# Spin-resolved spectroscopic evidence for spinarons in Co adatoms

Felix Friedrich  Artem Odobesko  Corresponding author for the experiments:
<EMAIL_ADDRESS>Physikalisches Institut, Experimentelle
Physik II, Julius-Maximilians-Universität Würzburg and Würzburg-Dresden
Cluster of Excellence ct.qmat, Am Hubland, 97074 Würzburg, Germany  Juba
Bouaziz  Peter Grünberg Institut and Institute for Advanced Simulation,
Forschungszentrum Jülich and JARA, 52425 Jülich, Germany  Samir Lounis
Corresponding author for the theory<EMAIL_ADDRESS>Peter Grünberg
Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and
JARA, 52425 Jülich, Germany  Faculty of Physics, University of Duisburg-Essen
and CENIDE, 47053 Duisburg, Germany  Matthias Bode  Physikalisches Institut,
Experimentelle Physik II, Julius-Maximilians-Universität Würzburg and
Würzburg-Dresden Cluster of Excellence ct.qmat, Am Hubland, 97074 Würzburg,
Germany  Wilhelm Conrad Röntgen-Center for Complex Material Systems (RCCM),
Julius-Maximilians-Universität Würzburg, Am Hubland, 97074 Würzburg, Germany

###### Abstract

Single cobalt atoms on the (111) surfaces of noble metals were for a long time
considered prototypical systems for the Kondo effect in scanning tunneling
microscopy experiments. Yet, recent first-principle calculations suggest that
the experimentally observed spectroscopic zero-bias anomaly (ZBA) should be
interpreted in terms of excitations of the Co atom’s spin and the formation of
a novel quasiparticle, the spinaron, a magnetic polaron resulting from the
interaction of spin excitations with conduction electrons, rather than in
terms of a Kondo resonance. Here we present state-of-the-art spin-averaged and
spin-polarized scanning tunneling spectroscopy measurements on Co atoms on the
Cu(111) surface in magnetic fields of up to 12 T, that allow us to
discriminate between the different theoretical models and to invalidate the
prevailing Kondo-based interpretation of the ZBA. Employing extended ab-initio
calculations, we instead provide strong evidence for multiple spinaronic
states in the system. Our work opens a new avenue of research to explore the
characteristics and consequences of these intriguing hybrid many-body states
as well as their design in man-made nanostructures.

spinaron, Kondo, many-body, STM, STS, spin-polarized STS

The collective screening of an impurity spin by conduction electrons in the
Kondo effect is one of the most fundamental manifestations of many-body
phenomena in physics [1]. Resulting in a resistance minimum at the Kondo
temperature in transport measurements of magnetically doped samples, the
experimental identification of the Kondo effect in individual atoms is linked
to the rise of a local electronic state at the Fermi level with a
characteristic temperature and field dependence [2, 3, 4]. Corresponding
spectroscopic signatures with a zero-bias anomaly (ZBA) in the differential
conductance signal were for the first time observed in scanning tunneling
spectroscopy (STS) experiments on Ce [5] and Co atoms [6, 7, 8, 9] on noble
metal (111) surfaces. While the ZBA detected on Ce/Ag(111) was later shown to
result from vibrational excitations of a hydrogen atom adsorbed to the Ce atom
[10], the anomaly found atop Co atoms on noble metal (111) surfaces is still
explained by a Fano resonance caused by interfering tunneling paths into the
Kondo state and atomic orbitals [11, 12, 13].

However, a different interpretation of the ZBA on Co atoms on the noble metal
(111) surfaces was recently proposed in terms of spin excitations (SEs) and
their interaction with conduction electrons, leading to the formation of a
localized quasiparticle, a magnetic polaron that was termed “spinaron” [14,
15]. This intriguing feature can be interpreted as the atomically localized
version of magnetic polarons emerging from the coupling of magnons and
electrons or holes [16].

In this contribution we investigate Co atoms on the Cu(111) surface in
magnetic fields of up to $12\text{\,}\mathrm{T}$ by means of spin-averaged and
spin-polarized STS, to experimentally assess this controversy. We find that
the ZBA on Co/Cu(111) consists of multiple spectroscopic features that show a
distinct field-dependent energy shift. Spin-resolved data reveal that features
with majority (minority) spin character shift to higher (lower) energies with
increasing field strength, a behavior which is opposite to the conventionally
expected field-dependent behavior. These findings invalidate the Kondo effect
as the origin of the ZBA, but instead provide evidence for the presence of the
predicted spinaron. This interpretation is supported by first-principles
calculations based on time-dependent density functional theory (TD-DFT) and
many-body perturbation theory (MBPT). Our findings are a crucial milestone in
all nano-physics fields gravitating around the interplay of many-body effects,
strong correlations and magnetic properties.

## Field dependence of magnetic states and excitations

In the past, significant experimental and theoretical effort was made to
investigate the properties of the ZBA in Co atoms on various surfaces. For
example, the influence of the Shockley-like surface state present on noble
metal (111) surfaces [17, 18, 19, 20, 21, 22, 12, 13], or of exchange [23, 24,
25, 26, 27] as well as spin-orbit coupling [28] onto the presumed Kondo
resonance were studied systematically. In contrast, two properties
characteristic for the Kondo effect have not yet been demonstrated for Co
atoms on the noble metal (111) surfaces: First, the broadening of the observed
resonance at temperatures above the Kondo temperature $T_{\mathrm{K}}$ [29,
30] could not be shown, because Co atoms on Cu, Ag and Au(111) become highly
mobile at temperatures much lower than $T_{\mathrm{K}}$ [31]. Second, no
magnetic field dependence of the tunneling conductance signal has been
experimentally reported to date [32, 33], contrary to, e.g., Co atoms on
$\mathrm{Cu}_{2}\mathrm{N}$ layers [30, 34], where a clear splitting of the
ZBA was observed. Yet, as will be shown below, it is precisely this field
dependence which allows us to unambiguously discriminate between the various
explanations for the ZBA and ultimately provides evidence for spinarons in the
system.

Figure 1: Schematic representation of the field-induced splitting of magnetic
resonances and excitations. Gray lines show the zero-field $\differential
I/\differential U$ (upper panels) and $\differential^{2}I/\differential U^{2}$
spectra (lower panels), red (blue) dashed lines represent the majority
(minority) contributions of the magnetic field-split resonances/excitations.
Arrows indicate the direction of the shift with increasing field strength. a
Fano curve originating from a Kondo resonance. b Bias-symmetric steps which
reflect the inelastic tunneling of electrons inducing an excitation of the
system’s spin. Equal to the features in a Kondo system, majority (minority)
SEs shift to lower (higher) energies. c Contrary, a spinaron, which
prominently appears as a peak in the $\differential^{2}I/\differential U^{2}$
signal of the majority channel, shifts to higher energies despite having
majority spin character, while all other features representing SEs, shift as
presented in b.

Figure 1 schematically illustrates the effect of an external magnetic field on
tunneling spectra expected for the three potential effects that might cause
the ZBA in Co/Cu(111): (i) a Kondo resonance, (ii) inelastic SEs, and (iii)
the newly proposed model of SEs in combination with the spinaron. (i) As the
Kondo resonance splits in an external magnetic field, the observed Fano curves
equally split symmetrically away from their zero-field position, see Fig. 1a.
Majority (red) and minority (blue) components shift to lower and higher
energies, respectively [34]. As can be seen from Fig. 1b, (ii) SEs caused by
the inelastic tunneling of electrons lead to steps in the differential
conductance which show a similar magnetic field-dependent behavior. Like the
Fano curves of a Kondo system, the steps in the majority (minority) spin
channel shift to lower (higher) energies [35, 36]. This behavior was also
confirmed for more intriguing manifestations of SEs found in recent tunneling
spectroscopy experiments [37].

