#  HyGNN: Drug-Drug Interaction Prediction via Hypergraph Neural Network

Khaled Mohammed Saifuddin*, Briana Bumgardner¶, Farhan Tanvir§, Esra Akbas†
*†Georgia State University, USA
¶ Rice University, USA
§ Oklahoma State University, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Drug-Drug Interactions (DDIs) may hamper the functionalities of drugs, and in
the worst scenario, they may lead to adverse drug reactions (ADRs). Predicting
all DDIs is a challenging and critical problem. Most existing computational
models integrate drug-centric information from different sources and leverage
them as features in machine learning classifiers to predict DDIs. However,
these models have a high chance of failure, especially for new drugs when all
the information is not available. This paper proposes a novel Hypergraph
Neural Network (HyGNN) model based on only the Simplified Molecular Input Line
Entry System (SMILES) string of drugs, available for any drug, for the DDI
prediction problem. To capture the drug chemical structure similarities, we
create a hypergraph from drugs’ chemical substructures extracted from the
SMILES strings. Then, we develop HyGNN consisting of a novel attention-based
hypergraph edge encoder to get the representation of drugs as hyperedges and a
decoder to predict the interactions between drug pairs. Furthermore, we
conduct extensive experiments to evaluate our model and compare it with
several state-of-the-art methods. Experimental results demonstrate that our
proposed HyGNN model effectively predicts DDIs and impressively outperforms
the baselines with a maximum F1 score, ROC-AUC, and PR-AUC of 94.61%, 98.69%,
and 98.68%, respectively. Finally, we show that our models also work well for
new drugs.

###### Index Terms:

Drug-Drug Interaction, Graph Neural Network, Hypergraph, Hypergraph Neural
Network, Hypergraph Edge Encoder

##  I Introduction

Many patients, especially those who suffer from chronic diseases such as high
blood pressure, cancer, and heart failure, often consume multiple drugs
concurrently for their disease treatment. Simultaneous usage of multiple drugs
may result in Drug-Drug Interactions (DDIs). These interactions may
unexpectedly reduce the efficacy of drugs and even may lead to adverse drug
reactions (ADRs) [1, 2]. Therefore, it is important to identify potential DDIs
early to minimize these adverse effects. However, since clinical trials to
identify DDIs are performed on a few patients for a brief period [3], many
potential new drug DDIs remain undiscovered before it is open to the market.
Also, it is too expensive to do clinical experiments with all possible drug
pairs. Thus, there is an obvious need for a computational model to detect DDIs
and mitigate unanticipated reactions automatically.

With the availability of public databases, including drug-related information
like DrugBank111https://go.drugbank.com/, STITCH222http://stitch.embl.de/,
SIDER333http://sideeffects.embl.de/,
PubChem444https://pubchem.ncbi.nlm.nih.gov/, KEGG555https://www.kegg.jp/,
etc., different computational models have been proposed to detect DDIs [4, 5].
Some of these models consider drug pairs’ chemical structure SMILES similarity
or binding properties [6]. SMILES is a specification that uses ASCII
characters to define molecular structures explicitly. With a string of
characters, SMILES may depict a three-dimensional chemical structure. On the
other hand, out of the entire chemical structure, only a few substructures are
responsible for chemical reactions between drugs, and the rest are less
relevant [7]. However, considering the whole chemical structure may create a
bias toward irreverent substructures and thus undermine the DDIs prediction
[8]. With the increasing ability of more relational information about drugs,
most of the current methods integrate multiple data sources to extract drug
features such as side effects, target protein, pathways, and indications [9,
10].

Network-based methods have recently been explored in this domain, where drug
networks are constructed based on drugs’ known DDIs. Most of the graph-based
methods consider a dyadic relationship between drugs. They operate on a simple
regular graph where each vertex is a drug, and each edge shows a connection
between two nodes. However, some methods also consider the relations of drugs
to other biological entities to create heterogeneous graphs. Then, different
topological information is extracted from the network to predict unknown links
(i.e., interactions) between drugs. With the current advancement of the graph
neural network (GNN), different GNN models for DDI prediction problems are
proposed [5, 11]. While some of them create heterogeneous graphs manually from
different resources, some of them create biomedical knowledge graphs by
extracting triples from raw data (e.g., DrugBank) [12, 13, 14]. In these
graphs, different entities, such as drugs, proteins, and side effects, are
represented as nodes, and relations between those entities are represented as
the edges. While including multiple drug-centric information from different
sources would help to learn about DDI and have achieved strong performance, it
is challenging to integrate data from different resources. It is also tough to
interpret which information is the most valuable in DDI prediction which needs
a strong knowledge of biomedical entities and this is challenging for drugs in
the early development stage. Moreover, multiple information may not always be
available for all drugs, specially new drugs; therefore, these models may fail
whenever any information is unavailable [15].

In this paper, to address these problems, we present a novel GNN-based
approach for DDI prediction by only considering the SMILES string of the
drugs, which is available for all drugs. Our method relies on the hypothesis
that similar drugs behave similarly, are likely to interact with the same
drugs, and two drugs are similar if they have similar substructures as
functional groups in their SMILES strings [16, 17]. Finding similarities
between SMILES strings based on their common substructures is a challenging
task. To properly depict the structural-based similarity between drugs, we
present them in a hypergraph setting, representing drugs as hyperedges
connecting many substructures as nodes. A hypergraph is a unique model of a
graph with hyperedges. Unlike a regular graph where the degree of each edge is
2, hyperedge is degree-free; it can connect an arbitrary number of nodes [18,
19, 20, 21]. After constructing the hypergraph, we develop a Hypergraph Neural
Network (HyGNN), a model that learns the DDIs by generating and using the
representation of hyperedges as drugs. HyGNN has an encoder-decoder
architecture. First, we present a novel hypergraph edge encoder to generate
the embedding of drugs. Afterward, the pair-wise representations of drugs are
passed through decoder functions to predict a binary score for each drug pair
that represents whether two drugs interact. The main contributions of this
paper are summarized as follows:

  * •

Hypergraph construction from SMILES strings:  We construct a novel hypergraph
to depict the drugs’ similarities. In the hypergraph, while each substructure
extracted from the drugs’ SMILES strings is represented as a node, each drug,
consisting of a set of unique substructures, is represented as a hyperedge.
This hypergraph represents the higher-level connections of substructures and
drugs, which helps us to define complex similarities between chemical
structures and drugs. Also, this helps us to learn better representation for
drugs with GNN models with a passing message not only between 2 nodes but
between many nodes and also between nodes and hyperedges.

  * •

Hypergraph GNN: To learn and predict DDIs, we propose a novel hypergraph GNN
model, called HyGNN, consisting of a novel hyperedge encoder and a decoder.
Encoder exploits two layers where the first layer generates the embedding of
nodes by aggregating the embedding of hyperedges. Then, the second layer
generates the embedding of hyperedges (i.e., drugs) by aggregating the
embedding of nodes. Since not all but a few substructures are mainly
significant in chemical reactions, we use attention mechanisms to learn the
significance of substructures (nodes) for drugs (edge) and chemical reactions.
Furthermore, a decoder is modeled to predict the DDIs by taking the pair-wise
drug representations as input. Our method solely utilizes drugs’ chemical
structure data to predict DDIs without requiring any other information or
strong knowledge of biomedical entities. Chemical structures are obtained from
SMILES strings that any drug has. Therefore, our model is applicable to any
drugs, including new drugs, without other information, such as side effects
and DDIs.

  * •

Extensive experiments:  We conduct extensive experiments to compare our
proposed model with the state-of-the-art models. The results with different
accuracy measures show that our method significantly outperforms all the
baseline models. Also, we show with case studies that our model can find not
only new DDIs for existing drugs but also DDIs for new drugs.

The rest of the paper is structured as follows. First, we briefly review the
related work on DDI and hypergraph GNN in Section II. Section III describes
our proposed HyGNN model, including hypergraph construction from SMILES
strings, encoder and decoder layers of the HyGNN model works for generating
the drug embedding and predicting DDIs. Next, in Section IV, we describe the
experimental results and discussion. Finally, a conclusion is given in Section
V.

##  II Related Work

Many works have been proposed for the DDI prediction problem over the years.
It can be categorized into similarity-based, classification-based, and
network-based methods. Previous works assume that similar drugs have similar
interaction profiles and define different similarities between drugs.
Traditionally, pharmacological, topological, or semantic similarity based on
statistical learning is utilized to predict DDIs. Vilar et al. [22] identify
the DDIs based on molecular similarities. They represent each drug by a
molecular fingerprint, a bit vector reflecting the presence/absence of a
molecular feature. [23] develops INDI that uses seven different drug-drug
similarity measures learned from drug side-effect, fingerprints, therapeutic
effects, etc. Another vital research is [24] that incorporates four different
biological information (e.g., target, transporter, enzyme, and carrier) of
drugs to measure the similarity of drug pairs.

Some models extract features of drugs from various biological entities and
drug interaction information and apply different machine learning (ML) methods
for DDI training [25, 8, 26, 27]. Davazdahemami and Delen [26] constructed a
graph containing both drug-protein and drug-side effect interactions. They
also employed a classification method on the feature set and produced many
similarities and centrality metrics based on the network. These features were
then fed into four machine models. Luo et al. [27] propose a DDI prediction
server that provides real-time DDI predictions based only on molecular
structure. The server docks a drug’s chemical structure across 611 human
proteins to create a 611-dimensional docking vector. The drug pair features
are created by concatenating the docking vectors for drug pairings. Finally,
utilizing these features, a logistic regression model for DDI prediction is
developed. Ibrahim et al.[25] first extract different similarity features and
employ logistic regression to pick the best feature; later, the best feature
is used in six different ML classifiers to predict DDIs. One primary problem
in DDI prediction is the lack of negative samples. A solution to address this
problem is proposed in [8], named DDI-PULearn. It first generates negative
samples using one-class SVM and kNN. Then positive and negative samples are
used to predict DDIs.

