11institutetext:  School of Mathematics and Natural Science, University of
Wuppertal, Wuppertal 42119, Germany
11email<EMAIL_ADDRESS>
# Assessing Non-Nested Configurations of Multifidelity Machine Learning for
Quantum-Chemical Properties

Vivin Vinod 11 0000-0001-6218-5053 Peter Zaspel 11 0000-0002-7028-6580

###### Abstract

Multifidelity machine learning (MFML) for quantum chemical (QC) properties has
seen strong development in the recent years. The method has been shown to
reduce the cost of generating training data for high-accuracy low-cost ML
models. In such a set-up, the ML models are trained on molecular geometries
and some property of interest computed at various computational chemistry
accuracies, or fidelities. These are then combined in training the MFML
models. In some multifidelity models, the training data is required to be
nested, that is the same molecular geometries are included to calculate the
property across all the fidelities. In these multifidelity models, the
requirement of a nested configuration restricts the kind of sampling that can
be performed while selection training samples at different fidelities.

This work assesses the use of non-nested training data for two of these
multifidelity methods, namely MFML and optimized MFML (o-MFML). The assessment
is carried out for the prediction of ground state energies and first vertical
excitation energies of a diverse collection of molecules of the CheMFi
dataset. Results indicate that the MFML method still requires a nested
structure of training data across the fidelities. However, the o-MFML method
shows promising results for non-nested multifidelity training data with model
errors comparable to the nested configurations.

###### Keywords:

multifidelity machine learning quantum chemistry DFT excitation energies

##  1 Introduction

Machine learning (ML) for Quantum Chemistry (QC) has become a rapidly
developing field of research for the prediction of various QC properties [7,
1, 18]. With a focus on reducing the time taken for new computation with such
ML-QC methods, research has recently begun focusing on reducing the time it
takes to generate a training set, for such ML models. The introduction of the
$\Delta$-ML method [10] has allowed for ML models to be trained on the
difference of two properties, one computationally expensive and other other
computationally cheaper, sometimes even semi-empirical.

Multifidelity Machine Learning (MFML) following refs.[20, 15] was introduced
to be a systematic methodological improvement of $\Delta$-ML. The term
fidelity is used to refer to the accuracy of the QC method used to train the
model. In general, a high accuracy QC method is also cost-expensive. Here,
multiple QC methods are used to create multiple sub-models, which are finally
summed up to predict QC properties such as excitation energies at the most
expensive QC method, or target fidelity. MFML was shown to produce low-cost
high-accuracy models. A methodological improvement of MFML was recently
introduced and shown to be superior to MFML in prediction atomization energies
of molecules and vertical excitation energies along molecular trajectories
[14].

Research in multifidelity methods has since increased to cover a wide range of
formulations including those that are different from $\Delta$-ML based
methods. Variations include the hierarchical-ML (h-ML) model which was shown
to significantly reduce training data generation time in predicting the
potential energy surface (PES) of $\rm CH_{3}Cl$ [2]. The h-ML method used
more than two QC methods of calculations and implemented an ad hoc
optimization procedure to select the number of training samples to be used at
each level or fidelity of QC calculation. Multi-task Gaussian processes
(MTGPR) have also begun to utilize the different fidelities to train one ML
model simultaneously for multiple fidelities [3, 11]. Here, each fidelity is
treated as inter-related to the others and a surrogate MTGPR model is created.
Further, a diverse multifidelity dataset consisting of 135,000 point
geometries has recently been made available [16, 17] with various QC
properties, such as vertical excitation energies, calculated with DFT
formalism. The fidelities are differentiated by the choice of basis set used
in the calculation. This dataset will be used in all benchmarks of this study.

In most variations of multifidelity methods, which have been used in
predicting a range of QC properties [9, 20, 8, 2, 15], there has been one
similarity: the use of nested training data. Simply put, it refers to using
the same molecular geometries across the different fidelities. If a molecular
geometry is used at the highest fidelity, then it is also used at the
subsequent lower fidelities. This approach was recommended in Refs. [20, 15]
based on research from sparse-grid combination techniques [5, 4, 12].
Recently, ref. [3] investigated building multitask surrogate models with
heterogeneous data. However, it is to be pointed out that the data used did
not ensure complete non-nestedness since some training data at the lower
fidelity included at least a few samples which were used as test data in the
final surrogate model.

A note is to be made on the motivation for the need of a non-nested
configuration of training data for MFML. The current state of MFML models
usually trains them on nested datasets restricting their ability to be
transferred across unrelated datasets. On the other hand, having non-nested
configurations of MFML methods could enable the use of disparate datasets
resulting in more flexible multifidelity methods without necessitating
calculations at costlier fidelities. Thus, it becomes relevant to inquire the
effectiveness of fully non-nested configurations of MFML for QC. In addition
to reducing training data generation for multifidelity methods, it would
result in more versatile MFML models which can combine across the molecular
space without restrictions.

This work compares the use of fully non-nested configurations of training data
versus the nested configurations for multifidelity methods. The assessment is
performed by using the MFML and optimized MFML (o-MFML) models from Ref. [14].
These models are built to predict the ground state energies and first vertical
excitation energies for molecules of the CheMFi dataset [16, 17]. The rest of
this manuscript is structured as follows: Section 2 discusses the various
methodological pre-requisites for this work, including the dataset used, the
ML methods employed, and details of MFML and o-MFML methods. Section 3
discusses the results of the experiments carried out for nested and non-nested
configurations of the training data. Finally, conclusions for multifidelity
methods for ML in QC are drawn based on this work and future outlook is
discussed.

##  2 Methods

###  2.1 Kernel Ridge Regression

Let $\mathcal{T}^{(f)}:=\\{(\boldsymbol{X}_{i},E^{(f)}_{i})\\}_{i=1}^{{N}_{\rm
train}}$ be a training set of molecular descriptors $\boldsymbol{X}_{i}$ and
their corresponding properties computed at some fidelity $f$. An assessment of
using different descriptors for single fidelity KRR was performed (see Fig. 1)
between the SLATM [6] and Coulomb Matrix (CM). The results are in favor of
unsorted CM and therefore, the molecular descriptor used for this work is the
unsorted CM. For each geometry, this is computed as

$C_{i,j}:=\begin{cases}\frac{Z_{i}^{2.4}}{2}~{},&i=j\\\ \frac{Z_{i}\cdot
Z_{j}}{\left\lVert\boldsymbol{R}_{i}-\boldsymbol{R}_{j}\right\rVert}~{},&i\neq
j~{},\end{cases}$ (1)

where $\boldsymbol{R}_{i}$ is the Cartesian coordinate of the $i$-th atom, and
$Z_{i}$ the corresponding atomic charge. Since the molecules have different
sizes, the molecular descriptors are padded by zeros to maintain a uniform
size of the CM.

A KRR model for the prediction of some property at a fidelity $f$, for a given
query descriptor $\boldsymbol{X}_{q}$, is computed as

$P^{(f)}_{\rm KRR}\left(\boldsymbol{X}_{q}\right):=\sum_{i=1}^{N^{(f)}_{\rm
train}}\alpha^{(f)}_{i}k\left(\boldsymbol{X}_{q},\boldsymbol{X}_{i}\right)~{},$
(2)

where $k(\cdot,\cdot)$ is the kernel function and the sum runs over the number
of training samples $N_{train}^{(f)}$ for some fidelity $f$. This work uses
the Matérn Kernel of first order with the discrete L-2 norm. This is given as:

$k\left(\boldsymbol{X}_{i},\boldsymbol{X}_{j}\right)=\exp{\left(-\frac{\sqrt{3}}{\sigma}\left\lVert\boldsymbol{X}_{i}-\boldsymbol{X}_{j}\right\rVert_{2}^{2}\right)}\cdot\left(1+\frac{\sqrt{3}}{\sigma}\left\lVert\boldsymbol{X}_{i}-\boldsymbol{X}_{j}\right\rVert_{2}^{2}\right)~{},$
(3)

where $\sigma$ denotes a length scale hyperparameter. The coefficients of KRR,
$\boldsymbol{\alpha}^{(f)}$, are calculated by solving
$(\boldsymbol{K}+\lambda\boldsymbol{I})\boldsymbol{\alpha}^{(f)}=\boldsymbol{E}^{(f)}$.
Here,
$\boldsymbol{K}=\left(k(\boldsymbol{X}_{i},\boldsymbol{X}_{j})\right)_{i,j=1}^{{N}_{\rm
train}}$ is the kernel matrix, $\boldsymbol{I}$ the identity matrix, and
$\boldsymbol{E}^{(f)}=\left(E^{(f)}_{1},E^{(f)}_{2},\ldots,E^{(f)}_{{N}_{\rm
train}}\right)^{T}$ is the vector of energies from the training set,
$\mathcal{T}^{(f)}$. The parameter $\lambda$ penalizes overfitting and is
called the Lavrentiev regularizer.

###  2.2 Multifidelity Machine Learning

In Refs. [20, 14], it has been discussed that the MFML model for a property of
interest can be built by iteratively combining sub-models which are individual
KRR models, trained at a specific fidelity for a specific number of training
samples. In such a scheme, each sub-model is identified by a composite index,
$\boldsymbol{s}=(f,\eta_{f})$, where $2^{\eta_{f}}=N_{train}^{(f)}$ implicitly
denotes the number of training samples used at that fidelity. In mathematical
formalism, a MFML model can then be written as:

$P_{\rm
MFML}^{(F,\eta_{F};f_{b})}\left(\boldsymbol{X}_{q}\right):=\sum_{\boldsymbol{s}\in\mathcal{S}^{(F,\eta_{F};f_{b})}}\beta_{\boldsymbol{s}}P^{(\boldsymbol{s})}_{\rm
KRR}\left(\boldsymbol{X}_{q}\right)~{},$ (4)

where, the linear combination is performed for the various sub-models built
with different fidelities. The summation runs over the set of selected sub-
models of MFML, $\mathcal{S}^{(F,\eta_{F};f_{b})}$. This selection is decided
by the choice of the baseline fidelity, $f_{b}$, and the number of training
samples at the highest fidelity (also called the target fidelity), $N_{\rm
train}^{(F)}=\eta_{F}$. The sub-models for a given MFML model are chosen based
on a scheme discussed extensively in ref. [14]. The different
$\beta_{\boldsymbol{s}}$ are the coefficients of the linear combination of
these sub-models. Based on ref. [20] the value of $\beta_{\boldsymbol{s}}$ are
chosen to be:

$\beta_{\boldsymbol{s}}^{\rm MFML}=\begin{cases}+1,&\text{if
}f+\eta_{f}=F+\eta_{F}\\\ -1,&\text{otherwise}\end{cases}~{}.$ (5)

Optimized MFML (o-MFML) rewrites Eq. (4) by allowing for values of the
coefficients different from that given in Eq. (5). This is achieved by an
optimization process carried out on a validation set, $\mathcal{V}^{F}_{\rm
val}:=\\{(\boldsymbol{X}_{q}^{\rm val},y^{\rm val}_{q})\\}_{q=1}^{N_{\rm
val}}$. For a target fidelity $F$, with $N^{(F)}_{\rm train}=2^{\eta_{F}}$
training samples and a given baseline fidelity $f_{b}$, the o-MFML model is
defined as

$P_{\rm
o-MFML}^{\left(F,\eta_{F};f_{b}\right)}\left(\boldsymbol{X}_{q}\right):=\sum_{\boldsymbol{s}\in\mathcal{S}^{(F,\eta_{F};f_{b})}}\beta_{\boldsymbol{s}}^{\rm
opt}P^{(\boldsymbol{s})}_{\rm KRR}\left(\boldsymbol{X}_{q}\right)~{},$ (6)

where $\beta_{\boldsymbol{s}}^{\rm opt}$ are the optimized coefficients. These
values are attained by solving the following optimization:

$\beta_{\boldsymbol{s}}^{\rm
opt}=\arg\min_{\beta_{\boldsymbol{s}}}\left\lVert\sum_{v=1}^{N_{\rm
val}}\left(y_{v}^{\rm val}-\sum_{\boldsymbol{s}\in
S^{(F,\eta_{F};f_{b})}}\beta_{\boldsymbol{s}}P^{(\boldsymbol{s})}_{\rm
KRR}\left(\boldsymbol{X}^{\rm val}_{v}\right)\right)\right\rVert_{p}\,,$

where one minimizes some $p$-norm on the validation set defined above. Based
on ref. [14], the ordinary least squares (OLS) method is used to optimize the
coefficients in this work, that is, this work used a $p=2$ norm in the
optimization procedure for o-MFML.

