# Centrality and Transverse Spherocity dependent study of charged-particle
production in Xe-Xe collisions at $\sqrt{s_{NN}}$ = 5.44 TeV using PYTHIA8
Angantyr and AMPT models

Randhir Singh1111E-mail<EMAIL_ADDRESS>Renu Bala1222E-mail:
<EMAIL_ADDRESS>Sanjeev Singh Sambyal1333E-mail:
<EMAIL_ADDRESS>Department Of Physics, University of Jammu,
Jammu, India1

###### Abstract

Transverse Spherocity is an event structure variable which provide an
effective way to disentangle the data into hard and soft components of the
processes corresponding to events with small and large numbers of multi-parton
interactions (MPI), respectively. Recent experimental results in small systems
from the LHC suggest the importacnce of transverse spherocity variable in the
classification of the events. In this contribution, we have studied the
dynamics of identified particle production in Xe-Xe collisions at
$\sqrt{s_{NN}}$ = 5.44 TeV using A Multi-Phase Transport Model (AMPT) and the
recently developed Angantyr model, which is incorporated within PYTHIA8. A
study of the transverse momentum spectra of the dentified particles are
presented for soft (isotropic) and hard (jetty-like) events in different
centrality intervals.

##  I Introduction

The study of the behaviour of the matter at extreme conditions of temperature
and energy density is important to understand the processes involved in the
phase transition of the matter in Quantum Chromodynamics (QCD). Experiments
like Relativistic Heavy Ion Collider (RHIC) at BNL, USA and Large Hadron
Collider at CERN, Geneva, Switzerland are specially designed to study the
matter at these extreme conditions. In these experiments, the results have
suggested that a strongly interacting de-confined state of quarks and gluons,
also called Quark-Gluon plasma (QGP), is formed in ultra-relativistic heavy-
ion collisions. This plasma is a very hot and dense state and exist for a very
short duration of time. The relevant degrees of freedom in the QGP phase are
quarks and gluons instead of mesons and baryons (which are color neutral
confined state). After the formation this QGP plasma cools and expands
hydrodynamically and reaches first to the chemical freeze-out (the hadron
abundances are fixed) phase and then reaches to kinetic freeze-out (the hadron
momenta are fixed) phase.

The observable which is important to cahracterize the properties of the
created matter in these collisions is the multiplicity of the charged
particles. This is due to the fact that the partcle production is effected by
the initial energy density reached in the collision. The colliding nuclei are
extended objects. Therefore, in a collision, the region of overlap between the
two colliding nuclei varies from collision to collision. This degree of
overlap is expressed by the term impact parameter (denoted as ”b”). The impact
parameter cannot be measured directly. To overcome the difficulty, a different
parameter called centrality is used. The centrality characterises the initial
overlapping region geometry, which corresponds to the number of participating
nucleons Npart and binary collisions Ncoll. Npart and Ncoll will be different
for similar overlap in collisions of nuclei of different sizes.

Transverse spherocity is an event shape variable which has been introduced
recently. The recent development in event shape variables provides an
alternate way to characterise the high multiplicity events in small systems
considering particle production, event multiplicity, and event shape variable
together. Event shape observables provide information about the energy
distribution in an event. The event shape observables, calculated event-by-
event allows one to isolate jetty-like (high pT jets) and isotropic (partonic
scattering with low Q2) events Morsch . The ALICE, CMS, and ATLAS experiments
have done the studies based on transverse spherocity Chatrchyan ; Abelev .
These studies involved the understanding of event shape as a function of
charged particle multiplicity of the event. The results from these studies
show that for high multiplicity, the events were more isotropic. After
studying extensively in small systems ALICE:2019dfi , the use of transverse
spherocity parameter in heavy-ion collisions systems may reveal new and unique
results from heavy-ion collisions where the production of a QGP is well
established fact.

Transverse momentum (pT) spectra of the charged particles produced in the
heavy-ion collisions carry essential information about the thermal nature of
the interacting system Cleymans:2012ya ; Cleymans:2011in ; Mallick:2020ium ;
Mallick:2021wop ; Mallick:2021hcs ; Prasad:2021bdq ; Prasad:2022zbr ;
Rath:2018ytr . The Maxwell-Boltzmann distribution law tells us that the pT
spectra are related to the temperature of the system formed in these
collisions. At low to intermediate pT (up to 10 GeV/$c$), the collective
expansion of the system governs the charged particle production, which is
observed in the shapes of single-particle transverse-momentum spectra
ALICE:2015dtd ; ALICE:2018vuu  and multi-particle correlations Heinz:2013th .
At high pT (typically above 10 GeV/$c$), parton fragmentation governs the
particle production. This production is effected by the amount of energy loss
that the partons suffer when propagating in the medium. Moreover, the number
of charged particles (Nch) produced in these ultra-relativistic collisions are
also related to the temperature and energy density of the system created. The
ratios of yields of identified hadrons are also important to understand to the
processes involved in the production of hadron. The ratio of proton to pion
(p/$\pi$) and kaon to pion (K/$\pi$) characterize the relative baryon and
meson production, respectively.

In this work, we have explored all the above-mentioned acpects related to
particle production. We have also used transverse spherocity for the first
time in Xe-Xe collisions at $\sqrt{s_{NN}}$ = 5.44 TeV using A Multi-Phase
Transport Model (AMPT)Lin:2004en  and Angantyr model which is incorporated in
PYTHIA8 . The purpose of the present analysis is to study the dynamics of
heavy-ion collisions using transverse spherocity and collision centrality in
Xe-Xe collisions. Thus, we have used the PYTHIA8 and the string melting
parameterization available in AMPT model. We hope that this would drive
experimentalists to pursue such study in experiments at RHIC and the LHC. The
contribution is arranged as follows. We begin with a brief introduction and
motivation for the study in Section-I. In Section-II, the detailed analysis
methodology along with brief description about AMPT and PYTHIA (Angantyr) are
given. Section-III discusses about the results for the charged-particle
pseudorapidity density, $p_{T}$ spectra of $\pi^{\pm}$ , $K^{\pm}$ and
$p^{\pm}$ in different centrality and transverse spherocity ranges. Finally
the results are summarized in Section-IV.

##  II EVENT GENERATION AND ANALYSIS METHODOLOGY

This section will be devoted to discussions related to the event generators,
AMPT and PYTHIA8 (Angantyr). Around 0.5 Million Xe-Xe events have been
generated for both event generators at $\sqrt{s_{NN}}$ = 5.44 TeV. A
discussion will also be presented for transverse spherocity.

###  II.1 PYTHIA8 (Angantyr)

PYTHIA8 is general purpose Monte Carlo event generator that has been quite
successful in the study of elementary particle collisions. It has been
extensively used to simulate proton-proton, proton-lepton collisions to
understand the dynamics of strong and electroweak processes extending from
high momemtum transfer regions (perturbative scales) to the scales around
$\Lambda_{QCD}$ (Lattice QCD scales). Recently efforts have been made to
modify PYTHIA8 that made it possible to simulate ultra-relativistic heavy-ion
collisins like, proton-nuclei (pA) and nuclei-nuclei (AA). PYTHIA8
incorporates what is called ”Angantyr” which basically superpose many pp
collisions to make a single heavy-ion collision. This way a bridge is formed
between heavy ion and high energy hadron phenomenology. An important point is
to note that the assumption of the formation of a hot thermalised medium is
not included in the Angantyr as the production mechanisms is same as in small
collision systems. Thus it can be used to differentiate the effects of
collective and non-collective behaviour in heavy-ion collisions.