Last decade, network-based models got great attention for drug-related
problems. Some researchers construct a drug network using known DDIs where
drugs are nodes, and interacting drugs are connected by a link [28]. Moreover,
heterogeneous information networks leveraging different biomedical entities,
such as proteins, side effects, pathways, etc., are also used to address
similar problems [9]. As a different model, [29] constructs a molecular graph
for each drug from its SMILES representation. Moreover, existing network-based
models often extract drug embedding and directly learn latent node embedding
using various embedding methodologies. As a result, their capacity to obtain
specific neighborhood information on any organization in KG is restricted.

Recently, GNN has shown promising performance in different fields that include
drug discovery [30], drug abuse detection [31], and drug-drug interaction [32,
33, 29], etc. Decagon [5] created a knowledge graph based on protein-protein,
drug-drug, and drug-protein interactions. They also created a relational graph
convolutional neural network for predicting multi-relational links in
multimodal networks. Furthermore, they used a novel graph auto-encoder
technique to create an end-to-end trainable model for link prediction on a
multimodal graph. CASTER [7] recently created a dictionary learning framework
for predicting DDIs based on drug chemical structures. They predict drug-drug
interactions using the drug’s molecular structure in a text format of SMILES
[34] strings representation and outperform numerous deep learning approaches
such as DeepDDI [35] and molVAE [36]. CASTER uses a sequential pattern mining
approach to identify the common substrings included in the SMILES strings
supplied during the training phase, which are then translated to an embedding
using an encoder module. These features are then transformed into linear
coefficients fed to a decoder and a predictor to yield DDI predictions. [33]
constructs a drug network where two drugs are connected if they share common
chemical substructures. Then, they apply different GNN models on the network
to get the representation of drugs, and drug pair representation is passed to
the ML classifier to predict interaction.

More recently, hypergraph and hypergraph neural network models have been
developed to capture higher-order relations between different objects [37, 19,
38, 39]. All these works learn the representation of nodes and use these for
node classification problems. Our HyGNN model is the first attempt to use
hypergraph structure for DDI problems and, in general, drug-related problems.
Also, our model aims to learn the representation of hyperedges and edge pair
classification, which is different from current hypergraph neural network
models.

##  III HyGNN model for DDI

In this section, we first define our DDI prediction problem and then summarize
the preliminaries, model, and settings (Section III-A). Then we explain our
hypergraph construction step with substructures extraction from Drugs (Section
III-B). After that, we introduce our proposed HyGNN model with attention-based
encoder and decoder layers (Section III-C).

###  III-A Problem Formulation

Our goal is to develop a computational model that takes a drug pair
$(D_{x},D_{y})$ as input and predicts whether there exists an interaction
between this drug pair. Each drug is represented by the SMILES string. SMILES
is a unique chemical representation of a drug that consists of a sequence of
symbols of molecules and the bonds between them.

Most of the graph-based existing DDI methods consider a dyadic relationship
between drugs. This simple graph type considers an edge that can connect a
maximum of two objects. However, there could be a more complex network in real
life where an arbitrary number of nodes may interact as a group, so they could
be connected through a hyperedge (i.e., triadic, tetradic, etc.). A hypergraph
can be used to formulate such a complex network. A formal definition of the
hypergraph is given below.

_Hypergraph:_ _A hypergraph is a special kind of graph defined as $G=(V,E)$
where $V=\\{v_{1},....,v_{m}\\}$ is the set of nodes and
$E=\\{e_{1}.....e_{n}\\}$ is the set of hyperedges. Each hyperedge $e_{j}$ is
degree-free and consists of an arbitrary number of nodes._

Like the adjacency matrix of a regular graph, a hypergraph can be denoted by
an incidence matrix $H$ with $H_{i,j}=1$ if the node $i$ is in the hyperedge
$j$ as $v_{i}\in e_{j}$ and $H_{i,j}=0$ otherwise.

Figure 1: System architecture of the proposed method. The First step is to
construct a hypergraph network of drugs where each drug is a hyperedge, and
frequent chemical substructures of drugs are the nodes. The second step is to
design a hypergraph neural network (HyGNN) model with an attention-based
encoder for hyperedge (drug) representation learning and decoder for DDI
learning

In this paper, we construct a hypergraph network of drugs where each drug is a
hyperedge, and the chemical substructures of drugs are the nodes. The chemical
structures of a drug can be obtained from the SMILES, a unique chemical
representation of a drug. We design a novel hypergraph neural networks model
as an encoder-decoder architecture to accomplish the DDI prediction task. The
encoder part exploits an attention mechanism to learn the representations of
hyperedges (drugs) by giving attention to edges and nodes (substructures). The
decoder predicts the interaction between drug pairs using latent learned drug
features. The whole system is trained in a semi-supervised fashion. The
functional architecture of this paper is shown in Figure 1. Our proposed model
consists of two steps:

  1. 1.

Hypergraph construction from SMILES,

  2. 2.

DDI prediction with hypergraph neural networks

    1. (a)

Encoder: Drug (Hyperedge) representation learning

    2. (b)

Decoder: DDI prediction

###  III-B Drug Hypergraph Construction

We construct a hypergraph to depict the structural similarities among drugs.
At first, we decompose all the drugs’ SMILES into a set of substructures. In
our drug hypergraph, these substructures are used as nodes. Moreover, each
drug with a certain number of substructures is represented by a hyperedge.
Drugs as hyperedges may connect with other drugs employing shared
substructures as nodes. This hypergraph represents the higher-level
connections of substructures and drugs, which may help to define complex
similarities between chemical structures and drugs. Also, this helps us to
learn better representation for drugs with GNN models with the passing message
not only between 2 nodes but between many nodes and also between nodes and
edges.

Substructures can be generated by utilizing different algorithms such as ESPF
[40], $k$-mer[41], strobemers [42], etc. In this project, we use ESPF and
$k$-mer to see the effect of substructures on the results. While $k$-mer use
all extracted substructures, ESPF selects the most frequent ones. Algorithm 1
briefly shows the hypergraph construction steps.

Input:  $SMILES\\_strings$

Output: Hypergraph incident matrix: $H$

Call $Substructure\\_Decom(SMILES\\_strings)$;

/* Substructure_Decom() could be ESPF or $k$-mer that decomposes SMILES into
substructures. It returns a list of unique substructures and drug dictionary
that will be used in the following for loop */

for  _each Substructure in Substructure_list_ do

if  _Substructure is in Drug_dict[SMILES]_ then

$H[i,j]=1$ ;

/* i,j is the id of substructure and drug, respectively. */

end if

end for

Output: $H$, Hypergraph incident matrix

Algorithm 1 Drug Hypergraph Construction

Input:  Set of initial SMILES tokens ${S}$ as atoms and bonds, set of
tokenized SMILES strings ${TS}$, frequency threshold $\alpha$, and size
threshold ${L}$ for $S$.

for  _${t=1\dots,L}$_ do

$(S1,S2)$, $f$ $\leftarrow$ scan $TS$ ;

/* $(S1,S2)$ is the frequentest consecutive tokens. */

if  _$f <\alpha$_ then

break ;

/* $(S1,S2)$’s frequency lower than threshold */

end if

$TS\leftarrow$ find $(S1,S2)\in$ $TS$, replace with $(S1S2)$ ;

/* update $TS$ with the combined token $(S1S2)$ */

$S\leftarrow S\cup(S1S2)$ ;

/* add $(S1S2)$ to the token vocabulary set $S$ */

end for

Output:  $TS$, the updated tokenized drugs; $S$, the updated token vocabulary
set.

Algorithm 2 Explainable Substructure Partition Fingerprint (ESPF)

ESPF: Like the concept of sub-word units in the natural language processing
domain, ESPF is a powerful tool that decomposes sequential structures into
interpretable functional groups. They consider that a few substructures are
mainly responsible for drug chemical reactions, so they extract frequent
substructures as influential ones. The ESPF algorithm is shown in Algorithm 2.
Given a set of drug SMILES strings $S$, ESPF finds the frequent repetitive
moderate-sized substructures from the set and replaces the original sequence
with the substructures. If the frequency of each substructure in $S$ is above
a predefined threshold, it is added to a substructure list as a vocabulary.
Substructures appear in this list in the most to least frequent order. We use
this vocabulary list as our nodes and decompose any drug into a sequence of
frequent substructures concerning those. For any given drug, we partition its
SMILES in order of frequency, starting from the highest frequency. An example
of partitioning a SMILES of a drug, DB00226, is as follows.

NC(N)=NCC1COC2(CCCCC2)O1

$\Downarrow$

N C(N) =N CC1 CO C2 (CCC CC2) O1.

k-mer: $k$-mer is a tool to decompose sequential structures into subsequences
of length $k$. It is widely used in biological sequence analysis and
computational genomics. Similar to n-gram in natural language processing, a
k-mer is a sequence of k characters in a string (or nucleotides in a DNA
sequence). To get all k-mers from a sequence, we need to get the first k
characters, then move just a single character to start the next k-mer, and so
on. Effectively, this will create sequences that overlap in k-1 positions.
Pseudocode for $k$-mer is shown in Algorithm 3.

For a sequence of length $l$, there are $l-k+1$ numbers of $k$-mers and
$n^{k}$ total possible number of $k$-mers, where $n$ is the number of
monomers. $k$-mers are like words of a sentence. $k$-mers help to bring out
semantic features from a sequence. For example, for a sequence NCCO, monomers:
{N, C, and O}, 2-mers: {NC, CC, CO} , 3-mers: {NCC, CCO}.

Input:  $SMILES\\_strings$, size threshold $k$

Substructure_list: []

Drug_dict:{}

for  _each $SMILES$ in $SMILES\\_strings$_ do

Lst=[]

for  _$x$ in range ($l$-$k$+1)_ do

/* $l$ is the length of $SMILES$ */

$C=SMILES[x:x+k]$

$Lst.append(C)$

$Substructure\\_list.append(C)$

end for

$Drug\\_dict[SMILES]=Lst$

end for

Output:  Drug_dict, Substructure_list

Algorithm 3 $k$-mer

###  III-C Hypergraph Neural Network (HyGNN) for DDI Prediction

We utilize the Hypergraph Neural Network (HyGNN) for DDI prediction. HyGNN
includes an encoder, which generates the embedding of drugs, and a decoder
that uses the embedding of drugs from the encoder to predict whether a drug
pair interact or not.

