# High Resolution Observations of H I in the IC 63 Reflection Nebula

Lars Bonne SOFIA Science Center/USRA, NASA Ames Research Center, M.S. N232-12,
Moffett Field, CA 94035, USA  B-G Andersson SOFIA Science Center/USRA, NASA
Ames Research Center, M.S. N232-12, Moffett Field, CA 94035, USA  Robert
Minchin SOFIA Science Center/USRA, NASA Ames Research Center, M.S. N232-12,
Moffett Field, CA 94035, USA  National Radio Astronomy Observatory, P.O. Box
O, Socorro, NM 87801, USA  Archana Soam Indian Institute of Astrophysics, II
Block, Koramangala, Bengaluru 560034, India  SOFIA Science Center/USRA, NASA
Ames Research Center, M.S. N232-12, Moffett Field, CA 94035, USA  Joshua
Yaldaei  Department of Physics, Santa Clara University, Santa Clara, 500 El
Camino Real, Santa Clara, CA, USA  Kristin Kulas Department of Physics, Santa
Clara University, Santa Clara, 500 El Camino Real, Santa Clara, CA, USA  Janik
Karoly Jeremiah Horrocks Institute, University of Central Lancashire, Preston
PR1 2HE, UK  SOFIA Science Center/USRA, NASA Ames Research Center, M.S.
N232-12, Moffett Field, CA 94035, USA  Lewis B.G. Knee Herzberg Astronomy and
Astrophysics Research Centre, National Research Council of Canada, 5071 West
Saanich Road, Victoria, BC, V9E 2E7, Canada  Siddharth Kumar Department of
Physics, Indian Institute of Science, Bangalore 560012,India  Nirupam Roy
Department of Physics, Indian Institute of Science, Bangalore 560012,India

###### Abstract

Photodissociation regions (PDRs), where the (far-)ultraviolet light from hot
young stars interact with the gas in surrounding molecular clouds, provide
laboratories for understanding the nature and role of feedback by star
formation on the interstellar medium. While the general nature of PDRs is well
understood - at least under simplified conditions - the detailed dynamics and
chemistry of these regions, including gas clumping, evolution over time etc.
can be very complex. We present interferometric observations of the 21 cm
atomic hydrogen line, combined with [C II] 158 $\mu$m observations, towards
the nearby reflection nebula IC 63. We find a clumpy H I structure in the PDR,
and a ring morphology for the H I emission at the tip of IC 63. We further
unveil kinematic substructure, of the order of 1 km s-1, in the PDR layers and
several legs that will disperse IC 63 in $<$0.5 Myr. We find that the dynamics
in the PDR explain the observed clumpy H I distribution and lack of a well-
defined H I/H2 transition front. However, it is currently not possible to
conclude whether H I self-absorption (HISA) and non-equilibrium chemistry also
contribute to this clumpy morphology and missing H I/H2 transition front.

(ISM:) photon-dominated region (PDR), techniques: interferometric

††journal: ApJ††facilities: GMRT, WSRT, DRAO, SOFIA(upGREAT), CFHT(WIRCam),
CADC††software: astropy (Astropy Collaboration et al., 2013), MIRIAD (Sault et
al., 1995), AIPS (Greisen, 2003)

##  1 Introduction

The formation of high mass (OB) stars leads to stellar feedback in the form of
radiation, stellar winds and eventually supernovae. These feedback processes
ionize the surrounding interstellar medium (ISM) and can have an important
impact on its dynamical and chemical evolution. At the interface of the fully
ionized phase with the molecular phase this creates photodissociation regions
(PDR) where FUV photons with energies between 6 eV and 13.6 eV dominate the
chemistry (Hollenbach & Tielens, 1999; Röllig et al., 2007; Wolfire et al.,
2022). This leads to a chemical evolution as a function of depth into the
cloud due to the decreasing amount of FUV photons. This chemical evolution
includes, among (many) others, the transition from atomic to molecular
hydrogen (H2) and from ionized carbon (C+) to carbon monoxide (CO) (e.g.
Sternberg & Dalgarno, 1995). These different tracers can then also provide a
view on the dynamics of the cloud as a function of depth in the PDR which
helps to constrain the effect of stellar feedback on molecular cloud evolution
(e.g. Schneider et al., 2020). In addition, probing a variety of tracers
allows to investigate the physical, dynamic and chemical structure of the PDR.
Often, PDR models are based on plane parallel geometries (e.g. Tielens &
Hollenbach, 1985; Kaufman et al., 1999; Le Petit et al., 2006), yet PDR
structure can be more complicated in the 3D turbulent ISM. Therefore,
dedicated clumpy PDR models have been developed (e.g. Gierens et al., 1992;
Stoerzer et al., 1996) as well as PDR codes, such as 3D-PDR (Bisbas et al.,
2012), that make predictions by post-processing turbulent ISM simulations. In
addition, it is possible that the evolution in some PDRs is driven by non-
equilibrium chemistry due to a rapid progression of the ionization front which
can particularly affect the atomic to molecular hydrogen (H I/H2) transition
front (Bertoldi & Draine, 1996; Störzer & Hollenbach, 1998; Maillard et al.,
2021).

Here, we will investigate whether the atomic to molecular hydrogen transition
in a PDR is governed by non-equilibrium chemistry or affected by the 3D
turbulent dynamics of the region. For this we will particularly make use of
observations of the H I 21 cm line. However, because of the relatively long
wavelength of this line, it is challenging to resolve the structure and
dynamics of atomic hydrogen in PDRs with single dish observations even for the
most nearby regions. Interferometric observations with incomplete u-v spacing
coverage, on the other hand, risk resolving out the diffuse emission. To
reliably probe the PDR of the reflection nebula/molecular cloud IC 63 in H I
we have combined observations of three radio interferometers and single dish
data to ensure both high spatial and spectral resolution and full u-v spacing
coverage. We have combined observations from the Giant Metrewave Radio
Telescope (GMRT), the Westerbork Synthesis Radio Telescope (WSRT), the
Dominion Radio Astrophysical Observatory (DRAO) Synthesis Telescope (ST)
(Landecker et al., 2000) and the DRAO John A. Galt 26m single-dish telescope
to probe the kinematics and structure of the PDR. Together these observations
cover baselines from 23 km (with the GMRT) down to zero-spacing single dish
data (DRAO) and provide the possibility to reach a resolution of 2.5″. In this
study we present these interferometric H I 21 cm observations of the IC 63
reflection nebula, complemented with the SOFIA [C II] 158 $\mu$m observations
from Caputo et al. (2023, submitted), as well as the H2 data from Andersson et
al. (2013) and Soam et al. (2021a). Since carbon is still ionized when
molecular hydrogen has already formed (Tielens & Hollenbach, 1985; Sternberg &
Dalgarno, 1995), the combined data set allows a study of the PDR structure and
dynamics as a function of depth in the PDR.
IC 63 is a nebula in the Sh2-185 H II region which is illuminated and ionized
by a B0 IV star, $\gamma$ Cassiopeia (Karr et al., 2005). This cloud is
classified as type-B bright-rimmed cloud by Sugitani et al. (1991). Gaia
parallaxes provide a distance of $\lesssim 200$ pc to this region which makes
it one of the closest H II regions to the Sun (Soam et al., 2021c). Because of
its proximity, the interferometric observations reach a high spatial
resolution ($\sim$1.5$\times$10-2 pc). The nebula has also been extensively
studied in many tracers, from the UV (e.g. Witt et al., 1989; France et al.,
2005) to the IR (e.g. Fleming et al., 2010; Andrews et al., 2018) and mm-waves
(e.g. Jansen et al., 1994; Soam et al., 2021c).

The atomic hydrogen emission from IC 63 has previously been studied by e.g.
Blouin et al. (1997) using the DRAO ST outside Penticton, British Columbia.
However, even with the 604.3m maximum fully sampled baseline of the DRAO ST,
the angular resolution at 1.42 GHz of H I data only reaches
1${}^{\prime}\times$ 1.14′ for IC 63 (Blouin et al., 1997). For comparison,
the SOFIA/upGREAT [C II]maps obtained by Caputo et al. (2023, submitted) have
a beam size of $\sim$14″ and the GBT/ARGUS CO(J=1-0) observations (in a
forthcoming paper) have a beam size of 8″. Therefore high sensitivity and high
resolution observations of the region require more extensive arrays.

##  2 Observations and Data Reduction

Because of the small angular size of the IC 63 PDR, very high spatial
resolution is needed to analyze the nebula. We therefore combine radio
interferometry from the Giant Metrewave Radio Telescope (GMRT) in India, the
Westerbork Synthesis Radio Telescope (WSRT) in the Netherlands and the
Dominion Radio Astrophysical Observatory (DRAO) Synthesis Telescope (ST) in
British Columbia, Canada. To ensure full u-v plane coverage, especially at
short spacings, single dish observations using the DRAO 26m antenna were also
added.

