# Finding Lensed Direct-Collapse Black Holes and Supermassive Primordial Stars

Anton Vikaeus11affiliation: Observational Astrophysics, Department of Physics
and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden , Daniel
J. Whalen22affiliation: Institute of Cosmology and Gravitation, Portsmouth
University, Dennis Sciama Building, Portsmouth PO1 3FX and Erik
Zackrisson11affiliationmark:

###### Abstract

Direct-collapse black holes (DCBHs) may be the seeds of the first quasars,
over 200 of which have now been detected at $z>$ 6\. The James Webb Space
Telescope (JWST) could detect DCBHs in the near infrared (NIR) at $z\lesssim
20$ and probe the evolution of primordial quasars at their earliest stages,
but only in narrow fields that may not capture many of them. Wide-field NIR
surveys by Euclid and the Nancy Grace Roman Survey Telescope (RST) would
enclose far greater numbers of DCBHs but only directly detect them at
$z\lesssim$ 6 - 8 because of their lower sensitivities. However, their large
survey areas will cover thousands of galaxy clusters and massive galaxies that
could gravitationally lense flux from DCBHs, boosting them above current
Euclid and RST detection limits and revealing more of them than could
otherwise be detected. Here, we estimate the minimum number density of
strongly lensed DCBHs and supermassive primordial stars required for detection
in surveys by Euclid, RST and JWST at $z\lesssim$ 20\. We find that for
reasonable estimates of host halo numbers RST, Euclid, and JWST could
potentially find hundreds of strongly-lensed DCBHs at $z=$ 7 - 20. RST would
detect the most objects at $z\lesssim 10$ and JWST would find the most at
higher redshifts. Lensed supermassive primordial stars could potentially also
be found, but in fewer numbers because of their short lifetimes.

###### Subject headings:

quasars: supermassive black holes — black hole physics — early universe — dark
ages, reionization, first stars — galaxies: formation — galaxies: high-
redshift

##  1\. Introduction

Direct-collapse black holes could be the seeds of the first quasars, nine of
which have now been found at $z>7$ (Mortlock et al., 2011; Bañados et al.,
2018; Wang et al., 2021). They are thought to form when primordial haloes in
unusual environments grow to masses of $\sim$ 107 $M_{\odot}$ (Agarwal et al.,
2016; Latif et al., 2015; Hirano et al., 2017) and collapse via atomic cooling
at rates of up to $\sim$ 1 $M_{\odot}$ yr-1 (e.g., Regan & Downes, 2018;
Patrick et al., 2020). Stellar evolution models indicate that collapse leads
to supermassive stars (SMSs) that reach a few 104 \- 105 $M_{\odot}$ before
collapsing to DCBHs (Hosokawa et al., 2013; Woods et al., 2017; Haemmerlé et
al., 2018a, b; Woods et al., 2021; Herrington et al., 2021). DCBHs are thought
to be the seeds of the first quasars because it is difficult for normal Pop
III star BHs to grow rapidly after birth (Whalen et al., 2004; Whalen & Fryer,
2012; Smith et al., 2018). They are born with much larger masses and in much
higher densities in halos that retain their fuel supply even when it is heated
by X-rays (Johnson et al. 2013 – see Woods et al. 2019 for recent reviews). A
DCBH must lie at the nexus of cold accretion flows (e.g., Di Matteo et al.,
2012) or merge with other gas-rich halos capable of fueling its rapid growth
(e.g., Li et al., 2007) to become a SMBH at $z>$ 6\. Radiation hydrodynamical
simulations show that 105 $M_{\odot}$ DCBHs at $z\sim 19$ can grow to $2\times
10^{9}$ $M_{\odot}$ by $z=$ 7.1 in cold accretion flows (Smidt et al., 2018;
Latif & Khochfar, 2020; Zhu et al., 2020; Valentini et al., 2021) and that
these flows can form DCBHs without the need for exotic environments or even
atomic cooling (Latif et al., 2022).

Figure 1.— Source frame spectra for the 105 $M_{\odot}$ DCBH (left) and 1.0
$M_{\odot}$ yr-1 SMSs (right) after reprocessing by the host gas envelope.

DCBHs could be found at $z\sim$ 8 - 10 by the Square Kilometer Array and next-
generation Very Large Array (Whalen et al., 2020a), at $z\sim$ 10 by future
X-ray missions such as the Advanced Telescope for High-Energy Astrophysics
(ATHENA) and Lynx, and at $z\sim$ 20 by the James Webb Space Telescope (JWST;
Pacucci et al., 2015, 2016; Natarajan et al., 2017; Barrow et al., 2018;
Whalen et al., 2020b). NIR from their SMS progenitors could be detected at
$z\sim$ 10 by JWST (Surace et al., 2018, 2019) and gravitational waves from
DCBH mergers might be found by the Laser Interferometer Space Antenna (LISA;
Hartwig et al., 2018; Latif et al., 2020).

JWST could detect primordial quasars at their earliest stages of evolution but
only in narrow fields of view that may not capture many of them. Wide-field
surveys by Euclid and RST could enclose far greater numbers of DCBHs but only
detect them at $z\lesssim$ 6 - 8 because of their lower sensitivities (Whalen
et al., 2020b). Furthermore, because the filters in Euclid and RST are limited
to $\lambda<2\,\mu$m and $\lambda<2.3\,\mu$m, they cannot detect SMSs or DCBHs
at $z\gtrsim$ 15 and $z\gtrsim 18$ respectively, because flux blueward of the
corresponding wavelength in the rest frame of the object is resonantly
scattered and absorbed by the neutral intergalactic medium (IGM) prior to the
end of cosmological reionization.

However, Euclid and RST could still find DCBHs or SMSs at $z\gtrsim$ 10
because their wide-field surveys (several tens or thousands of square degrees,
respectively) will enclose thousands of galaxy clusters and massive galaxies
that could gravitationally lense flux from these primordial sources. Strong
lensing could offset the lower sensitivities of wide-area surveys and reveal
more objects than could otherwise be detected (and at higher redshifts). Until
now, surveys have targeted individual well-resolved cluster lenses with
extreme magnifications, $\mu$, exceeding $\sim$ 1000 (Welch et al., 2022) to
search for faint, high-redshift objects. Lensed galaxies at $z\gtrsim 6$ have
already been discovered at high magnifications (e.g., Bradley et al. 2014;
Vanzella et al. 2014; Salmon et al. 2020). Wide-field surveys could produce
more detections of high-$z$ objects because of their much larger total lensing
areas, even though their magnifications are more modest (e.g., Vikaeus et al.,
2022). Here, we estimate the number of strongly-lensed DCBHs and SMSs that
could be found in wide-field surveys by Euclid and RST and in deep JWST
surveys. In Section 2 we discuss our DCBH and SMS source-frame spectra and
statistical model for strong lensing in wide fields. We show SMS and DCBH
number counts in Euclid, RST and JWST surveys in Section 3 and conclude in
Section 4.

##  2\. Numerical Method

We convolve rest frame spectra for DCBHs and red and blue SMSs with
statistical estimates of strong lensing in wide or narrow fields to calculate
the minimum number density of objects required for just one object to appear
in a given survey over a range of redshifts.

###  2.1. DCBH / SMS Spectra

Studies of SMS evolution show that they either follow cool red tracks at
temperatures of $\sim$ 104 K if they grow at the Hayashi limit or hot, blue
tracks with temperatures above 105 K. We take spectra for 105 $M_{\odot}$
DCBHs and 105 $M_{\odot}$ red and blue SMSs growing at 1 $M_{\odot}$ yr-1 from
Whalen et al. (2020b) and Surace et al. (2018, 2019) respectively, which are
shown in Figure 1. Reprocessing of radiation from the object by the dense
cocoon of atomically-cooled gas in which it forms was included in the spectrum
calculation, and absorption by the atmosphere of the blue SMS was included in
its spectrum. Absorption by the envelope of the DCBH is evident from the
attenuation of its flux blueward of the Lyman limit and the emission features
at longer wavelengths. Similar features are visible in the spectra of both
stars. Absorption and emission features are more prominent in the spectrum of
the blue star because of its ionizing UV flux and absorption by its
atmosphere. The spectral limit of the red SMS occurs at a longer wavelength
than for the SMS because of its lower temperature, but more flux emerges from
the red star at wavelengths above 3000 Å, which has important consequences for
its detectability at higher redshifts. Likewise, the flatter spectrum of the
DCBH makes it easier to detect than SMSs at most redshifts, as will be shown
later.

