# Electrical noise spectroscopy of magnons in a quantum Hall ferromagnet

Ravi Kumar1111equally contributed Saurabh Kumar Srivastav1222equally
contributed Ujjal Roy1333equally contributed Jinhong Park2,3444equally
contributed Christian Spånslätt4 K. Watanabe5 T. Taniguchi5 Yuval Gefen6
Alexander D. Mirlin2,3 and Anindya<EMAIL_ADDRESS>
Department of Physics, Indian Institute of Science, Bangalore, 560012, India.

Institute for Quantum Materials and Technologies, Karlsruhe Institute of
Technology, 76021 Karlsruhe, Germany.

Institut für Theorie der Kondensierten Materie, Karlsruhe Institute of
Technology, 76128 Karlsruhe, Germany.

Department of Microtechnology and Nanoscience (MC2), Chalmers University of
Technology, S-412 96 Göteborg, Sweden.

National Institute of Material Science, 1-1 Namiki, Tsukuba 305-0044, Japan.

Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot
76100, Israel.

Collective spin-wave excitations – magnons – in a quantum Hall ferromagnet are
promising quasi-particles for next-generation spintronics devices, including
platforms for information transfer. Detection of these charge-neutral
excitations relies on the conversion of magnons into electrical signals in the
form of excess electrons and holes, but if these signals are equal the magnon
detection remains elusive. In this work, we overcome this shortcoming by
measuring the electrical noise generated by magnons. We use the symmetry-
broken quantum Hall ferromagnet of the zeroth Landau level in graphene to
launch magnons. Absorption of these magnons creates excess noise above the
Zeeman energy and remains finite even when the average electrical signal is
zero. Moreover, we formulate a theoretical model in which the noise is
generated by equilibration (partial or full, depending on the bias voltage)
between edge channels and propagating magnons. Our model, which agrees with
experimental observations, also allows us to pinpoint the regime of ballistic
magnon transport in our device.

Introduction. The emergence of charge-neutral collective excitations presents
a powerful platform for developing data processing as well as information
transfer with small power consumption. Among these excitations, spin-wave
excitations, or their quanta ‘magnons’, in magnetic materials are promising.
An obvious important task is to develop new techniques for the detection of
these charge-neutral quasi-particles. So far, various experimental tools, such
as inelastic neutron scattering[1, 2], inelastic tunnelling spectroscopy[3,
4], terahertz spectroscopy[5, 6], microwave Brillouin light scattering[7, 8],
nitrogen-vacancy centre[9, 10], and superconducting qubits[11] have been used
to detect magnons in bulk magnetic materials. However, their detection in
device geometries, which is necessary for information processing applications,
has remained challenging until very recently. In particular, it was
demonstrated by Wei et al.[12] that magnons can be converted into electrical
signals in a quantum Hall ferromagnet (QHF) in graphene.

Graphene offers a very versatile platform for new kinds of electronic devices.
When subjected to a perpendicular magnetic field, graphene shows several
unique quantum Hall (QH) phases, related to its peculiar sequence of Landau
levels (LL), manifesting both spin and valley degrees of freedom[13, 14, 15].
In particular, the particle-hole symmetric zeroth LL (ZLL) has a rich variety
of QHF phases[16, 17, 18, 19, 20, 21, 22]: When the ZLL is partially filled,
Coulomb interactions break spin and valley symmetries, and for a quarter
($\nu=-1$) or three quarters ($\nu=1$) filling, the QH phases comprise
ferromagnetic insulator bulks with spin-polarized edge states[23, 24, 25, 26].
While the charge excitations in the bulk of these QHF insulators have a gap
determined by the exchange energy ($E_{X}\sim\frac{e^{2}}{\epsilon\ell_{B}}$,
where $e$, $\epsilon$ and $\ell_{B}$ are the elementary charge, dielectric
constant, and the magnetic length), the spin-waves (magnons) have instead a
gap determined by the Zeeman energy ($E_{Z}=g\mu_{B}B$, where $g$ is the Landé
g-factor, and $\mu_{B}$ is the Bohr magneton)[27] and are in fact the lowest
energy excitations of the system. However, magnons do not carry electrical
charge, and therefore do not have a large impact on electrical transport,
which in turn makes it a difficult task to detect them. There are a few
reported attempts of generating and detecting spin-wave excitations or magnons
in graphene-based QHF devices[12, 28, 29, 30, 31, 32]. While magnon generation
in these phases is based on an out-of-equilibrium occupation of edge channels
with opposite spin, the detection of the magnons relies on the absorption of
magnons by edge modes in the vicinity of ohmic contacts. The absorption of
magnons by the edge modes creates excess electrons or holes in different
corners of the graphene devices, and the measured electrical signal depends on
the relative difference between the electron and hole signal magnitudes,
which, in turn, critically depend on the device geometry. One may, therefore,
not be able to detect any electrical signal if both the excited electrons and
hole signals are equal. Thus, an alternative technique, which does not rely on
the difference between excess electron and hole signals, is necessary for
sensitive detection of magnons.

In this work, we demonstrate that electrical noise spectroscopy of magnons is
a powerful method that satisfies the detection sensitivity requirement. We
first establish that our device hosts symmetry-broken robust QH phases and
study the magnon transport when the bulk filling is kept at $\nu=1$. In order
to generate magnons, we inject an edge current through an ohmic contact. While
the injected current only flows in the downstream direction (as dictated by
the electron motion subject to external magnetic field), we measure the non-
local electrochemical potential of a floating ohmic contact placed upstream
from the source contact. Whenever the bias voltage of the injection contact
corresponds to an energy smaller than the Zeeman energy $E_{Z}$, no non-local
signal is detectable. As the bias energy exceeds $E_{Z}$, we measure a finite
non-local signal for negative bias voltages. By contrast, the non-local signal
remains zero for the entire positive bias voltages, which may naively suggest
that magnons are not generated in this bias regime. Next, we switch to
measuring the electrical noise and show that, as expected, no noise is
detected below $E_{Z}$. On the other hand, as soon as the bias energy exceeds
$E_{Z}$, the noise increases for both signs of the bias voltage. We show that
the noise contributions created due to magnon absorption at different corners
in our devices are additive, even when the average electron and hole currents
mutually cancel (which happens for positive bias voltages). This renders noise
spectroscopy a highly sensitive tool for magnon detection. Finally, our
theoretically calculated noise captures well the experimental data and further
suggests that the detected noise is a result of an increase of the effective
temperature of the system as a result of equilibration between edge channels
and magnons.

Figure 1: Device schematic, magnon generation and detection in quantum Hall
ferromagnet. (a) The device has a left, right, transverse, and a floating
contact. The device is set to $\nu=1$, whereas regions adjacent to the
contacts are tuned to $\nu=2$, as shown by the additional circulating inner
edges near the contacts. The spin polarization of the outer and inner edges
are orthogonal, denoted by up and down arrows, respectively. A dc plus ac
current ($I_{dc}+dI$) is injected through the upper red transverse contact,
and when the electrochemical potential ($\mu$) exceeds Zeeman energy
($E_{Z}$), magnons are generated near point ’A’ via a spin-flip process. These
magnons propagate through the QH bulk and are absorbed at other corners via
the reverse spin-flip process. The bottom transverse contact is used to
measure the voltage ($dV$) and noise ($S$) of the floating contact using
standard lock-in ($\sim 13Hz$) and LCR resonance circuit ($\sim 740kHz$),
respectively. (b) Magnon absorption at the different corners creates electron-
hole excitations, but only points ’B’ and ’D’ contribute excess electrons and
holes to the floating contact, respectively. (c) QH response at $B=1$T. The
inset shows the activation gap of $\nu=1$, which is $\sim 4$K. (d) 2D colour
map of the differential resistance ($dV_{L}/dI$) measured at the source
contact vs the dc bias voltage ($V_{S}=I_{dc}\times\frac{h}{e^{2}}$) and the
gate voltage around the centre of the $\nu=1$ plateau. A line cut at
$V_{BG}=0.079$ is shown in solid magenta. (e) Non-local $dV_{NL}/dI$ of the
floating contact vs source and gate voltages. (f) (upper panel) Line cuts from
e). Each plot is shifted vertically for clarity. (bottom panel) Non-local
$dV_{NL}/dI$ for bulk $\nu=2$. The vertical lines in d), e), and f) represent
the Zeeman energy at B = 1T.

Device and experimental principle. Figure 1(a) shows the schematics of our
device and measurement setup. The device consists of $hBN$ encapsulated
graphite-gated high-mobility single-layer graphene, fabricated by the standard
dry transfer technique[33, 34]. Device fabrication and characterization are
detailed in the Supplementary Information (SI-S1). The QH response of the
device at a magnetic field ($B$) of $1$ T is shown in Fig. 1(c), indicating
robust QH plateaus and the inset depicts the activation gap at $\nu=1$, which
is estimated to be $\sim 4K$ (see SI-S1). As seen in Fig. 1(a), the device has
left and right ground contacts, while the upper transverse contact is utilized
to inject current for magnon generation. The lower transverse contact is used
to detect the change in chemical potential of the floating contact (FC) due to
magnon absorption. The device’s bulk is tuned to the $\nu=1$ QHF state,
allowing it to host magnons. Importantly, the local doping due to the attached
metallic contacts increases the filling factor to $\nu=2$ near these contacts;
this is represented (shown only for the right side of the FC) by additional
loop-shaped edge modes at each contact, and are referred to as the “inner
edge”. In contrast, the outer edge propagates between contacts, as shown in
Fig. 1(a). A noiseless current, $I_{dc}+dI$, comprising a dc and an ac
component, is injected into the red-coloured source contact in Fig. 1(a). The
injected current flows along the outer edge with up-spin polarization. This
current exits the sample at the right-most grounded contact. The current along
the inner edge, which flows around the source contact, has a down-spin
polarization, does not contribute to the electrical conductance in the
circuit. The dc voltage drop at the source contact,
$V_{S}=I_{dc}\times\frac{h}{e^{2}}$, is shown as the electrochemical potential
$\mu$ in Fig. 1(a). The corresponding ac voltage that drops at the source
contact is $dV_{L}=dI\times\frac{h}{e^{2}}$. Whenever $\mu$ exceeds the Zeeman
energy $E_{Z}$, i.e., $|\mu\rvert\geq E_{Z}$, the electrons flowing along the
circulating inner edge can tunnel into the outer edge by flipping their spin
via magnon emission near point ‘A’, as shown in Fig. 1(a). This process does
not directly alter the electrical conductance since the tunnelling current
flows back and is absorbed by the same injection contact. The emitted magnons
propagate through the bulk of the device and can be absorbed at the device
corners (‘B’, ‘C’, ‘D’, ‘E’ and ‘F’) via tunneling of electrons from the outer
edge to the inner edge through the reverse spin flipping process. However,
only parts of the currents generated at the two corners ‘B’ and ‘D’ arrive at
the FC and contribute to the fluctuations of the electrochemical potential
$\delta\mu_{\text{FC}}$ of the FC. This is so since generated electron and
hole excitations are separated at points ‘B’ and ‘D’ into two respective
currents, only one of which flows towards the FC as shown in Figs. 1(a) and
1(b). The fluctuations $\delta\mu_{\text{FC}}$ are measured in the lower
transverse contact placed to the left of the FC on the lower edge, by using an
LCR resonance circuit at a frequency of $\sim 740kHz$, followed by an
amplifier chain and a spectrum analyzer (see Ref. 35, 36, Methods, and SI-S8).
We also measure the average chemical potential of the FC ($dV_{NL}$) via the
same transverse contacts with standard lock-in measurements. It should be
noted that the magnon generation in Fig. 1(a) is shown only for negative bias
voltage; for positive bias voltage, magnons are instead generated near point
‘E’, as shown in Fig. 2(b). We carried out measurements in two devices, where
for the second device (bilayer graphene), the filling near the contacts was
tuned by local gating, showing similar results (see SI-S6 and S7).

