††thanks: Corresponding author

# Single-layer Skyrmions in a van der Waals antiferromagnet

A. Pozzi<EMAIL_ADDRESS>M. Mostovoy  Zernike Institute for Advanced
Materials, University of Groningen, Nijenborgh 4,9747 AG Groningen, The
Netherlands

###### Abstract

Transition metal halides are quasi-two-dimensional van der Waals magnets that
can potentially host antiferromagnetic (AFM) skyrmions induced by competing
exchange interactions. We study theoretically magnetic states of Fe-doped
NiBr2 using spin interactions obtained by fitting experimental data. We find
that AFM interlayer interactions suppress skyrmion lattices but allow for
skyrmions with topological charges $1,2$ and $3$ confined to a single magnetic
layer. These single-layer skyrmions exist in three different magnetic phases
and their dynamics changes drastically at the transition from a uniform to a
non-collinear modulated state.

###### pacs:

75.70.-i, 75.10.-b 75.30.Kz

Since their first observation Muelbauer2009 ; Yu2010 , magnetic skyrmions are
in the focus of many theoretical and experimental studies. Electrically driven
dynamics and stability rooted in non-trivial topology make skyrmions promising
information carriers in new magnetic memory devices Nagaosa2013 ; Fert2013 ;
Fert2017 ; Back2020 . Current research on skyrmions is focused on magnets with
chiral crystal lattices and multilayer devices where inversion symmetry is
broken at interfaces between magnetic and heavy-metal materials Jiang2015 ;
Woo2016 . The lack of inversion leads to anisotropic spin interactions
favoring non-collinear magnetic textures Bogdanov1989 .

Inversion symmetry breaking is not strictly necessary: the competing
Heisenberg exchange interactions in Mott insulators Okubo2012 ; Leonov2015 ;
Batista2016  and long-ranged spin interactions mediated by itinerant electrons
in magnetic conductors Azhar2017 ; Hayami2017 ; Ozawa2017 ; Okumura2020  can
stabilize skyrmions and even more complex spin textures, such as hedgehog
crystals Ishiwata2018 . Tiny (2-3 nm in diameter) skyrmions recently found in
centrosymmetric intermetallics give rise to large Topological Hall and Nernst
effects Kurumaji2019 ; Hirshberger2019 ; Khanh2020 ; Hirshberger2020 .
Isotropic interactions in achiral magnets result in additional low-energy
skyrmion degrees of freedom, helicity and voriticity, which strongly modify
interactions and dynamics of these topological defects Leonov2015 ; Leonov2017
; Kharkov2017 . There is also a lot of interest in skyrmions in natural and
artificial AFM materials that may propagate with a higher speed and show no
Skyrmion Hall effect Barker2016 ; Zhang2016 ; Legrand2020 .

Here, we explore the possibility to find skyrmions in layered transition metal
halides, which are wide gap Mott insulators showing a variety of magnetic
orders McGuire2017 . These van der Waals materials allow for exfoliation down
to monolayers, which enables control of two-dimensional magnetism by doping,
gating, magnetic and electric fields Jiang2018 . We focus on NiBr2 that
undergoes a transition into the collinear A-type AFM state at $T_{\rm N}=52$
K, which transforms into a multiferroic spiral state below 23 K with six
energetically equivalent orientations of the in-plane wave vector Day1976 ;
Adam1980 ; Tokunaga2011 . The spin spiral results from the competition between
the nearest-neighbor ferromagnetic (FM) Heisenberg exchange interactions on
the $\sim 90^{\circ}$ metal-halogen-metal bonds and further-neighbor AFM
interactions enhanced by strong metal-ligand covalency and lattice geometry.
The sign of magnetic anisotropy can be controlled in this material by doping
Moore1985 .

Our study of a model of doped NiBr2 with parameters obtained by fitting
experimental data shows that the AFM interlayer interactions suppress skyrmion
crystals at all magnetic fields and concentrations of dopands, but allow for
topological defects with skyrmion topology, which are trully two-dimensional
as their topological charge is only nonzero in a single magnetic layer. These
single-layer skyrmions and bi-merons are stable in three different magnetic
phases at experimentally accessible magnetic fields. We discuss dynamics of
these defects induced by the spin-Hall torque.

The model.

The spin Hamiltonian reads,

$\displaystyle H=$ $\displaystyle-
J_{1}\sum_{\left<i,j\right>,\lambda}\mathbf{S}^{\lambda}_{i}\cdot\mathbf{S}^{\lambda}_{j}+J_{2}\sum_{\left\llangle
i,j\right\rrangle,\lambda}\mathbf{S}^{\lambda}_{i}\cdot\mathbf{S}^{\lambda}_{j}$
(1)
$\displaystyle+\sum_{i,\lambda}\left[J_{\perp}\mathbf{S}^{\lambda}_{i}\cdot\mathbf{S}^{\lambda+1}_{i}-\frac{K}{2}\left(\mathrm{S}^{\lambda}_{i}\right)_{z}^{2}-\mathbf{h}\cdot\mathbf{S}_{i}^{\lambda}\right],$

where $\mathbf{S}^{\lambda}_{i}$ denotes the unit spin vector at the site $i$
of a triangular lattice in the layer $\lambda$. Indices in single and double
brackets denote pairs of nearest-neighbor (NN) and next-nearest-neighbor (NNN)
sites in the triangular lattice, respectively. The NN Heisenberg exchange
interaction is ferromagnetic, whereas the NNN exchange that gives rise to
frustration and the interlayer interaction are antiferromagnetic
($J_{1},J_{2},J_{\perp}>0$); $K$ is the single-ion anisotropy and $\mathbf{h}$
is the magnetic field.

Equation (1) contains a few simplifications. First, NiBr2 has a CdCl2 crystal
structure (space group $R\bar{3}m$) with ABC stacking of three triangular
layers in the rhombohedral unit cell McGuire2017 . Second, the third-nearest-
neighbor AFM exchange in NiBr2 is stronger than the NNN one Regnault1982 ;
Day1984 . Our model is justified by the long period of the spiral modulation
($\sim 18$ in-plane lattice constants), which makes possible to ignore the
relative shifts of neighboring layers. Spin models with 2 and 3 exchange
interactions become equivalent in the continuum limit, the only difference
being the orientation of the spiral wave vector.

Figure 1:  Zero-temperature magnetic phase diagram of NiBr2 in the magnetic
field $H\perp c$. The black dots show the experimentally measured values of
the three critical fields, $H_{1}<H_{2}<H_{3}$, corresponding to the
transitions between the spiral, canted AFM, fan, and ferromagnetic spin states
Katsumata1983 . Red arrows indicate the critical fields obtained in our
numerical simulations, for $J_{1}=59.5\,\mathrm{K}$, $J_{2}=20.4\,\mathrm{K}$
and $J_{\perp}=11.0\,\mathrm{K}$. Spin configurations in two neighboring $ab$
layers are schematically shown with arrows.  Figure 2:  Magnetic field vs
anisotropy phase diagram of FexN1-xBr2 for $H\|c$, which includes the A-type
AFM, vertical spiral (VS), conical spiral (CS) and field-induced ferromagnetic
(FM) states. The area encircled by red dotted line is the region of stability
of defects with skyrmion charge $\pm 1$ confined to a single spin layer.

