#  $Z\to{\tau\tau}$ and $W\to\tau\nu_{\tau}$ Cross-Sections at the LHC

Philip Ilten
on behalf of the LHCb Collaboration
including results from ATLAS and CMS  School of Physics, University College
Dublin

###### Abstract

Measurements of the $Z\to{\tau\tau}$ and $W\to\tau\nu_{\tau}$ cross-sections
at the LHC with data taken at $\sqrt{s}=7$ TeV are reported for the ATLAS,
CMS, and LHCb experiments. All results are found to agree with the Standard
Model.

###### keywords:

LHC , ATLAS , CMS , LHCb , electroweak , tau production

††journal: Nuclear Physics B Proceedings

##  1 Introduction

The production of $Z$ and $W$ bosons from $pp$-collisions, and their
subsequent decays to $\tau$-leptons at the Large Hadron Collider (LHC) not
only provides important tests of the Standard Model (SM), but also lays the
groundwork for the study of beyond the SM physics using $\tau$-lepton
signatures.

Decays of $Z$ bosons to $\tau$-lepton pairs are both a mechanism for
experimentally measuring hadronic $\tau$-lepton identification efficiencies
[1, 2] and a calibration channel for neutral Higgs searches [3, 4]. The final
states of $\tau$-leptons produced from $W$ bosons can be used to measure the
polarization of the $W$ [5] boson or to search for charged Higgses [6, 7].

In this review, the complete set of $Z\to{\tau\tau}$ and $W\to\tau\nu_{\tau}$
cross-sections, as measured by the ATLAS, CMS, and LHCb experiments on the
LHC, are reported using $2010$ and $2011$ datasets taken at
$\sqrt{s}=7~{}\mathrm{TeV}$. A full summary of the results, including
corresponding references, is provided in Table 1, and a comparison of the
theoretical agreement of the results is given in Figure 1.

All three detectors are fully instrumented with charged particle trackers,
electromagnetic and hadronic calorimeters, and muon systems. Both ATLAS [8]
and CMS [9] are general purpose detectors designed to cover a central
pseudorapidity range of $|\eta|<2.4$, while the LHCb [10] detector is a
forward arm spectrometer, purpose built for $B$-hadron physics, covering the
forward pseudorapidity range $2.0<\eta<5.0$.

Exp. | Analysis | $N_{\mathrm{obs}}$ | $N_{\mathrm{bkg}}$ | $\sigma\pm\mathrm{stat.}\pm\mathrm{syst.}\pm\mathrm{lumi.}$ | $\mathcal{L}$ | Data | Ref.
---|---|---|---|---|---|---|---
ATLAS | $Z\to$ | $\tau_{\mu}\tau_{\mu}$ | $90$ | $47$ | $0.96$ | $\pm$ | $0.22$ | $\pm$ | $0.12$ | $\pm$ | $0.03$ | nb | $36$ | $\mathrm{pb}^{-1}$ | $2010$ | [11]
| $\tau_{\mu}\tau_{e}$ | $1035$ | $56$ | $0.96$ | $\pm$ | $0.03$ | $\pm$ | $0.09$ | $\pm$ | $0.04$ | nb | $1.55$ | $\mathrm{fb}^{-1}$ | $2011$ | [12]
| $\tau_{\mu}\tau_{h}$ | $5184$ | $793$ | $0.91$ | $\pm$ | $0.01$ | $\pm$ | $0.09$ | $\pm$ | $0.03$ | nb | $1.55$ | $\mathrm{fb}^{-1}$ |
| $\tau_{e}\tau_{h}$ | $2600$ | $449$ | $1.00$ | $\pm$ | $0.02$ | $\pm$ | $0.13$ | $\pm$ | $0.04$ | nb | $1.34$ | $\mathrm{fb}^{-1}$ |
$W\to$ | $\tau_{h}\nu_{\tau}$ | $2335$ | $411$ | $11.1$ | $\pm$ | $0.3$ | $\pm$ | $1.7$ | $\pm$ | $0.4$ | nb | $34$ | $\mathrm{pb}^{-1}$ | $2010$ | [13]
CMS | $Z\to$ | $\tau_{\mu}\tau_{\mu}$ | $58$ | $23$ | $1.14$ | $\pm$ | $0.27$ | $\pm$ | $0.04$ | $\pm$ | $0.05$ | nb | $36$ | $\mathrm{pb}^{-1}$ | $2010$ | [14]
| $\tau_{\mu}\tau_{e}$ | $101$ | $14$ | $0.99$ | $\pm$ | $0.12$ | $\pm$ | $0.06$ | $\pm$ | $0.04$ | nb |  |
| $\tau_{\mu}\tau_{h}$ | $517$ | $228$ | $0.83$ | $\pm$ | $0.07$ | $\pm$ | $0.19$ | $\pm$ | $0.03$ | nb |  |
| $\tau_{e}\tau_{h}$ | $540$ | $346$ | $0.94$ | $\pm$ | $0.11$ | $\pm$ | $0.22$ | $\pm$ | $0.04$ | nb |  |
$W\to$ | $\tau_{h}\nu_{\tau}$ | $372$ | $155$ | not measured | $18$ | $\mathrm{pb}^{-1}$ | $2010$ | [15]
LHCb | $Z\to$ | $\tau_{\mu}\tau_{\mu}$ | $124$ | $42$ | $77.4$ | $\pm$ | $10.4$ | $\pm$ | $8.6$ | $\pm$ | $2.7$ | pb | $1.03$ | $\mathrm{fb}^{-1}$ | $2011$ | [16]
| $\tau_{\mu}\tau_{e}$ | $421$ | $130$ | $75.2$ | $\pm$ | $5.4$ | $\pm$ | $4.1$ | $\pm$ | $2.6$ | pb | $1.03$ | $\mathrm{fb}^{-1}$ |
| $\tau_{e}\tau_{\mu}$ | $155$ | $57$ | $64.2$ | $\pm$ | $8.2$ | $\pm$ | $4.9$ | $\pm$ | $2.2$ | pb | $0.96$ | $\mathrm{fb}^{-1}$ |
| $\tau_{\mu}\tau_{h}$ | $189$ | $53$ | $68.3$ | $\pm$ | $7.0$ | $\pm$ | $2.6$ | $\pm$ | $2.4$ | pb | $1.03$ | $\mathrm{fb}^{-1}$ |
| $\tau_{e}\tau_{h}$ | $101$ | $37$ | $77.9$ | $\pm$ | $12.2$ | $\pm$ | $6.1$ | $\pm$ | $2.7$ | pb | $0.96$ | $\mathrm{fb}^{-1}$ |
Table 1: A complete summary of the individual $Z\to{\tau\tau}$ and
$W\to\tau\nu_{\tau}$ cross-section analyses of ATLAS, CMS, and LHCb using
$2010$ and $2011$ datasets. The experiment, final state, number of observed
events ($N_{\mathrm{obs}}$), number of background events ($N_{\mathrm{bkg}}$),
cross-section measurement ($\sigma$), integrated luminosity ($\mathcal{L}$),
dataset, and reference are given for each analysis.

##  2 $Z\to{\tau\tau}$

Four final states produced from the $\tau$-lepton decays of $Z\to{\tau\tau}$
events are considered by all three experiments: two muons
($\tau_{\mu}\tau_{\mu}$), a muon and an electron ($\tau_{\mu}\tau_{e}$), a
muon and a hadronic jet ($\tau_{\mu}\tau_{h}$), and an electron and a hadronic
jet ($\tau_{e}\tau_{h}$). Six backgrounds to these final states are
considered: Drell-Yan production of di-muon or di-electron pairs, $WW$ decays,
$t\bar{t}$ decays, $Z$ production with an associated jet, $W$ production with
an associated jet, and QCD multijet events.

###  2.1 Particle Selection

All final states are triggered by a muon with a $p_{\mathrm{T}}$ greater than
$9-15~{}\mathrm{GeV}$, except the $\tau_{e}\tau_{h}$ final state, which is
triggered by an electron with $E_{\mathrm{T}}$ greater than
$12-15~{}\mathrm{GeV}$ for ATLAS and CMS and $p_{\mathrm{T}}$ greater than
$10~{}\mathrm{GeV}$ for LHCb. An additional hadronic trigger is included by
ATLAS for the $\tau_{e}\tau_{h}$ final state, while the LHCb
$\tau_{\mu}\tau_{e}$ final state is split into a muon triggered final state
($\tau_{\mu}\tau_{e}$), and an electron and muon triggered final state
($\tau_{e}\tau_{\mu}$).

Muons are identified by requiring an isolated track associated with muon
system hits. Electrons are identified by requiring an isolated track
associated with electromagnetic calorimeter energy. Both one and three-pronged
hadronic $\tau$-lepton decays are identified within ATLAS and CMS using the
anti-$k_{\mathrm{T}}$ jet algorithm and requiring one or three charged
particles, whereas only single-pronged hadronic $\tau$-lepton decays are
identified within LHCb by requiring a single isolated track with an associated
hadronic calorimeter energy and minimal electromagnetic calorimeter energy.

###  2.2 Event Selection

The $Z\to{\tau\tau}$ signal produces a high mass back-to-back final state in
the transverse plane with missing energy (${{\not\mathrel{E}}_{T}}$) and a
$p_{\mathrm{T}}$ imbalance between the two $\tau$-lepton decay products due to
unreconstructed neutrinos. Additionally, the lifetime of the $\tau$-lepton
produces decay products with an enhanced impact parameter. Selection
requirements based on these five signatures are used by the three experiments
to separate signal from background.

For ATLAS, a visible mass selection requirement is placed on all four final
states, and a transverse mass requirement on the $\tau_{\mu}\tau_{h}$ and
$\tau_{e}\tau_{h}$ final states. A requirement on the transverse separation of
the $\tau$-lepton decay products and the ${{\not\mathrel{E}}_{T}}$ of the
event is used for all final states except $\tau_{\mu}\tau_{\mu}$, where
further requirements on the angular separation, $p_{\mathrm{T}}$ asymmetry,
and impact parameters of the two $\tau$-lepton decay products are applied.

For CMS, only a transverse mass requirement is applied to the
$\tau_{\mu}\tau_{e}$, $\tau_{\mu}\tau_{h}$, and $\tau_{e}\tau_{h}$ final
states. Due to the large Drell-Yan background to the $\tau_{\mu}\tau_{\mu}$
final state, requirements are placed on the visible mass, transverse
separation between the muons and ${{\not\mathrel{E}}_{T}}$, the muon
$p_{\mathrm{T}}$ asymmetry, and the impact parameter of the muons.

Unlike for ATLAS and CMS, missing energy cannot be measured within LHCb.
However, a high resolution vertex locator allows for strict requirements to be
placed on the $\tau$-lepton decay product impact parameters for the
$\tau_{\mu}\tau_{\mu}$, $\tau_{\mu}\tau_{h}$, and $\tau_{e}\tau_{h}$ final
states. Both a visible mass and transverse separation requirement is placed on
all final states, while an additional $p_{\mathrm{T}}$ asymmetry requirement
is also placed on the $\tau_{\mu}\tau_{\mu}$ final state.