Magnon detection using non-local resistance and noise spectroscopy. Figure
1(d) shows a 2D color map of the differential resistance $R_{L}=dV_{L}/dI$
(with $L$ denoting “local”) measured in the injection contact as a function of
the bias voltage ($V_{S}$) and gate voltage ($V_{BG}$) around the center of
the $\nu=1$ plateau. It can be seen that within $E_{Z}$ [white vertical dashed
lines in Fig. 1(d)], $R_{L}$ remains constant at $\frac{h}{e^{2}}\sim
25.8k\Omega$ and decreases on both sides above $E_{Z}$, as shown by the solid
magenta line in Fig. 1(d). This feature is similar to that in Ref. 12, and can
be understood as follows: For negative bias voltages, magnons are generated at
‘A’. Absorption at ‘B’ and ‘F’ reduce (via holes) and increase (via electrons)
the chemical potential ($dV_{L}$) of the source contact, respectively, and
thus affect $R_{L}$. However, since the absorption at ‘B’ dominates over that
at ‘F’, $R_{L}$ decreases. Note that to be absorbed at ‘F’, the magnons have
to bend around the injected contact in contrast to their straight propagation
when reaching ‘B’. Similarly, for positive bias voltage, the generated magnons
from ‘E’ [see Fig. 2(b)] are absorbed dominantly at ‘F’ in comparison to ‘B’
and thus $R_{L}$ decreases.

A more powerful approach to magnon detection, which permits to explicitly
demonstrate and to explore magnon transport through the system, is provided by
non-local measurements[12, 31]. Figure 1(e) shows a 2D color map of the non-
local differential resistance, $R_{NL}=dV_{NL}/dI$, vs bias and gate voltages,
where $dV_{NL}$ is the chemical potential of the FC. As seen from the line cut
in Fig. 1(f) (top panel), $R_{NL}$ remains zero within $E_{Z}$ (vertical,
dashed lines), and increases for negative bias voltage above $E_{Z}$. However,
$R_{NL}$ is almost zero for the entire positive bias voltage range. When the
bulk filling was set to $\nu=2$, no detectable non-local signal (Fig. 1(f),
lower panel) was observed as the ground state is then non-magnetic. The
$R_{NL}$ in Figs. 2(e) and 2(f) can be understood as follows: As schematically
shown in Fig. 1(b), the magnon absorption at ‘B’ and ‘D’ contributes to the
non-local signal of the FC via excess electrons and holes, respectively. For
negative bias voltage, the magnons are generated at ‘A’, but the absorption at
‘B’ dominates over ‘D’ due to shorter distance [Fig. 1(a)], and thus $R_{NL}$
takes a finite value. However, for positive bias voltage, the magnons are
generated at ‘E’ [Fig. 2(b)], and the absorption at ‘B’ and ‘D’ are almost
equal due to their similar distance from ‘E’. Thus, $R_{NL}$ becomes almost
zero.

Figure 2: Noise spectroscopy of magnons. Magnon generation for negative (a)
and positive (b) bias voltages, where magnons are generated at point ’A’ and
’E’, respectively. The generated magnons propagate through the QH bulk and are
absorbed at different corners. Only magnon absorption at points ’B’ and ’D’
generates noise at the floating contact. (c) 2D colour map of excess noise
generated at the floating contact for different bias and gate voltages. (d)
(top panel) Line cuts of excess noise from c) for different $V_{BG}$ around
the centre of the $\nu=1$ plateau. Each plot is shifted vertically for
clarity. (bottom panel) Noise spectra for bulk $\nu=2$. As expected, no excess
noise is visible. (e) Noise spectra and $dV/dI$ for bulk filling $\nu=1$ at
$B=2$T. The vertically dashed lines in c), d), and e) depict the Zeeman energy
$E_{Z}$. (f) Thresholds of the bias voltage (with error bars) vs magnetic
field. The thresholds are extracted both from noise spectroscopy (solid red
circles) and non-local differential resistance measurements (solid blue
circles). Plotted is also $E_{Z}=g\mu_{B}B$ (solid black line).

Figure 2(c) shows a 2D color map of the measured excess noise ($S_{I}$) in the
FC as a function of bias and gate voltages. The corresponding line cuts are
shown in Fig. 2(d) (upper panel). We see that that $S_{I}$ remains zero as
long as $|eV_{S}|\leq E_{Z}$, and keeps increasing for larger values of either
positive or negative bias voltage. This feature stands in stark contrast to
Figs. 1(e) and 1(f). No significant noise was detected for the non-magnetic
state at $\nu=2$, shown in Fig. 2(d) (lower panel), which establishes that the
contribution from other degrees of freedom in the bulk (e.g., phonons) are
negligible in our experiment. We have repeated this measurement at different
magnetic fields (see SI-S2). For example, Fig. 2(e) shows $R_{NL}$ and $S_{I}$
at $B=2T$, which display features very similar as the data at $B=1$T. The
threshold voltage, $V_{\rm th}$ for magnon detection, extracted from $S_{I}$
at different $B$, is plotted in Fig. 2(f) (solid red circles) along with
$E_{Z}=g\mu_{B}B$ (solid black line), and we see that the two are strongly
correlated. We also show the threshold voltage extracted from the non-local
resistance (for negative bias voltage) as solid blue circles. At a given
magnetic field, the threshold voltage was extracted for several points across
the plateau, and its mean value and standard deviation are shown in Fig. 2(f)
(see SI-S3). The key insight from our noise spectroscopy of magnons is that
although $R_{NL}$ is almost zero due to the competition between excess
electrons at ‘B’ and excess holes at ‘D’, the fluctuations (variance) are
instead additive, making $S_{I}$ very sensitive for magnon detection without
any significant device geometry dependence.

We further note that the threshold voltage above which the non-local
resistance arises is significantly higher than $E_{Z}$ [see Figs. 2(e) and
2(f)] except for $B=3$T. This behavior has been observed in previous works as
well[12, 32]. In contrast to the resistance data, however, the noise starts to
increase at bias voltage $|eV_{S}|\sim E_{Z}$ [see Figs. 2(d), 2(e), and
2(f)]. The difference in threshold voltages for the non-local resistance and
the noise can be understood if magnons are absorbed at ‘B’ and ‘D’ with equal
probabilities within the bias voltage window $E_{Z}<|eV_{S}|<eV_{\text{th}}$.
Hence, this absorption process is invisible in the non-local resistance data
while strikingly visible in the noise data. Such an equal magnon absorption
may arise from a ballistic magnon transport in the bias voltage window
$E_{Z}<|eV_{S}|<eV_{\text{th}}$, where generated magnons propagate with a long
wavelength $\lambda\gg\ell_{B}$. Such magnons experience little scattering
from other degrees of freedom, particularly phonons or skyrmions[29]. However,
the ballistic motion of magnons may not be possible at a higher magnetic
field, $B=3$T. At this magnetic field, the phonons play an important role
since a larger current is required in order to generate magnons due to the
higher $E_{Z}$. Thus, the increased dissipation near ‘A’ and ‘E’ [see Fig.
1(a)] is able to excite phonons (see SI-S5). As a result, the threshold
voltage for the non-local resistance at $3$T is reduced to the vicinity of
$E_{Z}$, and in fact, is even slightly lower than $E_{Z}$ due to the
temperature broadening effect. A similar result was observed at elevated bath
temperatures at $B=1$T, as shown in SI-S4.

Figure 3: Theoretical model and temperature dependence of excess noise. (a)
The magnon absorption (wiggly green lines with arrow) at any corner is
modelled as a line segment of co-propagating edges, where tunnelling of
electrons occur from outer to inner edge [Figs. 1(a), 2(a), and 2(b)]. The
noise from the total tunnelling current is dominantly generated in the
vicinity of $x=L$ (yellow circle), where the local equilibrium noise dominates
over the shot noise. (b) Comparison between the experimentally measured excess
noise at $20$ mK (black solid circles) with the theoretically calculated data:
The red solid line is the prediction for the strongly equilibrated regime and
the blue solid line is for the partially equilibrated regime. These regimes of
no magnons, partial and strong equilibration are further indicated by the
horizontal arrows at the top of the axis. The right-hand side axis indicates
the excess temperature, see Eq. (1) (c) Measured noise at different bath
temperatures. (d) Noise calculated from Eqs. (1)-(2) at different bath
temperatures $T_{0}$. (e) Comparison between experiment and theory for the
excess noise at $V_{S}=-0.3$ mV as a function of bath temperature. The right
side of the axis indicates the excess temperature. (f) Excess noise for bulk
filling $\nu=2$ at $20$ mK and $600$ mK, both at $B=1$T.

Theoretical model and comparison to experiment. We model the edge segments
where the magnon generation and absorption take place as line junctions of co-
propagating edges with length $L$, where electrons tunnel between edge
channels (with spin-$\uparrow$ and spin-$\downarrow$), see Fig. 3(a). Each
such tunnelling event is associated with the generation or absorption of
magnons. We identify two distinct transport regimes depending on a degree of
equilibration, characterized by the equilibration length $\ell_{\text{eq}}$; a
short-junction regime ($L<\ell_{\text{eq}}$) with partial equilibration of the
edge channels and the magnons, and a long-junction regime
($L>\ell_{\text{eq}}$) with strong equilibration, see Methods and SI-S9 for
details. In the strong equilibration regime, equilibration in the magnon-
generation region ‘A’ takes place until the chemical potential difference
between the edge channels equals $E_{Z}$. At this saturation point, further
magnon generation is strongly suppressed. All generated magnons propagate in
the bulk of the QH state and are eventually absorbed in one of the absorption
regions (‘B’, ‘C’, ‘D’, ‘E’ and ‘F’). Each absorption event creates an
electron-hole pair (an electron in the spin-$\downarrow$ channel and a hole in
the spin-$\uparrow$ channel). These pairs produce the measured excess noise.
In each absorption line junction, the excess noise is dominantly generated
near $x=L$ [yellow circle in Fig. 3(a)] while remaining contributions are
exponentially suppressed, see Refs. 37, 38, 39, 36, 40 for a similar noise-
generating mechanism. The excess noise $S_{I}$ reflects an increased
temperature of the edge channels during the magnon absorption, given by

$\displaystyle
S_{I}=\frac{1}{2}\left(\frac{e^{2}}{h}(T_{0}+T)-2\frac{e^{2}}{h}T_{0}\right)=\frac{1}{2}\left(\frac{e^{2}}{h}(T-T_{0})\right)\,,$
(1)

where $T_{0}$ is the bath temperature and

$\displaystyle
T=\sqrt{T_{0}^{2}+\frac{3(|eV_{S}|-E_{Z})(2E_{Z}+3|eV_{S}|)}{(5\pi)^{2}}\theta(|eV_{S}|-E_{Z})}$
(2)

is the effective temperature of the system as a result of equilibration.
Furthermore, $\theta(|eV_{S}|-E_{Z})$ is the step function, which reflects the
fact that no magnons can be absorbed for bias energies below $E_{Z}$. The
factor $1/2$ in Eq. (1) originates from the noise-measurement scheme, see
Methods. In Fig. 3(b), we compare our theoretically calculated excess noise
(solid red line), $S_{I}$, with the experimentally measured noise versus the
bias energy $eV_{S}$ (for simplicity, only the negative bias side is
displayed), at fixed $T_{0}=20$mK. Figure 3(c) shows the measured noise at
different bath temperatures ($T_{0}$), and the corresponding theoretical plots
are shown in Fig. 3(d). A comparison between the experiment (orange circles)
and theory (blue circles) for $S_{I}$ at $V_{S}=-0.3$ mV as a function of
$T_{0}$ is shown in Fig. 3(e). Our theoretical model captures well the
characteristic features of the noise. Note that as seen in Fig. 3(f), no
excess noise was detected even at higher temperatures ($600$mK) for $\nu=2$ at
$B=1$T.