The most up-to-date approach to both MFML and o-MFML uses the nestedness of
training data as noted in refs. [20, 15, 14]. The nestedness of training data
can be mathematically formulated as follows: Consider the collection of
molecular descriptors
$\mathcal{X}^{(f)}=\left\\{\boldsymbol{X}_{i}^{(f)}\lvert\left(\boldsymbol{X}_{i}^{(f)},y^{(f)}_{i}\right)\in\mathcal{T}^{(f)}\right\\}$.
Nestedness of training data for MFML requires
$\mathcal{X}^{(F)}\subseteq\ldots\subseteq\mathcal{X}^{(2)}\subseteq\mathcal{X}^{(1)}$.
That is, if a molecular geometry is chosen to be used as the training data at
the target fidelity, it should also be used to train the model at the
subsequent lower fidelities.

In contrast, for the non-nested configuration of multifidelity models of the
form discussed in refs. [20, 15, 14] require that
$\mathcal{X}^{(F)}\not\subseteq\ldots\not\subseteq\mathcal{X}^{(2)}\not\subseteq\mathcal{X}^{(1)}$.
More succinctly, $\mathcal{X}^{(f)}\cap\mathcal{X}^{(g)}=\phi$, where $\phi$
is the empty set, for all $f\neq g$ values of the fidelities. This implies
that if a molecular geometry is picked to train the model at fidelity $F$,
then this geometry is not used to train the model at the $F-1$ fidelity. This
work benchmarks the use of non-nested configuration of multifidelity methods
against the nested configuration. Throughout this work, model errors are
calculated on a holdout test set, $\mathcal{V}_{\rm
test}^{F}:=\\{(\boldsymbol{X}_{q}^{\rm test},y^{\rm
test}_{q})\\}_{q=1}^{N_{\rm test}}$, which consist of the CM representations
and their corresponding reference values for property of interest (for
example, excitation energy) calculated at the target fidelity $F$. Mean
Absolute Errors (MAEs) are used as the numerical indicator of model accuracy.
MAE are calculated using a discrete $L_{1}$ norm

$MAE=\frac{1}{N_{\rm test}}\sum_{q=1}^{N_{\rm test}}\left\lvert P_{\rm
ML}\left(\boldsymbol{X}_{q}^{\rm test}\right)-{y}^{\rm
test}_{q}\right\rvert~{}.$ (7)

The model $P_{\rm ML}$ can be either the single fidelity KRR model or any of
the MFML models used in this work.

###  2.3 Dataset

For this work, the multifidelity dataset CheMFi was used [16, 17]. CheMFi
contains various QC properties calculated at 5 fidelities for nine diverse
molecules, namely: acrolein, alanine, thymine, urea, urocaninc acid,
2-nitrophenol, DMABN, SMA, and o-HBDI. For each of these molecules 15,000
geometries are provided with properties such as ground state energies and
excitation energies calculated at different fidelities. These are all TD-DFT
calculations with varying basis sets constituting the different fidelities.
The hierarchy of fidelities is taken as follows in increasing order: STO3G,
321G, 631G, def2-SVP, def2-TZVP. For the remainder of this work, these are
referred to by their shortened nomenclature such as TZVP and SVP.

The CheMFi dataset contains a total of 135,000 point geometries of 9 diverse
molecules. To ensure that the data chosen would indeed be non-nested the
following strategy was employed: $1.5\cdot 2^{9}=768$ samples were randomly
chosen from the 135,000 for the TZVP fidelity. Of the remaining 134,288
samples, $1.5\cdot 2^{10}=1,536$ samples were chosen for the SVP fidelity. In
this way, the STO3G fidelity contains $1.5\cdot 2^{13}=12,288$ training
samples in total. Thus the total training set spans $768+\ldots+12,288=23,808$
training samples with the respective sampling for each fidelity. For the case
of nested training data, across five fidelities, the corresponding number of
training samples as mentioned above were chosen.

For the validation set to be used in o-MFML, 1,000 samples were chosen at
random from the CheMFi dataset after removing all the training data.
Similarly, a holdout test set was chosen consisting of 2,192 samples. In other
words, the test set is never used in any stage of training the multifidelity
models. The validation set and the test set are fixed and not changed during
the course of the experiments in this work.

##  3 Results

To numerically study the effect of nestedness of training data for
multifidelity methods, the two models from ref. [14] were built with details
as reported in Section 2. These were built to predict two QC properties from
the CheMFi dataset [17, 16], namely, the ground state energies, and the first
vertical excitation energies. In this section, a preliminary analysis as
recommended by ref. [15] is performed for the multifidelity data for ground
state and excitation energies. Following this, the multifidelity learning
curves for these two properties are analyzed for nested and non-nested
training data set-ups with MFML and o-MFML models.

###  3.1 Preliminary Data Analysis

Figure 1:  Comparison of the use of unsorted and row-norm sorted CM [13] and
SLATM [6] representations for the prediction of ground state and excitation
energies with single fidelity KRR at the TZVP fidelity. For both ground state
and excitation energies, the unsorted CM outperforms the other
representations.

To assess the best molecular descriptor for this work, a short test was
performed on the use of unsorted CM and SLATM representations for single
fidelity KRR models. The TZVP fidelity properties were predicted with these
models. The resulting learning curves are shown in Fig. 1 for both ground
state and excitation energies. The horizontal axis on the left-hand side is
scaled for the ground state energies, while the one on the right-hand side
corresponds to the excitation energies. It becomes evident from the learning
curves that the unsorted CM outperforms the SLATM representations for both
ground state and excitation energies. Based on these results, the unsorted CM
are used for the remainder of this work. All multifidelity and single fidelity
models hereon are built with unsorted CM representations.

Figure 2:  Preliminary analysis for the multifidelity structure of ground
state energies. The distribution of the ground state energies shows that it
covers a wider range of values. The absolute difference of the various
fidelities to the target fidelity of TZVP shows that for the most part this
decreases with increasing fidelity. A scatter plot of the various fidelity
energies with respect to TZVP shows a systematic distribution of the energies
as can be seen in the inset image.

Previous work on MFML for excitation energies in ref. [15] recommends that
preliminary analysis be performed for multifidelity data to determine clear
hierarchy of the fidelities and a systematic distribution of the fidelities to
the target fidelity. The first preliminary analysis is to observe the
multifidelity data distribution of the properties of interest, that is, to
look at how the values are distributed across the energy domain. The second
analysis measures the absolute difference of each fidelity to the target
fidelity (in this case, TZVP). It is anticipated for conventional MFML to work
that this decreases at least monotonically as one goes up the hierarchy of
fidelities. The final analysis for the preliminary assessment of the
multifidelity data is a scatter plot of the property calculated at different
fidelities with respect to the target fidelity.

This form of analysis for the ground state energies is depicted in Fig. 2 with
the energies being in Hartree units. Since the ground state energies belong to
a conglomerate of molecules, the different peaks of the values are visible in
the distribution plot. These peaks appear since the dataset consists of
multiple molecules, each with a significantly different ground state energy on
average. It is also to be noted that the energy distribution is also
occasionally zero at certain locations on the energy domain for all
fidelities. This will not be a challenge since the test set is also sampled
from the CheMFi dataset and would lack energies within these ‘dips’ of the
density plot. The difference in values for various fidelities, as seen in the
center pane of the figure, shows a nearly monotonically decreasing difference
to the target fidelity, with the exception of SVP which is slightly more than
its preceding fidelity of 6-31G. This minor deviation is not anticipated to
cause any break down on the MFML model. The right-hand side scatter plot does
not show any clear distributions since the energies are spread over a large
range of values. However, the inset shows that there is indeed a systematic
distribution of the fidelities with respect to the target fidelity of TZVP.
This systematic distribution of energies across different fidelities generally
results in a meaningful MFML model [15]. Therefore, one anticipates that the
conventional MFML model for nested configuration will indeed show decreasing
errors with addition of cheaper fidelities.

Figure 3:  The multifidelity structure of the first vertical excitation
energies are analyzed to confirm the assumption of hierarchy. The distribution
of the energies on the left-most plot shows distinct peaks which correspond to
the different molecules. The difference in fidelities to the TZVP fidelity
decreases on average with STO3G having a large standard deviation as can be
seen in the plot in the center. This is confirmed in the scatter-plot from the
left-most plot as well where the STO3G energies show a wide distribution with
respect to TZVP energies.

A similar analysis for the excitation energies sampled from CheMFi was
performed and the results are shown in Fig. 3 for the energies in eV. Since
the excitation energies are not as large as the ground state energies, in this
case, one is able to better observe the different fidelities. The distribution
plot shows various local variations for each fidelity. This is simply an
indicator of the diversity of the data. Here one observes that most of the
fidelity show a predominantly bimodal distribution. However, STO3G shows a
slightly different distribution of energies. The central pane of Fig. 3 shows
the fidelity difference plot where one observes that STO3G shows a larger
standard deviation with respect to the other fidelities. The fidelity
difference depicts a decreasing value as one increases the fidelity. This
means that the assumed hierarchy of methods is indeed correct. Finally, the
scatter plot of energies calculated at various fidelities with respect to the
TZVP fidelity shows a systematic distribution for all fidelities. Since STO3G
has a wider spread in comparison to the other fidelities, it could potentially
be less effective in a multifidelity model. However, as ref. [14] has shown,
the o-MFML method can still result in a fairly accurate model superseding
conventional MFML.

###  3.2 Ground State Energies

Figure 4:  Learning curves of the MFML and o-MFML models built for ground
state energies. The top row corresponds to nested training set case while the
bottom row shows the results when non-nested training sets are used to build
multifidelity models. Both conventional MFML and o-MFML are assessed here with
the help of learning curves. The reference single fidelity KRR is also shown.

The learning curves for the diverse multifidelity models for the prediction of
ground state energies are shown in Fig. 4 for the MFML and o-MFML models. The
first row of the plots corresponds to the case, where the training data across
various fidelities is from a nested data setup. These are identical in
methodology to the models created in Refs. [14, 15]. Multifidelity learning
curves are interpreted a little different from the conventional ML learning
curves. Consider the case of nested MFML as seen in the top left pane of Fig.
4. The top learning curve corresponds to the standard KRR single fidelity
method. Here, addition of training samples directly corresponds to the values
of the x-axis. The next line is a MFML learning curve for $f_{b}=$SVP. In this
case, if the x-axis shows $N_{train}^{TZVP}=4$, then it also includes $2\times
4=8$ samples at the SVP fidelity. Similarly as one goes down the baseline
fidelities, the number of training samples used in the different fidelities
are indicated by the number of training samples used at TZVP. For instance,
the point on the learning curve of the STO3G baseline fidelity with 8 training
samples at TZVP implies that the MFML model has $[8,16,32,64,128]$ training
samples at TZVP, SVP, 631G, 321G, and STO3G respectively.

The learning curves for decreasing baseline fidelity for ground state energies
are shown in Fig. 4 and show clearly lowered offsets. Consider the learning
curve of single fidelity KRR with 128 training samples. If one were to draw a
horizontal line at the corresponding MAE, it would intersect the multifidelity
learning curve corresponding to STO3G at around $N_{\rm train}^{TZVP}=8$. This
implies that MFML with STO3G baseline can be built with a lower number of
expensive training samples and achieve the same error as a standard KRR model.
For $N_{\rm train}^{\rm TZVP}=512$, both the models $P_{\rm MFML}^{STO3G}$ and
$P_{\rm o-MFML}^{STO3G}$ report an error 0.010 which shows that these two
models are close in performance. One reason these models perform nearly same
could be that the default MFML combination of the sub-models is already
optimized. Such results have been previously reported in ref. [14] for some
cases.

The second row of Fig. 6 shows the MFML and o-MFML learning curves for the
case of non-nested training data, as explained in Section 2.2. One immediately
notices that the conventional MFML model breaks down with a non-nested
multifidelity training dataset. It fails to provide any reasonable improvement
for the different baseline fidelities. Regardless of the training set sizes,
the conventional MFML models fail to reduce the MAE in comparison to the
single fidelity KRR model. On the other hand, for the o-MFML models built with
varying baseline fidelities shows improvement similar to the MFML and o-MFML
models built with nested training data. For $N_{\rm train}^{\rm TZVP}=512$,
the MAE of $P_{\rm o-MFML}^{STO3G}$ is 0.015 hE which is only negligibly
larger than it was for the case of the nested training data. However, it must
be noted that for the non-nested case of o-MFML, the reduction in error with
addition of cheaper fidelities is not as pronounced as it is for the nested
configuration. The learning curves for baseline fidelities of 321G and 631G
appear to be converging. This goes to show that non-nested configurations are
indeed a challenging task.

