# The nature of very-faint X-ray binaries: Near-infrared spectroscopy of 1RXH
J173523.7–354013 reveals a giant companion

A. W. Shaw,1,2 N. Degenaar,3 T. J. Maccarone,4 C. O. Heinke,5 R. Wijnands,3 J.
van den Eijnden6
1Department of Physics and Astronomy, Butler University, 4600 Sunset Ave,
Indianapolis, IN, 46208, USA
2Department of Physics, University of Nevada, Reno, 1664 N. Virginia Street,
Reno, NV, 89557, USA
3Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science
Park 904, 1098 XH, Amsterdam, the Netherlands
4Department of Physics & Astronomy, Texas Tech University, Box 41051, Lubbock,
TX, 79409-1051, USA
5 Department of Physics, University of Alberta, CCIS 4-181, Edmonton, AB, T6G
2E1, Canada
6Department of Physics, University of Warwick, Coventry CV4 7AL, UK E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted 2023 November 22. Received 2023 November 21; in original form 2023
September 30)

###### Abstract

Very-faint X-ray binaries (VFXBs) are a sub-class of black holes and neutron
stars in binaries that appear to be accreting at a very low rate. In addition
to providing interesting constraints on poorly understood forms of accretion,
elucidating the nature of VFXBs is particularly interesting for binary
evolution and population modeling. Through near-infrared (nIR) spectroscopy,
we here investigate the nature of the bursting neutron star and VFXB 1RXH
J173523.7–354013 (J1735), which persistently accretes at an X-ray luminosity
of $L_{X}\sim 10^{34}-10^{35}~{}\mathrm{erg~{}s}^{-1}$. Our analysis shows
that the nIR emission is dominated by that of the companion star, which we
find to be a late G or early K-type giant, making this the second neutron star
identified as a VFXB found to have a giant companion. We discuss how several
of the system properties are difficult to reconcile with a wind-fed symbiotic
X-ray binary. We therefore also propose an alternative scenario wherein J1735
is a wide binary system (supported by the discovery of a $7.5$ d modulation in
the nIR light curves) with a quiescent luminosity of $L_{X}\sim
10^{34}-10^{35}~{}\mathrm{erg~{}s}^{-1}$, in which the donor star is
overflowing its Roche lobe. This raises the possibility that J1735 may, every
century or more, exhibit very long and very bright outbursts during which it
reaches accretion rates around the Eddington limit like the neutron star Z
sources.

###### keywords:

accretion: accretion disks – X-rays: individual (1RXH J173523.7–354013) –
stars: neutron – X-rays: binaries – X-rays: bursts

††pubyear: 2021††pagerange: The nature of very-faint X-ray binaries: Near-
infrared spectroscopy of 1RXH J173523.7–354013 reveals a giant companion–The
nature of very-faint X-ray binaries: Near-infrared spectroscopy of 1RXH
J173523.7–354013 reveals a giant companion

##  1 Introduction

Among the brightest X-ray point sources in the sky are neutron stars (NSs) and
black holes (BHs) that are in a binary system accreting gas from a less
massive companion star. These low-mass X-ray binaries (LMXBs) can generate an
X-ray luminosity of $L_{X}\simeq 10^{36-39}~{}\mathrm{erg~{}s}^{-1}$ when
actively accreting. However, a significant sub-set of the LMXBs (comprising
$\simeq$25% of the currently known population; Bahramian & Degenaar, 2022) has
a much lower peak accretion output of $L_{X}\simeq
10^{34-36}~{}\mathrm{erg~{}s}^{-1}$: the very faint X-ray binaries (VFXBs;
Wijnands et al., 2006).

Due to their low luminosity, detailed studies of VFXBs were long hampered by
sensitivity limitations of astronomical instrumentation. Whereas there has
been a significant uprise in targeted studies to find and characterize VFXBs
over the past $\simeq$2 decades (e.g., Sakano et al., 2005; Muno et al., 2005;
in’t Zand et al., 2005; Wijnands et al., 2006; Wijnands, 2008; Degenaar &
Wijnands, 2010; Armas Padilla et al., 2011; Sidoli et al., 2011; Shaw et al.,
2020; Bahramian et al., 2021), many aspects of the accretion morphology and
binary configurations of this large LMXB sub-group remain to be explored.
There is significant interest in this, because VFXBs are a gateway to study
accretion processes and outflows in an under-explored mass-accretion regime
(e.g., Armas Padilla et al., 2013a; Knevitt et al., 2014; Wijnands et al.,
2015; Degenaar et al., 2017; Gusinskaia et al., 2020). Moreover, elucidating
the binary configuration of VFXBs allows to test and refine various binary
evolution and population synthesis models (e.g., Pfahl et al., 2002; King &
Wijnands, 2006; Yungelson et al., 2006; van Haaften et al., 2012; Maccarone &
Patruno, 2013; Maccarone et al., 2015).

A natural explanation for the X-ray faintness of VFXBs may be that these LMXBs
have small accretion disks (i.e. short orbital periods; e.g. King & Wijnands,
2006; in’t Zand et al., 2007; Heinke et al., 2015). This includes the sub-
population of ultra-compact X-ray binaries (UCXBs), which have degenerate
donor stars and hence require orbital periods $\lesssim$90 min to facilitate
accretion through Roche-lobe overflow (e.g. Nelson et al., 1986). Indeed, by
now several VFXBs have been conclusively identified as UCXBs (see Bahramian &
Degenaar, 2022, and references therein). Nevertheless, at least three VFXBs,
1RXH J173523.7–354013 (aka 1RXS J173524.4–353957, RX J1735.3–3540, IGR
J17353–3539; J1735 hereafter), M15 X-3, and SAX J1806.5$-$2215, must have non-
degenerate donor stars; strong H$\alpha$ emission was detected in the optical
spectrum of J1735 (Degenaar et al., 2010), the high optical/X-ray ratio of M15
X-3 is inconsistent with an UCXB interpretation Arnason et al. (2015), and SAX
J1806.5$-$2215 appears to exhibit absorption in Brackett $\gamma$, providing
evidence for hydrogen in the system and therefore excluding a hydrogen-
stripped companion (Shaw et al., 2017).

Another possible explanation for the faint X-ray emission of VFXBs is that the
accretion flow is radiatively-inefficient. Standard accretion theory predicts
that towards low accretion rates the radiative efficiency decreases (Rees et
al., 1982; Narayan & Yi, 1995). This should be particularly pronounced for BH
systems with short (though not ultra-compact) orbital periods (Wu et al.,
2010; Knevitt et al., 2014), of which indeed a handful of candidates have been
identified among the VFXBs (McClintock et al., 2001; Corral-Santana et al.,
2013; Stoop et al., 2021). In NS systems, the strong NS magnetic field may
also hamper efficient accretion onto the stellar surface (e.g., Wijnands,
2008; Degenaar et al., 2014; Heinke et al., 2015). Several VFXBs have been
identified as accreting millisecond pulsars (e.g., Altamirano et al., 2010;
Strohmayer & Keek, 2017; Sanna et al., 2018, 2022). For one of these the inner
accretion disk was found to be truncated and hints of outflowing material were
found, suggesting that some material might be propelled out Degenaar et al.
(2017); van den Eijnden et al. (2018).

Finally, recent systematic surveys to find and identify VFXBs have revealed
that symbiotic systems (e.g. Iben & Tutukov, 1996; Corbet et al., 2008; Bozzo
et al., 2022) could be a growing sub-class (Shaw et al., 2020; Bahramian et
al., 2021). In these systems the compact object (which can be a NS or even a
white dwarf) accretes from the weak wind of the low-mass companion, which is
inefficient, and hence can explain the low X-ray luminosity. A somewhat
surprising aspect of this is that the one VFXB identified as a symbiotic X-ray
binary (IGR J17445$-$2747; Shaw et al., 2020; Bahramian et al., 2021) displays
thermonuclear bursts (Mereminskiy et al., 2017). These bursts (see e.g.,
Galloway & Keek, 2021, for a review) are often regarded as exclusively
occurring in LMXBs, since bursts are suppressed or have a different
observational signature on high $B$ field NSs (Bildsten & Brown, 1997), and
most low-$B$ NSs are found in LMXBs. Symbiotic systems, on the other hand, are
a short-lived stage of binary evolution such that there is not enough time for
accretion to reduce the magnetic field of the NS to a level where bursts can
be produced. Bursts from a symbiotic system therefore strongly suggest that
the NS was born with a low $B$ field. In summary, the VFXBs appear to be an
inhomogeneous group, consisting of several sub-classes of LMXBs: UCXBs, short-
period (though not ultra-compact) BH systems, accreting millisecond X-ray
pulsars and symbiotic X-ray binaries.

