | 00030848005 | 15 | 9333 | No |
$UVM2$ | 00030848006 | 11 | 7798 | No |
$UVW1$ | 00030848007 | 7 | 8969 | No |
$U$ | 00030848008 | 7 | 2333 | No |
$B$ | 00030848009,00030848010 | 11,11 | 2410,4184 | Yes |
$V$ | 00030848009,00030848010 | 11,11 | 2725,4489 | Yes |
$UVW1$ | 00030848011 | 11 | 7101 | Yes |
$U$ | 00030848012 | 11 | 2061 | Yes | Co-added exposures 1–6
$U$ | 00030848012 | 11 | 1798 | Yes | Co-added exposures 7–11

∗ Co-addition was performed for this ObsID, but the source was not detected in
any of the individual exposures nor in the co-added exposure. Therefore, in
this table (and in Fig. 2) the exposures are not co-added.

Table 3: Magnitudes calculated from UVOT fluxes.

MJD | Days after peak | Filter | Magnitude
---|---|---|---
54099.79 | -0.13 | $UVW1$ | 19.1
54099.86 | -0.07 | $UVW1$ | 18.4
54099.93 | 0.00 | $UVW1$ | 18.1
54099.99 | 0.07 | $UVW1$ | 18.1
54100.06 | 0.13 | $U$ | 18.0
54100.13 | 0.20 | $U$ | 18.0
54100.19 | 0.27 | $U$ | 18.1
54100.26 | 0.33 | $U$ | 18.2
54100.33 | 0.40 | $U$ | 18.2
54100.39 | 0.47 | $U$ | 18.3
54100.46 | 0.54 | $U$ | 18.4
54100.53 | 0.60 | $U$ | 18.5
54100.59 | 0.67 | $U$ | 18.5
54100.66 | 0.74 | $U$ | 18.4
54100.73 | 0.80 | $U$ | 18.5
54100.80 | 0.87 | $U$ | 18.4
54100.86 | 0.94 | $U$ | 18.6
54100.93 | 1.00 | $U$ | 18.6
54100.99 | 1.07 | $U$ | 18.5
54101.06 | 1.14 | $UVW2$ | 19.9
54101.13 | 1.21 | $UVW2$ | 20.1
54101.20 | 1.27 | $UVW2$ | 20.1
54101.26 | 1.34 | $UVW2$ | 19.9
54101.33 | 1.41 | $UVW2$ | 20.0
54101.40 | 1.47 | $UVW2$ | 20.1
54101.46 | 1.54 | $UVW2$ | 20.0
54101.53 | 1.61 | $UVW2$ | 20.0
54101.60 | 1.67 | $UVW2$ | 20.0
54101.67 | 1.74 | $UVW2$ | 20.0
54101.73 | 1.81 | $UVW2$ | 19.9
54101.80 | 1.87 | $UVW2$ | 20.1
54101.87 | 1.94 | $UVW2$ | 20.0
54101.93 | 2.01 | $UVW2$ | 20.0
54102.00 | 2.07 | $UVW2$ | 20.1
54102.07 | 2.14 | $UVM2$ | 20.1
54102.13 | 2.21 | $UVM2$ | 20.0
54102.20 | 2.28 | $UVM2$ | 20.1
54102.27 | 2.34 | $UVM2$ | 20.0
54102.34 | 2.41 | $UVM2$ | 20.1
54102.40 | 2.48 | $UVM2$ | 20.2
54102.47 | 2.54 | $UVM2$ | 20.2
54102.74 | 2.81 | $UVM2$ | 20.2
54102.80 | 2.88 | $UVM2$ | 20.1
54102.87 | 2.95 | $UVM2$ | 20.1
54102.93 | 3.01 | $UVM2$ | 20.0
54103.48 | 3.55 | $UVW1$ | 20.2
54103.54 | 3.62 | $UVW1$ | 20.0
54103.61 | 3.69 | $UVW1$ | 20.2
54103.68 | 3.75 | $UVW1$ | 20.2
54103.74 | 3.82 | $UVW1$ | 20.1
54103.81 | 3.89 | $UVW1$ | 20.1
54103.88 | 3.96 | $UVW1$ | 20.0
54104.22 | 4.29 | $U$ | 19.6
54104.29 | 4.36 | $U$ | 19.8
54104.35 | 4.43 | $U$ | 20.0
54104.42 | 4.50 | $U$ | 19.9
54104.49 | 4.56 | $U$ | 19.9
54104.55 | 4.63 | $U$ | 20.0
54104.94 | 5.02 | $U$ | 20.0
54106.15 | 6.22 | $V$ | 19.5
54106.16 | 6.24 | $B$ | 20.5
54107.42 | 7.49 | $UVW1$ | 20.9
54108.28 | 8.35 | $U$ | 20.5
54108.63 | 8.71 | $U$ | 20.6

Notes. For details about how the fluxes were determined, see Sect. 2.2. The
peak time is defined as MJD=54099.90. All magnitudes are in the AB system.
Errors on magnitudes are all on the order of 0.1–0.2 mags. For the co-added
exposures, the MJD given is the average of the time stamps of each individual
exposures, weighted by exposure time.

