††thanks: Correspondence and requests for materials should be addressed to Q.
H. (email: q.he@fkf.mpg.de).††thanks: These authors contributed equally to
this work

# Real-space visualization of quasiparticle dephasing near the Planckian limit
in the Dirac line node material ZrSiS

Qingyu He† Max Planck Institute for Solid State Research, Heisenbergstrasse 1,
Stuttgart, 70569, Germany  Lihui Zhou  Max Planck Institute for Solid State
Research, Heisenbergstrasse 1, Stuttgart, 70569, Germany  Andreas W. Rost
SUPA, School of Physics and Astronomy, University of St Andrews, North Haugh,
St Andrews, Fife, KY16 9SS, UK  Dennis Huang  Max Planck Institute for Solid
State Research, Heisenbergstrasse 1, Stuttgart, 70569, Germany  Andreas
Grüneis  Institute for Theoretical Physics, Technische Universität Wien, 1040
Vienna, Austria  Leslie M. Schoop  Department of Chemistry, Princeton
University, Princeton, NJ 08544, USA  Hidenori Takagi  Max Planck Institute
for Solid State Research, Heisenbergstrasse 1, Stuttgart, 70569, Germany
Department of Physics, University of Tokyo, 113-0033 Tokyo, Japan  Institute
for Functional Matter and Quantum Technologies, University of Stuttgart, 70569
Stuttgart, Germany

###### Abstract

Dirac line node (DLN) materials are topological semimetals wherein a set of
symmetry protected crossing points forms a one-dimensional (1D) line in
reciprocal space. Not only are the linearly dispersing bands expected to give
rise to exceptional electronic properties, but the weak screening of the
Coulomb interaction near the line node may enhance electronic correlations,
produce new many-body ground states, or influence the quasiparticle lifetime.
We investigate the quasiparticle dynamics in the DLN material ZrSiS via
spectroscopic imaging scanning tunneling microscopy (SI-STM). By studying the
spatial decay of quasiparticle interference patterns (QPI) from point
scatterers, we were able to directly and selectively extract the phase
coherence length $l_{\textrm{QPI}}$ and lifetime $\tau_{\textrm{QPI}}$ for the
bulk DLN excitations, which are dominated by inelastic electron-electron
scattering. We find that the experimental $\tau_{\textrm{QPI}}(E)$ values
below $-$40 meV are very short, likely due to the stronger Coulomb
interactions, and lie at the Planckian limit $\hbar/|E|$. Our results
corroborate a growing body of experimental reports demonstrating unusual
electronic correlation effects near a DLN.

## Introduction

Over the past number of years, three-dimensional topological semimetals (TSMs)
have commanded intense experimental and theoretical attention Yan and Felser
(2017); Armitage  _et al._ (2018). Dirac and Weyl semimetals, which host band
crossing points and quasiparticles that obey relativistic Hamiltonians, have
been at the heart of these investigations Yan and Felser (2017); Armitage  _et
al._ (2018); Burkov  _et al._ (2011); Phillips and Aji (2014); Weng  _et al._
(2016). More recently, the notion of special crossing points (zero
dimensional, 0D) has been extended to crossing lines (1D) in several newly
discovered Dirac line node (DLN) materials. DLN materials can be considered as
a precursor that gives rise to Dirac/Weyl semimetals or topological insulators
upon breaking of symmetry and gapping of the excitation spectrum Burkov  _et
al._ (2011); Phillips and Aji (2014); Fang  _et al._ (2015); Weng  _et al._
(2015, 2016); Fang  _et al._ (2016).

ZrSiS has emerged as an ideal and prototypical DLN compound Schoop  _et al._
(2016); Neupane  _et al._ (2016). It has a layered crystal structure composed
of 2D square nets with nonsymmorphic symmetry, which gives rise to a 1D DLN
protected by crystal symmetry [Figs. 1(a)–1(c)]. The Fermi surface originating
from the DLN has a closed diamond shape with nearly perfectly nested, quasi-1D
segments [inset of Fig. 1(c)] Schoop  _et al._ (2016); Chen  _et al._ (2017);
Schilling  _et al._ (2017); Neupane  _et al._ (2016); Butler  _et al._ (2017);
Lodge  _et al._ (2017). A number of high-resolution angle resolved
photoemission spectroscopy (ARPES) Topp  _et al._ (2017); Schoop  _et al._
(2016); Fu  _et al._ (2019); Neupane  _et al._ (2016) and scanning tunneling
microscopy (STM) studies Butler  _et al._ (2017); Lodge  _et al._ (2017); Su
_et al._ (2018); Zhang  _et al._ (2020) have confirmed the presence of
quasi-1D DLN bands. As in other TSMs, ZrSiS shows unconventional
magnetotransport properties, such as a butterfly-shaped angle-dependent
magnetoresistance Ali  _et al._ (2016) and a nontrivial Berry phase extracted
from quantum oscillations Matusiak  _et al._ (2017). Overall, ZrSiS is
particularly attractive in that its DLN lies close to the Fermi level, the
linearly dispersing bands span a wide energy range, and the band structure
below the Fermi surface is particularly clean, free from other trivial bulk
bands Schoop  _et al._ (2016). ZrSiS also harbors a so-called “floating”
surface state Topp  _et al._ (2017) along the $\bar{X}$–$\bar{M}$ line that
arises due to broken nonsymmorphic crystal symmetry at the surface, which is
shown in the slab calculation band structure in Fig. 1(d).

Despite the already impressive display of electronic properties, further
unusual effects beyond the level of non-interacting electrons are anticipated
in ZrSiS. Due to the vanishing density of states around the nodal line, the
screening of the Coulomb interaction is weaker, resulting in enhanced
electronic correlations Kim  _et al._ (2015); Syzranov and Skinner (2017); Roy
(2017); Armitage  _et al._ (2018). In addition, the nearly perfectly nested
Fermi surface may also enhance electron correlation effects. Fingerprints of
correlation-driven or many-body effects include an enhanced effective mass
under high magnetic fields Pezzini  _et al._ (2018), tip-induced
superconductivity Aggarwal  _et al._ (2019), and in the closely related
compound ZrSiSe, photoinduced band renormalization Gatti  _et al._ (2020).
Theoretical studies have also predicted an instability towards excitonic order
Rudenko  _et al._ (2018). The weaker screening of the Coulomb interaction
should also enhance the inelastic electron-electron scattering, as alluded to
in several optical studies Weber  _et al._ (2017, 2018); Kirby  _et al._
(2020). We are thus motivated to probe the influence of weaker screening on
the quasiparticle dynamics, which may be hidden behind the above-mentioned
unusual properties.

STM imaging of standing electronic waves in real space provides a somewhat
uncommon yet powerful technique of probing the dynamics of quasiparticles
Bürgi  _et al._ (1999). By measuring the spatial decay of the standing wave, a
phase coherence length can be extracted, which, when coupled with knowledge of
the band structure, can yield estimates of the quasiparticle lifetimes. At low
measurement temperatures, the measured lifetime predominantly reflects
inelastic electron scattering, rather than inelastic phonon scattering.
Despite the potential of this technique, its application has been largely
limited to the surface states of a few noble and transition metals Bürgi  _et
al._ (1999); Braun and Hla (2010); Vitali  _et al._ (2003), as well as that of
the topological insulator Bi2Se3 Song  _et al._ (2015), where standing
electronic waves produced at step edges were analyzed. Its application to the
growing body of TSMs, including their bulk states, is highly warranted.

Here, we apply SI-STM to access the quasiparticle dynamics of the DLN material
ZrSiS. From QPI patterns surrounding point defects, rather than those at step
edges, we show that we can directly extract an energy-dependent coherence
length and quasiparticle lifetime. To do so, we introduce a numerical method
of combining QPI patterns around several defects to enhance the signal-to-
noise ratio. The estimated lifetime is governed by inelastic electron-electron
scattering and intriguingly, falls just at the Planckian limit, which may
reflect the weaker Coulomb screening in this nested DLN material.

## Results

### STM topography

We performed STM measurements on cold-cleaved single crystals of ZrSiS at 4.3
K (see Methods). ZrSiS belongs to a nonsymmorphic, tetragonal $P4/nmm$ space
group Schoop  _et al._ (2016) and is isostructural to the iron-based
superconductor LiFeAs Tapp  _et al._ (2008). It consists of a stack of planar
square layers with the sequence Si-Zr-S-S-Zr-Si and lends itself to being
easily cleaved between the S layers [shown as the dotted line in Fig. 1(a)],
which are only weakly bonded together Butler  _et al._ (2017); Lodge  _et al._
(2017); Su  _et al._ (2018); Zhang  _et al._ (2020). Figure 1(e) shows a
topographic image, which reveals an atomically flat and clean surface with
sparse defects and visible Friedel oscillation patterns. The inset shows a
square lattice within a 2.5 nm $\times$ 2.5 nm field of view. The lattice
constant is 3.45 Å, which would be consistent with either a S or Zr
termination. We can exclude the Si square net, since the atomic spacing
between the nearest neighbour Si should be $\frac{1}{\sqrt{2}}$ of the
observed lattice constant.

Figure 1(f) presents a d$I$/d$V$ spectrum, which is proportional to the local
density of states (LDOS), acquired over a clean area. Calculated LDOS for slab
models with S and Zr terminations are also plotted, and the former shows
better agreement with the experimental data. Therefore, we argue that the
cleaved surface is terminated by the S layer. First-principles calculations
suggest that the S atoms are invisible to tunneling and that the bright spots
coincide with the Zr atoms underneath Butler  _et al._ (2017). The inset of
Fig. 1(f) reveals a dip in the d$I$/d$V$ spectrum around $-$15 meV, which can
be ascribed to the small gap at the DLN that opens due to spin-orbit coupling
Schoop  _et al._ (2016); Schilling  _et al._ (2017).

### Band structure mapping

To gain more information about the electronic structure of ZrSiS, we acquired
differential conductance maps $g=$d$I$/d$V(x,y,E)$ over the same field of view
shown in Fig. 1(e). The measurement covered a large range of energies $E$ from
$-$500 up to 1310 meV. Figures 2(a)-2(c) show the raw conductance maps at
three representative energy layers: $-$300, $-$40, and 500 meV. QPI patterns
emanating from defects are clearly visible, and their behavior changes with
energy. To better analyze the QPI patterns, we take the Fourier transform (FT)
of the conductance maps, $\mathcal{F}[g]$ [Figs. 2(d)-2(f)]. Strong
intensities at specific wave vectors that evolve with energy are again clearly
evident.

QPI patterns arise from elastic scattering off defects. When a quasiparticle
is scattered elastically from an initial state with momentum $\mathbf{k}_{i}$
and energy $E$ to a final state with momentum $\mathbf{k}_{f}$ and energy $E$,
an interference pattern with wave vector
$\mathbf{q}=\mathbf{k}_{f}-\mathbf{k}_{i}$ arises. Considering all the
possible initial and final states with energy $E$, the wave vectors
$\mathbf{q}$ that carry the most intense interference patterns will be those
that connect well-nested regions of the band structure at energy $E$, i.e. the
equal energy surface Hoffman  _et al._ (2002); Wang and Lee (2003). We observe
QPI patterns from both surface states and DLN states. To understand the QPI
pattern from the bulk DLN bands, we refer to the equal energy surface at
$-$300 meV ($k_{z}=0$), which is shown in Fig. 2(g). The equal energy surface
consists of two diamonds and the QPI signal marked by $\mathbf{q}_{D}$ in
Figs. 2(g) and 2(h) can be assigned to the intrapocket scattering vector
within the outer diamond. Note that $\mathbf{q}_{D}=2\mathbf{k}_{D}$, where
$\mathbf{k}_{D}$ is the wave vector of the DLN bands measured along
$\Gamma$–$M$ direction. Figure 2(k) compares the theoretical $k$\- and
experimental $q$-dispersion of the DLN bands, showing a good agreement.

