11institutetext: Mullard Space Science Laboratory, UCL, Holmbury St Mary,
Dorking, Surrey, RH5 6NT, UK 11email<EMAIL_ADDRESS>22institutetext:
Department of Physics, Imperial College London, London, UK 33institutetext:
Space Research Institute, Austrian Academy of Sciences, Graz, Austria
44institutetext: Institute of Physics, University of Graz, Graz, Austria
55institutetext: Institut für Experimentelle und Angewandte Physik, Christian-
Albrechts-Universität zu Kiel, Kiel, Germany

# Solar origins of a strong stealth CME detected by Solar Orbiter

Jennifer O’Kane 11 Lucie M. Green 11 Emma E. Davies 22 Christian Möstl 33
Jürgen Hinterreiter 3344 Johan L. Freiherr von Forstner 55 Andreas J. Weiss
3344 David M. Long 11 Tanja Amerstorfer 33

###### Abstract

Aims. We aim to locate the origin of a stealth coronal mass ejection (CME)
detected in situ by the MAG instrument on board Solar Orbiter, and make
connections between the CME observed at the Sun, and the interplanetary CME
(ICME) measured in situ.

Methods. Remote sensing data are analysed using advanced image processing
techniques to identify the source region of the stealth CME, and the global
magnetic field at the time of the eruption is examined using Potential Field
Source Surface (PFSS) models. The observations of the stealth CME at the Sun
are compared with the magnetic field measured by the Solar Orbiter spacecraft,
and plasma properties measured by the Wind spacecraft.

Results. The source of the CME is found to be a quiet Sun cavity in the
northern hemisphere. We find that the stealth CME has a strong magnetic field
in situ, despite originating from a quiet Sun region with an extremely weak
magnetic field.

Conclusions. The interaction of the ICME with its surrounding environment is
the likely cause of a higher magnetic field strength measured in situ. Stealth
CMEs require multi-wavelength and multi-viewpoint observations in order to
confidently locate the source region, however their elusive signatures still
pose many problems for space weather forecasting. The findings have
implications for Solar Orbiter observing sequences with instruments such as
EUI that are designed to capture stealth CMEs.

###### Key Words.:

Sun: coronal mass ejections (CMEs) – Sun: corona – Sun: magnetic fields – Sun:
heliosphere

##  1 Introduction

Coronal mass ejections (CMEs) are large eruptions of the Sun’s plasma and
magnetic field into the heliosphere. If Earth-directed, these events have the
potential to drive space weather effects. A subset of Interplanetary CMEs
(ICMEs), known as magnetic clouds, exhibit enhanced magnetic field strength, a
rotation in at least one of the magnetic field components, and low plasma
density, temperature and plasma $\beta$ when detected in situ (Burlaga et al.,
1981b). The helical field configuration of a magnetic cloud is known as a flux
rope. The geoeffectiveness of a magnetic cloud (or indeed any CME) is highly
dependent on the strength and orientation of the magnetic field. In
particular, a strong southward Bz for a period of time longer than a few hours
typically produces a strong geomagnetic storm. In order to improve space
weather prediction capabilities, it is necessary then to study the CME source
region and use this knowledge to forecast the configuration and properties of
the magnetic field in the interplanetary medium. The source region may or may
not itself have a flux rope configuration (see Green et al., 2018; Patsourakos
et al., 2020, for reviews). However, if a flux rope is present it can be
identified using a range of remote sensing data and the knowledge used to
forecast its geoeffectiveness (see Palmerio et al., 2017, 2018, and references
therein).

Stealth CMEs lack the classic low coronal signatures typically associated with
CMEs, such as solar flares, Extreme Ultraviolet (EUV) dimmings and filament
eruptions (see overview by Howard & Harrison, 2013). However, it has been
shown that image processing techniques can be used to identify observational
signatures, although these can be very faint and hard to detect (Alzate &
Morgan, 2017; O’Kane et al., 2019). Studies carried out so far indicate that
stealth CMEs may originate at relatively high heights $\approx$ 1.3 R⊙ in the
corona (Robbrecht et al., 2009; O’Kane et al., 2021), implying that the
eruptive structure has relatively low magnetic field strength and low plasma
density, leading to a correspondingly low energy release, lack of strong
radiative emissions and lower propagation speed (¡500 km/s, D’Huys et al.,
2014). Despite these characteristics, stealth CMEs still have the potential to
be geoeffective and therefore present a challenge for space weather
forecasting (Kilpua et al., 2014; Nitta & Mulligan, 2017). Carrying out in
situ and remote sensing connection studies can help understand the origins,
evolution and impact of CMEs in the heliosphere.

Here we study the solar origins of the first CME detected in situ by Solar
Orbiter (Müller et al., 2020). The encounter took place on 19 April 2020 at a
heliocentric distance of 0.809 AU. The details of the CME’s structure and
propagation in the inner heliosphere are reported in Davies et al. (2021) and
the corresponding Forbush decrease in Freiherr von Forstner et al. (2021). The
interplanetary CME had a clear flux rope structure and its arrival at Earth
produced a Dst minimum of -60 nT on 20 April 2020 and a category G1
geomagnetic storm. At the Sun, the event was determined to be a streamer
blowout with no clear signatures in the lower corona. This places the CME in
the stealth category and we present here the identification and analysis of
the CME source region that was made possible using advanced image processing
techniques. The paper is organised as follows: Sect. 2 describes the in situ
and remote sensing data as well as the methods used within this study; Sect. 3
describes the analysis of the stealth CME connecting the in situ and the solar
source perspectives; and finally Sect. 4 discusses the results and
conclusions.

##  2 Data and methods

###  2.1 Data

For the in situ analysis, magnetic field data were obtained from the
magnetometer (MAG; Horbury et al., 2020) onboard Solar Orbiter (Müller et al.,
2020), which provides in situ observations of the CME at approximately 0.8 AU.
The Magnetic Field Investigation (MFI; Lepping et al., 1995) and Solar Wind
Experiment (SWE; Ogilvie et al., 1995) of the Wind spacecraft provide magnetic
field and solar wind plasma observations at 1 AU.

For the remote sensing observations, the white light CME was analysed using
data from the Large Angle and Spectrometric Coronagraph (LASCO; Brueckner et
al., 1995) on board the Solar and Heliospheric Observatory (SOHO; Domingo et
al., 1995) and the COR1 and 2 coronagraphs (part of the Sun Earth Connection
Coronal and Heliospheric Investigation instrument suite, SECCHI, Howard et
al., 2008) on board the Solar Terrestrial Relations Observatory Ahead
(STEREO-A, Kaiser et al., 2008) spacecraft. EUV data were taken from the
Atmospheric Imaging Assembly (AIA, Lemen et al., 2011) on board the Solar
Dynamics Observatory (SDO, Pesnell et al., 2011) spacecraft, and the Extreme
Ultraviolet Imager (EUVI; Howard et al., 2008) onboard STEREO-A. Line of sight
magnetic field data were taken from the Helioseismic and Magnetic Imager data
(HMI, Scherrer et al., 2012) onboard SDO.

