# Signature of an Ultrafast Photo-Induced Lifshitz Transition in the Nodal-
Line Semimetal ZrSiTe

Robert J. Kirby  Department of Chemistry, Princeton University, Princeton, New
Jersey 08544, USA  Lukas Muechler  Center for Computational Quantum Physics,
The Flatiron Institute, New York, New York, 10010, USA  Sebastian Klemenz
Department of Chemistry, Princeton University, Princeton, New Jersey 08544,
USA  Caroline Weinberg  Department of Chemistry, Princeton University,
Princeton, New Jersey 08544, USA  Austin Ferrenti  Department of Chemistry,
Princeton University, Princeton, New Jersey 08544, USA  Mohamed Oudah
Department of Chemistry, Princeton University, Princeton, New Jersey 08544,
USA  Quantum Matter Institute, University of British Columbia, Vancouver,
British Columbia, V6T 1Z4, Canada  Daniele Fausti  Department of Chemistry,
Princeton University, Princeton, New Jersey 08544, USA  Department of Physics,
Università degli Studi di Trieste, Trieste I-34127, Italy  Sincrotrone Trieste
S.C.p.A., Basovizza, Trieste I-34012, Italy  Gregory D. Scholes  Department of
Chemistry, Princeton University, Princeton, New Jersey 08544, USA  Leslie M.
Schoop<EMAIL_ADDRESS>Department of Chemistry, Princeton University,
Princeton, New Jersey 08544, USA

###### Abstract

Here we report an ultrafast optical spectroscopic study of the nodal-line
semimetal ZrSiTe. Our measurements reveal that, converse to other compounds of
the family, the sudden injection of electronic excitations results in a
strongly coherent response of an $A_{1g}$ phonon mode which dynamically
modifies the distance between Zr and Te atoms and Si layers. “Frozen phonon”
DFT calculations, in which band structures are calculated as a function of
nuclear position along the phonon mode coordinate, show that large
displacements along this mode alter the material’s electronic structure
significantly, forcing bands to approach and even cross the Fermi energy. The
incoherent part of the time domain response reveals that a delayed electronic
response at low fluence discontinuously evolves into an instantaneous one for
excitation densities larger than $3.43\times 10^{17}$ cm-3. This sudden change
of the dissipative channels for electronic excitations is indicative of an
ultrafast Lifshitz transition which we tentatively associate to a change in
topology of the Fermi surface driven by a symmetry preserving $A_{1g}$ phonon
mode.

††preprint: APS/123-QED

Light can be a powerful driving force in physical and chemical
transformations; photoexcitation above a certain threshold fluence can often
induce the same phase transitions at ambient conditions that would normally
require, for example, high temperature, such as the metal-to-insulator
transition in VO2 [1, 2, 3, 4] or charge-density wave melting in LaTe3 [5].
Importantly, most photo-induced phase transitions (PIPTs) are associated with
a structural symmetry breaking. Photoexcitation can also nonadiabatically
access regions of a material’s potential energy surface that equilibrium
stimuli cannot, leading to the possibility of new transformations that would
be inaccessible with other external stimuli [6, 7, 8]. Ultrafast laser
experiments give access to out-of-equilibrium PIPTs. Furthermore, ultrafast
pump-probe spectroscopic techniques allow one to follow the progress of the
PIPT, either directly (as with near-field optical techniques [9]) or
indirectly (e.g. through the contribution of certain coherent phonon modes and
their fluence dependence [3]). This can provide very powerful insight into the
mechanism through which the phase transition occurs, and can help to tailor
any potential applications to a certain phase transition and vice versa.

Figure 1:  (a) Transient response of ZrSiTe pumped at 1.24 eV at a fluence of
1.2 mJ/cm2. The black vertical dashed line indicates the probe energy
displayed in (d). (b) FFT spectrum of the oscillatory response in (a) as a
function of probe energy, showing a single frequency at 125 cm-1,
corresponding to an A1g phonon mode. (c) The crystal structure of ZrSiTe (blue
= Si, red = Zr, and yellow = Te) showing one phase of the A1g phonon mode, in
which the Zr and Te atoms approach the Si square net from either side; in the
other phase of the phonon mode (not shown), the Zr and Te atoms move further
away from the Si square net relative to their equilibrium positions. (d) The
transient response of ZrSiTe pumped at 1.24 eV, 1.2 mJ/cm2, at a probe energy
of 1.15 eV. In the first few periods of the coherent phonon, the amplitude of
the oscillatory part is quite large, tens-of-percent of the total magnitude.
This is in contrast with the same singular coherent phonon mode observed in
ZrSiSe ((e) pumped at 0.95 eV, 1.2 mJ/cm2, probed at 1.1 eV) and ZrSiS ((d)
pumped at 0.95 eV, 1.7 mJ/cm2, probed at 1.02 eV). The coherent phonon
response in ZrSiTe is significantly stronger than in either ZrSiSe or ZrSiS.

Recently, indications of a Lifshitz transition — a peculiar electronic phase
transition in which the topology of the Fermi surface changes without an
accompanying change in crystal symmetry [10] — have been observed in high-
pressure Raman and equilibrium reflectance experiments on the nodal-line
semimetal (NLSM) ZrSiTe, supported by density functional theory (DFT)
calculations [11, 12]. In these experiments, pressure was applied along the
$c$-axis of the material, forcing the atomic layers of the material closer
together. The evidence supporting the electronic phase transitions were
discontinuities in the screened plasma frequency (extracted from Kramers-
Kronig analysis of the equilibrium reflectance spectra) and the nonlinear
response of the frequency of an $A_{1g}$ phonon mode. The DFT calculations
indicated that at high pressure, bands cross the Fermi energy along $\Gamma$-X
and $\Gamma$-M.

The ZrSi$X$ ($X=$ S, Se, Te) family of materials are the archetypal NLSMs, and
have presented themselves as a particularly active area of research in recent
years, especially ZrSiTe. Crystallizing in the PbFCl structure type, in the
$P4/nmm$ space group, these materials consist of layers of $X$ and Zr on
either side of a Si square net, i.e. $X$-Zr-Si-Zr-$X$. Their band structures
are characterized by large-bandwidth nodal-line bands near the Fermi energy,
which are relatively free from interference from trivial bands. All three
ZrSi$X$ compounds also display non-symmorphic symmetry-protected Dirac
crossings at the X-point, and although these crossings are hundreds of meV
above and below the Fermi energy in ZrSiS and ZrSiSe, they occur at EF in
ZrSiTe [13]. ZrSiTe also has weaker interlayer bonding than the other ZrSi$X$
members [14], behaving like a quasi-2D material [15]; it can be exfoliated
down to few-layer sheets [16], and the monolayer is predicted to be a
topological insulator [14]. Additionally, topological drumhead surface states
were recently observed for the first time in ZrSiTe [17], distinguishing it
from ZrSiS where only trivial surface states are experimentally
observable[18]. Staying within the family, electronic correlations were
recently shown to exist in ZrSiSe [19], and a temperature-induced Lifshitz
transition has also been proposed [20] in the same compound.

Other topological semimetals (TSMs) have been shown to exhibit Lifshitz
transitions, including ZrTe5 [21], VAl3 [22], NbAs [23], TaP [24], TaAs [24],
WTe2 [25], NiTe2 [26], PtTe2 [27], and Td-MoS2 [28]. Most Lifshitz transitions
have been observed to occur in equilibrium, induced by a wide range of stimuli
including temperature, pressure and strain, and material composition. However,
the correlated Weyl semimetal Td-MoS2 stands out as being the only material
found to undergo a non-equilibrium Lifshitz transition [29]. An ultrafast
near-infrared (NIR) laser pulse was used to excite the material, and the
subsequent change in the Fermi surface was observed with time- and angle-
resolved photoemission spectroscopy (tr-ARPES). Ultrafast spectroscopy has
been shown to be such a useful tool in inducing and tracking the evolution of
other varieties of electronic and structural phase transitions; it is
surprising that there is a dearth of examples of its applicability to Lifshitz
transitions.