###  2.3 Background Estimation

Drell-Yan production of lepton pairs is the primary background to the
$\tau_{\mu}\tau_{\mu}$ final state for all three experiments, as well as the
$\tau_{e}\tau_{h}$ final state for ATLAS and CMS. The Drell-Yan visible mass
shape is determined for ATLAS from simulation, while for both CMS and LHCb the
template is obtained with a reversed impact parameter requirement. For ATLAS
and LHCb the template is normalized to the on-shell $Z$ mass peak, and for CMS
normalized to an impact parameter side-band.

The QCD multijet background is large in the $\tau_{\mu}\tau_{h}$ and
$\tau_{e}\tau_{h}$ final states, and a visible mass shape is determined from
data for all three experiments by requiring candidates with same-sign charge.
The normalization is also taken from data, scaling the number of same-sign
events by the estimated opposite-sign/same-sign event ratio for the
background.

The $W$ with jets background mass shape is determined from simulation for all
three experiments and normalized using transverse mass side-bands for ATLAS
and CMS, and a same-sign side-band for LHCb. The $Z$ with jets visible mass
shape is taken from simulation for ATLAS and LHCb, and from a reversed impact
parameter requirement for CMS. The background is normalized using a visible
mass side-band for ATLAS, an impact parameter side-band for CMS, and a same-
sign side-band for LHCb.

For all three experiments the visible mass distributions for the $WW$ and
$t\bar{t}$ backgrounds are estimated from simulation. The normalization for
these backgrounds is also taken from simulation for ATLAS and LHCb, and from a
transverse mass side-band for CMS. Both the $WW$ and $t\bar{t}$ background
contributions are minimal for all final states.

###  2.4 Systematics

For both ATLAS and CMS the hadronic $\tau$-lepton identification efficiency
and energy scale is the primary systematic uncertainty for the
$\tau_{\mu}\tau_{h}$ and $\tau_{e}\tau_{h}$ final states, ranging from
$8\%-23\%$. In the $\tau_{\mu}\tau_{\mu}$ final state the primary uncertainty
is between $2\%-9\%$ from muon efficiency and acceptance, and for the
$\tau_{\mu}\tau_{e}$ final state is between $2\%-6\%$ from electron
efficiency.

For LHCb, the Drell-Yan background provides the largest systematic uncertainty
of $8\%$ to the $\tau_{\mu}\tau_{\mu}$ final state. Electron reconstruction
efficiency contributes the primary uncertainty to the $\tau_{\mu}\tau_{e}$,
$\tau_{e}\tau_{\mu}$, and $\tau_{e}\tau_{h}$ final states of $4\%$, while the
impact parameter selection efficiency provides a $2\%$ uncertainty to the
$\tau_{\mu}\tau_{h}$ final state.

###  2.5 Results

The measured cross-section, number of observed events, and number of
background events for each final state of all three experiments is given in
Table 1. Note the reduced statistics of the LHCb results, due to the
acceptance of the detector, but the enhanced purity of the
$\tau_{\mu}\tau_{\mu}$ final state.

The combined $Z\to{\tau\tau}$ cross-section measurement for each experiment is
given in Table 2, including the fiducial definition and predicted theory
result. The ATLAS and CMS theory predictions were calculated using Fewz [17],
while the LHCb prediction was calculated with Dynnlo [18]. The ATLAS combined
result does not include the $\tau_{\mu}\tau_{\mu}$ final state.

Exp. | $\sigma\pm\mathrm{stat.}\pm\mathrm{syst.}\pm\mathrm{lumi.}$ |  $\sigma$ theory
---|---|---
ATLAS | $0.92$ | $\pm$ | $0.02$ | $\pm$ | $0.08$ | $\pm$ | $0.03$ | $\mathrm{nb}$ | $0.96$ | $\pm$ | $0.05$ | $\mathrm{nb}$
$66<M_{\tau\tau}<116~{}\mathrm{GeV}$ | Fewz
CMS | $1.00$ | $\pm$ | $0.05$ | $\pm$ | $0.08$ | $\pm$ | $0.04$ | $\mathrm{nb}$ | $0.97$ | $\pm$ | $0.04$ | $\mathrm{nb}$
$60<M_{\tau\tau}<120~{}\mathrm{GeV}$ | Fewz
LHCb | $71.4$ | $\pm$ | $3.5$ | $\pm$ | $2.8$ | $\pm$ | $2.5$ | $\mathrm{pb}$ | $74.3$ | $\pm$ | $2.1$ | $\mathrm{pb}$
$p_{\mathrm{T}}^{\tau}>20~{}\mathrm{GeV},2.0<\eta^{\tau}<4.5$ | Dynnlo
$60<M_{\tau\tau}<120~{}\mathrm{GeV}$
Table 2: Combined $Z\to{\tau\tau}$ and theoretical cross-section results for
the three experiments. The ATLAS result does not include the
$\tau_{\mu}\tau_{\mu}$ final state.

##  3 $W\to\tau\nu_{\tau}$

Due to the large $W\to\ell\nu_{\ell}$ background to leptonically decaying
$\tau$-leptons produced from $W$ bosons, only hadronic decays of the
$\tau$-lepton are considered for the ATLAS and CMS $W\to\tau\nu_{\tau}$
analyses. QCD jets, $W\to\ell\nu_{\ell}$, $W\to\tau_{\ell}\nu_{\tau}$, and $Z$
with jets events provide the primary backgrounds to this signal. Of the three
experiments, only ATLAS has performed a $W\to\tau\nu_{\tau}$ measurement. An
observation has been made with CMS, but without a cross-section measurement.

###  3.1 Particle Selection

A $p_{\mathrm{T}}>12~{}\mathrm{GeV}$ trigger on the hadronic $\tau$-lepton
combined with a ${{\not\mathrel{E}}_{T}}>20~{}\mathrm{GeV}$ trigger is used to
select events for ATLAS, while a $p_{\mathrm{T}}>20~{}\mathrm{GeV}$ hadronic
$\tau$-lepton trigger and ${{\not\mathrel{E}}_{T}}>25~{}\mathrm{GeV}$ trigger
is used for CMS.

Hadronic $\tau$-leptons are selected for ATLAS using a boosted decision tree
based on the collimation, impact parameter, and lead track $p_{\mathrm{T}}$
over electromagnetic calorimeter energy of the hadronic $\tau$-lepton
candidate. For the CMS selection, requirements are placed on the hadronic
$\tau$-lepton lead track $p_{\mathrm{T}}$, isolation, and associated muon
hits.

###  3.2 Event Selection

To eliminate $W$ and $Z$ with jets backgrounds, events with high
$p_{\mathrm{T}}$ leptons outside the hadronic $\tau$-lepton jet are rejected
for both ATLAS and CMS. A large ${{\not\mathrel{E}}_{T}}$ is also required for
both to reduce the QCD jet background. Additionally, a high
${{\not\mathrel{E}}_{T}}$ significance and transverse separation between the
$\tau$-lepton and ${{\not\mathrel{E}}_{T}}$ is required for ATLAS, while the
ratio of the $\tau$-lepton jet $p_{\mathrm{T}}$ to the $p_{\mathrm{T}}$ of the
remaining jets is required to be large for CMS.

###  3.3 Background Estimation

The $W$ and $Z$ background transverse mass shapes for both ATLAS and CMS are
determined and normalized using simulation. The QCD jet background transverse
mass shape and normalization is determined using an $ABCD$ method for both.
For ATLAS, requirements are made on the ${{\not\mathrel{E}}_{T}}$ significance
and hadronic $\tau$-lepton identification, while requirements on the
${{\not\mathrel{E}}_{T}}$ and ratio of $\tau$-lepton jet $p_{\mathrm{T}}$ to
the $p_{\mathrm{T}}$ of the remaining jets are used for CMS.

###  3.4 Systematics

The primary systematic uncertainty for the ATLAS $W\to\tau_{h}\nu_{\tau}$
cross-section measurement is due to the identification efficiency for the
hadronic $\tau$-lepton jets and estimated to be $10\%$. For CMS, no cross-
section measurement was made, and so no uncertainty analysis is available.

###  3.5 Results

The number of observed $W\to\tau\nu_{\tau}$ events and background events is
given in Table 1 for both ATLAS and CMS, as well as the measured ATLAS cross-
section. To determine the total cross-section, the ATLAS
$W\to\tau_{h}\nu_{\tau}$ cross-section is extrapolated from the measured
fiducial region to full acceptance, and divided by the experimentally known
branching fraction for $\tau$-leptons to hadrons. The result given in Table 1
agrees well with the theoretically predicted value of $10.5\pm
0.5~{}\mathrm{nb}$ calculated using Fewz.

##  4 Conclusion

A comparison of the measured cross-sections of Table 1 and Table 2 divided by
their predicted theoretical values is given in Figure 1. The red points
represent combined results and the black points individual final states. The
dark error bars correspond to statistical uncertainty, while the light error
bars correspond the combined systematic uncertainty and uncertainty due to the
integrated luminosity. The light yellow line represents a ratio of unity,
while the dark yellow band indicates the theoretical uncertainty of the
prediction.

There is good agreement between all measured cross-sections and their
theoretical values. Currently, ATLAS provides the most statistics for the
combined $Z\to{\tau\tau}$ cross-section measurement, while LHCb yields the
most precise cross-section measurement. The ATLAS $W\to\tau\nu_{\tau}$
measurement has a large systematic uncertainty and does not allow for a more
precise test of the $W\to\tau\nu_{\tau}$ to $W\to\mu\nu_{\mu}$ ratio measured
at LEP.

Figure 1: Ratios of the experimental cross-section measurements of Table 1 and
Table 2 to their expected theoretical values. The combined results are given
in red and the individual final states in black. The dark error bars are the
statistical uncertainty, while the light error bars are the combined
systematic uncertainty and uncertainty due to the integrated luminosity. The
dark yellow band indicates the theoretical uncertainty centered about the
light yellow line.

## References

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CC_Attribution_4.0_International

#  High-Pressure Autoignition of Binary Blends of Methanol and Dimethyl Ether

Hongfu Wang  School of Mechanical and Electrical Engineering, Nanchang
University, Jiangxi Province, P.R. China  Department of Mechanical
Engineering, University of Connecticut, Storrs, CT, USA  Bryan W. Weber
Ruozhou Fang  Department of Mechanical Engineering, University of Connecticut,
Storrs, CT, USA  Chih-Jen Sung  Department of Mechanical Engineering,
University of Connecticut, Storrs, CT, USA<EMAIL_ADDRESS>
###### Abstract

Reactivity Controlled Compression Ignition (RCCI) is a new advanced engine
concept that uses a dual fuel mode of operation to achieve significant
improvements in fuel economy and emissions output. The fuels that are
typically used in this mode include a low- and a high-reactivity fuel in
varying proportions to control ignition timing. As such, understanding the
interaction effects during autoignition of binary fuel blends is critical to
optimizing these RCCI engines. In this work, we measure the autoignition
delays of binary blends of dimethyl ether (C2H6O, DME) and methanol (CH4O,
MeOH) in a rapid compression machine. In these experiments, dimethyl ether and
methanol function as the high- and low-reactivity fuels, respectively. We
considered five fuel blends at varying blending ratios (by mole), including
$100\text{\,}\mathrm{\char 37\relax}$ DME-$0\text{\,}\mathrm{\char 37\relax}$
MeOH, $75\text{\,}\mathrm{\char 37\relax}$ DME-$25\text{\,}\mathrm{\char
37\relax}$ MeOH, $50\text{\,}\mathrm{\char 37\relax}$
DME-$50\text{\,}\mathrm{\char 37\relax}$ MeOH, $25\text{\,}\mathrm{\char
37\relax}$ DME-$75\text{\,}\mathrm{\char 37\relax}$ MeOH, and
$0\text{\,}\mathrm{\char 37\relax}$ DME-$100\text{\,}\mathrm{\char 37\relax}$
MeOH. Experiments are conducted at an engine-relevant pressure of
$30\text{\,}\mathrm{bar}$, for the stoichiometric equivalence ratio. In
addition, the experimental results are compared with simulations using a
chemical kinetic model for DME/MeOH combustion generated by merging
independent, well-validated models for DME and MeOH.

