# Disordered Hyperuniform Solid State Materials

Duyu Chen<EMAIL_ADDRESS>Materials Research Laboratory, University of
California, Santa Barbara, California 93106, United States  Houlong Zhuang
<EMAIL_ADDRESS>Mechanical and Aerospace Engineering, Arizona State
University, Tempe, AZ 85287  Mohan Chen  HEDPS, Center for Applied Physics and
Technology, College of Engineering and School of Physics, Peking University,
Beijing 100871, People’s Republic of China  Pinshane Y. Huang  Department of
Materials Science and Engineering, University of Illinois Urbana-Champaign,
Urbana, IL 61801, United States  Vojtech Vlcek  Department of Chemistry and
Biochemistry, University of California, Santa Barbara, CA 93106, United States
Yang Jiao<EMAIL_ADDRESS>Materials Science and Engineering, Arizona
State University, Tempe, AZ 85287  Department of Physics, Arizona State
University, Tempe, AZ 85287

###### Abstract

Disordered hyperuniform (DHU) states are recently discovered exotic states of
condensed matter. DHU systems are similar to liquids or glasses in that they
are statistically isotropic and lack conventional long-range translational and
orientational order. On the other hand, they completely suppress normalized
infinite-wavelength density fluctuations like crystals, and in this sense
possess a hidden long-range order. Very recently, there are several exciting
discoveries of disordered hyperuniformity in solid-state materials, including
amorphous carbon nanotubes, amorphous 2D silica, amorphous graphene, defected
transition metal dichalcogenides, defected pentagonal 2D materials, and
medium/high-entropy alloys. It has been found the DHU states of these
materials often possess a significantly lower energy than other disorder
models, and can lead to unique electronic and thermal transport properties,
which resulted from mechanisms distinct from those identified for their
crystalline counterparts. For example, DHU states can enhance electronic
transport in 2D amorphous silica; DHU medium/high-entropy alloys realize the
Vegard’s law, and possess enhanced electronic band gaps and thermal transport
at low temperatures. These unique properties open up many promising potential
device applications in optoelectronics and thermoelectrics. Here, we provide a
focused review on these important new developments of hyperuniformity in
solid-state materials, taking an applied and “materials” perspective, which
complements the existing reviews on hyperuniformity in physical systems and
photonic materials. Future directions and outlook are also provided, with a
focus on the design and discovery of DHU quantum materials for quantum
information science and engineering.

Disorder Hyperuniformity, Nanotubes, 2D Materials, High/Medium-Entropy Alloys,
Electronic Structures, Thermal Properties

##  I Introduction

Solid-state materials, in either bulk or thin-film forms, inevitably contain
disorder of various kinds Kittel et al. (1996). At finite temperatures, the
otherwise perfect crystal lattice is constantly distorted by the thermal
motions of the atoms. Besides this normal “thermal disorder”, which can be
well understood in the framework of statistical thermodynamics, there are a
wide spectrum of “frozen-in disorder”, including point defects, dislocations,
grain boundaries and surfaces, to name but a few. The physical properties of
the materials, such as electronic band structures, density of states,
electrical and thermal transport properties, heat capacity, magnetic and
optical properties, etc., strongly depend on the concentration, spatial
distribution and interactions among the defects, which have been the subjects
of intensive studies in condensed matter and solid-state physics Baranovskii
and Rubel (2017); Lee and Ramakrishnan (1985); Dugdale (1995); Kramer and
MacKinnon (1993); Sheng and van Tiggelen (2007). Traditionally, it was
believed that defects may have negative impacts on material properties, while
recently it has been shown that novel desirable material performance can be
achieved by proper defect engineering Lin et al. (2016); Zhang et al. (2019a);
Shi et al. (2021). For example, it has been demonstrated that quantum emitters
can be realized by introducing defects into two-dimensional (2D) hexagonal
boron nitride ($h$-BN) Tran et al. (2016).

Another type of “frozen-in disorder” is typically found in amorphous or
“glassy” states of the materials, where the disorder is so significant that
the material cannot be considered as perturbed or defected states of otherwise
perfect crystals Binder and Kob (2011a); Angell et al. (2000). Examples
include correlated disorder, which resulted mostly from fast cooling, in
metallic glasses Wang et al. (2004) and amorphous silicon Hejna et al. (2013);
Xie et al. (2013); compositional disorder in alloys due to entropy-driven
mixing Porter and Easterling (2009); and spin disorder in randomly diluted
magnetic materials (i.e., spin glasses) Mézard et al. (1987, 1984). Compared
to their crystalline counterparts, the amorphous materials lack conventional
long-range translational and orientational order, e.g., there are no Bragg
peaks in their scattering intensity. On the other hand, they can possess
significant short-range order manifested by clearly structured pair-
distributions Xie et al. (2013); Chen et al. (2021a), as a result of local
chemical bonding and electronic structures. Importantly, the amorphous
materials generally preserve the “isotropy” property of the high-symmetry
liquid phase, which is highly desirable for many applications (e.g., the
window glass) Binder and Kob (2011a).

The nature of the aforementioned “frozen-in disorder” generally depends on the
specific material systems of interest. An important yet very challenging step
to develop a comprehensive theory for amorphous materials is the systematic
and rigorous mathematical description of such disorder and the understanding
of its effects on different material properties. To this end, a variety of
stochastic morphological models have been developed including random point
processes, random sets and random field models Jeulin (2021). Statistical
descriptors such as the hierarchy of $n$-point correlation functions Torquato
(2002), which in their most general settings quantify the probability of
occurrence of a unique combination of local states at specific locations in
the material of interest, have been applied in a diverse class of disordered
condensed matter systems. Moreover, percolation models Sahimi (1994); Torquato
and Jiao (2013) have been widely employed to understand the structural and
transport properties of amorphous materials.

$\begin{array}[]{c}\\\
\includegraphics[width=411.93767pt]{Fig1.png}\end{array}$

Figure 1: Illustration of point configurations with different degrees of
disorder. From left to right: periodic configuration (crystal), quasi-periodic
configuration (quasi-crystal), correlated disordered configuration (disordered
material), totally random configuration (ideal gas). The lower panels show the
static structure factor $S({\bf k})$ for the corresponding point
configurations (see definition in Sec. II).

Recently, a new framework for quantifying and classifying disorder has been
proposed based on the concept of “hyperuniformity” Torquato (2018). The notion
of hyperuniformity was first coined by Torquato and Stillinger Torquato and
Stillinger (2003) in the development of universal order metrics to distinguish
point configurations with different degrees of order, including periodic
patterns, quasi-periodic patterns, correlated disordered patterns, and totally
random patterns (see illustrations in Fig. 1). These configurations
respectively correspond to distributions of atoms in crystals, quasicrystals,
amorphous materials and ideal gases. Yet, the notion of hyperuniformity can be
readily generalized to, e.g., distribution of chemical species in alloys
(i.e., chemical but not spatial disorder). There are also no fundamental
limitations of hyperuniformity to classical particles (and classical
properties) and it may offer a new apparatus to capturing correlations of
quantum states (i.e., probability amplitudes, densities, density matrices,
etc). The generalization of hyperuniformity to the aforementioned systems will
be discussed in detail in Sec. II.B.

The definition of hyperuniformity involves the measurement of density
fluctuations in the system across different length scales (see Fig. 2).
Specifically, one can consider placing spherical observation windows
$\Omega(R)$ of radius $R$ at random locations in the system. As the window
moves from one (randomly selected) location to another, the number of points
(atoms) falling into the window $N(R)$ also fluctuates. The variance
$\sigma^{2}_{N}(R)$ associated with $N(R)$ is also a function of window radius
$R$. For crystals and quasicrystals, as the windows moves in the system, one
can imagine that the “bulk” portion enclosed the window would not change,
especially for large windows, due to the long-range translational and/or
orientational order in the systems. Thus, the contributions to the number
fluctuations only come from the “surface layer” of the window, leading to
$\sigma^{2}_{N}(R)\sim R^{d-1}$ (where $d$ is the dimensionality of the system
and $R^{d-1}$ describes the scaling of the surface area of a spherical
window). On the other hand, it is well known that for ideal gases, which is a
totally un-correlated system, $\sigma^{2}_{N}(R)\sim R^{d}$.

A hyperuniform system is one whose density fluctuations are strongly
suppressed on large length scales and possesses vanishing normalized number
variance for infinite windows, i.e.,
$\lim_{R\rightarrow\infty}\sigma^{2}_{N}(r)/R^{d}=0$. Thus, all crystals and
quasicrystals are hyperuniform by definition. Interestingly, certain exotic
correlated disordered systems, which can be in both equilibrium and non-
equilibrium settings, and come in both quantum-mechanical and classical
varieties, are also found to be hyperuniform. These disordered hyperuniform
(DHU) systems are similar to liquids or glasses in that they are statistically
isotropic and lack conventional long-range translational order. Yet they
possess a hidden long-range order, manifested as complete suppression of
large-scale normalized density fluctuations like in crystals. Examples of DHU
systems include the density fluctuations in early universe Gabrielli et al.
(2002), disordered jammed packing of hard particles Donev et al. (2005a);
Zachary et al. (2011a); Jiao and Torquato (2011a); Chen et al. (2014);
Atkinson et al. (2012); Zachary et al. (2011b, c), certain exotic classical
ground states of many-particle systems Zachary and Torquato (2011); Torquato
et al. (2015a); Uche et al. (2004); Batten et al. (2008, 2009); Lebowitz
(1983); Zhang et al. (2015a, b), jammed colloidal systems Berthier et al.
(2011); Kurita and Weeks (2011); Hunter and Weeks (2012); Dreyfus et al.
(2015), driven non-equilibrium systems Hexner and Levine (2015a); Jack et al.
(2015); Weijs et al. (2015); Tjhung and Berthier (2015); Salvalaglio et al.
(2020), certain quantum ground states Torquato et al. (2008a); Feynman and
Cohen (1956a), avian photoreceptor patterns Jiao et al. (2014), organization
of adapted immune systems Mayer et al. (2015), amorphous silicon Hejna et al.
(2013); Xie et al. (2013), a wide class of disordered cellular materials Klatt
et al. (2019), dynamic random organizating systems Hexner and Levine (2017);
Hexner et al. (2017); Weijs and Bartolo (2017); Lei et al. (2019a); Lei and Ni
(2019), electron density distributions Gerasimenko et al. (2019); Sakai et al.
(2022), vortex distribution in superconductors Rumi et al. (2019); Sánchez et
al. (2019), and even the distribution of primes on the number axis Torquato et
al. (2019). Many of these systems were thoroughly discussed in the seminal
review article by Torquato (see Ref. Torquato (2018)).

