11institutetext:  Max-Planck-Institute für Radioastronomie, Auf dem Hügel 69,
D-53121 Bonn, Germany 22institutetext: University of Illinois, Urbana-
Champagne, USA 33institutetext: Italian ALMA Regional Centre, INAF-Istituto di
Radioastronomia, Via P. Gobetti 101, I-40129 Bologna, Italy 44institutetext:
Max Planck Institut for Extraterrestrial Physics, D-86 Garching bei Muenchen,
Germany 55institutetext: Physics and Astronomy Department, University of
California, Los Angeles, CA 90095-1547, USA<EMAIL_ADDRESS>66institutetext: Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden
Street, Cambridge, MA 02138, USA 77institutetext: Technical University of
Munich, 85747 Garching, Germany

#  General Relativistic effects and the NIR variability of Sgr A* II

A systematic approach to temporal asymmetry

Sebastiano D. von Fellenberg 11 Gunther Witzel 11 Michi Bauboeck 22 Hui-Hsuan
Chung 11 Nicola Marchili 33 Greg Martinez 55 Matteo Sadun-Bordoni 44 Guillaume
Bourdarot 44 Tuan Do 55 Antonia Drescher 44 Giovanni Fazio 66 Frank Eisenhauer
44 Reinhard Genzel 44 Stefan Gillessen 44 Joseph L. Hora 66 Felix Mang 4477
Thomas Ott 44 Howard A. Smith 66 Eduardo Ros 11 Diogo C. Ribeiro 44 Felix
Widmann 44 S. P. Willner 66 J. Anton Zensus 11

(Created ; accepted )

A systematic study, based on the third-moment structure function, of Sgr A*’s
variability finds an exponential rise time
$\tau_{1,\rm{obs}}=14.8^{+0.4}_{-1.5}\leavevmode\nobreak\ \mathrm{minutes}$
and decay time $\tau_{2,\rm{obs}}=13.1^{+1.3}_{-1.4}\leavevmode\nobreak\
\mathrm{minutes}$. This symmetry of the flux-density variability is consistent
with earlier work, and we interpret it as caused by the dominance of Doppler
boosting, as opposed to gravitational lensing, in Sgr A*’s light curve. A
relativistic, semi-physical model of Sgr A* confirms an inclination angle
$i\leq 45\degree$. The model also shows that the emission of the intrinsic
radiative process can have some asymmetry even though the observed emission
does not. The third-moment structure function, which is a measure of the
skewness of the light-curve increments, may be a useful summary statistic in
other contexts of astronomy because it senses only temporal asymmetry, i.e.,
it averages to zero for any temporally symmetric signal.

###### Key Words.:

Galactic center – Methods: statistical – Black hole physics

##  1 Introduction

Ever since the detection of the near-infrared (NIR) and X-ray counterpart
(Baganoff et al. 2001; Genzel et al. 2003) of the radio source Sgr A*, the
source’s variability at all wavelengths has been the subject of intense study.
Sgr A*’s NIR light curve shows occasional bright outbursts, typically
phenomenologically referred to as flares. A NIR flare is always present when
an X-ray outburst occurs (Eckart et al. 2008), but the converse is not true:
only a fraction of bright NIR flares are accompanied by an X-ray flare (e.g.,
Dodds-Eden et al. 2009). Even when flares occur in both bands, the respective
flux levels are not highly correlated (e.g., GRAVITY Collaboration et al.
2021). The spectral shape of NIR-to X-ray flares has now been established to
be rising in the NIR as $\nu L_{\nu}\propto\nu^{+0.5}$ and falling in the
X-ray as $\nu L_{\nu}\propto\nu^{-0.5}$ (Hornstein et al. 2007; Yusef-Zadeh et
al. 2009; Ponti et al. 2017; GRAVITY Collaboration et al. 2021).

Modern interferometric observatories can detect Sgr A* at all times (GRAVITY
Collaboration et al. 2020a), which reveals the Sgr A* is a constantly variable
source even in the absence of higher flux peaks (flares). This NIR variability
has been studied in detail: the source shows stochastic, red-noise-like
behavior akin to many other compact objects (e.g., Do et al. 2009). The power
spectrum shows no features and breaks into uncorrelated white noise on time
scales of ${\approx}150\leavevmode\nobreak\ \mathrm{minutes}$ (e.g., Witzel et
al. 2012, 2018). The RMS–flux relation is linear with no significant
difference in variability for bright and faint parts of the light curve, again
very similar to other compact objects (Witzel et al. 2012; GRAVITY
Collaboration et al. 2020a; Weldon et al. 2023). On the other hand, the NIR
flux distribution is log-right-skewed, i.e., it shows a tail with respect to a
log-normal distribution (Dodds-Eden et al. 2009; Witzel et al. 2012; GRAVITY
Collaboration et al. 2020a). This tail is typically used to motivate the
concept of flares as events in the accretion flow, contrasting with the
quiescent condition.

If flares are modeled as distinct events, two types of one-zone models can
describe the data. One type peaks at submillimeter (submm) wavelengths and
produces NIR and X-ray flux through synchrotron-self-Compton (SSC) emission
(e.g., Eckart et al. 2008; Yusef-Zadeh et al. 2009; Witzel et al. 2021). In
the other type, both the NIR and X-ray emission are caused entirely by
synchrotron emission, and the resulting but unobserved SSC emission peaks at
gamma-ray energies. Models of the first type require enormous electron
densities to obtain sufficient SSC flux to reproduce the X-ray spectral index
and therefore are often disfavored (Dodds-Eden et al. 2011; Ponti et al. 2017;
GRAVITY Collaboration et al. 2021).

An additional observational input to models is that the astrometry of NIR
flares can be tracked. Observed flares show close-to-circular proper motions
in the sky, the so-called orbital motions (GRAVITY Collaboration et al. 2018,
2020b). In addition, NIR flares show characteristic loop-like swings in linear
polarization (GRAVITY Collaboration et al. 2020c, 2023). These NIR
observations were corroborated by EHT–ALMA observations (Wielgus et al. 2022;
Event Horizon Telescope Collaboration et al. 2022a), which showed that a
similar polarization loop was detected in the $230\leavevmode\nobreak\
\mathrm{GHz}$ radio light curve, incident right after a bright X-ray flare.
The astrometric behavior supports the notion of an event-like character of
flares, where flares are interpreted as orbiting “hot spots” in the accretion
flow.

If the flare motion is interpreted as an accelerating point-particle orbiting
Sgr A*, orbital parameters such as the radius and the viewing angle can be
constrained. Both the NIR data and the submm polarization loop (Wielgus et al.
2022) are consistent with a face-on inclination. The Sgr A* EHT image also
indicates that the accretion flow is viewed face-on (Event Horizon Telescope
Collaboration et al. 2022a, b, c, d, e, f). Thus there are four independent
indications that Sgr A*’s accretion flow is oriented face-on: NIR astrometry
of flares, NIR polarization of flares, radio polarization of the light curve
after an NIR flare, and the EHT image of Sgr A*.

Paper I (von Fellenberg et al. 2023) analyzed light curves obtained in the NIR
($4.5\leavevmode\nobreak\ \mathrm{micron}$, Spitzer) and in the X-ray (2–8keV,
Chandra, XMM-Newton). On average, Sgr A* flares are well approximated by a
double-sided, symmetric, exponential profile with a rise and decay time
$\tau\approx 15\leavevmode\nobreak\ \mathrm{min}$. For a fully relativistic
and ray-traced orbiting hot-spot model (GRAVITY Collaboration et al. 2020b),
such a profile can only be explained if gravitational lensing in the light
curve is not important, i.e., if the orbits are viewed face-on. In this
picture, the symmetry of the average flare shape is explained by dominant
Doppler-boosting as the hot spot orbits. When averaged over several flares,
the different initial orbital positions lead to symmetric Doppler
amplification of the underlying emission. The symmetric flare shape is a
fifth, independent constraint on the viewing angle of Sgr A*. This constraint
uses only the variability and the assumption of moving hot spots close to Sgr
A*. This constraint therefore applies to all hot-spot models, as the
variability induced by such a moving hot spot is valid for any radiative model
of the underlying emission mechanism, as any close-to-circular motion close to
Sgr A* will be lensed and boosted.