In this work, we investigate the nature of the VFXB J1735. Degenaar et al.
(2010) identified this system as a NS-LMXB based on the detection of a
thermonuclear burst. Reprocessed X-ray burst emission was clearly detected at
optical and UV wavelengths, which allowed them to pin down a unique
counterpart for the X-ray source. Ultraviolet (UV), optical and near-infrared
(nIR) photometry indicated that the counterpart was rather red, while optical
spectroscopy revealed a strong H$\alpha$ emission line (Degenaar et al.,
2010). Deep radio observations of the source did not yield a detection (van
den Eijnden et al., 2021), which implies that it is unlikely that its bright
nIR emission is due to the contribution of a jet. Gaia Data Release 3 (DR3;
Gaia Collaboration et al., 2016, 2022) measured the parallax to the optical
counterpart to be $p=1.82\pm 0.56$ mas, and Bailer-Jones et al. (2021)
calculated the photogeometric distance to be $d=3.5^{+2.3}_{-1.3}$ kpc based
on Gaia Early Data Release 3 (EDR3; Gaia Collaboration et al., 2021). In this
paper we further investigate the nature of this VFXB using nIR spectroscopy.

##  2 Observations and Data Reduction

###  2.1 Near-infrared monitoring

We queried data release 5 (DR5) of the Visible and Infrared Survey Telescope
for Astronomy (VISTA) Variable in the Via Lactea Survey (VVV; Minniti et al.,
2010) for nIR imaging of the J1735 field. We found that the field had been
observed multiple times since 2010 Sept, with the densest coverage in the
$K_{s}$-band. A $K_{s}$-band image of the field is shown in the upper panel of
Fig. 1. We downloaded the source detection table for the nIR counterpart of
J1735 (known as VVV J173523.74-354016.58), which contains aperture photometry
for numerous values of aperture radius.111For an overview of VISTA science
archive data, see http://horus.roe.ac.uk/vsa/dboverview.html . We also
downloaded source detection tables for four comparison stars (see Fig. 1). We
extracted the 1″ radius (as recommended by Saito et al., 2012, for the crowded
field case) aperture photometry from the source detection tables for J1735 and
the comparison stars and assembled differential $K_{s}$-band light curves,
plotting them in Fig. 2.222For clarity, we do not include the third comparison
star in Fig. 2, but the observed variability is nearly identical to what is
shown. We choose to calculate differential magnitudes instead of simply
plotting the VVV apparent magnitudes, because this allows us to assess whether
any variability is intrinsic to the J1735 system, or is an artefact of
performing aperture photometry in a crowded field.

Figure 1: Images of the J1735 field. Upper panel: VVV $K_{s}$-band image of
the field, with red circles at the positions of the comparison stars we use to
construct the differential light curves in Fig. 2. The white circle is at the
position of the nIR counterpart to J1735, as identified by Degenaar et al.
(2010). The blue box represents the approximate FOV of the VLT/SINFONI
spectroscopic follow-up. Lower panel: SINFONI $H+K$-band median cube image of
the J1735 field, with red circles showing some of the same comparison stars
from the upper panel. The nIR counterpart to J1735 is highlighted in this
panel by a blue circle. White patches in the lower panel represent artefacts
from the dithering pattern of SINFONI and the subsequent sky subtraction, but
this does not affect the extracted spectrum of J1735. Figure 2: Lower panel:
VVV differential $K_{\rm s}$-band light curves of the nIR counterpart to J1735
using a variety of comparison stars with similar magnitudes (c1, c2 and c4).
Upper panel: VVV differential $K_{\rm s}$-band light curves of the comparison
stars, showing minimal variability. In both panels, light curves are offset
from each other for clarity.

###  2.2 Near-infrared spectroscopy

The J1735 field was observed with the Spectrograph for INtegral Field
Observations in the Near Infrared (SINFONI; Eisenhauer et al., 2003) on the
Very Large Telescope at Paranal, Chile on 2018 August 03. Target of
Opportunity (ToO) observations were obtained as part of proposal 0102.D-0650
(PI: Degenaar). We obtained 14 total images, the first ten of which consisted
of 5 co-added 20s exposures while each of the remaining 4 images consisted of
4 co-added 20s exposures, all of which were obtained with the $H+K$ grating
($1.45-2.45$ $\mu$m). We utilised the 0$.\\!\\!^{\prime\prime}$25 spatial
pixel scale, corresponding to an 8″$\times$8″field-of-view for each image. A
dithering pattern was applied to account for the variable nIR sky background.
We used a nearby bright star as a natural guide star (NGS) to correct for
atmospheric distortions with the SINFONI adaptive optics (AO).

Data were reduced with the ESO Recipe Execution tool (esorex)
pipeline,333https://www.eso.org/sci/software/cpl/esorex.html which performs
typical reduction steps including dark subtraction, non-linearity correction,
flat fielding, wavelength calibration and sky subtraction to provide a co-
added data cube, from which the median image in Fig. 1 was derived. We
extracted the spectrum of the nIR counterpart to J1735 with the esorex task
sinfo_utl_cube2spectrum, using a circular region of radius 2 pixels
(0$.\\!\\!^{\prime\prime}$5).

We corrected for telluric features in the $H+K$ spectrum of J1735 using the
software molecfit (Smette et al., 2015; Kausch et al., 2015), which uses the
source spectrum itself as an input to fit and remove telluric lines. The
advantage of molecfit over the typical method of observing a telluric standard
is that using the science spectrum as an input means the derived telluric
spectrum is representative of the nIR sky spectrum under the actual observing
conditions. Telluric standards are typically observed at different times,
airmasses and sky background conditions and this can affect the accuracy of
the telluric correction of the science spectrum.

We also flux calibrated the spectrum of J1735 using observations of HIP 084435
(HD 155888), a B4V star originally included in the calibration plan as a
telluric standard, obtaining a single image consisting of five, 3s exposures.
We followed the same reduction steps as for J1735, and corrected for telluric
features using molecfit. We used the Image Reduction and Analysis Facility
(iraf; Tody, 1986) to perform the flux calibration of J1735. We first removed
all hydrogen absorption lines from the telluric-corrected spectrum of HIP
084435 using iraf task splot, fitting a Voigt profile to the Brackett lines
and subtracting the fit from the spectrum. We then calculated the sensitivity
function by assuming a 17200K blackbody for the HIP 084435 spectrum, with
magnitudes of $H=8.494\pm 0.042$ and $K=8.459\pm 0.020$ (Skrutskie et al.,
2006). The resultant sensitivity function was applied to the spectrum of J1735
to derive the flux calibrated $H+K$ spectrum. We show this spectrum in Fig. 3,
along with the dereddened spectrum, obtained using the iraf task deredden,
assuming $E(B-V)=1.68$ (Degenaar et al., 2010) and a Cardelli et al. (1989)
extinction law, for a standard Milky Way ratio of total to selective
extinction, $R_{V}=3.1$.

Figure 3: Full $H+K$ spectrum of the nIR counterpart to J1735, flux
calibrated. The observed spectrum is shown in grey and the dereddened spectrum
in black, which was calculated assuming $E(B-V)=1.68$ (Degenaar et al., 2010)
and a Cardelli et al. (1989) extinction law.

##  3 Analysis and Results

###  3.1 Long-term nIR monitoring

The VVV $K_{s}$-band differential light curve of the nIR counterpart to J1735
is shown in Fig. 2. When comparing the target/comparison differential light
curves with the comparison/comparison differential light curves, we note that
J1735 appears to show nIR variability with an amplitude of up to $\sim 0.45$
mag. The absence of such variability in the comparison light curves implies
that it is real, and intrinsic to the J1735 system. We also find that the
excess variance (an estimator of the intrinsic source variance, see e.g.
Vaughan et al., 2003) is much stronger for the target light curve than for any
comparison star light curve. The average NIR magnitudes of J1735 (using the 1″
aperture photometry from the source detection table) are $J=15.249\pm 0.008$,
$H=14.229\pm 0.007$ and $K_{s}=13.694\pm 0.002$.

We performed timing analysis on the differential light curves in order to
search for any periodic variability that might be associated with the system.
We calculated Lomb-Scargle periodograms (Lomb, 1976; Scargle, 1982) for each
of the five differential light curves shown in Fig. 2 using the LombScargle
class included in the astropy Python package (Astropy Collaboration et al.,
2013, 2018). The periodograms are shown in Fig. 4. For each periodogram, we
calculated a 99% significance threshold for the power. We did this by randomly
shuffling the light-curve magnitudes but keeping the time stamps the same,
effectively creating a randomized light curve with the same sampling as the
original data. We calculated the peak Lomb–Scargle power for 10,000 of these
randomized light curves, from which we derived the 99% significance for each
periodogram.

The periodograms of the target/comparison differential light curves all show
significant peaks, the two strongest of which correspond to periods of 7.5 and
66.2d, with the strongest peak (and only one to be consistently above the 99%
significance line) at $7.5\pm 0.3$d. These peaks are persistent through the
three target/comparison periodograms, and do not appear in the
comparison/comparison periodograms, implying that the corresponding periods
are real and not an artefact of the light curve sampling, though their origin
is unclear.

Figure 4: Lomb-Scargle periodograms of the light curves presented in Fig. 2.
Two peaks are marked with dashed lines and their corresponding periodicities
are labelled. The dash-dotted lines in each panel represent the 99%
significance level.

