# Tolerance and breakdown of topological protection in a disordered waveguide

Kiyanoush Goudarzi<EMAIL_ADDRESS>Department of Electrical Engineering,
Pohang University of Science and Technology (POSTECH), 37673 Pohang, Korea
Department of Electrical and Computer Engineering, North Carolina State
University, Raleigh, NC 27695, USA  Moonjoo Lee<EMAIL_ADDRESS>Department of Electrical Engineering, Pohang University of Science and
Technology (POSTECH), 37673 Pohang, Korea

###### Abstract

We consider a disordered waveguide consisting of trivial dielectric and non-
trivial magnetically anisotropic material. A topologically-protected edge mode
appears owing to the broken time-reversal symmetry of the non-trivial lattice.
While the edge mode maintains under other position and radius disorders, the
protection is immediately broken by applying a radius disorder to the non-
trivial lattice. This breakdown originates from donor and acceptor modes
occupying the topological bandgap. Furthermore, via the calculation of the
Bott index, we show that Anderson localization occurs as a metal conducting
gap changes to a topological gap along with increasing disorders.

Topologically protected state is generally known to be stable and robust
against perturbations. This topological protection (TP) holds significant
importance in various disciplines, including photonics [1, 2], condensed
matter [3, 4], atomic physics [5, 6], and acoustics [7]. In particular, TP
provides a solution for certain problems in photonics through the unique
transport property of the topological edge mode. For instance, in a
nanophotonic device, the transmission of an electromagnetic (EM) wave is often
affected by backreflection and scattering losses. The topological waveguide
enables the transmission of reflection-free waves, even in the presence of
substantial structural disorder [8, 9]. Besides, photonic systems with TP have
been excellent testbeds for exploring non-Hermitian physics [10, 11], non-
linear optics [12, 13], higher-order band topology [14], and Floquet physics
[15, 16].

This TP, however, can be influenced or even broken by specific environmental
conditions. In condensed matter physics, extensive studies were performed
regarding the breakdown of the TP. For example, the breakdown was observed in
graphene edges due to antidots and upstream modes in an electrostatic regime
[17, 18]. The TP breakdown also occurred in a two-dimensional electron gas by
enhanced vacuum fluctuations [19], and a smooth edge potential broke the TP
via edge reconstruction [20]. In contrast, such investigations in photonics
are relatively scarce. One study indicated TP breaking by transitioning from
the amorphous to crystalline structural state of the constitutive material
Ge2Sb2Te2 [21]. Another study revealed a TP breakdown using rotational
symmetry in the unit cell of periodic Kekulé patterns [22].

Here, we explore the tolerance and breakdown of TP of an edge mode in a
disordered waveguide. Our topological waveguide (TW) consists of a trivial
lattice interfaced with a non-trivial lattice. When both the position and
radius disorders are applied to the trivial lattice, the unidirectionality of
the edge mode sustains, exhibiting the TP nature of the mode. The
unidirectionality also maintains under the position disorder of the non-
trivial lattice. However, when the radius disorder is applied to the non-
trivial lattice, all different localized defect modes prevent the formation of
the topological bandgap, which breaks the edge mode. Moreover, as the radius
disorder increases, a frequency domain of bulk modes becomes a non-trivial
bandgap, accompanying Anderson localization (AL) of EM waves in the
topological regime. Such topological nature is characterized with the Bott
index (BI), a topological invariant obtained from the real-space
wavefunctions.

Our TW includes a trivial and a non-trivial mirror. The trivial mirror
contains dielectric rods with a relative permittivity of
$\epsilon_{\textrm{t}}=16$, a relative permeability of $\mu_{\textrm{t}}=1$,
and a radius of $R_{\textrm{t}}=0.30a$ in air where $a$ is the lattice
constant. The trivial lattice corresponds to the upper half of Figs. 1(a) and
(b). The non-trivial lattice consists of the magneto-optical material of
yttrium-iron-garnet (YIG) ferrite rods with a relative permittivity of
$\epsilon_{\textrm{n}}=15$ and a permeability of $\mu_{\textrm{n}}=\bm{\mu}$
with a radius of $R_{\textrm{n}}=0.11a$ in air (lower half of Figs. 1(a) and
(b)). Near an operating frequency of 4.5 GHz, the anisotropic permeability
$\bm{\mu}$ under applying a static magnetic field is

$\bm{\mu}=\begin{bmatrix}\mu&i\kappa&0\\\ -i\kappa&\mu&0\\\
0&0&\mu_{0}\end{bmatrix},$ (1)

where $\mu=14\mu_{0}$ and $\kappa=12.4\mu_{0}$, where $\mu_{0}$ is the vacuum
permeability [8].

Fig. 1(c) shows the dispersion diagram of the trivial mirror with the three
trivial TM bandgaps of BG${}_{\textrm{t1}}$, BG${}_{\textrm{t2}}$, and
BG${}_{\textrm{t3}}$. The first two gaps of BG${}_{\textrm{t1}}$ and
BG${}_{\textrm{t2}}$ are Mie bandgaps with TM01 and TM11 modes, respectively.
The TM01 and TM11 modes constitute pure Mie bandgaps that exhibit high
tolerance to the position and radius disorderings, and thus the penetration of
EM waves into the trivial mirror is well suppressed. Differently from
BG${}_{\textrm{t1}}$ and BG${}_{\textrm{t2}}$, both Mie and Bragg scatterings
generate BG${}_{\textrm{t3}}$ with the TM21 mode, less tolerant to the
disorderings, causing that the EM waves would penetrate gently more into the
trivial lattice than BG${}_{\textrm{t1}}$ and BG${}_{\textrm{t2}}$ [23].

Fig. 1:  Normalized $E_{z}(x,y)$ at $a/\lambda=0.570$ in (a) and $0.630$ in
(b). Black arrows denote the location of light source. (c)–(d) Dispersion
diagrams of trivial and non-trivial lattices. Green regions show photonic
bandgaps and $\Gamma$, M, and X represent vortices of the irreducible
Brillouin zone. Lattice constant, wavelength, and Chern number are referred to
as $a$, $\lambda$, and C, respectively. (e) Dispersion diagram of TW. Red and
black curves show edge modes with positive and negative group velocities. Red
and black arrows indicate $a/\lambda=0.570$ and $0.630$.

