# Advances of Machine Learning in Molecular Modeling and Simulation

Mojtaba Haghighatlari<EMAIL_ADDRESS>Department of Chemical and
Biological Engineering, University at Buffalo, The State University of New
York, Buffalo, NY 14260, United States  Johannes Hachmann
<EMAIL_ADDRESS>Department of Chemical and Biological Engineering,
University at Buffalo, The State University of New York, Buffalo, NY 14260,
United States  Computational and Data-Enabled Science and Engineering Graduate
Program, University at Buffalo, The State University of New York, Buffalo, NY
14260, United States  New York State Center of Excellence in Materials
Informatics, Buffalo, NY 14203, United States

###### Abstract

In this review, we highlight recent developments in the application of machine
learning for molecular modeling and simulation. After giving a brief overview
of the foundations, components, and workflow of a typical supervised learning
approach for chemical problems, we showcase areas and state-of-the-art
examples of their deployment. In this context, we discuss how machine learning
relates to, supports, and augments more traditional physics-based approaches
in computational research. We conclude by outlining challenges and future
research directions that need to be addressed in order to make machine
learning a mainstream chemical engineering tool.

##  I Machine Learning from a Chemical Perspective

Over the past few years, data science has started to offer a fresh perspective
on tackling complex chemical questions, such as discovering and designing
chemical systems with tailored property profiles, revealing intricate
structure-property relationships (SPRs), and exploring the vastness of
chemical space Hachmann  _et al._ (2018a). Data-derived prediction models
serve as surrogates for physics-based models that are at the heart of
traditional modeling and simulation work. They are attractive, because they
are usually dramatically less demanding than physics-based models and can thus
be deployed in studies of correspondingly larger scope and scale. If trained
on experimental data, they are also not subject to the approximations made in
physics-based models and may thus not exhibit the resulting discrepancies with
respect to non-idealized experimental findings. Of course, data-derived models
have their own intrinsic errors and limitations, which we will address in the
course of this review.

Machine learning (ML) is a data mining technique and used to create data-
derived models. It enables us to extract complex and often hidden correlations
(and thus ideally insights, patterns, rules, and guidance) from given data
sets and to encapsulate them in mathematical form. ML is commonly categorized
into four types, i.e., supervised, semi-supervised, unsupervised, and
reinforcement learning. The main difference between these types is in essence
the amount of information (i.e., labeling, context) that is available for the
target variable that serves as the ground truth for the training of an ML
algorithm. While all ML types have found application in chemical research
Alberi  _et al._ (2018), supervised learning has so far been most commonly
used, and this review will thus focus on it. The popularity of supervised
learning may be due to its heuristic and intuitive approach to learning, which
is similar to a scientist’s way of gaining insights into SPR. A supervised
prediction model can be thought of as a function $f\colon X\rightarrow Y$ that
maps an input $x\in X$ to an output $y\in Y$, where $x$ in this context is the
feature representation of a chemical system and $y$ its target property. If
the variables $x$ and $y$ are continuous (numerical), then the mapping is a
regression; if they are discrete (categorical), then it is a classification.

We can utilize a host of supervised ML algorithms to train and optimize model
$f$ to approximate the output value for a given input. Two popular algorithms
that have been widely used are artificial neural networks (ANNs) and kernel
methods. Both can be thought of as transforming the input $x$ into a new
feature (latent variable) space, in which it becomes linearly correlated with
the output $y$ Rupp  _et al._ (2018). The transformation itself is typically
highly non-linear. A major advantage of the ANNs is their capacity to
transform features sequentially through several layers, which is referred to
as deep learning. Kernel methods, on the other hand, usually transform
features in a one-step process using kernel functions. Unlike in ANNs, this
process is predefined prior to the tuning of the model’s parameters, and is
thus less flexible to learn the best latent variable space. The advantage of
kernel methods is their superior performance in finding global solutions, even
for small-size data sets where ANNs have deficits. The support vector machines
and kernel ridge regression are two common examples of kernel-based
algorithms.

The relationship between a molecular structure and its properties is
deterministic, i.e., there exists an exact mapping from fundamental physics
(i.e., the Schrödinger equation). This mapping is ultimately the foundation
for traditional modeling and simulation techniques. The topology of ML models
is generally very flexible (as, e.g., expressed in the universal approximation
theorem for ANNs), so that they can learn and recover the underlying SPRs of a
problem, even from simple chemical representations (assuming no significant
loss of information within this representation).

We can consider a feature representation method as a function $g:M\rightarrow
X$ that maps a basic chemical representation $m\in M$ to a feature input $x\in
X$ (typically called a descriptor). The representation $m$ may contain spatial
or at least topological information that defines a molecule and is expressed,
e.g., in atomic coordinates, simplified molecular-input line-entry system
(SMILES) Weininger (1988), international chemical identifier (InChI), or other
formats.

A common feature space $X$ is spanned by structural descriptors. Some ML
approaches also utilize physical or (physico-)chemical properties as
descriptors, such that $g$ corresponds to a simulation or some other type of
calculation (including those from first principles). As this approach builds
physics into the feature space, it has a certain appeal and has gained
corresponding popularity. However, it is important to point out that the
computational cost of obtaining such descriptors (which include optimized
geometries) may easily make this the bottleneck of an ML approach, in which
case it will limit its utility as an efficient surrogate for the prediction of
$y$. This issue has to be considered as part of a cost-benefit analysis.

Another class of descriptors is designed to capture the local environment of
each atom in a molecule Bartók  _et al._ (2013). This approach considers a
molecule as a graph with atom and bond (i.e., node and edge) features. Each
atom can interact with all other atoms in its immediate vicinity, which
results in an update of the corresponding local atomic features. Incidentally,
this approach has its roots in both chemical and data sciences: In the context
of molecular simulations, cutoff radii have long been used to exploit the
short-ranged nature of intermolecular interactions. In data science, the idea
of dynamic irregular graphs provides the underpinning of graph convolutional
neural networks. The overlap of the two disciplines in this area has led to
many methodological advances for the generation of descriptors. Results from a
number of recent studies suggest that an ensemble of local features (rather
than a global representation), is able to provide a more robust solution to
the challenges involved with variant graph size and the order of atoms in
molecules Kearnes  _et al._ (2016); Hy  _et al._ (2018).

The descriptors discussed so far are essentially hand-crafted to explicitly
expose certain structural, physical, or (physico-)chemical information $x$
from $m$ and provide a structured (i.e., tabular) feature representation.
Alternatively, the feature generation $g$ can also be merged into the
prediction model $f$ and both will be jointly optimized, e.g., through hidden
layers (latent space) in deep learning. This class of descriptors is called
learned features Schütt  _et al._ (2016).

The overall ML workflow for chemical problems encompasses a number of steps as
shown in Fig. 1, including parsing, cleaning, and preprocessing a chemical
data set $\\{M,Y\\}$, compiling an array of descriptors via $g$, as well as
training, evaluation, and validation of the prediction model $f$.

Figure 1: The major tasks and mathematical setup of a supervised machine
learning workflow: For a given data set $\\{M,Y\\}$, in which for a number of
molecules in basic chemical representation $m\in M$ the target property $y\in
Y$ is given, we apply a feature representation method as a function
$g:M\rightarrow X$ that maps $M$ to a feature input space $X$. After cleaning
and other preprocessing steps, we use $\\{X,Y\\}$ to formulate an ML model
$f\colon X\rightarrow Y$ that maps the feature input space $X$ to the output
label space $Y$. The ML model is trained on the training subset of
$\\{X,Y\\}$, and subsequently validated and optimized on its testing subset,
so that it minimizes the prediction error for $Y$.

##  II Applications of Machine Learning in Chemical Research

In the following section, we summarize three application areas of ML in
chemical research, with particular consideration of the inherent structure of
the associated data sets, types of representation, and connections to
traditional modeling. We limit the scope of our discussion to molecular
systems, which still cover a broad range of use cases.

###  II.1 Discovery and Design of New Compounds

The application of ML for the exploration of chemical space and the creation
of new compounds (ranging from small molecules to polymers and materials) can
be divided into two distinct approaches: (i) discovery, i.e., ML is used to
generate fast prediction models for properties of interest, with which large-
scale surveys of chemical space can be conducted in order to identify
compounds that exhibit desired property profiles; (ii) design, i.e., ML is
used to develop a quantitative understanding of the SPRs of interest, which
can be inverted to pursue the targeted, rational design (or inverse
engineering) of compounds with particular properties. While the core activity,
i.e., the ML of SPRs, is the same in both cases, its use follows different
perspectives.

Discovery. The idea of employing data-derived prediction models instead of
physics-based models (or experimentation) as a means to characterize
candidates in the search for new molecules may be one of the earliest
applications in chemistry, for which the use of ML was proposed. Traditional
molecular modeling and simulations have been used for this purpose for many
years. More recently, they have also been employed in the context of virtual
high-throughput screening studies, in which they are tasked with assessing
entire libraries of candidate compounds (see Fig. 2 and, e.g., Ref. Hachmann
_et al._ (2011, 2014)). However, the computational footprint, in particular of
first-principles approaches, is limiting both individual as well as large-
scale studies that seek to identify compounds with specifically targeted
properties.