Figure 6: Part of the XRT image (ObsID 00035866004) in which a possible faint
detection was made at the position of TUVO-22albb (labelled with a red
circle). North is up and east is left.


# Giant proximity exchange and flat Chern band in 2D magnet-semiconductor
heterostructures

Nisarga Paul, Yang Zhang, Liang Fu  Department of Physics, Massachusetts
Institute of Technology, Cambridge, MA, USA

###### Abstract

Van der Waals (vdW) heterostructures formed by two-dimensional magnets and
semiconductors have provided a fertile ground for fundamental science and for
spintronics. We present first-principles calculations finding a proximity
exchange splitting of 14 meV equivalent to an effective Zeeman field of 120 T
in the vdW magnet-semiconductor heterostructure MoS2/CrBr3, leading to a 2D
spin-polarized half-metal with carrier densities ranging up to $10^{13}$ cm-2.
We consequently explore the effect of large exchange coupling on the
electronic bandstructure when the magnetic layer hosts chiral spin textures
such as skyrmions. A flat Chern band is found at a “magic” value of
magnetization $\overline{m}\sim 0.2$ for Schrödinger electrons, and it
generally occurs for Dirac electrons. The magnetic proximity induced anomalous
Hall effect enables transport-based detection of chiral spin textures, and
flat Chern bands provides an avenue for engineering various strongly
correlated states.

Flat band materials have recently emerged as an area of intensive study in
condensed matter physics [1, 2, 3]. Due to the dominance of many-body effects
over the quenched kinetic energy, such systems provide avenues for realizing
strongly correlated electronic states such as generalized Wigner crystals,
magnetic orders, and superconductivity. At partially filling of a topological
flat band, interaction effects can further lead to quantum anomalous Hall
states [4, 5, 6, 7] as proposed in twisted bilayer graphene (TBG) [8, 9] and
transition metal dichalcogenide (TMD) moiré heterostructures [10, 11, 12]. A
common element of moiré materials is the presence of a spatially varying
periodic modulation, e.g. of interlayer tunneling strength, which generates
narrow or flat minibands.

Such a modulation is also present in magnets hosting periodic spin textures,
which couple to electrons as a Zeeman splitting with spatially varying
orientation. Noncoplanar or chiral spin textures, including skyrmions and
canted spirals or vortices, are characterized by their nonzero spin chirality
defined as $\vec{S}_{i}\cdot(\vec{S}_{j}\times\vec{S}_{k})$. Chiral spin
textures have been the focus of recent interest due to the observation of
skyrmions in a host of compounds and a large anomalous Hall effect in
noncoplanar magnets [13, 14, 15, 16, 17, 18, 19, 15]. Electrons coupled to
skyrmions experience an emergent magnetic field which manifests in the
topological Hall effect [20, 21]. The presence of electron minibands due to a
superlattice modulation is common to both structural moirés and periodic spin
textures [22, 23, 24, 25], but the close analogy between the two types of
systems has not yet been explored.

Figure 1: (a) Lattice structure and magnetic order of MoS2/CrBr3
heterostructures. (b) Kohn-Sham wavefucntion for $K$ valley and $\Gamma$
valley maximum. The $K$ valley wavefunction is localized in the Mo layer,
while the $\Gamma$ valley wavefunction is spread out in the z direction. (c)
Band structure of the fully relaxed MoS2/CrBr3 heterostructure with a 0.9 meV
valley splitting at $K^{\prime}$/$K$ and 14 meV spin splitting at $\Gamma$.

A natural question is whether chiral spin textures can provide a route to
realizing flat bands. Indeed, the spatial profile of spin textures can be
tuned by externally controllable parameters, such as magnetization and period,
which may function similarly to twist angle or lattice mismatch in TBG or TMDs
in optimizing band flatness. Moreover, the emergent magnetic field may
generate mini-bands with Chern number [26], furthering the analogy with
structural moirés. A possible hurdle is that the exchange coupling between
localized spins and itinerant electrons must be large for emergent field to
have a strong effect on the electronic band structure. In reality, however,
most skrymion materials are high density metals in which the exchange
splitting is small compared to the Fermi energy.