###  3.2 nIR spectroscopy

The flux-calibrated nIR spectrum of J1735 is shown in Fig. 3. We also show the
spectrum divided into $H$\- and $K$-band segments and continuum normalised in
Fig. 5. An initial examination of the spectrum reveals numerous lines
associated with neutral metals, along with molecular 12CO bandheads, classic
nIR spectral signatures of late-type (K-M) stars (e.g. Kleinmann & Hall, 1986;
Wallace & Hinkle, 1997; Meyer et al., 1998; Ivanov et al., 2004; Rayner et
al., 2009). Utilising the National Institute of Standards and Technology –
Atomic Spectra Database (NIST-ASD Kramida et al., 2021) to obtain the most
accurate rest wavelengths of these features, we used the iraf task rvidlines
task to calculate the observed radial velocity, $v_{\rm obs}$ of the source.
This is a sum of the systemic and orbital radial velocity of the system, as we
are unable to disentangle the two from a single spectrum. We find $v_{\rm
obs}=45\pm 11$ km s-1. The spectra shown in Fig. 5 are velocity corrected
using the measured $v_{\rm obs}$.

The most striking lines in the nIR spectrum of J1735 are the strong CO
bandheads at $\lambda>2.29$ $\mu$m. The presence of these bandheads (we note
that CO is also present in the $H$-band), along with the relative weakness of
Ca i and Na i, is strong evidence for a late-type giant companion (see e.g.
Wallace & Hinkle, 1997). To test this hypothesis, we can apply 2D spectral
classification techniques to the $K$-band portion of the nIR spectrum.
Numerous studies have identified temperature-dependent and surface gravity-
dependent features in nIR spectra of late-type stars (e.g. Ramirez et al.,
1997; Ivanov et al., 2004; Comerón et al., 2004; Rayner et al., 2009; Messineo
et al., 2017; Ghosh et al., 2019; Messineo et al., 2021). Comparing the
strengths of these features allows one to derive the spectral type and
luminosity class of the target. In the case of J1735 we assume this to be the
donor star of the accreting binary. These techniques were successfully applied
by Bahramian et al. (2014) and Shaw et al. (2020) to identify and type giant
and subgiant donors in a number of accreting binaries.

We adopt the Na i (2.21$\mu$m) and Ca i (2.26$\mu$m) passbands as the
temperature dependent features (e.g. Ramirez et al., 1997), and the 12CO
$(2,0)$ (2.29$\mu$m) bandhead as the surface-gravity and temperature-dependent
feature. For each feature we calculate the equivalent width (EW),
approximating the continuum level with a 1D polynomial (fit using astropy’s
Polynomial1D function) using two nearby, featureless regions of the spectrum.
We follow the method of Shaw et al. (2020), who define the continuum and line
bandpasses according to those of Bahramian et al. (2014) and Comerón et al.
(2004) and are almost identical to the continuum and line definitions of later
studies (see e.g. Ghosh et al., 2019). As an additional check, we also measure
the EW of the Mg i (1.71$\mu$m) line which, when compared with 12CO $(2,0)$,
provides a strong diagnostic for luminosity class (Messineo et al., 2017,
2021). The line and continuum definitions, along with the measured EWs, are
reported in Table 1. We also include continuum and line definitions for Si i
(1.59 $\mu$m), 12CO $(6,3)$ (1.62 $\mu$m) and Mg i (2.28 $\mu$m), the EWs of
which can be used to investigate how much, if at all, the measured spectrum is
diluted by non-stellar sources (such as an accretion disc; Förster Schreiber,
2000). Uncertainties on the EWs were calculated using a bootstrap-with-
replacement method as follows: we determined 10,000 new EWs using continuum
levels from a random sample of the passbands described in Table 1 and computed
the standard error. For each new continuum passband, we allowed the same
wavelength-flux pair to be selected multiple times. We cross-checked our EW
calculations with the equivalent_width function provided by the astropy
package specutils (Earl et al., 2023) and found them to be consistent. In
Table 1, we also include the integrated line fluxes of each spectral feature
of interest, measured from the continuum subtracted dereddened spectrum.

Aside from the metal lines, we also note the possible presence of a Brackett
$\gamma$ (Br$\gamma$) emission line at 2.17 $\mu$m (continuum and line
definitions provided in Table 1). Though weak (we measure EW$=-1.64\pm 0.13$
Å), it is consistent with the presence of hydrogen emission in the optical
spectrum (Degenaar et al., 2010). We see no further evidence of hydrogen
emission in the nIR spectrum of J1735.

Table 1: Measured EWs and fluxes of spectral features of the nIR spectrum of
J1735. We also define the locations of the features and the passbands used to
estimate the continuum, references for which are provided in the last column
of the table

.  Line Blue Continuum Red Continuum Feature Centre $\Delta\lambda$ Centre
$\Delta\lambda$ Centre $\Delta\lambda$ EW Flux Reference ($\mu$m) ($\mu$m)
($\mu$m) ($\mu$m) ($\mu$m) ($\mu$m) (Å) (erg cm-2 s-1) Si i 1.5890 0.004
1.5859 0.0015 1.5937 0.0015 $1.03\pm 0.13$ $-2.5\times 10^{-15}$ (1) 12CO
$(6,3)$ 1.6198 0.0045 1.6160 0.003 1.6270 0.003 $2.63\pm 0.24$ $-5.8\times
10^{-15}$ (2) Mg i 1.7115 0.005 1.7035 0.001 1.7145 0.002 $1.81\pm 0.14$
$-3.3\times 10^{-15}$ (1) Br$\gamma$ 2.1661 0.009 2.1560 0.012 2.1745 0.005
$-1.64\pm 0.13$ $1.0\times 10^{-15}$ (3) Na i 2.2075 0.007 2.1940 0.006 2.2150
0.004 $2.13\pm 0.19$ $-1.3\times 10^{-15}$ (4) Ca i 2.2635 0.011 2.2507 0.0106
2.2710 0.002 $1.18\pm 0.24$ $-5.9\times 10^{-16}$ (4) Mg i 2.2807 0.0035 2.277
0.0040 12.2865 0.008 $0.85\pm 0.06$ $-4.3\times 10^{-16}$ (4) 12CO $(2,0)$
2.2955 0.013 2.2500 0.016 2.2875 0.007 $8.66\pm 0.35$ $-4.2\times 10^{-15}$
(4)
References for line and continuum passbands: (1) Messineo et al. (2017), (2)
Ghosh et al. (2019), (3) This work, (4) Comerón et al. (2004)

Figure 5: VLT/SINFONI spectrum of the nIR counterpart to J1735. In the upper
sub-figure we show the $H$-band portion of the spectrum and in the lower sub-
figure we show the $K$-band portion of the spectrum. In the top panel of both
sub-figures we show the telluric transmission spectrum derived with molecfit
and used to correct the target spectrum. The lower panel of each sub-figure
shows the continuum normalised, telluric corrected spectrum of the target. The
target has been velocity corrected to the rest frame using the observed
$v_{\rm obs}=45\pm 11$ km s-1 and the spectrum has been smoothed with a
3-point boxcar function. Some spectral lines have been identified and
labelled.

##  4 Discussion

###  4.1 Spectral type of the companion

####  4.1.1 Luminosity class

Table 1 shows the EWs of lines important for deriving the spectral type of the
donor star in J1735. The first thing to note is that we measure EW[12CO
(2,0)]$=8.66\pm 0.35$ Å. Comerón et al. (2004) note that supergiants typically
exhibit EW[12CO (2,0)]$\gtrsim 25$ Å, while all the supergiants in the
Infrared Telescope Facility (IRTF) spectral library of cool stars (Rayner et
al., 2009) show EW[12CO (2,0)]$\gtrsim 20$ Å (Messineo et al., 2017, 2021). We
are therefore able to confidently rule out a supergiant donor for J1735,
implying that it is instead a dwarf, giant, or subgiant.

We can compare the EWs of 12CO $(2,0)$ to those of Na i and Ca i in order to
determine the luminosity class of the donor star. Ramirez et al. (1997) showed
that dwarfs exhibit $-0.22<\log_{10}${EW[12CO (2,0)]/(EW[Na i]$+$EW[Ca
i])}$<0.06$ while giants show much higher ratios in the range
$0.37<\log_{10}${EW[12CO (2,0)]/(EW[Na i]$+$EW[Ca i])}$<0.61$. Villaume et al.
(2017) also note that EW[12CO (2,0)] alone is a strong discriminator between
giant and dwarf stars for all metallicities. We measure $\log_{10}${EW[12CO
(2,0)]/(EW[Na i]$+$EW[Ca i])}$=0.42\pm 0.04$, strongly indicative of a giant
companion. Another piece of evidence for a giant classification comes from
EW[Mg i (1.71$\mu$m)], which, when compared with EW[12CO (2,0)], is consistent
with known giants (see figure 3 of Messineo et al., 2021). The strength of the
Mg i (1.71$\mu$m) line alone is also a good indicator of a giant
classification, as late-type dwarfs typically show EW[Mg i (1.71$\mu$m)]$>2$
(Rayner et al., 2009). We can therefore confidently identify the companion
star of J1735 as a late-type giant.