The application of data-derived prediction models enables us to dramatically
accelerate the survey of chemical space, often by several orders of magnitude.
A speed-up of that magnitude allows a corresponding increase in the scale and
scope that is viable for screening efforts. (It is thus sometimes referred to
as hyperscreening.) The candidate libraries are typically generated from a
collection of moieties and patterns that are of interest in a given context
Gong  _et al._ (2018). The combination of such a set of building blocks leads
to a molecular library for a particular domain in chemical space, i.e., the
candidates belong to the same distribution Pyzer-Knapp  _et al._ (2015). A
number of experimental or high-level computational training sets have been
developed for specific classes of molecules Butler  _et al._ (2018); Li  _et
al._ (2018). Since these data sets focus on relatively similar compounds from
the same distribution, the choice of representation and the ML model training
are arguably less challenging compared to more universally applicable models.
The extrapolative use of data-derived prediction models outside the domain for
which they were trained has to be conducted with great care and caution, as
they are least reliable here. This is a conceptual challenge, as screening
studies are often interested in compounds with extreme properties that are
likely at the margins of the distribution, where the predictions are least
reliable. Iterative retraining of ML models allows us to shift the training
data distribution into particular areas of interest, thus making them more
robust for use in discovery.

A reasonably diverse collection of molecules can be found in the open-source
QM9 data set originally extracted from the GDB-17 chemical universe of 166
billion organic molecules Ramakrishnan  _et al._ (2014). The QM9 data
comprises computed geometries and properties for 134,000 molecules at density
functional theory (DFT) level. Due to the diversity of molecular structures
and broad range of calculated properties, QM9 plays an important role as a
benchmark data set for new models and methods Ferré  _et al._ (2017); Collins
_et al._ (2018). Its contribution to method developments can be compared to
the MNIST data set in the hand-written character recognition community LeCun
_et al._ (2013). In contrast to, e.g., data sets from first-principles
modeling, those from data-derived models have so far rarely been used for the
generation of new reference data. Yet, they have played an important role in a
number of methodological advances in the field. As a result, the reported
accuracies for many of the recent ML prediction models surpass those of
traditional molecular modeling and simulations Faber  _et al._ (2017).

Figure 2: Flowchart showing a computational funnel typical for high-throughput
virtual screening (HTVS) studies. The neural network schemes on the left and
right represent deep generative model architectures that can conceptually
replace different elements of the screening funnel. Both generative
adversarial networks (GANs) and variational autoencoders (VAEs) include two
networks that revolutionize the conventional generation and analysis steps by
probabilistic means. A deep reinforcement learning (RL) network can also be
trained to bias the generation towards promising candidates.

Design. While discovery is still based on a traditional trial-and-error
process – albeit one drastically accelerated by ML – the notion of a
deliberate, de novo design of new compounds represents a different research
paradigm. It addresses the problem that even rapid and efficient
hyperscreening studies can only scratch the surface of the practically
infinite molecular space. Instead, the design paradigm seeks to utilize
insights into the SPRs obtained from ML for the targeted creation of systems
with specific properties. The understanding of how changes in the molecular
structure (or a compound’s features) lead to changes in the desired properties
can be inverted to gain a property to structure mapping. The mathematical
structure of a data-derived SPR prediction model (e.g., the dominant features,
principal components, latent variables, or learned features) yields a
foundation for inverse design. Models that are less easy to interpret can be
projected onto surrogate models for which the extraction of guidelines is
easier. A key challenge is to realize the simultaneous enhancement of
different properties. The emerging design rules can be used to formulate
individual compounds Sanchez-Lengeling and Aspuru-Guzik (2018), but also to
identify high-value domains in chemical space, which can be enumerated in
screening libraries (e.g., by sampling compounds similar to a lead compound).
The latter approach is effectively interfacing the discovery and design
perspectives and allows both physics-based and data-derived modeling studies
to be more targeted.

Another approach that is very promising and has caused much excitement is the
application of generative models (see Fig. 2). For instance, Sanchez-Lengeling
et al.  have shown that a generative adversarial network (GAN) in tandem with
reinforcement learning can outperform evolutionary algorithms in order to bias
the generative process toward the extreme regions of a property distribution
Sanchez-Lengeling  _et al._ (2017); Putin  _et al._ (2018). The use of GANs
for molecular design and library generation is very recent and a number of
concerns and challenges still need to be overcome in their development. Two of
the principal challenges of GANs (and other generative approaches) are the
rate at which invalid (i.e., chemically irrelevant or non-sensical) structures
are generated, and their ability to produce topologically different molecules
compared to the underlying training data Kadurin  _et al._ (2016); Popova  _et
al._ (2018). Another example of generative models are variational autoencoders
(VAEs) that learn the distribution of embedded space and thus enables tuning
in that space Gómez-Bombarelli  _et al._ (2018). Recurrent neural networks
(RNNs) operate in a sequential manner similar to creating new molecules one
atom at a time. One benefit of RNNs is their memory mechanism that allows them
to remember the effects of previous sequences Segler  _et al._ (2018).

###  II.2 Creation of New Modeling Techniques

Instead of replacing physics-based with data-derived modeling entirely as
outlined in Sec. II.1, ML can also be used to (i) calibrate and correct the
results of physics-based models to account for some of their systematic
errors; (ii) complement traditional modeling and simulation approaches (i.e.,
employ combinations of physics-based and data-derived models); and (iii)
facilitate the development of new physics-based modeling techniques.

The calibration approach allows us to improve the predictive performance of
physics-based models and obtain high-quality results at the cost of lower-
quality methods. It can also help bridge the gap between experiment and theory
that results from the inherent approximations in the latter (see, e.g., Ref.
Hachmann  _et al._ (2014)). Transfer learning is an ML design methodology that
has been a particularly successful technique in this context Goh  _et al._
(2017); Sultan and Pande (2018). In the combination approach, we only utilize
ML for aspects for which no good physics-based models are available or where
their use is impractical (e.g., because of insufficient accuracy, prohibitive
cost, or other numerical issues). We thus retain as much of the physical
foundations and robustness of traditional modeling as possible, while being
pragmatic about the parts of a problem, where that is not possible (see, e.g.,
Refs. Afzal  _et al._ (2018); Ribeiro  _et al._ (2018); Wei  _et al._ (2018)).

The development of entirely new modeling techniques by means of ML has seen
encouraging pioneering efforts, in particular for force fields (FFs) and DFT.
The major driving force behind ML-generated FFs is the lack of
generalizability in the classical FFs and the interatomic potentials that
underpin them. This is an area where ML is apparently able to bridge the
accuracy and versatility typically seen from quantum chemistry and the
efficiency of molecular mechanics simulations. A recent line of research has
focused on learning interatomic potentials from quantum chemical data sets
Khorshidi and Peterson (2016). There are two specific challenges involved in
this application that make it distinct from prediction models for molecular
properties. One is the need for a diverse sampling of non-equilibrium chemical
conformations, as both ML and classical FFs perform poorly outside of their
applicability domain. Access to a diverse collection of high-quality training
samples is thus essential in creating ML FFs. For instance, Botu et al.  have
improved on previous work by diversifying their training data, e.g., by adding
more atomic environments and applying clustering methods Botu  _et al._
(2017). Smith et al.  have pushed the normal mode sampling method to obtain
single point energies for more than 20 million conformations generated for
58,000 small molecules Smith  _et al._ (2017). The results of these efforts
were shown to be efficiently generalizable, even for the simulation of more
complex phenomena. The second important challenge is to conserve the
consistency between potential energies and forces as discussed by Chmiela et
al.  Chmiela  _et al._ (2017). They provide a robust solution to this
challenge by developing gradient-domain ML models (which reproduce global FFs
by training in the force domain and incorporating both energies and forces) in
an automated fashion, thus learning accurate ML FFs.

In the DFT context, ML is used to create new functionals for different terms
in the electronic Hamiltonian. The exact form of several functionals (e.g.,
the kinetic energy functional for interacting electrons or the exchange-
correlation functional in the Kohn–Sham formalism) are unknown and otherwise
approximated by physical reasoning. The ML-generated functionals allow DFT to
avoid common failures, such as in accurately describing bond-breaking
processes. Different ML functionals for specific classes of molecules, target
properties, and electronic structure situations are being developed, as are
fast methods that, e.g., learn energy functionals directly without having to
solve the Kohn–Sham equations, thus making them a viable approach for ab
initio molecular dynamics simulations Brockherde  _et al._ (2017).

###  II.3 Predictions of Chemical Reactions and Catalyst Systems

Research on chemical reactions is another field that has been benefiting from
the advances in ML methodology. ML has been paving the way for a better
understanding of chemical transformations with numerous real-world
implications. SMILES are often the representation of choice for both the
inputs (reactants and reagents) and outputs (products) of data-derived models
for chemical reactions. These models are trained on known reactions to
recognize structural patterns that may undergo bond-breaking or -formation in
the course of a reaction or catalytic process Coley  _et al._ (2018). One
particularly important data set for this application domain is the result of
the US patent reaction extraction by Lowe Lowe (2014).

