we find a near-optimal policy $\pi$ on the stationary version of this domain,
which we use as the evaluation policy. The policy learns the CR and CF
parameters of a basal-bolus controller. Let $\pi^{\texttt{rand}}$ be a random
policy with uniform distribution over actions. Then we define the behavior
policy $\beta(o,a)\coloneqq 0.5\pi(o,a)+0.5\pi^{\texttt{rand}}(o,a)$ for all
states and actions.

##### MEDEVAC:

This domain stands for medical evacuation using air ambulances. This domain
was developed by Robbins et al. [2020] for optimally routing air ambulances to
provide medical assistance in regions of conflict. This domain divides the
region of conflict into $34$ mutually exclusive zones, and has $4$ air
ambulances to serve all zones when an event occurs. Based on real-data, this
domain simulates the arrival of different events, from different zones, where
each event can have $3$ different priority levels. Serving higher priority
events yields higher rewards. If an ambulance is assigned to an event, it will
finish the assignment in a time dependent on the distance between the base of
the ambulance and the zone of the corresponding event. While engaged in an
assignment, that ambulance is no longer available to serve other events. A
good controller decides whether to deploy, and which MEDEVAC to deploy, to
serve any event (at the risk of not being able to serve a new high-priority
event if all ambulances become occupied).

The original implementation of the domain assumes that the arrival rates of
the events and the time taken by an ambulance to complete an event follow a
Poisson process with a fixed rate. However, in reality, the arrival rates of
different events can change based on external incidents during conflict.
Similarly, the completion rate can also change based on how frequently an
ambulance is deployed. To simulate such non-stationarity, we oscillate the
arrival rate of the incoming high-priority events, which induces passive non-
stationarity. Further, to induce wear and tear, we slowly decay the rate at
which an ambulance can finish an assignment. This decay is proportional to how
frequently the ambulance was used in the past. This induces active non-
stationarity. The presence of both active and passive changes makes this
domain subject to hybrid non-stationarity.

Similar to other domains, we used an actor-critic algorithm [Sutton and Barto,
2018] we find a near-optimal policy $\pi$ on the stationary version of this
domain, which we use as the evaluation policy. Let $\pi^{\texttt{rand}}$ be a
random policy with uniform distribution over the actions. Then we define the
behavior policy $\beta(o,a)\coloneqq 0.5\pi(o,a)+0.5\pi^{\texttt{rand}}(o,a)$
for all states and actions.

###  E.4 Additional Results

While the primary focus of this chapter was to develop methods to handle
active/hybrid non-stationarity, we observed that the proposed method OPEN also
provides benefits over the earlier algorithm Pro-WLS even when it is known
that there is only passive non-stationarity in the environment.

####  E.4.1 Single Run

Similar to Figure 4, in Figure 8 we present a step by step breakdown of the
intermediate stages of a single run of OPEN on the RoboToy-Passive domain.
Here the trend in how the performance of the evaluation policy was changing in
the past remains the same in the future. When only passive non-stationarity is
present, the double counter-factual correction performed by OPEN is
superfluous. However, it can be observed that OPEN can still correctly
identify the trend and provide useful predictions of $\pi$’s future
performance.

The blue curve corresponds to the performances $J_{i}(\pi)$ for the past
episodes. As there is no active non-stationarity, the choice of actions
executed does not impact the underlying non-stationarity. Therefore,
$J_{i}(\pi)$ follows the same trend in future as it did in the past. The blue
and gray curves are unknown to the algorithm.

OPEN first uses historical data to obtain counterfactual estimates of
$J_{i}(\pi)$ for the past episodes. One can see the high-variance in these
estimates (notice the change in the y-scale) due to the use of importance
sampling.

Before naively auto-regressing, OPEN first aims to denoise the past
performance estimates using the first stage of instrument variable regression.
Since $p=300$, the first $300$ terms were not denoised. It can be observed
that OPEN successfully denoises the importance sampling estimates.

Using the denoised estimates of past performances, with the second use of
counterfactual reasoning, OPEN performs the second stage of regression to
forecast the future performance when $\pi$ will be deployed. In the passive
setting, use of double-counterfactual is superfluous but OPEN is still able to
correctly predict the future performance.

Figure 8: An illustrative step by step breakdown of the stages in the proposed
algorithm OPEN for the RoboToy-Passive domain.

####  E.4.2 Summary Plots

In Figure 9 we provide bias and MSE analysis of different algorithms on the
domains that exhibit passive non-stationarity. Except for the stationary
setting, where WIS has the best performance overall, we observe that for all
other settings in the plot, OPEN performs better than both Pro-WLS and WIS
consistently.

One thing that particularly stands out in these plots is the poor performance
of Pro-WLS, despite being designed for the passive setting. We observed that
because of the choice of parametric regression using the Fourier basis, Pro-
WLS tends to suffer from high bias when the number of Fourier terms is not
sufficient to model the underlying trend. Also, if the number of Fourier terms
is increased naively, then they overfit the data and extrapolate poorly,
thereby resulting in high-variance. In contrast, our method is based on an
auto-regressive based time-series forecast that is more robust to the model
choice (we kept the number of lag terms for auto-regression as $p=300$ for
OPEN for all our experiments).

To obtain all the results for Figure 5 and Figure 9, in total 30 different
seeds were used for each speed of each domain for each algorithm to get the
standard error. The authors had shared access to a computing cluster,
consisting of 50 compute nodes with 28 cores each, which was used to run all
the experiments.

Figure 9: Comparison of different algorithms for predicting the future
performance of evaluation policy $\pi$ on domains that exhibit passive non-
stationarity. On the x-axis is the speed, which corresponds to the rate of
non-stationarity; higher speed indicates a faster rate of change and a speed
of zero indicates a stationary domain. (TOP) On the y-axis is the absolute
bias in the performance estimate. (Bottom) On the y-axis is the mean squared
error (MSE) in the performance estimate. Lower is better for all of these
plots. For each domain, for each speed, for each algorithm, 30 trials were
executed. Discussion of these plots can be found in Section E.4.

###  E.5 Ablation Study

In this section we study the sensitivity to hyper-parameters for the proposed
method OPEN and the baseline method Pro-WLS [Chandak et al., 2020b]. The
hyper-parameter for OPEN corresponds to the number of past terms to condition
on for auto-regression, as discussed in rem:p. The hyper-parameter for Pro-WLS
corresponds to the order of Fourier bases required for parametric regression.
In Figure 10 we present the results for how the performance of the methods
vary for different choices of hyper-parameters.