###  2.2 Methods

Coronagraph data were processed using the Normalizing-Radial-Graded Filter
(NRGF, Morgan et al., 2006). EUV data were processed using both the Multi-
Scale Gaussian Normalization technique (MGN, Morgan & Druckmüller, 2014) and
the NRGF technique. Stack plots were created using COR1 and COR2 data by
taking a radial slice from Sun centre. The slice taken at 90∘ is presented in
this paper, as this most clearly showed the CME curve. The plane-of-sky
kinematics were obtained using the Savitzky-Golay bootstrapping technique
(Byrne et al., 2013), and the the 3D shape and orientation of the CME in the
COR2 and LASCO/C2 fields of view were modelled using the Graduated Cylindrical
Shell model (GCS; Thernisien et al., 2006; Freiherr von Forstner, 2021).
Longitude and Latitude are defined in Carrington co-ordinates.

##  3 The stealth CME

###  3.1 In situ observations

Figure 1: In situ observations of the CME. In each panel, the shock front is
delineated by the solid vertical line, and the magnetic flux rope boundaries
by the dashed vertical lines. Top panel: Solar Orbiter magnetic field data in
Radial-Tangential-Normal (RTN) coordinates, where the total magnetic field
magnitude is shown in black and the magnetic field components, RTN, in red,
green and blue, respectively. Middle panel: Wind magnetic field data in RTN
coordinates, displayed similarly to that of Solar Orbiter. Bottom panel: the
proton speed (black) and density (red) measured by Wind. Figure 2: Left:
Combined EUVI-A 171Å and COR1-A illustrating the connectivity between the high
altitude structures observed in EUVI-A and the CME observed in COR1-A. The
EUVI-A image is taken at 22:09 UT 13 April 2020, when the cavity is observed,
whilst the COR1-A image is taken at 03:15 UT 14 April 2020, when the CME
eruption is underway. The blue line represents the 90∘ slice taken for the
stack plot. Right: A stack plot inverse image of COR1-A, taken at an angle of
90∘, showing the CME curve. The blue crosses indicate the points selected to
track the CME curve.

In situ magnetic field signatures observed by both Solar Orbiter and Wind, and
the solar wind plasma data observed by Wind whilst the spacecraft were close
to radial alignment, separated in longitude by less than 5∘ are shown in Fig.
1. The shock front (delineated by the vertical solid line in Fig. 1) driven by
the CME was first observed in situ by Solar Orbiter at 05:06 UT on 19 April
2020 whilst the spacecraft was located at a heliocentric distance of 0.809 AU.
The shock was then observed by Wind (0.996 AU) at 01:34 UT on 20 April 2020,
arriving with a mean proton speed of 346 km s-1.

The in situ data reveal a clear magnetic flux rope structure, constrained by
the vertical dashed lines in Fig. 1, where the leading edge times of the
magnetic flux rope are 08:59 UT on 19 April 2020 at Solar Orbiter, and 07:56
UT on 20 April 2020 at Wind (defined by Davies et al., 2021). The CME can be
classified as a magnetic cloud, meeting criteria detailed by Burlaga et al.
(1981a). The configuration of the magnetic flux rope can be classified as
South-East-North (Bothmer & Schwenn, 1998; Mulligan et al., 1998) and
therefore has left-handed chirality and the flux rope axis is of low
inclination to the ecliptic plane.

The maximum magnetic field strength reached within the magnetic flux rope was
20.1 nT at Solar Orbiter and 16.3 nT at Wind. This is a relatively high field
strength for an ICME at solar minimum, and/or from quiet Sun regions. The
proton density of the magnetic flux rope measured at Wind was relatively low
throughout, with a mean density of 11.6 cm-3, however, this includes the rise
in density towards the trailing edge (discounting this rise, the mean density
in the flux rope is 8.3 cm-3). The rise in density towards the trailing edge
is likely due to a compression of the magnetic flux rope caused by the high
speed stream that follows, which we observe at Wind to have a mean proton
speed of $\sim$450 km s-1. The high speed stream may also be the reason we
observe a stronger magnetic field strength than expected within the CME, where
the mean magnetic field strength of the flux rope (18.4 nT at Solar Orbiter
and 13.6 nT at Wind) is much stronger than those calculated for ICMEs at
similar heliocentric distances (e.g. Janvier et al., 2019).

The magnetic field profiles observed at both spacecraft show a slight decline
in magnetic field strength as the flux rope passes over the spacecraft. Such
asymmetric profiles are consistent with the model of an expanding flux rope
(Nieves-Chinchilla et al., 2018; Janvier et al., 2019). However, the proton
speed profile of the CME observed at Wind shows only a small decline
throughout the magnetic flux rope. Asymmetric magnetic field profiles with a
relatively small expansion may correspond to ICMEs with distorted structures
(Nieves-Chinchilla et al., 2018), as observed in images taken by the STEREO-A
Heliospheric Imager (HI) of this event (Davies et al., 2021). The mean
propagation speed of the leading edge between Solar Orbiter and Wind was 341
km s-1. Assuming a constant transit velocity would suggest a CME onset time
$\sim$5 days prior to its arrival at Wind (i.e. on 15 April 2020).

We have applied a semi-empirical 3D flux rope model to the Solar Orbiter data
(3DCORE, Möstl et al., 2018; Weiss et al., 2021) in order to derive global
parameters of the flux rope within this ICME. Detailed results are given in
Davies et al. (2021) and Freiherr von Forstner et al. (2021), and work on a
paper on simultaneous multi-point fitting of the Solar Orbiter, Wind and
BepiColombo flux rope magnetic field data is underway (Weiss et al. 2021b, in
prep.). Here we present additional information on the magnetic flux derived
from the 3DCORE fit on the Solar Orbiter MAG data only. This is of particular
interest because for stealth CMEs, large-scale magnetic reconnection during
flares can be ruled out as a formation mechanism of the flux rope (e.g. Qiu et
al., 2007; Möstl et al., 2009), and yields parameters of a flux rope that was
pre-existing in the solar corona.