Earlier theoretical study of a single-layer frustrated triangular magnet with
an easy-axis anisotropy ($K>0$) showed that the skyrmion crystal occupies a
large part of the phase diagram Leonov2015 . NiBr2 is an easy-plane
antiferromagnet ($K<0$) Regnault1982 ; Day1984 , but its anisotropy changes
sign upon substitution of magnetic ions: $\mathrm{Fe_{x}Ni_{1-x}Br_{2}}$
becomes an easy-axis magnet for $x>9.9\%$ Moore1985 . AFM interlayer
interactions in quasi-two-dimensional materials govern symmetry of skyrmion
crystals and can suppress them LinBatista2018 . The question arises whether
the interlayer interactions in dihalides allow for skyrmions. To address this
question, we determine exchange parameters of our model using experimental
data on NiBr2. With $J_{2}/J_{1}=0.343$ we fit the magnitude of the in-plane
component of the spiral wave vector, $\mathbf{Q}=[0.027,0.027,3/2]$ Moore1985
(see Supplementary Information). The interlayer coupling $J_{\perp}$ is
obtained from the fit of the phase diagram of NiBr2 under an in-plane magnetic
field Katsumata1983 , which shows three phase transitions: from the spiral to
the canted AFM state at $H_{1}=2.7\,\mathrm{T}$, from the AFM to the fan state
at $H_{2}=31.5\,\mathrm{T}$ and from the fan to saturated state at
$H_{3}=32.9\,\mathrm{T}$ measured at 1.3 K (see Fig. 1). The
$\frac{H_{1}}{H_{3}}$ ratio is a measure of the ratio of the exchange energy
difference between the uniform AFM and spiral states in zero field, $\Delta
E_{\rm ex}$, and the interlayer exchange energy:

$\frac{\Delta E_{\rm
ex}}{J_{\perp}}\simeq\left(\frac{H_{1}}{H_{3}}\right)^{2}\approx 6.7\cdot
10^{-3}.$ (2)

Figure 1 shows the comparison between the critical fields obtained in our
simulations for $J_{\perp}/J_{1}=0.184$ (red arrows) and experimentally
measured values (black dots). We converted $h$ into $H$ measured in Tesla
using $H=\frac{J_{1}}{2\mu_{\rm B}}h$ with $J_{1}=59.5\,\mathrm{K}$,
$2\mu_{\rm B}$ being the magnetic moment of Ni2+ ion. Since spins in all three
magnetic phases are confined to the $ab$ plane, the critical fields are
independent of $K$.

Magnetic states and topological defects.

Next we calculate the $(H,K)$ phase diagram of the Fe-doped compound,
FexN1-xBr2, treating the anisotropy parameter $K$ as a variable that depends
on the concentration of Fe dopands $x$. Figure 2 shows the phase diagram for
$K>0$ (easy-axis anisotropy) and $H\|c$ including the vertical spiral (VS),
A-type AFM, conical spiral (CS) and the field-polarized (FM) phases. The small
energy difference between non-collinear and collinear spin orders [see Eq.(2)]
confines the VS state to a narrow region between the A-type AFM state with
alternating layers of spins parallel/antiparallel to the $c$ axis and the CS
phase, in which $S^{z}$ is the same in all layers and the sign of the in-plane
spiral component alternates from layer to layer. The skyrmion lattices, both
the triangular array of skyrmion tubes and a three-dimensional HCP/FCC crystal
of skyrmions LinBatista2018 , are suppressed for the realistic model
parameters alongside with all multiply-periodic magnetic phases found a single
triangular layer Okubo2012 ; Leonov2015 .

Although the AFM interlayer coupling suppresses skyrmion crystals, it allows
for isolated skyrmions with topological charge confined to a single layer.
Remarkably, these single-layer skyrmions are stable in three different phases:
the FM, AFM and CS states in the region encircled by red dotted line in Fig.
2. Figure 3a shows the skyrmion in the AFM state, which is only stable in
layers with the magnetization opposite to the applied field. The magnetic
moment in the skyrmion center is parallel to the field, in contrast to FM
skyrmions (Fig. 3b). The small disturbance in the spin configuration of
neighboring layers is topologically trivial (Fig. 3c) and its magnitude decays
quickly with the distance from the skyrmion (see Supplementary information).
It has a stabilizing effect on skyrmions which are unstable in the surface
layers.

Unexpectedly, a magnetic defect with skyrmion topology is also stable in the
CS state (Fig. 4a). It has the appearance of a bi-meron – the bound vortex-
antivortex pair Kharkov2017 , but its topological charge distribution has a
single maximum and is not divided into two halfs (see Fig. 4c). We also found
single-layer skyrmions with topological charges $Q=\pm 2$ and $\pm 3$ in the
FM state (Figs. 4b,c) stabilized by AFM interlayer interactions.

a b

c d

Figure 3:  Spin configurations in the layers containing the skyrmion in the
(a) A-type AFM state and (b) FM state. (c) and (d) The corresponding spin
configurations in neighboring layers. In-plane spin components are shown with
arrows; the out-of-plane spin components are color coded.

a b

c d

Figure 4:  (a) Bi-meron in the CS state, (b) skyrmion with topological charge
3 in the FM state, and the corresponding distributions of the topological
charge density (panels c and d). In-plane spin components are shown with
arrows; the out-of-plane spin components and the topological charge density
are color coded. Pink dot marks the (anti)vortex center.

Dynamics:

Next we discuss dynamics of single-layer skyrmions induced by the spin-Hall
torque Slonczewski1996 ; Khvalkovskiy2013

$\mathbf{T}_{\rm
SH}\propto\mathbf{S}\times\mathbf{S}\times\left(\hat{\mathbf{c}}\times\mathbf{j}_{\rm
c}\right),$ (3)

where jc is the charge current in a heavy metal layer generating a spin
accumulation at the interface with the antiferromagnet, $\mathbf{S}$ is a spin
in the interface layer (layer 1) and the unit vector $\hat{\mathbf{c}}$ is
normal to spin layers. Due to the instability of skyrmions at the interface,
we study their dynamics in layer 2 (next to the interface layer) by numerical
integration of Landau-Lifshitz-Gilbert equation with the torque (see
Suplementary information).

The skyrmion in the uniform AFM and FM states shows the coupled helicity and
center-of-mass dynamics Leonov2017  with the helicity angle growing linearly
with time, corresponding to rotation of the in-plane spins, and the rotational
center-of-mass motion Lin2016 ; Zhang2017  (see Fig. 5a and Supplementary
Movie 1). Surprisingly, the bi-meron motion in the CS state only depends on
the direction of ${\bf j}_{\rm c}$ in the initial stage. In the steady state,
the bi-meron moves along a straight line perpendicular to the spiral wave
vector $\mathbf{Q}$ and its helicity remains constant (see Fig. 5b and
Supplementary Movies 2–3). The spin-Hall torque results in a sinusoidal
modulation of $S^{z}$ in layer 1 and the interlayer interaction attracts bi-
merons into a ‘channel’ perpendicular to ${\bf Q}$, located under the
$S^{z}\approx+1$ line in layer 1. The channel position and the skyrmion
helicity do depend on ${\bf j}_{\rm c}$, but the direction of motion does not.
This peculiar dynamics is a consequence of the rotational invariance of
Heisenberg exchange interactions and the fact that the rotation of in-plane
spins is equivalent to a translation of the CS along ${\bf Q}$. The transverse
motion of bi-merons is accompanied by a slow drift along ${\bf Q}$, which is a
finite-size artifact that becomes smaller as the system size increases (see
Supplementary information).

a

b

Figure 5:  Trajectories of topological defects in (a) the A-type AFM and (b)
CS states moved by the spin-Hall current applied along the positive $x$
direction.

Conclusions:

In conclusion, centrosymmetric NiBr2 has a number of properties required to
host skyrmion lattices: frustrated exchange interactions, three-fold symmetry
axis and magnetic anisotropy tunable by Fe-doping. However, because of the
long magnetic modulation period, even the relatively weak AFM interlayer
interactions suppress the skyrmion crystal and other multiply-periodic states
in this material. Yet the interlayer interactions do not prevent the formation
of isolated single-layer skyrmions and even have a stabilizing effect on them.
The single-layer skyrmions are mobile, stable in several magnetic phases and,
in contrast to AFM skyrmion tubes, can be easily nucleated, which makes them
promising candidates for information carriers in antiferromagnetic
spintronics.