###  2.1 GMRT Observations

The GMRT H i observations were taken in January 2017. These used the GMRT
Software Backend (Roy et al., 2010) with 512 channels across a bandwidth of
2.083 MHz, giving a channel separation of 0.86 km s-1. The data were reduced
and imaged using AIPS (Greisen, 2003) with a restoring beam of 8.5″.

###  2.2 WSRT Observations

The WSRT H i observations were taken over $4\times 12$ hours with the legacy
WSRT system in the 36m+54m+72m+90m configurations in December 2008 using 1024
channels over a bandwith of 2.5 MHz, giving a channel separation of 0.52 km
s-1 and an angular resolution of 15.2″$\times$13.4″. They were reduced and
imaged using standard procedures in MIRIAD using the CLEAN algorithm (Sault et
al., 1995). To check whether the clumpiness of the H i in the resulting image
was a CLEAN artifact, the data was also imaged using the MAXEN algorithm to
carry out a maximum entropy deconvolution, resulting in a virtually identical
image.

###  2.3 DRAO Observations

The DRAO H I data utilized here were originally acquired as part of the
Canadian Galactic Plane Survey (CGPS Taylor et al., 2003, 2017). Those
observations were taken in Phase I of the CGPS in 1995–2000 and include single
dish observations taken to fill in the zero-spacing data. The observations
cover 272 channels with a channel separation of 0.82 km s-1 and an angular
resolution of 1′x1.14′.

Figure 1: Coverage of the UV plane for the IC 63 observations by the different observatories. Blue is GMRT, red is WSRT and yellow is CGPS. Table 1: Baselines for the different datasets Dataset | Baselines
---|---
CGPS | 0–26 m + 12.9–604.3 m
WSRT | 36 m–2.7 km
GMRT | 100 m–26 km

###  2.4 H I data Reduction

The three datasets were combined linearly in the Fourier domain using the
immerge task in MIRIAD (Sault et al., 1995), using the cleaned and primary
beam corrected WSRT and GMRT images and a UV range for relative flux
calibration of 150 to 500m, which is present and well sampled in all three
datasets (see Table 1; Fig. 1). Prior to combination, the WSRT and DRAO images
were regridded onto the spatial and spectral grid of the GMRT data. This gives
a final combined dataset with the 8.5″angular resolution and the 0.86 km s-1
spectral resolution of the GMRT data. In order to improve the S/N of the data
and work with a resolution similar to the [C II] data, we decided to proceed
with a data cube smoothed to an angular resolution of 15″.

###  2.5 SOFIA observations

The [C II] fine-structure line at 158 $\mu$m was observed with the upGREAT
receiver (Heyminck et al., 2012; Risacher et al., 2016, 2018) onboard the
Stratospheric Observatory for Infrared Astronomy (SOFIA) (Young et al., 2012).
The data is part of SOFIA project 05_0052 (PI: B-G Andersson). The IC 63
nebula was mapped in the total power on-the-fly mode (TP OTF), reaching an
angular resolution of 15′′. The observations were calibrated with the GREAT
calibration software (Guan et al., 2012) and the main beam efficiencies for
the 7 different pixels of upGREAT vary between 0.59 to 0.68. To create the
final data cubes, a second order baseline was fitted to the data with
CLASS111https://www.iram.fr/IRAMFR/GILDAS/doc/html/class-html/class.html and
the data was spectrally smoothed to a resolution of 0.4 km s-1. Caputo et al.
(2023, submitted) will present a more detailed description of the [C II]
observations and data reduction.

Figure 2: Left: H I integrated intensity map (without background subtraction)
of the IC 63 region. The yellow and magenta boxes indicate the regions used to
make an average spectrum for background subtraction. The black box indicates
the region associated with IC 63. Right: The spectra from the corresponding
colored boxes on the left after background subtraction. The emission from IC
63 (black) has a clearly defined profile while the background regions (yellow
& magenta) are relatively close to the baseline.

##  3 Results

###  3.1 Galactic background removal

Since IC 63 is located relatively close to the Galactic plane (b $\approx$
-2o), the observed H I spectra are heavily confused/contaminated by background
emission (Taylor et al., 2003; McClure-Griffiths et al., 2005). In order to
obtain the actual emission from IC 63, the galactic background has to be
removed. To do so, we take a relatively simple approach where we define an
average H I background spectrum from regions of the data cube that do not
contain emission associated with IC 63. We chose this approach because IC 63
is already clearly visible in the available data cube and the method gives
excellent control on what is removed to isolate the emission associated with
IC 63.
Two rectangular regions were used to define the average background spectrum
for IC 63. These two regions are indicated by the yellow and magenta boxes
presented in Fig. 2. They are the only two regions that do not contain
emission from IC 63 (which is located in the black box in Fig. 2). For the
rest of the map, i.e. the north-east of the map, it is not certain whether
there is a contribution from IC 63 or IC 59 in the north-west corner of the
map. As a result, these other regions are not taken into account for the
background subtraction. Subtracting the average background spectrum in each
spatial pixel of the full data cube then creates the data cube that contains
the emission associated with IC 63. The resulting H I integrated intensity map
is shown in Fig. 3 and the resulting average spectrum for IC 63 is presented
in the right panel of Fig. 2. This shows a Gaussian line profile. From this
point forward, we will work with the background subtracted data unless
mentioned otherwise.

###  3.2 The integrated intensity map

The integrated intensity map of the H I emission towards IC 63 is presented in
Fig. 3. Overall, it shows the same morphology for IC 63 that is also seen in
e.g. the WISE maps (e.g. Soam et al., 2017). When looking in more detail, it
shows a bright clump and a ring-like morphology at the south-west head of the
region with further H I emission in elongated structures pointing towards the
north, north-east and east of the map (which we here also call $legs$). The H
I emission in these legs show a quite pronounced clumpy morphology as well.
The [C II] map is more limited in size and only covers the head of IC 63.
Inspecting the [C II] emission in Fig. 3 shows that the brightest emission is
located towards the head of the nebula with lower brightness emission
extending further along the start of the legs in the region. It is interesting
to note that there are regions in Fig. 3 with [C II] emission that do not have
a counterpart in H I emission.
We do not expect that the clumpiness of the H I emission is the result of
filtering during the data reduction. Reducing the observations using the CLEAN
technique, which tends to create clumpy structures, and the MEM technique,
which produces smoother structures, we find that the clumpy structure of the H
I emission is maintained in both. This is shown in App. A. Inspecting the zoom
into the tip of IC 63 in Fig. 3, we also find that the H I emission shows a
remarkable ring-like feature with an intensity drop in the middle. This
behaviour is not seen in [C II], see Fig. 3, which rather shows bright
emission towards the front of the PDR followed by a gradually decreasing
intensity towards the back.
Given the relatively limited signal-to-noise ratio (S/N) of the H I
observations, it could be suggested that the clumpiness of the H I data is
simply due to the noise. To assess this, we determined the noise in the lower
region of the integrated map without any known emission (i.e. the yellow and
magenta boxes in Fig. 2). This indicates a noise rms of 3.3$\times$10-13 erg
s-1 cm-2 sr-1. However, inspecting the map we identify several
clumps/locations, including the ring, that have a contrast above 10-12 erg s-1
cm-2 sr-1 (i.e. higher than 3$\sigma$) compared to their surroundings. Even
though the limited S/N in the integrated map might have some effect, we find
that several clumps should not be a S/N effect and therefore we are relatively
confident that the H I emission really is rather clumpy. More sensitive
observations in the future should allow to explore this clumpiness with
greater confidence.

Figure 3: Left: The H I intensity map of IC 63 that is obtained from the
background-subtracted data cube. The white box outlines the zoomed-in region
presented on the right. The red lines indicate the three legs visible in H I
that are connected to the tip of IC 63 with their names. Right: A zoom into
the brightest region of H I intensity map with the [C II] intensity contours
overlaid (starting at 10-4 erg s-1 cm-2 sr-1 with increments of 5$\times$10-5
erg s-1 cm-2 sr-1). The red circle highlights the circular morphology seen in
H I towards the tip of IC 63. The white arrow indicates the direction of the
incident radiation from $\gamma$ Cas. The full white circle indicates the 15′′
beam size of the H I data.