Davies et al. (2021) report the fit results with the 3DCORE technique applied
to the Solar Orbiter data, propagated to 1 AU: an axial field strength
$B_{0}=14.3\pm 0.9$ nT, a flux rope diameter of $D_{1AU}=0.114\pm 0.022$ AU
and a twist of $\tau=-3.7\pm 0.6$ field line turns over the full torus
($\tau<0$ implies left handed chirality), or $\tau_{1AU}=-0.6\pm 0.1$ turns
per AU. We have numerically integrated the flux rope cross section to find an
axial flux of $\Phi_{ax}=0.30\pm 0.06\times 10^{21}$ Mx. The application of
the formula for a uniform twist cylindrical flux rope (e.g. equation 16 in
Vandas & Romashets, 2017), with the parameters derived from the 3DCORE fit,

$\Phi_{ax}=\frac{\pi B_{1AU}}{\tau_{1AU}}\mathrm{ln}(1+\tau_{1AU}D_{1AU}/2)$
(1)

gives $\Phi_{ax}=0.33_{-0.13}^{+0.16}\times 10^{21}$ Mx, which is consistent
with the numerical result, although this should in general be seen only as a
rough approximation because the flux rope cross section in 3DCORE is not
cylindrical but elliptical. The axial flux we determined for this event is
thus at the lower end of reported axial fluxes in ICME flux ropes (cf. Qiu et
al., 2007).

###  3.2 Stealth CME source region

Figure 3: Left: HMI image at 20:00 UT 13 April 2020, showing the magnetic
field just before the cavity begins to rise. The blue ’X’ marks the location
of the GCS approximated source region, whilst the red box marks the
approximate location of the cavity that was observed at the solar limb. Right:
The corresponding AIA 193Å image, with the GCS and cavity regions highlighted
as before. In both figures, the limb of EUVI-A is overplotted as the white
line.  Figure 4: Remote sensing observations of the CME. STEREO-A COR2 (direct
images) and SOHO LASCO C2 (running difference images). In the bottom panels,
the GCS fitting according to the parameters obtained by Freiherr von Forstner
et al. (2021) is indicated by the blue mesh. Figure 5: NRGF processed EUVI-A
171Å images, illustrating the structures contributing to the stealth CME. Left
panel: The red arrows point out a bright loop structure that is above filament
material. Middle panel: The large bright high altitude structure, illustrated
by the red arrows. Right panel: The cavity structure, highlighted by the red-
dashed circle.  Figure 6: MGN-processed AIA 171Å image at 19:00 UT 7 April
2020, showing the cavity structure observed on the limb (red-dashed circle)
$\sim$6 days before the eruption.

As seen in Fig.2, the stealth CME is observed in COR1-A as a faint enhancement
in density, and becomes clear in COR2-A by 03:54 UT 15 April 2020, following
the brightening and swelling of a streamer. The CME exhibits a bright
underside, and a clear concave-up structure, suggestive of a flux rope
configuration (Vourlidas et al., 2013) . In COR1-A, the underside of the CME
can be tracked clearly from 10:16 UT 14 April 2020 (Figure 2), using the 90∘
slice, and has an average velocity of 18 km s-1. Using this speed, and a
height of 2.5 R⊙, at 22:30 UT 14 April 2020, we estimate a more accurate CME
onset time $\sim$19:30 UT 13 April 2020. In COR2-A the CME has an average
speed of 53 km s-1. The increase in CME speed between that measured from
COR2-A and that measured in situ at Wind, is likely due to the high speed
stream following the CME, and the background solar wind.

Primary observations of the Sun during this time period show no active
regions, and the photospheric magnetic field exhibited only weak small-scale
field fragments (see Fig. 3). EUV and H-alpha data showed that no filaments or
filament channels were present either, making the source region of the CME
very challenging to identify. Using the images from COR2-A and C2 at 05:39 UT
and 05:36 UT 15 April 2020, respectively, the CME was fitted with the GCS
model (see Fig.4). The GCS fit found an approximate source region at
143$\pm$7∘ longitude, and 3$\pm$3∘ latitude (see Table 2 in Freiherr von
Forstner et al., 2021), and is indicated by the blue cross in Fig. 3. This
puts the approximate source region on-disk from SDO, and on-limb from
STEREO-A.

Although both the MGN and difference imaging techniques were applied to each
AIA passband, they did not reveal any clear eruption signatures on disk.
However, the NRGF and MGN processed 195 Å and 171 Å EUVI-A data revealed
multiple high-altitude dynamic structures off the solar limb; a bright looped
region, a large bright structure, and a cavity structure (Fig. 5). The large
bright structure is observed from 08:50 UT 13 April 2020, and then slowly
rises into COR1-A, coinciding with the brightening and swelling of the
streamer. The concave-up cavity structure is observed from 22:09 UT 13 April
2020, at 35.7∘ latitude, and is deflected southwards as it erupts from the
Sun, coinciding with initiation of the CME in COR1-A. We therefore deduce that
this is the cavity responsible for the CME that is later detected in situ. The
deflection of the cavity was measured by applying the GCS model to the
structure in 171 Å EUVI-A. The cavity is observed to deflect southwards by
$\sim$19∘ within the EUVI FOV, and a further $\sim$12∘ into the FOV of COR1.
It is noted that due to applying the GCS model to one spacecraft instead of
two, errors are larger. However, this is smaller for latitude such that the
deflection can be measured with reasonable confidence.

The cavity is an observational signature of a flux rope present in the low
corona as the eruption takes place. We track the region back to the eastern
limb of SDO, to look for evidence of the flux rope present prior to the
eruption. Three cavities are observed on the limb in AIA 171 Å between 6 and
10 April 2020. The first cavity at 10∘ latitude erupted off-limb on 8 April
2020, and therefore was ruled out as being the source. The second cavity at
40∘ latitude, highlighted by the red-dashed circle in Fig. 6 matches closely
with the erupting cavity observed in EUVI-A (at 35.7∘ latitude) and with its
approximate longitude. The third cavity at 30∘ latitude does not match in
longitudinal extent with the cavity observed at the limb of EUIV-A. Thus we
conclude the second cavity observed is the structure that later erupts as the
stealth CME. Using these two observations, we obtain an approximate source
region from the SDO perspective, as 139${}^{\circ}\pm$5∘ longitude,
38${}^{\circ}\pm$2∘ latitude, indicated by the red box in Fig. 3. The
difference between the source region and the estimated region obtained from
the GCS is due to the southwards deflection of the cavity as it erupts.

We further investigated the magnetic topology of the source region and its
surrounding environment with a Potential Field Source Surface (PFSS) model,
using the PFSS solver in Python (pfsspy, Yeates, 2018; Stansby, 2019). Fig. 7
shows the PFSS model as viewed from both the AIA (left) and STEREO-A (right)
perspectives, with the modelled magnetic field lines showing the positive
(red) and negative (blue) polarities. The CME originates in the northern
hemisphere, near central meridian (west-limb) the AIA (STEREO-A) perspective.
Open field lines are deflected southwards, towards the equator, similar to the
trajectory of the cavity observed in EUVI-A. Given the absence of active
regions and solar minimum phase at this time, the global magnetic field is
close to that of a dipole field.

Figure 7: PFSS model of Carrington rotation 2229, as observed from AIA
perspective (left) and STEREO-A perspective (right). Positive (negative)
polarity is coloured in red (blue), on 13 April 2020. The approximate source
region of the stealth CME is pointed out by the white arrows.