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11institutetext: Department of Physics, University of Naples Federico II, C.U.
Monte Sant’Angelo, Via Cinthia, I-80126 Naples, Italy 22institutetext: INFN -
Sezione di Napoli, 80126 Naples, Italy 33institutetext: INAF-Osservatorio
astronomico di Capodimonte, Via Moiariello 16, I-30131 Naples, Italy
44institutetext: INAF-Oss.dell’Abruzzo, 64100 Teramo, Italy 55institutetext:
Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA
66institutetext: Steward Observatory, University of Arizona, 933 N. Cherry
Ave., Tucson, AZ 85719, USA. 77institutetext: School of Astronomy and Space
Science, Nanjing University, China 88institutetext: Institute of Astrophysics,
Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, 7820436
Macul, Santiago, Chile 99institutetext: European Southern Observatory,
Santiago, Chile 1010institutetext: Research Centre for Astronomy, Astrophysics
& Astrophotonics, Macquarie University, Sydney, NSW 2109, Australia
1111institutetext: Department of Physics & Astronomy, Macquarie University,
Sydney, NSW 2109, Australia

# Properties of intra-cluster low-mass X-ray binaries in Fornax globular
clusters

G. Riccio 11 M. Paolillo 112233 M. Cantiello 44 R. D’Abrusco 55 X. Jin 66 Z.
Li 77 T. Puzia 88 S. Mieske 99 D.J. Prole 10101111 E. Iodice 22 G. D’Ago 88 M.
Gatto 1133 M. Spavone 22

###### Abstract

Aims. We present a study of the intra-cluster population of low-mass X-ray
binaries (LMXB) residing in globular clusters (GC) in the central 1 $deg^{2}$
of the Fornax galaxy cluster. Differently from previous studies, which were
restricted to the innermost regions of individual galaxies, this work is aimed
at comparing the properties of the intra-cluster population of GC-LMXBs with
those of the host galaxy.

Methods. The data used are a combination of VLT Survey Telescope (VST) and
Chandra observations.We perform a cross-match between the optical and X-ray
catalogue, in order to identify the LMXBs residing in GCs. We divide the GC-
LMXBs into host-galaxy and intra-cluster objects based on their distance from
the nearest galaxy in terms of effective radius ($R_{eff}$). We found 82
intra-cluster GC-LMXBs and 86 objects that are hosted in galaxies. As the
formation of LMXBs also depends on the host GC colour, we performed a Gaussian
mixture model to divide the population into red and blue GCs.

Results. As has been found for the innermost regions of galaxies, LMXBs tend
to form in red and bright GCs in intra-cluster space as well. We find,
however, that the likelihood of a red GC to host an LMXB decreases with
galactocentric distance, but it remains approximately constant for the blue GC
population. Investigating the X-ray properties of the LMXBs residing in GCs,
we find a difference in the X-ray luminosity function between the intra-
cluster and host-galaxy sample: both follow a power-law down to $\sim
8.5\times 10^{37}$ erg s-1, which is consistent with field LMXBs for the
intra-cluster sample, while the latter agree with previous estimates for LMXBs
in GCs. We observe a deficiency of bright LMXBs in blue intra-cluster GCs,
however. This might indicate a lack of black hole binaries in metal-poor
systems. We further investigated the spectral properties of the GC-LMXBs
through their hardness-ratio. We detect a tentative difference in the hardness
ratio of two populations, where the intra-cluster GC-LMXBs appear to have
harder spectra than the host-galaxy objects. We find the same trend when we
compare red and blue GC-LMXBs: the spectra of the blue sample are harder
spectra than those of the red sample. This result could suggest a relation
between the spectral properties of LMXBs and the host GC colour and therefore
its metallicity.

We discuss the possibilities of spatial biases due to uncertainties in the
X-ray spectral response correction and due to contamination by background
active galactic nuclei.

Conclusions.

###### Key Words.:

X-rays: binaries, galaxies: star clusters, galaxies: clusters: individual
(Fornax), X-rays: galaxies: clusters

##  1 Introduction

Low-mass X-ray binaries (LMXBs) are stellar systems composed of an extremely
dense object (a neutron star or black hole) that accretes mass from a
secondary star (a main-sequence star of about one solar mass). They represent
the dominant X-ray binary (XRB) population in early-type galaxies. It has been
shown that a significant fraction of LMXBs resides in GCs. This varies from
10%-20% in small galaxies and reaches $\sim 70$% in cD galaxies, depending on
the morphological type of the galaxy and on the specific abundance of the GCs
(e.g. Kim et al., 2009).

The GC-LMXB association is particularly interesting as the high stellar
density near the centre of GCs may trigger the formation of binaries either by
three-body process or by tidal capture. It was observed that LMXBs tend to
form in bright GCs, as expected if the luminosity is a proxy for the total
number of stars they contain (Fabbiano 2006, and references therein). On the
other hand, size and concentration reflect the efficiency of dynamical
interaction and favour binary formation in dense environments. Furthermore,
the formation can also be influenced by the mass, size, and metallicity of
GCs. Red metal-rich GCs are about three times more likely to host LMXB than
blue (metal-poor) GCs, in part because red GCs are denser on average than the
blue counterpart (Jordán et al. 2004, Fabbiano 2006, Paolillo et al. 2011,
D’Ago et al. 2014 and references therein). However, the role played by
metallicity in LMXB formation is still unclear, and understanding this
connection would help us to know how these objects are formed, and if the
properties of the environment (the host GC or the galaxy properties) can have
an impact on the structure and emission of the LMXB.

The fact that the spatial distribution of GC-LMXBs in early-type galaxies
usually follows the distribution of the host GC population well (Paolillo et
al. 2011) suggests that the likelihood of LMXB formation is mainly driven by
the internal GC properties. However, there have been claims that LMXBs may be
less concentrated than GCs around giant ellipticals, suggesting that
environmental effects may influence the formation and evolution of LMXBs.
Unfortunately, most studies so far were limited to the central regions of
galaxies because the area explored by past surveys (Hubble Space Telescope)
was limited and high-resolution imaging was lacking. This imaging is needed to
select and measure the structural parameters of extragalactic GCs. This has
prevented studies of the effect of the distance from the galaxy centre on the
formation of LMXBs inside GCs. Because the spatial distribution of the red and
blue GCs changes according to the distance from the galaxy, moreover, wide-
field observations are required to distinguish the different physical
processes at work in a scenario in which red GCs are associated with the main
body of the galaxy while the blue GCs are associated with the halo (Cantiello
et al. 2018).

Significant populations of intra-cluster GCs have been discovered in the Virgo
(Durrell et al. 2014) and in the Fornax galaxy cluster. Using the VLT Survey
Telescope wide-field imaging obtained within the Fornax Deep Survey (FDS,
Iodice et al. 2016) in the Fornax cluster, our collaboration has proved the
existence of a vast population of intra-cluster stellar systems extending out
to a significant fraction of the viral radius (D’Abrusco et al. 2016;
Cantiello et al. 2018; Cantiello et al. 2020). This was found to match the
distribution of the intra-cluster light (Iodice et al. 2017), thus tracing the
past dynamical evolution of the galaxy cluster itself. In addition, Jin et al.
(2019) discovered a population of field LMXBs throughout the cluster, around
the dominant cD galaxy NGC1399, by analysing Chandra archival images.

In this work, we investigate the properties of LMXBs residing in this extended
population of intra-cluster GCs making use of the latest FDS data release,
combined with Chandra X-ray data covering the core of the Fornax galaxy
cluster. Based on this, we study the GC-LMXB connection within the whole
Fornax cluster and its dependence on host galaxy, environment, galactocentric
distance, and metallicity.