The scheme in Fig. 1c displays what is expected for (iii) the combination of
SEs and the spinaron [15]. Within a simple scenario, the response of the
spinaron to the magnetic field is expected to be opposite to that of SEs in
the same spin channel. The spinaron is a magnetic polaronic state, involving
the hybrid of a SE and an electron or hole with a spin character opposite to
that of the spin channel hosting the spinaron. This electron (hole) induces a
potential which is capable of binding the hybrid state forming the spinaron.
As for the other conventional magnetic states, the majority (minority) spin
potential shifts to lower (higher) energies in the external magnetic field,
but consequently drags the bound hybrid spinaronic state in the opposite
energetic direction. Therefore, the spinaron found in the majority spin
channel in the calculations of Ref. [15] is expected to move to higher
energies with increasing field strength, i.e. in the opposite direction than
any other conventional feature in the same spin channel. The additional
intrinsic SEs predicted for Co/Cu(111) will follow the trend presented in
(ii). This fundamental difference is exploited in the following experiments to
determine the physical mechanism which causes the ZBA in Co atoms on the noble
metal (111) surfaces.

##  .1 Splitting of the zero-bias anomaly in an external magnetic field

Figure 2: Magnetic field-dependent splitting of the zero-bias anomaly. a Scan
of the Cu(111) surface with single adsorbed Co atoms. Scale bar is
$10\text{\,}\mathrm{n}\mathrm{m}$. b Differential conductance $\differential
I/\differential U$ spectra of a single Co atom at indicated fields. Spectra
are normalized and offset by
$0.05\text{\,}\mu_{\mathrm{0}}\mathrm{H}\mathrm{/}\mathrm{T}$. c Numerically
calculated derivative of the data presented in b. Curves are offset by
$0.01\text{\,}\mu_{\mathrm{0}}\mathrm{H}\mathrm{/}\mathrm{T}$. d Exemplary fit
of the $\differential^{2}I/\differential U^{2}$ signal at
$12\text{\,}\mathrm{T}$ consisting of four Gaussians #1–#4 as described in the
main text (red line). e-h Energy of #1–#4 as a function of the applied field
as extracted from measurements on five individual atoms. Lines indicate linear
fits to the data, where for peaks $\\#3$ and $\\#4$ fits including
($\\#3^{\prime}/\\#4^{\prime}$) and excluding ($\\#3/\\#4$) the zero-field
data are displayed. Stabilization parameters: a
$U=$-20\text{\,}\mathrm{m}\mathrm{V}$$,
$I=$0.4\text{\,}\mathrm{n}\mathrm{A}$$; b-d
$-20\text{\,}\mathrm{m}\mathrm{V}$, $1\text{\,}\mathrm{n}\mathrm{A}$.

To investigate the magnetic field-dependent splitting of the ZBA, we deposit
Co atoms onto clean Cu(111) (see Methods). The resulting surface is displayed
in Fig. 2a, where individual Co atoms surrounded by Friedel oscillations can
be identified. Spectra recorded atop single Co atoms in external out-of-plane
magnetic fields of up to $12\text{\,}\mathrm{T}$ are presented in Fig. 2b for
one representative Co atom [38]. At zero magnetic field, we observe the same
Fano resonance-like spectrum already reported in Ref. [7]. With increasing
magnetic field strength, the minimum of the differential conductance signal
appears to shift to lower bias voltages and the overall width of the ZBA
increases. At fields $\geq$10\text{\,}\mathrm{T}$$, a kink in the
$\differential I/\differential U$ signal at about
$+3\text{\,}\mathrm{m}\mathrm{V}$ becomes visible.

To emphasize these changes of the ZBA, we present the derivative of the
differential conductance, the $\differential^{2}I/\differential U^{2}$ signal,
in Fig. 2c. The falling edge of the ZBA at negative bias voltages is
translated into a dip, and the rising edge appears as a peak. The field-
induced kink in the $\differential I/\differential U$ signal at positive bias
is reflected in the splitting of the $\differential^{2}I/\differential U^{2}$
peak. To track the magnetic field-induced shift of the individual features, we
fit the $\differential^{2}I/\differential U^{2}$ signal by four Gaussian
functions (#1–#4), as exemplarily presented for the $12\text{\,}\mathrm{T}$
data in Fig. 2d [38]. Please note that the choice of a Gaussian function is
not motivated by the underlying physics, but merely the shape of the observed
features. The first Gaussian (#1, orange) describes the dip at negative bias
and the envelope of Gaussians #3/#4 (blue/red) accounts for the peak at
positive bias voltage which splits at high fields. To fully capture the data,
it is necessary to add the fourth Gaussian (#2, gray) to account for the
shoulder close to zero bias, that appears in all spectra. This fit enables a
reasonable description of the experimental data, sufficient to extract the
field-dependent position of the features. The respective energy positions of
features #1–#4 displayed in Fig. 2e-h, respectively, can be fitted with a
linear dependence on the magnetic field. The extracted values of the slope
$\differential U/\differential(\mu_{0}H)$ and the intersection with the
$y$-axis $U(H=0)$ are presented in Tab. 1.

In contrast to what is expected for a Kondo resonance or SEs, where all
features are shifted by the same absolute value to higher or lower energies
depending on their spin character, the features detected on Co/Cu(111) exhibit
a different behavior. We find that the $\differential^{2}I/\differential
U^{2}$ dip at negative bias (orange) and the outer peak at positive bias (red)
exhibit a very similar absolute shift of $\approx$
$0.25\text{\,}\mathrm{m}\mathrm{V}\mathrm{/}\mathrm{T}$, but in opposite
directions (cf. Fig 2e,h and Tab. 1). The peak at lower positive bias (blue),
however, shifts with a much weaker field dependence of
$-0.07\text{\,}\mathrm{m}\mathrm{V}\mathrm{/}\mathrm{T}$ to lower energies
($-0.02\text{\,}\mathrm{m}\mathrm{V}\mathrm{/}\mathrm{T}$ if the zero-field
data is excluded from the fit). Although different field dependencies are also
observed for SEs whenever spin states are mixed due to a transverse anisotropy
[39], in this case all features should appear symmetrically around zero bias,
which we do not observe in our experiments. Lastly, the peak close to zero
bias does not show any field dependence, leading to the conclusion that it is
of non-magnetic origin and can therefore be neglected in the following
discussion. While no further investigations were performed to assess the
origin of this feature, we speculate about a phononic mode of the
adatom–surface system [40], acknowledging that the potential related negative-
bias inelastic tunneling feature could be hidden by peak #1 of the ZBA.

