# Introducing PT-REX ,
the Point-to-point TRend EXtractor

A. Ignesti<EMAIL_ADDRESS>
###### Abstract

Investigating the spatial correlation between different emissions in an
extended astrophysical source can provide crucial insights into their physical
connection, hence it can be the key to understand the nature of the system.
The point-to-point analysis of surface brightness is a reliable method to do
such an analysis. In this work we present PT-REX, a software to carry out
these studies between radio and X-ray emission in extended sources. We discuss
how to reliably carry out this analysis and its limitation and we introduce
the Monte Carlo point-to-point analysis, which allows to extend this approach
to poorly-resolved sources. Finally we present and discuss the application of
our tool to study the diffuse radio emission in a galaxy cluster.

###### keywords:

Techniques: image processing , methods: observational, statistical , radio
continuum: general , radio continuum: general

The point-to-point analysis is a reliable method to investigate the physical
connection between different components in complex, extended astrophysical
sources;

We introduce the Monte Carlo point-to-point analysis which allows to reliably
perform the point-to-point analysis also to small or poorly-resolved sources;

We present PT-REX, a new software that allows to easily perform point-to-point
analysis between radio and X-ray emission. It features several statistical
tools to reliably evaluate the spatial correlation for a plethora of
scientific cases.

As an example, we present here how to use PT-REX to perform the point-to-point
analysis on a diffuse radio source in the galaxy cluster RX J1347.5-1145. Due
to its architecture, PT-REX can be used to conduct point-to-point analysis on
every kind of extended emission, as well as radio and X-ray.

##  1 Introduction

It is no wonder that a multi-wavelength analysis can be the best way to
investigate an astrophysical system. In the case of extended sources, the
study of the spatial correlation between the surface brightness, $I_{a}$ and
$I_{b}$, produced by two different emission mechanisms, which can be simply
expressed as $I_{a}\propto I_{b}^{k}$, can reveal the physical connection
between the processes that are taking place in the source. Observing a
positive spatial correlation indicates that the two components responsible for
the emissions occupy the same volume in the source, whereas the slope $k$ may
provide some insights into their physical link. These studies can be carried
out by using the point-to-point (ptp) analysis which is, basically, the
comparison of surface brightness measured by two different observations made
by sampling the extended source with a grid. Under the assumption that each
cell of the grid cover the same space of the celestial sphere in each
observation, the ptp analysis is more flexible than a comparison between
surface brightness radial profiles because it can be seamlessly performed on
asymmetrical sources and it can be more responsive to the presence of
substructures embed in the extended emission111The drawback is that, contrary
to radial profile which can be easily interpreted by assuming spherical
symmetry of the system, understanding the physical meaning of a ptp trend
could be not trivial, especially for non-spherical objects.. One of the first
applications of the ptp analysis could be found in Govoni et al. [1], where it
was featured to study the spatial correlation between radio and X-ray emission
in galaxy clusters hosting diffuse radio emission. In this context, the trend
between radio and X-ray emission indicates that the thermal plasma is
permeated with magnetic field and relativistic particles and it suggests that
the radio emission depends on the local properties of the thermal intra-
cluster medium (ICM) [e.g., 2, for a detailed discussion of this].

In this work we present PT-REX, a tool to easily carry out ptp analysis
between radio and X-ray emission. We also introduce a new approach, the Monte
Carlo ptp analysis, which extends the application of the point-to-point
analysis to small, poorly-resolved objects. PT-REX offers the possibility to
study the spatial correlation by exploiting a set of different fitting methods
to tackle different scientific problems. This paper is structured as follow.
In Section 2 we present the tool, we explain how to use it effectively and we
introduce the different fitting methods. In Section 3 we show how to conduct
an analysis with PT-REX on a real diffuse raadio source and how the different
statistical methods can produce different results.

##  2 PT-REX

We present the Point-to-point TRend EXtractor (PT-REX
)222https://github.com/AIgnesti/PT-REX/blob/master/PTREX.tar.xz, a flexible
Python script to easily carry out the ptp analysis on every kind of extended
radio source. PT-REX handles most of the operations with the the Common
Astronomy Software Applications (CASA) packages v6.0 [3] developed by the
National Radio Astronomy Observatory. We integrated the CASA tools with a
variety of Python libraries from Astropy [4, 5] and Scipy [6]. The code is
structured in a series of tasks to handle the individual steps of a ptp
analysis independently, from defining a grid to sample the radio emission to
accurately analyze the data with several statistical methods. A preliminary
version of PT-REX has already been used in Ignesti et al. [7], in which we
also present the concept of Monte Carlo ptp analysis. Here we present in
detail each task and we discuss how to run PT-REX to perform ptp analysis
effectively.

###  2.1 Data preparation

PT-REX works by combining radio and X-ray images of an extended source.
Therefore, in order to have reliable results, input images must have matching
coordinates systems to assure that the source is mapped by the same sky
coordinates. Radio images can be produced with any preferred software,
provided that they include information about the beam size and the
pixel/arcsec scale in their header to be read with CASA task imhead.
Concerning the X-ray images, multiple observations can be combined together to
improve the count statistic. The X-ray images can be provided as a single
exposure-corrected and background-subtracted image in units of surface
brightness (e.g., photons cm-2 $s^{-1}$) or by providing the count, background
and exposure images separately. CASA region files, which are necessary to
define the grid and the masks (see Section 2.2 and 2.5) can be defined while
running PT-REX by using the CASA imview. Finally, ancillary information about
the calibration error of the radio images and the preferred statistic method
(see Section 2.6) have to provided before running the analysis.

###  2.2 Sampling algorithm

The core of the ptp analysis is the sampling of the diffuse emission. We
developed a simple algorithm that converts a rectangular region into a grid
that follows the morphology of the radio source. The region is intended to
include the source which is going to be sampled, and thus we refer to is as
region of interest. The boundaries of the sampling grid, as the cell-size
(here intended as the size of the grid cells), the lower threshold in surface
brightness to be sampled, and the regions to exclude, have to be provided at
the begin of the analysis. In order to have a reliable reconstruction of the
radio flux density and to reduce the correlation between contiguous cells, the
cell-size has to be large at least as the resolution of the radio image. A
larger cell-size can be adopted to increase the signal-to-noise ratio of each
cell in the radio and X-ray observations. However, for a given source using
larger cells entails a lower number of points (i.e., a lower statistic) to
study the spatial correlation that can potentially jeopardize the analysis. A
rule of the thumb is that 15 cells, at least, are necessary to sample the
diffuse emission to assure a reliable outcome of the analysis, so a compromise
between resolution of the grid and signal-to-noise has to be found. Finally,
the threshold defines the lowest value of surface brightness that is going to
be sampled by the grid and it is expressed in unit of Jy beam-1.