The unique structural characteristics of DHU systems, i.e., the combination of
both high-symmetry liquid-like structures on small-scales and crystal-like
hidden order on large scales, endow them with many exotic and desirable
physical properties, including optical properties Florescu et al. (2009); Man
et al. (2013a, b), transport properties Zhang et al. (2016a), mechanical
properties Xu et al. (2017) as well as optimal multi-functionalities Torquato
and Chen (2018). One of the most striking discoveries is that certain
carefully designed DHU materials are able to realize large, complete and
isotropic photonic band gaps, blocking all directions and polarizations
Florescu et al. (2009); Man et al. (2013a, b), which was considered not
possible in traditional wisdom and has significant ramifications for
electronic and phononic band gaps in disordered materials Tang et al. (2022).
This discovery sets a new paradigm for engineered disorder in photonic
metamaterials Wu et al. (2017); Zhang and Hao (2019) and optical applications
Zhang et al. (2019b, 2021, 2022). The designer DHU materials have also been
successfully fabricated or synthesized using different techniques Haberko et
al. (2013); Zito et al. (2015). In a recent review article Yu et al. (2021),
the engineered disordered in photonic metamaterials within the framework of
hyperuniformity have been thoroughly discussed.

In light of the existing comprehensive reviews Torquato (2018) and Yu et al.
(2021), one may question the necessity for another review on hyperuniformity.
There are several important motivations for this focused review: Very
recently, there are many new discoveries of disordered hyperuniformity in
solid state materials, including amorphous carbon nanotubes Chen et al.
(2022a), amorphous 2D silica Zheng et al. (2020), amorphous graphene Chen et
al. (2021a), defected transition metal dichalcogenides Chen et al. (2021b),
defected pentagonal 2D materials Zheng et al. (2021), and medium/high-entropy
alloys Chen et al. (2021c). It has been found the DHU states of these
materials often possess a significantly lower energy than other disorder
models, suggesting their superior stability. Moreover, DHU can lead to unique
electronic and thermal transport properties, which resulted from mechanisms
distinct from those that have been identified for their crystalline
counterparts. For example, DHU states are found to enhance electronic
transport in 2D amorphous silica Zheng et al. (2020). On the other hand, DHU
medium/high-entropy alloys realize Vegard’s law Vegard (1921), and possess
enhanced electronic band gaps and superior thermal transport properties at low
temperatures Chen et al. (2021c). These unique properties open up many
promising potential device applications in optoelectronics and
thermoelectrics. Our current review nicely complements the existing ones by
focusing on these important new developments in solid state materials. In
addition, our discussion of the concept of hyperuniformity is approached from
an applied and “materials” perspective, while Ref. Torquato (2018) focuses
more on the physics aspects and Ref. Yu et al. (2021) focuses more on the
photonics aspects.

The rest of this focus review is organized as follows: In Sec. II, we discuss
the definition of hyperuniformity and its generalizations in the context of
solid state materials, and subsequently discuss different types of disordered
hyperuniformity in these materials. In Sec. III, we discuss the effects of
imperfections, including thermal fluctuations, defects, and stochastic
displacements in solid state materials on hyperuniformity. Such knowledge can
provide valuable guidance to identify potential candidates of novel disordered
hyperuniform solid state materials. In Sec. IV, we focus on DHU medium/high-
entropy alloys. In Sec. V, we thoroughly review the recent developments in DHU
2D materials. In Sec. VI, we discuss quasi-1D DHU material systems such as
carbon nanotubes. In Sec. VII, we provide concluding remarks and outlook of
the field, with an emphasis on novel DHU quantum materials discovery and
design.

##  II Disorder and Hyperuniformity

###  II.1 Definition of Hyperuniformity

As briefly mentioned in Sec. I, the concept of hyperuniformity involves
measuring density fluctuations in a material system on different length
scales. Direct measurement of density fluctuations requires knowledge of
atomic positions, which for example can be extracted from electron microscopy
micrographs for 2D materials Zheng et al. (2020); Chen et al. (2021a). It is
generally much more challenging to extract atomic positions for three-
dimensional bulk materials. However, the hyperuniformity properties of such
materials can be assessed from scattering intensity data Hejna et al. (2013);
Xie et al. (2013), as we will discuss in detail below.

$\begin{array}[]{c}\\\
\includegraphics[width=205.97214pt]{Fig2.png}\end{array}$

Figure 2: Illustration of the measurement of number density fluctuations in a
material system using a spherical observation window $\Omega(R,{\bf x})$ with
radius $R$ and centered at ${\bf x}$. Image courtesy of Pinshane Huang.

Without loss of generality, consider a transmission electron microscopy (TEM)
image of a 2D amorphous silica, where the Si atomic columns appear as dark
dots (see Fig. 2). The positions of Si atoms, represented as the centers of
the dark dots, can be readily extracted using standard imaging processing
Zheng et al. (2020). This allows one to quantify the local Si atom density
$n({\bf x})$ via

$n({\bf x})=\sum_{i}^{N}\delta({\bf x}-{\bf x}_{i}),$ (1)

where $N$ is the total number of atoms, $\\{{\bf x}_{i}\\}$ ($i=1,\ldots,N$)
is the positions of the atomic centers, and $\delta({\bf x})$ is the Dirac
delta function. In other words, $n({\bf x})$ is a collection of $N$
$\delta$-functions placed at the atom centers, which “pick up” the atoms.

Now we place spherical observation window $\Omega(R,{\bf x})$ of radius $R$ at
location ${\bf x}$ in the system. The window only registers the atoms within
itself, which is mathematically represented by the window indicator function:

$\displaystyle\omega(R,{\bf x},{\bf y})$
$\displaystyle=\left\\{\begin{array}[]{l l}1,&|{\bf y}-{\bf x}|\leq R\\\
0,&|{\bf y}-{\bf x}|>R.\end{array}\right.$ (4)

The number of atomic centers falling into the window is then given by

$N(R,{\bf x})=\int_{V}{\omega(R,{\bf x},{\bf y})}\cdot n({\bf y})d{\bf y}.$
(5)

Therefore, $N(R,{\bf x})$ is random variable that depends on the observation
window location ${\bf x}$ as the window is randomly placed in the system. This
allows one to compute the mean number $\langle N(R)\rangle$ by averaging
$N(R,{\bf x})$ over the window locations, where $\langle\cdot\rangle$ denotes
the volume average. We note that the volume average is equivalent to ensemble
average for ergodic systems Torquato and Stillinger (2003), which is the focus
here.

The local number variance $\sigma^{2}_{N}(R)$ is then defined as

$\sigma^{2}_{N}(R)=\langle N^{2}(R)\rangle-\langle N(R)\rangle^{2}.$ (6)

It has been shown that $\sigma^{2}_{N}(R)$ possesses the following asymptotic
expansion in the large-$R$ limit:

$\sigma^{2}_{N}(R)\sim A\cdot R^{d}+B\cdot R^{d-1}+\ell(R^{d-1}),$ (7)

where $A\geq 0$, $B>0$ and $\ell(x)$ indicates terms with orders lower than
$x$. In Ref.Torquato and Stillinger (2003), it was shown that

$A=\lim_{|{\bf k}|\rightarrow 0}S({\bf k}),$ (8)

where $S({\bf k})$ is the static structure factor associated with the atom
centers, which can be obtained via scattering experiments (by excluding the
forward scattering intensity). Specifically, it is defined as

$S({\bf k})=\frac{1}{N}\langle{\sum_{i\neq j=1}^{N}\exp[-i{\bf k}\cdot({\bf
x}_{i}-{\bf x}_{j})]}\rangle.$ (9)

Another relevant statistical descriptor for the atomic distribution is the
pair correlation function $g_{2}({\bf r})$, which is defined via

$\rho g_{2}({\bf r})=\frac{1}{N}\langle{\sum_{i\neq j=1}^{N}\delta({\bf
r}+{\bf x}_{i}-{\bf x}_{j})}\rangle,$ (10)

where $\rho=N/V$ is the atomic number density. Intuitively, $g_{2}$ measures
the average local number density fluctuations with respect to the global
number density $\rho$. Combining Eq. (9) and Eq. (10) leads to the following
relation:

$S({\bf k})=1+\rho\tilde{h}({\bf k}),$ (11)

where $\tilde{h}({\bf k})$ is the Fourier transform of the total correlation
function $h({\bf r})\equiv g_{2}({\bf r})-1$. Eq. (11) indicates that $S({\bf
k})$ and $g_{2}({\bf r})$ encode the same amount of information on pair
correlations in the material systems.

A $d$-dimensional material is hyperuniform if its atomic number variance
$\sigma_{N}^{2}(R)$ grows in the large-$R$ limit slower than $R^{d}$, i.e.,

$\lim_{R\rightarrow\infty}\frac{\sigma_{N}^{2}(R)}{R^{d}}=0.$ (12)

This behavior is to be contrasted with those of typical disordered systems,
such as Poisson point patterns, gases and liquids, where the number variance
scales as $R^{d}$, i.e., like the volume of the observation window. It follows
immediately from Eq. (7) and Eq. (12) that the necessary and sufficient
condition of hyperuniformity is that the zero-wavenumber limit of the static
structure factor vanishes, i.e.,

$A=\lim_{|{\bf k}|\rightarrow 0}S({\bf k})=0.$ (13)

In many applications, condition (13) is easier to check compared to the direct
computation of $\sigma_{N}^{2}$ and assessing the large-$R$ limit.

Hyperuniform systems can be further classified according to the small
wavenumber (small $|k|$) behaviors of the structure factor $S({\bf k})$, which
are equivalent to the asymptotic large $R$ behaviors of the number variance
$\sigma_{N}^{2}(R)$. In particular, consider systems that are characterized by
a structure factor with a radial power-law form in the vicinity of the origin,
i.e., $S({\bf k})\sim|{\bf k}|^{\alpha}$ for $|{\bf k}|\to 0$. For
hyperuniform point configurations, the exponent $\alpha$ is positive
($\alpha>0$) and its value determines three different large-$R$ scaling
behaviors of the number variance Torquato and Stillinger (2003); Zachary and
Torquato (2009); Torquato (2018):

$\displaystyle\sigma_{N}^{2}(R)$ $\displaystyle\sim\left\\{\begin{array}[]{l
l}R^{d-1},&\alpha>1\text{ (class I)}\\\ R^{d-1}\ln R,&\alpha=1\text{ (class
II)}\\\ R^{d-\alpha},&0<\alpha<1\text{ (class III).}\end{array}\right.$ (17)

These scalings of $\sigma_{N}^{2}(R)$ define three classes of hyperuniformity
Torquato (2018), with classes I and III describing the strongest and weakest
forms of hyperuniformity, respectively. States of matter that belong to class
I include all perfect crystals Torquato and Stillinger (2003); Zachary and
Torquato (2009), many perfect quasicrystals Zachary and Torquato (2009); Lin
et al. (2017); Oğuz et al. (2017), and “randomly” perturbed crystal structures
Gabrielli and Torquato (2004); Gabrielli et al. (2008); Kim and Torquato
(2018a), classical disordered ground states of matter Torquato and Stillinger
(2003); Torquato et al. (2015b) as well as systems out of equilibrium Zhang et
al. (2016b); Lei et al. (2019b). Class II hyperuniform systems include some
quasicrystals Oğuz et al. (2017), and many disordered classical Donev et al.
(2005b); Zachary et al. (2011d); Jiao and Torquato (2011b) and quantum Feynman
and Cohen (1956b); Reatto and Chester (1967); Torquato et al. (2008b) states
of matter. Examples of class III hyperuniform systems include random
organization models Hexner and Levine (2015b) and perfect glasses Zhang et al.
(2016b).