The Angantyr model is basically inspired by the old Fritiof model and the
notion of “wounded” nucleons, but with the inclusion of effects of hard
partonic interactions. To calculate the number of wounded nucleons the use of
Glauber model is made, with the consideration of fluctuations in the nucleon-
nucleon(NN) interactions to differentiate non-diffractively and diffractively
wounded nucleons. In the model, the fluctuations of neuclonic wavefunction are
accommodated as fluctuations in the nuclei radius. Nucleons inside nuclei are
distributed randomly according to a Glauber formalism. Every nucleon is then
identified as either wounded or spectator. The interections between wounded
nucleons in projectile and target are classified as elastic, ND, secondary ND,
SD and double-diffrative(DD) which basically depends upon the interaction
probability. All the sub-events generated at the parton level are stacked
together to represent one pA or AA collision event.

In PYTHIA8 hadronization is done via the Lund string fragmentation model. In
this model, the Lund area law Andersson:2001yu  provides the probability of
creating hadrons from an initial state of partons. The produced partons are
connected to the beam remnants through color fluxtubes or strings which stores
potential energy. As the partons move away from each other, the string breaks
causing the formation of new quark-antiquark pair. This process continues
until the string pieces reduces to very small pieces, which are recognized as
shell hadrons. In this scheme, reconnection of strings between the partons
happens in such a way that strings length decreases; resulting to the decrease
in the particle production and hence multiplicity.

###  II.2 A Multi-Phase Transport (AMPT) model

The AMPT model has four main components : the initial conditions, partonic
interactions, the conversion from the partonic to the hadronic matter, and
hadronic interactions. In case of default AMPT model, initial conditions for
heavy ion collisions at RHIC are carried from the HIJING model Wang:1990qp ;
Wang:1991hta ; Gyulassy:1994ew . The production of particles are described
either as hard and soft components. The hard components are the ones that
involve a momentum transfer larger than a cutoff momentum, $p_{0}$, value
while soft components are the ones having momentum transfer below the cutoff
value. The hard component is evaluated in the perturbative Quantum
Chromodynamics domain (pQCD) using parton distribution functions in a nucleus.
These processes contribute to the production of minijets partons. The soft
component is discussed in non-perturbative region and are modeled by the
formation of strings. The excited strings then decay independently following
Lund JETSET fragmentation model. The energy density in the default AMPT model
can be very high in heavy ion collisions. To incorporate this effect, the AMPT
model is extended to include the string melting mechanism. The interactions
between partons are described using equations of motion which can be
approximately written as the Boltzmann equations. The Boltzmann equations are
then solved by using Zhang’s parton cascade (ZPC) Zhang:1997ej , in which two
partons undergo scattering whenever they are within a minimum closest
distance. In the String Melting version of AMPT (AMPT-SM), colored strings are
melted to form low momentum partons which take place at the start of the ZPC.
Corresponding to two different initial conditions, hadronization is also done
via two different mechanisms. In case of default AMPT model, the minijets and
their remaining parent nucleons coexist and after partonic interactions they
together form new excited strings. Lund string fragmentation model describes
the hadronization of these strings. In case of the AMPT model with string
melting, the conversion of strings into soft partons takesplace, and after
that hadronization happens via simple quark coalescence model. In the
coalescence model, hadronization is done by combining three closest quarks
(antiquarks) into a baryon (antibaryon) and two closest partons into a meson.
The hadrons thus produced undergo final evolution via meson-meson, meson-
baryon and baryon-baryon interactions described by the relativistic transport
mechanism. The quark coalescence mechanism for hadronization well explains the
spectra at the mid-pT regions and particle flow Greco:2003mm ; Fries:2003vb ;
Fries:2003kq . Thus, for our work, we have used AMPT-SM mode (AMPT version
2.26t7) with the default settings.

Figure 1: Jetty and isotropic events in the transverse planeKhuntia:2018qox .

###  II.3 Transverse Spherocity

Transverse spherocity $(S_{o})$ is the property of an event and is defined
using a unit vector $\hat{n}(n_{T},0)$ which minimizes the ratio
Cuautle:2014yda ; Ortiz:2015ttf  and chosen from all possible unit transverse
vectors:

$\centering
S_{o}=\frac{\pi^{2}}{4}\underset{\widehat{n}=(n_{x},n_{y},n_{y})}{min}\Bigg{(}\frac{\sum_{i}|\vec{p}_{T_{i}}\times\widehat{n}_{T}|}{\sum_{i}p_{T_{i}}}\Bigg{)}^{2}\@add@centering$
(1)

The factor, $\pi^{2}/4$ normalises the minimised value to 1 for isotropic
case. As a result, the value of transverse spherocity run from 0 to 1, as the
distribution of particles deviates from the jetty-like to an isotropic
structure respectively, i.e.

$\centering S_{o}=\begin{cases}0&\text{“pencil-like” limit (hard events)}\\\
1&\text{“isotropic” limit (soft events)}\\\ \end{cases}\@add@centering$ (2)

Fig.1 shows the distribution of particles in both the scenarios.

Figure 2: Charged-particle pseudorapidity density for different centrality
classes over a broad $\eta$ range in Xe-Xe collisions at $\sqrt{s_{NN}}$ =
5.44 TeV calculated with AMPT and PYTHIA8. The results also compared with the
results from the ALICE Experiment at the LHC at CERN.

Centrality | AMPT | PYTHIA8
---|---|---
$b_{min}$ | $b_{max}$ | $E_{T}^{min}$ | $E_{T}^{max}$
0-10% | 0 | 4.507 | 1031.47 | 2000
10-20% | 4.507 | 6.37525 | 729.13 | 1031.47
20-30% | 6.37525 | 7.80479 | 502.604 | 729.13
30-40% | 7.80479 | 9.00297 | 330.139 | 502.604
40-50% | 9.00297 | 10.0606 | 202.442 | 330.139
50-60% | 10.0606 | 11.0097 | 111.821 | 202.442
60-70% | 11.0097 | 11.8943 | 53.8031 | 111.821

Table 1: Table showing the values of impact parameter for AMPT and summed
transverse energy for PYTHIA8 for different centrality bins.

##  III RESULTS

In figure 2, the charged-particle multiplicity density $dN_{ch}/d\eta$ as a
function of pseudorapidity for different centrality classes are presented for
AMPT and PYTHIA8. The centrality intervals for the AMPT are defined using the
impact parameter values as can be seen in table 1 whereas for PYTHIA8 we
define the centrality intervals which is based on the summed transverse energy
($\sum E_{T}$) in the pseudorapidity interval [-0.8, 0.8]. Figure 3(a) shows
the distribution of events as a function of impact parameter for AMPT
generator while Figure 3(b) shows event distribution for PYTHIA8 event
generator as a function of summed transverse energy ($\sum E_{T}$)
Bierlich:2018xfw . The $dN_{ch}/d\eta$ values are also compared with the
results from the experimental study with the ALICE Detector at the LHC at CERN
ALICE:2018cpu . From the figure, it can be seen that the AMPT describe the
data fairly well at low centralities, and a slight overestimate at high
centralities. On the other hand, PYTHIA8 produces the shape pretty well but
slightly overestimates the data in the small centrality bins but this
overestimation increases in higher centrality bins. For Angantyr, we find that
the fluctuations as well as the distinction between primary and secondary
absorption-damaged nucleons have a fairly significant impact on the final
state multiplicity. It is to be noted that AMPT presumes that a hot dense
medium is formed in which a collective expansion of a thermalised medium
occurs whereas PYTHIA lacks a mechanism to reproduce the collective effects
seen in pp collisions, and therefore the Angantyr model lacks the assumption
of production of a medium as well.