The progress in predicting organic reactions and their products has been
particularly noteworthy in recent years. Nam et al.  have introduced sequence-
to-sequence models to address the reaction prediction task similar to
linguistic translation problems Nam and Kim (2016). More recently, Schwaller
et al.  outperformed a similar approach in an end-to-end template-free model
with a focus on the attention mechanism and a new tokenization strategy
Schwaller  _et al._ (2018). Coley et al.  introduced a graph convolutional
neural network approach with competitive performance. It was used for the
prediction of reaction products as well as reactive sites of the reagents that
are most likely to initiate a reaction Coley  _et al._ (2019). One major
contribution of the last two studies is the development of web applications to
facilitate easy access to their models. These tools are available via the
IBMRXN and ASKCOS websites, respectively ibm (2018); ask (2018).

A promising direction of ongoing work is the prediction of reaction pathways
and mechanisms. All these efforts ultimately aim for a practical and more
generalizable implementation of retrosynthetic analysis, which has been a
grand challenge in organic chemistry for many years Klucznik  _et al._ (2018).
Insights regarding the synthetic feasibility of virtual compounds are also a
key concern for the screening library generation and molecular design efforts
discussed in Sec. II.1.

##  III Outlook on Future Directions

###  III.1 Feature Representations

As discussed in Sec. I, the descriptors of a given molecular system are an
abstraction of its detailed nature (as well as a numerical representation).
The choice of a suitable feature space is still our first and most effective
means to infuse physics into ML models. There have been efforts to define
criteria for the development of efficient descriptors Bartók  _et al._ (2013),
e.g., that they are (1) invariant to the symmetries of the underlying physics;
(2) easy to interpret; (3) expressed in a direct and concise form to avoid
redundancy and the curse of dimensionality; and (4) computationally efficient.
However, developing molecular representations that adhere to all these
criteria has been an exceedingly difficult task. More importantly, there is
now agreement that ML approaches may require different types of descriptors to
recover the entirety of SPRs of molecular systems. Further research into the
creation of new descriptors (including fingerprint schemes) as well as the
formulation of additional criteria will be necessary for the foreseeable
future. The accessibility and flexibility of deep learning models can
accelerate future developments via learned features and theory-informed
models.

###  III.2 Machine Learning for Small Data

While ML ideas became popular during the recent ’big data’ wave (i.e., in
chemistry with the emergence of large-scale screening result from high-level
first-principles modeling), large data sets are in practice more often than
not unavailable. In fact, problems for which data is (still) sparse tend to be
of particular interest. As the data generation (both from experiment and
modeling) is often a limiting factor, we will have to strive to reduce its
cost or the number of data points needed to obtain ML models of a desired
accuracy. It is thus essential to put an emphasis on developing ML methods
that achieve better performance on small data sets. As mentioned in Sec. II.2,
transfer learning is a promising approach in this context. We will also need
to employ smart sampling methods and identify data points that are most
important for the training of ML models. Active learning strategies offer a
path towards this goal Häse  _et al._ (2018); Tran and Ulissi (2018); Gubaev
_et al._ (2019). Many of these techniques are of general-purpose utility, but
some will have to be tailored towards the specific problem settings of data-
derived models for chemistry.

###  III.3 Software and Tool Development

The idea to utilize ML and other data mining techniques in the chemical domain
is so recent that much of the basic infrastructure has not yet been developed,
or is still in its early stages Hachmann  _et al._ (2018a). The majority of
tools and expertise tend to be technically involved, labor intensive, or
otherwise unavailable to the community at large. Many researchers are now
starting to pursue open-source software development projects to tackle this
situation Hachmann  _et al._ (2018b). However, the lack of rigorous
development guidelines remains a challenge that researchers from domain
science need to overcome to make their efforts lasting and sustainable. The
Molecular Sciences Software Institute (MolSSI) is one of the pioneers in
establishing best practices and guidance for early-stage software developments
in this field Krylov  _et al._ (2018); Wilkins-Diehr and Crawford (2018).

##  IV Conclusions

In this review, we discussed how ML can advance traditional modeling and
simulation by (partially) replacing them (i.e., choosing data-derived over
physics-based models or combining the two); calibrating, augmenting, or
otherwise correcting their results; targeting studies and their objectives;
and providing the means to effectively mine their results for a deeper
understanding of hidden SPRs. Many ML models are still built on data provided
by modeling and simulation – often as part of virtual high-throughput
screening studies – and combining ML and traditional modeling infuses physics
and robustness into the resulting data-derived prediction models. These and
other emerging ML techniques have been enabling accelerated discovery and
rational design in numerous areas of chemistry. Its early successes indicate
that ML is bound to become a mainstream tool in chemical research. Yet, there
is still much to (machine) learn on how to develop the full potential of ML in
chemistry.

## Competing Financial Interests

The authors declare to have no competing financial interests.

###### Acknowledgements.

MH gratefully acknowledges support by Phase-I and Phase-II Software
Fellowships (grant No. ACI-1547580-479590) of the National Science Foundation
(NSF) Molecular Sciences Software Institute (grant No. ACI-1547580) at
Virginia Tech. JH acknowledges supported by the NSF CAREER program under grant
No. OAC-1751161, the NSF Big Data Spokes program under grant No. IIS-1761990,
and funding by the New York State Center of Excellence in Materials
Informatics (grant No. CMI-1148092-8-75163).

## Annotations

  * •

Hachmann  _et al._ (2018a) This NSF workshop report compiles the opinions of a
group of active researchers in the field regarding the current challenges and
future opportunities offered by data-driven approaches in the chemical domain.

  * •

Sanchez-Lengeling and Aspuru-Guzik (2018) This review discusses the recent
advances in inverse molecular design using deep generative models.

  * •

Smith  _et al._ (2017) In this study, deep learning is used to fit interatomic
potentials and develop the so-called ANI model for transferable data-derived
potentials with comparable accuracy to the reference DFT calculations.

  * •

Coley  _et al._ (2018) This study surveys the role of ML in synthesis planning
and the prediction of reaction outcomes.

  * •

Hachmann  _et al._ (2018b) This paper presents a software ecosystem for the
development and broader dissemination of techniques at the different stages of
a molecular data mining workflow.

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# Critical behavior of Anderson transitions in higher dimensional Bogoliubov-
de Gennes symmetry classes

Tong Wang<EMAIL_ADDRESS>International Center for Quantum Materials,
School of Physics, Peking University, Beijing 100871, China  Collaborative
Innovation Center of Quantum Matter, Beijing 100871, China  Zhiming Pan
<EMAIL_ADDRESS>Institute for Theoretical Sciences, Westlake
University, Hangzhou 310024, Zhejiang, China  Keith Slevin  Department of
Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan  Tomi Ohtsuki
Physics Division, Sophia University, Chiyoda-ku, Tokyo 102-8554, Japan

###### Abstract

Disorder is ubiquitous in solid-state systems, and its crucial influence on
transport properties was revealed by the discovery of Anderson localization.
Generally speaking, all bulk states will be exponentially localized in the
strong disorder limit, but whether an Anderson transition takes place depends
on the dimension and symmetries of the system. The scaling theory and symmetry
classes are at the heart of the study of the Anderson transition, and the
critical exponent $\nu$ characterizing the power-law divergence of
localization length is of particular interest. In contrast with the well-
established lower critical dimension $d_{l}=2$ of the Anderson transition, the
upper critical dimension $d_{u}$, above which the disordered system can be
described by mean-field theory, remains uncertain, and precise numerical
evaluations of the critical exponent in higher dimensions are needed. In this
study, we apply Borel-Padé resummation method to the known perturbative
results of the non-linear sigma model (NL$\sigma$M) to estimate the critical
exponents of the Boguliubov-de Gennes (BdG) classes. We also report numerical
simulations of class DIII in 3D, and classes C and CI in 4D, and compare the
results of the resummation method with these and previously published work.
Our results may be experimentally tested in realizations of quantum kicked
rotor models in atomic-optic systems, where the critical behavior of dynamical
localization in higher dimensions can be measured.

##  I Introduction

Since the discovery of Anderson localization [1], the effects of disorder in
various media have been a constant focus of the physics community. The
disorder-driven Anderson transition (AT) is a second-order quantum phase
transition, around which physical observables show universal power-law
behaviors. The universality class of the AT depends on the dimensionality and
fundamental symmetries of the system: time-reversal symmetry, particle-hole
symmetry, and chiral symmetry [2, 3, 4]. Based on these symmetries, Altland
and Zirnbauer (AZ) completed the symmetry classification of non-interacting
disordered Hamiltonians known as the “10-fold way” [5]. The classification is
comprised of the three Wigner-Dyson classes (A, AI, and AII), the three chiral
classes (AIII, BDI, and CII), and the four Bogoliubov–de Gennes (BdG) classes
(D, C, DIII, and CI). The AZ classification is revelatory not only to the
study of localization phenomena, but to the study of topological materials [6,
7, 8].