In this work, we propose a new platform for a topological flat band in 2D
magnet/ TMD semiconductor heterostructures, in which the magnetic exchange
coupling introduces an emergent Zeeman field acting on a low density of
carriers in the semiconductor. Our study is motivated by the recent
theoretical [27, 28, 29, 30] and experimental works [31] suggesting the
existence of skyrmion phases and periodic chiral spin textures arising from a
spatially varying interlayer coupling in magnetic moirés such as twisted
bilayers of CrI3 or CrBr3. Our two main results are as follows. First, we
predict from first-principles calculations a giant exchange splitting of $14$
meV in MoS2/CrBr3, which far exceeds the values found so far in 2D TMD/magnet
heterostructures. As a result of this large exchange splitting, for a wide
range of hole densities up to $0.83\times 10^{13}$cm-2 only one spin-polarized
band is occupied, leading to a 2D half metal. Second, we use a continuum model
approach to study the effect of the emergent field from magnetic skrymion
crystals on the low-energy electronic structure of proximitized 2D
semiconductors. We find an almost completely flat Chern band for Schrödinger
electrons (as in MoS2) at a “magic” value of magnetization $\overline{m}=0.2$,
while for Dirac electrons (as in graphene), remarkably, a flat Chern band
always occurs.

Giant exchange splitting in MoS2/CrBr3.— Our proposal to realize flat Chern
bands requires a strong interfacial exchange coupling in 2D magnet-TMD
semiconductor heterostructures. In monolayer TMDs, strong Ising spin-orbit
coupling and inversion symmetry breaking gives rise to opposite spin states in
the $K^{\prime}$ and $K$ valleys. Previous studies have demonstrated
$K^{\prime}$/$K$ spin-valley splitting in TMDs proximity coupled to
ferromagnetic semiconductors. However, such splitting is generally small,
e.g., 3.5 meV in WSe2/CrI3 [32].

We performed density functional calculations using the generalized gradient
approximation [33] with SCAN+rVV10 van der Waals density functional [34], as
implemented in the Vienna Ab initio Simulation Package [35]. By scanning
through various 2D TMD-magnet heterostructures, we find a giant exchange
splitting in MoS2/CrBr3, where monolayer MoS2 is a TMD semiconductor, and
CrBr3 is an insulating vdW magnet. Magnetic orders and lattice constants are
taken from the Computational 2D Materials Database (C2DB) [36]. The 2x2
supercell of MoS2 with lattice constant 6.38 Å is chosen to match the lattice
constant of CrBr3 (6.44 Å) with less than 1% strain. Similar to previous
works, we find a small $K^{\prime}$/$K$ valley splitting on the valence band
side of 0.9 meV for MoS2/CrBr3.

In monolayer MoS2, the $\Gamma$ valley is very close in energy to the $K$
valley. In MoS2/CrBr3, we find that the $\Gamma$ valley of MoS2 becomes the
valence band maximum after taking into account the relaxed heterostructure
with layer spacing 6.76 Å. We observe a giant spin splitting of up to 14 meV
in the $\Gamma$ valley as shown in Fig. 1. Assuming a Landé $g$-factor equal
to 2, this is an effective Zeeman field of 120 T. The large difference in the
exchange splitting in the $K$ and $\Gamma$ valleys can be understood from the
spatial distribution of wavefunctions. At the $K$ valley, due to the large in-
plane momentum, electron wavefunctions are strongly localized within the vdW
layer. However, at the $\Gamma$ valley, due to the zero in-plane momentum, the
wavefunctions are more spread out in out-of-plane direction, which enhances
coupling to the magnetic layer. The large exchange splitting in this specific
material, which is the first main result of this work, provides efficient spin
control of the $\Gamma$ valley bands.

Emergent gauge field at large exchange coupling.— In the following, we explore
the consequences of a large exchange coupling on the electronic bandstructure.
This is motivated by both the giant exchange splitting found in the previous
section and theoretical and experimental results suggesting that twisted
bilayers of vdW magnets may host noncoplanar or skyrmionic phases [27, 28, 29,
30, 31]. To this end, we turn to a completely general Hamiltonian describing
electrons coupled to a spin texture via a large exchange coupling. It is
important to note that in low density systems such as TMD semiconductors or
graphene, the physics of itinerant electrons coupled to periodic spin textures
is universally described by a continuum theory, taking the form

$\displaystyle H\textsubscript{S}$
$\displaystyle=\frac{p^{2}}{2m}+J\vec{\sigma}\cdot\vec{S}(r)$ (1a)
$\displaystyle H\textsubscript{D}$
$\displaystyle=v\vec{p}\cdot\vec{\tau}+J\vec{\sigma}\cdot\vec{S}(r)$ (1b)

for the case of Schrödinger and Dirac electrons with mass $m$ and velocity
$v$, respectively. The Pauli $\vec{\sigma}$ matrices act on electron spin,
while the Pauli $\vec{\tau}$ matrices act on a psuedospin degree of freedom,
e.g., sublattice in the case of graphene. In the following, we will consider
smoothly varying periodic spin textures $\vec{S}(r)$ with magnetic wavelength
$\xi$ and unit norm $|\vec{S}|=1$. An SU(2) gauge transformation
$\mathbf{U}(r)$ allows us to instead consider a uniformly polarized spin
texture provided we use the gauge-transformed momentum
$\vec{p}\mathbf{1}-e\vec{\mathbf{A}}$, where