(a) AMPT: String Melting.

(b) PYTHIA8 Angantyr

Figure 3: (a) Event distribution as a function of impact parameter for AMPT
and (b) Event distribution as a function of Summed Transverse energy ($\sum
E_{T}$) for PYTHIA8 Angantyr model.

The $p_{T}$ spectra of particles has been studied in jetty and isotropic
limits of transverse spherocity intervals for centrality classes. In Figure 4
the transverse spherocity distribution for AMPT 4(a) and PYTHIA8 4(b)
generators in different centrality classes for Xe-Xe collisions at
$\sqrt{s_{NN}}$ = 5.44 TeV are presented. From the figure, it is observed that
small centrality (high-multiplicity) events are more towards isotropic in
nature whereas high centrality (low multiplicity) events are towards the jetty
side. The means that the peak of the transverse spherocity distribution shifts
towards jetty events with centrality, which shows that more central events
contribute to the softer events and vice-versa. The process of isotropization
in a many-particle final state occurs through multiple interactions between
the quanta of the system. If the final state multiplicity in an event is
higher, the probability that the event will become isotropic is also higher.
Therefore, the differential study of particle production as a function of
centrality and transverse spherocity classes has great importance to
understand the particle production mechanism. As transverse spherocity
distribution depends on centrality and ultimately to the charged-particle
multiplicity, therefore to define the jetty and isotropic limits of transverse
spherocity , the 20$\%$ events of the extremum of the spherocity distribution
are considered. The cuts for jetty and isotropic events vary for different
centrality classes are shown in Table.2.

(a) AMPT: String Melting.

(b) PYTHIA8 Angantyr

Figure 4: (a) Event distribution as a function of impact parameter for AMPT
and (b) Event distribution as a function of Summed Transverse energy ($\sum
E_{T}$) for PYTHIA8 Angantyr model.

(a) AMPT: $\pi^{\pm}$

(b) AMPT: $K^{\pm}$

(c) AMPT: $p^{\pm}$

(d) PYTHIA8: $\pi^{\pm}$

(e) PYTHIA8: $K^{\pm}$

(f) PYTHIA8: $p^{\pm}$

Figure 5: $p_{T}$ spectra of $\pi^{\pm}$ , $K^{\pm}$ and $p^{\pm}$ in the
$centrality$ bin 0-10$\%$ bin for AMPT (left) PYTHIA8 (right) event
generators. The lower panel of the respective figures show the ratio of
$p_{T}$ -spectra for isotropic and jetty events w.r.t $S_{o}$ integrated
events. The green markers are for isotropic events, red markers are jetty
events, and black represent $S_{o}$ integrated events.

(a) AMPT: $\pi^{\pm}$

(b) AMPT: $K^{\pm}$

(c) AMPT: $p^{\pm}$

(d) PYTHIA8: $\pi^{\pm}$

(e) PYTHIA8: $K^{\pm}$

(f) PYTHIA8: $p^{\pm}$

Figure 6: $p_{T}$ spectra of $\pi^{\pm}$ , $K^{\pm}$ and $p^{\pm}$ in the
$centrality$ bin 60-70$\%$ bin for AMPT (left) PYTHIA8 (right) event
generators. The lower panel of the respective figures show the ratio of
$p_{T}$ -spectra for isotropic and jetty events w.r.t $S_{o}$ integrated
events. The green markers are for isotropic events, red markers are jetty
events, and black represent $S_{o}$ integrated events.

Centrality | AMPT | PYTHIA8
---|---|---
jetty | isotropic | jetty | isotropic
0-10% | 0-0.87 | 0.95-1 | 0-0.93 | 0.97-1
10-20% | 0-0.82 | 0.92-1 | 0-0.92 | 0.96-1
20-30% | 0-0.78 | 0.90-1 | 0-0.91 | 0.96-1
30-40% | 0-0.76 | 0.89-1 | 0-0.89 | 0.95-1
40-50% | 0-0.75 | 0.89-1 | 0-0.87 | 0.94-1
50-60% | 0-0.74 | 0.88-1 | 0-0.83 | 0.93-1
60-70% | 0-0.72 | 0.87-1 | 0-0.77 | 0.93-1

Table 2: Table showing the values of transverse spherocity for jetty and
isotropic ranges for different centrality bins.

The event shape study is done using transverse spherocity in various
centrality classes as given in Table 2. In Figure 5 anf 6, the $p_{T}$ spectra
of $\pi^{\pm}$ , $K^{\pm}$ and $p^{\pm}$ for the centrality bin 0-10$\%$ and
60-70$\%$ are shown for various spherocity classes at the mid-rapidity
$|\eta|<0.8$. The different classes of centrality are jetty (which corresponds
to lower 20$\%$ of the events in spherocity distribution), isotropic (which
corresponds to higher 20$\%$ of the events in spherocity distribution) and
integrated(0-1 transverse spherocity ). The ratio of the
$p_{\mathrm{T}}$-spectra for isotropic and jetty events with respect to the
spherocity integrated events ($0<S_{o}<1$) is shown in the lower panels. From
the figures, we find that the low $p_{T}$ regions are dominated by isotropic
events rather than jetty events. However, this scenario is reversed when
moving to higher $p_{T}$ . At certain points, called _crossing point_ , the
jetty event dominates the isotropic event. Therefore, the study of ’crossover
points’ is of great interest with respect to feasible limits of event-type
dominance and hence related particle generation mechanisms. As we progress
from low-multiplicity to high-multiplicities, it has been observed from prior
investigations of the light-flavor sector that the ’crossing point’ relies on
the multiplicity greatly and that the isotropic events are populated over
numerous events Khuntia:2018qox ; Bencedi:2018ctm ; ALICE:2019bdw . According
to the observations, the QGP-like effects observed in high-multiplicity pp
collisions may not be caused by jet-bias effects but instead may be caused by
a potential system development, which should be investigated. In the present
work, similar studies of the identified particles ($\pi^{\pm}$ , $K^{\pm}$ and
$p^{\pm}$ ) in the central pseudorapidity region $|\eta|<0.8$ in the heavy-ion
collision system show that isotropic events are dominant at low centralities
(high-multiplicities) and vice-versa. Figure 7 shows the the ‘crossing point’
in different centrality ranges for PYTHIA8 and AMPT event generators. The
values are also presented in Table 3. This illustrates how, for two event
generators, the contribution of jets to the creation of charged particles
varies with centrality conditions. From the Table 3 and Figure 7, a comparison
shows the shift of the ‘crossing point’ towards lower-$p_{T}$ in case of
PYTHIA8 Angantyr model indicating the jet dominant contribution in particle
production for AMPT model.This means that the contribution of jets is higher
in AMPT as compared to the PYTHIA8. The dominance of isotropy in low
centrality (high-multiplicity) events and jettiness in high centrality (low-
multiplicity) events in charge particle production further suggests a
reduction and softening of the jet yields at high charged-particle
multiplicity. Therefore in PYTHIA8 Angantyr model, where the high multiplicity
events involve the large momentum transfer, the reduction in jet contribution
in the particle production may indicate a reduced production of back-to-back
jets. However we should keep in mind that the Angantyr does not include an
assumption of a hot thermalised medium, therefore collective effects are
absent PYTHIA (Angantyr) whereas AMPT considers collective effects. Also for
the Figures 5 and 6, it is clearly visible that Pythia is clearly able to
differentiate the ratio as compared the AMPT where the ratio of jetty and
isotropic $p_{T}$ spectra with respect to the integrated $S_{o}$ move along
the unity in large $p_{T}$ region. This indicates that the study of the jets
production with PYTHIA8 Angantyr and AMPT will form an interesting subject.