##  2 Dataset

This work is based on the combination of VST and Chandra data. We briefly
summarize the main properties of the datasets. We refer to the cited papers
for more details.

###  2.1 Optical data

The optical data used were acquired as part of the Fornax FDS based on
observations obtained in u, g, r, and i bands with the Survey Telescope (VST)
of the Very Large Telescope at the ESO Paranal Observatory. The VST is a wide-
field optical imaging telescope with a 2.6-meter aperture with a field of view
(FoV) of 1 $degree^{2}$. The telescope is equipped with the 268 megapixel
OmegaCAM with a pixel scale of $0^{\prime\prime}.21$ $pixel^{-1}$. The survey
was designed to map the entire Fornax cluster out to the virial radius, and it
also covered the NGC1316 subgroup (Iodice et al., 2017). Located at a distance
of $D=20.13\pm 0.4$ $Mpc$ (Blakeslee et al., 2009), this cluster is the second
nearest to us after the Virgo cluster and therefore represents an ideal target
for this type of study. The survey observed $\sim 27$ square degrees
approximately centred on NGC1399 and NGC1316 (Cantiello et al. 2020), reaching
magnitude limits for point-like sources of 24.1, 25.2, 24.6, and 23.6 in the
u, g, r, and i bands, respectively (AB mag photometric system), while the
median seeing of the observation ranges from 0.6 to 1.1 arcsec. We limit our
analysis to the central 1.5 square degrees around NGC1399, which overlaps with
the Chandra X-ray coverage of the cluster core. The data reduction was
performed using the Astro-WISE pipeline (see e.g. Venhola et al., 2019). This
is a tool for the reduction of large field data to perform pre-reduction
(subtraction of bias, correction of flat), illumination, and edge correction
and photometric and astrometric calibrations. To properly detect and study
GCs, we need to minimize the contamination from the brightest galaxies in the
cluster (e.g. NGC1399 and NGC1404). In order to model and subtract the
galaxies, ELLIPSE task in IRAF STSDAS is used during the catalogue preparation
(Jedrzejewski et al. 1987). To produce a complete catalogue of all sources
present in the VST field of view, a combination of SExtractor (Bertin &
Arnouts 1996) and DAOphot (Stetson, 1987) was used on the galaxy-subtracted
frame independently in each filter. For additional information on the data
reduction and source photometry, we refer to Cantiello et al. 2020.

Figure 1: Spatial distribution of the GCs (grey points) and the X-ray sources
(red crosses) centred on NGC1399. The solid black circles represent bright
galaxies in the cluster from Ferguson (1997). The size is proportional to
their effective radii. The dashed line highlights the FoV of Chandra
observations.

At the Fornax distance, GCs appear unresolved from the ground. They are
therefore hard to separate from stars and compact background galaxies. The
selection of the GC sample accordingly represents a crucial step in our
analysis. In this case, we adopted the approach presented in Cantiello et al.
(2018) applied to the improved FDS dataset (Cantiello et al. 2020). The GC
selection criteria were defined using the training set of spectroscopically
confirmed GCs published in Pota et al. (2018) and Schuberth et al. (2010). We
refer to figure 1 of Pota et al. (2018) for the region covered by the
confirmed GCs.

In order to minimize the contamination from foreground sources, we further
discarded all sources with a g magnitude brighter than
$M_{TO}-3\sigma_{GCLF}$, where $M_{TO}$ is defined as the absolute magnitude
turn-off of the Gaussian GC luminosity distribution, assuming
$\sigma_{GCLF}=1.4\pm 0.1$ $mag$ and $M_{TO}=-7.4\pm 0.2$ $mag$ (e.g. Villegas
et al., 2010). At the distance of Fornax, the turn-over magnitude corresponds
to $m_{TO}\sim 24.0$ $mag$. The adopted selection criteria are given in Table
1.

Table 1: Selection criteria for the GC candidates.

Parameters | Values
---|---
magnitude | $19.8\leq m_{g}\leq 26$
concentration index1 | $0.8\leq CI_{n}\leq 1.15$
colour | $0.6\leq g-i\leq 1.45$
| $1.35\leq u-r\leq 3.5$
Difference from model2 | $\leq 0.4$

Notes: 1$CI_{n}$: normalized concentration index based on the difference in
g-band magnitude between apertures of 6 and 12 pixels. 2Maximum distance from
the best-fit population synthesis model for spectroscopically confirmed GCs in
the colour-colour diagram.

The catalogue of candidate GCs contains 5178 sources. Their spatial
distribution is shown in Figure 1. The figure shows the extended distribution
of GCs that was reported by D’Abrusco et al. (2016). While a large fraction of
these GCs are clustered around the brightest galaxies, many GCs occupy the
intra-cluster space, several effective radii ($R_{eff}$) away from any cluster
galaxy member. The positions of cluster galaxies and their effective radii are
extracted from Ferguson (1997). In Table 2 we show the $R_{eff}$ of the
galaxies in the FoV of the Chandra observations. We point out that the
$R_{eff}$ of NGC1399 and NGC1404 are considerably smaller than those reported
in Iodice et al. (2016) and Iodice et al. (2019). These values were estimated
using very deep observations probing very low surface brightnesses where the
extended stellar halos of NGC1399 (and NGC1404) merge with the intra-cluster
light and encompass most of the Fornax cluster. To identify intra-cluster
sources, we therefore preferred to use the value provided by Ferguson (1997).

Table 2:  List of galaxies from Ferguson (1997) in the Chandra FoV. Name |  RA ($deg$) |  DEC ($deg$) |  $R_{eff}$ ($kpc$)
---|---|---|---
NGC1387 | 54.2370 | -35.5066 | 4.86
NGC1399 | 54.6216666 | -35.45055 | 3.61
NGC1427A | 55.0383333 | -35.62305 | 3.52
NGC1389 | 54.29875 | -35.74444 | 1.95
NGC1404 | 54.7170833 | -35.59388 | 1.94
NG115 | 54.1558333 | -35.38472 | 1.74
NG6 | 54.83125 | -35.72333 | 1.61
NGC1380B | 54.2870833 | -35.19361 | 1.58
NG5 | 55.0970833 | -35.27444 | 1.54
G72 | 54.8054166 | -35.36972 | 1.40
NGC1381 | 54.1316666 | -35.29527 | 1.25
NG8 | 54.26875 | -35.58861 | 1.17
NG21 | 54.5779166 | -35.52916 | 1.14
G79 | 54.22625 | -35.37305 | 1.10
NG23 | 54.3908333 | -35.82777 | 0.99
NGC1396 | 54.5266666 | -35.43833 | 0.95
NG87 | 54.96375 | -35.32055 | 0.89
NG7 | 54.5795833 | -35.1275 | 0.82
NG116 | 54.1779166 | -35.43416 | 0.75
NG72 | 54.68875 | -35.26444 | 0.74
NG117 | 54.65625 | -35.75555 | 0.72
NG111 | 54.98 | -35.66083 | 0.66
D117 | 54.3241666 | -35.69722 | 0.62
FCC197 | 54.29125 | -35.38527 | 0.61
NG114 | 54.4204166 | -35.29472 | 0.61
NG118 | 54.6520833 | -35.8325 | 0.60
NG71 | 54.58875 | -35.25833 | 0.54
D138 | 54.7291666 | -35.23527 | 0.54

The completeness of the original gri catalogue is $\sim 80\%$ for magnitudes
$g\leq 24$ mag. However, the selection criteria introduced above in order to
minimize the contamination of foreground/background sources combined with the
use of the shallower u band constrain the actual completeness of our sample,
as shown in Fig. 2. We note that because LMXBs tend to reside in bright GC,
the final detection limit of $m_{g}\sim 25$ does not represent a significant
limitation for our study.