$\vec{\mathbf{A}}=\frac{i\hbar}{e}\mathbf{U}^{\dagger}\vec{\nabla}\mathbf{U}.$
(2)

The large exchange coupling limits are, respectively,

$Jm\xi^{2}\gg 1\quad\text{ and }\quad J\xi/v\gg 1.$ (3)

In this limit the ground state manifold consists of electrons with spins
locally anti-aligned with the spin texture. The effective Hamiltonian
describing the locally spin-polarized Schrödinger electrons has been derived
before [20]:

${\tilde{H}}\textsubscript{S}=\frac{1}{2m}(\vec{p}-e\vec{A})^{2}+\frac{e^{2}}{8m}(\partial_{i}\vec{S})^{2}$
(4)

(summing $i=x,y$). The second term, originating from off-diagonal terms of
$\vec{\mathbf{A}}$, is a scalar potential that accounts for the reduction of
electron hopping by the spin gradient and
$\vec{A}=[\vec{\mathbf{A}}]_{\downarrow\downarrow}$ is an emergent $U(1)$
gauge field. The corresponding emergent field strength

$B^{e}=\frac{\hbar}{2e}\vec{S}\cdot(\partial_{x}\vec{S}\times\partial_{y}\vec{S})$
(5)

is proportional to the spin chirality. A skyrmion can be defined as a texture
with flux $h/e$. It follows from this equation that purely in-plane spin
textures have vanishing emergent field. Moreover, purely in-plane textures are
symmetric under the combination of time reversal and a spin-space $C_{2}$
rotation, while the Chern number is odd under this combination, implying such
textures do not yield topological bands. Accordingly, we will consider chiral
magnets with out-of-plane spin textures.

Figure 2: (a) Skyrmion texture based on generic ansatz Eq. (7) with different
magnetizations and $\alpha=1$. $(S^{x},S^{y})$ is plotted as a vector field
with $S^{z}$ in color, and $\overline{m}$ is the average of $S^{z}$. (b)
Bandstructures in the presence of the textures in (a), following the $\Gamma
MK\Gamma$ path in the hexagonal mini BZ. $H_{S}$ was diagonalized using a
plane-wave cutoff $N=10$, taking $J=100,m=1$. A flat first band is observed
when $\overline{m}=0.22$. All bands carry Chern number $C=1$. (c) Local
density of states summed over the first band (sampling $4^{2}$ states),
spatially aligned with (a).

We will also be interested in Dirac electrons at large exchange coupling, for
which the effective Hamiltonian takes the form

${\tilde{H}}\textsubscript{D}=v(\vec{p}-e\vec{A})\cdot\vec{\tau}.$ (6)

It is notable that Dirac electrons only feel a periodic gauge field in
contrast with the previously studied Schrödinger case involving both the gauge
field and a scalar potential.

The origin of the emergent gauge field is the local U(1) rotational freedom of
the electron spin about the axis of the local spin. In analogy with
conventional Landau levels, the emergent field $B^{e}$ generally endows the
minibands with Chern number when the average is nonzero. In the following, we
investigate the flatness of the induced Chern bands in the large exchange
coupling limit.

Flat bands in skyrmion crystals.— We present numerical results on the
bandstructure of Schrödinger and Dirac electrons in the presence of generic
skyrmion crystals using the full Hamiltonians in Eqs. (1a) and (1b), i.e.
without any approximations. $H_{S}$ and $H_{D}$ are explicitly diagonalized in
a plane wave basis with a large wavenumber cutoff $N$ (see Supplemental
Material).

As a model for a generic SkX, we introduce the simple three-parameter ansatz
$\vec{S}=\vec{N}_{r}/|\vec{N}_{r}|$ with

$\vec{N}_{r}=\frac{1}{\sqrt{2}}\sum_{j=1}^{6}e^{iq_{j}\cdot
r}\hat{e}_{j}+\mu\hat{z}$ (7)

where $\vec{q}_{j}=\xi(\cos\theta_{j},\sin\theta_{j})$ and
$\hat{e}_{j}=(i\alpha\sin\theta_{j},-i\alpha\cos\theta_{j},-1)/\sqrt{2}$ and
the angles satisfy
$\theta_{2}=\theta_{1}+2\pi/3,\theta_{3}=\theta_{1}+4\pi/3$, and
$\theta_{j+3}=\theta_{j}+\pi$. Eq. (7) represents a normalized sum of three
helical spirals forming a triangular SkX, as plotted in Fig. 2a. This ansatz
is widely adopted in studies of chiral magnets and magnetic skyrmion crystals.
It qualitatively reproduces the observed real-space images of skyrmion
crystals [37, 38, 39, 40, 41, 42, 43]. A different choice of relative phases
results in Néel-like skyrmions but doesn’t alter bandstructures. The
parameters $\alpha,\xi,$ and $\mu$ control coplanarity, wavelength, and out-
of-plane bias, respectively.

