11institutetext: University of Illinois Urbana Champaign, United States
11email<EMAIL_ADDRESS>
# AutoAM: An End-To-End Neural Model for Automatic and Universal Argument
Mining

Lang Cao 11

###### Abstract

Argument mining is to analyze argument structure and extract important
argument information from unstructured text. An argument mining system can
help people automatically gain causal and logical information behind the text.
As argumentative corpus gradually increases, like more people begin to argue
and debate on social media, argument mining from them is becoming increasingly
critical. However, argument mining is still a big challenge in natural
language tasks due to its difficulty, and relative techniques are not mature.
For example, research on non-tree argument mining needs to be done more. Most
works just focus on extracting tree structure argument information. Moreover,
current methods cannot accurately describe and capture argument relations and
do not predict their types. In this paper, we propose a novel neural model
called AutoAM to solve these problems. We first introduce the argument
component attention mechanism in our model. It can capture the relevant
information between argument components, so our model can better perform
argument mining. Our model is a universal end-to-end framework, which can
analyze argument structure without constraints like tree structure and
complete three subtasks of argument mining in one model. The experiment
results show that our model outperforms the existing works on several metrics
in two public datasets.

###### Keywords:

Argument Mining Information Extraction Natural Language Processing

##  1 Introduction

Argument mining (AM) is a technique for analyzing argument structure and
extracting important argument information from unstructured text, which has
gained popularity in recent years [12]. An argument mining system can help
people automatically gain causal and logical information behind the text. The
argument mining techniques benefit plenty of many fields, like legal [31],
public opinions [19], finance, etc. Argument mining is beneficial to human
society, but there is still much room for development. Argument mining
consists of several tasks and has a variety of different paradigms [12]. In
this paper, we focus on the most common argument structure of monologue. It is
an argumentative text from one side, not an argument from two sides. The
microscopic structure of argumentation is the primary emphasis of the
monologue argument structure, which primarily draws out the internal relations
of reasoning.

Figure 1: An example of argument mining result after extraction in the CDCP
dataset [19]. It forms an argument graph. In this graph, every node AC
represents an argument component. Fact, Value, and Policy are three types of
ACs. Every edge AR denotes argument relation, and Reason is one type of AR.

In this setting, an argumentative paragraph can be viewed as an argument
graph. An argument graph can efficiently describe and reflect logical
information and reasoning paths behind the text. An example of AM result after
extraction is shown in Figure 1. The two important elements in an argument
graph are the argument component (AC) and the argument relation (AR). ACs are
nodes in this graph, and ARs are edges. The goal of an AM system is to
construct this argument graph from unstructured text automatically. The
process of the AM system definition we use is as following steps:

  1. 1.

Argument Component Identification (ACI): Given an argumentative paragraph, AM
systems will detect ACs from it and separate this text.

  2. 2.

Argument Component Type Classification (ACTC): AM systems will determine the
types of these ACs.

  3. 3.

Argument Relation Identification (ARI): AM systems will identify the existence
of a relationship between any ACs.

  4. 4.

Argument Relation Type Classification (ARTC): AM systems will determine the
type of ARs, which are the existing relations between ACs.

Subtask 1) is a token classification task, which is also a named entity
recognition task. This task has a large amount of research work on it. Most of
the previous argument mining works [25] [10] [3] assume that the subtask 1)
argument component identification has been completed, which is the argument
component has been identified and can be obtained from the argumentative text.
Therefore, the emphasis of argument mining research is placed on other
subtasks. Following previous works, we also make such assumptions in this
paper. On this basis, we design an end-to-end model to complete ACTC, ARI, and
ARTC subtasks simultaneously.

ARI and ARTC are the hardest parts of the whole argument mining process. An AR
is represented by two ACs. It is difficult to represent AR precisely and
capture this relation. Most ACs pairs do not have a relationship at all, which
leads to a serious sample imbalance problem. Among the whole process, ARI and
ARTC are parts of ACI and ACTC, so the performance of these tasks will be
influenced. Due to these reasons, many previous works give up and ignore the
classification of ARs. Besides, much research imposes some argument
constraints to do argument mining. In most normal cases, they assume argument
information is a tree structure, and they can use the characteristic of the
tree to extract information. Tree structure argument information is common in
an argumentative essay. However, argument information with no constraints is
more normal in the real world, like a huge amount of corpus on social media.
This information is just like the general argument graphs mentioned before and
needs to be extracted in good quality.

In this paper, we solve the above problems with a novel model called AutoAM
(the abbreviation of Automactic and Universal Argument Mining Model). This is
an efficient and accurate argument mining model to complete the entire
argument mining process. This model does not rely on domain-specific corpus
and does not need to formulate special syntactic constraints, etc., to
construct argument graphs from argumentative text. To improve the performance
of non-tree structured argument mining, we first introduce the argument
component attention mechanism (ArguAtten) in this model, which can better
capture the relevant information of argument components in an argumentative
paragraph. It benefits the overall performance of argument mining. We use a
distance matrix to add the key distance feature to represent ARs. A stratified
learning rate is also a critical strategy in the model to balance multi-task
learning. To the best of our knowledge, we are the first to propose an end-to-
end universal AM model without structure constraints to complete argument
mining. Meanwhile, we combine our novelty and some successful experience to
achieve the state of the art in two public datasets.

In summary, our contributions are as follows:

  * •

We propose a novel model AutoAM for argument mining which can efficiently
solve argument mining in all kinds of the argumentative corpus.

  * •

We introduce ArguAtten (argument component attention mechanism) to better
capture the relation between argument components and improve overall argument
mining performance.

  * •

We conduct extensive experiments on two public datasets and demonstrate that
our method substantially outperforms the existing works. The experiment
results show that the model proposed in this paper achieves the best results
to date in several metrics. Especially, there is a great improvement over the
previous studies in the tasks of ARI (argument relation identification) and
ARTC (argument relation type classification).

##  2 Related Work

Since argument mining was first proposed [16], much research has been
conducted on it. At first, people used rule-based or some traditional machine
learning methods. With the help of deep learning, people begin to get good
performance on several tasks and start to focus on non-tree structured
argument mining. We discuss related work following the development of AM.

###  2.1 Early Argument Mining

The assumption that the argument structure could be seen as a tree or forest
structure was made in the majority of earlier work, which made it simpler to
tackle the problem because various tree-based methods with structural
restrictions could be used. In the early stage of the development of argument
mining, people usually use rule-based structural constraints and traditional
machine learning methods to conduct argumentative mining. In 2007, Moens et
al. [16] conducted the first argument mining research on legal texts in the
legal field, while Kwon et al. [11] also conducted relevant research on
commentary texts in another field. However, the former only identified the
content of the argument and did not classify the argument components. Although
the latter one further completed the classification of argument components, it
still did not extract the relationship between argument components, and could
not explore the argument structure in the text. It only completed part of the
process of argument mining.

###  2.2 Tree Structured Argument Mining with Machine Learning

According to the argumentation paradigm theory of Van Eemeren et al. [6],
Palau and Moens [15] modeled the argument information in legal texts as a tree
structure and used the hand-made Context-Free Grammar (CFG) to parse and
identify the argument structure of the tree structure. This method is less
general and requires different context-free grammars to be formulated for
different structural constraints of argument. By the Stab and Gurevych [27]
[28] tree structure of persuasive Essay (Persuasive Essay, PE) dataset has
been in argument mining has been applied in many studies and practices. In
this dataset, Persing and Ng [23] and Stab and Gurevych [28] used the Integer
Linear Programming (ILP) framework to jointly predict the types of argument
components and argument relations. Several structural constraints are defined
to ensure a tree structure. The arg-micro text (MT) dataset created by
Peldszus [21] is another tree-structured dataset. In studies using this
dataset, decoding techniques based on tree structure are frequently used, such
as Minimum Spanning tree (MST) [22], and ILP [1].

###  2.3 Neural Network Model in Argument Mining

With the popularity of deep learning, neural network models have been applied
to various natural language processing tasks. For deep learning methods based
on neural networks, Eger et al. [7] studied argument mining as a sequence
labeling problem that relies on parsing multiple neural networks. Potash et
al. [25] used sequence-to-sequence pointer network [30] in the field of
argument mining and identified the different types of argument components and
the presence of argument relations using the output of the encoder and
decoder, respectively. Kuribayashi et al. [10] developed a span
representation-based argumentation structure parsing model that employed ELMo
[24] to derive representations for ACs.