##  4 Discussion and conclusions

We present here a study of the first stealth CME observed in situ by Solar
Orbiter, with accompanying Wind in situ observations and STEREO-A and SDO
remote sensing data. The combined use of STEREO-A and SDO data revealed that
the origin of the CME was a quiet Sun cavity within small-scale magnetic field
and no clear polarity inversion line. The cavity identified in EUVI-A and AIA
171 Å passbands extended up to 1.3 R⊙ (from Sun centre) in the corona. Such
dark elliptical cavities in the quiet Sun indicate the presence of a magnetic
flux rope and have been found to have a median height of 1.2 R⊙ (Gibson,
2015).

Despite the quiet Sun origin of the stealth CME, the magnetic cloud still had
a very high field strength in the interplanetary medium (13.6 nT at Wind). The
$\sim$1 AU value is significantly higher than that found in a comparison study
of stealth CMEs at solar minimum by Kilpua et al. (2014), who showed an
average Bmax of 9.7 nT across a sample of 10 stealth CMEs, and the in situ
stealth CME study of Nieves-Chinchilla et al. (2013). This maximum field
strength and the South-East-North configuration of the flux rope may have been
the deciding factors in this event producing a category G1 geomagnetic storm.
The slow speed of stealth CMEs means we do not typically expect them to drive
shocks. In this event a forward shock is detected in situ, similar to the case
of the stealth CME studied in Nieves-Chinchilla et al. (2013). The shock is
likely due to the slower solar wind conditions at 1 AU ahead of the ICME
($\sim$300 km s-1). Also, the slight expansion in the unperturbed section of
the ICME means that a combination of relatively slow solar wind speed ahead of
the CME and the CME expansion may drive the shock as proposed by Lugaz et al.
(2017).

This study reinforces the need for multi-wavelength, multi-view point
observations of the solar corona, combined with image processing techniques,
to further our ability to detect and analyse stealth CME source regions. These
observations indicate that for Solar Orbiter EUI to successfully observe
stealth CME sources, observing sequences should include regular synoptic
images from the 174 Å waveband of the Full Sun Imager (FSI) telescope, which
will have a field-of-view of 4 R⊙ at perihelion.

###### Acknowledgements.

JO thanks the STFC for support via funding given in her PHD studentship. LMG
is grateful to the Royal Society, which supported this research through the
Royal Society University Research Fellowship scheme. DML is grateful to the
Science Technology and Facilities Council for the award of an Ernest
Rutherford Fellowship (ST/R003246/1). JvF thanks the German Space Agency
(Deutsches Zentrum für Luft- und Raumfahrt e.V., DLR) for their support of his
work on the Solar Orbiter EPD team under grant 50OT2002. CM, AJW, JH and TA
thank the Austrian Science Fund (FWF): P31521-N27, P31659-N27, P31265-N27. ED
is supported by funding from the Science and Technology Facilities Council
(STFC) studentship ST/N504336/1. SDO is a mission of NASA’s Living With a Star
Program. STEREO is the third mission in NASA’s Solar Terrestrial Probes
program. SOHO is a mission of international cooperation between ESA and NASA.
The authors thank the SDO, STEREO, and SOHO teams for making their data
publicly accessible.The Solar Orbiter magnetometer was funded by the UK Space
Agency (grant ST/T001062/1). Its data are available in the Solar Orbiter
Archive at http://soar.esac.esa.int/soar/.

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# Higgs and Goldstone spin-wave modes in striped magnetic texture

Matías Grassi  Université de Strasbourg, CNRS, Institut de Physique et Chimie
des Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France.  Moritz
Geilen  Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische
Universität Kaiserslautern, 67663 Kaiserslautern, Germany.  Kosseila Ait
Oukaci  Institut Jean Lamour, Université de Lorraine, UMR 7198, CNRS, F-54000
Nancy, France.  Yves Henry  Université de Strasbourg, CNRS, Institut de
Physique et Chimie des Matériaux de Strasbourg, UMR 7504, F-67000 Strasbourg,
France.  Daniel Lacour  Institut Jean Lamour, Université de Lorraine, UMR
7198, CNRS, F-54000 Nancy, France.  Daniel Stoeffler  Université de
Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg,
UMR 7504, F-67000 Strasbourg, France.  Michel Hehn  Institut Jean Lamour,
Université de Lorraine, UMR 7198, CNRS, F-54000 Nancy, France.  Philipp Pirro
Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität
Kaiserslautern, 67663 Kaiserslautern, Germany.  Matthieu Bailleul  Université
de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de
Strasbourg, UMR 7504, F-67000 Strasbourg, France.

###### Abstract

Spontaneous symmetry breaking is ubiquitous in physics. Its spectroscopic
signature consists in the softening of a specific mode upon approaching the
transition from the high symmetry side and its subsequent splitting into a
zero-frequency ‘Goldstone’ mode and a non-zero-frequency ‘Higgs’ mode.
Although they determine the whole system dynamics, these features are
difficult to address in practice because of their vanishing coupling to most
experimental probes and/or their strong interaction with other fluctuations.
In this work, we consider a periodic magnetic modulation occurring in a
ferromagnetic film with perpendicular-to-plane magnetic anisotropy and observe
its Goldstone and Higgs spin-wave modes at room temperature using microwave
and optical techniques. This simple system constitutes a particularly
convenient platform for further exploring the dynamics of symmetry breaking.

##  I Introduction

Upon spontaneous symmetry breaking, a system organizes itself in a state with
a lower symmetry than that of its constituting entities, as exemplified by
superconducting, magnetic or incommensurate structural phases [1, 2].
According to the Landau theory, such transition is conveniently visualized by
defining an order parameter $\psi$ and following the morphology of the free
energy surface $E(\psi)$ [3]. This is illustrated in Fig. 1(a) for a system
with $U(1)$ symmetry, where $\psi$ is a complex number (or, equivalently, a
two-dimensional real vector). In the high symmetry phase, $E$ presents a
single minimum at $\psi=0$. Upon driving the system through the transition,
the curvature around this point decreases, reaches zero at the critical point
and then changes sign. In the low symmetry phase, the energy surface
eventually takes the shape of a Mexican hat with a degenerate minimum
extending over a circle of radius $|\psi|=\psi_{0}$. The system has to
“choose” a phase arg$(\psi)$, which constitutes the symmetry break. This
particular energy landscape gives rise to characteristic low frequency dynamic
modes, conveniently viewed as the oscillations of a mass moving on such
surface [1]. Upon driving the system from the high symmetry phase, the
oscillations around the $\psi=0$ minimum [blue arrow in Fig. 1(a)] are
expected to soften gradually, reach zero frequency at the critical point, and
subsequently splits in two, a zero-frequency mode with azimuthal trajectory
along the rim and a non-zero frequency mode with radial trajectory across the
rim (solid and dashed red arrows in Fig. 1(a), respectively). These two modes,
referred as Goldstone and Higgs modes, respectively, dominate the whole
dynamics of the low symmetry phase, but also its coupling to external degrees
of freedom, in particular gauge ones. Originally explored in the context of
superconductivity, the latter is of particular importance for particle-
physics, as the finite masses of the W and Z weak bosons can only be explained
by their coupling to the symmetry-breaking Higgs field [4, 5].