The bias voltage dependence of the excess noise defines three regimes in Fig.
3(b); (i) Biases $|eV_{S}|<E_{Z}$ result in no magnon generation and thus no
excess noise. (ii) In a narrow region $0<|eV_{S}|-E_{Z}<\frac{1}{\gamma L}$,
the equilibration in magnon absorption and generation regions is only partial,
$L<\ell_{\textrm{eq}}\equiv\frac{1}{\gamma(|eV_{S}|-E_{Z})}$. Here, $\gamma$
is a parameter proportional to the tunneling strength in every tunnel junction
comprising the line junction. This lack of equilibration allows us to model
the magnon-generation and absorption regions as single tunnel junctions in
regime (ii). In such a model, the noise generation is of non-equilibrium
nature, resulting in $S_{I}\propto(|eV_{S}|-E_{Z})^{2}$, shown in Fig. 3(b) by
the solid blue line. (iii) For larger biases $|eV_{S}|>E_{Z}+\frac{1}{\gamma
L}$ and hence $\ell_{\textrm{eq}}<L$, the edge channels and magnons achieve
full equilibration in the magnon absorption and generation regions. We find
that our theoretical model is in good agreement with the experimental data. In
particular, at sufficiently large biases [regime (iii)], our equilibrated line
junction model correctly describes several experimental observations: the
sudden increase followed by (approximate) saturation of the non-local
conductance as a function of the bias voltage [see Fig. 1(f)], the linear
behavior of the noise as a function of the bias voltage [Fig. 2(d)], and the
temperature dependence of the excess noise [Fig. 3(d)-(e)]. In addition, our
single tunnel junction model [partial equilibration regime (ii)] properly
describes the crossover region of bias voltages close to $E_{Z}$. Note that
our theory assumes that magnons are absorbed in all the absorption regions
with the same probability, but in reality, there may be deviations. These can
explain some variations between experimental data curves.

Discussion. As we show in Eq. (1), the excess noise generated in the line
junction [regime (iii)] reflects the increase in temperature $T-T_{0}$ of the
edge due to heating. The temperature behavior extracted from the measured
excess noise data in Fig. 3(b) (right y-axis) is similar to the temperature
behavior in Fig. 3(d) of Ref. 32, a result which was obtained from $R_{xx}$
thermometry measurements. However, we have never observed any saturation of
the temperature at high bias voltage, which was argued to be indicative of
strong equilibration between free magnons and skyrmions. Furthermore, the
noise behavior as a function of the bias voltage appears to be correlated with
that of the visibility of the Mach-Zender interferometry measured in Ref. 30.
It would be interesting to make a detailed connection between those two
different quantities. Finally, we emphasize that our measurements were
performed for relatively small magnetic fields and lower ambient temperatures
than in previous works[12, 30, 32]. These small quantities allow us to fully
neglect the effect of phonons. We have observed a sizeable effect of phonons
only for magnetic fields $B>2$T (see SI-S5).

Conclusion. We have demonstrated the utility of electrical noise spectroscopy
as a highly sensitive tool for detecting and studying magnons in a quantum
Hall ferromagnet. Our new protocol overcomes non-universal (device geometry
dependent) features that screen out the presence of magnons, when other
detection tools are employed, most prominently non-local conductance
measurements. This robustness paves the way for utilizing magnons as low-power
information carriers in future quantum technologies. Intriguing
generalizations of our approach, with a promise of novel physics, include bulk
phases of the fractional quantum Hall regime as well as of integer and
fractional Chern insulator phases of twisted bilayer graphene[41, 42, 43].
Further implementations of our approach may include other ferromagnetic
materials and vdW magnets[44, 45].

## Methods.

###  0.1 Device and measurements scheme.

Utilizing the dry transfer pick-up approach, we fabricated encapsulated
devices consisting of a heterostructure involving hBN (hexagonal boron
nitride), single-layer graphene (SLG), and graphite layers. The procedure for
creating this heterostructure comprised the mechanical exfoliation of hBN and
graphite crystals onto an oxidized silicon wafer through the widely employed
scotch tape method. Initially, a layer of hBN, with a thickness of
approximately 25-30 nm, was picked up at a temperature of 90°C. This was
achieved using a Poly-Bisphenol-A-Carbonate (PC) coated Polydimethylsiloxane
(PDMS) stamp on a glass slide attached to a home-built micromanipulator. The
hBN flake was aligned over the previously exfoliated SLG layer picked up at
90°C. The subsequent step involved picking up the bottom hBN layer of similar
thickness. Following the same process, this bottom hBN was picked up utilizing
the previously acquired hBN/SLG assembly. After this, the hBN/SLG/hBN
heterostructure was employed to pick up the graphite flake. Ultimately, this
resulting heterostructure (hBN/SLG/hBN/graphite) was placed on top of a 285 nm
thick oxidized silicon wafer at a temperature of 180°C. To remove the residues
of PC, this final stack was cleaned in chloroform (CHCl3) overnight, followed
by cleaning in acetone and isopropyl alcohol (IPA). After this, Poly-methyl-
methacrylate (PMMA) photoresist was coated on this heterostructure to define
the contact regions using electron beam lithography (EBL). Apart from the
conventional contacts, we defined a region of $\sim$ $6\mu m^{2}$ area in the
middle of SLG flake, which acts as a floating metallic reservoir upon edge
contact metallization. After EBL, reactive ion etching (mixture of CHF3 and O2
gas with a flow rate of 40 sccm and 4 sccm, respectively, at 25°C with RF
power of 60W) was used to define the edge contact. The etching time was
optimized such that the bottom hBN did not etch completely to isolate the
contacts from the bottom graphite flake, which was used as the back gate.
Finally, the thermal deposition of Cr/Pd/Au (3/12/60 nm) was done in an
evaporator chamber with a base pressure of $\sim$ $1\times 10^{-7}$ mbar.
After deposition, a lift-off procedure was performed in hot acetone and IPA.
The device’s schematics and measurement setup are shown in Fig. 1(a). The
distance from the floating contact to the ground contacts was $\sim 5\mu m$,
whereas the transverse contacts were placed at a distance of $\sim 2.5\mu m$.

All measurements were done in a cryo-free dilution refrigerator with a $\sim$
20 mK base temperature. The electrical conductance was measured using the
standard lock-in technique, whereas the noise was measured using an LCR
resonant circuit at resonance frequency $\sim$ 740 kHz. The signal was
amplified by a homemade preamplifier at 4 K followed by a room temperature
amplifier, and finally measured by a spectrum analyzer. At zero bias, the
equilibrium voltage noise measured at the amplifier contact is given by

$S_{V}=g^{2}(4k_{B}TR+V_{n}^{2}+i_{n}^{2}R^{2})BW\,,$ (3)

where $k_{B}$ is the Boltzmann constant, $T$ is the temperature, $R$ is the
resistance of the QH state, $g$ is the gain of the amplifier chain, and $BW$
is the bandwidth. The first term, $4k_{B}TR$, corresponds to the thermal
noise, and $V_{n}^{2}$ and $i_{n}^{2}$ are the intrinsic voltage and current
noise of the amplifier. At finite bias above the Zeeman energy, due to magnon
absorption at points ‘B’ and ‘D’, chemical potential fluctuations of FC create
excess voltage noise at the amplifier contact. At the same time, the intrinsic
noise of the amplifier remains unchanged. Due to the white nature of the
thermal noise and the excess noise, we could operate at higher frequency
($\sim$740 kHz), which eliminates the contribution from flicker noise (1/f)
which usually becomes negligible for frequencies above few tens of Hz. The
excess noise ($\delta S_{V}$) due to bias current is obtained by subtracting
the noise value at zero bias from the noise at finite bias, i.e $\delta
S_{V}=S_{V}(I)-S_{V}(I=0)$. The excess voltage noise $\delta S_{V}$ is
converted to excess current noise $S_{I}$ according to $S_{I}=\frac{\delta
S_{V}}{R^{2}}$, where $R=\frac{h}{\nu e^{2}}$ is the resistance of the
considered QH edge.

###  0.2 Theoretical calculation of the non-local resistance and noise.

To compute the tunneling current, non-local resistance, and noise generated in
the magnon absorption regions, we model the magnon generation and absorption
regions as line junctions of length $L$. These line junctions are modelled as
extended segments with two co-propagating edge channels in which electrons
tunnel along a series of tunnel junctions, see Fig. 3(a). We identify two
distinct transport regimes: those of a short (partially equilibrated;
$L<\ell_{\text{eq}}$) and long (equilibrated; $L>\ell_{\text{eq}}$) junctions,
where $\ell_{\text{eq}}$ is the equilibration length. The short-junction
regime can be equivalently modelled as a single tunnel junction. Details of
the theoretical analysis are presented in SI, Sec. S9.

We first consider the partial-equilibration regime, treating the magnon
generation or absorption regions as a single tunnel junction (at position
$x=0$). The Hamiltonian describing this junction reads

$\displaystyle H$ $\displaystyle=-iv\sum_{s=\uparrow,\downarrow}\int
dx\psi_{s}^{\dagger}(x)\partial_{x}\psi_{s}(x)+\sum_{{\bm{q}}}(E_{Z}+\hbar\omega_{{\bm{q}}})b_{{\bm{q}}}^{\dagger}b_{{\bm{q}}}$
$\displaystyle+W\psi_{\uparrow}^{\dagger}(x=0)\psi_{\downarrow}(x=0)b^{\dagger}(x=0)+\textrm{h.c.}.$
(4)

Employing the Keldysh non-equilibrium formalism, we derive zero-temperature
expressions for the tunneling current $I_{\text{ab}}$, non-local resistance
$dV_{\text{ab}}/dI$, and noise $S_{\text{ab}}$ generated in an absorption
region, respectively:

$\displaystyle I_{\text{ab}}$
$\displaystyle=\gamma^{\prime}\frac{e}{2h}(|eV_{S}|-E_{Z})^{2}\theta(|eV_{S}|-E_{Z})\,,$
(5) $\displaystyle\left|\frac{dV_{\text{ab}}}{dI}\right|$
$\displaystyle=\gamma^{\prime}\frac{h}{e^{2}}(|eV_{S}|-E_{Z})\theta(|eV_{S}|-E_{Z})\,,$
(6) $\displaystyle S_{\text{ab}}$
$\displaystyle=\frac{e^{2}}{h}\gamma^{\prime}(|eV_{S}|-E_{Z})^{2}\theta(|eV_{S}|-E_{Z})\,.$
(7)

Here, $\gamma^{\prime}$ is a parameter associated with the tunneling strength
in the tunnel junction. While the non-local resistance increases linearly with
increasing bias voltage $eV_{S}$, the noise increases instead quadratically.
For finite temperature, we first numerically determine the $eV_{S}$-dependence
of the magnon chemical potential $\mu_{m}$, and thereby we obtain the
$eV_{S}$-dependence of the non-local resistance and noise. This finite
temperature result is used to fit the experimental data for regime (ii) in
Fig. 3(b).