Figure 2:  Luminosity function in g band for the sample of candidate GCs. The
dashed black line corresponds to the expected theoretical GC luminosity
function (with values $\mu=24$ and $\sigma=1.4$).

The colour distribution of GCs in the core of the Fornax cluster is known to
be bimodal, mainly due to metallicity effects (see e.g. D’Abrusco et al.,
2016; Cantiello et al., 2018, and references therein). In order to study the
possible differences between LMXBs formed in different environments, we used a
Gaussian mixture model (GMM; Muratov & Gnedin 2010) to divide the whole
population into red and blue GCs. Using a $g-i\sim 1.00$ threshold, we find
2085 red GCs ($\sim$ 40% of the total sample) and 3093 blue GCs ($\sim$60% of
the total sample). As discussed in more detail below, the relevance of each
population is dependent on several factors, including the host galaxy
properties, the galactocentric distance of the cluster, and the environment.
We adopted a fixed colour threshold as we intend to study the overall intra-
cluster population, independently of the individual galaxies. We therefore
ignored for the moment the variations in the GC colour distribution as a
function of the distance from the centres of the host galaxies (Kim et al.
2013; Cantiello et al. 2015) and the differences of the red/blue bimodality of
the individual GCs (Jordán et al. 2015). In figure 3 we plot the GMM model
over the g-i colour distribution for the entire sample of candidate GCs. We
point out that on cluster scales, the blue component dominates 3:2 of the
total GC population. This is different from what has been reported close to
individual galaxies, where red GCs are dominant (e.g. Puzia et al., 2014).
This agrees with the well-known trend for red and blue GC densities to have
different radial gradients, the latter of which are shallower than the former.
We note that the bulk of the blue GC population is redder than what was found
for Fornax dwarf galaxies in any case (Prole et al., 2019).

However, we tried to quantify the variation in the colour distribution as a
function of the distance by performing a GMM fit for different distances from
each galaxy in terms of effective radius. We find a slight shift in the
average value of each distribution towards more blue colours in progressively
outer regions (Fig.4). The slope of the fit is found to be $-0.0037\pm 0.0005$
for the blue sample and $-0.0037\pm 0.0017$ for the red sample. This is
inconsistent/marginally consistent ($\sim 2.2\sigma$), respectively, with a
constant trend. This means that not only does the intra-cluster space host a
larger fraction of blue GCs, but also that both distributions gradually become
bluer with increasing distance. For this result, we do not expect any
systematic effect from background galaxy light because it was removed during
the catalogue reduction process (Cantiello et al., 2020).

Figure 3: $g-i$ colour distribution of GC candidates in the central square
degree of the Fornax cluster. The smooth solid black, red, and blue lines show
the GMM model for the total, red, and blue populations. Figure 4: Mean values
calculated by the GMM for different distances from each galaxy in terms of
$R_{eff}$ for red and blue GCs, and the separation performed by the GMM. The
error bars correspond to the standard deviation of the mean.

###  2.2 X-ray data

The X-ray data were extracted from 29 archival Chandra observations (given in
Tab. 1 of Jin et al. 2019) that were obtained with the Advanced CCD Imaging
Spectrometer (ACIS) for a total exposure time of 1.3 Ms. This covers large
part of the central region of the Fornax cluster (Fig. 1). The details of the
source detection procedure can be found in Jin et al. (2019). We briefly
summarize the procedure here. The X-ray data were analysed using the Chandra
Interactive Analysis of Observation (CIAO) tool Wavdetect, producing counts
and exposure maps, at the native pixel scale of $0.49\arcsec$, in three
different bands: 0.5–2 (S band), 2–8 (H band), and 0.5–8 (F band) KeV. The
exposure maps were weighted by an assumed incident spectrum of an absorbed
power law with a photon index of 1.7 and an absorption column density
$N_{H}=10^{21}$ $cm^{-2}$. This latter value is higher than the Galactic
foreground absorption column ($\sim 1.5\times 10^{20}$ $cm^{-2}$), but takes
some intrinsic absorption of the LMXBs into account (see Jin et al. 2019)

The centroids of the sources were refined using a maximum likelihood method
that iterates over the positions of the individual counts within the 90% of
the enclosed counts radius (ECR). The photon fluxes in various bands were
calculated using a circular opening of 90% of the ECR in non-crowded zones and
50% for the crowded zones to avoid contamination between nearby sources. The
position uncertainty (PU) at the 68% confidence level was estimated following
the empirical relation between PU, source counts, and source position in terms
of the off-axis angle. At this point, a catalogue of 1279 independent sources
was obtained, of which 1177 are in F band (0.5-8 KeV), 924 in S band (0.5-2
KeV), and 713 in H band (2-8 KeV). To derive the unabsorbed energy flux in the
F band, a photon-to-flux conversion of $3.64\times 10^{-9}$ $erg/ph$ was used,
assuming the previously cited incident source spectrum.

In order to verify the quality and completeness of the catalogue, we compared
it to the Chandra Source Catalog 2.0 (CSC master catalogue,
http://cxc.harvard.edu/csc/). The CSC uses a different approach to compute the
source fluxes (based on the individual photon energy instead of assuming an
average conversion factor) and the exposure maps. In order to compare the two
catalogues, we therefore cross-matched the source positions and compared the
fluxes of the sources in common, deriving the rescaling factor ($r=1.41$) from
CSC to Jin et al. (2019). In figure 5 we show the cumulative X-ray luminosity
function of the two catalogues after correcting the CSC fluxes as explained
above. Both catalogues follow a similar truncated power-law distribution down
to $LF\simeq 3\times 10^{38}$ $erg/s$, after which the CSC LF flattens more
rapidly than the Jin catalogue. The higher completeness of the Jin catalogue
is due to two main factors: first, the CSC contains observations up to 2014,
while the catalogue of Jin et al. (2019) is based on all available Chandra
archived observations up to 2015. Of the 29 observations used in Jin et al.
(2019), the CSC lacks the two identified with ObsIDs 17549 and 14529, which
are centreed on NGC1399 and NGC1404, where a large fraction of the GC
population resides. Second, the CSC reports source properties in master,
stack, and per-observation tables. A single source might then not be included
in the master catalogue (used here) but might have entries at the stack or
observation level. We verified that when we include sources from all tables,
we obtain a comparable number of sources between the CSC and Jin catalogue
(1004 and 1177, respectively), and this considerably reduces the disagreement.
However, in this case, the choice for the measured properties that are used
becomes complex (see the detailed discussion at
https://cxc.cfa.harvard.edu/csc/organization.html). For this reason, we mainly
used the Jin et al. (2019) source catalogue and reverted to the CSC for
specific measurements and tests. Most of our result would be confirmed,
although with lower significance, using the CSC catalogue in any case, except
for the hardness-ratio differences discussed in Sec.4.2

Figure 5: Cumulative X-ray luminosity function for the sources from the
catalogue of Jin et al. (2019) and for the CSC catalogue. The shaded region
represents the $1\sigma$ error.

##  3 Identification of X-ray sources in GCs

In order to study the population of LMXBs residing in GCs, we performed a
cross-match between the optical and the X-ray catalogues. To minimize the
possibility of random matches, we employed the PU provided in the Jin et al.
(2019) catalogue as matching radius for the X-ray sources. This is different
for each object. For the optical sources, we chose a matching radius of 0.5
arcsec that is typical of the VST astrometric accuracy (see e.g. Capaccioli et
al. 2015). Two sources were considered to match when the separation between
their positions did not exceed the sum of the two matching radii for the
corresponding source. Based on the optical and X-ray source densities, we
expect three false matches over the entire FoV. In addition, compact
background galaxies hosting an active galactic nucleus (AGN) might be mistaken
for GCs hosting an LMXB. This type of contamination is discussed in section
4.2. We identify 168 X-ray sources that are positionally coincident with GCs.
We refer to these objects as GC-LMXBs.