Figure 10: (Top) Absolute bias in prediction of Pro-WLS for different choices
of its hyper-parameter. (Bottom) Absolute bias in prediction of OPEN for
different choices of its hyper-parameter. For all the plots, lower value is
better. Overall, we observe that OPEN being an auto-regressive method can
extrapolate/forecast better and is thus more robust to hyper-parameters than
Pro-WLS that uses Fourier bases for regression and is not as good for
extrapolation.


# Probing the superconducting ground state of the noncentrosymmetric
superconductors CaTSi3 (T = Ir, Pt) using muon-spin relaxation and rotation

R. P. Singh  Department of Physics, University of Warwick, Coventry, CV7 7AL,
UK  A. D. Hillier  ISIS facility, STFC Rutherford Appleton Laboratory, Harwell
Science and Innovation Campus, Oxfordshire, OX11 0QX, UK  D. Chowdhury  ISIS
facility, STFC Rutherford Appleton Laboratory, Harwell Science and Innovation
Campus, Oxfordshire, OX11 0QX, UK  J. A. T. Barker  Department of Physics,
University of Warwick, Coventry, CV7 7AL, UK  D. McK. Paul  Department of
Physics, University of Warwick, Coventry, CV7 7AL, UK  M. R. Lees  Department
of Physics, University of Warwick, Coventry, CV7 7AL, UK  G. Balakrishnan
Department of Physics, University of Warwick, Coventry, CV7 7AL, UK

###### Abstract

The superconducting properties of Ca$T$Si3 (where T = Pt and Ir) have been
investigated using muon spectroscopy. Our muon-spin relaxation results suggest
that in both these superconductors time-reversal symmetry is preserved, while
muon-spin rotation data show that the temperature dependence of the superfluid
density is consistent with an isotropic $s$-wave gap. The magnetic penetration
depths determined from our transverse-field muon-spin rotation spectra are
found to be 448(6) and 150(7) nm for CaPtSi3 and CaIrSi3 respectively.

###### pacs:

74.20.Rp, 74.70.Dd

##  I Introduction

There has been a great deal of interest in noncentrosymmetric superconductors
(NCS), instigated by the discovery of the noncentrosymmetric heavy-fermion
superconductor CePt3Si Bauer et al. (2004). The absence of a center of
inversion in the crystal structure along with a nontrivial anti-symmetric
spin-orbit coupling, leads to the intriguing possibility of a superconducting
state with an admixture of spin-triplet and spin-singlet pairs. Despite
intense theoretical and experimental efforts Gor’kov and Rashba (2001);
Frigeri et al. (2004, 2006); Sigrist (2009); Bauer and Sigrist (2012) the
study of the physics of noncentrosymmetric superconductors remains a dynamic
and active field.

One manifestation of unconventional superconductivity is the breaking of time-
reversal symmetry (TRS). The magnetic moments associated with the Cooper pairs
are non-zero for such superconductors. A local alignment of these moments
produces spontaneous, but extremely small, internal magnetic fields. Muon-spin
relaxation is especially sensitive to small changes in internal fields and can
easily measure fields of $\approx 0.1$ G which correspond to moments that are
just a few hundredths of a $\mu_{\mathrm{B}}$. Muon-spin relaxation is thus an
ideal probe with which to search for TRS breaking Schenck (1985); Lee et al.
(1998); Sonier (2007); Yaouanc and de Réotier (2011). Time-reversal symmetry
breaking is rare and is only observed directly in a few superconductors, e.g.
Sr2RuO4,Luke et al. (1998) UPt3,Luke et al. (1993) (although not without
controversy),de Réotier et al. (1995) (U,Th)Be13,Heffner et al. (1990)
PrOs4Sb12,Aoki et al. (2003) LaNiC2,Hillier et al. (2009) PrPt4Ge12,Maisuradze
et al. (2010) (PrLa)(OsRu)4Sb12,Shu et al. (2011) LaNiGa2,Hillier et al.
(2012) and more recently SrPtAs Biswas et al. (2013). The possibility of mixed
spin-singlet spin-triplet pairing in noncentrosymmetric superconductors makes
them prime candidates to exhibit TRS breaking Bauer and Sigrist (2012). To
date, the only truly NCS reported to show TRS breaking are LaNiC2 and
Re6Zr,Hillier et al. (2009); Singh et al. (2014) however, for LaNiC2, a mixing
of singlet and triplet states is forbidden due to the symmetry of the
structure Quintanilla et al. (2010), but is allowed for Re6Zr.

Many other NCS have been studied by magnetization, transport, and heat
capacity measurements e.g. Nb0.18Re0.82,Karki et al. (2010) LaRhSi3,Anand et
al. (2011) Mg10Ir19B16,Klimczuk et al. (2007) Mo3Al2CKarki et al. (2011) and
Re3W,Biswas et al. (2011) and several have been shown to exhibit
unconventional superconducting behavior including features such as upper
critical fields close to the Pauli limit, evidence for multiple gaps, or a
significant admixture of a triplet component to the superconducting order
parameter. Muon spectroscopy studies have been performed on some of these
compounds e.g. Re3W, LaRhSi3, LaPtSi3, LaPdSi3 Biswas et al. (2011); Anand et
al. (2011); Smidman et al. (2014). However, no spontaneous fields were
observed in the superconducting state in any of these materials. This
indicates that TRS breaking is either undetectable or not present in the
superconducting state of these compounds.

Muon-spin rotation can be used to accurately determine the magnetic
penetration depth $\lambda$ and hence the temperature dependence of the
superfluid density, yielding information on the symmetry of the
superconducting gap Schenck (1985); Lee et al. (1998); Sonier (2007); Yaouanc
and de Réotier (2011). This technique has been successfully applied to the
study of a number of NCS including Re3WBiswas et al. (2011) and LaPtSi3
Smidman et al. (2014). Using muon-spin relaxation and rotation ($\mu$SR)
together can provide crucial information on the nature and mechanisms of
superconductivity in new materials.

Figure 1:  (color online) Two possible crystal structures of the $RTX_{3}$
(113) compounds. Left is the noncentrosymmetric structure with the space group
$I4mm$ and right is the centrosymmetric structure with the space group
$Pm\bar{3}n$. The light (blue) spheres are the $R$ atoms, the black spheres
the $T$ atoms, and the dark (red) spheres the $X$ atoms.