Figure 5:  A study of the optimized coefficient values of o-MFML for both the
nested and non-nested cases in predicting the ground state energies. The
default coefficients of MFML are shown on the right-most plot for comparison.

To better comprehend the behavior of o-MFML and the corresponding results, one
can study the optimized coefficients, $\boldsymbol{\beta}^{\rm
opt}_{\boldsymbol{s}}$, of o-MFML. For the ground state energies, these are
shown in Fig. 5 with the default MFML coefficients shown on the right hand
side pane for reference. For each plot, the x-axis implicitly depicts the
value of training samples used at the fidelities, which are denoted on the
y-axis. Each box, therefore, represents a sub-model used to build the final
multifidelity model. The values of the coefficients are shown inside the
square boxes and correspond to $\beta_{\boldsymbol{s}}^{\rm opt}$ for o-MFML
and $\beta_{\boldsymbol{s}}^{\rm MFML}$ for conventional MFML.

The left-hand plot in Fig. 5 shows the values of the coefficients for the case
of nested training data. One observes that the values of the coefficients of
o-MFML for the nested configuration, both magnitude and sign, are close to the
default MFML coefficient values. This could be due to the conventional MFML
model already being optimal. Furthermore, the magnitudes of the coefficients
are similar implying that the each sub-model contributes almost similarly to
the overall multifidelity model.

For the non-nested configuration of o-MFML, the resulting values of
$\boldsymbol{\beta}^{\rm opt}_{\boldsymbol{s}}$ are shown in the center pane
of Fig. 5. One notices a significant deviation from the the default
coefficient values of MFML. Further, the values of the coefficients cover a
wider range of values in comparison to the nested configuration of o-MFML.
This could further confirm, in addition to the learning curves, that the non-
nested configuration does in fact pose a challenge to the multifidelity model.
A strong deviation of $\boldsymbol{\beta}^{\rm opt}_{\boldsymbol{s}}$ from the
values of $\boldsymbol{\beta}^{\rm MFML}_{\boldsymbol{s}}$ indicates a
significant change in the contribution of the corresponding sub-models in the
final multifidelity model. This could imply that with the non-nested
configuration, each sub-model being trained on distinct training data, cannot
be combined as in Eq. (4) but perhaps requires something different. However,
the flexibility of o-MFML in optimizing the coefficients allows it to
optimally combine the sub-models even if the samples are non-nested resulting
in a multifidelity model that still reduce errors with addition of cheaper
fidelities. The optimization over the validation set for this model makes it
superior over the conventional MFML model in the non-nested configuration.

###  3.3 Excitation Energies

Figure 6:  Multifidelity learning curves for the prediction of first vertical
excitation energies. The first row shows the results for MFML and o-MFML with
nested training set data. Similarly, the second row delineates the learning
curves for the case of non-nested training data. The learning curve for a
single fidelity KRR model is also shown for reference.

The prediction of excitation energies is in general considered to be more
challenging than predicting ground state energies [19, 1]. The multifidelity
learning curves for the prediction of excitation energies is shown in Fig. 6
for both nested and non-nested configurations of MFML and o-MFML models. The
nested configuration for both categories of multifidelity models shows
promising results for the prediction of excitation energies. A constantly
lowered offset is observed with addition of cheaper baselines in addition to a
negative slope of the learning curves. Similar to the case of ground state
energies, the negatively sloped learning curves indicate that the addition of
further training samples could potentially decrease the error of prediction.

However, for the non-nested configuration for MFML, the learning curves
indicate a poor performance of the models. For MFML, similar to the case of
ground state energies, the entire multifidelity structure seems to have broken
down with no meaningful model being formed for any baselines fidelity being
added. In fact, the addition of cheaper baselines worsens the model. It is a
ready conclusion that the non-nested configuration of MFML fails for the
prediction of excitation energies similar to the ground state energies. On the
other hand, with o-MFML, the non-nested configuration performs noticeably
better as in the case for ground state energies. The addition of SVP and 631G
fidelities in the multifidelity structure improves the model albeit not as
well as seen in the nested configuration of o-MFML. The learning curves show
lowered offsets for these baseline fidelities, although for SVP, the
difference is not as significant as it was for the nested configuration of
o-MFML. With the 321G and STO3G fidelities, the o-MFML model shows
improvements for small to medium training set sizes. But with $N_{\rm
train}^{\rm TZVP}=512$, the learning curve for the 321G baseline fidelity
converges to MAE values close to that corresponding to $f_{b}=$631G. This is
also observed for the STO3G baselines, where even for medium training set
sizes, the multifidelity learning curve converges to that with $f_{b}=$321G.
One possible reason for this could be that as larger training samples are used
at TZVP, the number of training samples at the lower fidelities scales by 2.
For large enough TZVP training samples, therefore, the multifidelity model has
a larger amount of non-nested data to combine and the OLS optimization
struggles to optimize these large and seemingly unrelated (due to non-
nestedness) sub-models.

Figure 7: o-MFML coefficient values for the prediction of excitation energies.
The values are shown for both nested and non-nested configurations. The
conventional MFML coefficient values are shown alongside for reference.

A further analysis of the o-MFML models can be performed with the study of the
value of $\boldsymbol{\beta}^{opt}_{\boldsymbol{s}}$ for nested and non-nested
configurations of o-MFML for the prediction of excitation energies. These are
shown in Fig. 7 where the right-most pane of the plot also shows the default
coefficients of MFML for reference. The coefficients for nested configuration
of o-MFML are delineated on the left-hand side of the figure. Here, one
observes that most of the coefficients lie in the same range of values and
closer to the default values of conventional MFML coefficients. As ref. [14]
argues, this could indicate that the combination of the sub-models was already
optimized with the default values of the coefficients. The values of
$\boldsymbol{\beta}^{opt}_{\boldsymbol{s}}$ for the non-nested configuration
of o-MFML are shown in the middle pane of the figure. The values of the
coefficients for the three cheapest fidelities change significantly in
comparison with the nested configuration. For 321G and STO3G the change is not
very significant. This could be due to the additional noise that these
fidelities include into the multifidelity model due to the non-nested training
data. As noted in the case for ground state energies, the large number of
training samples which are unrelated due to the non-nested configuration might
prove challenging to the OLS optimizer. Thus, it becomes additionally
difficult to discard the noise from this training data structure to optimally
combine the sub-models to provide a multifidelity model.

##  4 Conclusions and Outlook

Through the various numerical tests employed in this work, the effect of
nestedness of training data has been evaluated for multifidelity models. It is
seen that nested configurations of MFML and o-MFML generally out perform their
non-nested counterparts. However, the use of o-MFML with non-nested training
data shows promising outlooks. A focus on improving the optimization routine
and possibly including steps to account for the noise incorporated by the non-
nested training data could potentially make this a vital tool in ML for QC.
Future work on non-nested configurations of training data could include the
use of other multifidelity methods such as h-ML or multi-task methods.
Improved multifidelity models for non-nested configurations would allow for a
more flexible use of these models. For o-MFML, the optimization procedure and
the choice of a validation set implicitly decide the accuracy of the final
model. A refined choice of the validation set could improve the results and
provide a better optimized model even for the non-nested case. One example
case would be choosing validation set geometries corresponding to the largest
molecule of the CheMFi dataset, o-HDBI. However, this form of research lies
outside the scope of this article. Another potential research area for the use
of o-MFML for non-nested data is the use of non-linear combinations of sub-
models and what that would imply for multifidelity models.

Overall, the work presented here opens up areas for research in the use of
non-nested configurations of multifidelity models. The scope of using methods
such as o-MFML to tackle non-nested and heterogeneous multifidelity data has
become evident through the numerical examples in this work. Although using
non-nested training data seems to be a bottle neck for current multifidelity
models, further research in this direction can certainly improve them.

####  4.0.1 Acknowledgements

The authors acknowledge support by the DFG through the project ZA 1175/3-1 as
well as through the DFG Priority Program SPP 2363 on “Utilization and
Development of Machine Learning for Molecular Applications – Molecular Machine
Learning” through the project ZA 1175_4-1. The authors would also like to
thank for the support of the ‘Interdisciplinary Center for Machine Learning
and Data Analytics (IZMD)’ at the University of Wuppertal.

####  4.0.2

The authors declare that there is no conflict of interest or any competing
interests.

## References

  * [1] Pavlo O. Dral and Mario Barbatti.  Molecular excited states through a machine learning lens.  Nat. Rev. Chem., 5(6):388–405, 2021.  doi:10.1038/s41570-021-00278-1.
  * [2] Pavlo O Dral, Alec Owens, Alexey Dral, and Gábor Csányi.  Hierarchical machine learning of potential energy surfaces.  J. Chem. Phys., 152:204110, 2020.  doi:10.1063/5.0006498.
  * [3] Katharine Fisher, Michael Herbst, and Youssef Marzouk.  Multitask methods for predicting molecular properties from heterogeneous data [preprint], 2024.  arXiv:2401.17898.
  * [4] Helmut Harbrecht, Michael Peters, and Markus Siebenmorgen.  Combination technique based k-th moment analysis of elliptic problems with random diffusion.  J. Comput. Phys., 252:128–141, 2013.  URL: https://www.sciencedirect.com/science/article/pii/S0021999113004373, doi:10.1016/j.jcp.2013.06.013.
  * [5] Markus Hegland, Brendan Harding, Christoph Kowitz, Dirk Pflüger, and Peter Strazdins.  Recent developments in the theory and application of the sparse grid combination technique.  SPPEXA 2013-2015, pages 143–163, 2016.  doi:10.1007/978-3-319-40528-5_7.
  * [6] Bing Huang and O Anatole von Lilienfeld.  Quantum machine learning using atom-in-molecule-based fragments selected on the fly.  Nat. Chem., 12(10):945–951, 10 2020.  doi:10.1038/s41557-020-0527-z.
  * [7] Sergei Manzhos and Tucker Jr. Carrington.  Neural network potential energy surfaces for small molecules and reactions.  Chem. Rev., 121(16):10187–10217, 2021.  PMID: 33021368.  doi:10.1021/acs.chemrev.0c00665.
  * [8] Abhirup Patra, Rohit Batra, Anand Chandrasekaran, Chiho Kim, Tran Doan Huan, and Rampi Ramprasad.  A multi-fidelity information-fusion approach to machine learn and predict polymer bandgap.  Comput. Mat. Sci., 172:109286, 2020.  URL: https://www.sciencedirect.com/science/article/pii/S0927025619305853, doi:10.1016/j.commatsci.2019.109286.
  * [9] Ghanshyam Pilania, Chenchen Wang, Xun Jiang, Sanguthevar Rajasekaran, and Ramamurthy Ramprasad.  Accelerating materials property predictions using machine learning.  Sci. Rep., 3:1–6, 2013.  doi:10.1038/srep02810.
  * [10] Raghunathan Ramakrishnan, Pavlo O. Dral, Matthias Rupp, and O. Anatole von Lilienfeld.  Big data meets quantum chemistry approximations: The $\Delta$-Machine Learning approach.  J. Chem. Theory Comput., 11:2087–2096, 5 2015.  doi:10.1021/acs.jctc.5b00099.
  * [11] Kislaya Ravi, Vladyslav Fediukov, Felix Dietrich, Tobias Neckel, Fabian Buse, Michael Bergmann, and Hans-Joachim Bungartz.  Multi-fidelity Gaussian process surrogate modeling for regression problems in physics [pre-print], 2024.  arXiv:2404.11965.
  * [12] Christoph Reisinger.  Analysis of linear difference schemes in the sparse grid combination technique.  IMA J. Numer. Anal., 33:544–581, 09 2013.  doi:10.1093/imanum/drs004.
  * [13] Matthias Rupp, Alexandre Tkatchenko, Klaus-Robert Müller, and O. Anatole von Lilienfeld.  Fast and accurate modeling of molecular atomization energies with machine learning.  Phys. Rev. Lett., 108:05830–1 – 05830–5, 1 2012.  doi:10.1103/PhysRevLett.108.058301.
  * [14] Vivin Vinod, Ulrich Kleinekathöfer, and Peter Zaspel.  Optimized multifidelity machine learning for quantum chemistry.  Mach. learn.: sci. technol., 5(1):015054, 03 2024.  URL: https://dx.doi.org/10.1088/2632-2153/ad2cef, doi:10.1088/2632-2153/ad2cef.
  * [15] Vivin Vinod, Sayan Maity, Peter Zaspel, and Ulrich Kleinekathöfer.  Multifidelity machine learning for molecular excitation energies.  J. Chem. Theory Comput., 19(21):7658–7670, 2023.  PMID: 37862054.  doi:10.1021/acs.jctc.3c00882.
  * [16] Vivin Vinod and Peter Zaspel.  CheMFi: A Multifidelity Dataset of Quantum Chemical Properties of Diverse Molecules, 06 2024.  doi:10.5281/zenodo.11636903.
  * [17] Vivin Vinod and Peter Zaspel.  CheMFi: A Multifidelity Dataset of Quantum Chemical Properties of Diverse Molecules [pre-print], 2024.  arXiv:2406.14149.
  * [18] Julia Westermayr, Michael Gastegger, Kristof T. Schütt, and Reinhard J. Maurer.  Perspective on integrating machine learning into computational chemistry and materials science.  J. of Chem. Phys., 154(23):230903, 06 2021.  doi:10.1063/5.0047760.
  * [19] Julia Westermayr and Philipp Marquetand.  Machine learning for electronically excited states of molecules.  Chem. Rev., 121:9873–9926, 11 2020.  doi:10.1021/acs.chemrev.0c00749.
  * [20] Peter Zaspel, Bing Huang, Helmut Harbrecht, and O. Anatole Von Lilienfeld.  Boosting quantum machine learning models with a multilevel combination technique: Pople diagrams revisited.  J. Chem. Theory Comput., 15(3):1546–1559, 2019.  URL: https://doi.org/doi/10.1021/acs.jctc.8b00832, doi:10.1021/acs.jctc.8b00832.