####  III-C1 Drug Representation learning via HyGNN \- Encoder

To detect interacting pairs of drugs, we need features of drug pairs and,
thus, features of drugs that encode their structure information. To generate
features of drugs, we propose a novel hypergraph edge encoder. It creates
${d}^{{}^{\prime}}$ dimensional embedding vectors for hyperedges (drugs)
instead of nodes as in regular GNN models. Given the edge feature matrix,
$F\in R^{E*d}$ and incidence matrix $H\in R^{V*E}$, the encoder of the HyGNN
generates a feature vector of $d^{\prime}$ dimension through learning a
function $Z$. Any layer (e.g., ${(l+1)^{th}}$ layer) of HyGNN can be expressed
as

$F^{l+1}=Z(F^{l},H^{T}).$ (1)

We consider the hypergraph edge encoder with a memory-efficient self-attention
mechanism. It consists of two different levels of attention: (1) hyperedge-
level attention, (2) node-level attention.

While hyperedge-level attention aggregates the hyperedge information to get
the representation of nodes, node-level attention layer aggregates the
connected vertex information to get the representation of hyperedges. In
general, we define the HyGNN attention layers as

$\displaystyle p_{i}^{l}=\mathbf{A_{E}}^{l}(p_{i}^{l-1},{q_{j}^{l-1}}|\forall
e_{j}\in E_{i}),$ (2) $\displaystyle
q_{j}^{l}=\mathbf{A_{V}}^{l}(q_{j}^{l-1},{p_{i}^{l}}|\forall v_{i}\in e_{j})$
(3)

where AE is an edge aggregator that aggregates features of hyperedges to get
the representation $p_{i}^{l}$ of node $v_{i}$ in layer-$l$ and $E_{i}$ is the
set of hyperedges that are connected to node $v_{i}$. Similarly, AV is a node
aggregator that aggregates features of nodes to get the representation
$q_{j}^{l}$ of hyperedge $e_{j}$ in layer-$l$ and $v_{i}$ is the node that
connects to hyperedge $e_{j}$.

Hyperedge-level attention: In a hypergraph, each node may belong to multiple
numbers of edges. However, the contribution of hyperedges to a node may not be
equal. That is why we design an attention mechanism to highlight the crucial
hyperedges and aggregate their features to compute the node feature
$p_{i}^{l}$ of node $v_{i}$. With the attention mechanism, $p_{i}^{l}$ is
defined as

$p_{i}^{l}=\alpha\left(\sum_{e_{j}\in{E_{i}}}Y_{ij}W_{1}q_{j}^{l-1}\right)$
(4)

where $\alpha$ is a nonlinear activation function, $W_{1}$ is a trainable
weight matrix that linearly transforms the input hyperedge feature into a
high-level, $E_{i}$ is the set of hyperedges connected to node $v_{i}$, and
$Y_{ij}$ is the attention coefficient of hyperedge $e_{j}$ on node $v_{i}$.
The attention coefficient is defined as

$Y_{ij}=\frac{\exp({\mathbf{e_{j}}})}{\sum_{e_{k}\in
E_{i}}\exp(\mathbf{e_{k}})}$ (5)

$\mathbf{e_{j}}=\beta{(W_{2}q_{j}^{l-1}*W_{3}p_{i}^{l-1})}$ (6)

where $\beta$ is a LeakyReLU activation function, $W_{2}$, $W_{3}$ are the
trainable weight matrices, and $*$ is the element-wise multiplication.

Node-level attention: Each hyperedge in a hypergraph consists of an arbitrary
number of nodes. However, the importance of nodes in a hyperedge construction
may not be the same. We design a node-level attention mechanism to highlight a
hyperedge’s important nodes and aggregate their features accordingly to
compute the hyperedge feature $q_{j}^{l}$ of hyperedge $e_{j}$. With the
attention mechanism, $q_{j}^{l}$ is defined as

$q_{j}^{l}=\alpha\left(\sum_{v_{i}\in{e_{j}}}X_{ji}W_{4}p_{i}^{l}\right)$ (7)

where $W_{4}$ is a trainable weight matrix, and $X_{ji}$ is the attention
coefficient of node $v_{i}$ in the hyperedge $e_{j}$. The attention
coefficient is defined as

$X_{ji}=\frac{\exp({\mathbf{v_{i}}})}{\sum_{v_{k}\in
e_{j}}\exp(\mathbf{v_{k}})}$ (8)

$\mathbf{v_{i}}=\beta{(W_{5}p_{i}^{l}*W_{6}q_{j}^{l-1})}$ (9)

where $v_{k}$ is the node that belongs to hyperedge $e_{j}$, $W_{5}$, $W_{6}$
are the trainable weight matrices.

Our hypergraph edge encoder model works based on these two attention layers
that can capture high-order relations among data. Given the input hyperedge
features, we first gather them to get the representation of nodes with
hyperedge-level attention, then we gather the obtained node features to get
the representation of hyperedges with node-level attention.

####  III-C2 DDI prediction - Decoder

After getting the representation of drugs from the encoder layer, our target
is to predict whether a given drug pair interacts or not. To accomplish this
target, we design a decoder.

Given the vector representations $(q_{x},q_{y})$ of drug pairs $(D_{x},D_{y})$
as input, the decoder assigns a score, $p_{x,y}$ to each pair through a
decoder function defined as

$p_{x,y}=\gamma(q_{x},q_{y})$ (10)

We use two different types of decoder functions:

MLP: After concatenating the features of drug pairs, we pass it through a
multi-layer perceptron (MLP), which returns a scalar score for each pair

$\displaystyle\begin{aligned}
\gamma(q_{x},q_{y})=f_{2}(f_{1}(q_{x}\mathbin{\|}q_{y}))\\\ \end{aligned}$
(11)

where $f_{1}$ and $f_{2}$ are two different layers of MLP, and $||$ is the
concatenation operation.

Dot product: We compute a scalar score for each edge by performing element-
wise dot product between features of drug pairs using

$\gamma(q_{x},q_{y})=q_{x}\cdot q_{y}$ (12)

Afterward, we pass the decoder output through a sigmoid function
$\sigma({\gamma(q_{x},q_{y})})$ that generates predicted labels, $Y^{\prime}$,
within the range 0 to 1. Any output value closer to 1 implies a high chance of
interaction between two drugs.

####  III-C3 Training the whole model

We consider the DDI prediction problem as a binary classification problem
predicting whether there is an interaction between drug pairs or not. As a
binary classification problem, we train our entire encoder-decoder
architecture using a binary cross-entropy loss function defined as

$loss=-\sum_{i=1}^{N}\left({Y_{i}\log
Y_{i}^{{}^{\prime}}+(1-Y_{i})\log(1-Y_{i}^{{}^{\prime}})}\right)$ (13)

where $N$ is the total number of samples, $Y_{i}$ is the actual label, and
$Y_{i}^{{}^{\prime}}$ is the predicted label.

###  III-D Complexity

Our method is highly efficient with paralyzing across the edges and nodes
[43]. The hyperedge encoder generates ${d}^{{}^{\prime}}$ dimensional
embedding vector for each hyperedge with a given initial feature as $d$
dimensional vector using two-level attention. Hence, the time complexity in
the encoder part is the cumulative complexity of attention layers. According
to equation 4, the complexity in the hyperedge-level attention can be
expressed as: $O(|E|d{d}^{{}^{\prime}}+|V|D{d}^{{}^{\prime}})$, where $D$ is
the average degree of nodes. Similarly, the complexity in the node-level
attention can be expressed as:
$O(|V|d{d}^{{}^{\prime}}+|E|B{d}^{{}^{\prime}})$, where $B$ is the average
degree of hyperedges.

##  IV Experiment

In this section, we evaluate our proposed HyGNN model for DDI prediction with
extensive experiments on two different datasets. We use F1, ROC-AUC, and PR-
AUC accuracy metrics to compare our model’s performances with the state-of-art
baseline models. Making DDI predictions for new drugs could be more
challenging than existing drugs. Therefore, we assess our model’s performance
for both new and existing drugs as well. First, we describe our datasets,
TWOSIDES and DrugBank, then we explain our experiments, and present and
analyze our results.

###  IV-A Dataset

We evaluate the proposed model using two different sizes of datasets. One is a
small dataset, and another one is a large dataset. (1) TWOSIDES is our small
dataset. TWOSIDES was created using data from adverse event reporting systems.
Common adverse effects, such as hypotension and nausea, occur in more than a
third of medication combinations, but others, such as amnesia and muscular
spasms, occur in only a few. We extract 645 approved drugs’ information from
TWOSIDES. Each drug is linked to its chemical structure (SMILES). There are
63,473 DDI positive labels for the selected drugs. (2) DrugBank is our large
dataset and it is the largest dataset for drugs that is publicly available. It
is a drug knowledge database that includes clinical information about drugs,
such as side effects and (DDIs). DrugBank also includes molecular data, such
as the drug’s chemical structure, target protein, and so on. From DrugBank, we
retrieve information on 1706 approved drugs along with their SMILES strings
and 191,402 DDI information. Both datasets are publicly available on
Therapeutics Data Commons (TDC) 666https://tdcommons.ai/. The first unified
platform, TDC, was launched to comprehensively access and assess machine
learning across the entire therapeutic spectrum.

TABLE I: Statistics of Dataset Dataset | # of Drug | # of DDI
---|---|---
TWOSIDES | 645 | 63473
DrugBank | 1706 | 191402

All known DDIs in both datasets are our positive samples. However, to train
our model, we need negative samples as well. Therefore, we randomly sample a
drug pair from the complement set of positive samples for each positive
sample. Thus, we ensure a balanced dataset of equally positive and negative
samples for an individual dataset.

We apply the ESPF algorithm and $k$-mer separately to extract the
substructures from the SMILES string of drugs. For ESPF, we notice that when a
lower frequency threshold is set, it generates many substructures, some of
which may be unimportant. However, when a more significant threshold value is
set, it generates fewer substructures and may lose some critical
substructures. These substructures are used as nodes in the hypergraph. To
examine the impact of the frequency threshold and thus the number of nodes in
the hypergraph learning, we choose five different threshold values from 5 to
25. For $k$-mer, we notice that typically with the increment of $k$, the
number of substructures (i.e., nodes) also increases. Similarly, to examine
the impact of the $k$ and thus the number of nodes in the hypergraph learning,
we choose five different values of $k$ from 3 to 15. A statistic of both
datasets is shown in Table I. The number of nodes for different threshold
values of ESPF and $k$-mer is given in Table II and Table III for each
dataset.