####  4.1.2 Contamination of the donor star spectrum by the accretion disc

We can also use EW ratios of temperature-dependent features to assess whether
there is a source of diluting continuum emission that would affect the
conclusions we draw from the measured EWs. Given that there is evidence for
Br$\gamma$ emission in the $K$-band spectrum, it is possible that there could
be continuum emission from the accretion disc contributing the nIR spectrum.
Förster Schreiber (2000) demonstrated that ratios of temperature-dependent
lines close in wavelength can provide an indication of continuum dilution in
the $H$\- and $K$-bands. In their figure 8, Förster Schreiber (2000) show
plots of dilution-free EW ratios, with arrows indicating how measured ratios
would change with a source of dilution. For J1735, our measured values for
EW[12CO (2,0)], EW[Ca i] and EW[Mg i (2.28$\mu$m)] strongly suggest that there
is negligible dilution in the $K$-band, as our measured ratios all lie on the
loci for non-diluted sources. However, the ratio of the 12CO (6,3) and Si i
lines in the $H$-band lies below the observed distribution of non-diluted
stars presented by Förster Schreiber (2000), implying that there could be some
contamination in this portion of the nIR spectrum. We therefore focus on the
$K$-band spectrum when deriving the effective temperature of the donor star.

####  4.1.3 Effective temperature

We can use the measured EWs (typically of EW[12CO (2,0)]) to also derive the
effective temperature, $T_{\rm eff}$, and therefore the spectral subtype of
the donor star. However, metallicity can play a role in the relationship
between EW and $T_{\rm eff}$. Ghosh et al. (2021) derive a relationship
between $x=$EW[Na i] and metallicity $z=$[Fe/H] based on $R\sim 1200$ $K$-band
spectra of 260 late-type giants:

$z=m0+ax+cx^{2}+ex^{3},$ (1)

where $m0=-2.328\pm 0.080$, $a=1.863\pm 0.109$, $c=-0.485\pm 0.042$ and
$e=0.045\pm 0.005$. Though the standard error of estimate (SEE) on the
relation is large (SEE $\sim 0.25$ dex), it provides a first estimate of
[Fe/H]. Assuming EW[Na i]=$2.13\pm 0.19$, we find [Fe/H]$=-0.1\pm 0.4$,
propagating uncertainties and factoring in the SEE. Despite the large scatter,
the EW of the Na i line is indicative of a donor that is not unusually metal-
rich or metal-poor.

Ghosh et al. (2021) also derive a linear relationship between $T_{\rm eff}$
and EW[12CO]:

$z=m0+ax,$ (2)

where $z$ and $x$ in this instance refer to $T_{\rm eff}$ and EW[12CO (2,0)],
respectively. For the full sample, Ghosh et al. (2021) derive $m0=5370\pm 30$
and $a=-82.7\pm 2.0$, from which we infer $T_{\rm eff}=4650\pm 160$, factoring
in the large scatter (SEE $=149$). If, instead, we assume
$-0.3\leq{\rm[Fe/H]}\leq 0.3$, which is likely given our calculation of
[Fe/H]$=-0.1\pm 0.4$, then Ghosh et al. (2021) derive $m0=5651\pm 44$ and
$a=-99\pm 3$, which would increase the temperature to $T_{\rm eff}=4790\pm
140$. Both ranges of $T_{\rm eff}$ are indicative of a spectral type in the
range G6III – K1III according to the $T_{\rm eff}$ – spectral type relations
of van Belle et al. (2021), putting the radius of the companion in the range
$9R_{\rm\odot}\lesssim R_{\rm c}\lesssim 16R_{\rm\odot}$ (see table 17 of van
Belle et al., 2021).

Deriving the spectral type of the companion also allows us to further
investigate whether the Br$\gamma$ feature described in Section 3.2 is real or
chance coincidence. All giants in the IRTF spectral library of cool stars in
the range G6III – K1III show weak, but detectable, Br$\gamma$ absorption in
the $K$-band. Subtracting continuum-normalised IRTF standard spectra from our
(continuum-normalised) SINFONI spectrum of J1735 therefore highlights the
apparent Br$\gamma$ emission in the source spectrum even further. An F-test of
the Br$\gamma$ region of the spectrum of J1735, comparing the spectrum with
and without a best-fit Gaussian line at 2.166$\mu$m indicates that the line is
significant above 99.9% confidence for any IRTF standard in the G6III – K1III
spectral type range, and thus we believe it to be real.

####  4.1.4 Surface gravity and mass

Ghosh et al. (2019) derived a linear relationship between the surface gravity
($\log g$) and EW[12CO (2,0)] with the same form as Equation 2, where $z$ is
now $\log g$ and $x$ is EW[12CO (2,0)]. Using the best-fit values $m0=3.75\pm
0.13$ and $a=-0.16\pm 0.01$ we derive $\log g=2.4\pm 0.35$ cm s-2 (factoring
in SEE $=0.29$). Though the uncertainty is relatively large, we can use this
value to place a rough estimate on the mass of the donor star. Taking a median
$R_{\rm c}\sim 12.5\mathrm{M}_{\odot}$ (see Section 4.1.3, above) and $\log
g\sim 2.4$ cm s-2 we estimate that the mass of the companion is $M_{\rm c}\sim
1.4\mathrm{M}_{\odot}$ \- similar to the NS primary. We note that this is not
a detailed calculation of $M_{\rm c}$ as Ghosh et al. (2019) calibrated their
relation using only K and M giants, and we have established that the range of
$T_{\rm eff}$ we have derived for the donor of J1735 can include G stars. In
addition, the large range of valid stellar radii and $\log g$ mean this number
is also subject to a significant scatter. However, $M_{\rm c}\sim
1.4\mathrm{M}_{\odot}$ remains a useful estimate when we consider the nature
of accretion in the J1735 system.

###  4.2 Timing analysis

The periodograms presented in Fig. 4 imply at least two statistically
significant periodicities associated with the J1735 system, the strongest of
which is at $7.5\pm 0.3$d. In Fig. 6 we show the $K_{s}$-band light curve of
J1735, folded on this periodicity, along with the best-fit sinusoidal model as
determined by the astropy LombScargle method. Fig. 6 shows that the phase-
folded light curve follows a sinusoidal pattern with significant scatter about
the mean, possibly indicative of short-term flaring episodes. Despite the high
significance of the $P=7.5$d peak in the periodogram (a false alarm
probability of FAP=$0.02$%), the folded light curve does not provide any
insight into the nature of the variability. If $P=7.5$d is the orbital period
of the binary, then a future nIR spectroscopic radial velocity campaign will
be able to confirm this.

Figure 6: The VVV $K_{s}$-band light curve of the nIR counterpart to J1735,
folded on the strongest period from the timing analysis of $7.5$d. The grey
points represent the unbinned folded light curve; black points are grouped
into 10 bins per cycle. The dashed curve represents the best-fit sinusoidal
model as determined by the Lomb-Scargle method.

###  4.3 The nature of 1RXH J173523.7-354013: wind accretion or Roche-lobe
overflow?

Having identified the donor star in J1735 as a late-type giant through nIR
spectroscopy, we can now turn to the X-ray properties of the system to discuss
the binary configuration. Any plausible scenario must be able to explain i)
its low persistent X-ray luminosity of $\sim
10^{34}-10^{35}~{}\mathrm{erg~{}s}^{-1}$ (see Section 4.3.1), ii) its ability
to ignite thermonuclear bursts, and iii) the large optical increase observed
during its thermonuclear burst detection (Degenaar et al., 2010). It has
already been established that J1735 cannot be a UCXB, owing to the detection
of H$\alpha$ in its optical spectrum. This is further confirmed by the
detection of molecular lines in the nIR spectrum that indicate a giant
companion. We therefore explore two more potential scenarios here: Scenario 1
in which J1735 is an LMXB, accreting via Roche lobe overflow (RLOF) and
Scenario 2, in which J1735 is a wind-fed symbiotic X-ray binary.

####  4.3.1 A note on the distance

Degenaar et al. (2010) calculate an upper limit on the distance to J1735 to be
$d\lesssim 9.5$ kpc, inferred from the peak flux of the X-ray burst. However,
Degenaar et al. (2010) note that the peak flux was only measured in the hard
X-ray band, and extrapolated to lower energies, hence a large upper limit.
Bailer-Jones et al. (2021), on the other hand, calculated the photogeometric
distance to J1735 to be $d=3.5^{+2.3}_{-1.3}$ kpc, from Gaia EDR3.

Degenaar et al. (2010) calculate a persistent unabsorbed X-ray flux of $F_{\rm
X}\sim 1.9\times 10^{-11}$ erg cm-2 s-1 in the 0.5–10 keV band, and a
persistent bolometric flux of $F_{\rm bol}\sim 3.8\times 10^{-11}$ erg cm-2
s-1 (0.01–100 keV). Combining these measured fluxes with the Bailer-Jones et
al. (2021) distance range implies that J1735 has a persistent X-ray luminosity
in the range $1.1\times 10^{34}\lesssim L_{\rm X}\lesssim 7.7\times 10^{34}$
erg s-1, and bolometric luminosity $2.2\times 10^{34}\lesssim L_{\rm
bol}\lesssim 1.5\times 10^{35}$ erg s-1.

These values are much lower than the upper limits calculated by Degenaar et
al. (2010), which implies that the global mass-accretion rate on to the NS,
$\dot{M}$, is also lower. For a NS of mass and radius $M_{\rm
NS}=1.4\mathrm{M}_{\odot}$ and $R_{\rm NS}=10$ km, respectively, this would
imply $1.9\times 10^{-12}\lesssim\dot{M}\lesssim 1.3\times
10^{-11}\mathrm{M}_{\odot}$ yr-1.