###  II.2 Different Types of Disorder in Solid-State Materials and
Generalizations of Hyperuniformity

$\begin{array}[]{c}\\\
\includegraphics[width=411.93767pt]{Fig3.png}\end{array}$

Figure 3: Illustration of different types of disorder commonly seen in
amorphous solid state materials.

The notion of hyperuniformity was originally introduced for point
configurations that possess positional disorder Torquato and Stillinger
(2003), see Fig. 3(a). This type of disorder is also very commonly seen in
amorphous solid state materials, such as metallic glasses, with respect to the
distribution of the atomic centers. Another type of disorder commonly seen in
covalent materials is topological disorder (see Fig. 3(b)), such as in
amorphous silicon Hejna et al. (2013); Xie et al. (2013) and 2D materials
Zheng et al. (2020); Chen et al. (2021a). In these materials, local
topological defects that preserve the local chemical order (e.g., the
coordination shell) are accumulated, eventually leading to topologically
disordered networks, consisting of atomic loops of different sizes (e.g.,
5-fold, 6-fold and 7-fold rings). In these systems, the topological disorder
also induces positional disorder in atomic center distributions, which can
treated using the same framework for quantifying number variance discussed in
Sec. II.A.

Compositional disorder is typically found in solid solution states of alloys
(see Fig. 3(c)), in which atoms of different alloying elements occupy the
sites of certain crystal lattices (such as face-centered cubic and body-
centered cubic lattices) Porter and Easterling (2009). Different atomic
species can favor either like- or unlike-neighbors, as a result of the
competition between enthalpic and entropic effects, leading to local
composition fluctuations associated with different atomic species. Although
the compositional disorder can be characterized by analyzing the atomic number
variance for each component as well, it is also convenient to introduce scalar
fields to quantify, e.g., the local molar fractions of different atomic
species.

Magnetic disorder or spin disorder is a type of commonly seen disorder in
magnetic materials and randomly dilute magnetic alloys Mézard et al. (1987,
1984) (see Fig. 3(d)). Each atom in a magnet possesses an intrinsic magnetic
moment or spin, the orientation of which can be either binary (i.e., the Ising
model Peierls (1936); Aharoni et al. (2000)) or continuous (i.e., the
Heisenberg model Heisenberg (1985); Aharoni et al. (2000)). If the Hamiltonian
governing the interactions between the spins possesses simple forms (e.g.,
only including nearest neighbor interactions), the system can easily achieve
ground state with long-range spin orientation order such as the ferromagnetic
or anti-ferromagnetic state. On the other hand, if the Hamiltonian includes
non-trivial long-range spin-spin interactions, the system typically ends up in
a metastable state with frozen-in spin disorder, i.e., a spin glass. Spin
glasses can also be produced by alloying magnetic elements with non-magnetic
elements. In this case, the magnetic disorder is induced by the compositional
disorder in the alloy Binder and Kob (2011a). It is also convenient to
describe the magnetic system as a random scalar field, where the scalar values
indicate the local spin orientation states. In this case, the hyperuniform
framework based on point configuration cannot be applied and a proper
generalization of the notation to continuous random field is required.

The hyperuniformity concept was first generalized by Torquato and co-workers
to binary random fields Zachary and Torquato (2009) and subsequently to
continuous scalar fields Torquato (2016). Consider a statistically homogeneous
random scalar field $F({\bf x})$ in $d$-dimensional space that is real-valued.
Similar to the case of point configurations, we employ a spherical observation
window with radius $R$ and volume $v_{1}(R)\sim R^{d}$. In this case, the
quantity of interest is the local field intensity, defined as the integration
of field within the observation window over the window volume, which
fluctuates as the window randomly moves in the system. The associated variance
$\sigma_{F}^{2}(R)$ is given by

$\sigma_{F}^{2}(R)=\frac{1}{v_{1}(R)}\int_{\mathbb{R}^{d}}\psi({\bf
r})\alpha(r;R)d{\bf r},$ (18)

where $\alpha(r;R)$ is the scaled intersection volume, i.e., the intersection
volume of two spherical windows of radius $R$ whose centers are separated by a
distance $r$, divided by the volume $v_{1}(R)$ of the windows. A hyperuniform
scalar field is one whose $\sigma_{F}^{2}(R)$ decreases more rapidly than
$R^{-d}$ for large $R$, i.e.,

$\lim_{R\rightarrow\infty}\sigma_{F}^{2}(R)\cdot R^{d}=0.$ (19)

Equivalently, the hyperuniformity of scalar field can be defined via its
autocovariance function, i.e.,

$\psi({\bf r})=\langle{(F({\bf x}_{1})-\langle{F({\bf x}_{1})}\rangle)(F({\bf
x}_{2})-\langle{F({\bf x}_{2})}\rangle)}\rangle,$ (20)

where ${\bf r}={\bf x}_{2}-{\bf x}_{1}$. The associated spectral density
function $\hat{\psi}({\bf k})$ is given by

$\hat{\psi}({\bf k})=\int_{\mathbb{R}^{d}}\psi({\bf r})e^{-i{\bf k}\cdot{\bf
r}}d{\bf r},$ (21)

which is the Fourier transform of $\psi(r)$. The hyperuniform condition is
then given by

$\lim_{|{\bf k}|\rightarrow 0}\hat{\psi}({\bf k})=0,$ (22)

which implies that

$\int_{\mathbb{R}^{d}}\psi({\bf r})d{\bf r}=0.$ (23)

In the case of hyperuniform scalar field Torquato (2016), there are three
different scaling regimes (classes) that describe the associated large-$R$
behaviors of the local field intensity variance when the spectral density goes
to zero as a power-law scaling ${\tilde{\psi}}({\bf k})\sim|{\bf k}|^{\alpha}$
as $|\bf k|$ tends to zero:

$\displaystyle\sigma^{2}_{F}(R)\sim\begin{cases}R^{-(d+1)},\quad\quad\quad\alpha>1\qquad&\text{(Class
I)}\\\ R^{-(d+1)}\ln R,\quad\alpha=1\qquad&\text{(Class II)}\\\
R^{-(d+\alpha)},\quad 0<\alpha<1\qquad&\text{(Class III).}\end{cases}$ (24)

Classes I and III are the strongest and weakest forms of hyperuniformity,
respectively.

The above generalized framework for hyperuniform random fields can be readily
employed to quantify the aforementioned compositional disorder and spin
disorder in solid state materials. In addition, this framework also allows one
to analyze other important quantities in amorphous materials, such electron
density distributions Gerasimenko et al. (2019); Sakai et al. (2022). An
interesting case is when the electrons are sufficiently localized in the
vicinity of the atomic cores. In this case, the overall electron density
distribution $F({\bf x})$ can be considered as linear superposition of the
electron density distribution $K({\bf x},{\bf r}_{i})$ around the atomic core
$i$ centered at ${\bf r}_{i}$, i.e.,

$F({\bf x})=\sum_{i}K({\bf x},{\bf r}_{i}).$ (25)

It has been shown in Ref. Torquato (2016) that the spectral density for a
scalar field given by Eq. (25) can be written as

$\hat{\psi}({\bf k})=\rho\hat{K}^{2}({\bf k})S({\bf k}),$ (26)

where $\rho$ is the number density of the atoms, $\hat{K}({\bf k})$ is the
Fourier transform of the local electron density function $K({\bf x})$, and
$S({\bf k})$ is the structure factor associated with the atom distributions.
It follows from this analysis that the localized electrons possess a
hyperuniform density distribution in a material where the atoms are
hyperuniformly distributed, which has been demonstrated in an analogous
classical system Jiao (2022).

##  III Effects of Imperfections in Solid-State Materials on Hyperuniformity

Real materials inevitably contain structural defects and are subject to
thermal fluctuations, which could mask the the hyperuniformity nature of the
underlying materials. Understanding the effects of such defects, thermal
fluctuations and other perturbations could therefore provide valuable guidance
for the identification and discovery of new disordered hyperuniform solid
state materials. We will also discuss some general conditions that have been
identified under which hyperuniformity could arise or be preserved. An example
is the transformation of perfect crystalline state in 2D hexagonal materials
by introducing topological defects, which has been shown to preserve the
hyperuniformity property of the crystalline state Chen et al. (2021a).

According to the well-known fluctuation–compressibility theorem Hansen and
McDonald (1986), the isothermal compressibility of equilibrium single-
component many-particle ensembles $\kappa_{T}$ at temperature $T$ and number
density $\rho$ is linked to the structure factor $S(k)$ through the following
relationship:

$\rho k_{B}T\kappa_{T}=S({\bf k}={\bf 0}),$ (27)

where $k_{B}$ is the Boltzmann’s factor. Consequently, thermal fluctuations at
any finite temperature will destroy hyperuniformity in any compressible
equilibrium systems (i.e., $\kappa_{T}\neq 0$), regardless of the degree of
order in the system Torquato (2018). Moreover, strong theoretical arguments
Torquato (2018) indicate that the large-distance asymptotic behavior of the
direct correlation function $c({\bf r})$ of disordered phases is exactly
proportional to the potential $v({\bf r})$ and the volume integral of $c({\bf
r})$ does not exist for hyperuniform systems. Therefore, effective long-ranged
interactions are necessary to achieve incompressibility and drive an
equilibrium many-particle system to a hyperuniform state. However, this
condition is not necessary to achieve hyperuniformity in systems out of
equilibrium Torquato (2018). Here, the direct correlation function $c({\bf
r})$ is linked to the total correlation function $h({\bf r})$ defined above
via an integral equation Ornstein and Zernike (1914):

$h({\bf r})=c({\bf r})+\rho c({\bf r})\otimes h({\bf r}),$ (28)

which upon Fourier transformation evolves into

$\tilde{h}({\bf k})=\frac{\tilde{c}({\bf k})}{1-\rho\tilde{c}({\bf k})}.$ (29)

In the context of solid state materials, disorder often comes in the form of a
wide spectrum of defects and displacements introduced into crystals, and here
we discuss their effect on the preservation and generation of hyperuniformity.
For point defects such as vacancies and substitutions, it is known that they
will destroy hyperuniformity in proportion to the concentration $p$ of vacancy
or substitutions when they are introduced randomly into the crystals, i.e.,
there is no spatial correlation between vacancies or substitutions Kim and
Torquato (2018b):

$S({\bf k})=p+(1-p)S_{0}({\bf k}),$ (30)

where $S_{0}({\bf k})$ is the structure factor of the original cyrstalline
state. On the other hand, randomly introduced interstitials will destroy
hyperuniformity proportional to $p/(1+p)$ Kim and Torquato (2018b),

$S(k)=\frac{p}{1+p}+\frac{1}{1+p}S_{0}({\bf k}),$ (31)

where $p$ is the ratio of the number of interstitials over the number of
particles originally in the crystals. However, if vacancies, substitutions, or
interstitials are introduced in a hyperuniform manner, then the resulting
patterns will also preserve the hyperuniformity of the defect-free crystals
Kim and Torquato (2018b).