###### keywords:

chemical kineticsrapid compression machinebinary fuel blendsadvanced engines

##  1 Introduction

To reduce the environmental impact of combustion, future combustion processes
must feature substantially reduced pollutant emissions while maintaining high
efficiency. A promising concept in this respect is low-temperature combustion
(LTC). As an outstanding representative of LTC technologies, the dual-fuel
Reactivity Controlled Compression Ignition (RCCI) operation has great
potential in terms of combustion controllability. The general principle of
dual-fuel RCCI combustion requires two fuels with different reactivities, that
is using a high reactivity fuel (such as diesel or dimethyl ether (DME)) to
trigger the ignition and combustion of low-reactivity fuels (such as gasoline,
methanol, ethanol, or butanol).

DME is considered an efficient alternative fuel for use in diesel engines
because it has excellent autoignition characteristics. The low boiling point
($248\text{\,}\mathrm{K}$), low critical point ($400\text{\,}\mathrm{K}$), and
high cetane number ($>55$) of DME [1, 2] make it well suited for compression
ignition engines. In addition, the high oxygen content of DME
($34.8\text{\,}\mathrm{\char 37\relax}$ by mass) together with the absence of
C-C bonds contributes to ultra-low soot formation during DME combustion [1].

The promoting effect of DME blending on fuels with poor autoignition qualities
such as methane and propane is a promising feature with respect to the RCCI
concept. Therefore, some research has been conducted to reveal the promoting
potential of DME. Several researchers [3, 4, 5] have studied the ignition
delays of DME/methane blends in shock tubes. The results have shown that DME
has a strong promoting effect on the autoignition of methane, even when the
concentration of methane is much higher than that of DME. Further, [6] showed
the promoting effect of DME in DME/propane blends using a rapid compression
machine. The results of the work of [6] showed that propane combustion is
promoted due to the large amount of radicals produced by low-temperature DME
oxidation. These studies indicate the potential of DME as a combustion
promoter together with fuels with poor auto-ignition qualities.

For dual-fuel RCCI operation, a lower reactivity fuel should be studied in
combination with the high reactivity fuel. Methanol (MeOH) is well known as a
widely used alcoholic alternative fuel, but when operated in a single fuel
mode it tends to have poor autoignition quality [7]. Given the great potential
of DME as a combustion promoter, we expect that DME can significantly improve
the auto-ignition quality of methanol and the combination will be effective
for dual-fuel RCCI operation.

In this work, we explore the autoignition characteristics of DME/MeOH binary
blends. A set of ignition delay time data for DME/MeOH blends at different
blending ratios over a wide range of temperature at engine relevant pressure
conditions and the stoichiometric equivalence ratio is obtained in a rapid
compression machine (RCM). In addition, we compile a chemical kinetic model
for DME/MeOH combustion by merging independent models for the fuels.
Simulations utilizing this model are compared to experimental results and good
agreement is observed over the range of the experiments.

##  2 Experimental Methods

The RCM used in this study is a single piston arrangement and is pneumatically
driven and hydraulically stopped. The device has been described in detail
previously [8] and will be described here briefly for reference. The end of
compression (EOC) temperature and pressure ($T_{C}$ and $P_{C}$ respectively),
are independently changed by varying the overall compression ratio, initial
pressure, and initial temperature of the experiments. The primary diagnostic
on the RCM is the in-cylinder pressure. The pressure data is processed by a
Python package called UConnRCMPy [9], which calculates $P_{C}$, $T_{C}$, and
the ignition delay(s). The definition of the ignition delay is shown in Fig.
1. The time of the EOC is defined as the maximum of the pressure trace prior
to the start of ignition and the ignition delay is defined as the time from
the EOC until local maxima in the first time derivative of the pressure.

In addition to the reactive experiments, non-reactive experiments are
conducted to determine the influence of machine-specific behavior on the
experimental conditions and permit the calculation of the EOC temperature via
the isentropic relations between pressure and temperature [10]. The EOC
temperature is calculated by the procedure described in Section 3.1.

$\displaystyle-10$$\displaystyle 0$$\displaystyle 10$$\displaystyle
20$$\displaystyle 30$$\displaystyle 40$Time, ms$\displaystyle 0$$\displaystyle
10$$\displaystyle 20$$\displaystyle 30$$\displaystyle 40$$\displaystyle
50$$\displaystyle 60$Pressure,
bar$\displaystyle\tau$EOC$\displaystyle-0.25$$\displaystyle
0.00$$\displaystyle 0.25$$\displaystyle 0.50$$\displaystyle
0.75$$\displaystyle 1.00$$\displaystyle 1.25$$\displaystyle
1.50$$\displaystyle 1.75$$\displaystyle 2.00$Time Derivative of Pressure,
bar/msPressureNon-ReactivePressureTime Derivative

Figure 1: Definition of the ignition delay used in this work. The experiment
in this figure was conducted for a $25\text{\,}\mathrm{\char 37\relax}$ DME
blend with $P_{0}=$1.1528\text{\,}\mathrm{bar}$$,
$T_{0}=$338\text{\,}\mathrm{K}$$, $P_{C}=$29.98\text{\,}\mathrm{bar}$$,
$T_{C}=$748\text{\,}\mathrm{K}$$, and $\tau=$35.39\pm
1.93\text{\,}\mathrm{ms}$$.

The RCM is equipped with heaters to control the initial temperature of the
mixture. After filling in the components to the mixing tanks, the heaters are
switched on and the system is allowed $1.5\text{\,}\mathrm{h}$ to come to
steady state. The mixing tanks are also equipped with magnetic stir bars so
the reactants are well mixed for the duration of the experiments.

The mixtures considered in this study are shown in Table 1. The
“$\mathrm{\char 37\relax}$ DME” and “$\mathrm{\char 37\relax}$ MeOH” columns
indicate the molar percent of each component in the fuel blend. Mixtures are
prepared in stainless steel mixing tanks. The proportions of reactants in the
mixture are determined by specifying the absolute mass of the methanol in the
mixture (if present), the equivalence ratio, the oxidizer composition (in this
study, O2 and N2 in the ratio of $1:3.76$ are used throughout), and the molar
ratio of DME/MeOH in the fuel blend. Since MeOH is a liquid at room
temperature and pressure, it is injected into the mixing tank through a
septum. Proportions of DME, O2, and N2 are added manometrically at room
temperature.

Table 1: Mixtures considered in this work |  | Mole Fraction (purity)
---|---|---
$\mathrm{\char 37\relax}$ DME |  $\mathrm{\char 37\relax}$ MeOH | DME ($99.7\text{\,}\mathrm{\char 37\relax}$) | MeOH ($100.00\text{\,}\mathrm{\char 37\relax}$) |  O2 ($99.994\text{\,}\mathrm{\char 37\relax}$) |  N2 ($99.999\text{\,}\mathrm{\char 37\relax}$)
$100$ | $0$ | 0.0654 | 0.0000 | 0.1963 | 0.7383
$75$ | $25$ | 0.0556 | 0.0185 | 0.1945 | 0.7314
$50$ | $50$ | 0.0427 | 0.0427 | 0.1921 | 0.7225
$25$ | $75$ | 0.0252 | 0.0756 | 0.1889 | 0.7103
$0$ | $100$ | 0.0000 | 0.1229 | 0.1843 | 0.6928

##  3 Computational Methods

To the best of our knowledge, there are no chemical kinetic models for the
combustion of binary blends of DME and MeOH available in the literature.
Therefore, we compile a kinetic model in this work by combining two
independent models. The kinetics for DME are taken from the work of [3] while
the kinetics for MeOH are taken from the work of [11].

To combine the two models, duplicate reactions and species were taken from the
[3] model; however, the models were produced by the same research group
approximately one year apart, so we do not expect many differences in the
common chemistry. In the work of [6], it was found that for combined models of
high-reactivity fuels such as DME and low-reactivity fuels such as propane,
cross-reactions between the fuels do not strongly affect the ignition delay
and the fuels instead interact through radical species such as OH. Therefore,
we do not consider any cross-reactions between the high- and low-reactivity
fuels in this study (DME and MeOH, respectively).

###  3.1 RCM Modeling

All of the simulations in this work use the Python interface for Cantera 2.3.0
[12]. Two types of simulations are considered. The first is used to calculate
$T_{C}$. Detailed descriptions of the use of Cantera for these simulations can
be found in the work of [13] and [6]; a brief overview is given here. As
mentioned in Section 2, non-reactive experiments are conducted to characterize
the machine-specific effects on the experimental conditions in the RCM. This
pressure trace is used to compute a volume trace by assuming that the
reactants undergo a reversible, adiabatic, constant composition (i.e.,
isentropic) compression during the compression stroke and an isentropic
expansion after the EOC. The volume trace is applied to a non-reactive
simulation conducted in an `IdealGasReactor` in Cantera [12] and the
temperature at the end of compression is reported as $T_{C}$.

The second type of simulation uses a constant-volume, adiabatic reactor. This
method does not consider the effect of the compression stroke and post-
compression heat loss present in the experiments and the initial conditions in
the simulation are set equal to the EOC conditions in the experiment. The
ignition delay is defined as the time required for the simulated temperature
to increase by $400\text{\,}\mathrm{K}$ over the initial temperature in the
simulation.

##  4 Results and Discussion

Ignition delay results for the mixtures listed in Table 1 are shown in Fig. 3
for the stoichiometric equivalence ratio and $P_{C}=$30\text{\
}\mathrm{bar}$$. In the following, we use the shorthand of specifying the
molar percent of DME in the fuel blend to indicate the blending condition.

It can be seen in Fig. 3 that the $100\text{\,}\mathrm{\char 37\relax}$ DME
case ($0\text{\,}\mathrm{\char 37\relax}$ MeOH) is the most reactive while the
$0\text{\,}\mathrm{\char 37\relax}$ DME case ($100\text{\,}\mathrm{\char
37\relax}$ MeOH) is the least reactive. Interestingly, the change in
reactivity as MeOH is added to DME appears to be non-linear with respect to
the molar percent of MeOH added. In other words, the change in ignition delay
at a fixed $T_{C}$ is smaller going from $100\text{\,}\mathrm{\char 37\relax}$
DME to $50\text{\,}\mathrm{\char 37\relax}$ DME than going from
$50\text{\,}\mathrm{\char 37\relax}$ DME to $0\text{\,}\mathrm{\char
37\relax}$ DME.