In the non-trivial mirror, we find three bandgaps of BG${}_{\textrm{n1}}$,
BG${}_{\textrm{n2}}$, and BG${}_{\textrm{n3}}$ in the dispersion diagram of
Fig. 1(d). The lowest bandgap of BG${}_{\textrm{n1}}$ is a trivial Mie bandgap
with TM01 mode. The second and third bandgaps are non-trivial topological
gaps. Due to the broken time-reversal symmetry in the non-trivial lattice, the
high-symmetry points of M and $\Gamma$ in the irreducible Brillouin zone
break, resulting in the creation of BG${}_{\textrm{n2}}$ and
BG${}_{\textrm{n3}}$, respectively [8, 24]. Breaking the time-reversal
symmetry originates from the fact that the anisotropic material of YIG has
imaginary off-diagonal elements in the permeability tensor with
$\bm{\mu}\neq\bm{\mu}^{T}$, where $T$ is the transpose operation [25]. The
calculations of the Chern number and dispersion diagram are elaborated in Ref.
[26].

We proceed to the calculation of Chern numbers in both the trivial and non-
trivial lattices. The Chern numbers over the first Brillouin zone are zero for
all bands of the trivial lattice. The zero value of the Chern numbers is based
on preserving the time-reversal symmetries [8, 27, 24]. As contrary, we obtain
the Chern numbers of $0,1,-2,$ and $-1$ for the first four bands, from low to
high frequency, in the non-trivial lattice [26].

Each band’s Chern number constitutes the value of the Chern number for each
bandgap, i.e., the bandgap’s Chern number (C${}_{\textrm{g}}$) is defined as
the sum of bands’ Chern number below the bandgap [8, 27, 24]. As a
consequence, we obtain C${}_{\textrm{g}}=0,1,$ and $-1$ for
BG${}_{\textrm{n1}}$, BG${}_{\textrm{n2}}$, and BG${}_{\textrm{n3}}$,
respectively. The overlap between BG${}_{\textrm{t2}}$ and
BG${}_{\textrm{n1}}$ gives rise to the creation of the bandgap of
BG${}_{\textrm{w}}$ at normalized frequencies $0.353<a/\lambda<0.451$ in the
dispersion diagram of the TW: This results in no edge mode, because the
difference of C${}_{\textrm{g}}$ of BG${}_{\textrm{n1}}$ and that of
BG${}_{\textrm{t2}}$ is zero. Differently from BG${}_{\textrm{w}}$, the
overlap between BG${}_{\textrm{t3}}$ and the two bandgaps of
BG${}_{\textrm{n2}}$ and BG${}_{\textrm{n3}}$ creates two unidirectional edge
modes at $0.556<a/\lambda<0.578$ and $0.613<a/\lambda<0.637$ with positive and
negative group velocities, respectively, as shown in Fig. 1(e). The edge modes
are visualized in Figs. 1(a) and (b). The out-of-plane electric fields,
$E_{z}(x,y)$, at $a/\lambda=0.570$ and $0.630$ represent the propagation of EM
waves to the right and left directions.

Fig. 2:  (a)–(h) Normalized $E_{z}(x,y)$ under position and radius disorders.
(a), (e) Trivial mirror with $\eta_{\textrm{P}}=40\%$, (b), (f) non-trivial
mirror with $\eta_{\textrm{P}}=40\%$, (c), (g) trivial mirror with
$\eta_{\textrm{R}}=40\%$, and (d), (h) non-trivial mirror with
$\eta_{\textrm{R}}=40\%$. TM polarized dipole sources are denoted as black
arrows at the interface of the mirrors with a normalized frequency
$a/\lambda=0.570$. Transmission $T_{\textrm{LR}}$ when position disorder is
applied to (i) trivial and (j) non-trivial lattices, and radius disorder to
(k) trivial and (l) non-trivial lattices for several $\epsilon_{\textrm{t}}$.
$a/\lambda=0.570$ for $\epsilon_{\textrm{t}}=16,18$, and $20$ and
$a/\lambda=0.550$ for $\epsilon_{\textrm{t}}=22$.

Next, we explore the impact of disorders to the unidirectionality of the edge
modes. We consider two types of disorderings, in the position and radius of
the rods. The disordered position of the rod $i$ is defined as
$(x^{i},y^{i})$, where $x^{i}=x_{0}^{i}+\sigma_{\textrm{P}}F_{x}^{i}$ and
$y^{i}=y_{0}^{i}+\sigma_{\textrm{P}}F_{y}^{i}$ with the original position
$(x_{0}^{i},y_{0}^{i})$, the strength of the position disorder is
$\sigma_{\textrm{P}}$, and $F_{x}^{i}$ and $F_{y}^{i}$ are uniformly
distributed random variables between $-1$ and $1$ for the $i$th rod along the
$x$ and $y$ directions. The position disordering parameter is defined as
$\eta_{\textrm{P}}=\sigma_{\textrm{P}}/a$. In a similar way, the disordered
radius of the rod $i$ in the trivial (non-trivial) lattice is
$R_{\textrm{t(n)}}^{i}=R_{\textrm{t(n),0}}+\sigma_{\textrm{R}}F_{\textrm{R}}^{i}$,
where $R_{\textrm{t(n),0}}$, $\sigma_{\textrm{R}}$, and $F_{\textrm{R}}^{i}$
stand for the original radius of the rod in the trivial (non-trivial) lattice,
strength of the radius disordering, and the uniformly distributed random
parameter over the interval of $[-1,1]$. We define the parameter of the radius
disordering as $\eta_{\textrm{R}}=\sigma_{\textrm{R}}/R_{\textrm{t(n)},0}$.