The presence of two diamonds from the bulk bands at $-$300 meV [Fig. 2(g)]
could in principle produce additional QPI signals around $\mathbf{q}_{D}$
arising from the intrapocket scattering within the inner diamond and from the
interpocket scattering between the inner and outer diamonds. Within our
experimental resolution, however, we do not see them in the QPI patterns in
Figs. 2(d)-2(f) and the extracted dispersion relationship Fig. 2(k). Our
result is consistent with previous reports Zhang  _et al._ (2020); Butler  _et
al._ (2017). We argue that the distinct orbital characters of the two diamonds
account for the absence of QPI signal. While the inner diamond which grows
with increasing energy has predominant Zr $d_{xy}$ orbital character, the
outer diamond which shrinks has predominant Zr $d_{xz}/d_{yz}$ orbital
character (see SM). The different orbital characters should substantially
reduce interpocket scattering. As the overlap of the Zr $d_{xy}$ orbital with
the tip wave function should be considerably smaller compared to that of
$d_{xz}/d_{yz}$, the signal from intrapocket scattering within the inner
diamond with predominant Zr $d_{xy}$ character should also be suppressed.

The scattering vector from the surface states can be understood from the equal
energy surface from slab calculation at $-$300 meV, shown in Fig. 2(i), and
the observed scattering vectors in $q$-space are shown in Fig. 2(i) and 2(j).
See SM for detailed analysis of QPI patterns from surface states. In the
following, we focus on the QPI signals from the bulk DLN bands
($\mathbf{q}_{D}$) and discuss their dynamics.

### DLN dephasing dynamics and lifetime

As previously motivated, ZrSiS is an ideal playground to probe the low-energy
excitations of the DLN band. Over a wide energy range from roughly $-$40 to
$-$300 meV, the dispersion remains linear and there are no coexisting trivial
bulk bands. To explore the dynamics of DLN electrons through their standing
waves, i.e. QPI patterns, in real space, it only remains to disentangle the
QPI patterns of the bulk DLN electrons from those of the electrons in the
surface states. To do so, we performed a $q$-space filtering of the
$\mathbf{q}_{D}$ scattering [Figs. 3(a)-3(c)] and conducted the inverse
Fourier transform (IFT) [Figs. 3(d)-3(f)]. The resulting real-space images
show strong fourfold scattering patterns, consisting of quasi-1D rays with
indiscernible beam divergence (resembling the directed rays of a light house).
This comes from the nearly perfect nesting of the outer diamond with four-fold
symmetry.

The characteristic decay length over which the interference pattern extends
away from a defect shows strong energy dependence. At $E$ = $-$40 meV, very
close to the actual line node, the extension of quasi-1D rays is quite long-
ranged and, in some cases, exceeds the field of view [Fig. 3(d)], which
corresponds to the sharp $\mathbf{q}_{D}$ peak in $q$-space [Fig. 3(a)].
Moving away from the node to $E$ = $-$120 meV, we observe a faster decay of
the quasi-1D rays [Fig. 3(e)], reflecting the broadening of the
$\mathbf{q}_{D}$ peak in $q$-space [Fig. 3(b)]. Eventually, at $E$ = $-$200
meV, the real space QPI patterns are localized within a few unit cells from
the impurities in real space [Fig. 3(f)], in accord with the diffuse nature of
the $\mathbf{q}_{D}$ peak in $q$-space [Fig. 3(c)]. To conduct more
quantitative analysis, we carry out an IFT of the power spectral density;
i.e., we take the modulus of the complex masked FT,
$\left|\mathcal{F}_{\textrm{mask}}[g]\right|$, before applying an IFT
$\mathcal{F}^{-1}[\left|\mathcal{F}_{\textrm{mask}}[g]\right|]$. This
procedure removes the phase information encoding the real-space location of
the scatterers and effectively places them all at the center. Again, we
observe a clear shortening of “light house” QPI pattern in real space from $E$
= $-$40 to $-$120 and $-$200 meV [Fig. 3(g)–3(i)]. The shortening of the QPI
pattern is more clearly seen in Fig. 3(j), which shows the diagonal line cut
of the IFT of the power spectral density at different energies from $-$60 meV
to $-$260 meV. The enhanced signal-to-noise ratio of the real space
oscillations facilitates a fitting to a decay function.

After applying our method of effectively averaging the QPI patterns across all
defects in the field of view, we fit the spatial decay of the QPI to the
following equation Bürgi  _et al._ (1999):

$g\propto\frac{1}{r^{\alpha}}\exp\bigg{(}\frac{-2r}{l_{\textrm{QPI}}}\bigg{)}\sin(qr+\delta),$
(1)

where $r$ is the distance from the origin, $l_{\textrm{QPI}}$ is the QPI decay
length, $\delta$ is a scattering phase shift, and $\alpha$ is a geometric
factor. When the QPI pattern arises from well-nested segments of the equal
energy surface and is essentially 1D in nature, $\alpha$ = 0 Weismann  _et
al._ (2009); Lounis  _et al._ (2011); Kim  _et al._ (2020), which applies to
our case. Fig. 3(k) plots $l_{\textrm{QPI}}$ from $E$ = $-$220 to $-$40 meV.
We note that there are lower and upper limits in the extraction of
$l_{\textrm{QPI}}$ using this analysis. When $l_{\textrm{QPI}}$ decreases to a
length scale of few unit cells, Eq. (1) is no longer valid, as the impurity is
not a point scatterer, which sets the lower limit. The upper limit is set
roughly by the interference of QPI from different impurities. When
$l_{\textrm{QPI}}$ exceeds the average distance of the impurities, we observed
the rays from different impurities crossing one another or hitting other
impurities, which effectively reduces the apparent decay length. These limits
are represented by the shaded areas in Fig. 3(k).

Since the DLN dispersion is linear, the quasiparticle lifetime
$\tau_{\textrm{QPI}}$ = $l_{\textrm{QPI}}/v_{f}$ is estimated by using the
Fermi velocity $v_{f}$ = 8.5 $\times$ $10^{5}$ m/s, extracted from the
dispersion of the $q$-vector in Eq. (1). $v_{f}$ as measured by ARPES Schoop
_et al._ (2016) is 30% smaller, which does not qualitatively affect our
conclusions. As shown on the right axis of Fig. 3(k), $\tau_{\textrm{QPI}}$
ranges from 3 to 12 femtoseconds. As electron-phonon scattering is strongly
suppressed at the measurement temperature of 4.3 K (see SM), the estimated
$\tau_{\textrm{QPI}}$ should be governed by inelastic electron-electron
interactions. We note that $\tau_{\textrm{QPI}}$ is obtained only for $|E|$
$\geq$ 40 meV ($\frac{|E|}{k_{B}}\sim$ 470 K) due to the upper limit in the
estimate of $l_{\textrm{QPI}}$ and that we are dealing with rather “hot”
quasiparticles here.

The extracted $\tau_{\textrm{QPI}}$ for hot quasiparticles falls just at the
Planckian limit $\tau=\hbar/|E|$, as indicated by the dashed line in Fig.
3(k). The Planckian limit, a fundamental lower bound on the lifetime due to
the uncertainty principle, was discussed in connection with a universal
scattering lifetime with $\tau=\hbar/(\alpha_{s}k_{B}T)$,
$0.9<\alpha_{s}<2.2$, in metals displaying $T$-linear resistivity Bruin  _et
al._ (2013). The observation indicates that the high energy quasiparticles
with $|E|$ $\geq$ 40 meV are only marginally defined.

## Discussion

Figure 4 compares the energy dependence of $\tau_{\textrm{QPI}}(E)$ for ZrSiS
with those for a few other materials, extracted from STM d$I$/d$V$
measurements on standing waves extending from 1D step edges. The $\tau$ values
for the surface states of nonmagnetic Ag and Bi2Se3 lie well above the
Planckian limit Song  _et al._ (2015); Braun and Rieder (2002); Vitali  _et
al._ (2003). Ferromagnetic Ni has lower values of $\tau$ closer to the
Planckian limit Braun and Hla (2010), especially on the electron side near 100
meV, and this suppression relative to nonmagnetic Ag was attributed to spin-
flip scattering. ZrSiS has the shortest quasiparticle lifetimes, which lie
just at the Planckian limit. Clearly, we cannot attribute this suppression to
spin-flip processes. Instead, we argue that the Planckian limit behaviour in
ZrSiS originates from the enhanced electron-electron interactions due to the
poor Coulomb screening. The one-dimensionality of the band, which gives rise
to the nearly perfectly nested equal energy surfaces, may further enhance
electron correlation effects.

We emphasize that the Planckian-limit behavior of the quasiparticle lifetime
$\tau_{\textrm{QPI}}$ at such high energies does not necessarily mean the
breakdown of Fermi liquid theory, which holds only in the low energy $E=0$
limit. It would be extremely interesting if the Planckian-limit ($1/E$ linear)
behavior were to persist to the low-energy limit, reflecting the effects of
one-dimensionality and electron correlations. However, one might naturally
expect that $\tau_{\textrm{QPI}}$ is enhanced from the Planckian-limit-$1/E$
behavior and eventually crosses over to a Fermi-liquid-$1/E^{2}$ behavior at
energies much lower than 40 meV. The DLN bands are close to, but in fact, not
completely one-dimensional. The higher dimensionality should be particularly
pronounced below $|E|\approx 20$ meV. The location of the DLN in $k$-space is
also energy-dependent (though only very weakly) on the scale of 20 meV, even
within the $k_{z}=0$ plane, and is weakly gapped on the scale of 10–20 meV due
to spin-orbit coupling. Quantum oscillations reveal Fermi surface warping and
a breakdown of the quasi-1D approximation in the low-energy limit Schoop  _et
al._ (2016); Chen  _et al._ (2017); Schilling  _et al._ (2017). To fully
address the low-energy limit behavior, measurements on cleaner samples with
larger average impurity distances are highly desired.

In conclusion, we have used atomically resolved SI-STM to shed light on the
quasiparticle dynamics of bulk DLN excitations in ZrSiS. We introduced a
numerical method of enhancing the signal-to-noise ratio of QPI patterns around
defects, in order to extract their characteristic decay length
$l_{\textrm{QPI}}$ in real space. The measured values of $l_{\textrm{QPI}}$
are converted to quasiparticle lifetimes $\tau_{\textrm{QPI}}$, which, at the
measurement temperature of 4.3 K, are dominated by inelastic electron-electron
scattering. We find that $\tau_{\textrm{QPI}}$ at high energies $|E|$ $\geq$
40 meV falls just around $\tau=\hbar/|E|$, which may reflect enhanced
electronic correlations due to weaker Coulomb screening. In contrast to
previous STM studies that extracted the decay length from scattering off 1D
step edges Song  _et al._ (2015); Bürgi  _et al._ (1999), our method employs
scattering off sparsely distributed point defects. Because of the underlying
one dimensionality and the linear band dispersion, our case for ZrSiS is
essentially equivalent to those of the 1D step edge, which simplified our
analysis. We note, however, that there is in principle no need for the one-
dimensionality and the linear dispersion in analyzing the decay length from
point defects. We should be able to explore the full anisotropy of the
quasiparticle lifetime. A much wider range of exotic electrons, for example,
those in topological systems, await the successful application of the present
method using point defects.