Table 1: Magnetic field induced shift and zero-field energy of the four separate $\differential^{2}I/\differential U^{2}$ features of the ZBA extracted from the linear fits displayed in Fig. 2e-h. Peak # | Shift (mV/T) | Position at $0\text{\,}\mathrm{T}$ (mV)
---|---|---
1 ${\color[rgb]{1,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.5,0}\bullet}$ | $-0.22\pm 0.03$ | $-2.2\pm 0.3$
2 ${\color[rgb]{0.75390625,0.75390625,0.75390625}\definecolor[named]{pgfstrokecolor}{rgb}{0.75390625,0.75390625,0.75390625}\bullet}$ | $-0.001\pm 0.006$ | $0.86\pm 0.05$
3 ${\color[rgb]{0,0.0625,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.0625,0.5}\bullet}$ | $-0.024\pm 0.011$ | $3.2\pm 0.1$
3’ ${\color[rgb]{0,0.0625,0.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.0625,0.5}\bullet}$ | $-0.073\pm 0.007$ | $3.69\pm 0.06$
4 ${\color[rgb]{0.796875,0.01953125,0.01953125}\definecolor[named]{pgfstrokecolor}{rgb}{0.796875,0.01953125,0.01953125}\bullet}$ | $0.27\pm 0.02$ | $4.0\pm 0.1$
4’ ${\color[rgb]{0.796875,0.01953125,0.01953125}\definecolor[named]{pgfstrokecolor}{rgb}{0.796875,0.01953125,0.01953125}\bullet}$ | $0.14\pm 0.01$ | $5.3\pm 0.2$

The field-dependent data presented in Fig. 2 is clearly inconsistent with the
simple Kondo picture, as we find features exhibiting different energy shifts
as a function of the field strength. We can also exclude that the ZBA is
caused by pure SEs, as the observed features are not symmetric with respect to
zero bias. Yet, the origin of the ZBA still remains elusive. Especially a
direct hint at the predicted spinaron is so far missing.

##  .2 Spin polarization of the zero-bias anomaly

Figure 3: Implementation and results of spin-resolved measurements on
Co/Cu(111). a Topographic scan of the Cu(111) surface decorated with Fe
islands, Nc molecules and Co atoms. After picking up a small magnetic Fe
cluster, the tip polarization is determined by measuring the SE signal on top
of a Nc molecule. b A fit to the data reveals a spin polarization of the tip
of $\approx$ $60\text{\,}\%$. Insets illustrate the spin excitation process
detected with a spin-polarized tip. c Differential conductance $\differential
I/\differential U$ and d the $\differential^{2}I/\differential U^{2}$ signal
measured atop a Co atom (black) and the bare Cu(111) surface (gray) with the
same tip. Dashed lines indicate the position of the blue and red
$\differential^{2}I/\differential U^{2}$ feature as determined in Fig. 2.
Stabilization parameters: $\mu_{0}H=$12\text{\,}\mathrm{T}$$; a,b
$U=$-20\text{\,}\mathrm{m}\mathrm{V}$$,
$I=$0.1\text{\,}\mathrm{n}\mathrm{A}$$; c,d
$-20\text{\,}\mathrm{m}\mathrm{V}$, $0.4\text{\,}\mathrm{n}\mathrm{A}$. Scale
bar is $4\text{\,}\mathrm{n}\mathrm{m}$.

To elucidate the physical origin of the ZBA, we performed spin-polarized STS
measurements to identify the spin character of the observed spectroscopic
features. For this purpose, small magnetic clusters are transferred to the tip
apex by dipping the tip into Fe islands grown on the clean Cu(111) surface.
The tip magnetization is subsequently determined by assessing SEs in
nickelocene (Nc) molecules on the same surface (see Methods for Fe and Nc
deposition). Nc was shown to be a spin-1 system with easy-plane anisotropy,
exhibiting two SEs with opposite spin character in an external field, making
it ideally suited to probe the tip polarization [41, 42, 38].

Figure 3a shows the surface after the consecutive deposition of Fe islands, Nc
molecules, and Co atoms. The ring-shaped Nc molecules are preferentially found
at Cu vacancy edges which have formed during Fe deposition [43]. Fig. 3b shows
the $\differential^{2}I/\differential U^{2}$ signal recorded atop a Nc
molecule at $12\text{\,}\mathrm{T}$ with a spin-polarized tip. As sketched in
the insets, the asymmetry of the spectrum is caused by the spin-polarized
tunneling current and the transfer of angular momentum $\Delta S_{z}=\pm 1$
between the tunneling electrons and the molecule in the SE process [36]. Since
the excitation increasing (decreasing) the molecule’s spin is probed
predominantly at negative (positive) bias voltage, we can conclude that this
particular tip mainly probes the majority spin channel of the sample LDOS. The
value of the spin polarization can be obtained by a fit to the experimental
data displayed as a dashed line, yielding a polarization of $\approx$
$60\text{\,}\%$ [44].

This pre-characterized tip is then used to perform spin-resolved measurements
on Co atoms at $12\text{\,}\mathrm{T}$, where the splitting of the
$\differential^{2}I/\differential U^{2}$ features is maximal. The resulting
differential conductance and its derivative are displayed in Fig. 3c and d,
respectively. For comparison, measurements with the same tip on bare Cu(111)
are also displayed [38]. Dashed blue and red lines indicate the position of
the respective $\differential^{2}I/\differential U^{2}$ peaks extracted from
the spin-averaged measurements at the same field. It is evident, that the
high-energy feature at positive bias (red) is prominently present, whereas the
low-energy positive-bias feature (blue) is essentially absent. A clear
statement concerning the dip at negative bias voltage (orange) is not
possible, as the $\differential^{2}I/\differential U^{2}$ signal atop the bare
Cu(111) surface exhibits a dip of comparable intensity at a similar energy
[38]. In combination with the analysis of the tip polarization on the Nc
molecules, this finding allows us to ascribe the high-energy (red) peak a
majority spin character, whereas we find that the low-energy (blue) peak has a
minority spin character.

Against the background of theoretical and experimental work up to this date,
this finding is surprising: If solely judged based on their energy shift in
magnetic field, the various spectroscopic features in the Co atoms conductance
signal would be attributed to states or excitations in the majority spin
channel, as they shift to lower energies with increasing field strength
(orange and blue peak), and in the minority spin channel, as they shift to
higher energies with increasing field strength (red peak). Yet, at least for
the positive-bias features (blue and red peak), the spin-polarized
measurements reveal the opposite spin character of what is anticipated from
their field-induced energy shift. This behavior again clearly contradicts the
explanation of the ZBA’s origin by the Kondo effect or common inelastic SEs.

In contrast, the spinaron was explicitly predicted to shift to higher energies
in an increasing magnetic field, while having a majority spin character, in
agreement with the presented experimental data of the high-energy (red)
feature in our STS measurements. Yet, the low-energy positive-bias feature,
which similarly exhibits a field-induced shift contradicting its
experimentally established spin character within conventional models, remains
unexplained by the calculations published in Ref. [15]. Extending these
calculations, which focused on the $d_{z^{2}}$ orbital of the Co atom
satisfying the symmetry conditions that enable tunneling of electrons to a tip
positioned exactly atop the adatom, we here examine the electronic structure
of all orbitals pertaining to the Co adatom in more detail.

## Theory

Figure 4: Orbital-resolved local density of states and theoretical inelastic
tunneling spectra. a and b Orbital-resolved local density of states (LDOS) at
the adatom level of a Co atom on Cu(111) for the $d_{z^{2}}$ and $d_{yz}$
(degenerate with $d_{xz}$) state as function of the magnetic field for both
the majority and minority spin channel. The presence of spinarons and
intrinsic SEs can be identified by the characteristic field dependence of the
visible features, most clearly seen in the corresponding energy derivatives
plotted in d and e. The inset in e shows a zoom to the relevant features in
the energy derivative of the minority spin channel LDOS. The spin-resolved
tunneling spectra and their energy derivatives are shown in c and f, assuming
a tip shifted slightly away from the Co atom’s center.