The critical exponent $\nu$ of the AT characterizes the power-law divergence
of the correlation length $\xi$ on approaching the critical point,

$\displaystyle\xi\sim|x-x_{c}|^{-\nu},$ (1)

where $x$ is the tuning parameter and $x_{c}$ is the critical point.
Constrained by computational capacity, relatively few numerical studies have
gone beyond three-dimensions (3D) [9, 10, 11, 12, 13] into higher dimensions
where a stronger strength of disorder is required to drive the system into
localization. A strong-disorder renormalization group (RG) approach is in
development to provide theoretical insights [14, 15]. Recently, the potentials
of such efforts are revealed by the proposed superuniversality of ATs in
Hermitian and non-Hermitian systems [16], and the mapping between certain
disorder-free interacting systems and disordered non-interacting systems with
extra dimension [17, 18]. Moreover, the theory and numerical simulations are
applicable to experimental realizations of quantum kicked rotors with
synthetic dimensions [19, 20, 21].

While the lower critical dimension $d_{l}=2$ of the AT is well established by
the one-parameter scaling theory [3], the upper critical dimension $d_{u}$,
above which a mean-field description is accurate, remains debatable. The self-
consistent theory of AT by Vollhardt and Wölfle [22, 23] gives the critical
exponent of Anderson model (class AI) as

$\displaystyle\nu=\begin{cases}\displaystyle\frac{1}{d-2},&2<d<4,\\\
\displaystyle\ \ \frac{1}{2}\ \ ,&d\geq 4.\end{cases}$ (2)

The results that $d_{u}=4$ and the mean-field critical exponent $\nu=1/2$ are
reminiscent of the $\phi^{4}$ theory. A modified version of this theory that
considers the renormalization of the diffusion coefficient [9] gives

$\displaystyle\nu=\frac{1}{2}+\frac{1}{d-2},$ (3)

and $d_{u}=\infty$. The prediction of the limiting value

$\lim_{d\rightarrow\infty}\nu=\frac{1}{2},$ (4)

by both theories agree with the value from the Anderson model on an infinite-
dimensional Bethe lattice [24, 25, 26, 27, 28, 29, 30]. However, Eq. (3) is in
better agreement with numerical results[31, 32, 33, 11] of the orthogonal
symmetry class for $d=3,4,5,6$ than Eq.(2).

On the other hand, the nonlinear sigma model (NL$\sigma$M), an effective field
theory of Anderson localization, has been studied extensively in
$d=2+\epsilon$ dimensions [34, 35, 36, 37]. The $\beta$-function, which
describes the renormalization of the conductance with system size, can be
calculated analytically using perturbation techniques [38, 39, 40, 41]. From
the $\beta$-function one can derive relevant physical quantities including a
series in powers of $\epsilon$ for the critical exponent $\nu$. This method,
which is referred to as the $\epsilon$-expansion, is rigorous only when
$\epsilon\ll 1$. In this limit, the $\epsilon$-expansion gives
$\nu=1/\epsilon$ in agreement with Eq. (2) but not Eq. (3) and with numerical
simulations on fractals with spectral dimensions close to $2$ [31, 42]. To
obtain results for higher dimensions resummation methods are needed. However,
a straightforward resummation[37] of the power series for the critical
exponent yields $\nu\to 0$ in the limit $d\to\infty$ in disagreement with both
Eq. (2) and Eq. (3). For the Wigner-Dyson classes ressumations that
incorporate the correct asymptotic behaviour of the critical exponent for
$d\to\infty$ have been performed [11, 43] giving better agreement with
numerical simulations [33, 11, 12, 32] and experimental results [44, 20].
However, a comprehensive understanding of the dimensional-dependence of the AT
in different symmetry classes is still lacking.

In this paper, we focus on the BdG symmetry classes in 3D and 4D. The four BdG
classes appear naturally in the topological superconducting system [5, 4]. The
underlying BdG Hamiltonian $H$ is invariant under the antiunitary transform of
particle-hole symmetry (PHS) $\mathcal{C}=U_{C}K$,

$\displaystyle\mathcal{C}:\quad H\rightarrow-U_{C}^{\dagger}H^{T}U_{C},$ (5)

where $U_{C}$ is a unitary matrix and $K$ denotes the operation of complex
conjugation [8]. The BdG universality classes are realized at the particle-
hole symmetric point, $E=0$. The particle-hole symmetry can be classified into
two kinds, even ($\mathcal{C}^{2}=+1$) or odd ($\mathcal{C}^{2}=-1$). The
symmetry classes can further be characterized by time-reversal symmetry (TRS)
$\mathcal{T}$. There are four BdG classes: singlet/triplet SC (class D),
singlet SC (class C), singlet/triplet SC with TRS (class DIII) and singlet SC
with TRS (class CI). Class D and class C describes BdG systems with even or
odd PHS and broken TRS. Classes DIII and CI are characterized by a time-
reversal operator $\mathcal{T}:H\rightarrow U_{T}H^{T}U_{T}^{-1}$, where the
unitary matrix $U_{T}$ satisfies $U_{T}^{2}=\pm 1$. For classes DIII, one has
PHS $\mathcal{C}^{2}=+1$ and TRS $\mathcal{T}^{2}=-1$. For class CI, one has
PHS $\mathcal{C}^{2}=-1$ and TRS $\mathcal{T}^{2}=+1$. The symmetries of the
BdG classes are summarized in Table 1. Due to the absence of spin-rotation
invariance, class D and class DIII exhibit weak antilocalization.

Below we apply the resummation method previously employed [11, 43] for the
Wigner-Dyson symmetry classes to the BdG classes. We also report simulations
using the transfer matrix method for class DIII in 3D, and classes C and CI in
four dimensions (4D). We compare estimates of the critical exponent $\nu$
obtained by finite-size scaling analysis of the numerical simulations with the
results of the resummation method. Our results show the ability of this Borel-
Padé analysis to give quantitative predictions of critical exponents $\nu$ for
the BdG classes beyond 2D.

The rest of the paper is organized as follows. In Sec. II, we review briefly
the Borel-Padé resummation. In Sec. III, we apply the Borel-Padé method to the
$\epsilon$-series of the critical exponent $\nu$ for the BdG classes. In Sec.
IV, we apply the Borel-Padé method to the $\epsilon$-series of the
$\beta$-functions. In Sec. V we report our numerical simulations. In Sec. VI
we compare the Borel-Padé predictions with numerical results (both those
reported here and previously published work). A summary is given in Table 5.
In Sec. VII we discuss and conclude our findings.

Table 1: List of the BdG symmetry classes and their transformation behavior under time-reversal, particle-hole, chiral (sublattice) (SLS) symmetries, and the presence ($\triangle$) or absence ($\times$) of SU(2) spin-rotation symmetry. The penultimate column shows corresponding non-compact fermionic replica non-linear sigma-model (NL$\sigma$M) manifolds. The last column shows the $\beta$-function[38, 45, 4] of the four BdG symmetry classes. Here $\zeta$ is the Riemann zeta function. Class | TRS | PHS | SLS | SU(2) | NL$\sigma$M Manifold |  $\beta(t)$-function
---|---|---|---|---|---|---
D | $0$ | $+1$ | $0$ | $\times$ | $\mathrm{Sp}(2N)/\mathrm{U}(N)$ | $\epsilon t+t^{2}-2t^{3}+\displaystyle\frac{7}{2}t^{4}-\frac{47}{6}t^{5}+\mathcal{O}(t^{6})$
C | $0$ | $-1$ | $0$ | $\triangle$ | $\mathrm{O}(2N)/\mathrm{U}(N)$ | $\epsilon t-2t^{2}-8t^{3}-28t^{4}-\displaystyle\frac{376}{3}t^{5}+\mathcal{O}(t^{6})$
DIII | $-1$ | $+1$ | $1$ | $\times$ | $\mathrm{Sp}(2N)$ | $\epsilon t+t^{2}-\displaystyle\frac{1}{2}t^{3}+\frac{3}{8}t^{4}-\frac{1}{8}\Big{(}\frac{19}{6}+6\zeta(3)\Big{)}t^{5}+\mathcal{O}(t^{6})$
CI | $+1$ | $-1$ | $1$ | $\triangle$ | $\mathrm{O}(N)$ | $\epsilon t-2t^{2}-2t^{3}-3t^{4}-2\Big{(}\displaystyle\frac{19}{6}+6\zeta(3)\Big{)}t^{5}+\mathcal{O}(t^{6})$

##  II Borel-Padé Resummations

In the scaling theory of Anderson transition [3], the $\beta$-function is
defined as,

$\displaystyle\beta(g)=\frac{d\ln g}{d\ln L},$ (6)

where $g$ is the dimensionless conductance measured in units of $e^{2}/h$ and
summed over the spins, and $L$ is the length of a $d$-dimensional cubic
system. For the NL$\sigma$M description it is more convenient to work with the
inverse conductance $t=1/(\pi g)$ and