Figure 1: Flowchart of sampling algorithm.

The sampling algorithm is quite straightforward and it described in Figure 1.
After a preliminary check on the cell-size, the region of interest is
converted in a rectangular grid. At this step, the coordinates of each cell
are defined in the pixel units of the radio image. Then the radio surface
brightness and the position with respect to the mask are evaluated for each
cell of the grid, starting from the bottom-left to the top-right. These checks
are done with the CASA task imstat. All those cells that do not meet the
requirements (i.e., those measuring a radio surface brightness below the
threshold or overlapping with the masked regions) are excluded, whereas the
others are converted in J2000 coordinates and finally stored in a region file,
which is the final output of the routine. Every sampling grid can be displayed
on the top of the radio image by using the CASA viewer and it can be further
modified manually to better adapt to the science case. After having defined a
sampling grid, we can use it to compare the radio and X-ray emissions.

###  2.3 Single-mesh analysis

We define a single-mesh ptp (SMptp) analysis a ptp analysis carried out by
using only a mesh to sample the radio emission. It is composed of two steps:
first the radio and X-ray surface brightness are measured in each cell of the
grid, then they are compared as $I_{\text{R}}$ vs $I_{\text{X}}$ to evaluate
the spatial correlation. We present the flowchart of the process in Figure 2.
The routine that collect the values of $I_{\text{R}}$ and $I_{\text{X}}$ is
straightforward. For a given sampling grid, which was previously created by
using the sampling algorithm, $I_{\text{R}}$ and $I_{\text{X}}$ are evaluated
for each of its cells. Because the cells are defined in J2000 coordinates
according to the radio map coordinate system (see Section 2.2), by using maps
with matching coordinates each cell will sample the same direction of the sky
for both the radio and X-ray images. However, we note that if the X-ray map
pixel-size is larger than the radio map one, then there could be a mismatch
between the position of the grid cells in the two images due to the different
sampling of the sky coordinates. In this case, we suggest to check how the
sampling grid is positioned on the X-ray map before running the SMptp analysis
and, if possible, to regrid the X-ray map to match the size and the pixel-size
of the radio one.

For the radio image, the flux density in each cell is measured with the CASA
task imstat and the converted in $I_{\text{R}}$ by dividing for the area of
the cell, $\Omega_{\text{c}}$, in units of arcsec2. The associated error,
$\sigma_{\text{R}}$, is computed as;

$\sigma_{\text{R}}=\frac{\sqrt{\left(f\cdot
S\right)^{2}+\left(\text{RMS}\cdot\sqrt{\Omega_{\text{c}}/\Omega_{B}}\right)^{2}}}{\Omega_{\text{c}}}$
(1)

where $f$ and RMS are, namely, the amplitude calibration error and the root
mean square of the radio image provided by the user, $S$ is the flux density
measured in the cell and $\Omega_{B}$ is the beam area. Since every cell has
the same size, the second term of Equation 1 is the same for all and it
becomes significant only for those cells with a low radio surface brightness.
As for the X-ray images, when several Chandra observations of the same cluster
are involved, we compute the total $I_{\text{X}}$ of a cell as

$I_{\text{X}}=\frac{\sum N_{\text{cnt,i}}-\sum N_{\text{bkg,i}}}{\sum
q_{\text{exp,i}}}\frac{1}{\Omega_{\text{c}}}=\frac{\sum\left(S_{\text{X,i}}\cdot
q_{\text{exp,i}}\right)}{\sum q_{\text{exp,i}}}\frac{1}{\Omega_{\text{c}}},$
(2)

where

$S_{\text{X,i}}=\frac{N_{\text{cnt,i}}-N_{\text{bkg,i}}}{q_{\text{exp,i}}}$
(3)

is the flux measured for the $i$-th Chandra observation, $\Omega_{\text{c}}$
is the angular area of the cell in units of arcsec2 , and $N_{\text{cnt,i}}$,
$N_{\text{bkg,i}}$ (in units of counts), and $q_{\text{exp,i}}$ (in units of
counts cm2 s photons-1) are, respectively, the values measured on the counts,
the background, and the exposure map of the $i$-th Chandra observation. When
no background or exposure maps are provided, their values are set to,
respectively, 0 and 1 for each cell333This entails that, by providing only the
count image, this procedure is valid for every other kind of observation where
the signal can be divided into counts, e.g. an optical image. This makes PT-
REX virtually able to compare radio emission with other kinds of emissions, as
well as X-ray emission.. We derive the associated errors on $S_{\text{X,i}}$
by assuming a Poisson error for $N_{\text{cnt,i}}$ and $N_{\text{bkg,i}}$ and
computing the error propagation of Equation 2. During this phase, the sampling
is further refined by excluding all these cells that measure negative values
of $I_{\text{R}}$ and $I_{\text{X}}$ or upper limits in radio and X-ray, i.e.
those values with relative uncertainties greater than 100$\%$. As a caveat, we
note a limit of the current sampling algorithm. Due to the fact that the
sampling is mainly driven by the signal-to-noise ratio of the radio image, the
final grid may include cells with an X-ray emission close to the noise level
which may negatively impact on the fit. In this case, we suggest to either
increase the cell-size to reach a compromise between the X-ray signal-to-noise
and the resulting sampling cells number, or to manually remove the most
critical cells by using the CASA task imview to display and modify the grid.

Figure 2: Flowchart of the SMptp routine.

Once the values of $I_{\text{R}}$ and $I_{\text{X}}$ have been calculated, the
fitting can begin. We fit the $I_{\text{R}}$-$I_{\text{X}}$ distribution with
a power-law relation as:

$I_{\text{R}}=A\cdot I_{\text{X}}^{k}$ (4)

We propose a set of different fitting algorithms to measure $k$ and its
associated error $\sigma_{k}$ (see Section 2.6). We also provide a direct
estimate of the linear correlation in the logarithmic space by estimating the
Spearman and Pearson ranks with the scipy.stats library. At the end of this
routine a data file with $I_{\text{R}}$ and $I_{\text{X}}$ and relative errors
is produced as output. These values can be used for further analysis, e.g. to
be examined with a fitting method which is not included in PT-REX [for example
LIRA, 8] or, by combining multiple SMptp analysis of the same object observed
at different radio frequencies, to study the spatial correlation between
spectral index and $I_{\text{X}}$. In addition to the data file, PT-REX
produces a simple plot with the data and the best-fit line with the interval
which has the $95\%$ chance of containing the true regression line.