Table 1Filters, areas and photometric depths for proposed JWST (NIRCam/MIRI), Euclid, and RST surveys. Filter | Area (deg2) | Depth (AB mag)
---|---|---
|  | 5$\sigma$ point-source
deep NIRCam: |  |
F200W/F356W/F444W | 0.013 | 30.6/30.1/29.8
med. NIRCam: |  |
F200W/F356W/F444W | 0.053 | 29.7/29.3/29.0
deep MIRI (F770W) | 0.0022 | 27.4
med. MIRI (F770W) | 0.0039 | 25.6
wide RST (F184/F213) | 2000 | 26.3/25
wide Euclid (H) | 15000 | 24
ultra deep NIRCam: |  |
F200W/F356W/F444W | 0.0027 | 31.8/31.3/31.0
ultra deep MIRI (F770W) | 0.0006 | 27.8
deep RST (F184/F213) | 40 | 28.5/27.2
deep Euclid (H) | 40 | 26

Figure 2.— Minimum magnification ($\mu_{\mathrm{min}}$) required for
photometric detection of a gravitationally lensed DCBH (top), blue SMS
(center) and red SMS (bottom) as a function of redshift for the different
surveys. The filter used for each object and survey is specified in table 2.
Note that the lines are only plotted for $\mu>1$. Therefore the ultradeep
NIRCam exposures are not included for the DCBH because no magnification is
required in that particular case. The two dash-dotted black lines indicate the
magnification above which an object with a given intrinsic size (1 pc and 10
pc plotted here) could become resolved in imaging.

###  2.2. Minimum Number Densities

The minimum comoving number density ($n_{\mathrm{min}}$) of DCBHs or SMSs
required to detect just one object in a randomly-oriented sky survey is

$n_{\mathrm{min}}=\frac{1}{V_{c}(A,z,\Delta z)\times
P(\geq\mu_{\mathrm{min}},z)}\frac{\Delta t(z,\Delta z)}{\tau},$ (1)

where $P(\geq\mu_{\mathrm{min}},z)$ is the probability of achieving the
minimum magnification required to boost flux from the object above the
detection threshold for the survey and is calculated from the lensing model of
Zackrisson et al. (2015, see Figure 1 therein for plots of $P(\geq\mu,z)$ for
$\mu\geq$ 10, 100 and 1000). $V_{c}$ is the comoving volume spanned by the
survey area $A$ of the telescope at a redshift $z$ with a redshift bin $\Delta
z$ corresponding to those used in dropout selection techniques when observing
with photometric filters (here, we set $\Delta z=$ 1). The effect of $\Delta
z$ on $n_{\mathrm{min}}$ is partly compensated by the ratio of the cosmic time
spanned by the survey volume, $\Delta t$, and the time for which the object
remains uniquely identifiable as a DCBH or SMS, defined here as $\tau$. For
larger choices of $\Delta z$, the corresponding cosmic time $\Delta t$ is also
greater, mitigating the dependence of $n_{\mathrm{min}}$ on $\Delta z$. We
assume $\tau=$ 40 Myr for the DCBH, about one Salpeter e-folding time
(Salpeter, 1964), after which its luminosity would begin to change
significantly. This assumption is somewhat conservative as the luminosity is
expected to increase over time. The SMSs have $\tau=$ 200,000 yr because they
are much shorter lived.

Our lensing model uses the large scale dark matter distribution of the
Millennium simulation with a semi-analytical model (Baugh et al., 2005) for
the distribution of galaxies inside dark matter halos. From this distribution
the lensing potential for a sequence of lens planes along different lines of
sight can be calculated. Magnifications for randomly oriented sightlines are
then calculated as a function of redshift for the source to produce the
magnification probability function $P(\geq\mu,z)$. Meneghetti et al. (2020)
argued that failing to consider substructures in galaxy clusters could lead to
underestimates of the likelihood of strong gravitational lensing. Since our
dark matter distributions do not resolve these substructures, our estimates of
DCBH and SMS detections should be considered to be conservative.

As constructed in Equation 1, $n_{min}$ establishes the requirements on
minimum object number density for a 1$\sigma$ detection such that any
randomly-oriented sky survey will find at least one object in the survey
volume with a probability of $P_{\mathrm{det}}\sim 0.68$. This constraint is
set by $P_{\mathrm{det}}=1-[1-P(>\mu\mathrm{{}_{min}},z)]^{N}$, where $N$ is
the number of SMSs or DCBHs within the survey area with an age that does not
exceed their respective $\tau$. Requiring a specific probability
$P_{\mathrm{det}}$ therefore sets a relation between $N$ and $P(\geq\mu,z)$
that can be varied by requiring, for example, a 2$\sigma$ detection
($P_{\mathrm{det}}\sim 0.95$). This increases the required number of objects
inside the survey volume by a factor of 3 compared to the 1$\sigma$ detection
for any given $P(\geq\mu,z)$. Here we consider 1$\sigma$ detections to set a
lower limit on $n_{min}$.

###  2.3. Resolved/Unresolved Imaging

The DCBH would appear as a point source given the magnifications necessary to
detect it in the surveys here because we assume its spectrum is dominated by
its accretion disk and that nebular emission is negligible. The intrinsic size
of the DCBH is then tens of AU, not parsecs, which could be the case for the
nebula. The top panel of Figure 2 shows the magnification required to detect a
105 $M_{\odot}$ DCBH. The black dash-dotted lines indicate the magnifications
above which an object with an intrinsic size of 1 pc or 10 pc could become
resolved in imaging. Here, we make the assumption that the stretching of the
image due to lensing is only in one direction. In reality the object will also
be stretched by a non-negligible amount in other directions, which would allow
for higher magnifications, making this a conservative estimate. If there is
significant nebular emission, Figure 2 shows that, e.g., the deep RST survey
would only detect DCBHs that are resolved in imaging at $z\gtrsim 14.5$ if
they are 10 pc sources or $z\gtrsim 16.5$ if they are 1 pc sources. Note that
a DCBH or SMS with an intrinsic size of $\sim 10$ AU would remain point-like
(unresolved) in all surveys since $\mu\gtrsim 4\times 10^{6}$ would be
required throughout the redshift range to resolve such a small object.

Similar arguments apply to SMSs, which can also be taken to be point sources
if their intrinsic size is on the order of AU. Nebular emission in blue SMS
spectra can produce a larger object that could be resolved at higher
magnifications depending on the size of the nebula (center and bottom panels
of Figure 2). If DCBHs and SMSs are indeed tens of AU in size, their point-
like appearance even at high magnifications enables their detection at very
high redshifts. We find that JWST NIRCam imaging never requires significant
magnifications and would therefore be able to detect DCBHs and SMSs as point-
like sources throughout the redshift range, even for objects with intrinsic
sizes on the order of parsecs. When objects become resolved, additional
modelling of the sources on smaller scales is required to determine if the
detection is a DCBH or an SMS.

Table 2Photometric filters providing the lowest minimum number densities for the different surveys and their corresponding redshift ranges. JWST filters are selected from the surveys in Rieke et al. (2019). Object | Survey | Filter | Redshifts
---|---|---|---
DCBH | Euclid | H |  $z=$ 7 - 15
DCBH | RST | F184 |  $z=$ 7 - 13.8
DCBH | RST | F213 |  $z=$ 13.8 - 18
DCBH | NIRcam | F444W |  $z=$ 7 - 20
DCBH | MIRI | F770W |  $z=$ 7 - 20
BSMS | Euclid | H |  $z=$ 7 - 15
BSMS | RST | F184 |  $z=$ 7 - 14.5
BSMS | RST | F213 |  $z=$ 14.5 - 18
BSMS | NIRcam UD | F200W |  $z=$ 7 - 16.4
BSMS | NIRcam UD | F356W |  $z=$ 16.4 - 20
BSMS | NIRcam D | F356W |  $z=$ 7 - 9.5
BSMS | NIRcam D | F200W |  $z=$ 9.5 - 16.4
BSMS | NIRcam D | F356W |  $z=$16.4 - 20
BSMS | NIRcam M | F356W |  $z=$ 7 - 9.7
BSMS | NIRcam M | F200W |  $z=$ 9.7 - 16.3
BSMS | NIRcam M | F356W |  $z=$ 16.3 - 20
BSMS | MIRI | F770W |  $z=$ 7 - 20
RSMS | Euclid | H |  $z=$ 7 - 15
RSMS | RST | F184 |  $z=$ 7 - 13.4
RSMS | RST | F213 |  $z=$ 13.4 - 18
RSMS | NIRcam | F444W |  $z=$ 7 - 20
RSMS | MIRI | F770W |  $z=$ 7 - 20
Note: UD = ultra deep, D = deep, M = medium,
BSMS = blue SMS, RSMS = red SMS.