$\displaystyle\beta(t)=-\frac{dt}{d\ln L}=\frac{\beta(g)}{\pi g}.$ (7)

The critical point $t_{c}>0$ of the AT is a zero-crossing point of $\beta(t)$

$\displaystyle\beta(t_{c})=0,$ (8)

and the critical conductance is given by $g_{c}=1/(\pi t_{c})$. The critical
exponent $\nu$ is related to the derivative of the $\beta$-function at the
critical point,

$\displaystyle\frac{d\beta(t)}{dt}\Big{|}_{t=t_{c}}=-\frac{d\beta(g)}{d\ln
g}\Big{|}_{g=g_{c}}=-\frac{1}{\nu}.$ (9)

The $\beta$-functions of the BdG classes up to the 4-loop order [38, 45, 4]
are listed in Table 1. Note that the coefficient of $t^{5}$ for class C in
Table 1 differs from that given in Table III of Ref. [4]. 111We use the value
$-376/3$ for the coefficient of $t^{5}$ for class C, whereas in Table III of
Ref. [4] it is $-376/48$. We believe the latter is a typo and that the
coefficient $c_{3}(-2N)$ should be replaced by $16c_{3}(-2N)$ so that the
$\beta$-functions of classes D and C satisfy the duality relation $\beta_{\rm
Sp}(t)=-2\beta_{\rm O}(-t/2)$ of the underlying NL$\sigma$M manifolds. The
$\beta$-function of class D, which corresponds to the NL$\sigma$M manifold
$\mathrm{Sp}(2N)/\mathrm{U}(N)$, is given in Eq. (3.7) of Ref. [45]. We thank
Alexander D. Mirlin for private communication.. We also note in passing that
the $\beta$-functions of the chiral symmetry classes were found to be strictly
zero in all orders in perturbation theory [47, 48].

The Borel-Padé resummation method is a technique for dealing with truncated
and possibly divergent series. Given an infinite series $f$

$f(x)=\sum_{k}f_{k}x^{k},$ (10)

its Borel sum is defined as

$\tilde{f}(x)=\sum_{k}\frac{f_{k}}{k!}x^{k}.$ (11)

The original series in Eq. (10) can be recovered by calculating the Borel
transform

$f(x)=\frac{1}{x}\int_{0}^{\infty}e^{-y/x}\tilde{f}(y)\mathrm{d}y.$ (12)

Suppose the coefficients $f_{k}$ are known for order $k\leq l$. We approximate
$\tilde{f}$ on the r.h.s by a rational function

$\tilde{f}(x)\approx r(x)=\frac{p(x)}{q(x)},$ (13)

where $p(x)$, $q(x)$ are polynomials of order $m$ and $n$, respectively,

$\displaystyle p(x)=\sum_{k=0}^{m}p_{k}x^{k},\qquad
q(x)=\sum_{k=0}^{n}q_{k}x^{k},\ q_{0}\equiv 1.$ (14)

For choices of $[m,n]$ that satisfy $m+n=l$, the coefficients of the
polynomials $p$ and $q$ are uniquely determined. In some cases we require
$m<n$ so that the Padé approximant satisfies

$\lim_{x\rightarrow\infty}r(x)=0.$ (15)

Then, the rational function $r$ can be decomposed into a sum of partial
fractions

$r(x)=\sum_{j=1}^{n}\frac{a_{j}}{x-\lambda_{j}},$ (16)

where ${\lambda_{j}}$ are the roots of the polynomial $q(x)$. In general, the
$\lambda_{j}$ and ${a_{j}}$ are complex numbers. Substituting the above
equation into Eq. (12) and performing the integration, we obtain the Borel-
Padé approximation $F$ of the series for $f$

$F(x)=\frac{1}{x}\sum_{j=1}^{n}a_{j}B\left(\frac{\lambda_{j}}{x}\right).$ (17)

Here, the function $B$ is defined by

$\displaystyle
B(s)=\begin{cases}-\exp(-s)\mathrm{E_{i}}(s)\quad&s\in\mathbb{R},s\neq 0,\\\
\exp(-s)\mathrm{E_{1}}(-s)\quad&s\in\mathbb{C},\arg s\neq\pi,\end{cases}$ (18)

where

$\displaystyle\mathrm{E_{i}}(x)=$
$\displaystyle-\int_{-x}^{\infty}\frac{e^{-t}}{t}dt=\int_{-\infty}^{+x}\frac{e^{t}}{t}dt,$
$\displaystyle\mathrm{E_{1}}(z)=$
$\displaystyle\int_{z}^{\infty}\frac{e^{-t}}{t}dt,\quad|\arg z|<\pi.$ (19)

##  III Resummation of the series for $\nu(\epsilon)$

Series in powers of $\epsilon$ for the critical exponent $\nu$ can be derived
starting from the series for the $\beta$-function in powers of $t$ as follows.
We take symmetry class C as an example. We first find an approximation for
$t_{c}(\epsilon)$ by solving Eq. (8) using the available terms in the power
series for $\beta(t)$. For class C we find

$t_{c}(\epsilon)=\frac{1}{2}\epsilon-\epsilon^{2}+\frac{9}{4}\epsilon^{3}-\frac{77}{12}\epsilon^{4}+\mathcal{O}(\epsilon^{5}).$
(20)

Here we have chosen the root for which

$\lim_{\epsilon\rightarrow 0}t_{c}=0.$ (21)

If we then substitute the series for $t_{c}$ into Eq. (9), we obtain the
following series in powers of $\epsilon$ for the inverse of $\nu$

$\frac{1}{\nu}\left(\epsilon\right)=\epsilon+2\epsilon^{2}-\epsilon^{3}+\frac{15}{2}\epsilon^{4}+\mathcal{O}(\epsilon^{5}).$
(22)

Taking the reciprocal of this series we obtain

$\nu(\epsilon)=\frac{1}{\epsilon}-2+5\epsilon-\frac{39}{2}\epsilon^{2}+\mathcal{O}(\epsilon^{3}).$
(23)

Similarly, for symmetry class CI we find

$\displaystyle t_{c}(\epsilon)$
$\displaystyle=\frac{1}{2}\epsilon-\frac{1}{4}\epsilon^{2}+\frac{1}{16}\epsilon^{3}-\frac{1+9\zeta(3)}{24}\epsilon^{4}+\mathcal{O}(\epsilon^{5})$
$\displaystyle\frac{1}{\nu}\left(\epsilon\right)$
$\displaystyle=\epsilon+\frac{1}{2}\epsilon^{2}+\frac{1}{4}\epsilon^{3}+\frac{5+36\zeta(3)}{16}\epsilon^{4}+\mathcal{O}(\epsilon^{5})$
$\displaystyle\nu(\epsilon)$
$\displaystyle=\displaystyle\frac{1}{\epsilon}-\frac{1}{2}-\frac{3+36\zeta(3)}{16}\epsilon^{2}+\mathcal{O}(\epsilon^{3}).$
(24)

This approach works for symmetry classes C and CI because the coefficient of
the $t^{2}$ term in $\beta(t)$ is negative and the lower critical dimensions
for these classes is $d_{l}=2$. However, for symmetry classes D and DIII the
coefficient of the $t^{2}$ term in $\beta(t)$ is positive, so that when we
follow the procedure explained above we find

$\lim_{\epsilon\rightarrow 0}t_{c}\neq 0,$ (25)

and we are unable to obtain a useful series in powers of $\epsilon$ for $\nu$.
This reflects the possibility that the lower critical dimensions for these two
classes is below 2D ($d_{l}<2$), as thought to be the case for the symplectic
class AII.

Now we apply the Borel-Padé resummation introduced in the previous section. A
naive resummation tacitly assumes the limiting behavior

$\lim_{d\rightarrow\infty}\nu=0,$ (26)

which disagrees with self-consistent theories of the AT and the results for
the AT on the Bethe lattice, i.e., with Eq. (4). Instead, we rewrite

$\nu\left(\epsilon\right)=\frac{1}{2}+\frac{1}{\epsilon}f\left(\epsilon\right),$
(27)

and perform the resummation of $f(\epsilon)$ with the requirement $m\leq n$.
Such a treatment guarantees the limiting behavior given in Eq. (4). Of course,
the application of this restraint to the BdG symmetry classes needs to be
justified. For later reference, in Table 2, we compare the results given by
imposing Eq. (4) and Eq. (26) for the classes C and CI in 3D and 4D.