## Methods

### Sample synthesis

Single crystals of ZrSiS were grown in a two-step synthesis as previously
reported Schoop  _et al._ (2016). First, a polycrystalline powder was
obtained. Second, high quality single crystals were grown from the
polycrystalline powder via I${}_{\text{2}}$ vapour transport at a temperature
of 1100 ∘C with a 100 ∘C temperature gradient. The single crystals were
obtained at the cold end.

### Scanning tunneling microscopy

All experiments were conducted in a home-built ultrahigh vacuum (UHV) STM
operated at 4.3 K. Single crystals of ZrSiS were mounted onto the sample
holder using conducting epoxy (H20E from Epoxy Technology). The samples were
cleaved on a liquid-nitrogen-cooled cleaving stage in the UHV chamber, then
immediately transferred to the microscope head held at 4.3 K. The tip was
electrochemically etched from polycrystalline tungsten wire. After being
transferred into the UHV chamber, it was further conditioned with electron-
beam bombardment, then field emission on a gold surface. Topographic images
were recorded in the constant-current mode with a setpoint $I_{\text{s}}$ and
a bias voltage $V_{\text{s}}$ applied to the sample. Single-point differential
conductance curves $\mathrm{d}I/\mathrm{d}V$ were acquired using a standard
lock-in technique, wherein a small modulation voltage $V_{\text{m}}$ of 1–25
mV and frequency 736 Hz was added to the bias voltage $V_{\text{s}}$. The
feedback loop was disabled during the $\mathrm{d}I/\mathrm{d}V$ measurement.
For the conductance map, a modulation voltage of 7–25 mV was used and detailed
setup conditions are listed in the Table 1.

Table 1: Setup conditions for the conductance maps. Energy range (meV) | Bias voltage (mV) | Setpoint (pA) | Lock-in modulation (mV)
---|---|---|---
$-$500 to $-$300 | 50 | 20 | 10
$-$300 to 100 | 150 | 100 | 10
80 to 560 | 50 | 20 | 20
560 to 1010 | 50 | 7 | 25
1010 to 1160 | 50 | 6 | 25
1160 to 1310 | 50 | 5 | 25

### Band structure calculations

DFT calculations of the bulk band structure and the surface slab model were
carried out using the Vienna ab initio simulation package (VASP) Kresse and
Joubert (1999), which employs the projector augmented wave method. For Zr, the
4$s$, 4$p$, 5$s$, and 4$d$ were treated as valence states, whereas for S and
Si, the 3$s$, 3$p$ states were treated as valence states. The kinetic energy
cutoff for the plane wave basis was set to 450 (bulk) or 500 eV (slab). We
employed the PBE exchange correlation energy functional. The Brillouin zone
sampling was $25\times 25\times 11$ (bulk) and $4\times 4\times 1$ (slab)
$\Gamma$-centered $k$-mesh. The surface model has a slab thickness of 7 unit
cells. Experimental lattice parameters Schoop  _et al._ (2016) were used in
the calculations. Spin-orbit coupling was included in an additional non-self-
consistent step.

## Acknowledgements

The authors thank Raquel Queiroz, Andreas Topp, Andreas Schnyder and Christian
R. Ast for discussion and comments, and K. Pflaum and M. Dueller for technical
assistance. A.W.R. was supported by the Engineering and Physical Sciences
Research Council (grant number EP/P024564/1). This work has been supported in
part by the Alexander von Humboldt Foundation.

## Author contributions

Q.H., L.Z., and A.W.R. carried out the STM measurements. L.M.S. grew the
crystals. D.H. and A.G. performed electronic structure calculations. Q.H.,
L.Z., A.W.R. and D.H. performed the data analysis. Q.H., D.H., and H.T. wrote
the manuscript with input and feedback from all the authors.

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Figure 1: (a) Crystal structure of ZrSiS. The dashed line represents the
cleavage plane. (b) Bulk and surface-projected Brillouin zones (BZs). (c) Band
structure of ZrSiS derived from a bulk calculation. The inset shows the
diamond-shaped Fermi surface representing the DLN in the $k_{z}=0$ plane. (d)
Band structure of ZrSiS derived from a slab calculation. In addition to the
bulk bands, additional surface states (SS) are present (red arrows). The size
of the red circles represents the surface-projected spectral weight. The inset
shows the Fermi surface projected onto the surface BZ, with surface states
around $\bar{X}$. (e) STM topographic image of the S-terminated layer of ZrSiS
after cleavage (setup condition: 150 mV, 100 pA). The inset shows the atomic
lattice (2.5 nm $\times$ 2.5 nm; setup condition: 100 mV, 65 pA). (f)
Differential conductance curve of ZrSiS (black), along with the calculated
LDOS of a S-terminated (magenta) and Zr-terminated (orange) surface from slab
calculations. The d$I$/d$/V$ curve was averaged and rescaled from several
conductance map measurements covering different energy ranges. The inset shows
a spectrum from $-$100 to 100 meV (setup condition: 200 mV, 65 pA; lock-in
modulation: 10 mV).

Figure 2: Conductance maps $g$ and their FT $|\tilde{g}|$ at three selected
energies: (a, d) $-$300 meV, (b, e) $-$40 meV, and (c, f) 500 meV. These data
were acquired in the same area as in Fig. 1(e). (g) Equal energy surface for
bulk bands at $-$300 meV ($k_{z}=0$). $\mathbf{q}_{D}$ represents the main
observed scattering vector, as marked by the blue arrow. (h) Schematic of QPI
pattern in $q$-space from DLN bands, based on the equal energy surface in (g).
(i) Equal energy surface from slab calculation at $-$300 meV.
$\mathbf{q}_{s1}$ and $\mathbf{q}_{s2}$ represents the main scattering vector
from the surface states, as marked by the red arrows. (j) Schematic of QPI
pattern in $q$-space from the surface bands, based on the equal energy surface
in (i). (k) Comparison of experimental dispersions for the DLN bands in
$q$-space (right) with theoretical dispersions in $k$-space (left). The
experimental intensities were taken along line cuts of $|\tilde{g}|$, as
marked by the dashed line in (d). The overlays on the right panels of (k) are
guides for the eye.

Figure 3: (a)-(c) Masked FT of the conductance map,
$\mathcal{F}_{\textrm{mask}}[g]$, at three energies: $-$40, $-$120, and $-$200
meV. (d)-(f) Corresponding IFT,
$|\mathcal{F}^{-1}[\mathcal{F}_{\textrm{mask}}[g]]|$, of (a)-(c), showing
“light house” QPI patterns corresponding solely to DLN states emanating from
defects. (g)-(i) IFT of power spectral density,
$\mathcal{F}^{-1}[|\mathcal{F}_{\textrm{mask}}[g]|]$, which effectively places
all scatterers at the center and enhances the QPI signal. (j) The diagonal
linecut of the IFT of power spectral density at different energies from $-$60
meV to $-$260 meV. (k) Energy-dependent decay length $l_{\textrm{QPI}}$ (left
axis) and corresponding lifetime $\tau_{\textrm{QPI}}$ (right axis), extracted
from fits to Eq. (1). The shaded areas mark the lower and upper boundaries
determined by short-range scattering and the average impurity distance,
respectively. The dashed line represents the Planckian limit, $\tau=\hbar/|E|$
(or equivalently, $l=\hbar v_{f}/|E|$).

Figure 4: Comparison of the quasiparticle lifetime dependence on the
excitation energy, measured for ZrSiS, Ni(111) Braun and Hla (2010), Ag(111)
Vitali  _et al._ (2003), and Bi2Se3 Song  _et al._ (2015). The dashed line
denotes the Planckian limit.


# Combinatorial Persistent Homology Transform

Brittany Terese Fasy  School of Computing and Dept. of Mathematical Sciences,
Montana State University  Amit Patel  Department of Mathematics, Colorado
State University

###### Abstract

The combinatorial interpretation of the persistence diagram as a Möbius
inversion was recently shown to be functorial. We employ this discovery to
recast the Persistent Homology Transform of a geometric complex as a
representation of a cellulation on ${\mathbb{S}}^{n}$ to the category of
combinatorial persistence diagrams. Detailed examples are provided. We hope
this recasting of the PH transform will allow for the adoption of existing
methods from algebraic and topological combinatorics to the study of shapes.
††BTF is supported by the National Science Foundation under grant no. DMS
1664858. AP is supported by the Leverhulme Trust grant VP2-2021-008.

##  1 Introduction

Persistent homology (PH) captures the “shape” of a topological space often
arising from data. It takes as input a filtration of a space, usually
paramterized by the reals, and outputs a multiset of points in the extended
plane known as the persistence diagram. Each point in the persistence diagram
represents a homological “feature” of the topological space and its
coordinates provide the interval of parameters for which that feature is
present. Individually, a persistence diagram is a powerful data analysis tool;
however, a single descriptor is often not rich enough to capture the
intricacies of large, complex data. For example, the family of filtrations
giving rise to the same persistence diagram can be arbitrarily large. One
approach for capturing more information about a space is to consider not just
one filtration of the space but a family of filtrations of the space and
therefore a family of persistence diagrams.

One such family of descriptors and the focus of this paper is the _persistent
homology transform_ , introduced initially in [25], which takes as input a
space embedded in ${\mathbb{R}}^{N+1}$ and represents it—or transforms it—into
a family of persistence diagrams parameterized by the set of all directions
${\mathbb{S}}^{N}$. The idea behind this transform goes back the early 2000s
[1] and was recently generalized to weighted simplicial complexes in [17]. The
PH transform is a _complete_ representation of the original shape. That is, no
two different shapes have the same PH transform. Although ${\mathbb{S}}^{N}$
comprises an uncountable set of directions, if $K$ is a nicely embedded
geometric simplicial complex, then it has a finite representation [2, 9, 13,
22]. The PH transform and other topological transforms have been applied using
a sample of directions [8, 25, 17, 3, 16, 20], yet there is still a
substantial gap between the theory and this practice [12].