Recently a new family of superconducting materials has been discovered with
the formula RTX3 (where R = rare earth, T = transition metal, and X = Si or
Ge). These 113 materials can crystallize in either the noncentrosymmetric
BaNiSn3-type crystal structure (space group $I4mm$), or in the centrosymmetric
LaRuSn3-type cubic structure (space group $Pm\bar{3}n$) shown in Fig. 1.
Several 113 materials with a noncentrosymmetric structure such as CeCoSi3 Haen
et al. (1985); Iwamoto et al. (1995), CeRhSi3 Kimura et al. (2005, 2007), and
CeIrSi3 Sugitani et al. (2006); Mukuda et al. (2008) exhibit novel ground
states. For example, at ambient pressure CeCoSi3 becomes superconducting at
1.3 K, while CeRhSi3 and CeIrSi3 exhibit antiferromagnetic ordering at 1.5 and
5 K, respectively. The latter two compounds also reveal superconductivity
under applied pressure. In CeRhSi3 the superconducting transition temperature
$T_{c}$ increases from 0.45 to 1.1 K between 0.4 and 2.3 GPaKimura et al.
(2005, 2007), while for CeIrSi3 $T_{c}$ increases from 0.5 to 1.6 K between
1.8 and 2.5 GPa Sugitani et al. (2006). In these compounds, the competition
between the on-site (nonmagnetic) Kondo interaction and the oscillatory inter-
site long-range magnetic interactions, known as Rudermann-Kittel-Kasuya-Yosida
(RKKY) interaction, plays an important role in the observed and rather unusual
physical properties. Replacing Ce with a non-magnetic analogue such as La also
produces superconductors which exhibit unusual properties. For example,
LaRhSi3 and LaPdSi3 are type I superconductors, whereas LaPtSi3 is type II
Anand et al. (2011); Smidman et al. (2014).

In this paper we report on the properties of the new $f$-electron free 113
compounds, CaIrSi3 and CaPtSi3, both of which have the potential to be of
great interest. CaIrSi3 and CaPtSi3 are NCS with superconducting transition
temperatures of 3.6 and 2.3 K respectively and therefore do not require
pressure to induce the superconducting state unlike their Ce analogues Eguchi
et al. (2011); Equchi et al. (2010). Specific heat data are in general
agreement with these superconductors being fully gapped, however, CaIrSi3
appears to show a deviation from a pure $s$-wave gap, which has been suggested
as evidence for a multiband or anisotropic gap Eguchi et al. (2011). In the
presence of Ir and Pt it is expected that spin-orbit coupling will be
significant, strengthening the possibility that the mechanisms for
superconductivity might not be entirely conventional in these materials.
Therefore, it is timely and interesting to probe the superconducting state of
CaIrSi3 and CaPtSi3 using muon spectroscopy. In this work, muon-spin
relaxation is used to search for evidence of TRS breaking in these two
superconductors. Muon-spin rotation is used to determine the temperature
dependence magnetic penetration depth. Since $\lambda(T)$ is directly related
to the superfluid density, the pairing symmetry can then be determined.

##  II Experimental Details

###  II.1 Sample preparation

Polycrystalline samples of CaPtSi3 and CaIrSi3 were prepared by arc melting
stoichiometric quantities of high purity Ca ($5\%$ excess of Ca to compensate
for any weight loss), Pt/Ir and Si in a tri-arc furnace under an argon (5N)
atmosphere on a water-cooled copper hearth. In order to minimize the loss of
the Ca by evaporation, melting is done in two steps. In the first step, Pt/Ir
are melted with Si. The observed weight loss during the melting of binaries
Pt/Ir-Si is negligible. In the second step, Pt/Ir-Si binaries are melted with
$5\%$ excess of Ca. The sample buttons were melted and flipped several times
to improve phase homogeneity.

###  II.2 Sample characterization

Powder x-ray diffraction data were collected for both samples. Refinement of
the x-ray data (see table 1) confirmed both the samples had the tetragonal
structure (space group $I4mm$ (No. 107)) with lattice parameters which are in
good agreement with those reported earlier Eguchi et al. (2011). There are
some impurity peaks present in both the CaIrSi3 and CaPtSi3 samples at the
same positions observed by Eguchi et al Eguchi et al. (2011).

Table 1: Lattice parameters of noncentrosymmetric CaPtSi3 and CaIrSi3 determined from powder x-ray diffraction data collected at 298 K. | CaPtSi3 | CaIrSi3
---|---|---
Structure | Tetragonal | Tetragonal
Space group | $I4mm$ | $I4mm$
a (nm) | 0.42182(5) | 0.41957(2)
c (nm) | 0.9880(2) | 0.98711(7)

In order to confirm the superconducting transition temperatures of the
samples, dc magnetic susceptibility measurements were made using a Quantum
Design Magnetic Property Measurement System. Fig. 2 shows the magnetic
susceptibility as a function of temperature in an applied field of 5 Oe. The
observed superconducting transition temperatures $T_{C}$ for CaPtSi3 and
CaIrSi3 are approximately 2.3 and 3.5 K respectively (see Table 2). These
transition temperatures are in good agreement with previously reported results
measured by dc susceptibility on samples with the same composition Eguchi et
al. (2011). There is no evidence from the dc susceptibility data that the
impurities present in our samples order magnetically or become
superconducting. Since muon spectroscopy probes the full volume of the sample
the results presented below are representative of the majority superconducting
phases.

Figure 2: (color online) Temperature dependence of the dc magnetization for
CaIrSi3 (upper) and CaPtSi3 (lower). The samples were cooled in zero-field and
a field of 5 Oe was then applied. Data were collected on zero-field-cooled
warming (ZFCW) and during a subsequent field-cooled cooling (FCC) in the same
applied field.

###  II.3 Muon spectroscopy

The muon-spin relaxation measurements in zero field (ZF) and muon-spin
rotation experiments in transverse fields (TF), were carried out at the ISIS
pulsed neutron and muon facility using the MuSR spectrometer King et al.
(2013). At the ISIS facility, a pulse of protons with a full width at half
maximum of $\approx 70$ ns are produced every 20 ms, with 4 out of 5 pulses
going through the muon target. The muons produced are implanted into the
sample and decay with an average lifetime of $2.2~{}\mu$s into a positron
which is emitted preferentially in the direction of the muon spin axis along
with two neutrinos. These positrons are detected and time stamped in the 64
detectors which are positioned either before, $F$, or after, $B$ the sample
for longitudinal (relaxation) experiments. The asymmetry $A$ of the $\mu$SR
time spectrum is then obtained as $A(t)=(F(t)-\alpha B(t))/(F(t)+\alpha
B(t))$, where $\alpha$ represents a relative counting efficiency of the
forward and backward detectors Lee et al. (1998); Yaouanc and de Réotier
(2011). Using these counts the asymmetry in the positron emission can be
determined and, therefore, the muon polarization is measured as a function of
time.

