# Extraordinary Optical and Transport Properties of Disordered Stealthy
Hyperuniform Two-Phase Media

Jaeuk Kim1,3 Salvatore Torquato1,2,3,4

###### Abstract

Disordered stealthy hyperuniform two-phase media are a special subset of
hyperuniform structures with novel physical properties due to their hybrid
crystal-liquid nature. We have previously shown that the rapidly converging
strong-contrast expansion of a linear fractional form of the effective dynamic
dielectric constant $\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf
k}_{1},\omega\right)}}$ [Phys. Rev. X 11, 296 021002 (2021)] leads to accurate
approximations for both hyperuniform and nonhyperuniform two-phase composite
media when truncated at the two-point level for distinctly different types of
microstructural symmetries in three dimensions. In this paper, we further
elucidate the extraordinary optical and transport properties of disordered
stealthy hyperuniform media. Among other results, we provide detailed proofs
that stealthy hyperuniform layered and transversely isotropic media are
perfectly transparent (i.e., no Anderson localization, in principle) within
finite wavenumber intervals through the third-order terms. Remarkably, these
results imply that there can be no Anderson localization within the predicted
perfect transparency interval in stealthy hyperuniform layered and
transversely isotropic media in practice because the localization length
(associated with only possibly negligibly small higher-order contributions)
should be very large compared to any practically large sample size. We further
contrast and compare the extraordinary physical properties of stealthy
hyperuniform two-phase layered, transversely isotropic media, and fully 3D
isotropic media to other model nonstealthy microstructures, including their
attenuation characteristics, as measured by the imaginary part of
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{1},\omega\right)}}$,
and transport properties, as measured by the time-dependent diffusion
spreadability ${\cal S}(t)$. We demonstrate that there are cross-property
relations between them, namely, we quantify how the imaginary parts of
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{1},\omega\right)}}$
and the spreadability at long times are positively correlated as the
structures span from nonhyperuniform, nonstealthy hyperuniform, and stealthy
hyperuniform media. It will also be useful to establish cross-property
relations for stealthy hyperuniform media for other wave phenomena (e.g.,
elastodynamics) as well as other transport properties. Cross-property
relations are generally useful because they enable one to estimate one
property, given a measurement of another property.

1 Department of Chemistry, Princeton University, Princeton, NJ 08544, USA

2 Department of Physics, Princeton University, Princeton, NJ 08544, USA

3 Princeton Materials Institute, Princeton, NJ 08544, USA

4 Program in Applied and Computational Mathematics, Princeton University,
Princeton, NJ 08544, USA

Corresponding author: Salvatore Torquato; E-mail<EMAIL_ADDRESS>
##  1 Introduction

Hyperuniform many-body systems and materials are characterized by an anomalous
suppression of density fluctuations at large length scales compared to typical
disordered systems [1]. The broad importance of the hyperuniformity concept
for statistical physics, condensed matter physics and materials science was
introduced two decades ago in an investigation that focused on fundamental
theoretical issues pertaining to local density fluctuations, including how it
provides a unified means to classify and categorize crystals, quasicrystals,
and special disordered point configurations [1]. A hyperuniform (or
superhomogeneous [2]) many-particle system in $d$-dimensional Euclidean space
$\mathbb{R}^{d}$ possesses a structure factor $S({\bf k})$ (proportional to
the scattering intensity) that vanishes as the wavenumber $k\equiv|\bf k|$
tends to zero, i.e.,

$\lim_{|{\bf k}|\rightarrow 0}S({\bf k})=0.$ (1)

Equivalently, a hyperuniform system is one in which the number variance of
particles within a spherical observation window of radius $R$, denoted by
$\sigma^{2}_{{}_{N}}(R)$, grows, for large $R$, more slowly than the window
volume, i.e., $R^{d}$ [1]. Typical disordered systems, such as ordinary gases
and liquids, have the expected asymptotic volume scaling, i.e.,
$\sigma^{2}_{{}_{N}}(R)\sim R^{d}$. On the other hand, all perfect crystals
and many perfect quasicrystals are hyperuniform with the surface-area scaling
$\sigma^{2}_{{}_{N}}(R)\sim R^{d-1}$. Disordered hyperuniform systems are
exotic amorphous states of matter that lie between a crystal and liquid: they
can behave like perfect crystals in the way they suppress large-scale density
fluctuations and yet have characteristics of liquids or glasses at small
length scales such that they are statistically isotropic with no Bragg peaks.
Thus, disordered hyperuniform systems can be thought of as having a hidden
order (see Fig. 2 of Ref. [3] for a vivid example), and their hybrid crystal-
liquid nature can endow them with novel physical properties, as described
below. It is noteworthy that the hyperuniformity concept extends our
traditional notions of long-range order to not only include crystals and
quasicrystals but exotic disordered states of matter [3].

Two decades ago, only a few examples of disordered hyperuniform systems and
their manifestations were known [1, 2]. It has come to be discovered that
these special disordered systems arise in a variety of contexts, across the
physical, materials, mathematical, and biological sciences, including
disordered hard-sphere plasmas [4, 3, 5], classical disordered
(noncrystalline) ground states [6, 7, 8, 9, 10], maximally random jammed (MRJ)
hard-particle packings [11, 12, 13, 14, 15, 16], jammed bidisperse emulsions
[17], jammed thermal colloidal packings [18, 19], jammed athermal soft-sphere
models of granular media [20, 21], nonequilibrium phase transitions [22, 23,
24, 25, 26, 27, 28, 29, 30], dynamical processes in ultracold atoms [31],
avian photoreceptor patterns [32], receptor organization in the immune system
[33], vegetation patterns in arid ecosystems [34], certain quantum ground
states (both fermionic and bosonic) [35, 36], vortex structures in
superconductors [37], “perfect” glasses [38], the distribution of the
nontrivial zeros of the Riemann zeta function [35, 39], and the eigenvalues of
various random matrices [40, 41, 35, 3, 42]. Thus, it is apparent that
disordered hyperuniform states of matter can exist as both equilibrium and
nonequilibrium phases and come in classical and quantum-mechanical varieties.

The fundamental and practical importance of the hyperuniformity concept in the
context of condensed matter physics began to emerge when it was demonstrated
that classical many-particle systems with certain long-ranged pair potentials
could counterintuitively freeze into disordered hyperuniform states at
absolute zero with singular scattering patterns, such as the stealthy
hyperuniform one depicted in the right panel of Fig. 1 [6, 7, 8, 9]. Stealthy
hyperuniform many-particle systems possess a structure factor that is zero not
only at infinite wavelengths but also vanishes for a range of wavenumbers
around the origin, i.e.,

$S({\bf k})=0\qquad\mbox{for}\;0<|{\bf k}|\leq K,$ (2)

implying that there is no single scattering down to intermediate wavelengths
of the order of $2\pi/K$ [6, 7, 8, 9]. Mapping such particle configurations to
network enabled the discovery of the first disordered dielectric networks to
have large isotropic photonic band gaps comparable in size to photonic
crystals [43]. This computational study led to the design and fabrication of
disordered cellular solids with the predicted photonic band-gap
characteristics for the microwave regime, enabling unprecedented free-form
waveguide geometries that are robust to defects not possible with crystalline
structures [44, 45]. Afterward, stealthy hyperuniform materials were
demonstrated to possess singular wave propagation, transport, and elasticity
characteristics, including wave transparency [46, 47, 48, 49, 50, 51, 52, 53],
tunable localization and diffusive regimes [47, 54, 55], enhanced absorption
of waves [56], enhanced solar cell efficiency [57], phononic properties [58,
59, 60], Luneberg lenses with reduced backscattering [61], extraordinary
phased arrays [62, 63], optimal sampling array of 3D ultrasound imaging [64],
high quality factor optical cavity [65], and network materials with nearly
optimal effective electrical conductivities and elastic moduli [66].

Figure 1: Left: Scattering pattern for a crystal. Right: Scattering pattern
for a disordered “stealthy” hyperuniform material defined by relation (2).
Scattering intensity increases from zero (blue color) to the highest value
(red color). Notice that apart from forward scattering, there is a circular
region around the origin in which there is no scattering, a singularly exotic
situation for an amorphous state of matter. This figure is adapted from Fig. 1
of Ref. [67].

The hyperuniformity concept was generalized to describe other contexts,
including two-phase heterogeneous media [68, 69], and random scalar and vector
fields [67]. Two-phase media, of central interest in this paper, are
ubiquitous; examples include porous media, composites, cellular solids,
biological media, colloids, granular media, foams, and polymer blends [70,
71]. Here, the phase volume fraction fluctuates within a spherical window of
radius $R$, which is useful to describe by the volume-fraction variance
$\sigma_{{}_{V}}^{2}(R)$. For ordinary disordered two-phase media, the
variance $\sigma_{{}_{V}}^{2}(R)$ goes to zero like $R^{-d}$ for large $R$. By
contrast, for hyperuniform disordered two-phase media,
$\sigma_{{}_{V}}^{2}(R)$ goes to zero faster than the inverse of the window
volume in the large-$R$ asymptotic limit, i.e., faster than $R^{-d}$, which is
equivalent to the vanishing of the spectral density
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left({\bf k}\right)}}$ (defined in
Sec. 2) in the infinite-wavelength limit, i.e.,

$\displaystyle\lim_{|\mathbf{k}|\rightarrow
0}\tilde{\chi}_{{}_{V}}(\mathbf{k})=0.$ (3)

Similar to the instance of hyperuniform point configurations [1, 3, 68], three
different large-$R$ scaling regimes arise when the spectral density goes to
zero with the power-law form

${\tilde{\chi}}_{{}_{V}}({\bf k})\sim|{\bf k}|^{\alpha};$ (4)

specifically,

$\displaystyle\sigma^{2}_{{}_{V}}(R)\sim\cases{R^{-(d+1)},\qquad\qquad\alpha>1&\mbox{(Class
I)}\\\ R^{-(d+1)}\ln R,\qquad\alpha=1&\mbox{(Class II)}\\\
R^{-(d+\alpha)},\qquad 0<\alpha<1&\mbox{(Class III)},}$ (5)

where the exponent $\alpha$ is a positive constant. Classes I and III are the
strongest and weakest forms of hyperuniformity, respectively. Stealthy
hyperuniform media, the major focus of this paper, are also of class I and are
defined to be those that possess zero-scattering intensity for a range of
wavevectors in the vicinity of the origin [69], i.e.,

${\tilde{\chi}}_{{}_{V}}({\bf k})=0\qquad\mbox{for}\;0<|{\bf k}|\leq K.$ (6)

Examples of stealthy hyperuniform media are periodic packings of spheres as
well as unusual disordered sphere packings derived from stealthy point
patterns [69, 72]. The reader is referred to Ref. [73] for a review of the
extraordinary multifunctional characteristics of disordered hyperuniform
media.

On the other hand, for any nonhyperuniform two-phase medium, it is simple to
demonstrate that the local volume-fraction variance has the following
large-$R$ scaling behaviors [74]:

$\displaystyle\sigma^{2}_{{}_{V}}(R)\sim\cases{R^{-d},\qquad\qquad\alpha=0&\mbox{(typical
nonhyperuniform)}\\\
R^{-(d+\alpha)},\quad-d<\alpha<0&\mbox{(antihyperuniform)}.}$ (7)

For a “typical” nonhyperuniform system, ${\tilde{\chi}}_{{}_{V}}(0)$ is
bounded [3]. In antihyperuniform systems, ${\tilde{\chi}}_{{}_{V}}(0)$ is
unbounded, i.e.,

$\lim_{|{\bf k}|\to 0}{\tilde{\chi}}_{{}_{V}}({\bf k})=+\infty,$ (8)

and hence are diametrically opposite to hyperuniform systems. Antihyperuniform
systems include systems at thermal critical points (e.g., liquid-vapor and
magnetic critical points) [75, 76], fractals [77], disordered non-fractals
[3], and certain substitution tilings [78].

In this paper, we further contrast and compare the extraordinary physical
properties of stealthy hyperuniform two-phase layered, transversely isotropic
media and fully three-dimensional (3D) isotropic media to other model
microstructures, including the optical characteristics as measured by the
effective dynamic dielectric constant, and transport properties, as measured
by the time-dependent diffusion spreadability. We also show, for the first
time, that there are cross-property relations between the attenuation
characteristics (imaginary part of the dielectric constant) and the
spreadability. (Cross-property relations have been profitably used to link
seemingly disparate physical properties to one another [79, 80, 81, 82, 83,
84, 85, 86, 70, 87, 88, 48].) For these purposes, it is instructive to briefly
review the strong-contrast formalism for the effective dielectric constant and
the spreadability concept [89], as described immediately below.

Figure 2:  This schematic illustrates how electric waves propagate through a
2D stealthy hyperuniform medium consisting of identical disks (green circles)
of dielectric constant $\varepsilon_{2}$ embedded in the matrix phase of
dielectric constant $\varepsilon_{1}$ within a perfect transparency interval.
The upper panel depicts the spatial distribution of the electric field inside
the sample as the wave travels from the left to the right, assuming
$\varepsilon_{2}>\varepsilon_{1}$. Blue and red colors represent the regions
where the electric field becomes positive and negative, respectively. The
lower panel depicts the average electric field as a function of the
propagation distance. Importantly, the electric field is a perfect sine wave,
implying that the wave can propagate indefinitely without loss.

We have derived exact nonlocal strong-contrast expansions of the effective
dynamic dielectric constant tensor
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{1},\omega\right)}}$
that treat general three-dimensional (3D) two-phase composite microstructures
[50]. These expansions are rational functions of the effective dielectric
constant tensor $\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf
k}_{1},\omega\right)}}$, the terms of which depend on the microstructure via
functionals of the $n$-point correlation functions
$\mathinner{S_{n}^{(i)}\mathopen{\left({\bf x}_{1},\ldots,{\bf
x}_{n}\right)}}$ for all $n$ (see Sec. 2) and exactly treat multiple
scattering to all orders beyond the long-wavelength regime (i.e.,
$0\leq\left|{\bf k}_{1}\right|\xi\lesssim 1$). Due to the rapid convergence of
the strong-contrast expansions [50, 51], their truncations at low orders yield
highly accurate estimates of
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{1},\omega\right)}}$
suited for various microstructural symmetries, implying that this resulting
formula very accurately approximates multiple scattering through all orders.
Specifically, we previously showed that second-order truncations, which depend
on the microstructure via functionals involving the spectral density
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left({\bf k}\right)}}$, already
provide accurate approximations of
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{1},\omega\right)}}$
beyond the long-wavelength regime for transverse electric (TE) polarization in
transversely isotropic media [50] and transverse polarizations in layered [51]
and fully 3D isotropic media [50]. Among other results, we predicted that
stealthy hyperuniform two-phase dielectric composites possess the perfect
transparency intervals [50, 51, 90]. As illustrated in Fig. 2, within such
transparency intervals, the sinusoidal electric waves can propagate
indefinitely without decaying (i.e., no Anderson localization [91, 92, 93, 94,
95, 96], in principle).