####  4.3.2 Scenario 1: Roche Lobe Overflow

The first possibility that we must consider is that J1735 is an LMXB
undergoing RLOF, but with a giant companion rather than the typical main
sequence secondary. If we assume that the periodicity measured in Section 4.2
is the orbital period $P_{\rm orb}$ of the binary and that the binary
components are the same mass ($M_{\rm NS}=M_{\rm c}\sim
1.4\mathrm{M}_{\odot}$; Section 4.1.4) then we can estimate the orbital
separation, $a\sim 1.6\times 10^{12}$ cm, using Kepler’s Third Law. The
approximate radius of the Roche lobe in a binary system with equal masses is
$R_{\rm RL}=0.38a$ (Eggleton, 1983). Utilizing the same assumption above
regarding $P_{\rm orb}$, as well as $q\sim 1$ (where $q=M_{\rm c}/M_{\rm NS}$
is the binary mass ratio), we calculate $R_{\rm RL}\sim 6.1\times 10^{11}$ cm,
or $\sim 9\mathrm{R}_{\odot}$.

Given the range of stellar radii given by van Belle et al. (2021) for a G6III
– K1III star, it seems plausible that the companion would exceed its Roche
Lobe. However, this is heavily dependent on the assumption that $P_{\rm
orb}=7.5$ d. If the $P_{\rm orb}$ of J1735 is instead 66.2 d, where a peak is
also seen in some of the periodograms in Fig. 4, a late G/early K giant would
not fill the Roche Lobe. In addition, any change in component masses that
would result in the assumed $q\sim 1$ being higher, such as a more massive
companion, or lower mass NS, increases the size of the Roche lobe, such that
the lower end of the $R_{\rm c}$ range calculated in Section 4.1.3 would not
fill it in the $P_{\rm orb}=7.5$ d configuration.444We note that the opposite
can also be true, such that a lower value of $q$ could reduce the size of the
Roche Lobe.

If $P_{\rm orb}=7.5$ d and the system is indeed a RLOF system, then
$\dot{M}_{\rm RLO}\sim 10^{-8}\mathrm{M}_{\odot}$ yr-1 (Webbink et al., 1983),
and thus mass transfer would have to be highly non-conservative to match the
calculated $\dot{M}$ on to the NS. For several NS-LMXBs in compact orbits it
has been proposed that mass-transfer is highly non-conservative based on
timing studies of the X-ray pulsations (di Salvo et al., 2008; Burderi et al.,
2009; Marino et al., 2019), or by modeling the multi-wavelength spectral
energy distribution of the accretion disc (Hernández Santisteban et al.,
2019). However, if severe mass losses are indeed the explanation for these
observations, it is unclear what should be causing this and if the same could
be happening in J1735. Severe mass losses, in principle, can be observed
through future dedicated multi-wavelength studies of the system, e.g. as an
additional source of absorption in X-rays (see e.g. Iaria et al., 2021), or as
the origin of reprocessed emission in the mid – far IR (which would require
JWST).

One possibility worth considering then is if J1735 could be a RLOF system ‘in
quiescence’, in the sense that its accretion disk is currently cold and
neutral but steadily filling up until sufficient mass is accumulated for it to
become ionized and an outburst to occur (see Hameury, 2020, for a recent
review on this disk instability paradigm). There is an empirical relation
between the orbital period and quiescent X-ray luminosity of (BH) LMXBs
accreting via RLOF (Garcia et al. 2001; see also Carotenuto et al. 2022 for a
recent representation), where a longer $P_{\rm orb}$ is expected to lead to a
higher quiescent $L_{\rm X}$ due to the existence of a larger disk. In the
work of Garcia et al. (2001), there are no NSs with known orbital periods on
the order of $\sim 100$ hr and measured quiescent luminosities, but there are
a few transient BH-LMXBs with long periods that have $L_{\rm X}\sim
10^{33}-10^{34}~{}\mathrm{erg~{}s}^{-1}$. At the same time, we know that for
similar orbital periods, NS systems typically have a factor $\sim 10-100$
higher quiescent luminosities than the BH ones (due to NSs having a solid
surface; Menou et al., 1999). While speculative, it does not seem implausible
that J1735 is a quiescent LMXB with $L_{\rm X}\sim
10^{34}-10^{35}~{}\mathrm{erg~{}s}^{-1}$. One might then expect such a system
to spend a long time in this quiescent state, filling up the large disk,
before launching into a prolonged outburst. If true, such an outburst might
further be expected to be very bright, reaching mass-accretion rates around
the Eddington limit, as is seen in the NS Z-sources (e.g., Hasinger & van der
Klis, 1989; Homan et al., 2010) It is worth noting that the calculated
$\dot{M}$ lies well inside the unstable (transient) regime predicted by the
disk-instability model that reproduces many of the features of LMXB outbursts,
assuming $P_{\rm orb}=7.5$ d (Lasota, 2001; Coriat et al., 2012).

We briefly note that the X-ray spectral properties of J1735 do not exclude the
luminous quiescence scenario. An XMM-Newton observation of J1735 showed that
it consists of a $\simeq 0.3$ keV black-body component, plausibly emerging
from the NS surface, and a $\Gamma\simeq 1.4$ power law (Armas Padilla et al.,
2013b). This is consistent with many other transient NS-LMXBs that have lower
X-ray luminosities in quiescence (e.g., Rutledge et al., 2001; Wijnands et
al., 2002; Campana & Stella, 2003; Jonker et al., 2003). However, it is also
true that short-period NS-LMXBs accreting at low rates have similar X-ray
spectra (e.g., Degenaar et al., 2017; Armas Padilla et al., 2013b; Papitto et
al., 2013).

Following van Paradijs & McClintock (1994), we can also test whether the
observed burst properties work within the framework of optical reprocessing of
the X-ray emission by an accretion disk. The peak of the X-ray burst was a
factor $\sim 1000$ brighter than the persistent X-ray flux (Degenaar et al.,
2010). We would therefore expect an increase in optical flux by $\sim 3.4$
mag, provided it is dominated by an accretion disk (van Paradijs & McClintock,
1994). Degenaar et al. (2010) measured an optical increase of 3.2 mag, well
within the scatter of the van Paradijs & McClintock (1994) relation. The
observed increase in optical brightness is therefore consistent with the
reprocessing of the X-ray burst in the accretion disk that one might expect in
a RLOF LMXB.

Finally, we briefly consider the possibility that J1735 is a RLOF LMXB that is
not X-ray faint because it is accreting at very low rate, but due to a very
high inclination prohibiting us to see the bright central X-ray source. A
number of such LMXBs are known and are referred to as Accretion Disk Coronae
(ADC) sources because their X-ray emission comes from a small fraction of the
central source that is scattered in material around the disk (e.g., White &
Holt, 1982; Tomaru et al., 2023). If J1735 is an ADC source, the mass-
accretion rate inferred from the observed X-ray luminosity is highly
underestimated, and that it could in fact be as high as that expected for a
7.5-d binary. However, the multi-wavelength properties of J1735 do not show
indications of a high inclination (such as a high X-ray absorption, dipping
X-ray variability, or a flat optical/infrared spectral energy distribution).
Moreover, in this scenario it would be difficult to explain why the system
significantly brightened during its burst, by a factor of $\sim 1000$ in
X-rays and by $\sim 3$ mag in the UV/optical band (Degenaar et al., 2010).
Finally, we note that the long duration of that burst is consistent with that
seen in other low-$\dot{M}$ NS-LMXBs (e.g., Falanga et al., 2009; Bozzo et
al., 2015; Degenaar et al., 2011; Degenaar et al., 2013). The ADC scenario
therefore does not seem likely for J1735.

####  4.3.3 Scenario 2: A wind-fed system

Given the propensity for giant stars to drive dense stellar winds, it is worth
investigating if the companion of J1735 is able to drive a strong enough wind
to produce the observed X-ray emission. Schröder & Cuntz (2005) developed a
semi-empirical formula for the mass loss of cool stellar winds, building upon
Reimers (1975) law. The mass loss rate, $\dot{M}_{\rm w}$, according to this
relation, can be described as such:

$\dot{M}_{\rm w}=\eta\frac{L_{*}R_{*}}{M_{*}}\left(\frac{T_{\rm eff}}{4000{\rm
K}}\right)^{3.5}\left(1+\frac{g_{\odot}}{4300g_{*}}\right)$ (3)

where $L_{*}$, $R_{*}$ and $M_{*}$ are the star’s luminosity, radius and mass
in solar units, respectively, $g_{*}$ and $g_{\odot}$ and are the star’s
surface gravity and solar surface gravity, respectively and $\eta$ is a
fitting parameter given as $\eta\sim 8\times 10^{-14}\mathrm{M}_{\odot}$ yr-1.

As we have already established that a $12.5\mathrm{R}_{\odot}$ star would fill
its Roche Lobe, we can only investigate the wind scenario if we assume that
the radius of the companion star is on the lower end ($R_{\rm c}\sim
9\mathrm{R}_{\odot}$) of the range given in Section 4.1.3 (van Belle et al.,
2021). Making a simple assumption on the luminosity of the companion based
from the Stefan-Boltzmann law, and assuming the same values for mass and
surface gravity as in Section 4.1.4, we derive $L_{*}\sim 40{\rm L}_{\odot}$
and thus $\dot{M}_{\rm w}\sim 4\times 10^{-11}\mathrm{M}_{\odot}$ yr-1.