###  2.4 Recent Non-Tree Structured Argument Mining

Recently, more works have focused on the argument mining of non-tree
structures. The US Consumer Debt Collection Practices (CDCP) dataset [18] [19]
greatly promotes the development of non-tree structured argument mining. The
argument structures contained in this dataset are non-tree structures. On this
dataset, Niculae et al. [18] carry out a structured learning method based on a
factor graph. This method can also handle the tree structure of datasets. It
can also be used in the PE dataset, but the factor diagram needs a specific
design according to the different types of the argument structure. Galassi et
al. [8] used the residual network on the CDCP dataset. Mor IO et al. [17]
developed an argument mining model, which uses a task-specific parameterized
module to encode argument. In this model, there is also a bi-affine attention
module [5] to capture the argument. Recently, Jianzhu Bao et al. [2]tried to
solve both tree structure argument and non-tree structure argument by
introducing the transformation-based dependency analysis method [4] [9]. This
work gained relatively good performance on the CDCP dataset but did not
complete the ARTC task in one model and did not show the experiment results of
ARTC.

However, these methods either do not cover the argument mining process with a
good performance or impose a variety of argument constraints. There is no end-
to-end model for automatic and universal argument mining before. Thus, we
solve all the problems above in this paper.

##  3 Methodology

As shown in Figure 2, we propose a new model called AutoAM. This model adopts
the joint learning approach. It uses one model to simultaneously learn the
ACTC, ARI, and ARTC three subtasks in argument mining. For the argument
component extraction, the main task is to classify the argument component
type, and the argument component identification task has been completed by
default on both the PE and the CDCP datasets. For argument relation
extraction, the model regards ARI and ARTC as one task. The model classifies
the relationship between the argument components by a classifier and then
gives different prediction results for two tasks by post-processing prediction
labels.

Figure 2: The framework of our proposed model called AutoAM.

###  3.1 Task Formulation

The input data contains two parts: a) A set of $n$ argumentative text
$T=\\{T_{1},T_{2},...,T_{n}\\}$, b) for the $i$th argumentative text, there
are $m$ argument component spans $S=\\{S_{1},S_{2},...,S_{m}\\}$, where every
span marks the start and end scope of each AC $S_{i}=(start_{i},end_{i})$. Our
aim is to train an argument mining model and use it to get output data: a)
types of $m$ ACs provided in the input data
$ACs=\\{AC_{1},AC_{2},...,AC_{m}\\}$, b) $k$ existing ARs
$ARs=\\{AR_{1},AR_{2},...,AR_{k}\\}$ and their types, where
$AR_{i}=(AC_{a}\rightarrow AC_{b})$.

###  3.2 Argument Component Extraction

By default, the argument component identification task has been completed. The
input of the whole model is an argumentative text and a list of positional
spans corresponding to each argument component $S_{i}=(start_{i},end_{i})$.

We input argumentative text $T$ into pre-trained language models (PLMs) to get
contextualized representations $H\in\mathbb{R}^{m\times d_{b}}$, where $d_{b}$
is the dimension of the last hidden state from PLMs. Therefore, we represent
argumentative text as $H=(h_{1},h_{2},...,h_{m})$, where $h_{i}$ denotes the
$i$th token contextualized representation.

We separate argument components from the paragraph using argument component
spans $S$. In the PE dataset, the argument components do not appear
continuously. We use mean pooling to get the representation of each argument
component. Specifically, the $i$ argument component can be represented as:

$AC_{i}=\frac{1}{end_{i}-start_{i}+1}\sum^{end_{i}}_{j=start_{i}}h_{i},$ (1)

where $AC_{i}\in\mathbb{R}^{d_{b}}$. Therefore, all argument components in the
argumentative text can be represented as $ACs=(AC_{1},AC_{2},...,AC_{n})$. For
each argument component, we input it into AC Type Classifier $MLP_{a}$ in
order. This classifier contains a multi-layer perceptron. A Softmax layer is
after it. The probability of every type of argument component can be get by:

$p(y_{i}|AC_{i})=Softmax(MLP_{a}(AC_{i})),$ (2)

where $y_{i}$ represent the predicted labels of the $i$th argument component.
We get the final predicted label of its argument component as:

$\hat{y}_{i}=Argmax(p(y_{i}|AC_{i})).$ (3)

###  3.3 Argument Relation Extraction

This model views ARI and ARTC as having the same task and distinguish them by
post-processing predictions. We classify every argument component pair
$(AC_{i}\rightarrow AC_{j})$. Argument component pairs are different of
$(AC_{i}\rightarrow AC_{j})$ and $(AC_{j}\rightarrow AC_{i})$. We add a label,
‘none’ here. ‘none’ represents that there is no relation of $AC_{i}\rightarrow
AC_{j}$.

In the argument relation extraction part, we use the enumeration method. We
utilize output results from the ACTC step. We combine two argument components
and input them into AR Type Classifier to get the predicted output.

First, the model uses ArguAtten (Argument Component Attention mechanism) to
enhance the semantic representation of argument components. The self-attention
mechanism is first proposed in the Transformer [29]. The core of this
mechanism is the ability to capture how each element in a sequence relates to
the other elements, i.e., how much attention each of the other elements pays
to that element. When the self-attention mechanism is applied to natural
language processing tasks, it can often capture the interrelationship of all
lexical elements in a sentence and better strengthen the contextual semantic
representation. In the task of argument mining, all argument components in an
argumentative text also meet this characteristic. The basic task of argument
mining is to construct an argument graph containing nodes and edges, where
nodes are argument components and edges are argument relations. Before the
argument relation extraction task, the self-attention mechanism of argument
components can be used to capture the mutual attention of argument components.
It means that it can better consider and capture the argument information of
the full text. This mechanism is conducive to argument relations extraction
and the construction of an argument graph. We define ArguAtten as:

$ArguAtten(Q,K,V)=Softmax(\frac{QK^{T}}{\sqrt{d_{k}}})\times V,$ (4)

where $Q$, $K$, $V$ are got by multiplying ACs with $W_{Q}$, $W_{K}$, $W_{V}$.
They are three parameter matrices $W_{Q},W_{K},W_{V}\in\mathbb{R}^{d_{b}\times
d_{k}}$, and $d_{k}$ is the dimension of attention layer. Besides, we also use
ResNet and layer normalization (LN) after the attention layer to avoid
gradient explosion:

$ResNetOut=LN(ACs+ArgutAtten(ACs)).$ (5)

Through the self-attention of argument components, we obtain a better
contextualized representation of argument components and then begin to
construct argument pairs to perform argument relation extraction.

We consider that the relative distance between two argument components has a
decisive influence on the type of argument relations between them. By
observing the dataset, we can find that there is usually no argument relation
between the two argument components, which are relatively far apart. It can
significantly help the model to classify the argument relation types.
Therefore, we incorporate this feature into the representation of argument
relations. At first, the distance vector is introduced, and the specific
definition is shown as:

$V_{dist}=(i-j)\times W_{dist},$ (6)

where $(i-j)$ represents a relative distance, it can be positive or negative.
$W_{dist}\in\mathbb{R}^{1\times d_{dist}}$ is a distance transformation
matrix, and it can transform a distance scalar to a distance vector.
$d_{dist}$ is the length of the distance vector.

For each argument relation, it comes from the source argument component (Src
AC), the target argument component (Trg AC), and the distance vector (Dist
Vec). We concatenate them to get the representation of an argument relation
as:

$AR_{i,j}=[AC_{i},AC_{j},V_{dist}],$ (7)

where $AR_{i,j}\in\mathbb{R}^{d_{b}\times 2+d_{dist}}$, $d_{dist}$ is the
length of distance vector.

Therefore, argument relations in an argumentative text can be represented as
$ARs=(AC_{1,2},AC_{1,3},...,AC_{n,n-1},)$, contains $n\times(n-1)$ potential
argument relations in total. We do not consider self-relation like
$AR=(AC_{i}\rightarrow AC_{i})$.

For each potential argument relation, we separately and sequentially input
them into the AR Type Classifier $MLP_{b}$. The classifier uses a Multi-Layer
Perceptron (MLP) containing a hidden layer of 512 dimensions. The output of
the last layer of the Multi-layer Perceptron is followed by a Softmax layer to
obtain the probability of an argument relationship in each possible type
label, as shown in:

$p(y_{i,j}|AR_{i,j})=Softmax(MLP_{b}(AR_{i,j})),$ (8)

where $y_{i,j}$ denotes the predicted label of the argument relation from the
$i$th argument component to the $j$th argument component. The final predicted
labels are:

$\hat{y}_{i,j}=Argmax(p(y_{i,j}|AR_{i,j})).$ (9)

To get the predicted labels of ARI and ARTC, we post-processed the prediction
of the model. The existence of an argument relation in the ARI task is defined
as:

$\hat{y}_{ARI}=\begin{cases}0&\text{if }\hat{y}_{AR}=0\\\ 1&\text{if
}\hat{y}_{AR}\neq 0\end{cases}$ (10)

where $\hat{y}_{AR}$ is the predicted label from the model output.