###  3.3 The kinematics of IC 63

The channel maps in Fig. 4 provide a view of the H I kinematics in IC 63. At
the most blueshifted velocities ($\lesssim$ 0 km s-1), the most eastern part
of IC 63 is prominent. At slightly more redshifted velocities (0.8-1.7 km
s-1), the northern leg becomes bright with the north-eastern leg between 0.8
and and 3.4 km s-1. This clearly demonstrates that the different legs of IC 63
are velocity-coherent gas structures in the region.
For a more detailed look into the kinematics of IC 63, we also made several
position-velocity (PV) diagrams that cut through the IC 63 region. These PV
diagrams go along and perpendicular to north-eastern and northern legs, and
across the ring and brightest H I clump. The resulting PV diagrams, overlaid
with [C II] contours are presented in Fig. 5. Focusing first on the H I
emission alone, it is observed that there is a velocity gradient from more
blueshifted to more redshifted along the northern leg (PV 3) and towards the
north-east from the tip (PVs 5 + 1). In the cut perpendicular to the legs (PV
2), a hint of the velocity gradient west to east is seen in H I. It is however
tentative due to the limited spectral resolution and relatively high noise in
the H I data. Lastly, the cut through the ring at the tip of the nebula shows
a velocity gradient from south (1-2 km s-1) to the north (0 km s-1) (see PV 6)
while the clump to the east does not show any clear gradient (see PV 4).

When comparing the H I and [C II] kinematics -the [C II] kinematics are
presented and studied in more detail in Caputo et al. (2023, submitted)- both
differences and similarities are observed. The kinematics of the legs in H I
and [C II] appear to show the same behaviour, see Fig. 5, even though some
regions that are detected in [C II] are not detected in H I which is the
result of the clumpy H I structure. The velocity gradient from west to east
that was tentatively observed in the H I channel maps is not really clear in
[C II] (see PV 2 of Fig. 5). However, this could be the case because this
velocity gradient was tentative from the start. Most remarkable is probably
the difference of the centroid velocity for H I and [C II] towards the head of
IC 63. Fig. 5 shows that the [C II] velocity field remains constant towards
the ring whereas a gradient is observed in the H I emission. This velocity
difference of both lines is confirmed when fitting the [C II] and H I data
with a single Gaussian velocity profile, which finds velocity difference of
the order of 1-2 km s-1, see Fig. 6. From this figure it is observed that [C
II] is predominantly blueshifted with respect to H I in most parts of the
ring. On the other hand, [C II] is slightly redshifted with respect to H I in
the bright clump to the east of the ring (in particular at the edge of the
clump) and in the eastern region of the ring itself. Note that there is some
uncertainty on the fitted central velocities because of the low S/N of the H I
data. However, the velocity typically has an uncertainty of 0.1 km s-1 and
does not go over 0.2 km s-1. The velocity shifts thus appear to be
statistically significant.

Figure 4: Channel maps between -4.3 and 7.7 km s-1 of the IC 63 region in
steps of $\sim$ 0.8 km s-1.

Figure 5: Left: The H I moment zero map for IC 63 with the colored arrows
indicating the cuts that were made to construct position-velocity (PV)
diagrams on the right. Right: The H I PV diagrams along the axes/arrows
indicated in the figure on the left. The contours indicate the corresponding
[C II] PV diagrams starting at 2 K with increments of 2 K. The darkest
contours indicate the weakest emission and the white contour indicates the
brightest emission. Figure 6: The difference in central velocity of the
emission in [C II] (Caputo et al. 2023, submitted) and H I at the locations
where both lines are detected. The black contours indicate the integrated [C
II] emission starting at 10-4 erg s-1 cm-2 sr-1 with increments of
5$\times$10-5 erg s-1 cm-2 sr-1.

##  4 Discussion

###  4.1 The dynamic evolution and timescales of IC 63

The three major $legs$, defined in Fig. 3, of IC 63 are each seen at a
different velocity, confirming they are coherently moving substructures of IC
63. From the H I channel maps it was observed that the most eastern leg is the
most blueshifted. This leg is proposed to be the closest to us (Andersson et
al., 2013), which would imply that the different legs are currently being
dispersed. Based on the 0.5 pc physical size of the H I legs at a distance of
$\sim$ 170 pc (van Leeuwen, 2007), the velocity difference of $\sim$1-2 km s-1
implies that IC 63 might be fully dissolved in $\leq$ 0.5 Myr. Interestingly,
this is similar, and even slightly shorter, than the dispersal timescales that
are found with [C II] for denser regions around high-mass star formation
(Tiwari et al., 2021; Bonne et al., 2022). As IC 63 is likely at a more
evolved stage where most of the dense gas has already been dispersed, this
indicates that the more diffuse gas around star forming regions can be
dispersed on relatively similar timescales. Another interpretation for this
velocity field could be a relatively rapid rotational motion of the legs if IC
63 is spinning, although it is unclear what would drive the spinning of this
region. It is also observed that the legs to the north of the map are found to
be generally more redshifted in Fig. 4. This velocity gradient could be
associated with the dispersal of the region, but it is difficult to reach a
firm conclusion.
Based on Fig. 6 we noted that H I and [C II] show up at slightly different
velocities, in particular towards the bright clump and ring at the head of IC
63. As H I is expected to trace the outer regions of the PDR compared to [C
II], because H2 can form while carbon is still ionized (Tielens & Hollenbach,
1985; Sternberg & Dalgarno, 1995), this suggests that there is a velocity
gradient over the PDR. The bright PDR in IC 63 is thus a dynamic structure
with local velocity differences up to 1 km s-1. As these clumps have a size of
only 0.05 pc, the structure of these bright PDRs would thus be changing on a
relatively short timescale of only $\sim$ 5$\times$104 yr. The potential
effect of these dynamics on the chemical structure of the PDR will be
discussed in more detail in the next section.

###  4.2 The ring and clumpy PDR structure of H I in IC 63

In order to assess the emission profile of the H I ring-like structure at the
tip of the nebula, we created 8 radial cuts through the ring. The resulting H
I profiles as a function of radius are presented in App. B. This confirms an
axial symmetry of the emission with the peak emission in the 8 cuts only
varying within 30% among the different profiles, which is expected based on
the noise rms. The dip towards the center of the ring is however larger than
3$\times$ the noise rms. In App. B it is also verified that the observed ring-
like morphology is retrieved from the data of the different interferometers.
With this evidence of a ring-like structure, we folded these 8 H I profiles
(i.e. average the profile at negative offset with the profile at positive
offset) and then averaged all 8 profiles into a single H I emission profile
for the ring. This average emission profile, fitted with a Gaussian
distribution, is presented in Fig. 8. The central intensity is
1.4$\times$10-13 erg s-1 cm-2 sr-1 and the peak intensity of the ring is
9.3$\times$10-13 erg s-1 cm-2 sr-1, resulting in a peak to center ratio of
$\sim$7\. The question arises whether this ring-like structure is the result
of self-absorption or rather a result of the hydrogen chemistry or dynamics in
the PDR. Investigating the individual spectra for self-absorption
unfortunately does not provide conclusive insight because of the limited S/N
for the currently available observations. We can however constrain this
possibility, as well as chemical effects, with a toy model. This model assumes
a spherical geometry for the optically thin atomic hydrogen that is organized
in a shell with a constant spin temperature, see Fig. 7. In the interior of
the shell we consider either the lack of H I emission due to chemistry, e.g.
the formation of H2 which can be supported by the H2 fluorescence observations
in Andersson et al. (2013), or the presence of cold atomic H I self-absorbing
(HISA) gas. With this toy model we can use the observed size and thickness of
the ring, and compare it to the ratio of brightness temperature at the center
and rim. We can also evaluate whether the required H I absorbing layer is
consistent with the AV (i.e. extinction in the V band) of IC 63.

If we assume that the shell is basically devoid of H I, due to the formation
of H2, then the ratio at the center and the rim will be proportional to the
path lengths through the shell, see Figure 7. If we designate the inner and
outer radii of the shell ri and ro, the radial path length at the center of
the shell is simply C=2$\cdot$(ro \- ri) and the distance from the center to
the middle of the rim is $\frac{1}{2}\cdot(r_{o}+r_{i})$. The path length
through the middle of the shell is then
Scent=2$\cdot\sqrt{r_{o}^{2}-0.25\times(r_{o}+r_{i})^{2}}$. For further use in
equations we here define the size ($\Delta$) and thickness ($\delta$) with the
following relations:

$\delta=r_{o}-r_{i};\Delta=\frac{1}{2}\cdot(r_{o}+r_{i})$ (1)

The largest pencil-beam column density through the shell occurs for at r=ri
with a resulting path length ratio

$\frac{S_{long}}{C}=\frac{\Delta+\frac{\delta}{2}}{\delta}\sqrt{1-\left(\frac{\Delta-\frac{\delta}{2}}{\Delta+\frac{\delta}{2}}\right)^{2}}$
(2)

However, we are comparing to observations with a finite-sized beam, and the
predicted shell intensity profile is not symmetric at the peak (see Fig. 14 in
Kabanovic et al., 2022). This can thus affect the observed S/C ratio and
slightly shift the radius of the observed peak emission in a non-trivial way.
Therefore, we also use the center path length through the shell which gives

$\frac{S_{cent}}{C}=\frac{\sqrt{(\Delta+\frac{\delta}{2})^{2}-\Delta^{2}}}{\delta}$
(3)

This provides an expected range of observed S/C ratios that accounts for the
fact that we are observing at finite resolution.