###  IV-B Parameter Settings

Each dataset is randomly split into three parts: train (80%), validation
(10%), and test (10%). We repeat this five times and report the average
performances in terms of F1-score, ROC-AUC, and PR-AUC. The optimal hyper-
parameters are obtained by grid search based on the validation set. The ranges
of grid search are shown in Table IV.

We employ a single-layer HyGNN having two levels of attention. We use a
LeakyReLU activation function in the encoder side and a ReLU activation
function in the MLP predictor of the decoder side. During training, we
simultaneously optimize the encoder and decoder using adam optimizer. Each
model is trained for 2000 epochs with an early stop if there is no change in
validation loss for 200 consecutive epochs.

For the baselines in subsection: IV-C, each GNN model is used as a two-layer
architecture. All other parameters of each GNN are set by following their
sources. For DeepWalk and node2vec, the walk length, number of walks, and
window size are set to 100, 10, and 5, respectively. We use Logistic
Regression as a simple ML classifier.

TABLE II: # of Nodes (N) in the Hypergraph based on parameters of the methods, ESPF and $k$-mer, for TWOSIDES Dataset ESPF | $|N|$ |  $k$-mer | $|N|$
---|---|---|---
5 | 555 | 3 | 822
10 | 324 | 6 | 7025
15 | 249 | 9 | 14002
20 | 208 | 12 | 17351
25 | 187 | 15 | 18155
TABLE III: # of Nodes (N) in the Hypergraph based on parameters of the methods, ESPF, and $k$-mer, for DrugBank Dataset ESPF | $|N|$ |  $k$-mer | $|N|$
---|---|---|---
5 | 1266 | 3 | 1296
10 | 729 | 6 | 11849
15 | 550 | 9 | 29443
20 | 462 | 12 | 43634
25 | 400 | 15 | 51315
TABLE IV: Hyper-parameter Settings Parameter | Values
---|---
Learning rate | 1e-2, 5e-2, 1e-3, 5e-3
Hidden units | 32, 64, 128
Dropout | 0.1, 0.5
Weight decay | 1e-2, 1e-3

###  IV-C Baselines

We compare our model performance with different types of state-of-the-art
models. We categorize the baseline models into five groups based on the data
representation and methodology.

1\. Random walk-based embedding (RWE) on DDI graph  We construct a regular
graph based on the drug interaction information called a DDI graph. Drugs are
represented as nodes, and two drugs share an edge if they interact. After
constructing the graph, we apply the random walk-based graph embedding methods
to get the representations of drugs. DeepWalk [44] and Node2vec [45] are two
well-known graph embedding methods. They are both based on a similar mechanism
of ‘walk’ on the graph traversing from one node to another. We apply DeepWalk
and Node2Vec on the DDI graph and generate the embedding of nodes. Afterward,
we concatenate drug embeddings to get the drug pair features and feed that
into a machine learning classifier for binary classification.

2\. GNN on DDI graph: After constructing the DDI graph as explained above, we
apply three different GNN models with unsupervised settings; graph convolution
network (GCN) [46], graph attention network (GAT) [43], and GraphSAGE [47] to
get the representations of drugs. These GNN models are obtained from
DGL777https://docs.dgl.ai/. After getting the representations of drugs, we
concatenate them pair-wise and use them as the features of drug pairs in the
ML classifier for binary classification.

3\. GNN on substructure similarity graph (SSG)  We follow [33] to create the
substructure similarity graph (SSG). We construct an edge between two drugs if
they have at least a predefined number of common substructures. We apply the
ESPF algorithm to the SMILES strings of drugs to get the frequent
substructures. Afterward, we apply three different GNN models, GCN, GAT, and
GraphSAGE, to the constructed graph to get the representations of drugs. The
drug representations are then concatenated pair-wise and fed into a classifier
to predict the DDI.

4\. CASTER We apply the Caster algorithm [7] for DDI prediction. It takes
SMILES strings as input and employs frequent sequential pattern mining to
discover the recurring substructures. They use the ESPF algorithm to extract
frequent substructures. Then, they generate a functional representation for
each drug using those frequent substructures. Further, the functional
representation of drug pairs is used to predict DDIs. We reproduce CASTER
results for our datasets.

5\. Decagon Decagon [5] uses a multi-modal graph consisting of protein-protein
interactions and drug-protein targets interactions for DDI prediction. It has
an encoder-decoder architecture. The encoder exploits a graph convolution
network to generate the representation of drugs by embedding all drug
interactions with other entities in it. Then, the decoder takes drug pair
representations as input and predicts DDIs with an exact side effect. The same
TWOSIDES drug-drug interactions network is used in Decagon. That is why we
directly compare our model performances with their reported results for
TWOSIDES data instead of reproducing Decagon. However, we do not consider
DrugBank data for Decagon as we do not have the additional information (e.g.,
side effects and target protein) in our DrugBank dataset to construct the
multi-modal graph.

TABLE V:  Performance comparisons of HyGNN with baseline models on TWOSIDES dataset.  Model | Method | F1 | ROC-AUC | PR-AUC
---|---|---|---|---
| DeepWalk | 80.35 | 80.36 | 85.19
RWE on DDI Graph | Node2vec | 84.50 | 84.52 | 88.33
| GCN | 85.34 | 85.38 | 88.87
GNN on DDI graph | GraphSage | 85.83 | 85.80 | 89.28
| GAT | 82.67 | 82.68 | 86.86
| GCN | 53.85 | 54.04 | 66.94
GNN on SSG graph | GraphSage | 60.19 | 60.18 | 70.34
| GAT | 54.25 | 54.37 | 66.85
CASTER | - | 82.35 | 90.45 | 90.58
Decagon | - | - | 87.20 | 83.20
| ESPF & MLP | 88.79 | 96.01 | 96.30
| ESPF & Dot | 76.79 | 91.12 | 93.37
HyGNN |  $k$-mer & MLP | 89.21 | 96.25 | 96.53
|  $k$-mer & Dot | 78.55 | 91.80 | 93.88
TABLE VI:  Performance comparisons of HyGNN with baseline models on DrugBank dataset.  Model | Method |  F1 |  ROC-AUC | PR-AUC
---|---|---|---|---
| DeepWalk | 73.34 | 73.35 | 80.05
RWE on DDI Graph | Node2vec | 79.52 | 79.54 | 84.56
| GCN | 77.05 | 77.06 | 82.78
GNN on DDI graph | GraphSage | 80.83 | 80.88 | 85.51
| GAT | 63.84 | 69.75 | 78.52
| GCN | 58.00 | 58.04 | 69.11
GNN on SSG graph | GraphSage | 61.10 | 61.15 | 70.64
| GAT | 58.20 | 58.24 | 69.25
CASTER | - | 87.36 | 94.27 | 94.20
| ESPF & MLP | 92.42 | 97.63 | 97.53
| ESPF & Dot | 83.94 | 95.80 | 96.57
HyGNN |  $k$-mer & MLP | 94.61 | 98.69 | 98.68
|  $k$-mer & Dot | 87.38 | 97.99 | 98.28
Figure 2: Performance comparison of models for different frequency thresholds
of ESPF. Figure 3: Performance comparison of models for different sizes of
$k$-mer.

###  IV-D Results

####  IV-D1 Model Performances

We conduct detailed experiments on our proposed models for two different
datasets with different threshold values of $k$-mer and ESPF. Both MLP and dot
predictor-based decoder functions are employed individually for each setup to
compare their performances. The overall performances are illustrated in Fig. 2
and Fig. 3. Fig. 2 depicts the model’s performance for different ESPF
frequency thresholds ranging from 5 to 25. This figure shows that it has a
more significant impact on the TWOSIDES dataset, especially for the Dot
decoder function than DrugBank. On TWOSIDES with MLP, it gives similar results
till 25, and then it has a considerable decrease for 25. Since we get a
significantly less number of substructures, it would not be enough to learn
with those. For DrugBank, it gives similar results for different thresholds of
ESPF. In general, frequency threshold 5 gives the best performance for
TWOSIDES and DrugBank with MLP and DOT. As with the increment of the frequency
threshold, the number of substructures (i.e., nodes) decreases, which could be
a potential reason for performance degradation. The best performance for each
dataset and decoder is recorded for a threshold value of 5.

Figure 4: Performance comparison of models for different training sizes where
$x$-axes represent the training percentages.

Fig. 3 presents the models’ performances for five different $k$ values of
$k$-mer ranges from 3 to 15. Similar to ESPF, the effect of the parameter on
the results is higher for TWOSIDES than DrugBank. The reason for this could be
that it is smaller than DrugBank, so the graph’s size is affecting its
results. However, DrugBank is a large dataset with enough training data to get
good results, even with a small graph. Here, we can see that with the
increment of the size of $k$-mer, the performance of the model increases,
especially for TWOSIDES. As with increasing $k$, the number of substructures
(i.e., nodes) increases which could be the reason for overall performance
improvement. After some point, we will get too many substructures that could
put noise into the data and decrease the model’s performance. The best
performance for each dataset and decoder are reported with $k=9$ for $k$-mer.

####  IV-D2 Comparison with Baselines

We evaluate our models by comparing their performances with several baseline
models and present the results in Table V for the TWOSIDES, and Table VI for
the DrugBank dataset. As we see in these tables, our models comprehensively
outperform all the baseline models. More precisely, in Table V for the
TWOSIDES dataset, HyGNN achieves at least 7% on F1, 6% on ROC-AUC, and PR-AUC
better performance than other baseline models. While the best model among all
the baselines, CASTER, achieves an F1 score of 82.35%, our HyGNN with $k$-mer
& MLP scores 89.21% with almost 7% gain. A similar situation also happens for
the other two accuracy measures: ROC-AUC and PR-AUC.