When we gain the type of an existing argument relation in the ARTC task, we
assign the probability of ‘none’ to zero and select the other label with the
higher probability. They are represented as:

$\hat{y}_{ARTC}=Argmax(p(y_{AR}|AR_{i,j})),\quad y^{none}=0,$ (11)

where $y^{none}$ is the model output of the label ‘none’.

###  3.4 Loss Function Design

This model jointly learns the argument component extraction and the argument
relation extraction. By combining these two tasks, the training objective and
loss function of the final model is obtained as:

$L(\theta)=\sum_{i}log(p(y_{i}|AC_{i}))+\sum_{i,j}p(y_{i,j}|AR_{i,j})+\frac{\lambda}{2}||\theta||^{2},$
(12)

where $\theta$ represents all the parameters in the model, and $\lambda$
represents the coefficient of L2 regularization. According to the loss
function, the parameters in the model are updated repeatedly until the model
achieves better performance results to complete the model training.

##  4 Experiments

###  4.1 Datasets

We evaluate our proposed model on two public datasets: Persuasive Essays (PE)
[28] and Consumer Debt Collection Practices (CDCP) [18].

The PE dataset only has tree structure argument information. It has three
types of ACs: Major-Claim, Claim, and Premise, and two types of AR: support
and attack.

The CDCP dataset has general structure argument information, not limited to a
tree structure. It is different from the PE dataset and is more difficult. The
argument information in this dataset is more similar to the real world. There
are five types of ACs (propositions): Reference, Fact, Testimony, Value, and
Policy. Between these ACs, there are two types of ARs: reason and evidence.

We both use the original train-test split of two datasets to conduct
experiments.

###  4.2 Setups

In the model training, roberta-base [13] was used to fine-tune, and AdamW
optimizer [14] was used to optimize the parameters of the model during the
training. We apply a stratified learning rate to obtain a better semantic
representation of BERT context and downstream task effect. The stratified
learning rate is important in this task because this multi-task learning is
complex and have three subtasks. The ARI and ARTC need a relatively bigger
learning rate to learn the data better. The initial learning rate of the BERT
layer is set as 2e-5. The learning rate of the AC extraction module and the AR
extraction module is set as 2e-4 and 2e-3, respectively. After BERT output,
the Dropout Rate [26] is set to 0.2. The maximum sequence length of a single
piece of data is 512. We cut off ACs and ARs in the over-length text. The
batch size in each training step is set to 16 in the CDCP dataset and 2 in the
PE dataset. The reason is that there are more ACs in one argumentative text
from the PE dataset than in the CDCP dataset.

In training, we set an early stop strategy with 5 epochs. We set the minimum
training epochs as 15 to wait for the model to become stable. We use
$MacroF1_{ARI}$ as monitoring indicators in our early stop strategy. That is
because AR extraction is our main improvement direction. Furthermore, the ARI
is between the ACTC and the ARTC, so we can better balance the three tasks’
performance in the multi-task learning scenario.

The code implementation of our model is mainly written using PyTorch [20]
library, and the pre-trained model is loaded using Transformers [32] library.
In addition, model training and testing were conducted on one NVIDIA GeForce
RTX 3090.

###  4.3 Compared Methods

We compare our model with several baselines to evaluate the performance:

  * •

Joint-ILP [28] uses Integer Linear Programming (ILP) to extract ACs and ARs.
We compare our model with it in the PE dataset.

  * •

St-SVM-full [18] uses full factor graph and structured SVM to do argument
mining. We compare our model with it in both the PE and the CDCP datasets.

  * •

Joint-PN [25] employs a Pointer Network with an attention mechanism to extract
argument information. We compare our model with it in the PE dataset.

  * •

Span-LSTM [10] use LSTM-based span representation with ELMo to perform
argument mining. We compare our model with it in the PE dataset.

  * •

Deep-Res-LG [8] uses Residual Neural Network on AM tasks. We compare our model
with it in the CDCP dataset.

  * •

TSP-PLBA [17] introduces task-specific parameterization and bi-affine
attention to AM tasks. We compare our model with it in the CDCP dataset.

  * •

BERT-Trans [2] use transformation-based dependency analysis method to solve AM
problems. We compare our model with it in both the PE and the CDCP datasets.
It is also the state of the art on two datasets.

###  4.4 Performance Comparison

Methods | ACTC | ARI | ARTC | AVG
---|---|---|---|---
Macro | Value | Policy | Testi. | Fact | Refer. | Macro | Rel. | Non-rel. | Macro | Reason | Evidence
St-SVM-strict | 73.2 | 76.4 | 76.8 | 71.5 | 41.3 | 100.0 | - | 26.7 | - | - | - | - | -
Deep-Res-LG | 65.3 | 72.2 | 74.4 | 72.9 | 40.3 | 66.7 | - | 29.3 | - | 15.1 | 30.2 | 0.0 | -
TSP-PLBA | 78.9 | - | - | - | - | - | - | 34.0 | - | - | - | 18.7 | -
BERT-Trans | 82.5 | 83.2 | 86.3 | 84.9 | 58.3 | 100.0 | 67.8 | 37.3 | 98.3 | - | - | - | -
AutoAM (Ours) | 84.6 | 85.0 | 86.8 | 86.1 | 65.9 | 100.0 | 68.4 | 38.5 | 98.4 | 71.3 | 98.1 | 44.4 | 74.8

Table 1: The results of comparison experiments on the CDCP dataset. All
numbers in the table are f1 scores (%). The best scores are in bold. ‘-’
represents that the original paper does not report.

Methods | ACTC | ARI | ARTC | AVG
---|---|---|---|---
Macro | MC | Claim | Premise | Macro | Rel. | Non-rel. | Macro | Support | Attack
Joint-ILP | 82.6 | 89.1 | 68.2 | 90.3 | 75.1 | 58.5 | 91.8 | 68.0 | 94.7 | 41.3 | 75.2
St-SVM-strict | 77.6 | 78.2 | 64.5 | 90.2 | - | 60.1 | - | - | - | - | -
Joint-PN | 84.9 | 89.4 | 73.2 | 92.1 | 76.7 | 60.8 | 92.5 | - | - | - | -
Span-LSTM | 85.7 | 91.6 | 73.3 | 92.1 | 80.7 | 68.8 | 93.7 | 79.0 | 96.8 | 61.1 | 81.8
BERT-Trans | 88.4 | 93.2 | 78.8 | 93.1 | 82.5 | 70.6 | 94.3 | 81.0 | - | - | 83.4
AutoAM (Ours) | 88.7 | 91.9 | 80.3 | 93.9 | 81.6 | 65.8 | 98.5 | 75.4 | 97.6 | 53.2 | 81.9

Table 2: The results of comparison experiments on the PE dataset. All numbers
in the table are f1 scores (%). The best results are in bold. The second best
results are in italics. ‘-’ represents that the original paper does not
report.

The evaluation results are summarized in Table 1 and Table 2. In both tables,
‘-’ indicates that the original paper does not measure the performance of this
metric for its model. The best results are in bold, and the second-best
results are in italics.

On the CDCP dataset, we can see our model achieves the best performance on all
metrics in ACTC, ARI, and ARTC tasks. We are the first to complete all the
tasks and get ideal results on the CDCP dataset. Our model outperforms the
state of the art with an improvement of 2.1 in ACTC and 0.6 in ARI. The method
BERT-Trans does not perform ARTC with other tasks at the same time, and it
does not report results of ARTC, maybe due to unsatisfactory performance. In
particular, compared with the previous work, we have greatly improved the task
performance of ARTC and achieved ideal results.

On the PE dataset, our model also gets ideal performance. However, we get the
second-best scores in several metrics. The first reason is that the PE dataset
is tree-structured, so many previous work impose some structure constraints.
Their models incorporate more information, and our model assumes they are
general argument graphs in contrast. Another reason is that the models BERT-
Trans, Span-LSTM, and Joint-PN combine extra features to represent ACs, like
paragraph types, BoW, position embedding, etc. This information will change in
the different corpus, and we want to build an end-to-end universal model. For
example, there is no paragraph type information in the CDCP dataset.
Therefore, we do not use them in our model. Even if our model does not take
these factors into account, we achieve similar results to the state of the
art.

###  4.5 Ablation Study

Methods | ACTC | ARI | ARTC | AVG
---|---|---|---|---
Macro | $\bigtriangledown$ | Macro | $\bigtriangledown$ | Macro | $\bigtriangledown$ | Macro | $\bigtriangledown$
AutoAM (Ours) | 84.6 | - | 68.4 | - | 71.3 | - | 74.8 | -
w/o stratified learning rate | 76.9 | -7.7 | 66.2 | -2.2 | 49.3 | -22.0 | 64.2 | -10.6
w/o ArguAtten | 82.9 | -1.7 | 69.8 | +1.4 | 57.7 | -13.6 | 70.1 | -4.7
w/o Distance Matrix | 82.9 | -1.7 | 62.9 | -5.5 | 59.3 | -12.0 | 68.3 | -6.5
Table 3: The results of ablation experiments on the CDCP dataset. All numbers
in the table are f1 scores (%). The best results are in bold.