To study the photometric properties of the intra-cluster population, we
divided the GC-LMXBs sources into host-galaxy and intra-cluster objects based
on their projected distance from the nearest galaxy in terms of $R_{eff}$. The
choice on where to place the separation between the regions associated with
the galaxy and the intra-cluster space is somewhat arbitrary.

Several authors considered 5 $R_{eff}$ to be the upper limit for bound systems
(Kartha et al. 2014, Forbes 2017, Caso et al. 2019). In order to obtain a
nearly equal number of host-galaxy/intra-cluster sources, we considered as
intra-cluster GCs those lying more than 6 $R_{eff}$ from the nearest galaxy.
In this way, we found 86 host-galaxy and 82 intra-cluster GC-LMXBs.
Coincidentally for NGC1399, this limit roughly corresponds to the separation
between the central region covered by HST data (Paolillo et al., 2011) and the
outer cluster region that has only been studied with ground-based data so far.
Due to projection effects, some intra-cluster sources will be included in the
host-galaxy sample. However, our conservative choice should ensure that most
of the sources that are classified as intra-cluster objects are loosely bound
to individual galaxies. In Fig. 6 we show the distribution of intra-cluster
and host-galaxy GC-LMXBs in our field of view. The majority of intra-cluster
sources are located in the central cluster region around NGC1399.

In figure 7 we show the fraction of GC hosting LMXBs (XGC) as a function of
the distance from the nearest galaxy in terms of $R_{eff}$ for the entire GC
sample, as well as red and blue GC subsamples. These fractions are normalized
to the number of all, red, and blue GCs in the corresponding distance bin. We
refer to this as $f_{XGC}$, $f_{rXGC}$, and $f_{bXGC}$ for the total, red, and
blue fraction of XGCs, respectively. We selected only GCs with a magnitude
$g<23.5$ (the 90% completeness limit of the GC catalogue, see Fig. 2) in order
to minimize the radial dependence on the optical completeness. We further
verified that the same trends were confirmed when only LMXBs above the median
sensitivity limit $L_{x}>1.7\times 10^{38}$ $erg$ $s^{-1}$ of Jin et al.
(2019) were selected (corresponding to a photon flux of $1.5\times 10^{-6}$ ph
s-1 cm-2 in their Fig.2). In this way, we also minimized the dependence on the
variable X-ray completeness across the FoV.

Figure 7 shows that the likelihood of a red GC to host an LMXB decreases with
galactocentric distance, but remains approximately constant for the blue GC
population. We find the host-galaxy fraction of red XGC $f_{rXGC}=0.154\pm
0.017$ to be significantly different from the respective intra-cluster value
$f_{rXGC}=0.062\pm 0.008$. On the other hand, the host-galaxy fraction of blue
XGCs $f_{bXGC}=0.021\pm 0.006$ appears consistent within the errors but is
still higher than that of the intra-cluster counterparts $f_{bXGC}=0.013\pm
0.003$. Because of the normalization we adopted, this result is independent of
the known different radial distribution of red and blue GCs. The former follow
the galaxy light more closely than the latter (which represent the dominant
intra-cluster GC population). Thus, the clustering of LMXBs around bright
galaxies is due to the combined effects of the steeper profile of the red GC
population and the increased likelihood to host LMXBs close to the galaxy
centre. Previous studies targeting nearby elliptical galaxies on spatial
scales comparable to ours, such as Kim et al. (2006), show that $f_{rXGC}$ can
range from 2.7% up to 13% from one galaxy to the next (the latter value refers
to NGC1399), while $f_{bXGC}$ remain relatively constant at $\sim$ 2%, except
for NGC1399, where it reaches 5.8%. We thus find a strong agreement between
our host-galaxy $f_{rXGC}$ and the measurement carried out by Kim et al.
(2006) on NGC1399, suggesting that our sample of host-galaxy red XGC is
dominated by the population associated with the central cluster galaxy. In the
case of blue XGCs, the fraction we measure is closer to the average,
suggesting that blue GCs are more related to the overall cluster environment
than to a single galaxy, as discussed in more detail below.

Because the formation of LMXBs is strongly influenced by the luminosity of the
host GC, the drop in the fraction of GC-LMXB could be associated with an
average decrease in the GC luminosity towards the outer regions. In Fig. 8 we
show the GC g -band magnitude as a function of galactocentric distance. The
dashed blue and red lines represent the mean magnitude for the blue and red
sample, respectively. We can observe no clear decrease in the mean magnitude
towards the outer regions, which might justify the drop of the GC-LMXB
fraction. This might suggest that the LMXB formation is favoured in the
proximity of galaxies, or that it might also depend on other factors, such as
the environment of the host galaxy.

Figure 6: Spatial distribution of intra-cluster (cyan crosses) and host-galaxy
(magenta crosses) GC-LMXBs. Cluster galaxies are the solid black circles, and
the sizes are proportional to their effective radii. The dashed circle shows
the 6 $R_{eff}$ distance from the galaxy centre. The dashed line depicts the
FoV of the Chandra observations. Figure 7: Fraction of XGC as a function of
the distance in units of $R_{eff}$ for the total, red, and blue sample of GCs.
The fraction is defined as the number of total, red, or blue XGC divided by
the respective total number of GCs in the distance bin. The distances on the
X-axis are from the nearest galaxy to the object. Figure 8: Magnitude in g
band as a function of the distance in terms of $R_{eff}$. Grey circles and
triangles represent red and blue GCs, respectively. Black crosses are GCs that
host an LMXB. Dashed lines represent the mean g magnitude of the GC in each
distance range for the red and blue sample. The dashed lines show a computer
cut of the g magnitude at g¡23.5.

##  4 Properties of GC-LMXBs

###  4.1 Optical properties of the host GCs

Because most of previous works that focused on the GC-LMXB connection were
restricted to the innermost regions of the galaxies, we verified whether the
intra-cluster sample possesses the same properties as its host-galaxy
counterpart. As a starting point, we investigated the dependence of the LMXB
formation on the host GC luminosity. In Fig. 9 we present the fraction of GC-
LMXB as a function of the GC apparent magnitude in the g band for three
different galactocentric distances. We note a declining trend in the host-
galaxy sample ($R_{eff}\leq 6$) as well as in the intra-cluster sample
($R_{eff}>6$), suggesting that LMXBs tend to form in bright GCs in both
environments. Fig. 10 (upper panel) confirms this result by comparing the
magnitude distribution of GCs and GC-LMXBs for host-galaxy and intra-cluster
sources. In both galactic and intra-cluster cases, a Kolmogorov-Smirnov (K-S)
test confirms the difference between the distributions of GCs that hosting an
LMXB or are without one at the 99.9% confidence level.

Figure 9: Fraction of GCs hosting LMXBs as a function of luminosity. The
different types of lines correspond to different galactocentric distances. The
magnitude range is restricted to $g\leq 23.5$. The fraction is defined as the
number of XGCs divided by the number of GCs in the magnitude bin. The result
for the inner region agrees with the results obtained in Paolillo et al.
(2011). These authors covered almost the same FoV with HST.  Figure 10:
Magnitude distribution of GCs hosting (grey shaded) and GCs without (black
line) LMXBs in intra-cluster space (upper panel) and in the host-galaxy
environment (lower panel).

The LMXB formation efficiency was also shown to be dependent on the colour of
the host GC (see Sec.1), with a greater tendency for LMXBs to form in red GCs.
To verify whether this trend also holds for intra-cluster GCs, we present in
Fig. 11 the g-i colour distribution of intra-cluster and galactic GCs. Again,
a K-S test confirms the difference in colour between GCs with and without an
LMXB at the 99.9% level in both environments.