As a caveat, we note that the reliability of the SMptp analysis is limited by
the assumption that any sampling grid provides an unambiguous reconstruction
of the real surface brightness of the source. Such an assumption is valid for
those sources that can be sampled by large number of cells, i.e. where radio
emission is well resolved by the observation, but it may not be true for the
other smaller or poorly-resolved objects. In the next section we discuss how
the ptp analysis can be extended also to these cases.

###  2.4 Monte-Carlo analysis

Extended sources may contain smaller substructures (e.g. bright filaments
embed in the extended emission or surface brightness gradients). If the
resolution of the observation is sufficient to fully resolve them, i.e. the
angular resolution is lower than half of the angular scale of these features,
the result of the SMptp analysis will not depend on the sampling grid because
the surface brightness will be reliably reconstructed by every possible
combination of cells. Otherwise, if these substructures can not be properly
sampled by the observation/grid, the resulting SMptp analysis will be fatally
biased by the choice of the sampling grid. In this latter case, an approach
more complex than the SMptp is required to estimate the trend of the spatial
correlation.

The major feature introduced by PT-REX  is the possibility to use an
automatic, randomly-generated sampling routine to combine several SMptp
analysis into a Monte Carlo ptp (MCptp) analysis. By repeating several cycles
of SMptp analysis with randomly-generated grids, PT-REX produces a
distribution of values of $k$ that describe its parameter space, thus it
allows us to reliably estimate the trend (and its uncertainties). This routine
makes uses of the sampling algorithm and the SMptp routine presented in the
previous sections. We present the flowchart of the process in Figure 3. After
setting the number of Monte Carlo iterations ($N$), the region of interest,
the cell-size and the $I_{\text{R}}$ lower threshold are set, the Python
function numpy.random is used to generate a number $N$ of coordinates $(x,y)$
within the region of interest. These coordinates are used to define $N$
different, rectangular regions centered in $(x,y)$ and large enough to include
the original region of interest, hence the radio source. Then these $N$
regions are converted in sampling grids by the sampling algorithm and used to
carry out the corresponding SMptp analysis to measure $k\pm\sigma_{k}$.

Figure 3: Flowchart of the MCptp routine.

At the end of each cycle, a random value $k_{b}$ is extracted from a normal
distribution centered in $k$ with sigma $\sigma_{k}$ to be finally stored.
This boot-strapping procedure enables us to transpose the error of each
individual fit in the final $k_{b}$ distribution. After all the $N$ different
grids have been exploited, the result of the MCptp analysis, $k$, is computed
as:

$k=\overline{k}_{\text{b}}\pm\sigma_{k_{\text{b}}}$ (5)

where $\overline{k}_{\text{b}}$ and $\sigma_{k_{\text{b}}}$ are the mean and
the standard deviation of the distribution of boot-strapped $k_{b}$ obtained
at the end of each cycle. Usually SMptp and MCptp performed with the same
parameters (mask, starting region, cell-size and lower threshold) should be
consistent within 1 $\sigma$. If not, this could be due to 1) a very peculiar
choice of the SMptp mesh or 2) the presence of unmasked field source that are
erroneously, yet consistently, included in the sampling. An histogram of the
$k_{b}$ distribution is produced in output, which may point out additional
information about the source. On the one hand, observing a dispersion
significantly larger than the SMptp uncertainties indicates that the random
sampling affected the estimate, which is sign of a poor sampling of the radio
emission. On the other hand, an asymmetrical distribution may indicate the
presence of a secondary component in the radio source that, for fortuitous
combinations of cells have pivoted the fit. For instance, low-brightness
components or strong X-ray point-sources embed in the radio emission can
produce a negative skewness in the distribution, whereas the presence of point
sources with a strong radio and X-ray emission can induce a positive skewness.

MCptp analysis is advised for those sources that can be sampled with $<30$
cells. We suggest setting the number $N$ to a minimum of 100 to adequately
probe the parameter space of $k$. We note that the number of iterations, the
sizes of the region of interest and of the cells and the number of X-ray maps
involved in the analysis affect severely the duration of the procedure. As a
basic rule, mind that adjusting the cell-size by a factor $f$ changes the
number of sampling cells of $\sim 1/f^{2}$, which, in turns, modifies the
processing time of the same amount. PT-REX does not feature parallel
processing, thus it performance in terms of processing time may vary depending
on the available hardware. A scientific application of MCptp analysis to study
radio mini-halos in galaxy clusters is presented in Ignesti et al. [7].

###  2.5 Generate a mask

Field sources and those embed in the radio emission not associated with the
extended source (e.g. central radio galaxies) can jeopardize the results of
the ptp analysis. When the subtraction of those sources from the data is not
possible, they can be masked and excluded from the ptp analysis. PT-REX
includes a tool that allows the user to produce masks for the analysis simply
by providing the regions intended to be excluded. The regions are used to
define a matrix with the same size of the radio image that allows the sampling
routine to recognize and exclude any unwanted sources. Typically, there are
two kind of sources that have to be masked, 1) those embed in the extended
emission, as physically located within the source but its emission is not
related it, or background and foreground sources along the line of sight or 2)
those that are outside the diffuse emission but close to the region of
interest. As a simple rule of the thumb, we suggest to define carefully the
regions to be masked within the source and the region of interest. On the on
hand, define a mask that exceeds the size of the unwanted source can lead the
sampling routine to exclude a number of cells larger than the necessary, and
thus to reduce the number of point to evaluate the spatial correlation. On the
other hand, a mask smaller than the source should not contain the contribution
of the source in the extracted surface brightness, thus jeopardizing the
analysis. As for the sources close to the region of interest, they can be
problematic during a MCptp analysis. At the begin of the cycle a new region of
interest is defined and some of them can be erroneously included and sampled.
So we suggest to adopt large masks to safely account for their presence within
$2\times$, both in width and in height, the size of the region of interest.

###  2.6 Fitting algorithms

Fitting $I_{\text{R}}\propto I_{\text{X}}^{k}$ is a crucial part of the ptp
analysis, and different scientific problems may require different statistic
methods to evaluate the trend. For this reason, PT-REX  includes a range of
fitting algorithms:

  * 1.