Figure 8 shows that $\Delta\approx$25.2″and
$\sigma_{\delta}\approx$11.4″($\sigma_{\delta}$ is the observed width of the
shell). Deconvolving the shell-width with the 6.4″ beam width (=
FWHMbeam/2$\sqrt{2\text{ln}2}$ with FWHMbeam = 15″), $\sigma_{\delta}$ becomes
9.4″. The width of the shell is probably better approximated by the FHWM,
rather than the Gaussian width $\sigma$ which yields $\delta\approx$22.1″.
Inserting these values into Eq. 2 and 3, we find path length ratios S/C in the
range of 1.2 to 1.5. However, the observed brightness temperature ratio for
the shell and center of the ring is TS/TC=6.9$\pm$1.5, which is significantly
larger than the geometric ratio. It might thus be that the H I shell is not
spherical, see e.g. Fig. 7, or that there is HISA at the center of the ring.

Figure 7: Left: The geometry of a simple model to estimate whether the
observed H I ring is the projection of a spherical ring (see Eqs. 2 & 3). In
this specific schematic presentation, S corresponds to the path length through
the center of the shell (Scent). If we assume that all of the H I emission is
optically thin, we can compare the geometric ratio of ring size and rim/shell
thickness with the ratio of the related brightness temperatures. The observed
ring radius is $\Delta$=25.2″ and the shell thickness (FWHM) yields
$\delta$=22.1″. Right: The observed intensity ratio could be recovered from
this model if we assume that the shell is anisotropic, and/or that the center
of the region is filled with cold, self-absorbing (HISA), gas.

To calculate the potential cold H I column density, and compare it to the
visual extinction in the cloud, we calculate the HISA optical depth

$\tau_{HISA}(v)=-{\rm ln}\left(1-\frac{T_{on-
off}(v)}{T_{HISA}-pT_{off}(v)-T_{cont}}\right)$ (4)

where Ton-off is the difference between the HISA (Ton) and background (Toff)
brightness, THISA is the HISA temperature, $p$ a dimensionless parameter in
the range [0-1] which accounts for foreground emission, and Tcont the
continuum brightness (which we ignore here). We here use $p$ = 1 which assumes
no foreground emission for Toff. From this it is then possible to calculate
the HISA column density using (Wilson et al., 2009)

$N_{HISA}\,[cm^{-2}]=1.8224\times
10^{18}\,\,T_{HISA}\,[K]\int\tau_{HISA}(v)dv\,[km\,s^{-1}]$ (5)

Here, we are however hampered by the noisy spectra. Therefore, we assume a
constant $\tau_{HISA}$ with Ton-off = -137 K and Toff = 159 K. Ton-off and
Toff were calculated based on the central and peak intensity of the ring
mentioned above with an assumed FWHM of 2 km s-1. Assuming THISA = 20 K this
gives rise to a required NHISA values of 3$\times$1020 cm-2. For THISA = 10 K
this drops to 9$\times$1019 cm-2, but at THISA = 30 K no solution can be
found. For HISA temperatures of only 20 K or less it is thus possible to find
a solution. In addition, the column density associated with HISA is well below
the total column density at the tip of IC 63 (AV $\approx$ 2 (Van De Putte et
al., 2020), accounting for fore- and background extinction). With an
A${}_{V}\approx$ 2, or N${}_{H}\approx$ 4.4$\times$1021 cm-3 (using Güver &
Özel (2009)), about 2-7% of the gas in the line-of-sight would be associated
to HISA. However, it has to be noted that these are significant column
densities for such cold HISA (Wang et al., 2020; Kabanovic et al., 2022;
Seifried et al., 2022). In addition, for a path length of 0.05 pc the typical
density would be of the order of 0.6-3$\times$103 cm-3 which is relatively
high. It is thus possible to explain the ring morphology in IC 63 with H I
self-absorption, but the HISA conditions would have to be quite extreme
compared to typical values in the Galactic ISM. More sensitive observations
will be able to properly address this question.

Another option to explain the ring as well as the clumpy structure of H I in
the PDR would be a chemical effect that removes the presence of a layered PDR.
Work by Störzer & Hollenbach (1998) proposed a criterion $\chi$/n $<$ 0.2vIF
222With $\chi$ the FUV field strength (Draine, 1978), n the density and vIF
the ionization front (IF) velocity. where a PDR is dominated by non-
equilibrium chemistry that removes the presence of a H I/H2 transition front.
The recent semi-analytical solution by Maillard et al. (2021) proposes a more
involved criterion for the transition to non-equilibrium PDR chemistry and no
clearly defined H I/H2 transition front (see their Eq. 16). To examine the
potential role of non-equilibrium chemistry in the IC 63 PDR, we compute the
typical ionization front velocity (vIF) for the head of this region using

$\text{v}_{IF}=\frac{\text{n}_{e}}{\text{n}_{PDR}}\text{v}_{II}$ (6)

with ne the electron density on the H II side of the ionization front, a
velocity v${}_{II}\approx$ c${}_{II}\approx$ 10 km s-1 (i.e. the sound speed
of the ionized gas; Störzer & Hollenbach, 1998) and nPDR the density of the
PDR. To calculate the electron density (ne), we use the 1.4 GHz NRAO VLA Sky
Survey (NVSS) radio continuum data (Condon et al., 1998). At the tip of IC 63,
we typically find values of 0.05 Jy beam-1 with a beam size of 45′′. Since the
radio continuum spectral index analysis from Blouin et al. (1997) indicates
that the emission at 1.4 GHz is optically thin in IC 63, we can use

${\scriptstyle\left(\frac{\text{EM}}{\text{pccm}^{-6}}\right)=3.217\times
10^{7}\left(\frac{F_{\nu}}{Jy}\right)\left(\frac{\nu}{GHz}\right)^{0.1}\left(\frac{T_{e}}{K}\right)^{0.35}\left(\frac{\theta_{\text{source}}}{\text{arcsec}}\right)^{-2}}$
(7)

taken from Schmiedeke et al. (2016) to calculate the emission measure (EM).
Here, Fν (= 0.05 Jy) is the flux density, Te (= 8000 K) the electron
temperature, $\nu$ (= 1.4 GHz) the observed frequency and
$\theta_{\text{source}}$ (= 45′′) the studied aperture. This results in EM =
1.9$\times$103 pc cm-6. Using a size of 0.05 pc results in ne = 2.0$\times$102
cm-3. To calculate the density in the PDR, Van De Putte et al. (2020) found an
AV $\approx$ 2 for IC 63. Combining this with a typical size of 0.05 pc for
the clumps gives a proton density nH = 2.8$\times$104 cm-3 using the relation
from Güver & Özel (2009) that links AV to column density. Combining this gives
vIF = 7.0$\times$10-2 km s-1. To determine whether the PDR is governed by non-
equilibrium processes we have to estimate the FUV field strength. For the FUV
field strength, the original literature proposes a value of $\chi\approx$
6$\times$102 (Jansen et al., 1994) at a distance of 1.3 pc from the ionizing
star $\gamma$ Cas. We verified this value, using a distance of 1.3 to 2 pc
from $\gamma$ Cas (Caputo et al. 2023, submitted) with the method presented in
Bonne et al. (2020). This directly calculates the FUV field strength at a
specific distance from an OB star (or cluster) using the emitted radiation as
a function of wavelength from the OB stars in the Kurucz (1979) models. This
results in G0 = 1.1-2.6$\times$102 (with G0 = 1.7$\chi$, this leads to $\chi$
= 0.65-1.5$\times$102) which is significantly lower than the prediction by
Jansen et al. (1994). However, the value calculated here is consistent with
more recent work by Andrews et al. (2018) who proposed an FUV field strength
$\chi$ $\approx$ 90 and Soam et al. (2021b) who found $\chi$ =
0.6-2.4$\times$102. To investigate this in more detail, we use the [C II]
intensity presented in Caputo et al. (2023, submitted), which typically is
2.5-3.0$\times$10-4 erg s-1 cm-2 sr-1, and the PDR Toolbox (Pound & Wolfire,
2008, 2023). At densities of 1-5$\times$104 cm-3 this [C II] intensity
corresponds to a typical FUV field of $\chi$ = 0.6-2.0$\times$102. This is
consistent with the estimate based on the method in Bonne et al. (2020). Using
the calculated vIF, the prescription by Störzer & Hollenbach (1998) provides a
critical ($\chi$/n)crit = 1.4$\times$10-2 and the prescription by Maillard et
al. (2021) gives ($\chi$/n)crit = 2.4$\times$10-3. Using the determined $\chi$
= 0.6-2.0$\times$102 range from the [C II] observations gives $\chi$/n =
2.1-7.1$\times$10-3 for IC 63. The results based on both criteria provide
different conclusions. The Maillard et al. (2021) criterion suggests there
likely still is a H I/H2 transition front whereas the Störzer & Hollenbach
(1998) criterion suggests IC 63 no longer has a H I/H2 transition front which
might provide an explanation for the more complex or clumpy H I structure.
However, it has to be noted that the Maillard et al. (2021) solution should be
the more accurate one as it includes the dynamics induced by the photo-
evaporation and dust shielding (Sternberg et al., 2014).
A last thing to consider is the dynamics in the PDR itself which might be
significant when considering the apparent 1 km s-1 offset in velocity between
H I and [C II], in particular towards the tip of the region, in Fig. 6. In the
previous section we obtained a dynamical timescale of 5$\times$104 yr for the
PDR which we can compare with the H2 dissociation timescale. Based on Draine &
Bertoldi (1996) and Störzer & Hollenbach (1998), this is given by