Figure 3.— Minimum number densities $n_{\mathrm{min}}$ required for
photometric detection of one single gravitationally lensed DCBH (top), blue
SMS (center), and red SMS (bottom) as a function of redshift. The gray shaded
area (extending down to $10^{-9}$ Mpc-3 in all figures) shows the expected
range of number densities of halos that can host SMSs and DCBHs from
simulations (Habouzit et al., 2016; Valiante et al., 2017).

###  2.4. Survey Parameters

JWST NIRcam and MIRI filters range from 0.6 - 5 $\mu$m and 5.6 - 25.5 $\mu$m
and Euclid covers 0.9 - 2 $\mu$m and RST covers 0.5 - 2.3 $\mu$m. As noted
earlier, these filters allow JWST in principle to detect SMSs and DCBHs out to
$z\sim$ 20 but restrict detections by Euclid and RST to at most $z\sim$ 15 and
$z\sim$ 18 respectively. Point-source $5\sigma$ photometric detection
thresholds in proposed JWST, RST and Euclid surveys and their filters are
shown in Table 1 (Rieke et al., 2019; Akeson et al., 2019; Laureijs et al.,
2011). To maximize detections over the range of redshifts in our study, we
consider two filters for RST since an object at higher redshifts (e.g., at
$z\gtrsim 13.8$ for DCBHs) may be brighter in the F213 filter than the F184
filter (at $z\gtrsim 15$ the flux entering the F184 filter originates blueward
of Ly$\alpha$ and is thus attenuated by the neutral IGM). We likewise consider
four JWST filters to optimize detections over specific ranges in redshift. The
redshift ranges for which the filters in each survey were used to minimize
$n_{\mathrm{min}}$ for each type of object are summarized in Table 2. Note
that in most of the cases the redshift ranges for the optimum filter does not
depend on the depth of the survey. In this paper we consider the JWST surveys
summarized in Rieke et al. (2019), in which the filters that are available for
the designated survey depth and areal coverage are limited. Because of the
photometric depth of JWST, DCBHs can be detected in several of its filters
without gravitational lensing up to very high redshifts in ultra deep fields.
When gravitational lensing is not required, the survey area alone determines
the minimum required number density.

##  3\. Results

###  3.1. DCBHs

We show the minimum required number density ($n_{\mathrm{min}}$) for one
single detection from $z=$ 7 - 20 for a 105 $M_{\odot}$ DCBH in the top panel
of Figure 3. The deep field survey with RST will detect more strongly-lensed
DCBHs than its wide survey at $z\lesssim 12$, above which the wide RST survey
will perform equally well. The wide Euclid survey will detect more DCBHs than
its deep survey at all redshifts. The deep RST survey performs better than all
other surveys at $z\lesssim 10.5$. However, JWST medium NIRcam imaging would
find the most DCBHs at $10.5\lesssim z\lesssim 13.2$ and deep NIRCam imaging
would detect the most at $13.2\lesssim z\lesssim 17.5$. Single, ultra deep
NIRCam pointings with 100 hr exposures will detect the most DCBHs at $z\gtrsim
17.5$. We note that $n_{\mathrm{min}}$ for ultra deep NIRCam imaging varies
with redshift only with $V_{c}$ and $\Delta t$ because DCBHs are visible in
these exposures without gravitational lensing. In contrast, lensing is
required for all detections by wide and deep surveys with RST and Euclid, even
at lower redshifts. In these surveys, $n_{\mathrm{min}}$ rises rapidly at
$z\sim$ 14 - 15 because of absorption of DCBH flux at corresponding
wavelengths ($\lambda\leq\lambda_{\mathrm{Ly\alpha}}$) in the source frame by
the neutral IGM, as discussed earlier. Although the NIRcam plots in Figure 3
are for the F444W filter, the other NIRCam filters yield similar numbers of
detections because the DCBH spectrum is relatively flat (see Figure 1).

We overlay the range of number densities of halos expected from simulations
(defined here as $n_{\mathrm{sim}}$) to host DCBHs as the gray regions in
Figure 3 (Habouzit et al., 2016; Valiante et al., 2017). They vary by many
orders of magnitude with redshift because of uncertainties in Lyman-Werner and
UV self-shielding in the halos but have a lower limit of $\sim$ $10^{-9}$
Mpc-3 that is set by the number of observed quasars at $z\sim 6$. For none of
the surveys to detect a DCBH the number density of host halos would have to be
less than $\sim 10^{-7}$ Mpc-3 at $z\sim 7$ and $\sim 10^{-4.5}$ Mpc-3 at
$z\sim 20$, which are fairly severe limits. As $n_{\mathrm{min}}$ sets the
required minimum number density for one single detection, the ratio
$n_{\mathrm{sim}}/n_{\mathrm{min}}$ yields the number of objects expected to
be detected at a given redshift. In the optimistic limit this indicates that
these surveys could detect several hundred to several thousand lensed DCBHs
per unit redshift at lower $z$ and up to ten even at the highest redshifts –
but these predictions are very uncertain due to the large range of
$n_{\mathrm{sim}}$ at any given redshift. Note however that the regions marked
by gray in Figure 3 may somewhat underestimate the number of DCBHs expected to
be present at a given redshift because the most recent simulations of the
formation of atomically-cooled halos have shown that 1 - 5 DCBHs can form per
halo (e.g., Patrick et al., 2020).

###  3.2. SMSs

The $n_{\mathrm{min}}$ for the blue and red SMSs are shown as a function of
redshift in the center (blue SMS) and bottom (red SMS) panel of Figure 3. SMSs
are intrinsically brighter than the DBCH at some redshifts in some of the
filters, therefore one might naively expect that their $n_{\mathrm{min}}$
could be less than those for DCBHs. However, their shorter lifetimes result in
minimum required number densities that are orders of magnitude greater than
those for DCBHs and would be detected at most to $z\sim$ 14, even by ultradeep
NIRcam exposures. The breaks in some of the lines in the center panel of
Figure 3 are due to switching to the optimum filter at those redshifts and to
significant variations in the brightness of the blue SMS spectrum at certain
wavelengths. The troughs in the MIRI plots at $z\sim$ 8 - 13 in the center
panel are due to strong Balmer emission lines from the blue star. The MIRI
F770W filter probes the rest frame spectrum between $5000-9000$ Åwhich
contains these lines. This brightens the star and enhances the prospects for
detections over this redshift range by MIRI.

If we take into account the numbers of SMS host halos per redshift predicted
by simulations in gray in Figure 3, deep NIRcam imaging would detect the most
blue SMSs at $z=$ 7 - 8 and wide RST exposures would find the most at $z=$ 8 -
13.5, followed closely by ultradeep NIRcam pointings. The wide Euclid survey
is the only other one that would detect blue lensed stars, at $z=$ 10.5 -
12.5. Medium NIRcam imaging would find the most red SMSs at $z=$ 7 - 10.2 and
deep NIRcam exposures would find the most at $z=$ 10.2 - 13.5. Over both
ranges in redshift F444W is the optimum NIRCam filter. We again note that the
regions in gray in Figure 3 somewhat underestimate the number of SMSs expected
to exist per unit redshift because more than one star can form per halo.
Unlike DCBHs, we find that SMSs generally must be gravitationally lensed at
higher redshifts in order to be detected, even by ultradeep NIRCam imaging
(see fig. 2).

##  4\. Conclusion

Our calculations show that RST, Euclid, and JWST would detect far more lensed
DCBHs than lensed SMSs, primarily because the stars have such short lifetimes.
Indeed, the best way to constrain SMS numbers at high redshifts is to detect
their DCBHs since SMS detections could effectively undercount their true
numbers by orders of magnitude. Considering the surveys included in this
paper, RST will detect the most lensed DBCHs for $z\lesssim 10$ and JWST will
find the most at $z\sim$ 10 - 20. These surveys will find up to a few tens
more red SMSs than blue ones because more of their flux is redshifted into the
NIR today. In principle, RST, Euclid, and JWST may find no DCBHs if the number
density of source halos is small enough at high redshifts. However, the
failure to detect these objects would impose severe constraints on DCBH number
densities, less than 10-6 Mpc-3 from $z=$ 7 - 20. For reasonable estimates of
progenitor halo numbers, RST, Euclid, and JWST could find hundreds or even
thousands of strongly lensed DCBHs at $z=$ 7 - 20 in the coming decade.