In Fig. 2 we plot bandstructures and local density of states (LDOS) for
Schrödinger electrons in the presence of our chosen SkX. We set $\alpha=1$ and
vary the magnetization $\overline{m}$ (i.e. the unit cell average of $S^{z}$),
which is monotonic with $\mu$ ($\xi$ can be scaled out of the problem). We can
observe the flattening of lower bands and a near perfectly flat, well-
separated first band as we tune the magnetization past $\overline{m}=0.2$, the
second main result of this work. All bands shown carry Chern number $C=1$ and
spins are anti-aligned with the skyrmion texture.

In Fig. 3 we systematically plot band flatness, defined as the ratio of
bandwidth to bandgap for the first miniband, with varying $\alpha$ and
$\overline{m}$. Complete flatness is an ideal pathway for strongly correlated
matter since the kinetic energy is strongly quenched and band separation
remains large. We plot the branch $\overline{m}>0$ which corresponds to the
typical case of separated skyrmion bubbles. The plot indicates a quite robust
“magic” value of magnetization near $\overline{m}=0.2$ where flatness goes to
zero.

Figure 3: First band flatness, defined as the ratio of bandwidth to bandgap,
of $H_{S}$ in the presence of a generic skyrmion crystal (defined in Eq. (7))
with varying magnetization $\overline{m}$ and coplanarity $\alpha$. Near
perfect flatness occurs around $\overline{m}=0.2$. Flatness was calculated by
solving $H_{S}$ with plane-wave cutoff $N=10$ and $J=100$ over $10^{2}$ points
in the mini BZ.

Next we turn to the case of Dirac electrons under periodic Zeeman field, which
may be realized in graphene or twisted bilayer graphene in proximity with a 2D
magnet. Remarkably, we observe a flat band for arbitrary skyrmion textures in
Fig. 4. This can be understood from the effective Hamiltonian in the large $J$
limit.

${\tilde{H}}\textsubscript{D}=v\begin{pmatrix}0&\Pi\\\
\overline{\Pi}&0\end{pmatrix}$ (8)

where $\overline{\Pi}=(\Pi_{x}+i\Pi_{y})/2$ and $\vec{\Pi}=\vec{p}-e\vec{A}$.
It is known that the Dirac equation in the presence of a spatially varying
magnetic field admits an extensive set of zero-mode solutions [44]. The number
of zero modes is equal to the total number of flux quanta through the system
(which in our case is given by the total topological charge of the skrymion
crystal), regardless of the spatial profile of the magnetic field. This
remarkable property follows from the Atiyah-Singer index theorem. In the limit
of a uniform field, these zero modes reduce to the well-known $n=0$ Landau
level.

In Fig. 4a we plot the periodic emergent magnetic fields arising from the
skrymion crystal and in Fig. 4c we plot the LDOS of the flat band of Dirac
electrons under magnetic proximity effect. The LDOS shows significant spatial
variation, which can be detected by scanning tunneling microscopy.

Figure 4: (a) Emergent magnetic fields corresponding to (and spatially aligned
with) the SkX shown in Fig. 2a. Dirac electrons in the large exchange coupling
limit move in this emergent field. (b) Bandstructures analogous to Fig. 2b for
Dirac electrons. A flat band is pinned at $E=-J$ whenever there is net
emergent flux. (c) Local density of states summed over the $E=-J$ band
(sampling $150$ states), spatially aligned with (a).

Small exchange coupling and anomalous Hall effect.— While the large exchange
coupling regime is realistic for MoS2/CrBr3, other systems may have
constraints on magnetic wavelength or intrinsic exchange coupling not
conducive to this limit. Additionally, many chiral magnets host noncoplanar
states which do not enclose a net spin chirality. Examples of such states
include the canted vortex lattice, multiple-Q conical spiral, multilayers of
single-Q helices, and meron-antimeron lattice [29, 30].

Remarkably, we find that neither net spin chirality nor large exchange
coupling is necessary to attain Chern bands and even generic noncoplanar
textures may generate an anomalous Hall effect at small coupling. This goes
beyond the mean-field semiclassical result [20] that the anomalous Hall effect
in topological textures is proportional to the net spin chirality. We provide
examples of spin textures illustrating this point in the Supplemental
Material. The observation of marked Hall transport features suggests that
proximity-coupled electrons can serve as a reliable sensor for noncoplanar
textures in magnetic materials (Fig. 5). A transport-based sensor would be
advantageous because it can be done on tabletop while bulk probes such as
Lorentz TEM and neutron diffraction have a low sensitivity for 2D devices. We
also note that in the presence of a point-group symmetry, the leading Fourier
modes of the a spin texture may be inferred from the miniband Chern numbers.
This provides a useful direct mapping between real-space textures and
transport properties. In the Supplemental Material, we present the details for
the case of six-fold rotational symmetry.