We first obtain $E_{z}(x,y)$ under the influence of the position disordering.
The parameters of the trivial mirror are $R_{\textrm{t,0}}=0.30a$,
$\epsilon_{\textrm{t}}=18$, and $\mu_{\textrm{t}}=1$ and those of the non-
trivial mirror are $R_{\textrm{n,0}}=0.11a$, $\epsilon_{\textrm{n}}=15$, and
$\mu_{\textrm{n}}=\bm{\mu}$. The normalized frequency of the TM polarized
dipole source is $a/\lambda=0.570$ where the edge mode of a positive group
velocity appears. The calculation results are shown when the disorder is
$\eta_{\textrm{P}}=40$% in either the trivial or non-trivial mirror (Figs.
2(a), (b), (e), and (f)): The unidirectionality of the EM waves is preserved
at this substantial strength of disorder.

We explore the unidirectionality more quantitatively by calculating the
transmittance $T_{\textrm{LR}}$ of the edge mode. We judge that the EM wave
has the unidirectionality transmission when $T_{\textrm{LR}}>0.5$. Figs. 2(i)
and (j) show $T_{\textrm{LR}}$ at several disorderings and
$\epsilon_{\textrm{t}}$. Each transmittance is obtained by averaging the
results of 100 numerical simulations. Under applying the position disorders to
both the trivial and non-trivial mirrors, the unidirectionality is preserved
and $T_{\textrm{LR}}$ moderately decreases as the disorder increases from
$\eta_{\textrm{P}}=0$ to $50\%$.

Similar behavior is observed when the radius disorder is applied to the
trivial lattice (Figs. 2(c), (g), and (k)). However, the result is entirely
different when the radius of non-trivial lattice is disordered. Figs. 2(d) and
(h) show that the EM waves propagate to $+x,-x$, and $-y$ directions and the
penetration to the trivial lattice ($+y$ direction) is suppressed.
$T_{\textrm{LR}}$ abruptly decreases from 1 to 0.4 at
$\eta_{\textrm{R}}=10\%$, and to $\sim 0$ at $\eta_{\textrm{R}}=20\%$—the
unidirectionality is broken completely (Fig. 2(l)).

Fig. 3:  Dispersion diagrams of TW with $5\times 8$ dielectric rods and
$5\times 8$ YIG rods. (a) $\eta_{\textrm{P}}=40\%$ to trivial lattice, (b)
$\eta_{\textrm{P}}=40\%$ to non-trivial lattice, (c) $\eta_{\textrm{R}}=40\%$
to trivial lattice, and (d) $\eta_{\textrm{R}}=40\%$ to non-trivial lattice.
Blue regions are bulk modes. Yellow arrow indicates $a/\lambda=0.570$.

We describe the origin of the tolereance and breakdown of the edge mode. Fig.
3 shows the dispersion diagrams of the TW, containing a supercell with
$5\times 8$ of dielectric rods with $\epsilon_{\textrm{t}}=18$,
$\epsilon_{\textrm{n}}=15$, and $5\times 8$ YIG rods of $\bm{\mu}$ [26]. When
the position disorder $\eta_{\textrm{P}}=40\%$ is applied to either the
trivial or non-trivial lattice (Figs. 3(a) and (b)), the edge mode with a
positive group velocity still exists and thus the unidirectionality of the EM
wave maintains. This point is associated with the mode profile in the rods and
the interference of the EM waves. In Figs. 2(e) and (f), we identify that the
localized modes inside each dielectric and YIG rod under the position
disordering are TM21 and TM11, respectively. The unidirectionality to the $+x$
direction is a result of preserving the localized mode inside each rod in the
non-trivial lattice. The gentle penetration of EM waves to the $+y$ (Figs.
2(a) and (e)) and $-y$ (Figs. 2(b) and (f)) directions is owing to the random
increase in the distances between some rods. The increased distances result in
decreasing the coupling between quasi-bound states, which makes the EM waves
penetrate between the rods [28]. However, all identical TM11 modes in YIG rods
sustain the slope of the edge mode.

In case of radius disorders, defect modes affect the directionality of the
edge mode. In this configuration, each disordered rod corresponds to a point
defect of the waveguide. The point defects with larger (smaller) radii,
increase (decrease) the effective refractive index, resulting in the
appearance of donor (acceptor) modes. The donor (acceptor) modes, defect
modes, fall into the adjacent bandgap from the upper edge (lower edge) of the
gap, resulting in filling the bandgap with the defect modes. In other words,
donor and acceptor modes experience red and blue shifts, respectively; the
acceptor modes between BG${}_{\textrm{t2}}$ and BG${}_{\textrm{t3}}$ (radius
disorder to trivial lattice), acceptor modes between BG${}_{\textrm{n1}}$ and
BG${}_{\textrm{n2}}$ (radius disorder to non-trivial lattice), and donor modes
between BG${}_{\textrm{n2}}$ and BG${}_{\textrm{n3}}$ (radius disorder to non-
trivial lattice) generate the bulk modes that occupy the gap.

Note that this description holds for the cases of weak or moderate radius
disorders. As the disorder increases, the impact of donor mode becomes more
dominant than that of accepter mode, because more modes can exist in a rod of
larger radius (see the explanation for Fig. 4(b) below); as the radius
disorder increases strongly, the bandgap shifts to red.

The impact of the defect modes is different between occurred in the trivial
and non-trivial lattice. Given radius disorder in the trivial lattice, the
acceptor modes between BG${}_{\textrm{t2}}$ and BG${}_{\textrm{t3}}$ would
fill up the gap at $0.556<a/\lambda<0.578$. However, the localized eigenmode
in YIG rods does not change, which persists the directionality of the edge
mode. All identical mode profiles in YIG rods are identified in Figs. 2(c) and
(g), and accordingly the positive sign of the group velocity is preserved, as
shown in Fig. 3(c).

In contrast, when the radius disorder is applied to the non-trivial lattice,
the topological nature is broken totally. As shown in Fig. 3(d), both the
acceptor modes from the frequencies between BG${}_{\textrm{n1}}$ and
BG${}_{\textrm{n2}}$ and the donor modes between BG${}_{\textrm{n2}}$ and
BG${}_{\textrm{n3}}$ occupy the gap, and the resulting bulk modes exhibit no
directionality. The underlying reason of this breakdown is that all the
eigenmodes in YIG rods become different under the radius disorder. All
distinct these modes disturb the formation of the topological bandgap of the
non-trivial lattice, which herein causes the TP breakdown. This feature is
also found in Figs. 2(d) and (h), showing that different modes are localized
in every YIG rod.