AV and EZ acknowledge funding from the Swedish National Space Board.

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# Effective spin-1 breathing kagome Hamiltonian induced by the exchange
hierarchy in the maple leaf mineral bluebellite

Pratyay Ghosh  Institut für Theoretische Physik und Astrophysik and Würzburg-
Dresden Cluster of Excellence ct.qmat, Universität Würzburg, Am Hubland Campus
Süd, Würzburg 97074, Germany  Tobias Müller  Institut für Theoretische Physik
und Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat, Julius-
Maximilians-Universität Würzburg, Am Hubland Campus Süd, Würzburg 97074,
Germany  Yasir Iqbal  Department of Physics and Quantum Centers in Diamond and
Emerging Materials (QuCenDiEM) group, Indian Institute of Technology Madras,
Chennai 600036, India  Ronny Thomale  Institut für Theoretische Physik und
Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat, Julius-
Maximilians-Universität Würzburg, Am Hubland Campus Süd, Würzburg 97074,
Germany  Department of Physics and Quantum Centers in Diamond and Emerging
Materials (QuCenDiEM) group, Indian Institute of Technology Madras, Chennai
600036, India  Harald O. Jeschke  Research Institute for Interdisciplinary
Science, Okayama University, Okayama 700-8530, Japan  Department of Physics
and Quantum Centers in Diamond and Emerging Materials (QuCenDiEM) group,
Indian Institute of Technology Madras, Chennai 600036, India

###### Abstract

As a highly frustrated model Hamiltonian with an exact dimer ground state, the
Heisenberg antiferromagnet on the maple leaf lattice is of high theoretical
interest, and a material realization is intensely sought after. We determine
the magnetic Hamiltonian of the copper mineral bluebellite using density
functional theory based energy mapping. As a consequence of the significant
distortion of the spin $S=1/2$ maple leaf lattice, we find two of the five
distinct nearest neighbor couplings to be ferromagnetic. Solution of this
Hamiltonian with density matrix renormalization group calculations points us
to the surprising insight that this particular imperfect maple leaf lattice,
due to the strongly ferromagnetic Cu2+ dimer, realizes an effective $S=1$
breathing kagome Hamiltonian. In fact, this is another highly interesting
Hamiltonian which has rarely been realized in materials. Analysis of the
effective model within a bond-operator formalism allows us to identify a
valence bond solid ground state and to extract thermodynamic quantities using
a low-energy bosonic mean-field theory. We resolve the puzzle of the apparent
one-dimensional character of bluebellite as our calculated specific heat has a
Bonner-Fisher-like shape, in good agreement with experiment.

Triangular motifs in quantum antiferromagnets are a source of geometric
frustration and lead to highly nontrivial emergent phenomena like quantum spin
liquids [1]. Starting with the triangular lattice, site depletion leads to new
lattices which have lower coordination number but potentially more frustration
[2, 3]. For example, the kagome lattice is obtained by a $1/4$ site-depletion
of the triangular lattice and has coordination number of four; it also hosts
some of the most intensively studied spin liquid candidates [4]. In a somewhat
more exotic manner, the maple leaf lattice can be viewed as a one-seventh
site-depleted triangular lattice and has a coordination number of five [5].
The uniform nearest-neighbor Heisenberg antiferromagnetic model on this
lattice has been solved via exact diagonalization and other techniques [6, 7]
and found a magnetically ordered ground state. Recently, in an analytic work
on the model with a bond anisotropy, Ghosh _et. al._ established an exact
dimer ground state [8], making the maple leaf lattice the only other two-
dimensional lattice with uniform tiling that admits an exact dimer ground
state besides the widely known Shastry-Sutherland model [9] which has an
extremely rich phenomenology and phase diagram [10, 11, 12, 13]. While
SrCu2(BO3)2 has been found to be an extremely good representation of the
Shastry-Sutherland Hamiltonian, a material realizing the model proposed by
Ghosh _et. al._ on the maple leaf lattice has yet to be identified. Candidates
involving quantum spins are the copper minerals [14, 15] spangolite
Cu6Al(SO4)(OH)12Cl.3H2O [16], sabelliite Cu2ZnAsO4(OH)3 [17], mojaveite
Cu6TeO4(OH)9Cl [18], fuetterite Pb3Cu6TeO6(OH)7Cl5 [19] and finally
bluebellite Cu6IO3(OH)10Cl [18]. Magnetic properties have been characterized
experimentally for spangolite [20] and bluebellite [21], but the magnetic
Hamiltonians for any of these maple leaf compounds remains to be established.

Figure 1: (a) DFT based energy mapping: First eight exchange interactions of
bluebellite as function of on-site interaction strength $U$, at fixed $J_{\rm
H}=1$ eV. The vertical line indicates the $U$ value at which the Heisenberg
Hamiltonian parameters yield the experimentally observed Curie-Weiss
temperature [21]. (b) bluebellite structure with the five ”nearest neighbor”
exchange paths defining the maple leaf lattice. The lattice vectors are given
by $\mathbf{a}_{1}=\sqrt{7}/2(\hat{x}+\sqrt{3}\hat{y})$ and
$\mathbf{a}_{2}=\sqrt{7}\hat{x}$. (c) Effective $S=1$ breathing kagome model
with renormalized interactions. (d) Spin-spin correlations for all nearest-
neighbor bonds obtained from DMRG on a 108 site maple-leaf cluster. The
thickness of the bonds indicates the strength of the correlation and the color
red (blue) indicates positive (negative) correlation. Note the clear
dimerization in the ground state. (e) Enlargement from (d) with values of
spin-spin correlations.

Here, we focus on bluebellite and resolve the most pressing issues for this
prime example of a maple leaf antiferromagnet: we determine all relevant
exchange interactions and solve the resulting Hamiltonian employing numerical
and semi-analytical techniques. In particular, we address the question raised
by experiment: why do the susceptibility and specific heat of bluebellite
appear to have a Bonner-Fisher type shape, suggestive of one-dimensional
systems? An answer to this question based on order-by-disorder was attempted
without knowledge of the Hamiltonian [22]. Methodologically, we apply the
energy mapping technique which has proved valuable in extracting the
Hamiltonian couplings for many important quantum spin systems [23, 24, 25,
26]. By virtue of a statistical approach and by extracting more than the
apparently important exchange interactions, this method has led to surprising
insights and met with much success for many materials [27, 28, 29, 30, 31, 32,
33]. We then perform density matrix renormalization group (DMRG) calculations
[34] on the resulting maple leaf Hamiltonian. This method has been
instrumental in furthering the comprehension of the physics of the kagome
lattice antiferromagnet [35, 36]. Our DMRG calculations show that bluebellite
has a gapped valence bond solid ground state. In order to deepen our
understanding, we develop a low-energy theory by implementing the standard
bond operator formalism [37]. This theory permits one to perform calculations
directly in the thermodynamic limit, obtain static and dynamical structure
factors, and assess the behavior of thermodynamic quantities. The analytical
and numerical calculations reveal that the bluebellite magnetic interactions
very closely emulate an effective $S=1$ kagome Hamiltonian with a strong
breathing anisotropy. So far, the possible $S=1$ kagome candidates, e.g.,
KV3Ge2O9 [38], NaV6O11 [39], $\mathrm{m\mbox{--}MPYNN\cdot BF_{4}}$ [40], all
undergo lattice distortions at low temperatures. By establishing the
connection between maple leaf and kagome, our work paves the pathway to
possible realizations and synthesis of new effective $S=1$ kagome compounds
emerging out of $S=1/2$ maple-leaf systems. As an experimental outlook, this
enables the study of integer-spin highly frustrated kagome antiferromagnets,
notably magnetization plateaus, excitations, and topological properties.