Figure 7: The dependence of ‘crossing point’ on the centrality of the event
for PYTHIA8 (filled circles) and AMPT (Square boxes) respectively.

Centrality | $\pi^{\pm}$ | $K^{\pm}$ | $p^{\pm}$
---|---|---|---
PYTHIA | AMPT | PYTHIA | AMPT | PYTHIA | AMPT
0-10% | 2.61 | 5.83 | 3.56 | 5.97 | 3.63 | 6.09
10-20% | 2.59 | 5.2 | 2.82 | 5.41 | 2.93 | 5.53
20-30% | 2.32 | 5.02 | 2.41 | 5.18 | 2.65 | 5.31
30-40% | 2.16 | 4.79 | 2.31 | 4.91 | 2.63 | 4.99
40-50% | 2.02 | 4.63 | 2.04 | 4.83 | 2.35 | 4.90
50-60% | 1.89 | 4.21 | 1.90 | 4.43 | 2.24 | 4.57
60-70% | 1.73 | 3.9 | 1.83 | 3.99 | 1.88 | 4.1

Table 3: Table showing the crossing point ($p_{T}$ in GeV/c) of jetty and
isotropic events at mid-rapidity for Xe-Xe at $\sqrt{s_{NN}}$ = 5.44 TeV
collisions for PYTHIA8 and AMPT.

##  IV Summary

In conclusion, using A Multi-Phase Transport Model (AMPT) and PYTHIA8
Angantyr, we present the first application of transverse spherocity analysis
for Xe-Xe collisions at $\sqrt{s_{NN}}$ = 5.44 TeV . The findings demonstrate
that transverse spherocity successfully distinguishes between high-$S_{o}$ and
low-$S_{o}$ heavy-ion collision event topologies. From the results, one can
see that crossing points occurs at relatively smaller $p_{T}$ as we go from
low centrality to high centrality events. This means that for low centrality
events jettiness has a dominance over the isotropiness whereas for high
centrality events isotropiness has dominance. Also for PYTHIA8 Angantyr
crossing point occurs at much lower $p_{T}$ as compared to AMPT indicating
that jets production occurs at much lower $p_{T}$ for PYTHIA8 Angantyr as
compared to the AMPT. As we know that AMPT considers the formation of medium
as well as the collective effects whereas PYTHIA8 Angantyr does not take into
account such happenings. Therefore, jet production studies with both models
will be an interesting study to do. The results, in our opinion, are extremely
positive, and an experimental investigation in this direction would be very
beneficial to comprehend the event topology dependence of system dynamics in
heavy-ion collisions.

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# Majorana corner states in the dice lattice

Narayan Mohanta<EMAIL_ADDRESS>Department of Physics and
Astronomy, The University of Tennessee, Knoxville, TN 37996, USA  Materials
Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN
37831, USA  Department of Physics, Indian Institute of Technology Roorkee,
Roorkee 247667, India  Rahul Soni  Department of Physics and Astronomy, The
University of Tennessee, Knoxville, TN 37996, USA  Materials Science and
Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
Satoshi Okamoto  Materials Science and Technology Division, Oak Ridge National
Laboratory, Oak Ridge, TN 37831, USA  Elbio Dagotto  Department of Physics and
Astronomy, The University of Tennessee, Knoxville, TN 37996, USA  Materials
Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN
37831, USA

###### Abstract

Lattice geometry continues providing exotic new topological phases in
condensed matter physics. Exciting recent examples are the higher-order
topological phases, manifesting via localized lower-dimensional boundary
states. Moreover, flat electronic bands with a non-trivial topology arise in
various lattices and can hold a finite superfluid density, bounded by the
Chern number $C$. Here we provide a general route to analyze the topological
properties of flat bands in the superconducting state. We argue that the
topological superconductivity induced in a flat band with $C\\!=\\!n$ is of
order $n$ and the associated boundary Majorana states have a complex structure
when $n\\!>\\!1$. As example, we show that an attractive interaction in the
dice lattice, that exhibits flat bands with $C\\!=\\!2$, gives rise to second-
order topological superconductivity with mixed singlet-triplet pairing. The
second-order nature of the topological superconducting phase is revealed by
the zero-energy Majorana bound states at the lattice corners. These findings
suggest that flat bands with a finite Chern number provide feasible platforms
for inducing topological superconductivity of various orders.

## Introduction

Higher order topology in quantum matter has recently generated a flurry of
activity in several broad areas, including the field of superconductivity
Schindler_SciAdv2018 ; PhysRevLett.119.246401 ; PhysRevLett.120.026801 ;
PhysRevB.97.205136 ; PhysRevB.98.201114 ; PhysRevB.97.205135 ;
PhysRevX.9.011012 ; PhysRevB.99.041301 ; Li_2DMater2021 ; Ghosh_PRB2021 ;
PhysRevResearch.2.012060 ; PhysRevResearch.2.043300 . At the boundaries and
vortex cores, a topological superconductor harbors Majorana quasiparticles,
with potential value in the long-sought area of decoherence-free quantum
computing Kitaev_2001 ; RevModPhys.80.1083 ; Sarma2015 ; PhysRevX.6.031016 ;
RevModPhys.87.137 ; Mohanta_EPL2014 . Topological superconductivity can be
induced, for example, by a Rashba spin-orbit coupling together with a magnetic
field PhysRevLett.105.077001 ; Mourik_Science2012 ; Rokhinson2012 ;
Deng_Science2016 , and also by a spatially-modulated spin texture in proximity
to a conventional superconductor Desjardins2019 ; PhysRevApplied.12.034048 ;
Mohanta_CommPhys2021 ; Herbrych2021 . A $n^{\rm th}$-order topological
superconductor in $d$ dimensions hosts $(d\\!-\\!n)$-dimensional Majorana
states Song_PRL2017 . The corner-localized Majorana bound states (MBS) in a
second-order two-dimensional topological superconductor are particularly
interesting because a two-dimensional array of corner MBS – useful for
demonstrating non-Abelian statistics – can be easily achieved
PhysRevB.98.165144 ; PhysRevResearch.2.032068 ; PhysRevResearch.2.043025 .
These corner MBS have been proposed in many platforms including a topological
insulator in proximity to a $d$-wave or $s^{\pm}$-wave superconductor, and a
Josephson junction bilayer PhysRevLett.121.096803 ; PhysRevLett.121.186801 ;
PhysRevLett.122.126402 .

Following the discovery of unconventional superconductivity in twisted-bilayer
graphene Cao2018 , a series of studies suggested the possibility of electronic
pairing from repulsive interaction in materials with a high density of states
at the Fermi level, such as in a flat electronic band, leading to
superconductivity with a high critical temperature PhysRevB.101.014501 ;
PhysRevLett.126.027002 ; Volovik2016 ; Aoki2020 ; PhysRevB.106.125155 . When a
flat band is topologically nontrivial, the topological invariant places a
lower bound on the superfluid weight $D_{s}$ i.e. $D_{s}\\!\geq\\!C$, where
$C$ is the Chern number of the flat band Peotta2015 ; PhysRevLett.124.167002 .
In this case, near a band-inversion wavevector, the Berry phase can convert a
repulsive interaction between two oppositely-moving electrons into an
effective attraction.