The ablation study results are summarized in Table 3. We conduct an ablation
study on the CDCP dataset to see the impact of key modules in our model. It
can be observed that the stratified learning rate is the most critical in this
model. It verifies the viewpoint that multi-task learning is complex in this
model and ARs extraction module needs a bigger learning rate to perform well.
We can see ArguAtten improve the ACTC and ARTC performance by 1.7 and 13.6.
However, the ARI matric decreases a little bit. Even though the numbers are
small, we think that the reason is the interrelationship between ACs has
little impact on the prediction of ARs’ existence. ArguAtten mainly plays an
effect in predicting the type of ARs. From this table, we can also find that
the distance matrix brings the important distance feature to AR representation
with an overall improvement of 6.5.

##  5 Conclusion and Future Work

In this paper, we propose a novel method for argument mining and first
introduce the argument component attention mechanism. This is the first end-
to-end argument mining model that can extract argument information without any
structured constraints and get argument relations of good quality. In the
model, ArguAtten can better capture the correlation information of argument
components in an argumentative paragraph so as to better explore the
argumentative relationship. Our experiment results show that our method
achieves the state of the art. In the future, we will continue to explore
designing a better model to describe and capture elements and relationships in
argument graphs.

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# Universality classes of the Anderson transitions driven by quasiperiodic
potential
in the three-dimensional Wigner-Dyson symmetry classes

Xunlong Luo<EMAIL_ADDRESS>Science and Technology on Surface Physics
and Chemistry Laboratory, Mianyang 621907, China  Tomi Ohtsuki
<EMAIL_ADDRESS>Physics Division, Sophia University, Chiyoda-ku, Tokyo
102-8554, Japan

###### Abstract

Quasiperiodic system is an intermediate state between periodic and disordered
systems with unique delocalization-localization transition driven by the
quasiperiodic potential (QP). One of the intriguing questions is whether the
universality class of the Anderson transition (AT) driven by QP is similar to
that of the AT driven by the random potential in the same symmetry class.
Here, we study the critical behavior of the ATs driven by QP in the three-
dimensional (3D) Anderson model, Peierls phase model, and Ando model, which
belong to the Wigner-Dyson symmetry classes. The localization length and two-
terminal conductance have been calculated by the transfer matrix method, and
we argue that their error estimations in statistics suffer from the
correlation of QP. With the correlation under control, the critical exponents
$\nu$ of the ATs driven by QP are estimated by the finite size scaling
analysis of conductance, which are consistent with $\nu$’s of the ATs driven
by the random potential. Moreover, the critical conductance distribution and
the level spacing ratio distribution have been studied. We also find that a
convolutional neural network trained by the localized/delocalized
wavefunctions in a disordered system predicts the localized/delocalized
wavefunctions in quasiperiodic systems. Our numerical results strongly support
that the universality classes of the ATs driven by QP and random potential are
similar in the 3D Wigner-Dyson symmetry classes.

##  I Introduction

In the seminal work by Anderson, the extended wavefunction could become
localized driven by random potential because of the quantum interference [1].
This delocalization-localization transition is called the Anderson transition
(AT), which can be described by the single parameter scaling theory [2]. As a
second-order phase transition, the universality class of AT, which is
characterized by the critical exponents, depends on the symmetry and
dimension, but not the detail of the system [3].

On the other hand, the quasiperiodic potential (QP) can also drive the
Anderson localization, which has been extensively studied both theoretically
[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]
and experimentally [25, 26, 27, 28, 29, 30, 31, 32]. Quasiperiodic systems are
ubiquitous in nature and can be realized in the quasicrystals [33], photonic
lattices [25, 26, 27], ultracold quantum gases [28, 29, 30, 34, 35, 36, 37],
and Moiré superlattices (i.e., the twisted bilayer graphene at
$30^{\circ}$[38, 39]). One of the famous examples is the one-dimensional (1D)
Aubry-André-Harper (AAH) model [40, 4], where there is a delocalization-
localization transition with the same critical point at any Fermi energy due
to the self-duality. Generalized AAH models involve the exact mobility edges
[8, 9, 10, 11, 12, 13], connections to higher-dimensional topological matter
phases [41, 42, 43, 44, 45], and higher-dimensional ATs [17, 19, 18, 20].

Table 1: Polynomial fitting results for the two-terminal conductance around the Anderson transition points for three dimensional Anderson model (AM), Peierls phase model (PPM) and Ando model with quasiperiodic potential. The goodness of fit (GOF), critical quasiperiodic potential strength $V_{c}$, critical exponent $\nu$, and the scaling dimension of the least irrelevant scaling variable $-y$ are shown for various system sizes and for different orders of the Taylor expansion: $(m_{1},n_{1},m_{2},n_{2})$. The square bracket is the 95% confidence interval.  Model | $L$ | $m_{1}$ | $n_{1}$ | $m_{2}$ | $n_{2}$ | GOF | $V_{c}$ | $\nu$ | $y$
---|---|---|---|---|---|---|---|---|---
AM | 10-24 | 3 | 3 | 0 | 1 | 0.106 | 2.229[2.226, 2.233] | 1.599[1.569, 1.637] | 3.9[3.3, 4.6]
| 10-28 | 3 | 3 | 0 | 1 | 0.101 | 2.233[2.230, 2.236] | 1.613[1.587, 1.639] | 3.5[3.0, 4.1]
PPM | 10-24 | 3 | 3 | 0 | 1 | 0.105 | 2.335[2.331, 2.341] | 1.43[1.39, 1.49] | 3.0[2.2, 3.8]
| 10-28 | 3 | 3 | 0 | 1 | 0.162 | 2.339[2.336, 2.342] | 1.45[1.42, 1.49] | 2.8[2.3, 3.4]
Ando | 10-20 | 2 | 3 | 0 | 1 | 0.112 | 2.449[2.445, 2.455] | 1.37[1.35, 1.39] | 4.3[3.6, 4.9]
| 10-20 | 3 | 3 | 0 | 1 | 0.139 | 2.449[2.445, 2.455] | 1.36[1.34, 1.38] | 4.3[3.7, 4.9]

Both random potential and QP can drive the ATs, and a natural question is
whether the universality classes of the ATs driven by QP and those by random
potential are the same or not [17]. For 1D and two-dimensional (2D) systems in
orthogonal class, the infinitesimal random potential will localize the
wavefunctions and there is no AT at all [2]. However, the 1D AAH model shows a
delocalization-localization transition driven by QP with exact critical
exponent $\nu=1$ [4], and the critical exponent is estimated as $\nu\approx 1$
[19] for the generalized 2D AAH model. Both of them belong to the orthogonal
class. Therefore the ATs in 1D and 2D AAH models belong to new universality
classes without counterparts in the disordered systems. However, it is
reported that the universality class of AT in the generalized three-
dimensional (3D) AAH model is similar to that of the AT driven by random
potential in 3D orthogonal class [17, 18]. Considering the uniqueness of 3D
case, it is necessary to study the critical behavior by independent methods,
such as the localization length, conductance, level statistics, and deep
learning. Moreover, it is also interesting to check the conclusion in other
symmetry classes, such as unitary and symplectic symmetry classes [46, 47,
48].

In this paper, we focus on the critical behavior of the ATs driven by QP in
the 3D Anderson model (AM) [1], Peierls phase model (PPM) [49, 50], and Ando
model [51, 52], which belong to orthogonal, unitary, and symplectic classes,
respectively. The localization length and conductance have been calculated by
the transfer matrix method [53, 54, 55]. We find that the estimates of their
statistical error are difficult since the samples are not independent due to
QP. Because the estimates of critical exponents $\nu$ by the polynomial
fitting method are sensitive to the error bar of the physical quantities [56],
the incorrect error bar for the localization length or conductance will result
in incorrect estimates of $\nu$. We have chosen the proper number of samples
so that they are regarded as independent, and the critical exponents $\nu$
have been estimated based on the conductance, which are similar to those of
the ATs driven by random potential (see Table 1). Based on the critical points
estimated by conductance, we also calculated the critical conductance
distributions for the three models, which are independent of the system size
and significantly different from each other. By properly taking account of the
anisotropy, they are shown to be the same as those of disordered systems in
the same symmetry class. Although the level spacing ratio distribution $P(r)$
in the delocalized phase of 1D AAH model is special [57, 58], we find that
$P(r)$ in the delocalized states of the three models in this paper follow the
Wigner-like surmises [59]. To further understand the localization properties,
we constructed a convolutional neural network (CNN), whose parameters have
been trained by the localized/delocalized wavefunctions in a disordered system
[60]. We find this CNN can predict the localized/delocalized wavefunctions in
quasiperiodic systems, which indicates that the localization properties in the
disordered and quasiperiodic systems are similar.