Figure 2:  Band structures for ZrSiTe for various displacements along the
coherent A1g phonon mode. (a) shows the effects of displacements which bring
the Zr and Te atoms closer to the Si square net, relative to the equilibrium
positions, and (b) shows the effect of the Zr and Te atoms moving further away
from the Si square net, relative to their equilibrium positions. The
displacement direction in (a) is that displayed in Fig. 1 (c). At high
displacements — measured by the change in the Si-Te distance — bands approach
and cross EF; ZrSiTe is predicted to undergo Lifshitz transitions along both
phases of the coherently-excited A1g phonon mode.

In this letter, we present a combined experimental and computational study of
the ultrafast optical response of ZrSiTe, showing that ZrSiTe is only the
second topological semimetal to undergo a PIPT. Transient reflectivity shows
that a strong relationship exists between the electronic subsystem and the
lattice through a particular phonon mode. DFT “frozen phonon” calculations of
the band structure as a function of nuclear position along the phonon mode
coordinate show that intense excitation of this phonon mode can have drastic
effects on the material’s band structure, potentially resulting in a Lifshitz
transition. Returning to the transient reflectivity experiments, the signature
of an ultrafast PIPT is observed in the fluence dependence of the incoherent
part of the transient response, that is, the response of bulk thermalized
electrons upon which the oscillatory motion of the coherent phonon is
overlaid. While these experiments cannot cast light on the precise nature of
the PIPT and the topology of the resulting Fermi surface, the strong
excitation and computations suggest that we might be witnessing an ultrafast
non-equilibrium Lifshitz transition, as well as a new mechanism by which the
transition is mediated by fully-symmetric coherent phonons.

Transient reflectivity experiments were performed with a commercial pump-probe
setup (Ultrafast System Helios, Sarasota, Florida). The 1.55 eV output from a
1 kHz regeneratively-amplified Ti:sapphire laser (Coherent Libra, Santa Clara,
California) was split with a 50:50 beamsplitter to produce both the pump and
probe pulses. Narrowband pump pulses centered at 1.24 and 0.95 eV were
produced in an optical parametric amplifier (OPerA Solo, Vilnius, Lithuania).
Broadband NIR probe pulses covering 0.8 - 1.35 eV were produced by focusing
the 1.55 eV light in a sapphire crystal. A $\lambda/2$-waveplate, a polarizer,
and an iris control the polarization, intensity, and shape of the beam used to
produce the probe pulse. The pulses strike the sample at near-normal
incidence, and the reflected probe light is collected by a fiber optic and
sent to the detector. All samples were measured in the $ab$ plane, and all
measurements were performed at room temperature. Additional details of the
sample handling and data processing can be found in the SI.

The transient response of ZrSiTe pumped at 1.24 eV with a fluence of 1.2
mJ/cm2 can be seen in Fig. 1 a. In this response, large oscillations due to
the excitation of coherent phonons are superimposed on top of the bulk
incoherent electronic response of the material. The FFT spectrum of the
oscillations as a function of probe energy is presented in Fig. 1 b, showing
that a single phonon mode is being excited; the FFT amplitude increases with
increasing probe energy — possibly approaching a resonance — but does not
completely disappear at the lower end of the probe window. The coherent
phonons dephase after 2 ps.

Figure 3:  The incoherent parts extracted from the transient data for ZrSiTe
pumped at 1.24 eV with fluences of 0.17 mJ/cm2 (a) and 1.2 mJ/cm2 (b). Arrows
emphasize the minima in these incoherent parts at 1.15 eV probe energy. (c)
illustrates how these incoherent minima fall into two different fluence
regimes. Below 0.75 mJ/cm2, the delay time at which the minimum is observed
decreases linearly with increasing pump fluence; there is a discontinuity at
the threshold fluence 0.75 mJ/cm2, after which the minima are observed at 0 ps
delay time, independent of increasing pump fluence. This is the signature of
an ultrafast PIPT discussed in the text.

The phonon frequency extracted from the FFT spectrum, 125 cm-1, corresponds to
an A1g phonon mode also observed in steady-state Raman experiments (Fig. S.
2); the mode is depicted in Fig. 1 c. This mode brings the Zr and Te atoms
closer to (Fig. 1 c), and farther away from (not shown), the Si square net
relative to their equilibrium positions. Almost an order of magnitude of pump
fluences were used, ranging from 0.17 mJ/cm2 to 1.36 mJ/cm2. These experiments
were also repeated using a lower energy pump pulse, 0.95 eV. These additional
data are presented in Figs. S. 3, 4, 6, and 7. At both pump energies and at
all fluences, the same sole 125 cm-1 Raman mode was observed, with similar
relative amplitudes. No additional modes appear above or below a certain
fluence, which suggests that ZrSiTe does not undergo a photo-induced change in
symmetry within our fluence range. The frequency of the 125 cm-1 A1g phonon
mode is roughly 15 cm-1 higher than that observed in bulk samples in other
studies [16, 12]. This is possibly due to slightly different compositions or
carrier concentrations resulting from crystals grown in different labs.
However, the frequency is consistent between our transient reflectivity and
steady-state Raman measurements (Fig. S. 2).

The coherent phonon response can be seen more clearly in the transient data at
a single probe energy, 1.15 eV, in Fig. 1 d. The selfsame A1g coherent phonon
mode has also been observed in the transient reflectivity spectra of ZrSiSe
and ZrSiS when pumped with 0.95 eV photons (Fig. 1 e, f). In these materials,
the mode has a frequency of 150 cm-1 and 220 cm-1, respectively. However, the
amplitude of the oscillations in these materials is minuscule compared to
those in ZrSiTe, and in experiments in which ZrSiSe and ZrSiS were pumped with
1.24 eV photons, no coherent oscillations were observed. Furthermore, these
materials only exhibit coherent phonons when pumped at high fluence — 1.2
mJ/cm2 for ZrSiSe and 1.7 mJ/cm2 for ZrSiS — which is not the case for ZrSiTe.
Although Te’s large polarizability, relative to those of Se or S, is
undeniably a contributing factor in the enhanced Raman response compared to
the other two materials, it is highly unlikely that this wholly accounts for
the significant increase in amplitude. The large amplitude of the coherent
phonons relative to the incoherent electronic response — tens of percent in
the first period — suggests that this mode is being strongly excited.

A recent study predicts that the geometry of this phonon mode, in which the
distance between the Zr and Te atoms and the Si square net is changing, should
have a strong impact on the stability of the nodal-line structure [30]. To
explore this, “frozen phonon” DFT calculations were performed using the VASP
and phonopy packages [31, 32] with the standard pseudopotentials for Zr, Si,
and Te. The experimental geometries were taken from Ref. [33]. For the self-
consistent calculations, the reducible BZ was sampled by a $9\times 9\times 7$
k-mesh and spin-orbit coupling was included. The one-electron energies were
calculated as a function of the $A_{1g}$ normal mode coordinate, assuming the
adiabatic Born-Oppenheimer approximation.

The results of the calculations are separated by the phase of the phonon
motion. The effects of the Te and Zr atoms moving towards the Si square net
(compared to their equilibrium positions) on the band structure are shown in
Fig. 2 a, herein referred to as “negative distortions”, and the effect of the
atoms moving further away from the Si square net (again compared to
equilibrium), or “positive distortions”, in Fig. 2 b. The change in Si-Te
distance in the crystal structure due to displacement along the phonon mode is
used as a metric for the amount of distortion.