The diffusion spreadability, developed recently by Torquato [89], is a
dynamical probe that directly links the time-dependent diffusive transport
with the microstructure of heterogeneous media across length scales. Here, one
investigates the time-dependent problem of mass transfer of a solute in a two-
phase medium where all of the solute is initially contained in phase 2, and it
is assumed that the solute has the same diffusion coefficient $\cal D$ in each
phase. The spreadability $\mathcal{S}(t)$ is defined as the total solute
present in phase 1 at time $t$; see Fig. 3 for a schematic that shows
diffusion spreadability at different times for the very special case, for
purposes of illustration, in which phase 2 consists of a spatial distribution
of particles. For two different microstructures at some time $t$, the one with
the larger value of $\mathcal{S}(t)$ spreads diffusion information more
rapidly [89]. Torquato showed that the spreadability is exactly determined by
the spectral density ${\tilde{\chi}}_{{}_{V}}({\bf k})$ and its small-,
intermediate-, and long-time behaviors of $\mathcal{S}(t)$ are directly
determined by the small-, intermediate-, and large-scale structural
characteristics of the two-phase medium. Notably, the spreadability of
hyperuniform media always approaches their long-time asymptotic behaviors
substantially faster than those in nonhyperuniform ones. Remarkably,
disordered stealthy hyperuniform media approach their long-time asymptotic
behaviors exponentially faster among any hyperuniform variety [89]; see Sec.
2.2 for relevant formulas and more detail.

(a)  (b)

(c)

Figure 3: For purposes of illustration, this schematic shows diffusion
spreadability at different times for the special case in which phase 2 is
comprised of a spatial distribution of particles. The left panel (a) depicts
the uniform concentration of the solute species within phase 2 (dark blue
regions) at time $t=0$. The middle panel (b) depicts the spreading of
diffusion information at short times. The right panel (c) depicts the uniform
concentration of the solute species throughout both phases (light blue region)
in the infinite-time limit. The behavior of the spreadability ${\cal S}(t)$ as
a function of time is intimately related to the underlying microstructure.
This figure is adapted from Ref. [89].

Among other results, we further study their transparency characteristics for
different microstructural symmetries (Sec. 6) as well as provide detailed
proofs that stealthy hyperuniform layered and transversely isotropic media are
perfectly transparent (i.e., no Anderson localization, in principle) within
finite wavenumber intervals through the third-order terms; see Sec. 5 and B.
Remarkably, these results imply that there can be no Anderson localization
within the predicted perfect transparency interval in stealthy hyperuniform
layered and transversely isotropic media in practice because the localization
length (associated with only possibly negligibly small higher-order
contributions) should be very large compared to any practically large sample
size.

We further contrast and compare the extraordinary physical properties of 3D
stealthy hyperuniform two-phase media with various symmetries (layered,
transversely isotropic media, and fully 3D isotropic media) to other model
nonstealthy microstructures, including their attenuation characteristics, as
measured by the imaginary part of
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{1},\omega\right)}}$,
and transport properties, as measured by the time-dependent diffusion
spreadability ${\cal S}(t)$. Specifically, for layered media, we consider two
prototypical nonhyperuniform models: Debye random media and equilibrium hard
rods in a matrix; and two hyperuniform ones: nonstealthy hyperuniform
polydisperse packings in a matrix and stealthy hyperuniform packings in a
matrix (see Sec. 3). 111 Transport of waves and mass in layered and
transversely isotropic media can be rigorously considered to be those
transporting in one-dimensional ($d=1$) and two-dimensional ($d=2$) systems,
and so we sometimes refer to them as 1D and 2D media, respectively. For
transversely isotropic and fully 3D isotropic media, we study the two models
that attenuate the most and least, i.e., Debye random media and stealthy
hyperuniform ones, respectively (see Sec. 3). 222In the present work, we focus
on the imaginary parts of the effective dielectric constants since their
behaviors, as opposed to the real parts, are highly sensitive to the
microstructures. The reader is referred to Ref. [50] for estimates of the real
parts as a function of the wavenumber of various hyperuniform and
nonhyperuniform models.

For all models, we estimate the imaginary part of
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{q},\omega\right)}}$
using the second-order strong-contrast formulas presented in Sec. 4. In Sec.
5, we elaborate on the perfect transparency intervals of stealthy hyperuniform
media predicted by the second-order formulas for all symmetries considered
here, including proofs for certain cases of the same transparency intervals
through the third-order terms in the strong-contrast expansion. We present
results for the attenuation characteristics for all of our models considered
here in Sec. 6.1. We also provide in Sec. 6.2 comparisons of the spreadability
as a function of time for all of our models. In Sec. 6.3, we demonstrate that
there are cross-property relations between
$\mathrm{Im}[\mathinner{\varepsilon_{e}\mathopen{\left(k_{1}\right)}}]$ and
${\cal S}(t)$, namely, we quantify how the imaginary parts of the dielectric
and the spreadability at long times are positively correlated as the
structures span from nonhyperuniform, nonstealthy hyperuniform and stealthy
hyperuniform media. We utilize the inverse of specific surface (i.e., the mean
interface area per volume) $1/s$, as a natural characteristic inhomogeneity
length scale $\xi$ to scale distance or wavenumbers in order to compare
properties of different models of two-phase media, as discussed in Ref. [97].
In Sec. 7, we discuss our results and future avenues of research.

##  2 Definitions and Background

###  2.1 Two-Phase Media

A two-phase random medium is a domain of space
$\mathcal{V}\subseteq\mathbb{R}^{d}$ of volume $V$ that is partitioned into
two disjoint regions that make up $\mathcal{V}$: a phase 1 region
$\mathcal{V}_{1}$ with volume fraction $\phi_{1}$ and a phase 2 region
$\mathcal{V}_{2}$ with volume fraction $\phi_{2}$ [70].

####  2.1.1 $n$-Point Correlation Functions

The phase indicator function ${\cal I}^{(i)}({\bf x})$ for a given realization
is defined as

${\cal I}^{(i)}({\bf x})=\cases{1,\qquad{\bf x}\in{\cal V}_{i},\\\
0,\qquad{\bf x}\notin{\cal V}_{i}.}$ (9)

The two-phase medium is fully statistically characterized by the $n$-point
correlation $S^{(i)}_{n}({\bf x}_{1},{\bf x}_{2},\ldots,{\bf x}_{n})$
associated with phase $i$ defined by [70]

$S^{(i)}_{n}({\bf x}_{1},{\bf x}_{2},\ldots,{\bf
x}_{n})\equiv\left\langle{{\cal I}^{(i)}({\bf x}_{1}){\cal I}^{(i)}({\bf
x}_{2})\cdots{\cal I}^{(i)}({\bf x}_{n})}\right\rangle$ (10)

for all $n\geq 1$, where angular brackets denote an ensemble average and ${\bf
x}_{1},{\bf x}_{2},\ldots,{\bf x}_{n}$ are position vectors. The function
$S^{(i)}_{n}({\bf x}_{1},{\bf x}_{2},\ldots,{\bf x}_{n})$ can also be
interpreted to be the probability of simultaneously finding $n$ points with
positions ${\bf x}_{1},{\bf x}_{2},\ldots,{\bf x}_{n}$ in phase $i$.

For statistically homogeneous media, there is no preferred origin, and hence
the $n$-point correlation function depends on their relative displacement
vectors between the $n$ points, i.e.,

$S^{(i)}_{n}({\bf x}_{1},{\bf x}_{2},\cdots,{\bf x}_{n})=S^{(i)}_{n}({\bf
x}_{12},\cdots,{\bf x}_{1n}),$ (11)

where ${\bf x}_{ij}={\bf x}_{j}-{\bf x}_{i}$, where we have chose the origin
be at position ${\bf x}_{1}$. In such instances, the one-point correlation
function is a constant, namely, the phase volume fraction $\phi_{i}$, i.e.,

$\phi_{i}=\langle{\cal I}^{(i)}({\bf x})\rangle,$ (12)

such that $\phi_{1}+\phi_{2}=1$, and two-point correlation function depends on
the relative displacement vector ${\bf r}\equiv{\bf x}_{2}-{\bf x}_{1}$ and
hence $S_{2}^{(i)}({\bf x}_{1},{\bf x}_{2})=S_{2}^{(i)}({\bf r})$ [70, 67].
The autocovariance function $\chi_{{}_{V}}({\bf r})$ is defined as

$\displaystyle\chi_{{}_{V}}({\bf
r})\equiv\left\langle\mathinner{\mathcal{J}^{(i)}\mathopen{\left({\bf x}+{\bf
r}\right)}}\mathinner{\mathcal{J}^{(i)}\mathopen{\left({\bf
x}\right)}}\right\rangle=S^{(i)}_{2}({\bf r})-{\phi_{i}}^{2},$ (13)

where the fluctuating part of Eq. (9) is defined as

$\displaystyle\mathinner{\mathcal{J}^{(i)}\mathopen{\left({\bf
x}\right)}}\equiv\mathinner{\mathcal{I}^{(i)}\mathopen{\left({\bf
x}\right)}}-\phi_{i},$ (14)

and thus is identical for each phase $i=1,2$. At the extreme limits of its
argument, $\chi_{{}_{V}}$ has the following asymptotic behavior

$\chi_{{}_{V}}({\bf r}=0)=\phi_{1}\phi_{2},\qquad\lim_{|{\bf
r}|\rightarrow\infty}\chi_{{}_{V}}({\bf r})=0,$ (15)

the latter limit applying when the medium possesses no long-range order. If
the medium is statistically homogeneous and isotropic, then the autocovariance
function ${\chi}_{{}_{V}}({\bf r})$ depends only on the magnitude of its
argument $r=|\bf r|$, and hence is a radial function. In such instances, its
slope at the origin is directly related to the specific surface $s$ (interface
area per unit volume); specifically, we have in any space dimension $d$, the
asymptotic form [70],

$\chi_{{}_{V}}({\bf r})=\phi_{1}\phi_{2}-\beta(d)s\;|{\bf r}|+{\cal O}(|{\bf
r}|^{2}),$ (16)

where

$\beta(d)=\frac{\Gamma(d/2)}{2\sqrt{\pi}\Gamma((d+1)/2)}.$ (17)

####  2.1.2 Spectral Density

The nonnegative spectral density ${\tilde{\chi}}_{{}_{V}}({\bf k})$, which can
be obtained from scattering experiments [98, 99], is the Fourier transform of
$\chi_{{}_{V}}({\bf r})$, i.e.,

${\tilde{\chi}}_{{}_{V}}({\bf k})=\int_{\mathbb{R}^{d}}\chi_{{}_{V}}({\bf
r})e^{-i{\bf k\cdot r}}{\rm d}{\bf r}\geq 0,\qquad\mbox{for all}\;{\bf k}.$
(18)

For isotropic media, the spectral density only depends on $k=|{\bf k}|$ and,
as a consequence of (16), its decay in the large-$k$ limit is controlled by
the exact following power-law form [70]:

${\tilde{\chi}}_{{}_{V}}({\bf k})\sim\frac{\gamma(d)\,s}{k^{d+1}},\qquad
k\rightarrow\infty,$ (19)

where

$\gamma(d)=2^{d}\,\pi^{(d-1)/2}\,\Gamma((d+1)/2)$ (20)

is a $d$-dimensional constant.

For the special case of packing of identical spheres of radius $a$ that
comprises phase 2 with packing fraction $\phi_{2}$, the spectral density is
simply related to the structure factor [100, 70]:

$\displaystyle{\tilde{\chi}}_{{}_{V}}({\bf k})$ $\displaystyle=$
$\displaystyle\phi_{2}{\tilde{\alpha}}(k;a)S({\bf k}).$ (21)

where

${\tilde{\alpha}}(k;a)=\frac{1}{v_{1}(a)}\left(\frac{2\pi
a}{k}\right)^{d}J_{d/2}^{2}(ka),$ (22)

where $v_{1}(a)=\pi^{d/2}a^{d}/\Gamma(1+d/2)$ is the volume of a
$d$-dimensional sphere, and $\mathinner{J_{\nu}\mathopen{\left(x\right)}}$ is
the Bessel function of the first kind of order $\nu$. Since
${\tilde{\alpha}}(k;a)$ is a positive, bounded well-behaved function in the
vicinity of the origin, it immediately follows from expression (21) that if
the underlying point process is hyperuniform with a power-law structure factor
$S({\bf k})\sim|{\bf k}|^{\alpha}$ in the limit $|{\bf k}|\to 0$, then the
spectral density ${\tilde{\chi}}_{{}_{V}}({\bf k})$ inherits the same power-
law form [cf. (4)] only through the structure factor, not
${\tilde{\alpha}}(k;a)$ [69]. Moreover, it is clear that for stealthy
hyperuniform packings, relation (21) dictates that
${\tilde{\chi}}_{{}_{V}}({\bf k})$ is zero for the same wavenumbers at which
$S({\bf k})$ is zero [i.e., Eqs. (2) and (21) yield Eq. (6)], but the spectral
density also vanishes at the zeros of the function ${\tilde{\alpha}}(k;a)$,
which is determined by the zeros of $J_{d/2}(ka)$ [69].

###  2.2 Diffusion Spreadability

Torquato demonstrated that the time-dependent diffusion spreadability
$\mathcal{S}(t)$ in any $d$-dimensional Euclidean space $\mathbb{R}^{d}$ is
exactly related to the microstructure via the autocovariance function
$\chi_{{}_{V}}(\mathbf{r})$ in direct space, or equivalently, via the spectral
density $\tilde{\chi}_{{}_{V}}(\mathbf{k})$ in Fourier space [89]:

$\mathcal{S}(\infty)-\mathcal{S}(t)=\frac{1}{(2\pi)^{d}\phi_{2}}\int_{\mathbb{R}^{d}}\tilde{\chi}_{{}_{V}}(\mathbf{k})\exp[-k^{2}{\cal
D}t]d\mathbf{k}.$ (23)

Here, $\phi_{2}$ is the volume fraction of phase 2,
$\mathcal{S}(\infty)=\phi_{1}$ is the infinite-time limit of $\mathcal{S}(t)$,
and $\mathcal{S}(\infty)-\mathcal{S}(t)$ is called the excess spreadability
333Larger (smaller) spreadability $\mathcal{S}(t)$ implies smaller (larger)
excess spreadabilty $\mathcal{S}(\infty)-\mathcal{S}(t)$.. This situation is a
rare example of transport in two-phase media that is exactly described by only
the first two correlation functions, namely, $\phi_{1}$ and two-point
statistics via either $\chi_{{}_{V}}(\mathbf{r})$ or
$\tilde{\chi}_{{}_{V}}(\mathbf{k})$. The reader is referred to Ref. [89] for a
description of remarkable links between the spreadability ${\cal S}(t)$,
covering problem of discrete geometry, and nuclear magnetic resonance (NMR)
measurements.