Model Hamiltonian.- Before we could extract the parameters of the Heisenberg
Hamiltonian $H=\sum_{i<j}J_{ij}\mathbf{\hat{S}}_{i}\cdot\mathbf{\hat{S}}_{j}$
for bluebellite, we had to perform a relaxation of the internal positions of
H, O and Cl while keeping Cu and I positions, as well as the lattice
parameters fixed; this is necessary to sort out some obvious distortions in
the experimentally determined room temperature structure [21, 41]. Note, that
the maple leaf lattice in the structure of synthetic bluebellite is much more
regular compared to the structure of the mineral [18]. In applying the energy
mapping technique, we make no assumptions as to which are the important
exchange paths. Rather, we determine 20 couplings up to a Cu-Cu distance of
5.9 Å which is about twice the nearest neighbor distance. In Fig. 1 (a), we
show how the first eight Heisenberg Hamiltonian parameters evolve with on-site
Coulomb interaction $U$ applied to the strongly correlated Cu2+ $3d$ orbitals.
The relevant $U$ value is determined by calculating the Curie-Weiss
temperature as $\theta_{\rm CW}=-\frac{1}{3}S(S+1)\sum_{i}z_{i}J_{i}$ where
$z_{i}$ are the coordination numbers of $J_{i}$ and demanding that it matches
the experimentally observed value $\theta_{\rm CW}=-34.7$ K [21]. The result
for the five maple leaf couplings is $J_{1}=-120.8(1.3)$ K, $J_{2}=88.6(1.2)$
K, $J_{3}=-93.7(1.0)$ K, $J_{4}=147.6(1.3)$ K, $J_{5}=61.3(7)$ K. At first
glance, it is disappointing that two of the five couplings are ferromagnetic,
making this Hamiltonian very distant from the ideal maple leaf antiferromagnet
[8]. But the mixed ferro- and antiferromagnetic (FM and AFM) bluebellite
Hamiltonian may attain very attractive properties. First of all, the strongest
AFM $J_{4}$ would try to enforce a $120^{\circ}$ order on the $J_{4}$
triangles. However, it might not be sufficient to establish a full magnetic
order on the system, as the $J_{5}$ bonds, which forms the other set of
triangles on the lattice, is the weakest. On the other hand, the second
strongest AFM $J_{2}$ bonds form hexagons of alternating interactions with the
FM $J_{3}$ bonds, which might promote a dimerized singlet state on the
hexagons. Finally, the strong FM $J_{1}$ bond might try to project an
effective spin-$1$ on them and reduce the quantum fluctuations of the system.
The complex interplay of all these effects makes the properties of bluebellite
extremely intricate and intriguing, which we attempt to understand in the rest
of this communication.

Ground State.- First, we study the ground state of bluebellite with DMRG by
using the iTENSOR library [42]. On three distinct lattice sizes—$48$, $108$,
and $192$ site clusters—we conduct $24$–$30$ sweeps while maintaining a
maximum bond dimension of $2048$ [41]. Based on finite-size-scaling, we reveal
a magnetically disordered ground state with an energy per site
$E_{0}/\widetilde{J}=-0.217(1)$, which is gapped by
$\Delta/\widetilde{J}=0.106(2)$
($\widetilde{J}=\sqrt{J_{1}^{2}+J_{2}^{2}+J_{3}^{2}+J_{4}^{2}+J_{5}^{2}}=238\text{
K}$ set as the energy scale) [41]. We show the spin-spin correlations,
$\langle\vec{S}_{i}\cdot\vec{S}_{j}\rangle$, for all nearest-neighbor spin
pairs, $\vec{S}_{i}$ and $\vec{S}_{j}$, in Fig. 1 (d), which shows strong
singlet formation in the spin pairs across the AFM $J_{2}$ bonds. This
dimerization propensity on the $J_{2}$ bonds is indicative of a valence bond
solid (VBS) state.

For such a dimerized system, further critical insights can be obtained from
the bond operator formalism [37], which provides a suitable way to construct
an effective low-energy bosonic theory in the thermodynamic limit. In bond
operator representation, one transforms the spin basis to a dimer basis by
writing the singlet and three triplet states on a bond as bosons (deemed as
bond operators). In our analysis, the $J_{2}$ bonds define the dimers. We
assume a singlet background (a product state of singlets on the $J_{2}$ bonds)
as a mean-field, and the triplon (dispersing triplet) excitations on the
singlet are treated systematically while ignoring any triplon-triplon
interaction (details in the Supplemental Material [41] which contains
additional Refs. [43, 44, 45]). This method has been successful in describing
the ground state as well as thermodynamic properties of several magnetic
materials [46, 47, 48, 49, 50]. The results from this bosonic theory agree
well with the DMRG calculations. It finds a stable VBS ground state with
energy $E_{0}/\widetilde{J}=-0.22494$ and a spin gap of
$\Delta/\widetilde{J}=0.13474$. The spin-spin correlations on all the bonds
also agree with the DMRG results; the strongest, on the $J_{2}$ bonds, for
example, is found to be $-0.57570$ (compare with Fig. 1 (e)).

Apart from the strong singlets forming on the AFM $J_{2}$ bonds, the spins
connected by FM $J_{1}$ interactions develop strong ferromagnetic
correlations. The total spin moment on these FM bonds is estimated to be
approximately $1.8$ from both methods, signalling the system’s tendency to
satisfy the strongest FM bond $J_{1}$ by forming triplets so that the $S=1/2$
astride the $J_{1}$ bonds would project onto an effective $S=1$ (for a full
$S=1$ projection, the total spin moment $S(S+1)=2$ on a bond). Thus, we
discover a simple but elegant property of the bluebellite Heisenberg
Hamiltonian – it emerges as an effective $S=1$ kagome system (see Fig. 1 (c)).
This effective kagome lattice Hamiltonian will have two distinct first
neighbor interactions for smaller triangles $J_{\rm eff}\propto J_{2}+J_{5}$,
and for slightly larger triangles $J^{\prime}_{\rm eff}\propto J_{3}+J_{4}$,
leading to a breathing anisotropy (see Fig. 1 (b) and (e)) [51]. We have
precisely evaluated the interactions in the effective $S=1$ breathing kagome
lattice by repeating the energy mapping with $S=1/2$ moments connected by
$J_{1}$ bonds constrained to $S=1$ (for details, see Ref. [41]). We find
antiferromagnetic couplings $J_{\rm eff}=49(2)$ K and $J^{\prime}_{\rm
eff}=18(2)$ K, and the breathing anisotropy is $J_{\rm eff}/J^{\prime}_{\rm
eff}\approx 2.7$ (Fig. 1 (c)).

An analysis of the $S=1$ breathing kagome AFM due to Ghosh et. al. revealed
that the ground state of such a system is a trimerized singlet [52]. To
understand this ground state one can start with isolated $J_{\rm eff}$
triangles (see Fig. 1 (c)), i.e., set $J^{\prime}_{\rm eff}$ to zero. In this
limit, the ground state is a product of pure singlets on each _down_ triangle.
When the $J^{\prime}_{\rm eff}$ is turned on, triplet fluctuations develop on
this singlet, which for $J^{\prime}_{\rm eff}\leq J_{\rm eff}$ remain
insufficient to conjure a phase transition, forming a stable trimerized
singlet ground state for the system [52, 36, 53]. In the present case, a
similar ‘trimerized’ state is also realized, which translates as a dimerized
state. If one tunes $J_{1}\rightarrow\infty$, this dimerized state on the
maple-leaf lattice should continuously evolve to the trimerized singlet state
of $S=1$ breathing kagome. The trimerized state here is akin to the valence-
bond solid state for the AFM spin-$1$ chain due to Affleck, Kennedy, Lieb and
Tasaki [54]. In both cases, one sees the spin-$1$ as a combination of two
spin-$1/2$, tries to satisfy the AFM interactions locally, and then projects a
spin-$1$ out of two spin-$1/2$. Also note that for an AFM spin-$1$ chain with
coupling equal to $J_{2}$, the spin-gap is $0.15$ [55], which is slightly
larger compared to the spin gap we obtain. A similar physics was seen for
$S=1$ kagome in Ref. [49].

Figure 2: Static spin structure factor, $S_{\mathbf{q}}$, (a) obtained from
DMRG on a 192 site maple-leaf cluster, (b) and (e) obtained from the bond-
operator mean-field theory at the thermodynamic limit. (c) The
$S_{\mathbf{q}}$ for the maple-leaf lattice obtained by doing a basis
expansion of the $\mathbf{q}=(0,0)$ order on the classical kagome system. We
also introduce a small canting between the spins on the $J_{1}$ bonds (see the
bottom panel). We find Bragg peaks on the $M$ points of the fourth BZ of the
system. (d) The $S_{\mathbf{q}}$ for product state of pure singlets forming on
the $J_{2}$ bonds. The maxima shifts significantly away from the $M$ points.
Using the notation same as Fig. 1 (d), the bottom panels of both (c) and (d)
depicts the NN spin-spin correlations for corresponding states.