Although these dynamics are of crucial interest, their direct observation is a
serious challenge as it requires to drive the system through the transition
while keeping experimental access to the relevant low frequency excitations,
particularly prone to overdamping due to defects and thermal/quantum
microscopic fluctuations [2, 6]. This difficulty could be avoided in low-
temperature inelastic neutron scattering studies of very specific pressure-
induced structural and magnetic transitions [7, 8]. More recently, it was
proposed to use artificial systems, namely ultra-cold bosons lattices, which
can be driven through a quantum phase transition, their excitations being
characterized via real time optical spectroscopy [9, 10].

In this article, we show that an archetypal system of micromagnetism, the so-
called magnetic weak stripes, allows for a room-temperature observation of the
Higgs/Goldstone dynamics by inelastic light scattering and microwave
spectroscopy. Magnetic weak stripes consist of a field-tunable modulation at a
mesoscopic scale occurring in ferromagnetic films possessing a moderate
perpendicular magnetic anisotropy [11]. The stability of this texture was
predicted in the early days of theoretical micromagnetism [3, 12, 13, 14], and
later confirmed by static magnetic imaging [15, 16]. Several studies have
explored the corresponding dynamics evidencing a complex set of vibration
modes varying to a very large extent with the magnetic parameters of the film
and upon application of a control magnetic field. Initially described within
local resonance models (domain/domain-wall-resonance) [17, 18, 19], this
complex phenomenology has recently been rephrased in the vocabulary of
magnonics, as a complex set of spin-wave modes localized/scattered by the
periodic modulation [20, 21, 22]. Moving ahead in that direction, we provide
here a unified description of both statics and dynamics of magnetic stripes
based on the identification of a specific spin-wave mode which, upon stripe
nucleation, softens and then splits into a Goldstone/Higgs pair. For this
purpose, we first formulate an elementary analytical model of the stripe
statics and related spin-wave dynamics based on the Landau theory of phase
transitions. Then, we report inelastic light scattering measurements conducted
down to spin-wave wavelength of the order of the modulation period for
different points across the critical region. Confronting them with
complementary ferromagnetic resonance measurements and micromagnetic
simulations, we arrive at an unprecedented global picture of the low frequency
dynamics related to such symmetry breaking.

Figure 1: (a) Sketch of the characteristic dynamic modes associated with a
$U(1)$ symmetry breaking. The blue arrow shows the degenerate modes in the
high-symmetry phase (yellow potential surface). The solid and dashed red
arrows show the Goldstone and Higgs modes in the low-symmetry phase (sombrero-
shape blue potential surface). (b) Magnetization loop measured for our
$180~{}$nm Co40Fe40B20 film with a magnetic field $\bm{H}$ in the plane. (c)
Magnetic force microscopy image of the weak stripes magnetic texture existing
below the critical field $H_{c}$ (shown here at remanence). (d) Sketch of the
three-dimensional normalized magnetization distribution $\bm{M}(x,y,z)/M_{S}$
within the weak stripes texture, together with its minimal description in
terms of a U(1) symmetry breaking, with associate amplitude $|\psi|$ and phase
arg$(\psi)=k\,\delta$ (see details in the text).

##  II Results

The system studied consists of an amorphous Co40Fe40B20 film of thickness
$D=180~{}$nm deposited on intrinsic silicon and initialized by applying a
saturating magnetic field $\bm{H}$ in the film plane. Upon reducing the
magnitude of the field below about $12~{}$mT, the average magnetization of the
film starts to decrease in a roughly linear fashion [Fig. 1(b)]. In this
regime, magnetic force microscopy allows one to identify a modulation periodic
in one-dimension with a wavenumber of about $2\pi/(300$ nm$)=21~{}$rad/$\mu$m
[Fig. 1(c)] [23], identified with the archetypical ‘magnetic weak stripes’
arising in films which possess a moderate perpendicular magnetic anisotropy
$K$ [11]. In the following, we shall revisit this texture from the point of
view of its dynamics. For this purpose, we start by describing a minimal model
for stripe nucleation, as sketched in Fig. 1(d). The mechanism consists of a
competition between the in-plane magnetic field $\bm{H}$, which tends to
maximize the component of the magnetization distribution $\bm{M}(x,y)$ along
its direction $\bm{\hat{z}}$ (the system is assumed to be invariant along
$z$), and the perpendicular magnetic anisotropy, which tends to maximize its
(out-of-plane) $y$ component. The inhomogeneity of the texture is induced by
the dipolar interaction. In order to avoid the large demagnetizing energy
density that would be associated with a uniform out-of-plane excursion of
magnetization ($\frac{\mu_{0}}{2}M_{y}^{2}$, $\mu_{0}$ being the permeability
of vacuum), the latter favors an alternation of sign of $M_{y}$ in the film
interior, along the transverse direction $x$, together with a closure of the
resulting magnetic flux via quadrature sign changes of the transverse
component $M_{x}$ at both film surfaces. Finally, the overall distribution is
smoothed out by the exchange energy density $A\nabla^{2}\bm{M}$, where $A$ is
the exchange stiffness constant. For the magnetic parameters of our film
($M_{S}=1330~{}$kA/m, $K=32.7~{}$kJ/m3 $A=16.6~{}$pJ/m) [24] the comparison of
the different energy scales deduced from a dimensional analysis
$\mu_{0}M_{S}^{2}>>K\sim\frac{A}{D^{2}}$ suggests the use of a stray-field-
free ansatz of the magnetization distribution, which cancels the dominant
demagnetizing energy while reducing the magnetic anisotropy contribution with
respect to an in-plane saturated state. Following Hubert [25], we write this
ansatz as a combination of two sinusoidal functions in quadrature with each
other:

$\displaystyle\frac{\bm{M}}{M_{S}}(x,y)=$
$\displaystyle|\psi|{\Big{\\{}}\text{sin}\left[k(x-\delta)\right]\text{sin}\left(\frac{\pi
y}{D}\right)\bm{\hat{x}}+$ (1)
$\displaystyle\frac{kD}{\pi}\text{cos}\left[k(x-\delta)\right]\text{cos}\left(\frac{\pi
y}{D}\right)\bm{\hat{y}}{\Big{\\}}}.$