For statistically homogeneous microstructures whose spectral densities exhibit
the following power-law form

$\tilde{\chi}_{{}_{V}}(\mathbf{k})\sim B|\mathbf{k}|^{\alpha}$ (24)

in the limit $|\mathbf{k}|\to 0$, Torquato [89] showed that the long-time
excess spreadability for two-phase materials in $\mathbb{R}^{d}$ is given by
the inverse power-law

${\cal S}(\infty)-{\cal S}(t)=\frac{C}{({\cal
D}t/a^{2})^{(d+\alpha)/2}}+{o}\left(({\cal
D}t/a^{2})^{-(d+\alpha)/2}\right)\quad({\cal D}t/a^{2}\gg 1),$ (25)

where $a$ is a characteristic heterogeneity length scale, ${o}(x)$ signifies
all terms of order less than $x$, and

$C=B\,\Gamma((d+\alpha)/2)\,\phi_{2}/[2^{d}\,{\pi}^{d/2}\Gamma(d/2)]$ (26)

is a microstructure-dependent coefficient. Thus, it is seen that the long-time
asymptotic behavior of ${\cal S}(t)$ is determined by the exponent $\alpha$
and the space dimension $d$, i.e., at long times, ${\cal S}(t)$ approaches the
value $\phi_{1}$ with a power-law decay $1/t^{(d+\alpha)/2}$, implying a
faster decay as $\alpha$ increases for some dimension $d$. Thus, compared to a
nonhyperuniform medium with a power-law decay $t^{-d/2}$, a hyperuniform
medium with a decay rate $1/t^{(d+\alpha)/2}$ can be viewed as having an
effective dimension that is higher than the space dimension, namely,
$d+\alpha$. When $\alpha$ is bounded and positive, this result means that
class I hyperuniform media has the fastest decay, followed by class II and
then class III, which has the slowest decay among hyperuniform media. Of
course, antihyperuniform media with $\alpha\to-d$ has the slowest decay among
all translationally invariant media. In the stealthy limit in which
$\alpha=+\infty$, the predicted infinitely-fast inverse-power-law decay rate
implies that the infinite-time asymptote is approached exponentially fast.
Indeed, for disordered stealthy media defined by (6), it was shown [89] that
long-time behavior is exactly given by

${\cal S}(\infty)-{\cal S}(t)\sim\frac{\exp(-K^{2}{\cal D}t)}{K^{2}{\cal
D}t}\qquad({\cal D}t/a^{2}\gg 1).$ (27)

Figure 4: “Phase diagram” that schematically shows the spectrum of
spreadability regimes in terms of the exponent $\alpha$ (taken from Ref.
[89]). As $\alpha$ increases from the extreme antihyperuniform limit of
$\alpha\to-d$, the spreadability decay rate gets faster, i.e., the excess
spreadability follows the inverse power law $1/t^{(d+\alpha)/2}$, except when
$\alpha=+\infty$, which corresponds to stealthy hyperuniform media with a
decay rate that is exponentially fast.

##  3 Model Microstructures

In what follows, we describe nonhyperuniform and hyperuniform models of 3D
two-phase media with certain microstructural symmetries for which we
investigate the aforementioned optical and transport properties. Specifically,
we describe 3D model microstructures of layered, transversely isotropic media,
and fully isotropic media, , including a description of their spectral
densities, which are used to compute their effective dynamic dielectric
constants and spreadabilities. Representative images of the models are shown
in Fig. 5. In Fig. 5(a), we depict representative images of the four models of
layered media: Debye random media, equilibrium hard rods in a matrix,
nonstealthy hyperuniform polydisperse sphere packing in a matrix, and stealthy
hyperuniform sphere packing in a matrix. For transversely isotropic and fully
isotropic media, we depict representative images of the two models for each
symmetry: Debye media and stealthy hyperuniform packings in a matrix; see Fig.
5(b) and (c). Among four models, three, except for Debye random media, are
derived from packings of nonoverlapping particles such that the particles of
dielectric constant $\varepsilon_{2}$ are distributed throughout a matrix of
dielectric constant $\varepsilon_{1}$, and thus $\phi_{2}$ indicates the
packing fraction. We take $\phi_{2}=0.2$ for all models considered here. In
what follows, we describe all of the models.

Figure 5:  Representative images of 3D two-phase media with $\phi_{2}=0.20$
possessing three different microstructural symmetries that are considered in
the present study. (a) Four models of layered media, consisting of infinite
parallel slabs of phases 1 and 2 whose thicknesses are derived from 1D models.
From the leftmost to the rightmost, we present Debye random media (black),
equilibrium hard rods in a matrix (green), nonstealthy hyperuniform
polydisperse sphere packing in a matrix (red), and stealthy hyperuniform
sphere packing in a matrix (blue), respectively. (b) Two models of
transversely isotropic media with cylindrical symmetry obtained from 2D
models: Debye media (black) and stealthy hyperuniform sphere packings in a
matrix (blue). (c) Two models of fully isotropic media: Debye media (black)
and stealthy hyperuniform sphere packings (blue). In each subfigure, the
bright and dark colors represent phases 1 and 2, respectively.

###  3.1 Models

####  3.1.1 Debye Random Media

Debye et al. [99] hypothesized that the following autocovariance function
characterizes isotropic random media in which the phases form domains of
“random shape and size”:

$\chi_{{}_{V}}(r)=\phi_{1}\phi_{2}\exp(-r/a),$ (28)

where $a$ is a characteristic length scale. This functional form for
$\chi_{{}_{V}}(r)$ was later shown to be realizable by two-phase media across
space dimensions [101, 102, 103] and was dubbed Debye random media [101, 70].
The Taylor expansion of (28) about $r=0$ and comparison to (16) reveals that
the specific surface $s$ of a Debye random medium in any space dimension is
given by

$s=\frac{\phi_{1}\,\phi_{2}}{\kappa(d)\,a},$ (29)

The spectral density of Debye random media in any space dimension is given by
[102]

${\tilde{\chi}}_{{}_{V}}(k)=\frac{\phi_{1}\phi_{2}\,c_{d}\,a^{d}}{[1+(ka)^{2}]^{(d+1)/2}},$
(30)

where $c_{d}=2^{d}\pi^{(d-1)/2}\Gamma((d+1)/2)$. The representative images of
this model shown in Fig. 5 are obtained using the construction techniques of
Ref. [103].

####  3.1.2 Equilibrium Hard Rods in a Matrix

We also examine equilibrium (Gibbs) ensembles of identical hard spheres of
radius $a$ with packing fraction $\phi_{2}$ [104, 105] in a matrix. In
particular, we consider such disordered packings along the stable disordered
fluid branch in the phase diagram [70, 105]. All such states are
nonhyperuniform. In the case of 1D equilibrium hard rods in a matrix, pair
statistics are known exactly [106]. In particular, using the exact solution of
the direct correlation function [107, 106] and the Ornstein-Zernike integral
equation, one can obtain the exact structure factor
$\mathinner{S\mathopen{\left(k\right)}}$. This solution for
$\mathinner{S\mathopen{\left(k\right)}}$, together with Eq. (21), yields the
corresponding spectral density
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}$ [70, 89]:

$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}=\phi_{2}\left[\frac{2\sin(ka)^{2}}{k^{2}a}\right]\left[1-\frac{2\phi_{2}\left\\{\phi_{2}\left[\cos(2ak)-1\right]+2ak\sin(2ak)(\phi_{2}-1)\right\\}}{(1-\phi_{2})^{2}(2ak)^{2}}\right]^{-1}.$
(31)

The representative image of this model in Fig. 5(a) is generated via the Monte
Carlo method [70]. Henceforth, we call this model equilibrium (Equil.) media.

####  3.1.3 Nonstealthy Hyperuniform Polydisperse Sphere Packings in a Matrix

Nonstealthy hyperuniform (NSHU) packings of spheres with a polydispersity in
size in a matrix can be constructed from nonhyperuniform progenitor point
patterns via a tessellation-based procedure [108]. Specifically, we begin with
the Voronoi tessellation [70] of the progenitor point patterns, which are the
centers of 1D equilibrium media (see section 3.1.2) with packing fraction
$0.20$ in this work. We then move the particle center in a Voronoi cell to its
centroid and rescale the particle such that the packing fraction inside this
cell is identical to a prescribed value $\phi_{2}<1$. The same process is
repeated over all cells. The final packing fraction is
$\phi_{2}=\sum_{j=1}^{N}\mathinner{v_{1}\mathopen{\left(a_{j}\right)}}/V_{\mathfrak{F}}=\rho\mathinner{v_{1}\mathopen{\left(a\right)}}$,
where $\rho$ is the number density of particle centers and $a$ represents the
mean sphere radius. In the thermodynamic limit, the spectral densities of the
resulting particulate composites, which are NSHU media, exhibit a power-law
scaling $\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left({\bf
k}\right)}}\sim\left|{\bf k}\right|^{4}$ for small wavenumbers [108], which
are of class I.

####  3.1.4 Stealthy Hyperuniform Sphere Packings in a Matrix

Stealthy hyperuniform (SHU) two-phase media have
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left({\bf k}\right)}}=0$ for the
finite range $0<\left|{\bf k}\right|\leq K$, called the exclusion-region
radius. Specifically, we consider $d$-dimensional SHU sphere packings in a
matrix of packing fraction $\phi_{2}$ that can be numerically generated in the
following two-step procedure. First, we generate point configurations
consisting of $N$ particles in a fundamental cell $\mathfrak{F}$ under
periodic boundary conditions via the collective-coordinate optimization
technique [6, 7, 9], which finds numerically the ground-state configurations
of the following potential energy;

$\mathinner{\Phi\mathopen{\left({\bf
r}^{N}\right)}}=\frac{1}{V_{\mathfrak{F}}}\sum_{{\bf
k}}\mathinner{\tilde{v}\mathopen{\left({\bf
k}\right)}}\mathinner{S\mathopen{\left({\bf
k}\right)}}+\sum_{i<j}\mathinner{u\mathopen{\left(r_{ij}\right)}},$

where $V_{\mathfrak{F}}$ is the volume of $\mathfrak{F}$,
$\mathinner{\tilde{v}\mathopen{\left({\bf
k}\right)}}=\mathinner{\Theta\mathopen{\left(K-\left|{\bf k}\right|\right)}}$,
$\Theta(x)$ (equal to 1 for $x>0$ and zero otherwise) is the Heaviside step
function, soft-core repulsion
$\mathinner{u\mathopen{\left(r\right)}}=(1-r/\sigma)^{2}\mathinner{\Theta\mathopen{\left(\sigma-r\right)}}$
[10]. The consequent ground-state configurations, if they exist, are still
stealthy and hyperuniform, and their nearest-neighbor distances are larger
than the length scale $\sigma$. Finally, to create packings in a matrix, we
follow Ref. [72] by circumscribing the points by identical spheres of radius
$a<\sigma/2$ under the constraint that they cannot overlap. Importantly, one
cannot obtain disordered SHU packings in a matrix with $\phi_{2}\geq 0.2$ for
$d=1,2$ without the soft-core repulsion
$\mathinner{u\mathopen{\left(r\right)}}$ [9].

For such SHU media, the degree of stealthiness $\chi$ is measured by the ratio
of the number of the wave vectors within the exclusion-region radius in the
Fourier space to the total degrees of freedom [6, 7, 9], i.e.,
$\chi=K/(2\pi\rho)$ in one dimension, and $\chi=K^{2}/(16\pi\rho)$ in two
dimensions. These SHU media are highly degenerate and disordered if $\chi<1/3$
in one dimension or $\chi<1/2$ in two and three dimensions [9]. In the present
work, we consider the cases of $\chi=0.3$ in the first three space dimensions.

###  3.2 Spectral Densities

Figure 6:  Log-log plots of dimensionless radial spectral densities
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}s^{d}$ as functions
of dimensionless wavenumber $k/s$ for all two-phase media considered here with
three structural symmetries, each with $\phi_{2}=0.20$: (a) layered, (b)
transversely isotropic, and (c) fully isotropic media. Representative images
of each model are shown in Fig. 5.

Estimation of the attenuation and spreadability behaviors reported in section
6 require the computation of the spectral density for the models. Figure 6
shows the dimensionless radial spectral density
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}s^{d}$ for all
models considered here with three microstructural symmetries, each with
$\phi_{2}=0.20$. For all nonhyperuniform media (Debye for $d=1,2,3$ and
equilibrium for $d=1$),
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}$ is evaluated by
using the analytic expressions given in Eqs. (30) and (31). For all
hyperuniform media (NSHU for $d=1$ and SHU for $d=1,2,3$), we obtain
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}$ from numerically
generated configurations.

Across all length scales, the spectral densities for all four models of
layered media are considerably different from one another; see Fig. 6(a). In
the long-wavelength regime ($k/s\ll 1$), the SHU medium suppresses volume-
fraction fluctuations [i.e.,
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}=0$] to a greater
degree than the NSHU medium over a wider range of wavelengths. By contrast,
Debye media depart the most from hyperuniformity. In the small-wavelength
regime ($k/s\gg 10$),
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}$ of SHU and
equilibrium media commonly show sharp oscillations because these two media
consist of particles of identical size. By contrast, for the same wavelength
regime, Debye and NSHU media do not exhibit any oscillations because they
consist of phase domains with a broad range of sizes. For the other two
symmetries ($d=2,3$), the behaviors of
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}$ are qualitatively
similar to those observed in layered media, as seen in Fig. 6(b) and (c).

##  4 Strong-Contrast Formulas for the Effective Dynamic Dielectric Constant

In what follows, we present the key formulas for the effective dynamic
dielectric constant tensor of layered and transversely isotropic media, which
were extracted from the general strong-contrast expansion in Refs. [50, 51,
90]. We present relevant formulas at the three-point level to prove that SHU
layered and transversely isotropic media are perfectly transparent within
finite wavenumber intervals through the third-order terms in section 5. At the
two-point level, these formulas involve
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}$, which are
applied in section 6 to estimate the effective attenuation behaviors of the
various models due to their simplicity and great accuracy. We also employ the
corresponding formula for 3D isotropic media at the two-point level, which was
previously derived in Ref. [50], and apply it in the present work. The reader
is referred to A for the problem setup and assumptions from which the formulas
below are obtained.