Figure 5: An anomalous Hall effect is generically expected in semiconductor /
chiral magnet heterostructures with noncoplanar textures even at small
coupling and without net spin chirality.

Conclusions.— We investigated the emergence of flat bands in semiconductors
proximity-coupled to spin textures in the limit of large exchange coupling.
This limit was motivated by the giant exchange of 14 meV we found using first-
principles calculations in the TMD/ vdW magnet heterostructure MoS2/CrBr3, the
latter of which is proposed to host skyrmions in a moiré [28, 27]. For
Schrödinger electrons coupled to a generic SkX we find a flat Chern band
emerges at out-of-plane magnetization $\overline{m}=0.2$ while a flat Chern
band is generically present for Dirac electrons. A large exchange coupling and
long electron mean free path (for sharply defined minibands) are prerequisites
for this route to flat bands. These continuum results are universal at low
densities, only making use of effective mass or Dirac velocity in the band
structure. We show explicitly that anomalous Hall effect is generically
present in TMD semiconductors or graphene proximitized by noncoplanar magnets.
Our results also suggest the possibility of fractional Chern insulators (FCIs)
in 2D magnet-semiconductor or graphene heterostructures.

Acknowledgements— We thank Joe Checkelsky, Takashi Kurumaji, Kin Fai Mak, Jie
Shan, Riccardo Comin and Xiaodong Xu for helpful discussions, and Junkai Dong
for collaboration on a related project. This work is funded by the Simons
Foundation through a Simons Investigator Award. LF is partly supported by the
David and Lucile Packard Foundation.

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##  Appendix A Supplemental Material

###  A.1 Bandstructure

Figure 6: Band structure of (a) MoS2/CrBr3, (b) $2\times 2$ MoS2, (c) CrBr3.

We present the separate band structures of $2\times 2$ MoS2 and CrBr3 in Fig.
6. The K valley is 30 meV higher than the $\Gamma$ valley in monolayer MoS2.
Due to the type II band alignment, the band gap of the heterobilayer is
reduced to 1.2 eV.

The plane wave approximation for the continuum model is as follows. Let
$\vec{S}(r)=\vec{S}(r+a_{1})=\vec{S}(r+a_{2})$ be a doubly periodic spin
texture and let $b_{1},b_{2}$ be the reciprocal vectors, $b_{i}\cdot
a_{j}=2\pi\delta_{ij}$. We can write any wavefunction with momentum
$k=(k_{x},k_{y})$ as

$\psi(r)=e^{ikr}\sum_{m_{1},m_{2}}e^{i(m_{1}b_{1}+m_{2}b_{2})r}\chi_{m_{1}m_{2}}$
(9)

where $m_{1},m_{2}$ are integers and
$\chi_{m_{1}m_{2}}=\begin{pmatrix}\chi^{\uparrow}_{m_{1}m_{2}}\\\
\chi^{\downarrow}_{m_{1}m_{2}}\end{pmatrix}$. The Hamiltonian can be written
as

$H_{S}=\frac{p^{2}}{2m}\mathbf{1}+J\sum_{m_{1},m_{2}}e^{i(m_{1}b_{1}+m_{2}b_{2})r}\vec{S}_{m_{1}m_{2}}\cdot\vec{\sigma}$
(10)

where $\vec{S}_{m_{1}m_{2}}=\vec{S}^{*}_{-m_{1},-m_{2}}$ is a Fourier
component of $\vec{S}(r)$. We then obtain the matrix equation

$\frac{(k+n_{1}b_{1}+n_{2}b_{2})^{2}}{2m}\chi_{n_{1}n_{2}}+J\sum_{m_{1}m_{2}}\vec{S}_{m_{1}m_{2}}\cdot\vec{\sigma}\chi_{n_{1}-m_{1},n_{2}-m_{2}}=E\chi_{n_{1}n_{2}}$
(11)

We apply a large momentum cutoff $|n_{1}|,|n_{2}|\leq N$ to approximate the
spectrum and eigenfunctions, with an analogous procedure for the Dirac case.
In the Dirac case we omit spurious modes which appear on the lattice edges and
whose support is measure zero as $N\to\infty$, which can be easily identified
as they break the $C_{6}$ symmetry. The plane wave method was used for Fig.s
1, 3, and 4 while for Fig. 7 a square lattice nearest neighbor tight-binding
Hamiltonian was used.