For the transverse-field experiments, a magnetic field is applied
perpendicular to the initial muon spin direction and momentum. In this
configuration, the signals from the instruments 64 detectors are normalized
and reduced to two orthogonal components which are then fitted simultaneously.

Powder samples of each material were mixed with GE varnish and mounted onto
silver holders. Any muons which stop in silver give a time independent
background for ZF-$\mu$SR experiments and a non-decaying precession signal in
the TF-$\mu$SR. The sample holder and sample were mounted into a helium-3
cryostat with a temperature range of 0.3 to 50 K. The samples were cooled to
base temperature in zero field and the relaxation spectra were collected at
fixed temperature upon warming while still in zero field. The stray fields at
the sample position were canceled to within 10 mG by a flux-gate magnetometer
and an active compensation system controlling three pairs of correction coils.
The TF-$\mu$SR experiments were conducted in a range of applied fields from 50
to 600 Oe. The field was applied above the superconducting transition before
cooling.

##  III Results and Discussion

###  III.1 Zero-field muon-spin relaxation

Firstly, let us consider the zero-field muon-spin relaxation results (see Fig.
3). The absence of an oscillation in the ZF-$\mu$SR asymmetry data at all
temperatures for both samples confirms that there are no coherent magnetic
fields, usually, associated with long-range magnetic order. In the absence of
atomic moments, in CaTSi3 the muon-spin relaxation is expected to arise from
the local fields associated with nuclear moments. These moments are usually
static on the timescale of the muon and are randomly orientated. In a case
such as this the depolarization function can be described by a Kubo-Toyabe
function Lee et al. (1998); Yaouanc and de Réotier (2011). In Fig. 3, we can
see that for both CaIrSi3 and CaPtSi3 the data is relatively flat and does not
have the characteristic shape of the aforementioned Kubo-Toyabe function. This
indicates that the fields from the nuclear moments are small. Moreover, the
$\mu$SR signals for temperatures above and below the superconducting
transition overlay and the depolarization rate is the same. This indicates
that time-reversal symmetry is preserved, as would be expected in a
conventional singlet superconductor, or at least any symmetry breaking field
is not observable by $\mu$SR.

Figure 3:  (color online) Zero-field muon-spin relaxation spectra for CaIrSi3
(upper panel) and CaPtSi3 (lower panel) at temperatures above (open symbols)
and below (closed symbols) $T_{c}$. Figure 4:  Typical muon-spin rotation
spectra for CaPtSi3 in a transverse field of 100 Oe at temperatures of 0.2 K
(upper), 1.1 K(middle) and 1.95 K (lower). The lines are fits to the data
using Eq. 1 as described in the text.

###  III.2 Transverse-field muon-spin rotation

Transverse-field muon-spin rotation can be used to determine the magnetic
penetration depth $\lambda$. Figure 4 shows typical spectra with a transverse
applied field of 100 Oe at $T=0.2$, 1.1, and 1.95 K after being cooled through
$T_{c}$. The TF-$\mu$SR spectra were fit as a sum of sinusoidally oscillating
components, each within a Gaussian relaxation envelope,

$G_{x}(t)=\sum_{i=1}^{n}A_{i}\exp\left(-\frac{\sigma_{i}^{2}t^{2}}{2}\right)\cos\left(\gamma_{\mu}B_{i}t+\varphi\right),$
(1)

Figure 5:  Field dependence of the muon depolarization rate at $T=100$ mK for
CaIrSi3 (open symbols) and CaPtSi3 (closed symbols). The lines are fits to the
data using Eq. 4 as described in the text.

where $A_{i}$ is the initial asymmetry, $\sigma_{i}$ is the Gaussian
relaxation rate, and $B_{i}$ is the first moment for the $i$-th component of
the field distribution Maisuradze et al. (2009); Weber et al. (1993). There is
a common phase offset $\varphi$ and $\gamma_{\mu}$ is the muon gyromagnetic
ratio. In these fits, $\sigma_{n}$ for the $n$-th component is set to zero and
corresponds to a background term arising from those muons which are implanted
into the silver sample holder producing an oscillating signal that has no
depolarization, as silver has a negligible nuclear moment. Using Eq. 1 is
equivalent to assuming a field distribution $P\left(B\right)$ within the
sample given by

$P(B)=\gamma_{\mu}\sum_{i=1}^{n-1}\frac{A_{i}}{\sigma_{i}}\exp\left(-\frac{\gamma^{2}_{\mu}\left(B-B_{i}\right)^{2}}{2\sigma^{2}_{i}}\right).$
(2)

The second moment of the field distribution within the sample
$\left\langle\Delta B^{2}\right\rangle$ is

$\left\langle\Delta
B^{2}\right\rangle=\left(\frac{\sigma}{\gamma_{\mu}}\right)^{2}=\sum_{i=1}^{n-1}\frac{A_{i}}{A_{\mathrm{tot}}}\left[\left(\frac{\sigma_{i}}{\gamma_{\mu}}\right)^{2}+\left(B_{i}-\left\langle
B\right\rangle\right)^{2}\right],$ (3)

where $A_{\mathrm{tot}}=\sum_{i=1}^{n-1}A_{i}$ and $\left\langle
B\right\rangle=\sum_{i=1}^{n-1}\frac{A_{i}B_{i}}{A_{\mathrm{tot}}}$. The
superconducting component of the second moment $\sigma_{\mathrm{sc}}$ is then
given by $\sigma^{2}_{\mathrm{sc}}=\sigma^{2}-\sigma^{2}_{\mathrm{nm}}$ where
$\sigma^{2}_{\mathrm{nm}}$ is the signal in the normal state due to the
nuclear moments.