$\text{t}_{H_{2}}\approx 0.6\frac{\text{N(H}_{2}\text{)}^{3/4}}{\chi}\text{s}$
(8)

Using N(H2) = $0.5\times 10^{21}$ cm-2 (Störzer & Hollenbach, 1998) and $\chi$
= 0.6-2.0$\times$102 results in a dissociation timescale of 0.3-1.0 Myr. It
thus appears that the dynamical timescale is significantly shorter than the H2
dissociation timescale. As a result the dynamics in the PDR can reorganize the
internal structure and so remove a plane-parallel H I/H2 structure which could
explain the observed clumpy H I structure.

Figure 8: The average H I emission profile of the ring as a function of the
offset from the center of the ring with a Gaussian profile fitted to it (red
dashed line). The error bars indicate the standard deviation of the emission
over the ring at each offset.

##  5 Conclusions

We have presented combined interferometric H I observations from the GMRT,
WSRT and DRAO of IC 63 with an effective angular resolution of 15′′ which
allows to resolve the structure of the PDR in IC 63. These observations are
further complemented with SOFIA [C II] data. To properly analyze the H I
observations at the low Galactic latitude of IC 63 (b=-2), we subtract the
galactic background from the spectrum using nearby regions devoid of H I
emission associated with IC 63. Although the S/N is somewhat limited we
conclude that the H I structure of IC 63 appears to be clumpy, and that the
tip of IC 63 appears to have a ring-like morphology in H I. With analytical
models, we show that this ring-like structure is likely not the projection of
an unfilled spherical shell. However, more sensitive observations with new and
future observatories can help to confirm these results and explore them in
further detail. We also obtain a view on the dynamics. This shows dynamic
structure with $\Delta$v $\approx$ 1 km s-1 inside the tip of the PDR. In
addition, the full region consists of several legs that appear to be moving
apart. This suggests that IC 63 will completely disperse within less than 0.5
Myr.
Analyzing the H I spectra and PDR characteristics, we conclude that the
dynamical timescale in the PDR is too short for there to be a plane-parallel H
I/H2 structure which could explain the clumpy H I structure. The S/N is not
sufficient to conclude on the contribution of HISA to the spectra, but if the
HISA temperature is below 20 K then HISA can contribute to the apparent clumpy
H I structure. We also investigated whether the ionization front velocity
leading to non-equilibrium chemistry could explain the clumpy structure.
Different prescriptions for the transition to non-equilibrium chemistry due to
the ionization front velocity lead to different conclusions, but the most
recent and accurate prescription by Maillard et al. (2021) concludes that non-
equilibrium chemistry should not play a major role. We thus tentatively
propose that the apparent clumpy H I PDR structure is predominantly explained
by the short dynamical timescale in the PDR, but it cannot be excluded that
different mechanisms contribute to this observed clumpy H I structure.

We thank the anonymous referee for detailed comments that helped to improve
the clarity of the paper significantly. We thank G. Barentsen for fruitful
exchange on the use of the IPHAS dataset. We acknowledge the professional and
patient support of the WSRT staff, in particular Gyula Jozsa and Adriaan
Renting. We thank the staff of the GMRT that made these observations possible.
GMRT is run by the National Centre for Radio Astrophysics of the Tata
Institute of Fundamental Research. This research used the facilities of the
Canadian Astronomy Data Centre operated by the National Research Council of
Canada with the support of the Canadian Space Agency. L.B. was supported by a
USRA postdoctoral fellowship, funded through the NASA SOFIA contract
NNA17BF53C. B-G A., A.S. and J.K. gratefully acknowledge the support from the
National Science Foundation (NSF) under grant AST-1715876 to USRA. K.K. and
J.Y., gratefully acknowledge the support from NSF under grant AST-1715060 to
SCU. J.Y. was supported by a Fox Fellowship from Santa Clara University. R.M.
acknowledges support from the National Radio Astronomy Observatory (NRAO). The
National Radio Astronomy Observatory is a facility of the National Science
Foundation operated under cooperative agreement by Associated Universities,
Inc. Based, in part, on observations made with the NASA/DLR Stratospheric
Observatory for Infrared Astronomy (SOFIA). SOFIA is jointly operated by the
Universities Space Research Association, Inc. (USRA), under NASA contract
NAS2-97001, and the Deutsches SOFIA Institut (DSI) under DLR contract 50 OK
0901 to the University of Stuttgart.

##  Appendix A CLEAN vs. MEM reduction

Figure 9 displays the moment zero map for the H I emission between -2.5 and
4.5 km s-1, which is not background subtracted, when reduced using the MEM
technique (left) and the CLEAN technique (right). Both methods create a very
similar map which indicates that the clumpy morphology for the region is not
an artefact.

Figure 9: Left: The moment zero map for IC 63 between -2.5 and 4.5 km s-1
obtained using the MEM reduction technique. Right: The same using the CLEAN
reduction technique, showing a similar clumpy morphology.

##  Appendix B The individual H I profiles and spectrum over the ring

To obtain a better view of the ring-like feature, we created 8 cuts through
the ring with in steps of 20o, which is shown in Fig. 10. The resulting
profiles along the cut are presented in Fig. 10. The center was defined by
making sure that the offset from the center to the maximum intensity is equal
in the positive and negative direction along the vertical and horizontal axes.
Within a typical offset of $\pm$ 40′′, the profiles show similar behaviour
with peak values varying between 0.7$\times$10-12 and 1.4$\times$10-12 erg s-1
cm-2 sr-1 while the minimal value towards the center of the ring is relatively
close to 1.0$\times$10-13 erg s-1 cm-2 sr-1.
To further verify that the ring-like feature at the tip is not an image
artefact, the moment 0 map was created for the WSRT and GMRT maps individually
which both show a dip towards the same location that is seen in the full
combined map, see Fig. 11. This gives further confidence that this drop in
emission towards the center of the tip is a real feature. Additionally, Fig.
11 also indicates that the weaker emission in the eastern part of the ring
might be attributed to an artefact in the GMRT data cube.
Lastly, we also present the average spectrum with the best possible S/N
extracted from inside the ring, see Fig. 12. This spectrum shows a non-
Gaussian skewed line profile, but because of the relatively high noise rms in
the data it is not possible to assess whether this non-Gaussian profile is due
to self-absorption or not.

Figure 10: Left: The different paths that were defined over the H I ring-like
feature on top of the H I integrated intensity map. Right: The H I emission
profiles over the ring for the 8 individual paths defined on the left.

Figure 11: Left: The ring seen in the moment 0 map from the combined cube with
contours from 0.1 to 0.225 Jy km s-1 in steps of 0.025 Jy km s-1. Center: The
same field in the moment 0 map from the WSRT data with the contours from the
combined cube overlaid. Right: The same field in the moment 0 map from the
GMRT data with the contours from the combined cube overlaid. These plots show
that the ring is seen in both interferometric datasets and so is not an
artefact from a single telescope that has propagated into our combined cube.
They also reveal that there appears to be some substructure in the ring.

Figure 12: The average H I spectrum obtained from inside the ring at the tip
of IC 63. As expected the noise is fairly high, making it challenging to
conclude on the presence of self-absorption.