This is also demonstrated by Fig. 3, which shows the $T_{C}$ values for
ignition delays near $20\text{\,}\mathrm{ms}$ at the range of mixtures
considered in this study. As the $\mathrm{\char 37\relax}$ DME decreases in
the blend, the temperature required to achieve the same ignition delay
increases. However, the temperature increase from $100\text{\,}\mathrm{\char
37\relax}$ to $50\text{\,}\mathrm{\char 37\relax}$ DME is much smaller than
the increase from $50\text{\,}\mathrm{\char 37\relax}$ to
$0\text{\,}\mathrm{\char 37\relax}$ DME.

$\displaystyle 1.1$$\displaystyle 1.2$$\displaystyle 1.3$$\displaystyle
1.4$$\displaystyle 1.5$$\displaystyle 1.6$$\displaystyle 1.7$$\displaystyle
1000/T_{C}$, 1/K110100Ignition Delay, ms100% DME75% DME50% DME25% DME0% DME900
K800 K700 K600 K

Figure 2: Ignition delays of blends of DME and MeOH as a function of inverse
temperature, for an equivalence ratio of $\phi=1.0$ and $P_{C}=$30\text{\
}\mathrm{bar}$$. Constant volume, adiabatic simulations are shown as the solid
lines.

$\displaystyle 0$$\displaystyle 20$$\displaystyle 40$$\displaystyle
60$$\displaystyle 80$$\displaystyle 100$% DME$\displaystyle 600$$\displaystyle
650$$\displaystyle 700$$\displaystyle 750$$\displaystyle 800$$\displaystyle
850$$\displaystyle T_{C}$, K

Figure 3: $T_{C}$ values for ignition delays near $20\text{\,}\mathrm{ms}$ at
the range of blends considered in this study

Also shown on Fig. 3 are constant volume, adiabatic simulations computed
according to the procedure laid out in Section 3.1. In general, the agreement
between the model and the experiments is quite good over the entire range of
the experiments. It can be seen in Fig. 3 that at low temperatures for a given
mixture composition the ignition delay tends to be under-predicted, while at
the higher temperatures the ignition delay is over-predicted.

As discussed by [14], this is likely due in part to the modeling procedure
used in this work. In general, we expect constant volume simulations to have
shorter ignition delays than the experiments for long ignition delays because
they do not include the effect of post-compression heat loss; conducting
simulations that include the post-compression heat loss are very likely to
improve agreement in this region. Furthermore, for short ignition delays, we
expect constant volume simulations to over-predict the experimental ignition
delay because they do not include the effect of radical pool buildup during
the compression stroke. Therefore, conducting simulations that include the
compression stroke are very likely to improve the agreement for short ignition
delays.

##  5 Conclusions

In this study, we have measured ignition delays for binary blends of dimethyl
ether and methanol for engine-relevant pressure, temperature, and equivalence
ratio conditions using a heated rapid compression machine. The ignition delay
results show that pure DME is more reactive than pure MeOH, and that the
increase in ignition delay as DME is replaced by MeOH is non-linear as a
function of the blending fraction. The ignition delays are also compared to a
chemical kinetic model compiled by combining independent models for the two
fuels. This model does not consider cross reactions between DME and MeOH.
Nonetheless, the model gives quite good agreement with the data, supporting
the hypothesis that the fuels do not interact via cross reactions but instead
through common radicals such as OH. In addition, this further demonstrates
that models for low-reactivity fuels such as methanol and high-reactivity
fuels such as DME can be constructed by simple concatenation and deduplication
of their respective independent models.

##  6 Acknowledgements

This work was supported by the National Science Foundation under Grant No.
CBET-1402231.

## References

  * [1] Constantine Arcoumanis, Choongsik Bae, Roy Crookes and Eiji Kinoshita  “The Potential of Di-Methyl Ether (DME) as an Alternative Fuel for Compression-Ignition Engines: A Review”  In _Fuel_ 87.7, 2008, pp. 1014–1030  DOI: 10.1016/j.fuel.2007.06.007
  * [2] Ho Teng, James C. Mccandless and Jeffrey B. Schneyer  “Thermochemical Characteristics of Dimethyl Ether - An Alternative Fuel for Compression-Ignition Engines”, 2001  DOI: 10.4271/2001-01-0154
  * [3] Ultan Burke, Kieran P. Somers, Peter O’Toole, Chis M. Zinner, Nicolas Marquet, Gilles Bourque, Eric L. Petersen, Wayne K. Metcalfe, Zeynep Serinyel and Henry J. Curran  “An Ignition Delay and Kinetic Modeling Study of Methane, Dimethyl Ether, and Their Mixtures at High Pressures”  In _Combust. Flame_ 162.2, 2015, pp. 315–330  DOI: 10.1016/j.combustflame.2014.08.014
  * [4] Chenglong Tang, Liangjie Wei, Jiaxiang Zhang, Xingjia Man and Zuohua Huang  “Shock Tube Measurements and Kinetic Investigation on the Ignition Delay Times of Methane/Dimethyl Ether Mixtures”  In _Energy Fuels_ 26.11, 2012, pp. 6720–6728  DOI: 10.1021/ef301339m
  * [5] Zheng Chen, Xiao Qin, Yiguang Ju, Zhenwei Zhao, Marcos Chaos and Frederick L. Dryer  “High Temperature Ignition and Combustion Enhancement by Dimethyl Ether Addition to Methane–air Mixtures”  In _Proc. Combust. Inst._ 31.1, 2007, pp. 1215–1222  DOI: 10.1016/j.proci.2006.07.177
  * [6] Enoch E. Dames, Andrew S. Rosen, Bryan W. Weber, Connie W. Gao, Chih-Jen Sung and William H. Green  “A Detailed Combined Experimental and Theoretical Study on Dimethyl Ether/Propane Blended Oxidation”  In _Combust. Flame_ 168, 2016, pp. 310–330  DOI: 10.1016/j.combustflame.2016.02.021
  * [7] Dennis L. Siebers and C.. Edwards  “Autoignition of Methanol and Ethanol Sprays under Diesel Engine Conditions”, 1987  DOI: 10.4271/870588
  * [8] Gaurav Mittal and Chih-Jen Sung  “A Rapid Compression Machine for Chemical Kinetics Studies at Elevated Pressures and Temperatures”  In _Combust. Sci. Technol._ 179.3, 2007, pp. 497–530  DOI: 10.1080/00102200600671898
  * [9] Bryan William Weber, Ruozhou Fang and Chih-Jen Sung  “UConnRCMPy”, 2017  DOI: 10.5281/zenodo.269678
  * [10] Daeyup Lee and Simone Hochgreb  “Rapid Compression Machines: Heat Transfer and Suppression of Corner Vortex”  In _Combust. Flame_ 114, 1998, pp. 531–545  DOI: 10.1016/S0010-2180(97)00327-1
  * [11] Ultan Burke, Wayne K. Metcalfe, Sinead M. Burke, K. Heufer, Philippe Dagaut and Henry J. Curran  “A Detailed Chemical Kinetic Modeling, Ignition Delay Time and Jet-Stirred Reactor Study of Methanol Oxidation”  In _Combust. Flame_ 165, 2016, pp. 125–136  DOI: 10.1016/j.combustflame.2015.11.004
  * [12] David G. Goodwin, Harry K. Moffat and Raymond L. Speth  “Cantera: An Object-Oriented Software Toolkit for Chemical Kinetics, Thermodynamics, and Transport Processes”, 2017  DOI: 10.5281/zenodo.170284
  * [13] Bryan William Weber and Chih-Jen Sung  “UConnRCMPy: Python-Based Data Analysis for Rapid Compression Machines”  In _Proceedings of the 15th Python in Science Conference_ , 2016, pp. 36–44
  * [14] Gaurav Mittal, Marcos Chaos, Chih-Jen Sung and Frederick L. Dryer  “Dimethyl Ether Autoignition in a Rapid Compression Machine: Experiments and Chemical Kinetic Modeling”  In _Fuel Process. Technol._ 89.12, 2008, pp. 1244–1254  DOI: 10.1016/j.fuproc.2008.05.021


11institutetext: Centre for mathematical Plasma-Astrophysics, Department of
Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium
22institutetext: Solar-Terrestrial Centre of Excellence—SIDC, Royal
Observatory of Belgium, 1180 Brussels, Belgium
33institutetext: Rays of Space, Finland-Belgium
44institutetext: Institute of Physics, University of Maria Curie-Skłodowska,
ul. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland

# Updates on the FRi3D CME model in EUHFORIA

A. Maharana 1122 K. Plets 11 A. Isavnin 33 S. Poedts 1144

A magnetised flux rope model, ”Flux Rope in 3D” (FRi3D) is used in the
framework of European Heliospheric Forecasting Information Asset (EUHFORIA)
for studying the evolution and propagation of coronal mass ejections (CME). In
this paper, we rectify the mistake in the mentioned magnetic field profile of
the FRi3D model used in Maharana et al. (2022), and we clarify the actual
profile used in that work. In addition, we provide the recent updates
introduced to the FRi3D implementation in EUHFORIA like optimising the
”ai.fri3d” python package to reduce computational time and exploring different
CME leg disconnection methods to make the numerical implementation more
stable.

##  1 A non-Lundquist formula

It was reported in Maharana et al. (2022) that the FRi3D model implemented in
EUHFORIA followed the Lundquist (Lundquist, 1951) magnetic field profile from
ai.fri3d package as follows:

$B_{\rho}=0\text{,}\qquad
B_{\varphi}=B_{0}J_{1}(\alpha\rho),\qquad\text{and}\qquad
B_{z}=B_{0}J_{0}(\alpha\rho),\qquad$ (1)

where $\rho$ is the poloidal distance from the axis in cylindrical
coordinates, $B_{0}$ is the strength of the core field, $J_{0}$ and $J_{1}$
are the Bessel functions of first and second order, respectively, and
$\alpha\rho$ gives the first zero of $J_{0}$ at the edge of the flux rope
($\alpha$ is a free parameter).

However, we noticed that the python package was already updated in 2020 to
follow a bi-variate normal distribution, instead of the mentioned Lundquist
configuration at the time the work for Maharana et al. (2022) was carried out.
In this paper, we clarify the details of the updated FRi3D magnetic field
configuration. The bi-variate distribution is able to reproduce the in situ
magnetic field profiles in the observed CME events shown in Maharana et al.
(2022, 2023); Palmerio et al. (2023).