First-principles simulations indicate that the Co adatom carries a large spin
(orbital) magnetic moment of $2\mu_{\rm B}$ ($0.47\mu_{\rm B}$), which points
out-of-plane as favored by a significant magnetic anisotropy energy of
$4.3\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$. The latter induces gapped
intrinsic SEs, which are expected to emerge in the adatom spectral function
symmetrically around the Fermi energy $\varepsilon_{\rm F}$. The interaction
of the electronic states and the SEs renormalizes the electrons’ self-energy
as well as the electronic properties, which can be quantified via MBPT
combined with relativistic TD-DFT (see Methods section).

The resulting orbital-resolved local density of states (LDOS) of the Co adatom
is presented in Fig. 4a,b in red (blue) for the $d_{z^{2}}$ ($d_{xz,yz}$)
orbitals for different magnetic fields of up to 20 T. The respective energy
derivatives $\differential n/\differential\varepsilon$ are plotted in Fig.
4d,e. From there, it is possible to deduce the characteristic response of each
feature to the magnetic field, which allows to distinguish SEs and spinarons
in our calculations as was discussed in the beginning of this paper.

In the majority spin channel, Fig. 4a,d, we identify the previously reported
prominent $d_{z^{2}}$ spinaron at 4 meV above $\varepsilon_{\rm F}$, which
overlaps with the expected SE at $\varepsilon-\varepsilon_{\rm F}\approx-4$
meV. In the minority spin channel, Fig. 4b,e, the $d_{z^{2}}$ orbital
exclusively hosts a SE which is found at positive energy. The $d_{xz,yz}$
orbitals, equally exhibiting an intrinsic SE that shifts to higher energies
with increasing field strength, additionally host a spinaron that shifts to
lower energies [38]. At zero field, both energetically coincide at $\approx
2.5$ meV and visibly split in large magnetic fields due to their opposing
response.

Based on the LDOS at the adatom level presented in Fig. 4a,b, we present the
theoretical tunneling spectrum and its derivative obtained within the Tersoff-
Haman approximation in Fig. 4c,f, assuming a slightly shifted tip to capture
the contribution of the $d_{xz,yz}$ orbitals [45]. In the experiment, tip
orbitals deviating from the theoretically assumed $s$-wave tip could equally
result in an enhancement of signatures from the $d_{xz,yz}$ orbitals in the
tunneling spectrum. The resulting curves exhibit the known shape of the ZBA
and will be compared to the experimentally obtained conductance in the
following.

##  .3 Discussion

Based on our ab-initio calculations and their interpretation, we can now re-
dissect our experimental results. Staggeringly, our calculations support the
presence of two spinaronic features in the LDOS of the Co atom on Cu(111), one
being found in the majority spin channel of the $d_{z^{2}}$ orbital and the
other one originating from the minority spin channel of the $d_{xz,yz}$
orbitals. Matching the experimentally determined spin character and direction
of the magnetic field-induced energy shift as well as the energetic position
at zero field of both features, we attribute the high-energy (red) feature in
the experimental $\differential^{2}I/\differential U^{2}$ signal to the
$d_{z^{2}}$ spinaron in the majority channel and the low-energy (blue) feature
to the $d_{xz,yz}$ spinaron in the minority channel.

Additional signatures of intrinsic SEs are predicted in all orbitals and spin
channels as discussed above. Experimentally, only the negative-bias (orange)
$\differential^{2}I/\differential U^{2}$ dip can presumably be related to a
majority SE, although its spin character remains unresolved in our
measurements. The two SEs predicted in the minority channel remain unresolved,
potentially due to their energetic overlap with the intense spinaronic
signals, which are also found in the calculated vacuum LDOS atop the Co atom
and in the corresponding derivative in Fig. 4c,f.

Bearing in mind the ab-initio nature of the theoretical framework that
refrains from any fitting adjustment, the identified agreement between
experimental and theoretical data is astounding, establishing the experimental
findings as strong evidence for the presence of two spinarons in the LDOS of
Co atoms on the Cu(111) surface. Our work motivates further investigations to
untwist the intriguing nature of the unraveled spinaronic states and to
identify their presence in other sub-nanoscopic systems (e.g., other noble
metal (111) surfaces, see Ref. [38]). The presented study demands for the
reassessment of previous work studying the interaction of the magnetic states
in Co atoms with their environment [46, 47] and inspires the exploration of
the potential of engineering the spinarons’ properties such as amplitude and
spin-polarization via confinement effects or in atom-by-atom constructed
nanostructures [48, 24, 27, 49]. Finally, future research to establish the
consequences of the presence of spinarons for spin-resolved transport could
challenge the effects of magnetic impurities far beyond the field of surface
physics.

##  .4 Methods

Experiment. The Cu(111) crystal is prepared by cycles of Ar-sputtering and
annealing to $500\text{\,}\degreecelsius$. Co atoms are deposited onto the
surface from a Co rod at a substrate temperature of $4.2\text{\,}\mathrm{K}$.
For spin-polarized measurements, first, bilayer high Fe islands are grown at
room temperature with a surface coverage of $25\text{\,}\%$. After cooling the
sample to liquid helium temperature, Nc molecules are deposited by briefly
placing the sample in front of a leak valve connected to the Nc container.
Finally, Co atoms are deposited in-situ. Gently dipping the tip into the Fe
islands leads to a transfer of Fe onto the tip apex, forming a magnetic
cluster that results in a spin-polarization of the tunneling current. All
measurements are performed at $1.4\text{\,}\mathrm{K}$. The differential
conductance data are recorded via a lock-in amplifier with a modulation
frequency of $f=$890\text{\,}\mathrm{H}\mathrm{z}$$ and a modulation amplitude
of $U_{\mathrm{mod}}^{\mathrm{rms}}=$0.5\text{\,}\mathrm{m}\mathrm{V}$$.
Fitting of the spectra on Nc is performed within the dynamical scattering
model introduced in Ref. [44].

First-principles simulations. The ground state properties of the Co adatom
deposited on Cu(111) were explored with density functional theory (DFT) as
implemented in the scalar-relativistic full-electron Korringa-Kohn-Rostoker
(KKR) Green function augmented self-consistently with spin-orbit interaction
[50]. An angular momentum cutoff at $l_{\text{max}}=3$ is assumed for the
orbital expansion of the Green function and when extracting the local density
of states a k-mesh of $300\times 300$ is considered. The adatom relaxes
towards the surface by $14\%$ of the Copper lattice parameter. The intrinsic
SEs were obtained utilizing time-dependent DFT (TD-DFT) [51, 52, 53] including
spin-orbit interaction, while the renormalized electronic states due to
interaction with the SEs are evaluated within many-body perturbation theory
(MBPT) accounting for relativistic effects [14, 15]. Our results are found
stable with both adiabatic local spin density and generalized gradient
approximations.

The intrinsic spin excitation spectra are extracted from the dynamical
magnetic susceptibility after solving the Dyson-like equation:

$\underline{\chi}(\omega)=\underline{\chi}_{\text{KS}}(\omega)+\underline{\chi}_{\text{KS}}(\omega)\,\underline{\mathcal{K}}\,\underline{\chi}(\omega)\quad,$
(1)

where the bare Kohn-Sham susceptibility $\underline{\chi}_{\text{KS}}(\omega)$
describes uncorrelated electron-hole excitations and $\underline{\mathcal{K}}$
corresponds to the exchange-correlation kernel. The electronic self-energy
renormalized by the coupling of electronic states and SEs is extracted from
MBPT, and consists of a convolution of the dynamical spin-susceptibility
obtained from TD-DFT and Green functions calculated with DFT. More details are
given in Refs. [14, 15].