A limitation in paper I was the arbitrary selection of flares, their flux
normalization, and the shifting and adding applied. This paper generalizes the
analysis by first establishing summary statistics to measure asymmetry based
on flux differences. Section 3 introduces a statistical model to measure
asymmetry in a stochastic process, and Section 4 applies the method to fit the
observed Sgr A* light curves (Section 2). Section 5 introduces a semi-physical
general-relativistic model and uses it to derive intrinsic Sgr A* properties
consistent with the observed light curves, and Section 6 puts the results in
context. All fits to data make use of the approximate Bayesian Computation
(ABC) method as described in subsection 4.3.

##  2 Data

Our data consist of eight NIR observations by Spitzer/IRAC, each $\sim$24
continuous hours, observed from 2013 to 2017. The timing resolution is 8.4
seconds, which we re-binned to a cadence of 1 minute. Throughout this paper we
will use data that are not corrected for extinction, i.e., we will use
observed flux densities. This choice makes no difference to our results except
for overall normalization of light curves. The data are public (Witzel et al.
2018, 2021), and the calibration approach was explained by Hora et al. (2014).
As those authors explain, the flux density zeropoint is unknown, but that
makes no difference for this paper.

##  3 Light curves and structure functions

To characterize the statistical properties of light curves, this paper uses
flux-density differences of pairs of points separated by a given time lag
$\tau$. This is a common approach in astronomy (e.g., Simonetti et al. 1985).
Figure 1 illustrates the underlying concept: for each time-lag $\tau$, all
available flux pairs in a light curve are measured and histogrammed. Then the
statistical moments of each histogram, here the mean ($\mu_{1}$), the variance
($\mu_{2}$), and the (Fischer) skewness ($\mu_{3}$), are calculated. The value
of any of these moments as a function of the time lag $\tau_{1}$ is the
structure function:

$\mathrm{SF}(\tau,\mu_{i})=c\times\mu_{i}(\tau)\quad,$ (1)

where $c$ is a normalization constant. For practical reasons, and to increase
the fidelity of the estimate, time lags are grouped in bins $\tau_{i}$ of
suitable width in $\tau$.

Figure 1: Cartoon showing the concept of moment-based summary statistics for
light-curve analysis. The bottom panel shows an example light curve, the
orange and blue annotation indicate the flux differences for two given time
lags $\tau_{1}$ and $\tau_{2}$. The top panel shows the histogram of flux
differences for every time lag $\tau_{i}$. Each histogram has its respective
statistical moments: mean ($\mu_{1}$), variance ($\mu_{2}$), and skewness
($\mu_{3}$). At upper right, variance and skewness are plotted as functions of
time lag.

###  3.1 The second-moment structure function

The vast majority of astronomical works use the second-moment (variance)
structure function $\mu_{2}=\sigma^{2}$. Different normalizations have been
used in the literature, but a standard formula for the unnormalized SF
($\rm{SF_{\mu_{2}}}$, e.g., Simonetti et al. 1985) is

$\mathrm{SF^{\prime}}_{\mu_{2}}(\tau_{i})=\frac{1}{N_{i}}\sum_{t_{j},t_{k}}[F(t_{j})-F(t_{k})]^{2}\leavevmode\nobreak\
,\leavevmode\nobreak\ \tau_{i}<(t_{j}-t_{k})\leq\tau_{i+1}\leavevmode\nobreak\
.$ (2)

Here, $N$ is the number of data pairs in each time-lag bin $i$, and
$F(t_{j})-F(t_{k})$ is the difference in flux density between times $t_{j}$
and $t_{k}$.

For this work, we will normalize the structure-function to the interval
$\mathrm{SF}(\tau)\in[0,1]$:

$\mathrm{SF}_{\mu_{2}}(\tau_{i})=\frac{1}{4N_{i}\sigma^{2}(F)}\mathrm{SF^{\prime}}_{\mu_{2}}(\tau_{i})\quad,$
(3)

which is identical to definition in Equation 2 up to the factor
$1/4\sigma^{2}(F)$, where $\sigma^{2}(F)$ is the variance of the light curve.
This choice of normalization renders the $\mathrm{SF_{\mu_{2}}}$
dimensionless, does not depend on the absolute flux levels, and measures only
the correlation in the data. This can be understood when a mathematical
property of the second moment structure function is taken into account. The
structure function forms a Fourier pair of the power spectrum of a light curve
(a consequence of the Parceval’s theorem (des Chênes 1799)), and this
normalization removes the absolute value of the power spectrum. Therefore
$\rm{SF_{\mu_{2}}}$ is sensitive only to the slope of the power spectrum.

###  3.2 The third-moment structure function

Following the logic of $\rm{SF_{\mu_{2}}}$, the third moment of the flux
density differences, the skewness $\mu_{3}$ defines the unnormalized third-
moment structure function

$\mathrm{SF^{\prime}}_{\mu_{3}}(\tau_{i})=\frac{1}{N_{i}}\sum_{t_{j},t_{k}}[F(t_{j})-F(t_{k})]^{3}$
(4)

with the same symbols as in Equation 2. To obtain a unit-free normalization in
the interval $\mathrm{SF}_{\mu_{3}}(\tau)\in[0,1]$ we apply

$\mathrm{SF}_{\mu_{3}}(\tau_{i})=\dfrac{\sqrt{N_{\rm{tot}}}}{\mu_{3}(F)}\times\dfrac{\rm{SF^{\prime}_{\mu_{3}}(\tau_{i})}}{\mathrm{(SF^{\prime}}_{\mu_{2}}(\tau_{i}))^{3/2}}\leavevmode\nobreak\
,$ (5)

where $\sigma^{2}(F)$ denotes the variance of the light curve, and
$\mu_{3}(F)$ denotes its Fischer skewness (e.g., Joanes & Gill 1998) defined
as

$\mu_{3}=\dfrac{E[(X-\mu)^{3}]}{E[(X-\mu)^{2}]^{3/2}}\quad,$ (6)

where $E$ stands for the expectation value. As for $\rm{SF_{\mu_{2}}(\tau)}$,
this dimensionless normalization ensures that the third moment structure
function is sensitive only to the asymmetry in the flux differences and is not
dependent on the slope of the power spectrum.

The third-moment structure functions possess several useful properties
illustrated in Figure 2.111There have been attempts to use modifications of
the structure-function to measure asymmetry in light curves (e.g., Kawaguchi
et al. 1998; Hawkins 2002; Chen & Wang 2015), non of which probe temporal
asymmetry directly. Shishov & Smirnova (2005) used an asymmetry function to
probe the flux density distributions of pulsars for interstellar
scintillation, which, however, does not probe temporal asymmetry. The third
moment structure function shows a negative sign for positively skewed flares
and a negative sign for negatively skewed flares. For symmetrical flares, the
skewness function is approximately zero for all time lags. Similarly, white
noise, which is added to all curves, is approximately zero, on average, in any
one realization.

Thus the $\rm{SF_{\mu_{3}}(\tau)}$ is sensitive only to asymmetric signals,
and white noise averages to zero when there are enough data points. These
properties make the third-moment structure function a versatile metric for
astronomical signals that are intrinsically asymmetric (e.g. QPEs Arcodia et
al. (2021), eclipsing binaries Gautam et al. (2024), …).