In Table V, for the DDI graph, from the GNN models, GraphSage gives the best
results with 85.83%, 85.80%, and 89.29% on F1, ROC-AUC, and PR-AUC,
respectively. Also, from random walk-based embedding models, Node2Vec gives
the best results, which is very similar to GraphSage results with 84.50% on
F1, 84.52% on ROC-AUC, and 88.33% PR-AUC scores. For the SSG graph, again,
GraphSage gives the best result. In our result comparison table, CASTER is the
best competitor of HyGNN. Out of all baseline models, CASTER shows the best
performance with ROC-AUC and PR-AUC of above 90%. Decagon is a multi-modal
graph that also exhibits better performance than GNN on SSG.

Table VI presents the performance comparison of HyGNN with baselines for the
DrugBank dataset. As for TWOSIDES, here again, out of three different GNN
models on DDI and SSG graphs, GraphSage yields the best result. Similarly,
Node2Vec performs better than DeepWalk. CASTER is still the best performer
among all baselines, with 87.36%, 94.27%, and 94.20% on F1, ROC-AUC, and PR-
AUC, respectively. However, our HyGNN with $k$-mer & MLP significantly
surpasses CASTER with 94.61% on F1, 98.69% on ROC-AUC, and 98.68% on PR-AUC.
As Decagon depends on the drug and other drug-centric information, we could
not experiment with it on the DrugBank dataset.

In summary, HyGNN with $k$-mer gives better results than ESPF. The reason for
this could be that with the ESPF, we eliminate many substructures but keep
just frequent ones. This may result in losing important ones that are not
frequent. However, with $k$-mer, we get all and let the attention models in
HyGNN learn which substructures are more important for DDI.

Moreover, we take the best-performing method from each baseline model, namely
Node2Vec from random walk-based embedding, GraphSage from GNN on DDI,
GraphSage from GNN on SSG, CASTER, and $k$-mer & MLP from our HyGNN models.
Then, we compare their performances by changing the training sizes from 10% to
80% for both datasets. A comparison of performance is outlined in Fig 4.
Results indicate HyGNN to be the best-performing model, and it still gives
very good results with small training data. However, decreasing the training
size affects the baseline models significantly, especially GraphSage on the
SSG model. It is worthy of mention that based on our results, all graph-based
models, especially different variants of GNNs, including HyGNN and baselines,
have performed fairly well on our data.

TABLE VII:  Novel DDI Predictions by HyGNN on TWOSIDES Dataset  Drug1 | Drug2 | TWOSIDES Label | Predicted DDI Score | DrugBank Label
---|---|---|---|---
Desvenlafaxine | Paroxetine | 0 | 0.9989 | 1
Probenecid | Metformin | 0 | 0.9931 | 1
Fluvastatin | Metronidazole | 0 | 0.9212 | 1
Loratadine | Isradipine | 0 | 0.9703 | 1
Glyburide | Bosentan | 0 | 0.9068 | 1
Salmeterol | Dicycloverine | 0 | 0.9189 | 1
Valdecoxib | Sodium sulfate | 0 | 0.9105 | 1
Lisinopril | Naratriptan | 0 | 0.9336 | 1
Bexarotene | Maprotiline | 0 | 9.9993e-10 | 0
Amoxapine | Econazole | 0 | 6.8256e-09 | 0
Nabilone | Oxaprozin | 0 | 4.1440e-08 | 0
Dexmedetomidine | Carbachol | 0 | 1.2417e-08 | 0
TABLE VIII:  Novel DDI Predictions by HyGNN on DrugBank Dataset  Drug1 | Drug2 | DrugBank Label | Predicted DDI Score | TWOSIDES Label
---|---|---|---|---
Hydroxychloroquine | Loratadine | 0 | 0.9879 | 1
Dextromethorphan | Ofloxacin | 0 | 0.9772 | 1
Midazolam | Warfarin | 0 | 0.9884 | 1
Benzthiazide | Fentanyl | 0 | 5.6989e-14 | 0
Labetalol | Levonorgestrel | 0 | 9.1049e-07 | 0
Cefprozil | Disulfiram | 0 | 1.0882e-11 | 0
TABLE IX:  Performance of HyGNN for New Drugs  Dataset | Unseen Node |  F1 |  ROC-AUC | PR-AUC
---|---|---|---|---
TWOSIDES | 5% | 72.75 | 78.25 | 85.64
DrugBank | 5% | 65.23 | 70.84 | 78.04

Hypergraphs are used in a wide range of scientific fields. Hypergraphs are a
natural method to illustrate shared group relationships. Through a hypergraph
structure, HyGNN is able to capture higher-order correlations between data
(i.e., triadic, tetradic, etc.). Furthermore, employing an attention mechanism
makes it more robust by giving more weight to important substructures while
learning representations of drugs. Though GAT has attention architecture as
well, it can not discover the important edges. The main strength of our HyGNN
is the proposed hypergraph edge encoder that has two levels of attention
mechanism. At first, it aggregates the hyperedges to generate the
representation of the node. While aggregating, it imposes more attention on
the important hyperedges. Similarly, to generate the representation of a
hyperedge, it aggregates the nodes’ information with much attention to the
important ones.

Moreover, HyGNN has a decoder function, and we learn all the parameters of the
encoder and decoder simultaneously during training. From Table V and Table VI,
we can see HyGNN with $k$-mer & MLP performs better than dot product. $k$-mers
are $k$-length substrings included inside a biological sequence. A bigger
$k$-mer is preferable since it ensures greater uniqueness in the base
sequences that will create the string. Larger $k$-mer sizes aid in the
elimination of repetitive substrings. Moreover, MLP Predictors are well-suited
for classification problems in which data is labeled. They are extremely
adaptable and may be used to learn a mapping from inputs to outputs in
general. Additionally, it generates superior results compared to dot predictor
since it has trainable parameters that are learned throughout the training.

####  IV-D3 Case Study - Prediction and Validation of Novel DDIs

We evaluate the effectiveness of HyGNN model on novel DDIs prediction. We
select some drug pairs from TWOSIDES. None of these drug pairs have DDI info
in TWOSIDES but have DDI info in DrugBank. Then we train our HyGNN using
TWOSIDES and make predictions for those pairs. Following that, we collect
predicted scores for those drug pairs as shown in Table VII. From this table,
we see that for the first eight drug pairs, though the TWOSIDES label for each
of these pairs is zero, we get predicted scores above 90% for each pair, which
shows there is a high chance that each pair will interact between them. To
further validate it, we cross-check our predicted score with DrugBank, which
says each of these eight drug pairs interacts between them. Moreover, for the
last four-drug pairs of Table VII the predicted scores are minimal, and
TWOSIDES, and DrugBank both say they don’t interact. Similarly, six drug pairs
are selected from DrugBank having no DDI info in DrugBank but in TWOSIDES as
shown in Table VIII, then HyGNN is trained using DrugBank data and validated
the predicted scores by TWOSIDES.

####  IV-D4 Case Study- DDI Prediction for New Drugs

Making DDI predictions for new drugs could be more challenging than existing
drugs. Since the model does not learn based on the SMILES strings of new
drugs. To show the effectiveness of our model for new drugs, at first, we
randomly select a 5% drug from a dataset and completely remove these drugs’
information from the corresponding train set and keep those drugs’ information
only in the test set. These selected 5% drugs can be considered new drugs. The
experimental results for both datasets with new drugs are shown in Table IX.
As we see in the table, our model predicts DDIs effectively for both datasets.

##  V Conclusion

In this paper, we propose a novel GNN-based framework for DDI prediction based
on the chemical structures (SMILES) of drugs. In contrast to existing graph-
based models, we utilize a novel hypergraph structure to depict higher-order
structural similarities between drugs. Following that, we propose a Hypergraph
GNN model with an encoder-decoder architecture to learn the drug
representation for DDI prediction. We develop a hypergraph edge encoder to
construct drug embeddings and a decoder with drug representations to predict a
score for each drug pair, indicating whether two drugs interact. Finally, with
several experiments, we show that our method outperforms different types of
baseline models and also it is able to predict DDIs for new drugs.

As future work, we plan to extend our model to address other problems in
bioinformatics, like protein-protein interaction prediction, drug repurposing,
and sequence classification.

## Acknowledgment

This work is funded partially by National Science Foundation under Grant No
2104720.

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# Optical conductivity of the threefold Hopf semimetal

Tetsuro Habe  Nagamori Institute of Actuators, Kyoto University of Advanced
Science, Kyoto 615-8577, Japan

###### Abstract

A multifold Hopf semimetal is a topological point node semimetal possessing an
anisotropy in the internal electronic structure, e.g., the dipole structure of
the Berry curvature. In this paper, the unique features of threefold Hopf
semimetals in terms of the optical conductivity are theoretically investigated
with a minimal theoretical model by using linear response theory for a
linearly polarized photon. The frequency spectrum of the optical conductivity
shows an anisotropic dependence on the polarization angle of the incident
photon even if the electronic band structure is completely isotropic. The
longitudinal optical conductivity linearly depends on the photon frequency and
possesses step-like changes in the frequency spectrum. The anisotropic
electronic structure has a varying the number of steps with the orientation of
the photon polarization axis. We reveal that the anisotropy is attributed to
symmetries preserving the point node in threefold Hopf semimetals. The
linearly polarized photon also induces a Hall current but it vanishes with the
photon polarization axis parallel to the Berry dipole axis. The numerical
calculations show that these characteristic features can be observed even with
a non-zero temperature and disorder.

##  I Introduction

Topological nodal semimetals are characterized by electronic bands with point
nodes or line nodes, on which two or more bands are touching.Murakami (2007);
Burkov  _et al._ (2011); Young  _et al._ (2012); Bradlyn  _et al._ (2016)
Around the nodes, the electronic states present an unusual structure of the
Berry curvature in the wave number space. For instance, the vector field of
the Berry curvature is the same as that produced by a monopole in Weyl
semimetals.Murakami (2007) Recently, a novel type of topological nodal
semimetal, the so-called Hopf semimetal, has been theoretically proposed and
predicted to possess multifold point nodes at which three or more bands are
attached.Graf and Piéchon (2022) Around the nodes, the dispersion of energy
bands is flat or linear, and it is qualitatively the same as that of multifold
point node semimetalsWieder  _et al._ (2016); Bradlyn  _et al._ (2016).
However, the Berry curvature possesses a dipole structure different from the
isotropic monopole in these multifold point-node semimetalsChang  _et al._
(2018); Flicker  _et al._ (2018). Since the anisotropic Berry curvature
remains even with an isotropic band structure, the multifold Hopf semimetals
possess internal anisotropy in electronic states.