Focusing first on the effects that negative distortions have on the band
structure (Fig. 2 a), we can see that there are two noticeable changes in the
vicinity of the Fermi energy. At high displacement, Te $p$-states increase in
energy towards EF between $\Gamma$ and X, almost touching EF at the highest
amplitude studied. Additionally, the nodal-line bands between R and Z,
primarily composed of Si $p$\- and Zr $d$-states, shift closer to R and to
higher energy even at modest distortions. Both of these band shifts would
alter the Fermi surface. Other changes to the band structure occur further
away from EF and are of little consequence to our experimental results.

Moving now to the effects of the positive distortions (Fig. 2 b), two striking
features stand out. The first can be seen at $\Gamma$, where a band approaches
EF from above, eventually crossing below it at the highest displacement
studied. The second is at Z, where bands increase in energy through EF. Other
shifts in the energy- and momentum-position of the nodal-line bands can be
seen between R and Z, similar to those observed at negative distortions.

These data indicate that ZrSiTe undergoes Lifshitz transitions at the extrema
of both phases of the phonon mode observed in the transient data. Due to the
strong excitation of this mode, these distortions along the phonon mode are
not believed to be unrealistic. The higher negative distortions were chosen to
be the Si-$X$, $X=$ (S, Se), distances in ZrSi$X$ (3.532 Å for ZrSiS and 3.615
Å for ZrSiSe, compared with 3.89105 Å for ZrSiTe), which is a chemically
realistic assumption of the range of distortion.

In order to show that the large phonon displacement might drive a Lifshitz
transition, we analyze the bulk incoherent electronic response that we
mentioned earlier. To extract the incoherent portion from the overall response
— comprised of the coherent oscillations and the underlying incoherent part —
we fit the transient at each probe energy with a low-order polynomial, which
was then subtracted from the overall response leaving just the non-oscillatory
part. Fig. 3 a and b show these extracted incoherent responses for low (a) and
high (b) fluences when pumped at 1.24 eV; the incoherent parts of the other
fluences and 0.95 eV experiments can be found in Figs. S. 5 and 8.

The incoherent part of the low fluence experiment (Fig. 3 a) reveals a minimum
in the data roughly 1 ps after the pump and probe pulses overlap, at a probe
energy of 1.15 eV. However, no such delayed-minimum appears in the high
fluence incoherent part (Fig. 3 b); here the minimum occurs right at 0 ps.
Plotting the delay time at which these minima occur (Fig. 3 c), we can see
there are two distinct regimes: for fluences below 0.75 mJ/cm2, the delay time
minimum decreases linearly with increasing fluence, whereas above 0.75 mJ/cm2
the minimum occurs during excitation by the pump pulse, independent of
fluence. The same phenomenon is observed in the 0.95 eV-pump experiments,
however here the threshold fluence is lower, 0.42 mJ/cm2 (Fig. S. 6). This
behavior, in which there are distinct regimes of a quantity (here the
incoherent minimum delay time) separated by a threshold fluence or excitation
density, is indicative of a PIPT. The threshold fluences correspond to
excitation densities of $3.43\times 10^{17}$ cm-3 (1.24 eV) and $2.71\times
10^{17}$ cm-3 (0.95 eV). The increase in threshold excitation density with
increasing pump energy is almost perfectly mirrored by a decrease in
absorbance: 0.0315 (1.24 eV) compared with 0.0417 (0.95 eV), meaning fewer
photons need to be absorbed to provide the required energy to complete the
phase transition immediately.

To conclude, transient reflectivity measurements on the NLSM ZrSiTe show that
NIR pulses can strongly excite one particular 125 cm-1 A1g phonon mode —
coherently — on top of the bulk incoherent electronic response. DFT studies
showed that large displacements along this phonon mode adjust the band
structure so significantly that bands approach and cross EF, abruptly changing
the Fermi surface such that the material undergoes a Lifshitz transition.
Furthermore, the fluence dependence of the incoherent part of the transient
response has the hallmark of an ultrafast PIPT, with a threshold fluence of
0.75 mJ/cm2 when pumping with 1.24 eV photons and 0.42 mJ/cm2 when pumping
with 0.95 eV photons. The large-amplitude coherent phonons give credence to
the displacements required to undergo a Lifshitz transition by DFT; however,
further studies are necessary to show whether the signature of an ultrafast
PIPT observed in the transient data corresponds to this Lifshitz transition or
arises from some other ultrafast electronic transition. Moreover, we
elucidated this picture within the intuitive Born-Oppenheimer approximation,
but in reality the nuclear and electronic degrees of freedom are likely
coupled. The possibility of an ultrafast photo-induced Lifshitz transition
could make ZrSiTe an ideal candidate material for high-frequency photo-
switchable optoelectronic devices, but also presents ZrSiTe as a potential
system from which to gain a better understanding of this exciting class of
ultrafast PIPT.

###### Acknowledgements.

This work was supported by NSF through the Princeton Center for Complex
Materials, a Materials Research Science and Engineering Center DMR-1420541, by
Princeton University through the Princeton Catalysis Initiative, and by the
Gordon and Betty Moore Foundation through Grant GBMF9064 to L.M.S. G.D.S. is a
CIFAR Fellow in the Bio-Inspired Energy Program. D.F. was supported by the
European Commission through the European Research Council (ERC), Project
INCEPT, Grant 677488. The authors also acknowledge the use of Princeton’s
Imaging and Analysis Center, which is partially supported by the Princeton
Center for Complex Materials, a National Science Foundation (NSF)-MRSEC
program (DMR-1420541). The Flatiron Institute is a division of the Simons
Foundation.