# Electronic structures near unmovable nodal points and lines in two-
dimensional materials

V Damljanović1 and N Lazić2 1Institute of Physics Belgrade, University of
Belgrade, Pregrevica 118, 11000 Belgrade, Serbia<EMAIL_ADDRESS>2NanoLab,
Faculty of Physics, University of Belgrade, Studentski Trg 12, 11001 Belgrade,
Serbia

###### Abstract

Unmovable nodal points (UNP) and lines (UNL) are band crossings which
positions in the Brillouin zone are unaltered by symmetry preserving
perturbations. Not only positions but also the band structure in the vicinity
is determined by the little group of a wave vector and it’s irreducible
(co)representations. In this paper we give the full set of electronic
dispersions near all UNPs and UNLs in non-magnetic, quasi two-dimensional (2D)
materials both with and without spin-orbit coupling (SOC). Analysis of all
layer gray single and double groups gives nineteen different quasiparticles,
great majority of which are unavoidable for a 2D material which belongs to
certain layer groups. These include Weyl and Dirac nodal lines, dispersions
with quadratic or cubic splitting, anisotropic Weyl and Dirac cones which
orientation can be varied by _e.g._ strain _etc_. We indicated qusiparticles
that are robust to SOC. For convenience, our results are concisely presented
graphically - as a map, not in a tabular, encyclopedia form. They may be of
use as checkpoints or for fitting of experimentally (via _e.g._ ARPES) and
numerically obtained electronic band structures as well as for deeper
theoretical investigations.

  * December 2022

##  1 Introduction

The Fermi-Dirac distribution dictates that most of the material properties
depend on the electronic dispersions shape near the Fermi energy in the
Brillouin zone (BZ). On the other hand, these shapes are solely determined by
the crystal symmetry and, for non-magnetic materials, the time-reversal
symmetry (TRS). Presence of Dirac cones around the BZ $K-$points of the
honeycomb lattice is the most notable example. Experimental discovery of the
electric field effect in graphene, fabrication of graphene’s three dimensional
(3D) analogs accompanied with the explanation of certain band structure
properties using topological laws, brought topological materials into the
focus of solid-state science [1, 2, 3]. Non-trivial band topology, such as
non-zero Berry phase, originates from singularities of eigenvectors in the
reciprocal space. Such singularities often appear at points where two or more
bands touch.

In two-dimensional (2D) materials, bands can touch at isolated points or at
lines. Band touchings caused by the increased dimensionality of
corepresentation (correp) of little group do not move if the Hamiltonian
parameters change, as long the symmetry is preserved. Such touchings we call
unmovable nodal points (UNP) and lines (UNL). Weyl/Dirac nodal lines
(WNL)/(DNL) are examples of UNL, which presence in a band structure leads to
interesting physical properties [4, 5, 6, 7]. These include graphene-like
density of states (DOS) [8, 9] and induction of drumhead states which flat
bands may cause strong electron correlations [5, 10]. In 3D, WNL and DNL were
determined in achiral gray double space groups [11], in 230 ordinary/gray
single/double space groups [12], in certain black-and-white magnetic space
groups [13] and in the presence and absence of spin-orbit coupling (SOC) [14].
Refs. [15, 16] report DNL in the presence of SOC in 2D. On the other hand,
full enumeration of layer groups’ UNPs hosting fully linear dispersion in non-
magnetic 2D materials is completed task [17]. In Refs. [17, 18, 19] the
Hamiltonians are distinguished from each other by the form of their
eigenvalues. The term Dirac was reserved for four-component Hamiltonians with
graphene-like, isotropic or anisotropic eigenvalues [17, 20, 21]. There are
several physical reasons for this distinction, _e.g._ the non-vanishing DOS of
some dispersion [18, 19] near zero energy. On the other hand, analysis of all
dispersions other than linear in 2D materials has not been reported thus far
[22]. Recently published reports for (3D) space groups [23, 24, 25, 26] could
be (in principle) used for layer groups, if one subduces the results for
corresponding space groups to 2D BZ. Ref. [27] suggest that this method is
indeed applied for layer groups but, in all fairness, it is published (at the
moment of submission of this manuscript) without the Supplementary Material
which contains the results announced in the main part of the paper. Ref. [28]
has classified all massless excitations in 2D and 3D.

In this paper we calculated exact shapes of electronic bands near all high-
symmetry points (HSP) and lines (HSL) in the BZ of all layer groups, including
cases without and with SOC, in the presence of TRS. It turns out that in the
Taylor expansion of Hamiltonians, the relevant terms are up to maximally third
order in momentum. We found in total nineteen quasiparticles, almost none of
which were investigated in the present literature. Our results are presented
graphically, as an atlas (map) rather than in a tabular, encyclopedia form,
and fit into two figures and a few pages of text/formulas. For all symmetry
constrained Hamiltonians, the eigenvalues are given. We single out
quasiparticles that are not destroyed by inclusion of SOC.

##  2 Method

To find all possible HSPs and HSLs with multidimensional correps, one has to
search through all little groups of all gray layer groups - 80 for spinless
and 80 for spinful cases. Unlike correps of space groups, which were tabulated
and made available long time ago [29, 30], layer group representations became
publicly available only recently [31]. Ref. [31] was a starting point of our
search.

Symmetry adapted, low-energy Hamiltonians were derived from the following
formula [17, 32]:

$\hat{H}\left(\mathbf{k}_{0}+\mathbf{q}\right)=\hat{D}^{{\dagger}}\left(g\right)\hat{H}\left(\mathbf{k}_{0}+\hat{h}^{\prime}\mathbf{q}\right)\hat{D}\left(g\right),$
(1)

where $\mathbf{k}_{0}$ denotes HSP or belongs to HSL,
$g=\left\\{\hat{h}\right|\left.\mathbf{\tau}_{\hat{h}}\right\\}$ is an element
(in Seitz notation) of the little group $G_{\mathbf{k}_{0}}$
($\hat{h}^{\prime}\mathbf{k}_{0}$ is equivalent to $\mathbf{k}_{0}$),
$\hat{h}^{\prime}$ is an operator reduction of $\hat{h}$ to 2D BZ and
$\hat{D}\left(g\right)$ is the irrep matrix that corresponds to $g$. If
$-\mathbf{k}_{0}$ belongs to the star of $\mathbf{k}_{0}$, the full symmetry
group of $\mathbf{k}_{0}$ is $G_{\mathbf{k}_{0}}+\theta
g_{0}G_{\mathbf{k}_{0}}$, where $\theta$ denotes time-reversal and the element
$g_{0}=\left\\{\hat{h}_{0}\right|\left.\mathbf{\tau}_{\hat{h}_{0}}\right\\}$
does not belong to $G_{\mathbf{k}_{0}}$, if $\mathbf{k}_{0}$ is not the time
reversal invariant momentum ($\hat{h}^{\prime}_{0}\mathbf{k}_{0}$ is
equivalent to $-\mathbf{k}_{0}$). The additional constrain on contribution
$\hat{H}^{\prime}$ appearing in the Taylor expansion
$\hat{H}\left(\mathbf{k}_{0}+\mathbf{q}\right)\approx\hat{H}\left(\mathbf{k}_{0}\right)+\hat{H}^{\prime}\left(\mathbf{q}\right)$
is [17, 32]:

$\hat{H}^{\prime*}\left(\mathbf{q}\right)=\hat{R}^{{\dagger}}\left(\theta
g_{0}\right)\hat{H}^{\prime}\left(-\hat{h}_{0}^{\prime}\mathbf{q}\right)\hat{R}\left(\theta
g_{0}\right),$ (2)

where $\hat{R}$ is correp matrix (linear part - without complex conjugation)
that corresponds to $\theta g_{0}$. As opposed to dispersions, which are
invariants uniquely determined by the symmetry, the Hamiltonians can be
changed by different choice of basis wave-functions. However, once the
particular Hamiltonian $\hat{H}(\mathbf{q})$ in the vicinity of
$\mathbf{k}_{0}$ is chosen, the Hamiltonian $\hat{H}_{p}(\mathbf{q})$ near
$\mathbf{k}_{p}\equiv\hat{h}\mathbf{k}_{0}$ from the star of $\mathbf{k}_{0}$,
is fixed by the following formula:

$\hat{H}_{p}\left(\mathbf{q}\right)=\hat{R}\left(g_{p}^{-1}g\right)\hat{H}\left(\hat{h}^{-1}\mathbf{q}\right)\hat{R}^{{\dagger}}\left(g_{p}^{-1}g\right),$
(3)

where $g_{p}=\left\\{\hat{h}_{p}|\mathbf{\tau}_{p}\right\\}$ is the
representative of the coset that contains the element
$g=\left\\{\hat{h}|\mathbf{\tau}\right\\}$ of the layer group, and $\hat{R}$
is the correp matrix of the little group of $\mathbf{k}_{0}$. In (3) one has
to use the very same correp that gives $\hat{H}(\mathbf{q})$ near
$\mathbf{k}_{0}$, not just an equivalent one. Similarly, for dispersions
around $\mathbf{k}_{p}$:

$E_{p}(\hat{h}\mathbf{q})=E(\mathbf{q}).$ (4)

Also, dispersions around two points in the reciprocal space, that differ by a
vector from the reciprocal lattice are identical. All necessary matrices are
available in PolSym code in addition to the web page [31, 33].

To avoid unnecessary complication, we do not list here all Hamiltonians that
correspond to each particular correp from [31], since many of them can be
transformed into each other by the similarity transformation. If two
$n$-component Hamiltonians $\hat{H}_{1}$ and $\hat{H}_{2}$, have all
eigenvalues the same, then
$\hat{H}_{2}=\hat{U}\hat{H}_{1}\hat{U}^{{\dagger}}$, with
$\hat{U}=\sum_{l=1}^{n}\left|\Psi_{l}\right\rangle\left\langle\Phi_{l}\right|$
being an unitary matrix and $\left|\Psi_{l}\right\rangle$
($\left|\Phi_{l}\right\rangle$) the normalized eigenvector of $\hat{H}_{2}$ (
$\hat{H}_{1}$) that corresponds to the eigenvalue $E_{l}$, $(l=1,2,...,n)$.
Therefore, of all Hamiltonians having the same eigenvalues, only one
representative is shown.

For distinction between anisotropic Dirac and Weyl dispersions which
orientation depends on the parameters of the model, and the ones with fixed
orientation, we use the following theorem. The quadratic form
$f=c_{1}q_{1}^{2}+c_{3}q_{1}q_{2}+c_{2}q_{2}^{2}=\left\langle\mathbf{q}\right|\hat{A}\left|\mathbf{q}\right\rangle$,
can be reduced to $f=\lambda_{1}q_{1}^{\prime}+\lambda_{2}q_{2}^{\prime}$,
where
$q_{j}^{\prime}=\left\langle\lambda_{j}\right.\left|\mathbf{q}\right\rangle$,
while $\lambda_{j}$ and $\left|\lambda_{j}\right\rangle$ are eigenvalues and
eigenvectors of the matrix $\hat{A}=\left(\begin{array}[c]{cc}c_{1}&c_{3}/2\\\
c_{3}/2&c_{2}\end{array}\right)$, $(j=1,2)$. The $\lambda_{j}$ and
$\left|\lambda_{j}\right\rangle$ are well known and won’t be presented here.
The cross-section of $f$ is an ellipse whose orientation (in principle)
depends on parameters $c_{1}$, $c_{2}$ and $c_{3}$. The orientation of the
ellipse is fixed if eigenvectors are independent on $c_{1}$, $c_{2}$ and
$c_{3}$. This is the case if either $c_{3}=0$ or:

$c_{2}-c_{1}=uc_{3},$ (5)

where $u$ is a real constant, independent of $c_{1}$, $c_{2}$ and $c_{3}$.