Figure 2: Illustration of the properties of $\rm{SF_{\mu_{3}}(\tau)}$. The
left panel shows four artificial light curves of different types as labeled,
and the right panel shows their skewness functions defined by Equation 5.

##  4 Quantifying asymmetry in light curves

###  4.1 An intuitive overview of the method

While $\mu_{3}$ (Equation 6) easily quantifies asymmetry in a distribution of
measurements (or random numbers), it is not immediately apparent how asymmetry
can be understood for a time series. For a singular event, asymmetry can
easily be defined by whether the left and right flanks around the event’s peak
are identical (symmetric) or not (asymmetric). While an event is not a
probability distribution, the mathematical formalism to calculate the third
moment can be applied, and positive or negative skew will translate to
positive or negative $\rm{SF_{\mu_{3}}(\tau)}$ (Equation 5) values. What works
for a single peak must also work for repeating, isolated events, and asymmetry
is easily probed in that case by the sign of $\rm{SF_{\mu_{3}}(\tau)}$, as
illustrated in Figure 2. We discuss a proper definition of symmetry under time
reversal in Appendix C and proof that in the case of symmetry the third moment
structure function is zero. The sign of the $\rm{SF_{\mu_{3}}(\tau)}$ also
probes asymmetry for different light curve types, including when individual
flares are less isolated and overlap or when no easy definition of an
individual event is apparent.

The difficult part of using $\rm{SF_{\mu_{3}}(\tau)}$ to measure asymmetry in
a light curve is to quantify the asymmetry and its statistical significance.
Any light curve can be split up into segments, some of which will, by chance,
show positive or negative values of $\rm{SF_{\mu_{3}}(\tau)}$. As for any
statistic, with fewer data pairs, the moment $\mu_{3}$ of the flux difference
distribution becomes less well determined. The following sections establish a
method to quantify the significance of asymmetry in the light curve based on
the moving-average (MA) model for time series. This model allows generic
symmetric and asymmetric light-curve generation as well as an easy definition
of the magnitude of asymmetry and its significance. This approach has largely
been inspired by the discussion of Scargle (2020), who discussed the utility
of MA models for astronomy.

###  4.2 Moving average model

The MA model is a generic way of generating a stationary light curve (e.g.,
Priestley 1988; Scargle 1981, 2020). Following the notation of Scargle (2020),
a light-curve point $X(n)$ at time step $n$ is given by

$X(n)=\sum_{k}c_{k}R(n-k)+D(n)\quad,$ (7)

where $c_{k}$ are the MA coefficients called the impulse response of the
process. $R(n)$ are uncorrelated random numbers at each time step $k$,
typically called the innovation, and $D(n)$ is a deterministic function.
Effectively, this can be expressed as the convolution of the random process
$R(n)$ with the impulse response $C$:

$X(n)=R(n)*C+D(n)\quad.$ (8)

It can be shown that, for a suitable choice of $c_{k}$ and $R(n)$, the MA
model is capable of representing any stationary light curve (Wold theorem:
Wold 1964; Scargle 2020). MA is therefore suitable to generate mock light
curves depending only on the choice of impulse response, the chosen
distribution of random numbers, and the luck of the actual random numbers
generated.

In generating mock light curves, the random numbers $R(n)$ need not be drawn
from a uniform distribution. Again following Scargle (2020), we chose a
$\beta$-distribution, i.e., a random variable $x$, drawn from uniformly
distributed random numbers $\mathcal{U}\in[0,1]$, generates

$R(x)=x^{\alpha}\leavevmode\nobreak\ ,\quad\alpha>0\quad.$ (9)

Choosing $\alpha=0$ gives white noise. Figure 11 demonstrates the effect of
larger values of $\alpha$: $\alpha=10$ leads to a non-uniform, but not highly
skewed, distribution of random values $R$. This, when convolved with the
kernel $C$, leads to a light curve where individual events are hardly stand
out. Higher values of $\alpha$ lead to more skewed innovations and more
distinct individual events in the light curve.

The choice of kernel $C$ is arbitrary, and while it can be measured from the
data, it is difficult to find a good kernel a priori. Paper I showed that
flares of Sgr A* are well represented, at least on average, by a double
exponential function. In particular, Paper I used a PCA decomposition of
stacked flares to obtain a de-noised kernel, derive a $\tau_{+/-}\sim
15\leavevmode\nobreak\ \mathrm{minutes}$, and show that PCA can differentiate
different kernel shapes for a broad range of MA model parameters. This
motivates our choice of a doubled-sided exponential parameterized by
$\tau_{1}$ and $\tau_{2}$ for the impulse response:

$C(t,\tau_{1},\tau_{2})=\cases{e}^{|t|/\tau_{1}}&t\leq 0\\\
e^{-|t|/\tau_{2}}t>0\leavevmode\nobreak\ .\\\ {}$ (10)

This choice has the advantage that 1) the MA model can be fit to the light-
curve data using the structure functions discussed in section 3, and 2) the
asymmetry in the light curve can be expressed as
$\Delta\tau=\tau_{1}-\tau_{2}$. The uncertainties of the values $\tau_{1}$,
$\tau_{2}$, and $\Delta\tau$ can be derived from the posteriors of the model
fit.

Figure 3: Top: one real and three mock light curves. Colored lines show mock
light curves generated with the MA model and best-fit posteriors. The black
line shows a segment of an observed Sgr A* light curve. Bottom: kernel
functions constructed using $\tau_{1,2}$ drawn from the MA-model posterior.

###  4.3 Parameter estimation by ABC

To find what values of $\tau_{1}$ and $\tau_{2}$ best fit the observed light
curves, we used the approximate Bayesian computation (ABC) approach. The model
requires two additional parameters, $\alpha$ and an overall flux-density
normalization $f_{\rm scale}$. Specifically,

$\displaystyle F$
$\displaystyle{}_{\rm{model}}(t;\tau_{1,\rm{obs}},\tau_{2,\rm{obs}},f_{\rm{scale}},\alpha)=$
(11) $\displaystyle f_{\rm{scale}}\cdot
C_{\rm{obs}}(\tau_{1,\rm{obs}},\tau_{2,\rm{obs}})*\mathcal{U}(t)^{\alpha}+\mathcal{N}(\sigma_{\rm{obs.}})-{M}(F)\leavevmode\nobreak\
,$

where $\mathcal{U}^{\alpha}$ is the innovation shown here convolved with
$C_{\rm{obs}}(\tau_{1,\rm{obs}},\tau_{1,\rm{obs}})$ the observed impulse
response as defined above, $f_{\rm{scale}}$ is the normalization constant. The
amplitude of the measured noise in the rebinned light curves
$\sigma_{\rm{obs}}$ and process median ${M}(F)$ are known in advance and need
not be solved for, but each model light curve requires a random realization of
the measured noise $\mathcal{N}(\sigma_{\rm{obs.}})$.

The model parameters were constrained using the pyabc code (Klinger et al.
2018; Schälte et al. 2022), which minimizes the distance between the summary
statistics of the data and of the model. Our choice of unit-free normalization
for both structure functions causes a selective parameter sensitivity in the
numerical problem

  * •

$\rm{SF_{\mu_{2}}(\tau)}$ mostly constrains the average width of the kernel
$\langle\tau\rangle\equiv(\tau_{1}+\tau_{2}$)/2;

  * •

$\rm{SF_{\mu_{3}}(\tau)}$ mostly constrains the impulse response difference
$\Delta\tau\equiv(\tau_{2}-\tau_{1})$.

To constrain $\alpha$ and $f_{\rm{scale}}$, we included a third summary
statistic, namely the KS-test, to estimate the difference between model and
data flux distributions. Table 1 gives an overview of model parameters and the
chosen priors.