## References

  * Cavalleri  _et al._ [2004] A. Cavalleri, T. Dekorsy, H. H. Chong, J.-C. Kieffer, and R. W. Schoenlein, Evidence for a structurally-driven insulator-to-metal transition in VO2: A view from the ultrafast timescale, Phys. Rev. B 70, 161102 (2004).
  * Kübler  _et al._ [2007] C. Kübler, H. Ehrke, R. Huber, R. Lopez, A. Halabica, R. Haglund Jr, and A. Leitenstorfer, Coherent structural dynamics and electronic correlations during an ultrafast insulator-to-metal phase transition in VO2, Phys. Rev. Lett. 99, 116401 (2007).
  * Wall  _et al._ [2012] S. Wall, D. Wegkamp, L. Foglia, K. Appavoo, J. Nag, R. Haglund, J. Stähler, and M. Wolf, Ultrafast changes in lattice symmetry probed by coherent phonons, Nat. Commun. 3, 1 (2012).
  * Wegkamp  _et al._ [2014] D. Wegkamp, M. Herzog, L. Xian, M. Gatti, P. Cudazzo, C. L. McGahan, R. E. Marvel, R. F. Haglund Jr, A. Rubio, M. Wolf, and J. Stähler, Instantaneous band gap collapse in photoexcited monoclinic VO2 due to photocarrier doping, Physical review letters 113, 216401 (2014).
  * Zong  _et al._ [2019] A. Zong, A. Kogar, Y.-Q. Bie, T. Rohwer, C. Lee, E. Baldini, E. Ergeçen, M. B. Yilmaz, B. Freelon, E. J. Sie, H. Zhou, J. Straquadine, P. Walmsley, P. E. Dolgirev, A. V. Rozhkov, I. R. Fisher, P. Jarillo-Herrero, B. V. Fine, and N. Gedik, Evidence for topological defects in a photoinduced phase transition, Nat. Phys. 15, 27 (2019).
  * Ichikawa  _et al._ [2011] H. Ichikawa, S. Nozawa, T. Sato, A. Tomita, K. Ichiyanagi, M. Chollet, L. Guerin, N. Dean, A. Cavalleri, S.-i. Adachi, T.-H. Arima, H. Sawa, Y. Ogimoto, M. Nakamura, R. Tamaki, K. Miyano, and S.-Y. Koshihara, Transient photoinduced ‘hidden’phase in a manganite, Nat. Mater. 10, 101 (2011).
  * Stojchevska  _et al._ [2014] L. Stojchevska, I. Vaskivskyi, T. Mertelj, P. Kusar, D. Svetin, S. Brazovskii, and D. Mihailovic, Ultrafast switching to a stable hidden quantum state in an electronic crystal, Science 344, 177 (2014).
  * Han  _et al._ [2015] T.-R. T. Han, F. Zhou, C. D. Malliakas, P. M. Duxbury, S. D. Mahanti, M. G. Kanatzidis, and C.-Y. Ruan, Exploration of metastability and hidden phases in correlated electron crystals visualized by femtosecond optical doping and electron crystallography, Sci. Adv. 1, e1400173 (2015).
  * Donges  _et al._ [2016] S. A. Donges, O. Khatib, B. T. O’Callahan, J. M. Atkin, J. H. Park, D. Cobden, and M. B. Raschke, Ultrafast nanoimaging of the photoinduced phase transition dynamics in VO2, Nano. Lett. 16, 3029 (2016).
  * Lifshitz  _et al._ [1960] I. Lifshitz _et al._ , Anomalies of electron characteristics of a metal in the high pressure region, J. Exptl. Theoret. Phys. (U.S.S.R.) 11, 1130 (1960).
  * Ebad-Allah  _et al._ [2019] J. Ebad-Allah, M. Krottenmüller, J. Hu, Y. Zhu, Z. Mao, and C. Kuntscher, Infrared spectroscopy study of the nodal-line semimetal candidate zrsite under pressure: Hints for pressure-induced phase transitions, Phys. Rev. B 99, 245133 (2019).
  * Krottenmüller  _et al._ [2020] M. Krottenmüller, M. Vöst, N. Unglert, J. Ebad-Allah, G. Eickerling, D. Volkmer, J. Hu, Y. Zhu, Z. Mao, W. Scherer, and C. A. Kuntscher, Indications for Lifshitz transitions in the nodal-line semimetal ZrSiTe induced by interlayer interaction, Phys. Rev. B 101, 081108 (2020).
  * Topp  _et al._ [2016] A. Topp, J. M. Lippmann, A. Varykhalov, V. Duppel, B. V. Lotsch, C. R. Ast, and L. M. Schoop, Non-symmorphic band degeneracy at the Fermi level in ZrSiTe, New J. Phys. 18, 125014 (2016).
  * Xu  _et al._ [2015] Q. Xu, Z. Song, S. Nie, H. Weng, Z. Fang, and X. Dai, Two-dimensional oxide topological insulator with iron-pnictide superconductor LiFeAs structure, Phys. Rev. B 92, 205310 (2015).
  * Hu  _et al._ [2016] J. Hu, Z. Tang, J. Liu, X. Liu, Y. Zhu, D. Graf, K. Myhro, S. Tran, C. N. Lau, J. Wei, and Z. Mao, Evidence of topological nodal-line fermions in ZrSiSe and ZrSiTe, Phys. Rev. Lett. 117, 016602 (2016).
  * Yuan  _et al._ [2019] H. Yuan, X. Zhou, Y. Cao, Q. Bian, Z. Zhang, H. Sun, S. Li, Z. Shao, J. Hu, Y. Zhu, Z. Mao, W. Ji, and M. Pan, Raman detection of hidden phonons assisted by atomic point defects in a two-dimensional semimetal, npj 2D Mater Appl 3, 1 (2019).
  * Muechler  _et al._ [2020] L. Muechler, A. Topp, R. Queiroz, M. Krivenkov, A. Varykhalov, J. Cano, C. R. Ast, and L. M. Schoop, Modular arithmetic with nodal lines: Drumhead surface states in ZrSiTe, Phys. Rev. X 10, 011026 (2020).
  * Topp  _et al._ [2017] A. Topp, R. Queiroz, A. Grüneis, L. Müchler, A. W. Rost, A. Varykhalov, D. Marchenko, M. Krivenkov, F. Rodolakis, J. L. McChesney, B. V. Lotsch, L. M. Schoop, and C. R. Ast, Surface floating 2D bands in layered nonsymmorphic semimetals: ZrSiS and related compounds, Phys. Rev. X 7, 041073 (2017).
  * Shao  _et al._ [2020] Y. Shao, A. Rudenko, J. Hu, Z. Sun, Y. Zhu, S. Moon, A. Millis, S. Yuan, A. Lichtenstein, D. Smirnov, Z. Mao, M. Katnelson, and D. Basov, Electronic correlations in nodal-line semimetals, Nat. Phys. , 1 (2020).
  * Chen  _et al._ [2020] F. Chen, Y. Fei, S. Li, Q. Wang, X. Luo, J. Yan, W. Lu, P. Tong, W. Song, X. Zhu, L. Zhang, H. B. Zhou, F. w. Zheng, P. Zhang, A. L. Lichtenstein, M. I. Katnelson, Y. Yin, N. Hao, and Y. P. Sun, Temperature-Induced Lifshitz Transition and Possible Excitonic Instability in ZrSiSe, Phys. Rev. Lett. 124, 236601 (2020).
  * Zhang  _et al._ [2017] Y. Zhang, C. Wang, L. Yu, G. Liu, A. Liang, J. Huang, S. Nie, X. Sun, Y. Zhang, B. Shen, J. Liu, H. Weng, L. Zhao, G. Chen, X. Jia, C. Hu, Y. Ding, W. Zhao, Q. Gao, C. Li, S. He, L. Zhao, F. Zhang, S. Zhang, F. Yang, Z. Wang, Q. Peng, X. Dai, Z. Fang, Z. Xu, C. Chen, and X. J. Zhou, Electronic evidence of temperature-induced Lifshitz transition and topological nature in ZrTe5, Nat. Commun. 8, 1 (2017).
  * Liu  _et al._ [2020] Y. Liu, Y.-F. Liu, X. Gui, C. Xiang, H.-B. Zhou, C.-H. Hsu, H. Lin, T.-R. Chang, W. Xie, and S. Jia, Bond-breaking induced Lifshitz transition in robust Dirac semimetal VAI3, Proc. Natl Acad. Sci. USA 117, 15517 (2020).
  * Yang  _et al._ [2019] H. Yang, L. Yang, Z. Liu, Y. Sun, C. Chen, H. Peng, M. Schmidt, D. Prabhakaran, B. A. Bernevig, C. Felser, B. H. Yan, and Y. L. Chen, Topological Lifshitz transitions and Fermi arc manipulation in Weyl semimetal NbAs, Nat. Commun. 10, 1 (2019).
  * Caglieris  _et al._ [2018] F. Caglieris, C. Wuttke, S. Sykora, V. Süss, C. Shekhar, C. Felser, B. Büchner, and C. Hess, Anomalous Nernst effect and field-induced Lifshitz transition in the Weyl semimetals TaP and TaAs, Phys. Rev. B 98, 201107 (2018).
  * Wu  _et al._ [2015] Y. Wu, N. H. Jo, M. Ochi, L. Huang, D. Mou, S. L. Bud’ko, P. C. Canfield, N. Trivedi, R. Arita, and A. Kaminski, Temperature-induced Lifshitz transition in WTe2, Phys. Rev. Lett. 115, 166602 (2015).
  * Qi  _et al._ [2020] M. Qi, C. An, Y. Zhou, H. Wu, B. Zhang, C. Chen, Y. Yuan, S. Wang, Y. Zhou, X. Chen, R. Zhang, and Z. Yang, Pressure-driven Lifshitz transition in type-II Dirac semimetal NiTe2, Phys. Rev. B 101, 115124 (2020).
  * Liu  _et al._ [2019] F. Liu, J. Li, K. Zhang, S. Peng, H. Huang, M. Yan, N. Li, Q. Zhang, S. Guo, X. Lü, P. Cai, L. Yin, S. Zhou, W. Duan, J. Shen, and W. Yang, Pressure-induced Lifshitz transition in the type II Dirac semimetal PtTe2, Sci. China-Phys. Mech. Astron. 62, 48211 (2019).
  * Xu  _et al._ [2018] N. Xu, Z. Wang, A. Magrez, P. Bugnon, H. Berger, C. E. Matt, V. N. Strocov, N. C. Plumb, M. Radovic, E. Pomjakushina, K. Conder, J. H. Dil, J. Mesot, R. Yu, H. Ding, and M. Shi, Evidence of a Coulomb-Interaction-Induced Lifshitz Transition and Robust Hybrid Weyl Semimetal in Td-MoTe2, Phys. Rev. Lett. 121, 136401 (2018).
  * Beaulieu  _et al._ [2020] S. Beaulieu, S. Dong, N. Tancogne-Dejean, M. Dendzik, T. Pincelli, J. Maklar, R. P. Xian, M. A. Sentef, M. Wolf, A. Rubio, L. Rettig, and R. Ernstorfer, Ultrafast Light-Induced Lifshitz Transition, arXiv preprint arXiv:2003.04059 (2020).
  * Klemenz  _et al._ [2020] S. Klemenz, A. K. Hay, S. M. Teicher, A. Topp, J. Cano, and L. M. Schoop, The role of delocalized chemical bonding in square-net-based topological semimetals, J. Am. Chem. Soc. 142, 6350 (2020).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mat. Sci. 6, 15 (1996).
  * Togo and Tanaka [2015] A. Togo and I. Tanaka, First principles phonon calculations in materials science, Scr. Mater. 108, 1 (2015).
  * Bensch and Dürichen [1994] W. Bensch and P. Dürichen, Structure of the layer compound ZrSiTe, Acta Cryst. C 50, 346 (1994).