“Uncorrelated” stochastic displacements in perfect lattices, i.e., the
displacements of distinct particles are uncorrelated and the joint probability
density function can be reduced into a product of two singlet probability
densities Kim and Torquato (2018b), have also been employed to model perturbed
crystalline states of solid materials. It has been shown that they can never
destroy the hyperuniformity but it can be degraded such that the perturbed
lattices fall into class-III hyperuniform systems Gabrielli (2004); Kim and
Torquato (2018b); Klatt et al. (2020). On the other hand, correlated
displacements (e.g., thermal excitation in compressible crystals or mechanical
vibrations) can destroy hyperuniformity Kim and Torquato (2018b); Chen et al.
(2021b). However, under certain special conditions correlated displacements
can also preserve hyperuniformity Chen et al. (2021d). For example, it has
been shown that hyperuniformity is preserved when the following three
conditions are met: (i) sources of the displacements are randomly introduced
and sparse; (ii) the displacements are sufficiently localized (i.e., the
volume integrals of the displacements and squared displacements caused by
individual defect are finite); (iii) the displacement-displacement correlation
matrix of the system is diagonalized and isotropic.

It is also noteworthy that a variety of hyperuniformity-generating operations
that can convert nonhyperuniform point patterns into hyperuniform materials
have also been reported. For example, Klatt et al. Klatt et al. (2019)
demonstrated that one can convert a wide spectrum of nonhyperuniform and
hyperuniform point patterns into effectively stealthy hyperuniform patterns by
applying a sequence of correlated displacements that leads to the generation
of centroidal voronoi tessellations. Kim and Torquato Kim and Torquato (2019)
have discovered that by decorating each point in a point pattern with spheres
of different sizes so that the local packing fraction is identical across the
sample, one can convert a variety of point patterns into perfectly
hyperuniform materials. This operation shared some similarities with the
“equal-volume tessellation” operation Gabrielli et al. (2008) that can be used
to obtain hyperuniform point patterns.

##  IV Disordered Hyperuniform Medium/High-Entropy Alloys

High-entropy alloys (HEAs) are those composed of mixtures of equal or
relatively large proportions of five or more elements. Alloys containing three
elements are usually referred to as medium-entropy alloys (MEAs) Zhang et al.
(2014). These MEAs and HEAs are distinct from traditional metallic alloys
which contain one or two major components with smaller amounts of other
elements in that they are stabilized by entropy effects as solid solution
phases. Studies have shown that MEAs and HEAs can possess superior mechanical
performance and resistance to corrosion compared to traditional alloys Miracle
and Senkov (2017), and can have unusual thermal and electronic transport
properties Mu et al. (2018), opening up both new structural and functional
applications.

Detailed atomic structures of medium/high-entropy alloys largely remain a
mystery, due to challenges in direct imaging of bulk atomic packing
arrangements in three dimensions (3D). Over the past three decades, the random
mixture model and special quasirandom structure (SQS) Zunger et al. (1990);
Van de Walle et al. (2013) have been widely used for HEA/HEA structures,
assuming that atoms of different types in the alloys are randomly distributed
on the sites of an underlying crystalline lattice. Recently, a number of
studies (e.g., on chromium-cobalt-nickel (CrCoNi) MEAs and silicon-germanium-
tin (SiGeSn) MEAs Jin et al. (2022)) have discovered significant short-range
order (SRO). These SROs are often manifested as suppressed clustering of atoms
of the same type, which are shown to lead to energetically favorable states of
the alloys and are not captured by random mixture and SQS models Ding et al.
(2018); Zhang et al. (2020); Walsh et al. (2021); Besson (2021). A direct
consequence of the existence of such short-range order is the suppression of
composition (or density) fluctuations. Since the atoms prefer certain neighbor
arrangements, it can be expected that as an observation window moves randomly
within the system, the number of atoms of a particular element that fall
within the window would not fluctuate significantly, especially for large
window sizes, which is consistent with the concept of hyperuniformity.

###  IV.1 Multi-hyperuniform Structural Model

$\begin{array}[]{c}\\\
\includegraphics[width=411.93767pt]{figIII_1.png}\end{array}$

Figure 4: (a) Three-dimensional visualizations (bottom panel) and their two-
dimensional slices (top panel) along the (001) plane of representative
multihyperuniform (with an exclusion region of $K_{0}a=3.6$ for targeted
structure factor, where $a$ is the lattice constant of a diamond cubic unit)
SiGeSn alloys on a 9 $\times$ 9 $\times$ 9 diamond lattice. (b) Structure
factors and (c) Warren-Cowley short-range order parameters $\alpha_{ij}^{\nu}$
of multihyperuniform alloy model.

In Ref. Chen et al. (2021c), it was shown that SROs can lead to significant
suppression of large-scale composition/density fluctuations, using SiGeSn as
an example. In particular, Monte Carlo (MC) quenching simulations using
Stillinger-Weber empirical potential calibrated for SiGeSn alloys Tomita et
al. (2018): Starting from a random solution state, a random selected pair of
atoms of different types are swapped and the resulting energy change is
calculated. Only swaps leading to decreasing energy are accepted, which
amounts to a stochastic steepest descant in the energy landscape. Quenching
simulations were used because the solid solution phases of the alloy, even
with significant SROs, are not thermodynamically stable at low temperatures
Jin et al. (2022). The quenched alloys possess significant SROs, quantified
via the Warren-Cowley SRO parameter $\alpha_{ij}^{\nu}$, which is defined as

$\alpha_{ij}^{\nu}=1-p_{ij}^{\nu}/c_{j},$ (32)

where $p_{ij}^{\nu}$ is the probability of finding atomic species $j$ around
an atom of type $i$ in the $\nu$-th neighboring shell. The systems also
exhibit very strong (although not complete) suppression of large-scale
composition fluctuations for all elements, a key feature of a
multihyperuniform system. It is intuitive that the existence of SROs implies
suppression of large scale fluctuations (and vice versa), as SROs indicate the
elements are more homogeneously spread and thus, local clustering of same
elements leading to large fluctuations is very rare.

Based by these observations, a multi-hyperuniform structural model for
MEAs/HEAs with short-range orders were proposed in Ref. Chen et al. (2021c).
In particular, a highly efficient generic Fourier-space construction technique
was devised to generate large realizations of multihyperuniform MEAs (or
HEAs): A fictitious “energy” $E$ of the system was defined as the squared
differences between the target and constructed structure factors associated
with different types of elements, i.e.,

$E=\sum_{i}\sum_{k}[S_{i}(k)-S_{i,0}(k)]^{2},$ (33)

where $S_{i,0}(k)$ and $S_{i}(k)$ respectively are the angular-averaged
structure factor associated with type-$i$ atoms in the target and constructed
structures. The simulated annealing procedure was employed to evolve the
system under construction (by randomly swapping atoms of different types)
towards one with targeted structure factors via a stochastic optimization of
the “energy” $E$. To obtain multihyperuniform MEAs or HEAs, the target
structure factors $S_{i,0}(k)$ was set to be

$S_{i,0}(k)=0~{}~{}~{}~{}\textnormal{for}~{}~{}k<K_{i,0},$ (34)

where $K_{i,0}$ is the range of exclusion region in the Fourier space for the
target structure factor associated with element $i$. Cubic simulation boxes
with length $L$ and periodic boundary conditions were employed, for which the
wavevector $\bf{k}$ only takes discrete values ${\bf
k}=\frac{2\pi}{L}(n_{x},n_{y},n_{z})$, where $n_{x}$, $n_{y}$, and $n_{z}$ are
integers. To mitigate the statistical noise of $S({\bf k})$ at each individual
${\bf k}$, angular average of $S({\bf k})$ was performed to obtain $S(k)$ by
binning ${\bf k}$ according to its magnitude $k\equiv|{\bf k}|$.

A representative example of equimolar SiGeSn alloys (i.e.,
$c_{\mathrm{Si}}=c_{\mathrm{Ge}}=c_{\mathrm{Sn}}=1/3$) with an underlying
diamond lattice structure, obtained from the inverse construction (by setting
$K_{i,0}=K_{0}=3.6/a$ for all atom species, where $a$ is the lattice constant
of the diamond cubic unit cell) is shown in Fig. 4(a). The structure factors
associated with different elements are shown in Fig. 4(b), which decrease as
$k$ decreases at small $k$ and essentially approaches zero as $k$ goes to
zero, indicating hyperuniformity of the atomic distribution for all element
types. The SRO parameters are shown in Fig. 4(c), which indicates the
multihyperuniform structures possess substantial SRO, i.e., atoms of the same
type are disfavored in the first three neighbor shells around any atom in the
system. The emergent SRO in MEAs is induced by multihyperuniformity (i.e., no
constraints were imposed to target the realization of SRO), suggesting that
these two types of order may be coupled. This suggests in order to achieve
suppression of composition fluctuations on large length scales for each
species on a lattice, the system needs to suppress local clustering of the
same type of atoms. This rationale is not unique to the SiGeSn alloy, and
given the ubiquitous nature of SRO discovered previously in MEAs and HEAs Ding
et al. (2018); Zhang et al. (2020); Walsh et al. (2021); Besson (2021),
strongly indicates the possibility of hidden hyperuniformity in other MEAs or
HEAs with SROs.

###  IV.2 Physical Properties

Figure 5: Physical properties of multihyperuniform SiGeSn alloys: Band
structures of the multihyperuniform (a), SQS (b), and random (c) models for
SiGeSn MEAs. (d) Phonon density of states (PDOS) and partition ratio (PR) of
SiGeSn MEA with a multihyperuniform structure.

####  IV.2.1 Stability and lattice distortions

In Ref. Chen et al. (2021c), the authors calculated the formation energy of
the multihyperuniform SiGeSn MEAs with respect to a linear combination of the
energies of pure Si, Ge, and Sn to be 0.089 eV/atom using DFT calculations of
systems with 216 total atoms. The positive values indicate the solution phases
of the alloy are thermodynamically unstable to phase separation at 0 K, and
thus are metastable states that only can be obtained by quenching Jin et al.
(2022). It was also discovered that the multihyperuniform structures with
suppressed composition fluctuations should lead to a smaller deviation from
Vegard’s law and smaller lattice distortion that is intrinsic to all HEA or
MEA systems He and Yang (2018), compared to other solution models.
Specifically, the structures of different models were optimized using DFT
simulations and the lattice constants of the optimized multihyperuniform,
random, and SQS structures with $3\times 3\times 3$ supercells were determined
to be 17.903, 17.883, and 17.883 Å, respectively. According to Vegard’s law
Vegard (1921), the lattice constant of an equimolar SiGeSn alloy is predicted
to be 17.904 Å, the weighted average of the lattice constants of pure Si
(16.406 Å), Ge (17.346 Å), and Sn (19.960 Å) crystals with $3\times 3\times 3$
supercells, which clearly indicates that the multihyperuniform system exhibits
an minimal amount of deviation from Vegard’s law and reflects the nearly ideal
mixing of the three constituent elements in the multihyperuniform SiGeSn MEA.
The lattice distortion was also computed using the vector dissimilarity metric
Zhuang (2022), i.e., the root mean squared displacement (RMSD) between the
reference lattice (a diamond lattice with a lattice constant as predicted by
Vegard’s law) for the multihyperuniform, random, and SQS systems, which are
2.14, 3.21, and 3.31 Å, respectively. Clearly, the multihyperuniform system
possesses the smallest lattice distortion, which leads to lower energy of such
system than the (quasi)random systems.