This ensures the vanishing of both surface magnetic charges ($M_{y}(y=\pm
D/2)=0$) and volume magnetic charges ($\frac{\partial M_{x}}{\partial
x}+\frac{\partial M_{y}}{\partial y}=0$). Here $k$ is the wavenumber of the
modulation ($k=2\pi/\lambda$ for which we will select later the value
minimizing the total energy). With the two other parameters defining the
modulation, namely its amplitude $|\psi|$ (taken as the maximum in-plane
excursion of the magnetization) and its lateral positioning $\delta$ [measured
with respect to an arbitrary reference point, see Fig. 1(d)], we build a
complex number $\psi=|\psi|e^{ik\delta}$ that we identify with the order
parameter of the stripe texture. Then, following Landau theory, [3] we develop
the spatially averaged magnetic energy density $\left\langle
E\right\rangle=E_{0}+a(k,H)|\psi|^{2}+b(k,H)|\psi|^{4}+O(|\psi|^{6})$, the
terms with odd powers being zero by symmetry (see [24] for details). This
simple form allows us to derive an analytical estimate of the field and
wavenumber at the critical point
($\mu_{0}H_{c}=\frac{2\,K}{M_{S}}-\frac{4\,\pi\sqrt{A\,K}}{D\,M_{s}}=10.5~{}$mT,
$k_{c}=\frac{\pi}{D}\sqrt{\frac{2\,K+\mu_{0}M_{S}H_{c}}{2\,K-\mu_{0}\,M_{S}\,H_{c}}}=21.6~{}$rad/$\mu$m,
respectively, as deduced from the conditions $a=\frac{\partial a}{\partial
k}=0$), and the modulation amplitude below nucleation
$\psi_{0}=\sqrt{\frac{-a(k_{c},H)}{2\,b(k_{c},H)}}$ . Despite its simplicity,
this explicit model captures most of the physics of the weak stripes observed
experimentally. Despite a small underestimate of critical field of $1.3~{}$mT,
it is also in good agreement with micromagnetic simulations (see [24]).

Figure 2: (a) Sketch of the Damon-Eshbach geometry, magnetization precession
(black ellipses) and basis functions used for modelling (red and blue bars).
The thin bars can be combined to form the stray-field-free ansatz of Eq. (1),
while the thick bars, orthogonal to them, carry non-zero magnetic pseudo-
charges, as shown at the bottom of the film. (b) Spin-wave dispersion
relations calculated analytically using the basis of panel (a) (see details in
the text and in [24]), for an external field of 30, 20, 14, 11, 9 and
$7~{}$mT. For the last two values, the saturated state is unstable, and the
spin-wave frequency becomes imaginary over a certain range of wavenumbers (see
the distinct vertical scale at the bottom of the graph). (c,d), Color maps of
the Brillouin light scattering intensity measured as function of the
transferred wave-vector (scattering geometry shown by the green arrows in
panel (a) and transferred frequency $f$ under an external field $\mu_{0}H=30$
and $14~{}$mT, respectively. The lines show the calculated soft mode frequency
[same as in panel (b)].

Our minimal model of stripe nucleation forms the skeleton of a description of
the spin wave dynamics in this regime: We place ourselves in the saturated
state and consider a plane-wave of angular frequency $\omega$ and wavenumber
$k$ propagating along $\bm{\hat{x}}$ , $\bm{m}(y)\,e^{i(\omega t-kx)}$ [spin-
wave configuration referred to as Damon-Eshbach, Fig. 2(a)] [26]. Its complex
amplitude distribution $\bm{m}(y)$ is written as a linear combination of four
vector functions : the two functions sin$(\pi y/D)\bm{\hat{x}}$ and cos$(\pi
y/D)\bm{\hat{y}}$ appearing in the static ansatz of Eq.(1) and two extra ones
sin$(\pi y/D)\bm{\hat{y}}$ and cos$(\pi y/D)\bm{\hat{x}}$ obtained through a
local 90∘ rotation and necessary for describing the precession of the
magnetization. Unlike the former, the latter pair of functions carries
magnetic pseudo-charges [Fig. 2(a)], so that the precession will lead to
sizeable stray fields. We identify the associated demagnetizing energy with a
kinetic energy, which, combined with the potential energy $\left\langle
E\right\rangle$ described above, will determine the mode frequency, in analogy
with the Döring mass term of magnetic domain-wall dynamics [11]. More
specifically, we shall project the linearized equation of motion of the
magnetization, $i\omega\bm{m}=\gamma M_{S}\bm{\hat{z}}\times\frac{\partial
E}{\partial\bm{m}}$, where $\gamma$ is the gyromagnetic ratio, onto this basis
set and diagonalize the resulting 4$\times$4 matrix (Eq. S9) to obtain
eigenfrequencies and eigenmodes [27]. The spin-wave dispersion relation
$\omega(k)/(2\,\pi)$ obtained for the lowest frequency mode is shown in the
top panel of Fig. 2(b) for different values of the applied field. Far above
nucleation, one distinguishes clearly a non-monotonic wave-vector dependence
with a minimum frequency at a non-zero wave vector of about $21~{}$rad/$\mu$m.
This minimum constitutes the dynamic precursor of stripe nucleation: its
wavenumber is the critical one $k_{c}$ of the stripe modulation and its
frequency tends to zero as $H$ approaches $H_{c}$. This allows us to
reinterpret stripe domain nucleation as the freezing of the lowest frequency
spin-wave of the system [28].

To observe this characteristic mode softening, we now resort to Brillouin
light scattering (BLS), an inelastic light scattering technique capable of
probing thermally excited spin waves over a broad range of wave-vectors. The
measurement geometry is sketched in Fig. 2(a): the film is illuminated with a
laser beam under an angle of incidence $\theta_{i}$ in the presence of a
magnetic field $\bm{H}$ perpendicular to the incidence plane. The
backscattered light is collected and frequency-analyzed with a high finesse
Fabry-Pérot interferometer. Due to the conservation of energy and in-plane
linear momentum, the frequency shift of the scattered light and the
transferred wave-vector
($k=4\pi/\lambda_{\text{laser}}\,\text{sin}(\theta_{i})$) are to be identified
with those of the quasi-particles absorbed/emitted during the scattering
process. In our case, these are the spin-waves which couple to light via
magneto-optical effects [29].