##  3 Results

All groups and all BZ points and lines hosting electronic dispersions in the
absence of SOC are shown in the Figure 1, those with SOC are shown in the
Figure 2. Dispersions are denoted with different symbols, as indicated in the
captions and explained in the subsections below. Dispersions of the same type
but with different parameters are denoted with the same symbols but of
different size.

Figure 1:  Dispersions near UNP and UNL in the BZ of layer groups without SOC
(notation as in [34]). Dispersion acronyms in the lowest panel and the
corresponding subsections in the main text are: DNL - Dirac nodal line -
3.1.2, FT - fortune teller - 3.2.4, IDD - isotropic Dirac - 3.2.8, CQ - corner
quadratic - 3.3.1, ICQ - isotropic corner quadratic - 3.3.2, QD - quadratic -
3.3.3, IQD - isotropic quadratic - 3.3.4 and SB - simple band - 3.3.5. The
green color denotes more than one possibility for a dispersion around a given
BZ point. Other colors denote spinful degeneracy at $\mathbf{q}=0$: red
(4-fold) and yellow (8-fold).

[capbesideposition=left,top,capbesidewidth=0.4]figure[]

Figure 2: Dispersions near UNP and UNL in the BZ of layer groups with SOC
(superscript $D$ is omitted, notation as in [34]). Dispersion acronyms in the
lowest panel and the corresponding subsections in the main text are: WNL -
Weyl nodal line - 3.1.1, DNL - Dirac nodal line - 3.1.2, IWD - isotropic Weyl
- 3.2.1, AWDv -anisotropic Weyl with variable orientation - 3.2.2, AWDf
-anisotropic Weyl with fixed orientation - 3.2.3, FT - fortune teller - 3.2.4,
PF-poppy flower - 3.2.5, ADDv -anisotropic Dirac with variable orientation -
3.2.6, ADDf -anisotropic Dirac with fixed orientation - 3.2.7, IDD - isotropic
Dirac - 3.2.8, CQ - corner quadratic - 3.3.1, ICQ - isotropic corner quadratic
- 3.3.2, SB - simple band - 3.3.5, CD - cubic - 3.4.1 and TC - triangular
cubic - 3.4.2. The green color denotes more than one possibility for a
dispersion around a given BZ point. Other colors denote spinful degeneracy at
$\mathbf{q}=0$: blue (2-fold) and red (4-fold).

We used Eq. (4) for graphical presentation of dispersions in the whole BZ.
Primitive vectors of reciprocal lattice are $\mathbf{k}_{1}$ and
$\mathbf{k}_{2}$. For oblique and rectangular-p groups with all non-symmorphic
elements having fractional translations parallel to one direction (for group
45), $\mathbf{k}_{1}$ is perpendicular to that direction (to the symmorphic
reflection plane). For rectangular-c groups $35^{D}$ and $36^{D}$, WNL is
along symmorphic axis of order two. For groups 27 and 34 (32, 33 and 43),
$\mathbf{k}_{1}$ is along axis (screw axis) of order two. This convention is
of use when directions along $\mathbf{k}_{1}$ and $\mathbf{k}_{2}$ differ
significantly in dispersions they host. For rectangular-c groups in Figure 2,
BZ is shown for the angle between primitive vectors greater than $\pi/2$. The
results are equally valid if that angle is sharp, just the BZ border would be
rotated. Groups not shown in Figure 1 (Figure 2) do not have UNP and/or UNL in
the case without (with) SOC.

The studies are performed for linear, quadratic and, when it was relevant, for
the third order in $\mathbf{q}$, and all together we single out 19 types of
dispersions. In the following, they are described in more detail. Letters $a$
and $b$ denote real numbers, $z$ denotes complex numbers, the superscript $D$
denotes layer double group, $\hat{\sigma}_{1}$, $\hat{\sigma}_{2}$,
$\hat{\sigma}_{3}$ ($\hat{\sigma}_{0}$) are Pauli matrices (is the two-
dimensional unit matrix), while $q_{1}$ and $q_{2}$ are projections of
$\mathbf{q}$ along orthonormal vectors $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$
respectively.

###  3.1 Nodal line dispersions

####  3.1.1 Weyl nodal line (WNL)

WNL is line of double spinful degeneracy which splits linearly in the
direction perpendicular to the line. It appears only in SOC case and it is
denoted with blue lines in the Figure 2. The dispersion and the Hamiltonian
are:

$E_{1,2}=E_{0}+a_{1}q_{||}\pm c\left|q_{\bot}\right|,$ (6a)
$\hat{H}=\left(E_{0}+a_{1}q_{||}\right)\hat{\sigma}_{0}+q_{\bot}\sum_{j=1}^{3}b_{j}\hat{\sigma}_{j},$
(6b)

where $c=\sqrt{\sum_{j=1}^{3}b_{j}^{2}}>0$ and $q_{\bot}$ ($q_{||}$) is
projection of $\mathbf{q}$ perpendicular to (along) the line. For groups
$20^{D}$, $21^{D}$, $24^{D}$, $25^{D}$, $54^{D}$, $56^{D}$, $58^{D}$ and
$60^{D}$, $b_{3}=0$. For groups $27^{D}$, $28^{D}$, $29^{D}$(WNLs
perpendicular to $\mathbf{k}_{1}$), $30^{D}$, $31^{D}$, $32^{D}$, $33^{D}$(WNL
perpendicular to $\mathbf{k}_{2}$), $34^{D}$, $35^{D}$, $36^{D}$, $78^{D}$ and
$79^{D}$, $b_{3}=b_{2}=0$. For points in the middle and at both ends of WNL,
if not occupied by any other dispersion, the following substitution should be
made in the last two equations above: $E_{0}+a_{1}q_{||}\rightarrow
E_{0}+a_{1}q_{||}^{2}$. In these points, one of the bands achieves extremum at
$\mathbf{q}=0$ with the effective mass:
$\hat{m}_{eff}=\left(\begin{array}[c]{cc}0&0\\\ 0&m_{||}\end{array}\right)$,
which means that excitations are massless along the direction perpendicular to
the WNL.

####  3.1.2 Dirac nodal line (DNL)

DNL is a line of 4-fold spinful degeneracy, which splits into two double
degenerate bands in the direction perpendicular to the line. DNL appears both
in SOC and non-SOC case and it is denoted by a red line. Dispersion is:

$E_{1,2,3,4}=E_{0}+aq_{||}\pm c\left|q_{\bot}\right|,$ (6ga) where $q_{\bot}$
($q_{||}$) is projection of $\mathbf{q}$ perpendicular to (along) the line.
For layer single groups, the total Hamiltonian is
$\hat{H}_{tot}=\hat{\sigma}_{0}\otimes\hat{H}$, with $\hat{H}$ given by (6b),
while $E_{1}^{\uparrow}=E_{1}^{\downarrow}=E_{1}=E_{2}$,
$E_{2}^{\uparrow}=E_{2}^{\downarrow}=E_{3}=E_{4}$ and
$c=\sqrt{\sum_{j=1}^{3}b_{j}^{2}}$. For groups 15, 16, 17, 20, 21, 24, 25, 40,
43(DNLs perpendicular to $\mathbf{k}_{1}$), 44, 45(DNLs perpendicular to
$\mathbf{k}_{1}$), 54, 56, 58, 60 and 63, $b_{3}=0$. For groups 28, 30, 31,
32, 33(DNLs perpendicular to $\mathbf{k}_{2}$), 34, 38, 39, 41, 42, 43(DNLs
perpendicular to $\mathbf{k}_{2}$), 45(DNLs perpendicular to
$\mathbf{k}_{2}$), 46, 62 and 64, $b_{3}=b_{2}=0$. For points in the middle
and at both ends of DNL, if not occupied by any other dispersion, the
following substitutions should be made in the equations for Hamiltonian (6b)
and dispersion (6ga): $E_{0}+a_{1}q_{||}\rightarrow E_{0}+a_{1}q_{||}^{2}$ and
$b_{2}=b_{3}=0$. For layer double groups, the Hamiltonian is:
$\hat{H}=(E_{0}+aq_{||})\hat{I}_{4}+q_{\bot}\left(\begin{array}[c]{cccc}b&0&z&0\\\
0&-b&0&-z\\\ z^{*}&0&-b&0\\\ 0&-z^{*}&0&b\end{array}\right),$ (6gb)

with $c=\sqrt{b^{2}+|z|^{2}}$. For points in the middle and at both ends of
DNL, if not occupied by any other dispersion, the following substitutions
should be made in (6gb): $E_{0}+a_{1}q_{||}\rightarrow E_{0}+a_{1}q_{||}^{2}$
(endpoints of $43^{D}$ and $45^{D}$) and (in addition) $b=0$ (for remaining
such points).

###  3.2 Linear dispersions

####  3.2.1 Isotropic Weyl dispersion (IWD)

IWD consists of two spinful non-degenerate isotropic cones touching each other
at $\mathbf{q}=0$ and it appears only in the case with SOC. IWD is denoted
with a blue circle in Figure 2 (when no other dispersion is possible around
that point), with a green ring with a dot inside (when simple band (SB)
described latter is an alternative), or with a green square (when cubic
dispersion (CD) described latter is an alternative). The dispersion and
Hamiltonian are:

$E_{1,2}=E_{0}\pm|z|\left|\mathbf{q}\right|,$ (6gha)
$\hat{H}=E_{0}\hat{\sigma}_{0}+\left(\begin{array}[c]{cc}0&z(q_{1}-iq_{2})\\\
z^{*}(q_{1}+iq_{2})&0\end{array}\right),$ (6ghb)

for groups $54^{D}$, $56^{D}$, $58^{D}$, $60^{D}$, $z=(1-i)b_{1}$, for groups
$67^{D}$, $69^{D}$, $z=(1+i\sqrt{3})b_{1}$, for the remaining groups
$z=b_{1}$, except for $49^{D}$, $50^{D}$, $65^{D}$, $73^{D}$, where $z$
remains a complex number.

####  3.2.2 Anisotropic Weyl dispersion with variable orientation (AWDv)

AWDv consists of two anisotropic, spinful non-degenerate cones and appears
only when SOC is included. The cross section of the cones is an ellipse whose
orientation (major axis direction) changes when the parameters of the
dispersion change. It is represented by a blue ellipse in Figure 2, such that
its orientation is neither strictly horizontal, nor strictly vertical nor
along a BZ edge. This is just a symbol indicating that the orientation of AWDv
is not pinned to any particular direction. The actual orientation can be
different than the one adopted in Figure 2, although eq. (4) is used for
presenting AWDv on the same star of the wave-vector. Dispersion and
Hamiltonian are:

$E_{1,2}=E_{0}\pm\sqrt{c_{1}q_{1}^{2}+c_{2}q_{2}^{2}+c_{3}q_{1}q_{2}},$
(6ghia)
$\hat{H}=E_{0}\hat{\sigma}_{0}+\left(\begin{array}[c]{cc}b_{1}q_{1}+b_{2}q_{2}&z_{1}q_{1}+z_{2}q_{2}\\\
z_{1}^{*}q_{1}+z_{2}^{*}q_{2}&-b_{1}q_{1}-b_{2}q_{2}\end{array}\right),$
(6ghib)

where $c_{1}=b_{1}^{2}+|z_{1}|^{2}$, $c_{2}=b_{2}^{2}+|z_{2}|^{2}$,
$c_{3}=2b_{1}b_{2}+z_{1}z_{2}^{*}+z_{1}^{*}z_{2}$. For groups $3^{D}$,
$22^{D}$, $26^{D}$, $49^{D}$, $50^{D}$ and $73^{D}$, $b_{1}=b_{2}=0$. The
parameters $c_{1}$, $c_{2}$ and $c_{3}$ do not satisfy eq. (5).