# Structure of the codimesion one gradient flows with at most six singular
points on the Möbius strip

Maria Loseva, Alexandr Prishlyak, Kateryna Semenovych, Yuliia Volianiuk

###### Abstract

We describe all possible topological structures of Morse flows and typical
one-parametric gradient bifurcation on the Möbius strip in the case that the
number of singular point of flows is at most six. To describe structures, we
use the separatrix diagrams of flows. The saddle-node bifurcation is specified
by selecting a separatrix in the diagram of the Morse flow befor the
bifurcation and the saddle connection is specified by a separatrix, which
connect two saddles on the diagram.

## Introduction

We consider gradient flows on the Möbius strip. Since the function increases
along each trajectory, the flow has no cycles and polycycles. In general
position, a typical gradient flow is a Morse flow (Morse-Smale flow without
closed trajectories). In typical one-parameter families of gradient flows, two
types of bifurcations are possible: saddle-node and saddle connection. The
vector fields at the moment of the bifurcation completely determine the
topological type of the bifurcation in our case. To classify Morse flows, we
use a separatrix diagram, in which separetrices are trajectories of one-
dimensional stable or unstable manifolds.

Without loss of generality, we assume that under bifurcation (as the parameter
increases), the number of singular points does not increase. The saddle-node
bifurcation is defined by a separatrix, which is contructed to a point. We
mark this separatrix on the diagram. A saddle connection bifurcation in the
diagram corresponds to a separatrix, which conect two saddles.

We colar stable separatrices in red, unstable separatrices in green and saddle
connections in black.

Reeb [40] construct topological invariants of functions oriented 2-maniofolds.
It was generlized in [17] for the case of non-orientable two-dimensional
manifolds and in [7, 11, 12] for manifolds with boundary, in [33] for non-
compact manifolds.

Since Morse flows as gradient flows of Morse functions, we can fix the value
of functions in singular points. Then the flow determinate the topological
structure of the function [17, 42]. Therefore, Morse flows structure with
order of critical values determinates the structure of the functions.

Possible structures of smooth function on closed 2-manifolds was described in
[5, 12, 11, 33, 32, 37, 17, 22, 48, 45, 1, 41, 49], on 2-manifolds with the
boundary in [9, 12, 10] and on closed 3- and 4-manifolds in [31, 21, 13].

In [6, 14, 18, 19, 20, 38, 27, 30, 43, 47, 28, 29, 36, 26, 16], the structures
of flows on closed 2- manifolds and [4, 15, 30, 27, 23, 35, 20, 36] on
manifolds with the boundary were investigated. Topological properties of
Morse-Smale flows on 3-manifolds was considered in [39, 44, 46, 34, 22, 47,
24, 25, 8, 3, 2].

The purpose of this paper is to describe all possible topological structures
of the Morse flows and typical bifurcations with no more than six singular
points (a saddle-node point we consider as two points) on the Mb̈ious strip.

##  1 Typical one-parameter bifurcations of gradient flows on a Möbius strip

Typical vector fields (flow) on compact 2-manifolds are Morse-Smale fields
(flow). Morse fields (or Morse-Smale gradient-like fields) are not containing
closed trajectories. They satisfy following properties:

1) it contain a finite number of singular points and its are nondegenerate;

2) there are no separatric connections between saddle points;

3) $\alpha$-limiting ($\omega$-limiting) set of each trajectory is a singular
point.

In typical one-parameter field families, one of these conditions is violated.
If violation of the first condition is, then we have a saddle-node
bifurcation. The third condition cannot be violated for gradient fields.

###  1.1 Internal bifurcations

According to the theory of bifurcations, there are only two typical
bifurcations of gradient flows: a saddle-node and a saddle connection.

####  1.1.1 Saddle-node bifurcation

The saddle-node bifurcation, when the node is the source, is shown in Fig. 1.

Figure 1: SN – saddle-node bifurcation

It can be described by the equation $V(x,y,a)=\\{x,y^{2}+a\\}$ if the node is
the source and the equation $V(x,y,a)=\\{-x,-y^{2}a\\}$ if the node is a sink.
Here, $a$ is a parameter. If $a<0$ we get the flow before the bifurcation, if
$a>0$ we get the flow after the bifurcation, and if $a=0$ we get the flow at
the moment of the bifurcation (the flow of codimensionality 1). In order to
determine the saddle-node bifurcation it is necessary to select a separatrix
on the seperatrix diagram.

####  1.1.2 Saddle connection

The bifurcation of the saddle connection is shown in Fig. 2. It can be
described by the equation $V(x,y,a)=\\{x^{2}-y^{2}-1,-2xy+a\\}$.

Figure 2: SC – saddle connection

###  1.2 Bifurcations of singular points on the boundary

Depending on the types of points that stick together, different options for
bifurcations are possible.

Figure 3:  BSN – saddle-node (source) bifurcation at the surface boundary

1) In the first case, the sourse and saddle are glueded together at a point.
In Fig. 3 a) we show the flow before the bifurcation ($a=-1$), in Fig. 3 b) –
flow at the moment of bifurcation ($a=0$), in Fig. 3 c) – flow after
bifurcation ($a=1$).

2) In the second case, the saddle and the sink merge into a point, which then
disappears (Fig.4).

Figure 4:  BSN – saddle-node bifurcation (sink) at the surface boundary

If one of the two saddle-node bifurcation points is internal, and the other
lies on the boundary, then we have two types of semi-boundary saddle-node
bifurcations: at the moment of bifurcation, a saddle (HS) or a node (HN).