Table 1: Notation of physical quantities used in the lib.pyx code in ai.fri3d package. Code variable | Symbol | Physical quantity
---|---|---
xx[0] | $r$ | radial distance
xx[1] | $\theta$ | polar angle
xx[2] | $\phi$ | toroidal axis angle
xx[3] | $R_{t}$ | toroidal height
xx[4] | $\varphi_{\text{hw}}$ | half-width
xx[5] | $\varphi_{\text{hh}}$ | half-height
xx[6] | $n$ | flattening
xx[7] | $\theta_{p}$ | pancaking coefficient
xx[8] | $\tau$ | twist
xx[9] | $\sigma$ | Gaussian standard deviation
xx[10] | $X$ | $\int_{-\varphi_{\text{hw}}}^{\varphi_{\text{hw}}}R_{t}\cos^{n}(\frac{\pi\phi}{2\varphi_{\text{hw}}})d\phi$

The magnetic field does not follow Lundquist as reported in Maharana et al.
(2022), but rather is distributed along a bi-variate normal distribution. The
distribution is bound by the boundaries of the geometrical shell defined for
the CME. The direction of the magnetic field $\mathbf{B}$ is along the
magnetic axis line. The magnetic field is primarily defined as:

$B(r)=B_{0}\,\exp\Bigg{(}-\frac{r^{2}}{2\sigma^{2}}\Bigg{)}=B_{0}\,\exp\Bigg{(}-\frac{x^{2}}{2\sigma^{2}}-\frac{y^{2}}{2\sigma^{2}}\Bigg{)},$
(2)

where $x=r\cos(\theta)$ and $y=r\sin(\theta)$ considering at a particular
$\phi$, $B_{0}$ is the axial magnetic field strength, and $\sigma$ is the
standard deviation coefficient of the Gaussian distribution of the total
magnetic field in the cross-section of the CME, which is set to a default
value of 2. The description of the notation of various symbols is given in
Table 1. The magnetic field strength ($|B|$) at any point in the flux rope is
given by:

$\displaystyle\frac{|B|}{B_{0}}$
$\displaystyle=\exp\left(-\frac{(r\cos{\theta})/r_{y})^{2}}{2\sigma^{2}}-\frac{(r\sin{\theta})/r_{y})^{2}}{2\sigma^{2}}\right)\times\sin\left(\tan^{-1}\left(s^{\prime}(\phi)X\text{
,
}\frac{R^{2}(\phi)2\pi\tau}{1-(1-\theta_{p})\sin(\theta)}\right)\right)=f_{1}\times
f_{2},$ (3)

where

$r_{y}=R(\phi)\tan(\varphi_{\text{hh}})=R(\phi)R_{p}/R_{t}=R_{p}cos^{n}{(\frac{\pi}{2}{\phi}/\varphi_{\text{hw}})},$
$s^{\prime}(\phi)=\sqrt{R(\phi)^{2}+(R^{\prime}(\phi))^{2}},$
$R(\phi)=R_{t}\cos^{n}{(\frac{\pi}{2}{\phi}/\varphi_{\text{hw}})},$

and

$X=\int_{-\varphi_{\text{hw}}}^{\varphi_{\text{hw}}}R_{t}\cos^{n}(\frac{\pi\phi}{2\varphi_{\text{hw}}})d\phi.$

In the implementation file, $r$ is normalised with $r_{y}$, which incorporates
the toroidal/longitudinal variability in the magnetic field distribution. The
above factors can be simplified as follows:

$\displaystyle f_{1}$
$\displaystyle=\exp\left(-\frac{1}{2}\left(\frac{r}{r_{y}\sigma}\right)^{2}\right),$
(4) $\displaystyle f_{2}$
$\displaystyle=\sin(\tan^{-1}(A,B))=A/\sqrt{A^{2}+B^{2}},$ (5)

where $A(\phi)=s^{\prime}(\phi)X$, and
$B(\theta,\phi)=\frac{R^{2}(\phi)2\pi\tau}{1-(1-\theta_{p})\sin(\theta)}$.
Here, $f_{1}$ and $f_{2}$ incorporate the tapering and bending the magnetic
axis as per the geometry of the shell.

##  2 Optimisation of calculations

We profiled the ai.fri3d package to find the bottlenecks in the code. Then we
simplified some computations and discarded some of the repeating computations
to optimise the code.

Figure 1: EUHFORIA simulated magnetic field and plasma properties, depicted
against the corresponding measurements recorded by the Wind spacecraft. The
panels show, from top to bottom: Magnetic field magnitude, magnetic field
components in Cartesian coordinates, solar wind bulk speed, proton density,
proton temperature, plasma beta, and Dst index (WDC Kyoto data is shown in red
and the black profile displays the modeled Dst based on Wind in-situ
measurements using the algorithm of O’Brien & McPherron, 2000). The observed
arrival time of the shock is marked with the solid vertical line, and the
leading and trailing edges of the ICME ejecta are marked with dashed vertical
lines.

We first simplify the computation of the axial magnetic field strength
($B_{0}$) which is defined as a function of the longitude $\phi$:

$\displaystyle B_{0}(\phi)$
$\displaystyle=\Phi\left/\int_{0}^{2\pi}d\theta\int_{0}^{\rho(\theta,\phi)}dr^{\prime}r^{\prime}\exp\left(-\frac{1}{2}\left(\frac{r^{\prime}}{r_{y}\sigma}\right)^{2}\right)\frac{A}{\sqrt{A^{2}+B^{2}}}\right.,$
(6)
$\displaystyle=\Phi\left/\int_{0}^{2\pi}d\theta\frac{A}{\sqrt{A^{2}+B^{2}}}\int_{0}^{\rho(\theta,\phi)}dr^{\prime}r^{\prime}\exp\left(-\frac{1}{2}\left(\frac{r^{\prime}}{r_{y}\sigma}\right)^{2}\right)\right.,$
(7)

where $\Phi$ is the total magnetic flux constrained from the observations,

$\rho(\theta,\phi)=r_{y}^{2}\alpha_{p}\left/\sqrt{r_{y}^{2}\cos^{2}(\theta)+r_{y}^{2}\alpha_{p}^{2}\sin^{2}(\theta)}=r_{y}\alpha_{p}\right/\sqrt{1-(1-\alpha_{p}^{2})\sin^{2}(\theta)}=\alpha_{p}R_{p}\frac{\cos^{n}(\frac{\pi\phi}{2\varphi_{\text{hw}}})}{\sqrt{1-(1-\alpha_{p}^{2})\sin^{2}(\theta)}},$

and

$\alpha_{p}=1-\left[(1-\theta_{p})\left/\sqrt{1+\left(\frac{n\pi\tan(\frac{\pi\phi}{2\varphi_{\text{hw}}})}{2\varphi_{\text{hw}}}\right)^{2}}\right.\right].$

Here, $\rho(\theta,\phi)$ is the radial coordinate in the local FRi3D
coordinates, and $\alpha_{p}$ is a longitudinal implementation of pancaking
($\theta_{p}$) in the flux rope. Finally, $B_{0}$ is multiplied with the unit
position vector of the magnetic axis at each point to obtain the magnetic
field vector. First, we reduced the 2D integral in the computation of $B_{0}$
into a 1D integral by analytically integrating the Gaussian function:

$\displaystyle B_{0}(\phi)$
$\displaystyle=\Phi\left/\int_{0}^{2\pi}d\theta\frac{(r_{y}\sigma)^{2}A\left[1-\exp\left(-\frac{1}{2}\left(\frac{\rho(\theta,\phi)}{r_{y}\sigma}\right)^{2}\right)\right]}{\sqrt{A^{2}+B^{2}}}\right.,$
(8)
$\displaystyle=\frac{\Phi}{s^{\prime}(\phi)X(1-(1-\theta_{p})(r_{y}\sigma)^{2}}\left/\int_{0}^{2\pi}d\theta\sin(\theta)\frac{1-\exp\left(-\frac{1}{2}\left(\frac{\rho(\theta,\phi)}{r_{y}\sigma}\right)^{2}\right)}{\sqrt{\left(R^{2}(\phi)2\pi\tau\right)^{2}+\left[s^{\prime}(\phi)X(1-(1-\theta_{p})\sin(\theta))\right]^{2}}}\right..$
(9)

This step contributed significantly in reducing the computation time in
EUHFORIA simulations as the double integral had to be computed at at every
point in the CME cross-section through the inner boundary throughout its
injection.

The second optimisation was achieved by defining a separate function
‘integralphi’ to compute the coordinates of the magnetic field axis. This
method replaced the repeated computation of the integral in $X$ (mentioned in
Section 1) in the functions to compute the twist or axial magnetic field
required for the computation of magnetic field components. Instead, the result
of the integral was efficiently used in the above-mentioned functions. This
step contributed significantly in reducing the computation time in EUHFORIA
simulations as the integral had to be computed at multiple times while
checking the position of grid points on the EUHFORIA inner boundary with
respect to the FRi3D magnetic axis. These optimisations are implemented in the
latest version of ai.fri3d package version 0.0.17
(https://pypi.org/project/ai.fri3d/). Using the updated package to obtain the
magnetic field configuration at a particular point takes almost 60$\%$ less
time. The EUHFORIA simulation with the optimised version takes $\sim 55\%$
less time for the case shown in Maharana et al. (2022), and we obtain the
nearly the same CME arrival time and magnetic field profiles at Earth, as
shown in Fig.1 and discussed below.

##  3 Updates on the FRi3D implementation in EUHFORIA

###  3.1 CME leg disconnection

The FRi3D CME leg disconnection follows the process in which the solar wind
plasma and magnetic field variables are reassigned at the inner boundary of
the EUHFORIA heliosphere within the cross-section where the CME was injected.
In Maharana et al. (2022), the CME legs were disconnected point by point when
the CME speed dropped below the ambient solar wind speed in the grid points
where the CME was injected. We introduced a new leg disconnection method whose
numerical implementation consumes less computational time and is more stable.
The flux rope was fully disconnected from the boundary (i.e., all points in
the flux rope cross-section were replaced with solar wind properties at the
same time) when the magnetic axis of the CME reached 1 au. The updated method
has been used in Maharana et al. (2023); Palmerio et al. (2023) to obtain
satisfactory results.

###  3.2 Addition of skew parameter in EUHFORIA implementation

We included the skew parameter of the FRi3D model in the framework of
EUHFORIA. The implementation was tested to model the propagation of a flux
rope resulting from an extended asymmetric filament eruption as presented in
Lynch et al. (2021). The results of EUHFORIA simulation with FRi3D model are
reported in Palmerio et al. (2023). The FRi3D model performed better than the
CME models with spherical/spheroidal geometry in predicting CME arrival time
and magnetic field upon impact.

##  4 Discussion

The results of EUHFORIA simulations with different versions of FRi3D
implementation are shown in Fig.1. All the simulations were performed on 144
parallel processors of the Vlaams Supercomputer Centrum (VSC), a Belgian high-
performance computing facility111http://www.vscentrum.be. The time series
shown by the legend Maharana+2022 (blue profile) is the first ever
implementation of FRi3D in EUHFORIA, and consumed a wall-clock time of 24
hours to produce a forecast of 5 days. Updating the leg disconnection method,
the simulation was completed in 9 hours to produce a forecast of 7 days (green
profile). Adding the optimisations discussed in Section 2, the forecast of 7
days was achieved in only 4 hours and 9 minutes, that is $\sim 55\%$ less
computational time expenditure (magenta profile). The two updated versions of
FRi3D implementation are qualitatively similar to the first implementation in
Maharana et al. (2022), and are much more stable and less time consuming.