Static and Dynamical Structure Factors.- We first calculate the static
structure factor,
$S(\mathbf{q})=\frac{1}{N}\sum_{ij}e^{\imath\mathbf{q}\cdot(\mathbf{r}_{i}-\mathbf{r}_{j})}\langle\mathbf{\hat{S}}_{i}\cdot\mathbf{\hat{S}}_{j}\rangle$,
using DMRG and bosonic theory. The DMRG result (Fig. 2 (a)), shows diffused
peaks at the $M$ points of the fourth Brillouin zone (BZ). In contrast, the
peaks from the low-energy bosonic theory (Fig. 2 (b)) are significantly
shifted from the $M$ points. To understand this further, we exploit the
relation between the kagome lattice and the maple-leaf lattice (see Fig. 1
(c)), to transform the static spin structure factor of kagome $\mathbf{q}=0$
order into the $S_{\mathbf{q}}$ for a maple-leaf system (see Ref. [41]) and
find Bragg peaks at the $M$ points (see Fig. 2 (c)). We also introduce a
relative spin canting across $J_{1}$ bonds (see Fig. 2 (c)) and find it to be
incapable of changing the Bragg peak positions. Therefore, a shift in the
peaks in Fig. 2 (b) must have a quantum mechanical origin. To confirm this, we
assume a state which is a product state of pure singlets on the $J_{2}$ bonds.
Note that this state is similar to the mean-field wavefunction for our bosonic
theory. This state reproduces the shift in question (see Fig. 2 (d)). Thus, we
infer that the shift in the maximum of $S_{\mathbf{q}}$ away from the $M$
points is due to the stabilization of the spin-singlets on the $J_{2}$ bonds.
The other modifications seen in Fig. 2 (b) are ascribed to triplet
fluctuations.

It is apparent from Fig. 2 (b), that a shift in the peak results in an
enlargement of the BZ of the $S_{\mathbf{q}}$. A form factor calculation
reveals that by traversing along either of the reciprocal vectors of the
lattice, the structure factor is only periodic in seven reciprocal lattice
spacing. Thus, the extended BZ of $S_{\mathbf{q}}$ is seven times larger than
the actual BZ of the lattice (see Fig. 2 (e)). This significantly enhanced
extended BZ is the reason behind the finite-size DMRG and bond-operator mean-
field theory in the thermodynamic limit not agreeing in the calculation of
$S(\mathbf{q})$.

We further calculate the powder averaged static spin structure factor,
$S(Q)=\frac{1}{4\pi}\int\mathrm{d}\Omega_{\hat{\mathbf{q}}}S(\mathbf{q})$,
with $Q=|\mathbf{q}|$, from both DMRG and the bond operator mean-field theory.
We show the magnetic form factor, $F(Q)$, modulated $S(Q)$ in Fig. 3 (a).
Apart from the static spin structure factor, we also calculate the dynamical
spin structure factor,
$S(\mathbf{q},\omega)=\frac{1}{N}\sum_{ij}e^{\imath\mathbf{q}\cdot(\mathbf{r}_{i}-\mathbf{r}_{j})}\int_{-\infty}^{\infty}\mathrm{d}te^{\imath\omega
t}\langle\mathbf{S}_{j}(t)\cdot\mathbf{S}_{i}(0)\rangle$, for this system
using the bosonic mean-field theory. We use the extended BZ of the system and
plot $S(\mathbf{q},\omega)$ along the high-symmetry lines in Fig. 3 (b).

Figure 3: (a) Form factor modulated powder averaged static structure factor.
(b) Dynamical spin structure factor calculated via bond-operator mean-field
theory. (c) Magnetic specific heat $C_{\rm mag}$ obtained from bond-operator
mean-field theory. Inset: Magnetic entropy $S_{\rm mag}$.

Thermodynamic properties.- Capturing the finite temperature behavior precisely
down to low temperature is a hard problem, but we can obtain qualitative
information from our bond operator theory. We calculate specific heat using
the bosonic mean-field theory (see [41]) and present it in Fig. 3 (c) together
with the experimental result for the magnetic contribution to specific heat
[21]. It shows a broad peak around $22$ K and matches the overall behavior of
the experimental data. The specific heat shape that has been interpreted in
terms of the Bonner-Fisher fit for the 1D Heisenberg chain [21] is captured
well by our entangled 2D wavefunction. This may be rationalized by the fact
[56] that zero-, one- and two-dimensional magnets can all have the same
overall shape of specific heat, and magnetic susceptibility. The magnetic
entropy calculation further validates effective spin-$1$ behavior of the
system by approaching $\frac{1}{2}\ln{3}$ at high temperature (the factor of
$1/2$ is due to two spin-$1/2$ combining to form a spin $1$), instead of
$\ln{2}$ which would be the case for a fully AFM system. The residual entropy
comes from the interlocking of the spins across the $J_{1}$ bonds at low
temperature.

However, the sharp peak seen in the experiment at $17$ K, which was identified
as the onset of a magnetic order [21], is not seen in our computation, and we
find a magnetically disordered ground state instead. The appearance of a
magnetic order is indicative of substantial structural changes in bluebellite
at low temperatures. However, we believe that this anomalous behavior can also
be attributed to the thermal activation of other excited states which live
above the triplet excitations [41].

Conclusions.- We have determined the Heisenberg Hamiltonian for bluebellite by
DFT energy mapping and found that two of the five couplings in the slightly
distorted maple leaf lattice are ferromagnetic. Based on the DMRG result that
the second largest antiferromagnetic exchange leads to very strong
antiferromagnetic spin correlations, we have developed a bond operator mean
field theory for bluebellite which gives us access to thermodynamic
properties. Our calculated specific heat does not show a sharp ordering peak
as the ground state for the Hamiltonian parameters determined for the room
temperature crystal structure is magnetically disordered. However, there is
excellent agreement in the overall shape of the specific heat, and we obtain
the apparent Bonner-Fisher curve that was interpreted in terms of 1D physics
[21] from our fully two-dimensional maple leaf Hamiltonian. Furthermore,
focusing on a pattern of strong ferromagnetic spin correlations in both DMRG
and bond operator theory, we find that bluebellite realizes an effective
spin-1 breathing kagome lattice. We predict both static and dynamic spin
structure factors and show how they can be understood based on the classically
expected $\mathbf{q}=\mathbf{0}$ order of the effective kagome model. We would
like to point out that the maple leaf Hamiltonian and the effective spin-1
kagome model are determined from a room temperature crystal structure; they
are both expected to show a 1/3 magnetization plateau which was not observed
in the $T=4.2$ K magnetization process [21]. Therefore, it will be very
interesting future work to determine a low temperature crystal structure of
bluebellite experimentally and to study its Hamiltonian.

###### Acknowledgements.

We thank Y. Haraguchi for allowing us to replot the magnetic specific heat.
The work in Würzburg is supported by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) through Project-ID 258499086-SFB 1170 and the
Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum
Matter – ct.qmat Project-ID 390858490-EXC 2147. R. T. and H. O. J. thank IIT
Madras for a visiting faculty fellow position under the IoE program which
facilitated the completion of this research work. Y.I. acknowledges financial
support by the Science and Engineering Research Board (SERB), DST, India
through the MATRICS Grant No. MTR/2019/001042 and CEFIPRA Project No. 64T3-1.
This research was supported in part by the National Science Foundation under
Grant No. NSF PHY-1748958, ICTP through the Associates Programme and from the
Simons Foundation through grant number 284558FY19, IIT Madras through the IoE
program for establishing the QuCenDiEM group (Project No.
SB20210813PHMHRD002720), the International Centre for Theoretical Sciences
(ICTS), Bengaluru, India during a visit for participating in the program
“Frustrated Metals and Insulators” (Code: ICTS/frumi2022/9), and the faculty
associate program. Y.I. thanks IIT Madras for IoE travel grant to Germany
which facilitated progress on the research work. Y.I. acknowledges the use of
the computing resources at HPCE, IIT Madras.

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# Stabilization of Controller-Driven Nonuniformly Sampled Systems via Singular
Value Assignment

Ufuk Sevim  Leyla Goren-Sumer  Control and Automation Engineering Department,
Istanbul Technical University, Istanbul, Turkey (e-mail:
ufuk.sevim@itu.edu.tr).  Control and Automation Engineering Department,
Istanbul Technical University, Istanbul, Turkey (e-mail:
leyla.goren@itu.edu.tr).

###### Abstract

The stabilization problem of nonuniformly sampled systems is considered,
provided that the controller/scheduler can select the sampling period. We
proposed a method to construct the stabilizing state feedback controller using
singular value assignment. The closed-loop system is guaranteed to be stable
for arbitrary selection of bounded sampling periods. This method also provides
the upper bound for the maximum sampling period.