Table 2: Comparison of the critical exponents $\nu$ for classes C and CI in 3D and 4D obtained from Borel-Padé resummations of the series for $\nu(\epsilon)$ when imposing different limiting conditions, i.e., Eq. (26) compared with Eq. (4). Numbers in the square brackets indicate the orders of polynomials, $m$ and $n$ [Eq. (14)]. (a) 3D | $\displaystyle\lim_{d\rightarrow\infty}\nu=0$ | $\displaystyle\lim_{d\rightarrow\infty}\nu=\frac{1}{2}$
---|---|---
class | $[0,3]$ | $[1,2]$ | $[0,3]$ | $[1,2]$
C | 0.357 | 0.227 | 0.773 | 0.360
CI | 0.555 | 0.776 | 0.924 | 1.226
(b) 4D | $\displaystyle\lim_{d\rightarrow\infty}\nu=0$ | $\displaystyle\lim_{d\rightarrow\infty}\nu=\frac{1}{2}$
class | $[0,3]$ | $[1,2]$ | $[0,3]$ | $[1,2]$
C | 0.111 | -0.050 | 0.580 | 0.527
CI | 0.185 | 0.329 | 0.633 | 0.876

##  IV Resummation of the series for $\beta(t)$

An alternative to the approach above is to apply the Borel-Padé method
directly to the series for the $\beta$-function.[11]. All the series take the
form

$\beta(t)=\epsilon t-tf(t),$ (28)

where $f$ is a power series in $t$. In terms of $f(t)$ the critical exponent
is

$\frac{1}{\nu}=t\frac{df(t)}{dt}\Big{|}_{t=t_{c}}.$ (29)

We need to impose the limiting behaviour at infinite dimension given in
Eq.(4). We first note that in high dimensions the Anderson transition takes
place at strong disorder and, moreover, that

$\lim_{d\rightarrow\infty}t_{c}=\infty$ (30)

This means that we can obtain the correct limiting behaviour by arranging that

$\lim_{t\rightarrow\infty}t\frac{df}{dt}=A,$ (31)

with $A=2$. To do so, we define $h$, a polynomial in $t$, by

$h(t)=t\frac{df(t)}{dt}-A.$ (32)

Applying the Borel-Padé method to $h$, we obtain an approximation $H$ for $h$
that satisfies

$\lim_{t\rightarrow\infty}H(t)=0,$ (33)

so that Eq. (31) is satisfied. To obtain the corresponding approximation $F$
for $f$, a further integration is needed,

$\displaystyle f(t)\approx F(t)=\int_{0}^{t}\frac{A+H(t)}{t}dt.$ (34)

The result can be expressed in the form[11]

$\displaystyle F(t)=\sum_{j=1}^{n}c_{j}B(\lambda_{j}/t),\quad
c_{j}=\frac{a_{j}}{\lambda_{j}}.$ (35)

Finally, the $\beta$-function is approximated as

$\displaystyle\beta(t)\approx\epsilon t-tF(t).$ (36)

We show the resulting Borel-Padé approximations for $\beta(g)$ in 3D for
classes C, CI in Fig. 1 and Fig. 2, respectively, together with the series
without resummation. We omit the $[m,n]=[1,3]$ resummation for class C because
the resulting $\beta$-function is not monotonic and has two unphysical fixed
points. The limiting behavior $\beta(g)\sim 2\ln g$ at $g\ll 1$ guaranteed by
the constraint $A=2$ in Eq. (31) is observed only at $\ln g$ much smaller than
the range plotted in Fig. 1 .

We show the resulting Borel-Padé approximations for $\beta(g)$ in 2D for
classes D and DIII in Fig. 3 and Fig. 4, respectively, together with the
series without resummation. In classes D and DIII, for $d<2$, two fixed points
appear: a critical fixed point, and a stable fixed point. At the lower
critical dimension $d_{l}$, these two fixed points annihilate, e.g., the
dashed curve in Fig. 3 and Fig. 4, and the value of the $\beta$-function at
its maximum is zero

$\displaystyle\max_{d=d_{l}}\beta(g)=0.$ (37)

This leads directly to an estimate for $d_{l}$,

$\displaystyle d_{l}\approx 2-\max\beta(g,\epsilon=0).$ (38)

Estimates of the lower critical dimension obtained from the Borel-Padé
resummations are summarized in Table 3.

Figure 1: Comparison of the approximations for the $\beta(g)$-function before and after Borel-Padé resummation of the series for class C in 3D. Numbers in the square brackets indicate the orders of polynomials, $m$ and $n$ [Eq. (14)]. Figure 2: Comparison of the approximations for the $\beta(g)$-function before and after Borel-Padé resummation of the series for class CI in 3D. Figure 3: Comparison of the approximations for the $\beta(g)$-function before and after Borel-Padé resummation of the series for class D in 2D. The $[1,3]$ Bore-Padé resummation of the $\beta(g)$-function at the corresponding estimate $d_{l}=1.76$ of the lower critical dimension is plotted with a dashed line. Figure 4: Comparison of the approximations for the $\beta(g)$-function before and after Borel-Padé resummation of the series for class DIII in 2D. The $[1,3]$ Bore-Padé resummation of the $\beta(g)$-function at the corresponding estimate $d_{l}=1.21$ of the lower critical dimension is plotted with a dashed line. class | no resummation | Borel-Padé
---|---|---
|  | $[0,4]$ | $[1,3]$
D | $1.85$ | $1.88$ | $1.76$
DIII | $1.66$ | $1.70$ | $1.21$
Table 3: Lower critical dimension of the BdG symmetry classes D and DIII
obtained from the $\beta$-function without resummation and with order $[m,n]$
Borel-Padé resummation.

##  V Numerical Simulations

To evaluate the effectiveness of the Borel-Padé resummation in estimating the
critical exponents of the BdG symmetry classes, especially in high spatial
dimensions $d\geq 3$, we perform simulations for 3D class DIII, 4D class C,
and 4D class CI. We set the energy $E$ to the particle-hole symmetric point,
$E=0$, and vary the disorder strength $W$.

###  V.1 3D class DIII

This symmetry class describes time-reversal symmetric superconductors with
broken spin-rotational symmetry. We study a four-band tight-binding model on
cubic lattice [49, 50],

$\displaystyle\mathcal{H}_{\rm DIII}$
$\displaystyle=\sum_{\mathbf{r},\mathbf{r^{\prime}}}c_{\mathbf{r}}^{\dagger}[H_{\rm
DIII}]_{\mathbf{r}\mathbf{r}^{\prime}}c_{\mathbf{r}^{\prime}}$
$\displaystyle=\sum_{\mathbf{r}}\sum_{\mu=1}^{3}\left[\frac{it}{2}c_{\mathbf{r}+\mathbf{e}_{\mu}}^{\dagger}\alpha_{\mu}c_{\mathbf{r}}-\frac{m_{2}}{2}c_{\mathbf{r}+\mathbf{e}_{\mu}}^{\dagger}\beta
c_{\mathbf{r}}+\text{ H.c. }\right]$
$\displaystyle+\sum_{\mathbf{r}}\left(m_{0}+3m_{2}+v_{\mathbf{r}}\right)c_{\mathbf{r}}^{\dagger}\beta
c_{\mathbf{r}}$ (39)

where $c^{\dagger}_{\mathbf{r}}$ ($c_{\mathbf{r}})$ is the 4-component
creation (annihilation) operator on a cubic-lattice site $\mathbf{r}$. For
convenience we set the lattice constant $a$ to be unity. The
$\mathbf{e}_{\mu=1,2,3}$ are the primitive lattice vectors along the $x,y,z$
directions, respectively. The matrices $\alpha_{\mu}$ and $\beta$ are defined
as

$\displaystyle\alpha_{\mu}=\left(\begin{array}[]{cc}0&\sigma_{\mu}\\\
\sigma_{\mu}&0\end{array}\right),\quad\beta=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right),$ (44)

where $\sigma_{\mu}$ and $\tau_{\mu}$ are Pauli matrices acting on different
degrees of freedom (e.g., spin and orbital). Parameter $m_{0}$ is a mass, and
parameters $m_{2}$ and $t$ are hopping amplitudes. This Hamiltonian has time-
reversal symmetry $U_{T}^{\dagger}\;H_{\rm DIII}^{*}\;U_{T}=H_{\rm DIII}$
where

$\displaystyle
U_{T}=\delta_{\mathbf{rr}^{\prime}}(\sigma_{2}\otimes\tau_{0}),\quad
U_{T}^{T}=-U_{T},$ (45)

and a particle-hole symmetry $U_{S}^{\dagger}\;H_{\rm DIII}\;U_{S}=-H_{\rm
DIII}$ where

$\displaystyle U_{S}=\delta_{\mathbf{rr}^{\prime}}(\tau_{0}\otimes\tau_{2}).$
(46)

This model depicts a 3D $\mathbb{Z}$ topological insulator (TI) when $m_{0}<0$
and a trivial insulator when $m_{0}>0$.