Therefore, an important question that emerges is whether topological
superconductivity can be realized in a topologically non-trivial flat band.
Here, we address this general question and argue that there is a connection
between the Chern number of the topological flat band and the order of the
induced topological superconductivity. As an example, we consider the
topological flat bands with $C\\!=\\!2$ of the dice lattice PhysRevB.84.241103
; PhysRevB.102.045105  in the presence of an attractive interaction and show
that second-order topological superconductivity is induced by populating these
flat band at the Fermi level. A hallmark of the induced second-order
topological superconducting phase is found via the zero-energy MBS localized
at the lattice corners, as shown schematically in Fig. 1e.

Fig. 1: Majorana corner states and topological flat bands in the dice lattice.
a Dice lattice with open boundary conditions (OBC) and the schematic
description of the corner-bound Majorana states in the second-order
topological superconducting phase. There are three inequivalent sites in the
unit cell, shown by the dashed lines. The triangles denote three-coordination
sites and the hexagrams denote six-coordination sites. The black arrows
surrounding the six-coordination site (middle layer) represent the vectors of
the Rashba spin-orbit coupling, with clockwise sense of rotation for the upper
triangles (bottom layer) and counter-clockwise for the lower triangles (top
layer). b Band dispersion of the dice lattice with periodic boundary
conditions (PBC), a finite spin-orbit coupling and a magnetic field, showing
the topological nearly-flat bands with Chern number $C\\!=\\!\pm 2$, close to
the Fermi level ($E\\!=\\!0$). The topological superconducting phase is
obtained by populating the lower topological flat band at the Fermi level.

The bipartite nature of the dice lattice, which can be envisaged as two merged
triangular lattices, protects two degenerate flat bands coexisting with four
dispersive bands. Such a geometry can be realized using a few layers of
transition-metal oxides, dichalcogenides, and graphene. In the simplest
realization of the dice lattice involving three (111) layers of cubic
transition-metal oxides, such as in a SrTiO3/SrIrO3/SrTiO3 trilayer, the cubic
symmetry is reduced to trigonal symmetry. The strong spin-orbit coupling from
the Ir4+ ion and the broken inversion symmetry produces a Rashba spin-orbit
coupling. In the reduced $D_{3d}$ symmetry of the trilayer, the Rashba spin-
orbit coupling vectors lie in the plane parallel to the trilayer and have
opposite senses of rotation for the top and the bottom layers of the three-
coordination sites, surrounding the middle layer of six-coordination sites. In
the presence of this Rashba spin-orbit coupling, the flat bands become
isolated from the dispersive bands. Repulsive interactions in the flat bands
then spontaneously generate ferro/ferri-magnetic order on the Kramer’s pair of
flat bands PhysRevB.84.241103 ; PhysRevB.102.045105 , especially when they are
close to half filling, and split them into two nearly-flat bands with Chern
number $C\\!=\\!\pm 2$, as shown in Fig. 1b.

Besides the topological origin of superconductivity in the flat bands,
superconductivity in the transition-metal oxide trilayer can also be induced
by doping SrTiO3 PhysRevLett.12.474 ; PhysRev.163.380 . Our calculations
reveal that the superconducting state realized in the dice-lattice topological
flat bands, exhibits both singlet and triplet pairings. Such a singlet-triplet
mixing is allowed by broken inversion symmetry in this oxide trilayer. Using
symmetry analysis, we observed that the possible nearest-neihgbor pairing
channels allowed in this lattice geometry are $d_{xy}$, $d_{x^{2}-y^{2}}$,
$p_{x}$ and $p_{y}$. The nearest-neighbor pairing amplitudes were found to be
complex numbers, illustrating the chiral nature of the induced topological
superconducting phase. Besides finding corner MBS, supporting the second-order
nature of the topological superconducting phase, we perform a Chern number
analysis and obtain phase diagrams for the topological phase.

The dice lattice is a special case of the commonly-known
$\alpha\\!-\\!\tau_{{}_{3}}$ lattice which interpolates between the dice
lattice ($\alpha\\!=\\!0$, pseudospin 1) and the honeycomb lattice
($\alpha\\!=\\!1$, pseudospin 1/2) PhysRevB.94.125435 . It is also possible to
transform the honeycomb lattice into the dice lattice, and vice versa, in an
experimentally-simulated ultracold atomic gas platform PhysRevB.73.144511 .
Also, the results presented here are relevant to possible topological
superconducting phases in twisted bilayer/multilayer graphene
PhysRevLett.121.087001 .

## Results

Model and set up
The electron pairing in the topological flat bands of the dice lattice,
realizable in a transition-metal oxide trilayer as discussed above, can be
described by the following tight-binding Hamiltonian

$\displaystyle{\cal H}\\!=\\!$ $\displaystyle-t\\!\sum_{\langle
i\alpha,j\beta\rangle,\sigma}\\!(c_{i\alpha\sigma}^{\dagger}c_{j\beta\sigma}^{\phantom{\dagger}}+{\rm
H.c.})\\!-\mu\\!\sum_{i,\alpha,\sigma}\\!c_{i\alpha\sigma}^{\dagger}c_{i\alpha\sigma}^{\phantom{\dagger}}$
$\displaystyle-\lambda\\!\sum_{\langle
i\alpha,j\beta\rangle,\sigma,\sigma^{\prime}}\\!(i[\mathbf{\hat{D}}_{ij}\cdot\bm{\sigma}]_{{}_{\alpha\beta}}^{\sigma\sigma^{\prime}}c_{i\alpha\sigma}^{\dagger}c_{j\beta\sigma^{\prime}}^{\phantom{\dagger}}+{\rm
H.c.})$ $\displaystyle-
B_{z}\\!\sum_{i,\alpha,\sigma,\sigma^{\prime}}\\!([\sigma_{z}]^{\sigma\sigma^{\prime}}c_{i\alpha\sigma}^{\dagger}c^{\phantom{\dagger}}_{i\alpha\sigma^{\prime}}+{\rm
H.c.})$ $\displaystyle-
U_{0}\sum_{i,\alpha}\\!n_{i\alpha\uparrow}n_{i\alpha\downarrow}-\sum_{i,j,\alpha,\beta,\sigma,\sigma^{\prime}}U_{1}n_{i\alpha\sigma}n_{j\beta\sigma^{\prime}},$
(1)

Fig. 2: Emergence of the corner MBS. a Real-space quasiparticle spectrum with
varying chemical potential $\mu$, revealing the range
$-1.16t\lesssim\mu\lesssim-0.82t$ (green region), where two pairs of low-
energy eigenstates come close to zero energy, while other eigenstates move to
higher energies, creating a topological energy gap that protects the Majorana
bound states. b, c Plots of the local density of states $\rho(\bf{r})$ (in
arbitrary units) in the topological superconducting phase ($\mu\\!=\\!-t$) and
in the trivial superconducting phase ($\mu\\!=\\!-0.2t$). Other parameters are
$\lambda\\!=\\!0.1t$ and $B_{z}\\!=\\!0.1t$. In b, the localization of the MBS
at the lattice corners indicates the second-order nature of the realized
topological superconducting phase.