Least squares (LS): data can be fitted with a power-law relation
$I_{\text{R}}=A\cdot I_{\text{X}}^{k}$ by using the least-squares method with
the scipy.optimize.curve$\\_$fit method. Only the uncertainties on
$I_{\text{R}}$ are taken into account. This method estimates the best-fitting
parameter of the power-law which minimize the distance from the data, under
the assumption that the data intrinsically follow a power-law distribution and
the scatter is only due to observational errors. Due to its assumptions, this
method can be biased by outliers that can pivot the fit. For a physical point
of view, the assumption of an intrinsic, perfect correlation between the two
quantities may be questionable. In a complex, physical system the apparent
correlation between two quantities can depend on a third, unknown factor. In
this case, an internal scatter of the data is expected regardless of the
quality of the observations and, thus, the base assumptions of this method can
lead to biases in the scientific conclusions. Therefore, although we include
this method for the sake of completeness because it has been largely used in
literature, we advise to use it cautiously;

  * 2.

BCES orthogonal and bisector: The fitting method is the bivariate correlated
errors and intrinsic scatter (BCES) presented in Akritas and Bershady [9]. The
BCES regression offers several advantages compared to ordinary least squares
fitting, as measurement errors on both variables and it can account for an
intrinsic internal scatter of the data. The fitting is performed by the bces
module444https://github.com/rsnemmen/BCES. $I_{\text{R}}$ and $I_{\text{X}}$
are transposed in the logarithmic space before being fitted as
$\text{log}I_{\text{R}}=k\text{log}I_{\text{R}}+\text{log}A$. By default, we
assume that the errors on $I_{\text{R}}$ and $I_{\text{X}}$ are not
correlated. We included in PT-REX both the orthogonal method and the bisector
method, which compute the symmetric lines constructed from the BCES best-fits
of $(I_{\text{R}}|I_{\text{X}})$ and $(I_{\text{X}}|I_{\text{R}})$;

  * 3.

LinMix: this is a Bayesian method to account for measurement errors in linear
regression introduced in Kelly [10]. This method allows for heteroscedastic
and possibly correlated measurement errors and intrinsic scatter in the
regression relationship. The method is based on deriving a likelihood function
for the measured data, especially for the case when the intrinsic distribution
of the independent variables can be approximated using a mixture of Gaussian
functions. LinMix incorporates multiple independent variables, nondetections,
and selection effects (e.g., Malmquist bias). We implemented this algorithm
with the linmix module555https://github.com/jmeyers314/linmix. This method
derives a likelihood function for the data, thus the best-fit slope is
estimated from the mean of the posterior distribution. To run this method, a
number of chains for the bayesian algorithm, n$\\_$chain, and the number of
gaussian to build the prior, K, has to be defined by the user. This method is
significantly more time-consuming than the other options. We advise to use it
when a large number of cells is involved and the chose of the lower threshold
is expected to impact on the fit.

A detailed discussion about the best fitting strategy to adopt for different
science cases can be found in Isobe et al. [11], Akritas and Bershady [9],
Kelly [10], Hogg et al. [12].

##  3 Application to a scientific case

We present here the application of PT-REX to study the correlation between
radio and X-ray emission in the mini-halo in the RX J1347.5-1145 galaxy
cluster. Radio mini-halos are diffuse radio sources observed at the center of
relaxed clusters and whose origins are still debated. Although they strongly
differ, all the models which have been proposed to explain the origin of the
relativistic, radio-emitting electrons agree on linking the properties of the
non-thermal particles to the thermal gas [e.g., 2, 13, for reviews].
Therefore, studying the spatial correlation between radio and X-ray emission
is crucial to investigate this physical connection and to discriminate between
the different scenarios. The mini-halo in RX J1347.5-1145 is elongated, where
the brightest part located toward north with a fainter extension toward south.
The diffuse radio emission surrounds the bright radio galaxy GALEX
J134730.-114509, and two large extended radio sources are placed side by side
with it. The X-ray emission exhibits an elliptical shape, although more
symmetric than the radio emission, with the major axis aligned along
northwest-southeast. Previous X-ray studies unveiled the complex dynamical
status of this cluster where shocks and merging sub-clusters coexist with a
sloshing cool-core [e.g., 14, 15, 16, 17]. Therefore, due to the complex
morphology of this system, the ptp analysis is the best suited to study the
connection between radio and X-ray emission.

Figure 4: X-ray image of RXJ1347.5-1145 in the 0.5-2.0 keV band with the -3,
3, 24, 96$\sigma$ level countours of the VLA radio image from Gitti et al.
[18] (1$\sigma$= 0.04 mJy beam-1). We report here the region of interest
(green dashed), a random smpling grid (green continuous) and the mask (grey).

We combined the Very Large Array observation at 1.4 GHz presented in Gitti et
al. [18] (RMS=0.04 mJy beam-1, beam 17.7${}^{\prime\prime}\times
13.6^{\prime\prime}$) with an X-ray image in the 0.5-2.0 keV band produced
from the archival Chandra observation 2222 (PI Khan, exposure time 100 ks),
which we processed following the standard analysis
procedure666https://cxc.cfa.harvard.edu/ciao/threads/index.html. We used the
3$\times$RMS level contours of the VLA image to define the region of interest,
which indicates to PT-REX the size and the position of the radio source, and
the mask (Figure 4). We carefully defined the mask within the region of
interest by making sure to remove the emission of the central galaxy and of
the field sources, whereas we adopted a more crude approach adopted outside
the region by masking the other sources with large squares. Then we define a
first grid to sample the emission above the 3$\sigma=0.12$ mJy beam-1 level of
the radio image. The beam area is 273.6 arcsec2 ( 23 pixel), so we use cell
with a 17.5′′ size and a total area of 289 arcsec2 (25 pixel). The resulting
grid shown in Figure 4 is composed of 16 cells which were then used in a SMptp
analysis. The Spearman and Pearson ranks that we measured are 0.79 and 0.8,
respectively, which indicate a strong linear correlation between
$I_{\text{R}}$ and $I_{\text{X}}$. This indicates that the ICM component
responsible of the radio and X-ray emission occupy the same volume, which is
in agreement to what is observed in radio mini-halos. In Figure 5 we report
the resulting SMptp analysis carried out with the algorithms implemented in
PT-REX. Albeit the large errors, they are all consistent within 1 $\sigma$ and
indicate a sub-linear correlation between radio and X-ray emission.