####  3.2.3 Anisotropic Weyl dispersion with fixed orientation (AWDf)

AWDf consists of two anisotropic, spinful non-degenerate cones, whose
orientation is unaltered by the symmetry preserving perturbations. It appears
only when SOC is included and it is presented by a blue ellipse whose
orientation is strictly vertical, strictly horizontal, along the BZ edge or
perpendicular to it. Again, in actual orientation major and minor axes may
interchange, and eq. (4) is used for presentation in Figure 2. The dispersion
can be reduced to:

$E_{1,2}=E_{0}\pm\sqrt{c_{1}q_{1}^{2}+c_{2}q_{2}^{2}}.$ (6ghij)

Hamiltonians of interest can be derived from (6ghib). For groups $8^{D}$,
$9^{D}$, $10^{D}$, $11^{D}$, $12^{D}$, $13^{D}$, $31^{D}$, $32^{D}$, $33^{D}$
and $34^{D}$, $b_{2}=0$ and $z_{1}=0$ so that $c_{1}=b_{1}^{2}$ and
$c_{2}=|z_{2}|^{2}$. For groups $19^{D}$, $20^{D}$, $22^{D}$, $23^{D}$,
$24^{D}$, $26^{D}$, $53^{D}$, $55^{D}$, $57^{D}$ and $59^{D}$,
$b_{1}=b_{2}=0$, $z_{1}=ib_{3}$ and $z_{2}=b_{4}$ so that $c_{1}=b_{3}^{2}$
and $c_{2}=b_{4}^{2}$. For groups $21^{D}$ and $25^{D}$, $b_{2}=0$, $z_{1}=0$
and $z_{2}=ib_{3}$ so that $c_{1}=b_{1}^{2}$ and $c_{2}=b_{3}^{2}$. For groups
$67^{D}$, $68^{D}$, $69^{D}$ and $70^{D}$, $b_{1}=\sqrt{3}b_{2}$ and
$z_{2}=-\sqrt{3}z_{1}$ so that
$E_{1,2}=E_{0}\pm\sqrt{|z_{1}|^{2}(q_{1}-\sqrt{3}q_{2})^{2}+b_{2}^{2}(q_{2}+\sqrt{3}q_{1})^{2}}$.
For groups $76^{D}$ and $77^{D}$, $b_{1}=b_{2}=0$ and
$z_{2}=z_{1}^{r}/\sqrt{3}-i\sqrt{3}z_{1}^{i}$ so that
$E_{1,2}=E_{0}\pm\sqrt{(z_{1}^{i})^{2}(q_{1}-\sqrt{3}q_{2})^{2}+(z_{1}^{r})^{2}(q_{1}+q_{2}/\sqrt{3})^{2}}$.
Last two dispersions are examples where eq. (5) applies with $u=1/\sqrt{3}$.

####  3.2.4 Fortune teller dispersion (FT)

FT appears both in SOC and non-SOC cases, with the degeneracies being doubled
in the latter [18, 19]. It is denoted by the quad arrow, yellow for non-SOC
and red for SOC case. The dispersion is:

$E_{1,2,3,4}=E_{0}\pm\left|\left|b_{1}q_{2}\right|\pm
c\left|q_{1}\right|\right|.$ (6ghika) For layer double groups the Hamiltonian
is:
$\hat{H}=E_{0}\hat{I}_{4}+\left(\begin{array}[c]{cccc}b_{2}q_{1}&ib_{1}q_{2}&0&zq_{1}\\\
-ib_{1}q_{2}&b_{2}q_{1}&-zq_{1}&0\\\ 0&-z^{*}q_{1}&-b_{2}q_{1}&ib_{1}q_{2}\\\
z^{*}q_{1}&0&-ib_{1}q_{2}&-b_{2}q_{1}\end{array}\right),$ (6ghikb)

so that $c=\sqrt{\left|z\right|^{2}+b_{2}^{2}}>0$ and $\hat{I}_{4}$ is the
unit matrix. For layer single groups the total Hamiltonian is
$\hat{H}_{tot}=\hat{\sigma}_{0}\otimes\hat{H}$, with $\hat{H}$ given by
(6ghikb) and $\left(\forall
j\in\left\\{1,2,3,4\right\\}\right)E_{j}^{\uparrow}=E_{j}^{\downarrow}=E_{j}$.
For groups 43 and 45, $b_{2}=0$.

####  3.2.5 Poppy flower dispersion (PF)

PF appears only in some non-centrosymmetric groups when SOC is included [19].
It is represented in Figure 2 by a red double-ellipse. Degeneracy at
$\mathbf{q}=0$ is 4-fold (spin included) and it splits into spinful non-
degenerate bands away from $q_{1}=0$ and $q_{2}=0$ lines. Dispersion and
Hamiltonian are:

$E_{1,2,3,4}=E_{0}\pm\sqrt{c_{1}q_{1}^{2}+c_{2}q_{2}^{2}\pm
c_{3}|q_{1}q_{2}|}.$ (6ghikla)

For groups $21^{D}$ and $25^{D}$:

$\hat{H}=E_{0}\hat{I}_{4}+\left(\begin{array}[c]{cccc}b_{1}q_{1}&b_{2}q_{2}&z_{1}q_{1}&z_{2}q_{2}\\\
b_{2}q_{2}&-b_{1}q_{1}&z_{2}q_{2}&-z_{1}q_{1}\\\
z_{1}^{*}q_{1}&z_{2}^{*}q_{2}&-b_{1}q_{1}&-b_{2}q_{2}\\\
z_{2}^{*}q_{2}&-z_{1}^{*}q_{1}&-b_{2}q_{2}&b_{1}q_{1}\end{array}\right),$
(6ghiklb)

here
$c_{3}=\sqrt{4b_{1}^{2}|z_{2}|^{2}+4b_{2}^{2}|z_{1}|^{2}-(z_{1}z_{2}^{*}-z_{1}^{*}z_{2})^{2}-4b_{1}b_{2}(z_{1}z_{2}^{*}+z_{1}^{*}z_{2})}$,
$c_{2}=b_{2}^{2}+|z_{2}|^{2}$, $c_{1}=b_{1}^{2}+|z_{1}|^{2}$. For groups
$28^{D}$, $30^{D}$, $32^{D}$ and $34^{D}$:

$\hat{H}=E_{0}\hat{I}_{4}+\left(\begin{array}[c]{cccc}b_{1}q_{1}+b_{2}q_{2}&0&z_{1}q_{1}+z_{2}q_{2}&0\\\
0&b_{1}q_{1}-b_{2}q_{2}&0&z_{1}q_{1}-z_{2}q_{2}\\\
z_{1}^{*}q_{1}+z_{2}^{*}q_{2}&0&-b_{1}q_{1}-b_{2}q_{2}&0\\\
0&z_{1}^{*}q_{1}-z_{2}^{*}q_{2}&0&-b_{1}q_{1}+b_{2}q_{2}\end{array}\right),$
(6ghiklc)

here $c_{1}=b_{1}^{2}+|z_{1}|^{2}$, $c_{2}=b_{2}^{2}+|z_{2}|^{2}$,
$c_{3}=|2b_{1}b_{2}+z_{1}z_{2}^{*}+z_{1}^{*}z_{2}|$. For groups $54^{D}$,
$56^{D}$, $58^{D}$ and $60^{D}$:

$\displaystyle\hat{H}=E_{0}\hat{I}_{4}$ (6ghikld) $\displaystyle+$
$\displaystyle\left(\begin{array}[c]{cccc}0&(1-i)b_{1}(q_{1}-iq_{2})&z^{*}(q_{1}+iq_{2})&0\\\
(1+i)b_{1}(q_{1}+iq_{2})&0&0&z^{*}(q_{2}+iq_{1})\\\
z(q_{1}-iq_{2})&0&0&(1-i)b_{1}(q_{2}-iq_{1})\\\
0&z(q_{2}-iq_{1})&(1+i)b_{1}(q_{2}+iq_{1})&0\end{array}\right),$ (6ghikli)

here $c_{1}=c_{2}=2b_{1}^{2}+|z|^{2}$, $c_{3}=4\sqrt{2}|b_{1}z|$ (called
isotropic PF in [19]).

####  3.2.6 Anisotropic Dirac dispersion with variable orientation (ADDv)

ADDv is analogous to AWDv with degeneracy of each band doubled. It appears
only in SOC case. ADDv is represented by a red ellipse with the same
convention as for AWDv described in Section 3.2.2. Dispersion and Hamiltonian
are:

$E_{1,2,3,4}=E_{0}\pm\sqrt{c_{1}q_{1}^{2}+c_{2}q_{2}^{2}+c_{3}q_{1}q_{2}},$
(6ghiklma)
$\hat{H}=E_{0}\hat{I}_{4}+\left(\begin{array}[c]{cccc}0&b_{1}q_{1}+b_{2}q_{2}&z_{1}q_{1}+z_{2}q_{2}&0\\\
b_{1}q_{1}+b_{2}q_{2}&0&0&z_{1}q_{1}+z_{2}q_{2}\\\
z_{1}^{*}q_{1}+z_{2}^{*}q_{2}&0&0&-b_{1}q_{1}-b_{2}q_{2}\\\
0&z_{1}^{*}q_{1}+z_{2}^{*}q_{2}&-b_{1}q_{1}-b_{2}q_{2}&0\end{array}\right),$
(6ghiklmb)

here $c_{1}=b_{1}^{2}+|z_{1}|^{2}$, $c_{2}=b_{2}^{2}+|z_{2}|^{2}$,
$c_{3}=2b_{1}b_{2}+z_{1}z_{2}^{*}+z_{1}^{*}z_{2}$.

####  3.2.7 Anisotropic Dirac dispersion with fixed orientation (ADDf)

ADDf is analogous to AWDf with degeneracy of each band doubled. It appears
only in SOC case and it is represented by a red ellipse. The convention
follows the one for AWDf in Section 3.2.3. Dispersion and Hamiltonian are:

$E_{1,2,3,4}=E_{0}\pm\sqrt{c_{1}q_{1}^{2}+c_{2}q_{2}^{2}},$ (6ghiklmna) For
groups $15^{D}$, $16^{D}$ and $17^{D}$:
$\hat{H}=E_{0}\hat{I}_{4}+\left(\begin{array}[c]{cccc}b_{2}q_{2}&ib_{1}q_{1}&zq_{2}&0\\\
-ib_{1}q_{1}&-b_{2}q_{2}&0&-zq_{2}\\\ z^{*}q_{2}&0&-b_{2}q_{2}&ib_{1}q_{1}\\\
0&-z^{*}q_{2}&-ib_{1}q_{1}&b_{2}q_{2}\end{array}\right),$ (6ghiklmnb) here
$c_{1}=b_{1}^{2}$, $c_{2}=b_{2}^{2}+|z|^{2}$. For groups $38^{D}$,
$39^{D}$(points $\left(\pm\pi/a_{1},0\right)$ and
$\left(0,\pm\pi/a_{2}\right)$), $41^{D}$, $42^{D}$, $43^{D}$, $45^{D}$,
$46^{D}$(points $\left(\pm\pi/a_{1},0\right)$ and
$\left(0,\pm\pi/a_{2}\right)$), $62^{D}$(points $\left(\pm\pi/a,0\right)$ and
$\left(0,\pm\pi/a\right)$) and $64^{D}$(points $\left(\pm\pi/a,0\right)$ and
$\left(0,\pm\pi/a\right)$), $b_{2}=0$. For groups $39^{D}$(BZ-corners) and
$46^{D}$(BZ-corners), $z=0$.

####  3.2.8 Isotropic Dirac dispersion (IDD)

IDD is analogous to IWD with degeneracy of each band doubled. It appears in
cases with and without SOC. An example of IDD is dispersion near $K$-points in
graphene. IDD is represented by a red circle (when no other dispersions near
this point are possible) or by a green ring (when SB is an alternative).
Dispersion and Hamiltonian are:

$E_{1,2,3,4}=E_{0}\pm|z||\mathbf{q}|$ (6ghiklmnoa) For group $52^{D}$:
$\hat{H}=E_{0}\hat{I}_{4}+\left(\begin{array}[c]{cccc}0&0&z(q_{1}-iq_{2})&0\\\
0&0&0&z(q_{1}-iq_{2})\\\ z^{*}(q_{1}+iq_{2})&0&0&0\\\
0&z^{*}(q_{1}+iq_{2})&0&0\end{array}\right),$ (6ghiklmnob)

for groups $62^{D}$ and $64^{D}$, $z=ib_{1}$. For layer single groups
$\hat{H}_{tot}=\hat{\sigma}_{0}\otimes\hat{H}$, with $\hat{H}$ given by (6ghb)
and $E_{1}^{\uparrow}=E_{1}^{\downarrow}=E_{1}=E_{2}$,
$E_{2}^{\uparrow}=E_{2}^{\downarrow}=E_{3}=E_{4}$. For layer single groups
$z=b_{1}$ except for 66, 73 and 75.