###  4.1 Multiple-Scattering Approximations for Layered Media

We consider our multiple-scattering approximations for the effective dynamic
dielectric constant tensor
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{1}\right)}}$ of
layered media [Fig. 5(a)] when waves are normally incident, i.e., ${\bf
k}_{1}=k_{1}\hat{{\bf z}}$, where $\hat{{\bf z}}$ is a unit vector along the
$z$-direction, which is the symmetry axis of the layered media. (The reader is
referred to Supplement1 of Ref. [90] for the formulas of obliquely incident
waves.) Thus, these formulas depend on the wavenumber $k_{1}$. Due to the
symmetries of layered media, one can decompose
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left(k_{1}\right)}}$ into two
orthogonal components
$\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}$ and
$\mathinner{\varepsilon_{e}^{z}\mathopen{\left(k_{1}\right)}}$ for the
transverse and longitudinal polarizations, respectively, as follows:
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left(k_{1}\right)}}=\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}\left(\bm{I}-\hat{{\bf
z}}\hat{{\bf
z}}\right)+\mathinner{\varepsilon_{e}^{z}\mathopen{\left(k_{1}\right)}}\hat{{\bf
z}}\hat{{\bf z}}$.

Here, we focus on the formulas for
$\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}$. (The
reader is referred to A.1 for the formula for the longitudinal polarization.)
We first present the formula at the three-point level, which is used to prove
that SHU layered media are perfectly transparent at the three-point level; see
section 5. The _scaled_ strong-contrast approximation for
$\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}$ at the
three-point level is given as

$\displaystyle\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}=$
$\displaystyle\varepsilon_{1}\left[1+\frac{{\phi_{1}}^{2}\varepsilon_{2}/\varepsilon_{1}\beta^{\mathrm{(1D)}}_{21}}{\phi_{2}-(\varepsilon_{2}\beta^{\mathrm{(1D)}}_{21})\mathinner{A_{2}^{\perp}\mathopen{\left(k_{*};\left\langle\varepsilon\right\rangle\right)}}-(\varepsilon_{2}\beta^{\mathrm{(1D)}}_{21})^{2}\mathinner{A_{3}^{\perp}\mathopen{\left(k_{*};\left\langle\varepsilon\right\rangle\right)}}}\right],$
(32)

where $k_{*}\equiv
k_{1}\sqrt{\left\langle\varepsilon\right\rangle/\varepsilon_{1}}$,
$\beta^{\mathrm{(1D)}}_{pq}\equiv 1-\varepsilon_{q}/\varepsilon_{p}$ is the 1D
counterpart of the _dielectric polarizability_ , and the second- and third-
order terms are defined, respectively, as

$\displaystyle\mathinner{A_{2}^{\perp}\mathopen{\left(k;\varepsilon\right)}}\equiv$
$\displaystyle\frac{1}{\varepsilon}\left\\{\frac{{k}^{2}}{\pi}\mathrm{p.v.}\int_{0}^{\infty}\mathinner{\mathrm{d}q}\frac{\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(q\right)}}}{{q}^{2}-{(2k)}^{2}}+\frac{ik}{4}[\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(0\right)}}+\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(2k\right)}}]\right\\},$
(33)
$\displaystyle\mathinner{A_{3}^{\perp}\mathopen{\left(k;\varepsilon\right)}}\equiv$
$\displaystyle\frac{-1}{\phi_{2}}\left(\frac{{k}^{2}}{2\pi\varepsilon}\right)^{2}\int_{-\infty}^{\infty}\mathinner{\mathrm{d}q_{1}}\frac{1}{(k+q_{1})^{2}-{k}^{2}}\int_{-\infty}^{\infty}\mathinner{\mathrm{d}q_{2}}\frac{1}{(k+q_{2})^{2}-{k}^{2}}\mathinner{\tilde{\Delta}_{3}^{(p)}\mathopen{\left(q_{1},q_{2}\right)}},$
(34)

where $\mathrm{p.v.}$ stands for the Cauchy principal value, and
$\mathinner{\tilde{\Delta}_{3}^{(2)}\mathopen{\left(q_{1},q_{2}\right)}}$ is
the Fourier transform of
$\mathinner{\Delta_{3}^{(2)}\mathopen{\left(z_{21},z_{31}\right)}}\equiv\mathinner{S_{2}^{(2)}\mathopen{\left(z_{21}\right)}}\mathinner{S_{2}^{(2)}\mathopen{\left(z_{32}\right)}}-\phi_{2}\mathinner{S_{3}^{(2)}\mathopen{\left(z_{21},z_{32}+z_{21}\right)}}$.

Note that $\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}$
given in (32) is complex-valued, implying that the media can be lossy due to
forward scattering and backscattering from fluctuations in the local
dielectric constant. The static limit of Eq. (32) is the arithmetic mean of
the local dielectric constants, i.e.,
$\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(0\right)}}=\left\langle\varepsilon\right\rangle\equiv\phi_{1}\varepsilon_{1}+\phi_{2}\varepsilon_{2}$,
which is exact for any 1D microstructure [70].

In the long-wavelength regime ($k_{1}/s\ll 1$), assuming a power-law scaling
of $\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}$ for small $k$
as specified by Eq. (24), the imaginary part of Eq. (32) has the following
asymptotic behaviors:

$\displaystyle\mathrm{Im}[\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}]\sim(\varepsilon_{2}-\varepsilon_{1})^{2}\mathrm{Im}[\mathinner{A_{2}^{\perp}\mathopen{\left(k_{*};\left\langle\varepsilon\right\rangle\right)}}]\sim\cases{k_{1},\quad\alpha=0~{}~{}\mbox{(typical
nonhyperuniform)}\\\
{k_{1}}^{1+\alpha},~{}\alpha>0~{}~{}\mbox{(hyperuniform)}}.$ (35)

It is seen that hyperuniform media attenuate less than nonhyperuniform ones in
the long-wavelength regime. Furthermore, hyperuniform systems can exhibit a
wide range of behaviors by tuning the exponent $\alpha$. The attenuation
behavior of the SHU medium is elaborated in section 5.

###  4.2 Multiple-Scattering Approximations for Transversely Isotropic Media

We obtain our multiple-scattering approximations for the effective dynamic
dielectric constant tensor
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{1}\right)}}$ of
transversely isotropic media [see Fig. 5(b)] for the situation in which waves
are normally incident to the symmetry axis of the media, i.e., ${\bf
k}_{1}=k_{1}\hat{{\bf y}}$, where $\hat{{\bf y}}$ is a unit vector along the
$y$-direction. (The reader is referred to Supplement1 of Ref. [90] for the
formulas obliquely incident waves.) These formulas depend on the wavenumber
$k_{1}$ in the reference phase $q~{}(=1)$. Due to the symmetries of the
problems, one can decompose
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{1}\right)}}$ into two
orthogonal components
$\mathinner{\varepsilon_{e}^{TM}\mathopen{\left(k_{1}\right)}}$ and
$\mathinner{\varepsilon_{e}^{TE}\mathopen{\left(k_{1}\right)}}$ for transverse
magnetic (TM) and TE polarizations 444The electric field for TM (TE)
polarization is parallel (normal) to the symmetry axis., respectively, as
follows:
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left(k_{1}\right)}}=\mathinner{\varepsilon_{e}^{TM}\mathopen{\left(k_{1}\right)}}\hat{{\bf
z}}\hat{{\bf
z}}+\mathinner{\varepsilon_{e}^{TE}\mathopen{\left(k_{1}\right)}}(\bm{I}-\hat{{\bf
z}}\hat{{\bf z}})$.

In this paper, we also consider
$\mathinner{\varepsilon_{e}^{TM}\mathopen{\left(k_{1}\right)}}$. (Formulas for
TE polarization are given in A.2.) We begin by presenting the three-point
formula, which is used to prove that SHU transversely isotropic media are also
perfectly transparent at the three-point level; see section 5. The _scaled_
strong-contrast approximation for
$\mathinner{\varepsilon_{e}^{TM}\mathopen{\left(k_{1}\right)}}$ at the three-
point level is given as

$\displaystyle\frac{\mathinner{\varepsilon_{e}^{TM}\mathopen{\left(k_{1}\right)}}}{\varepsilon_{1}}=$
$\displaystyle
1+\frac{{\phi_{2}}^{2}\left[(\varepsilon_{2}+\varepsilon_{1})\beta^{\mathrm{(2D)}}_{21}\right]}{\phi_{2}-\mathinner{A_{2}^{TM}\mathopen{\left(k_{*}^{TM},\left\langle\varepsilon\right\rangle\right)}}\left[(\varepsilon_{2}+\varepsilon_{1})\beta^{\mathrm{(2D)}}_{21}\right]-\mathinner{A_{3}^{TM}\mathopen{\left(k_{*}^{TM},\left\langle\varepsilon\right\rangle\right)}}\left[(\varepsilon_{2}+\varepsilon_{1})\beta^{\mathrm{(2D)}}_{21}\right]^{2}},$
(36)

where
$\beta^{\mathrm{(2D)}}_{pq}\equiv(\varepsilon_{p}-\varepsilon_{q})/(\varepsilon_{p}+\varepsilon_{q})$
is the 2D counterpart of the dielectric polarizability, and $k_{*}^{TM}\equiv
k_{1}\sqrt{\left\langle\varepsilon\right\rangle/\varepsilon_{1}}$ is
wavenumber in the optimal reference phase for TM polarization. The second- and
third-order coefficients are defined as

$\displaystyle\mathinner{A_{2}^{TM}\mathopen{\left(k;\varepsilon\right)}}=\frac{1}{\varepsilon}\left\\{\frac{{k}^{2}}{\pi^{2}}\int_{0}^{\pi/2}\mathinner{\mathrm{d}\phi}\left[\mathrm{p.v.}\int_{0}^{\infty}\mathinner{\mathrm{d}q}\frac{2q\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(q\right)}}}{q^{2}-(2k\cos\phi)^{2}}+i\pi\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(2k\cos\phi\right)}}\right]\right\\},$
(37)
$\displaystyle\mathinner{A_{3}^{TM}\mathopen{\left(k;\varepsilon\right)}}=\frac{-1}{\phi_{2}}\left[\frac{{k}^{2}}{\varepsilon(2\pi)^{2}}\right]^{2}\int_{\mathbb{R}^{2}}\mathinner{\mathrm{d}{\bf
q}_{1}}\int_{\mathbb{R}^{2}}\mathinner{\mathrm{d}{\bf
q}_{2}}\frac{\mathinner{\tilde{\Delta}_{3}^{(2)}\mathopen{\left({\bf
q}_{1},{\bf q}_{2}\right)}}}{[\left|{\bf q}_{1}+k\hat{{\bf
y}}\right|^{2}-{k}^{2}][\left|{\bf q}_{2}+k\hat{{\bf
y}}\right|^{2}-{k}^{2}]},$ (38)

where $\mathinner{\tilde{\Delta}_{3}^{(2)}\mathopen{\left({\bf q}_{1},{\bf
q}_{2}\right)}}$ is the Fourier transform of
$\mathinner{\Delta_{3}^{(2)}\mathopen{\left({\bf x}_{21},{\bf
x}_{31}\right)}}\equiv\mathinner{S_{2}^{(2)}\mathopen{\left({\bf
x}_{21}\right)}}\mathinner{S_{2}^{(2)}\mathopen{\left({\bf
x}_{32}\right)}}-\phi_{p}\mathinner{S_{3}^{(2)}\mathopen{\left({\bf
x}_{21},{\bf x}_{32}+{\bf x}_{21}\right)}}$. The static limit of Eq. (36) is
the arithmetic mean of the local dielectric constant:

$\mathinner{\varepsilon_{e}^{TM}\mathopen{\left(0\right)}}=\left\langle\varepsilon\right\rangle=\varepsilon_{1}\phi_{1}+\varepsilon_{2}\phi_{2}.$
(39)

In the long-wavelength regime ($k_{1}/s\ll 1$), assuming a power-law scaling
of $\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}$ for small $k$
as specified by Eq. (24), the imaginary part of Eq. (36) has the following
asymptotic behaviors:

$\mathrm{Im}[\mathinner{\varepsilon_{e}^{TM}\mathopen{\left(k_{1}\right)}}]\sim(\varepsilon_{2}-\varepsilon_{1})^{2}\mathrm{Im}[\mathinner{A_{2}^{TM}\mathopen{\left(k_{*}^{TM};\left\langle\varepsilon\right\rangle\right)}}]\sim\cases{{k_{1}}^{2},\quad\alpha=0~{}\mbox{(typical
nonhyperuniform)}\\\ {k_{1}}^{2+\alpha},\alpha>0~{}\mbox{(hyperuniform)}},$
(40)

which are identical to those for TE polarization [50]. Thus, in the long-
wavelength regime, hyperuniform media attenuate less than nonhyperuniform ones
and can exhibit a wide range of behaviors by tuning the exponent $\alpha$. The
attenuation behavior of the SHU medium is elaborated in section 5.

##  5 Perfect Transparency Intervals of Stealthy Hyperuniform Media

We have previously shown that second-order strong-contrast formulas for SHU
two-phase composites, i.e., those in which the spectral density obeys the
condition (6), predict perfect transparency or, equivalently, a zero imaginary
part of the effective dielectric constant, in a finite range of wavenumbers
that is determined by the exclusion-region radius $K$. This was done
specifically for TM [90] and TE [50] polarization in transversely isotropic
media and transverse polarization in fully isotropic media [50] and layered
media [51]. Across space dimensions, the predictions of such perfect
transparency intervals have a similar form 555For layered media, the perfect
transparency interval (41) at the two-point level was derived from the
condition that the second-order formula (32) has
$\mathrm{Im}[\mathinner{A_{2}^{\perp}\mathopen{\left(k_{*};\left\langle\varepsilon\right\rangle\right)}}]=0$.
The corresponding intervals for TM and TE polarizations in transversely
isotropic media were derived by finding the intervals in which
$\mathinner{A_{2}^{TM}\mathopen{\left(k_{*}^{TM};\left\langle\varepsilon\right\rangle\right)}}$
[Eq. (36)] and
$\mathinner{A_{2}^{TE}\mathopen{\left(k_{*}^{TE};\varepsilon_{BG}^{(2D)}\right)}}$
[Eq. (50)] have zero imaginary parts, respectively.:

$0\leq k_{1}<K_{T}\equiv\frac{K}{2\sqrt{\varepsilon_{*}/\varepsilon_{1}}},$
(41)

where the dielectric constant of the optimal reference phase $\varepsilon_{*}$
is given as

$\displaystyle\varepsilon_{*}=$
$\displaystyle\left\langle\varepsilon\right\rangle,\qquad~{}~{}\mbox{Transverse
polarization in layered media},$ (42) $\displaystyle\varepsilon_{*}=$
$\displaystyle\left\langle\varepsilon\right\rangle,\qquad~{}~{}\mbox{TM
polarization in transversely isotropic media },$ (43)
$\displaystyle\varepsilon_{*}=$
$\displaystyle\varepsilon_{BG}^{(2D)},\qquad\mbox{TE polarization in
transversely isotropic media},$ (44) $\displaystyle\varepsilon_{*}=$
$\displaystyle\varepsilon_{BG}^{(3D)},\qquad\mbox{Transverse polarization in
fully isotropic media},$ (45)

and $\varepsilon_{BG}^{(dD)}$ is the Bruggeman approximation [109, 70] of the
effective static dielectric constant of $d$-dimensional two-phase composites
666For a contrast ratio $\varepsilon_{2}/\varepsilon_{1}\leq 4$, the Bruggeman
formulas are very close to the Maxwell-Garnett formulas employed in Ref. [50].
The former offers a slightly better renormalization, since it uses the fact
that the mean of the depolarization tensors is exactly zero; see Ref. [90]. .
Within the finite perfect transparency interval (41), the rapid convergence of
strong-contrast expansions and high predictive power of their second-order
truncations, as validated in Refs. [50, 51, 90], imply that third- and higher-
order contributions to the imaginary part
$\mathrm{Im}[\mathinner{\varepsilon_{e}\mathopen{\left(k_{1}\right)}}]$ are
negligibly small for relatively large contrast ratios
($\varepsilon_{2}/\varepsilon_{1}\lesssim 10$).