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[1,10]Mathias Kläui

1]Institute of Physics, Johannes Gutenberg-University Mainz, Staudingerweg 7,
Mainz, 55128, Germany

2]Peter Grünberg Institut and Institute for Advanced Simulation,
Forschungszentrum Jülich and JARA, Jülich, 52425, Germany

3]Department of Physics, Yeshiva University, , New York, NY, USA

4]Department of Applied Physics, Universidad de Salamanca, Plaza de la Merced,
Salamanca, 37008, Spain

5]Physical Science and Engineering Division, King Abdullah University of
Science and Technology, Thuwal, 23955-6900, Saudi Arabia

6]Institute of Materials Research and Engineering (IMRE), Agency for Science,
Technology and Research (A*STAR), Singapore, 138634, Singapore

7]Department of Materials Physics, Nagoya University, Nagoya, 464-8603, Japan

8]Beijing Institute of Technology, Beijing, 100081, China

9]Aix-Marseille Université, CNRS, CINaM, Marseille, France

10]Centre for Quantum Spintronics, Norwegian University of Science and
Technology, Trondheim, 7491, Norway

# Revealing the higher-order spin nature of the Hall effect in non-collinear
antiferromagnet $\mathrm{Mn_{3}Ni_{0.35}Cu_{0.65}N}$

Adithya Rajan  Tom G. Saunderson  Fabian R. Lux  Rocío Yanes Díaz  Hasan M.
Abdullah  Arnab Bose  Beatrice Bednarz  Jun-Young Kim  Dongwook Go  Tetsuya
Hajiri  Gokaran Shukla  Olena Gomonay  Yugui Yao  Wanxiang Feng  Hidefumi
Asano  Udo Schwingenschlögl  Luis López-Díaz  Jairo Sinova  Yuriy Mokrousov
Aurélien Manchon<EMAIL_ADDRESS>[  [  [  [  [  [  [  [  [  [

###### Abstract

Ferromagnets generate an anomalous Hall effect even without the presence of a
magnetic field, something that conventional antiferromagnets cannot replicate
but noncollinear antiferromagnets can. The anomalous Hall effect governed by
the resistivity tensor plays a crucial role in determining the presence of
time reversal symmetry and the topology present in the system. In this work we
reveal the complex origin of the anomalous Hall effect arising in noncollinear
antiferromagnets by performing Hall measurements with fields applied in
selected directions in space with respect to the crystalline axes. Our
coplanar magnetic field geometry goes beyond the conventional perpendicular
field geometry used for ferromagnets and allows us to suppress any magnetic
dipole contribution. It allows us to map the in-plane anomalous Hall
contribution and we demonstrate a 120° symmetry which we find to be governed
by the octupole moment at high fields. At low fields we subsequently discover
a surprising topological Hall-like signature and, from a combination of
theoretical techniques, we show that the spins can be recast into dipole,
emergent octupole and noncoplanar effective magnetic moments. These co-
existing orders enable magnetization dynamics unachievable in either
ferromagnetic or conventional collinear antiferromagnetic materials.

###### keywords:

Anomalous Hall effect, Noncollinear Antiferromagnets, Spintronics, Berry
curvature

##  1 Introduction

Broken time reversal symmetry ($\mathcal{T}$) and its interplay with the spin-
orbit coupling (SOC) result in the transverse flow of electrons generating an
anomalous Hall voltage, a signature that distinguishes a ferromagnet (FM) from
conventional antiferromagnets (AFM) [1, 2]. However, it has been predicted
that a certain class of AFMs with noncollinear spin textures can lead to an
unusual type of anomalous Hall effect (AHE) that can exist even in the absence
of a net magnetization due to spin-lattice coupling [3, 4]. Whilst in
noncollinear antiferromagnets (NC-AFM) it has been shown that their weak
magnetization is insufficient to account for the AHE [5, 6, 7, 8, 9], further
work to experimentally substantiate the fundamental mechanism responsible for
the observed AHE signals has still not been undertaken. This is because any
studies of the AHE have, so far, only been performed with the electrical
measurements in plane while driving a magnetic field out of the plane,
entangling the magnetization signal to any novel contribution to the AHE. In
order to understand the origin of the AHE signal, one must go beyond the
perpendicular field geometry and apply the field in different directions in
space to identify the symmetry, and from this the mechanism leading to the AHE
signal. Here we show the full dependence of the AHE in the NC-AFM
Mn3Ni0.35Cu0.65N ($\mathrm{Mn_{3}NiCuN}$) when the magnetic field is swept not
only out-of-plane but also in the plane. The in-plane field experimental
configuration, by construction, does not allow for the conventional dipole
(magnetization) component of the AHE signal to contribute. We can therefore
show, using a variety of theoretical techniques, that at high fields the in-
plane AHE comes purely from the octupole moment and, surprisingly, an
additional topological Hall-like (THE) signal occurs at low fields. Our
results demonstrate that in NC-AFMs one must expand beyond the dipole
contribution to the AHE to explain their rich, coexisting orders. Such
coexisting orders go beyond the magnetization dynamics achievable in
conventional FMs and AFMs. Harnessing these coexisting orders paves the way
for spintronic devices that go beyond the state-of-the-art.

Figure 1: Cubic unit cell of $\mathrm{Mn_{3}NiCuN}$ with Mn atoms in orange,
Ni and Cu in blue, and N in off-white. The blue arrows in (a) indicate the Mn
spin arrangement in the (111) plane (blue). (b) The eight octupoles which
point perpendicular to the 8 equivalent (111) planes. The out-of-plane
components of the octupole moment are labelled as $\Upsilon_{0}$ and
$\overline{\Upsilon}_{0}$, pointing along the [111] and
[$\bar{1}\bar{1}\bar{1}$] directions respectively. (c) The Mn spin
configurations (blue arrows) and the corresponding octupoles (black bold
arrows) when a magnetic field is applied along different crystallographic
directions in the (111) plane. The particular directions labelled represent
the specific angles where the magnetic field is parallel to the in-plane
projections of the octupoles $\Upsilon_{i}$ and $\overline{\Upsilon}_{i}$. The
central panel of (c) shows the spin configuration when the magnetic field is
swept out the kagome plane ($\overline{\Upsilon}_{0}$) and that corresponds to
$\vec{Q}_{T_{1g}}$ along [111] direction. (d) Graphical representation of the
evolution of $\vec{Q}_{T_{1g}}$ in spherical coordinates as a function of the
applied external magnetic field in the (111) Kagome plane (shown in color
bar). It shows $120^{\circ}$ oscillations of the projection of
$\vec{Q}_{T_{1g}}$ along [111] direction, parallel to the $\Upsilon_{0}$
octupole. When the magnetic field is swept from positive to negative in-plane
field along certain in-plane projections of the octupole, for example
$\Upsilon_{1}$ to $\overline{\Upsilon}_{1}$, there is switching of the out-of-
plane component of $\vec{Q}_{T_{1g}}$ with the maximum strength (shown in
black stars), which will manifest as a step in the Hall resistance (generating
AHE) and follow $120^{\circ}$ angular dependence.

The antisymmetric AHE in the NC-AFM arises from the octupole moment produced
by the noncollinear frustrated spin structure in the kagome plane [3, 10]
which is in the (111) plane in the case of Mn3NiCuN, the material we are
investigating in this work (Fig. 1(a)). This can be visualized as an emergent
octupole moment [11, 12, 10, 13, 14, 15, 16], $\vec{Q}_{T_{1g}}$ (the first
term in equation (1)) that purely originates from the coplanar orientation of
the spin texture lying in the kagome plane (Fig. 1(a)). In
$\mathrm{Mn_{3}NiCuN}$, the direction of $\vec{Q}_{T_{1g}}$ points out of the
kagome plane, along [111] direction. Upon expanding the AHE signal in the
local spins, beyond the conventional ferromagnetic case, the octupole and
topological Hall effects [17, 18, 19, 20] emerge as higher order contributions
to the Hall effect in noncollinear compounds. Hence, the total contributions
to the AHE are,

$\sigma_{xy}=\gamma_{\mathrm{Oct}}~{}\left[\vec{Q}_{T_{1g}}\right]_{z}+\gamma_{\mathrm{dip}}~{}\left[\vec{M}_{T_{1g}}\right]_{z}+\gamma_{\mathrm{SSC}}~{}\left[\sum_{ijk}\vec{S}_{i}\cdot(\vec{S}_{j}\times\vec{S}_{k})\right],$
(1)

where equation (1) is obtained by using representation theory to derive the
irreducible representations of $\mathrm{Mn_{3}NiCuN}$ as described in section
‘2 Representation theory’ of the supplementary material [21]. The second term
of equation (1) represents the conventional AHE, proportional to the net
magnetic dipole moment, $\vec{M}_{T_{1g}}$, as observed in a regular
ferromagnet such as Fe [1], and that can arise in our system due to the
canting of the spins upon the application of an external magnetic field.
Generally, in this type of NC-AFM [22], including our system (see Fig. S2(a)
in ‘1.1 Sample characterization’ [21]), the induced moment is very small [22]
and therefore has a very small effect on the AHE.