###### keywords:

Sampled-data control, Nonuniform sampling, Networked control systems,
Stability, Singular value assignment

##  1 Introduction

Study of nonuniformly sampled systems may originate from networked control
systems (Suh (2008), Hespanha et al. (2007), Baillieul and Antsaklis (2007)),
event based control (Lunze and Lehmann (2010)), or they can be treated as a
subset of discrete switching or hybrid systems (Naghshtabrizi et al. (2006)).

The stability results and control techniques are well-established when the
sampling time is uniform, i.e. constant for all times (Kuo (1995), Chen and
Francis (1995)). But since nonuniformly sampled systems are time-varying,
finding stabilizing controllers for these systems is proved to be difficult
and many efforts have been made in recent years Lin and Antsaklis (2009), Kao
and Fujioka (2013), Fujioka and Nakai (2010), Seuret (2012), Zhang et al.
(2001).

While most of the literature focus on stabilizing a nonuniformly sampled
system when the sampling intervals are unknown using a time-invariant
controller, we consider a different setting. We assume that the
controller/scheduler also selects the next sampling interval for the system
online. This setting is also discussed in Haimovich and Osella (2013) and
stabilizing controllers are found when the number of inputs is at least 1 less
than the unstable eigenvalues of the open loop system.

In our previous work Sevim and Goren-Sumer (2016), we proposed a sampling-
period-varying state feedback controller that guarantees stability for bounded
and arbitrarily varying sampling intervals using digital redesign technique.
The key idea of that study was to use eigenvector matrix of the desired
closed-loop system matrix to construct an induced norm that guarantees
stability.

The same idea of constructing an induced matrix norm using similarity
transformation is also used in this paper to establish an alternative method
to design a stabilizing state feedback controller using singular value
assignment.

The proposed method is based on the main theorem given in Martin and Wang
(2009), which gives the necessary and sufficient conditions for the existence
of a state feedback gain to assign the desired singular values of the closed-
loop system. The authors also provide an algorithm to calculate that state
feedback gain.

To the best of our knowledge, this is the first work that uses singular value
assignment to stabilize non-uniformly sampled systems.

###  1.1 Notation and Terminology

For $x\in\mathbb{C}^{n}$, $\left\lvert x\right\rvert$ denotes any vector norm
and for $A\in\mathbb{C}^{n\times n}$, $\left\lVert A\right\rVert$ denotes a
norm induced by some vector norm $\left\lvert\cdot\right\rvert$, i.e.
$\left\lVert A\right\rVert:=\max_{x\in\mathbb{C}^{n}}\left\lvert
Ax\right\rvert/\left\lvert x\right\rvert$. $\bar{\sigma}(A)$ denotes the
maximum singular value of $A$.

##  2 Problem Formulation

Consider the continuous time system

$\dot{x}(t)=Ax(t)+Bu(t)$ (1)

where $x(t)\in\mathbb{R}^{n}$, $u(t)\in\mathbb{R}^{m}$,
$A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$,
$\operatorname{rank}B=m\leq n$, and $(A,B)$ is stabilizable.

Define the sequence of time instances $\\{t_{k}\\}_{k\in\mathbb{N}}$ where

$0=t_{0}<t_{1}<\dots<t_{k}<\dots$

with $\lim_{k\to\infty}t_{k}=\infty$. We assume that
$\\{t_{k}\\}_{k\in\mathbb{N}}$ are known a priori and sampling intervals
$h_{k}:=t_{k+1}-t_{k}$ are bounded, i.e.
$h_{k}\in[h_{\operatorname{min}},h_{\operatorname{max}}],\forall
k\in\mathbb{N}$ for some $h_{\operatorname{max}}>h_{\operatorname{min}}>0$.

The objective is to design a sampled state feedback controller
$K(t_{k})\in\mathbb{R}^{m\times n}$ such that the system

$\dot{x}(t)=Ax(t)+BK(t_{k})x(t_{k}),~{}~{}\forall t\in[t_{k},t_{k+1})$ (2)

is stable for any selection of $\\{t_{k}\\}_{k\in\mathbb{N}}$ as long as
$h_{k}\in[h_{\operatorname{min}},h_{\operatorname{max}}]$.

Consider the discretized model of (1)

$x_{k+1}=F_{k}x_{k}+G_{k}u_{k}$ (3)

where $u_{k}:=u(t_{k})$,

$F_{k}:=e^{Ah_{k}}~{}~{}\text{and}~{}~{}G_{k}:=\left(\int_{0}^{h_{k}}e^{A\tau}d\tau\right)B.$

The following result is standard:

###### Lemma 1

Define $K_{k}:=K(t_{k})$ where $K(t_{k})$ is given in (2). Then the solutions
of (2) and

$x_{k+1}=(F_{k}+G_{k}K_{k})x_{k}$ (4)

are same at the sampling instances, i.e. $x_{k}=x(t_{k}),\forall
k\in\mathbb{N}$, assuming $x(t_{0})=x_{0}$.

We prove with induction. $x(t_{0})=x_{0}$ by assumption. Assume that
$x(t_{k})=x_{k}$, so

$\displaystyle x(t_{k+1})$
$\displaystyle=e^{A(t_{k+1}-t_{k})}x(t_{k})+\int_{t_{k}}^{t_{k+1}}e^{A(t_{k+1}-\eta)}Bu(\eta)d\eta$
$\displaystyle=e^{Ah_{k}}x(t_{k})+\left(\int_{0}^{h_{k}}e^{A\tau}d\tau\right)Bu(t_{k})$
$\displaystyle=F_{k}x_{k}+G_{k}K_{k}x_{k}$ $\displaystyle=x_{k+1}$

So the problem becomes finding $K_{k}$ and $h_{\operatorname{max}}$ such that
system (4) is guaranteed to be stable for
$h_{k}\in[h_{\operatorname{min}},h_{\operatorname{max}}]$. We are going to use
singular value assignment to guarantee the stability of (4).

##  3 Background

It is well-known that the stability conditions for nonuniformly sampled
systems are not trivial (Liberzon and Morse (1999)). There are many sufficient
stability results exist, such as Lin and Antsaklis (2009), Kao and Fujioka
(2013), Seuret (2012), Liberzon et al. (1999).

We use the following definition and sufficient condition for stability.

###### Definition 2

The system

$x_{k+1}=F_{k}x_{k}$ (5)

is globally asymptotically stable if, for any $x_{0}\in\mathbb{R}^{n}$

$\lim_{k\to\infty}\left\lvert x_{k}\right\rvert=0$

for some vector norm $\left\lvert\cdot\right\rvert$.

###### Theorem 3

If there exists a matrix norm $\left\lVert\cdot\right\rVert$ induced over some
vector norm $\left\lvert\cdot\right\rvert$ such that $\left\lVert
F_{k}\right\rVert<1,\forall k\in\mathbb{N}$, then the system (5) is globally
asymptotically stable.

Let $V(x_{k}):=\left\lvert x_{k}\right\rvert$. Then,

$V(x_{k+1})=\left\lvert x_{k+1}\right\rvert\leq\left\lVert
F_{k}\right\rVert\left\lvert x_{k}\right\rvert<\left\lvert
x_{k}\right\rvert=V(x_{k}).$

Hence $V(\cdot)$ is a Lyapunov function for the system (5).

###### Lemma 4

Let $\left\lVert\cdot\right\rVert$ be a matrix norm induced over some vector
norm $\left\lvert\cdot\right\rvert$. Then for any $A\in\mathbb{C}^{n\times n}$

$\left\lVert A\right\rVert_{T}:=\left\lVert T^{-1}AT\right\rVert$ (6)

is an induced norm over the vector norm

$\left\lvert x\right\rvert_{T}:=\left\lvert T^{-1}x\right\rvert$ (7)

where $x\in\mathbb{C}^{n}$ and $T\in\mathbb{C}^{n\times n}$ is an invertible
matrix.

It is easy to see that $\left\lvert\cdot\right\rvert_{T}$ is a vector norm.
Then,

$\displaystyle\max_{x\in\mathbb{C}^{n}}\frac{\left\lvert
Ax\right\rvert_{T}}{\left\lvert x\right\rvert_{T}}$
$\displaystyle=\max_{y\in\mathbb{C}^{n}}\frac{\left\lvert
ATy\right\rvert_{T}}{\left\lvert Ty\right\rvert_{T}}$
$\displaystyle=\max_{y\in\mathbb{C}^{n}}\frac{\left\lvert
T^{-1}ATy\right\rvert}{\left\lvert T^{-1}Ty\right\rvert}$
$\displaystyle=\left\lVert A\right\rVert_{T}$

###### Corollary 5

If there exists an invertible matrix $T\in\mathbb{C}^{n\times n}$ such that

$\bar{\sigma}(T^{-1}F_{k}T)<1,~{}~{}\forall k\in\mathbb{N},$ (8)

then the system (5) is globally asymptotically stable, where
$\bar{\sigma}(\cdot)$ is the maximum singular value.