In the present work, we now also prove that the interval (41) for layered
media and transversely isotropic media (with TM polarization) applies not only
at the two-point but the three-point level as well. We begin by proving that
the perfect transparency interval specified by Eq. (41) in conjunction with
Eq. (42) for layered media is exact through the third-order terms. Accounting
for the perfect transparency at the two-point level, from the third-order
formula (32), we obtain
$\mathrm{Im}[\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}]\propto(\varepsilon_{2}-\varepsilon_{1})^{3}\mathrm{Im}[\mathinner{A_{3}^{\perp}\mathopen{\left(k_{*},\left\langle\varepsilon\right\rangle\right)}}]$
for $0\leq k_{*}<K/2$. Thus, it is sufficient to show that

$\mathrm{Im}[\mathinner{A_{3}^{\perp}\mathopen{\left(k_{*},\left\langle\varepsilon\right\rangle\right)}}]=0,\quad\mbox{for
}k_{*}<K/2.$ (46)

The reader is referred to B.1 for a detailed proof of Eq. (46).

We now show that the perfect transparency interval for TM polarization in
transversely isotropic media is specified by Eq. (41) together with Eq. (43)
through the third-order terms. Due to the perfect transparency at the two-
point level, the third-order formula of Eq. (36) yields
$\mathrm{Im}[\mathinner{\varepsilon_{e}^{TM}\mathopen{\left(k_{1}\right)}}]\propto(\varepsilon_{2}-\varepsilon_{1})^{3}\mathrm{Im}[\mathinner{A_{3}^{TM}\mathopen{\left(k_{*}^{TM};\left\langle\varepsilon\right\rangle\right)}}]$
for $0\leq k_{*}^{TM}<K/2$. Therefore, it is sufficient to show that

$\displaystyle\mathrm{Im}[\mathinner{A_{3}^{TM}\mathopen{\left(k_{*}^{TM},\left\langle\varepsilon\right\rangle\right)}}]=0,\quad\mbox{for
}k_{*}^{TM}<K/2.$ (47)

The detailed proof of (47) is presented in B.2.

The fact that the above results show perfect transparency through third-order
terms in the strong-contrast expansion implies that the localization length
for such stealthy hyperuniform media should be very large compared to any
practically large sample size, and thus, there can be no Anderson localization
within the predicted perfect transparency intervals in stealthy hyperuniform
layered and transversely isotropic media in practice, as noted in Refs. [90].
This prediction for layered media is especially remarkable because the
traditional understanding is that localization must occur for any type of
disorder in 1D systems [93, 94, 95, 96, 110].

Figure 7:  Prediction (41) of the dimensionless sizes $K_{T}/K$ of the perfect
transparency intervals for stealthy hyperuniform media in the four cases:
transverse polarization in layered media, TM and TE polarizations in
transversely isotropic media, and transverse polarization in fully isotropic
media. (a) The sizes as functions of the contrast ratios
$\varepsilon_{2}/\varepsilon_{1}$ with the volume fraction $\phi_{2}=0.20$ of
the higher dielectric constant. Here, $\delta^{+}$ denotes an infinitesimally
small positive number. (b) The sizes as functions of $\phi_{2}$ with
$\varepsilon_{2}/\varepsilon_{1}=4.0$.

It is noteworthy that the dimensionless size of the perfect transparency
interval, $K_{T}/K$, defined by Eq. (41), tends to decrease as the wave
propagation is disturbed further for the four cases considered in Eqs.
(42)-(45); see Fig. 7. Specifically, Fig. 7(a) shows that the dimensionless
interval size $K_{T}/K$ decreases with the contrast ratio
$\varepsilon_{2}/\varepsilon_{1}$, since a higher contrast ratio strengthens
scattering. Similarly, in Fig. 7(b), we see that $K_{T}/K$ decreases with the
volume fraction $\phi_{2}$ at a fixed contrast ratio
$\varepsilon_{2}/\varepsilon_{1}=4$, since the frequency of scattering events
is proportional to $\phi_{2}$. Figure 7 also reveals the relative ranking of
the four cases for $K_{T}/K$: $K_{T}/K$ is the lowest for TM polarization in
transversely isotropic media (and layered media), followed by fully isotropic
ones, and is the highest for TE polarization in transversely isotropic media.
This ranking is again due to the strength of reflectance (i.e., scattering) on
an interface between two dielectric materials. Specifically, the Fresnel
equations [111] dictate that among $s$/$p$-polarized lights, in which the
electric field is normal/parallel to the incident plane, the $s$-polarized one
is always reflected more than the $p$-polarized one. Thus, for transversely
isotropic media, the dimensionless size of $K_{T}$ for TM is lower than TE
because the TM/TE wave is always $s$/$p$-polarized. For layered media, the
ratio $K_{T}/K$ is identical to $K_{T}/K$ for the TM polarization because the
transverse waves are always $s$-polarized. Similarly, the dimensionless
interval size $K_{T}/K$ for isotropic media is in between those for TE and TM
polarizations, since the reflected waves in a 3D isotropic media have both $s$
and $p$ polarizations due to interfaces that are isotropically oriented with
respect to the incident radiation.

##  6 Results for Attenuation and Spreadability Characteristics and Their
Correlations

###  6.1 Attenuation Characteristics

Figure 8:  Predictions of the scaled strong-contrast approximation [Eq. (32)]
of the imaginary part of the effective dynamic dielectric constant
$\mathrm{Im}[\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}]$
as a function of the dimensionless incident wavenumber $k_{1}/s$ for
disordered layered media with $\phi_{2}=0.20$ and
$\varepsilon_{2}/\varepsilon_{1}=4.0$, where $s$ is the specific surface. We
consider the four models: Debye, Equil., NSHU, and SHU media. The plot is
shown on a log-log scale.

Here, we compare the extraordinary effective attenuation characteristics of
SHU media to those of their nonstealthy counterparts. For this purpose, we
compute the imaginary part of the effective dielectric constant
$\mathrm{Im}[\mathinner{\varepsilon_{e}\mathopen{\left(k_{1}\right)}}]$ (which
is related to the imaginary part of the effective wavenumber $k_{e}\equiv
k_{1}\sqrt{\varepsilon_{e}/\varepsilon_{1}}$), as predicted from the scaled
second-order strong-contrast approximations. We begin by showing in Fig. 8,
the imaginary part
$\mathrm{Im}[\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}]$
for the four models of layered media ($d=1$) (see Sec. 3) with $\phi_{2}=0.20$
and $\varepsilon_{2}/\varepsilon_{1}=4.0$. As noted in the previous section,
the stealthy hyperuniform medium exhibits perfect transparency (i.e.,
$\mathrm{Im}[\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}]=0$)
in the finite interval specified by Eq. (41), whereas the three nonstealthy
media, whether hyperuniform or not, always exhibit nonzero attenuation with
power-law scalings [i.e.,
$\mathrm{Im}[\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}]\sim{k_{1}}^{1+\alpha}$]
for small wavenumbers ($k_{1}/s<0.3$). Furthermore, for $k_{1}/s<0.3$, the
relative ranking of their degree of attenuation is identical to their
corresponding rankings of the large-scale volume-fraction fluctuations, which
are proportional to the scattering intensity [i.e.,
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}$ for small $k$],
because the attenuation comes solely from scattering due to dielectric
inhomogeneity (not absorption). Thus, the Debye medium exhibits the strongest
attenuation, followed by the equilibrium and then the NSHU media, down to the
SHU one, in which the attenuation is zero.

For transversely isotropic and fully 3D isotropic media, we study the
imaginary parts of the effective dielectric constant for two of the
aforementioned models that exhibit the greatest and least attenuation, namely,
Debye and SHU media (see Sec. 3), respectively, with $\phi_{2}=0.20$ and
$\varepsilon_{2}/\varepsilon_{1}=4$. We do not report corresponding results
for the equilibrium and the NSHN media for $d=2$ and $d=3$, since
qualitatively they are similar to the layered media models. The predictions
shown in Fig. 9(a) are obtained from the second-order formula of Eq. (36) for
TM polarization in transversely isotropic media ($d=2$), whereas those in Fig.
9(b) are obtained from the scaled approximation derived in Ref. [50]; see Eq.
(73) for $d=3$ therein. For these two symmetries, the perfect transparency
interval (41) of the SHU medium is singularly different from the
nonhyperuniform Debye medium with a substantial degree of attenuation, i.e.,
with a power-law scaling
$\mathrm{Im}[\mathinner{\varepsilon_{e}\mathopen{\left(k_{1}\right)}}\sim{k_{1}}^{d}]$
for small wavenumbers ($k_{1}/s<0.3$). As for layered-media cases for
$k_{1}/s<0.3$, the relative ranking of the attenuation behaviors of these two
models is consistent with the corresponding rankings of their large-scale
volume-fraction fluctuations.

(a)  (b)

Figure 9:  Predictions of the scaled strong-contrast approximation of the
imaginary part of the effective dynamic dielectric constant as a function of
the dimensionless incident wavenumber $k_{1}/s$ for disordered two-phase media
in three dimensions with $\phi_{2}=0.20$ and
$\varepsilon_{2}/\varepsilon_{1}=4.0$ with two distinct symmetries: (a) TM
polarization in transversely isotropic media ($d=2$) and (b) transverse
polarization in fully isotropic media ($d=3$), where $s$ is the specific
surface. For each symmetry, we consider Debye and SHU media. The plots shown
in (a) are obtained from Eq. (36), whereas the plots shown in (b) are obtained
from the scaled approximation for $d=3$ in Ref. [50]. Both plots are shown on
log-log scales.

###  6.2 Spreadability Characteristics

Figure 10:  Comparison of the excess spreadabilities for disordered layered media ($d=1$) with $\phi_{2}=0.20$ as functions of the dimensionless time ${\cal D}ts^{2}$, where $s$ is the specific surface. We consider four models: Debye, Equil., NSHU, and SHU media. The long-time behaviors of $\mathinner{\mathcal{S}\mathopen{\left(\infty\right)}}-\mathinner{\mathcal{S}\mathopen{\left(t\right)}}$ for each of these models is indicated.  Table 1:  Values of the coefficients $B$ and $C$ for all models considered here in three distinct structural symmetries with $\phi_{2}=0.20$. We consider four models for layered media: Debye, Equil., NSHU, and SHU media. For the other two symmetries, we focus on Debye and SHU media. Here, the exponent $\alpha$ and $B$ are specified by Eq. (24) in units of the specific surface $s$, and $C$ is specified by Eq. (26). For SHU, we denote $\alpha=+\infty$ because its spectral density is smaller than any power-law scaling.  Symmetry | Model | $B$ | $C$
---|---|---|---
Layered ($d=1$) | Debye ($\alpha=0$) | 1.01$\times 10^{-1}$ | 1.44$\times 10^{-1}$
Equil. ($\alpha=0$) | 5.12$\times 10^{-2}$ | 7.22$\times 10^{-2}$
NSHU ($\alpha=4$) | 5.13$\times 10^{-2}$ | 4.37$\times 10^{-2}$
SHU ($\alpha=+\infty$) | 0.0 | 0.0
Transversely Isotropic ($d=2$) | Debye ($\alpha=0$) | 4.94$\times 10^{-1}$ | 2.01$\times 10^{-1}$
SHU ($\alpha=+\infty$) | 0.0 | 0.0
Isotropic ($d=3$) | Debye ($\alpha=0$) | 2.50 | 2.86$\times 10^{-1}$
SHU ($\alpha=+\infty$) | 0.0 | 0.0

We begin by considering the four models of layered media ($d=1$) with
$\phi_{2}=0.2$ to demonstrate the extraordinary property of diffusion
spreadability for the SHU medium; see Fig. 10. At the intermediate times
($0.01<{\cal D}ts^{2}<1$), we find that the excess spreadability for the Debye
medium shows the slowest decay, followed by the equilibrium one, and those for
two hyperuniform media commonly shows the fastest decay, as predicted in Ref.
[89]. Consistent with previous theoretical results [89], we also find that the
excess spreadability for nonhyperuniform Debye medium has the slowest long-
time decay to its infinite-time behavior among the four models, followed by
the equilibrium media with the same decay rate, i.e., $1/t^{1/2}$. The NSHU
medium has the second fastest decay rate ($1/t^{5/2}$). Of course, the SHU
medium is distinguished among all of the nonstealthy models, since its excess
spreadability decays exponentially fast, as predicted by the asymptotic
formula (27). More specifically, referring to the asymptotic formula (25), the
two nonhyperuniform layered media exhibit a common power-law decay with a
coefficient $C$, specified by Eq. (26). Table 1 summarizes the values of the
coefficients $B$, defined in Eq. (24), and $C$ for all models of layered media
considered here, along with their values of the exponent $\alpha$. We take SHU
medium to be one with $\alpha=+\infty$, roughly speaking, because its decay
rate of $\mathcal{S}(t)$ is faster than any inverse power law. The long-time
decay rate of the excess spreadability
$\mathinner{\mathcal{S}\mathopen{\left(\infty\right)}}-\mathinner{\mathcal{S}\mathopen{\left(t\right)}}$
is faster for media with smaller large-scale volume-fraction fluctuations
(i.e., small values of
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}$ for small $k$
around the origin) [112], i.e., it takes less time to reach the state of
uniform concentration of the solute in the infinite-time limit in media when
such fluctuations are smaller; see Fig. 3.