Figure 5:  Results of the SMptp analysis carried out with the grid shown in
Figure 4 with the different fitting algorithm: Least squares (top-left), BCES
orthogonal (top-right), BCES bisector (bottom-left), LinMix (bottom-right). We
report the 95$\%$ confidence interval.

However, the low number of cells indicates that the source is poorly resolved,
so our results may be biased by the grid which we used. Therefore we performed
also a MCptp analysis to test this possibility and to better constrain $k$. We
present here the results of 500 iterations of MCptp analysis carried out with
the BCES orthogonal fit. We used the mask and region of interest presented in
Figure 4 with the same cell size and threshold adopted for the SMptp analysis.
In Figure 6 we report the mean and the standard deviation of the final
distribution and of the first 50, 100 and 200 iterations.

Figure 6:  Histogram with the distribution of $k$ produced by 500 iterations
of the MCptp routine. We individually report the mean and standard deviation
of the first 50, 100, 200 iterations and of the total distribution.

In Table 1 we report the results of the all the ptp analysis carried out. All
the method produced estimates of $k$ that are in agreement within 1$\sigma$,
although the MCptp estimates a linear trend. All of the SMptp analysis
resulted in a sub-linear trend and, amongst them, the LS fit estimated the
flattest trend because it was pivoted by the two points at the extremes of the
distribution (Figure 5, top-left panel). The two BCES estimators produced
similar results, whereas the the LinMix fit (which was performed with 100
chains and a prior composed of two gaussians) returned a slightly flatter
value of $k$, albeit with larger errors. By comparing the SMptp analysis
performed with the BCES orthogonal algorithm with the MCptp, we can conclude
that the grid we used in the former was slightly biased toward flatter values,
most likely by the large errors of the low-brightness points. The
uncertainties of the MCptp are larger than the ones of the SMptp, regardless
of the fitting method. This indicate that the random sampling has
significantly affected the estimate of $k$, i.e. it confirms that the
observation does not fully resolve the mini-halo and there may be sub-
structures embed in the emission likely associated with the features (such as
shocks, sloshing and merging sub-clumps) of the thermal ICM. Therefore, by
featuring PT-REX we could comprehensively study the spatial correlation
between diffuse radio and X-ray emission in RXJ1347.5-1145, finding that they
are linked by a linear trend. This implies that the distribution of magnetic
field and relativistic particles is deeply bounded to the distribution of the
thermal plasma and, therefore, it may suggest that the diffusion and re-
acceleration of the radio-emitting electrons depends mostly on the local
properties of the thermal gas (e.g. its turbulence). Our result also suggests
that the current observations may be missing some components of this radio
source, which are instead glimpsed by the random sampling. The scientific
interpretation of this result in the context of origin of the radio emission
and the connection with the complex dynamical state of this cluster is not
trivial, and it is beyond the scope of this work. Here we just mention that
this result is in agreement to what is observed in general for radio mini-
halos [7].

SMptp: fitting method | $k$
---|---
Least squares | 0.66$\pm$0.19
BCES orthogonal | 0.85$\pm$0.14
BCES bisector | 0.88$\pm$0.12
LinMix | 0.75$\pm$0.19
MCptp: number of iterations | $k$
50 | 1.09$\pm$0.28
100 | 1.03$\pm$0.28
200 | 1.07$\pm$0.26
500 | 1.06$\pm$0.26
Table 1:  Results of the different ptp analysis.

##  4 Conclusions

We introduced PT-REX, a software to estimate the point-to-point trend between
radio and X-ray surface brightness in extended sources. We presented here a
set of tools to perform this analysis and we discussed how to run them
effectively for a variety of scientific problems. We also introduced the Monte
Carlo ptp analysis, which allows to explore the parameter space of the scaling
by combining numerous random sampling of the source. This method is advised
when studying small or poorly-resolved sources. Finally, we showed how to use
PT-REX by studying the $I_{\text{R}}$-$I_{\text{X}}$ trend in the cluster
RXJ1347.5-1145. Despite the low resolution of the radio image, we could
reliably estimate a linear scaling between radio and X-ray emission
($k=1.06\pm 0.26$) that indicates a strong bond between the thermal gas and
the radio-emitting plasma. With PT-REX similar studies can be easily conducted
on a variety of science cases to derive some useful (and sometimes unexpected)
insights into their physics.

## Acknowledgments

AI thanks G. Brunetti and M. Sereno for the useful discussions, M. Gitti for
providing the VLA image of RXJ1347.5-1145, and L. Bruno for having tested the
code with infinite patience. This research made use of
Astropy777http://www.astropy.org, a community-developed core Python package
for Astronomy [4, 5], SciPy [6] and APLpy, an open-source plotting package for
Python [19].

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# Damped Dirac magnon in a metallic kagomé antiferromagnet FeSn

Seung-Hwan Do  Materials Science and Technology Division, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, USA  Koji Kaneko  Materials and Life
Science Division, J-PARC Center, Tokai, Ibaraki, 319-1195, Japan  Materials
Sciences Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki,
319-1195, Japan  Ryoichi Kajimoto  Materials and Life Science Division, J-PARC
Center, Tokai, Ibaraki, 319-1195, Japan  Kazuya Kamazawa  Neutron Science and
Technology Center, Comprehensive Research Organization for Science and
Society, Tokai, Ibaraki, 319-1106, Japan  Matthew B. Stone  Neutron Scattering
Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
Shinichi Itoh  Materials and Life Science Division, J-PARC Center, Tokai,
Ibaraki, 319-1195, Japan  Institute of Materials Structure Science, High
Energy Accelerator Research Organization, Tsukuba, Ibaraki, 305-0081, Japan
Takatsugu Masuda  Institute for Solid State Physics, The University of Tokyo,
Chiba 277-8581, Japan  Institute of Materials Structure Science, High Energy
Accelerator Research Organization, Tsukuba, Ibaraki, 305-0081, Japan  German
D. Samolyuk  Materials Science and Technology Division, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, USA  Elbio Dagotto  Materials Science
and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee
37831, USA  Department of Physics and Astronomy, University of Tennessee,
Knoxville, Tennessee 37996, USA  William R. Meier  Materials Science and
Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee
37831, USA  Brian C. Sales  Materials Science and Technology Division, Oak
Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA  Hu Miao  Materials
Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge,
Tennessee 37831, USA  Andrew D. Christianson  Materials Science and Technology
Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

###### Abstract

The kagomé lattice is a fertile platform to explore topological excitations
with both Fermi-Dirac and Bose-Einstein statistics. While relativistic Dirac
Fermions and flat-bands have been discovered in the electronic structure of
kagomé metals, the spin excitations have received less attention. Here we
report inelastic neutron scattering studies of the prototypical kagomé
magnetic metal FeSn. The spectra display well-defined spin waves extending up
to 120 meV. Above this energy, the spin waves become progressively broadened,
reflecting interactions with the Stoner continuum. Using linear spin wave
theory, we determine an effective spin Hamiltonian that reproduces the
measured dispersion. This analysis indicates that the Dirac magnon at the
K-point remarkably occurs on the brink of a region where well-defined spin
waves become unobservable. Our results emphasize the influential role of
itinerant carriers on the topological spin excitations of metallic kagomé
magnets.