By adopting a Bondi-Hoyle-Lyttleton formalism (Hoyle & Lyttleton, 1941; Bondi
& Hoyle, 1944), we can estimate the mass accretion rate on to the NS in the
wind-accreting scenario:

$\dot{M}=\frac{3}{16}\left(\frac{R_{\rm
c}}{a}\right)^{2}\frac{q^{2}}{\beta_{\rm w}^{4}}\left(1+\frac{1+q}{2\beta_{\rm
w}^{2}}\frac{R_{\rm c}}{a}\right)^{-\frac{3}{2}}|\dot{M}_{\rm w}|.$ (4)

Where $a$ is the orbital separation, $q$ is the binary mass ratio ($M_{\rm
NS}/M_{\rm c}$ in this case, as opposed to the inverse in Section 4.3.2), and
$\beta_{\rm w}$ is the wind velocity parameterized as a fraction of the
companion star’s escape velocity (see Willems & Kolb, 2003). As discussed
above, if the 7.5 d periodicity measured in Section 4.2 is $P_{\rm orb}$, then
$a\sim 1.6\times 10^{12}$ cm, assuming the mass of each star is
$1.4\mathrm{M}_{\odot}$.

Using $q=1$, as well as $R_{\rm c}\sim 9$ $\mathrm{M}_{\odot}$ and
$\dot{M}_{\rm w}\sim 4\times 10^{-11}\mathrm{M}_{\odot}$ yr-1, we estimate
$\dot{M}\sim 7\times 10^{-13}\mathrm{M}_{\odot}$ yr-1, assuming that the wind
velocity is equivalent to the escape velocity of the companion ($\beta_{\rm
w}=1$). This is inconsistent with the $\dot{M}$ calculated in Section 4.3.1.
Based on our assumptions, we find that Bondi-Hoyle-Lyttleton accretion is
highly unlikely to be driving the observed properties of J1735.

Considering that a 9$\mathrm{R}_{\odot}$ star may not quite fill the Roche
Lobe, especially if some of the assumptions regarding the mass ratio change
even slightly, it is possible that its stellar wind is gravitationally focused
in the direction of the NS (Friend & Castor, 1982; de Val-Borro et al., 2009).
de Val-Borro et al. (2017) calculate that the mass-accretion rate on to the
compact object can be up to 5–20% of the total wind mass-loss rate from the
donor in the focused wind case. For J1735, this would imply that $\dot{M}$ can
be as high as $8\times 10^{-12}$ $\mathrm{M}_{\odot}$ yr-1, adopting the
derived $\dot{M_{\rm w}}$ for a $9\mathrm{R}_{\odot}$ companion above. This is
well within the calculated range of $\dot{M}$ in Section 4.3.1. We do
emphasize that this calculation depends heavily on many assumptions about the
binary components, and this scenario only works if the companion star is a
relatively small giant.

##  5 Conclusions

In this work, we have examined the $H$\- and $K$-band spectrum of the nIR
counterpart to the VFXB 1RXH J173523.7$-$354013\. We find that the $K$-band
spectrum is dominated by the companion star, which we derive to be a late G or
early K giant. This makes J1735 the second NS system originally identified as
a VFXB now known to have a giant companion, after IGR J17445$-$2747 (Shaw et
al., 2020). Timing analysis of nIR photometry of J1735 has revealed a strong
modulation on a period of $7.5\pm 0.3$d, which we postulate to be the orbital
period of the system. Other periods are present in the periodograms, but only
the $7.5$d period is consistently detected above 99% confidence, regardless of
the choice of comparison star.

Given the most likely properties of the binary, we consider the most likely
accretion mode for J1735 to involve Roche lobe overflow, rather than accretion
from the giant companion’s wind. However, we cannot rule out accretion from a
focused wind. RLOF requires a large mass-transfer rate ($\dot{M}_{\rm RLO}\sim
10^{-8}$ $\mathrm{M}_{\odot}$ yr-1) and thus mass-transfer on to the NS would
have to be highly non-conservative in order to be consistent with the
calculated $\dot{M}$ on to the NS (Section 4.3.1). Alternatively, we
hypothesize that J1735 could be a transient system that is currently in a
state in which its disk is building up to an outburst. The fact that it is
much brighter than typical LMXBs in such a ‘quiescent’ state could be
explained by its large accretion disk. This scenario predicts that the source
at one point would exhibit a very bright and long outburst, appearing like a Z
source.

In order to solidify our conclusions surrounding J1735, the next step will be
to confirm $P_{\rm orb}$ with a nIR radial velocity campaign, which will not
only confirm the orbital period, but will also allow us to derive the mass
function of the binary, providing more constraints on the true mass of the
companion. If the source is indeed the RLOF LMXB we claim it to be, then we
may also be able to detect ellipsoidal modulations in time-series photometry,
which will allow us to model the inclination of the binary.

Regardless of the true nature of accretion in J1735, it is becoming clear that
compact objects with evolved companions are emerging as a significant
contributor to the growing population of faint X-ray sources. Shaw et al.
(2020) uncovered one symbiotic X-ray binary (IGR J17445$-$2747), as well as a
cataclysmic variable with a subgiant donor (3XMM J174417.2$-$293944) in their
follow-up of sources discovered in a systematic X-ray survey of the Galactic
bulge (Bahramian et al., 2021).

##  6 Acknowledgments

We thank the anonymous referee for their insightful comments that helped
improve this manuscript. AWS thanks the hospitality of the Anton Pannekoek
Institute, where part of this work was carried out. COH is supported by NSERC
Discovery Grant RGPIN-2016-04602.

Based on observations collected at the European Organisation for Astronomical
Research in the Southern Hemisphere under ESO programme 0102.D-0650(C). This
research made use of astropy (https://www.astropy.org), a community-developed
core python package for Astronomy (Astropy Collaboration et al., 2013, 2018).

## Data Availability

The SINFONI observations of 1RXH J173523.7$-$354013 are publicly available in
the European Southern Observatory Science Archive Facility, along with the
corresponding raw and processed calibration data. The VVV data used in this
work are publicly available from the VISTA Science Archive.

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#  Geometric origins of topological insulation in twisted layered
semiconductors

Hao Tang  Department of Physics, Peking University, Beijing, 100871, P.R.China
Stephen Carr  Brown Theoretical Physics Center and Department of Physics,
Brown University, Providence, Rhode Island 02912-1843, USA  Efthimios Kaxiras
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
John A. Paulson School of Engineering and Applied Sciences, Harvard
University, Cambridge, Massachusetts 02138, USA

###### Abstract

Twisted bilayers of two-dimensional (2D) materials are proving a fertile
ground for investigating strongly correlated electron phases. This is because
the moiré pattern introduced by the relative twist between layers introduces
long-wavelength effective potentials which lead to electron localization.
Here, we develop a generalized continuum model for the electronic structure of
moiré patterns, based on first-principles calculations and tailored to capture
the physics of twisted bilayer 2D semiconductors. We apply this model to a
database of eighteen 2D crystals covering a range of atomic relaxation and
electronic structure features. Many of these materials host topologically
insulating (TI) moiré bands in a certain range of twist angles, which
originate from the competition between triangular and hexagonal moiré
patterns, tuned by the twist angle. The topological phases occur in the same
range as the maximally flat moiré bands.

Quantum materials, engineered by creative manipulation of the features of
conventional crystals, offer new possibilities for breakthroughs in
understanding electron correlations and superconductivity. What is intriguing
is that these phenomena emerge at a length scale much larger than the
underlying crystal lattice constant (by a factor of 10 to 1000), through
features due to strain patterns, controlled by geometric constraints. A new
platform for such studies are systems comprising few-layer two-dimensional
(2D) crystals, like twisted bilayers of graphene or transition metal
dichalcogenides (TMDCs) [1, 2, 3, 4, 5]. The slight lattice mismatch between
two layers of a 2D material at a relative twist angle results in a moiré
superlattice (MSL) and a long-wavelength periodic modulation of the effective
electronic potential [6, 7]. In a narrow range of the twist angle, the moiré
potentials act as confining wells for the electrons of the constituent
monolayers, causing isolated flat bands and localized wave functions near the
Fermi surface [8, 9].

In the moiré flat bands, the kinetic energy is heavily suppressed and
electronic interactions play a dominant role, with the intensity of the
interactions controlled by the twist angle; this effect has been dubbed
“twistronics” [10]. Compared to twisted bilayer graphene (TBG), the twisted
bilayer semiconductors can host flat bands in a large range of twist angles
[11, 12] instead of at precisely a magic angle [1]. This makes it possible to
overcome some experimental challenges in twisted bilayers of semiconductors;
thus, the twist angle becomes an additional degree of freedom for fine-tuning
other physical effects in the strongly correlated regime [13]. Intriguingly,
topological insulator (TI) moiré bands were predicted in a twisted bilayer of
MoTe2 [14]; this work introduced a possible candidate for observing concurrent
correlated and TI phases in the same material.