The electronic properties of topological nodal semimetals have been
investigated in terms of several measurements. The optical conductivity
represents the electronic transport properties associated with excited
electrons by photon irradiation and has been investigated theoretically and
experimentally for topological nodal semimetals.Xu  _et al._ (2016); Neubauer
_et al._ (2016); Tabert and Carbotte (2016); Carbotte (2017); Mukherjee and
Carbotte (2017); Ahn  _et al._ (2017); Barati and Abedinpour (2017); Schilling
_et al._ (2017); Habe and Koshino (2018); Sánchez-Martínez  _et al._ (2019);
Habe (2019); Xu  _et al._ (2020) The frequency dependence of the optical
conductivity describes the characteristic energy dispersion in topological
nodal semimetals. For instance, point node semimetals show a linear dependence
on the frequency and line-node semimetals present a frequency independent flat
spectrum.Xu  _et al._ (2016); Neubauer  _et al._ (2016) Moreover, recent work
suggests that the internal structure of the wave function also affects the
optical conductivity and can be observed as the variation of the spectrum.Habe
(2019) In topological nodal semimetals, the low-energy excitation is
associated only with the band structure around the nodes even if there are
other Fermi pockets. The optical conductivity is a useful observable for
investigating the electronic property attributed to the electronic structure
around the node in topological nodal semimetals.

In this paper, the optical conductivity of threefold Hopf semimetals is
theoretically investigated, and the characteristic spectrum attributed to the
unusual electronic structure is presented. In Sec.II, a minimal model and the
electronic structure are reviewed. In Sec.III, the optical conductivity is
presented analytically in the clean limit at zero temperature and numerically
in the disordered case at non-zero temperature. Moreover, the anisotropic
behaviors in the spectrum are discussed in terms of symmetries of multifold
Hopf semimetals. The conclusion is given in Sec.IV.

Figure 1: Schematic band structure of a threefold Hopf semimetal in (a). The
arrows indicate the optical excitation processes with a frequency of the
photon. The left and right panels show the cases with the Fermi energy below
and above the point node, respectively. In (b), schematics of the incident
electric field with respect to the direction of dipole $\boldsymbol{d}$.

##  II Theoretical model

In this section, the electronic structure of a threefold Hopf semimetal is
briefly reviewed in terms of the electronic states and Berry curvature around
the point node. The theoretical analysis is performed by using a minimal model
proposed in Ref. Graf and Piéchon, 2022. Around the point node, the electronic
states are described by

$\displaystyle
H^{\xi}(\boldsymbol{q})=\hbar{v}\begin{pmatrix}0&q^{\xi}_{-}&-iq_{z}\\\
q^{\xi}_{+}&0&0\\\ iq_{z}&0&0\end{pmatrix},$ (1)

where $\boldsymbol{q}$ is a wave vector with respect to the nodal point with
$q^{\xi}_{\pm}=\xi q_{x}\pm iq_{y}$, and the isotropic speed $v$ is assumed.
Here, $\xi=\pm 1$ indicates the direction of the dipole structure,
$\boldsymbol{d}=(0,0,\xi)$. In previous workGraf and Piéchon (2022), two
symmetries were proposed for the protection of the point node: axial rotation
symmetry and chiral symmetry. The axial rotation operator $L^{d}+\Sigma^{d}$
is defined with the angular momentum operator,
$L^{d}=\boldsymbol{d}\cdot(i\nabla_{\boldsymbol{q}}\times\boldsymbol{q})$,
projected on the $\boldsymbol{d}$-vector and the effective spin projection
operator $\Sigma^{d}=(1/3)\mathrm{diag}[1,-2,1]$. The symmetry for the
rotation is given by

$\displaystyle[L^{d}+\Sigma^{d},H^{\xi}(\boldsymbol{q})]=0.$ (2)

The chiral operator for the minimal model is represented by
$S=\mathrm{diag}[1,-1,-1]$ where chiral symmetry is given by
$S^{-1}H^{\xi}(\boldsymbol{q})S=-H^{\xi}$. In addition to the two symmetries,
the mirror symmetry along the $\boldsymbol{d}$-vector, the $z$-direction in
the present case, is necessary for the point node to be protected. For the
minimal model, the mirror symmetry is represented by
$M_{z}^{\dagger}H^{\xi}(q_{x},q_{y},-q_{z})M_{z}=H^{\xi}(\boldsymbol{q})$ with
$M_{z}=\mathrm{diag}[1,1,-1]$. In practice, a matrix with a constant scalar
$c$,

$\displaystyle H^{\prime}=\begin{pmatrix}0&0&c\\\ 0&0&0\\\
c&0&0\end{pmatrix},$ (3)

removes the node without breaking the axial rotation symmetry and chiral
symmetry. In the other axes, there is no mirror symmetry because of the
presence of axial mirror rotation symmetry.

The minimal model provides a characteristic band structure of threefold point
node semimetals, i.e., one flat band ${\varepsilon}_{0,\boldsymbol{q}}=0$ and
two cone-shaped bands,

$\displaystyle{\varepsilon}_{\pm,\boldsymbol{q}}=\pm\hbar vq,$ (4)

with $q=|\boldsymbol{q}|$, where these three bands are attached at
$\boldsymbol{q}=0$ as shown in Fig. 1. The electronic states are represented
by scale-free wave functions,

$\displaystyle|\psi^{\xi}_{\pm,\boldsymbol{q}}\rangle=\frac{1}{\sqrt{2}q}\begin{pmatrix}q\\\
\pm q^{\xi}_{+}\\\ \pm iq_{z}\end{pmatrix},$ (5)

and

$\displaystyle|\psi^{\xi}_{0,\boldsymbol{q}}\rangle=\frac{1}{q}\begin{pmatrix}0\\\
iq_{z}\\\ q^{\xi}_{-}\end{pmatrix},$ (6)

where the subscript $\alpha$ of $\psi_{\alpha,\boldsymbol{q}}^{\xi}$
corresponds to the band index. The simple representation enables us to
analytically describe the fascinating electronic properties of threefold Hopf
semimetals.

The Berry curvature is theoretically given by a Kubo-formulaYao  _et al._
(2004),

$\displaystyle\Omega_{\alpha,\rho}(\boldsymbol{q})=-\epsilon_{\mu\nu\rho}\sum_{\alpha^{\prime}\neq\alpha}\frac{2\mathrm{Im}[\langle\psi_{\alpha,\boldsymbol{q}}^{\xi}|v_{\mu}|\psi_{\alpha^{\prime},\boldsymbol{q}}^{\xi}\rangle\langle\psi_{\alpha^{\prime},\boldsymbol{q}}^{\xi}|v_{\nu}|\psi_{\alpha,\boldsymbol{q}}^{\xi}\rangle]}{|{\varepsilon}_{\alpha,\boldsymbol{q}}-{\varepsilon}_{\alpha^{\prime},\boldsymbol{q}}|^{2}},$
(7)

with Levi-Civita symbol $\epsilon_{\mu\nu\rho}$ in the orthogonal coordinate,
where the velocity operator is defined by $v_{\mu}=(1/\hbar)\partial
H^{\xi}/\partial q_{\mu}$. In the case of the threefold Hopf semimetal, the
inter-band components of the velocity operator are scale-free and highly
anisotropic,

$\displaystyle\begin{aligned}
\langle\psi_{0,\boldsymbol{q}}^{\xi}|\boldsymbol{v}|\psi_{\pm,\boldsymbol{q}}^{\xi}\rangle&=\frac{v}{\sqrt{2}q}\left(-i\xi{q_{z}},{q_{z}},i{q_{\pm}^{\xi}}\right)\\\
\langle\psi_{+,\boldsymbol{q}}^{\xi}|\boldsymbol{v}|\psi_{-,\boldsymbol{q}}^{\xi}\rangle&=\frac{v}{q}\left(-i\xi
q_{y},iq_{x},0\right).\end{aligned}$ (8)

The anisotropy leads to the dipole structure of the Berry curvature,

$\displaystyle\begin{aligned}
\boldsymbol{\Omega}_{0}(\boldsymbol{q})&=2\frac{\xi}{\hbar^{2}}(\boldsymbol{d}\cdot\boldsymbol{q})\frac{\boldsymbol{q}}{q^{4}}\\\
\boldsymbol{\Omega}_{\pm}(\boldsymbol{q})&=-\frac{\xi}{\hbar^{2}}(\boldsymbol{d}\cdot\boldsymbol{q})\frac{\boldsymbol{q}}{q^{4}},\end{aligned}$
(9)

i.e., the Berry curvature vanishes in a plane of $q_{z}=0$. The anisotropic
Berry curvature shows that the internal anisotropy can be found in electronic
properties even if the energy dispersion is completely isotropic. In previous
workGraf and Piéchon (2022), it was proposed that the anisotropy of the
internal electronic structure can be confirmed by the Hall conductivity.
However, the band structure of a multifold nodal semimetal cannot be confirmed
in the measurement simultaneously. Moreover, the Hall conductivity is also
affected by the presence of other nodes with the opposite
$\boldsymbol{d}$-vector and/or conventional Fermi pockets. On the other hand,
the optical conductivity can represent both characteristics of internal
anisotropy and a multifold nodal semimetal, and it does not vary even in the
presence of other Fermi pockets. The frequency dependence is related to the
energy dispersion of the multifold nodal semimetal, and the different number
of step-like structures can be found in the spectrum by changing the
polarization axis of the photon.

##  III Optical conductivity

The optical conductivity is a response function representing the induction of
electric current by an oscillating electric field with a unique frequency,
e.g. , the photo-induced electric current. In this paper, the electric field
is considered to be polarized along a unique direction for simulating a
linearly polarized photon. The external field is represented by
$E_{\mu}(t)=E^{0}_{\mu}\cos(\omega t)$ with the polarization axis in the $\mu$
direction and the frequency $\omega$. In the presence of the oscillating
field, electrons are excited by absorbing the photon energy corresponding to
the frequency and the negligibly small change in wave number. Since the
excited electrons mainly carry the electric current, the frequency dependence
of the optical conductivity characterizes the distribution of the excitation
energy for electronic states in the wave number space.