In the limit of a long line junction, the last term in Eq. (4) is modified to
describe tunneling in the spatial region $0\leq x\leq L$. In the equilibrated
regime, $L>\ell_{\text{eq}}$, this model yields non-local resistance and noise
characteristics distinct from those in the single tunnel-junction model.
Specifically, the equilibrated line-junction model predicts the following
tunneling current, non-local resistance, and the excess noise in each
individual absorption region,

$\displaystyle I_{\text{ab}}$
$\displaystyle=\frac{e}{2hM}(|eV_{S}|-E_{Z})\theta(|eV_{S}|-E_{Z})\,,$ (8)
$\displaystyle\left|\frac{dV_{\text{ab}}}{dI}\right|$
$\displaystyle=\frac{h}{2Me^{2}}\theta(|eV_{S}|-E_{Z})\,,$ (9) $\displaystyle
S_{\text{ab}}$ $\displaystyle=\frac{e^{2}}{h}(T-T_{0})\,,$ (10)

with the increased temperature $T$ of the system, Eq. (2). Here $M=5$ is the
number of absorption regions. Notably, the non-local resistance (9) is
constant in $eV_{S}$, whereas the noise [Eqs. (1)-(2)] instead increases
linearly in $eV_{S}$ at sufficiently large bias voltage $eV_{S}$. In the
calculation of Eqs. (2), (8), and (9), we have assumed for simplicity that the
magnons are absorbed in each individual absorption region with equal
probabilities. Note that the measured excess noise $S_{I}$ in Eq. (1) has the
additional factor $1/2$ compared with the excess noise generated in an
absorption region, i.e.,
$S_{I}=\frac{1}{2}S_{\text{ab}}=2\times\frac{1}{4}S_{\text{ab}}$. The factor
of $2$ reflects contributions from two noise spots (‘B’ and ‘D’) and the
factor $1/4=(1/2)^{2}$ originates from that only one channel out of the two
emanating from the FC is measured at the bottom transverse contact, see Fig.
1(a).

We also calculate the dependence of the equilibration length
$\ell_{\text{eq}}$ on the bias voltage $eV_{S}$. We do this by using the
results for the partial-equilibration regime (short $L$) and inspecting at
what $L$ the equilibration becomes strong. The result reads

$\ell_{\text{eq}}=\frac{1}{\gamma(|eV_{S}|-E_{Z})},\,\quad\text{for
}|eV_{S}|>E_{Z}\,.$ (11)

This equation implies a partial-equilibration regime for $|eV_{S}|$ slightly
exceeding $E_{Z}$ and a strong-equilibration regime for larger $|eV_{S}|$, as
discussed in the “Discussion” section above, and also illustrated in Fig.
3(b). Equation (11) shows that $\ell_{\text{eq}}$ increases significantly as
the bias energy approaches $E_{Z}$, indicating that the equilibration process
takes place very slowly near $|eV_{S}|\sim E_{Z}$. This happens because the
absorption rate per unit length is proportional to $(|eV_{S}|-E_{Z})^{2}$ [Eq.
(5)] whereas the total tunneling current in the equilibrated regime scales as
$(|eV_{S}|-E_{Z})$ [Eq. (8)].

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## Acknowledgements

A.D. thanks the Department of Science and Technology (DST) and Science and
Engineering Research Board (SERB), India, for financial support
(SP/SERB-22-0387) and acknowledges the Swarnajayanti Fellowship of the
DST/SJF/PSA-03/2018-19. A.D. also thanks financial support from CEFIPRA:
SP/IFCP-22-0005. S.K.S. and R.K. acknowledge the Prime Minister’s Research
Fellowship (PMRF), Ministry of Education (MOE), and Inspire fellowship, DST
for financial support, respectively. A.D.M., J.P., and Y.G. acknowledge
support by the DFG Grant MI 658/10-2 and by the German-Israeli Foundation
Grant I-1505-303.10/2019. Y.G. acknowledges support by the Helmholtz
International Fellow Award, by the DFG Grant RO 2247/11-1, by CRC 183 (project
C01), and by the Minerva Foundation. C.S. acknowledges funding from the 2D
TECH VINNOVA competence Center (Ref. 2019-00068). This project has received
funding from the European Union’s Horizon 2020 research and innovation
programme under grant agreement No 101031655 (TEAPOT). K.W. and T.T.
acknowledge support from the Elemental Strategy Initiative conducted by the
MEXT, Japan and the CREST (JPMJCR15F3), JST.

## Author contributions

R.K., S.K.S., and U.R. contributed to device fabrication, data acquisition,
and analysis. A.D. contributed to conceiving the idea and designing the
experiment, data interpretation, and analysis. K.W. and T.T. synthesized the
hBN single crystals. J.P., C.S., Y.G., and A.D.M. contributed to the
development of theory and data interpretation, and all the authors contributed
to writing the manuscript.

## Competing interests

The authors declare no competing interests.

## Data and materials availability:

The data presented in the manuscript are available from the corresponding
author upon request.

See pages 1-31 of SI.pdf


# Censored EM algorithm for Weibull mixtures: application to arrival times of
market orders

Markus Kreer  CSSM, Faculty of Sciences, School of Physical Sciences,
Department of Physics, University of Adelaide, 5005, Adelaide, Australia.
CAMPUSERVICE GmbH, Servicegesellschaft der Johann Wolfgang Goethe-Universitat
Frankfurt, Rossertstrasse 2, 60323 Frankfurt am Main, Germany  Feldbergschule,
Oberhochstadter Str., 20, 61440, Oberrursel (Taunus), Germany  Ayşe Kızılersü
CSSM, Faculty of Sciences, School of Physical Sciences, Department of Physics,
University of Adelaide, 5005, Adelaide, Australia.
<EMAIL_ADDRESS>Anthony W. Thomas  CSSM, Faculty of Sciences,
School of Physical Sciences, Department of Physics, University of Adelaide,
5005, Adelaide, Australia.

###### Abstract

In a previous analysis the problem of “zero-inflated” time data (caused by
high frequency trading in the electronic order book) was handled by left-
truncating the inter-arrival times. We demonstrated, using rigorous
statistical methods, that the Weibull distribution describes the corresponding
stochastic dynamics for all inter-arrival time differences except in the
region near zero. However, since the truncated Weibull distribution was not
able to describe the huge “zero-inflated” probability mass in the
neighbourhood of zero (making up approximately 50% of the data for limit
orders), it became clear that the entire probability distribution is a mixture
distribution of which the Weibull distribution is a significant part. Here we
use a censored EM algorithm to analyse data for the difference of the arrival
times of market orders, which usually have a much lower percentage of zero
inflation, for four selected stocks trading on the London Stock Exchange.

††preprint: ADP-20-34/T1144

##  I Introduction

Electronic Order Book (EOB) trading is now common to most stock exchanges. A
set of EOB data from the FTSE for the period June to September 2010 was
described and partly analysed in Ref.[1]. In this data set timestamps of
changes in the EOB were given in milliseconds and because of the huge volume
of EOB trading by (ultra-)high frequency trading algorithms, taking place on a
microsecond scale, differences of timestamps appeared to be zero. This is why
the timestamp time series appeared to be “zero-inflated”. Because of this
rounding error, all differences in timestamps, $\Delta t$, are mapped to our
actual data set as follows

$\displaystyle\Delta t\in(0.0,0.5)=\mathcal{I}_{1}$
$\displaystyle\longrightarrow$ $\displaystyle 0=c_{1}$ $\displaystyle\Delta
t\in[0.5,1.5)=\mathcal{I}_{2}$ $\displaystyle\longrightarrow$ $\displaystyle
1=c_{2}$ $\displaystyle\Delta t\in[1.5,2.5)=\mathcal{I}_{3}$
$\displaystyle\longrightarrow$ $\displaystyle 2=c_{3}$ $\displaystyle etc.$

where the integer numbers $c_{1},c_{2},...$ are the rounded time stamp
differences to the precision of milliseconds.

In Ref.[1] it was demonstrated that sets of non-negative time differences
between subsequent market orders (MO) with $\Delta t>10$ milliseconds describe
a random variable $X$ that could be fitted to parametric distributions, such
as a left-truncated Weibull distribution, which passed a rigorous set of
goodness-of-fit tests, such as Kolmogorof-Simirnov, Anderson-Darling, Cramer
von Misses and Kuiper tests111The analysis in Ref.[1] also included limit
orders (LO) but this will not be the topic of our letter..

Here we analyse the entire set of observations of the random variable $X$,
including the ”zero-inflated” ones, by starting from a Weibull distribution
for the whole range

$\displaystyle
f_{i}(x|\alpha_{i},\beta_{i})=\frac{\beta_{i}}{\alpha_{i}}\left(\frac{x}{\alpha_{i}}\right)^{\beta_{i}-1}e^{-\left(\frac{x}{\alpha_{i}}\right)^{\beta_{i}}}\,,$
(1)

where $\alpha_{i}>0$ is the scale parameter and $\beta_{i}>0$ the shape
parameter. Note that for $\beta_{i}=1$ we recover the exponential
distribution. We denote the parameter vector by
$\theta_{i}=(\alpha_{i},\beta_{i})$ and use the subscript index $i$ to denote
various Weibull or exponential distributions with different parameter vectors
$\theta_{i}$.

Now, the attempt to analyse the entire data set including the “zero-inflated”
part (which sometimes was even more than half of the observed data set)
requires a different procedure. One promising approach is to apply interval
censoring to the observed data for the intervals of small time differences
$\Delta t$: we use the above intervals
$\mathcal{I}_{1},\mathcal{I}_{2},\mathcal{I}_{3},...$ together with the
information of how many observations belong to them, i.e.
$N_{1},N_{2},N_{3},...$. In this notation we can write an observed sample of a
positive random variable $X$ as

$\displaystyle x_{1},$ $\displaystyle x_{2},\cdots$
$\displaystyle,x_{n},x_{n+1},x_{n+2},...,x_{N}$ (2) $\displaystyle=$
$\displaystyle
x_{1},x_{2},...,x_{n},\underbrace{c_{1},c_{1},...,c_{1}}_{N_{1}\text{times}},...,\underbrace{c_{L},c_{L},...,c_{L}}_{N_{L}\text{times}}\,,$

where, after grouping, all observations with index bigger than $n$ have been
censored (with $L$ censoring intervals).