Equation (1) predicts that $\vec{Q}_{T_{1g}}$ produces the maximum AHE when
the magnetic field is swept out of the plane while the electrical measurements
are performed in the kagome plane. This arrangement is similar to the
measurement of AHE in a regular ferromagnet (FM) where the direction of the
applied electric current ($J_{xx}$), voltage measurement ($V_{xy}$) and
magnetic moment ($M_{z}$) are orthogonal to each other
($V_{xy}\propto(J_{xx}M_{z})$). Alternatively, in NC-AFMs with a vanishing
magnetic moment, $\vec{Q}_{T_{1g}}$ plays the role of a fictitious
magnetization, akin to $M_{z}$ in ferromagnets. Nonetheless our symmetry
analysis reveals a fundamental difference between the octupole driven AHE and
the conventional dipole driven AHE: the magnetic octupole supports a specific
finite AHE response when the external field is rotated in the plane of the
electrical measurements, whereas the dipole driven AHE remains unaffected.
Thus, in plane fields have to be used to reveal this contribution.

The octupole vector $\vec{Q}_{T_{1g}}$ hosts a total of eight poles in
$\mathrm{Mn_{3}NiCuN}$ as shown with the grey vectors in Fig. 1(b), two of
which point out of the plane, along the [111] direction and six others have
projections both in the plane and out of the plane with 120∘ in-plane
rotational symmetry enforced by the crystal structure (space group
$Pm\overline{3}m$ (No. 221), point group $m\overline{3}m$ ($O_{h}$)). It is
evident that when an external field is applied out-of-plane, along the [111]
axis, one expects to obtain an AHE from the combination of octupole
($\vec{Q}_{T_{1g}}$) and dipole ($\vec{M}_{T_{1g}}$) contributions. However,
our symmetry analysis additionally shows (see ‘2.6 Approximate ground state’
[21]) that when the in-plane applied magnetic field is collinear to the in-
plane projections of any one of the six components of these $\vec{Q}_{T_{1g}}$
vectors, it immediately couples to the octupoles by reorienting the spin-
configuration into one of the equivalent (111) planes as shown in Fig. 1(c).
This coupling provides direct access to the out-of-plane projection of
$\vec{Q}_{T_{1g}}$ which can lead to an AHE as per equation (1) when
electrical measurements are performed in the plane. Due to this, one expects a
120∘ angular dependence of the measured AHE as theoretically calculated in
Fig. 1(d). We point out here that we plot all components of the
$\vec{Q}_{T_{1g}}$ vector in Fig. 1(d) and find that its dependence on the
magnetic field resembles that of the magnetization, $\vec{M}_{T_{1g}}$, which
illustrates its role as a fictitious magnetization; however, the key
difference is that the octupole produces finite contributions to the in-plane
AHE, unlike $\vec{M}_{T_{1g}}$. This defines a distinct difference from the
conventional (dipole-driven) AHE in a FM as the AHE is not allowed to occur
when electrical measurements and applied magnetic field are coplanar.

In addition, our theoretical calculations suggest a third term in equation (1)
that predicts a nontrivial Hall effect originating from the scalar chirality
of the spin textures. Such a quantity was first discussed in high temperature
superconductors [23, 24] when considering contributions to the Hall
conductivity, whose underlying physics can similarly be attributed to the THE
in a skyrmion [25, 17, 18, 19, 20]. This can be observed in the low field
regime when the spins reorient, maximizing the scalar spin chirality.

##  2 Experimental Results

To test these theoretical predictions, $\mathrm{Mn_{3}NiCuN}$ (111) thin film
of 20 nm thickness are grown on an MgO (111) substrate, and then capped with a
3 nm thin Pt layer to prevent oxidation. After the thin film growth, Hall bar
devices were patterned on the (111) kagome plane as shown in Fig. 2(g). The
details of the sample preparation can be found in the method section and in
reference [22]. First, we verify the presence of the $\vec{Q}_{T_{1g}}$
component’s contribution to the AHE signal by measuring the standard Hall
effect in the kagome plane while sweeping the magnetic field out-of-plane
field along the [111] direction (z-axis) at 100 K, well below the Néel
temperature ($T_{N}\sim$ 200 K) [22]. The longitudinal resistivity of the film
is $\sim 150~{}\mu\Omega\mathrm{cm}^{-1}$ at a temperature 100 K. Fig. 2(a)
shows the measured AHE data for the z-field sweep that can be fitted by using
$R_{xy}=A\tanh(B)$. We also verify that this AHE signal disappears above the
Néel temperature (see Fig. S2(b) in ‘1.1 Sample characterization’ [21]). This
result is consistent within the framework of the octupole moment
($\vec{Q}_{T_{1g}}$) induced AHE (equation (1), first term), but additionally
a small magnetic moment is also present in our system (see Fig. S2(a) in ‘1.1
Sample characterization’ [21]), meaning that the magnetic dipole component
($\vec{M}_{T_{1g}}$) also contributes to the AHE. The measured anomalous Hall
resistance ($R_{xy}$) is comparable to the previous reports [26, 9, 27, 8].

Now we perform the measurements of transverse resistance on the (111) plane
while sweeping the magnetic field in the plane along different in-plane
crystallographic directions (Figs. 2(b-c)). We observe two important features:
(1) AHE-like signals, representing a step in the measured resistance, and (2)
topological Hall-like (THE) signals, which are additional features only
appearing in the low field regime which we will discuss later. Our data can be
fitted well to equation (2),

$R_{xy}(B)=R^{\mathrm{Oct}}_{xy}\tanh(a(B-B_{\mathrm{c}}))+R^{\mathrm{SSC}}_{xy}\exp\left(-c(B-B_{0})^{2}\right),$
(2)

where $R^{\mathrm{Oct}}_{xy}$ represents the maximum contribution coming from
the octupole moment, and $R^{\mathrm{SSC}}_{xy}$ the maximum contribution of
the scalar spin chirality, $B$ is the magnitude of the magnetic field, $B_{c}$
is the coercivity, $a$ is related to the slope of $R^{\mathrm{Oct}}_{xy}$
switching, and c is related to the width of the scalar spin chirality signal.

Figure 2:  (a)-(c) The measured Hall resistance as a function of applied
magnetic field out of the kagome plane (a) and within the kagome plane along
two different crystallographic angles, $\phi=-75^{\circ}$ (b) and
$\phi=205^{\circ}$ (c). (d)-(f) Schematics of the direction of the applied B
field corresponding to cases shown in (a)-(c) respectively. The black symbols
are the measured values of $R_{xy}$ in the experiment which is fit to equation
(2) (green curves) using a $\tanh(B)$ function (red curves) and a Gaussian
function (blue curves). Small back arrows indicate the direction of the
hysteresis loop-shift. (g) Schematics of the experiment with the coordinate
system. (h) Red and black points correspond to data quantified values of
$R^{\mathrm{Oct}}_{xy}$ and $R^{\mathrm{SSC}}_{xy}$ fit by sinusoidal curves
with 120∘ periodicity. (i) Theoretically determined scalar spin chirality from
symmetry analysis showing the same periodicity as the octupole contribution
but differing by a sign change.

The observation of AHE-like signal in such coplanar geometry is a nontrivial
finding as it is strictly prohibited in regular FMs. However, it is a
signature that applied magnetic fields can access one of the components of
$\vec{Q}_{T_{1g}}$ depending on the direction of the magnetic field-sweep with
respect to the crystal axis (Figs. 1(c-d)) which is further evident following
the red curve in Fig. 2(h) that exhibits a roughly 120∘ angular dependence,
consistent with our theoretical predictions (Fig. 1). The error-bars in the
data points represent the standard deviation in five different measurements.

In these experiments surprising additional features appear at low field values
which can be fitted by a Gaussian function that changes sign for positive and
negative applied field cycling direction. Features like these are often
attributed to THE as they occur in the presence of skyrmions whose nontrivial
topology can induce an emergent effective magnetic field [25, 18]. The spin
textures in our system exhibit no such topology [25], however it is possible
that in the low field regime the spins rotate out of their coplanar
orientation and maximize the scalar spin chirality
($\vec{S}_{i}\cdot(\vec{S}_{j}\times\vec{S}_{k})$) [17] thereby producing THE-
like signals as predicted by the third term in equation (1). A consequence of
this is that for certain $\phi$ where the THE-like component is larger than
the $\vec{Q}_{T_{1g}}$ component, $R_{xy}$ switches before B changes polarity
as shown in Fig. 2(b) for $\phi=-75^{\circ}$. This would be thermodynamically
prohibited if the AHE has contributions only from $\vec{Q}_{T_{1g}}$ or
$\vec{M}_{T_{1g}}$. To verify the source of the additional contribution, we
compute the components of the scalar spin chirality by parameterizing a free
energy expression of the irreducible representations with first principles
calculations and solve a Landau-Lifshitz-Gilbert equation to provide the
dependence of the scalar spin chirality
($\vec{S}_{i}\cdot(\vec{S}_{j}\times\vec{S}_{k})$) and octupole
($\vec{Q}_{T_{1g}}$) moment as a function of in-plane magnetic field angle
$\phi$. The simulation suggests that the non-coplanar spin-textures are
possible and predicts that both the octupole signal (arising from
$\vec{Q}_{T_{1g}}$) and the THE-like signal (arising from
$\vec{S}_{i}\cdot(\vec{S}_{j}\times\vec{S}_{k})$) possess an angular
dependence with 120∘ rotational symmetry (Fig. 2(i)). This result is
qualitatively consistent with the angular dependence of the both the signals
in our experiment (Fig. 2(h)) complete with their opposing signs.