###  4.4 Light-curve asymmetry results

Sgr A*’s observed light curve in the NIR can be successfully modeled with an
almost symmetric kernel, with an average rise and decay time $\tau\approx
15\leavevmode\nobreak\ \mathrm{minutes}$. Table 1 lists the model parameters,
their uninformative priors, and the resulting posteriors. Figure 3 illustrates
example light curves and impulse responses, which are all nearly symmetric.
The result is consistent with that of paper I, where only the shape of the
average flares in the light curve was modeled. In particular, the constraints
on the rise and fall time are

$\displaystyle\tau_{1,\rm{obs}}=14.8^{+0.4}_{-1.5}\leavevmode\nobreak\
,\leavevmode\nobreak\ \tau_{2,\rm{obs}}=13.1^{+1.3}_{-1.4}.$ (12)

and there is no significant difference between the rise and fall times of Sgr
A*’s observed impulse response $C_{\rm{obs}}$.

Figure 4: relativistic magnification in Sgr A*’s light curve. The left graph
shows the magnification kernel computed from a library of ray-traced images
GRAVITY Collaboration et al. (2020b) for two different inclination angles as
indicated by the line colors. The time interval plotted is two orbits. Both
lensing and boosting peak at orbit phase near $1.5\pi$ (Hamaus et al. 2009).
The right graph shows $\rm{SF_{\mu_{3}}(\tau)}$ computed for each
magnification curve using Equation 13.

##  5 General relativistic effects

###  5.1 A semi-physical relativistic Sgr A* model

The intrinsic variability of Sgr A* is modified by relativistic effects, such
as Doppler boosting and relativistic lensing. To explore these, we use the so-
called “orbiting hot-spot model” in which bright flares are caused by a
localized zone moving in the accretion flow. The flares are assumed to move on
bound orbits around the black hole and will be boosted and lensed depending on
the observer viewing angle and the position in the orbit. Magnifying effects
comprise both Doppler boosting and lensing. Both are inclination-angle
dependent and are minimum for face-on orbits. Increasing the inclination angle
increases both effects but by differing amounts. At $i<30\degree$, lensing is
negligible, but for $i>50\degree$, lensing dominates. In addition, lensing is
“faster,” leading to a distinct magnification peak which is broadened by the
“slower” boosting contribution.

For this work, magnifications were calculated from a ray-traced image library
(GRAVITY Collaboration et al. 2018). The model is relativistic, albeit with a
fast-light approximation (for discusion see e.g., Porth et al. 2019). Figure 4
illustrates the effects for two inclination angles. High inclination (i.e.,
edge-on view) leads to asymmetric peaks in the magnification and hence in the
light curve, and the asymmetry shows up in a characteristic dependence of
$\rm{SF_{\mu_{3}}(\tau)}$ on $\tau$. Low inclination leads to smaller,
symmetric boosts in magnification, and $\rm{SF_{\mu_{3}}(\tau)}$ is near zero
for all $\tau$.

In order explore the relativistic effects, we constructed a semi-physical
model for Sgr A*’s light curve:

$\displaystyle F_{\rm{model}}$
$\displaystyle(t,i,r_{\rm{f}},\tau_{1,\rm{int}},\tau_{2,\rm{int}},\alpha,\beta,\kappa_{\rm{ou}},\sigma_{\rm{ou}},\theta_{\rm{ou}})=$
(13) $\displaystyle\sum_{n}a_{n}(\alpha,\beta)\cdot GR_{n}(t,i)\cdot
C_{\rm{int}}(\tau_{1,\rm{int}},\tau_{2,\rm{int}})$
$\displaystyle+AR_{1}(t,\kappa_{\rm{ou}},\sigma_{\rm{ou}},\theta_{\rm{ou}})+\mathcal{N(\sigma_{\rm{obs}})}$

The first part in Equation 13 represents a shot-noise model. This models the
light curves as a sequence of flares, where the number of flares is drawn from
a Poisson distribution with rate $r_{{f}}$. The flare amplitudes are drawn
from an inverse gamma distribution222The choice of a gamma distribution was
motivated by the observed flux-density distribution GRAVITY Collaboration et
al. (2020a). with parameters $\alpha$, $\beta$. $GR_{n}(t,i)$ is the
relativistic magnification kernel. To obtain the relativistic magnification
for each flare, we assumed that the flares occur uniformly around the black
hole with distance from the black hole drawn from a Gaussian distribution
(mean $8R_{g}$, SD $1R_{g}$, bounded at [$6R_{g}$, $10R_{g}$]). The intrinsic
emission of the flare was modeled with a double-exponential kernel function
$C_{\rm{int}}(t,\tau_{1,\rm{int}},\tau_{2,\rm{int}})$ as in Equation 10, but
the intrinsic rise and fall times in general will differ from the observed
ones. The second part in Equation 13 is added to account for low level
variance in the data, typically referred to as the quiescence emission (e.g.,
Dodds-Eden et al. 2009). In order to model this quiescence emission, we used a
generic time series model: the exponentiated Ohrnstein–Uhlenbeck
($\mathrm{AR}_{1}$) process, which is identical to an first order
autoregressive model ($\mathrm{AR}_{1}$). This process models the overall data
structure well and is by definition temporally symmetric (Witzel et al. 2018).

Our model does not try to model the radiative emission mechanism. This is a
limitation because the radiative process itself may well produce asymmetry in
the light curve. Examples include adiabatic expansion (e.g., Yusef-Zadeh et
al. 2009) and radiative cooling (e.g., Aimar et al. 2023). Nevertheless, the
model constrains the asymmetry of the intrinsic emission (modeled by
$C_{\rm{int}}[t,\tau_{\rm{1,int}},\tau_{\rm{2,int}}]$), though the intrinsic
emission could be more asymmetric than the observed emission if the asymmetry
induced by relativistic effects happens to cancel the intrinsic asymmetry.

The actual model fit, like the one in subsection 4.3, was via the ABC
algorithm.

###  5.2 Derived Sgr A* properties

The cumulative flux distribution, as well as the second- and third-moment
structure function, are well described by the model, as illustrated in Figure
6. Light curves generated from the posterior distribution qualitatively match
the observed light curves of Sgr A* (Figure 7).

Figure 5: Inclination posterior of the relativistic model. Blue line shows the
derived posterior, grey line shows the inclination posterior of GRAVITY
Collaboration et al. (2023), and the gray shaded region shows the constraints
derived from the 230 GHz image of the EHT Collaboration Event Horizon
Telescope Collaboration et al. (2022a).

In the context of the model, high-inclination viewing angles are ruled out
(Figure 5). This is because $\rm{SF_{\mu_{3}}}(\tau)$ shows no significant
deviation from zero, inconsistent with the characteristic swing for high
inclinations (Figure 4).

Figure 6: Summary statistics from posterior samples (blue) of the converged
model fit compared to real observations (black). Statistics are shown
separately for the eight observed light curves. The left panel shows the
cumulative flux-density distribution, and the other two panels show the two
structure-function distributions as labeled.

Furthermore, the best fit shows no dip in $\rm{SF_{\mu_{2}}(\tau)}$, which
would correspond to a peak in the power spectrum. This means the fit does not
require significant periodicity, despite the flares being modeled as orbiting
hot-spots. This is a consequence of the low flare rate and the different
initial orbital positions. In addition, the preferred low inclinations cause
only weak relativistic boosting and therefore little orbital modulation.
Finding an orbiting-hot-spot model with no periodicity prove that the absence
of significant periodicity is not evidence against the orbiting-hot-spot model
itself but is rather a consequence of the model parameters.

Figure 7: Simulated light curve draw from the model posterior compared to a
differential light curve measured by Spitzer. The blue line shows simulated
flares (first part in Equation 13 ) drawn from the flare-model posterior. The
brown line shows the full model light curve including the quiescence emission
and observational noise, and the black line shows a Spitzer observation. The
curves are vertically offset for clarity, but the y-axis has the same linear
scale for both curves.