Despite censoring, the random variable $X$ itself will be assumed to come from
a mixed distribution with a mixture of $M$ components. As a consequence of the
findings in Ref.[1], we expect that a significant component of this mixed
distribution should come from a Weibull distribution. Similar results where
found in Ref.[3]: the authors found that the differences between subsequent
MO’s for 30 DJIA stocks from the NYSE in October 1999 can be well described by
Weibull distributions. Thus we want to investigate here mixtures of the
following kind

$\displaystyle f(x|\theta)=\sum_{i=1}^{M}\Pi_{i}f_{i}(x|\theta_{i})$ (3)

where the weights, $\Pi_{i}$, satisfy $\Pi_{i}\geq 0$ with
$\sum_{i=1}^{M}\Pi_{i}=1$ and the probability density functions
$f_{i}(\cdot|\theta_{i})$ are given by Eq.(1), where the parameters
$\theta_{i}$ need to be estimated by maximum likelihood estimation or other
methods222In the following the index ”i” stands for the mixture components and
the index ”j” for the number of observation.. Defining
$\Theta=(\theta_{1},...,\theta_{M})$, the log-likelihood function
$\mathcal{L}$ for $L$ censoring intervals
$\mathcal{I}_{1},...,\mathcal{I}_{L}$ is now given as in Ref.[5] by

$\displaystyle\mathcal{L}=f(x_{1}|\Theta)\cdots\
f(x_{n}|\Theta)\cdot\left(\int_{\mathcal{I}_{1}}~{}dyf(y|\Theta)\right)^{N_{1}}\cdots\left(\int_{\mathcal{I}_{L}}~{}dyf(y|\Theta)\right)^{N_{L}}$
(4)

which is usually a difficult expression to handle. However, given our data
sample Eq.(2), the estimation problem would be a standard maximum likelihood
estimation (MLE) problem if we knew by some indicator $z_{ij}$ (taking values
0 or 1) denoting which observation $x_{j}$ belongs to which probability
density function $f_{i}(\cdot|\theta_{i})$ and likewise some indicator
$\tilde{z}_{i\ell}$ (also taking values 0 or 1) which censored observation
$c_{\ell}$ belongs to which distribution. In this case, we would just group
the observations and thus factorize the likelihood
$\mathcal{L}=\mathcal{L}_{1}\cdots\mathcal{L}_{M}$ and separately maximize
each group likelihood $\mathcal{L}_{i}$ corresponding to one mixture component
$f_{i}(\cdot|\theta_{i})$.

The expectation-maximization (EM) algorithm Ref.[6, 7, 8] will be used to
iteratively generate estimates for the indicators $z_{ij}$, for uncensored
observations, $x_{j}$ and $\tilde{z}_{i\ell}$ for censored observations,
$c_{\ell}$, and solve the estimation problem (provided it converges). The
problem for censored mixtures has been discussed in some detail in Ref.[9]. In
our study we apply this analysis for mixtures of exponential and Weibull
distributions and study market order (MO) inter-arrival times. In section 2 we
give a very brief review of the censored EM algorithm as given by Ref.[9] and
in section 3 the relevant equations for Weibull distributions are given.
Section 4 summarizes our results and conclusions.

##  II Censored EM algorithm for mixtures in a nutshell

The EM algorithm is an iterative procedure where an expectation-step (E-step)
tries to estimate the unobservable indicators $z_{ij}$ and
$\tilde{z}_{i\ell}$, respectively, and then uses the result in the
maximization step (M-step) to estimate the parameters of the mixtures by an
MLE. After the M-step there is another E-step and so on. Here we follow
closely Ref.[9], where a hint is given that this iteration converges for
certain function families (including exponential and Weibull distribution
functions) in a certain limited parameter range. In the following sections the
integer index $k$ denotes the number of the iteration.

###  II.1 The E-step

Using the above notation and assuming that the parameter vector
$\theta_{i}^{(k-1)}$ for each mixture component $i$ are known from a previous
step, Eq.(3.3) in the Ref.[9] provides the following term for the uncensored
observations, $j=1,...,n$ for the mixture component $i$

$\displaystyle z_{ij}^{(k)}$ $\displaystyle=$
$\displaystyle\frac{f_{i}(x_{j}|\theta_{i}^{(k-1)})}{\sum_{i=1}^{M}\Pi_{i}^{(k-1)}f_{i}(x_{j}|\theta_{i}^{(k-1)})}\cdot\Pi_{i}^{(k-1)}$
(5)

and for the censored observations in the censoring intervals
$\mathcal{I}_{\ell}=[\xi_{\ell-1},\xi_{\ell})$ with $\ell=1,2,...,L$, Eq.(3.4)
in the Ref.[9] provides the following expression for the mixture $i$th
component

$\displaystyle\tilde{z}_{i\ell}^{(k)}$ $\displaystyle=$
$\displaystyle\frac{\int_{\mathcal{I}_{\ell}}dy~{}f_{i}(y|\theta_{i}^{(k-1)})}{\sum_{i=1}^{M}\Pi_{i}^{(k-1)}\int_{\mathcal{I}_{\ell}}dy~{}f_{i}(y|\theta_{i}^{(k-1)})}\cdot\Pi_{i}^{(k-1)}$
(6)

###  II.2 The M-step

The sample size, $N=n+\sum_{\ell=1}^{L}N_{\ell}$, is the number of all
observations (uncensored and censored) and the new weights after the E-step
are given by the following update-rule (Eq.(3.7) of Ref.[9]):

$\displaystyle\Pi_{i}^{(k)}$ $\displaystyle=$ $\displaystyle\hskip
8.5359pt\frac{1}{N}\sum_{j=1}^{n}\frac{f_{i}(x_{j}|\theta_{i}^{(k-1)})}{\sum_{i=1}^{M}\Pi_{i}^{(k-1)}f_{i}(x_{j}|\theta_{i}^{(k-1)})}\cdot\Pi_{i}^{(k-1)}$
$\displaystyle+\frac{1}{N}\sum_{\ell=1}^{L}N_{\ell}\frac{\int_{\mathcal{I}_{\ell}}dy~{}f_{i}(y|\theta_{i}^{(k-1)})}{\sum_{i=1}^{M}\Pi_{i}^{(k-1)}\int_{\mathcal{I}_{\ell}}dy~{}f_{i}(y|\theta_{i}^{(k-1)})}\cdot\Pi_{i}^{(k-1)}\,.$

Note that this update-rule is just same as the following formula

$\displaystyle\Pi_{i}^{(k)}=\frac{1}{N}\left(\sum_{j=1}^{n}z_{ij}^{(k)}+\sum_{\ell=1}^{L}N_{\ell}\tilde{z}_{i\ell}^{(k)}\right)$
(7)

Now, the following expression needs to be maximised with respect to the
parameter vectors $\theta_{i}^{(k)}$, where the index $i$ refers to the
mixture and $k$ to the actual number in the iterative proceedure

$\displaystyle\sum_{i=1}^{M}\sum_{j=1}^{n}z_{ij}^{(k)}\log{f_{i}(x_{j}|\theta_{i}^{(k)})}+\sum_{i=1}^{M}\sum_{\ell=1}^{L}N_{\ell}~{}\tilde{z}_{i\ell}^{(k)}\int_{\mathcal{I}_{\ell}}dy~{}\log{f_{i}(y|\theta_{i}^{(k)})}h_{i}(y|c_{\ell},\theta_{i}^{(k-1)})$
(8)

Here, the conditional density function on the censoring interval
$\mathcal{I}_{\ell}$ is given by Ref.[9] Eq.(3.5)

$\displaystyle
h_{i}(y|c_{\ell},\theta_{i}^{(k-1)})=\frac{f_{i}(y|\theta_{i}^{(k-1)})}{\int_{\mathcal{I}_{\ell}}dy~{}f_{i}(y|\theta_{i}^{(k-1)})}$
(9)

The MLE equations will be obtained by differentiating Eq.(8) with respect to
the components of the parameter vector $\theta_{i}^{(k)}$. Note that in this
expression terms like $\sum_{i,j}z_{ij}^{(k)}\log{\Pi_{i}^{(k)}}$ and the
normalization condition $\mu\cdot\left(\sum_{i}\Pi_{i}^{(k)}-1\right)$ are not
given because they depend only on the parameter vector $\theta_{i}^{(k-1)}$
from the previous iteration and are irrelevant for the maximization with
respect to the parameter vector $\theta_{i}^{(k)}$. Note also that the
maximization problem decouples into independent maximization problems for each
individual probability distribution.

##  III Implementation of the censored Weibull mixtures

For simplicity we now consider a 2-component mixture consisting of an
exponential distribution, denoted by $i=1$, and a general Weibull
distribution, denoted by $i=2$, as given in Eq.(1). Note that here the
$\alpha$ and $\beta$ have a subscript $i$ for the mixture and a superscript
$(k)$ for the iteration step in the EM algorithm. The expression corresponding
to Eq.(8) is given in Appendix A. Our algorithmic results extend the results
as given by Ref.[10] for exponential mixtures. It is clear how to modify our
computations for arbitrary mixtures, e.g. $(p+r)$ mixtures consisting of $p$
exponentials and $r$ Weibulls,with $p,r=0,1,2,...$

###  III.1 MLE equations for M-step

Maximising the expression in Appendix A, Eq.(LABEL:eq:MLE), with respect to
the first parameter, $\alpha_{1}^{(k)}$ leads for the exponential distribution
(index $i=1$) to an explicit solution

$\displaystyle\alpha_{1}^{(k)}=\frac{\sum_{j=1}^{n}z_{1j}^{(k)}x_{j}+\sum_{\ell=1}^{L}N_{\ell}\tilde{z}_{1\ell}^{(k)}C_{1\ell}^{(k-1)}}{\sum_{j=1}^{n}z_{1j}^{(k)}+\sum_{\ell=1}^{L}N_{\ell}\tilde{z}_{1\ell}^{(k)}}$
(10)

where the quantity $C_{1\ell}^{(k-1)}$ is defined in Appendix A, Eq.(15).

The MLE equations for the Weibull distribution (index $i=2$) are obtained by
computing $\frac{\partial}{\partial\alpha_{2}^{(k)}}$ and equating the
expression to 0,

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}z_{2j}^{(k)}\left[-1+\left(\frac{x_{j}}{\alpha_{2}^{(k)}}\right)^{\beta_{2}^{(k)}}\right]$
$\displaystyle+$
$\displaystyle\sum_{\ell=1}^{L}N_{\ell}~{}\tilde{z}_{2\ell}^{(k)}\left\\{-1+\left(\frac{\alpha_{2}^{(k-1)}}{\alpha_{2}^{(k)}}\right)^{\beta_{2}^{(k)}}\frac{\Gamma\left(\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1,\zeta_{\ell-1}\right)-\Gamma\left(\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1,\zeta_{\ell}\right)}{e^{-\zeta_{\ell-1}}-e^{-\zeta_{\ell}}}\right\\}$

and likewise for $\frac{\partial}{\partial\beta_{2}^{(k)}}$

$\displaystyle
0=\sum_{j=1}^{n}z_{2j}^{(k)}\left[\frac{1}{\beta_{2}^{(k)}}+\log{\frac{x_{j}}{\alpha_{2}^{(k)}}}-\left(\frac{x_{j}}{\alpha_{2}^{(k)}}\right)^{\beta_{2}^{(k)}}\log{\frac{x_{j}}{\alpha_{2}^{(k)}}}\right]$
$\displaystyle+\sum_{\ell=1}^{L}N_{\ell}~{}\tilde{z}_{2\ell}^{(k)}\left\\{\frac{1}{\beta_{2}^{(k)}}+\log{\frac{\alpha_{2}^{(k-1)}}{\alpha_{2}^{(k)}}}+\frac{D_{2\ell}^{(k)}}{e^{-\zeta_{\ell-1}}-e^{-\zeta_{\ell}}}\right\\}\,\,$
(12)

The auxiliary functions $D_{2\ell}^{(k)}$, as well as $\zeta_{\ell}$, are
defined in the Appendix A. Reference [11] has obtained similar results for the
complete finite mixture of Weibull distributions albeit without the censoring
terms displayed here.