Figure 11: Colour distribution of GCs hosting (grey shaded) and GCs without
(black line) LMXBs in intra-cluster space (upper panel) and in the host-galaxy
environment (lower panel).

###  4.2 X-ray properties of the GC-LMXB

Previous works (Jordán et al. 2004, Kim & Fabbiano 2004, Paolillo et al. 2011)
showed that the X-ray properties of LMXBs, such as their X-ray luminosity
function, do not depend on the properties of the stellar environment in which
they form, such as the colour, magnitude, or density of the host GC. In this
section we try to understand if these statements are valid for the intra-
cluster population of GC-LMXBs. In order to correct the completeness of our
X-ray source catalogue for the variable detection limit across the FoV due to
hot gas emission near the brightest galaxies, the variable PSF with off-axis
angle, and the different exposure time across the X-ray mosaic, we used the
X-ray sensitivity map produced by Jin et al. (2019) to weight X-ray sources
according to the fraction of the GCs in which they could have been detected.
In Figure 12 we show the observed and completeness-corrected X-ray luminosity
function of the intra-cluster and host-galaxy samples of GC-LMXBs. We found
that the completeness-corrected intra-cluster and host-galaxy populations
follow a single power law down to $L_{x}\simeq 8.5\times 10^{37}$ erg
s-1111This limit yields the best power-law fit according to the algorithm of
Newman (2005) as implemented in the R routine powerlaw.R.. The intra-cluster
sample is more affected by incompleteness because the sensitivity at the edges
of the FoV is lower. Fitting the completeness-corrected distributions, we
obtain an LF slope of $\alpha=2.04\pm 0.13$ for the host-galaxy sample (in
agreement with the slopes found in previous works for GC-LMXBs belonging to
galaxies, e.g. Kim et al. 2006, Paolillo et al. 2011, Jin et al. 2019), and
$\alpha=2.44\pm 0.13$ for the intra-cluster sample. This is considerably
steeper and consistent with the slope found for field LMXBs in Jin et al.
(2019) ($\alpha=2.30\pm 0.12$) and in previous works (Paolillo et al. 2011). A
K-S test confirms the difference of the two completeness-corrected LFs at the
99% confidence level.

Then, we studied the $L_{x}$ distribution by dividing the sample of GC-LMXBs
according to the host GC colour (Fig. 13). The completeness-corrected red GC-
LMXB follows a power-law distribution with a slope $\alpha=2.37\pm 0.12$ down
to $L_{x}\simeq 8.5\times 10^{37}$ erg s-1 . A possible lack of bright LMXBs
is observed in the blue systems, which is found to be significant at the $\sim
2\sigma$ (97%) level according to a Poisson statistics. Fitting the entire
sample of blue GC-LMXBs down to $L_{x}\simeq 8.5\times 10^{37}$ erg s-1 yields
$\alpha=2.21\pm 0.18$. To further investigate this trend, we separated the red
and blue sample into intra-cluster and host-galaxy sources (lower panel of
Fig. 13). The population of blue intra-cluster GCs does not host LMXBs above
$\sim 6\times 10^{38}$ $erg/s$, although again the number of object is small
for strong conclusions. The majority of objects above $10^{39}$ $erg/s$
resides in red GCs possibly because multiple LMXBs reside in a single GC
because it is easier to form LMXBs in these systems. However, previous studies
often detected variability in systems like this, which argues against this
possibility. On the other hand, this high luminosity break has already been
observed in literature (Fabbiano 2006, and reference therein) and may reflect
the transition between the most massive neutron stars and low-mass black hole
systems, suggesting that blue environments are less likely to form binary
systems with massive black holes. We point out, however, that considering the
small statistics at high $L_{x}$, a small contamination by background AGNs
randomly matched with GCs could be enough to cause the difference (see below).
To verify that this is not linked to the different host GC distribution, we
show in Fig. 14 the correlation between $L_{x}$ and the colour/galactocentric
distance of the GCs for the red and blue sample of GC-LMXB. While red GC-LMXB
are more centrally concentrated, as discussed before, the brightest sources
with $L_{X}>10^{39}$ erg s-1 are uniformly distributed throughout the whole
distance range. The small number of sources with $L_{X}>10^{39}$ erg s-1 does
not allow us to draw definitive conclusions, and therefore prevents us from
excluding the effect of a background AGN contamination.

Figure 12:  Cumulative X-ray luminosity function for the host-galaxy and
intra-cluster sample of GC-LMXBs. The solid lines represent the completeness-
corrected LFs, and the dotted lines show the observed LFs. The shaded region
represents the $1\sigma$ error.

Figure 13: Cumulative X-ray luminosity functions comparison. Upper panel. Red
and blue sample of GC-LMXB. Lines and shaded areas are represented as in
Figure 12. Lower panel. Completeness-corrected luminosity functions for red
and blue intra-cluster and host-galaxy sources separately.  Figure 14: X-ray
luminosity as a function of distance from each galaxy. Grey dots represent the
whole sample of X-ray sources. Blue triangles and red crosses represent the
respective GC-LMXBs. The mean errors in each X-ray luminosity order of
magnitude are shown in black.

We further investigated the spectral properties of the LMXB population through
their hardness-ratio (HR), defined as
$HR=\frac{S_{Hard}-S_{Soft}}{S_{Hard}+S_{Soft}}$, with $S_{hard}$ the photon
flux in the hard band (2-8 KeV) and $S_{soft}$ the photon flux in the soft
band (0.5-2 Kev). In Fig. 15 we present the HR distribution of host-galaxy and
intra-cluster sources. It is readily apparent that intra-cluster GC-LMXBs have
harder spectra. A K-S test confirms that the two samples are drawn from
different distributions. To understand this trend, we considered whether this
systematic difference is related to the LMXB population itself or to their
host GCs because we know from Sec.2.1 and from previous works that the GC
population becomes increasingly blue with increasing galactocentric distance
(Jordán et al., 2006; D’Abrusco et al., 2016; Cantiello et al., 2018, 2020).

In our case, 694 blue GCs (containing 10 LMXBs, i.e. 1.4% of the population)
and 618 red GCs (72 LMXBs, i.e. 11.6% of the population) are associated with
the main cluster galaxies, while the intra-cluster sample contains 2400 blue
GCs (26 LMXBs, i.e. 1.1% of the population) against 1467 red (60 LMXBs, i.e.
4.1% of the population). The number of blue GCs therefore increases in the
outer regions. As shown in Fig. 4, we also observe a shift in the average
value of the red distribution towards bluer colours, however. In Fig. 16 we
show that LMXBs in blue GCs seem to have a harder spectrum than those in red
GCs; the result is confirmed when red and blue GC-LMXBs are compared
restricted to the intra-cluster sample alone222Comparing red and blue GC-LMXBs
restricted to the host-galaxy sample alone does not allow us to draw
definitive conclusions because there are too few blue GC-LMXBs..

To explain this result, we explored three possibilities: 1) Intra-cluster
sources are more heavily contaminated by harder-backgound sources than the
host-galaxy population. 2) The difference is due to uncertainties in the X-ray
spectral response correction, which in turn is due to the combination of the
multiple Chandra exposures with different off-axis angles and the presence of
diffuse gas, which may affect the background estimate. Finally, 3) the
difference is real.

To address the first possibility, we estimated the number of expected
contaminants in our FoV. To this end, we used the COSMOS catalogues by Civano
et al. (2016) and Laigle et al. (2016) to select X-ray sources within the same
X-ray flux limit as was used for the catalogue of Jin et al. (2019), and with
optical counterparts obeying the same selection criteria in terms of limiting
optical magnitude, colours, and concentration index as we adopted in Sec.2.1.
We predict a contamination of about ten background X-ray sources with optical
counterparts over our FoV. In addition, we expect three random superpositions
of background sources without optical counterparts, as discussed in Sec.3.
These contaminants do have harder average HR than the bulk of GC-LMXB sources,
but they do not account for the entire excess observed in our sample. When we
removed the ten hardest sources from the sample of intra-cluster GC-LMXBs, the
result was unchanged.