In linear response theory, the induced electric current in the $\nu$ direction
is represented by the superposition of those attributed to two complex fields,
$E_{\pm\omega,\mu}(t)=E^{0}_{\mu}\exp[\mp i\omega t]$,

$\displaystyle
J_{\nu}(t)=\frac{1}{2}\left(J_{\omega,\nu}(t)+J_{-\omega,\nu}(t)\right),$ (10)

with

$\displaystyle J_{\omega,\nu}(t)=-\int_{-\infty}^{t}{dt^{\prime}}\
\mathrm{Tr}\left\\{\rho_{0}[\hat{j}_{\nu}(t),\hat{j}_{\mu}(t^{\prime})]\right\\}\frac{E_{\omega,\mu}(t^{\prime})}{\omega}.$
(11)

By using the eigenstate $|m\rangle$, the optical conductivity
$\sigma_{\nu\mu}(\omega)$ is given by

$\displaystyle\sigma_{\nu\mu}(\omega)=$ $\displaystyle
J_{\omega,\nu}(t)/E_{\omega,\mu}(t)$ $\displaystyle=$
$\displaystyle\sum_{{\varepsilon}_{m}\leq{\varepsilon}_{n}}\frac{ie^{2}}{\omega}\left(n_{F}({\varepsilon}_{m})-n_{F}({\varepsilon}_{n})\right)$
$\displaystyle\times\left\\{\frac{v_{\nu}^{mn}v_{\mu}^{nm}}{\hbar\omega+{\varepsilon}_{m}-{\varepsilon}_{n}+i\delta}-\frac{(v_{\nu}^{mn}v_{\mu}^{nm})^{\ast}}{\hbar\omega-{\varepsilon}_{m}+{\varepsilon}_{n}+i\delta}\right\\},$
(12)

with a small imaginary factor $i\delta$, where $n_{F}(\varepsilon)$ and
$v_{\mu}^{mn}=\langle m|v_{\mu}|n\rangle$ are Fermi distribution function and
the matrix component of the velocity operator, respectively. Here, the
summation over the electronic states is performed under the condition of
${\varepsilon}_{m}\leq{\varepsilon}_{n}$. The response function satisfies
$\sigma_{\nu\mu}(-\omega)=\sigma_{\nu\mu}(\omega)^{\ast}$. Thus, the electric
current can be represented by a real function with two oscillating functions,

$\displaystyle
J_{\nu}(t)=\mathrm{Re}[\sigma_{\nu\mu}(\omega)]E_{\mu}^{0}\cos\omega
t+\mathrm{Im}[\sigma_{\nu\mu}(\omega)]E_{\mu}^{0}\sin\omega t.$ (13)

Since the external electric field oscillates with $\cos\omega t$, the first
term represents the dissipative electric current, i.e., the observable. Thus,
the coefficient
$\sigma^{(1)}_{\nu\mu}(\omega)=\mathrm{Re}[\sigma_{\nu\mu}(\omega)]$ gives the
optical conductivity to the linearly polarized electromagnetic wave with a
frequency $\omega$.

###  III.1 Clean limit

In the clean limit, the optical conductivity can be obtained by introducing an
infinitesimal into $\delta$ in Eq. (12). In this limit, the analytic
representation is given by

$\displaystyle\sigma^{(1)}_{\nu\mu}(\omega)=$
$\displaystyle\frac{e^{2}}{\hbar}\sum_{\varepsilon_{m}\leq\varepsilon_{n}}\frac{\Delta
n^{mn}_{F}}{\omega}$
$\displaystyle\times\left\\{\mathrm{Re}[v_{\nu}^{mn}v_{\mu}^{nm}](\delta(\omega-\Delta\omega_{nm})-\delta(\omega+\Delta\omega_{nm}))\right.$
$\displaystyle\left.-\mathrm{Im}[v_{\nu}^{mn}v_{\mu}^{nm}]\left(\mathcal{P}\frac{1}{\omega-\Delta\omega_{nm}}+\mathcal{P}\frac{1}{\omega+\Delta\omega_{nm}}\right)\right\\},$
(14)

where $\Delta\omega_{nm}=(\varepsilon_{n}-\varepsilon_{m})/\hbar$ and $\Delta
n_{F}^{mn}=n_{F}(\varepsilon_{m})-n_{F}(\varepsilon_{n})$ are the energy and
distribution differences, respectively, between two states $|m\rangle$ and
$|n\rangle$. In threefold Hopf semimetals, the electronic energy
$\varepsilon_{m}$ and the velocity component $v_{\nu}^{mn}$ are given in Eqs.
(4) and (8), respectively. For the bulk material, the summation can be
replaced by an integration in the wave number $\boldsymbol{q}$. In the optical
excitation process, the wave number is almost unchanged with a negligibly
small momentum of the photon, and thus two electronic states $|m\rangle$ and
$|n\rangle$ can be assumed to possess the same wave number. Since the energy
is isotropic and the velocity is scale free, the angle and amplitude
integration can be performed separately,

$\displaystyle\sigma^{(1)}_{\nu\mu}(\omega)=$
$\displaystyle\frac{e^{2}}{\hbar}\sum_{\alpha<\alpha^{\prime}}\left\\{\mathrm{Re}[A_{\nu\mu}^{\alpha\alpha^{\prime}}]F^{\alpha\alpha^{\prime}}_{1}(\omega)+\mathrm{Im}[A_{\nu\mu}^{\alpha\alpha^{\prime}}]F^{\alpha\alpha^{\prime}}_{2}(\omega)\right\\},$
(15)

where $\hat{\theta}$ represents the solid angle, with

$\displaystyle A^{\alpha\alpha^{\prime}}_{\nu\mu}=\int d^{2}\hat{\theta}\
v_{\nu}^{\alpha\alpha^{\prime}}v_{\mu}^{\alpha^{\prime}\alpha},$ (16)

and

$\displaystyle F^{\alpha\alpha^{\prime}}_{1}(\omega)=$
$\displaystyle\frac{\pi}{\omega}\int^{q_{c}}_{0}\frac{dq}{(2\pi/a)^{3}}{q}^{2}\delta(\omega-\Delta\omega_{\alpha\alpha^{\prime}}(q))\Delta
n_{F}^{\alpha\alpha^{\prime}}(q),$ (17) $\displaystyle
F^{\alpha\alpha^{\prime}}_{2}(\omega)=$
$\displaystyle\frac{1}{\omega}\int^{q_{c}}_{0}\frac{dq}{(2\pi/a)^{3}}\mathcal{P}\frac{2\omega
q^{2}\Delta
n_{F}^{\alpha\alpha^{\prime}}(q)}{(\Delta\omega_{\alpha\alpha^{\prime}}(q))^{2}-\omega^{2}}.$
(18)

Here, the cut-off wave number $q_{c}$ indicates the radius of the sphere in
which the cone and flat band are preserved, and $a$ is introduced as a
normalization factor for the integral.

The anisotropy of the optical conductivity is described by
$A_{\nu\mu}^{\alpha\alpha^{\prime}}$ which can be easily calculated by using
Eq. (8). The non-zero components are given by

$\displaystyle A_{xx}^{+-}$
$\displaystyle=A_{yy}^{+-}=A_{zz}^{0\pm}=\frac{4\pi}{3}{v}^{2},$ (19)
$\displaystyle A_{xx}^{0\pm}$
$\displaystyle=A_{yy}^{0\pm}={i\xi}A_{xy}^{0\pm}=\frac{2\pi}{3}{v}^{2}.$ (20)

and the other components vanish due to the angle dependence. Since the inter-
cone component is absent for $A_{zz}^{\alpha\alpha^{\prime}}$, the linearly
polarized photon in the $z$ axis does not induce the electronic excitation
between the two cone-shaped bands. When the polarization axis is parallel to
the $x$ or $y$ axis, any electronic excitation is allowed among the three
bands. The angle integrals also show the presence of optical Hall conductivity
in the $xy$ plane, i.e., linearly polarized photons within the $xy$ plane
induce an electric current perpendicular to the polarization axis.

At zero temperature, the analytic representation of the response function can
be obtained. In this condition, the Fermi distribution function can be
replaced with a step function,
$n_{F}(\varepsilon)\simeq\theta(\varepsilon_{F}-\varepsilon)$, in the
calculation. Thus, the optical response is non-zero if the photon energy is
larger than the Fermi energy, $|\varepsilon_{F}|<\hbar\omega$, between the
flat-band and a cone, and twice the Fermi
energy,$|\varepsilon_{F}|<\hbar\omega$, between two cones. The analytic
formula of optical conductivity is represented by

$\displaystyle\begin{aligned}
\sigma^{(1)}_{xx}(\omega)=&\frac{e^{2}}{h}\omega\frac{a^{3}}{6}\left(\frac{1}{2v}\theta(\hbar\omega-2\varepsilon_{F})+\frac{1}{v}\theta(\hbar\omega-\varepsilon_{F})\right),\\\
\sigma^{(1)}_{zz}(\omega)=&\frac{e^{2}}{h}\omega\frac{a^{3}}{3v}\theta(\hbar\omega-\varepsilon_{F}),\\\
\sigma^{(1)}_{xy}(\omega)=&\xi\frac{e^{2}}{h}\frac{a^{3}}{3\pi{v}}\left\\{(q_{c}-q_{F})+\omega\log\left|\frac{1+\gamma_{F}}{1-\gamma_{F}}\right|\right\\},\end{aligned}$
(21)

with the Fermi wave number $q_{F}$ and $\gamma_{F}=\hbar\omega/vq_{F}$. Here,
the longitudinal conductivity is isotropic in the $xy$ plane, i.e.,
$\sigma^{(1)}_{yy}(\omega)=\sigma^{(1)}_{xx}(\omega)$. In the third equation,
it is assumed that the frequency is much smaller than the cut-off energy,
i.e., $\hbar\omega/vq_{c}\ll 1$. This assumption works for investigating the
characteristic property of the electronic structure around the point node.