###  III.2 Algorithmical implementation

We use one censoring interval only, $\mathcal{I}_{1}=(0,0.5)$ and $L=1$ to
handle our zero-inflated data sets (see Ref.[1]). Also in the spirit of
Ref.[12], as a practical approximation to simplify the solution of these non-
linear equations, we use a self-consistency assumption by setting the ratios
$\alpha_{2}^{(k-1)}/\alpha_{2}^{(k)}$ and $\beta_{2}^{(k-1)}/\beta_{2}^{(k)}$
equal to 1 in the MLE equations for the Weibull parameters. This considerably
simplifies the MLE equations, Eqs. (LABEL:eq:Weibull_alpha)–(12), and in our
experience leads to rapid convergence in most cases.333A comparison to the
direct maximisation of the terms in Eq.(8) as given in the Appendix A leads to
comparable results and might justify our trick to speed-up convergence. Note
that if we already knew the exact solution $\alpha_{2}^{\infty}$ and
$\beta_{2}^{\infty}$, i.e. if we started with the true fixed point, these
equations would be trivially satisfied. For the EM-algorithm we proceed as
follows444We describe the censored EM-algorithm for a mixture of 1 exponential
+ 1 Weibull. The generalisation to arbitrary mixtures is obvious:

  1. 1.

At $k=0$ initialize mixture weights $\Pi_{i}^{(0)}$ and initial values
$\alpha_{1}^{(0)},\alpha_{2}^{(0)},\beta_{2}^{(0)}$

  2. 2.

Compute $z_{ij}^{(k)},\tilde{z}_{i\ell}^{(k)},\Pi_{i}^{(k)}$ using Eq.(5),
(6), (7) for $k=1,2,...$

  3. 3.

Compute $\alpha_{1}^{(k)}$ using Eq.(10) for exponential distribution for
$k=1,2,...$

  4. 4.

Compute $\alpha_{2}^{(k)}$ using Eq.(LABEL:eq:Weibull_alpha) for Weibull
distribution putting here only $\beta_{2}^{(k)}=\beta_{2}^{(k-1)}$ for
$k=1,2,...$

  5. 5.

Compute $\beta_{2}^{(k)}$ using Eq.(12) for Weibull distribution for
$k=1,2,...$

  6. 6.

Compute the current log-likelihood from Eq.(3)–(4)

  7. 7.

If the absolute value of “log-likelihood at step $k$ minus log-likelihood at
step $k-1$” is bigger than $\varepsilon>0$, then put $k\rightarrow k+1$ and go
back to step 2., otherwise terminate.

In our experience this version of the censored EM algorithm (based on the MLE
equations) converges sufficiently fast to a desired maximum solution for
suitable initial conditions. We have taken $\varepsilon=10^{-5}$ and start
with equal mixtures setting $\beta=1$ and take values of $\alpha$ motivated by
our previous study Ref.[1]. When testing the algorithm, its results have been
cross-checked by the corresponding algorithm which uses a direct maximisation
of the objective function given in Eq.(8) in the M-step. Usually the results
obtained both way agree fairly well. In Ref.[9] it is claimed that there is
local convergence almost surely.

##  IV Analysis of MO arrival times and model selection

###  IV.1 A first approach: “naive” analysis of entire data set

Here we take all time stamps of a given stock from 1st June to 30th September
2010 into one large sample and fit various mixture models to the arrival times
(i.e. the difference of subsequent timestamps) using the algorithm described
above. This analysis would be sensible if the stochastic process were
stationary and the sample data were independent identically distributed
(i.i.d). In Table 1 we provide the log-likelihood per data point to obtain a
first idea about the best model. The percentage numbers in round brackets
denote the proportion of those data points which have been censored. Note that
the quantity “log-likelihood per data point”, or “average log-likelihood”,
corresponds to a negative Shannon entropy per event because for large sample
size $N$ and i.i.d. data we have under the usual consistency property of the
maximum likelihood estimator

$\displaystyle\frac{1}{N}\log{\mathcal{L}}\sim\mathbb{E}(\log{p})=\int\log{p}\cdot
dp$

The physical meaning of this quantity is the “information content” or
“surprisal” when a new MO enters the EOB.

Table 1: Average log-likelihood |  | RIO | BARC | RRLN | ABFLN
---|---|---|---|---|---
|  | (2.5%) | (2.3%) | (2.5%) | (2.1%)
Model | dof |  |  |  |
1 exp + 1 wbl | 4 | -8.357 | -8.294 | -9.510 | -10.074
0 exp + 2 wbl | 5 | -8.349 | -8.288 | -9.492 | -10.060
3 exp + 0 wbl | 5 | -8.422 | -8.394 | -9.792 | -10.323
2 exp + 1 wbl | 6 | -8.350 | -8.290 | -9.494 | -10.061
1 exp + 2 wbl | 7 | -8.348 | -8.288 | -9.489 | -10.057
4 exp + 0 wbl | 7 | -8.360 | -8.300 | -9.511 | -10.081
0 exp + 3 wbl | 8 | -8.348 | -8.288 | -9.489 | -10.054
3 exp + 1 wbl | 8 | -8.349 | -8.289 | -9.491 | -10.055
5 exp + 0 wbl | 9 | -8.351 | -8.291 | -9.497 | -10.063

We see from Table 1 that all models seem to yield very similar Shannon entropy
measures for an individual stock. Also, depending on the trading activity, the
entropies differ: for a heavily traded stock such as RIO or BARC the
“surprise” is lower than for less actively stocks such as RRLN or ABFLN. A
model selection criterion here would be the model with least entropy and thus
a mixture of 3 Weibull distributions seems to be the best choice.

However, we know from Ref.[1] that the data are only i.i.d. for smaller
subsamples and thus the scale parameters in the exponential and Weibull
distribution will exhibit a time dependence. When looking at smaller samples
it became customary for model selection rather looking at the above Shannon
entropy to put the sample size in relation to the degrees of freedom (dof)
(see e.g. Ref.[15]). We have decided to use the Bayesian Information Criterion
(BIC) (Ref.[16], Ref.[15])

$\displaystyle\text{BIC}=-2\log{\mathcal{L}}+d\log{N}$ (13)

where $\mathcal{L}$ are likelihoods and $d$ is the number of degrees of
freedom of the individual model. From Table 1 we can generate hypothetically
Table 2 by using Eq.(13) with $N=200$ for “larger stocks”, with higher trading
activity (RIO, BARC), and $N=100$ for “smaller stocks”, with lower trading
activity (RRLN, ABFLN). These values for $N$ correspond to time intervals of
approximately 10 minutes and we have seen in Ref.[1] that these time intervals
yield samples for which the asumption of stationarity of the data set can be
somehow justified. We clearly see that now mixtures with low dof are favored,
in particular the mixture “1 exponential + 1 Weibull”. Note also that the
suggestion of Ref.[17] to model the arrival time distribution as a suitable
mixture of exponential waiting times, will be excluded in the model selection
using BIC by a too high value for the dof.

Table 2: Expected BIC from Table1 with different sample size $N$ |  | RIO | BARC | RRLN | ABFLN
---|---|---|---|---|---
|  | (2.5%) | (2.3%) | (2.5%) | (2.1%)
Model | dof | $N=200$ | $N=200$ | $N=100$ | $N=100$
1 exp + 1 wbl | 4 | 3363.99 | 3338.79 | 1923.19 | 2035.99
0 exp + 2 wbl | 5 | 3366.09 | 3341.69 | 1924.89 | 2038.49
3 exp + 0 wbl | 5 | 3395.29 | 3384.09 | 1984.89 | 2091.09
2 exp + 1 wbl | 6 | 3371.79 | 3347.79 | 1930.59 | 2043.99
1 exp + 2 wbl | 7 | 3376.29 | 3352.29 | 1934.89 | 2048.49
4 exp + 0 wbl | 7 | 3381.09 | 3357.09 | 1939.29 | 2053.29
0 exp + 3 wbl | 8 | 3381.59 | 3357.59 | 1940.19 | 2053.19
3 exp + 1 wbl | 8 | 3381.99 | 3357.99 | 1940.59 | 2053.39
5 exp + 0 wbl | 9 | 3388.08 | 3364.08 | 1947.08 | 2060.28

###  IV.2 A second approach: Analysis of stationary subsamples

The previous analysis was naive as we assumed the entire data sample would
consist of i.i.d. random variables. This is clearly not the case. As already
noticed in Ref.[1] the volume of trading changes a great deal during a trading
day, so that the scale parameter $\alpha$ must also vary. However, in Ref.[1]
Kizilersü et al. argued that for “small” subsamples the assumptions of
stationarity and i.i.d. might be expected to be justified. We take as
subsample size $N=200$ for “larger stocks” with higher trading activity (RIO,
BARC) and $N=100$ for “smaller stocks” with lower trading activity (RRLN,
ABFLN). Motivated by Table 2, our candidates for possible models are the
following mixtures

  * •

1 exp + 1 wbl (dof=4)

  * •

0 exp + 2 wbl (dof=5)

  * •

3 exp + 0 wbl (dof=5)

  * •

2 exp + 1 wbl (dof=6)

To quote Ref.[18] “[t]he practice of using the same data set to select a best-
fitting model and to assess the significance of model parameter estimates or
interpret the model structure is based on the often implicit assumption that
the selected model is the true model that generated the data […]. However,
this assumption does not hold in general. The sampling error related to model
selection is ignored if the same data are used for inference.” Thus, we
separate the task of model selection from the best fit of the parameters.

####  IV.2.1 Model selection and bootstrapping

For the model selection we take for every trading day in the months of June
and July 2010 a random time stamp for each individual stock. From this time
stamp onward we take 200 successive time stamps for the big stocks (RIO and
BARC) and 100 successive time stamps for the small stocks (RRLN and ABFL). For
each of these original samples we generate additional 999 bootstrap samples
out of the original sample (e.g. for more details on bootstrapping the
standard reference [19]) . For each ensemble of 1000 bootstrap samples we run
the censored EM-algorithm and compute the log-likelihood and the BIC. Hence,
for each model we have obtained a BIC distribution which is approximately
normal. We then perform a Welch t-test with 5% confidence level on the
following hypothesis

“Can the alternative model beat 1 exp + 1 Weibull using BIC?”