Several factors can affect the derived hardness ratios in the spectral
response, including different off-axis angles, detectors (ACIS-I versus
ACIS-S), and observing epoch (due to the efficiency degradation of ACIS at low
energies with time). We point out, however, that Jin et al. (2019) only used
ACIS I0, I1, I2, I3, S2, and S3 chips in order to ensure an optimal source
sensitivity. In addition, most intra-cluster sources are still located in the
central region of the cluster, although at a greater distance from the main
galaxies, and they are covered by multiple observations spanning several years
(see Table 1 in Jin et al. 2019). These properties should reduce the potential
systematic effects in the X-ray catalogue. To evaluate how the catalogue
properties might affect our results, we compared the HR estimates used in this
work with the estimate reported in the CSC catalogue. In Fig. 17 we show the
HR as a function of the distance from the nearest galaxy for X-ray sources
with a counterpart in the CSC master catalogue. We note a difference in the
mean HR, mainly driven by host-galaxy sources, both considering GC-LMXBs and
field X-ray sources. The higher HR in the CSC is higher on average than the
HRs reported in the catalogue used in this work. This suggests that the
spectral corrections differ between the two catalogues, possibly due to the
different data reduction strategy, source extraction method (see Sec.2.2), and
the treatment of the contribution of the central diffuse emission.

Concerning the possibility that the observed difference might be real, it is
known that the colour mainly depends on metallicity in old GCs, where metal-
rich GCs are significantly redder (Cantiello et al. 2018). This might suggest
a relation between the spectral properties of LMXBs and the metallicity of the
host GC. One model that discusses such a trend was presented by Maccarone et
al. (2004), who proposed irradiation-induced winds in metal-poor stars that
would cause absorption, mainly of the soft part of the X-ray spectra, which
would produce higher HR values. The authors reported this trend for NGC4472
(Maccarone et al. 2003). In this case Fornax, would be the only other system
in which this effect is directly observed so far.

Figure 15: Hardness ratios between host-galaxy (grey) and intra-cluster
(black) GC-LMXBs. The error bar is the median value of the errors associated
with the HR of the sources. Figure 16: Hardness ratios between red (grey) and
blue (black) GC-LMXBs. The error bar is the median value of the errors
associated with the HR of the sources.

Figure 17:  Hardness ratio as a function of the distance for field X-ray
objects (left panel) and GC-LMXBs (right panel) with counterparts in the CSC
catalogue. Black dots represent the HR estimates in the CSC catalogue, and
grey triangles show the HR we used. Dashed green and magenta lines again
represent the mean value in each distance bin.

##  5 Summary and conclusions

We performed an analysis of the properties of LMXBs residing in the population
of intra-cluster GCs in the Fornax cluster. The main goal of this work was to
study the GC-LMXB connection within the core of the Fornax cluster and its
dependence on the environment in terms of galactocentric distance and host GC.
For this purpose, we used optical photometry of 5178 candidate GCs from the
Fornax Deep Survey (Cantiello et al., 2020). In order to study the possible
differences between LMXBs that formed in different GC environments, we
performed a Gaussian Mixture Model fit on the g-i colour distribution, finding
a separation threshold of $g-i\sim 1.00$ between red and blue that agrees with
the threshold reported in previous works (D’Abrusco et al. 2016, Cantiello et
al. 2020). We obtained 2085 red GCs and 3093 blue GCs. We further observe a
tendency in the average $g-i$ colour of becoming bluer with increasing
galactocentric distance. Because the GC colours mainly depend on metallicity,
this result suggests that the intra-cluster GCs have a lower metallicity on
average.

The X-ray data employed in this work were extracted from archival Chandra
observations. The source detection procedure together with the extraction of
photometry was performed by Jin et al. (2019). In order to study the
connection between LMXBs and GCs, we performed a cross-match between the
optical and the X-ray catalogues. In this way, we identified 168 X-ray sources
that are positionally coincident with GCs. We divided this population into
host-galaxy and intra-cluster objects based on their projected distance from
the nearest galaxy. We considered as intra-cluster objects the GC-LMXBs within
more than 6 $R_{eff}$ from the nearest galaxy and found 82 intra-cluster and
86 host-galaxy GC-LMXBs. Furthermore, considering the colour division
performed for the GC sample, we found 36 LMXBs to be associated with blue GCs
and 132 LMXBs to be associated with red GCs.

We find the fraction of GC-LMXBs to be dependent on the galactocentric
distance; this effect is particularly evident for the red population. Because
the GC magnitude seems to be independent of the distance from the galaxies, we
conclude that this result may suggest that the LMXBs formation channel in GCs
may also depend on the host-galaxy environment. In the past, the evidence for
a dependence of the LMXB formation likelihood on the local galaxy environment
has been debated (see e.g. Kim et al. 2006 and Paolillo et al. 2011 for an
opposing view), but these studies were essentially limited to the inner region
of galaxies within a few $R_{eff}$. Even in this work, there is little
evidence for a dependence like this within $\sim 6\leavevmode\nobreak\
R_{eff}$ , which is visible only when the analysis is extended to the intra-
cluster population. If this is confirmed, the enhanced LMXB formation rate in
red GCs might be related to their orbital parameters, as suggested by Puzia et
al. (2014), for instance, leading to a stronger influence of the external
tidal field. Webb et al. (2016) found that the sizes of red GCs in NGC1399 are
consistent with more radial orbits compared to blue GCs.

We confirm that intra-cluster LMXBs tend to form in red and bright GCs, as has
been found for their host-galaxy couterparts. Furthermore, we studied the
X-ray properties of the intra-cluster population of GC-LMXB to test whether
they are independent of the properties of the host GC as for the host-galaxy
sources. We find that the completeness-corrected X-ray luminosity function of
the intra-cluster population of GC-LMXBs follows a power law with a slope that
is marginally consistent ($\sim 2.2\sigma$) with the slope of the host-galaxy
population, and it is consistent with the slope found for field LMXBs in the
literature. A Kolmogorov-Smirnov test indicates a statistically significant
difference between the $LF$ of the intra-cluster and that of the host-galaxy
sample. We cannot confirm any difference between the completeness-corrected
LFs of the red and blue populations. We find a lack of bright LMXBs in blue
GCs, however, which agrees with what has been found for host-galaxy sources
and possibly indicates a lack of black hole binaries in metal-poor systems.

Finally, we observed a puzzling difference in hardness ratio between intra-
cluster and host-galaxy GC-LMXBs: the spectra of the intra-cluster sample are
harder than those of the host-galaxy sample. Because intra-cluster GCs are
bluer on average than the host-galaxy GCs, this result might suggest a
relation between HR and the colour of the host GC, and hence the metallicity.
We explored different explanations for this difference, including residual
systematics in the data and background contamination or a possible physical
origin. We found that contamination alone seems unable to explain the observed
trend. This result is still tentative, however, and a final conclusion will
have to wait for a full spectral analysis of the host-galaxy and intra-cluster
GC-LMXB populations.

###### Acknowledgements.

GR acknowledges support from the National Science Centre
(UMO-2018/30/E/ST9/00082, UMO-2018/30/M/ST9/00757 and
UMO-2020/38/E/ST9/00077). GD acknowledges support from FONDECYT REGULAR
1200495, and ANID project Basal FB-210003. This research has made use of data
obtained from the Chandra Data Archive and the Chandra Source catalogue, and
software provided by the Chandra X-ray Center (CXC) in the application
packages CIAO and Sherpa.

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