We further investigate this breakdown of TP with another topological variable,
the BI [29, 30, 31, 32]. The BI, obtained from the real-space electric-field
distribution, is particularly useful for characterizing disordered structures
where the Chern number cannot be defined. The BI and Chern number manifest the
topological nature of a band with nonzero values, having the same absolute
value with the opposite sign. More details of the BI calculation are provided
in Ref. [26]

The calculation results of the BI are presented in Fig. 4. We consider a
supercell of $6\times 6$ YIG rods in air under position and radius
disorderings over the frequency interval of $0.430\leq a/\lambda\leq 0.660$.
The BI at each frequency and disordering is obtained by averaging the results
of 50 simulations. In case of position disorder, we find two topological
bandgaps of BG${}_{\textrm{n2}}$ and BG${}_{\textrm{n3}}$ with the BI of $-1$
and $+1$, respectively, which do not change as the position disorder increases
(Fig. 4(a)). This shows that the edge mode is not influenced by the position
disorder of the trivial and non-trivial lattices, agreeing with the results in
Figs. 2(a) and (b), and Figs. 3(a) and (b). The localization of
TM${}_{\textrm{11}}$ mode in all YIG rods gives the same BI, which is
independent of position disordering: This supports the unidirectionality of
the edge mode.

As shown in Fig. 4(b), the behavior of the topological bandgap, under the
radius disorder of the non-trivial lattice, is very different from that under
the position disorder: Both the bandgaps of BG${}_{\textrm{n2}}$ and
BG${}_{\textrm{n3}}$ undergo red shift as $\eta_{\textrm{R}}$ increases. This
frequency shift is attributed to the donor modes in the point defects of
larger radii. Under the radius disorder, defects with both larger and smaller
radii are present. While many bulk modes can appear in the rods of larger
radii, the number of modes that can exist in smaller defects would be much
less; for instance, certain modes cannot survive if the size of a rod is
smaller than the critical radius. Dominated by the impact of defects with
larger radii, the effective index of refraction of the lattice increases,
causing the red shift of overall defect modes—the bandgaps shift to red as
well. More quantitative explanation is offered with the calculations of
$E_{z}(x,y)$ and the mode frequencies in Ref. [26].

Fig. 4:  Bott index of $6\times 6$ YIG rods over a frequency range of
$0.430\leq a/\lambda\leq 0.660$ under (a) position and (b) radius disorders.
(c) $|E_{z}(x,y)|$ at the location (1) of bulk modes. (d) $|E_{z}(x,y)|$ at
(2) where TAL happens. Source position of EM wave is the origin. We choose
$\epsilon_{\textrm{t}}=24$ so that BG${}_{\textrm{t3}}$ covers
$a/\lambda=0.470$. (e) $|E_{z}(a,y)|$ (blue solid line) and $|E_{z}(3a,y)|$
(red solid line), and fitting with exponential decay functions (dashed lines).
(f) $|E_{z}(x,-a)|$.

This red shift brings about two phenomena: The breakdown of the TP and the
emergence of AL [33, 34]. First, we consider an edge mode at
$a/\lambda=0.570$. As $\eta_{\textrm{R}}$ increases, the topological bandgap
is occupied with bulk modes, resulting in the disappearance of the edge mode
(BI changes from $-1$ to $0$): This is the reason for the breakdown of the
unidirectionality as discussed above. Second, we concentrate on a region of
$0.430<a/\lambda<0.528$ in Fig. 4(b). While this domain is occupied with bulk
modes at $\eta_{\textrm{R}}=0$, these modes gradually disappear and the
topological bandgap emerges as $\eta_{\textrm{R}}$ grows. Accordingly the edge
mode appears in this frequency region, which accompanies not only the
unidirectionality of the mode but also the localization of the EM waves under
the disorder, which corresponds to AL. The behavior of AL is revealed in Fig.
4(e), where $|E_{z}(a,y)|$ and $|E_{z}(3a,y)|$ are fitted with the exponential
decay function of $E_{0}\cdot\exp{(-|y|/\xi)}$. The localization length is
given by $\xi$. From the fitting, we obtain $\xi\simeq 3.3a$ for
$|E_{z}(a,y)|$ and $\xi\simeq 3.8a$ for $|E_{z}(3a,y)|$, and this
exponentially decaying feature proves the genuine nature of AL. Summarizing
the phenomena here, as the disorder increases, transition from the metal
conducting gap to non-trivial bandgap happens, in tandem with a disorder-
induced emergence of AL in this topological regime.

We finally remark two points regarding our work. First, as far as we are
aware, our work is for the first time to explore the impact of radius disorder
in a topological waveguide. While previous studies were done with position
disorders [35, 36, 37], we study how the radius disorders affect the edge
mode, making it possible to manifest the TP breakdown. Second, we herein
envision the possibility of a similar study in the optical and infrared (IR)
frequency domain. The deployed YIG material exhibits the nature of broken
time-reversal frequency near 4.5 GHz [8, 9]. In order to investigate such
topological behavior in the optical domain, one can utilize a honeycomb
lattice including a dielectric-helical waveguide, in an ambient medium with a
refractive index of $1.45$ [15]. The symmetry breaks in the helical direction,
which acts like time-reversal symmetry breaking in YIG, leading to the
unidirectional EM wave around the lattice at optical frequencies. In the IR
region, one can make use of dielectric rods with a refractive index of $3.42$
in a honeycomb pattern in air, revealing pseudo-time reversal symmetry
preservation. This would result in the generation of unidirectional chiral
edge states in the IR domain [38].

In conclusion, we have studied the tolerance and breakdown of TP in a
disordered waveguide. Both the position and radius disorders are applied to
either the trivial or non-trivial lattice. The edge mode disappears under the
radius disorder of non-trivial lattice, because all different donor and
acceptor modes prevent the creation of the topological gap. Moreover, through
the calculation of a topological variable of the BI, we show that EM waves are
localized in a certain frequency region, which is AL effect associated with
the topological gap. Our work offers new understanding on the one-way light
propagation in nanophotonics, and also gives an insight to the development of
topological optical circuits and integrated photonic devices.