The fit also allows for the intrinsic timescales to be longer than the
observed time scales (subsection 4.2), as shown in Figure 8. This illustrates
the “speeding up” of timescales by relativistic magnification. The speedup can
also be quantified by the overall variance of the model light curves, which
are altered by the magnification to show more and sharper peaks. This can be
quantified by comparing posteriors with the same random seeds but relativistic
effects set to either zero or one. The overall light curve variance is
decreased by $\sim$20% by the relativistic effects.

Figure 8: Posteriors of intrinsic time scales. Colored curves show posteriors
of $\tau_{1,\rm{int}}$ (orange) and $\tau_{2,\rm{int}}$ (blue) derived from
fitting the relativistic model. Vertical dashed lines show the timescales
($\tau_{1,\rm{obs}}$ and $\tau_{2,\rm{obs}}$, colors as labeled) in the
observed light curves as derived from fitting the MA model.

##  6 Discussion and Summary

This paper extends the results from Paper I, which showed that the average
shape of Sgr A*’s NIR and X-ray flares is symmetric. The present work confirms
this symmetry in the entire light curve. The average observed rise and fall
times are $\tau_{\rm{1,obs}}\approx\tau_{\rm{2,obs}}\approx
15\leavevmode\nobreak\ \mathrm{minutes}$.

It is surprising that Sgr A*’s overall light curve shows no significant
asymmetry because the radiative processes thought relevant for Sgr A* are
typically asymmetric. For instance, variability caused by electron-synchrotron
cooling or adiabatic expansion after an initial acceleration event produces
skewed light curves (e.g., Dodds-Eden et al. 2009; Aimar et al. 2023; Michail
et al. 2024).

Paper I demonstrated that Doppler boosting of an orbiting hot spot can
plausibly explain the symmetry in Sgr A*’s flares. While individual events may
have strong asymmetry, depending on the starting position of the hot spot in
the orbit, the boosting averaged over multiple independent hot spots will be
symmetric. Paper I also showed that hot spots in edge-on orbits produce
asymmetric flares because of lensing. Such orbits are therefore inconsistent
with the observed flare symmetry. This paper corroborates these results by
modeling the light curve with a semi-physical, relativistic model, composed of
orbiting flares on top of a correlated red-noise-like quiescence emission. The
intrinsic process may be more asymmetric than the observed emission because of
the averaging over orbits, but our model fits the observations only if $i\la
45\degr$ (Figure 5). This is consistent with other observations, which give
stricter constraints.

Lastly, Sgr A*’s light curves have a red-noise-like character with no
significant periodicity. Nevertheless, our models show that orbiting hot spots
in the accretion flow are consistent with these light curves. The absence of
significant periodicity is explained by a combination of short intrinsic time
scales, low inclinations, a low flare rate, and flares erupting at random
orbit phases (i.e., at uniformly distributed orbit positions).

The third-moment structure function is a useful statistic for a wide variety
of astronomical studies because many astronomical signals are temporally
asymmetric. Examples include astrometric signatures of orbiting black holes
(El-Badry et al. 2023a, b), light curves of eclipsing binaries (Ott et al.
1999; Gautam et al. 2024), and even gravitational-wave signals (Abbott et al.
2019). These examples illustrate the utility of the third-moment structure
function well beyond the topic of this paper.

###### Acknowledgements.

The authors would like to thank the referee, Dr. Jeffrey Scargle, for his very
constructive and quick review, which significantly improved the manuscript.
H.-H. Chung is supported for this research by the International Max-Planck
Research School (IMPRS) for Astronomy and Astrophysics at the University of
Bonn and Cologne. This publication is part of the M2FINDERS project which has
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 Research and Innovation Programme (grant agreement No
101018682).

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##  Appendix A Properties of the third-moment structure function

Figure 9: Third-moment structure functions $\mathrm{SF}_{\mu_{3}}$ for
different kernels constructed from double-exponential profiles (Equation 10).
Insets show example kernels, and the curves in the main panel show the
resulting $\mathrm{SF}_{\mu_{3}}$. Left: Symmetric (grey) and asymmetric
(positive in orange, negative in blue) kernels. Middle: Five kernels with a
fixed rise time $\tau_{1}=15$ and fall times $\tau_{2}=15$ (symmetric), 20,
25, 30, and 35. The larger $\mathrm{SF}_{\mu_{3}}$ values correspond to longer
fall times as indicated by the curves’ color saturation. Right: Asymmetric
kernels of different amplitudes. All have rise times $\tau_{1}=15$ and fall
times $\tau_{2}=30$, but amplitudes are multiplied by 1, 2, 3, 4, and 5. The
structure function is the same for all amplitudes.

The third-moment structure function $\mathrm{SF}_{\mu_{3}}$ measures
asymmetry. Figure 9 demonstrates that for an exponential impulse-response
kernel, the structure function has a characteristic functional form. In
particular, the larger the asymmetry (difference between rise and fall time),
the stronger the signal in $\mathrm{SF}_{\mu_{3}}$. Because of the
normalization, the structure function depends only on the time difference in
the rise and decay times, not the amplitude of the kernel.

Figure 10 demonstrates how the $\rm{SF}_{\mu_{2}}(\tau)$ function reacts to
correlated noise. The three panels show light curve realizations generated
with the same asymmetric impulse response and identical random seeds but
different exponents $\alpha$ of the innovation (Equation 9). The longer the
light curve, the better different $\rm{SF}_{\mu_{3}}(\tau)$ functions
approximate the functional form of the skewness function of the impulse
response. For shorter light curves, the errors are largest at large lag times,
where there are fewer sample pairs, and fewer samples suffice when $\alpha$ is
large because events are less blended. Regardless of the correlation in the
noise, if the sufficient data are available, the $\rm{SF}_{\mu_{3}}(\tau)$
function recovers the temporal asymmetry in the data. This demonstrates its
utility: even correlated noise does not bias the results, especially at short
lag times.

Figure 10:  Simulated light curves and derived structure functions. Upper
panels: Light curves for different values of $\alpha$ (Equation 9) as labeled.
The light curves were generated by the MA model (Equation 7) and include
random and correlated noise with the random seed the same for all panels. The
kernels are double-exponential profiles (Equation 10) having $\tau_{1}=?$ and
$\tau_{2}=?$, and the innovation $R=U^{\alpha}$ was drawn from uniformly
distributed samples in [0, 1). Colors at the left mark sections of the light
curve with 5 000 and 50 000 samples with the full curves containing 500 000.
Lower panels: Structure functions of the simulated light curve above each
panel. The black curve shows the true structure function without noise. The
colored curves show the structure function derived from each light curve with
a finite number of samples as labeled.

##  Appendix B Models and fitting

Figure 11 illustrates the properties of light curves generated from a moving
average model. The figure shows the kernel function ($C(t,\tau_{1},\tau_{2})$
(impulse response) in the top panel. The middle panel shows three realizations
of the innovation $\mathcal{U}^{\alpha}$ for $\alpha$=$1000$, $100$, and $10$
respectively. The higher the value of $\alpha$, the more the distribution is
skewed. The bottom panel shows the light curves generated from such a process.
The higher $\alpha$ the more “isolated” flares appear in the light curve,
while for low $\alpha$ the concept of flare seems ill defined. Nevertheless,
the underlying impulse response is the same.

Figure 11:  Simulated light curves generated based on the MA model. Top:
temporally symmetric kernel response function. Middle: innovation $R(\alpha)$
for sparse, intermediate, and dense random impulses with $\alpha=1000$, 100,
and 10, respectively. Bottom: the light-curve products convolved from the
kernel and the corresponding innovation processes. Light curves are normalized
by their maximal flux value for demonstration.