###  3.3 Quadratic dispersions

####  3.3.1 Corner quadratic dispersion (CQ)

CQ appears at BZ corners of rectangular groups where two DNL meet. It appears
both in SOC and non-SOC cases and it is denoted by a red rectangle. The 4-fold
spinfull degenerate band at $\mathbf{q}=0$, splits into two double degenerate
bands for $\mathbf{q}\neq 0$ and away from DNLs. The splitting causing term is
of the second order (quadratic) in $\mathbf{q}$. The dispersion is:

$E_{1,2,3,4}=E_{0}+a_{1}q_{1}^{2}+a_{2}q_{2}^{2}\pm c\left|q_{1}q_{2}\right|.$
(6ghiklmnopa) For layer single groups
$E_{1}^{\uparrow}=E_{1}^{\downarrow}=E_{1}=E_{2}$,
$E_{2}^{\uparrow}=E_{2}^{\downarrow}=E_{3}=E_{4}$ and the Hamiltonian is:
$\hat{H}=\hat{\sigma}_{0}\otimes\left[(E_{0}+a_{1}q_{1}^{2}+a_{2}q_{2}^{2})\hat{\sigma}_{0}+q_{1}q_{2}\left(b_{1}\hat{\sigma}_{1}+b_{2}\hat{\sigma}_{2}\right)\right],$
(6ghiklmnopb) so that $c=\sqrt{b_{1}^{2}+b_{2}^{2}}$. For groups 32, 34, 39,
42, 46, $b_{2}=0$. For layer double groups:
$\hat{H}=(E_{0}+a_{1}q_{1}^{2}+a_{2}q_{2}^{2})\hat{I}_{4}+q_{1}q_{2}\left(\begin{array}[c]{cccc}0&ib_{1}&0&z\\\
-ib_{1}&0&-z&0\\\ 0&-z^{*}&0&-ib_{1}\\\ z^{*}&0&ib_{1}&0\end{array}\right),$
(6ghiklmnopc)

so that $c=\sqrt{b_{1}^{2}+|z|^{2}}$.

####  3.3.2 Isotropic corner quadratic dispersion (ICQ)

In the BZ corners of square groups where two DNL meet, the axis of order four
imposes $a_{1}=a_{2}$ in CQ and gives ICQ. The term _isotropic_ denotes that
disperions along $q_{1}=0$ and $q_{2}=0$ are identical. ICQ appears in both
SOC and non-SOC cases and it is marked by a red square. The dispersion is:

$E_{1,2,3,4}=E_{0}+aq^{2}\pm c\left|q_{1}q_{2}\right|.$ (6ghiklmnopqa) For
layer single groups $E_{1}^{\uparrow}=E_{1}^{\downarrow}=E_{1}=E_{2}$,
$E_{2}^{\uparrow}=E_{2}^{\downarrow}=E_{3}=E_{4}$ and the Hamiltonian is:
$\hat{H}=\hat{\sigma}_{0}\otimes\left[(E_{0}+aq^{2})\hat{\sigma}_{0}+q_{1}q_{2}\left(b_{1}\hat{\sigma}_{1}+b_{2}\hat{\sigma}_{2}\right)\right],$
(6ghiklmnopqb)

so that $c=\sqrt{b_{1}^{2}+b_{2}^{2}}$. For correps that remain irreducible
when TRS becomes broken, $b_{2}=0$. For layer double group $63^{D}$,
$a_{1}=a_{2}=a$ and $b_{1}=0$ in (6ghiklmnopc) so that $c=|z|$.

####  3.3.3 Quadratic dispersion (QD)

QD appears near isolated BZ points only in non-SOC case [35]. It is denoted by
the green star in Figure 1, since SB can appear instead. The splitting in all
directions away from the point is quadratic, as indicated by dispersion and
Hamiltonian:

$E_{1,2}^{\uparrow}=E_{1,2}^{\downarrow}=E_{0}+aq^{2}\pm\sqrt{[b_{3}(q_{1}^{2}-q_{2}^{2})+b_{1}q_{1}q_{2}]^{2}+[b_{4}(q_{1}^{2}-q_{2}^{2})+b_{2}q_{1}q_{2}]^{2}},$
(6ghiklmnopqra)
$\hat{H}=\hat{\sigma}_{0}\otimes\left\\{(E_{0}+aq^{2})\hat{\sigma}_{0}+[b_{3}(q_{1}^{2}-q_{2}^{2})+b_{1}q_{1}q_{2}]\hat{\sigma}_{1}+[b_{4}(q_{1}^{2}-q_{2}^{2})+b_{2}q_{1}q_{2}]\hat{\sigma}_{2}\right\\}.$
(6ghiklmnopqrb)

Except for groups 49, 50, 51 and 52, in (6ghiklmnopqra) and (6ghiklmnopqrb)
$b_{1}=b_{4}=0$. If $b_{1}b_{4}=b_{2}b_{3}$, two lines of accidental
degeneracy intersect at $\mathbf{q}=0$ symmetrically under rotation by
$\pi/2$, so that $\mathbf{q}=0$ is not an isolated point any more.

####  3.3.4 Isotropic quadratic dispersion (IQD)

IQD is QD which is fully isotropic. It appears only in non-SOC case [35] and
it is indicated with a green snowflake, since the appearance of SB is other
possibility. Dispersion and Hamiltonian are:

$E_{1,2}^{\uparrow}=E_{1,2}^{\downarrow}=E_{0}+\left(a\pm\sqrt{b_{1}^{2}+b_{2}^{2}}\right)q^{2},$
(6ghiklmnopqrsa)
$\hat{H}=\hat{\sigma}_{0}\otimes\left\\{(E_{0}+aq^{2})\hat{\sigma}_{0}+[2b_{2}q_{1}q_{2}+b_{1}(q_{1}^{2}-q_{2}^{2})]\hat{\sigma}_{1}+[b_{2}(q_{1}^{2}-q_{2}^{2})-2b_{1}q_{1}q_{2}]\hat{\sigma}_{2}\right\\},$
(6ghiklmnopqrsb)

with $b_{2}=0$ except for groups 65, 66, 73, 74 and 75. Unlike QD, IQD can not
be turned into a line of degeneracies by suitable choice of non-zero
parameters.

####  3.3.5 Simple band (SB)

SB is a band (spinful non-degenerate, with SOC and double spinful degenerate
in non-SOC case) which disperses quadratically near $\mathbf{q}=0$ without any
splitting (_i.e._ it is neither UNP nor UNL). It is introduced here only when
it appears as an alternative to other dispersions. These cases are denoted by
green symbols in Figures 1 and 2. The dispersion and Hamiltonian are:

$E_{1}=E_{0}+aq^{2},$ (6ghiklmnopqrsta) $\hat{H}=E_{0}+aq^{2}.$
(6ghiklmnopqrstb)

For layer single groups $\hat{H}_{tot}=(E_{0}+aq^{2})\hat{\sigma}_{0}$ and
$E_{1}^{\uparrow}=E_{1}^{\downarrow}=E_{1}$.

###  3.4 Cubic dispersions

####  3.4.1 Cubic disperson (CD)

CD appears only when SOC is included [35]. It consists of spinful double
degenerate band at $\mathbf{q}=0$ which splits cubically away from this point.
It is denoted with a green square, since IWD is other dispersion possible at
groups/BZ points form Figure 2. Dispersion and Hamiltonian are:

$\displaystyle E_{1,2}$ $\displaystyle=E_{0}+a|\mathbf{q}|^{2}$
(6ghiklmnopqrstua)
$\displaystyle\pm\sqrt{\left|z_{1}q_{1}(q_{1}^{2}-3q_{2}^{2})+z_{2}q_{2}(q_{2}^{2}-3q_{1}^{2})\right|^{2}+[b_{1}q_{1}(q_{1}^{2}-3q_{2}^{2})+b_{2}q_{2}(q_{2}^{2}-3q_{1}^{2})]^{2}},$
$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle(E_{0}+a|\mathbf{q}|^{2})\hat{\sigma}_{0}$ (6ghiklmnopqrstud)
$\displaystyle+$
$\displaystyle\left(\begin{array}[c]{cc}b_{1}q_{1}(q_{1}^{2}-3q_{2}^{2})+b_{2}q_{2}(q_{2}^{2}-3q_{1}^{2})&z_{1}q_{1}(q_{1}^{2}-3q_{2}^{2})+z_{2}q_{2}(q_{2}^{2}-3q_{1}^{2})\\\
z_{1}^{*}q_{1}(q_{1}^{2}-3q_{2}^{2})+z_{2}^{*}q_{2}(q_{2}^{2}-3q_{1}^{2})&-b_{1}q_{1}(q_{1}^{2}-3q_{2}^{2})-b_{2}q_{2}(q_{2}^{2}-3q_{1}^{2})\end{array}\right).$

If vectors
$\left\\{(z_{1}^{r},z_{2}^{r})^{T},(z_{1}^{i},z_{2}^{i})^{T}\right\\}$ are
linearly dependent and vectors
$\left\\{(z_{1}^{r},z_{2}^{r})^{T},(b_{1},b_{2})^{T}\right\\}$ are also
linearly dependent, then three accidental nodal lines intersect symmetrically
under rotation by $\pi/3$ at $\mathbf{q}=0$, so that $\mathbf{q}=0$ is not an
isolated point any more. For groups $67^{D}$, $68^{D}$, $69^{D}$ and $70^{D}$,
$b_{1}=0$ and $z_{2}=0$. For group $73^{D}$, $b_{1}=b_{2}=0$. For groups
$76^{D}$ and $77^{D}$, $b_{1}=b_{2}=0$, $z_{1}=ib_{3}$ and $z_{2}=b_{4}$.

####  3.4.2 Triangular cubic dispersion (TC)

TC appears only in SOC case, when three WNL intersect at a single point. It is
denoted by a blue triangle in Figure 2. The splitting is caused by cubic terms
in the Taylor expansion of Hamiltonian. Dispersion and Hamiltonian are:

$E_{1,2}=E_{0}+aq^{2}\pm\left|b\right|\left|q_{2}(q_{2}^{2}-3q_{1}^{2})\right|,$
(6ghiklmnopqrstuva)
$\hat{H}=(E_{0}+aq^{2})\hat{\sigma}_{0}+b\>\\!q_{2}(q_{2}^{2}-3q_{1}^{2})\hat{\sigma}_{3}.$
(6ghiklmnopqrstuvb)

TC appears in Hexagonal groups $78^{D}$ and $79^{D}$.

##  4 Discussion and conclusions

Comparison of Figs. 1 and 2, allows conclusions on how _e.g._ inclusion of SOC
affects dispersions from Figure 1. Dispersions in groups 40, 44 and 63 are
unaltered, up to numerical values of parameters, by the inclusion of SOC. In
groups 43 and 45, DNLs perpendicular to $\mathbf{k}_{1}$ are also preserved
when SOC is included. On the other hand, SOC induces DNL to split into two
WNLs in groups 9, 12, 20, 21, 24, 25, 28, 29, 30, 32 (DNL perpendicular to
$\mathbf{k}_{1}$), 33 (DNL perpendicular to $\mathbf{k}_{1}$), 34 (DNL
perpendicular to $\mathbf{k}_{1}$), 54, 56, 58 and 60. In groups 21 and 25,
SOC induces transition from CQ to PF, while in square groups 54, 56, 58 and
60, SOC turns ICQ to isotropic PF. Other SOC-induced transitions are CQ to
ADDf (groups 39 and 46) and ICQ to IDD in groups 62, 64. SOC induced
splittings at HSP in non-magnetic layers are also discussed in [17].

Dispersions presented in Section 3 are exact, in the sense that if we knew how
to solve the one-electron Schroedinger equation for a single crystal exactly,
we would get the very same formulas (with the numerical values of parameters
as an additional result). The dispersions are unaltered when electron
correlations are included, as long as they are weak enough as to preserve the
quasiparticle picture. For these reasons, each dispersion presented here can
be a research topic of its own. The synthesis of a new 2D material with the
right placement of the Fermi level, would likely trigger new research
directions.

Besides band contacts treated here, there are accidental band contacts which
appear along HSLs or away from them in the BZ. Some of these contacts are
guaranteed to exist by the topological laws. Such guaranteed contacts can be
moved but not removed by the symmetry preserving perturbations and are
tabulated for (3D) hexagonal [36], tetragonal [37] and orthorhombic [38] space
groups. These results could be used for layer groups, by subduction from
corresponding space groups to 2D BZ. On the other hand, truly accidental band
contacts can be destroyed by a perturbation that keeps the symmetry, which in
real materials can be caused by the change of temperature, pressure, strain
_etc_. Four-fold spinless degenerate points that belong to HSL, treated in
[39, 40, 41], are examples of truly accidental band contacts.