####  IV.2.2 Electronic bandgaps

The effect of multihyperuniform long-range order (MHLRO) on the electronic
structure of the SiGeSn MEA was investigated by calculating the band structure
of the SiGeSn MEA with the multihyperuniform, random, and SQS structure.
Figures 5(b), (c), and (d) reveal direct band gaps of 0.28, 0.36, and 0.50 eV
at the $\Gamma$ point for the random, SQS, and multihyperuniform systems,
respectively. For pure crystalline Si, Ge, and Sn, the band gaps are 1.29,
0.33, and 0.15 eV, respectively. In the random and SQS model for the SiGeSn
alloy, the clustering of Ge and Sn atoms significantly reduces the band gap,
while in the multihyperuniform model the presence of leads to suppressed local
clustering of atoms of the same type, resulting in a band gap that is nearly
the average band gap (0.59 eV) of the three constituent elements. This is
another manifestation that the MHLRO SiGeSn alloy approximately realizes
Vegard’s law.

####  IV.2.3 Low-temperature thermal conductivity

The thermal conductivity of the multihyperuniform system at low temperatures
is generally higher than that of the random solution models, which are 1.53
and 0.87 W/(m$\cdot$ K) at 10 K for the multihyperuniform and random systems,
respectively. The improved thermal conductivity makes the multihyperuniform
SiGeSn MEA a better candidate for electronic device applications such as
micro-chips that require a stricter management in heat dissipation. The higher
thermal conductivity of the multihyperuniform system can be attributed to the
smaller lattice distortions of such system compared to the random mixture,
since large lattice distortions would enhance the scattering of phonons and
lead to low thermal conductivity Fan et al. (2016). However, as the
temperature increases, the effect of phonon-phonon coupling becomes more
prominent, which leads to similarly decreased thermal conductivities for
multihyperuniform and random mixtures. For example, it is found that the
computed thermal conductivities at 400 K for the multihyperuniform and random
systems are 0.53 and 0.57 W/(m$\cdot$ K), respectively. Figure 5(e) shows the
phonon density of states (PDOS) and partition ratio (PR, which for each
vibrational mode $n$ is calculated as
$PR_{n}=(\sum_{i}e_{i,n}^{2})^{2}/N\sum_{i}e_{i,n}^{4}$, where $e_{i,n}$
denotes the phonon eigenvector of mode $n$ and $N$ refers to the total number
of atoms Lee and Hwang (2017); Guo and Huang (2015)) of the multihyperuniform
mixture. Although the overall shape of the PDOS is similar to that of the
random system Wang et al. (2020), a sudo gap around 280 cm-1 (corresponding to
$T\sim 400$ K) was observed, leading to diminishing PR modes in the vicinity
of this frequency, and a reduction in the thermal conductivity.

##  V Disordered Hyperuniform 2D Materials

###  V.1 Modeling Amorphous 2D Materials via Hyperuniform Networks Containing
Stone-Wales Topological Defects

Figure 6: (a) Structural model of amorphous 2D material obtained by
introducing SW defects in a perfect honeycomb network. (b) Illustration of a
SW defect, which changes the local network topology and leads to a cluster of
two pentagons and two heptagons.

2D materials, or single layers of van der Waals materials, are solid state
materials a single unit cell thick, which can be structurally modeled as 2D
networks. A commonly seen type of disorder in 2D materials results from
topological defects, such as the Stone-Wales (SW) defects, which can be
introduced in the system via, e.g., proton radiation (see Fig. 6) Stone and
Wales (1986). In the case of hexagonal 2D materials, the resulting structures
contain “flipped” bonds that change the local topology of the original
honeycomb network, leading to paired clusters of two pentagons and two
heptagons. The Stone-Wales-like defects have been observed in a number of 2D
material systems Huang et al. (2012, 2013a); Eder et al. (2014); Zhang et al.
(2015c); Toh et al. (2020), and their effects on the materials’ physical
properties have been numerically investigated Li et al. (2011); Van Tuan et
al. (2012a).

Figure 7: Representative amorphous structural models at different defect
fractions $p=0.02$ (a), $p=0.04$ (b), $p=0.12$ (c), $p=0.14$ (d), with the SW
defect bonds highlighted in blue. Note that at small $p$, the defect bonds
form small isolated clusters, while the defect bonds percolate and form large
interconnected clusters at large $p$. (e) The static structure factor $S(k)$
of these networks possesses the scaling $S(k)\sim k^{\alpha}$ for small wave
number $k$, where $1\leq\alpha(p)\leq 2$ is monotonically decreasing as the SW
defect concentration $p$ increases. (f) The corresponding number variance
$\sigma_{N}^{2}(R)$ possesses the scaling $\sigma_{N}^{2}(R)\sim R^{\gamma}$
with $\gamma<2$. (g) The scaling exponent $\alpha\in[1,2]$ in $S(k)$ decreases
as $p$ increases, reaches a value of 1.0 at $p\approx 0.12$, and then remains
relatively flat as $p$ increases beyond 0.12. These results indicate that the
Stone-Wales defects preserve hyperuniformity in amorphous 2D networks, and the
existence of of hyperuniformity class transition from Class I to Class II at
$p\sim 0.12$.

The global structures of amorphous 2D materials resulted from these local
defects were comprehensively and systematically investigated in Ref. Chen et
al. (2021a). Specifically, the structural models for amorphous 2D network
structures were generated by randomly introducing SW defects (i.e., bond-
ration induced topological transformations) into the perfect honeycomb lattice
until a specific defect concentration $p$ is achieved, which was followed by
structural relaxation to minimize a harmonic energy that regularizes the bond
lengths and angles. It was found that all of the generated structures are
hyperuniform, manifested as $\lim_{k\rightarrow 0}S(k)=0$ and the scaling
$\sigma_{N}^{2}(R)\sim R^{\gamma}$ with $\gamma<2$ (see Figs. 7). These
results indicate that the SW transformation preserve hyperuniformity in the
resulting amorphous network materials. This is consistent with the observation
that SW defects are local perturbations, and thus, do not fundamentally change
the nature of density fluctuations on large length scales compared to the
original honeycomb lattice, which is hyperuniform Chen et al. (2021d).

The static structure factor of the amorphous networks possesses the scaling
$S(k)\sim k^{\alpha}$ for small wave number $k$ for all defect concentrations
studied here, with the scaling exponent $\alpha\in[1,2]$. $\alpha$ first
decreases as $p$ increases, reaches 1.0 at $p\approx 0.12$, and then remains
relatively flat as $p$ increases beyond 0.12 (see Fig. 7). The initial
decrease of $\alpha$ is due to the increasing disorder, which is saturated as
the defects percolate at $p_{c}\approx 0.11$. This is also equivalently
reflected in the scaling of the number variance $\sigma_{N}^{2}(R)$ from $\sim
R$ to $R\ln(R)$ for large $R$ as $p$ increases. Since the generation of these
amorphous network material models didn’t assume any material specific details,
they can be employed to model the structures of a variety of hexagonal 2D
materials. For example, the amorphous network structure with $p=0.06$ matches
previously synthesized experimental amorphous graphene sample Toh et al.
(2020) well in terms of various statistics, which we discuss below.

###  V.2 Amorphous Graphene

Figure 8: Amorphous 2D materials containing the Stone-Wales topological
defects. (a) Scanning transmission electron microscopy (STEM) image of 2D
amorphous graphene. Image courtesy of the Barbaros Oezyilmaz group. (b)
Structure factor $S(k)$ of the amorphous graphene sample obtained in
experiments and our structural model at $p=0.06$. (c) Polygonal shape
distributions of the experimental graphene sample (red) and our structural
model at $p=0.06$ (blue). Density of states (DOS) of different concentrations
(0$\leq$$p$$\leq$0.14) of Stone-Wales defects (d) in the energy window of -5
to 5 eV and (e) at the Fermi level denoted as $\Omega({p}$). Middle panels:
Atomic energy distribution (in eV) of graphene with (f) $p$ = 0.02 and (g) $p$
= 0.12. Lower panels: Electron densities at the Fermi level of graphene with
(h) $p$ = 0.02 and (i) $p$ = 0.12. The green surface represents isosurface of
1.0$\times 10^{-5}$ atomic unit.

Recently, single-layer amorphous graphene has been successfully synthesized
Toh et al. (2020), which possesses a structure distinctly different from the
random network model as revealed by detailed transmission electron microscopy
characterization. In a wider context, amorphous carbon-based systems have been
extensively studied, including graphene sheets Harris (2013); Dai et al.
(2016); Martin et al. (2019), cross-linked graphene network Hu et al. (2017),
amorphous glassy carbon and carbon fibers Jenkins and Kawamura (1971); Sarac
et al. (2008), to name a few. The generic structural networks discussed in
Sec. IV.A can be converted into 2D amorphous material by decorating each
vertex (as in the case of graphene) and/or the mid-point of each bond (as in
the case of amorphous silica) in the network with an atom of a particular
type. Examples of resulting 2D materials include graphene and graphene-like
materials such as boron nitride (BN), molydenum disulphide (MoS2), and silica
(SiO2).

The amorphous graphene model derived from our hyperuniform network structure
at $p=0.06$ (class I hyperuniform) matches previously synthesized experimental
amorphous graphene sample Toh et al. (2020) well in terms of various
statistics (see Fig. 8(b) and (c)). Specially, the static structure factor
$S(k)$ directly computed from the TEM image data reveals a degree of
hyeruniformity in experimentally synthesized amorphous graphene, which agrees
very well with the corresponding structure factor computed from the network
model. Among other properties, the electronic structure properties of a wide
spectrum of disordered hyperuniform graphene materials derived from our
network models at different SW concentration values $p$ were investigated Chen
et al. (2021a).

Fig. 8(d) shows the density of states (DOS) of the DHU graphene materials,
which revels that the Dirac cone, i.e., the semi-metal nature of crystalline
graphene, is destructed in in DHU graphene systems, which possess increasingly
higher DOS at the Fermi level as $p$ increases. The DOS values $\Omega(p)$ at
the Fermi level as a function of $p$ is shown in Fig. 8(e), which exhibits
monotonically increasing behavior as $p$ increases. The increased DOS at the
Fermi level are also manifested in the energies and charge densities: The
carbon atoms at the flipped C-C bonds and their adjacent regions exhibit
higher energies. This is illustrated in Fig. 8(f) and (g), respectively
showing the atom-resolved total energies for two representative systems with
$p$ = 0.02 and 0.12. These two amorphous graphene systems correspond class-I
hyperuniformity ($p=0.02$) and class-II hyperuniformity ($p=0.12$),
respectively. Figure 8(h) and (i) respectively show the charge density at the
Fermi level for these two systems. It can be seen that the electrons in
class-I DHU graphene spread out in the entire system, while the electrons in
class-II DHU graphene are localized in separate islands. These patches are
similar to the localization regions found by Tuan et al. and shown to degrade
the electrical transport of graphene Van Tuan et al. (2012b).