###  B.1 Model parameters, priors, and posteriors

We give model parameters, priors and posteriors in Table 1.

Table 1: Model parameters, priors, and posteriors for the MA model and semi-physical Sgr A* model. parameters | prior | posterior | notes
---|---|---|---
MA model |  |  |
$\alpha$ | uniform $\mathcal{U}\in[1,1000]$ | $218^{+14.3}_{-13.4}$ | exponent of innovation
$f_{\rm{scale}}$ | uniform $\mathcal{U}\in[-100,100]$ | $22.4^{+1.3}_{-1.1}$ | flux density normalization (mJy)
$\tau_{1}$ | uniform $\mathcal{U}\in[1,100]$ | $14.8^{+0.4}_{-1.4}$ | rise time (minutes)
$\tau_{2}$ | uniform $\mathcal{U}\in[1,100]$ | $13.8^{+1.3}_{-1.4}$ | fall time (minutes)
Sgr A*-model |  |  |
$i$ | uniform $\mathcal{U}\in[5,90]$ | $32.7^{21.7}_{44.3}$ | inclination (degrees)
$\tau_{1,\rm{int}}$ | uniform $\mathcal{U}\in[10,35]$ | $23.6^{30.5}_{16.3}$ | intrinsic rise time (minutes)
$\tau_{2,\rm{int}}$ | uniform $\mathcal{U}\in[10,35]$ | $18.7^{24.8}_{12.6}$ | intrinsic fall time (minutes)
$r_{f}$ | uniform $\mathcal{U}\in[5,300]$ | $10.1^{15.5}_{4.8}$ | flare rate parameter (Poisson)
$\mu_{\rm{amps}}$ | uniform $\mathcal{U}\in[0.1,1000]$ | $525^{738}_{313}$ | amplitude distribution parameter 1 (inv. $\gamma$)
$\sigma_{\rm{amps}}$ | uniform $\mathcal{U}\in[0.1,1000]]$ | $6259^{8549}_{3968}$ | amplitude distribution parameter 2 (inv. $\gamma$)
$\sigma_{\rm{measurement}}$ | uniform $\mathcal{U}\in[0.05,0.6]]$ | $0.16^{0.20}_{0.12}$ | measurement noise (mJy)
$\kappa_{\rm{O.U.}}$ | uniform $\mathcal{U}\in[0.01,0.1]]$ | $0.06^{0.08}_{0.05}$ | exp. Ornstein Uhlenbeck parameter 1
$\theta_{\rm{O.U.}}$ | uniform $\mathcal{U}\in[0.1,0.4]]$ | $0.6^{0.9}_{0.35}$ | exp. Ornstein Uhlenbeck parameter 2
$\sigma_{\rm{O.U.}}$ | uniform $\mathcal{U}\in[0.001,0.1]]$ | $0.08^{0.10}_{0.07}$ | exp. Ornstein Uhlenbeck parameter 3

###  B.2 Distance functions and model convergences

We used three distance function for the likelihood interference scheme. The
second moment structure function of the simulated data was compared to the
observed data using a distance function akin to the $\log\chi^{2}$ function:

$\log\chi^{2}=0.5\left(\log\rm{SF}_{\rm{obs_{i},}\leavevmode\nobreak\
\mu_{2}}/\log\rm{SF}_{\rm{sim_{i},}\leavevmode\nobreak\ \mu_{2}}\right)^{2}.$
(14)

The third moment structure function of the simulated data was compared to
observed data using a distance function akin to the $\chi^{2}$ function:

$\chi^{2}=0.5\left(\rm{SF}_{\rm{obs_{i},}\leavevmode\nobreak\
\mu_{2}}-\rm{SF}_{\rm{sim_{i},}\leavevmode\nobreak\ \mu_{2}}\right)^{2},$ (15)

which allows to account for the factor that the third moment structure
function can have positive as well as negative values. The time lags were
binned using $3\leavevmode\nobreak\ \mathrm{min}$ linear time bins, which was
the empirically determined optimum between time sampling and noise in the
structure functions.

For the MA model we used to the classical KS-test to compare observed and
simulated flux distribution, which turned out to be too insensitive to tail
outliers for the semi-physical Sgr A*. For the latter, we used cumulative
(flux distribution) histogram fitting as a distance function, where we
calculated the $\chi^{2}$ distance between the histogram’s bins.

We employ the Population Monte Carlo (PMC) algorithm (Beaumont 2010)
implemented in abcpy, which converges using ‘populations’ of random samples,
which are iteratively generated, with each generation posing tight constraints
on the parameters. One practical convergence critirum is that more generations
do not lead to substainly different constraints on the derived parameters. We
illustrate this is the case for both the generic MA model, as well as for the
semi-physical relativistic Sgr A* in the following.

####  B.2.1 MA model

Figure 12 illustrates that we have run the abcpy’s PMC suffiecently long to
derive meaning full parameter constraints. The quantilies of the distribution
do not change substainly between the last generation and the second last.

Figure 12: Posterior for the model (Equation 11) generated using the
Approximate Bayesian Computation (ABC) method. The different colored lines
indicate different “sample generations”, with darker colors representing more
converged chains.

####  B.2.2 Relativistic Sgr A* model

As for the MA model, Figure 13 illustrates that the alogrithm has run for
sufficient time to be considered converged.

Figure 13: Similar as Figure 12, but for the convergences behavior of the of
the relativistic Sgr A* model. Dark colors indicate later populations.

##  Appendix C Proof of $\rm{SF}_{\mu_{3}}$ asymmetry properties

We would like to show that for a temporally symmetric light curve
$\rm{SF_{\mu_{3}}}$ is zero.

Let $\\{F_{1}(t)\\}$ be a stationary random process with $t=0,\pm 1,\pm 2,...$
and $\\{F_{2}(t)\\}$ as second stationary random process with
$F_{2}(-t)=F_{1}(t)$ for all $t$. Let us define $\rm{SF}_{\mu_{3}}(\tau)$ and
$\widehat{\rm{SF}}_{\mu_{3}}(\tau)$ to be the respective structure functions
(see definition Equation 2).

Then,

$\displaystyle-\widehat{\rm{SF}}_{\mu_{3}}(\tau)=-1/N_{i}\sum_{t_{j},t_{i}}(F_{2}(t_{i})-F_{2}(t_{j}))^{3}$
$\displaystyle=1/N_{i}\sum_{t_{j},t_{i}}(F_{2}(t_{j})-F_{2}(t_{i}))^{3}$
$\displaystyle=1/N_{i}\sum_{t_{j},t_{i}}(F_{1}(t_{i})-F_{1}(t_{j}))^{3}=\rm{SF}_{\mu_{3}}(\tau),$

i.e., time reversal leads to a change of sign of the third moment structure
function.

In this paper, we define temporal symmetry not in the sense of a deterministic
function (i.e., $F(t)=F(-t)$), but in a statistical sense. This can be
achieved by adopting a definition similar to the definition of stationarity,
as given in Priestley (1988):

Definition. The random process $\\{F_{1}(t)\\}$ is said to be statistically
time-symmetric up to order m if, for any admissible $t_{1},t_{2},...,t_{n}$
all the joint moments up to order $m$ of
$\\{F_{1}(t_{1}),F_{1}(t_{2}),...,F_{1}(t_{n})\\}$ exist and equal the
corresponding joint moments up to order m of
$\\{F_{2}(t_{1}),F_{2}(t_{2}),...,F_{2}(t_{n})\\}$, with $F_{2}(t)=F_{1}(-t)$
for all $t$.

Then also all the corresponding moments on time lag selected differences are
equal, e.g., $\rm{SF}_{\mu_{3}}$ = $\widehat{\rm{SF}}_{\mu_{3}}$.

If the third order structural functions are equal and negative to one another,
their values have to be zero.