For numerical calculations, we specify the parameters $t=2$, $m_{2}=1$,
$m_{0}=-2.5$, and use independent uniform distributions for the random on-site
potential

$\displaystyle v_{\mathbf{r}}\in[-W/2,W/2],\quad\langle
v_{\mathbf{r}}v_{\mathbf{r}^{\prime}}\rangle=\delta_{\mathbf{rr}^{\prime}}W^{2}/12.$
(47)

Here, $\langle\cdots\rangle$ indicates a disorder average. We use the transfer
matrix method to calculate the localization length of the model [33] and
impose periodic boundary conditions in the transverse direction. We simulate a
semi-infinite bar with a cross section of size $L\times L$ and estimate the
quasi-one-dimensional (Q1D) localization length $\lambda$ at disorder strength
$W$ and linear size $L$. A dimensionless ratio $\Lambda$ is defined as

$\displaystyle\Lambda(W,L)=\lambda(W,L)/L.$ (48)

The results are shown in Fig. 5 where $\Lambda$ is plotted versus $W$ for
various $L$. Curves for different $L$ have an approximate common crossing
point. This point indicates the Anderson transition between the TI (localized)
phase and the metallic (extended) phase.

To estimate the critical exponent, we fit the data to the following scaling
form that includes corrections to single parameter scaling due to an
irrelevant scaling variable [51, 33]

$\displaystyle\Lambda=F\big{(}\phi_{1},\phi_{2}\big{)}=F\big{(}u_{1}(w)L^{1/\nu},u_{2}(w)L^{-y}\big{)},$
(49)

where

$\displaystyle\omega=(W-W_{c})/W_{c},$ (50)

and $\phi_{1}=u_{1}L^{1/\nu}$ is the relevant scaling variable that encodes
the power-law divergence of correlation length $\xi\sim|u_{1}(w)|^{-\nu}$
around the critical point. The second scaling variable $\phi_{2}=u_{2}L^{-y}$
with exponent $-y<0$ is the leading irrelevant correction, and vanishes in the
limit $L\to\infty$. We approximate the scaling function $F$ using a truncated
Taylor series near the critical point ($|w|\ll 1$),

$\displaystyle F\left(\phi_{1},\phi_{2}\right)$
$\displaystyle=\sum_{j=0}^{n_{2}}F_{j}(\phi_{1})\phi_{2}^{j}=\sum_{i=0}^{n_{1}}\sum_{j=0}^{n_{2}}f_{ij}\phi_{1}^{i}\phi_{2}^{j},$
(51)

and

$\displaystyle u_{1}=\sum_{k=1}^{m_{1}}b_{k}w^{k},\quad
u_{2}=\sum_{k=0}^{m_{2}}c_{k}w^{k}.$ (52)

We set $b_{1}=c_{0}=1$ to remove the arbitrariness of the expansion
coefficients. The numerical data are fitted to the scaling function by
minimizing the $\chi$-squared statistic

$\displaystyle\chi^{2}=\sum_{n=1}^{N_{\rm
D}}\frac{(\Lambda_{n}-F_{n})^{2}}{\sigma_{n}^{2}}.$ (53)

Here, $N_{\rm D}$ is the number of data points, $\Lambda_{n}$ is the value of
$\Lambda$ for $n$th data point, $\sigma_{n}$ its standard error, and $F_{n}$
the value of the scaling function for the $n$th data point. To assess whether
or not the fit is acceptable, we use the goodness of fit probability. Here,
this is well approximated by [33]

$\displaystyle\mathrm{GoF}\approx 1-\frac{1}{\Gamma(N_{\rm
F}/2)}\int_{0}^{\chi_{\rm min}^{2}/2}\mathrm{d}t\,e^{-t}\,t^{\chi_{\rm
min}^{2}/2-1},$ (54)

where $N_{\mathrm{F}}=N_{\mathrm{D}}-N_{\mathrm{P}}$ is the degrees of freedom
(with $N_{\mathrm{P}}$ the number of fitting parameters),
$\chi_{\mathrm{min}}^{2}$ is the minimum value of the $\chi$-squared
statistic, and $\Gamma$ is the Gamma function. The fitting results are shown
in Table 4 (a). Our estimate of the critical exponent for 3D Class DIII is

$\displaystyle\nu=0.96\pm 0.01$ (55)

Figure 5: the dimensionless ratio $\Lambda$ near the Anderson transition for
the 3D class DIII model. The expansion order is $(n_{1},n_{2})=(3,1)$
$(m_{1},m_{2})=(2,0)$ as defined in Eqs. (51, 52). The solid lines are the
fitting functions, and the black dots with error bars are simulation data
points. Inset: the scaling function  Table 4: FSS results for class DIII in
3D, and classes C and CI in 4D. The orders of the expansion of the scaling
function are fixed at $n_{1}=3$ and $n_{2}=1$. Here, $m_{1}$ and $m_{2}$ and
the orders, respectively, of the expansions of $u_{1}$ and $u_{2}$ (see Eq.
(51,52). The values enclosed in square brackets are 95% confidence intervals
determined from 1000 Monte Carlo samples.

Figure 6: Dimensionless Landauer conductance as a function of disorder $W$
around the Anderson transition. The expansion order is
$(n_{1},n_{2},m_{1},m_{2})=(3,1,2,0)$. Left panel: 4D symmetry class C. Right
panel: 4D symmetry class CI. The colored solid lines are fitting functions and
black dots with error bars are the numerical data.

###  V.2 4D class C

Symmetry class C describes disordered superconductors with spin-rotational
symmetry but broken time-reversal symmetry. For this symmetry class the spin
quantum Hall effect occurs in two-dimensions [52]. We extend the 3D tight-
binding model for class C of Ref. [53] to 4D,

$\displaystyle\mathcal{H}_{\rm C}$
$\displaystyle=\sum_{\mathbf{r},\mathbf{r^{\prime}}}c_{\mathbf{r}}^{\dagger}[H_{\rm
C}]_{\mathbf{r}\mathbf{r}^{\prime}}c_{\mathbf{r}^{\prime}}$
$\displaystyle=\sum_{\mathbf{r}}\left[\sum_{\mu=1}^{3}tc_{\mathbf{r}+\mathbf{e}_{\mu}}^{\dagger}c_{\mathbf{r}}+t_{\parallel}c_{\mathbf{r}+\mathbf{e}_{4}}^{\dagger}c_{\mathbf{r}}\right.$
$\displaystyle+\left.\mathrm{i}t_{\perp}\bigg{(}c_{\mathbf{r}+\mathbf{e}_{1}}^{\dagger}\sigma_{1}c_{\mathbf{r}}+\sum_{\mu=2,3}c_{\mathbf{r}+\mathbf{e}_{\mu}}^{\dagger}\sigma_{2}c_{\mathbf{r}}\bigg{)}+\mathrm{H.c.}\right]$
$\displaystyle+\sum_{\mathbf{r}}(v_{\mathbf{r}}+\Delta)c_{\mathbf{r}}^{\dagger}\sigma_{3}c_{\mathbf{r}}.$
(56)

Here, $c_{\mathbf{r}}^{\dagger}$ is the creation operator on lattice site
$\mathbf{r}=(x_{1},x_{2},x_{3},x_{4})$ where the two components act on spin,
orbital or Nambu space depending on the nature of the system. The Hamiltonian
has a particle-hole symmetry $U_{P}^{\dagger}H_{\rm C}^{*}U_{P}=-H_{\rm C}$
with

$\displaystyle
U_{P}=\delta_{\mathbf{r}\mathbf{r}^{\prime}}e^{\mathrm{i}\pi\sum_{\mu=1}^{4}\mathbf{r}\cdot\mathbf{e}_{\mu}}\sigma_{2},\quad
U_{P}^{T}=-U_{P}.$ (57)

In the clean limit the Fourier transformation of the Hamiltonian is

$\displaystyle h_{\rm C}(\bm{k})$ $\displaystyle=2t_{\parallel}\cos
k_{4}+2t\sum_{\mu=1}^{3}\cos k_{\mu}+\Delta\sigma_{3}$
$\displaystyle-2t_{\perp}\left[\sin k_{1}\sigma_{1}+\left(\sin k_{1}+\sin
k_{2}\right)\sigma_{2}\right].$ (58)

For numerical simulations, we set $\Delta=0.5$, $t_{\perp}=t=1$ and
$t_{\parallel}=0.8$ so that the clean system has a finite Fermi surface at
$E_{F}=0$. We calculate the two-terminal Landauer conductance $G$ using the
transfer matrix method [54],

$\displaystyle G=\frac{e^{2}}{h}g,\quad
g=\mathrm{Tr}\left[\tilde{t}^{\dagger}\tilde{t}\right],$ (59)

where $\tilde{t}$ is the transmission matrix of the hypercubic samples of size
$L^{4}$ along $w$ axis. We impose periodic boundary conditions in directions
the transverse to the current. While the dimensionless conductance $g$
exhibits fluctuations, various disorder average are well described by a
scaling function like Eq. (49) [55, 56]. We calculate $\ln\langle g\rangle$,
and use the same nonlinear fitting procedures as described through Eq.
(49-54). Each data point $\langle g\rangle$ is averaged over 5000–20000
samples to ensure a relative error smaller than 1%. The results for the
critical exponent $\nu$ and other quantities are shown in Table 4 (b). The
fitting results are stable against change of expansion order $m_{1},m_{2}$ and
the range of system size. Our estimate of the critical exponent for 4D class C
is

$\displaystyle\nu=0.72\pm 0.02.$ (60)

Note that the critical disorder $W_{c}$ and critical conductance $g_{c}$ are
model-dependent, i.e.not universal.