The spectra from the CaIrSi3 were best described by three oscillating
functions whereas the spectra from the CaPtSi3 could be described by just two
Maisuradze et al. (2009). The field dependence of the superconducting
depolarization rates $\sigma_{\mathrm{sc}}$ are shown in Fig. 5. As the field
increases the depolarization rates decrease as may be expected for a
superconductor when the applied field is a significant fraction of the upper-
critical field $B_{\mathrm{c2}}$ Sonier (2007). The field dependence of
$\sigma_{\mathrm{sc}}$ can be used to determine the magnetic penetration depth
and to give an estimate for the upper-critical field (see Fig. 5).
$B_{\mathrm{c2}}$ can be independently verified using other measurements. The
$\sigma_{\mathrm{sc}}\left(B\right)$ data shown in Fig. 5 were fit using Eq. 4

$\sigma_{sc}\left[\mu
s^{-1}\right]=A\times(1-b)(1+1.21(1-\sqrt{b})^{3})\lambda^{-2}\left[\mathrm{nm}\right],$
(4)

where $\lambda$ is in nm, $b=B/B_{\mathrm{C2}}$ is the ratio of applied field
to the upper critical field, and $A$ is a prefactor related to the structure
of the flux-line lattice ($A=4.83\times 10^{4}$ for a hexagonal lattice)
Brandt (1988, 2003). Assuming the penetration depth follows either a two-fluid
model
$\left(\lambda^{-2}\left(T\right)/\lambda^{-2}\left(0\right)=\left[1-\left(T/T_{c}\right)^{4}\right]\right)$
or can be described using the local (London) approximation for an $s$-wave gap
superconductor (see below) gives penetration depths $\lambda(0)$ of 448(6) and
150(7) nm for CaPtSi3 and CaIrSi3 respectively. The upper critical fields
estimated from the $\sigma_{\mathrm{sc}}$ versus $B$ data and from
extrapolations to zero kelvin of the $B_{\mathrm{c2}}\left(T\right)$ curves
determined from $M\left(H\right)$ loops collected at temperatures above 1.5 K
(data not shown) are in good agreement with those reported by Eguchi et al.
Eguchi et al. (2011) from magnetic and transport data, although as in Ref.
Eguchi et al., 2011 there is considerable uncertainty associated with these
estimates. More comprehensive data sets down to low temperatures are required
to accurately determine $B_{\mathrm{c2}}(0)$ for both materials.

Figure 6:  Temperature dependence of the muon depolarization rate $\sigma$ for
CaPtSi3 (closed symbols) and CaIrSi3 (open symbols) .

The temperature dependence of the muon depolarization rate $\sigma$ for
CaPtSi3 and CaIrSi3 are given in Fig. 6. For both samples, the data shows a
plateau and then decrease as the temperature is increased. $\sigma$ then
levels off at a temperature slightly less than $T_{c}$. The depolarization
rates at higher temperatures ($T\geq T_{c}$) are small, in agreement with the
zero-field data discussed above. This depolarization $\sigma_{\mathrm{nm}}$ is
associated with the nuclear moments. Below $T_{c}$, the depolarization rates
are related to the magnetic penetration depth (see Eq. 4) and therefore the
structure of the superconducting gaps for the two materials can be
investigated. After subtracting $\sigma_{\mathrm{nm}}$ from $\sigma$ to give
$\sigma_{\mathrm{sc}}$ as described above, $\lambda$ can be calculated at each
temperature, with a correction for the strong field dependence of the
depolarization rates made using the $B_{\mathrm{c2}}(T)$ data from Ref.
[Eguchi et al., 2011] and Eq. 4. The temperature dependence of
$\lambda\left(T\right)$ determined in this way for CaPtSi3 and CaIrSi3 and
plotted as $\lambda^{-2}\left(T\right)/\lambda^{-2}\left(0\right)$ versus the
reduced temperature $T/T{\mathrm{c}}$ are shown in Fig. 7. The $\lambda(T)$
curves can then be fit within the local (London) approximation Tinkham (1996)
for an $s$-wave gap superconductor in the clean limit using the following
expression:

$\frac{\lambda^{-2}(T)}{\lambda^{-2}(0)}=1+2\int_{\Delta(T)}^{\infty}\left(\frac{\partial
f}{\partial E}\right)\frac{E}{\sqrt{E^{2}-\Delta^{2}(T)}}dE,$ (5)

where $f=[1+\exp(E/k_{\mathrm{B}}T)]^{-1}$ is the Fermi function. The
temperature dependence of the gap is approximated by
$\Delta(T)=\Delta(0){\tanh}[1.82(1.018(T_{\mathrm{C}}/T-1))^{0.51}]$
Carrington and Manzano (2003). As can be seen from Fig. 7 the temperature
dependence of $\sigma_{\mathrm{sc}}$ for both samples is very well described
by this isotropic $s$-wave model giving $\Delta(0)=0.81(1)$ and
$\Delta(0)=0.38(1)$ meV, and BCS ratios $2\Delta\left(0\right)/k_{B}T_{c}$ of
3.8(2) and 5.4(2), for CaPtSi3 and CaIrSi3 respectively. The value for the BCS
ratio is CaPtSi3 is slightly higher that the 3.5 expected in the weak-coupling
limit. The higher value obtained for CaIrSi3, along with the strong field
dependence of $\sigma_{\mathrm{sc}}$ shown in Fig. 5, could be evidence of a
strong-coupling and/or multigap behavior with each gap having a similar
temperature dependence. Such a suggestion is consistent with the departure
from a pure $s$-wave behavior seen in the specific heat of CaIrSi3, although
the specific heat results for CaIrSi3 are generally well explained by a weak-
coupling BCS theory Eguchi et al. (2011).

Figure 7:  Normalized inverse square of the London penetration depth (the superfluid density) $\lambda^{-2}\left(T\right)/\lambda^{-2}\left(0\right)$ versus the reduced temperature $T/T{\mathrm{c}}$ for CaPtSi3 and CaIrSi3. The lines are fits to the data as described in the text. The closed symbols and solid line are the data and fit for the CaPtSi3, the open symbols and dashed line are the data and fit for CaIrSi3. Table 2: Superconducting parameters of noncentrosymmetric CaPtSi3 and CaIrSi3 determined from the dc magnetization and muon spectroscopy data. | CaPtSi3 | CaIrSi3
---|---|---
$T_{c}$, (K) | 2.30(5) | 3.50(5)
$\lambda(0)$, (nm) | 448(6) | 150(7)
$\Delta(0)$, (meV) | 0.38(1) | 0.81(1)
BCS ratio | 3.8(2) | 5.4(2)

##  IV Summary

In summary, we have investigated the superconducting compounds CaPtSi3 and
CaIrSi3 by using muon-spin relaxation and rotation. There is no evidence of
time-reversal symmetry breaking in either material, at least within the
sensitivity of $\mu$SR. The superconducting parameters determined from this
study are summarized in Table 2. The temperature dependence of the penetration
depths for both materials are consistent with $s$-wave isotropic gaps. The BCS
ratios place both materials in the intermediate to strong-coupling limit.