We then count the success rate for the winning distribution. Our results are
depicted in Table 3, displaying the proportion of winnings. Our first
intuition is confirmed: the mixture “1 exponential + 1 Weibull” is the clear
winner.

Table 3: BIC-winners from bootstrapping ensembles in June and July 2010 Model | dof | RIO | BARC | RRLN | ABFLN
---|---|---|---|---|---
1 exp + 1 wbl | 4 | 0.77 | 0.77 | 0.77 | 0.76
0 exp + 2 wbl | 5 | 0.09 | 0.14 | 0.09 | 0.12
3 exp + 0 wbl | 5 | 0.14 | 0.09 | 0.14 | 0.12
2 exp + 1 wbl | 6 | 0 | 0 | 0 | 0

####  IV.2.2 Results for the preferred model: “1 exponential + 1 Weibull”

In this subsection we use the convention that for the exponential contribution
of the mixture $\beta_{1}=1$ will be suppressed and for the Weibull
contribution we write $\beta$ rather than $\beta_{2}$ for easier reading. In
Table 4 the results of the estimated parameters are summarised. We find for
the “complete” data samples that the Weibull shape parameter $\beta$ takes on
a universal value of approximately 0.57 as already found in Ref.[1] (where
using a left-truncation was the way of handling the “zero-inflated” data).

Table 4:  Weibull component for “1 exp + 1 Weibull” | $\frac{1}{N}\left(\log{L}\pm\Delta\log{L}\right)$ | weight | median $\beta$ | $\beta\pm\Delta\beta$ | No. samples
---|---|---|---|---|---
RIO | -8.29$\pm$0.63 | 0.82 | 0.55 | 0.57 $\pm$ 0.11 | 3107
BARC | -8.24$\pm$0.72 | 0.81 | 0.57 | 0.58 $\pm$ 0.11 | 3346
RRLN | -9.35$\pm$0.83 | 0.78 | 0.48 | 0.50 $\pm$ 0.12 | 535
ABFLN | -9.91$\pm$0.78 | 0.81 | 0.47 | 0.49 $\pm$ 0.12 | 291
Table 5: Scale parameters $\alpha$ in [ms] for “1 exp + 1 Weibull” | median $\alpha_{2}$ | range | median $\alpha_{1}$ | range | No. samples
---|---|---|---|---|---
RIO | 2499 | [1099,5491] | 17.2 | [6.5,70.8] | 3107
BARC | 2452 | [1078,5310] | 19.0 | [9.0,64.0] | 3346
RRLN | 13846 | [6079,28988] | 16.0 | [6.6,47.1] | 535
ABFLN | 23095 | [10374,45580] | 18.7 | [7.0,49.1] | 291

From Table 5 we see that the median of the Weibull scale parameter
$\alpha_{2}$ varies for different stocks (depending on their trading
activity), whereas the median of the exponential scale parameter $\alpha_{1}$
seems to be the same for different stocks. From Table 5 we suspect a stronger
time dependence for the Weibull scale parameter. To investigate this time
dependence, we divide the trading hours from 9:00 to 17:30 UK summer time in
intervals of 10 (respectively 30) minutes for RIO and BARC (RRLN and ABFLN
respectively) and average the values of $\alpha_{2}$, $\alpha_{1}$ and $\beta$
over the trading days from June to September 2010. Due to this partitioning of
the data, the sample size will vary significantly, depending on the time of
the day, with $N\gg 200$ at some times and $N\ll 200$ at some other times. As
we have already remarked in Ref.[1] the Weibull scale parameter $\alpha_{2}$
will exhibit a strong time dependence during the trading day. The more
actively a stock is traded, the smaller the Weibull scale parameter
$\alpha_{2}$ will be. We see from Figure 1 that the typical scale parameter
for the big stocks RIO and BARC is well below 10 seconds, and both curves as
function of time are nearly identical. This can be explained by index
arbitrage in the FTSE100, which requires trading in big stocks such as RIO and
BARC at the same time to exploit the arbitrage. Obviously at lunch time there
is less activity and the $\alpha_{2}$ becomes larger.

Figure 1: Weibull $\alpha_{2}$ in milliseconds for various tickers during
trading hours. Figure 2: $\beta$ for various tickers during trading hours.

Since the Fisher information matrix is not diagonal for the MLE problem for
Weibull distributions a bias in the estimated scale parameter $\alpha_{2}$
will also result in a bias of the estimated shape parameter $\beta$. Thus, we
see in Figure 2 that the value of $\beta$ becomes slightly larger at lunch
time, when the estimated value of $\alpha_{2}$ is larger than that found at
the opening or closing of trading hours. Despite this, the high level of
stability found for $\beta$ suggests that it is reasonable to assume that the
true shape parameter is universal with $\beta=0.57$ as conjectured in Ref.[1].

Figure 3: $\alpha_{1}$ in milliseconds for various tickers during trading
hours.

Figure 4: Mixture weights for various tickers during trading hours.

Finally we point the reader’s attention to Fig. 3 which displaying the time
dependence of the scale parameters $\alpha_{1}$ of the exponential
contribution and Fig. 4 displaying the time dependence of the mixture weights:
the time dependence of $\alpha_{1}$ is mainly in the region of 10 to 20
milliseconds for all stocks with exceptions of the “smallest” and sometimes
illiquid stock ABFLN. Also the weights of this exponential contribution in the
two-component mixture is nearly independent of time and consistently around 20
%, valid for all stocks.

##  V Conclusion

The censored EM-algorithm in combination with a bootstrapping argument applied
to the Baysean Information Criterion (BIC) allows us to choose as a model for
the MO arrival times a two-component mixture distribution consisting of “1
exponential + 1 Weibull”. Of course, this conclusion is not applicable in the
exceptional case of extraordinary trading activity in a stock when critical
information is disclosed to the market participants. The first component of
this mixed distribution is an exponential distribution with a relative weight
of approximately 20% and a rather short scale parameter, in the range of 10 to
20 milliseconds. This result is independent of the stock under consideration
and almost constant during the trading day. The second component, with a
relative weight of approximately 80%, is a Weibull distribution for which the
scale parameter varies intra-day with trading activity and lies typically
between 1000 and 25000 milliseconds, albeit with universal shape parameter
$\beta\approx 0.57$.

This result can be understood with a view to a typical stock exchange computer
architecture (e.g. Ref.[20]): market orders are captured via various order
gateways which forward the electronic orders to the so called accumulator,
where timestamps are given. Low latency order gateways typically cater for
high-frequency traders (e.g. algorithmic hedge funds) whose buy- and sell-
orders are generated by computers located in the stock exchange’s immediate
neighbourhood to minimize the transmission time. Here an exponential waiting
time between these signals (with a short time scale parameter) is to be
expected resulting in a Poisson process for high-frequency buy- and sell-
orders. On the other hand, VSAT gateways cater for stock brockers whose
customers include both institutional and private clients and whose orders are
usually transmitted via an internet connection. It has been shown that
internet traffic sends information as TCP-“parcels” with arrival times that
can be described by Weibull distributions with a shape parameter less than 1
(see Ref.[21], Ref.[22]). In Ref.[21] it was shown that a shape parameter
$\beta=0.569$ provided an excellent fit to the observed data. This is close to
the value that we find for the Weibull distribution. The natural conclusion
would be that for the time period we have studied trading on the LSE
approximately 10% of the trading was done by market participants whose orders
were generated electronically by a computer in the immediate neighborhood,
whereas the other 90% of market participants were using the internet to
generate their buy- and sell-orders.

Finally we want to emphasize that by Ref.[23] any Weibull distribution
function with scale parameter $\beta$ smaller than 1 can be expressed as a
superposition of exponential waiting time distributions. Thus our favorite
model “1 Weibull + 1 exponential” is equivalent to a suitable mixture of
exponential waiting time distributions. Consequently, the proposed censored EM
algorithm for finite Weibull mixtures maybe compared to the problem discussed
firstly in Ref.[17] and later expanded in Ref.[24] of how to fit a waiting
time distribution consisting of a finite number of exponential waiting time
distributions to the observed tick-by-tick data. Although exponential mixtures
with more than 5 components seem to describe our actual data sets very well,
they are excluded by the BIC due to their too high dof-value.

##  Appendix A Expression of objective function in M-step and further auxilary
functions

First define the censoring intervals
$\mathcal{I}_{\ell}=[\xi_{\ell-1},\xi_{\ell})$. For simplicity introduce a
transformed quantity
$\zeta_{\ell}=(\xi_{\ell}/\alpha_{i}^{(k-1)})^{\beta_{i}^{(k-1)}}$. Then we
find after some lengthy computations in the spirit of section 2 the following
version of Eq.(8)

$\displaystyle\sum_{j=1}^{n}z_{1j}^{(k)}\left[\log{\frac{1}{\alpha_{1}^{(k)}}}-\frac{x_{j}}{\alpha_{1}^{(k)}}\right]+\sum_{\ell=1}^{L}N_{\ell}~{}\tilde{z}_{1\ell}^{(k)}\Bigg{[}\log{\frac{1}{\alpha_{1}^{(k)}}}-\frac{1}{\alpha_{1}^{(k)}}C_{1\ell}^{(k-1)}\Bigg{]}$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{n}z_{2j}^{(k)}\left[\log{\frac{\beta_{2}^{(k)}}{\alpha_{2}^{(k)}}}+(\beta_{2}^{(k)}-1)\log{\frac{x_{j}}{\alpha_{2}^{(k)}}}-\left(\frac{x_{j}}{\alpha_{2}^{(k)}}\right)^{\beta_{2}^{(k)}}\right]$
$\displaystyle+$
$\displaystyle\sum_{\ell=1}^{L}N_{\ell}~{}\tilde{z}_{2\ell}^{(k)}\Bigg{\\{}\log{\frac{\beta_{2}^{(k)}}{\alpha_{2}^{(k)}}}+(\beta_{2}^{(k)}-1)\log{\frac{\alpha_{2}^{(k-1)}}{\alpha_{2}^{(k)}}}$
$\displaystyle+$
$\displaystyle\left.\frac{\beta_{2}^{(k)}-1}{\beta_{2}^{(k-1)}}~{}\frac{e^{-\zeta_{\ell-1}}\log{\zeta_{\ell-1}}-e^{-\zeta_{\ell}}\log{\zeta_{\ell}}+\Gamma(0,\zeta_{\ell-1})-\Gamma(0,\zeta_{\ell})}{e^{-\zeta_{\ell-1}}-e^{-\zeta_{\ell}}}\right.$
$\displaystyle{\color[rgb]{1,0,0}-}$
$\displaystyle\left(\frac{\alpha_{2}^{(k-1)}}{\alpha_{2}^{(k)}}\right)^{\beta_{2}^{(k)}}\frac{\Gamma\left(\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1,\zeta_{\ell-1}\right)-\Gamma\left(\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1,\zeta_{\ell}\right)}{e^{-\zeta_{\ell-1}}-e^{-\zeta_{\ell}}}\Bigg{\\}}$

where

$\displaystyle
C_{1\ell}^{(k-1)}=\alpha_{1}^{(k-1)}+\frac{\xi_{\ell-1}e^{-\frac{\xi_{\ell-1}}{\alpha_{1}^{(k-1)}}}-\xi_{\ell}e^{-\frac{\xi_{\ell}}{\alpha_{1}^{(k-1)}}}}{e^{-\frac{\xi_{\ell-1}}{\alpha_{1}^{(k-1)}}}-e^{-\frac{\xi_{\ell}}{\alpha_{1}^{(k-1)}}}}$
(15)

Note that in Eq.(LABEL:eq:MLE) we have for the lower censoring interval
boundary $\zeta_{0}=0$ that the term with $\ell=1$ is well-behaved and reduces
to

$\displaystyle\frac{e^{-\zeta_{\ell-1}}\log{\zeta_{\ell-1}}-e^{-\zeta_{\ell}}\log{\zeta_{\ell}}+\Gamma(0,\zeta_{\ell-1})-\Gamma(0,\zeta_{\ell})}{e^{-\zeta_{\ell-1}}-e^{-\zeta_{\ell}}}$
$\displaystyle\vspace*{5cm}=-\frac{\gamma+e^{-\zeta_{1}}\log{\zeta_{1}}+\Gamma(0,\zeta_{1})}{1-e^{-\zeta_{1}}}$

We need to maximise expression Eq.(LABEL:eq:MLE) with respect to the
parameters $\alpha_{1}^{(k)},\alpha_{2}^{(k)},\beta_{2}^{(k)}$, all other
quantities being known from previous steps.