where $t$ is the electron hopping amplitude, $i$ and $j$ are indices of
different unit cells, $\alpha$ and $\beta$ represent indices of the three
inequivalent sites within a unit cell, $\sigma=\uparrow,\downarrow$ labels the
electron spin projection along the $z$ axis, $\langle\rangle$ represents
nearest-neighbor sites, $\mu$ is the chemical potential, $\lambda$ is the
strength of the Rashba spin-orbit coupling, $\mathbf{\hat{D}}_{ij}$ is the
unit vector between unit cells $i$ and $j$, $\bm{\sigma}$ represents the Pauli
matrices, $B_{z}$ is the strength of the magnetization field, the last two
terms represent the onsite and non-local density-density attractive
correlations with $U_{0}$ and $U_{1}$ as the strengths of the interactions,
respectively, and
$n_{i\alpha\sigma}=c_{i\alpha\sigma}^{\dagger}c_{i\alpha\sigma}$ is the
electron density at the unit cell $i$, site $\alpha$ and spin $\sigma$. The
density-density correlations are treated at the mean-field level and they give
pairing amplitudes in different channels after decomposition. The pairing
amplitudes between electrons at the same and nearest-neighbor (NN) sites are
defined by $\Delta_{i\alpha\uparrow\downarrow}^{\rm onsite,s}=-U_{0}\langle
c_{i\alpha\uparrow}c_{i\alpha\downarrow}\rangle$ (onsite singlet),
$\Delta_{ij\alpha\beta\uparrow\downarrow}^{\rm NN,s}=-U_{1}\langle
c_{i\alpha\uparrow}c_{j\beta\downarrow}\rangle$ (NN singlet), and
$\Delta_{ij\alpha\beta\sigma\sigma}^{\rm NN,t}=-U_{1}\langle
c_{i\alpha\sigma}c_{j\beta\sigma}\rangle$ (NN triplet). We choose $U_{0}$ and
$U_{1}$ to be of the same value $U$; a different choice does not change the
conclusions presented here because the pairing amplitudes are calculated self-
consistently. For this self-consistent determination of the pairing
amplitudes, we solve the Bogoliubov-de Gennes (BdG) equations, derived by
performing the unitary transformation
$c_{i\alpha\sigma}=\sum_{n}u_{i\alpha\sigma}^{n}\gamma_{n}^{\phantom{\dagger}}+v_{i\alpha\sigma}^{n*}\gamma_{n}^{\dagger}$
on the Hamiltonian (1), where $\gamma_{n}^{\phantom{\dagger}}$ is a fermionic
annihilation operator in the $n^{\rm th}$ eigenstate, $u_{i\alpha\sigma}^{n}$
and $v_{i\alpha\sigma}^{n}$ are respectively the quasi-particle and quasi-hole
amplitudes (see Methods for details). We use, throughout, lattice spacing
$a\\!=\\!1$, hopping scale $t\\!=\\!1$, and attractive potential $U\\!=\\!2t$.
The triplet pairing amplitude is dynamically generated in the presence of
Rashba spin-orbit coupling (broken inversion symmetry) and magnetic field
(broken time-reversal symmetry) PhysRevLett.87.037004 ; PhysRevB.97.214507 .
Odd-parity, equal-spin pairing in the triplet channel is considered because
such pairing is favored by parity fluctuations in the presence of Rashba spin-
orbit coupling and a time-reversal symmetry-breaking Zeeman exchange field
PhysRevLett.115.207002 . Alternate routes to obtain spin-triplet topological
pairing in similar systems include forward electron-phonon scattering which
also suggests a robust equal-spin pairing 2022arXiv220709443L .

Corner-localized MBS
We investigate the emergence of the zero-energy MBS by inspecting the
quasiparticle spectrum, obtained by numerically solving the Hamiltonian (1)
while varying the chemical potential $\mu$. This procedure is repeated for
many values of $\lambda$ and $B_{z}$, to search for signatures of the MBS in
the quasiparticle spectrum. As shown in Fig. 2a, at
$(\lambda,B_{z})\\!=\\!(0.1t,0.1t)$ and within the range
$-1.16t\lesssim\mu\lesssim-0.82t$, two pairs of lowest-energy quasiparticle
states remain close to zero energy while other low-energy levels move away
towards higher energies, thus creating an energy gap. This energy gap provides
topological protection to the near-zero-energy MBS preventing them from
hybridizing with the higher-energy ordinary quasiparticle states. To study the
real-space localization of these zero-energy MBS in a two-dimensional lattice,
in Fig. 2b-c we plot the local density of states, obtained via
$\rho(\mathbf{r})\\!=\\!\sum_{\alpha,\sigma}(|u_{i\alpha\sigma}^{n}|^{2}+v_{i\alpha\sigma}^{n}|^{2})$
with $n$ corresponding to the lowest-positive energy eigenstate. We use two
values for the chemical potential: $\mu\\!=\\!-t$, where the zero-energy
states appear with a topological energy gap, and $\mu\\!=\\!-0.2t$, where the
lowest-energy states are away from zero energy. At $\mu\\!=\\!-t$, the lowest-
energy eigenstate is localized at the four lattice corners, while at
$\mu\\!=\\!-0.2t$ it is distributed inside the bulk. The corner-localized
zero-energy states provide a strong indication of the appearance of the MBS,
and hence of the induced second-order topological superconducting phase.

Pairing symmetry
The dice lattice has sites with coordination numbers three or six, and this
feature distinguishes it from the triangular and hexagonal lattices. As
discussed below, the presence of these two types of sites also determines the
pairing symmetry in the superconducting state. Furthermore, the Rashba spin-
orbit coupling enforces its symmetry in the superconducting pairing.

Fig. 3: Pairing symmetry in the dice lattice. Possible pairing symmetries
around a six-coordination site in the dice lattice with Rashba spin-orbit
coupling and induced magnetization. $\Delta$ is the pairing amplitude used for
the illustration of the amplitudes along different neighbors.

In this $D_{3d}$ crystal symmetry with broken both inversion symmetry (due to
Rashba spin-orbit coupling) and time-reversal symmetry (due to the induced
magnetization), the possible pairing symmetries are {$d_{xy}$,
$d_{x^{2}-y^{2}}$} in the singlet channel, and {$p_{x}$, $p_{y}$} in the
triplet channel, as shown in Fig. 3. Mixing of the singlet and triplet
components is allowed by the broken structural inversion symmetry in the
discussed oxide trilayers PhysRevLett.87.037004 . Therefore, a linear
combination of these four types of pairing symmetry is stabilized. Figure 4
shows the profiles of the pairing amplitudes on the dice lattice at the same
set of parameters where the corner MBS are found. The imaginary components of
the nearest-neighbor pairing amplitudes are nonzero, implying a chiral mixed-
parity topological superconducting state. The real part of the onsite singlet
pairing amplitude Re($\Delta_{i\alpha\uparrow\downarrow}^{\rm onsite,s}$), in
Fig. 4a top panel, clearly reveals a difference between the three and six
coordination sites. While the onsite singlet pairing amplitude
Re($\Delta_{i\alpha\uparrow\downarrow}^{\rm onsite,s}$) at the six-
coordination sites is slightly smaller than that at the three-coordination
sites, the nearest-neighbor singlet pairing amplitude
Re($\Delta_{ij\alpha\beta\uparrow\downarrow}^{\rm NN,s}$), displayed in Fig.
4b top panel, shows the opposite behavior.