# Probing the Manipulation of Antiferromagnetic Order in CuMnAs Films Using
Neutron Diffraction

S. F. Poole  School of Physics and Astronomy, University of Nottingham,
University Park, Nottingham, NG7 2RD, United Kingdom  L. X. Barton  School of
Physics and Astronomy, University of Nottingham, University Park, Nottingham,
NG7 2RD, United Kingdom  M. Wang  School of Physics and Astronomy, University
of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom  P. Manuel
ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, OX11 0QX,
United Kingdom  D. Khalyavin  ISIS Facility, Rutherford Appleton Laboratory,
Chilton, Didcot, OX11 0QX, United Kingdom  S. Langridge  ISIS Facility,
Rutherford Appleton Laboratory, Chilton, Didcot, OX11 0QX, United Kingdom  K.
W. Edmonds  School of Physics and Astronomy, University of Nottingham,
University Park, Nottingham, NG7 2RD, United Kingdom  R. P. Campion  School of
Physics and Astronomy, University of Nottingham, University Park, Nottingham,
NG7 2RD, United Kingdom  V. Novák  Institute of Physics, Czech Academy of
Sciences, 162 00 Praha 6, Czech Republic  P. Wadley
<EMAIL_ADDRESS>School of Physics and Astronomy, University of
Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom

###### Abstract

We describe measurements of the uniaxial magnetic anisotropy and spin-flop
rotation of the Néel vector in antiferromagnetic CuMnAs thin films using
neutron diffraction. The suppression of the magnetic (100) peak under magnetic
fields is observed for films as thin as 20 nm indicating that they undergo a
spin-flop transition. Good agreement is found between neutron diffraction and
electron transport measurements of the spin-flop rotation in the same layer,
with a similar shape and hysteresis of the obtained curves, while the neutron
measurements provide a quantitative determination of the spin flop extent
throughout the antiferromagnet layer.

††preprint: AIP/123-QED

There is growing interest in antiferromagnetic (AF) materials for applications
in spintronics. Predictions and demonstrations of current-induced switching of
the AF Néel vector have stimulated the development of all-electrical AF memory
devices, with potential for high speed operation, robustness against external
fields, and multi-level neuromorphic outputs [1, 2, 3, 4, 5]. Electrical
readout of the AF state typically utilizes anisotropic magnetoresistance (AMR)
or spin Hall magnetoresistance, but such effects are small and can easily be
obscured by structural changes induced by Joule heating [6]. Direct
determination of the Néel vector is therefore important. This has been
achieved by imaging local modifications of AF domains and domain walls
affected by current pulses or magnetic fields using x-ray photoemission
electron microscopy [7, 8, 9, 10, 11], and more recently by optical
birefringence [12], spin-Seebeck [13] or magneto-Seebeck [14] microscopies.

Neutron diffraction is a well-established technique to directly measure the
atomic arrangement of magnetic moments in a bulk AF crystal. It provides a
quantitative determination of the Néel vector and, unlike electron microscopy
techniques, is routinely performed in high magnetic fields. It is therefore
well-suited to studying spin-flop transitions, which typically occur in
magnetic fields of several Tesla. In AF thin films and devices, spin-flop
rotations of the Néel vector are invaluable for characterizing extraordinary
magnetotransport [11] or magnon transport phenomena [15, 16], and exploring
multi-stability of magnetic memory states [17, 18]. However, there are few
direct studies of spin-flop rotation in AF thin films using neutron
diffraction due to the very small volume of the sample film compared to a
typical neutron experiment, which is a flux limited technique.

Here, we demonstrate that neutron diffraction can be used in an unambiguous
total volume determination of the spin-flop rotation in CuMnAs films with
thickness down to 20 nm. We also show good agreement between neutron
diffraction and magnetotransport measurements performed on the same sample.
CuMnAs is a collinear antiferromagnet with a Néel temperature of around 480 K
[19]. It can exhibit pronounced modifications of its AF domain structure in
response to applied electrical or optical pulses, which may be accompanied by
significant electrical readout signals [5]. Understanding of its magnetic and
transport properties is therefore important for the design of spintronic
memory devices.

The CuMnAs 20 nm and 45 nm films were grown on GaP(001) substrates using
molecular beam epitaxy [20]. X-ray diffraction measurements confirmed the
tetragonal crystal structure of the films, illustrated in Fig. 1(a), with the
CuMnAs[001] axis perpendicular to the plane and the CuMnAs principal axes at
$45^{\circ}$ to the substrate principal axes. Previous neutron diffraction and
x-ray magnetic linear dichroism measurements on similar films have shown that
the magnetic moments align in the $ab$ plane, with a competition between a
uniaxial CuMnAs[010] anisotropy and a biaxial CuMnAs[110]/[1$\bar{1}$0]
anisotropy [19]. The uniaxial in-plane anisotropy arises because of the
substrate-film interface where the symmetry is broken by the P-terminated unit
cell of the GaP substrate.

The neutron diffraction measurements utilized the time of flight cold neutron
beam line WISH at the ISIS facility[21]. Figure 1(b) represents the geometry
of the experiment with horizontal scattering plane and vertical magnetic field
generated by a superconducting split-pair solenoid magnet. For both films, the
samples were mounted in two orientations: in one case, the uniaxial easy axis
([010]) was parallel to the applied magnetic field, and in the other the field
was parallel to the [100] axis. These corresponded to the (h0l) and (0kl)
scattering plane, respectively. The position of the samples in respect of the
incident beam was chosen to optimize the neutron flux for the (100) and (010)
magnetic reflections as well as to access the (001) reflection at the same
time. It was achieved when the surface plane of the samples was at
$45^{\circ}$ to the beam direction. The measurements were conducted at a
sample temperature of 100 K and the sample position was the same for both
films investigated.

For a magnetic field applied parallel to the easy axis of a uniaxial
antiferromagnet (Fig. 1(c)), there is a threshold field strength known as the
spin flop field, $H_{\textup{sf}}$, that causes a 90∘ reorientation of the
antiferromagnetic Néel vector. Above this field, the moments are canted around
the external field direction (Fig. 1(d)). Due to higher-order magnetic
anisotropies as well as domain walls and other inhomogeneities, the sharp
transition at $H_{\textup{sf}}$ is typically broadened into an s-shaped curve.

Nuclear contribution to the (100) and (010) reflections is forbidden in
tetragonal CuMnAs (P4/nmm space group) and therefore they contain only
magnetic scattering which is sensitive to the component of magnetic moments
perpendicular to the scattering vector. Consider the case where the [010]
direction is a uniaxial easy magnetic axis and the field is to be applied
along this direction. In the zero field state where $H<H_{\textup{sf}}$, the
magnetic moments are parallel to the easy axis and the (100) peak is
observable. Under a field $H>H_{\textup{sf}}$, the moments are perpendicular
to the easy axis and aligned along the [100] direction with small-angle
canting of the moments in the direction of the field and therefore the (100)
peak is extinguished. Hence, the magnitude of the peak provides a direct
measurement of the spin flop transition in the sample. If only a partial
rotation of the magnetic moments occurs, the magnitude of the peak will reduce
in proportion, so the ratio of the peak amplitude at field compared to the
original state will give a direct measurement of the extent of spin flop in
the sample.

Figure 1:  a) CuMnAs unit cell with arrows representing the expected uniaxial
anisotropy along the [010] crystal direction. (b) Schematic diagram of the
neutron diffraction experiment showing the horizontal scattering plane and the
vertical magnetic field. The incident beam is represented by the black arrow,
and the detector arrays covering in-plane scattering angles from 10∘ to 170∘
are depicted as gray semi-circles. (c,d) Sketches showing the magnetic moment
orientations for an uniaxial antiferromagnet with respect to an applied
magnetic field below (c) and above (d) the spin-flop field $H_{sf}$. e) Plots
of the (100) peak for the 45 nm film with three different field strengths
applied along the easy axis. The integration regions used in background
subtraction are shown.