For the MLE equations the following expression arises during the lengthy
computations

$\displaystyle D_{2\ell}^{(k)}$ $\displaystyle=$
$\displaystyle\frac{e^{-\zeta_{\ell-1}}\log{\zeta_{\ell-1}}-e^{-\zeta_{\ell}}\log{\zeta_{\ell}}+\Gamma(0,\zeta_{\ell-1})-\Gamma(0,\zeta_{\ell})}{\beta_{2}^{(k-1)}}$
$\displaystyle-\frac{1}{\beta_{2}^{(k-1)}}\left(\frac{\alpha_{2}^{(k-1)}}{\alpha_{2}^{(k)}}\right)^{\beta_{2}^{(k)}}\Bigg{[}\sum_{p=0}^{\infty}\frac{(-1)^{p}}{p!}\frac{\zeta_{\ell-1}^{\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1+p}-\zeta_{\ell}^{\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1+p}}{\left(\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1+p\right)^{2}}$
$\displaystyle-\Gamma\left(\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1\right)\left(\log{\zeta_{\ell-1}}-\log{\zeta_{\ell}}\right)$
$\displaystyle+\Gamma\left(\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1,\zeta_{\ell-1}\right)\log{\zeta_{\ell-1}}-\Gamma\left(\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1,\zeta_{\ell}\right)\log{\zeta_{\ell}}\Bigg{]}$
$\displaystyle-\log{\frac{\alpha_{2}^{(k-1)}}{\alpha_{2}^{(k)}}}\left(\frac{\alpha_{2}^{(k-1)}}{\alpha_{2}^{(k)}}\right)^{\beta_{2}^{(k)}}\Bigg{\\{}\Gamma\left(\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1,\zeta_{\ell-1}\right)-\Gamma\left(\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1,\zeta_{\ell}\right)\Bigg{\\}}$

and for $\ell=1$

$\displaystyle D_{21}^{(k)}$ $\displaystyle=$
$\displaystyle\frac{-\gamma-e^{-\zeta_{1}}\log{\zeta_{1}}-\Gamma(0,\zeta_{1})}{\beta_{2}^{(k-1)}}$
$\displaystyle-\frac{1}{\beta_{2}^{(k-1)}}\left(\frac{\alpha_{2}^{(k-1)}}{\alpha_{2}^{(k)}}\right)^{\beta_{2}^{(k)}}\Bigg{[}-\sum_{p=0}^{\infty}\frac{(-1)^{p}}{p!}\frac{\zeta_{1}^{\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1+p}}{\left(\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1+p\right)^{2}}$
$\displaystyle+\Gamma\left(\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1\right)\log{\zeta_{1}}-\Gamma\left(\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1,\zeta_{1}\right)\log{\zeta_{1}}\Bigg{]}$
$\displaystyle-\log{\frac{\alpha_{2}^{(k-1)}}{\alpha_{2}^{(k)}}}\left(\frac{\alpha_{2}^{(k-1)}}{\alpha_{2}^{(k)}}\right)^{\beta_{2}^{(k)}}\Bigg{\\{}\Gamma\left(\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1\right)-\Gamma\left(\frac{\beta_{2}^{(k)}}{\beta_{2}^{(k-1)}}+1,\zeta_{1}\right)\Bigg{\\}}$

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##  Appendix C Measurements of stellar metallicity

The two datasets used in this work were observed with different spectral
resolution, and analyzed using different spectral grids. The AO spectra (R
$\sim$ 5,000) were fitted with the MARCS spectral grid, while the seeing-
limited spectra (R $\sim$ 3310 - 4660) were fitted with the PHOENIX spectra
grid. We compare the metallicity measurements for the 27 common stars from
both datasets. See Figure 23. The median difference is 0.31 dex with a
standard deviation of 0.35 dex. We note that very metal-rich stars with
$[M/H]$ $>$ +0.5 dex generally show a larger discrepancy between the two
measurements owing to the greater systematic uncertainties in the high-
metallicity range. See Appendix A.2 for further discussion on how these metal-
rich stars may affect the fitting results on the cluster age.

We further investigated the effects of spectral resolution and spectral grids
on the metallicity measurements by re-fitting the original AO spectra using
the PHOENIX grid (low-spectral-resolution) with the same settings as the
seeing-limited dataset. See Figure 23 for the measurements. The median
difference between the MARCS-grid and PHOENIX-grid AO measurements is 0.14
dex. The median difference between the PHOENIX-grid AO measurements and the
PHOENIX-grid seeing-limited measurements is 0.17 dex. The re-fitted
metallicity measurements show that both the resolution of the spectra and the
grids have about the same effect on the overall difference between the two
datasets. Adding them together results in the total difference between the two
datasets. The 27 common stars between the two surveys have consistent
metallicity measurements within the uncertainties of each of the method,
indicating that the two datasets with different spectral resolution and grids
are in reasonable agreement.

##  Appendix D Comparison between the AO & seeing-limited results

The star formation history results that are modeled from two datasets with
different detection depth and spatial coverage are in great agreement. The
star formation history of the NSC shows no substantial discrepancy at
different distances to the Galactic center within the central $\sim$1.5 pc. In
total we have included $\sim$25% of the total cluster mass covering the
central $\sim$4 pc2 at a distance of 8 kpc. By modeling the two datasets
independently, we can investigate if there are systematic differences between
the datasets and assess the accuracy of our results. We fitted independently
the two datasets that were observed using different telescopes and
instruments, and analyzed using different spectral grids. Importantly, we
obtain consistent age estimates for both bursts from the star formation
history modelings. The possible systematic uncertainties we discussed (see
section 4.5) have been well represented in our reported 68% confidence
interval, and thus our reported results are robust and confident.

The deeper AO dataset seems to be more useful in modeling the star formation
history. The NIFS AO observations with higher spatial and spectral resolution
are more sensitive to differentiate supersolar and subsolar metallicity stars
(see details in Do et al. 2015; Feldmeier-Krause et al. 2017). Furthermore,
the deeper AO observations are able to detect the fainter and low-metallicity
stars below K = 14 mag, enabling a more intrinsic estimate of the mass
fraction of each burst and a better constraint on the age of the metal-poor
population. In the future, AO observations with wider area coverage, as well
as a higher spectral resolution, will be helpful to further constrain the star
formation history. In addition, the James Webb Space telescope will have the
ability to obtain spectra with increased depth and wavelength coverage, and
thus will largely increase the number of the observed subsolar metallicity
stars and help to place constraints on the origin of their progenitors.

Table 10: Summary of AO Observations

Namea | R.A. | Dec. | $K_{S}^{b}$ | $K_{S,err}$ | $H^{b}$ | $H_{err}$ | $A_{K_{S}}^{b}$ | $T_{eff,CO}^{c}$ | $\sigma_{T_{eff,CO}}$ | $T_{eff,*}^{d}$ | $\sigma_{T_{eff,*}}$ | [M/H]d | $\sigma_{[M/H]}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| (∘) | (∘) | (mag) | (mag) | (mag) | (mag) | (mag) | (K) | (K) | (K) | (K) |  |
E5-1-001 | 266.421656 | -29.007947 | 12.01 | 0.01 | 14.19 | 0.01 | 2.66 | 3260 | 171 | 3497 | 413 | 0.96 | 0.32
E5-1-002 | 266.421449 | -29.007402 | 12.61 | 0.01 | 14.51 | 0.01 | 2.47 | 3673 | 168 | 3671 | 414 | 0.55 | 0.32
E5-1-003 | 266.421601 | -29.007516 | 13.15 | 0.01 | 15.11 | 0.01 | 2.50 | 3611 | 170 | 3597 | 414 | 0.85 | 0.32

aafootnotetext: Name from Støstad et al. (2015).

bbfootnotetext: $K_{S}$\- and $H$-band photometry and $A_{K_{S}}$ extinction
taken from Schödel et al. (2010); see section 2.1. This is the first time the
reported matches to those stars were made between the catalogs.

ccfootnotetext: Effective temperature as derived from the calibrated
$T_{eff}$-$EW_{CO}$ relation; see Appendix B for details.

ddfootnotetext: Effective temperature and metallicity derived from full
spectrum fitting using the STARKIT code, Kerzendorf & Do (2015).

††footnotetext: (The full table is available online)

Table 11: Summary of seeing-limited Observations

Namea | R.A. | Dec. | $K_{S}^{b}$ | $K_{S,err}$ | $H^{b}$ | $H_{err}$ | $A_{K_{S}}^{b}$ | $T_{eff,CO}^{c}$ | $\sigma_{T_{eff,CO}}$ | $T_{eff,*}^{d}$ | $\sigma_{T_{eff,*}}$ | [M/H]d | $\sigma_{[M/H]}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| (∘) | (∘) | (mag) | (mag) | (mag) | (mag) | (mag) | (K) | (K) | (K) | (K) |  |
1 | 266.41675 | -29.010296 | 9.95 | 0.01 | 12.07 | 0.01 | 2.51 | 2982 | 251 | 3190 | 209 | 0.87 | 0.31
5 | 266.41571 | -29.012167 | 10.54 | 0.01 | 12.80 | 0.01 | 2.66 | 3119 | 164 | 3301 | 206 | 0.13 | 0.25
6 | 266.42401 | -29.003611 | 10.57 | 0.90 | 12.65 | 0.90 | 2.78 | 3408 | 181 | 3374 | 205 | 0.14 | 0.25

aafootnotetext: Name from Feldmeier-Krause et al. (2017).

bbfootnotetext: $K_{S}$\- and $H$-band photometry and $A_{K_{S}}$ extinction
taken from Schödel et al. (2010); Nogueras-Lara et al. (2018a) and Nishiyama
et al. (2006); see section 2.2. This is the first time the reported matches to
those stars were made between the catalogs.

ccfootnotetext: Effective temperature as derived from the calibrated
$T_{eff}$-$EW_{CO}$ relation; see Appendix B for details.

ddfootnotetext: Effective temperature and metallicity derived from full
spectrum fitting using the STARKIT code, Kerzendorf & Do (2015).

††footnotetext: (The full table is available online)