###  V.3 4D class CI

Symmetry class CI describes disordered superconductors with both time-reversal
symmetry and spin-rotational symmetry. Again, we extended the 3D class CI
model of Ref. [53] to 4D

$\displaystyle H_{\rm CI}$
$\displaystyle=\sum_{\mathbf{r},\mathbf{r^{\prime}}}c_{\mathbf{r}}^{\dagger}[H_{\rm
CI}]_{\mathbf{r}\mathbf{r}^{\prime}}c_{\mathbf{r}^{\prime}}$
$\displaystyle=\sum_{\mathbf{r}}\bigg{[}\sum_{\mu=1}^{3}t_{\perp}c_{\mathbf{r}+\mathbf{e}_{\mu}}^{\dagger}c_{\mathbf{r}}+t_{\parallel}c_{\mathbf{r}+\mathbf{e}_{4}}^{\dagger}\sigma_{3}c_{\mathbf{r}}$
$\displaystyle+t^{\prime}_{\parallel}c_{\mathbf{r}+\mathbf{e}_{4}}^{\dagger}\sigma_{1}c_{\mathbf{r}}+\mathrm{H.c.}\bigg{]}+\sum_{\mathbf{r}}(v_{\mathbf{r}}+\Delta)c_{\mathbf{r}}^{\dagger}\sigma_{1}c_{\mathbf{r}}.$
(61)

The Hamiltonian is time-reversal symmetric since $H_{\rm CI}^{*}=H_{\rm CI}$,
and has particle-hole symmetry $U_{P}^{\dagger}H_{\rm CI}^{*}U_{P}=-H_{\rm
CI}$ given by

$\displaystyle
U_{P}=\delta_{\mathbf{r}\mathbf{r}^{\prime}}e^{\mathrm{i}\pi\sum_{\mu=1}^{3}\mathbf{r}\cdot\mathbf{e}_{\mu}}\sigma_{2},\quad
U_{P}^{T}=-U_{P}.$ (62)

In the clean limit the Fourier transformation of the Hamiltonian is

$\displaystyle h_{\rm CI}(\bm{k})$
$\displaystyle=2t_{\perp}\sum_{\mu=1}^{3}\cos
k_{\mu}+2t_{\parallel}^{\prime}\cos k_{4}\sigma_{3}$
$\displaystyle+(\Delta+2t_{\parallel}\cos k_{4})\sigma_{1}.$ (63)

In numerical simulations of the two-terminal Landauer conductance, we chose
$\Delta=1.2$, $t_{\perp}=1$ and $t_{\parallel}=t^{\prime}_{\parallel}=0.5$.
Following the same procedures as described in the previous section, we
estimate the critical exponent $\nu$ and other quantities. The results are
shown in Table 4 (b). Our estimate of the critical exponent $\nu$ for 4D class
CI is

$\displaystyle\nu=0.83\pm 0.04$ (64)

##  VI Comparison of Borel-Padé predictions with numerical results

Referring to Table 5, we see that for classes C and CI in both 3D and 4D, the
estimates of the critical exponent obtained with the $[0,4]$ Borel-Padé
resummations are in good agreement with the numerical estimates. For 3D class
D the discrepancy is relatively large and even larger for 3D class DIII. These
are also the two symmetry classes where $d_{l}<2$ (see Table 3). In addition
we notice an inconsistency between our estimation of the critical exponent for
3D class DIII $\nu=0.96\pm 0.01$ and that in Ref. [57] $\nu=0.85\pm 0.05$. The
model used in Ref. [57] is essentially the same as here, but the data set of
Ref. [57] is of smaller size and of lower numerical precision. However, we
note the possibility that the weak topological indices may change the critical
behavior of Anderson transition [58].

(a) 3D
---
| Borel-Padé with $A=2$ | numerical
class | $[0,4]$ | $[1,3]$ | $\nu$ | Ref.
C | 1.056 | - | $0.996\pm 0.012$ | [53, 59]
CI | 1.107 | 1.822 | $1.17\pm 0.02$ | [53]
D | 0.823 | 0.858 | $0.87\pm 0.03$ | [53]
DIII | 0.751 | 0.674 | $0.85\pm 0.05$ | [57]
$0.96\pm 0.01$ | *
(b) 4D
| Borel-Padé with $A=2$ | numerical
class | $[0,4]$ | $[1,3]$ | $\nu$ | Ref.
C | 0.714 | - | $0.70\pm 0.02$ | *
CI | 0.729 | 1.103 | $0.83\pm 0.04$ | *
D | 0.640 | 0.666 | - | -
DIII | 0.616 | 0.589 | - | -
Table 5: Critical exponents $\nu$ of the BdG symmetry classes in 3D and 4D
obtained from order $[m,n]$ Borel-Padé resummation of the $\beta$-function
with $A=2$, and numerical simulations. Here, $*$ indicates the numerical
estimates in this paper, whereas $-$ indicates that the value is yet to be
determined. We omit the $[1,3]$ resummation with $A=2$ for class C because the
resummed $\beta$-function is not monotonic and has two unphysical fixed
points. This is also the case for $A=1$.

We have resummed the series for the $\beta$-function in such a way that Eq.
(31) is satisfied. This resummation means that in the localised regime the
$\beta$-function will behave like $A\ln g$ up to a constant. It would then
seem more natural to set $A=1$ rather than $A=2$. However, the former choice
does not yield the correct limiting behavior Eq. (4). For reference, we also
tabulate the estimates of the critical exponents calculated from the truncated
$\beta$-function series without resummation and from the Borel-Padé analysis
with $A=1$ in Table 6. Without resummation, we obtain estimates that violate
the Chayes inequality $\nu\geq 2/d$ [60]. With $A=1$, the estimates satisfy
the Chayes inequality but are in poorer agreement with the numerical estimates
compared with $A=2$.

(a) 3D
---
| no resummation | Borel-Padé with $A=1$
class |  | $[0,4]$ | $[1,3]$
C | 0.471 | 1.446 | -
CI | 0.555 | 1.478 | 2.131
D | 0.187 | 1.254 | 1.249
DIII | 0.151 | 1.202 | 1.088
(b) 4D
| no resummation | Borel-Padé with $A=1$
class |  | $[0,4]$ | $[1,3]$
C | 0.200 | 1.122 | -
CI | 0.217 | 1.129 | 1.602
D | 0.103 | 1.075 | 1.079
DIII | 0.091 | 1.062 | 1.026
Table 6: Critical exponents $\nu$ of the BdG symmetry classes in 3D and 4D
obtained from $\beta$-function series without resummation and order $[m,n]$
Borel-Padé resummation with $A=1$.

##  VII Summary and Discussion

In this paper, we have studied the Anderson transition in the BdG symmetry
classes both analytically and numerically. We applied the Borel-Padé
resummation method to the known perturbative results for the NL$\sigma$M to
estimate the critical exponents in 3D and 4D. We also reported numerical
simulations of class DIII in 3D, and classes C and CI in 4D, and compared the
results of the resummation method with the results of the resummations and
previously published work. We find that the results of the Borel–Padé analysis
provide estimates of the critical exponent with the numerical estimates
provided the limiting behaviour Eq. (4) is imposed during the resummation. In
principal, the NL$\sigma$M theory of Anderson localization and its
renormalization analysis in $d=2+\epsilon$ dimensions are valid only when
$\epsilon$ is small, i.e., the Anderson transition occurs under weak disorder.
Nonetheless, our results show that the perturbative $\beta$-functions can
provide useful information concerning critical properties in 3D and 4D.

The estimations of the critical exponents in BdG symmetry classes based on the
Borel-Padé resummation methods with the assumption of infinite upper critical
dimension match the numerical results better. This suggest that the upper
critical critical dimension $d_{u}$ may be infinite for the Anderson
localization in BdG symmetry classes. Previous theoretical works have argued
that in noncompact NL$\sigma$M, the upper critical dimension is infinite[61,
62], which seems to be consistent with the numerical results and estimation of
Borel-Padé resummation method in this work. Further theoretical efforts are
needed to conform these observations.

Recently, it has been pointed out that the NL$\sigma$M model characterizes the
measurement-induced phase transition in quantum circuits[63]. This scenario
involves a replica number $N$ equal to 1. The resummation method discussed in
this paper is also applicable to that case, allowing for the prediction of
critical exponents in quantum circuit systems.

Acknowledgments We thank and Ryuichi Shindou, Ferdinand Evers and Alexander D.
Mirlin for fruitful discussions. T.W. was supported by the National Basic
Research Programs of China (Grant No. 2019YFA0308401) and the National Natural
Science Foundation of China (Grants No. 11674011 and No. 12074008). Z.P. was
supported by National Natural Science Foundation of China (No. 12147104). T.O.
and K.S. were supported by JSPS KAKENHI Grants 19H00658, and T.O. was
supported by JSPS KAKENHI 22H05114.

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