††preprint: APS/123-QED

This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-
AC05-00OR22725 with the U.S. Department of Energy. The United States
Government retains and the publisher, by accepting the article for
publication, acknowledges that the United States Government retains a non-
exclusive, paid-up, irrevocable, world-wide license to publish or reproduce
the published form of this manuscript, or allow others to do so, for United
States Government purposes. The Department of Energy will provide public
access to these results of federally sponsored research in accordance with the
DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

The interplay between charge, spin, and geometric frustration is an important
underlying theme to problems at the forefront of condensed matter physics [1,
2, 3, 4, 5, 6, 7, 8, 9]. Kagomé magnets, consisting of a corner shared
transition-metal triangular-network (Fig. 1(a)), are an ideal platform to
explore correlated topological states, including the fractional quantum Hall
effect [1, 2, 3, 4], the intrinsic Chern state [10, 11, 12, 9] and magnetic
Weyl semimetals [13]. While the charge excitations of kagomé magnets have been
extensively investigated [14, 8, 15, 6, 16, 5, 5, 9, 13], their magnetic
counterparts and the intertwined correlations between charge and spin degrees
of freedom have not yet been investigated in detail.

Similar to the electronic structure, a spin model with nearest-neighbor
magnetic exchange, $J_{1}$, yields a Dirac magnon at the K-point and a flat
magnon band, as shown in Fig. 1(c) [17, 18, 19]. Time-reversal symmetry
breaking interactions, such as the Dzyaloshinskii-Moriya interaction in
magnetic insulators, introduce a gap at the Dirac point and can induce a
topological thermal Hall effect [18, 20, 19, 21, 17]. In addition, magnon-
magnon interactions may modify the dispersion to realize interaction-
stabilized topological magnons [22, 23]. This simplified picture is, however,
challenged in a metallic kagomé magnet, where the presence of itinerant
electrons will introduce long-range magnetic interactions through, $e.g.$ RKKY
(Ruderman–Kittel–Kasuya–Yosida) interactions, that dramatically change the
magnon dispersion as shown in Fig. 1(d). Moreover, the high-energy spin wave
excitations will interact with the particle-hole continuum of the Stoner
excitation (Fig. 1(e)), resulting in mode decay.

Figure 1:  (a) (b) Crystal and magnetic structure of FeSn. The exchange paths
between Fe spins are indicated. The spin wave dispersions of a ferromagnetic
kagomé lattice with $J_{1}$=-1 meV and (c) $J_{2}$=0 and (d)
$J_{2}$=0.2$J_{1}$ ($J_{2}$=-0.2$J_{1}$), are displayed with black and gray
(blue) curves, respectively. High symmetry points are indicated in the inset
to (d). (e) Schematic of a Stoner excitation spectra (continuum) and magnon
(sharp dispersion) as a function of momentum (Q) and energy ($E$). The spin
wave mode decays into a particle-hole pair near the Fermi energy ($E_{F}$)
when it enters the Stoner continuum. The continuum boundary shifts with gap
$\Delta$ ($\delta$), reflecting the direct (indirect) electronic transitions,
as shown in the inset.

To explore the effects of itinerant carriers on the magnons in the
ferromagnetic kagomé spin-lattice, we study the spin excitation spectra of the
metallic kagomé magnet FeSn using inelastic neutron scattering (INS). The
measured spectra show relatively sharp spin waves of the ferromagnetic kagomé
spin-lattice below 120 meV. At higher energies, the spin waves exhibit decay
due to interactions with the Stoner continua. Interestingly, we find that
while the Dirac magnon remains, the upper branch of the Dirac band is heavily
damped, uncovering a non-trivial interplay between magnon and continuum.

FeSn crystallizes in a hexagonal structure (P6$/mmm$) with the Fe atoms
forming a two-dimensional kagomé spin-lattice(Fig. 1(b)). Below
$T_{\text{N}}$=365 K, the Fe spins form ferromagnetic kagomé layers which are
stacked antiferromagnetically along the $c$-axis with an ordering wave vector
of $\text{{Q}}_{\text{m}}$=(0,0,1/2). As we show in this letter, the dominant
in-plane ferromagnetic interactions allow the behavior of the quasi two-
dimensional ferromagnetic kagomé spin-lattice to be probed. For the INS
measurements, 4.43 g of FeSn single crystals were grown using the flux method
[24] and co-aligned on aluminum plates with a [$H,0,L$] horizontal scattering
plane. The INS data were obtained at $T$=100 K using HRC [25] (incident
energies $E_{\text{i}}$=40 and 153 meV) and 4SEASONS [26] ($E_{\text{i}}$=27
meV, 46 meV, 96 meV, and $E_{\text{i}}$=300 meV) spectrometers at the Japan
Proton Accelerator Research Complex (J-PARC). Additional data were collected
with the SEQUOIA [27] spectrometer ($E_{\text{i}}$=500 meV) at the Spallation
Neutron Source (SNS) at Oak Ridge National Laboratory (see [28] for additional
details).