#### Our Contributions

Consider a space $X$ with an embedding $\phi:X\to{\mathbb{R}}^{N+1}$. The PH
transform of $\phi$ is the assignment to every direction
$\mu\in{\mathbb{S}}^{N}$ the sublevel set persistence diagram of the height
function $\phi_{\mu}:X\to{\mathbb{R}}$ in that direction. As mentioned above,
the persistence diagram is the assignment to every interval of ${\mathbb{R}}$
a natural number often called its multiplicity or, in our parlance, a
_charge_. Unfortunately, there is a major downside to this approach. In
practice, $\phi$ is finitely generated; for example, $\phi$ is a PL embedding
of a finite simplicial complex. Its PH transform consists of an uncountable
set of distinct persistence diagrams; however, when ignoring exact birth/death
coordinates and keeping track of only the relations between birth/death
coordinates, we observe that there are distinct places where _combinatorial_
changes in the diagrams happen. We call two persistence diagrams
_combinatorially equivalent_ when the diagrams are related by a homeomorphism
from ${\mathbb{R}}$ to itself. In this setting, we expect that most directions
$\mu\in{\mathbb{S}}^{N}$ have an open neighborhood
$U\subseteq{\mathbb{S}}^{N}$ such that the persistence diagrams associated to
any two directions $\nu,\nu^{\prime}\in U$ are combinatorially equivalent.

In that light, we take a combinatorial approach to the PH transform by
adopting the combinatorial framework of persistent homology developed by
McClearly and Patel [21]. In this setting, filtrations are indexed by an
abstract totally ordered, finite poset $P$ as opposed to the real numbers. The
combinatorial persistence diagram is then the assignment to every non-empty
(order-theoretic) interval $[a,b]\subseteq P$ a charge. Given a PL embedding,
we show that there is a cellulation of ${\mathbb{S}}^{n}$ such that the
combinatorial persistence diagrams associated to any two directions on a
common cell are exactly the same. This cellular decomposition is the same
decomposition observed in [2, 14, 9] for the PH transform and in [7] for
vineyards. Further, we harness functoriality of McClearly and Patel to express
the PH transform as a representation of the cellulation on ${\mathbb{S}}^{n}$
to the category of combinatorial persistence diagrams and charge-preserving
morphisms. The cellulation is finite and every combinatorial persistence
diagram is finite making our _combinatorial PH transform_ a finite object. We
hope that this recasting of the PH transform will allow for the adoption of
existing methods from algebraic and topological combinatorics to the study of
shapes.

##  2 Preliminiaries

We use this section to establish background and notation. Section 2.1
introduces geometric complexes and their height functions. Section 2.2
introduces the combinatorial persistent homology pipeline of McClearly and
Patel. See also Appendix A for a discussion of our notation on posets and
categories. As mentioned in Section 1, the combinatorial persistence diagram
is indexed by an abstract totally ordered, finite poset as opposed to the
traditional persistence diagram that is indexed by the real numbers. Since we
are departing from traditional persistent homology and focusing solely on its
combinatorial version, we drop the word “combinatorial” from our discussion of
combinatorial persistent homology.

###  2.1 Embeddings of Simplicial Complexes

We start with an introduction to our main object of study: geometric complexes
and their height functions. We adopt notation similar to [15, Section 2.1].

Fix a simplicial complex $K$, let $K^{0}\subseteq K$ be its vertices, and
denote by $|K|$ its underlying space. For the purposes of this paper, we
assume all simplicial complexes are finite. Given a function
$\hat{\phi}:K^{0}\to{\mathbb{R}}^{N+1}$ defined on its vertices, we extend it
to a PL function on the entire underlying space $|K|$ as follows. For a
simplex $\sigma=[v_{0},\cdots,v_{k}]$, express every point $x\in|\sigma|$ by
its barycentric coordinates $(x_{0},x_{1},\cdots,x_{k})$. Then, the linear
extension of $\hat{\phi}$ to $|\sigma|$ is the map that sends $x$ to the point
$x_{0}\hat{\phi}(v_{0})+x_{1}\hat{\phi}(v_{1})+\cdots+x_{k}\hat{\phi}(v_{k})$.
In this way, the function $\hat{\phi}$ extends to a PL function
$\phi:|K|\to{\mathbb{R}}^{N+1}$ on the entire complex. If such an extension is
injective, we call $\phi$ a _geometric complex_.

We are interested in examining the geometric complex
$\phi:|K|\to{\mathbb{R}}^{N+1}$ from all directions. Denote by
${\mathbb{S}}^{N}:=\\{\mu\in{\mathbb{R}}^{N+1}:||\mu||=1\\}$ the sphere of all
directions in ${\mathbb{R}}^{N+1}$. The _height function_ on $\phi$ along a
direction $\mu\in{\mathbb{S}}^{N}$ is the continuous function
$\phi_{\mu}:|K|\to{\mathbb{R}}^{N+1}$ defined as the dot product
$\phi_{\mu}(x):=\phi(x)\cdot\mu$. For $L\subseteq|K|$, we let
$\phi_{\mu}(L):=\sup_{x\in L}\phi_{\mu}(x)$. In particular, if $L$ is a
simplex in $K$, then $\phi_{\mu}(L)$ is the maximum height of the vertices
that define $L$.

###  2.2 Persistent Homology Pipeline

We now give a brief introduction to the persistent homology pipeline of
McCleary and Patel [21] restricted to the special case of combinatorial one-
parameter filtrations and to a fixed simplicial complex $K$. This pipeline
consists of three categories ${\mathsf{Fil}}(K)$, ${\mathsf{Mon}}(n)$, and
${\mathsf{Fnc}}(n)$ and two functors ${{\mathsf{ZB}}}_{\ast}$ and
${{\mathsf{MI}}}$:

${{\mathsf{Fil}}(K)}$${{\mathsf{Mon}}(n)}$${{\mathsf{Fnc}}.}$$\scriptstyle{{{\mathsf{PH}}}_{\ast}:={{\mathsf{MI}}}\circ{{\mathsf{ZB}}}_{\ast}}$$\scriptstyle{{{\mathsf{ZB}}}_{\ast}}$$\scriptstyle{{{\mathsf{MI}}}}$

We call ${{\mathsf{ZB}}}_{\ast}$ the _birth-death functor_ , ${{\mathsf{MI}}}$
the _Möbius inversion functor_ , and the composition ${{\mathsf{PH}}}_{\ast}$
the _persistent homology functor_.

#### Filtrations

Let $\Delta K$ be the poset consisting of subcomplexes of $K$ ordered by
inclusion. Given a finite, totally ordered poset $P$, a _$P$ -filtration_ of
$K$ is a functor ${F}:P\to\Delta K$ such that ${F}(\top)=K$. Let $P$ and $Q$
be finite, totally ordered posets. A _filtration-preserving morphism_ is a
triple $({F},{G},\alpha)$ where ${F}:P\to\Delta K$ and ${G}:Q\to\Delta K$ are
$P$\- and $Q$-filtrations of $K$, respectively, and $\alpha:P\to Q$ is a
bounded monotone function satisfying the following axiom. For all $a\in Q$,
${G}(a)={F}(a^{\star})$, where $a^{\star}:=\max\alpha^{-1}[\bot,a]$, i.e., the
maximal element in $P$ that maps to $a$:

If $({F},{G},\alpha)$ and $({G},{H},\beta)$ are filtration-preserving
morphisms, then the composition $({F},{H},\beta\circ\alpha)$ is a filtration-
preserving morphism:

${P}$${Q}$${R}$${\Delta
K.}$$\scriptstyle{{F}}$$\scriptstyle{\alpha}$$\scriptstyle{{G}}$$\scriptstyle{\beta}$$\scriptstyle{{H}}$

######  Definition 2.1:

Fix a simplicial complex $K$. Let ${\mathsf{Fil}}(K)$ be the category whose
objects are $P$-filtrations of $K$, over all finite, totally ordered posets
$P$, and whose morphisms are filtration-preserving morphisms. We call
${\mathsf{Fil}}(K)$ the category of filtrations of $K$.

#### Monotone Integral Functions

Assume the usual total ordering on the integer ${\mathbb{Z}}$. Let $P$ and $Q$
be two finite, totally ordered posets and let $f:\overline{P}\to{\mathbb{Z}}$
and $g:\overline{Q}\to{\mathbb{Z}}$ be two monotone integral functions on
their poset of intervals; see Appendix A. A _monotone-preserving morphism_ is
a triple $(f,g,\bar{\alpha})$, where $f:\overline{P}\to{\mathbb{Z}}$ and
$g:\overline{Q}\to{\mathbb{Z}}$ are monotone integral functions and
$\bar{\alpha}:\overline{P}\to\overline{Q}$ is a bounded monotone function
induced by a bounded monotone function $\alpha:P\to Q$ satisfying the
following axiom. For all $I\in\overline{Q}$,
$g(I)=f(\max\bar{\alpha}^{-1}[\bot,I])$. In other words, for every interval
$I=[a,b]$, $g(I)=f\big{(}[a^{\star},b^{\star}]\big{)}$ where $a^{\star}=\max
f^{-1}[\bot,a]$ and $b^{\star}=\max f^{-1}[\bot,b]$.

${\overline{P}}$${\overline{Q}}$${{\mathbb{Z}}.}$$\scriptstyle{f}$$\scriptstyle{\bar{\alpha}}$$\scriptstyle{g}$

If $(f,g,\bar{\alpha})$ and $(g,h,\bar{\beta})$ are two monotone-preserving
morphisms, then the composition $(f,h,\bar{\beta}\circ\bar{\alpha})$ is a
monotone-preserving morphism:

${\overline{P}}$${\overline{Q}}$${\overline{R}}$${{\mathbb{Z}}.}$$\scriptstyle{f}$$\scriptstyle{\bar{\alpha}}$$\scriptstyle{g}$$\scriptstyle{\bar{\beta}}$$\scriptstyle{h}$

######  Definition 2.2:

Let ${\mathsf{Mon}}$ be the category whose objects are monotone integral
functions, over all finite, totally ordered posets, and whose morphisms are
monotone-preserving morphisms. We call ${\mathsf{Mon}}$ the category of
monotone functions.

#### Integral Functions

Let $P$ and $Q$ be finite, totally ordered posets and let
$\sigma:\overline{P}\to{\mathbb{Z}}$ and $\tau:\overline{Q}\to{\mathbb{Z}}$ be
two _integral functions_ on their poset of intervals. Let $\alpha:P\to Q$ be a
bounded poset function such that for all $I\in\overline{Q}$,

$g(I)=\sum_{J\in\alpha^{-1}(I)}f(J).$ (1)

Here, if $\alpha^{-1}(I)$ is empty, then we interpret the sum as $0$. We call
the triple $(\sigma,\tau,\bar{\alpha})$ a _charge-preserving morphism_ , where
$\bar{\alpha}:\overline{P}\to\overline{Q}$ is the bounded poset function
induced by $\alpha$. If $(\sigma,\tau,\bar{\alpha})$ and
$(\tau,\upsilon,\bar{\beta})$ are charge-preserving morphisms, then
$(\sigma,\upsilon,\bar{\beta}\circ\bar{\alpha})$ is a charge-preserving
morphism:

${\overline{P}}$${\overline{Q}}$${\overline{R}}$${{\mathbb{Z}}.}$$\scriptstyle{\sigma}$$\scriptstyle{\bar{\alpha}}$$\scriptstyle{\tau}$$\scriptstyle{\bar{\beta}}$$\scriptstyle{\upsilon}$

######  Definition 2.3:

Let ${\mathsf{Fnc}}$ be the category whose objects are integral functions over
finite, totally ordered posets, and whose morphisms are charge-preserving
morphisms. We call ${\mathsf{Fnc}}$ the category of integral functions.