###### Acknowledgements.

We thank H. G. Maragheh for helpful discussions. We acknowledge the support
from BK21 FOUR program and Educational Institute for Intelligent Information
Integration, National Research Foundation (Grant No. 2019R1A5A1027055),
Samsung Electronics Co., Ltd (IO201211-08121-01), and Samsung Science and
Technology Foundation (SRFC-TC2103-01).

Our data are available at https://doi.org/10.5281/zenodo.8339612.

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Supplemental Material: Tolerance and breakdown of topological protection in a
disordered waveguide

##  I Dispersion diagram of topological waveguide

Fig. S1:  The TM band structure of the non-trivial mirror consisting of YIG
rods with $R_{\textrm{n,0}}=0.11a$, $\epsilon_{\textrm{n}}=15$,
$\mu=14\mu_{0}$, and $\kappa=12.4$ in air and a trivial mirror of dielectric
rods in air with $R_{\textrm{t,0}}=0.3a$ and (a) $\epsilon_{\textrm{t}}=14$,
(b) $\epsilon_{\textrm{t}}=16$, (c) $\epsilon_{\textrm{t}}=18$, (d)
$\epsilon_{\textrm{t}}=20$, (e) $\epsilon_{\textrm{t}}=22$, and (f)
$\epsilon_{\textrm{t}}=24$. The blue color represents bulk modes. Solid black
and red curves are edge modes with negative and positive group velocities,
respectively.

We discuss the transverse magnetic (TM) dispersion diagrams, photonic band
structures, of a topological waveguide (TW) containing a trivial mirror with
$R_{\textrm{t,0}}=0.3a$ and $\epsilon_{\textrm{t}}$ in air and a non-trivial
mirror consisting of YIG rods with $R_{\textrm{n,0}}=0.11a$,
$\epsilon_{\textrm{n}}=15$, $\mu=14\mu_{0}$, and $\kappa=12.4\mu_{0}$ in air.
Fig. S1 shows the dispersion diagrams of non-trivial bandgaps at $0.5\leq
a/\lambda\leq 0.68$ with either positive or negative group velocities. The
positive or negative group velocities over the wavevector $k$x represent the
unidirectionality of the electromagnetic waves in $+x$ or $-x$ directions,
respectively, at the interface between the trivial and non-trivial mirrors.
Note that the frequency region of the bandgap changes as
$\epsilon_{\textrm{t}}$ varies. The dispersion diagrams of the TW show an edge
mode with negative group velocity and the bandgap’s Chern number Cg = -1 at
$\epsilon_{\textrm{t}}=14$ (Fig. S1(a)) and an edge mode with positive group
velocity and Cg = +1 at $\epsilon_{\textrm{t}}=18$ and 20 (Fig. S1(c) and
(d)). Also, the dispersion diagrams of the TW show two edge modes at
$\epsilon_{\textrm{t}}=16$, 22, and 24 (Fig. S1(b), (e), and (f)). The first
mode with higher frequency, negative group velocity and Cg = -1, and the
second mode with lower frequency, positive group velocity and Cg = +1. The
blue color in the band structure diagrams stands for bulk modes that are of no
unidirectionality. This calculation is performed using COMSOL Multiphysics
software.

##  II The Chern number and Berry curvature calculations

The Chern number for the band $n$ is calculated in the wavevector space over
the first Brillouin zone (BZ). BZ is calculated for a unit cell in Fig. S2(a).
The Chern number is presented as

$C_{n}=\frac{1}{2\pi i}\int_{\textrm{BZ}}\textrm{d}^{2}kF^{n}(\mathbf{k}),$
(S1)

where

$F^{n}(\mathbf{k})=\left(\frac{\partial A_{y}^{nn}}{\partial
k_{x}}-\frac{\partial A_{x}^{nn}}{\partial k_{y}}\right),$ (S2)

and

$\mathbf{A}^{nn^{\prime}}(\mathbf{k})\equiv
i\int\textrm{d}^{2}r\,\epsilon_{r}(\mathbf{r})\mathbf{E}_{n\mathbf{k}}^{*}(\mathbf{r})\cdot\left[\nabla_{\mathbf{k}}\mathbf{E}_{n^{\prime}\mathbf{k}}(\mathbf{r})\right].$
(S3)

Fig. S2:  (a) Unit cell of the periodic structure containing a 2D cylindrical
rod in air and (b) the first BZ. $a$ and $r$ are the lattice constant and the
radius of the rod, respectively.

In the above equations, $F$, A, and $\textbf{E}_{n\textbf{k}}$ are the Berry
curvature, Berry connection, and electric field (Bloch function),
respectively. $\epsilon_{\textrm{r}}(\mathbf{r})$ is the position-dependent
dielectric constant and the asterisk stands for the complex conjugation.
Quantification of the Chern number and Berry curvature requires calculating
the eigenfrequencies and eigenvalues over the first BZ bounded at
$-\frac{\pi}{a}<k_{x,y}<\frac{\pi}{a}$ as shown in Fig. S2(b), where $a$ is
the lattice constant. To calculate the Chern number, Eqs. (S1)–(S3) are
replaced with the following equations

$C_{n}=\frac{1}{2\pi
i}\sum_{\mathbf{k}_{l}\in\textrm{BZ}}F_{xy}^{n}(\mathbf{k}_{l})\delta\mathbf{k}_{x}\delta\mathbf{k}_{y}.$
(S4)

$\displaystyle F_{xy}(\mathbf{k}_{l})\delta\mathbf{k}_{x}\delta\mathbf{k}_{y}$
$\displaystyle=(\partial_{x}A_{y}(\mathbf{k}_{l})-\partial_{y}A_{x}(\mathbf{k}_{l}))\delta\mathbf{k}_{x}\delta\mathbf{k}_{y}$
(S5)
$\displaystyle\approx(\Delta_{x}A_{y})\delta\mathbf{k}_{y}-(\Delta_{y}A_{x})\delta\mathbf{k}_{x}.$