These results indicate that increasing $p$ in the DHU graphene materials
significantly populates the number of electron states $\Omega(p)$ at the Fermi
level, which is a result of the increasing number of high-energy states
induced by the topological defects, and the Fermi-level charge densities
indicate different electronic transport mechanisms associated with different
classes of disorder, from patch-spreading to highly localized states. Such
knowledge strengthens our fundamental understanding of change of behaviors in
density fluctuations as defects are gradually introduced into ordered
structures, and may have important implications for novel applications in
photovoltaics, semiconductors, electrodes, batteries, water purification and
multi-functional composites Bhimanapati et al. (2015); Miró et al. (2014); Xu
et al. (2013). The degradation of mechanical properties in DHU graphene as $p$
increase was also investigated in Ref. Chen et al. (2021a).

###  V.3 Amorphous Silica

Figure 9: Hyperuniformity in 2D amorphous silica. (a) TEM image of 2D
amorphous silica. Reproduced from Ref. Huang et al. (2013b). (b) Angle-
averaged spectral density $\psi({\bf k})$ associated with the TEM micrograph
and the simulated amorphous silica network. Insets show the full spectral
density. Density of states of (c) the supercells of 2D crystalline and (d)
hyperuniform SiO2 calculated using density functional theory. The inset shows
a zoom-in view of the DOS near the Fermi level. Electron densities (e) at the
HOMO level of the crystalline structure and (f) at the Fermi level of the DHU
structure. The isosurface values is 0.5 $\times 10^{-8}$ $e$/Bohr3.

Recently, the discovery of hyperuniformity in amorphous 2D silica
(conventionally modeled as “continuous random networks” Binder and Kob
(2011b)), based on the analysis of aberration corrected transmission electron
microscopy (TEM) images of the material was reported Zheng et al. (2020), see
Fig.9. The amorphous silica samples were fabricated using chemical vapor
deposition (CVD) and the procedure for obtaining the imaging data set was
reported in detail in Ref. Huang et al. (2013b). As shown in Fig.9, the black
spots represent the silicon atoms. The TEM images are processed to retain the
distribution information of the silicon atoms by thresholding the grayness
intensity distribution associated with each silicon atom and the associated
spectral density $\hat{\psi}({\bf k})$ (where ${\bf k}$ is the wave-vector) is
then directly computed from the processed image following Ref. Ma and Torquato
(2017) and shown in the inset of Fig.9(b). The angularly averaged
$\hat{\psi}(k)$ (with $k=|{\bf k}|$) is shown Fig.9(b). The spectral density
analysis used here is equivalent to the structure. It can be seen that
$\hat{\psi}({\bf k})$ is fully isotropic and the scattering is completely
suppressed at infinite wavelength, i.e., $\lim_{k\rightarrow
0}\hat{\psi}(k)=0$ with $\hat{\psi}({\bf k})\sim|{\bf k}|$ for small $k$
values, which indicates that the 2D amorphous silica samples analyzed are
class-II hyperuniform. The origin of hyperuniformity in these systems also
result from the hyperuniformity-preserving nature of the Stone-Wales
topological defects Zheng et al. (2020).

The density of states (DOS) for both the hyperuniform amorphous silica systems
and their crystalline counterpart were calculated at the DFT-PBE level of
theory Perdew et al. (1996); Chen et al. (2010); Li et al. (2016), see Fig.
9(c) and (d), respectively. These calculations revealed the insulator nature
of the 2D crystalline SiO2, with a predicted band gap of 5.31 eV. By contrast,
in the hyperuniform amorphous silica a small but finite number of states
occupy the Fermi level of the DHU structure, showing metallic behavior of the
electrons with a typical band gap of $\sim 0.05$ eV. This is comparable to the
thermal fluctuations at room temperature $\sim 0.025$ eV. The density $\rho$
of the electrons that contribute to the electrical conductivity of the
amorphous silica at room temperature was estimated as $\rho$ as 2.33 $\times
10^{12}$ cm-2, by integrating the number of states in the energies ranging
from 25 meV (corresponding to the thermal energy) to the Fermi level. This
value belongs to the category of “high doping” regime (e.g., between 6.0
$\times 10^{11}$ and 9.2 $\times 10^{12}$ cm-2) in common semi-metals Yang et
al. (2014). In other words, the disordered hyperuniformity fundamentally
changes the electrical transport behavior of 2D SiO2, from an effective
insulator at room temperature (as in the crystalline form) to an effective
metal (as in the disordered hyperuniform form).

The metallic behavior of disordered hyperuniform SiO2 can also be seen from
the distribution of charge densities within an energy window of 0.5 eV below
the Fermi level (see Fig. 9(f)). For purpose of comparison, the charge density
distribution for the crystalline structure within an energy window of 0.5 eV
below the highest occupied molecular orbital (HOMO) level is shown in Fig.
9(e). The electrons in the crystalline structure fully occupy the valence
bands, but they cannot be thermally excited at room temperature to the
conduction bands due to the large band gap, leading to zero electrical
conductivity for pure 2D crystalline SiO2. On the other hand, the number of
valence electrons in the disordered hyperuniform system in the same energy
window is much less, resulting in a low carrier density, but nevertheless, a
non-zero conductivity. A closer look at the slightly wider energy window
associated with lowest density of states (e.g. -2$\sim$0 eV) reveals that the
distribution of states still forms an almost continuous spectrum of peaks, see
the inset in Fig. 9(d). Figure 9(f) reveals that the valence electrons
contributing to the conductivity originate from a small portion of the Si and
O atoms in the DHU system.

###  V.4 Pentagonal 2D Materials

Very recently, a class of pentagonal 2D materials have been discovered, which
can be derived from the Cairo tessellation with type-II pentagons (see Fig.
10(a)), and realized by single-layer $AB_{2}$ pyrite structures (see Fig.
10(b)). An example of such 2D material is single-layer PtP2, in which each
fundamental cell contains two Pt atoms and four P atoms forming two pairs of
pentagons touching through a vertex. Unlike their 2D hexagonal counterparts,
2D pentagonal materials, in particular, PdS2 that have been successfully
synthesized exhibit intrinsic in-plane anisotropy useful for various (e.g.,
thermoelectric) applications Lu et al. (2020).

Figure 10: Illustration of the Cairo tessellation formed with type II
pentagons filling the plane (a), the pentagonal 2D $AB_{2}$ material mapped
from the Cairo tessellation (b), and the Stone-Wales (SW) topological defect
in the Cairo tessellation (c). (d) and (e) respectively illustrate two
distinct topological pathways through which the pentagonal Cairo tiling (P5)
respectively transforms into a crystalline rhombus-hexagon (C46) tiling (d)
and random rhombus-pentagon-hexagon (R456) tilings (e) by continuously
introducing the SW defects. The intermediate configurations along the P5-C46
pathway contain rhombi and hexagons whose orientations of the are all aligned
in the same direction. (f) and (g) respectively show the structure factor
$S(k)$ of the structures along the P5-C46 pathway (f) and P5-R456 pathway (g)
as a function of the dimensionless wavenumber $k/(2\pi\rho^{1/2})$, where
$\rho$ is the number density of the system. Figure 11: Band structures of 2D
PtP2 with the (a) P5 Liu and Zhuang (2018) and (b) C46 structures computed at
the HSE06 level of theory. (c) Density of states of four intermediate
structures ($p$ = 0.2, 0.4, 0.6, and 0.8) in the P5-C46 transition pathway
calculated at the same level of theory. (d) and (e) respectively show partial
charge densities that are $\pm 0.1$ eV about the Fermi level of two
intermediate structures ($p$ = 0.5) in the P5-C46 (d) and P5-R456 (e)
transition pathways calculated at the PBE level of theory. The isosurface
value of the charge density plots is set to 0.0002$e$/$a_{0}^{3}$ (where
$a_{0}$ is the Bohr radius).

Similar to the hexagonal 2D materials discussed above, Stone-Wales defects can
be introduced into the crystalline pentagonal materials as a source of
disorder Zheng et al. (2021). A single SW defect in the fundamental cell of
the Cairo tilling induces a local topological transformation which converts
two pairs of pentagons into a pair of rhombi and a pair of hexagons touching
through a common edge (See Fig. 10(c)). Unlike the SW defects in hexagonal 2D
materials such as graphene, which cause distortions, the defects in pentagonal
2D materials preserve the shape and symmetry of the fundamental cell of P5
tiling and are associated with a minimal energy cost, making the intermediate
R456 tilings realizable metastable states at room temperature Zheng et al.
(2021).

In Ref. Zheng et al. (2021), two distinct topological pathways were reported
through which the pentagonal Cairo tiling (P5) (i.e., single-layer $AB_{2}$
pyrite materials) respectively transforms into a crystalline rhombus-hexagon
(C46) tiling and random rhombus-pentagon-hexagon (R456) tilings, by
continuously introducing the Stone-Wales (SW) topological defects, see Fig.
10(d) and (e). It was found that these topological transformations are
controlled by the orientation correlations among neighboring $B$-$B$ bonds,
and exhibit a phenomenological analogy of the (anti)ferromagnetic to
paramagnetic transition in two-state Ising systems. Moreover, the intermediate
structures along the two pathways are neither crystals nor quasicrystals, and
yet these random tilings preserve hyperuniformity of the P5 or C46 crystal
(i.e., the infinite-wavelength normalized density fluctuations are completely
suppressed, see Fig. 10(f) and (g)), and can be viewed as 2D analogs of
disordered Barlow packings in three dimensions.

The resulting 2D materials possess metal-like electronic properties, making
them promising candidates for forming Schottky barriers with the
semiconducting P5 material. Figure 11(a) and (b) compare the band structures
of the 2D materials derived from the P5 and C46 structures. As reported in
Ref. Liu and Zhuang (2018), the P5 structure is semiconducting with a direct
band gap of 0.52 eV. By contrast, the C46 structure is metallic with Dirac-
cone-like dispersion in the conduction bands near the X point. This dispersion
is caused by the presence of the hexagonal six-membered rings in the C46
structure. The density of states for four disordered intermediate structures
with p = 0.2, 0.4, 0.6, and 0.8 along the P5-C46 transition pathway,
calculated at the HSE06 level of theory, are shown in Fig. 11(c). The closure
of band gaps (marking the apparent continuous transition from semiconducting
to metallic behaviors) as seen from the calculated DOS is already very evident
in the structures with $p=0.2$, and all band gaps are completely closed for
structures associated with larger $p$. Additional calculations showed that the
electron states around the Fermi level in these hyperuniform intermediate
structures are itinerant rather than localized (see Fig. 11(d) and (e)),
further indicating the metallic nature of these systems.