##  .5 Acknowledgments

This research was supported by the DFG through SFB 1170 “ToCoTronics” and the
Würzburg-Dresden Cluster of Excellence ct.qmat, EXC2147, project-id 390858490.
M.B., A.O. and F.F. thank Paolo Sessi (Max-Planck-Institut für
Mikrostrukturphysik, Halle, Germany) for bringing this scientific topic to our
attention. S.L. acknowledges Sascha Brinker and Alexander Weismann for
fruitful discussions.

##  .6 Author contributions

F.F., A.O., M.B., and S.L. conceived the experiments. F.F. and A.O. conducted
the measurements and analyzed the data. J.B. performed the first-principle
simulations. S.L. and M.B. supervised the project. All authors discussed the
results. F.F. and S.L. wrote the manuscript with input from all authors.

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# Dark Matter Effects on Compact Binary Stars

Ebrahim Hassani
Department of Physics, Faculty of Sciences, University of Birjand, Birjand,
Iran
ebrahim.hassani<EMAIL_ADDRESS>
&Amin Rezaei Akbarieh
Faculty of Physics, University of Tabriz, Tabriz, Iran
<EMAIL_ADDRESS>
&Yousef Izadi
Department of Physics and Applied Physics, University of Massachusetts,
Lowell, MA 01854, USA
<EMAIL_ADDRESS>
&Reza Pazhouhesh
Department of Physics, Faculty of Sciences, University of Birjand, Birjand,
Iran
<EMAIL_ADDRESS>

###### Abstract

Abstract

As compact binary star systems move inside the halo of the galaxies, they
interact with dark matter particles. The interaction between dark matter
particles and baryonic matter causes dark matter particles to lose some part
of their kinetic energy. Once dark matter particles lose part of their kinetic
energy, they are gravitationally bound to stars, and stars start to absorb
dark matter particles from the halo. The absorption of dark matter particles
inside compact binary systems increases the mass of the binary components, and
then the total mass of the binary systems increases as well. According to
Kepler’s third law, increased mass through this way can affect other physical
parameters (e.g., semi-major axes and orbital periods) of these systems too.
We estimate the period change of some known compact binary systems due to the
accretion of dark matter particles into them. We investigate the effects of
different dark matter particle candidates with masses in the range $\simeq
10^{-12}-10^{5}\>(GeV.c^{-2})$ and dark matter density as high as the dark
matter density near the Galactic central regions. We find that the estimated
change of period due to the accretion of dark matter particles into compact
binary systems can be as large as the measured values for these systems.

##  1 Introduction

Rotation curves of galaxies reveal the non-uniform distribution of dark matter
(DM) inside them [1]. From this, one can infer that all astronomical objects
inside galaxies are immersed inside DM. Therefore, one can argue that the
presence of DM affects the physics of every astronomical object, including
stars, binary star systems, and stellar clusters inside galaxies.
The interaction of DM particles with baryonic matter through weak interaction
can affect the internal structure and the evolutionary courses of stars.
According to the known properties of DM particles, DM particles can affect the
physics of stars mostly in two ways [2, 3]: 1) DM particles can transfer
energy (whether DM particles annihilate or not) between different layers of
stars. 2). If DM particles annihilate inside stars, they can act as a new
source of energy (beside the energy that comes from baryonic matter energy
production cycles). In other words, considering the effects of DM on the
physics of stars causes them to follow different evolutionary paths compared
to the standard stellar evolutionary models.
In 1978, Steigman proposed DM effects as a solution to the solar neutrino
problem [4] (though, after discovering the neutrino oscillations by the Super-
Kamiokande experiment in 1998 [5] this problem has been considered as a solved
problem by the neutrino oscillations assumption, instead of the DM assumption
[6]). Simulation of dwarf galaxies also supports the interaction between DM
particles and individual stars. In evolved dwarf galaxies, DM halo around
dwarf galaxies heats up by the stars inside them. Then, the more the dwarf
galaxy evolves, the more the DM halo heats up by the stars inside them [7]. In
addition, the effects of DM on stars can be used to solve the paradox of youth
problem for stars that are located near the central regions of our galaxy [8].
In addition to normal stars, DM effects on other celestial bodies like the
moon [9, 10], planets [11, 12], neutron stars (NS) [13, 14, 15, 16, 17, 18,
19, 20, 21], white dwarf stars (WD) [22, 23, 24, 25], black holes [26, 27, 28]
and binary star systems [29] have been investigated in the literature.
As time passes, stars absorb and gather DM particles from the halo of
galaxies. As a result, the mass of the stars increases. According to the
definition, the capture rate (CR) of DM particles by a massive spherical body
(like Earth, Sun, neutron stars, etc.) is the number of DM particles that are
gravitationally bound to that body per unit time [30]. For the first time,
Press and Spergel estimated the CR relation of weakly interacting massive
particles (WIMP) particles that are accreting into the sun [31]. In the next
step, Gould obtained a general relation for the capture rate of DM particles
by other round bodies like planets and stars [30]. Kouvaris used Press and
Spergel relation to derive CR relation for NSs [32]. Hurst et al. used Gould
relation to obtain CR relation for WDs [33].
As a compact binary system moves inside a DM halo, it gets affected by the
dynamical friction that is imposed by the DM halo. Induced dynamical friction
by the DM halo can alter the physical parameters (e.g. orbital period and
semi-major axis) of these systems. Dynamical friction that is imposed in this
way can be used to constrain parameters of different DM models [34]. In a
series of studies, people have tried to estimate the change of period of
compact binary systems due to the dynamical friction that is imposed by DM
[35, 36, 37, 34, 38, 39, 40, 41, 42]. As an example, Gabriel and Rueda
compared the observed period change of some known compact binary systems with
the estimated period change due to dynamical friction and also the estimated
period change due to the gravitational waves emissions from these systems
[35]. One of their results is that, in some regions of parameter space of DM,
the orbital decay due to dynamical friction could be comparable to the orbital
decay due to the gravitational wave emission.
In addition to "dynamical friction" and "gravitational wave emission," the
accretion of DM particles inside compact binary systems can be the source of
period change (and other physical parameters, like semi-major axis) in these
systems. As a compact binary system moves inside the DM halo, each star of the
system can capture DM particles from the DM halo. The captured DM particles
increase the total mass of the system as well. Then, according to Kepler’s
third law ( i.e., $P^{2}\varpropto a^{3}/M$), the increased total mass of the
binary systems through this way, alters the semi-major axis and orbital period
of these systems.
In Sec. (5), we estimate the period change of some known compact binary
systems due to the accretion of DM particles inside them and then compare the
results with the period change of these systems due to dynamical friction and
gravitational waves emission (see table (1) for a summary of our results). The
rest of the paper is structured as follows, in Sec. (2), we present and
discuss CR relations for white dwarfs and neutron stars. In Sec. (3), a
relation for the CR by compact binary systems is obtained. In Sec. (4), we
explain DM density distribution inside galaxies. Finally, Sec. (5) is devoted
to the results and conclusions.