$\displaystyle A_{\lambda}(\mathbf{k}_{l})\delta\mathbf{k}_{\lambda}$
$\displaystyle=\langle
u(\mathbf{k}_{l})|\partial_{\lambda}|u(\mathbf{k}_{l})\rangle\delta\mathbf{k}_{\lambda}$
(S6) $\displaystyle=\langle
u(\mathbf{k}_{l})|u(\mathbf{k}_{l}+\delta\mathbf{k}_{\lambda})-u(\mathbf{k}_{l})\rangle$
$\displaystyle=\langle
u(\mathbf{k}_{l})|u(\mathbf{k}_{l}+\delta\mathbf{k}_{\lambda})\rangle-1.$

where $n$, $A_{x}$, $A_{y}$, $|u(\mathbf{k}_{l})\rangle$ are the band number
(we drop it in the rest of the context), the Berry connection in the $x$
direction, the Berry connection in the $y$ direction, and the Bloch function.
Utilizing the Taylor series, the Berry connection reads

$\displaystyle e^{A_{\lambda}(\mathbf{k}_{l})\partial\mathbf{k}_{\lambda}}$
$\displaystyle=e^{\langle
u(\mathbf{k}_{l})|u(\mathbf{k}_{l}+\delta\mathbf{k}_{\lambda})\rangle-1}$ (S7)
$\displaystyle\approx 1+\langle
u(\mathbf{k}_{l})|u(\mathbf{k}_{l}+\delta\mathbf{k}_{\lambda})\rangle-1$
$\displaystyle=\langle
u(\mathbf{k}_{l})|u(\mathbf{k}_{l}+\delta\mathbf{k}_{\lambda})\rangle,$

$A_{\lambda}(\mathbf{k}_{l})\partial\mathbf{k}_{\lambda}=\ln\langle
u(\mathbf{k}_{l})|u(\mathbf{k}_{l}+\delta\mathbf{k}_{\lambda})\rangle.$ (S8)

$\Delta_{\mu}A_{\lambda}(\mathbf{k}_{l})\partial\mathbf{k}=A_{\lambda}(\mathbf{k}_{l}+\partial\mathbf{k}_{\mu})\partial\mathbf{k}-A_{\lambda}(\mathbf{k}_{l})\partial\mathbf{k};(\mu,\lambda:x,y),$
(S9)

By substituting Eqs. (S8) and (S9) into Eq. (S5), the following equation
appears

$\displaystyle
F_{xy}(\mathbf{k}_{l})\partial\mathbf{k}_{x}\partial\mathbf{k}_{y}=$
$\displaystyle\ln(\langle
u(\mathbf{k}_{l}+\partial\mathbf{k}_{x})|u(\mathbf{k}_{l}+\delta\mathbf{k}_{x}+\delta\mathbf{k}_{y})\rangle)$
(S10) $\displaystyle-\ln(\langle
u(\mathbf{k}_{l})|u(\mathbf{k}_{l}+\delta\mathbf{k}_{y})\rangle)$
$\displaystyle-\ln(\langle
u(\mathbf{k}_{l}+\delta\mathbf{k}_{y})|u(\mathbf{k}_{l}+\delta\mathbf{k}_{x}+\delta\mathbf{k}_{y})\rangle)$
$\displaystyle+\ln(\langle
u(\mathbf{k}_{l})|u(\mathbf{k}_{l}+\delta\mathbf{k}_{x})\rangle).$

Because the Chern number is real and integer and the Berry connection is
purely imaginary, as a result, the link variables are written as

$\begin{split}&U_{12}(\mathbf{k}_{l})=\ln\left(\frac{\langle
u(\mathbf{k}_{l})|u(\mathbf{k}_{l}+\delta\mathbf{k}_{x})\rangle}{|\langle
u(\mathbf{k}_{l})|u(\mathbf{k}_{l}+\delta\mathbf{k}_{x})\rangle|}\right),\\\
&U_{23}(\mathbf{k}_{l})=\ln\left(\frac{\langle
u(\mathbf{k}_{l}+\partial\mathbf{k}_{x})|u(\mathbf{k}_{l}+\delta\mathbf{k}_{x}+\delta\mathbf{k}_{y})\rangle}{|\langle
u(\mathbf{k}_{l}+\partial\mathbf{k}_{x})|u(\mathbf{k}_{l}+\delta\mathbf{k}_{x}+\delta\mathbf{k}_{y})\rangle|}\right),\\\
&U_{34}(\mathbf{k}_{l})=\ln\left(\frac{\langle
u(\mathbf{k}_{l}+\partial\mathbf{k}_{x}+\delta\mathbf{k}_{y})|u(\mathbf{k}_{l}+\delta\mathbf{k}_{y})\rangle}{|\langle
u(\mathbf{k}_{l}+\partial\mathbf{k}_{x}+\delta\mathbf{k}_{y})|u(\mathbf{k}_{l}+\delta\mathbf{k}_{y})\rangle|}\right),\\\
&U_{41}(\mathbf{k}_{l})=\ln\left(\frac{\langle
u(\mathbf{k}_{l}+\delta\mathbf{k}_{y})|u(\mathbf{k}_{l})\rangle}{|\langle
u(\mathbf{k}_{l}+\delta\mathbf{k}_{y})|u(\mathbf{k}_{l})\rangle|}\right),\end{split}$
(S11)

Substituting link variables in Eq. (S4), the Chern number reads

$C_{n}=\frac{1}{2\pi
i}\sum_{\mathbf{k}_{l}\in\textrm{BZ}}\ln(U_{12}U_{23}U_{34}U_{41}).$ (S12)

As obvious from Eq. (S12), to calculate the Chern number at
$\textbf{{k}}_{l}$, a counterclockwise loop is considered as represented in
Fig. S2(b).