Fig. 4: Real-space profile of singlet and triplet pairing amplitudes.
Superconducting pairing amplitudes for different pairing channels: a onsite
singlet, b nearest-neighbor singlet, and c nearest-neighbor triplet. Top and
bottom rows show the real and imaginary parts of the pairing amplitudes. Model
parameters are $\lambda\\!=\\!0.1t$, $B_{z}\\!=\\!0.1t$, and $\mu\\!=\\!-t$.
The results indicate that the topological superconducting phase exhibits both
singlet and triplet pairing components. The pairing amplitudes in the nearest-
neighbor channel are complex numbers, implying a chiral pairing symmetry. OBC
are used. The slightly different boundary effects among horizontal and tilted
edges, as in the red vs white colored edges in lower panel a, originate in the
different site endings along those edges, as can be seen in Fig. 1a. Fig. 5:
Quasiparticle band dispersion and topological phase diagram. a-b Bulk
quasiparticle bands, obtained with periodic boundary conditions, along the
momentum path $\Gamma(0,0)-{\rm K}(\frac{4\pi}{3\sqrt{3}},0)-{\rm
M}(\frac{\pi}{\sqrt{3}},\frac{\pi}{3})-\Gamma$, below energy $E\\!=\\!0$, in
the trivial ($\mu\\!=\\!-0.2t$) and in the topological ($\mu\\!=\\!-t$)
superconducting phases. In b, one band (in thick green) is isolated and
carries a Chern number $C\\!=\\!2$. c-d Quasiparticle spectrum, obtained in a
cylindrical geometry, at the same sets of parameters as in a, b, respectively.
The pair of lowest-energy edge states (in red) crosses in the case of the
topological superconducting phase in d, near $k_{1}\\!=\\!0.95\pi$. Parameters
used are $\lambda\\!=\\!0.1t$, $B_{z}\\!=\\!0.1t$, $\Delta^{\rm
s}_{1,3}\\!=\\!0.5t$, and $\Delta^{\rm s}_{2}\\!=\\!0.45t$. e-f Plots of the
Chern number of the occupied bands in the $\mu\\!-\\!B_{z}$ plane at two
Rashba coupling strengths $\lambda\\!=\\!0.1t$ and $\lambda\\!=\\!0.3t$.

The slight variation in the pairing amplitudes near the corners and edges of
the lattice is due to the open boundary conditions used in the calculations.
The real part of the nearest-neighbor triplet pairing amplitude
Re($\Delta_{ij\alpha\beta\sigma\sigma}^{\rm NN,t}$) vanishes inside the bulk,
but remains finite at the boundaries along the $\pm x$ directions. Such an
exact cancellation of the pairing amplitude inside the bulk implies a
$p_{x}$-wave pairing component. The imaginary part of the nearest-neighbor
triplet pairing amplitude Im($\Delta_{ij\alpha\beta\sigma\sigma}^{\rm NN,t}$)
is finite in the bulk and it is different in magnitude and changes its sign at
the six and three coordination sites. This implies a $p_{y}$-wave pairing
component because only a partial cancellation of the pairing amplitudes is
possible along the $y$ direction in the dice lattice due to the lattice
asymmetry along the $x$ and $y$ directions.

Topological superconducting transition
The transition to the second-order topological superconducting phase can be
understood by inspecting the quasiparticle band dispersion in momentum space.
The realized topological superconducting phase has broken time-reversal
symmetry, but it respects particle-hole symmetry, and hence it belongs to the
Altland-Zirnbauer symmetry class D and can be identified by evaluating the
Chern number PhysRevB.78.195125 . We calculate the topological invariant of
the superconducting phase as the sum of the Chern numbers of the quasiparticle
bands below $E\\!=\\!0$. Isolated quasiparticle bands in the dice lattice have
Chern number $C\\!=\\!\pm 2$. Therefore, the Chern number of the total
occupied states becomes finite whenever an odd number of topological
quasiparticle bands appear as isolated at a given set of parameters. The
momentum-space quasiparticle bands are obtained by diagonalizing the following
BdG Hamiltonian at wavevector $\mathbf{k}$

$\displaystyle{\cal H}_{\rm
BdG}(\mathbf{k})\\!=\\!\begin{pmatrix}\Psi_{\mathbf{k}}^{\dagger}&\Psi_{-\mathbf{k}}\end{pmatrix}\begin{pmatrix}H_{e}&H_{\Delta}\\\
H_{\Delta}^{*}&H_{h}\end{pmatrix}\begin{pmatrix}\Psi_{\mathbf{k}}\\\
\Psi_{-\mathbf{k}}^{\dagger}\end{pmatrix}$ (2)

where $H_{e}$, $H_{h}$ and $H_{\Delta}$ are the matrices representing,
respectively, the electron, the hole, and the pairing sectors of the
Hamiltonian. The basis wave vector for $H_{e}$ is
$\Psi_{\mathbf{k}}=[c_{\mathbf{k}1\uparrow},c_{\mathbf{k}2\uparrow},c_{\mathbf{k}3\uparrow},c_{\mathbf{k}1\downarrow},c_{\mathbf{k}2\downarrow},c_{\mathbf{k}3\downarrow}]^{T}$
(see Methods section for more information). Figures 5a-b show the
quasiparticle band dispersion at two values of the chemical potential
$\mu\\!=\\!-0.2t$ and $\mu\\!=\\!-t$. In the latter, an isolated topological
band with $C\\!=\\!2$ appears, resulting in a second-order topological phase.
In Fig. 5c-d, we plot the energy spectra using now a cylindrical geometry,
with periodic boundary condition along the $x$ direction and open boundary
condition along the $y$ direction, for the same two values of $\mu$ as above.
In the trivial superconducting phase, Fig. 5c, an energy gap prevails in the
spectrum, while in the case of the topological superconducting phase, Fig. 5d,
two pairs of doubly-degenerate low energy quasiparticle bands cross near
$k_{1}\\!=\\!0.95\pi$. The Chern number, used to construct a phase diagram for
the second-order topological superconducting phase in the $\mu\\!-\\!B_{z}$
plane, is shown in Fig. 5e-f, for two values of the Rashba spin-orbit coupling
strengths $\lambda\\!=\\!0.1t$ and $\lambda\\!=\\!0.3t$. The induced
topological phase appears within a range of $\mu$, and this range tends to
decrease with increasing the Zeeman field or the Rashba coupling.

## Discussion

Analogies between the topological superconductivity in flat bands, as found
here, and the quantum-Hall insulator/superconductor interfaces can be drawn.
Theoretically, it is known that a quantum Hall state with Chern number
$C\\!=\\!1$, in proximity to a fully gapped $s$-wave superconductor, generates
a topological (first-order) superconducting phase PhysRevB.82.184516 ;
PhysRevResearch.2.023063 . Likewise, the fractionalized MBS, i.e. some
realizations of parafermions, have been proposed in fractional quantum Hall
states when in proximity to an $s$-wave superconductor Clarke2013 ;
PhysRevX.4.031009 ; PhysRevResearch.2.013232 . In the $C\\!=\\!2$ flat band
considered here with an attractive pairing, the topological superconductivity
is of order two. These findings suggest a close connection between the Chern
number $C$ of the normal state and the degree of the induced topological
superconductivity.