##  2 Capture Rate by Compact Stars

CR of DM particles by hydrogen atoms is different from the CR relation for
elements heavier than hydrogen atoms. Since in hydrogen atoms, the role of the
spin of the outermost electron is not insignificant, it should be taken into
account in scattering cross section relations. CR by Hydrogen atoms is given
by [29]

$C_{\chi,H}=\left[4\sqrt{6\pi}\frac{\rho_{\chi}}{m_{\chi}}\frac{1}{\overline{v}_{\chi}v_{\ast}}exp(-\frac{3v^{2}_{\ast}}{2\overline{v}^{2}_{\chi}})\right]\times\left[\sigma_{\chi,SI}+\sigma_{\chi,SD}\right]\times\left[\int_{0}^{R_{\ast}}n_{H}(r)r^{2}dr\right]\times\\\
\left[\int_{0}^{\infty}exp(-\frac{3u^{2}}{2\overline{v}^{2}_{\chi}})sinh(\frac{3uv_{\ast}}{\overline{v}^{2}_{\chi}})(v_{e}^{2}-\frac{\mu_{-,H}^{2}}{\mu_{H}}u^{2})\theta(v_{e}^{2}-\frac{\mu_{-,H}^{2}}{\mu_{H}}u^{2})du\right],$
(1)

and for elements heavier than Hydrogen it has the following form [29]

$C_{\chi,i}=\left[8\sqrt{6\pi}\frac{\rho_{\chi}}{m_{\chi}^{2}}\frac{E_{0}}{\overline{v}_{\chi}v_{\ast}}\frac{\mu^{2}_{+,i}}{\mu_{i}}exp(-\frac{3v^{2}_{\ast}}{2\overline{v}^{2}_{\chi}})\right]\times\left[\sigma_{\chi,SI}A_{i}^{2}(\frac{m_{\chi}m_{n,i}}{m_{\chi}+m_{n,i}})^{2}(\frac{m_{\chi}+m_{p}}{m_{\chi}m_{p}})^{2}\right]\times\left[\int_{0}^{R_{\ast}}n_{i}(r)r^{2}dr\right]\times\\\
[\int_{0}^{\infty}exp(-\frac{3u^{2}}{2\overline{v}^{2}_{\chi}})sinh(\frac{3uv_{\ast}}{\overline{v}^{2}_{\chi}})\>\times\left\\{exp(-\frac{m_{\chi}u^{2}}{2E_{0}})-exp(-\frac{m_{\chi}u^{2}}{2E_{0}}\frac{\mu_{i}}{\mu^{2}_{+,i}})exp(-\frac{m_{\chi}v_{e}^{2}}{2E_{0}}\frac{\mu_{i}}{\mu^{2}_{-,i}}(1-\frac{\mu_{i}}{\mu^{2}_{+,i}}))\right\\}du].$
(2)

In Eqs. (1) and (2), $\rho_{\chi}$ is the DM density that surrounds the star,
$m_{\chi}$ is the mass of DM particles, $\overline{v}_{\chi}$ is the
dispersion velocity of DM particles, $v_{\ast}$ is the speed of star relative
to the DM halo, $\sigma_{\chi,SI}$ is the spin-independent scattering cross-
section, $\sigma_{\chi,SD}$ is the spin-dependent scattering cross-section,
$n_{H}$ is the number density of Hydrogen atoms in different locations of
stars, $n_{i}$ is the number density of heavier (i.e. heavier than Hydrogen
atoms) elements in different locations of stars, $r$ is the distance from the
center of the star, $m_{p}$ is the mass of protons, $m_{i}$ is the nuclear
mass of the element i, $A_{i}$ is the atomic number of element i,
$E_{0}=3\hslash^{2}/(2m_{n,i}(0.91m_{n,i}^{1/3}+0.3)^{2})$ is the
characteristic coherence energy (for more details see Ref. [30]) and
$\mu_{i}$, $\mu_{+,i}$ and $\mu_{-,i}$ are defined in the
form:$\mu_{i}=m_{\chi}/m_{n,i}$ and $\mu_{\mp,i}\equiv(\mu_{i}\mp 1)/{2}$.
As mentioned above, in Eqs. (1) and (2), quantities $\sigma_{\chi,SI}$ and
$\sigma_{\chi,SD}$ represents spin-independent and spin-dependent scattering
cross sections respectively. Scattering cross section is a measure of the rate
(or probability) of the scattering of DM particles from baryonic matter
particles. In practice, scattering cross section depends on many physical
parameters, like the structure of the particles (atoms, molecules, photons,
etc.) under interaction, whether the interaction is elastic or inelastic, the
relative velocity of the particles under the interaction, charge, spin, mass
of the particles under interaction, and etc. In addition, due to the lack of
knowledge about the nature of DM particles, the exact definition of scattering
cross section for every DM particle candidate is highly model-dependent. For
example, the definition of scattering cross section for SuperWIMP (one of DM
candidates) differs from model to model. To surmount these difficulties, we
select maximal DM particle-nucleon scattering cross sections allowed by the
direct DM detection experiments. Similar studies used the same reasoning to
chose numerical values for DM particle-nucleon scattering cross-section [43,
44, 45, 46]. For DM candidates that is used in this study (i.e. Neutrinos,
SuperWIMP, WIMP), we choose scattering cross sections to be:
$\sigma_{Neutrino,SI}=10^{-46}$ for Neutrinos [47] and
$\sigma_{SuperWIMP,SI}=\sigma_{WIMP,SI}=10^{-48}$ for SuperWIMP and WIMP
particles [48].
In evolved stars, like white dwarfs and neutron stars, stars consume almost
all of their initial Hydrogen content. Thus, in compact stars, the
contribution of Hydrogen atoms in capturing DM particles is insignificant
compared to the heavier elements. Therefore, we only use Eq. (2) to calculate
the CR of DM particles by white dwarfs and neutron stars. In addition, since
in elements heavier than Hydrogen, the contribution of the spin of the atoms
in the scattering process is not significant (in comparison with the spin part
of the scattering cross-section), we only consider the spin-independent part
of the scattering cross section (i.e. $\sigma_{\chi,SI}$) in our calculations.
Eq. (2) is consisted of four different bracket. The First three brackets can
be calculated analytically. We calculate fourth bracket using numerical
integration. In third bracket the amount of number density is supposed to be a
constant value for both white dwarfs and neutron stars. The average number
density of a typical white dwarf (e.g. sirius WD star with mass
$M_{siriusB}=1.018\>M_{\odot}$ [49] and radius $R_{radius}=0.0084\>R_{\odot}$
[50]) is approximately

$\bar{n}_{WD}=\dfrac{N}{V}=\dfrac{(\dfrac{M_{WD}}{m_{c}})}{(4/3)\pi
R^{3}_{WD}}=1.2\times 10^{35}\>(Kg.m^{-3}).$ (3)

In Eq. (3) we suppose that a white dwarf consists of carbon atoms with atomic
mass $m_{c}=12u=1.99\times 10^{-26}(Kg)$. Similar to Eq. (3), number density
of a typical neutron star can be calculated as

$\bar{n}_{NS}=\dfrac{N}{V}=\dfrac{(\dfrac{M_{NS}}{m_{n}})}{(4/3)\pi
R^{3}_{WD}}=3.46\times 10^{48}\>(Kg.m^{-3}),$ (4)

in which $m_{n}=1.67\times 10^{-27}(Kg)$ is the mass of a neutron.