The frequency dependence of the longitudinal optical conductivity shows the
characteristic features of internal anisotropy in threefold Hopf semimetals.
In the $z$ direction, the spectrum is similar to that of threefold point node
semimetalSánchez-Martínez  _et al._ (2019), where the spectrum is zero for
$\hbar\omega<\varepsilon_{F}$ and shows the linear dependence on the frequency
for $\hbar\omega>\varepsilon_{F}$ with a step-like increment at
$\hbar\omega=\varepsilon_{F}$. In the case of a threefold point node
semimetal, the optical conductivity shows the same spectrum in any direction
but, in the threefold Hopf semimetal, the conductivity strongly depends on the
polarization axis of the incident photon. The optical conductivity within the
$xy$-plane shows another characteristic feature attributed to the electronic
state in the Hopf semimetal. In the spectrum, there are two step-like
increments found at $\hbar\omega=\varepsilon_{F}$ and $2\varepsilon_{F}$. The
first step and the second step correspond to the electronic excitation between
the flat band and one cone, and that between two cones, respectively. The
second one is a unique feature in a Hopf semimetal because the inter-cone
excitation is prohibited in conventional threefold point node semimetals.

The optical Hall conductivity $\sigma_{\mu\nu}^{(1)}(\omega)$ can be observed
in the $xy$-plane but it vanishes if the polarization axis or electric current
is parallel to the $\boldsymbol{d}$-vector. The analytic representation of
$\sigma_{xy}^{(1)}(\omega)$ includes the frequency-independent term
proportional to $q_{c}-q_{F}$. Although this term depends on the cut-off
parameter $q_{c}$ explicitly, it does not mean that this term is an
unrealistic one. The frequency-independent term represents the conventional
Hall current following the oscillating electric field. The amplitude of this
component decreases linearly with the Fermi energy and vanishes with it
reaching $q_{F}=q_{c}$. The disappearance corresponds to that in association
with full occupation of all electronic bands in a crystalline material. In
crystalline materials, the electronic states form bands bounded in the
Brillouin zone and the summation of the Berry phase over all bands must be
zero.

###  III.2 Relation to symmetry

In the previous section, it was shown that the optical conductivity visualizes
the internal anisotropy in the spectrum within the minimal model. The
anisotropic behavior in the spectrum is independent of the minimal model and
directly attributed to the electronic property of the Hopf semimetal. In this
section, the number of step-like structure is discussed in terms of the three
symmetries protecting the multifold point node. The symmetries introduce a
restriction on the electronic excitation responsible for the optical response
represented by the general expression in Eq. (12). Since there are three bands
attaching at the point node, there are two possible excitation processes, the
excitation from the lower cone to the upper cone, and that between the flat
band and one cone. The former excitation process introduces a step-like
structure at $\hbar\omega=2\epsilon_{F}$ and the latter produce another step-
like structure at $\hbar\omega=\epsilon_{F}$ (see Figs. 2 and 3).
Theoretically, the absence of electronic excitation is described by the
absence of off-diagonal velocity component
$v_{\mu}^{mn}=\langle\psi_{m}|v_{\mu}|\psi_{n}\rangle$. The off-diagonal
component vanishes only for the $z$-direction, the parallel axis to the
$d$-vector, between the upper and lower cones as shown in Eq. (8).

The prohibition of inter-cone excitation can be described by the symmetries
protecting the multifold point node. The chiral and $z$-axis mirror symmetries
are relevant for the inter-cone excitation. For the energy eigenstates, the
chiral operator exchanges the electronic states,
$|\psi_{\pm,\boldsymbol{q}}^{\xi}\rangle$, between the upper and lower cones,

$\displaystyle
S|\psi_{\pm,\boldsymbol{q}}^{\xi}\rangle=c_{S}|\psi_{\mp,\boldsymbol{q}}^{\xi}\rangle,$
(22)

with $c_{S}=1$ or $-1$ due to $S^{2}=I$. On the other hand, the $z$-axis
mirror operator $M_{z}$ causes electronic states to remain in the same band,

$\displaystyle
M_{z}|\psi_{\pm,\boldsymbol{q}}^{\xi}\rangle=c_{M_{z}}^{\pm}|\psi_{\pm,\boldsymbol{q}}^{\xi}\rangle.$
(23)

Here, the coefficient is the same, $c_{M_{z}}^{+}=c_{M_{z}}^{-}=1$ or $-1$,
because the chiral and mirror operators are commutable $[S,M_{z}]$. Thus, in
the presence of the two symmetries, the velocity operator $v_{z}$ never
induces the inter-cone excitation because of

$\displaystyle\langle\psi^{\xi}_{+}|v_{z}|\psi^{\xi}_{-}\rangle$
$\displaystyle=c_{M_{z}}^{2}\langle\psi^{\xi}_{+}|(M_{z}^{\dagger}v_{z}M_{z})|\psi^{\xi}_{-}\rangle$
$\displaystyle=\langle\psi^{\xi}_{+}|(-v_{z})|\psi^{\xi}_{-}\rangle=0.$ (24)

Therefore, the different number of steps in Eq. (21) can be a signature of the
internal anisotropy in threefold Hopf semimetals.

The absence of inter-cone excitation can also be observed in conventional
threefold point node semimetalsSánchez-Martínez  _et al._ (2019); Habe (2019)
but the mechanism is different from that in threefold Hopf semimetals. In the
conventional threefold point node semimetals, electronic states can be
described by a model of spin-1 particles and those in the upper and lower
cones possess $\pm 1$ spin angular momentum. The difference in angular
momentum leads to the prohibition of optical excitation for any direction of
the electric field. In the threefold Hopf semimetals, on the other hand, the
absence of inter-cone excitation is attributed to the symmetries protecting
the point node.

###  III.3 Numerical analysis

In this section, the optical conductivity is numerically investigated
including the effects of disorder and temperature. The numerical analysis for
the optical conductivity is performed by using Eq. (12). The non-zero
temperature changes the distribution function
$n_{F}(\varepsilon)=(1+\exp[(\varepsilon-\varepsilon_{F})/(k_{B}T)])^{-1}$ and
the disorder is considered a reduction of the electronic relaxation time in
the calculation where the relaxation time $\tau$ is introduced by replacing
$\delta$ with $\hbar/(2\tau)$ in Eq. (12). The summation is performed over the
band index and the wave number within the sphere possessing a diameter
$q_{c}$. Since the electronic states are scale free in the wave number space,
the Fermi energy $\varepsilon_{F}$ is set to unity and the other parameters,
$k_{B}T$, $\hbar vq_{c}$, and $\delta=\hbar/2\tau$, are normalized by
$\varepsilon_{F}$. The cut-off energy is ten times the Fermi energy, $\hbar
vq_{c}=10\varepsilon_{F}$. In the numerical results, some constants, $k_{B}$,
$\hbar$, and $v$, are set to unity for simplicity. The Drude peak, the
electronic conductivity without inter-band excitation, is omitted in the
calculation.

Figure 2: The frequency dependence of the optical conductivity of a Hopf
semimetal with changing electronic relaxation time $\tau=\hbar/2\delta$. The
other parameter is fixed as $T=1\times 10^{-3}$.

The effect of disorder is analyzed numerically as shown in Fig. 2. The
longitudinal optical conductivity shows a linear dependence on the frequency
larger than the Fermi energy in both the $x$ and $z$ directions, where the
longitudinal conductivity in the $y$ direction is equivalent to that in the
$x$ direction. In the $x$ and $y$ directions, another step can be observed in
the spectrum at twice the Fermi energy and it does not appear in that of the
$z$ direction. The anisotropy is consistent with the analytic results in the
clean limit in Eq. (21). These steps are broadened with an increase in
$\delta$, the decrease in relaxation time,but they can be observed for
$\delta=1\times 10^{-1}$, which is equal to 10meV for
$\varepsilon_{F}=100$meV. The optical Hall conductivity $\sigma_{xy}^{(1)}$
shows a negative peak structure around $\omega=\varepsilon_{F}$. It is
represented by the second term of $\sigma_{xy}^{(1)}$ in Eq. (21). The
amplitude decreases with the frequency and it is also consistent with the
analytic result.

Figure 3: The frequency dependence of the optical conductivity of a Hopf
semimetal with changing temperature $T$. The other parameter is fixed as
$\delta=1\times 10^{-3}$.

The temperature dependence is presented in Fig. 3. The characteristic
structures in the spectrum change to be smooth but they can be observed even
at non-zero temperature.

These results by using the minimal model show the ideal spectra of optical
conductivity. Since the spectra are modified in a realistic material due to
the more complex electronic structure, possible corrections are discussed in
what follows. Firstly, the linear and flat bands are not preserved in the
whole Brillouin zone. The characteristic spectrum can be observed for the
frequency within the excitation energy among the bands which can be described
by Eq. (4). Secondly, there can be other Fermi pockets where electronic states
cannot be described as multifold fermions. However, in the case of the optical
response, such Fermi pockets do not contribute to the low-frequency spectrum
unless low-energy excited states accompany them. In practice, the optical
conductivity of conventional multifold fermion is reproduced in realistic
materials of a multifold point node semimetal with such additional Fermi
pocketsSánchez-Martínez  _et al._ (2019); Habe (2019).

##  IV conclusion

In this paper, a theoretical analysis of the optical conductivity of threefold
Hopf semimetals is presented by using a simplified theoretical model
possessing an isotropic nodal band structure and anisotropy represented by the
$d$-vector in the electronic states. In the clean and zero-temperature limit,
the analytic representations of the optical conductivity are obtained in Eq.
(21). The theoretical calculations reveal that the effect of internal
anisotropy can be observed in the spectrum of the optical conductivity. The
spectrum of the longitudinal optical conductivity shows a linear dependence on
the frequency and step-like behaviors at some frequencies. For an optical
polarization axis perpendicular to the $d$-vector, two steps can be found at
the frequency corresponding to the Fermi energy and twice the Fermi energy. On
the other hand, for the polarization axis parallel to the $d$-vector, the step
at twice the Fermi energy vanishes and a single step is found in the spectrum.
The qualitative anisotropy is attributed to the prohibition of inter-cone
electronic excitation due to chiral and mirror symmetries protecting the point
node. Moreover, the optical Hall conductivity can be observed only if the
current and electric field are perpendicular to the $d$-vector. The numerical
calculations show that the anisotropic behaviors are stable even with the non-
zero temperature and electronic relaxation time.

This work was supported by JSPS KAKENHI Grant No. JP20K05274.

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