##  3 Discussion and Conclusion

Whilst attempts have been made to quantify the behavior of the complex spin
structures present in NC-AFMs using simple pictures to explain the sign
change, for example, in the AHE [5, 6, 8, 9, 28, 29, 30, 31, 32, 33, 7, 34,
35, 36], our work provides a full understanding of the signal of the AHE as a
function of the magnetic field. Our analysis of the underlying spin dynamics
elucidates a clear interplay of coexisting orders within a noncollinear
antiferromagnet, which sets this class of magnetic order apart from
conventional ferromagnets and antiferromagnets. These collinear spin
structures host only one order parameter apiece, the magnetization and the
Néel vector, whereas noncollinear antiferromagnets exhibit further
complexities that have direct consequences on the AHE that can only be
understood through expansion in local spins beyond the ferromagnetic case. For
$\mathrm{Mn_{3}NiCuN}$, and subsequently the whole class of
$\mathrm{Mn_{3}XY}$ compounds, one can uncover the coexistence of three
distinct orders: the magnetization, a quantity in part responsible for the
conventional AHE in this NC-AFM, the octupole contribution, that provides both
conventional and also in-plane AHE contributions and finally the THE-like
component generated when the spins are non-coplanar. Although this particular
material is merely a prototypical example of the vast array of other NC-AFM
orders, we present a very clear paradigm shift in the way these materials are
to be viewed: that they have multiple unconventional order parameters which
behave like a magnetization, yet they affect the Hall transport in unique
ways.

In summary, we systematically study the symmetry of the transverse Hall
resistance in $\mathrm{Mn_{3}NiCuN}$ (111) by applying magnetic fields in the
kagome plane along related crystallographic directions. We find that whenever
the external applied magnetic field is collinear to the in-plane projections
of any one of the octupoles, an AHE can be detected even in the limit of
vanishing magnetization. The six octupoles that have in-plane components lead
to a 120∘ symmetric AHE signal even when magnetic field and electrical
measurements are coplanar, a remarkable contrast from the conventional AHE in
FMs. We observe THE-like features and decompose the signal into octupole and
THE-like components, finding the THE-like components to behave with a similar
120∘ angular dependence as the octupole component while sweeping the magnetic
field in the kagome plane. We attribute this effect to a scalar-chirality
emerging during the spin-reorientation at low magnetic fields. Our
experimental results are well supported by theoretical predictions. Our work
represents a significant step in uncovering the complex, non-trivial phenomena
in NC-AFMs, greatly enhancing the understanding of the octupole moments, and
distilling the complex spin textures into clear, coexisting orders. This
understanding opens the possibility to explore multiple means to harness NC-
AFMs for novel spintronic technological applications.

Acknowledgments

We acknowledge funding from King Abdullah University of Science and Technology
(KAUST) under award 2020 $-$ CRG8 $-$ 4048\. A.M was supported by the
Excellence Initiative of Aix-Marseille University - A*MIdex, a French
”Investissements d’Avenir program”. In addition, the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) $-$ Grant No. TRR
173/2 $-$ 268565370 Spin+X (projects A01, BO2 and A11). A.R, B.B, L.L.D and
M.K acknowledge funding from the European Union’s Framework Programme for
Research and Innovation Horizon 2020 (2014 $-$ 2020) under the Marie
Skłodowska-Curie Grant Agreement No. 860060 (ITN MagnEFi). Y.M., J.S., W.F.,
and Y.Y. acknowledge the funding under the Joint Sino-German Research Projects
(Chinese Grant No. 12061131002 and German Grant No. 1731/10-1) and the Sino-
German Mobility Programme (Grant No. M-0142). R.Y.D. and L.L-D. acknowledge
funding from Spanish ”Ministerio de Ciencia e Innovacion” under project
PID2020-117024GD-C41. T.G.S., F.R.L., D.G. and Y.M. gratefully acknowledge the
Jülich Supercomputing Centre for providing computational resources under
project jiff40. T.H and H.A acknowledge funding from the Japan Society for the
Promotion of Science (KAKENHI Grant Nos. 20H02602 and 19K15445).

###  3.1 Contributions

Adithya Rajan performed the experimental measurements, Tom G. Saunderson
performed the first principles calculations and wrote the paper with Adithya
Rajan and Arnab Bose. Fabian R. Lux performed the symmetry analysis, Rocío
Yanes Díaz and Luis López-Díaz performed atomistic spin dynamics simulations.
Hasan M. Abdullah performed tight binding calculations and provided insights
on the scalar spin chirality with Gokaran Shukla, Udo Schwingenschlögl and
Aurélien Manchon. Jun-Young Kim assisted in the experimental discussions,
Tetsuya Hajiri and Hidefumi Asano performed thin film sample deposition,
Beatrice Bednarz assisted with device fabrication. Yugui You, Wangxiang Feng,
Yuriy Mokrousov and Dongwook Go provided insights in the transport from the
first principles perspective, Olena Gomonay and Jairo Sinova provided insights
on the symmetry analysis. Tom G. Saunderson, Fabian R. Lux and Yuriy Mokrousov
coordinated the theory effort. Mathias Kläui provided experimental insight and
has been the principal investigator, supervising the whole project.

###  3.2 Corresponding author

Correspondence to Mathias Kläui.

##  4 Ethics declarations

###  4.1 Conflict of Interest

The Authors declare that there is no conflict of interest.

##  5 Methods

###  5.1 Material and Device fabrication

On MgO (111) substrate, 20 nm Mn3NiCuN thin film was deposited using reactive
magnetron sputtering at 375 °C substrate temperature under 2.0 Pa with 4% N2
\+ 96% Ar gas mixtures. After growth, the sample was annealed in-situ at 500∘C
under the same atmosphere as film growth. Please refer to [22] for further
details on thin film deposition. A 3 nm Pt layer was deposited in-situ to
prevent oxidation. Epitaxial Mn3NiCuN growth was confirmed by X-ray
diffraction measurements with Cu Kα radiation (See Fig. S1 ‘1.1 Sample
characterization’ in the Supplementary [21]). After the thin film growth, Hall
bar devices with current line of width 10 $\mu$m and voltage line of width 3
$\mu$m were patterned using electron beam lithography, and the surrounding
area removed using Ar+ ion etching.

###  5.2 Transport measurements

The sample with patterned Hall bars was mounted on a standard PCB with Au
contacts contacted to the Hall bars by wire bonding using TPT Hybond 572-40.
The PCB with contacted sample was then mounted on attocube ANRv51/RES piezo-
controlled rotatable sample holder which was inserted in a variable
temperature cryostat from Oxford Instruments to perform magneto-transport
measurements. Keithley 2400 was used as a current source, and the Hall voltage
was measured using Keithley 2182A nano-voltmeter.

###  5.3 Data processing

The acquired Hall signal was anti-symmetrized
($R_{xy}(B)=(V_{xy}(B)-V_{xy}(-B)/2I_{xx})$ to extract the anomalous Hall
effect, and to remove any contributions from the longitudinal signal. The
anomalous Hall signal was centered and the contribution linear with B
corresponding to the ordinary Hall effect subtracted. The planar hall effect
was also analyzed and removed in Fig. S3 and is plotted in Fig. S4 in section
‘1.2 Data processing’ of the supplementary [21].

###  5.4 Computational details

We performed Density Functional Theory calculations for bulk Mn3NiCuN using
the experimental lattice constant $3.9012\AA$ from Zhao et al [22]. We perform
calculations using the FLEUR code [37] which implements the full potential
linear augmented plane wave method (FP-LAPW) [38], employing the generalized
gradient approximation (GGA) [39]. In order to parameterize the symmetry
analysis, full self-consistencies were performed at multiple non-coplanar
canting angles from the $\Upsilon_{0}$ spin configuration in Fig. 1(a). In
order to quantify the AHE from first principles, we employ the use of Wannier
interpolation [40, 41] to efficiently compute the Berry curvature using the
Kubo formalism [1]. Full details are provided in section ‘3 Numerical details’
of the supplementary information [21].

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