Experimental realization of a 2D material with prescribed layer symmetry is
far from being trivial, so one could search across database of 3D materials
which have already been synthesized and exfoliate easy into layers [42]. It
contains materials of almost all 3D symmetries obtained by periodic repetition
of layers perpendicularly. Dispersions presented in Section 3 are of
significance if in a 2D material the Fermi level crosses bands near $E_{0}$.
Although this cannot be guaranteed solely by group-theoretical arguments,
there are some symmetry based results which are helpful. Refs. [43, 44, 45]
give electron fillings necessary for a crystal belonging to certain space or
layer group to be insulating. When not fulfilled, the crystal is metallic (or
insulating if the electron-electron interaction is strong). For example, if in
a 2D material belonging to layer group 33, the sum of atomic numbers of all
nuclei in the primitive cell is not divisible by eight, the material is a
metal [43, 44, 45]. It would be possible then to dope the material with
electrons or holes by _e.g._ gating, especially if the material belongs to
layer group which is also a symmetry of a surface (a wallpaper group).

Authors acknowledge funding by the Ministry of Science, Technological
Development and Innovation of the Republic of Serbia provided by the Institute
of Physics Belgrade (V.D.) and Faculty of Physics (N.L.).

## References

## References

  * [1] Binghai Yan and Claudia Felser.  Topological materials: Weyl semimetals.  Annual Review of Condensed Matter Physics, 8(1):337–354, 2017.
  * [2] N. P. Armitage, E. J. Mele, and Ashvin Vishwanath.  Weyl and $\mathrm{Dirac}$ semimetals in three-dimensional solids.  Rev. Mod. Phys., 90:015001, Jan 2018.
  * [3] Heng Gao, Jörn W.F. Venderbos, Youngkuk Kim, and Andrew M. Rappe.  Topological semimetals from first principles.  Annual Review of Materials Research, 49(1):153–183, Jul 2019.
  * [4] Shuo-Ying Yang, Hao Yang, Elena Derunova, Stuart S. P. Parkin, Binghai Yan, and Mazhar N. Ali.  Symmetry demanded topological nodal-line materials.  Advances in Physics: X, 3(1):1414631, 2018.
  * [5] Chen Fang, Hongming Weng, Xi Dai, and Zhong Fang.  Topological nodal line semimetals.  Chinese Physics B, 25(11):117106, nov 2016.
  * [6] Si Li, Zhi-Ming Yu, Yugui Yao, and Shengyuan A. Yang.  Type-$\mathrm{II}$ topological metals.  Frontiers of Physics, 15(4):43201, 2020.
  * [7] Min-Xue Yang, Wei Luo, and Wei Chen.  Quantum transport in topological nodal-line semimetals.  Advances in Physics: X, 7(1):2065216, 2022.
  * [8] A. A. Burkov, M. D. Hook, and Leon Balents.  Topological nodal semimetals.  Phys. Rev. B, 84:235126, Dec 2011.
  * [9] Michael Phillips and Vivek Aji.  Tunable line node semimetals.  Phys. Rev. B, 90:115111, Sep 2014.
  * [10] G. E. Volovik.  Flat band in topological matter.  Journal of Superconductivity and Novel Magnetism, 26:2887 – 2890, Sep 2013.
  * [11] Ying-Ming Xie, Xue-Jian Gao, Xiao Yan Xu, Cheng-Ping Zhang, Jin-Xin Hu, Jason Z. Gao, and K. T. Law.  Kramers nodal line metals.  Nature Communications, 12:3064, May 2021.
  * [12] Lin Wu, Feng Tang, and Xiangang Wan.  Symmetry-enforced band nodes in 230 space groups.  Phys. Rev. B, 104:045107, Jul 2021.
  * [13] Jian Yang, Chen Fang, and Zheng-Xin Liu.  Symmetry-protected nodal points and nodal lines in magnetic materials.  Phys. Rev. B, 103:245141, Jun 2021.
  * [14] Chen Fang, Yige Chen, Hae-Young Kee, and Liang Fu.  Topological nodal line semimetals with and without spin-orbital coupling.  Phys. Rev. B, 92:081201, Aug 2015.
  * [15] Deping Guo, Pengjie Guo, Shijing Tan, Min Feng, Limin Cao, Zheng-Xin Liu, Kai Liu, Zhong-Yi Lu, and Wei Ji.  Two-dimensional $\mathrm{Dirac}$-line semimetals resistant to strong spin–orbit coupling.  Science Bulletin, 67(19):1954–1957, 2022.
  * [16] Pengjie Guo, Chen Peng, Zheng-Xin Liu, Kai Liu, and Zhong-Yi Lu.  Symmetry-enforced two-dimensional $\mathrm{Dirac}$ node-line semimetals.  Materials Futures, 2022.
  * [17] N Lazić, V Damljanović, and M Damnjanović.  Fully linear band crossings at high symmetry points in layers: classification and role of spin–orbit coupling and time reversal.  Journal of Physics A: Mathematical and Theoretical, 55(32):325202, aug 2022.
  * [18] Vladimir Damljanović, Igor Popov, and Radoš Gajić.  Fortune teller fermions in two-dimensional materials.  Nanoscale, 9:19337–19345, 2017.
  * [19] V Damljanović, N Lazić, A Šolajić, J Pešić, B Nikolić, and M Damnjanović.  Peculiar symmetry-protected electronic dispersions in two-dimensional materials.  Journal of Physics: Condensed Matter, 32(48):485501, sep 2020.
  * [20] V Damljanović and R Gajić.  Existence of $\mathrm{Dirac}$ cones in the $\mathrm{Brillouin}$ zone of diperiodic atomic crystals according to group theory.  Journal of Physics: Condensed Matter, 28(8):085502, feb 2016.
  * [21] V Damljanović and R Gajić.  Addendum to ‘$\mathrm{Existence}$ of $\mathrm{Dirac}$ cones in the $\mathrm{Brillouin}$ zone of diperiodic atomic crystals according to group theory’.  Journal of Physics: Condensed Matter, 28(43):439401, sep 2016.
  * [22] Xiaolong Feng, Jiaojiao Zhu, Weikang Wu, and Shengyuan A. Yang.  Two-dimensional topological semimetals.  Chinese Physics B, 30(10):107304, nov 2021.
  * [23] Feng Tang and Xiangang Wan.  Exhaustive construction of effective models in 1651 magnetic space groups.  Phys. Rev. B, 104:085137, Aug 2021.
  * [24] Zhi-Ming Yu, Zeying Zhang, Gui-Bin Liu, Weikang Wu, Xiao-Ping Li, Run-Wu Zhang, Shengyuan A. Yang, and Yugui Yao.  Encyclopedia of emergent particles in three-dimensional crystals.  Science Bulletin, 67(4):375–380, 2022.
  * [25] Gui-Bin Liu, Zeying Zhang, Zhi-Ming Yu, Shengyuan A. Yang, and Yugui Yao.  Systematic investigation of emergent particles in type-$\mathrm{III}$ magnetic space groups.  Phys. Rev. B, 105:085117, Feb 2022.
  * [26] Zeying Zhang, Gui-Bin Liu, Zhi-Ming Yu, Shengyuan A. Yang, and Yugui Yao.  Encyclopedia of emergent particles in type-$\mathrm{IV}$ magnetic space groups.  Phys. Rev. B, 105:104426, Mar 2022.
  * [27] Zeying Zhang, Weikang Wu, Gui-Bin Liu, Zhi-Ming Yu, Shengyuan A. Yang, and Yugui Yao.  Encyclopedia of emergent particles in 528 magnetic layer groups and 394 magnetic rod groups, 2022.
  * [28] Feng Tang and Xiangang Wan.  Complete classification of band nodal structures and massless excitations.  Phys. Rev. B, 105:155156, Apr 2022.
  * [29] Mois I. Aroyo, Asen Kirov, Cesar Capillas, J. M. Perez-Mato, and Hans Wondratschek.  Bilbao Crystallographic Server. II. Representations of crystallographic point groups and space groups.  Acta Crystallographica Section A, 62(2):115–128, Mar 2006.
  * [30] Luis Elcoro, Barry Bradlyn, Zhijun Wang, Maia G. Vergniory, Jennifer Cano, Claudia Felser, B. Andrei Bernevig, Danel Orobengoa, Gemma de la Flor, and Mois I. Aroyo.  Double crystallographic groups and their representations on the Bilbao Crystallographic Server.  Journal of Applied Crystallography, 50(5):1457–1477, Oct 2017.
  * [31] Božidar Nikolić, Ivanka Milošević, Tatjana Vuković, Nataša Lazić, Saša Dmitrović, Zoran Popović, and Milan Damnjanović.  Irreducible and site-symmetry-induced representations of single/double ordinary/grey layer groups.  Acta Crystallographica Section A, 78(2):107–114, Mar 2022.
  * [32] J. L. Mañes.  Existence of bulk chiral fermions and crystal symmetry.  Phys. Rev. B, 85:155118, Apr 2012.
  * [33] Milan Damnjanović and Ivanka Milošević.  Full symmetry implementation in condensed matter and molecular physics—modified group projector technique.  Physics Reports, 581:1 – 43, 2015.
  * [34] V Kopsky and D. B. Litvin.  International Tables of Crystallography Volume E: Subperiodic Groups.  Kluwer Academic Publishers, 2002.
  * [35] Weikang Wu, Ying Liu, Zhi-Ming Yu, Y. X. Zhao, Weibo Gao, and Shengyuan A. Yang.  Higher-order nodal points in two dimensions, 2021.
  * [36] J. Zhang, Y.-H. Chan, C.-K. Chiu, M. G. Vergniory, L. M. Schoop, and A. P. Schnyder.  Topological band crossings in hexagonal materials.  Phys. Rev. Materials, 2:074201, Jul 2018.
  * [37] Moritz M. Hirschmann, Andreas Leonhardt, Berkay Kilic, Douglas H. Fabini, and Andreas P. Schnyder.  Symmetry-enforced band crossings in tetragonal materials: Dirac and $\mathrm{Weyl}$ degeneracies on points, lines, and planes.  Phys. Rev. Materials, 5:054202, May 2021.
  * [38] Andreas Leonhardt, Moritz M. Hirschmann, Niclas Heinsdorf, Xianxin Wu, Douglas H. Fabini, and Andreas P. Schnyder.  Symmetry-enforced topological band crossings in orthorhombic crystals: Classification and materials discovery.  Phys. Rev. Materials, 5:124202, Dec 2021.
  * [39] Peng-Jie Guo, Yi-Wen Wei, Kai Liu, Zheng-Xin Liu, and Zhong-Yi Lu.  Eightfold degenerate fermions in two dimensions.  Phys. Rev. Lett., 127:176401, Oct 2021.
  * [40] Peng-Jie Guo, Yi-Wen Wei, Kai Liu, Zheng-Xin Liu, and Zhong-Yi Lu.  Erratum: Eightfold degenerate fermions in two dimensions [$\mathrm{Phys.Rev.Lett.}$ 127, 176401 (2021)].  Phys. Rev. Lett., 129:049901, Jul 2022.
  * [41] Jialin Gong, Jianhua Wang, Hongkuan Yuan, Zeying Zhang, and Xiaotian Wang.  Dirac phonons in two-dimensional materials, 2022.
  * [42] Nicolas Mounet, Marco Gibertini, Philippe Schwaller, Davide Campi, Andrius Merkys, Antimo Marrazzo, Thibault Sohier, Ivano Eligio Castelli, Andrea Cepellotti, Giovanni Pizzi, and Nicola. Marzari.  Two-dimensional materials from high-throughput computational exfoliation of experimentally known compounds.  Nature Nanotechnology, 13:246 – 252, Mar 2018.
  * [43] Haruki Watanabe, Hoi Chun Po, Michael P. Zaletel, and Ashvin Vishwanath.  Filling-enforced gaplessness in band structures of the 230 space groups.  Phys. Rev. Lett., 117:096404, Aug 2016.
  * [44] Haruki Watanabe, Hoi Chun Po, Ashvin Vishwanath, and Michael Zaletel.  Filling constraints for spin-orbit coupled insulators in symmorphic and nonsymmorphic crystals.  Proceedings of the National Academy of Sciences, 112(47):14551–14556, 2015.
  * [45] Benjamin J. Wieder and C. L. Kane.  Spin-orbit semimetals in the layer groups.  Phys. Rev. B, 94:155108, Oct 2016.