###### Acknowledgements.

We acknowledge the EPSRC, UK for providing funding (grant number
EP/I007210/1). We thank T. E. Orton for technical support. Some of the
equipment used in this research at the University of Warwick was obtained
through the Science City Advanced Materials: Creating and Characterizing Next
Generation Advanced Materials Project, with support from Advantage West
Midlands (AWM) and partly funded by the European Regional Development Fund
(ERDF).

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discussion in Liu  _et al._ (2021)), and since the theorem holds for any $t$
and $\alpha$ these can be optimized over.

###  E.3 Protocol with biased local random numbers (Protocol 2)

To derive the randomness rates, we use Theorem 2 with $C_{i}=A_{i}B_{i}$ and
$D_{i}=X_{i}Y_{i}$, as in the previous subsection. What remains is to derive
the min-tradeoff function and error terms. In this section, we compute these
quantities and derive the expression for the completeness error in terms of
the biasing parameters $\zeta_{A},\zeta_{B}$ and statistical error $\delta$.

####  E.3.1 Deriving the min-tradeoff function

We seek a min-tradeoff function suitable for using with Protocol 2. To
construct it we write the EAT channel in a slightly different way that is
explicit in the input distribution $p_{XY}$:

$\displaystyle\mathcal{M}_{p_{XY}}(\rho)=\sum_{abxy}p_{XY}(x,y)|a\rangle\\!\langle
a|_{A}\otimes|b\rangle\\!\langle b|_{B}\otimes|x\rangle\\!\langle
x|_{X}\otimes|y\rangle\\!\langle
y|_{Y}\otimes|(x,y,w)\rangle\\!\langle(x,y,w)|_{U}\otimes\mathcal{M}^{x,y}_{a,b}(\rho)\,,$
(56)

where $\mathcal{M}^{x,y}_{a,b}$ are subnormalized channels. We can also
consider the analogous channel where the $U$ register only stores $w$ (we use
$\tilde{\mathcal{M}}$ to indicate this case). Next consider the entropy
$H(AB|X=0,Y=0E)$, this entropy is calculated for the normalization of the
state

$(|0\rangle\\!\langle 0|_{X}\otimes|0\rangle\\!\langle
0|_{Y}\otimes\openone_{ABUE})(\mathcal{M}_{p_{XY}}\otimes\mathcal{I}_{E})(\rho_{RE})(|0\rangle\\!\langle
0|_{X}\otimes|0\rangle\\!\langle 0|_{Y}\otimes\openone_{ABUE})\,.$

For fixed $\\{\mathcal{M}^{x,y}_{a,b}\\}$, this is independent of $p_{XY}$ (it
is defined provided $p_{XY}(0,0)\neq 0$).

We next note that for $q$ as the distribution on the score ($U$) register

$\displaystyle(\mathcal{M}_{p_{XY}}\otimes\mathcal{I}_{E})(\rho_{RE})_{U}$
$\displaystyle=$
$\displaystyle\sum_{abxyw}p_{XY}\mathrm{tr}(\mathcal{M}^{x,y}_{a,b}(\rho_{R}))|(x,y,w)\rangle\\!\langle(x,y,w)|=\sum_{xyw}q(x,y,w)|(x,y,w)\rangle\\!\langle(x,y,w)|$
$\displaystyle(\mathcal{M}_{1/4}\otimes\mathcal{I}_{E})(\rho_{RE})_{U}$
$\displaystyle=$
$\displaystyle\sum_{abxyw}\frac{1}{4}\mathrm{tr}(\mathcal{M}^{x,y}_{a,b}(\rho_{R}))|(x,y,w)\rangle\\!\langle(x,y,w)|=\sum_{xyw}\frac{q(x,y,w)}{4p_{XY}}|(x,y,w)\rangle\\!\langle(x,y,w)|\,,$

and hence

$(\tilde{\mathcal{M}}_{1/4}\otimes\mathcal{I}_{E})(\rho_{RE})_{U}=\sum_{xyw}\frac{q(x,y,w)}{4p_{XY}}|w\rangle\\!\langle
w|\,.$

It follows that

$\displaystyle\biggl{\\{}H(AB|X=0,Y=0,E)_{(\tilde{\mathcal{M}}_{1/4}\otimes\mathcal{I}_{E})(\rho_{RE})}:(\tilde{\mathcal{M}}_{1/4}\otimes\mathcal{I}_{E})(\rho_{RE})_{U}=(1-s)|0\rangle\\!\langle
0|+s|1\rangle\\!\langle 1|,\ s=\sum_{xy}\frac{q(x,y,1)}{4p_{XY}}\biggr{\\}}=$
$\displaystyle\biggl{\\{}H(AB|X=0,Y=0,E)_{(\mathcal{M}_{p_{XY}\otimes\mathcal{I}_{E})}(\rho_{RE})}:(\mathcal{M}_{p_{XY}}\otimes\mathcal{I}_{E})(\rho_{RE})_{U}=\sum_{xyw}q(x,y,w)|(x,y,w)\rangle\\!\langle(x,y,w)|\biggr{\\}}\,.$

Let $\mathfrak{G}_{\zeta^{A},\zeta^{B}}$ be the set of channels for which $X$
and $Y$ are independent, $X$ is 1 with probability $\zeta^{A}$ and $Y$ is 1
with probability $\zeta^{B}$.