######  Remark 2.4:

An important observation is that ${\mathsf{Fnc}}$ does not have arbitrary
colimits. For example, the following two integral functions do not have a
colimit. Let $P=Q$ be totally-ordered posets on four elements; in particular,
we write $P=Q=\\{1<2<3<4\\}$. Let $\sigma\colon\overline{P}\to{\mathbb{Z}}$ be
the function

$\sigma([i,j])=\begin{cases}1,\text{ if }i=1,j=3\\\ 1,\text{ if }i=2,j=4\\\
0,\text{ otherwise},\end{cases}$

and let $\tau\colon\overline{Q}\to{\mathbb{Z}}$ be the function

$\tau([i,j])=\begin{cases}1,\text{ if }i=1,j=4\\\ 1,\text{ if }i=2,j=3\\\
0,\text{ otherwise}.\end{cases}$

If a colimit existed, then it would come from gluing elements in $P$ and
gluing elements in $Q$ until there is a common integral function. This leaves
two contenders for the colimit:

  1. 1.

The integral function $\mu:\overline{Z}\to{\mathbb{Z}}$ on the poset
$Z=\\{a<b<c\\}$ that maps $[a,b]\mapsto 1$, $[a,c]\mapsto 1$, and everything
else to zero. This is obtained by mapping $1$ and $2$ in both $P$ and $Q$ to
$a$, $3$ in $P$ and $Q$ to $b$, and $4$ in $P$ and $Q$ to $c$.

  2. 2.

The integral function $\nu:\overline{Z}\to{\mathbb{Z}}$ on the poset
$Z=\\{a<b<c\\}$ that maps $[a,c]\mapsto 1$, $[b,c]\mapsto 1$, and everything
else to zero. This is obtained by mapping $1$ in both $P$ and $Q$ to $a$, $2$
in $P$ and $Q$ to $b$, and $3$ and $4$ in $P$ and $Q$ to $c$.

We note that neither of these integral functions can be obtained from the
other by gluing elements. This means that arbitrary colimits do not exist in
${\mathsf{Fnc}}$.

#### Birth-Death Functor

Fix a field $\mathsf{k}$. The $d$-th birth-death functor ${{\mathsf{ZB}}}_{d}$
assigns to a filtration ${F}:P\to\Delta K$ the monotone integral function
${{\mathsf{ZB}}}_{d}({F}):\overline{P}\to{\mathbb{Z}}$ defined as follows. For
every interval $[a,b]\in\overline{P}$, where $b\neq\top$,
${{\mathsf{ZB}}}_{d}[a,b]$ is the dimension of the $\mathsf{k}$-vector space
of $d$-cycles in the simplicial complex ${F}(a)$ that are $d$-boundaries in
the larger simplicial complex ${F}(b)$. For an interval $[a,\top]$,
${{\mathsf{ZB}}}_{d}[a,\top]$ is simply the dimension of the vector space of
$d$-cycles in ${F}(a)$. The birth-death functor takes a filtration-preserving
morphism $({F},{G},\alpha)$ to the monotone-preserving morphism
$\big{(}{{\mathsf{ZB}}}_{i}({F}),{{\mathsf{ZB}}}_{d}({G}),\bar{\alpha}\big{)}$,
which turns out to be a monotone-preserving morphism.

#### Möbius Inversion Functor

Given a monotone integral function $f:\overline{P}\to{\mathbb{Z}}$, there is a
unique integral function $\sigma:\overline{P}\to{\mathbb{Z}}$, called the
Möbius inversion of $f$, such that for all $J\in P$,

$f(J)=\sum_{I\in\overline{P}:I\leq J}\sigma(I).$ (2)

The functor ${{\mathsf{MI}}}$ assigns to every monotone-integral function its
Möbius inversion. For every monotone-preserving morphism $(f,g,\bar{\alpha})$
in ${\mathsf{Mon}}$,
$\big{(}{{\mathsf{MI}}}\,f,{{\mathsf{MI}}}\,g,\bar{\alpha})$ is a charge-
preserving morphism in ${\mathsf{Fnc}}$.

######  Definition 2.5:

We call the composition
${{\mathsf{PH}}}_{\ast}:={{\mathsf{MI}}}\circ{{\mathsf{ZB}}}_{\ast}:{\mathsf{Fil}}(K)\to{\mathsf{Fnc}}$
the persistent homology functor. It assigns to every filtration $P$ in
${\mathsf{Fil}}(K)$ its persistence diagram ${{\mathsf{PH}}}_{\ast}(F)$ and to
every filtration-preserving morphism $(F,G,\alpha)$ in ${\mathsf{Fil}}(K)$ the
charge-preserving morphism
$({{\mathsf{PH}}}_{\ast}(F),{{\mathsf{PH}}}_{\ast}(G),\bar{\alpha})$.

An obvious but important observation is the following proposition, which
follows immediately from the definition of the birth-death function and
Equation (2).

######  Proposition 2.6:

Let $F$ be a filtration in ${\mathsf{Fil}}(K)$. Then, the total charge of its
$d$-th persistence diagram,
$\Sigma_{I\in\overline{P}}{{\mathsf{PH}}}_{d}(F)(I)$, is the dimension of the
$d$-th cycle space of $K$.

##  3 Combinatorial Transform for Geometric Complexes

Fix a geometric embedding $\phi:|K|\to{\mathbb{R}}^{N+1}$ of a simplicial
complex $K$.

###  3.1 Constructing the Cellulation

Let $E:=\\{(v,v^{\prime}):v,v^{\prime}\in K^{0}\\}$ be the set of all
unordered, distinct pairs of vertices of $K$. For each pair of vertices
$e=(v,v^{\prime})\in E$, there is a unique plane in ${\mathbb{R}}^{N+1}$
perpendicular to the vector $\phi(v)-\phi(v^{\prime})$ through the origin; we
denote that plane $H_{e}$. The intersection of $H_{e}$ with the $N$-sphere,
$S_{e}:=H_{e}\cap{\mathbb{S}}^{N}$, is a great $(N-1)$-sphere of
${\mathbb{S}}^{n}$. By the Jordan-Brouwer separation theorem,
${\mathbb{S}}^{N}\setminus S_{e}$ has two connected components (often called
_sides_ or _hemispheres_) denoted arbitrarily by $S_{e}^{+}$ and $S_{e}^{-}$;
see [19, 5, 15]. Let ${{\mathcal{A}}}:=\\{S_{e}:e\in E\\}$ be the multiset of
all such great $N$-spheres. We call ${{\mathcal{A}}}$ along with an assignment
of a sign to each side of $S_{e}$, over all $e\in E$, a _signed arrangement_
of spheres. The signed arrangement ${{\mathcal{A}}}$ is _essential_ if the
following condition holds: $\bigcap_{e\in{{\mathcal{A}}}}S_{e}=\emptyset$.

The signed arrangement ${{\mathcal{A}}}$ induces a cell complex on
${\mathbb{S}}^{N}$ where every cell is described by whether it lies in
$S_{e}^{-}$, $S_{e}$, or $S_{e}^{+}$ over every $S_{e}\in{{\mathcal{A}}}$. To
define this cell complex, let $\\{-,0,+\\}^{E}$ denote the set of all
functions from $E$ to $\\{-,0,+\\}$. Consider the following partial order on
$\\{-,0,+\\}$: $0<-$ and $0<+$. This partial order extends to a partial order
on the set of functions $\\{-,0,+\\}^{E}$, where $f\leq g$ if $f(e)\leq g(e)$,
for all $e\in E$. Let $\Phi:{\mathbb{S}}^{N}\to\\{-,0,+\\}^{E}$ be the
function that assigns to every direction $\mu\in{\mathbb{S}}^{N}$ the
following vector indexed by elements $e\in E$:

$\Phi(\mu)_{e}:=\begin{cases}-&\text{if $\mu\in S_{e}^{-}$}\\\ 0&\text{if
$\mu\in S_{e}$}\\\ +&\text{if $\mu\in S_{e}^{+}$}.\end{cases}$ (3)

Consider the subposet
${{\mathcal{P}}}({{\mathcal{A}}}):=\\{\Phi(\mu):\mu\in{\mathbb{S}}^{N}\\}\subseteq\\{-,0,+\\}^{E}$.
The following proposition is a rewording of [4, Proposition 5.1.5].

######  Proposition 3.1:

Let $\phi:|K|\to{\mathbb{R}}^{N+1}$ be geometric embedding and let
${{\mathcal{A}}}$ be an induced signed arrangement of spheres on
${\mathbb{S}}^{N}$. If ${{\mathcal{A}}}$ is essential, then the sets
$\Phi^{-1}(f)\subseteq{\mathbb{S}}^{N}$, over all
$f\in{{\mathcal{P}}}({{\mathcal{A}}})$, are the cells of a cell complex
$({\mathbb{S}}^{N},{{\mathcal{C}}})$. Furthermore, for every pair of such
cells $C_{1}:=\Phi^{-1}(f)$ and $C_{2}:=\Phi^{-1}(g)$ in ${{\mathcal{C}}}$,
$C_{1}\leq C_{2}$ if and only if $f\leq g$ in
${{\mathcal{P}}}({{\mathcal{A}}})$.

The signed arrangement ${{\mathcal{A}}}$ associated to the piecewise linear
embedding $\phi$ is essential if and only if
$\bigcap_{e\in{{\mathcal{A}}}}S_{e}\neq\emptyset$ if and only if there is not
a direction $\mu\in{\mathbb{S}}^{n}$ such that the height function
$\phi_{\mu}$ is constant on all of $K$. For example, if $\phi$ embedds $K$
into a linear $N$-subspace of ${\mathbb{R}}^{N+1}$, then ${{\mathcal{A}}}$ is
not essential and we cannot apply Proposition 3.1. However, we can easily fix
this problem by adding one more $(N-1)$-great sphere to the arrangement that
is different from any of the $(N-1)$-great spheres already in
${{\mathcal{A}}}$. In general, for
$\dim\left(\bigcap_{e\in{{\mathcal{A}}}}S_{e}\right)=k$, we have to add $k+1$
great spheres to the arrangement ${{\mathcal{A}}}$.

###  3.2 Filtrations over Cellulation

Fix a geometric embedding $\phi:|K|\to{\mathbb{R}}^{N}$ and let
$({\mathbb{S}}^{N},{{\mathcal{C}}})$ be the induced cellulation on the
$N$-sphere as constructed above. Recall ${{\mathcal{C}}}$ is a poset where
$C_{1}\leq C_{2}$ if $C_{1}$ is a face of $C_{2}$. For every direction
$\mu\in{\mathbb{S}}^{n}$, the height function $\phi_{\mu}$ gives rise to an
object of ${\mathsf{Fil}}(K)$. By construction of the cellulation, any two
directions $\mu_{1}$ and $\mu_{2}$ give rise to the same object in
${\mathsf{Fil}}(K)$. Further, for every face relation $C_{1}\leq C_{2}$, there
is a natural filtration-preserving morphism from the filtration associated to
$C_{2}$ to the filtration associated to $C_{1}$. We now formalize this data as
a functor ${F}:{{\mathcal{C}}}\to{\mathsf{Fil}}(K)$.