For both transversely isotropic ($d=2$) and fully isotropic ($d=3$) media, we
study the two models with $\phi_{2}=0.20$ whose
$\mathinner{\mathcal{S}\mathopen{\left(\infty\right)}}-\mathinner{\mathcal{S}\mathopen{\left(t\right)}}$
at long times decay slowest and fastest, i.e., Debye media and SHU ones,
respectively; see Fig. 11. Even at the intermediate times,
$\mathinner{\mathcal{S}\mathopen{\left(\infty\right)}}-\mathinner{\mathcal{S}\mathopen{\left(t\right)}}$
for SHU media decays much faster than those for the Debye ones. While the
long-time
$\mathinner{\mathcal{S}\mathopen{\left(\infty\right)}}-\mathinner{\mathcal{S}\mathopen{\left(t\right)}}$
for Debye media decays like $1/t^{d/2}$, as predicted by Eq. (25), those for
the SHU media decay exponentially fast, as predicted by Eq. (27). The
corresponding coefficients $B$ and $C$ are also listed in Table 1. For the
same reasons given for layered media, the relative rankings of these two
models for the asymptotic decay rate of the long-time behavior of
$\mathinner{\mathcal{S}\mathopen{\left(\infty\right)}}-\mathinner{\mathcal{S}\mathopen{\left(t\right)}}$
is also identical to that of their large-scale volume-fraction fluctuations
[112].

(a)  (b)

Figure 11:  Comparison of the excess spreadabilities as functions of the
dimensionless time ${\cal D}ts^{2}$ for (a) disordered transversely isotropic
media ($d=2$) and (b) disordered fully isotropic media ($d=3$) with
$\phi_{2}=0.20$, where $s$ is the specific surface. We consider two models for
each symmetry: Debye and SHU media. The long-time behavior of
$\mathinner{\mathcal{S}\mathopen{\left(\infty\right)}}-\mathinner{\mathcal{S}\mathopen{\left(t\right)}}$
for each of these models is indicated.

###  6.3 Cross-Property Relations

Figure 12:  Cross-property relation between the imaginary part of the effective dielectric constant $\mathrm{Im}[\varepsilon_{e}]$ at a small dimensionless wavenumber $k_{1}/s=0.1$ within the transparency interval and the excess spreadability $\mathinner{\mathcal{S}\mathopen{\left(\infty\right)}}-\mathinner{\mathcal{S}\mathopen{\left(t\right)}}$ at a large dimensionless time ${\cal D}ts^{2}=20$ for the four models of layered media with $\phi_{2}=0.20$. We consider Debye, Equil., NSHU, and SHU media. The data point for the SHU medium is shown in the gray shaded region, since it has a zero imaginary part, which cannot be shown in the log scale. The Debye medium has the largest attenuation and spreadability, while the SHU has zero attenuation and the fastest (exponentially fast) spreadability decay rate.  Table 2:  Values of the imaginary part of the effective dielectric constant $\mathrm{Im}[\varepsilon_{e}]$ at a small dimensionless wavenumber $k_{1}/s=0.1$ within the transparency interval and the excess spreadability $\mathinner{\mathcal{S}\mathopen{\left(\infty\right)}}-\mathinner{\mathcal{S}\mathopen{\left(t\right)}}$ at a large dimensionless time ${\cal D}ts^{2}=20$ for all models considered here for three distinct structural symmetries with $\phi_{2}=0.20$. For layered media ($d=1$), we consider Debye, Equil., NSHU, and SHU media. For other symmetries, we consider Debye and SHU media. While we arbitrarily choose the values for $k_{1}/s$ and ${\cal D}ts^{2}$, the relative rankings for the models are independent of the specific values if $k_{1}/s<0.3$ and ${\cal D}ts^{2}>10$.  Symmetry | Model | $\mathrm{Im}[\varepsilon_{e}(k_{1})]$ | $\mathcal{S}(\infty)-\mathcal{S}(t)$
---|---|---|---
Layered ($d=1$) | Debye | 3.60$\times 10^{-2}$ | 3.22$\times 10^{-2}$
Equil. | 1.81$\times 10^{-2}$ | 1.61$\times 10^{-2}$
NSHU | 2.33$\times 10^{-5}$ | 1.41$\times 10^{-5}$
SHU | 0.0 | 1.97$\times 10^{-11}$
Transversely Isotropic ($d=2$) | Debye | 5.77$\times 10^{-3}$ | 9.87$\times 10^{-3}$
SHU | 0.0 | 7.74$\times 10^{-22}$
Fully Isotropic ($d=3$) | Debye | 1.58$\times 10^{-4}$ | 3.05$\times 10^{-3}$
SHU | 0.0 | 2.75$\times 10^{-23}$

We now show that the attenuation behaviors of electromagnetic waves for
sufficiently small wavenumbers and the long-time decaying behaviors of the
excess spreadabilities are positively correlated with one another, i.e., we
demonstrate that there are cross-property relations between them. We begin by
considering our four models of layered media (i.e., $d=1$): Debye,
equilibrium, NSHU, and SHU media. Figure 12 depicts the values of the
imaginary part of the effective dielectric constant
$\mathrm{Im}[\varepsilon_{e}(k_{1})]$ at a dimensionless wavenumber
$k_{1}/s=0.1$ and the excess spreadability
$\mathinner{\mathcal{S}\mathopen{\left(\infty\right)}}-\mathinner{\mathcal{S}\mathopen{\left(t\right)}}$
at a large dimensionless time ${\cal D}ts^{2}=20$. This plot clearly
demonstrates that as the degree of attenuation increases from zero attenuation
and the associated smallest long-time excess spreadability (fastest decay) for
the SHU media, long-time excess spreadability increases (slower decay rate)
until the largest attenuation and long-time excess spreadability (smallest
decay rate) is achieved for the Debye media. We see that the NSHU medium has a
substantially smaller attenuation and excess spreadability than that of the
equilibrium media. Table 2 lists the values of the imaginary part of the
effective dielectric constant
$\mathrm{Im}[\mathinner{\varepsilon_{e}\mathopen{\left(k_{1}\right)}}]$ at a
dimensionless wavenumber $k_{1}/s=0.1$ and the excess spreadability
$\mathinner{\mathcal{S}\mathopen{\left(\infty\right)}}-\mathinner{\mathcal{S}\mathopen{\left(t\right)}}$
at a large dimensionless time ${\cal D}ts^{2}=20$ corresponding to the plotted
values in Fig. 12. While our choices for the values of the small dimensionless
wavenumber $k_{1}/s$ and the dimensionless time ${\cal D}ts^{2}$ are
arbitrary, the relative rankings of the models are independent of the specific
values if $k_{1}/s<0.3$ and $10<{\cal D}ts^{2}$.

Table 2 lists corresponding values the imaginary part of the effective
dielectric constant
$\mathrm{Im}[\mathinner{\varepsilon_{e}\mathopen{\left(k_{1}\right)}}]$ at a
dimensionless wavenumber $k_{1}/s=0.1$ and the excess spreadability
$\mathinner{\mathcal{S}\mathopen{\left(\infty\right)}}-\mathinner{\mathcal{S}\mathopen{\left(t\right)}}$
for Debye and SHU media in both the transversely isotropic ($d=2$) and fully
isotropic case ($d=3$). The analogous positive correlations between the
attenuation behaviors and the long-time spreadability behaviors seen in for
$d=1$ are observed in these instances, even if we do not report corresponding
results of the equilibrium and NSHN media for $d=2$ and $d=3$, since
qualitatively, they are similar to the layered media models. Specifically, the
SHU media achieve both zero attenuation (i.e., perfect transparency) and the
smallest long-time excess spreadability (i.e., exponential decay), while Debye
media exhibit the strongest attenuation and largest long-time excess
spreadability among all four models. Furthermore, for a $d$-dimensional NSHU
model, the power-law scaling of its small-wavenumber attenuation behavior
[i.e.,
$\mathrm{Im}[\mathinner{\varepsilon_{e}\mathopen{\left(k_{1}\right)}}]\sim{k_{1}}^{d+\alpha}$,
as predicted by Eqs. (35) and (40)] determines that of its long-time excess
spreadability [i.e.,
$\mathinner{\mathcal{S}\mathopen{\left(\infty\right)}}-\mathinner{\mathcal{S}\mathopen{\left(t\right)}}\sim(\mathcal{D}ts^{2})^{-(d+\alpha)/2}$,
as predicted by Eq. (25)], where the exponent $\alpha$ is specified by Eq.
(24).

##  7 Discussion

In this paper, we have further elucidated the extraordinary optical and
transport properties of 3D disordered SHU media with three distinct
microstructural symmetries: layered, transversely isotropic, and fully
isotropic cases. We had previously shown that the accurate truncations of the
strong-contrast expansions of the effective dynamic dielectric constant
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{1}\right)}}$[50]
predicted the perfect transparency interval defined by Eq. (41) for SHU
layered [51], transversely isotropic [50, 90] and fully isotropic media [50].
In B, we provided detailed proofs that the same perfect transparency interval
exactly applies through the third-order truncations for such layered media and
transversely isotropic in the case of TM polarization. Given the rapid
convergence of the strong contrast expansion, the previously demonstrated
accuracy of the second-order approximations [50, 51, 90] leads to the
conclusion that there can be no Anderson localization within the predicted
perfect transparency interval in such SHU media in practice because the
localization length (associated with only possibly negligibly small higher-
order contributions) should be very large compared to any practically large
sample size, as noted in Refs. [90]. This prediction for layered media is
especially remarkable because the traditional understanding is that
localization must occur for any type of disorder in 1D media.

Additionally, we further revealed the singular physical properties of 3D SHU
two-phase media with three distinct symmetries (layered, transversely
isotropic media, and fully isotropic media) by comparing and contrasting them
to those of other model nonstealthy microstructures. Specifically, we studied
their attenuation characteristics, as measured by the imaginary part of
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{1},\omega\right)}}$,
and transport properties, as measured by the time-dependent diffusion
spreadability ${\cal S}(t)$. Among the possible hyperuniform states of
composite media, the SHU ones stand out as having singular physical
properties, such as exponentially fast decay of excess spreadability and
finite perfect transparency intervals for all symmetries. We demonstrated that
there are cross-property relations between the attenuation and spreadability
behaviors by quantifying how the imaginary parts of the effective dielectric
constant and the excess spreadability at long times are positively correlated
as the structures span from nonhyperuniform, NSHU, and SHU media. Such cross-
property relations are useful because they enable one to estimate one of the
properties, given a measurement of the other [70].

There are a variety of outstanding open problems for future research. For
example, while our results imply that there can be no Anderson localization
for 1D disordered SHU medium in practice, its rigorous and direct proof in the
thermodynamic limit remains an open problem. To resolve this problem, we will
systematically study the localization length of 1D SHU media as a function of
system size [113]. Another interesting direction is to extend the cross-
property relations between diffusion spreadability and attenuation behaviors
for general cases, such as arbitrary incidence direction or two-phase media
with absorbing phases. It will also be useful to establish cross-property
relations for SHU media for other wave phenomena, such as elastodynamics
characteristics [48, 49], as well as other transport properties. Concerning
the latter possibility, it has recently been shown that the fluid permeability
associated with fluid transport in porous media is bounded from above in terms
of a functional involving the spectral density
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}$[102]. Since the
second-formulas for $\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf
k}_{1}\right)}}$ and the formulas for the spreadability ${\cal S}(t)$ also
depend on the spectral density, it would be relatively straightforward to
determine whether useful cross-property relations exist between these three
different physical properties for nonhyperuniform and hyperuniform media.

##  Appendix A Scaled Strong-Contrast Formulas at Three-Point Levels

Here, we provide detailed descriptions, including the problem setup and
assumptions, of the strong-contrast formulas for the effective dynamic
dielectric constant, omitted in section 4. We consider a plane electric wave
of an angular frequency $\omega$ and a wavevector ${\bf k}_{q}$ in the
reference phase $q$, which is taken as the matrix phase (i.e., $q=1$) of the
composite unless otherwise stated. We make three assumptions about both
phases: (a) they have constant real-valued dielectric constants, (b) they are
dielectrically isotropic, and (c) they are nonmagnetic. Thus, such a
dielectric composite cannot absorb waves but attenuate them solely due to
multiple scattering from fluctuations in the local dielectric constant. Since
these assumptions give a simple relation between the wavenumber $k_{1}$ in the
reference phase $q=1$ and angular frequency $\omega$ [i.e.,
$\mathinner{k_{1}\mathopen{\left(\omega\right)}}\equiv\left|{\bf
k}_{1}(\omega)\right|=\sqrt{\varepsilon_{1}}\omega/c$], where $c$ is the speed
of light in vacuum, we henceforth do not explicitly indicate the $\omega$
dependence, i.e., $\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf
k}_{1},\omega\right)}}=\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf
k}_{1}\right)}}$.

Under these three assumptions, we derived the following general strong-
contrast expansion for the linear fractional form of
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{1}\right)}}$ in Ref.
[50]:

$\displaystyle\phi_{2}\bm{L}_{2}^{(1)}\cdot\left(\left\\{\bm{I}+\bm{D}^{(1)}\cdot\left[\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf
k}_{1}\right)}}-{\varepsilon}_{1}\bm{I}\right]\right\\}\cdot\left[\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf
k}_{1}\right)}}-{\varepsilon}_{1}\bm{I}\right]^{-1}\right)\cdot\phi_{2}\bm{L}_{2}^{(1)}$
$\displaystyle\qquad=\phi_{2}\bm{L}_{2}^{(1)}-\mathinner{\bm{\mathcal{A}}_{2}^{(2)}\mathopen{\left({\bf
k}_{1}\right)}}-\mathinner{\bm{\mathcal{A}}_{3}^{(2)}\mathopen{\left({\bf
k}_{1}\right)}}-\ldots,$ (48)

where $p~{}(=2)$ indicates the polarized phase,
$\mathinner{\bm{\mathcal{A}}_{n}^{(2)}\mathopen{\left({\bf k}_{1}\right)}}$ is
a wavevector-dependent second-rank tensor that is a functional involving the
set of correlation functions $S_{1}^{(2)},S_{2}^{(2)},\ldots,S_{n}^{(2)}$,
defined in Sec. 2.1.1, and products of the principal part of the dyadic
Green’s function, and the two constant second-rank tensors $\bm{L}_{2}^{(1)}$
and $\bm{D}^{(1)}$ are associated with an infinitesimal region around the
singularity in the Green’s function [50, 114]. The linear fractional form of
the strong-contrast expansion (48) results in a rapidly converging series
whose lower-order truncations lead to accurate approximation formulas for
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{1}\right)}}$, even
for large contrast ratios. This is to be contrasted with standard weak-
contrast expansions that do not converge rapidly for large contrast ratios;
see Refs. [50, 90] for such a quantitative explanation. Indeed, its truncation
at the two-point level yields a highly accurate estimation of
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{1}\right)}}$ due to
the fact that higher-order contributions are negligibly small, implying that
this resulting formula very accurately approximates multiple scattering
through all orders [50, 51]. Furthermore, this series takes into account the
singularity of the dyadic Green’s function and thus is valid even for large
values of phase contrast ratio $\varepsilon_{2}/\varepsilon_{1}$ and various
microstructural symmetries. In what follows, we briefly outline the
derivations of the key formulas for the effective dynamic dielectric constant
tensor of layered and transversely isotropic media, which were extracted from
the general strong-contrast expansion (48) in Refs. [50, 51, 90].