It is well-known that the maximum singular value of a matrix is the induced
matrix norm over Euclidean vector norm. Then the result follows from Lemma 4.

The necessary and sufficient conditions for assigning singular values using
state feedback is given in Martin and Wang (2009). The following theorem is
given without proof for completeness.

###### Theorem 6

For any given $n\times n$ matrix $A$ and $n\times m$ full rank matrix $B$
[over reals] with $1\leq m\leq n$, let the singular values of

$(I-B(B^{T}B)^{-1}B^{T})A$ (9)

be

$0=a_{1}=\dots=a_{m}\leq a_{m+1}\leq\dots\leq a_{n}.$

For any given set of values

$0\leq s_{1}\leq s_{2}\leq\dots\leq s_{n}$

there exists a real $F$ such that the singular values of $A+BF$ are
$\\{s_{1},\dots,s_{n}\\}$ if and only if

$a_{j}\leq s_{j}\leq a_{j+m}$ (10)

for all $j=1,\dots,n$ (with the convention that $a_{j}=\infty$ if $j>n$).

##  4 Main Results

The following Corollary is immediate from Theorem 6 and Corollary 5.

###### Corollary 7

If there exists an invertible matrix $T\in\mathbb{R}^{n\times n}$ such that

$\bar{\sigma}\left(\\{I-\hat{G}(h)[\hat{G}^{T}(h)\hat{G}(h)]^{-1}\hat{G}^{T}(h)\\}\hat{F}(h)\right)<1$
(11)

for all $h\in(0,h_{\operatorname{max}})$ for some $h_{\operatorname{max}}>0$,
where

$\hat{F}(h):=T^{-1}e^{Ah}T~{}~{}\text{and}~{}~{}\hat{G}(h):=T^{-1}\left(\int_{0}^{h}e^{A\tau}d\tau\right)B,$

then there exists $\hat{K}(h)\in\mathbb{R}^{m\times n}$ such that

$\bar{\sigma}\left(\hat{F}(h)+\hat{G}(h)\hat{K}(h)\right)<1$

for all $h\in(0,h_{\operatorname{max}})$. Also the state feedback gain

$K_{k}:=\hat{K}(h_{k})T^{-1}$ (12)

guarantees the stability of (4) for any given $\\{t_{k}\\}_{k\in\mathbb{N}}$
such that $h_{k}\in(0,h_{\operatorname{max}})$.

###### Theorem 8

Let $(A,B)$ is given in (1) and there exists $K\in\mathbb{R}^{m\times n}$ such
that $A+BK$ is stable and has real distinct eigenvalues. Let $T$ be the matrix
such that

$D:=T^{-1}(A+BK)T$

is diagonal. Then, there exists an interval $(0,h_{\operatorname{max}})$ such
that (11) is satisfied with this $T$. Moreover, $h_{\operatorname{max}}$ is
determined by the eigenvalues of $D$.

Let $F_{d}(h):=e^{(A+BK)h}$. As a result of Theorem 9 in Sevim and Goren-Sumer
(2016) we can state that there exists a $K(h)$ such that

$\left\lVert F(h)+G(h)K(h)-F_{d}(h)\right\rVert<\delta$

for $h\in(0,h_{\operatorname{max}})$ and arbitrary $\delta>0$. Then,

$\displaystyle\left\lVert T^{-1}[F(h)+G(h)K(h)-F_{d}(h)]T\right\rVert$
$\displaystyle\qquad\qquad=\left\lVert\hat{F}(h)+\hat{G}(h)\hat{K}(h)-e^{Dh}\right\rVert$
$\displaystyle\qquad\qquad\leq\left\lVert T^{-1}\right\rVert\left\lVert
F(h)+G(h)K(h)-F_{d}(h)\right\rVert\left\lVert T\right\rVert$
$\displaystyle\qquad\qquad<\delta\left\lVert T^{-1}\right\rVert\left\lVert
T\right\rVert:=\epsilon$

This implies

$\displaystyle\left\lVert\hat{F}(h)+\hat{G}(h)\hat{K}(h)\right\rVert<\left\lVert
e^{Dh}\right\rVert+\epsilon$

By selecting

$\delta=\frac{1-\left\lVert e^{Dh}\right\rVert}{\left\lVert
T^{-1}\right\rVert\left\lVert T\right\rVert}$

we can see that there exists a $\hat{K}(h)$ that assigns the maximum singular
value of $\hat{F}(h)+\hat{G}(h)\hat{K}(h)$ less than 1. From Theorem 6, we can
conclude that

$\bar{\sigma}\left(\\{I-\hat{G}(h)[\hat{G}^{T}(h)\hat{G}(h)]^{-1}\hat{G}^{T}(h)\\}\hat{F}(h)\right)<1$

###### Algorithm 1

The following design algorithm is proposed based on Theorem 8.

  1. 1.

Find a $K\in R^{m\times n}$ such that $A+BK$ is Hurwitz and has real distinct
eigenvalues.

  2. 2.

Find $T$ such that $D:=T^{-1}(A+BK)T$ is diagonal.

  3. 3.

Calculate

$\hat{F}(h):=T^{-1}e^{Ah}T~{}~{}\text{and}~{}~{}\hat{G}(h):=T^{-1}\left(\int_{0}^{h}e^{A\tau}d\tau\right)B.$

  4. 4.

Calculate the singular values of

$\hat{P}(h):=\left(I-\hat{G}(h)[\hat{G}^{T}(h)\hat{G}(h)]^{-1}\hat{G}^{T}(h)\right)\hat{F}(h).$

Due to Theorem 8, there exists $h_{\operatorname{max}}$ such that
$\bar{\sigma}(\hat{P}(h))<1$ for all $h\in(0,h_{\operatorname{max}})$.

  5. 5.

Determine the desired singular values according to Theorem 6.

  6. 6.

Calculate $\hat{K}(h)$ such that $\hat{F}(h)+\hat{G}(h)\hat{K}(h)$ has desired
singular values using the algorithm given in Martin and Wang (2009).

  7. 7.

The controller $K_{k}:=\hat{K}(h_{k})T^{-1}$ is a stabilizing controller for
the system (4).

##  5 Numerical Example

###### Example 9

We consider the example given in Haimovich and Osella (2013) with an
additional unstable eigenvalue. This example is also given in our previous
work Sevim and Goren-Sumer (2016).

$A=\begin{bmatrix}1&-2&0\\\ 2&1&0\\\
0&0&0.5\end{bmatrix},~{}~{}B=\begin{bmatrix}0.5\\\ 2\\\ 1\end{bmatrix}$

We selected the following state feedback gain to stabilize the system:

$K=\begin{bmatrix}\frac{1128}{289}&\frac{-1064}{289}&\frac{-105}{34}\end{bmatrix}$

The nonzero singular values of $\hat{P}(h)$ are calculated as in Figure 1.
From the figure, we can conclude that $h_{\operatorname{max}}\approx 0.62$ and
the singular values of $\hat{F}(h)+\hat{G}(h)\hat{K}(h)$ can be arbitrarily
assigned anywhere between the solid curves. We selected the dashed lines to
assign the singular values, which guarantees the stability of the closed loop
system for any selection of sampling time between
$(0,h_{\operatorname{max}})$.

In Figure 2, the state response of the closed loop system
$\dot{x}(t)=Ax(t)+BK(t_{k})x(t_{k}),\forall t\in[t_{k},t_{k+1})$ under random
sampling is given.

Figure 1: The nonzero singular values of $\hat{P}(h)$. Figure 2: The state
response of the closed loop system $\dot{x}(t)=Ax(t)+BK(t_{k})x(t_{k}),\forall
t\in[t_{k},t_{k+1})$ under random sampling.

##  6 Conclusions

We considered the stabilization problem of nonuniformly sampled systems with
the assumption that sampling periods are known to the controller. We presented
a theorem for the existence of a state feedback controller that assigns the
singular values of the closed loop system matrix under similarity
transformation, to desired values. Also, we proposed an algorithm to calculate
the sampling-period-varying state feedback controller that guarantees
stability for bounded and arbitrarily varying sampling intervals. We
illustrated the accuracy of the method in numerical examples.

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