It is difficult to model twistronic systems at small twist angles
($\theta\simeq 1\degree$) using first-principles calculations, because the
number of atoms in the MSL scales as $\theta^{-2}$. To overcome this
limitation, continuum models with a low-energy effective Hamiltonian based on
first-principles (density-functional theory, DFT) calculations were developed
for electronic structures of TBG; this approach can accurately describe flat
bands and magic angles [15, 9, 16, 17]. Although continuum models have also
been applied to the twisted bilayer semiconductors [18, 14], they have yet to
include the effect of atomic relaxations which play an important role at small
angles [19, 20, 12, 11].

In this letter, we present results from a DFT-based generalized continuum
method designed specifically for twisted bilayer semiconductors. The computed
electronic structures are consistent with full-DFT results [12, 11, 2] but
only require a relatively inexpensive set of bilayer calculations. These
calculations involve systems with only a handful of atoms per unit cell, in
contrast to the many thousands of atoms necessary in the calculation of a full
MSL. We derive a database of relaxation and the corresponding coefficients of
tight-binding electronic structure hamiltonians for eighteen materials with
various lattice symmetries and band edge momenta. These coefficients do not
capture the material’s entire band structure, but rather focus on the details
of the parabolic band edges. Each layer contributes one band to the full
twisted bilayer model, and these bands are coupled through a set of stacking-
dependent electronic interactions.

In Fig. 1 we provide an overview of the different phases and their geometric
origins, as the twist angle is changed in a moiré bilayer (Fig. 1a). The full
interaction between the bands can be decomposed into two complementary parts.
The first describes the tendency for electrons in one layer to tunnel to the
other, as shown in Fig. 1b – labeled $\Delta T$ for “Tunneling”. The tunneling
coefficients are strongest at $AA$ (aligned) stacking regions and form a
triangular (TR) lattice across the MSL. The second contribution captures the
stacking dependence of the monolayer bands’ on-site energies, which depends on
the electrostatic potential from the opposite layer, shown in Fig. 1c) –
labeled $\Delta t$ and $\Delta b$ for “top/bottom” layer. The electrostatic
potentials have maxima at the $AB$ and $BA$ stacking regions, forming a
honeycomb (HC) lattice.

Figure 1:  (a) A moiré pattern of a twisted TMDC with areas of aligned ($AA$)
stacking and eclipsed ($AB$ or $BA$) stacking highlighted. (b) At $AA$
stacking, the local electronic Hamiltonian is described by interlayer
tunneling between the layers ($\Delta T$), forming a triangular (TR) lattice.
(c) At $AB$ or $BA$ stacking, the electronic Hamiltonian is described by
interlayer electrostatic potentials ($\Delta t,\Delta b$), forming a honeycomb
(HC) lattice. (d) Dependence of the dominant moiré interaction on twist angle,
with atomic relaxation of the superlattice (red line) or without it (blue
line): evolution of the lattice character from TR to HC with decreasing
$\theta$ induces TI states and flat bands at small angles, leading to Mott
insulator (MI) and superconducting (SC) behavior. (e) The TI phase is caused
by band hybridization as the electrons transition between quantum harmonic
oscillator (QHO) states on a TR to a HC lattice.

In the majority of materials studied here, the in-plane atomic relaxation of
the twisted bilayers enhances the effects of the electrostatic potential
fluctuations and reduces that of the interlayer tunneling, especially at small
twist angles. Electron localization transitions, from the tunneling-dominated
TR lattice to the electrostatic-dominated HC lattice, occur at low twist angle
in various materials, as illustrated in Fig. 1d. In the low-angle region,
ultra-flat moiré bands are predicted, likely to host superconducting (SC) and
Mott insulating (MI) phases. The topologically insulating (TI) moiré bands
appear for intermediate values of the twist angle, during the transition
between the two types of lattice geometries for the interlayer electronic
interactions (Fig. 1e). The electronic potentials naturally lead to quantum
harmonic oscillator (QHO) states at small twist angles [21, 22, 23], whose
presence underlies the appearance of the moiré flat bands and whose
competition explains the large number of TI phases possible in twisted
semiconductors.

We briefly introduce the methodology here. To capture the stacking-dependent
electronic and atomic details, we perform DFT calculations on aligned bilayers
over a grid sampling all possible interlayer displacements. Using the
effective mass approximation, we treat the dispersion around the monolayer
band extrema as a kinetic energy term in a continuum Hamiltonian. Including
the three stacking-dependent potentials described in the introduction, we
obtain the effective Hamiltonian:

$H=\left(\begin{matrix}-\frac{\hbar^{2}(\nabla-
ik_{0})^{2}}{2m^{*}}+\Delta_{t}(\bm{r})&\Delta_{T}^{*}(\bm{r})\\\
\Delta_{T}(\bm{r})&-\frac{\hbar^{2}(\nabla-
ik_{0})^{2}}{2m^{*}}+\Delta_{b}(\bm{r})\end{matrix}\right)$ (1)

where $m^{*}$ is the effective mass, $\Delta_{t,b}$ are the electrostatic
potentials for the top and bottom layer, and $\Delta_{T}$ is the interlayer
tunneling strength. At small twist angles we also include the all-important
relaxation effects by minimizing the total mechanical energy of the moiré
patterns [20]. The local electronic structures can then be derived from the
DFT through the expansion:

$E^{(\pm)}(\bm{r},\bm{k})=E^{(\pm)}(\bm{\theta}\times\bm{r}+2\bm{u})+\frac{\hbar^{2}(\bm{k}-\bm{k}_{0})^{2}}{2m^{*}}$
(2)

To determine the $\Delta_{t/b},\Delta_{T}(\bm{r})$ from the local electronic
structure, the Bloch wave functions at band extrema are extracted from DFT
calculations to assess the layer polarization of each band. The moiré bands
can then be calculated by diagonalizing the matrix form of the Hamiltonian for
a truncated basis set in $k$-space (see SM for details).

The aligned ($AA$) stacking configuration has higher energy than the partially
eclipsed ($AB$/$BA$) configuration for TMDCs and hBN homo-bilayers
($\theta\simeq 0\degree$). Consequently, relaxation tends to reduce the in-
plane area of $AA$ stacking region and to increase that of the $AB$/$BA$
stacking region, thus minimizing the total energy [16, 20]. Upon relaxation,
the large values of $\Delta_{t,b}$, at $AB$ and $BA$ stacking, expand to cover
a larger area, while the peak regions of $\Delta_{T}$, at $AA$ stacking,
shrink, with relaxed structures showing stronger electrostatic potential
effects and weaker tunneling effects, which causes a clear angle-dependent
transition of the moiré electronic structure.

Figure 2: Angle-dependent band structures and real-space localization. (a, b)
The band average energy (ave[E${}_{\textrm{n}}$]), bandwidth (color
saturation, deeper color for flatter bands), and localization (red for
$AB$/$BA$ stacking and blue for $AA$ stacking) of the three top moiré valence
bands as a function of the twist angle $\theta$ for twisted bilayers of MoTe2.
A fitted QHO model is given by the dashed lines, and the first twist angles
with bandwidths less than 10 meV are denoted with green stars. The $1\degree$
band structures are shown in the insets, with the first two QHO levels
highlighted in color. (c) LDOS of the top six moiré bands of unrelaxed bilayer
MoTe2 twisted at 0.5$\degree$. The quantum numbers in the QHO model are
denoted as n = ($n_{x},n_{y}$). (d) LDOS of the moiré bands for a 0.7$\degree$
and a 2$\degree$ twisted bilayer of MoTe2. The average LDOS in the MSL are
normalized to one in both (c) and (d).

In Fig. 2a,b we present the angle-dependent bands and real-space localization
of twisted bilayer MoTe2 as a representative case. The top-most valence bands
for both the relaxed and unrelaxed moiré systems have a bandwidth less than 10
meV when the twist angle is below $2.5\degree$. Using the Coulomb repulsion
energy ($U$) in TBG and twisted hBN as a guide [4, 11], this small kinetic
energy implies that strongly correlated states could exist for any twist angle
below a certain critical value in some TMDC materials [2, 24]. The relaxation
effects drive the small-angle ($<2\degree$) valence moiré bands from TR-type
to HC-type (Fig. 2). In the unrelaxed case, the band structure at $1\degree$
shows a single uppermost flat band and a pair of flat bands under it,
consistent with a $AA$ stacking QHO model [21, 22]. The energy levels of the
2D TR QHO model are given by $E_{n}=-\hbar\omega(n_{1}+n_{2}+1)$,
$\omega=\omega_{\theta}\theta$, and the first three levels
$-\hbar\omega$,$-2\hbar\omega$, and $-3\hbar\omega$ are accurately represented
by the computed band averages for $\theta<1\degree$ [21]. In contrast, a pair
of top moiré bands and the four lower bands at $0.7\degree$ correspond to the
first ($n=0$) and second ($n=1$) QHO states, respectively of the HC lattice
[22].

The real space localization of the tunneling-dominated (TR) QHO state is
evident in the local density of states (LDOS) of the moiré bands for the
unrelaxed condition, as shown in Fig. 2c. The electronic states localize
around the $AA$ stacking center and show s, p, and d-orbital distribution for
the first three energy levels, respectively. The six moiré bands shown
correspond to the QHO ground state, single excitation states (2-fold
degenerate), and double excitation states (3-fold degenerate).