##  3 Capture rate by compact binary systems

According to Kepler’s third law, the square of the period in binary systems is
proportional to the inverse of the mass of the system and also is proportional
to the cube of the semi-major axis of the orbit [51]

$P^{2}=(\dfrac{4\pi^{2}}{GM})\>a^{3},$ (5)

where $M=M_{1}+M_{2}$ . If we suppose that all binary systems inside the
galaxies are immersed inside the DM halos, then one can assume that both
components of the compact binary systems absorb and accrete DM particles.
Therefore, as time passes, the total mass of the binary systems will increase.
At this point, we estimate the effects of the increased mass on other
parameters of the compact binary systems. Taking differential from both sides
of the Eq. (5) (and assuming that there is not a net external force on the
system and assuming the absence of other physical effects (e.g. dynamical
friction, gravitational waves emission, mass transfer between the components
of the binary and, etc) that can affect the parameters of the binary systems)
and after some simplifications, we have

$\frac{\dot{P}}{P}=\frac{3}{2}\frac{\dot{a}}{a}-\frac{1}{2}\frac{\dot{M}}{M}=\frac{3}{2}\frac{\dot{a}}{a}-\frac{1}{2}\frac{\dot{M_{1}}+\dot{M_{2}}}{(M_{1}+M_{2})},$
(6)

where $\dot{P}=dp/dt$, $\dot{a}=da/dt$, $\dot{M}=dM/dt$,
$\dot{M_{1}}=dM_{1}/dt$ and $\dot{M_{2}}=dM_{2}/dt$. $\dot{M_{1}}$ and
$\dot{M_{2}}$ signifies the mass variation of the binary components due to the
accretion of DM particles inside them. To calculate $\dot{M_{1}}$ and
$\dot{M_{2}}$ in Eq. (6) we need to multiply Eq. (1) and Eq. (2) to the mass
of the DM particles $m_{\chi}$ (i.e. $\dot{M_{1}}=CR\times m_{\chi}$ where
$M_{1}$ is the mass of the primary compact star).

Figure 1: The calculated period decay rate for various types of compact binary
systems and different DM particle candidates. Sub-plots (a)-(c) are for WD-WD
type, sub-plots (d)-(f) for WD-NS type, and sub-plots (g)-(i) are for NS-NS
type compact binary systems. In all sub-plots, horizontal axes are the mass
range of different DM particle candidates, and vertical axes are the period
decay rate of the compact binary systems. Only SuperWIMP particles have the
potential to change the period of the compact binary systems in the range of
the measured values (see Table (1) for the measured period change values of
some of the known compact binary systems). Table 1: Physical parameters of the
compact binary systems that are used in this study. Columns include: Type:
types of the components of the compact binary systems; Name: name of the
systems; $M_{p}$ and $M_{c}$: mass of primary and companion stars; P: orbital
period of the systems; d: distance from the galactic center;
$\dot{P}^{obs}/P$: measured period change; $\dot{P}^{GW}/P$ and
$\dot{P}^{DF}_{NFW}/P$: period change due to the gravitational wave emission
and dynamical friction; $\dot{P}_{{\rm Neutrino}}^{\rm DM}/P$, $\dot{P}_{{\rm
SuperWIMP}}^{\rm DM}/P$ and $\dot{P}_{{\rm WIMP}}^{\rm DM}/P$ are calculated
period changes of the systems due to the capture of DM particles. All data
(except the last three columns) are from Table 1 of the Ref. [35]. By
comparing measured $\dot{P}^{obs}/P$ values with the calculated
$\dot{P}^{DM}/P$ values, we infer that for some DM candidates (e.g.
SuperWIMPs), measured $\dot{P}^{obs}/P$ values can be as large as the
predicted $\dot{P}^{DM}/P$ values. So, capturing DM particles that have
similar properties like SuperWIMPs, by compact binary systems can be
considered as one of the reasons for the observed period decay in these
systems.

Type | Name |  $m_{p}$ $[M_{\odot}]$ |  $m_{c}$ $[M_{\odot}]$ |  $P$ $[days]$ |  d $[kpc]$ |  $\dot{P}^{\rm obs}/P$ $\times(10^{-16})$ $[Sec^{-1}]$ |  $\dot{P}^{\rm GW}/P$ $\times(10^{-16})$ $[Sec^{-1}]$ |  $\dot{P}_{{\rm NFW}}^{\rm DF}/P$ $\times(10^{-24})$ $[Sec^{-1}]$ |  $\dot{P}_{{\rm Neutrino}}^{\rm DM}/P$ $[Sec^{-1}]$ |  $\dot{P}_{{\rm SuperWIMP}}^{\rm DM}/P$ $[Sec^{-1}]$ |  $\dot{P}_{{\rm WIMP}}^{\rm DM}/P$ $[Sec^{-1}]$
---|---|---|---|---|---|---|---|---|---|---|---
NS-NS | J0737-3039 | 1.3381(7) | 1.2489(7) | 0.104 | 1.15(22) | -1.393 | -1.38874 | -1.168 | $10^{-10}-10^{-6}$ | $10^{-30}-10^{-18}$ | $10^{-31}-10^{-29}$
B1534+12 | 1.3330(4) | 1.3455(4) | 0.421 | 0.7 | -0.052905 | -0.037554 | -6.7126 |  |  |
J1756-2251 | 1.312(17) | 1.258(17) | 0.321 | 2.5 | -0.0757 | -0.0793 | -0.009771 |  |  |
J1906+0746 | 1.323(11) | 1.290(11) | 0.166 | 5.4 | -0.3939 | -0.363 | -0.18512 |  |  |
B1913+16 | 1.4398(2) | 1.3886(2) | 0.325 | 9.9 | -0.8533 | -0.8556022 | -0.2828 |  |  |
NS-WD | J0348+0432 | 2.01(4) | 0.172(3) | 0.104 | 2.1(2) | \- 0.3038 | -0.2871 | -0.04441 | $10^{-9}-10^{-5}$ | $10^{-30}-10^{-17}$ | $10^{-30}-10^{-29}$
J1012+5307 | 1.64(22) | 0.16(2) | 0.60 | 0.836(80) | -0.289 | -0.0212 | -0.06566 |  |  |
J1141-6545 | 1.27(1) | 1.02(1) | 0.20 | 3.7 | -0.2321 | -0.2332 | -0.2071 |  |  |
J1738+0333 | 1.46(6) | 0.181(7) | 0.354 | 1.47(10) | -0.0084680 | -0.009155 | -0.0693 |  |  |
WD-WD | WDJ0651+2844 | 0.26(4) | 0.50(4) | 0.008 | 1 | -140 | -120 | -0.02025 | $10^{-9}-10^{-5}$ | $10^{-29}-10^{-17}$ | $10^{-30}-10^{-29}$

##  4 Dark matter density profile

N-body simulations of galaxies reveals the non-uniform distribution of DM
inside galaxies. These results are in agreement with observations (i.e. the
rotation curves of galaxies). Navarro, Frenk and White (NFW) DM profile
describes the radial distribution of DM density inside galaxies and is written
as [52, 53]

$\rho_{NFW}(r)=\dfrac{\rho_{0}}{\dfrac{r}{r_{s}}(1+\dfrac{r}{r_{s}})^{2}},$
(7)

where $\rho_{0}$ is the local DM density and $r_{s}$ is the size of the DM
halo [54]. Both $\rho_{0}$ and $r_{s}$ vary from a galaxy to a galaxy. In the
case of the Milky way galaxy, these parameters are estimated to be about
$\rho_{0}=0.26\>(GeV.cm^{-3})$ and $r_{s}=20\>(Kpc)$ [54] and for the case of
M31 galaxy (andromeda galaxy) the estimation for these parameters are about
$\rho_{0}=0.42\>(GeV.cm^{-3})$ and $r_{s}=16.4\pm 1.5\>(Kpc)$ [55].
In addition to the NFW DM density profile, Einasto, Moore, and isothermal DM
density profiles can be used to describe the distribution of DM inside
galaxies. These profiles have the following expressions [54]

$\rho_{\chi,Isothermal}=\begin{cases}&\rho_{0}\frac{1+(r_{\odot}/r_{s})^{2}}{1+(r/r_{s})^{2}}\\\
&r_{s}=5\>(kpc),\end{cases}$ (8)
$\rho_{\chi,Einasto}=\begin{cases}&\rho_{0}\>exp\left\\{\frac{-2}{\alpha}\left[(r^{\alpha}-r^{\alpha}_{s})/r^{\alpha}_{s}\right)]\right\\}\\\
&\alpha=0.17,r_{s}=25\>(kpc),\end{cases}$ (9)
$\rho_{\chi,Moore}=\begin{cases}&\rho_{0}(\frac{r_{\odot}}{r})^{1.16}(\frac{1+r_{\odot}/r_{s}}{1+r/r_{s}})^{1.84}\\\
&r_{s}=30\>(kpc).\end{cases}$ (10)

Fig. (2) depicts the 2D representation of the distribution of DM density
according to the NFW, Einasto, Moore, and isothermal DM density profiles and
for the central region of our galaxy. From Fig. (2), it is clear that DM
density near the central regions of the galaxies is the highest, and by moving
toward the outer regions, its value decreases. Hence, one can conclude that DM
affects the physics of astronomical objects (especially compact binary
systems, which is the subject of this study) in the central regions of
galaxies more than the objects in the outer regions.