Fig. 2(c) shows a color plot of the spectra recorded in the saturated state
with $\mu_{0}H=30~{}$mT for different transferred wave vector, thus providing
a direct picture of the spin-wave dispersions, up to a wave-vector of about
$k=18~{}$rad/$\mu$m. One recognizes clearly three spin-wave branches. The two
highest ones with nearly constant frequency can be assigned to perpendicular
standing spin waves with an increasing number of nodal planes across film
thickness [24]. On the other hand, the lowest frequency branch clearly shows a
negative group velocity (frequency decreases as wave-vector increases) for
wave vector above a few rad/µm, which fits very well with the dispersion
relation calculated for the stripe precursor mode (red line). This negative
velocity can appear surprising at first glance since spin-waves in the Damon-
Eshbach configuration normally have positive group velocity [26]. However, it
was already observed in the presence of a perpendicular magnetic anisotropy
[20] and it finds a natural explanation here: The perpendicular magnetic
anisotropy favors the out-of plane component of the magnetization precession
with respect to the in-plane one, which allows for a certain degree of dipolar
field cancellation at sufficiently short length-scale. Decreasing the field to
$14~{}$mT, i.e. about $2~{}$mT above stripe nucleation, leads to a clear
frequency decrease [Fig. 2(d)], which can be extrapolated to a perfect
softening at ($k_{c},H_{c}$). Let us now examine the spin-wave dispersion
below nucleation. Symbols in Fig. 3(a) show the positions of the Brillouin
light scattering peaks measured at $7~{}$mT (see raw data in Fig. S5 in the
Supplementary Information [24]). We distinguish clearly two branches. The
frequency of the bottom one decreases rapidly as a function of wavenumber down
to an extrapolate $f\sim 0$ at $k_{c}$. The frequency of the top one decreases
much slower and extrapolates to a value of about $3.5~{}$GHz at $k_{c}$. To
help interpret these observations, we have performed mumax3 finite difference
micromagnetic simulations [30] of spin-wave propagation. Fig. 3(a)-(c) show
color plots of the amplitude spectral density obtained upon Fourier
transforming the spatio-temporal evolution of the surface magnetization
following a localized pulse excitation (see [24]) for field values of 7, 10
and $11.7~{}$mT, respectively. Right below nucleation [Fig. 3(c)], one
distinguishes a secondary branch with a non-zero minimum frequency emerging
from the characteristic $(k,f)=(k_{c},0)$ cusp. Upon reducing further the
field, the minimum frequency of this secondary branch gradually increases,
while the main branch remains soft [Fig. 3(b)]. These two branches account
very well for the measured inelastic peaks positions [Fig. 3(a)]. This
phenomenology can be understood as the mesoscopic counterpart of the one
occurring at the microscopic level for charge density waves [31] and
incommensurate displacive phases [7] whose nucleation is also described by the
softening of a dynamic mode which splits into amplitude and phase modes upon
symmetry breaking. Fig. 3(d,e) show maps of the out-of-plane component of the
dynamic magnetization $m_{y}(x,y)$ at $k_{c}$ for these two branches, together
with the distribution of the transverse static magnetization, sketched as a
vector plot. One clearly recognizes two similar patterns phase-shifted by
$\pi/2$. For the zero frequency mode, the antinodes of the dynamic
magnetization are aligned with the nodes of the static distribution [Fig.
3(d)]. In contrast, for the non-zero frequency mode, dynamic and static
antinodes are aligned with each other [Fig. 3(e)]. The evolution with respect
to the spectrum above saturation is explained as follows: The phase transition
being second order, the overall spin wave spectrum changes smoothly upon
stripe nucleation. The soft spin waves actually adapt in the form of intensity
modulations in phase or in quadrature with respect to the nucleated texture.

Figure 3: (a,b,c) Color plot of the simulated spin-wave spectrum amplitude
density as function of wave-vector and frequency, for a field of 7, 10 and
$11.7~{}$mT, respectively. In panel (a), the positions of the measured
Brillouin light scattering peaks are reported as circles and squares and the
Higgs mode frequency determined analytically is shown as a star. (d) Color
plot of the distribution of the out-of-plane component of the dynamic
magnetization for the Goldstone mode [$k=k_{c}$, $f=0.1~{}$GHz in panel (b)].
The vector plot shows the distribution of the transverse magnetization of the
underlying static stripe modulation. (e) Same for the Higgs mode [$k=k_{c}$,
$f=3.5~{}$GHz in panel (b)]. The green arrows sketch the motion of the stripe
structure, namely a rigid displacement and an amplitude oscillation in panels
(d) and (e), respectively. (f) Color plot of the microwave absorption measured
in the stripe phase as function of field and frequency, for the longitudinal
pumping geometry sketched in the inset. (g) Color plot of the Brillouin light
scattering intensity measured in the saturated phase as function of field and
frequency for a transferred wave-vector $k_{c}=21~{}$rad/$\mu$m. The solid and
dashed lines in panels (f) and (g) show the soft/Higgs mode frequency
extracted from our analytical approach (see details in the text and the
Supplementary Information [24]) and from micromagnetic simulations,
respectively. The (min,max) values for color plots are (4,20), (-1,-18)
$\times 10^{-3}$ and (100,300) for panels (a-c), (f) and (g), respectively.

We shall now identify these two types of modulations with the Goldstone and
Higgs modes of the stripe texture. According to Fig. 1(a), in the low symmetry
phase, one should distinguish phase oscillations
$\delta=\delta_{0}+\delta_{1}\text{cos}(\omega t)$ and amplitude oscillations
$|\psi|=\psi_{0}+\psi_{1}\text{cos}(\omega t)$. As these time-oscillations
occur around an equilibrium which is oscillating in space (e.g.
$M_{y}(x,0)\propto\text{cos}\left[k_{c}(x-\delta_{0})\right]$), they
correspond to non-zero wavenumber spin waves. More precisely, the dynamic
magnetization profiles (e.g. $\delta_{1}\frac{\partial M_{y}}{\partial
x}(x,0)\propto\text{sin}\left[k_{c}(x-\delta_{0})\right]$ and
$\psi_{1}M_{y}(x,0)\propto\text{cos}\left[k_{c}(x-\delta_{0})\right]$) can be
interpreted as two standing wave patterns formed by the interference between
counterpropagating spin waves with $k=\pm k_{c}$ and a well-defined phase
difference of 0 or $\pi/2$ with respect to the equilibrium modulation. This
corresponds exactly to the modal distributions of Fig. 3(d,e), to be
identified with the Goldstone and Higgs modes, respectively. The zero
frequency of the former is associated with the translation invariance of the
whole stripe texture: the total energy is exactly the same whatever the value
of the lateral shift $\delta$ in Fig. 1(d), as already noticed by [20]. The
non-zero frequency of the Higgs mode arises from the finite curvature of the
energy potential along the radial direction. We can evaluate this frequency
via a suitable extension of the description of the dynamics above saturation
[7, 10]. From the expression of the Landau potential, it can be shown that the
positive curvature around the stable equilibrium value
$\psi_{0}=\sqrt{\frac{-a}{2\,b}}$ is related to the negative curvature around
the unstable equilibrium value $\psi=0$, namely
$\left(\frac{\partial^{2}E}{\partial|\psi|^{2}}\right)_{\psi_{0}}=-2\left(\frac{\partial^{2}E}{\partial|\psi|^{2}}\right)_{0}$.
Then, we can write the frequency of the amplitude mode as
$\omega\propto\sqrt{\frac{\partial^{2}E}{\partial|\psi|^{2}}\frac{\partial^{2}E}{\partial\tau^{2}}}$
where $\frac{\partial^{2}E}{\partial\tau^{2}}$ is a “kinetic” term accounting
for the extra energy generated by magnetization precession (Fig. S3 in [24]).
This is essentially a strong demagnetizing contribution [Fig. 2(a)] which does
not depend on the subtle energy balance that governs nucleation. It can
therefore be assumed to be the same at $|\psi|=0$ and $\psi_{0}$. Accordingly,
we obtain $\omega|_{\psi_{0}}=-i\sqrt{2}\omega|_{0}$, which allows us to
relate the frequency of the Higgs mode in the low symmetry phase to the growth
rate of the unstable mode in a fictitious high symmetry state below
nucleation. Using the growth rate calculated from our spin wave ansatz [bottom
panel in Fig. 2(b)], we obtain the value shown as a star in Fig. 3(a), in good
agreement with the numerical simulations.