The rest of the paper is organized as follows. In the next section, we will
first introduce the three models with QP belonging to the Wigner-Dyson
symmetry classes. Next, we will study the phase diagram spanned by eigenenergy
and QP strength, and the mobility edge in section III. Then we will discuss
the finite size scaling of conductance in section IV. Moreover, the critical
conductance distribution (section V) and the level spacing ratio distribution
(section VI) are studied. In section VII, we apply the CNN trained for the
random potential to the wavefunctions in QP, and show that the CNN correctly
predicts the critical strength of QP. The final section is devoted to the
summary and concluding remarks.

(a) Anderson model

(b) Peierls phase model

(c) Ando model

Figure 1: Normalized participation ratio in the logarithmic form $\eta$ for
different eigenenergies $E$ as a function of quasiperiodic potential strength
$V$ for the three quasiperiodic models. $\eta\approx 3$ stands for the well
delocalized state and $\eta\approx 0$ for the strongly localized state. $\eta$
is simulated with cubic system with linear size $L=10$.

##  II Model and quasiperiodic potential

We first introduce the following tight-binding models defined on a 3D cubic
lattice,

$\displaystyle{\cal
H}=\sum_{\bm{r}}\varepsilon_{\bm{r}}c^{\dagger}_{\bm{r}}c_{\bm{r}}+\sum_{\langle{\bm{r}},{\bm{r}^{\prime}}\rangle}e^{2\pi{\rm
i}\theta_{{\bm{r}},{\bm{r}^{\prime}}}}c^{\dagger}_{\bm{r}}c_{\bm{r}^{\prime}},$
(1)

where $c^{\dagger}_{\bm{r}}$ ($c_{\bm{r}}$) is the creation (annihilation)
operator. $\bm{r}$ and $\bm{r}^{\prime}$ specify the cubic lattice site;
${\bm{r}}=(r_{x},r_{y},r_{z})$ and $r_{\mu}=1,2,\cdots,L_{\mu}$ with
$\mu=x,y,z$. $\langle{\bm{r}},{\bm{r}^{\prime}}\rangle$ means that $\bm{r}$
and $\bm{r}^{\prime}$ are the nearest neighbor lattice sites. $\rm
i=\sqrt{-1}$ is the imaginary unit. The model in Eq. (1) is the Anderson model
with $\theta_{{\bm{r}},{\bm{r}^{\prime}}}$ = 0 [1]. When
$\theta_{{\bm{r}},{\bm{r}}+{\bm{e}}_{z}}=\Phi\\!\ r_{x}$ with ${\bm{e}}_{z}$
the unit vector in $z$ direction and $\theta_{{\bm{r}},{\bm{r}^{\prime}}}$ = 0
in the hoppings along $x$ and $y$ directions, the model is the Peierls phase
model [49, 50]. $\Phi$ is a magnetic gauge flux that penetrates through every
square plaquette in the $z$-$x$ plane of the cubic lattice in units of $h/e$,
where $h$ is the Planck constant and $e$ is the elementary charge. We set
$\Phi=0.1$ in the following calculations.

The Ando model can be constructed in the 3D cubic lattice as follows [51, 52],

$\displaystyle{\cal
H}=\sum_{\bm{r}}\varepsilon_{\bm{r}}c_{{\bm{r}},\sigma}^{\dagger}c_{{\bm{r}},\sigma}+\sum_{\langle{\bm{r}},{\bm{r}^{\prime}}\rangle,\sigma,\sigma^{\prime}}R({\bm{r}},{\bm{r}^{\prime}})_{\sigma,\sigma^{\prime}}c_{{\bm{r}},\sigma}^{\dagger}c_{{\bm{r}^{\prime}},\sigma^{\prime}},$
(2)

where $c_{{\bm{r}},\sigma}^{\dagger}$($c_{{\bm{r}},\sigma}$) is the creation
(annihilation) operator with $\sigma=\uparrow,\downarrow$. We set the spin-
orbit coupling between the nearest neighbor lattice sites as
$R({\bm{r}},{\bm{r}}+{\bm{e}}_{\mu})=e^{{\rm i}\theta\sigma_{\mu}}$, where
${\bm{e}}_{\mu}$ is the unit vector in the $\mu=x,y,z$ directions and
$\sigma_{\mu}$ is the Pauli spin matrix. The parameter $\theta$, which is set
to be $\theta=\pi/6$ in our simulations, represents the strength of spin-orbit
coupling.

$\varepsilon_{\bm{r}}$ in Eq. (1) and (2) is the onsite potential. Here we
study the critical behavior of ATs driven by QP, so the QP is added into these
three models instead of random potential. QP has already been directly
realized in the photonic lattices [25, 26], ultracold atomic experiments [28,
36, 29, 34, 35, 30, 37], and Moiré superlattices [38, 39]. Following Ref. 17,
we set

$\displaystyle\varepsilon_{\bm{r}}=2V\sum_{i=1}^{3}\cos(2\pi{\bm{b}}_{i}\cdot{\bm{r}}+\phi_{i}),$
(3)

where $V$ is the strength of QP, $\\{{\bm{b}}_{i}\\}$ are vectors determining
the quasiperiodicity, and $\\{\phi_{i}\\}$ are the arbitrary phases. We write
${\bm{b}}_{i}$ in the matrix form ${\bm{B}}=[{\bm{b}}_{1}\ \ {\bm{b}}_{2}\ \
{\bm{b}}_{3}]^{\rm T}$. In order to keep quasiperiodicity and make the model
nonseparable, we set $\bm{B}=\beta{\bm{R}}(\alpha)$ with
$\beta=\frac{\sqrt{5}-1}{2}$ and

$\displaystyle{\bm{R}}(\alpha)=\begin{bmatrix}c^{2}+s^{3}&cs&cs^{2}-cs\\\
cs&-s&c^{2}\\\ cs^{2}-cs&c^{2}&c^{2}s+s^{2}\\\ \end{bmatrix},$ (4)

where $c=\cos(\alpha)$ and $s=\sin(\alpha)$. $\beta=\frac{\sqrt{5}-1}{2}$ is
the golden mean, which is an irrational number characterizing the
quasiperiodicity. $\bm{R}(\alpha)$ is a symmetric orthonormal matrix which
makes the model nonseparable, as long as we avoid $\alpha=0$, $\alpha=\pi/2$,
and multiples thereof. In the following calculations, we set $\alpha=\pi/7$.
We always impose open boundary condition in the calculations because of the
quasiperiodicity. In order to simulate many samples, we randomize the
arbitrary phases $\\{\phi_{i}\\}$ in the cosines within $(0,2\pi]$. These
systems will be equivalent by an overall translation in the thermodynamic
limit.

##  III Phase diagram and mobility edge

We first study the phase diagrams of these models spanned by the eigenenergy
$E$ and the QP strength $V$. Here, we diagonalize the Hamiltonians in cubic
systems with linear size $L$ and obtain the eigenenergies $\\{E_{i}\\}$ and
the normalized eigenfunctions $\\{\psi_{i}(\bm{r})\\}$. The participation
ratio for the eigenfunction $\psi_{i}(\bm{r})$ is defined as

$\displaystyle{\rm
PR}(E_{i})\equiv\frac{1}{\sum_{\bm{r}}|\psi_{i}(\bm{r})|^{4}}.$ (5)

The delocalization/localization states can be characterized by the normalized
participation ratio [61, 62, 13] in the logarithmic form, which is defined as

$\displaystyle\eta(E_{i})\equiv\frac{\ln{\rm PR}(E_{i})}{\ln L}.$ (6)

When the wavefunction is well delocalized, ${\rm PR}\sim L^{3}$ and
$\eta\approx 3$. On the contrary, when the wavefunction is localized, ${\rm
PR}\sim\xi^{3}$ with $\xi$ the localization length, and $\eta\approx 0$.
$\eta$ is plotted in the plane spanned by $E$ and $V$ (Fig. 1) for the three
quasiperiodic models introduced in Section II. As seen in Fig. 1, the values
of $\eta$ change rapidly, indicating the existence of mobility edges which
fluctuate significantly as we vary QP strength $V$.