We start by defining ${F}$ on the cells of ${{\mathcal{C}}}$. Choose a cell
$C\in{{\mathcal{C}}}$ and a direction $\mu\in C$. Two vertices
$v,v^{\prime}\in K^{0}$ are related, denoted $v\sim_{\mu}v^{\prime}$, if
$\phi_{\mu}(v)=\phi_{\mu}(v^{\prime})$. The relation $\sim_{\mu}$ is an
equivalence relation. Note that $v\sim_{\mu}v^{\prime}$ if and only if $\mu\in
S_{(v,v^{\prime})}$. Denote by ${{P}}_{\mu}$ the set of equivalence classes
$K^{0}/\sim_{\mu}$ union the singleton $\\{\top\\}$. For two equivalence
classes $[v]_{\mu},[v^{\prime}]_{\mu}$ in ${{P}}_{\mu}$, let
$[v]_{\mu}\leq[v^{\prime}]_{\mu}$ if
$\phi_{\mu}(v)\leq\phi_{\mu}(v^{\prime})$. Make $\top$ the top element. Thus
${{P}}_{\mu}$ is a finite, totally ordered lattice that is independent of the
choice of $\mu\in C$. Note that $[v]_{\mu}\neq[v^{\prime}]_{\mu}$ if and only
if $\mu\in S_{(v,v^{\prime})}^{+}$ or $\mu\in S_{(v,v^{\prime})}^{-}$. We now
define the filtration ${F}(C):={F}_{\mu}:P_{\mu}\to{\mathsf{Fil}}$. For every
$[v]_{\mu}$, let

${F}_{\mu}\big{(}[v]_{\mu}\big{)}:=\big{\\{}\sigma\in
K:\phi_{\mu}(\sigma)\leq\phi_{\mu}(v)\big{\\}}.$ (4)

Let ${F}_{\mu}(\top)$ be the entire simplicial complex $K$. Note that
${F}_{\mu}\big{(}[v]_{\mu}\big{)}$ is a subcomplex of $K$ and for
$[v]_{\mu}\leq[v^{\prime}]_{\mu}$, ${F}_{\mu}\big{(}[v]_{\mu}\big{)}$ is a
subcomplex of ${F}_{\mu}\big{(}[v^{\prime}]_{\mu}\big{)}$. In other words,
${F}_{\mu}$ is an object of ${\mathsf{Fil}}(K)$.

We now define ${F}$ on the face relations $C_{1}\leq C_{2}$ of
${{\mathcal{C}}}$. Choose directions $v\in C_{1}$ and $\mu\in C_{2}$. There is
a canonical bounded lattice function $\alpha:{{P}}_{\mu}\to{{P}}_{\nu}$ that
turns out to be a filtration-preserving morphism
$\big{(}{F}(C_{2}),{F}(C_{1}),\alpha\big{)}$ as follows. Since
$\Phi(v)\leq\Phi(\mu)$, $[v]_{\mu}\subseteq[v]_{\nu}$ for every vertex $v\in
K^{0}$. Let $\alpha\big{(}[v]_{\mu}\big{)}:=[v]_{\nu}$. Again, since
$\Phi_{(v,v^{\prime})}(\mu)\leq\Phi_{(v,v^{\prime})}(v)$, we have that
$[v]_{\mu}\leq[v^{\prime}]_{\mu}$ implies $[v]_{\nu}\leq[v^{\prime}]_{\nu}$.
Let $\alpha(\top):=\top$. Thus $\alpha$ is a bounded lattice function. By
piecewise linearity of $\phi$, a simplex
$\sigma\in{F}_{\mu}\big{(}[v]_{\mu}\big{)}$ if and only if
$\phi_{\mu}(v_{i})\leq\phi_{\mu}(v)$ for every vertex $v_{i}$ of $\sigma$.
Since $\alpha$ is order-preserving, $\phi_{\mu}(v_{i})\leq\phi_{\mu}(v)$
implies $\phi_{\nu}(v_{i})\leq\phi_{\nu}(v)$ and thus
$\sigma\in{F}_{\nu}\big{(}[v]_{\nu}\big{)}$. Thus
$\big{(}{F}_{\mu},{F}_{\nu},\alpha\big{)}$ is a filtration-preserving
morphism. Now consider a third cell $C_{3}$ such that $C_{1}\leq C_{2}\leq
C_{3}$ and choose a direction $w\in C_{3}$. Let
$\beta:{{P}}_{w}\to{{P}}_{\mu}$ be the corresponding bounded lattice function.
Then, the assignment $[v]_{w}\mapsto[v]_{\mu}$ is the composition
$\alpha\circ\beta\big{(}[v]_{w}\big{)}$. Thus the composition ${F}(C_{1}\leq
C_{2})\circ{F}(C_{2}\leq C_{3})$ of filtration-preserving morphisms equals
${F}(C_{1}\leq C_{3})$.

###  3.3 Persistence Diagrams over Cellulation

Finally, we define the combinatorial PH transform associated to a geometric
complex.

######  Definition 3.2:

Let $\phi:|K|\to{\mathbb{R}}^{N+1}$ be a geometric complex,
$({\mathbb{S}}^{N},{{\mathcal{C}}})$ the induced cellulation on the
$N$-sphere, and ${F}:{{\mathcal{C}}}\to{\mathsf{Fil}}$ the functor as
constructed above. The $d$-th combinatorial PH transform of $\phi$ is the
functor ${{\mathsf{PH}}}_{d}\circ{F}:{{\mathcal{C}}}\to{\mathsf{Fnc}}$. The
combinatorial PH transform of $\phi$ is the indexed set
$\big{\\{}{{\mathsf{PH}}}_{d}\circ F\big{\\}}_{d\in{{\mathbb{N}}}}$, over all
dimensions $d$.

Fix an embedding $\phi:|K|\to{\mathbb{R}}^{N+1}$ and let
$\tilde{F}_{d}:{{\mathcal{C}}}\to{\mathsf{Fnc}}$ be its $d$-th combinatorial
PH transform. The _display locale_ [10, 11] of $\tilde{F}_{d}$, or generalized
vineyard [7], has an interesting structure. By Proposition 2.6, there is a
constant $n\in{{\mathbb{N}}}$ such that, for each cell $C\in{{\mathcal{C}}}$,
the total charge of the persistence diagram $\tilde{F}_{d}(C)$ is $n$.
Further, for every cell relation $C_{1}\leq C_{2}$, the morphism
$\tilde{F}(C_{1}\leq C_{2})$ taking the persistence diagram $\tilde{F}(C_{2})$
to the persistence diagram $\tilde{F}(C_{1})$ is charge-preserving. Thus,
there is a unique charge associated to each connected component of the display
locale.

##  4 Examples

In this section, we walk through two examples, one in ${\mathbb{R}}^{2}$ and
one in ${\mathbb{R}}^{3}$.

###  4.1 The V Example

We start with a simple example to step through the construction of the
combinatorial PH transform of Definition 3.2. Specifically, our abstract
simplicial complex consists of two edges connected along a single vertex:
$K=\\{\\{v_{1}\\},\\{v_{2}\\},\\{v_{3}\\},\\{v_{1},v_{2}\\},\\{v_{2},v_{3}\\}\\}$.
This complex is piecewise linearly embedded into ${\mathbb{R}}^{2}$ through
the function $\phi\colon|K|\to{\mathbb{R}}^{2}$ with $\phi(v_{1})=(-1,1)$,
$\phi(v_{2})=(0,0)$, and $\phi(v_{3})=(1,1)$. See Figure 1.

Figure 1: The V, embedded in ${\mathbb{R}}^{2}$. This simplicial complex has
three vertices and two edges. By exploring the combinatorial PH transform for
this example, we illustrate each step of the construction.

#### Cellulation

The embedding $\phi$ induces a cellulation on the unit circle
${\mathbb{S}}^{1}$ of directions in ${\mathbb{R}}^{2}$. We now walk through
the construction of this cellulation. Let $E$ be the set of unique pairs of
vertices in $K$. In other words, let
$E:=\\{(v_{1},v_{2}),(v_{1},v_{3}),(v_{2},v_{3})\\}$. For every
$(v_{i},v_{j})\in E$, let $H_{(i,j)}$ be the unique line through the origin
that is perpendicular to the vector $\phi(v_{i})-\phi(v_{j})$. That is,
$H_{(i,j)}$ is the line perpendicular to the line through the embeddings of
$v_{i}$ and $v_{j}$, as shown in Figure 2(2(a)). The intersection
$S_{(i,j)}:=H_{(i,j)}\cap{\mathbb{S}}^{1}$ is a zero-sphere separating
${\mathbb{S}}^{1}$ into two components. As an arbitrary choice of signs, we
denote the connected component containing $(1,0)$ as $S_{(i,j)}^{+}$ and the
other connected component as $S_{(i,j)}^{-}$. We now have a function
$\Phi\colon{\mathbb{S}}^{1}\to\\{-,0,+\\}^{E}$ as in Equation (3). Finally,
let $({\mathbb{S}}^{1},{{\mathcal{C}}})$ be the cell complex as defined in
Proposition 3.1; in particular, ${{\mathcal{C}}}$ has six zero-cells and six
one-cells. See Figure 2(2(b)), where the $12$ cells are labeled with their
images under $\Phi$. Notice that in this example, we have an essential
arrangement. Since no two cells have the same label, we use $C_{X}$ to denote
the cell of ${{\mathcal{C}}}$ labeled $X\in\\{-,0,+\\}^{3}$ (if such a cell
exists). Again, by Proposition 3.1, we also have a partial order on the cells
(e.g., $C_{(0++)}<C_{(+++)}$).

(a) Lines Intersecting ${\mathbb{S}}^{1}$

(b) Cellulation of ${\mathbb{S}}^{1}$

Figure 2:  In (2(a)), we see the three linear subpaces of ${\mathbb{R}}^{2}$
that are used to define the cellulation over ${\mathbb{S}}^{1}$. In (2(b)),
each cell is labeled by a vector in $\\{-,0,+\\}^{3}$ according to which side
of $S_{1,2}$, $S_{1,3}$, and $S_{2,3}$ the cell falls. For example, the vector
$(-++)$ labels the one-cell whose points are all in $S_{1,2}^{-}$,
$S_{1,3}^{+}$, and $S_{2,3}^{+}$. The vector $(0++)$ labels the zero-cell that
is in $S_{1,2}$, $S_{1,3}^{+}$, and $S_{2,3}^{+}$. In fact,
$C_{(0++)}=S_{1,2}\cap S_{1,3}^{+}\cap S_{2,3}^{+}$. Note that no label is
$(000)$, and that all labels are distinct. The partial order of the cells is
denoted by arrows (where $a\to b$ indicates that $b<a$).

#### Objects of Filtration Functor

The geometric embedding $\phi$ induces a functor
${F}:{{\mathcal{C}}}\to{\mathsf{Fil}}$ of filtrations and filtration-
preserving morphisms. We now walk through the construction of ${F}$ on the
four cells in the highlighted region of Figure 2(2(b)).

Figure 3: The bounded lattice functions for the three highlighted face
relations in Figure 2(2(b)). Notice that the two maps into ${{P}}_{(0++)}$ are
nearly bijections, except for two vertices ($v_{1}$ and $v_{2}$) that map to
the same equivalence class for both maps. This corresponds to the
transposition of the two vertices between directions in $C_{(+++)}$ and
$C_{(-++)}$. In fact, this property holds more generally for any face relation
between codimension-one cells.