To characterize directly the approach towards the critical point from both
sides, we finally combine two techniques [Fig. 3(f,g)]. The mode softening in
the saturated phase ($H>H_{c}$) is followed by Brillouin light scattering,
placing ourselves exactly at $k_{c}$ [Fig. 3(g)]. There, one distinguishes
clearly a gradual drop which follows precisely the characteristic softening
predicted for the precursor mode by our analytical approach (solid line) or by
micromagnetic simulations (dashed line). This technique becomes less efficient
in the stripe phase because of the dephasing induced by inhomogenities of the
stripe phase across the several tens of $\mu$m of the focal spot of the laser.
This results in a sizeable drop of the light scattering signal at high
wavenumber (Fig. S5(b) in [24]). Rather, we resort to another technique able
to probe the stripe texture in a scalar way [32] (i.e. regardless the phase of
the nucleated texture [6]) namely ferromagnetic resonance under longitudinal
pumping. The measurement configuration is shown in the inset of Fig. 3(f). The
film is placed on top of a broadband transmission line (see Fig. S6(a) in
[24]), which generates a (mostly in-plane and homogeneous) microwave magnetic
field $\bm{h}_{1}$, the static field $\bm{H}$ being oriented parallel to it.
This can be viewed as an analog of the lattice depth modulation technique used
in cold atoms systems [9]: During a microwave cycle, the pumping field
alternatively increases and decreases the total external field, which
translates into an oscillation of the Zeeman energy and, in turn, into an
oscillation of the amplitude of the order parameter. Fig. 3(f) shows the
imaginary part of the effective magnetic susceptibility $\chi_{\text{eff}}$ of
the loaded transmission line (which is proportional to the microwave
absorption coefficient) as function of both the microwave frequency and the
static field intensity. One recognizes clearly a strong absorption feature
below $H_{c}$, with a frequency increasing from about $1.5~{}$GHz in good
agreement with the frequency upturn predicted by our analytical approach
(solid line) and simulations (dashed line)[33]. This absorption is associated
to the excitation torque $\bm{M}(x,y)\times\bm{h}_{1}$ which is zero in the
saturated state but increases gradually below nucleation due to the transverse
static components of the stripe modulation ($\bm{M}\perp\bm{h}_{1}$). This
allows us to shed a new light onto previous experiments of ferromagnetic
resonance in stripe domains, traditionally interpreted in terms of distinct
domain- and domain-wall resonances [17, 18, 19, 21]. Our analysis indicates
that the mode probed by longitudinal pumping corresponds to a spin-wave
already present in the saturated state at $k=k_{c}$ and made accessible to a
$k=0$ experiment by a Bragg scattering process induced by the nucleated
texture.

##  III Conclusion

To conclude, we show that both the close-to-nucleation statics and the low-
frequency dynamics of magnetic stripe domains are entirely determined by the
specific behaviour of flux-closure Damon-Eshbach spin-waves around a certain
wave vector $k_{c}$. The evolution of the whole spin-wave dispersion upon the
high to low-symmetry transition can be analyzed in universal terms invoking
the softening of the low frequency spin-wave branch, its freezing in the form
of the translation-symmetry-breaking static stripe modulation, and its
subsequent splitting into a Goldstone and a Higgs branch. The identified
intimate relationship between the statics and the dynamics of a magnetic
texture is a generic feature that could be taken advantage of in future
developments of magnonics [34, 35, 22, 36, 37]. More importantly, the
described system constitutes a particularly simple and explicit implementation
of the dynamics around symmetry-breaking phase transitions, paving the way for
further exploration, including time-resolved imaging studies, extension to the
non-linear regime, or the quest for a possible Higgs-Anderson mechanism for
magnons.

###### Acknowledgements.

We thank S. Cherifi-Hertel and H. Majjad for complementary magnetic imaging,
T. Böttcher and M. Ruhwedel for support with the BLS measurements, and J.V.
Kim, J. P. Adam, T. Devolder, J. Solano and G. DeLoubens for fruitful
discussions. This work was funded primarily by the French National Research
Agency (ANR) through the project SWANGATE (ANR-16-CE24-0027). We acknowledge
the technical support of the STnano clean-room platform partly funded by the
Interdisciplinary Thematic Institute QMat, as part of the ITI 2021-2028
program of the University of Strasbourg, CNRS and Inserm, IdEx Unistra (ANR 10
IDEX 0002), SFRI STRAT’US project (ANR 20 SFRI 0012) and ANR-17-EURE-0024
under the framework of the French Investments for the Future Program. We
acknowledge support from the High Performance Computing Center of the
University of Strasbourg which provides access to computing resources, partly
funded by the Equipex Equip@Meso project (Programme Investissements d’Avenir)
and the CPER Alsacalcul/Big Data. We also acknowledge financial support from
Region Grand Est through its FRCR call (NanoTeraHertz and RaNGE projects),
from the impact project LUE-N4S part of the French PIA project “Lorraine
Université d’Excellence”, reference ANR-15IDEX-04-LUE and from the “FEDER-FSE
Lorraine et Massif Vosges 2014–2020”, a European Union Program. MG and PP
acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - TRR 173 - 268565370 (“Spin + X”, Project B01) and
Priority Programme SPP2137 Skyrmionics. M.Gr., D. S., Y.H. and M.B. designed
the experiments. M. H. fabricated the magnetic films. K. A. and D. L.
performed magnetometry and magnetic imaging. M. Ge. performed Brillouin light
scattering measurements. M. Gr. performed ferromagnetic measurements and
analysed the experimental data, with the help of M. Ge., P. P. and M. B. M.
Gr. and M. B. developed the analytical model. D. S. performed the
micromagnetic simulations with the help of Y. H. and M. Gr. M. Gr. and M. B.
wrote the manuscript with the help of all authors.

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