# Scattering theory of delicate topological insulators

Penghao Zhu  Department of Physics and Institute for Condensed Matter Theory,
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA  Jiho
Noh  Department of Mechanical Science and Engineering, University of Illinois
at Urbana–Champaign, Urbana, IL 61801 USA  Yingkai Liu  Department of Physics
and Institute for Condensed Matter Theory, University of Illinois at Urbana-
Champaign, Urbana, Illinois 61801, USA  Taylor L. Hughes  Department of
Physics and Institute for Condensed Matter Theory, University of Illinois at
Urbana-Champaign, Urbana, Illinois 61801, USA

###### Abstract

We study the scattering theory of delicate topological insulators (TIs), which
are novel topological phases beyond the paradigm of the tenfold way,
topological quantum chemistry, and the symmetry indicator method. We
demonstrate that the phase of the reflection amplitude can probe the delicate
topology by capturing a characteristic feature of a delicate TI. This feature
is the returning Thouless pump, where an integer number of charges are pumped
forward and backward in the first and second half of the adiabatic cycle
respectively. As a byproduct of our analysis we show that requiring additional
symmetries can stable the boundary states of a delicate TI beyond the
conventional requirement of a sharply defined surface. Furthermore, we propose
a photonic crystal experiment to implement a delicate TI and measure its
reflection phase which reveals the delicate topology.

††preprint: APS/123-QED

##  I Introduction

Delicate topology has been introduced recently as a fine-grained
classification of phases [1, 2] that were previously considered trivial in the
tenfold way [3, 4], topological quantum chemistry [5, 6, 7], and symmetry
indicator [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] classification
schemes. The characteristic feature of a class of delicate topological
insulators (TIs) is a $2\pi$-quantized difference between two Berry phases
defined over a pair of high symmetry lines in the Brillouin zone (BZ), and it
is delicate in the sense that this quantization can be nullified by adding
trivial bands to either the occupied or unoccupied subspace.

As we review below, since nontrivial Chern numbers are precluded in a delicate
TI by assumption, the $2\pi$-quantized difference in the Berry phase does not
indicate an adiabatic pumping of charge [20], but instead generates a
returning Thouless pump (RTP). Indeed, if we regard the momentum in the
direction perpendicular to the high symmetry lines as an adiabatic parameter,
an RTP indicates that an integer number of charges are pumped toward one
direction by an integer number of unit cells in the first half of the cycle,
and are pumped back to their starting point in the second half of the cycle
[1, 2, 21]. A non-vanishing RTP in a delicate TI guarantees that gapless
surface states will be localized on a _sharp_ boundary with no
deformations/reconstructions, which is weaker than the conventional TI bulk-
boundary correspondence. Such a sharp boundary is perhaps difficult to achieve
naturally in solid-state materials, but could be engineered in metamaterials
such as photonic or acoustic crystals.

Besides the gapless surface states on a sharp boundary and previous work by
some of us on surface magnetism [22], few robust experimental observables of
delicate TIs have been proposed. Here we plan to leverage the scattering
theory of topological phases to provide a practical route to extract delicate
topological data from a reflection matrix [23, 24, 25, 26]. Previous studies
have comprehensively discussed the scattering theory for stable topological
phases protected by internal symmetries, i.e., phases in the tenfold way
periodic table[24]. Recently, the study of scattering theory has been extended
to a wider range of topological phases, e.g., 2D higher-order TIs that have
gapped edges but gapless corners [27, 28]. However, the scattering theory of
delicate TIs, which may provide new insight and experimental observables for
delicate topology, is still absent. To address this, here we study the
scattering theory of delicate TIs protected by rotation/mirror symmetry and
propose a photonic experiment to measure our predicted reflection-phase
observable.

The remainder of the article is organized as follows. In Sec. II, we give a
brief review of delicate TIs. Then, in Sec. III, we investigate the
relationship between the number of pumped charges during an adiabatic process
and the reflection phase. We show that the pumped charges that lead to an RTP
in a delicate TI can be detected by the phase of the reflection amplitude on a
sharp boundary. We also show, for the first time, that if there are extra
symmetries on the high symmetry lines in the BZ, then the boundary modes, and
the nontrivial behavior of the reflection phase, can remain robust even when
the requirement of a sharp boundary is relaxed. Finally, in Sec. IV we propose
a photonic experiment to implement a delicate TI in a synthetic dimension
where measurements of the reflection phase are possible and can be used to
detect the bulk RTP and hence delicate topology. We end our article with
conclusions and remarks about open questions for future research in Sec. V.