Figure 2:  (color on-line) 2D illustration of the different DM density
profiles that are discussed in the text. For distances far from the galactic
center (e.g., distances more than 1 kpc), the differences between different DM
density profiles are less than one order of magnitude. For this reason, the
NFW DM density profile is selected for this study.

For distances far from the central black hole of the galaxy (e.g., farther
than 1 kpc), the differences between different DM density profiles are less
than one order-of-magnitude. For this reason, we choose the NFW DM density
profile throughout the rest of the study. To obtain the results of the
calculations, we consider DM density around compact binary systems to be
$\rho_{\chi}=10^{2}\>(GeV.cm^{-3})$ which correspond to the distance about
$r\simeq 1\>(kpc)\simeq 3300\>(ly)$ from the central black hole of the galaxy.

##  5 Results and Discussions

We have used Eqs. (2) and (6) to estimate the accretion of DM particles inside
compact binary systems. In calculations of this paper, stars with masses lower
than $1.24\>M_{\odot}$, and stars with masses higher than $1.24\>M_{\odot}$
have been considered to be white dwarfs and neutron stars, respectively. The
masses of the DM particles have been taken to be in the range
$10^{-12}-10^{5}\>(Gev.c^{-2})$ corresponding to the different DM particle
candidates that we have used (see horizontal axes of Fig. (1) for the mass
range of different DM particles candidates).
From Eq. (5) we can see that semi-major axis and period of a binary system has
the $a\propto M^{1/3}$ and $P\propto M^{-1/2}$ proportionality with the total
mass of the system. This means by increasing the total mass of the binary
system $M$, the period of the system will be affected more than the semi-major
axis. For this reason the $\dot{a}/a$ term can be approximately omitted from
the right hand side of the Eq. (6). Then, Eq. (6) can be written in the form

$\frac{\dot{M}}{M}\simeq-2\frac{\dot{P}}{P},$ (11)

which means $\dot{M}/M$ has a linear relation with $\dot{P}/P$. After
estimating $\dot{M}/M$ values theoretically (through the procedure that is
explained in Sec. (3)), we have compared the results with the observed
$\dot{P}/P$ values (in the rest of this section we use $\dot{P}^{obs}/P$
notation instead of $\dot{P}/P$ for the observed period changes of compact
binary systems). The results of the calculations are presented in Fig. (1) and
table (1). Our main results are:

  * •

According to table (1), observed values of $\dot{P}/P$ are of the
$10^{-16}-10^{-17}$ ($sec^{-1}$) order of magnitude which are almost in the
same order of magnitude of the estimated $\dot{P}^{GW}/P$ values. Therefore,
we can infer that gravitational wave emission from compact binary systems can
be considered as one of the main reasons for period decay in these systems.

  * •

In table (1), estimated values of $\dot{P}^{DF}/P$ are of the order of
magnitude $10^{-24}-10^{-27}$ ($sec^{-1}$) which are much less than the
measured $\dot{P}^{obs}/P$ values. So, we can conclude that dynamical friction
that is imposed by DM halo can not be considered as the main reason for the
period change in compact binary systems.

  * •

According to the estimated $\dot{P}^{DM}/P$ values in Fig. (1), period change
due to the accretion of DM particles is highly model dependent. Here, we only
consider three DM particles models which are Neutrinos with mass in the range
$10^{-12}-10^{-10}(GeV.c^{-2})$, SuperWIMPs with mass in the range
$10^{-6}-10^{4}(GeV.c^{-2})$, and WIMPs with mass in the range
$10^{-1}-10^{5}(GeV.c^{-2})$. Though there are many other DM candidates in the
literature (for instance WIMPzilla particles with mass in the range
$10^{12}-10^{15}(GeV.c^{-2})$, [56]), for the purposes of this study, using
the above mentioned candidates is sufficient.
From table (1) and Fig. (1), it is conceivable that, between the selected DM
candidates for this study (i.e., Neutrinos, SuperWIMPs, and WIMPS), low-mass
candidates have the highest potential to change the period of compact binary
systems. However, this does not mean that, in reality, low-mass DM affects the
physics of compact binary systems more than other candidates. This is because
there are many other DM-related parameters (e.g., scattering cross sections of
DM particles with electrons and different atom species, the velocity of stars
relative to the DM halo, DM density around the binary system, and etc.) that
have the potential to affect the CR amounts by each component of a compact
binary system. In fact, we aim to show that the CR of DM particles has the
potential to affect the period of compact binary systems as much as the
observed values. But because of the lack of our knowledge about the exact
physical nature of DM, we are not able to claim confidently which DM candidate
can be considered as the best candidate for the observed period decay in
compact binary systems.

  * •

Period decay of compact binary systems due to the accretion of DM particles is
not a monotonic function of just DM particles mass $m_{\chi}$. This behaviour
can be conceivable from Eq. 2 and Fig. 1 too. As an example consider
SuperWIMPs in Fig. 1. When the mass of SuperWIMPs is in the range
$m_{\chi}\varpropto 10^{-1}-10^{3}(GeV.c^{-2})$, the effects of other physical
parameters (e.g. spin-dependent and spin-independent scattering cross-section,
DM density at the location of binary system, velocity of stars with respect to
the DM halo, and etc.) causes deviation from the monotonic behaviour in the
diagram.

Our main result from the above discussions is that the accretion of DM
particles inside compact binary systems can be considered as one of the main
reasons for the observed period change in compact binary systems. The results
of this paper prefer DM candidates with mass in the range $m_{\chi}\varpropto
10^{-1}-10^{5}(GeV.c^{-2})$ as the best DM candidates for this observed period
decays. The further investigation of the possible consequences of the
accretion of DM candidates into compact binary systems requires that other
physical parameters that affect the CR relation in Eq. (2) be taken into
account. This can be the subject of the next studies in this respect.

##  6 Data availability

No new data were generated or analyzed in support of this research.

##  7 Acknowledgements

The authors would like to express their special thanks to Prof. Joakim Edsjö
from the University of Stockholm, Sweden, and Prof. Gianfranco Bertone from
the University of Amsterdam, Netherlands, and Marco Taoso from the National
Institute of Nuclear Physics (INFN) Turin, Italy, for their helpful
discussions. We appreciated the comments of the anonymous referee, which
resulted in an improved paper. Finally, we acknowledge the use of the Python
scientific computing packages NumPy [57], SciPy [58] and Pandas [59], as well
as the graphics environment Matplotlib [60].

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