Figure 5:  HS – semi-boundary saddle bifurcation

Figure 6: HN – semi-boundary node bifurcation

In addition, the following options are possible:

3) both points that stick together are saddles: $\\{x^{2}-y^{2}+a,-2xy\\}$
(Fig. 7),

Figure 7: BDS – double saddle bifurcation at the boundary

On a set of flows with fixed points on the boundary and without closed
trajectories, the bifurcation in a typical family is given either by the
initial flow and a compressible trajectory for local bifurcations on the
boundary or by the flow at the moment of bifurcation in the case of a saddle
connection.

Therefore, the following types of gradient bifurcations are possible on Möbius
strip:

SN – internal saddle node;

SC – internal saddle connection;

BSN - boundary saddle-node;

BDS – boundary double saddle;

HN – semi-boundary saddle node (node);

HS – semi-boundary saddle-node (saddle);

HSC – semiboundary saddle connection;

BSC – saddle connection of boundary saddle.

In the case of saddle-node bifurcations, such a bifurcation is given by a
separatrix diagram to the bifurcation, on which the trajectory (separatrix)
between the saddle and the node is highlighted, which is compressed to a
point. To specify the bifurcation of the saddle connection, a separatrix
diagram at the moment of bifurcation is sufficient.

##  2 The structure of typical flows and bifurcations with no more than 4
singular points on the Möbius strip

To find all possible structures of Morse flows on the Möbius strip, we use the
Poincaré-Hopf theorem for doubling the vector field. Since doubling the Möbius
strip results in a Klein bottle with an Euler characteristic equal to zero,
the sum of the Poincaré indices of the doubled field is also zero. Since the
saddle index is -1, and the source and drain indices are 0, we have the
following statement: the total number of Morse flow sources and sinks on the
Klein bottle is equal to the number of saddles. Note that when doubling,
internal points are doubled, but boundary points are not. Therefore, the
formula for the Morse flow on the Möbius strip is:

$2N_{i}+N_{b}=2S_{i}+S_{b}.$ (1)

Here $N_{i}$ is the total number of internal sources and sinks, $N_{b}$ is the
total number of sources and sinks on the boundary, $S_{i}$ is the number of
internal saddles, and $S_{b}$ is the number of saddles on the boundary.

123

Figure 8: Morse flows with no more than 4 singular points

Еach Morse flow has a source and a sink, therefore, to fulfill the formula
(1), it must contain more saddles. If the flow has three singular points, then
according to (1), the only possible variant is a flow with one source and one
sink at the boundary and one internal saddle. For flows with four singular
points, the following options are possible: 1) internal sink and saddle,
boundary source and saddle, 2) internal source and saddle, boundary sink and
saddle, 3) all singular points (source, sink and two saddles) lie on the
boundary.

In Fig. 8, we show all possible (with accuracy to homeomorphism) separatrix
Morse flow diagrams with no more than 4 singular points.

Diagrams 1) and 3) are the same if we reverse orientations on the
trajectories, but for diagram 2) we obtain other flow diagram. In the
following figures, we will depict only one of such pair of diagrams.

For saddle-node bifurcations, it is only necessary to note how many different
separatrixes and limit trajectories connecting a saddle and a node exist (with
homeomorphism accuracy) on each diagram.

If the separatrix is one of the multiple edges on the separatrix diagram, then
when it is pulled to a point, other multiple edges will form loops, which is
not possible for gradient flows. Therefore, only those separatrices that are
not one of the multiple edges should be selected.

On the diagram 8-1, all the separatrices and trajectories of the boundary are
multiple edges, so it does not specify a bifurcation. In the 8-2 diagram, the
only non-multiple edge is the separatrix between the saddle on the boundary
and the sink. It also specifies a single bifurcation of the HN type. We get a
bifurcation of the same type if we consider the reverse flow. On the 8-2
diagram, only the boundary trajectories set bifurcations – two BSN
bifurcations and one BDS.

12

Figure 9: Saddle connections in flows with 4 singular points

Saddle bifurcation is not possible for flows with three singular points,
because such flows have only one saddle. For flows with two saddles, two types
of saddle bifurcations are possible: HSC, if one saddle is internal and the
other on the boundary, and BSC, if both saddles belong to the boundary. All
possible diagrams of such flows are shown in fig.9. If we reverse the
direction of movement in the first flow, we get a new flow, and for the second
flow, the flow is equivalent to itself.

Summarizing all of the above, we have the following:

###### Theorem 1

On the Möbius strip, there exists, up to topological equivalence, a single
Morse flow with three critical points. With four critical points, there are
four Morse streams and the following bifurcations: two HN bifurcations, two
BSNs, one BDS, two HSCs, and one BSC.

##  3 Flows and bifurcations with 5 singular points

Next, we consider flows with five singular points (Fig. 10).

12345678

Figure 10: Morse flows with 5 singular point

Since there can be only an even number of singular points on the boundary,
there are either 2 or 4 of them. Let us first consider the flows with two
points on the boundary. If one of these points is a saddle, and the other is
not, then formula (1) is not fulfilled. Either both of these points are saddle
points, or both are not saddle points. If both saddle points on the boundary
are saddles, then the three interior points are the source, saddle and sink.
Since in this case the separatrices can be uniquely drawn from the saddles to
the source and sink (with accuracy up to homeomorphism), there is a single
Morse flow, the diagram of which is shown in Fig. 10-1.

Let us now consider the case when both points on the boundary are not saddle
points. Then one of them is a source, and the other is a sink. Internal
singular points are two saddles and a node (source or sink). If three
separatrices out of the node, then two of them are separatrixes of the same
saddle, and therefore form a loop which is central line of Möbius strip 10-8,
othewise there is another singular point inside of the loop, which is
impossible. If two separators come out of the node, then the only possible
flow has the diagram in Fig. 10-1. If the node is connected to the saddle by
one separatrix, then it lies inside the loop opened by the other two
separatrixes of this saddle. Two options are possible: this loop lies inside
the corner adjacent to the boundary 10-3 or does not lie in such a corner
10-4.

Let’s consider the case of four singular points on the boundary. Then it
follows from formula (1) that one or three saddles lie on the boundary. If
there is one saddle on the boundary, then the other saddle is internal, and
three more non-saddle points lie on the boundary. Let, for certainty, two of
them are sources, and one is a sink. There are two possibilities for red
separatrixes entering the inner saddle: 1) they start at a point (Fig. 10-5),
or 2) at different points (Fig. 10 -6).

Let’s consider the case of three saddle points on the boundary. Let the fourth
point on the boundary be the source. Then the interior singular point is a
sinkeds. The only possible flow has the diagram in Fig. 10-7. Since we have
exhausted all possible options, there are no other flows with five special
points on the Möbius strip.

The following saddle-node bifurcations are possible for these flows:

1) 2 HN;

2) SN, 1 HS; ($\times$2)

3) 1 SN, 1 HS; ($\times$2)

4) 1 SN, 1 HS; ($\times$2)

5) 2 BSN; ($\times$2)

6) 1 HS, 1 BSN; ($\times$2)

7) 1 HN, 1 BSN, 1 BDS; ($\times$2)

8) 1 SN, 1 HS ($\times$2).

Note that only 1) of the considered diagrams will turn into itself when the
flow is reversed. Therefore, we leave it unchanged the total number of
bifurcations in this case, we multiplied it by 2 in other seven cases.

All possible separatrix connections on Möbius stri with no more than 5
singular points are shown in fig. 11.