##  II Review of delicate topology, RTP, and surface states

Before discussing the scattering theory, we shall first review delicate
topology and its characteristic RTP protected by rotation/mirror symmetry. For
clarity, let us first set up our notation. We separate 3D momentum
$\mathbf{k}$ in the BZ into $\mathbf{k}_{\perp}$ and $k_{\parallel}$ that are
respectively perpendicular or parallel to the direction of the _symmetry axis_
(i.e., the rotation axis in 3D and/or mirror axis in 2D). We call the subspace
spanned by $\mathbf{k}_{\perp}$ at each given $k_{\parallel}$ the reduced
Brillouin zone (rBZ). Then, we consider tight-binding Hamiltonians written in
a basis $\\{\left|\phi_{l_{i}}(\mathbf{R})\right\rangle\\}$ where $\mathbf{R}$
represents the unit cell coordinates, and $i=1,2,\ldots$ labels the orbitals
with _symmetry eigenvalues_ $l_{i}$ in each unit cell (i.e., eigenvalues of
the rotation/mirror operator). We assume that all
$\left|\phi_{l_{i}}\right\rangle$’s with different $l_{i}$ in one unit cell
have their center localized on the same symmetry axis, with respect to which
the symmetry eigenvalues are defined.

If a point in the rBZ remains invariant under a rotation/mirror transformation
up to a reciprocal lattice vector, we then call this point a high symmetry
point and denote it as $\bm{\Lambda}$. At each high symmetry point, we define
the Berry phase of the Bloch bands in sector $l$ along the direction of the
symmetry axis as

$\gamma_{l}(\bm{\Lambda})=\int_{0}^{2\pi}dk_{\parallel}\sum_{n}i\langle
u^{l}_{n}(k_{\parallel},\bm{\Lambda})|\partial_{k_{\parallel}}u^{l}_{n}(k_{\parallel},\bm{\Lambda})\rangle,$
(1)

where $\left|u_{n}^{l}(k_{\parallel},\bm{\Lambda})\right\rangle$ is the $n$-th
eigenband with rotation/mirror eigenvalue $l$ at
$\mathbf{k}=(k_{\parallel},\bm{\Lambda})$, and $i\langle
u^{l}_{n}(k_{\parallel},\bm{\Lambda})|\partial_{k_{\parallel}}u^{l}_{n}(k_{\parallel},\bm{\Lambda})\rangle\equiv\mathcal{A}_{l}(\mathbf{k})$
is the Berry connection, and $n$ is summed over the whole Hilbert space, i.e.,
occupied and unoccupied states. The Berry phase $\gamma_{l}(\bm{\Lambda})$ has
the physical meaning of polarization along the symmetry axis contributed by
all states with momentum $\bm{\Lambda}$ and symmetry eigenvalue $l$. We note
that in this work we always assume the Bloch Hamiltonian is periodic in
momentum space, i.e., $H(\mathbf{k}+\mathbf{G})=H(\mathbf{k})$ where
$\mathbf{G}$ is a reciprocal lattice vector. For simplicity we also ignore the
internal structure of the unit cell, i.e., we assume all orbitals in one unit
cell have the same position coordinates within the unit cell, which we take to
be the position reference point. More detailed discussions which relax this
assumption can be found in Ref. 2.

With $\gamma_{l}(\bm{\Lambda})$ defined in Eq. (1), let us now discuss the
$2\pi$-quantized difference between $\gamma_{l}$ at a pair of high symmetry
points in the rBZ. We denote the set of symmetry eigenvalues of all occupied
(unoccupied) bands at $\bm{\Lambda}$ as $l_{v}(\bm{\Lambda})$
($l_{c}(\bm{\Lambda})$). If the intersection of $l_{v}(\bm{\Lambda})$ and
$l_{c}(\bm{\Lambda})$ is empty, i.e., the mutually disjoint condition
$l_{v}(\mathbf{\Lambda})\cap l_{c}(\mathbf{\Lambda})=\varnothing$ is
satisfied, then: (i) the Berry phase $\gamma_{v}(\bm{\Lambda})$ of all
occupied bands at a high symmetry point $\bm{\Lambda}$ can be expressed as
$\gamma_{v}(\bm{\Lambda})=\sum_{l\in l_{v}}\gamma_{l}(\bm{\Lambda})$, and (ii)
$\gamma_{l}(\bm{\Lambda})\equiv 2\pi n$ with $n$ an integer for any $l$. Point
(i) is obviously true, so let us now explain point (ii). At a high symmetry
point $\bm{\Lambda}$ in the rBZ, the Bloch Hamiltonian is block-diagonal. Each
of its blocks, $H_{l}(k_{\parallel},\bm{\Lambda})$, can be viewed as a 1D
tight-binding Hamiltonian in the symmetry sector labeled by $l$, i.e., the
Hilbert space spanned by all $\left|\phi_{l_{i}}\right\rangle$’s with
$l_{i}=l$. The Berry phase of _all_ Bloch bands (i.e., both occupied and
unoccupied Bloch bands) of such a 1D tight-binding model is always an integer
multiple of $2\pi$, i.e., $2\pi n(\mathbf{\Lambda})$ where
$n(\mathbf{\Lambda})$ is the integer multiple at $\mathbf{\Lambda}$. This is
because the Wannier orbitals constructed by inverse-Fourier transforming all
Bloch bands are just the basis orbitals centered at the reference point up to
an integer number of lattice constants.