######  Lemma 26.

The function $F_{AB|00E}$ as defined in the main text can be used to define a
rate function for $\mathfrak{G}_{\zeta^{A},\zeta^{B}}$ by taking
$\mathrm{rate}_{\zeta^{A},\zeta^{B}}(q)=F_{AB|00E}(\omega(q))$ for
$q\in\mathcal{Q}_{\mathfrak{G}_{\zeta^{A},\zeta^{B}}}$ where

$\omega(q)=\frac{1}{4}\sum_{xy}\frac{1}{p_{X}(x)p_{Y}(y)}q((x,y,1))\,.$ (57)

###### Proof.

We have

$\displaystyle F_{AB|00E}$
$\displaystyle(\omega(q)):=\\!\\!\\!\inf_{(\tilde{\mathcal{M}},\rho_{RE})}\\!\left\\{H(AB|X=0,Y=0,E)_{(\tilde{\mathcal{M}}_{1/4}\otimes\mathcal{I}_{E})(\rho_{RE})}\\!:\\!(\tilde{\mathcal{M}}_{1/4}\otimes\mathcal{I}_{E})(\rho_{RE})_{U}=(1-\omega(q))|0\rangle\\!\langle
0|+\omega(q)|1\rangle\\!\langle 1|\right\\}$
$\displaystyle=\\!\\!\\!\inf_{(\mathcal{M},\rho_{RE})}\\!\biggl{\\{}\\!H(AB|X=0,Y=0,E)_{(\mathcal{M}_{p_{X}p_{Y}}\otimes\mathcal{I}_{E})(\rho_{RE})}\\!:\\!(\mathcal{M}_{p_{X}p_{Y}}\otimes\mathcal{I}_{E})(\rho_{RE})_{U}\\!=\\!\sum_{xyw}q(x,y,w)|(x,y,w)\rangle\\!\langle(x,y,w)|\\!\biggr{\\}}$
$\displaystyle\leq\inf_{(\mathcal{M},\rho_{RE})}\left\\{H(AB|XYE)_{(\mathcal{M}_{p_{X}p_{Y}}\otimes\mathcal{I}_{E})(\rho_{RE})}:(\mathcal{M}_{p_{X}p_{Y}}\otimes\mathcal{I}_{E})(\rho_{RE})_{U}=\sum_{xyw}q(x,y,w)|(x,y,w)\rangle\\!\langle(x,y,w)|\right\\},$

and hence $F_{AB|00E}(\omega(q))$ is a rate function for
$q\in\mathcal{Q}_{\mathfrak{G}_{\zeta^{A},\zeta^{B}}}$. ∎

We can hence form min-tradeoff functions suitable for using with Protocol 2 by
taking affine lower bounds to $F_{AB|00E}$. Taking the tangent to $F_{AB|00E}$
at $t$ we have min-tradeoff function

$f_{t}(q):=F_{AB|00E}(t)+F^{\prime}_{AB|00E}(t)\left(\frac{1}{4}\sum_{x,y}\frac{1}{p_{X}(x)p_{Y}(y)}q((x,y,1))-t\right),$

or, in other words, considering deterministic distributions on $U=(x,y,w)$

$f_{t}(\delta_{(x,y,w)})=\begin{cases}\frac{F^{\prime}_{AB|00E}(t)}{4p_{X}(x)p_{Y}(y)}+F_{AB|00E}(t)-tF^{\prime}_{AB|00E}(t)&\text{if}\
w=1\\\ F_{AB|00E}(t)-tF^{\prime}_{AB|00E}(t)&\text{if}\ w=0\end{cases}$

We have

$\displaystyle\mathrm{Max}(f_{t})$
$\displaystyle=\frac{F^{\prime}_{AB|00E}(t)}{4\zeta^{A}\zeta^{B}}+F_{AB|00E}(t)-tF^{\prime}_{AB|00E}(t)$
$\displaystyle\mathrm{Min}_{\mathcal{Q}_{\mathfrak{G}_{\zeta^{A},\zeta^{B}}}}(f_{t})$
$\displaystyle=F_{AB|00E}(t)-F^{\prime}_{AB|00E}(t)\left(t-\frac{1}{2}\left(1-\frac{1}{\sqrt{2}}\right)\right)$

We now find a bound on $\mathrm{Var}_{q}(f_{t})$ using the Bhatia-Davis bound
Bhatia and Davis (2000).

######  Lemma 27 (Bhatia-Davis bound).

Let $X$ be a real-valued random variable with $\max(X)=M$, $\min(X)=m$ and
$\mathbb{E}(X)=\mu$, then

$\displaystyle\mathrm{Var}_{X}\leq(M-\mu)(\mu-m)\,.$ (58)

In our case, $M=\mathrm{Max}(f_{t})$,
$m=F_{AB|00E}(t)-tF^{\prime}_{AB|00E}(t)$ and
$\mu=\mathbb{E}_{q}(f_{t})=F_{AB|00E}(t)+F^{\prime}_{AB|00E}(t)\left(\omega(q)-t\right)$,
where $\omega(q)$ is defined in (57). Thus,

$\displaystyle\mathrm{Var}_{q}(f_{t})$ $\displaystyle\leq$
$\displaystyle(F^{\prime}_{AB|00E}(t))^{2}\omega(q)\left(\frac{1}{4\zeta^{A}\zeta^{B}}-\omega(q)\right)$
$\displaystyle\leq$
$\displaystyle\begin{cases}(F^{\prime}_{AB|00E}(t))^{2}\left(\frac{1}{4\zeta^{A}\zeta^{B}}-1\right)&\text{if}\
\zeta^{A}\zeta^{B}<1/8\\\
\left(\frac{F^{\prime}_{AB|00E}(t)}{8\zeta^{A}\zeta^{B}}\right)^{2}&\text{if}\
\zeta^{A}\zeta^{B}\geq 1/8\end{cases}$

where we have optimized over $\omega(q)\in[0,1]$ for the second inequality.

####  E.3.2 Completeness error

We can form a bound on the completeness error using Hoeffding’s inequality
Hoeffding (1963).