12345678

Figure 11: Separatrix connections with no more than 5 singular points

Consider flows with an internal separatrix connection (SC). In addition to the
two saddles, there is another internal singular point. Let, for certainty, it
be a sink. Then the singular points on the boundary are the source and the
sink. Consider the case when one separatrix enters the internal sink. Diagrams
of three possible flows in this case are shown in fig. 11.1–3. If this source
includes two separators, then the possible options are shown in fig. 11. 6, 7.
In the case of a separatrix connection between the internal and boundary
saddle points (HSC), two cases are possible: 1) the boundary contains two
singular points 11.5; 2) the boundary contains 4 singular points 11.4.

The only possible case of a saddle connection between points on the boundary
(BSC) is shown in fig. 11.8.

In all 8 cases, the inverted fields are not topologically equivalent to the
original ones, so the total number of bifurcations is multiplied by 2.

###### Theorem 2

On the Möbius strip, there exists, up to topological equivalence, 15 Morse
flows with five critical points and following numbers of bifurcations:

10 SN bifurcations, 14 SC, 6 BSN, 2 BDS, 4 HN, 10 HS, 4HSC and 2 BSC.

##  4 Flows and bifurcations with 6 singular points

In fig. 12 shows all possible separatrix Morse flow diagrams with 6 singular
points.

12345678910111213141516171819202122

Figure 12: Morse flows with 6 singular points

The flow can have two (12.1–6), four (12.7–19) or six (12.20–22) singular
points on the boundary.

Figure 13: SC flows with 6 singular point  Figure 14: HSC and BSC flows with 6
singular point

Depending on the number of separatrixes on each of the diagrams, we get the
following possible bifurcations for each of them:

1) 3 SN, 1 HN, 1 HS, (x2);

2) 2 SN, 1 HN (x2);

3) 2 SN, 1 HN, (x2);

4) 1 SN, 1 HN, 1 HS, (x2);

5) 2 SN, 1 HN, (x2);

6) 1 SN, 1 HN, 1 HS, (x2);

7) 2 BSN,1 HN, (x2);

8) 2 BSN, 1 BDS, 1 HN, (x2);

9) 2 BSN, 1 BDS, 1 HN, (x2);

10) 4 HS;

11) 1 SN, 1 BSN, 1 HN, 1 HS, (x2);

12) 4 HS, (x2);

13) 1 BSN, 1 HN, 1 HS, (x2);

14) 1 BDS, 2 HN;

15) 1 SN, 2 BSN, 2 BDS, 1 HN, 1 HS, (x2);

16) 2 SN, 1 HN, (x2);

17) 1 SN, 1 HN, 1 HS, (x2);

18) 1 SN, 2 BSN, 1 BDS, 1 HS, (x2);

19) 1 SN, 2 BSN, 1 BDS, 1 HS, (x2);

20) 4 BSN, 1 BDS (x2);

21) 2 BSN, 2 BDS, (x2);

22) 4 BSN, 1 BDS, (x2).

When changing the direction of the flow, diagrams 10) and 14) will not change,
therefore, in the general calculation of the number of bifurcations, the above
values will not change. For the rest of the charts, these values will be
doubled.

Internal saddle connections are shown in Figure 13. In last three diagram,
reverse of oreintation leads to the same diagram. Seven HSC and one BSC
bifurcation are shown in Figure 14. In all of them, reverse of oreintation
leads to the new diagrams.

We went through all the possible options, and therefore it is fair

###### Theorem 3

The following possible structures of typical one-parameter gradient
bifurcations with 6 singular points exist on the Möbius strip:

36SN, 15 SC, 48 BSN, 21 BDS, 30 HN, 14 HSC, 2 BSC.

## Conclusion

All possible structures of Morse flows and typical one-parameter bifurcations
on Möbius strip in which no more than six singular points are found (see Table
1). We hope that the research carried out in this paper can be extended to
other surfaces and with a larger number of singular points.

Number of points | Morse | SN | SC | BSN | BDS | HN | HS | HSC | BSC
---|---|---|---|---|---|---|---|---|---
3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
4 | 4 | 2 | 0 | 2 | 1 | 2 | 0 | 2 | 1
5 | 15 | 10 | 14 | 6 | 2 | 4 | 10 | 4 | 2
6 | 42 | 36 | 15 | 48 | 21 | 30 | 30 | 14 | 2
Table 1: Number of Morse flows and bifurcations on Mö (number of points befor
bifurcation)