###  A.1 Layered Media

Under the condition specified in section 4.1, the scaled strong-contrast
approximations associated with layered media are extracted from exact strong-
contrast series (48) by choosing a disk-like exclusion volume normal to the
symmetry axis $\hat{{\bf z}}$ that involves the singularity of the dyadic
Green’s function. Due to the symmetries of layered media, one can decompose
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left({\bf k}_{1}\right)}}$ into two
orthogonal components
$\mathinner{\varepsilon_{e}^{\perp}\mathopen{\left(k_{1}\right)}}$ and
$\mathinner{\varepsilon_{e}^{z}\mathopen{\left(k_{1}\right)}}$ for the
transverse and longitudinal polarizations, respectively. Thus, we derive two
independent approximations by truncating the strong-contrast expansion and
solving for $\mathinner{\bm{\varepsilon}_{e}\mathopen{\left(k_{1}\right)}}$
from this linear fractional form. Renormalization of the resulting formulas
with the reference phase for the optimal convergence, equivalent to using the
effective Green’s function in Ref. [50], yields scaled strong-contrast
approximations for disordered layered media at the three-point level. The
reader is referred to Refs. [51, 90] for derivations.

Such a formula for the transverse polarization is given in Eq. (32). The
analogous formula for longitudinal polarization is given as

$\displaystyle\mathinner{\varepsilon_{e}^{z}\mathopen{\left(k_{1}\right)}}=$
$\displaystyle\varepsilon_{1}(1-\phi_{2}\beta^{\mathrm{(1D)}}_{21})^{-1}.$
(49)

Note that $\mathinner{\varepsilon_{e}^{z}\mathopen{\left(k_{1}\right)}}$ is
independent of $k_{1}$, since a traveling longitudinal wave cannot exist under
the aforementioned conditions. The static limit of Eq. (49) is the harmonic
mean of the local dielectric constants, i.e.,
$\mathinner{\varepsilon_{e}^{z}\mathopen{\left(0\right)}}=(\phi_{2}/\varepsilon_{2}+\phi_{1}/\varepsilon_{1})^{-1},$
which is exact for any 1D microstructure [70].

###  A.2 Transversely Isotropic Media

Under the condition specified in section 4.2, the scaled strong-contrast
approximations associated with transversely isotropic media are extracted from
exact strong-contrast series (48) by choosing a needle-like exclusion volume
aligned along $\hat{{\bf y}}$ that involves the singularity of the dyadic
Green’s function. Due to the symmetries of the problems, one can decompose
$\mathinner{\bm{\varepsilon}_{e}\mathopen{\left(k_{1}\right)}}$ into two
orthogonal components
$\mathinner{\varepsilon_{e}^{TM}\mathopen{\left(k_{1}\right)}}$ and
$\mathinner{\varepsilon_{e}^{TE}\mathopen{\left(k_{1}\right)}}$ for TM and TE
polarizations, respectively. Thus, we extract two independent approximations
of $\mathinner{\varepsilon_{e}^{TE}\mathopen{\left(k_{1}\right)}}$ and
$\mathinner{\varepsilon_{e}^{TM}\mathopen{\left(k_{1}\right)}}$ by truncating
the strong-contrast series at third-order terms. Solving for the effective
dielectric constants from these linear fractional forms and then renormalizing
them with the optimal reference phase [115, 116, 50], we finally obtain the
scaled strong-contrast approximations at the three-point level for disordered
transversely isotropic media.

Such a formula for the TM polarization is given in Eq. (36). Its TE analog is
given as

$\displaystyle\frac{\mathinner{\varepsilon_{e}^{TE}\mathopen{\left(k_{1}\right)}}}{\varepsilon_{1}}=$
$\displaystyle
1+\frac{2{\phi_{2}}^{2}\beta^{\mathrm{(2D)}}_{21}}{\phi_{2}(1-\phi_{2}\beta^{\mathrm{(2D)}}_{21})-\mathinner{A_{2}^{TE}\mathopen{\left(k_{*}^{TE};\varepsilon_{BG}^{(2D)}\right)}}\left[2\varepsilon_{1}\beta^{\mathrm{(2D)}}_{21}\right]-\mathinner{A_{3}^{TE}\mathopen{\left(k_{*}^{TE};\varepsilon_{BG}^{(2D)}\right)}}\left[2\varepsilon_{1}\beta^{\mathrm{(2D)}}_{21}\right]^{2}},$
(50)

where $k_{*}^{TE}\equiv k_{1}\sqrt{\varepsilon_{BG}^{(2D)}/\varepsilon_{1}}$
is wavenumber in the optimal reference phase for TE polarization, and
$\varepsilon_{BG}^{(2D)}$ is the Bruggeman approximation for 2D two-phase
media [109, 70]. Here, the second-order coefficient is given as
$\mathinner{A_{2}^{TE}\mathopen{\left(k;\varepsilon\right)}}=\mathinner{A_{2}^{TM}\mathopen{\left(k;\varepsilon\right)}}/2$;
see Eq. (37), and the third-order coefficient is defined as

$\displaystyle\mathinner{A_{3}^{TE}\mathopen{\left(k;\varepsilon\right)}}=\frac{-1}{\phi_{2}}\left[\frac{1}{2\varepsilon(2\pi)^{2}}\right]^{2}\int_{\mathbb{R}^{2}}\mathinner{\mathrm{d}{\bf
q}_{1}}\int_{\mathbb{R}^{2}}\mathinner{\mathrm{d}{\bf
q}_{2}}\mathinner{\tilde{\Delta}_{3}^{(2)}\mathopen{\left({\bf q}_{1},{\bf
q}_{2}\right)}}$
$\displaystyle\qquad\times\frac{\Big{[}({k}^{2})^{2}-\left|k\hat{{\bf y}}+{\bf
q}_{1}\right|^{2}\left|k\hat{{\bf y}}+{\bf q}_{2}\right|^{2}+2[(k\hat{{\bf
y}}+{\bf q}_{1})\cdot(k\hat{{\bf y}}+{\bf
q}_{2})]^{2}\Big{]}}{\left(\left|k\hat{{\bf y}}+{\bf
q}_{1}\right|^{2}-{k}^{2}\right)\left(\left|k\hat{{\bf y}}+{\bf
q}_{2}\right|^{2}-{k}^{2}\right)}.$ (51)

We note that Eq. (50) is identical to the 2D formula derived in Ref. [50],
except for the reference phase. The current choice for the reference
dielectric constant $\varepsilon_{BG}^{(2D)}$ offers slightly better
renormalization than 2D Maxwell-Garnett approximation employed in Ref. [50]
because $\varepsilon_{BG}^{(2D)}$ makes the mean of the depolarization tensors
exactly zero, i.e.,
$\phi_{1}\bm{L}_{1}^{(*)}+\phi_{2}\bm{L}_{2}^{(*)}=\bm{0}$. The static limit
($k_{1}\to 0^{+}$) of Eq. (50) is the third-order static strong-contrast
approximation for $d=2$ [70]:

$\mathinner{\varepsilon_{e}^{TE}\mathopen{\left(0\right)}}=\varepsilon_{1}\frac{\phi_{2}(1+\phi_{2}\beta^{\mathrm{(2D)}}_{21})-(1-\phi_{2})\zeta_{2}[\beta^{\mathrm{(2D)}}_{21}]^{2}}{\phi_{2}(1-\phi_{2}\beta^{\mathrm{(2D)}}_{21})-(1-\phi_{2})\zeta_{2}[\beta^{\mathrm{(2D)}}_{21}]^{2}},$
(52)

where $\zeta_{2}$ is the three-point parameter that lies in the closed
interval $[0,1]$ [70].

##  Appendix B Perfect Transparency at the Three-Point Level

Here, we analytically show that the finite perfect transparency interval given
in (41) for SHU two-phase media with
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}=0$ for $k<K$ is
exact through the third-order terms. The detailed proofs presented here were
only very briefly sketched in Ref. [90].

We first show the proof for transverse polarization in the SHU layered medium
(or, equivalently, 1D SHU medium) in B.1. We then show the proof for TM
polarization in the SHU transversely isotropic medium (or, equivalently, 2D
SHU medium) in B.2. For these proofs, we utilize the three general properties
of $\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left({\bf k}\right)}}$ for
disordered SHU two-phase media in $\mathbb{R}^{d}$:

Stealthy hyperuniform condition:
$\displaystyle\left|\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left({\bf
k}\right)}}\right|=0,\quad\qquad\mathrm{for~{}}0\leq\left|{\bf k}\right|<K,$
(53) No Bragg peaks:
$\displaystyle\left|\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left({\bf
k}\right)}}\right|<C,\quad\qquad\mathrm{for~{}any~{}}{\bf k},$ (54)
Large-$\left|{\bf k}\right|$ scaling from Eq. (19):
$\displaystyle\left|\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left({\bf
k}\right)}}\right|\leq\frac{C^{\prime}}{\left|{\bf
k}\right|^{(d+1)/2}},\qquad\mathrm{for~{}}Q\leq\left|{\bf k}\right|,$ (55)

where $C^{\prime}$ and $C$ are some finite positive numbers, and $Q$ is a
large wavenumber much greater than $K$.

###  B.1 Layered Media

In Sec. 5, we analytically show that for SHU layered medium with
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}=0$ for $k<K$, the
finite perfect transparency interval given in Eq. (41) conjunction with Eq.
(42) is exact through the second-order terms. Here, we provide detailed proof
that this transparency interval is exact through the third-order terms
$A_{3}^{\perp}$.

Without losing generality, one can rewrite the third-order condition for
perfect transparency as follows:

$\displaystyle\mathrm{Im}[\mathinner{A_{3}^{\perp}\mathopen{\left(k;1\right)}}]=0,\quad(k<K/2).$
(56)

Substituting
$\mathinner{\tilde{\Delta}_{3}^{(p)}\mathopen{\left(q_{1},q_{2}\right)}}=-\phi_{p}\left\langle\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(q_{1}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(q_{2}-q_{1}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-q_{2}\right)}}\right\rangle-{\phi_{p}}^{2}(2\pi)\mathinner{\delta\mathopen{\left(q_{1}-q_{2}\right)}}\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(q_{2}\right)}}+\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(q_{1}\right)}}\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(q_{2}\right)}}$
into the Fourier representation of $A_{3}^{\perp}$ given in Eq. (34) allows us
to separate the third-order coefficient into three terms:

$\displaystyle\mathinner{A_{3}^{\perp}\mathopen{\left(k;1\right)}}$
$\displaystyle=\frac{-1}{\phi_{p}}\left(\frac{k^{2}}{2\pi}\right)^{2}\int_{-\infty}^{\infty}\mathinner{\mathrm{d}q_{1}}\int_{-\infty}^{\infty}\mathinner{\mathrm{d}q_{2}}\frac{1}{(k+q_{1})^{2}-k^{2}}\frac{1}{(k+q_{2})^{2}-k^{2}}$
$\displaystyle\times\Bigg{[}\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(q_{1}\right)}}\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(q_{2}\right)}}-{\phi_{p}}^{2}(2\pi)\mathinner{\delta\mathopen{\left(q_{1}-q_{2}\right)}}\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(q_{2}\right)}}-\phi_{p}\left\langle\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-q_{1}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(q_{1}-q_{2}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(q_{2}\right)}}\right\rangle\Bigg{]}$
(57)
$\displaystyle=\frac{-1}{\phi_{p}}\frac{{k}^{4}}{(2\pi)^{2}}\Bigg{\\{}\mathinner{C_{1}^{\perp}\mathopen{\left(k\right)}}-{\phi_{p}}^{2}(2\pi)\mathinner{C_{2}^{\perp}\mathopen{\left(k\right)}}-{\phi_{p}}\mathinner{C_{3}^{\perp}\mathopen{\left(k\right)}}\Bigg{\\}},$

where $\mathinner{C_{1}^{\perp}\mathopen{\left(k\right)}}$ and
$\mathinner{C_{2}^{\perp}\mathopen{\left(k\right)}}$ depend on the two-point
statistics, whereas $\mathinner{C_{3}^{\perp}\mathopen{\left(k\right)}}$
depends on the three-point statistics:

$\displaystyle\mathinner{C_{1}^{\perp}\mathopen{\left(k\right)}}\equiv$
$\displaystyle\left[2\pi{k}^{-2}\mathinner{F^{\mathrm{(1D)}}\mathopen{\left(k\right)}}\right]^{2},$
(58) $\displaystyle\mathinner{C_{2}^{\perp}\mathopen{\left(k\right)}}\equiv$
$\displaystyle\int_{-\infty}^{\infty}\mathinner{\mathrm{d}q_{1}}\left[(k+q_{1})^{2}-{k}^{2}\right]^{-2}\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(q_{1}\right)}},$
(59) $\displaystyle\mathinner{C_{3}^{\perp}\mathopen{\left(k\right)}}\equiv$
$\displaystyle\int_{-\infty}^{\infty}\mathinner{\mathrm{d}q_{1}}\int_{-\infty}^{\infty}\mathinner{\mathrm{d}q_{2}}\frac{\left\langle\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(q_{1}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-q_{1}+q_{2}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-q_{2}\right)}}\right\rangle}{[(k+q_{1})^{2}-{k}^{2}][(k+q_{2})^{2}-{k}^{2}]}.$
(60)

Therefore, we can prove Eq. (56) by showing that for $n=1,2,3,$

$\displaystyle\mathrm{Im}[\mathinner{C_{n}^{\perp}\mathopen{\left(k\right)}}]=0\quad(k<K/2),$
(61)

where $\mathinner{C_{n}^{\perp}\mathopen{\left(k\right)}}$ are defined in Eqs.
(58)-(60), respectively. We prove (61) by using the three general properties
of $\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(k\right)}}$ for 1D
disordered SHU two-phase media, given in Eqs. (53)-(55).