###  V.5 Defected Two-Dimensional Transition Metal Dichalcogenides

While SW defects are prevalent in graphene-like 2D materials, other types of
defects such as chalcogen vacancies, metal vacancies and trefoil defects are
dominant in monolayer transition metal dichalcogenides (TMDCs) such as MoS2
and WSe2 Lin et al. (2015); Lee et al. (2020) (see Fig. 12(a)). For example,
Lin and coworkers Lin et al. (2015) have elaborated how the local structures
evolve as various types of defects are introduced into MoS2. Experimental
studies Lehnert et al. (2019); Leiter et al. (2020); Komsa et al. (2013)
examining the evolution of structures as various types of defects are
introduced into samples of TMDCs have also been reported.

Figure 12: (a) A schematic of the 2H-WSe2 monolayer (upper panel) and its
projection (lower panel) when seen from above. Note that the Te-doped 2H-WSe2
monolayer (at low Te concentration can be viewed as effectively the 2H-WSe2
monolayer), when projected from above, is mapped into a perfect honeycomb
lattice, with each “particle” in the projected plane possessing three bonds.
(b) A schematic illustrating the mapping from a raw image (top) of a defected
monolayer crystalline transition metal dichalcogenide obtained using the
annular dark-field scanning transmission electron microscopy (ADF-STEM)
technique to the extracted atomic positions (bottom). (c) Chemical-bonding
informed coordination networks overlaid with raw images of different frames
during the evolution of the material system obtained using the ADF-STEM
technique. Static structure factor $S(k)$ of the evolving system in nearly
hyperuniform (d) and non-hyperuniform (e) regimes.

In Ref. Chen et al. (2021b), a comprehensive characterization of the evolution
of global structures as double chalcogen vacancies are gradually introduced
into experimental samples of TMDCs (see Fig. 12(b)) was carried out within the
hyperuniformity framework. Deep-learning algorithms were first employed to
extract the atomic positions in a sequence of image frames obtained from ADF-
STEM, as the scanning electron probe continuously introduces defects into a
monolayer crystalline 2D transition metal dichalcogenide alloy, Te-doped
2H-WSe2. The chemical-bonding informed coordination networks were subsequently
constructed and refined for this evolving system (see Fig. 12(c)). In the
perfect crystalline state, the coordination network is represented as a
honeycomb network similar to crystalline graphene and silica.

Calculations of static structure factor $S(k)$ of the derived coordination
networks indicated the systems are nearly hyperuniform at low defect
concentrations (see Fig. 12(d)), and the hyperuniformity of the crystalline
network is completely destroyed even when there is still a significant portion
of crystalline sites in the system (specifically, less than $20\%$ defects,
see Fig. 12(e)). The effects of temperature on detecting hyperuniformity from
experimental images were also investigated, which was found to be equivalent
to inducing correlated displacement of lattice sites in a perfect crystalline
honeycomb network. Moreover, it was shown that the structure factor of the
system at relatively low concentration $p$ of double chalcogen vacancies
assumes the analytical expression

$S(k)\approx\frac{p}{2}+(1-\frac{p}{2})S_{0}(k),$ (35)

where $S_{0}(k)$ is the structure factor of the vacancy-free honeycomb
network. These results reveal the level of degradation of hyperuniformity that
one should expect due to the finite experimental measurement precision in STEM
experiments. In addition, the analysis procedures can be readily adapted to
characterize the structures of other ordered and disordered two-dimensional
materials.

##  VI Disordered Hyperuniform Quasi-1D Materials: Amorphous Nanotubes

Carbon nanotubes, a class of quasi-1D materials, can be conceptually
constructed by rolling up one or multiple graphene sheets, and in this sense
can be viewed as derivative structures of 2D materials. In particular, single-
walled carbon nanotubes (SWNTs) can be formed by rolling up a long strip of a
single graphene sheet along different directions Sinnott and Andrews (2001).
Conventionally, the type of SWNTs can be specified by a rolling vector $(n,m)$
(with $n>0$, $m\geq 0$, and $n\geq m$) in the basis of two linearly
independent vectors (e.g. ${\bf u}$ and ${\bf v}$ in Fig. 13(a)) that connect
a carbon atom in the graphene sheet to either two of its nearest atoms with
the same bond directions. Two most common types of SWNTs are: (i) zigzag
nanotubes with $n>0$ and $m=0$ (see an example in Fig. 13(b)), and (ii)
armchair nanotubes with $n=m$ (see an example in Fig. 13(c)). Multi-walled
nanotubes (MWNTs) consisting of purely zigzag nanotubes, or purely armchair
nanotubes, or a mixture of both are also of great interest.

$\begin{array}[]{c}\\\
\includegraphics[width=411.93767pt]{fig_nano_1.png}\end{array}$

Figure 13: (a) Illustration of two linearly independent vectors ${\bf u}$ and
${\bf v}$ that form the basis to specify the rolling direction. A rolling
vector $n\mathbf{u}+n\mathbf{v}$ uniquely determines the resulting nanotube
structure from a given graphene sheet. (b) Illustration of a defected (3,0)
zigzag nanotube formed by rolling up a graphene sheet with randomly
distributed Stone-Wales defects along the rolling vector $3\mathbf{u}$. (c)
Illustration of a defected (3,3) armchair nanotube formed by rolling up a
graphene sheet with randomly distributed Stone-Wales defects along the rolling
vector $3\mathbf{u}+3\mathbf{v}$.

$\begin{array}[]{c}\\\
\includegraphics[width=424.94574pt]{fig_nano_2.png}\end{array}$

Figure 14: Local number variances $\sigma_{N}^{2}(R)$ and structure factor
$S(k)$ of defected singled-walled and multi-walled nanotubes at different
defect concentrations $p$. (a) $\sigma_{N}^{2}(R)$ of (5,0) zigzag nanotubes.
(b) $\sigma_{N}^{2}(R)$ of (5,5) armchair nanotubes. (c) $\sigma_{N}^{2}(R)$
of double-walled nanotubes consisting of a (3,0) zigzag nanotube and a (5,0)
zigzag nanotube. (d) $\sigma_{N}^{2}(R)$ of double-walled nanotubes consisting
of a (3,3) armchair nanotube and a (5,5) armchair nanotube. (e)
$\sigma_{N}^{2}(R)$ of double-walled nanotubes consisting of a (5,0) zigzag
nanotube and a (5,5) armchair nanotube. (f) $S(k)$ of (5,0) zigzag nanotubes.
(g) $S(k)$ of (5,5) armchair nanotubes. (h) $S(k)$ of double-walled nanotubes
consisting of a (3,0) zigzag nanotube and a (5,0) zigzag nanotube. (i) $S(k)$
of double-walled nanotubes consisting of a (3,3) armchair nanotube and a (5,5)
armchair nanotube. The results are all averaged over 10 configurations.

$\begin{array}[]{c}\\\
\includegraphics[width=411.93767pt]{fig_nano_3.png}\end{array}$

Figure 15: Density of states (DOS) of (10, 0) zigzag DHU carbon nanotubes
computed by PBE functional at defect concentration $p$: (a) 0.00, (b) 0.01,
(c) 0.0167, (d) 0.02, (e) 0.03, (f) 0.04, (g) 0.05, (h) 0.06.

Similar to the cases of 2D materials, SW defects are commonly seen in carbon
nanotubes (see Fig. 13(b) and (c) for illustrations of carbon nanotubes with
randomly introduced SW defects), which generally can be introduced via high-
energy radiations, during the synthesis of materials, or by applying strains
Zhou and Shi (2003). Very recently, Chen and coauthors Chen et al. (2022b)
have systematically quantified the density fluctuations along the propagation
direction in amorphous carbon nanotubes containing randomly distributed Stone-
Wales defects. By effectively looking at 1D projections of the graphene sheets
along the axial/propagation direction, they demonstrated that all of these
amorphous nanotubes are hyperuniform, i.e., the infinite-wavelength density
fluctuations of these systems are completely suppressed, regardless of the
diameter, rolling axis, number of rolling sheets, and defect fraction of the
nanotubes.

Specifically, the authors Chen et al. (2022b) considered a generalized concept
of hyperuniformity by taking into account that multiple points in the graphene
sheets can be mapped to the same point in the projections. They generalized
the definition of the structure factor $S({\bf k})$ to be:

$S({\bf k})=\frac{1}{G}\left|{\sum_{j=1}^{M}g_{j}\exp(i{\bf k}\cdot{\bf
x}_{j})}\right|^{2}\quad({\bf k}\neq{\bf 0}),$ (36)

where $M$ is the number of distinguishable points in the projections, the
multiplicity $g_{j}$ is defined as the number of points in the higher-
dimensional structures that are mapped to the given point ${\bf x}_{j}$ in the
projections, and $G=\sum_{j=1}^{M}g_{j}$. Accordingly, the concept of
$N(R)\equiv\langle N(R;{\bf x}_{0})\rangle$ associated with
$\sigma_{N}^{2}(R)$ was generalized to be:

$N(R;{\bf x}_{0})=\sum_{j=1}^{M}g_{j}m({\bf x}_{j}-{\bf x}_{0};R),$ (37)

where $\langle\cdots\rangle$ denotes ensemble average, $m({\bf x}-{\bf
x}_{0};R)$ is the indicator function of the observation window centered at
${\bf x}_{0}$ with radius $R$ and is defined as

$m({\bf
x;R})=\left\\{\begin{array}[]{c@{\hspace{0.3cm}}c@{\hspace{0.3cm}}c}1,\hfil\hskip
8.5359pt&|{\bf x}|\leq R,\hfil\hskip 8.5359pt&\\\ 0,\hfil\hskip
8.5359pt&\textnormal{otherwise}.\hfil\hskip 8.5359pt&\end{array}\right.$ (38)

It has been demonstrated Chen et al. (2022b) that the local number variance
$\sigma_{N}^{2}(R)$ of defect-free $(n,0)$ zigzag SWNTs or MWNTs consisting of
$K$ zigzag nanotubes with $(n_{1},0)$, $(n_{2},0)$, $\cdots$, $(n_{K},0)$
rolling vectors, respectively, can be determined analytically as

$\sigma_{N}^{2}(R)=(n_{1}+n_{2}+\cdots+n_{K})^{2}\sigma_{N,D}^{2}(R),$ (39)

where $\sigma_{N,D}^{2}(R)$ is the local number variance of the projected
periodic two-scale 1D point patterns Torquato and Stillinger (2003) with
$\zeta=\frac{1}{3}$, and $K=1$ is the special case for SWNTs. Here $\zeta$ is
defined as the ratio of the nearest-neighbour distance over the length of the
smallest repeating unit in the projected 1D point pattern. Similarly, the
local number variance $\sigma_{N}^{2}(R)$ of defect-free $(n,n)$ armchair
SWNTs or MWNTs consisting of $K$ armchair nanotubes with $(n_{1},n_{1})$,
$(n_{2},n_{2})$, $\cdots$, $(n_{K},n_{K})$ rolling vectors, respectively, can
be determined as

$\sigma_{N}^{2}(R)=4(n_{1}+n_{2}+\cdots+n_{K})^{2}\sigma_{N,S}^{2}(R),$ (40)

where $\sigma_{N,S}^{2}(R)$ is the local number variance of the projected
periodic single-scale 1D pattern Torquato and Stillinger (2003), and $K=1$ is
the special case for SWNTs. On the other hand, as SW defects are introduced
into the nanotubes, the structures gradually transition into amorphous ones,
which are reflected in their local number variance $\sigma_{N}^{2}(R)$ shown
in Fig. 14. At low $p$, $\sigma_{N}^{2}(R)$ exhibits “periodicity” in the
window radius $R$, indicating that the crystalline order is reminiscent in the
systems; at large $p$, the oscillations of $\sigma_{N}^{2}(R)$ become much
more damped as $R$ increases, suggesting the emergence of truly amorphous
states. The variance $\sigma_{N}^{2}(R)$ of these SWNTs and MWNTs fluctuate
around certain constants as $R$ increases at all investigated $p$, indicating
that these structures are class-I hyperuniform. The results of ensemble-
averaged $S(k)$ of the aforementioned zigzag and armchair nanotubes are also
shown in Fig. 14, which all decreases to zero as $k$ goes to zero, regardless
of defect concentration $p$. These results further confirm the hyperuniformity
of these nanotubes, and are consistent with the results of
$\sigma_{N}^{2}(R)$.