## References

  * [1] S. Bilun and A. Prishlyak.  The closed morse 1-forms on closed surfaces.  Visn., Mat. Mekh., Kyv. Univ. Im. Tarasa Shevchenka, 2002(8):77–81, 2002.
  * [2] S. Bilun and A. Prishlyak.  Visualization of morse flow with two saddles on 3-sphere diagrams.  arXiv preprint arXiv:2209.12174, 2022. doi:10.48550/ARXIV.2209.12174.
  * [3] S. Bilun, M. Loseva, O. Myshnova, A. Prishlyak  Typical one-parameter bifurcations of gradient flows with at most six singular points on the 2-sphere with holes.  arXiv preprint arXiv:2303.14975, 2023. doi:10.48550/arXiv.2303.14975.
  * [4] S. Bilun, M. Hrechko, O. Myshnova, A. Prishlyak.  Structures of optimal discrete gradient vector fields on surface with one or two critical cells  arXiv preprint arXiv:2303.07258, 2023. doi:10.48550/arXiv.2303.07258.
  * [5] S. Bilun, A. Prishlyak, S. Stas, A. Vlasenko.  Topological structure of Morse functions on the projective plane  arXiv preprint arXiv:2303.03850, 2023. doi:10.48550/arXiv.2303.03850.
  * [6] S. Bilun, B. Hladysh, A. Prishlyak, V Sinitsyn.  Gradient vector fields of codimension one on the 2-sphere with at most ten singular points  arXiv preprint arXiv:2303.10929, 2023. doi:10.48550/arXiv.2303.10929.
  * [7] A.V. Bolsinov and A.T. Fomenko.  Integrable Hamiltonian systems. Geometry, Topology, Classification.  A CRC Press Company, Boca Raton London New York Washington, D.C., 2004\.  724 p.
  * [8] C. Hatamian and A. Prishlyak.  Heegaard diagrams and optimal morse flows on non-orientable 3-manifolds of genus 1 and genus 2.  Proceedings of the International Geometry Center, 13(3):33–48, 2020\. doi:10.15673/tmgc.v13i3.1779.
  * [9] B. I. Hladysh and A. O. Pryshlyak.  Functions with nondegenerate critical points on the boundary of the surface.  Ukrainian Mathematical Journal, 68(1):29–41, 2016. doi:10.1007/s11253-016-1206-5.
  * [10] B. I. Hladysh and A. O. Pryshlyak.  Deformations in the general position of the optimal functions on oriented surfaces with boundary.  Ukrainian Mathematical Journal, 71(8):1173–1185, 2020. doi:10.1007/s11253-019-01706-8.
  * [11] B.I. Hladysh and A.O. Prishlyak.  Topology of functions with isolated critical points on the boundary of a 2-dimensional manifold.  SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 13:050, 2017. doi:0.3842/SIGMA.2017.050.
  * [12] B.I. Hladysh and A.O. Prishlyak.  Simple morse functions on an oriented surface with boundary.  Журнал математической физики, анализа, геометрии, 15(3):354–368, 2019. doi:10.15407/mag15.03.354.
  * [13] B.Hladysh, M.Loseva, A. Prishlyak.  Topological structure of functions with isolated critical points on a 3-manifold.  Proceedings of the International Geometry Center, 16(3):231–243, 2023. doi:10.15673/pigc.v16i3.2512.
  * [14] Z. Kybalko, A. Prishlyak, and R. Shchurko.  Trajectory equivalence of optimal Morse flows on closed surfaces.  Proc. Int. Geom. Cent., 11(1):12–26, 2018. doi:10.15673/tmgc.v11i1.916.
  * [15] M. Losieva and A. Prishlyak.  Topology of morse–smale flows with singularities on the boundary of a two-dimensional disk.  Pr. Mizhnar. Heometr. Tsentr, 9(2):32–41, 2016. doi:10.15673/tmgc.v9i2.279.
  * [16] M. Loseva, A. Prishlyak, K.Semenovych, D. Synieok  Structure of Morse flows with at most six singular points on the torus with a hole  arXiv preprint arXiv:2404.02223 , 2024. doi:10.48550/arXiv.2404.02223.
  * [17] D.P. Lychak and A.O. Prishlyak.  Morse functions and flows on nonorientable surfaces.  Methods of Functional Analysis and Topology, 15(03):251–258, 2009\.
  * [18] A.A. Oshemkov and V.V. Sharko.  Classication of morse-smale flows on two-dimensional manifolds.  Matem. Sbornik, 189(8):93–140, 1998.
  * [19] M.M. Peixoto.  On the classication of flows of 2-manifolds.  Dynamical Systems (Proc. Symp. Univ. of Bahia, Salvador, Brasil, 1971), pages 389–419, 1973.
  * [20] A.O. Prishlyak.  On graphs embedded in a surface.  Russian Mathematical Surveys, 52(4):844, 1997. doi:10.1070/RM1997v052n04ABEH002074.
  * [21] A.O. Prishlyak.  Conjugacy of Morse functions on 4-manifolds.  Russian Mathematical Surveys, 56(1):170, 2001.
  * [22] A.O. Prishlyak.  Morse–smale vector fields without closed trajectories on-manifolds.  Mathematical Notes, 71(1-2):230–235, 2002. doi:10.1023/A:1013963315626.
  * [23] A.O. Prishlyak.  On sum of indices of flow with isolated fixed points on a stratified sets.  Zhurnal Matematicheskoi Fiziki, Analiza, Geometrii [Journal of Mathematical Physics, Analysis, Geometry], 10(1):106–115, 2003.
  * [24] A.O. Prishlyak.  Complete topological invariants of morse–smale flows and handle decompositions of 3-manifolds.  Fundamentalnaya i Prikladnaya Matematika, 11(4):185–196, 2005.
  * [25] A.O. Prishlyak.  Complete topological invariants of morse-smale flows and handle decompositions of 3-manifolds.  Journal of Mathematical Sciences, 144:4492–4499, 2007.
  * [26] A. Prishlyak and L. Di Beo.  Flows with minimal number of singularities on the Boy’s surface  Proceedings of the International Geometry Center, 15(1):32–49, 2020\.
  * [27] A. Prishlyak and M. Loseva.  Topological structure of optimal flows on the girl’s surface.  Proceedings of the International Geometry Center, 15(3-4):184–202, 2022.
  * [28] A. Prishlyak, A. Prus, and S. Huraka.  Flows with collective dynamics on a sphere.  Proc. Int. Geom. Cent, 14(1):61–80, 2021. doi:10.15673/tmgc.v14i1.1902.
  * [29] A. Prishlyak and M. Loseva.  Topology of optimal flows with collective dynamics on closed orientable surfaces.  Proceedings of the International Geometry Center, 13(2):50–67, 2020\. doi:10.15673/tmgc.v13i2.1731.
  * [30] A. Prishlyak and A. Prus.  Morse-smale flows on torus with hole.  Proc. Int. Geom. Cent., 10(1):47–58, 2017. doi:10.15673/tmgc.v1i10.549.
  * [31] A.O. Prishlyak.  Equivalence of morse function on 3-manifolds.  Methods of Func. Ann. and Topology, 5(3):49–53, 1999.
  * [32] A.O. Prishlyak.  Conjugacy of morse functions on surfaces with values on a straight line and circle.  Ukrainian Mathematical Journal, 52(10):1623–1627, 2000. doi:10.1023/A:1010461319703.
  * [33] A.O. Prishlyak.  Morse functions with finite number of singularities on a plane.  Meth. Funct. Anal. Topol, 8:75–78, 2002.
  * [34] A.O. Prishlyak.  Topological equivalence of morse–smale vector fields with beh2 on three-dimensional manifolds.  Ukrainian Mathematical Journal, 54(4):603–612, 2002.
  * [35] A.O. Prishlyak.  Topological classification of m-fields on two-and three-dimensional manifolds with boundary.  Ukrainian Mathematical Journal, 55(6):966–973, 2003.
  * [36] A.O. Prishlyak and M.V. Loseva.  Optimal morse–smale flows with singularities on the boundary of a surface.  Journal of Mathematical Sciences, 243:279–286, 2019.
  * [37] A.O. Prishlyak and K.I. Mischenko.  Classification of noncompact surfaces with boundary.  Methods of Functional Analysis and Topology, 13(01):62–66, 2007\.
  * [38] A.O. Prishlyak and A.A. Prus.  Three-color graph of the morse flow on a compact surface with boundary.  Journal of Mathematical Sciences, 249(4):661–672, 2020. doi:10.1007/s10958-020-04964-1.
  * [39] A.O. Prishlyak, S.V.Bilun, A.A. Prus.  Morse Flows with Fixed Points on the Boundary of 3-Manifolds.  Journal of Mathematical Sciences, 274(6):881–897, 2023. doi:10.1007/s10958-023-06651-3.
  * [40] G. Reeb.  Sur les points singuliers d’une forme de pfaff complétement intégrable ou d’une fonction numérique.  C.R.A.S. Paris, 222:847—849, 1946.
  * [41] V.V. Sharko.  Functions on manifolds. Algebraic and topological aspects., volume 131 of Translations of Mathematical Monographs.  American Mathematical Society, Providence, RI, 1993.
  * [42] S. Smale.  On gradient dynamical systems.  Ann. of Math., 74:199–206, 1961.
  * [43] В.М. Кузаконь, В.Ф. Кириченко, and О.О. Пришляк.  Гладкi многовиди. Геометричнi та топологiчнi аспекти.  Працi Iнcтитуту математики НАН України.—2013.—97.—500 с, 2013.
  * [44] А.О. Пришляк.  Векторные поля Морса–Смейла с конечным числом особых траекторий на трехмерных многообразиях.  Доповiдi НАН України, (6):43–47, 1998.
  * [45] А.О. Пришляк.  Сопряженность функций Морса.  Некоторые вопросы совр. математики. Институт математики АН Украины, Киев, 1998.
  * [46] А.О. Пришляк.  Топологическая эквивалентность функций и векторных полей Морса—Смейла на трёхмерных многообразиях.  Топология и геометрия. Труды Украинского мат. конгресса, pages 29–38, 2001.
  * [47] А.О. Пришляк.  Векторные поля Морса–Смейла без замкнутых траекторий на трехмерных многообразиях.  Математические заметки, 71(2):254–260, 2002\.
  * [48] О.О. Пришляк.  Топологія многовидів.  Київський університет, 2015.
  * [49] О.О. Пришляк.  Теорія Морса.  Київський університет, 2002.

Taras Shevchenko National University of Kyiv

Maria Loseva Email<EMAIL_ADDRESS>Orcid ID: 0000-0002-2282-206X

Alexandr Prishlyak Email address<EMAIL_ADDRESS>Orcid ID:
0000-0002-7164-807X

Kateryna Semenovych Email<EMAIL_ADDRESS>Orcid ID:
0000-0001-9717-1524

Yuliia Volianiuk Email<EMAIL_ADDRESS>Orcid ID: 0009-0006-7203-9467