Let $\mu$ be a direction in $C_{(+++)}$. Using the height function
$\phi_{\mu}\colon|K|\to{\mathbb{R}}$, we obtain the following totally ordered
lattice $[v_{1}]_{\mu}<[v_{2}]_{\mu}<[v_{3}]_{\mu}$; see Figure 3 (far left).
To this, we add an additional equivalence class $[\top]_{\mu}$ as the top
element of the lattice; in other words, we have the totally ordered lattice:

$[v_{1}]_{\mu}<[v_{2}]_{\mu}<[v_{3}]_{\mu}<[\top]_{\mu}.$

Denote this lattice ${{P}}_{\mu}$. Associated to ${{P}}_{\mu}$ is the
filtration ${F}_{\mu}$ of Equation (4). Specifically,

$\displaystyle{F}_{\mu}([v_{1}]_{\mu})$ $\displaystyle:=\\{v_{1}\\}$
$\displaystyle{F}_{\mu}([v_{2}]_{\mu})$
$\displaystyle:=\\{v_{1},v_{2},\\{v_{1},v_{2}\\}\\}$
$\displaystyle{F}_{\mu}([v_{3}]_{\mu})$
$\displaystyle:=\\{v_{1},v_{2},v_{3},\\{v_{1},v_{2}\\},\\{v_{2},v_{3}\\}\\}=K$
$\displaystyle{F}_{\mu}([\top]_{\mu})$ $\displaystyle:=K.$

Notice that
${F}_{\mu}([v_{1}]_{\mu})\subset{F}_{\mu}([v_{1}]_{\mu})\subset{F}_{\mu}([v_{1}]_{\mu})\subseteq{F}_{\mu}([\top]_{\mu})$.
Moreover, this lattice and filtration are independent of the choice of $\mu$.
Therefore, we can define $[v_{i}]_{(+++)}:=[v_{i}]_{\mu}$,
$[\top]_{(+++)}:=[\top]_{\mu}$, ${{P}}_{(+++)}:={{P}}_{\mu}$, and
${F}_{(+++)}:={F}_{\mu}$.

Next, consider the cell $C_{(-++)}$. We proceed as above and observe that
${{P}}_{(-++)}$ is the totally ordered lattice
$[v_{2}]_{(-++)}<[v_{1}]_{(-++)}<[v_{3}]_{(-++)}<[\top]_{(-++)}$; see Figure 3
(middle right). For simplicity, we drop the subscripts of the equivalence
classes when the cell (or direction) is clear from context and simply write
$[v_{2}]$ from here on. Again, we use Equation (4) to define the filtration
${F}_{(-++)}$ is defined by

$\displaystyle{F}_{(-++)}([v_{2}])$ $\displaystyle:=\\{v_{2}\\}$
$\displaystyle{F}_{(-++)}([v_{1}])$
$\displaystyle:=\\{v_{1},v_{2},\\{v_{1},v_{2}\\}\\}$
$\displaystyle{F}_{(-++)}([v_{3}])$
$\displaystyle:=\\{v_{1},v_{2},v_{3},\\{v_{1},v_{2}\\},\\{v_{2},v_{3}\\}\\}=K$
$\displaystyle{F}_{(-++)}([\top])$ $\displaystyle:=K.$

We note the partial order on ${{P}}_{(-++)}$ given by the heights of vertices
in direction $\mu$ induces the following partial order on the simplicial
complexes:
${F}_{(-++)}([v_{2}])\subset{F}_{(-++)}([v_{1}])\subset{F}_{(-++)}([v_{3}])\subseteq{F}_{(-++)}([\top])$.

The zero-cells of ${{\mathcal{C}}}$ are exactly the directions that two
vertices are seen at the same height. For example, in $C_{(0++)}$,the heights
of $v_{1}$ and $v_{2}$ are the same (since
$\phi_{\mu}(v_{1})=\phi_{\mu}(v_{2})$), and this height is less than
$\phi_{\mu}(v_{3})$. Thus, the lattice ${{P}}_{(0++)}$ is the following total
order on the induced equivalence classes: $[v_{1}]<[v_{3}]<[\top]$. In
addition, the filtration ${F}_{(0++)}$ is defined by

$\displaystyle{F}_{(0++)}([v_{1}])$
$\displaystyle:=\\{v_{1},v_{2},\\{v_{1},v_{2}\\}\\}$
$\displaystyle{F}_{(0++)}([v_{3}])$
$\displaystyle:=\\{v_{1},v_{2},v_{3},\\{v_{1},v_{2}\\},\\{v_{2},v_{3}\\}\\}=K$
$\displaystyle{F}_{(0++)}([\top])$ $\displaystyle:=K.$

Again, we notice the inclusion of subcomplexes:
${F}_{(0++)}([v_{1}])\subset{F}_{(0++)}([v_{3}])\subseteq{F}_{(0++)}([\top])$.
Finally, ${{P}}_{(-0+)}$ is the lattice $[v_{2}]<[v_{3}]<[\top]$, with
corresponding filtration ${F}_{(-0+)}$ defined by
${F}_{(-0+)}([v_{2}])=\\{v_{2}\\}$ and
${F}_{(-0+)}([v_{3}])=\\{v_{1},v_{2},v_{3},\\{v_{1},v_{2}\\},\\{v_{2},v_{3}\\}\\}=K={F}_{(-0+)}([\top])$.
See Figure 3.

#### Arrows of the Filtration Functor

We now walk through the construction of ${F}$ on the three arrows in the
highlighted region of Figure 2(2(b)). Consider the face relation
$C_{(0++)}\leq C_{(+++)}$. We define the filtration-preserving morphism
$\big{(}{F}(C_{(+++)}),{F}(C_{(0++)},\alpha\big{)}$ as follows. For each
vertex $v_{i}$, we have the inclusion
$[v_{i}]_{(+++)}\subseteq[v_{i}]_{(0++)}$. Indeed,

$\displaystyle[v_{1}]_{(+++)}$
$\displaystyle=\\{v_{1}\\}\subseteq\\{v_{1},v_{2}\\}=[v_{1}]_{(0++)}$
$\displaystyle[v_{2}]_{(+++)}$
$\displaystyle=\\{v_{2}\\}\subseteq\\{v_{1},v_{2}\\}=[v_{2}]_{(0++)}$
$\displaystyle[v_{3}]_{(+++)}$
$\displaystyle=\\{v_{3}\\}\subseteq\\{v_{3}\\}=[v_{3}]_{(0++)}$
$\displaystyle[\top]_{(+++)}$
$\displaystyle=\emptyset\subseteq\emptyset=[\top]_{(0++)}$

As a consequence, $F([v]_{(+++)}))\subset{F}([v]_{(0++)})$ for all objects
$[v]\in{{P}}_{(+++)}$. Thus, we can define
$\alpha({F}([v]_{(+++)})):={F}([v]_{(0++)})$. In other words, we have
collapsed two equivalence classes in the lattice and consequently the
filtration, and left the rest untouched. See Figure 3.

#### Combinatorial PH Transform

Now that we have a filtration functor, we can apply the persistent homology
pipeline of [21] and described in Section 2.2. Our next step is to apply the
birth-death functor ${{\mathsf{ZB}}}\colon{\mathsf{Fil}}\to{\mathsf{Mon}}$. To
keep the notation as simple as possible, we use the field
$\mathsf{k}={\mathbb{Z}}/2{\mathbb{Z}}$. We explain this functor by looking at
how it behaves on the objects ${F}_{(+++)}$ and ${F}_{(0++)}$, and the arrow
${F}_{(+++)}\to{F}_{(0++)}$. Recall from above that ${{P}}_{(+++)}$ is the
totally ordered lattice $[v_{1}]<[v_{2}]<[v_{3}]<[\top]$. For simplicity of
notation, we use $[i,j]$ to represent $\big{[}[v_{i}],[v_{j}]\big{]}$. Thus,
$\overline{P_{(+++)}}$ is the nine element poset shown on the left hand side
of Figure 4(4(a)). Similarly, $\overline{P_{(0++)}}$ is the poset shown on the
right hand side of Figure 4(4(a)).

(a) Intervals

(b) Image of ${{\mathsf{ZB}}}_{0}$

Figure 4: The maps $\overline{P_{(+++)}}\to\overline{P_{0++}}$ and
${{\mathsf{ZB}}}_{0}(\overline{P_{(+++)}}\to\overline{P_{0++}})$. In both
maps, the objects in the same pink region get mapped to the same object in the
codomain. Using the field $\mathsf{k}={\mathbb{Z}}/2{\mathbb{Z}}$, for
$b\neq\top$, ${{\mathsf{ZB}}}_{0}[a,b]$ counts the number of zero-cycles in
the simplicial complex ${F}(a)$ that are zero-boundaries in the larger
simplicial complex ${F}(b)$. ${{\mathsf{ZB}}}_{0}[a,\top]$ is a count of the
number of vertices in ${F}(a)$.

The bounded lattice function $\alpha\colon{{P}}_{(+++)}\to{{P}}_{(0++)}$ (the
leftmost map in Figure 3) induces a bounded lattice function
$\bar{\alpha}\colon\overline{P_{(+++)}}\to\overline{P_{(0++)}}$. In
particular,

$\displaystyle[1,1]_{(0++)}$
$\displaystyle=\bar{\alpha}([1,1]_{(+++)})=\bar{\alpha}([1,2]_{(+++)})=\bar{\alpha}([2,2]_{(+++)}),$
$\displaystyle[1,3]_{(0++)}$
$\displaystyle=\bar{\alpha}([1,3]_{(+++)})=\bar{\alpha}([2,3]_{(+++)}),$
$\displaystyle[1,\top]_{(0++)}$
$\displaystyle=\bar{\alpha}([1,\top]_{(+++)})=\bar{\alpha}([2,\top]_{(+++)}),$
$\displaystyle[3,3]_{(0++)}$ $\displaystyle=\bar{\alpha}([3,3]_{(+++)}),$
$\displaystyle[3,\top]_{(0++)}$
$\displaystyle=\bar{\alpha}([3,\top]_{(+++)}),\text{ and }$
$\displaystyle[\top,\top]_{(0++)}$
$\displaystyle=\bar{\alpha}([\top,\top]_{(+++)}).$

These three equivalences are illustrated with pink shading in Figure 4(4(a)).
If $i\neq 1$, then $K$ has no $i$-cycles and hence ${{\mathsf{ZB}}}_{i}$ is
trivial everywhere. The values of ${{\mathsf{ZB}}}_{0}$ on the objects in
$\overline{P_{(+++)}}$ and $\overline{P_{0++}}$ are illustrated in Figure
4(4(b)), with the induced monotone-preserving morphism.