## References

  * Lundquist (1951) Lundquist, S. 1951, Physical Review, 83, 307
  * Lynch et al. (2021) Lynch, B. J., Palmerio, E., DeVore, C. R., et al. 2021, ApJ, 914, 39
  * Maharana et al. (2022) Maharana, A., Isavnin, A., Scolini, C., et al. 2022, Advances in Space Research, 70, 1641
  * Maharana et al. (2023) Maharana, A., Scolini, C., Schmieder, B., & Poedts, S. 2023, A&A, 675, A136
  * O’Brien & McPherron (2000) O’Brien, T. P. & McPherron, R. L. 2000, J. Geophys. Res., 105, 7707
  * Palmerio et al. (2023) Palmerio, E., Maharana, A., Lynch, B. J., et al. 2023, arXiv e-prints, arXiv:2310.05846


††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.††thanks: These authors contributed
equally to this work.

# Theory of $d+id$ Second-Order Topological Superconductors

Zi-Ming Wang  Department of Physics and Chongqing Key Laboratory for Strongly
Coupled Physics, Chongqing University, Chongqing 400044, China  Meng Zeng
Department of Physics, University of California, San Diego, California 92093,
USA  Chen Lu  New Cornerstone Science Laboratory, Department of Physics,
School of Science, Westlake University, Hangzhou 310024, Zhejiang, China  Da-
Shuai Ma  Department of Physics and Chongqing Key Laboratory for Strongly
Coupled Physics, Chongqing University, Chongqing 400044, China  Rui-Xing Zhang
Department of Physics and Astronomy, University of Tennessee, Knoxville,
Tennessee 37996, USA  Department of Materials Science and Engineering,
University of Tennessee, Knoxville, Tennessee 37996, USA  Lun-Hui Hu
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Tennessee, Knoxville, Tennessee 37996, USA  Center for Correlated Matter and
School of Physics, Zhejiang University, Hangzhou 310058, China  Department of
Applied Physics, Aalto University School of Science, FI-00076 Aalto, Finland
Dong-Hui Xu<EMAIL_ADDRESS>Department of Physics and Chongqing Key
Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing
400044, China

###### Abstract

Topological superconductors are a class of unconventional superconducting
materials featuring sub-gap zero-energy Majorana bound modes that hold promise
as a building block for topological quantum computing. In this work, we study
the realization of second-order topology that defines anomalous gapless
boundary modes in a two-orbital superconductor with spin-orbital couplings. We
reveal a time-reversal symmetry-breaking second-order topological
superconducting phase with $d+id$-wave orbital-dependent paring without the
need for the external magnetic field. Remarkably, this orbital-active $d$-wave
paring gives rise to anomalous zero-energy Majorana corner modes, which is in
contrast to conventional chiral $d$-wave pairing, accommodating one-
dimensional Majorana edge modes. Our work not only reveals a unique mechanism
of time-reversal symmetry breaking second-order topological superconductors
but also bridges the gap between second-order topology and orbital-dependent
pairings.

_Introduction._ — Topological superconductors (TSCs) are exotic quantum
condensed phases of matter with topologically nontrivial structures of Cooper
pair wavefunctions. As perhaps the most remarkable consequence of TSCs,
spatially localized Majorana zero modes (MZMs) can be trapped in vortex cores
of a two-dimensional (2D) $p$-wave TSC [1, 2, 3] or be formed at the ends of a
one-dimensional $p$-wave superconductor [4]. MZMs manifest non-Abelian quantum
statistics, which naturally encode topological qubits that pave the way for
fault-tolerant quantum computation [5, 6]. Although naturally occurring TSC
materials appear rare and elusive, the past few decades have witnessed a
tremendous effort to discover artificial topological superconductivity in
various quantum materials [7, 8, 9, 10], following the pioneering theories
[11, 12, 13]. So far, evidence of MZMs has been experimentally reported in
several systems, ranging from one-dimensional superconducting hybrids [14, 15,
16] to vortex cores on a proximitized topological insulator surface [17] or on
an iron-based superconductor surface [18].

The recent advances of topological band theory have unveiled an entirely new
category of “higher-order” TSCs with an unprecedented bulk-boundary relation
[19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56,
57, 58, 59, 60, 61, 62, 63]. For example, in two dimensions (2D), a second-
order TSC generally binds 0D MZMs around the geometric corners of a finite-
size system, which is in contrast with a “conventional” TSC. In pursuit of
corner MZMs, a crucial conceptual question to look for new simple and feasible
recipes that are applicable to real-world superconductors. Given the important
role of orbital degrees of freedom in unconventional superconducting systems,
a comprehension of whether multi-orbiral pairing can enable higher-order TSC
is certainly necessary but still largely incomplete [64, 65, 66, 67].

In this work, we show that orbital-active pairing can stabilize second-order
class-D TSC that is protected by a $C_{4}$ rotation symmetry. The topological
nature of the TSC phase is confirmed by numerically revealing the Majorana
corner modes, as well as a topological quantum chemistry analysis. Besides
sample corners, lattice dislocations are also found to trap MZMs as a result
of the inherent weak topology, which makes our system a viable platform for
designing and building Majorana qubits.

_Model of $d+id$ second-order TSCs and symmetry analysis._—We consider a
normal state two-orbital $\\{d_{xz},d_{yz}\\}$ tight-binding model on the
square lattice with both asymmetric SOC (i.e. Rashba type) and on-site SOC,

$\displaystyle H_{\text{n}}=$
$\displaystyle\sum_{\bm{k}}\Psi^{\dagger}\left(\bm{k}\right)\\{\epsilon\left(\bm{k}\right)\sigma_{0}s_{0}+\widetilde{\epsilon}\left(\bm{k}\right)\sigma_{z}s_{0}+\epsilon^{\prime\prime}\left(\bm{k}\right)\sigma_{x}s_{0}$
$\displaystyle+$
$\displaystyle\lambda_{\text{I}}\sigma_{y}s_{z}+\lambda_{\text{R}}\sin
k_{x}\sigma_{0}s_{y}-\lambda_{\text{R}}\sin
k_{y}\sigma_{0}s_{x}\\}\Psi\left(\bm{k}\right),$ (1)

where
$\Psi^{\dagger}\left(\bm{k}\right)=({c}^{\dagger}_{d_{xz},\uparrow},{c}^{\dagger}_{d_{xz},\downarrow},{c}^{\dagger}_{d_{yz},\uparrow},{c}^{\dagger}_{d_{yz},\downarrow})$,
$\epsilon_{\textbf{k}}=-2t\cos k_{x}-2t\cos k_{y}+4t-\mu$,
$\widetilde{\epsilon}\left(\bm{k}\right)=-2\widetilde{t}\cos
k_{x}+2\widetilde{t}\cos k_{y}$, and
$\epsilon^{\prime\prime}\left(\bm{k}\right)=2t^{\prime\prime}\sin k_{x}\sin
k_{y}$. ${\sigma}_{x,y,z}$ and ${s}_{x,y,z}$ are the Pauli matrices for the
orbital and spin degrees of freedom, $\sigma_{0}$ and $s_{0}$ are two $2\times
2$ identity matrices. $t$ describes the intra-orbital nearest-neighbor
hopping, $\widetilde{t}$ depicts the hopping anisotropy along the different
direction of $d_{xz}$, $d_{yz}$ orbitals and $t^{\prime\prime}$ is the inter-
orbital next-nearest-neighbor hopping. The $\lambda_{\text{I}}$ and
$\lambda_{\text{R}}$ are the strength of intrinsic and Rashba SOCs,
respectively. The intrinsic SOC originates from the atomic spin-orbit coupling
[68, 69, 70, 71], while Rashba SOC can be intrinsic or externally induced by
substrate effect. This normal Hamiltonian breaks inversion symmetry but
preserves TRS:
${\mathcal{T}}{H_{\text{n}}}\left(\bm{k}\right){\mathcal{T}}^{-1}={H_{\text{n}}}\left(\bm{-k}\right)$,
where $\mathcal{T}=i\sigma_{0}s_{y}\mathcal{K}$ with $\mathcal{K}$ the complex
conjugation operator. In addition, the normal Hamiltonian has the four-fold
rotation symmetry ${r}_{4z}=i\sigma_{y}e^{-i\pi s_{z}/4}$.

To study the superconductivity, we define the Nambu basis
$\left\\{\Psi^{\dagger}\left(\bm{k}\right),\Psi^{T}\left(\bm{-k}\right)\right\\}$
and construct the Bogoliubov-de-Gennes (BdG) Hamiltonian as

$H=\left[\begin{array}[]{cc}H_{\text{n}}\left(\bm{k}\right)&\Delta\left(\bm{k}\right)\\\
\Delta^{\dagger}\left(\bm{k}\right)&-H_{\text{n}}^{T}\left(\bm{-k}\right)\end{array}\right].$
(2)

Here, the pairing potential $\Delta\left(\bm{k}\right)$ consists of both
orbital-independent and orbital-dependent pairings, and can be generally
written as,

$\displaystyle\Delta\left(\bm{k}\right)=\left[\Delta_{\text{i}}\Phi\left(\bm{k}\right)\sigma_{0}+\Delta_{\text{o}}(\bm{d}_{\text{o}}\left(\bm{k}\right)\cdot\bm{\sigma})\right]is_{y}.$
(3)

Here $\Delta_{\text{i}}$ and $\Delta_{\text{o}}$ are pairing amplitudes in
orbital-independent and orbital-dependent channels, respectively. For our
purpose, we are particularly interested in the d-wave pairing sector, thus the
orbital-independent part is typically chosen as $\Phi\left(\bm{k}\right)=-\cos
k_{x}+\cos k_{y}$. Without breaking the crystalline symmetry, a uniform
orbital-dependent pairing $\bm{d}_{\text{o}}\left(\bm{k}\right)=(0,0,1)$ is
also allowed. For example, they belong to the same irreducible representation
($B_{1}$) of the $C_{4v}$ point group [72, 73, 74]. In the Supplementary
Materials (SM), we self-consistently calculate the above gap function by using
random phase approximation in the absence of SOCs and further find a
spontaneous TRS breaking $d+id$ pairing by minimizing the free energy.
Different from the traditional $d_{xy}+id_{x^{2}-y^{2}}$ (or $B_{1}+iB_{2}$),
this $d+id$ pairing (or $B_{1}+iB_{1}$) preserves mirror symmetry, which
enforces the vanishing of the BdG Chern number for Eq. (2). While the first-
order topology has thus been ruled out, we will show below that 2nd-order TSC
can emerge naturally [75, 76, 77, 78, 79].