From now we focus on the Fermi energy level $E=0$ and analyze the critical
behavior. In order to characterize the delocalization-localization
transitions, the localization length and conductance are calculated by the
transfer matrix method. Then the critical exponents are extracted by the
polynomial fitting method. We note that the fitting is sensitive to the
statistical error estimates of the fitting data, and the incorrect error
estimates result in incorrect evaluation of critical exponents. Therefore we
should be careful of the statistical error estimates of the localization
length and conductance in quasiperiodic systems. We argue that there is a
correlation between samples for localization length and conductance resulting
from the QP, which result in incorrect error estimates of them. However, the
correlation can be controlled by the limited number of samples (see the
detailed analysis in Appendices B and C). In this paper, we take conductance
as an example to study the critical behavior. We think the localization length
analysis also gives a similar result if the error estimates are correctly
done.

In the next section, we will introduce the calculation of conductance and
perform the finite size scaling analysis to extract the critical parameters,
such as critical exponent $\nu$ and critical quasiperiodic potential strength
$V_{c}$.

##  IV Finite size scaling of the conductance

The zero temperature two-terminal conductance in units of $e^{2}/h$ is given
by

$\displaystyle g={\rm tr}(tt^{\dagger})$ (7)

with $t$ the transmission matrix. The transmission matrix $t$ can be
calculated by the transfer matrix method [63]. The two-terminal conductance is
calculated along $z$ direction with ideal lead in the two side of the cubic
system ($L_{x}=L_{y}=L_{z}=L$).

The mean conductance $\langle g\rangle$ is averaged over $50000$ samples for
the AM and PPM, $10^{5}$ samples for the Ando model with random phase
$\\{\phi_{i}\\}$. We note that $g(\\{\phi_{i}\\})$ and
$g(\\{\phi_{i}+\delta_{\phi_{i}}\\})$ are correlated for small
$\delta_{\phi_{i}}$, but uncorrelated when $\delta_{\phi_{i}}$ is large enough
(see Appendix B), just like the magnetoconductance in mesoscopic systems [64,
65, 66, 67, 68]. The behavior of the mean conductance $\langle g\rangle$ or
logarithmic average $\langle\ln g\rangle$ with error bar are shown in Fig. 2
for various $V$ and $L$. The scale invariant point of conductance indicates
the critical point $V_{c}$ of the AT driven by QP. With $V<V_{c}$ ($V>V_{c}$),
the conductance increases (decreases) with system size $L$, indicating it is
in the delocalized (localized) phase. To extract the universal critical
parameters near the critical point, the polynomial fitting method based on the
finite size scaling theory is introduced as follows.

Figure 2: The two-terminal conductance as a function of the strength of
quasiperiodic potential $V$ for Anderson, Peierls phase, and Ando models (from
top to bottom). The black points with the error bars are the raw data and
lines with different colors are the polynomial fitting results with expansion
order ($m_{1}$, $n_{1}$, $m_{1}$, $n_{1}$) = (3, 3, 0, 1).

(a) Three quasiperiodic models

(b) Anderson model

Figure 3: (a) Critical conductance distributions (CCDs) along $z$ direction
for the Anderson model (AM), Peierls phase model (PPM), and Ando model at
critical quasiperiodic potential $V_{c}$ estimated in TABLE 1. (b) The CCDs of
quasiperiodic AM along $x$, $y$, and $z$ directions. For comparison, the CCD
of disordered AM at critical disorder $W_{c}$ is also shown in circles. The
two-terminal conductance $g$ is calculated in cubic system with linear size
$L$ and $L=16$ for (b). Data for each distribution comes from $10^{6}$
samples.

The single parameter scaling theory [2] assumes that the dimensionless
physical quantity $\Gamma$, such as the normalized localization length and the
two-terminal conductance, obeys the universal scaling function,

$\displaystyle\Gamma=f(\phi_{1}L^{1/\nu}),$ (8)

where $\phi_{1}$ is the relevant scaling variable, $L$ is the system size, and
$\nu$ is the critical exponent describing the power-law divergence of the
length scales near the critical point. For the actual fitting, we often need
to take into account the irrelevant scaling variables because of the
corrections to the single parameter scaling [69, 56]. Here we just consider
the least-irrelevant scaling variable $\phi_{2}$, then the universal scaling
function can be written as

$\displaystyle\Gamma=F(\phi_{1}L^{1/\nu},\phi_{2}L^{-y}),$ (9)

where $-y$ is the scaling dimension for the irrelevant scaling variable. The
term $\phi_{2}L^{-y}$ will decrease to zero in power-law with $L$ increasing
and the scaling function returns to the single parameter scaling form in Eq.
(8) in the thermodynamic limit. The relevant or irrelevant scaling variables
depend on the system parameter driving the AT. In this paper, QP strength $V$
is this system parameter, and a normalized distance of $V$ from the critical
point $V_{c}$ is define as $w\equiv(V-V_{c})/V_{c}$. Then scaling variables
$\phi_{1}$ and $\phi_{2}$ are functions of $w$ with $\phi_{1}(w=0)=0$ by
definition. Considering the possible non-linearity of $w$ near the critical
point, $\phi_{1}$ and $\phi_{2}$ are expanded in powers of small $w$ when $V$
is sufficiently close to $V_{c}$;

$\phi_{i}(w)\equiv\sum^{m_{i}}_{j=0}b_{i,j}w^{j}$ (10)

with $i=1,2$, $b_{1,0}=0$. When $V$ is sufficiently close to $V_{c}$ and the
values of $\Gamma$ are controlled in a finite range, the universal scaling
function $F$ in Eq. (9) can also be expanded in powers of its small arguments,

$F=\sum^{n_{1}}_{j_{1}=0}\sum^{n_{2}}_{j_{2}=0}a_{j_{1},j_{2}}(\phi_{1}L^{1/\nu})^{j_{1}}(\phi_{2}L^{-y})^{j_{2}}.$
(11)

To reduce the redundant two degrees of freedom in Eq. (11), we set
$a_{0,1}=a_{1,0}=1$. We note that the assumption here is a posteriori
justified by the numerical analysis.

Given $(m_{1},n_{1},m_{2},n_{2})$ in Eqs. (10) and (11), $F$ is a finite-order
of polynomial of $L$ and $w\equiv(V-V_{c})/V_{c}$. Numerical data of $\Gamma$
for different $L$ and $V$ are fitted by the polynomial with fitting parameters
$V_{c}$, $\nu$, $y$, $a_{i,j}$ and $b_{i,j}$. We minimize $\chi^{2}$ in terms
of the fitting parameters,

$\chi^{2}\equiv\sum^{N_{D}}_{k=1}\frac{(\Gamma_{k}-F_{k})^{2}}{\sigma^{2}_{\Gamma_{k}}}.$
(12)

Here $k$ counts the data points ($k=1,\cdots,N_{D}$), and each data point is
specified by $L$ and $V$; $k=(L,V)$. $\Gamma_{k}$ and $\sigma_{\Gamma_{k}}$
are a mean value of dimensionless physical quantity and its standard deviation
at $k=(L,V)$, respectively, while $F_{k}$ is the fitting value from the
polynomial $F$ at $k=(L,V)$. $F_{k}$ depends on the fitting parameters and
$\chi^{2}$ is minimized in terms of them. The minimization is carried out for
several different choices of $(m_{1},n_{1},m_{2},n_{2})$. The quality of the
fitting is characterized by the goodness of fit in statistics [70, 71]. The
95$\%$ confidence intervals for the fitting results are determined by 1000
sets of $N_{D}$ synthetic data points, which are statistically generated from
the fitting value $F_{k}$ with the same standard deviation of $\Gamma_{k}$ at
each point $k$.

Here the dimensionless physical quantity conductance [$\langle g\rangle$ or
$\langle\ln g\rangle$] will be fitted by the polynomial fitting method. The
fitting results with the goodness of fit greater than 0.1 are shown in Table
1. $V_{c}$ and $\nu$ are shown to be robust against the change of the
expansion orders and various system sizes in Table 1. The critical exponents
$\nu$ are estimated as $\nu=1.613~{}[1.587,1.639]$ for the AM,
$\nu=1.45~{}[1.42,1.49]$ for the PPM, and $\nu=1.37~{}[1.35,1.39]$ for the
Ando model, which are consistent with the critical exponents of ATs driven by
random potential, i.e., $\nu\approx 1.57$ [56] for 3D orthogonal class,
$\nu\approx 1.43$ [69] for 3D unitary class, and $\nu\approx 1.36$ [72] for 3D
symplectic class, respectively. We also find that the critical QP strength
$V_{c}$ increases for the PPM and Ando model compared with that for the AM,
which is similar to that in disordered systems. This increase of $V_{c}$ by
the magnetic field (Peierls phase model) and spin-orbit interaction (Ando
model) is consistent with the weak localization [73]. In conclusion, the
universality classes of ATs driven by QP are similar to those of ATs driven by
the random potential in the 3D Wigner-Dyson symmetry classes [17].