To summarize, we showed that topological flat bands in the dice lattice with
attractive interaction among electrons harbor second-order topological
superconductivity. A signature of this exotic topological phase is revealed by
the presence of MBS at the lattice corners. We propose that topological flat-
bands provide a promising platform for realizing a general $n^{\rm th}$-order
topological superconductor, where $n$ is the Chern number of the flat band.
Topological flat bands with higher Chern numbers are found not only in the
dice lattice, but also in kagomé and Lieb lattices PhysRevB.104.235115 ;
Okamoto_CommPhys2022 . Other than the examples of a few-layer graphene and a
transition-metal-oxide trilayer, another candidate compound is CsV3Sb5
PhysRevLett.125.247002 ; Zhao2021 ; HU2022495 , where lattice geometry, flat-
band topology and superconductivity may also produce Majorana states such as
those discussed here. Hence, we expect future research will unveil topological
superconductivity in a variety of compounds that exhibit topological flat
bands. Furthermore, the superconducting transition temperature is proportional
to the density of states at the Fermi level which is large for these flat-band
systems. Therefore, when looking forward, topological flat-bands with higher
Chern numbers provide an opportunity to search for higher-order topological
superconductivity at high temperatures.

## Methods

Self-consistent BdG formalism
The attractive interaction term in the Hamiltonian (1) is decoupled into
different pairing channels (singlet and triplet, onsite and nearest-neighbor).
The mean-field Hamiltonian is then diagonalized using the BdG transformation
$c_{i\alpha\sigma}\\!=\\!\sum_{n}u_{i\alpha\sigma}^{n}\gamma_{n}+v_{i\alpha\sigma}^{n*}\gamma_{n}^{\dagger}$,
where $\gamma_{n}$ is a fermionic annihilation operator at the $n^{\rm th}$
eigenstate, $u_{i\alpha\sigma}^{n}$ and $v_{i\alpha\sigma}^{n}$ are the quasi-
particle and quasi-hole amplitudes, respectively. The quasi-particle and
quasi-hole amplitudes are obtained by solving the BdG equations $\sum_{j}{\cal
H}_{ij}\psi_{j}^{n}\\!=\\!\epsilon_{n}\psi_{i}^{n}$, where
$\psi_{i}^{n}\\!=\\![u_{i\alpha\uparrow}^{n},u_{i1\downarrow}^{n},v_{i\alpha\uparrow}^{n},v_{i\alpha\downarrow}^{n}]^{T}$
with
$u_{i\alpha\uparrow}^{n}=[u_{i1\uparrow}^{n},u_{i2\uparrow}^{n},u_{i3\uparrow}^{n}]$
and similarly for other components, while $\epsilon_{n}$ is the energy
eigenvalue of the $n^{\rm th}$ eigenstate. Then, the pairing amplitudes are
calculated via $\Delta_{i\alpha\uparrow\downarrow}^{\rm
onsite,s}\\!=\\!-U_{0}\langle c_{i\alpha\uparrow}c_{i\alpha\downarrow}\rangle$
(onsite singlet), $\Delta_{ij\alpha\beta\uparrow\downarrow}^{\rm
NN,s}\\!=\\!-U_{1}\langle c_{i\alpha\uparrow}c_{j\beta\downarrow}\rangle$ (NN
singlet), and $\Delta_{ij\alpha\beta\sigma\sigma}^{\rm
NN,t}\\!=\\!-U_{1}\langle c_{i\alpha\sigma}c_{j\beta\sigma}\rangle$ (NN
triplet). The self-consistency iterations continue until all the pairing
amplitudes converge at all lattice sites, within a tolerance of 10-8.

Momentum-space Hamiltonian
The Hamiltonian (2) is expressed in the basis
$\Psi_{\mathbf{k}}=[c_{\mathbf{k}1\uparrow},c_{\mathbf{k}2\uparrow},c_{\mathbf{k}3\uparrow},c_{\mathbf{k}1\downarrow},c_{\mathbf{k}2\downarrow},c_{\mathbf{k}3\downarrow}]^{T}$,
where 1, 2, 3 denote the three inequivalent sites within a unit cell, and is
given by

$\displaystyle{\cal H}_{e}(\mathbf{k})\\!=\\!$
$\displaystyle\begin{pmatrix}-B_{z}-\mu&-t\gamma_{\mathbf{k}}^{*}&0&0&-i\lambda\gamma_{\mathbf{k+}}^{*}&0\\\
-t\gamma_{\mathbf{k}}&-B_{z}-\mu&-t\gamma_{\mathbf{k}}^{*}&i\lambda\gamma_{\mathbf{k-}}&0&i\lambda\gamma_{\mathbf{k+}}^{*}\\\
0&-t\gamma_{\mathbf{k}}&-B_{z}-\mu&0&-i\lambda\gamma_{\mathbf{k-}}&0\\\
0&-i\lambda\gamma_{\mathbf{k-}}^{*}&0&B_{z}-\mu&-t\gamma_{\mathbf{k}}^{*}&0\\\
i\lambda\gamma_{\mathbf{k+}}&0&i\lambda\gamma_{\mathbf{k-}}^{*}&-t\gamma_{\mathbf{k}}&B_{z}-\mu&-t\gamma_{\mathbf{k}}^{*}\\\
0&-i\lambda\gamma_{\mathbf{k+}}&0&0&-t\gamma_{\mathbf{k}}&B_{z}-\mu\end{pmatrix},$
(3)

where $\gamma_{\mathbf{k}}\\!=\\!1\\!+\\!e^{ik_{1}}\\!+\\!e^{ik_{2}}$,
$\gamma_{\mathbf{k\pm}}\\!=\\!1+e^{ik_{1}\pm 2\pi/3}+e^{ik_{2}\pm 4\pi/3}$,
$k_{1,2}\\!=\\!\mathbf{k}.\cdot\mathbf{e}_{1,2}$, and $\mathbf{e}_{1,2}$ are
lattice translational vectors. The topological flat bands in Fig. 1B are
obtained by diagonalizing ${\cal H}_{e}(\mathbf{k})$ at $t\\!=\\!1$,
$\mu\\!=\\!0$, $\lambda\\!=\\!0.3t$, and $B_{z}\\!=\\!0.4t$. The hole part of
the Hamiltonian (2) is given by ${\cal H}_{h}(\mathbf{k})\\!=\\![-{\cal
H}_{e}(-\mathbf{k})]^{T}$, and the pairing part is given by

$\displaystyle{\cal
H}_{\Delta}(\mathbf{k})\\!=\\!\begin{pmatrix}0&0&0&\Delta_{1}^{\rm s}&0&0\\\
0&0&0&0&\Delta_{2}^{\rm s}&0\\\ 0&0&0&0&0&\Delta_{3}^{\rm s}\\\
-\Delta_{1}^{\rm s}&0&0&0&0&0\\\ 0&-\Delta_{2}^{\rm s}&0&0&0&0\\\
0&0&-\Delta_{3}^{\rm s}&0&0&0\\\ \end{pmatrix},$ (4)

where $\Delta_{\alpha}^{\rm s}$ ($\alpha\\!=\\!1,2,3$) represents the onsite
singlet pairing amplitude at site index $\alpha$.

## Acknowledgements

This work was supported by the U.S. Department of Energy, Office of Science,
Basic Energy Sciences, Materials Sciences and Engineering Division.

## Data availability

All data obtained from numerical calculations have been presented in the
paper.

## Code availability

Simulation codes are available from the corresponding author upon reasonable
request.

## Author contributions

NM planned the work, performed numerical calculations and wrote the manuscript
with inputs from all coauthors. All authors contributed to the analysis of the
results, data presentation and manuscript writing.

## Competing interests

The authors declare no competing interests.

## Correspondence

Correspondence should be addressed to NM.

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