Figure 3: Topological phase diagram for moiré bands. (a) Phase of the moiré
bands as a function of the intensity of potential and tunneling fluctuations
(1st Fourier components). The light (dark) green areas represent X (Y) type TI
phase, while the orange and light blue areas represent topologically trivial
HC and TR QHO states, respectively. The twist-angle dependence of the relaxed
band structures is shown by the solid lines for selected materials. The band
structures and Chern numbers for unrelaxed and relaxed moiré bands in
$1\degree$ twisted bilayer WSe2 are compared. (b) Berry curvature of the
uppermost band for X-type ($1\degree$) and Y-type ($1.5\degree$) twisted
bilayer WSe2 in the MBZ.

After relaxation, the competition between the two types of lattices is greatly
altered, as shown in Fig. 2d. The two top bands in the low-angle case
($\theta=0.7\degree$) localize in the $AB$/$BA$ HC potential wells to from
2-fold degenerate ground states of the HC QHO model, while the third band is
one of the 4-fold degenerate first excited states. In contrast, the
localization of the uppermost band gradually changes to the $AA$ stacking in
the high-angle case ($\theta=2\degree$) with the second band localized in the
$AB$/$BA$ region. This indicates that the energy level of the HC QHO model and
the triangular QHO model cross as the angle changes, leading to a reordering
of the bands’ localization. Therefore, the transition between the HC and TR
electronic states can be controlled by varying the twist angle in the presence
of atomic relaxations. The intermediate twist angles correspond to competing
real-space distribution and band reordering, making these materials excellent
candidates for hosting non-trivial topological properties.

To study the topological properties associated with the TR-HC transition, we
calculate the Chern numbers of the moiré bands for a generic twisted bilayer
of a hexagonal semiconductor, and use these results to generate a TI phase
diagram. The condition for the existence of topological bands is represented
by the electrostatic potential and tunneling coefficients in Fig. 3a for a
generic system, which are approximated by their first order Fourier
coefficients labeled $V$ and $w$, respectively. In this simplified model, the
topological properties can be completely described by just two parameters,
$\alpha=Vm^{*}a^{2}/\theta^{2}$ and $\beta=wm^{*}a^{2}/\theta^{2}$, with the
Hamiltonian

$H\approx\hat{K}+\hat{V}+\hat{W}=\frac{\theta^{2}}{m^{*}a^{2}}\left(\hat{K_{0}}+\alpha\hat{V_{0}}+\beta\hat{W_{0}}\right)$
(3)

where $a$ is the lattice parameter, $\hat{K}$, $\hat{V}$, $\hat{W}$ are the
kinetic, potential, and tunneling energy, respectively, and $\hat{K_{0}}$,
$\hat{V_{0}}$, $\hat{W_{0}}$ are the material-independent quantities. The
topologically trivial TR and HC phases appear in the $w$-dominated and
$V$-dominated regions of the phase space of Fig. 3a, respectively. Interesting
band structures occur in the intermediate region, which includes two types of
topological non-trivial valence moiré bands. The Chern numbers of the three
top-most moiré bands are (1, -1, 0) and (1, 1, -2) for X-type and Y-type
topological insulators, respectively. Transitions between topological and
trivial insulators for the uppermost band will occur on the boundary between
the Y-type topological phase and TR phase where the top two bands overlap and
then reopen, hosting protected edge modes at the transition. Similar
topological transitions between the second and the third bands appears in the
boundary between the X-type and Y-type topological phases. The transition on
the boundary of the X-type topological phase and the HC phase is a TI-to-
semimetal transition.

In Fig. 3a we show the curves in the phase diagram for 4 TMDC’s, indicating
transitions between phases with increasing twist angle (without relaxation
these curves would be straight lines like the one shown for WSe2). For
comparison, we include the curve for hBN, which remains in one phase (TR) as
it shows very weak relaxation. An explicit example of the topological
transitions is also shown Fig. 3a for WSe2, including the unrelaxed case.

The band structures and Chern numbers of the bands confirm that the
topological transition is primarily driven by atomic relaxation. In the
unrelaxed 1$\degree$ twisted WSe2, although the uppermost valence band is
topologically trivial, the second valence band has a nonzero Chern number. The
topological features of these lower valence bands are not reflected by the
phase diagram, but we list them in our database for each material. In
realistic experimental conditions, atomic-scale defects, bending, and local
strain introduced during fabrication cause twisted bilayers to include
different twist angles in different areas within a single sample [25].
Therefore, different domains of twist angle occur and introduce both
topological and trivial bands. Topological edge states can appear along domain
boundaries in the twist angle, according to the critical values in Tab. 1.
These edge states would not be defined by a sharp change from one crystal to
another, or from material to vacuum, but rather by slow variation in the twist
angle. For this reason, these “internal” protected edge states would be
excellent candidates for observing spin or valley-polarized states, as the
only disorder comes from twist-angle variations, which are unlikely to induce
a spin or valley swapping.

Table 1:  $\theta_{\rm HC/X},\theta_{\rm X/Y},$ and $\theta_{\rm Y/TR}$ are the critical twist angles (in degrees) for transitions of the top bands between the indicated phases. $E_{\rm gap}$ is the maximum band gap between the top and lower bands (in meV), and $\omega_{\rm top}$ is the minimum band width of the top bands in the topologically non-trivial regime (in meV); $\theta_{\rm gap}$ and $\theta_{\rm top}$ are the angles where these extrema occur. Material | $\theta_{\rm HC/X}$ | $\theta_{\rm X/Y}$ | $\theta_{\rm Y/TR}$ | $E_{\rm gap}(\theta_{\rm gap})$ | $\omega_{\rm top}(\theta_{\rm top})$
---|---|---|---|---|---
MoSe2 | 0.9 | 1.4 | 2.4 | 2.98 ($1.67\degree$) | 0.18 ($1.21\degree$)
MoTe2 | 0.7 | 1.1 | 1.5 | 1.43 ($1.20\degree$) | 0.30 ($1.10\degree$)
WSe2 | 0.8 | 1.2 | 2.1 | 3.24 ($1.40\degree$) | 0.19 ($1.15\degree$)
WTe2 | 0.7 | 1.4 | 2.4 | 3.70 ($1.68\degree$) | 0.18 ($1.31\degree$)

In Fig. 3b we show the Berry curvature of the uppermost valence band for two
values of the twist angle. In the HC phase in the small-angle region, the
first transition appears at $\theta_{\rm HC/X}=1\degree$ where the two top
bands separate at the MBZ’s $K$ points, indicated by the Berry curvature’s
concentration there. With increasing twist angle, the Berry curvature of the
top band gradually transfers to the $\Gamma$ point accompanied by an X/Y
transition at $\theta_{\rm X/Y}=1.5\degree$. During this process, the
topologically non-trivial top-most band takes on its minimal bandwith ,
$\omega_{\rm top}\simeq 0.19$ meV, and its maximal topological band gap
$E_{\rm gap}\simeq 3.24$ meV. At larger twist angle, the Berry curvature
concentrates at the $\Gamma$ point during the transition to the TR phase,
where the two top-most bands merge. The numerical results for the critical
angles, maximal gap, and minimal band width for the TMDCs with topological
valence bands are presented in Table 1. We have also verified that the bands
and real space localization of MoS2 [12], WSe2 [2], and hBN [11] from our
model are consistent with previous full DFT calculations. Twisted bilayers of
WTe2 show the highest gap and one of the lowest bandwidths, making this
material the best candidate for experimental study.

Including spin degrees of freedom, the spin-dependent moiré Hamiltonian
decomposes into two copies [14]. At the $K$ valleys of the aligned bilayer,
the two copies will be split by the spin-orbit coupling term,
$\Delta_{\textrm{SOC}}$, which for most TMDCs is on the scale of $100$ meV,
and the opposite spin-ordering will occur at the $K^{\prime}$ valley. The
$\Gamma$ and $M$ points tend to have very weak spin-splitting, leading to two
copies of the spin-independent Hamiltonian, as is the case in continuum
treatments of twisted bilayer graphene [8]). However, as the two nonequivalent
$K$ valleys are related by time reversal symmetry, the spin-up bands at $K$
valley and spin-down bands at $K^{\prime}$ valley have opposite Berry
curvature and Chern numbers. Topologically non-trivial uppermost bands with
opposite spin and Chern number give rise to the helical edge states protected
by the TR symmetries at the boundaries, and could lead to observable quantum
spin Hall (QSH) effects [26].

The origin of the TI phases has been ascribed [14] to a skyrmionic texture in
the moiré $\Delta$ potentials, which is an alternate interpretation of the
TR/HC competition we presented here but captures the same key features. As the
flatness of the bands causes stronger correlation effects, the topological
phases in twisted bilayer semiconductors could exhibit the combination of TI
and superconductivity, which has been the subject of an intense decades-long
search for fractional statistics and Majorana fermions [26].

The generality of the relaxation-induced TI phases is partially explained by
the connection between the stacking-dependence of the electronic structure and
that of the ground state energy: the lowest stacking configuration of these
materials seem to correspond to maxima of the electrostatic interaction and
minima of the tunneling strength. We note that this trend might not be true
across all 2D materials, and finding exceptions to the rule could be a
valuable endeavor.

###### Acknowledgements.

We thank Ziyan Zhu and Daniel Larson for useful discussions. This work was
supported by the National Science Foundation under grant No. OIA-1921199. The
calculations in this work were performed in part on the FAS Research Computing
cluster of Harvard University.

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