Fig. S3:  Berry curvature distribution over $k_{\textrm{x}}$ and
$k_{\textrm{y}}$ for the (a) first, (b) second, (c) third, and (d) forth bands
of a YIG rod with radius $R_{d}^{0}=0.11a$, $\epsilon_{\textrm{n}}=15$, and
$\mu=\bm{\mu}$. The dashed triangles with the vortices of $\Gamma$, X, and M
represent the irreducible BZ. Fig. S4:  Dispersion diagrams of the TW with 5 ×
8 dielectric rods with $\epsilon_{t}=16$ and 5 × 8 YIG rods. (a)
$\eta_{\textrm{P}}=40\%$ to the trivial lattice, (b) $\eta_{\textrm{P}}=40\%$
to the non-trivial lattice, (c) $\eta_{\textrm{R}}=40\%$ to the trivial
lattice, and (d) $\eta_{\textrm{R}}=40\%$ to the non-trivial lattice. Black
and red curves represent topological modes. Blue and green colors show bulk
and defect modes, correspondingly. The orange arrow shows $a/\lambda=0.57$.
Fig. S5:  The structure of the TW containing trivial lattice of $5\times 6$
dielectric rods in air and non-trivial lattice of the $5\times 6$ YIG rods in
air. The TW under applying position disordering to the trivial lattice (a) and
(b) and to the non-trivial lattice (c) and (d). The TW under applying radius
disordering to the trivial lattice (i) and (j) and to the non-trivial lattice
(k) and (l). For (a), (c), (i), and (k), Gaussian waves propagate from left to
right and for (b), (d), (j), and (l), Gaussian waves propagate from right to
left. (e)–(h) are the simulation results for different $\epsilon_{\textrm{t}}$
for (a)–(d), respectively; (m)–(p) are the simulation results for different
$\epsilon_{\textrm{t}}$ for (i)–(l), respectively. $a/\lambda=0.570$ for
$\epsilon_{\textrm{t}}=16,18$, and $20$ and $a/\lambda=0.550$ for
$\epsilon_{\textrm{t}}=22$.

The Berry curvature of the first four bands for the non-trivial lattice
containing the YIG rod in air represents $C_{4}$ rotation symmetry over the
first BZ; the $C_{4}$ rotation symmetry of the Berry curvatures is inherited
from the same symmetry of the lattice (Fig. S3). Also, due to the broken time-
reversal symmetry and preserving inversion symmetry, the Berry curvatures
reveal an even function over the wavevector, $F(-\textbf{k})$ =
$F(\textbf{k})$. The integral of the Berry curvature for each band over the
first BZ is proportional to the Chern number. The Chern number for Band 1,
Band 2, Band 3, and Band 4 are $0,+1,-2$, and $-1$, respectively.

##  III Dispersion diagram calculation

The dispersion diagram is calculated by discretization of one of the master
equations (Eqs. (S13) and (S14)) using the finite-element method (FEM) under
applying Bloch boundary conditions (Eq. (S15)) as follows

$\displaystyle\frac{1}{\epsilon_{\textrm{r}}(\textbf{{r}})}\nabla\times\left[\frac{1}{\mu_{\textrm{r}}(\textbf{{r}})}\nabla\times\textbf{{E}}\right]$
$\displaystyle=\left(\frac{\omega}{c}\right)^{2}\textbf{{E}},$ (S13)
$\displaystyle\frac{1}{\mu_{\textrm{r}}(\textbf{{r}})}\nabla\times\left[\frac{1}{\epsilon_{\textrm{r}}(\textbf{{r}})}\nabla\times\textbf{{H}}\right]$
$\displaystyle=\left(\frac{\omega}{c}\right)^{2}\textbf{{H}},$ (S14)

$\begin{split}\textbf{{E}}(\textbf{{r}})&=\Phi_{\textbf{{E}}}(\textbf{{r}})e^{j\textbf{{K}}\cdot\textbf{{r}}},\\\
\textbf{{H}}(\textbf{{r}})&=\Phi_{\textbf{{H}}}(\textbf{{r}})e^{j\textbf{{K}}\cdot\textbf{{r}}},\\\
\end{split}$ (S15)

where E, H, K,
$\epsilon_{\textrm{r}},\mu_{\textrm{r}},\omega,\Phi_{\textbf{{E}}}$, and
$\Phi_{\textbf{{H}}}$ are the electric field, magnetic field, Bloch
wavevector, relative permittivity, relative permeability, frequency, electric-
periodic part of Bloch function and magnetic-periodic part of Bloch function,
respectively.

The FEM works in real-space for calculating the dependent variables of E, H,
$\omega$, and K by discretizing the master equation (either Eq. (S13) or
(S14)). This method is useful for calculating photonic dispersion diagrams of
disorder structures that are of experimental interest. Also, the method is
used to calculate the dispersion diagram of arbitrary complex geometries with
the material dispersion. In addition, discontinuities in the dielectric
function do not affect the convergence of the method. The disadvantage of this
method is requiring extensive computer memory. The main part of the FEM is
dividing the structure into small units which are called mesh elements. In
two-dimensional FEM simulations, mesh elements are triangular based on the
Delaunay algorithm. The FE method approximates independent variables using a
linear combination of basis functions (shape functions) that are solved using
an appropriate solver. Here, for calculating the dispersion diagrams of
trivial and non-trivial lattices and the TW, FEM method embedded in COMSOL
Multiphysics software is utilized.

To calculate the dispersion diagrams, a TW containing a supercell of $5\times
8$ dielectric rods with $\epsilon_{\textrm{t}}=16$ in air of a structure
consisting of the $5\times 8$ YIG rods in air is considered. The boundary
conditions of supercells for $x_{\textrm{min}}$ and $x_{\textrm{max}}$ are
Bloch boundary conditions and for $y_{\textrm{min}}$ and $y_{\textrm{max}}$
are continuity conditions. The dispersion diagrams of the TW under applying
$\eta_{\textrm{P}}=40\%$ to the trivial and non-trivial lattices (Figs. S4(a)
and (b)) and $\eta_{\textrm{R}}=40\%$ to the trivial lattice (Fig. S4(c))
represents two types of non-trivial mirrors. The non-trivial mirrors at
$a/\lambda<0.6$ and $a/\lambda>0.6$ represent edge modes with positive and
negative group velocities and Cg of $+1$ and $-1$, respectively. The TW under
applying $\eta_{\textrm{R}}=40\%$ to the non-trivial lattice represents no
unidirectional bandgap as shown in Fig. S4(d), because defect modes occupying
the bandgaps.