##  V Critical conductance distribution

Figure 4: Comparison of the critical conductance distributions (CCDs) between
quasiperiodic and disordered systems. The lines with points stand for the CCDs
of disordered systems at critical disorder $W_{c}$, i.e., Anderson model (AM),
U(1) model, and SU(2) model in cubic systems with linear size $L$. The lines
with circles are the CCDs for quasiperiodic systems at critical quasiperiodic
potential $V_{c}$, i.e., AM, Peierls phase model (PPM) and Ando model, with
tuned geometry (cross-section linear size $L$ and transmission length
$L_{z}$). Data for each distribution comes from $10^{6}$ samples.

To reinforce our conclusion that the universality classes in 3D quasiperiodic
systems are the same as the cases in disordered systems, we have also studied
the critical conductance distributions (CCDs) and compared them with those in
disordered systems.

At the critical point, the correlation length will diverge and the conductance
distribution will not depend on the system size $L$, which has been shown in
Fig. 3 (a). This also indicates that the estimates of critical points are
correct.

The CCDs along $x$ and $y$ directions are also calculated for the 3D AM. We
notice that the CCDs along $x$, $y$, and $z$ directions are significantly
different from each other [Fig. 3 (b)], which indicates the system is
anisotropic because of the QP we use. We also find that the average
conductance along $x$, $y$, and $z$ directions, i.e., $\langle
g_{x}\rangle\approx 0.24$, $\langle g_{y}\rangle\approx 0.47$, $\langle
g_{z}\rangle\approx 0.26$, and the average conductance of AM with random
potential $\langle g\rangle\approx 0.3$ satisfy the relationship $\langle
g_{x}\rangle\langle g_{y}\rangle\langle g_{z}\rangle\approx\langle
g\rangle^{3}$. We note that this relation is similar to that for the quasi-1D
localization length [74].

To test the universality classes of the ATs driven by QP, we compare the CCDs
between disordered and quasiperiodic systems. Besides the symmetry and
dimension, CCD also depends on the topology [75] (i.e., the boundary
conditions) and effective geometry [76] (i.e., the ratio of the linear sizes
between transverse direction and transmission direction, or the mean value of
conductance). In order to compare the CCDs, we need to keep the symmetry,
dimension, boundary condition, and effective geometry the same.

For the disordered systems, we impose the open boundary condition in the
transverse direction and calculate the conductance in the cubic geometry.
Anderson model, U(1) model [77], and SU(2) model [72] with random potential
are chosen as the examples of Wigner-Dyson symmetry classes because of their
isotropic property. We add the random potential with a box distribution
$[-W/2,W/2]$ to these models. The critical disorder $W_{c}$ is believed to be
independent of the boundary conditions [75] so here we just use $W_{c}$
estimated in systems with periodic boundary condition for the CCD
calculations.

The effective geometry in quasiperiodic systems is different from the real
geometry because of the anisotropic conductance caused by the QP. Following
the idea in Ref. 76, we can fine-tune the geometry ratio of these
quasiperiodic models; $\rho\equiv L/L_{z}$ with $L$ the size in the transverse
direction and $L_{z}$ the transmission length, so that the mean value of
conductance for quasiperiodic and isotropic disordered systems is almost the
same. The ratios are set as $\rho=19/18$, $1$, and $23/24$, for the AM, PPM,
and Ando model, respectively. Fig. 4 shows that CCDs of quasiperiodic and
disordered systems are almost similar, which again indicates that the
universality classes of the ATs in quasiperiodic and disordered systems are
the same.

Figure 5: Level spacing ratio distribution $P(r)$. The solid lines stand for
the distributions of the random matrix in the Gaussian orthogonal ensemble
(GOE), Gaussian unitary ensemble (GUE), and Gaussian symplectic ensemble
(GSE). The circles are the distributions in the delocalized states of the
Anderson model (AM), Peierls phase model (PPM), and Ando model.

(a) Anderson model

(b) Peierls phase model

(c) Ando model

Figure 6: The probability of localization $P_{loc.}$ for wavefunctions in
quasiperiodic systems as a function of quasiperiodic potential strength $V$
predicted by a convolutional neural network trained by the
localized/delocalized wavefunctions in the disordered Anderson model. The
critical points $V_{c}$ estimated in Table 1 are indicated by the vertical
dashed lines as a guide to the eye. The probability of each data point is
averaged by 10 samples. One wavefunction is selected near $E=0$ in a cubic
system with linear size $L=42$ for each sample with random $\\{\phi_{i}\\}$.

##  VI Level spacing ratio distribution

The level spacing ratio has been introduced to characterize the energy level
repulsion in Ref. 78. Let $\\{E_{i}\\}$ be a series of energy levels indexed
in ascending order and $s_{i}=E_{i}-E_{i-1}$ is the nearest-neighbor spacing.
Then the level spacing ratio is defined as

$\displaystyle
r_{i}\equiv\frac{\min\\{s_{i},s_{i-1}\\}}{\max\\{s_{i},s_{i-1}\\}}.$ (13)

According to the Wigner-Dyson random matrix theory [46, 47, 48], the level
spacing ratio distribution $P(r)$ follows the Wigner-like surmises for all the
three random matrix ensembles [59], i.e., Gaussian orthogonal ensemble (GOE),
Gaussian unitary ensemble (GUE), and Gaussian symplectic ensemble (GSE).
$P(r)$ in the delocalized states of the 1D AAH model doesn’t follow the
distribution of GOE [57, 58]. On the other hand, $P(r)$ for the non-
interacting Hamiltonians defined on the 2D quasiperiodic Ammann-Beenker tiling
[79], 2D Moiré lattice [19], and 3D generalized AAH model [17] show the
distribution expected from GOE.

Here we study the level spacing ratio distribution in the three quasiperiodic
models. We diagonalize the Hamiltonian in a cubic system with linear size
$L=10$. We set $V=0.5$ for the three models and take half of the eigenenergy
near $E=0$ so that all of the states are delocalized (see Fig. 1). $6\times
10^{4}$ samples with random phase $\\{\phi_{i}\\}$ have been simulated for the
distribution. We observe that $P(r)$ in the delocalized states of the AM, PPM,
and Ando model well coincide with the distributions of GOE, GUE, and GSE,
respectively (Fig. 5).

##  VII Convolutional neural network study of the wavefunction

Deep neural networks have been trained to predict the
delocalization/localization probability and topological/non-topological
probability of the wavefunctions in the 2D [80] and 3D [81, 82] disordered
systems. Here we use a convolutional neural network (CNN) whose parameters
have been trained by the localized/delocalized wavefunctions in the 3D
disordered Anderson model. The details of the network structure is found in
Ref. [60]. We input the wavefunctions of quasiperiodic systems at $E=0$ to
this CNN, and let it determines the localization/delocalization probability.
The predicted critical points (Fig. 6) well coincide with those estimated by
conductance, which means that the localization behavior in 3D QP systems is
similar to the ones in random systems.

##  VIII Summary and concluding remarks

In this paper, we have used three models in the 3D Wigner-Dyson symmetry
classes to study the critical behavior of the ATs driven by QP. The two-
terminal conductance and localization length have been calculated by the
transfer matrix method. We find there is a correlation between samples, which
results in the wrong estimation of their statistical errors. By carefully
taking account of the statistical independence of samples, the critical
exponents $\nu$ and critical points $V_{c}$ have been estimated by the finite
size scaling analysis of two-terminal conductance. The estimated critical
exponents $\nu$ coincide with those of the ATs driven by random potential with
the same symmetries. Our estimations of $\nu$ for the AM and Ando model are
consistent with those reported in Ref. 17, 18. Moreover, the critical
conductance distribution and the level spacing distribution have also been
studied. We also find that a convolutional neural network trained by the
localized/delocalized wavefunctions in a disordered system can determine
whether the wavefunctions in quasiperiodic systems are localized or not. These
numerical results strongly support that the universality classes of ATs driven
by QP and random potential are the same in the 3D Wigner-Dyson symmetry
classes.

Considering the recent works about the universality classes of the Anderson
transitions in non-Hermitian disordered systems [83, 84, 85, 86], we can also
extend the present work to check the universality classes in the non-Hermitian
systems with QP, which is an interesting topic left for the future.

###### Acknowledgements.

X. L. was supported by National Natural Science Foundation of China (Grants
No.12105253 and 11934020) and the Project supported by CAEP Foundation (Grant
No. CX20210035). T. O. was supported by JSPS KAKENHI Grants No. 19H00658 and
22H05114.