Figure 2:  Ratio of in plane (IP) with the sample surface to out-of-plane
(OOP) integrated peak amplitudes versus magnetic field, for CuMnAs films of
thickness (a) 45 nm and (b) 20 nm. The red circles and blue diamonds
correspond to to the measurements of the (100) and (010) in-plane peaks, with
the field along the [010] and [100] axes respectively. Open and filled symbols
represent decreasing and increasing field sweeps.

The (100) peak for a 45 nm CuMnAs film is shown in Fig. 1(e) for three
different magnetic field strengths. The peak is significantly reduced at 1 T
applied field compared to its zero-field amplitude, and has almost vanished at
8 T. To quantify the spin-flop transition, the measured intensity is
integrated across the peak and background regions indicated in Fig. 1(e). The
combined magnetic and structural (001) peak is also measured, to allow
calibration of the alignment and detection efficiency. Because the moments are
expected to stay within the (001) plane, the (001) peak is unaffected by spin
flop and is used as a control measurement. Consistent with this, the peak
amplitude is found to be independent of the magnetic field within the
experiment uncertainty (see Supplementary Information).

Plots of the ratios of integrated peak amplitudes versus magnetic field for 45
nm and 20 nm thick CuMnAs films are shown in Fig. 2. For the 45 nm sample
(Fig. 2a), the (100) peak is larger than the (010) peak by a factor of $5.4\pm
0.9$. This indicates that the magnetic moments in the sample are predominantly
aligned with the [010] axis. A sharp decrease in intensity is observed between
zero and 1 T for the (010) peak, and between zero and 2 T for the (100) peak,
due to the spin flop rotation of the Néel vector into an axis which is
perpendicular to the field. In high fields, the (100) peak is decreased to
$2.2\pm 0.7$ % of its zero-field amplitude, showing conclusively that the vast
majority of the CuMnAs layer undergoes spin flop in this field range. After
the high field measurements the sample recovers its original state. This
observation in combination with the well-defined critical fields strongly
suggests that the reorientation of the magnetic moments is a flop-transition
rather than due to continuous moving domain walls.

The 20 nm CuMnAs layer (Fig. 2b), exhibited a (100) to (010) peak amplitude
ratio of $10.5\pm 3.4$ and therefore demonstrates even stronger anisotropy
than the thicker layer. As with the thicker layer, the (100) peak intensity
drops sharply between 1 T and 2 T applied field. However, there is a
significant intensity remaining between 2 T and 6 T, which may be attributed
to the presence of $180^{\circ}$ magnetic domain walls and defect-driven
domain pinning. The intensity then drops to zero between 6 T and 8 T. Due to
time constraints and the weak signal from this thin film, the (010) peak was
not measured over the whole magnetic field range.

Figure 3:  (a) Schematic of an L-bar device. The orange represents the 100 µm
wide CuMnAs bar and the gold squares represent contact pads to which wires are
bonded. The arrows represent the current flow in the device and the voltages
drawn correspond to the voltages described in the text where 1 and 2
correspond to the first and second arm the current flows through respectively.
(b) Transverse AMR and (c) longitudinal AMR, as defined by Eq. 1 and Eq. 2
respectively, versus magnetic field for an L-bar device fabricated from the 45
nm thick CuMnAs film. Red and blues lines are for field along the [010] and
[100] axes, respectively. The plots are offset for clarity to show zero AMR at
zero field.

Figure 4:  (a) Spin flop extent extracted from neutron diffraction, and b)
transverse AMR, for the 45nm thick CuMnAs film with external magnetic field
applied along the [010] axis. Arrows indicate the direction of the magnetic
field sweep.

Complementary electrical transport measurements were conducted on the 45 nm
sample which was patterned into an L-shaped Hall bar structure as illustrated
in Fig. 3a. This allows currents to be applied simultaneously along two in-
plane crystal axes, at $45^{\circ}$ and $-45^{\circ}$ to an applied magnetic
field respectively, while recording the longitudinal and transverse voltages
$V_{\parallel}$ and $V_{\perp}$. Each arm of the structure is of width
$\bm{100}\mu$m. The magnetoresistance measurements were conducted with the
sample submerged in liquid helium at 4K inside a superconducting magnet.
Magnetic fields were applied either parallel to [010] or [100], with applied
current $I_{probe}=2$ mA.

Due to anisotropic magnetoresistance (AMR), the longitudinal and transverse
voltages depend on the angle $\phi$ between the applied current and the
orientation of the Néel vector. Neglecting crystalline terms, this dependence
is given by $V_{\parallel}=V_{0}+V_{1}\cos(2\phi)$, and
$V_{\perp}=V_{1}\sin(2\phi)$. Hence, the rotation of the Néel vector from
$+45^{\circ}$ to $-45^{\circ}$ with respect to the current direction should
generate no change in the longitudinal voltages, and a maximum change in the
transverse voltages.

Figures 3b and 3c show the measured normalized transverse and longitudinal
anisotropic magnetoresistances, defined as

$\displaystyle AMR_{\perp}$ $\displaystyle=\frac{2(V_{\perp 1}-V_{\perp
2})}{V_{\parallel 1}+V_{\parallel 2}}$ (1) $\displaystyle AMR_{\parallel}$
$\displaystyle=\frac{V_{\parallel 1}-V_{\parallel 2}}{V_{\parallel
1}+V_{\parallel 2}}$ (2)

where the factor of 2 for $AMR_{\perp}$ accounts for the number of squares
between the longitudinal voltage contacts. For both field directions, a broad
s-shaped change in $AMR_{\perp}$ is observed below 2 T. A hysteresis is
observed between the increasing and decreasing field sweeps, but the values
return to the same point at zero field. The variations in $AMR_{\parallel}$
are much smaller and are ascribed to small variations in the sample
temperature during the field sweep.

Figure 4 compares the neutron diffraction and $AMR_{\perp}$ measurements over
the same magnetic field range. The spin flop extent is defined as
$1-\frac{P(H)}{P(0)}$ where $P(H)$ is the integrated peak amplitude at field
$H$ and ${P(0)}$ is the initial value at zero field. It can be seen that the
transport measurement gives a qualitatively accurate measure of the spin flop
extent, with the shape, hysteresis and spin flop field all matching well. Note
that the neutron diffraction measurements were performed at 100 K while the
AMR was measured at 4 K. Previous studies have shown that this difference in
temperature creates a small change in amplitude of the AMR signal due to
changes in base resistance, but does not qualitatively change the signal [11].

We have demonstrated the use of magnetic neutron scattering as a tool for
probing the antiferromagnetic spin-flop regime in CuMnAs. From our results we
were able to show that 98% of the magnetic order is reoriented under action of
an applied magnetic field along the magnetic easy axis in a strongly uniaxial
sample. Any remnant signal could be attributed to $180^{\circ}$ domain walls
and domain pinning due to defects. Complementary electrical transport
measurements were congruent with the neutron scattering signal in both shape
and hysteretic behaviour. Overall, we propose that this technique should be
applicable for measuring the magnetic Néel vector magnitude and direction of
any such compatible magnetic crystals.

## Supplementary Material

See supplementary material for plots of the peak amplitudes that were used in
the normalization process to make Fig. 2. The supplementary material also
includes waterfall plots of the diffraction peaks used to find the amplitudes
in the aforementioned plots.

###### Acknowledgements.

The work was supported by the EU FET Open RIA Grant no 766566 and the UK
Engineering and Physical Sciences Research Council [grant number
EP/P019749/1]. We wish to acknowledge ISIS Neutron and Muon Source for the
beamtime on WISH. Data is available here:
https://doi.org/10.5286/ISIS.E.RB1920313

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