Figure 2:  (a) Contour map of the INS intensity along high symmetry directions
(given in (f)). The data ((a),(c)) were measured using HRC with $E_{i}$=153
meV. The spectrum above (below) the horizontal line at 30 meV was obtained
from the BZ for $\Gamma$ at Q=(0,0,1/2) ((0,0,3/2)), integrating over Q=0.22
Å-1 along the vertical direction. Horizontal (vertical) error bars of pink
(green) circles indicate the fitted peaks full width at half maxima (FWHM),
and vertical (horizontal) error bars indicate the range of energy (momentum)
integration. (b) INS data (left) and spin wave calculations (right) as
described in the text along the out-of-plane direction through the ZC,
measured using the 4SEASONS spectrometer with $E_{\text{i}}$=46 meV. The solid
line is the calculated magnon dispersion. (c) Constant energy slice of the
magnon spectra in the [$H$,0,$L$] plane and the calculated spectra. (d) Low-
energy spectrum of $I(\textbf{Q},E)$ near the ZC measured using
$E_{\text{i}}$=27 meV at 4SEASONS, and (e) the corresponding calculation
including an easy-plane anisotropy of $D_{z}$=0.2 meV. Figure 3:  (a) High-
energy INS spectra (plotted as $E\times I(\text{{Q}},E)$) and spin wave
calculations ((b)) along high symmetry directions as indicated in the right
panel of the $HK$-reciprocal space map. Data were obtained by integrating over
Q=0.19 Å-1 and -4$\leq L\leq$4\. The calculation was performed for an
identical Q-integration range and convoluted with the instrumental resolution
of SEQUOIA. The black solid lines display the magnon dispersion for $L$=0.5.
Horizontal (vertical) error bars of white filled circles indicate the fitted
peaks FWHM (range of energy integration). (c) Constant energy cut along the
high symmetry directions, integrated over energy $\pm$5 meV. Solid lines are
Gaussian fits described in the text with fitted values displayed in (a). (d)
INS spectra obtained from HRC ($E_{i}$=153 meV), integrated over -3$\leq
L\leq$3\.

Figure 2 shows the spectra in the three-dimensional hexagonal Brillouin zone
(BZ), measured by INS. The acoustic magnons emanate from
$\textbf{Q}_{m}$=$\Gamma$(0,0,1/2), and disperse throughout the entire BZ.
Strongly dispersive magnons in the $HK$-plane extend well above 80 meV,
whereas the magnon dispersion along the out-of-plane direction has a bandwidth
of less than 20 meV indicating the dominant spin-spin interactions are within
the kagomé-lattice planes. The nearly two-dimensional character of the spin
excitation spectrum is further evidenced by the rod-like scattering shown in
Fig. 2(c).

Table 1: Hamiltonian parameters determined from the spin wave theory analysis.  Label (number of paths) |  $J_{1}$ (4) |  $J_{int1}$ (2) |  $J_{2}$ (4) |  $J_{int2}$ (8) |  $J_{3}$ (2) |  $J_{4}$ (4) | $D_{z}$
---|---|---|---|---|---|---|---
$J_{ij}^{\text{Fit}}$ (meV) | -44.33 $\pm 1.56$ | 4.51$\pm 1.00$ | 12.23 $\pm 1.06$ | 1.27 $\pm 0.24$ | -5.28 $\pm 2.32$ | -4.60 $\pm 0.90$ | 0.1
Distance (Å) | 2.65 | 4.45 | 4.59 | 5.18 | 5.30 | 5.30 | -

The high-energy spectra were measured using the SEQUOIA spectrometer with
$E_{i}$=500 meV. We integrate the INS data over $-4\leq L\leq$4 r.l.u. to
enhance statistics. Note that due to momentum and energy conservation, high-
energy transfer data is obtained from a larger magnitude $L$-region, which
results in lower scattering intensity from the magnetic form factor
contribution. As shown in Fig. 3, the excitations extend to at least 200 meV.
Two individual magnon branches are observed corresponding to the lower- and
mid-magnon bands in Fig.1(c) of the ferromagnetic kagomé spin-lattices through
the M- and K-points in the BZ. The higher energy spectral weight above
$\sim$120 meV is diffuse, and becomes indiscernible from background above
$\sim$200 meV. Figure 3(c) shows momentum scans through
$\Gamma$’-$M$-$\Gamma$-K-X for increasing energy transfer. Along both the
$\Gamma$-M and $\Gamma$-K directions, the peak linewidths broaden as a
function of Q near the zone boundary (ZB), and the peak-positions are intact
over a wide energy range 120$<E<$170 meV (80$<E<$120 meV) near the K
(M)-point. These Q-, $E$\- peak broadenings indicate the decay of the magnons,
resulting from the quasiparticle scattering [29, 30, 31]. Considering the
metallicity of FeSn along with the collinear spin configuration, FeSn
presumably has a large magnon-electron interaction, which results in strong
damping of the magnon spectra.

Figure 4:  (a)-(d) Constant energy slices of the INS data ($E\times
I(\text{{Q}},E)$) and spin wave calculations. Dashed lines indicate the first
BZ in the $HK$-plane. The color bar for (a)(b) ((c),(d)) is shown in right of
(b) ((d)). INS spectra through the K-point along (e) transverse- and (f)
radial-directions (see arrows in insets). (g) Momentum scans at constant
energy through the K-point along the transverse direction. The dispersion was
extracted by fitting the spectra to Gaussian functions (solid lines) and the
results are displayed as circles in (e). The lines in (e)(f) represent the
linearly crossing magnons for $L$=0.5 at the Dirac node. Horizontal (vertical)
error bars in (e)(f) indicate the fitted FWHM (range of energy integration).
(h) Constant wave-vector scan at the Dirac point. Data are shown as symbols
and the spectral weight from LSWT (shaded region) is described in the text.
The line is a guide to eye. The data was obtained by integrating over the
momentum region [$H$,0,0]=$\pm$0.05, [2$K$,-$K$,0]=$\pm$0.06, and
[0,0,$L$]=$\pm$4\. (e),(f),(h) For clarity, the nonmagnetic background was
obtained from the scattering at Q=(2/3,2/3,0) and subtracted from the measured
intensities [28].

To understand the observed spin wave spectra and the underlying spin-spin
interactions, we use linear spin wave theory (LSWT) with the Hamiltonian,
${\cal H}=J_{n}\sum_{i,j}S_{i}S_{j}-D_{z}\sum_{i}(S_{i}^{z})^{2}$, as
implemented in the SpinW software package [32]. We set $S$=1 considering the
measured effective moment of 3.4 $\mu_{B}$ (2.8 $\mu_{B}$ for $S$=1, where
$g$=2) [24]. $J_{n}$ and $D_{z}$ correspond to Heisenberg exchange couplings
for the $n$th nearest-neighbor and a single-ion anisotropy, respectively
111Indeed, Dzyalloshinskii-Moriya interaction (DMI) along $z$-axis ($c$) is
symmetrically allowed. However, since the DMI on the spins aligning in the
plane ($z\perp S$) does not change the dispersion we exclude the DMI in the
model Hamiltonian.. Interactions up to fourth (second) nearest-neighbor in-
the-plane (out-of-plane) direction (see Fig. 1(a)(b)) were considered. Note
that $J_{3}$ and $J_{4}$ have the same distance but different paths. Hence,
the distinction of these parameters is maintained due to the potential effects
on the RKKY interaction of the complicated band structure near the fermi
surface [34]. The measured dispersion is fitted to the calculated dispersion
(see [28]), yielding the parameters listed in Table 1. The parameters indicate
a dominant nearest-neighbor ferromagnetic interaction $J_{1}$ responsible for
the ferromagnetic kagomé spin-lattice. We also determine non-negligible
further neighbor exchanges, $J_{2}(\sim-0.28J_{1})$, $J_{3}(\sim 0.12J_{1})$,
and $J_{4}(\sim 0.1J_{1})$, are present. The sign and relative size of the
parameters from the spin wave analysis are largely consistent with parameters
determined from first principles calculations (see Supplemental materials [28]
and [24]). Furthermore, the symmetry-allowed easy-plane single-ion anisotropy
($D_{z}>$0) reproduces the peaked intensity data near 4 meV shown in Fig 2(d)
[28].