Because of the $d$-wave pairing $r_{4z}\Delta({\bf
k})r_{4z}^{T}=-\Delta(C_{4}^{-1}{\bf k})$, the BdG Hamiltonian preserves
${{C}}_{4z}=r_{4z}\oplus-r_{4z}^{\ast}$, together with other symmetries
$C_{2x}=i{\tau}_{z}{\sigma}_{0}s_{z}$ and
$M_{x}=i{\tau}_{z}{\sigma}_{z}s_{x}$, we are capable of diagnosing the
topology of the superconducting spectrum once it is fully gapped. In Fig. 1,
we present the band structures of superconductors with fully gapped trivial
and higher-order topological superconductor phases. In order to diagnose
spatial symmetry-protected topological states, we employ the topological
quantum chemistry theory to obtain symmetry-data vector $B$ [80, 81, 82, 83,
84] that is constituted by irreducible representations (irreps) of little
groups at the maximal momenta in the first Brillouin Zone, as shown in TABLE 1
and inserted in Fig. 1. Referring to TABLE 1, we can see that the topological
trivial system, as shown in Fig. 1(a), is equivalent to that of a $s$ and
$p_{z}$ orbitals at Wyckoff position of $1a$. In sharp contrast, the higher-
order topological phase in Fig. 1(b), is equivalent to that of two $p_{z}$
orbitals at the $1a$ and $1b$ Wyckoff positions. Notice that the $1b$ site is
at the center of the square lattice and cannot be occupied by any orbitals in
real space. Thus, the potential spatial symmetry-protected topological states
fall within the scope of a superconducting analogue of obstructed atomic
insulator (OAI), whose BdG Wannier orbitals are displaced from the lattice
sites. The AOI can be effectively diagnosed by the real space invariant (RSI)
defined at the $1b$ site. The RSI method determines the non-trivial second-
order topology as $\left(\delta_{1},\delta_{2}\right)=\left(-1,\pm 1\right)$,
and the trivial phase is represented as
$\left(\delta_{1},\delta_{2}\right)=\left(0,0\right)$. The RSI
$\left(\delta_{1},\delta_{2}\right)$ with SOC and broken TRS is defined in the
SM.

Figure 1:  Bulk band structure and elementary band representations (EBRs) at high-symmetry points of the TRS breaking $d+id$ superconductor. (a) Topologically trivial superconducting phase for $\Delta_{\text{o}}=0.6$. (b) ${C}_{4}$ protected second-order TSC for $\Delta_{\text{o}}=0.3$. Using EBRs analysis from double space group $P4mm$, We can obtain ${C}_{4}$ index $J_{z}$ at $\Gamma$-point and M-point which is also $z$-direction angular momentum, where $J_{z}=\pm\frac{3}{2}$ of ${\Gamma}_{6}$ or ${M}_{6}$ and $J_{z}=\pm\frac{1}{2}$ of ${\Gamma}_{7}$ or ${M}_{7}$. Common parameters: $t=1$, $\widetilde{t}=0.2$, $t^{\prime\prime}=0.1$, $\mu=0.3$, $\lambda_{I}=0.5$, $\lambda_{R}=0.3$, $\Delta_{\text{i}}=0.1$. Table 1:  The EBRs of space group $P4mm$ with SOC. The first row labels the Wyckoff positions. The second row is the irreducible representation of the double space group. The third row represents the orbits that induce the irreducible representation. The fourth row is the irreducible representations at high-symmetry points. WPs |  $1a$ $(000)$ |  $1b$ $(\frac{1}{2}\frac{1}{2}0)$
---|---|---
EBRs | ${\overline{E}}_{1}\uparrow G(2)$ | ${\overline{E}}_{2}\uparrow G(2)$ | ${\overline{E}}_{1}\uparrow G(2)$ | ${\overline{E}}_{2}\uparrow G(2)$
Orbitals | $s$ | $p_{z}$ | $s$ | $p_{z}$
${\overline{\Gamma}}$ $(000)$ | ${\overline{\Gamma}}_{7}(2)$ | ${\overline{\Gamma}}_{6}(2)$ | ${\overline{\Gamma}}_{7}(2)$ | ${\overline{\Gamma}}_{6}(2)$
${\overline{M}}$ $(\frac{1}{2}\frac{1}{2}0)$ | ${\overline{M}}_{7}(2)$ | ${\overline{M}}_{6}(2)$ | ${\overline{M}}_{6}(2)$ | ${\overline{M}}_{7}(2)$
${\overline{X}}$ $(\frac{1}{2}00)$ | ${\overline{X}}_{5}(2)$ | ${\overline{X}}_{5}(2)$ | ${\overline{X}}_{5}(2)$ | ${\overline{X}}_{5}(2)$

_Superconducting phase diagram._ —In Fig. 2(a), we present the superconducting
phase diagram, which contains nodal superconductor, second-order TSC, and
trivial phases, on the plane of $\mu$ and $\Delta_{\text{o}}$. The gap closing
and reopening of bulk dispersion at off-high-symmetry points (${\bf
k}\notin\\{\Gamma,X,Y,M\\}$) distinguishes a nodal superconductor from a fully
gapped one. In the fully gapped phase, we use the RSI method discussed above
to determine its bulk topology. First, the gap function changes sign with
respect to the reflection line along the [11] or [$1\bar{1}$] directions,
suggesting a mirror symmetry-protected nodal superconductor [85], which is
highlighted in blue in Fig. 2 (a). For example, it must be a nodal $d$-wave
superconductor in the limit $\Delta_{\text{o}}=0$. More details can be found
in the SM. Furthermore, the fully gapped superconductor can be either
topologically trivial or nontrivial as we discussed above. When
$\Delta_{\text{o}}$ is large enough, it is a fully gapped but trivial phase
(the white region in Fig. 2 (a)). The red region represents the second-order
TSC phase. On the other hand, the TRS-breaking nodal superconductor is also
topological, whose bulk nodes are protected by the mirror symmetry. Namely,
the topological nodes are stable along the mirror-invariant lines (i.e.,
movable but irremovable by local perturbations). To show that, we perform a
slab calculation with open boundary condition along the [$1\bar{1}$] direction
and find the Majorana flat band states connecting two bulk nodes, as shown in
$E_{k_{x}^{\prime}}$ of Fig. 2 (b). At fixed $k_{x}^{\prime}$, the 1D
Hamiltonian $H(k_{y}^{\prime})$ has its own particle-hole symmetry
(mirror$\otimes$PHS), leading to the $\mathbb{Z}_{2}$ topology invariant, as
discussed in the SM.

Figure 2:  (a) Superconducting phase diagram on the plane of
$\Delta_{\text{o}}$ and $\mu$. The white region represents the trivial phase,
the blue region is the nodal superconducting phase and the red region is the
second-order TSC phase. (b) The spectral function of the nodal superconductor
for open boundary condition along [1,1] when $\Delta_{\text{o}}=0.01$. There
are two pairs of bulk nodes, and each pair of nodal points is connected by
Majorana flat-band edge states. All the nodes are gapped out when
$\Delta_{\text{o}}\neq 0$, as shown in (c-d). The energy gap increases as
$\Delta_{\text{i}}$ is increased. (c) The spectral function for open boundary
condition along [01] when $\Delta_{\text{i}}=0.05$, and (d) is
$\Delta_{\text{i}}=0.1$. (e) The wave function profile of the four corner
modes, where the inset shows the finite system spectrum near zero energy and
four Majorana zero modes (red dots). The sample size is $120\times 120$. (f)
The density of states near zero energy with the density distributions of bulk,
edge, and corner. Common parameters unless otherwise specified: $t=1$,
$\widetilde{t}=0.2$, $t^{\prime\prime}=0.1$, $\mu=0.3$, $\lambda_{I}=0.5$,
$\lambda_{R}=0.5$, $\Delta_{\text{o}}=0.3$, $\Delta_{\text{i}}=0.1$.

We next consider the second-order TSC phase when $\mu$ is “approaching” the
band bottom or top. First, we use Green’s function method to calculate the
spectral function along the [10] or [01] direction, which shows fully gapped
features in Figs. 2 (c) and (d), where the gap opening for the in-gap edge
states is increased by increasing $\Delta_{\text{i}}$ (the gap of edge states
$\sim 0.06$ in (c) and $\sim 0.12$ in (d)). This will be explained later by
constructing a low-energy edge theory. Then, we take the full tight-binding
simulation on a square lattice to show the Majorana corner states in Fig. 2
(e), where the inset shows the energy spectra with four zero energy states. We
further calculate the local density of states (DOSs) in Fig. 2 (f) to detect
the Majorana zero modes. As we expect, both bulk and edge DOSs show a “U”
shape, while a share zero bias peak is found for the DOS measured at corners.

Figure 3:  Bulk energy spectrum evolution under increasing $\Delta_{\text{o}}$
from $\Delta_{\text{o}}=0$ to $0.5$ of $\Gamma$-M line, where
$k^{\prime}=k_{x}=k_{y}$. (a-e) Display the bulk energy spectrum evolution
when $\mu=0.3$. The nodal superconductor can transition to second-order SC
when $\Delta_{\text{o}}\approx 0.17$ in (b), and the critical point between
second-order and trivial SC phase when $\Delta_{\text{o}}\approx 0.4$ in (d).
(f-j) Exhibit the bulk energy spectrum evolution when $\mu=4$. The nodal
superconductor can transition to a trivial superconductor when
$\Delta_{\text{o}}\approx 0.38$ in (i). Common parameters: $t=1$,
$\widetilde{t}=0.2$, $t^{\prime\prime}=0.1$, $\lambda_{\text{I}}=0.5$,
$\lambda_{R}=0.5$, $\Delta_{\text{i}}=0.1$.

Starting from a nodal superconductor, the appearance of the second-order TSC
or trivial superconductor can be understood in terms of the annihilation of
the nodes. As we show in Fig. 3 (a) and (b), the annihilation of a pair of
nodes results in a phase transition from a nodal superconductor to a second-
order TSC (Fig. 3 (c)). The gap-closing at the $\Gamma$ point in Fig. 3 (d)
further leads to a trivial superconductor. However, two pairs of nodes
annihilate simultaneously corresponding to the transition from a nodal
superconductor directly to a trivial superconductor, as shown in Figs. 3f-3j.

_Helical TSC and second-order TSC._ —In the spirit of “boundary of boundary”,
we aim to derive an edge theory to illustrate the occurrence of corner states
as Jackiw-Rebbi modes [86]. We start with an interesting observation that when
$\Delta_{\text{i}}=0$, the system is a first-order helical TSC [87]. The
topological condition is given by
$-\sqrt{\lambda_{\text{I}}^{2}-\Delta_{\text{o}}^{2}}<\mu<\sqrt{\lambda_{\text{I}}^{2}-\Delta_{\text{o}}^{2}}$.
For a ${\bf k}\cdot{\bf p}$-type Hamiltonian in Eq. (2) in the long-wave
limit, we consider the edge along the $y$-axis as an example. Hence, $k_{x}$
can be replaced by $-i\partial_{x}$ while $k_{y}$ remains a good quantum
number in the BdG Hamiltonian, $H(-i\partial_{x},k_{y})=H_{0}+H^{\prime}$,
where the first part
$H_{0}=-i\lambda_{R}\partial_{x}\tau_{z}\sigma_{0}s_{y}-\mu\tau_{z}\sigma_{0}s_{0}+\lambda_{\text{I}}\tau_{0}\sigma_{y}s_{z}-\Delta_{\text{o}}\tau_{y}\sigma_{z}s_{y}$
needs to be solved analytically for the Majorana zero modes, and the second
part
$H^{\prime}=-\partial_{x}^{2}(t\tau_{z}\sigma_{0}s_{0}+\Delta_{\text{i}}\tau_{x}\sigma_{0}s_{y})-\lambda_{R}k_{y}\tau_{0}\sigma_{0}s_{x}$
is treated as a perturbation in the zero modes basis.