##  IV Unidirectionality transmissions and scatterings

We consider a TW containing $5\times 6$ dielectric rods as a trivial lattice
and $5\times 6$ YIG rods as a non-trivial lattice. The FEM module of COMSOL
Multiphysics software is utilized to numerically simulate the TW. Given
disorderings to the trivial and non-trivial lattices, transmissions are
recorded at the top (TTLR and TTRL) and bottom (TBLR and TBRL) of the TW,
respectively. Under applying position disordering to the trivial lattice
(Figs. S5 (a) and (b)), the scattering transmissions of TTLR and TTRL are
approximately zero (Figs. S5(e) and (f)). TBLR is higher than TBRL under
applying position disordering to the non-trivial lattice that results in
blocking the Gaussian wave that launched from the right side of the waveguide
(Figs. S5(c), (d), (g), and (h)). Under applying radius disordering to the
trivial lattice TTLR shows a higher value than TTRL (Figs. S5 (i), (j), (m),
and (n)). In contrast, applying radius disordering to the non-trivial lattice
represents approximately the same values for TBLR and TBRL, revealing
unidirectionality breaking (Figs. S5(k), (l), (o), and (p)).

##  V Bott index

The Bott index ($C_{B}$) is an integer number that can characterize both the
ordered and disordered systems (with the broken time-reversal symmetry). The
calculation procedure of the Bott index is as follows. Here we propose a
supercell of $6\times 6$ YIG rods in air with anti-periodic boundary
conditions in the $x$ and $y$ directions. The supercell is discretized into
$M^{2}$ coordinated points. $N$ eigenmodes, $[E_{z}(x,y)]_{M^{2}\times N}$,
with eigenfrequencies below specific frequency $f$ are calculated using COMSOL
Multiphysics software. Two Hermitian diagonal matrices of $[X]_{M^{2}\times
M^{2}}$ and $[Y]_{M^{2}\times M^{2}}$ with $x$ and $y$ coordinates as diagonal
elements are defined. Two diagonal matrices of $[U_{x}]_{M^{2}\times
M^{2}}=\textrm{exp}(i2\pi X/L_{x})$ and $[U_{y}]_{M^{2}\times
M^{2}}=\textrm{exp}(i2\pi Y/L_{y})$ are considered, where $L_{x}$ and $L_{y}$
are the lengths of the supercell in the $x$ and $y$ directions, respectively.
Band-projected position matrices of $[\tilde{U}_{X}]_{N\times
N}=[P^{\prime}]_{N\times M^{2}}[U_{x}]_{M^{2}\times M^{2}}[P]_{M^{2}\times N}$
and $[\tilde{U}_{Y}]_{N\times N}=[P^{\prime}]_{N\times
M^{2}}[U_{y}]_{M^{2}\times M^{2}}[P]_{M^{2}\times N}$ are defined, where prims
sign represents a transpose of the matrix. $P$ is the projection matrix to all
eigenmodes with frequencies below $f$. The square matrix is defined as
$[A]_{N\times N}=[\tilde{U}_{Y}]_{N\times N}[\tilde{U}_{X}]_{N\times
N}[\tilde{U}^{\prime}_{Y}]_{N\times N}[\tilde{U}^{\prime}_{X}]_{N\times N}$. A
diagonal matrix $[B]_{N\times N}$ is introduced with diagonal elements
containing the eigenvalues of $[A]_{N\times N}$. The Bott index at the
frequency $f$ is introduced as

$C_{B}=\frac{1}{2\pi}\textrm{Im}\\{\textrm{tr}(\textrm{ln}\\{[B]_{N\times
N}\\})\\},$ (S16)

where Im, tr, and ln are the imaginary part, trace, and Neperian logarithm,
respectively.

##  VI Electric field distribution

We study the electric field distribution $|E_{z}(x,y)|$ of a structure
consisting of the supercell containing $5\times 10$ YIG rods in air at the
bottom and a metal zone as a reflector at the top of it. The $|E_{z}(x,y)|$
distribution at each frequency and disordering is the avearge of 50
simulations. In Fig. S6, $|E_{z}(x,y)|$ distributions show unidirectionality
independent of the position disordering for the topological bandgaps of BGn2
and BGn3; the directionality is broken at the other frequencies.
$|E_{z}(x,y)|$ distributions for the structure under radius disordering
demonstrate the unidirectionality of BGn2 and BGn3 at low radius disordering
and unidirectionality breaking for the bandgaps at higher radius disordering,
as presented in Fig. S7. To some extent of increase in the radius disordering,
the unidirectionality appears at the lower frequencies, revealing TAL. These
results confirm the calculation results of Bott index in Fig. 4.

Fig. S6:  The $|E_{z}(x,y)|$ distributions of a structure consists of the TW
containing a non-trivial mirror of the $5\times 10$ YIG rods in air and a
metal bar as a trivial mirror at the top of it under position disordering at
different frequencies. Fig. S7:  The $|E_{z}(x,y)|$ distributions of a
structure consists of the TW containing a non-trivial mirror of the $5\times
10$ YIG rods in air and a metal bar as a trivial mirror at the top of it under
radius disordering at different frequencies.

##  VII Eigenfrequency analysis

This section describes the eigenfrequency analysis of the non-trivial lattice.
The lattice consists of $6\times 6$ YIG rods in air. The eigenfrequency of the
lattice under applying the radius disorder to the YIG rods as a function of
eigenfrequency is computed using the FE method embedded in COMSOL Multiphysics
software. The eigenfrequencies are computed for the average of an ensemble of
50 simulations. The boundary conditions are anti-periodic boundary conditions
in both the $x$ and $y$ directions. The computed eigenfrequencies illustrate
three main bandgaps of BGn1, BGn2, and BGn3 are in gray with no disorder as
illustrated in Fig. S8. By increasing the radius disorder, the
eigenfrequencies from the upper edge of the bandgaps of BGn1 and BGn2 fall
down to the bandgaps and experience red shifts. BGn1 and BGn2 disappear at
$\eta_{\textrm{R}}=50\%$ and $\eta_{\textrm{R}}=20\%$. By increasing the
radius disorder, eigenfrequencies experience red and blue shifts for BGn3.
Also, BGn3 disappears at $\eta_{\textrm{R}}=10\%$.

Fig. S8:  The eigenfrequencies of a non-trivial lattice containing of the
$6\times 6$ YIG rods in air under applying the radius disorder. Gray regions
show the bandgaps of BGn1, BGn2, and BGn3.