Interestingly, MWNTs consisting of both zigzag and armchair nanotubes exhibit
different behaviors in their density fluctuations from SWNTs or MWNTs
consisting of purely zigzag or armchair nanotubes Chen et al. (2022b). For
example, as shown in Fig. 14(e), $\sigma_{N}^{2}(R)$ of MWNT consisting of a
(5,0)-zigzag nanotube and a (5,5)-armchair nanotube exhibit no “periodicity”
in the window radius $R$ even at low $p$. This is a direct result of the fact
that the length of the smallest repeating unit in a defect-free (5,0)-zigzag
nanotube and that of the smallest repeating unit in a defect-free
(5,5)-armchair nanotube do not have an integer common multiple, and the 1D
projections of the two nanotubes collectively onto the cylinder axis are no
longer periodic.

The electronic structures of DHU nanotubes with SW defects have also been
investigated. For example, as shown in Fig. 15 Chen et al. (2022b), the
density of states (DOSs) of (10, 0) zigzag DHU carbon nanotubes at different
$p$ as well as the DOS of defect-free carbon nanotubes. While it is well known
Azadi et al. (2010); Partovi-Azar et al. (2013) that (10,0) nanotubes possess
a well-defined band gap at Fermi level, increasing disorder in the DHU system
(i.e., increasing the amount of SW defects) results in two observed effects on
the computed DOS Chen et al. (2022b); Crespi et al. (1997): (i) closure of the
band gap at Fermi level and (ii) broadening and flattening of the DOS. This
trend is also similar to the observations in DHU 2D materials Zheng et al.
(2020); Chen et al. (2021a). Specifically, the band gap is closed at
$p=0.0167$, suggesting the presence of the semiconductor-to-metal transition
around this defect concentration. Moreover, as $p$ increases the DOS becomes
more and more extended, converging to a metallic characteristic.

##  VII Outlook on DHU Quantum Materials

Although still in its infancy, the discoveries of disordered hyperuniform
(DHU) solid-state materials, in particular the 2D and quasi-1D materials, shed
light on the feasibility of identifying and engineering novel DHU quantum
materials and DHU quantum states. There are a variety of potential device
applications for disordered hyperuniform solid state materials. For example,
quantum emitters that are capable of generating single photons have been
proposed, which could be realized by introducing defects into two-dimensional
(2D) materials such as hexagonal boron nitride ($h$-BN) Tran et al. (2016).
These materials are promising platforms to realize quantum bits (“qubits”) at
room temperature Liu and Hersam (2019). It is noteworthy that quantum
decoherence Liu and Hersam (2019), a process in which the state of the qubit
loses it intrinsic quantum properties via interactions with the environment.
This inherently changes the state leading to the quantum information loss and
ultimately translates to loss of quantum operation fidelity and ultimately
error generation. Naturally, the routes to control and suppress quantum
decoherence constitute the crucial issue that needs to be addressed in the
material research for quantum information science and they are critically
determine the applicability of the current and future quantum computers Liu
and Hersam (2019).

In this context DHU materials offer a potentially unique platform for
realizing decoherence protection by restricting the available collective
lattice vibrational modes (i.e., presence of phononic gaps) and electron
localization that limits long-range couplings. To this end, a comprehensive
understanding of DHU in the context of quantum correlations and excited states
is required, which is currently lacking. Many fundamental questions need to be
addressed: Can addressable local excitations (such as quantum defects) be
hosted by DHU systems suppressing coupling to the lattice excitations
(phonons)? Similarly, can such decoupling be realized in a fully quantum
regime similar to driven decoupling that increases the coherence times in
solid state qubits, such that spatial (and spatiotemporal) noise is employed
instead of the time domain environmental noise for decoupling Joos et al.
(2022)? How to control and engineer such states? How to properly generalize
the notion of DHU to quantum states? How do DHU states emerge in single or
correlated electron systems? How can the properties of individual DHU
materials be leveraged to create interfaced DHU quantum materials?

Addressing these questions requires tackling the fundamental underlying
physics of DHU systems and (quantum) states. This knowledge would feed into a
design of DHU quantum materials targeting suppression or enhancement of
particular types of couplings, mitigating some of the outstanding problems
associated with quantum information manipulation and storage. In practice, the
various types of correlated disorder intrinsic to 2D quantum material systems
need to be revealed, quantified and categorized within the hyperuniformity
framework that is distinctly different from the notion of disorder treated by
currently existing theories and models. The distinguishable nature of the
variety of hyperuniform disorder, in particular the unique quasi-long range
(QLR) correlations, and their effects on quantum decoherence should be
investigated and elucidated. Here, first principles computations and materials
simulations are indispensable for novel compound discovery and design.

In particular, the nuclear, electronic, and their coupled degrees of freedom
are clearly a fruitful area for exploration. We envision the following three
specific research directions that will advance our fundamental understanding
of the novel quantum DHU states and the fundamental roles played by the
defects and disorder in quantum materials:

(i) Understanding the electronic structure in DHU systems and exploration of
strong electron-electron correlations that result in new DHU quantum states.
Generalization of the notion of DHU states to quantum systems could be
achieved by exploring two superficially similar yet very distinct realizations
of quantum hyperuniformity in electronic states: 1) those supported by a DHU
atomic lattice and (2) those emerging from the mutual quantum interactions
regardless of the atomic geometric arrangement. Realizing quasiparticle states
for which the long-range oscillations are effectively near-sighted are of
immense potential application for protecting quantum states from dielectric
noise, which is typically long-ranged. We surmise that this investigation will
provide a unified picture of how electronic states may, under favorbale
conditions, be protected against decoherence through favorable (de)coupling of
the electronic fluctuations.

(ii) DHU-driven phonon decoherence protection of quantum defect states. DHU
materials behave as a crystalline solid in the quasi-long range and an
amorphous material in the short range. If such systems harbor strongly
localized electronic states, they will naturally couple to the lattice only
locally (in analogy to the nearsightedness mentioned in the preceding section)
and hence possibly exhibit advantages over other types of solid-state qubits.
In the idealized extreme case, such quantum states will behave like behaving
like trapped atoms/ions, are completely isolated from the external environment
and thereby maximizing the coherence time. Further, in contrast to delocalized
electronic states in a periodic system, which enhance the entanglements among
qubits. DHU platform based qubits are positioned between the two extremes and
likely to share both advantages, leading to ideal optimal coherent time,
strong inter-qubit coupling, and maximally compact physical platform. This
context creates a clear and strong incentive to design DHU systems with
optimal phonon degrees of freedom and electron-phonon couplings to realize
effective decoherence protection. Note that the phononically gaped systems
would also constitute an ideal encapsulation layer for various quantum
materials protecting them from environmental couplings.

While the DHU order is distinct from ideally periodic and completely random
arrangements, it is not simply an interpolation between the two “conventional”
states of matter. One of the most promising areas of DHU application is the
realization of wide phononic gaps and phonon dispersion narrowed to localized
modes. Both effects hold potential to decouple the electronic and vibrational
degrees of freedom. Such a scenario is particularly appealing for cases when
the DHU system “embeds” a localized and addressable quantum state (i.e., a
qubit in practice). To an extent, this hypothesized mechanism can be
paralleled with microscale acoustic optomechanical (de)coupling, which has
been explored on multiple fronts in quantum transductions Noguchi et al.
(2017); Mirhosseini et al. (2020). Here, however, the interactions with
mechanical (acoustic) modes are governed by patterning the systems at the
atomistic level, i.e., by introducing additional structural defects toward the
DHU limit.

(iii) Emergent DHU ordering at the interfaces of 2D DHU heterostructures. The
DHU state of matter has been associated with a single 2D DHU quantum material
so far. Yet, the fundamental concept is general and relates, in principle, to
DHU heterostructures. Conventional 2D heterostructures have important quantum
information science and engineering implications. For example, twisting one
sheet of a 2D material (e.g., graphene) with respect to another causes an
angle-dependent Moire superlattice generally with an extremely large unit cell
and localized electronic states near the Fermi level exhibiting all hallmarks
of strong correlations: when applying a strong electric field, bilayer
graphene exhibits an insulating behavior at half-filling of the localized
Moire states (paralleled with Mott insulator physics) and a related
superconducting regime upon doping Wang et al. (2021). This behavior occurs at
a small twist “magic” angle of 1.1 degree, which leads to strong coupling
between two graphene monolayers and formation of flat (localized) states. The
behavior of electrons at other superlattice arrangements can drastically
differ, even in the large inter-layer coupling limit (e.g., at high pressures)
Romanova and Vlček (2022). The mechanisms behind unconventional
superconductivity is not fully understood. These and similar pioneering
experiments on van der Waals bonded multi-layer systems with mutual rotational
”mis-arrangement” have led to a burgeoning field called “twistronics”, which
has recently morphed into a new playground for exploring intriguing physics
owing to strong electron-electron correlations. Given the intriguing quantum
phenomena in conventional 2D heterostructures, it is keen to discover 2D DHU
heterostructures that could serve as a novel platform to explore exotic
quantum effects.

###### Acknowledgements.

The authors are very grateful to Dr. Sef Tongay for his kind invitation and
inspiration for this focused review, and to Dr. Yu Zheng, Dr. Lei Liu, Dr. Duo
Wang, Chia-Hao Lee, Dr. Sangmin Kang, Dr. Wenjuan Zhu, Xinyu Jiang, Yu Liu,
Justine Ilyssa Vidallon for their contributions to the work on disordered
solid-state materials discussed in this review, and to Dr. Sal Torquato, Dr.
Ge Zhang, and Dr. Jaeuk Kim for very helpful discussions. Y. J. was supported
by the Army Research Office under Cooperative Agreement Number
W911NF-22-2-0103. P. H. was supported by U.S. Department of Energy, Office of
Science, Office of Basic Energy Sciences, Division of Materials Sciences and
Engineering under award number DE-SC0020190

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