###  A.2  Small exchange coupling and anomalous Hall effect

Figure 7: (a) Weak-coupling bandstructures for the SkX
($\overline{m}=0.1,\alpha=1$) and the canted vortex lattice (or meron-
antimeron lattice). Each band in the former has Chern number 1, and the lowest
bands in the latter have Chern numbers [0,0,1,-1,1]. (b) Hall conductances.
The SkX exhibits a quantized anomalous Hall effect from the lowest bands, and
the canted vortex lattice has a nonzero anomalous Hall response despite its
vanishing net spin chirality. (c) Rescaled Berry curvature
$|b_{1}|^{2}\Omega(\mathbf{k})/2\pi$ corresponding to the lowest bands in in
Fig. 2b plotted in a square of size $2|b_{1}|$. Berry curvature becomes more
uniform near the flat band at $\overline{m}=0.22$. (d) Similarly for an SkX
(left) and canted vortex lattice (right) at weak exchange coupling,
corresponding to the lowest bands of (a)

In the large exchange coupling regime, we found well-separated flat Chern
bands (which imply a quantized anomalous Hall effect) in the presence of
skyrmion-like textures. If we drop these two prerequisites and instead
consider small exchange coupling and general noncoplanar textures (i.e. not
necessarily enclosing a net spin chirality), we find numerically that an
anomalous Hall conductance is still generically present. We illustrate this
point with an SkX at small exchange coupling (Fig. 7a) and a canted vortex
lattice spin texture, which takes the form

$\vec{S}_{r}=(\sqrt{1-(S^{z})^{2}}\sin\theta_{r},\sqrt{1-(S^{z})^{2}}\cos\theta_{r},S^{z})$
(12)

with $\theta_{r}$ the angle relative to the unit cell center and $S^{z}$ is a
smooth periodic function which equals unity at the unit cell centers. We chose
$S^{z}(r)=\frac{1}{3}\sum_{i}\cos^{2}(q_{i}\cdot r/2)$ where
$q_{1}=b_{1},q_{2}=b_{2},q_{3}=-b_{1}-b_{2}$ and $b_{1,2}$ are reciprocal
lattice vectors. This texture has alternating regions of $\pm 1/2$ emergent
flux, so it may also be considered to be a meron-antimeron lattice. Because
the net spin chirality vanishes, and naïvely we may expect no momentum-space
topology. However, we observe miniband Chern numbers and a corresponding
anomalous Hall effect (Fig. 7b).

In the following, we connect the leading Fourier modes of point-group
symmetric spin textures to the Chern number of associated electron bands. We
approach this by using perturbation theory in a small exchange coupling $J$,
in which case Chern numbers can be computed from the point-group symmetry
eigenvalues of the states at the high-symmetry points in the Brillouin zone
[45]. These eigenvalues in turn depend only on the leading Fourier modes of
the spin texture. We assume a six-fold rotational symmetry of the Hamiltonian,
i.e. $[\hat{C}_{6},H]=0$ where $\hat{C}_{6}=e^{\frac{-2\pi
i}{6}L^{z}}e^{\frac{-2\pi i}{6}\frac{\sigma^{z}}{2}}$ (other point-groups can
be analyzed similarly). The Chern number $C$ is given by

$e^{2\pi iC/6}=\prod_{i\in
occ.}(-1)^{F}\eta_{i}(\Gamma)\theta_{i}(K)\zeta_{i}(M)$ (13)

where $\eta(\Gamma)$ is the $C_{6}$ eigenvalue at the $\Gamma$ point,
$\theta(K)$ is the $C_{3}$ eigenvalue at the $K$ point, and $\zeta(M)$ is the
$C_{2}$ eigenvalue at the $M$ point. The Chern number for the first band
follows from degenerate perturbation theory, and we obtain

$C\bmod 6=\begin{cases}1&S_{0}^{z}>0,S_{1}^{z}>0\\\ 2\text{ or
}0&S_{0}^{z}>0,S_{1}^{z}<0\\\ -2\text{ or }0&S_{0}^{z}<0,S_{1}^{z}>0\\\
-1&S_{0}^{z}<0,S_{1}^{z}<0\\\ \end{cases}$ (14)

assuming $J>0$. For $J<0$ the Chern number changes sign mod 6. Here
$\vec{S}_{0},\vec{S}_{1}$ are the zeroth and first Fourier components of
$\vec{S}(r)$, so in particular $S_{0}^{z}=\overline{m}$. For fixed
$S_{1}^{z}$, we may note that the first band undergoes a topological
transition as the magnetization changes sign, which can be traced to the gap
closing at $\Gamma$. Likewise, for fixed magnetization there is a topological
transition as $S_{1}^{z}$ changes sign.

$\mathit{\Gamma}$ point.—The two states $\ket{\Gamma,+\frac{1}{2}}$ and
$\ket{\Gamma,-\frac{1}{2}}$ have $\hat{C}_{6}$ eigenvalues $e^{-i\pi/6}$ and
$e^{i\pi/6}$ respectively. First order perturbation theory gives an energy
correction

$\braket{\Gamma,s}{J\vec{\sigma}\cdot\vec{S}}{\Gamma,s}=2sJS^{z}_{0}$ (15)

where $s=\pm\frac{1}{2}$. Therefore

$\eta(\Gamma)=\begin{cases}e^{-i\pi/6}&JS_{0}^{z}<0\\\
e^{i\pi/6}&JS_{0}^{z}>0\end{cases}.$ (16)