One can immediately see that (61) is true for $n=1$. For $k<K/2$, (33) gives
$\mathrm{Im}[\mathinner{F^{\mathrm{(1D)}}\mathopen{\left(k\right)}}]=0$, and
thus $\mathrm{Im}[\mathinner{C_{1}^{\perp}\mathopen{\left(k\right)}}]=0$. We
then prove (61) for $n=2$. The integrand in (59) is real-valued and
nonnegative except at two singular points ($q_{1}=0$ and $-2k$) because it is
the product of the spectral density
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(q_{1}\right)}}$ and the
square of the Green’s function [i.e.,
$(\left|k+q_{1}\right|^{2}-{k}^{2})^{-2}$]. For $k<K/2$, however, this
integrand has no singularity because the stealthy hyperuniform condition
[(53)] can completely remove these two singular points that the Green’s
function can arise. Furthermore, due to the properties of
$\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}$ [(54) and (55)],
this nonnegative integrand of (59) is bounded from above and decays rapidly
like $\left|q_{1}\right|^{-6}$ for large $\left|q_{1}\right|$. Therefore, for
$k<K/2$, the real-valued integrand of (59) is absolutely integrable, leading
to $\mathrm{Im}[\mathinner{C_{2}^{\perp}\mathopen{\left(k\right)}}]=0$.

We then prove (61) for $n=3$. The crucial part of the proof is to show that
for $k<K/2$, the integrand in (60) is absolutely integrable (i.e., the
integral of its absolute value is well-defined and finite) such that the
Fubini-Tonelli theorem [117] enables one to interchange the integration order
in (60). Once we show the integration order is interchangeable, using the
general property of the Fourier transform [i.e.,
$\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(k^{\prime}\right)}}^{*}=\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-k^{\prime}\right)}}$],
we can immediately show that for $k<K/2$,
$\mathinner{C_{3}^{\perp}\mathopen{\left(k\right)}}$ is real-valued because

$\displaystyle[\mathinner{C_{3}^{\perp}\mathopen{\left(k\right)}}]^{*}$
$\displaystyle=\int_{-\infty}^{\infty}\mathinner{\mathrm{d}q_{1}}\int_{-\infty}^{\infty}\mathinner{\mathrm{d}q_{2}}\frac{\left\langle[\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(q_{1}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-q_{1}+q_{2}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-q_{2}\right)}}]^{*}\right\rangle}{[(k+q_{1})^{2}-{k}^{2}][(k+q_{2})^{2}-{k}^{2}]}$
(62)
$\displaystyle=\int_{-\infty}^{\infty}\mathinner{\mathrm{d}q_{1}}\int_{-\infty}^{\infty}\mathinner{\mathrm{d}q_{2}}\frac{\left\langle\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-q_{1}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(q_{1}-q_{2}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(q_{2}\right)}}\right\rangle}{[(k+q_{1})^{2}-{k}^{2}][(k+q_{2})^{2}-{k}^{2}]}$
$\displaystyle=\int_{-\infty}^{\infty}\mathinner{\mathrm{d}q_{2}}\int_{-\infty}^{\infty}\mathinner{\mathrm{d}q_{1}}\frac{\left\langle\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(q_{2}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-q_{2}+q_{1}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-q_{1}\right)}}\right\rangle}{[(k+q_{2})^{2}-{k}^{2}][(k+q_{1})^{2}-{k}^{2}]}=\mathinner{C_{3}^{\perp}\mathopen{\left(k\right)}}.$

Now let us show that the integrand in Eq. (60) is absolutely integrable if
$k<K/2$. We first reduce the absolute value of this integrand in a simpler
form:

$\displaystyle\left|\frac{1}{(k+q_{1})^{2}-{k}^{2}}\frac{1}{(k+q_{2})^{2}-{k}^{2}}\left\langle\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(q_{1}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-q_{1}+q_{2}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-q_{2}\right)}}\right\rangle\right|$
$\displaystyle\leq$
$\displaystyle\left|\frac{1}{(k+q_{1})^{2}-{k}^{2}}\right|\left|\frac{1}{(k+q_{2})^{2}-{k}^{2}}\right|\left\langle\left|\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(q_{1}\right)}}\right|\left|\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-q_{1}+q_{2}\right)}}\right|\left|\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-q_{2}\right)}}\right|\right\rangle$
$\displaystyle\leq$
$\displaystyle\left|\frac{1}{(k+q_{1})^{2}-{k}^{2}}\right|\left|\frac{1}{(k+q_{2})^{2}-{k}^{2}}\right|C\left\langle\left|\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(q_{1}\right)}}\right|\left|\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-q_{2}\right)}}\right|\right\rangle,$
(63)

where we have used Eq. (54). If the integral of (63) is bounded from above by
a finite constant, then we prove that the integrand in (60) is absolutely
integrable.

$\displaystyle\int_{-\infty}^{\infty}\mathinner{\mathrm{d}q_{1}}\int_{-\infty}^{\infty}\mathinner{\mathrm{d}q_{2}}\left|\frac{1}{(k+q_{1})^{2}-{k}^{2}}\right|\left|\frac{1}{(k+q_{2})^{2}-{k}^{2}}\right|C\left\langle\left|\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(q_{1}\right)}}\right|\left|\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-q_{2}\right)}}\right|\right\rangle$
$\displaystyle=C\left\langle\left[\int_{-\infty}^{\infty}\mathinner{\mathrm{d}q_{1}}\frac{\left|\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(q_{1}\right)}}\right|}{\left|(k+q_{1})^{2}-{k}^{2}\right|}\right]^{2}\right\rangle.$
(64)

We then compute the integral that appears in the upper bound on Eq. (64):

$\displaystyle\int_{-\infty}^{\infty}\mathinner{\mathrm{d}q_{1}}\frac{\left|\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(q_{1}\right)}}\right|}{\left|(k+q_{1})^{2}-{k}^{2}\right|}=\left[\int_{\left|q_{1}\right|<K}+\int_{K\leq\left|q_{1}\right|<Q}+\int_{Q\leq\left|q_{1}\right|}\right]\mathinner{\mathrm{d}q_{1}}\frac{\left|\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(q_{1}\right)}}\right|}{\left|(k+q_{1})^{2}-{k}^{2}\right|}$
$\displaystyle\leq
0+\int_{K\leq\left|q_{1}\right|<Q}\mathinner{\mathrm{d}q_{1}}\frac{C}{\left|(k+q_{1})^{2}-{k}^{2}\right|}+\int_{Q\leq\left|q_{1}\right|}\mathinner{\mathrm{d}q_{1}}\frac{C^{\prime}/\left|q_{1}\right|}{\left|(k+q_{1})^{2}-{k}^{2}\right|},$
(65)

where we have applied Eqs. (53)-(55) sequentially in the three integrals. It
is important to note that the first integral over $-K<q_{1}<K$ is divergent
unless the system is stealthy hyperuniform. The second integral in (65) is
positive and bounded from above, as shown below:

$\displaystyle\int_{K\leq\left|q_{1}\right|<Q}\mathinner{\mathrm{d}q_{1}}\frac{C}{\left|(k+q_{1})^{2}-{k}^{2}\right|}\leq
2C\int_{-Q}^{-K}\mathinner{\mathrm{d}q_{1}}\frac{1}{q_{1}(q_{1}+2k)}=\frac{C}{k}\mathinner{\ln\mathopen{\left(\frac{K}{K-2k}\frac{Q-2k}{Q}\right)}}.$
(66)

The third integral in (65) is also positive and bounded from above, as shown
below:

$\int_{Q\leq\left|q_{1}\right|}\mathinner{\mathrm{d}q_{1}}\frac{C^{\prime}/\left|q_{1}\right|}{\left|(k+q_{1})^{2}-{k}^{2}\right|}\leq
2C^{\prime}\int_{-\infty}^{-Q}\mathinner{\mathrm{d}q_{1}}\frac{-1}{{q_{1}}^{2}(q_{1}+2k)}=\frac{-C^{\prime}}{k}\left[\frac{1}{Q}+\frac{1}{2k}\mathinner{\ln\mathopen{\left(1-\frac{2k}{Q}\right)}}\right].$
(67)

Combining Eqs. (62)-(67) shows that the integrand in (60) is absolutely
integrable, leading (61) for $n=3$ to be true as noted in (62).

###  B.2 Transversely Isotropic Media

In Sec. 5, we analytically show that for SHU transversely isotropic systems
with $\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(k\right)}}=0$ for $k<K$,
the finite perfect transparency interval, Eq. (41) conjunction with Eq. (43),
is exact through the second-order terms. Here, we provide detailed proof that
this transparency interval is exact through the third-order terms.

Without losing generality, we can rewrite the third-order condition for the
perfect transparency as follows:

$\displaystyle\mathrm{Im}[\mathinner{A_{3}^{TM}\mathopen{\left(k;1\right)}}]=0,\quad(k<K/2).$
(68)

As in Eq. (57), we decompose $A_{3}^{TM}$ given in Eq. (38) into three terms:

$\displaystyle\mathinner{A_{3}^{TM}\mathopen{\left(k;1\right)}}$
$\displaystyle=\frac{-1}{\phi_{p}}\frac{{k}^{4}}{(2\pi)^{2}}\left[\mathinner{C_{1}^{TM}\mathopen{\left(k\right)}}-{\phi_{p}}^{2}(2\pi)^{2}\mathinner{C_{2}^{TM}\mathopen{\left(k\right)}}-{\phi_{p}}\mathinner{C_{3}^{TM}\mathopen{\left(k\right)}}\right],$
(69)

where $\mathinner{C_{1}^{TM}\mathopen{\left(k\right)}}$ and
$\mathinner{C_{2}^{TM}\mathopen{\left(k\right)}}$ depend on the two-point
statistics, whereas $\mathinner{C_{3}^{TM}\mathopen{\left(k\right)}}$ depends
on the three-point statistics:

$\displaystyle\mathinner{C_{1}^{TM}\mathopen{\left(k\right)}}\equiv$
$\displaystyle\left[-\frac{2\pi^{3}}{{k}^{2}}\mathinner{F^{\mathrm{(2D)}}\mathopen{\left(k\right)}}\right]^{2},$
(70) $\displaystyle\mathinner{C_{2}^{TM}\mathopen{\left(k\right)}}\equiv$
$\displaystyle\int_{\mathbb{R}^{2}}\mathinner{\mathrm{d}{\bf
q}_{1}}\left[\frac{1}{\left|k\hat{{\bf y}}+{\bf
q}_{1}\right|^{2}-{k}^{2}}\right]^{2}\mathinner{\tilde{\chi}_{{}_{V}}\mathopen{\left(q_{1}\right)}},$
(71) $\displaystyle\mathinner{C_{3}^{TM}\mathopen{\left(k\right)}}\equiv$
$\displaystyle\int_{\mathbb{R}^{2}}\mathinner{\mathrm{d}{\bf
q}_{1}}\int_{\mathbb{R}^{2}}\mathinner{\mathrm{d}{\bf
q}_{2}}\frac{\left\langle\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left({\bf
q}_{1}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-{\bf
q}_{1}+{\bf
q}_{2}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-{\bf
q}_{2}\right)}}\right\rangle}{(\left|k\hat{{\bf y}}+{\bf
q}_{1}\right|^{2}-{k}^{2})(\left|k\hat{{\bf y}}+{\bf
q}_{2}\right|^{2}-{k}^{2})},$ (72)

which are the 2D counterparts of Eqs. (58)-(60). We now prove (68) by showing
that for $n=1,2,3,$

$\displaystyle\mathrm{Im}[\mathinner{C_{n}^{TM}\mathopen{\left(k\right)}}]=0\quad(k<K/2).$
(73)

Because the proof of Eq. (73) is analogous to (61), we outline how to prove it
by using the three general properties of
$\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left({\bf k}\right)}}$ for 2D
disordered SHU two-phase media, given in Eqs. (53)-(55).

One can immediately see that (73) is true for $n=1$ because
$\mathrm{Im}[\mathinner{F^{\mathrm{(2D)}}\mathopen{\left(k\right)}}]=0$ for
$k<K/2$, which is identical to the perfect transparency at the two-point
level. One can also immediately show (73) for $n=2$ because the integrand in
(71) is (i) real-valued and (ii) absolutely integrable for $k<K/2$. The
condition (ii) can be confirmed because this integrand has no singularity due
to the stealthy hyperuniform condition [(53)], and its absolute value is also
bounded from above and decays rapidly as $\left|{\bf q}_{1}\right|$ increases
due to Eqs. (54) and (55).

We now outline the proof of (73) for $n=3$. Similar to B.1, the crucial part
of this proof is to show that for $k<K/2$, the absolute value of the integrand
in (72) is integrable (i.e., absolutely integrable), and thus the value of
$\mathinner{C_{3}^{TM}\mathopen{\left(k\right)}}$ is independent of the
integration order due to the Fubini-Tonelli theorem [117]. Once we enable the
interchange of the integration order, we can immediately show that
$\mathrm{Im}[C_{3}^{\perp}(k)]=0$ for $k<K/2$, as in Eq. (62).

One can show that the integrand in (72) is absolutely integrable, as we did so
in B.1. Roughly speaking, the proof is based on Eqs. (53)-(55). We first show
that the integrand in Eq. (72) has no singularity because while all possible
singularities of the Green’s functions in this integrand lie along two circles
[i.e., $\left|{\bf q}_{1}+k\hat{{\bf y}}\right|=k$ and $\left|{\bf
q}_{2}+k\hat{{\bf y}}\right|=k$],
$\left\langle\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left({\bf
q}_{1}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-{\bf
q}_{1}+{\bf
q}_{2}\right)}}\mathinner{\tilde{\mathcal{J}}^{(p)}\mathopen{\left(-{\bf
q}_{2}\right)}}\right\rangle=0$ along these singularities because of (53). In
addition, due to Eq. (54), the absolute value of the integrand in (72) is
bounded from above for any ${\bf q}_{1}$ and ${\bf q}_{2}$. Furthermore,
because of Eq. (55), the absolute value of the integrand vanishes as fast as
or faster than $\left|{\bf q}_{1}\right|^{-7/2}\left|{\bf
q}_{2}\right|^{-7/2}$ for large $\left|{\bf q}_{1}\right|$ and $\left|{\bf
q}_{2}\right|$. These three observations directly show that the integrand is
absolutely integrable.

The research was sponsored by the U.S. Army Research Office and was
accomplished under Cooperative Agreement No. W911NF-22-2-0103.

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