(a) $\phi_{1}=0.5,\phi_{2}=0.5$

(b) $\phi_{1}=0.5,\phi_{2}=1.5$

(c) $\phi_{1}=1.5,\phi_{2}=0.5$

(d) $\phi_{1}=1.5,\phi_{2}=1.5$

(e) $\phi_{1}=0.5,\phi_{2}=0.5$

(f) $\phi_{1}=0.5,\phi_{2}=1.5$

(g) $\phi_{1}=1.5,\phi_{2}=0.5$

(h) $\phi_{1}=1.5,\phi_{2}=1.5$

Figure 7: The upper panel of (a)-(h): the conductance as a function of
$\phi_{3}$ with $\phi_{3}$ changed from zero to $2\pi$ with the interval
$\delta_{\phi}=0.04\pi$ for (a)-(d), and $\delta_{\phi}=0.01\pi$ for (e)-(h).
$\phi_{1}$ and $\phi_{2}$ are fixed. The conductance is calculated along the
$z$ direction for the Anderson model with quasiperiodic potential strength
$V=2$ in the cubic system $L_{x}=L_{y}=L_{z}=20$. The lower panel of (a)-(h):
the autocorrelation $r_{k}$ as a function of lag $k$ for the conductance in
the upper panel. The red-filled circles are the autocorrelation and the two
blue lines are the two standard error limits indicating the $95\%$ confidence
interval under the assumption that the series is completely random.

##  Appendix A Autocorrelation

To test whether a series of data is correlated or not, let us review how the
autocorrelation is defined and calculated. Suppose there is a series of data
sorted by time $t$, i.e., $\\{y_{t}\\}$ with $t=1,2,\cdots,T$. The
autocorrelation measures the correlation between $y_{t}$ and $y_{t+k}$ with
$k=0,1,2,...,K$. According to Ref. 87, the autocorrelation for lag $k$ is
defined as

$\displaystyle r_{k}\equiv\frac{c_{k}}{c_{0}},$ (14)

where

$\displaystyle c_{k}\equiv\frac{1}{T}\sum_{t=1}^{T-k}(y_{t}-\langle
y\rangle)(y_{t+k}-\langle y\rangle).$ (15)

$\langle\cdot\rangle$ stands for the mean value in statistics and $c_{0}$ is
the sample variance of the time series.

Suppose that $q$ is the lag beyond which the theoretical autocorrelation is
effectively zero. Then, according to Ref. 87, the estimated standard error of
the autocorrelation at lag $k>q$ is

$\displaystyle\sigma_{r_{k}}=\sqrt{\frac{1}{T}(1+\sum_{j=1}^{q}r_{j}^{2})}.$
(16)

If the series is completely random, then the standard error reduces to
$1/\sqrt{T}$. Two standard error limits determine the $95\%$ confidence
interval under the assumption that the series is completely random. If all of
the autocorrelation for lag $k>0$ is within the $95\%$ confidence interval,
then we can accept the assumption, otherwise, we should reject the assumption.

In this paper, the time is replaced with the sample index, and the time $T$
will be replaced by the number of samples $N$.

##  Appendix B Autocorrelation of conductance

The conductance has been averaged by $N$ samples with random phase
$\\{\phi_{i}\\}$;

$\displaystyle\langle g\rangle\equiv\frac{1}{N}\sum_{i}^{N}g_{i}.$ (17)

Then the standard error of $\langle g\rangle$ has been estimated as follows,

$\displaystyle\sigma_{\langle g\rangle}=\frac{\sigma_{g}}{\sqrt{N}},$ (18)

where $\sigma_{g}$ is the standard deviation for the single sample,

$\displaystyle\sigma_{g}=\sqrt{\frac{\sum_{i=1}^{N}(g_{i}-\langle
g\rangle)^{2}}{N-1}}.$ (19)

Eq. (18) is correct when each $g_{i}$ is independent and obeys the same
probability distribution. However, in the quasiperiodic systems of this paper,
the conductance will be almost the same when the two sets of $\\{\phi_{i}\\}$
are close to each other. Then it would produce the correlation between the
conductance when $\\{\phi_{i}\\}$’s are close to each other. We use
$\\{\delta_{\phi_{i}}\\}$ to characterize the distance between
$\\{\phi_{i}\\}$ and its nearest neighbor. $\\{\delta_{\phi_{i}}\\}$ should be
large enough to guarantee there is no correlation between samples. The allowed
minimum distance is labeled as $\delta_{\phi}$. In the actual calculations, we
simulate $N$ samples in the space of $\phi_{1}-\phi_{2}-\phi_{3}$ with
$\phi_{i}\in[0,2\pi),i=1,2,3$, then the allowed simulated samples satisfy

$\displaystyle N<(2\pi/\delta_{\phi})^{3}.$ (20)

In order to estimate $\delta_{\phi}$, we check the correlation of conductance
with $\\{\phi_{i}\\}$ in a linear line in the space of
$\phi_{1}-\phi_{2}-\phi_{3}$. For example, we fix $\phi_{1}$ and $\phi_{2}$,
and change $\phi_{3}$. Then the conductance $g$ is a continuous function of
$\phi_{3}$, which varies smoothly within $\delta_{\phi}$. We use the
autocorrelation introduced in Appendix A to test whether there is any
correlation. From Fig. 7 (a)-(d), it is safe to say there is no correlation
when $\delta_{\phi}=0.04\pi$. However, the correlation is obvious when
$\delta_{\phi}=0.01\pi$ [Fig. 7 (e)-(h)]. Then the allowed number of samples
should satisfy $N<1.25\times 10^{5}$. We, therefore, chose $N=50000$ for the
AM. We also checked the $\delta_{\phi}$ for the PPM and Ando model and found
that the number of samples set in this paper satisfies Eq. (20).

(a) $L=12,L_{z}=10^{4}$

(b) $L=12,L_{z}=10^{5}$

(c) $L=12,L_{z}=10^{6}$

(d) $L=16,L_{z}=10^{6}$

Figure 8: The autocorrelation $r_{k}$ as a function of lag $k$ for the
consecutive series of $\\{\gamma_{i}\\}$ in the quasi-1D Anderson model with
cross-section size $L$ and transmission length $L_{z}$. The red-filled circles
are the autocorrelations and the two blue lines are the two standard error
limits determines the $95\%$ confidence interval under the assumption that the
series is completely random. Insets in (c) and (d): the enlarged pictures to
show the detail except $r_{0}=1$.

##  Appendix C Autocorrelation of Lyapunov exponent

The localization length $\lambda\equiv 1/\gamma$ with $\gamma$ the Lyapunov
exponent is also a good quantity to characterize the Anderson transition.
However, we find there is also a problem with the error estimation because of
the long-range correlation of QP.

Let us first review the method to estimate the error of the Lyapunov exponent
[56]. We focus on the wavefunction decay in a quasi-1D system with
$L_{x}=L_{y}=L\ll L_{z}$. The localization length is the slowest decay of the
amplitude of wavefunction and is related to the smallest positive Lyapunov
exponent $\gamma$ by $\lambda\equiv 1/\gamma$. In practice, we do the $QR$
decomposition every $q$ layer of multiplications of the transfer matrix. We
treat $p$ ($=n\times q$ with $n$ integer) layers as a statistics independent
sample, and the $i$th sample gives the estimated smallest positive Lyapunov
exponent $\gamma_{i}$. Then the smallest positive Lyapunov exponent is
estimated as

$\displaystyle\gamma=\frac{1}{N}\sum_{i}^{N}\gamma_{i}$ (21)

with $N=L_{z}/p$. Then their sample standard deviation is estimated as,

$\displaystyle\sigma=\sqrt{\frac{\sum_{i=1}^{N}(\gamma_{i}-\gamma)^{2}}{N-1}}$
(22)

and the standard deviation of $\gamma$ is estimated as,

$\displaystyle\sigma_{\gamma}=\frac{\sigma}{\sqrt{N}}.$ (23)

Similar to the conductance, $\\{\gamma_{i}\\}$ should be independent and
without correlations in order to estimate $\sigma_{\gamma}$ correctly
according to Eq. (23). However, we find there are correlations for
$\\{\gamma_{i}\\}$ in our quasiperiodic systems. We use the autocorrelation
introduced in Appendix A to test the correlation of a series of
$\\{\gamma_{i}\\}$. We just take the AM with QP as an example and set $V=2$
and $p=50$. In Fig. 8 (a)-(c), the autocorrelation for cross-section size
$L=12$ have a tendency to go beyond the 95$\%$ confidence interval with
increasing the transmission length $L_{z}$, which suggests the correlation
would be strong for $\\{\gamma_{i}\\}$ with larger $L_{z}$. This means $L_{z}$
should be small if we want to guarantee that there is no correlation. However,
to extract the critical exponent precisely from localization length, the
transmission length $L_{z}$ should be large enough, such as $10^{7}$. Then the
correlation of the Lyapunov exponent will limit the precise estimations of the
critical exponents in quasiperiodic systems. We also note that a larger cross-
section size $L=16$ could reduce the correlation [Fig. 8(d)], but the cost of
computation will increase as $L^{7}$.

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