Finally, we apply the Möbius Inversion Functor
${{\mathsf{MI}}}\colon{\mathsf{Mon}}\to{\mathsf{Fnc}}$. Again, we illustrate
this functor by describing how it behaves on
${{\mathsf{ZB}}}_{0}(\overline{P_{(+++)}}\to\overline{P_{(0++)}})$. First, we
define
${{\mathsf{MI}}}_{(+++)}:={{\mathsf{MI}}}({{\mathsf{ZB}}}_{0}(\overline{P_{(+++)}})$
to be the unique function $\sigma\colon\overline{P_{(+++)}}\to{\mathbb{Z}}$
that satisfies Equation (2), where
$f={{\mathsf{ZB}}}_{0}(\overline{P_{(+++)}}$. In particular, for
$x\in\overline{P_{(+++)}}$

${{\mathsf{MI}}}_{(+++)}(x)=\begin{cases}1&x\in\big{\\{}[2,2],[3,3],[1,\top]\big{\\}}\\\
0&\text{otherwise}.\end{cases}$

In other words, the augmented persistence diagram (defined in [13, Def. 1])
has three points: $(2,2),(3,3)$, and $(1,\infty)$. This corresponds to the
persistence diagram with one off-diagonal point, namely $(1,\infty)$, which
corresponds to the connected component born at the height of $v_{1}$. Next, we
consider the cell $S_{(0++)}$. Let $x\in\overline{P_{(0++)}}$. Then,

${{\mathsf{MI}}}_{(0++)}(x)=\begin{cases}1&x\in\left\\{[1,1],[3,3],[1,\top]\right\\}\\\
0&\text{otherwise}.\end{cases}$

Moreover, the map ${{\mathsf{MI}}}_{(+++)}\to{{\mathsf{MI}}}_{(0++)}$
corresponds to mapping the points in the following way:
$[2,2]_{(+++)}\mapsto[1,1]_{(0++)}=[2,2]_{(0++)}$,
$[3,3]_{(+++)}\mapsto[3,3]_{(0++)}$, and $[1,T]_{(+++)}\mapsto[1,T]_{(0++)}$.
We note that while $[2,2]$ and $[1,T]$ have distinct birth heights in
${{\mathsf{MI}}}_{(+++)}$, their images $[1,1]$ and $[1,T]$ have the same
birth height in ${{\mathsf{MI}}}_{(0++)}$.

###  4.2 3D Example

We now consider a geometric complex embedded in ${\mathbb{R}}^{3}$.
Specifically, let $K$ be the following abstract simplicial complex known as a
book with two pages (see Figure 5(5(a))):

$K=\big{\\{}\\{v_{1}\\},\\{v_{2}\\},\\{v_{3}\\},\\{v_{1},v_{2}\\},\\{v_{2},v_{3}\\},\\{v_{1},v_{3}\\},\\{v_{1},v_{2},v_{3}\\}\\{v_{1},v_{4}\\},\\{v_{2},v_{4}\\},\\{v_{1},v_{2},v_{4}\\}\big{\\}}.$

This complex is piecewise linearly embedded into ${\mathbb{R}}^{3}$ through
the function $\phi\colon|K|\to{\mathbb{R}}^{2}$ with $\phi(v_{1})=(1,0,0)$,
$\phi(v_{2})=(0,1,0)$, $\phi(v_{3})=(0,0,1)$ and $\phi(v_{0})=(0,0,0)$. Note
that the affine space spanned by the points $\\{v_{1},v_{2},v_{3},v_{4}\\}$ is
${\mathbb{R}}^{3}$, indicating that the vertices of $K$ are in general
position.

(a) Complex in ${\mathbb{R}}^{3}$

(b) Cellulation of ${\mathbb{S}}^{2}$

Figure 5: A geometric complex in ${\mathbb{R}}^{3}$ with the induced
cellulation of ${\mathbb{S}}^{2}$. The cellulation depends only on the vetices
of the complex. Since the vertices are in general position, each great circle
on ${\mathbb{S}}^{2}$ is distinct.

Rather than go through the whole pipeline again, we investigate the cell
complex and the poset of equivalence classes of vertices associated to each
cell.

Let $({\mathbb{S}}^{2},{{\mathcal{C}}})$ be the cell complex defined in
Proposition 3.1. This cellulation has $24$ two-cells, one for each permutation
of the vertices. (Note that if we were to add one more vertex to $K$ that
there would then be more permutations of the vertices than cells of the
cellulation due to geometric constraints). Each of the $36$ one-cells
correspond to where two vertices are at the same height and the remaining two
vertices are at distinct heights. There are two types of zero-cells: (type-1)
eight have degree six and correspond to exactly three vertices at the same
height, and (type-2) six have degree four and correspond to two vertices at
one height and the other two vertices at another height. See Figure 5(5(b)).

For each vector $X\in\\{-,0,+\\}^{6}$, we use $C_{(X)}$ to denote the cell of
${{\mathcal{C}}}$ labeled $X$ according to which side of $S_{1,2}$, $S_{1,3}$,
$S_{2,3}$, $S_{1,4}$, $S_{2,4}$, and $S_{3,4}$ the cell falls (if such a cell
exists). For example, $C_{(+---++)}=S_{1,2}^{+}\cap S_{1,3}^{-}\cap
S_{2,3}^{-}\cap S_{1,4}^{-}\cap S_{2,4}^{+}\cap S_{3,4}^{+}$. The faces of
$C_{(+---++)}$ are:

$\\{C_{(+---++)},C_{(0---++)},C_{(+0--++)},C_{(+---+0)},C_{(+0-0+0)},C_{(0---+0)},C_{(000-++)}\\}.$

In fact, every two-cell in this cellulation has seven faces (itself, three
one-cells, and three zero-cells). In other words, every two-cell is a
topological triangle. Some labelings of zero-cells do not exist. For example,
no cell is labeled ${(00--++)}$. By contradiction, suppose there exists
$\mu\in C_{(00--++)}$. Then, since the first coordinate is zero, $\mu\in
S_{1,2}$, which implies that $[v_{1}]_{\mu}=[v_{2}]_{\mu}$. In addition, since
the second coordinate is zero, $\mu\in S_{2,3}$, which implies that
$[v_{2}]_{\mu}=[v_{3}]_{\mu}$. Thus, $[v_{1}]_{\mu}=[v_{2}]_{\mu}$, which
means that $\mu\in S_{2,3}$, a contradiction.

To see what happens at a type-$1$ zero-cell, consider ${{P}}_{(000-++)}$. In
particular, the equivalence classes for the vertices induced by a direction
vector in $C_{(000-++)}$ is:

$\displaystyle[v_{1}]_{(000-++)}=[v_{2}]_{(000-++)}=[v_{3}]_{(000-++)}$
$\displaystyle[v_{4}]_{(000-++)}$ $\displaystyle[\top].$

In general, all type-$1$ zero-cells have three vertices sharing an equivalence
class. To see what happens at a type-$2$ zero-cell, consider
${{P}}_{(0---+0)}$. Here, we have the following equivalence classess:

$\displaystyle[v_{1}]_{(0---+0)}=[v_{2}]_{(0---+0)}$
$\displaystyle[v_{3}]_{(0---+0)}=[v_{4}]_{(0---+0)}$ $\displaystyle[\top].$

In general, all type-$2$ zero-cells partition the vertices into two
equivalence classes, each of size two (and one additional equivalence class
for $\top$).

Figure 6: Face relations induce surjective poset maps. These maps, in turn,
induce arrows in ${\mathsf{Fil}}$.

For geometric complexes in ${\mathbb{R}}^{3}$ or higher, we can look at
compositions of proper face relations. Consider the following two face
relations: $C_{(000-++)}<C_{(0---++)}<C_{(+---++)}$. These face relations
induced posets of equivalence classes of the vertices, as well as the maps
connecting these posets, are given in Figure 6. These two maps corresponds to
the equivalences classes $[v_{1}]_{(+---++)}\neq[v_{2}]_{(+---++)}$ being
mapped to the same equivalence class $[v_{1}]_{(0---++)}$ in $P_{(0---++)}$,
followed by the equivalences classes
$[v_{1}]_{(0---++)}\neq[v_{3}]_{(0---++)}$ equivalence classes in
$P_{(+---++)}$ being mapped to the same equivalence class $[v_{1}]_{(000-++)}$
in $P_{(000-++)}$. Observe that the map ${{P}}_{(+---++)}\to{{P}}_{(000-++)}$
is exactly this composition resulting in collapsing three equivalence classes
in $P_{(+---++)}$ (namely, $[v_{1}]$, $[v_{2}]$, and $[v_{3}]$) to one
equivalence class.

##  5 Discussion

By taking a combinatorial approach to persistent homology, we are able to
express the PH transform of a geometric complex as a functor from the
cellulation of a sphere to the category of combinatorial persistence diagrams.
This point of view may be applied to any application that involves a
parameterized family of persistence diagrams. For example, consider time-
varying data [23, 18, 7, 6]. Assuming the data is finite, there is a
cellulation of the real line (time parameter) such that any two times on the
same cell have the same combinatorial persistence diagram. The resulting
family of combinatorial persistence diagrams connected by charge-preserving
morohisms provides a categorical framework in which to talk about all the
persistence diagrams at once in a cohesive manner.

We hope this paper will start conversations on parameterized families of
combinatorial persistence diagrams in general and not just for the PH
transform. There are many questions to ask. For example, consider a cell
complex $(X,{{\mathcal{C}}})$ on any space $X$ and a functor
$F:{{\mathcal{C}}}\to{\mathsf{Fnc}}$. What are its global co-sections? Does
the edit distance of McClearly and Patel lead to a stability result for such
functors? These are just a few questions worth asking.

#### Acknowledgements

The first author acknowledges Vin de Silva for a late night conversation about
counting cells and David L. Millman for generating Figure 5(5(b)).

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##  Appendix A Notation and Definitions

#### Posets

Given a finite, totally ordered poset $P$, denote by $\bot\in P$ its smallest
element and by $\top\in{{P}}$ the largest element. A function
$\alpha:{{P}}\to{{Q}}$ between two finite, totally ordered posets is a
_bounded monotone function_ if $f(\top)=\top$, $f(\bot)=\bot$, and for all
$a\leq b$, $\alpha(a)\leq\alpha(b)$.

For a finite, totally ordered poset ${{P}}$, let
$\overline{P}:=\big{\\{}[a,b]\subseteq{{P}}:a\leq b\big{\\}}$ be its set of
non-empty intervals with the following partial relation: $[a,b]\preceq[c,d]$
if $a\leq c$ and $b\leq d$. Its bottom element is $[\bot,\bot]$ and its top
element is $[\top,\top]$. A bounded monotone function $\alpha:{{P}}\to{{Q}}$
between two finite, totally ordered posets induces a bounded monotone function
$\bar{\alpha}:\bar{P}\to\bar{Q}$ where
$\bar{\alpha}\big{(}[a,b]\big{)}:=\big{[}\alpha(a),\alpha(b)\big{]}$; see [21]
for a proof.

#### Categories and Functors

In this paper, we assume that the reader is familiar with categories and
functors. We defer to [24] for an introduction to category theory. In Table 1,
we provide notations used in this paper.

Table 1: Notations for categories and functors. We assume ${\mathsf{C}}$ is a category and $a,b,c\in{\mathsf{ob}\,{\mathsf{C}}}$. ${\mathsf{ob}\,{\mathsf{C}}}$ |  objects in ${\mathsf{C}}$
---|---
${\mathrm{Hom}}_{\mathsf{C}}(a,b)$ |  the set of morphisms or arrows between $a$ and $b$ in ${\mathsf{C}}$
$\circ$ |  composition of morphisms
$1_{a}$ |  the identity morphism in ${\mathrm{Hom}}_{{\mathsf{C}}}(a,a)$