$\mathit{K}$ point.—We start by defining the states

$\ket{K,\nu,s}=\frac{1}{\sqrt{6}}\sum_{j=0}^{5}e^{\frac{-i\pi}{6}j(2\nu+1)}\hat{C}_{6}^{j}\ket{K,s}\qquad\nu=0,1,\ldots,5.$
(17)

The six eigenspaces can be labelled by $\nu$:

$\displaystyle\hat{C}_{6}\ket{K,\nu,s}$
$\displaystyle=e^{\frac{i\pi}{6}(2\nu+1)}\ket{K,\nu,s}$

The first-order effective $2\times 2$ Hamiltonian in each eigenspace can be
written
$[H^{\text{eff.}}_{\nu}]_{ss^{\prime}}=J\braket{K,\nu,s}{\vec{\sigma}\cdot\vec{S}}{K,\nu,s^{\prime}}$,
giving

$\displaystyle H^{\text{eff.}}_{0}=H^{\text{eff.}}_{3}$
$\displaystyle=J\begin{pmatrix}S_{0}^{z}+2S_{1}^{z}&0\\\
0&-S_{0}^{z}+S_{1}^{z}\end{pmatrix},$ (18a) $\displaystyle
H^{\text{eff.}}_{1}=H^{\text{eff.}}_{4}$
$\displaystyle=J\begin{pmatrix}S_{0}^{z}-S_{1}^{z}&0\\\
0&-S_{0}^{z}+S_{1}^{z}\end{pmatrix},$ (18b) $\displaystyle
H^{\text{eff.}}_{2}=H^{\text{eff.}}_{5}$
$\displaystyle=J\begin{pmatrix}S_{0}^{z}-S_{1}^{z}&0\\\
0&-S_{0}^{z}-2S_{1}^{z}\end{pmatrix}.$ (18c)

Let

$\mu_{K}=\min[\pm J(S_{0}^{z}-S_{1}^{z}),\pm J(S_{0}^{z}+2S_{1}^{z})].$ (19)

It follows that

$\theta(K)=\begin{cases}e^{i\pi/3}&\mu_{K}=J(S_{0}^{z}+2S_{1}^{z})\\\
e^{-i\pi/3}&\mu_{K}=-J(S_{0}^{z}+2S_{1}^{z})\\\ -1\text{ or
}e^{-i\pi/3}&\mu_{K}=J(S_{0}^{z}-S_{1}^{z})\\\ e^{i\pi/3}\text{ or
}-1&\mu_{K}=-J(S_{0}^{z}-S_{1}^{z})\end{cases}.$ (20)

The latter two cases require 2nd order perturbation theory, and thus higher
Fourier harmonics, to resolve.

$\mathit{M}$ point.—Define the states

$\ket{M,\nu,s}=\frac{1}{\sqrt{6}}\sum_{j=0}^{5}e^{\frac{-i\pi}{6}j(2\nu+1)}\hat{C}_{6}^{j}\ket{M,s}\qquad\nu=0,1,\ldots,5.$
(21)

Once again the six eigenspaces can be labelled by $\nu$:

$\hat{C}_{6}\ket{M,\nu,s}=e^{\frac{i\pi}{6}(2\nu+1)}\ket{M,\nu,s}.$ (22)

The first-order effective Hamiltonian in each eigenspace is given by
$[H^{\text{eff.}}_{\nu}]_{ss^{\prime}}=\braket{M,\nu,s}{V}{M,\nu,s^{\prime}}$,
yielding

$\displaystyle H^{\text{eff.}}_{0}=H^{\text{eff.}}_{2}=H^{\text{eff.}}_{4}$
$\displaystyle=J\begin{pmatrix}S_{0}^{z}+S_{1}^{z}&0\\\
0&-S_{0}^{z}+S_{1}^{z}\end{pmatrix},$ (23a) $\displaystyle
H^{\text{eff.}}_{1}=H^{\text{eff.}}_{3}=H^{\text{eff.}}_{5}$
$\displaystyle=J\begin{pmatrix}S_{0}^{z}-S_{1}^{z}&0\\\
0&-S_{0}^{z}-S_{1}^{z}\end{pmatrix}.$ (23b)

Let

$\mu_{M}=\min[\pm J(S_{0}^{z}+S_{1}^{z}),\pm J(S_{0}^{z}-S_{1}^{z})].$ (24)

It follows that

$\zeta(M)=\begin{cases}i&\mu_{M}=J(\pm S_{0}^{z}+S_{1}^{z})\\\
-i&\mu_{M}=J(\pm S_{0}^{z}-S_{1}^{z})\end{cases}.$ (25)

###  A.3 Berry curvature

We have produced plots of the Berry curvatures of the lowest bands in the
presence of a generic SkX texture at large exchange coupling (Fig. 7c) and for
the SkX and canted vortex lattice textures at weak exchange couplings (Fig.
7d).