######  Lemma 28 (Hoeffding’s inequality).

Let $X_{i}$ be $n$ i.i.d. random variables with $a\leq X_{i}\leq b$,
$a,b\in\mathbb{R}$. If $S=\sum_{i}X_{i}$ and $\mu=\mathbb{E}(S)$. Then for
$t>0$

$\displaystyle\mathbb{P}(S-\mu\geq
t)\leq\operatorname{e}^{-\frac{2t^{2}}{n(b-a)^{2}}}\,.$ (59)

######  Theorem 4.

Suppose Protocol 2 is run using honest devices that behave in an i.i.d.
fashion and that have an expected CHSH score $\omega_{\exp}$. The probability
that the protocol aborts is no greater than

$\displaystyle\operatorname{e}^{-32n(\delta\zeta^{A}\zeta^{B})^{2}}\,.$ (60)

###### Proof.

Recall the abort condition in the protocol, which states that
$\omega<\omega_{\exp}-\delta$ where

$\omega=\frac{1}{4}\sum_{x,y}\frac{|\\{i:U_{i}=(x,y,1)\\}|}{np_{X}(x)p_{Y}(y)}\,.$

We can write this as $\sum_{i}J_{i}$, where

$\displaystyle J_{i}((x,y,w))$ $\displaystyle=$
$\displaystyle\begin{cases}0&\text{ if }w=0\\\ 1/(4np_{X}(x)p_{Y}(y))&\text{
if }w=1\end{cases}$ (61)

This construction gives
$\mathbb{E}\left[\sum_{i}J_{i}\right]=n\mathbb{E}[J_{i}]=\sum_{xy}\frac{1}{4p_{X}(x)p_{Y}(y)}\mathbb{P}(U=(x,y,1))$.
In an honest implementation of the protocol, the distribution on the register
$U$ takes the form

$\displaystyle\mathbb{P}(U=(x,y,w))=\begin{cases}p_{X}(x)p_{Y}(y)(1-\omega_{xy})&\text{
if }w=0\\\ p_{X}(x)p_{Y}(y)\omega_{xy}&\text{ if }w=1\end{cases}$ (62)

where $\sum_{xy}\omega_{xy}=4\omega_{\exp}$, and hence
$\mathbb{E}\left[\sum_{i}J_{i}\right]=\omega_{\exp}$. The abort condition can
be expressed as $\omega_{\exp}-\sum_{i}J_{i}>\delta$. We have

$\displaystyle\mathbb{P}(\omega_{\exp}-\sum_{i}J_{i}>\delta)$
$\displaystyle=\mathbb{P}(\sum_{i}(-J_{i})-(-\omega_{\exp})>\delta)$
$\displaystyle\leq\operatorname{e}^{-32n(\delta\zeta^{A}\zeta^{B})^{2}}\,,$

where we have used Hoeffding’s inequality for the random variable $-J_{i}$
with $a=-1/(4n\zeta^{A}\zeta^{B})$ and $b=0$. ∎

###  E.4 Error parameters

Both Theorems 2 and 3 are stated in terms of the probability that the protocol
does not abort is $p_{\Omega}$, which is unknown to the users of the protocol.
However, if we replace $p_{\Omega}$ by $\epsilon_{\text{EAT}}$, then if
$p_{\Omega}\geq\epsilon_{\text{EAT}}$ we have a correct bound on the entropy.
On the other hand, if $p_{\Omega}<\epsilon_{\text{EAT}}$ then the protocol
aborts with probability greater than $1-\epsilon_{\text{EAT}}$. In other
words, prior to running the protocol the probability that it will both not
abort and that the entropy is not valid is at most $\epsilon_{\text{EAT}}$.
The soundness error of the protocol is
$\epsilon_{S}=\max(\epsilon_{\text{EAT}},2\epsilon_{h}+\epsilon_{\text{EXT}})$,
where $\epsilon_{\text{EXT}}$ is the extractor error (essentially the
probability that the extraction fails). A summary of the aspects of extraction
relevant to the present discussion and in the same notation as used here can
be found in (Liu  _et al._ , 2021, Supplementary Information I C).

###  E.5 Application to $H(AB|E)$ and $H(A|E)$

Note that the EAT as stated in Theorem 2 cannot be directly used in
conjunction with $H(AB|E)$ and $H(A|E)$. The basic reason is that the event
$\Omega$ should be an event on ${\bf U}$, which in turn should be a
deterministic function of ${\bf C}$ and ${\bf D}$. To use $H(AB|E)$ and
$H(A|E)$ we need ${\bf D}$ to be empty and ${\bf C}$ to be either ${\bf AB}$
or ${\bf A}$. This means the register ${\bf U}$ cannot depend on the inputs,
${\bf XY}$, but without a score that depends on the inputs we cannot certify
non-classicality let alone randomness.

Since we do not have strong use cases for $H(AB|E)$ and $H(A|E)$ (cf. Section
II), we do not consider possible extensions of the EAT in this work.

##  Appendix F Discussion of composability

Throughout this work we consider a composable security definition, which means
that, except with some small probability, the output randomness can be treated
as a perfect random string as part of any larger protocol. Composable security
definitions involve a distinguisher who tries to guess whether the real
protocol or a hypothetical ideal protocol are being run. This distinguisher is
allowed access to all the systems an eavesdropper has access to, but is also
assumed to learn whether or not the protocol was successful and the length of
the final random string. This allows a small amount of information leak in the
protocol and is the reason why the input strings ${\bf X}$ and ${\bf Y}$
cannot be reused without prior processing Colbeck and Kent (2011). For
instance, it would be easy to set up the devices such that they abort unless
$X_{1}=0$ and so if the protocol passes it is known that $X_{1}=0$. If $X$
directly forms part of the output, then with probability $1/2$ one bit of the
final output is known, which contradicts the security claim that the
probability of distinguishing the output from perfect randomness is at most
the soundness error. We also need to subtract a few bits from the size of the
output to account for the number of bits of information conveyed by the length
of the final string.

Conservatively, in a CHSH-based protocol with $n$ rounds, the concatenation
${\bf ABXY}$ comprises at most $4n$ bits. Equation (3.21) of Renner (2005)
implies $H_{\min}^{\epsilon_{h}}(A|BL)\geq H_{\min}^{\epsilon_{h}}(A|B)-\log
d_{L}$, where $d_{L}$ is the dimension of $L$. Therefore, if $L$ is a random
variable conveying the length of the final output we have

$H_{\min}^{\epsilon_{h}}({\bf ABXY}|EL)\geq H_{\min}^{\epsilon_{h}}({\bf
ABXY}|E)-\log d_{L}\geq H_{\min}^{\epsilon_{h}}({\bf ABXY}|E)-\log(4n)\,.$

Hence if we reduce the length of the extractor output by $\log(4n)$ bits we
can recycle the input randomness. Since the leading order term in
$H_{\min}^{\epsilon_{h}}({\bf ABXY}|E)$ is proportional to $n$, this reduction
is minor.

Because we are working with device-independent protocols, the ongoing security
of any randomness generated can be compromised if the devices used for one
instance of the protocol are subsequently reused Barrett  _et al._ (2013).
Hence, our discussion of security assumes devices are not reused (possible
modifications to protocols that aim to allow restricted reuse are also
discussed in Barrett  _et al._ (2013)).

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