The refined spin wave scattering intensity is compared to the experimental
data in Figs. 2, 3, and 4. The calculations reproduce the low-energy spectra.
However, the scattering intensity and dispersion deviate from the calculation
at the zone boundary and well-defined modes are essentially absent above 200
meV in the measurements. This discrepancy in the scattering intensity is
ascribed to interactions with the Stoner continuum [35, 36, 37, 38], and
indicates the energy scale of the Stoner excitations. Due to the large number
of electronic bands in FeSn, it is challenging to make direct comparisons to
the magnetic spectra measured here. However, electronic band structure
calculations do indicate splitting of majority and minority spin bands near
the fermi energy [39, 24]. The minimum energy of an indirect inter-band
transition for these bands near Q=$\Gamma$ is $\sim$0.1-0.2 eV, which results
in a gap of the Stoner excitations with finite momenta (see Fig.1(e)), and is
consistent with the energy scale above which damping begins to dominate the
INS spectra.

The determined spin Hamiltonian and the symmetry of the spin configuration
preserves time reversal symmetry, and permit the existence of a Dirac point in
the magnon spectrum. LSWT presents a sharp linear magnon band crossing at
$E\sim$120 meV at the K-point (see dispersion line in Fig. 3(a)(b)). However,
due to interactions with electron, the spin waves near the Dirac node are
susceptible to decay. Figure 4(a)-(d) presents constant energy slices measured
up to 180 meV. The low-energy spectrum below the Dirac node are reproduced by
LSWT. The Dirac node is evident at the K-point at 120 meV as shown in Fig.
4(b). Above 120 meV, the excitations significantly broaden. The is
particularly evident near the zone boundary and the broadening increases with
increasing energy transfer. Figures 4(e) and (f) highlight the dispersion in
the vicinity of the Dirac nodes along transverse and radial directions,
respectively. Figure 4(g) shows constant energy scans along the transverse
direction through the Dirac nodal point as having two clear peaks below 100
meV and above 150 meV, but only a single peak between 100 meV and 150 meV in
the vicinity of the two crossing bands. Peak positions extracted from Gaussian
fits compare well to the LSWT dispersion curve in Figs. 4(e). We note that
finite spectral weight likely due to damping from interactions with the
continuum is present between the two peaks above the Dirac node. In contrast,
the momentum scan along the radial direction deviates from the calculated
dispersion above 120 meV, as shown in Fig. 4(f). Rather than two peaks,
constant energy scans along this direction show a broadened spectral weight
centered near the Dirac node. These results demonstrate that the scattering
with itinerant electrons reconstruct the upper Dirac cone dispersion, but also
the diffusive continuum from the decay fills in the Dirac cone. Figure 4(h)
shows an energy scan at the Dirac node compared to the calculated spectral
weight of the LSWT model convoluted with the instrumental energy resolution.
The decayed spectral weight is visible above 150 meV and extends well beyond
the LSWT model of the scattering.

Additionally, the LSWT completely fails to explain the observed upper spectral
weight above 120 meV along $\Gamma$ to M (see Fig 3(a)(b)). We note that
adjusting the exchange values of the Hamiltonian to have a large
antiferromagnetic $J_{4}\sim$3.5 meV (with ferromagnetic $J_{3}$) decreases
the upper magnon branch down to 120 meV and reproduces the observed spectral
weight near M. However, this results in a large discrepancy in the other
magnon bands (see Supplemental Material [28]). It is worth noting that a
fluctuation continua is also present at the top of the lower magnon branch at
Q=M (zone boundary) above 80 meV (see Fig. 3(d)). It connects the lower magnon
branch to the upper spectral weight without a gap in the spectrum. This in
turn generates a band touching at M around the Dirac node, resulting in a weak
ring-shaped spectral weight in the all constant energy slices between 80 meV
and 150 meV (see Fig. 4). This continuous scattering confirms that the
excitation near M is not simply due to a spin wave excitation. Therefore, a
likely component of the measured spectral weight near M is the decayed spectra
of the upper magnon band. To explain this may require a comparison to the
itinerant band model [40, 38], a more sophisticated approach which includes
the correction from the interactions with itinerant electrons [41, 42], or
spin-fermion model [43, 44, 45].

In summary, we have found that the spin excitation spectrum in the
ferromagnetic kagomé metal FeSn is quasi-two-dimensional with progressively
stronger damping of the spin waves with increasing energy transfer. The
determined exchange terms for the spin Hamiltonian provide for a symmetry
allowed magnon Dirac nodal point near the electronic continua. The interaction
with the itinerant electrons is large near the nodal point, resulting in a
significant spectral broadening with momentum dependence. The interactions are
also large near the M-point, which results in continuous spectral weight
between the lower and upper magnon bands. A more complete understanding of
these observations require calculations which account for the electron-magnon
interactions. It will be particularly interesting to check if the spin-charge
coupled spectra in the kagomé metallic magnet possesses the topology arising
from correlation effects.

###### Acknowledgements.

We acknowledge M. Lumsden for useful discussions. This research was supported
by the U.S. Department of Energy, Office of Science, Basic Energy Sciences,
Materials Science and Engineering Division. Work at the Oak Ridge National
Laboratory Spallation Neutron Source was supported by U.S. DOE, Office of
Science, BES, Scientific User Facilities Division. The neutron experiment at
the Materials and Life Science Experimental Facility of the J-PARC was
performed under a user program (Proposal No. 2019B0248 and 2020A0217).

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