To solve the zero modes, a domain wall between a topologically trivial
superconductor ($x<0$) and a topologically non-trivial superconductor ($x>0$)
is created along the $y$-axis. For simplicity, we set the chemical potential
$\mu=0$, which gives rise to a non-trivial (trivial) superconductor region
with $\Delta_{\text{o}}<\lambda_{\text{I}}$
($\Delta_{\text{o}}>\lambda_{\text{I}}$). Taking the ansatz
$\psi(x)=\mathcal{N}e^{-\kappa x}\chi$ for the zero modes, where
$\kappa_{R}\equiv\kappa(x>0)>0$ and $\kappa_{L}\equiv\kappa(x<0)<0$ since the
bulk on both sides of the domain wall is gapped, and $\chi$ is the spinor
part. After solving $H_{0}\psi=0$, we find only
$\kappa=(\Delta_{\text{o}}+\lambda_{\text{I}})/\lambda_{R}$ satisfies the sign
condition given the topological condition on both sides of the domain wall.
The corresponding spinor parts are
$\chi_{1}=(-i,0,0,-1,-i,0,0,1)^{\text{T}}/2$ and
$\chi_{2}=(0,1,-i,0,0,1,i,0)^{\text{T}}/2$. Therefore, we arrive at the
following two corresponding zero modes,
$\psi_{1,2}(x)=\mathcal{N}e^{-\kappa(x)x}\chi_{1,2}$ with the normalization
constant $\mathcal{N}=\sqrt{2\kappa_{R}\kappa_{L}/(\kappa_{L}-\kappa_{R}})$.
Then, we project $H^{\prime}$ onto this Majorana basis
$\\{\psi_{1},\psi_{2}\\}$, and find the effective edge theory

$\displaystyle
H_{\text{edge}}(k_{y})=\mathcal{N}^{2}\lambda_{R}k_{y}\tau_{y}-m_{\text{meff}}\tau_{x},$
(4)

where the effective mass is
$m_{\text{eff}}=-|\mathcal{N}|^{2}\Delta_{\text{i}}\delta\Delta_{\text{o}}/(2\lambda_{R})$.
Below, we use Pauli matrices $\tau_{x,y,z}$ for this Majorana basis. Please
notice that the $d$-wave pairing naturally leads to a sign-changing feature
for
$\delta\Delta_{\text{o}}\equiv\Delta_{\text{o}}(x<0)-\Delta_{\text{o}}(x>0)$
between two neighboring edges ($\Delta_{\text{o}}\to-\Delta_{\text{o}}$ under
$C_{4z}$). More details can be found in the SM. Due to the sign flipping of
the mass term at each corner, there will be localized zero modes at the four
corners of the system, which establishes the $d+id$ second-order TSC.

Figure 4:  Topological defect on a sample of square shape. (a) The single side
edge dislocation of Volterra cut-and-glue. The black orthogon displays a
dislocation line. The orange and blue arrow lines show the Burgers circuit and
the Burgers vector $\bm{b}=(-1,0)$. (b) Phase diagram of the trivial phase and
$\mathcal{Z}_{2}$ non-trivial phase. The red and blue color regions are marked
by different weak topological indices as $(0,0)$ or $(1,1)$ for the first-
order TSC. (c) The energy spectrum of the TRS-protected first-order
topological superconductor with $\Delta_{\text{i}}=0$. The orange dots are the
fourfold dislocation Majorana zero modes. (d) The wave function profile of the
four dislocation modes. Common parameters: $t=1$, $\widetilde{t}=0.2$,
$t^{\prime\prime}=0.1$, $\mu=7.7$, $\lambda_{\text{I}}=0.5$,
$\lambda_{R}=0.5$, $\Delta_{\text{o}}=0.3$.

_Topological defects in TSCs._ — The Majorana corner modes may be hard for
experimental detection, for example, the superconductivity may lose its phase
coherence near the sample boundary. A bulk-defect correspondence may help to
solve this issue if the bulk TSC has a non-zero weak index [88, 89, 90, 91].
For a 2D square lattice, an edge dislocation is illustrated in Fig. 4 (a) with
a Burger vector ${\bm{b}}=(-1,0)$. It causes an effective $\pi$-flux to trap
Majorana zero modes once

$\bm{b}\cdot\bm{M_{\nu}}=1\textbf{ mod }2,$ (5)

where $\bm{M_{\nu}}$ is defined by
$\bm{M_{\nu}}=\nu_{1}\bm{G}_{1}+\nu_{2}\bm{G}_{2}$. Here, $\bm{G_{i}}$ are
reciprocal lattice vectors, and the vector $(\nu_{1},\nu_{2})$ are the weak
topological indices, which can be calculated via the position of Wannier
center in our 2D system [92]. We first focus on the $\Delta_{\text{o}}=0$
case, the model in Eq. (2) preserves TRS and thus belongs to class DIII of the
A-Z classification [93], which can be characterized by the $\mathcal{Z}_{2}$
topological invariant [94]. The phase diagram in Fig. 4 (b) shows
$\mathcal{Z}_{2}=1$ for both blue and red regions, while only the helical TSC
phase in the blue region ($\mu\sim 8$) carries a weak index $(1,1)$, which is
related to the polarization [95]. Our system has the $C_{4z}$ symmetry, thus
the band inversion happens simultaneously at $(\pi,0)$ and $(0,\pi)$ points.
Thus, the orbital-fluctuation-driven nematicity breaks the $C_{4}$ symmetry,
which can be detected via this bulk-defect correspondence.

We then study the bulk-defect correspondence to show the dislocation Majorana
zero modes. Performing a full tight-binding model calculation with an edge
dislocation, the energy spectrum is shown in Fig. 4 (c) and the wave function
in Fig. 4 (d). A periodic boundary condition for both $x$ and $y$ directions
has been assumed, such that there are no Majorana corner states. Due to the
presence of TRS, Majorana Kramers pairs (MKPs) are trapped at each dislocation
core. Note a dislocation with ${\bm{b}}=(0,1)$ leads to the same result. As
shown in the SM, we use the cut-and-glue progress to derive the effective 1D
Hamiltonian for the MKPs. We first consider the “cutting” step. As shown in
Fig. 4 (a), the edge dislocation cuts the square lattice into two parts. Thus,
the low-energy edge Hamiltonian for the left part is given by Eq. (4) based on
the Majorana basis $\\{\psi_{1},\psi_{2}\\}$. The mirror $M_{x}$ symmetry
leads to that for the right part. In terms of
$\\{\psi_{1},\psi_{2},M_{x}\psi_{2},M_{x}\psi_{1}\\}$, the low-energy
Hamiltonian for this dislocation reads

$\displaystyle
H_{\text{dis}}(k_{y})={\lambda}_{R}k_{y}\varrho_{z}\tau_{y}+m_{\text{eff}}\varrho_{0}\tau_{x},$
(6)

where $\varrho_{x,y,z}$ is for left and right part related by $M_{x}$. The
matrix representation of $M_{x}$, TRS and particle-hole symmetry become
$i\varrho_{y}\tau_{x}$, $i\varrho_{z}\tau_{y}\mathcal{K}$, and
$\varrho_{z}\tau_{z}\mathcal{K}$ respectively. The $m_{\text{eff}}$ term
breaks TRS because of the TRS-breaking $d$-wave pairing $\Delta_{\text{i}}$.
Next, we consider the effect of the “gluing” step on $H_{\text{dis}}$. This
gives rise to the hybridization between the left edge and right edge, and
$m_{\text{eff}}^{\prime}(y)\varrho_{x}\tau_{y}$ is found due to the Rashba SOC
[see details in the SM]. This term is allowed to preserve $M_{x}$ and TRS.
Once Eq. (5) is satisfied, the mass $m_{\text{eff}}^{\prime}(y)$ changes sign
around the dislocation core [96, 97, 98, 99, 100, 101, 102]. When
$m_{\text{eff}}=0$,
$H_{\text{dis}}={\lambda}_{R}k_{y}\varrho_{z}\tau_{y}+m_{\text{eff}}^{\prime}(y)\varrho_{x}\tau_{y}$
corresponds to the KMP at each dislocation core.

Figure 5: (a) Energy spectrum of the square sample for TRS-breaking second-
order TSC. The green dots energy are the fourfold degenerate zero-energy
Majorana corner modes. The two pair of orange dots are gapped out dislocation
modes. (b) The wave function profile of the four Majorana corner modes and
four dislocation modes. Common parameters: $t=1$, $\widetilde{t}=0.2$,
$t^{\prime\prime}=0.1$, $\mu=7.7$, $\lambda_{I}=0.5$, $\lambda_{R}=0.5$,
$\Delta_{i}=0.1$ and $\Delta_{o}=0.3$.

On the other hand, the dislocation KMPs will be gapped out once TRS is broken
by $\Delta_{\text{i}}\neq 0$, as shown in Fig. 5 (a). This gap is caused by
$m_{\text{eff}}\propto\Delta_{\text{i}}$ in Eq. (6), since it anti-commutes
with the $m_{\text{eff}}^{\prime}$ term. This is inherited from the anti-
commutation relations between bulk terms in Eq. (2), where the Rashba term
($\lambda_{R}$) anti-commutes with the $d$-wave pairing ($\Delta_{\text{i}}$).

_Conclusions._ —We have established symmetry-protected 2nd-order topology as a
new topological possibility for d+id superconductors. A few candidate
materials with $d$-wave pairing and spontaneous TRS breaking may be used to
realize our theory, including Sr2RuO4 [103], LaPt3 [104], and SrPtAs [105]. In
the absence of TRS caused by the $d+id$ pairing, both of the two possible
topological phases, the nodal TSC and the second-order TSC, can realize
Majorana-bound states. We find that the nodal TSC is protected by mirror
symmetry, while the second-order TSC is protected by the $C_{4z}$ symmetry,
consistent with the OAI phase. The bulk-defect correspondence is also
investigated using a non-zero weak index.

_Acknowledgments._ — This work was supported by the National Natural Science
Foundation of China (NSFC, Grants No. 12074108, No. 12147102, and No.
12204074), the Natural Science Foundation of Chongqing (Grant No.
CSTB2022NSCQ-MSX0568), and the Fundamental Research Funds for the Central
Universities (Grant No. 2023CDJXY-048). L.-H. H. at Aalto is funded by the
Jane and Aatos Erkko Foundation and the Keele Foundation as part of the SuperC
collaboration. R.-X. Z. acknowledges the start-up fund at the University of
Tennessee. D.-S. M. also acknowledges the funding from the China National
Postdoctoral Program for Innovative Talent (Grant No. BX20220367).

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