After establishing points (i) and (ii), we can conclude that if at all
$\mathbf{\Lambda}$’s, the mutually disjoint condition is satisfied, then the
difference between two Berry phases at a pair of high symmetry points, i.e.,
$\gamma_{v}(\bm{\Lambda}_{1})-\gamma_{v}(\bm{\Lambda}_{2})$, should be
quantized to $2\pi m$ with $m$ an integer. Given that the polarization along
the symmetry axis should change continuously from $\bm{\Lambda}_{1}$ to
$\bm{\Lambda}_{2}$ in the rBZ, the $2\pi m$ quantization of
$\gamma_{v}(\bm{\Lambda}_{1})-\gamma_{v}(\bm{\Lambda}_{2})$ necessarily
indicates an RTP if the system has vanishing Chern numbers, as will be the
case for delicate topology111Remember that to discuss the delicate topology in
a system, the nontrivial Chern number is precluded by assumption.. Explicitly,
the Berry phase difference indicates that $m$ charges are pumped toward one
direction along the symmetry axis as we move from $\bm{\Lambda}_{1}$ to
$\bm{\Lambda}_{2}$, and they are then pumped back when we complete the other
half the loop in the rBZ from $\bm{\Lambda}_{2}$ to $\bm{\Lambda}_{1}$. In the
following, we say the system has an RTP $m$ along the path
$\bm{\Lambda}_{1}\to\bm{\Lambda}_{2}\to\bm{\Lambda}_{1}$ if
$\gamma_{v}(\bm{\Lambda}_{1})-\gamma_{v}(\bm{\Lambda}_{2})=2\pi m$. The RTP is
protected as long as the bulk energy gap, the rotation/mirror symmetry, and
the Hilbert space restriction $l_{v}(\mathbf{\Lambda})\cap
l_{c}(\mathbf{\Lambda})=\varnothing$ are preserved. This kind of topology is
delicate in the sense that adding extra trivial bands (in either the occupied
or unoccupied subspaces) that make the intersection of $l_{v}$ and $l_{c}$ at
any $\mathbf{\Lambda}$ non-empty, can destabilize the RTP.

If we apply Stokes theorem we find

$\gamma_{v}(\bm{\Lambda}_{1})-\gamma_{v}(\bm{\Lambda}_{2})=\int
d\mathbf{k}_{\parallel}\int_{\bm{\Lambda}_{1}}^{\bm{\Lambda}_{2}}dk_{\perp}\operatorname{Tr}\Omega_{v}(\mathbf{k}),$
(2)

where $\Omega_{v}(\mathbf{k})$ is the Berry curvature in the occupied
subspace, and the integral $dk_{\perp}$ is on a 1D path from
$\bm{\Lambda}_{1}$ to $\bm{\Lambda}_{2}.$ Similarly, we can also define the
Berry phase along the direction of the symmetry axis of all unoccupied bands
as $\gamma_{c}(\bm{\Lambda})=\sum_{l\in l_{c}}\gamma_{l}(\bm{\Lambda})$. We
denote the Berry curvature in the unoccupied subspace as
$\Omega_{c}(\mathbf{k}).$ For any Bloch Hamiltonian
$\operatorname{Tr}\Omega_{v}(\mathbf{k})=-\operatorname{Tr}\Omega_{c}(\mathbf{k})$,
so we can conclude that the RTP in the unoccupied subspace is opposite to that
in occupied subspace, i.e.,
$\gamma_{v}(\bm{\Lambda}_{1})-\gamma_{v}(\bm{\Lambda}_{2})=-(\gamma_{c}(\bm{\Lambda}_{1})-\gamma_{c}(\bm{\Lambda}_{2})).$

We also recall that delicate TIs with nonzero RTP generate gapless boundary
modes on a sharp boundary. A sharp boundary means that the Hamiltonian of a
system with boundaries is almost the same as the corresponding periodic
Hamiltonian except that all hopping matrix elements across the boundaries are
turned off. Intuitively, the bulk RTP implies that states with specific
symmetry eigenvalues protrude from the bulk at the boundary, and these states
must be “compensated” by surface states with complementary symmetry
eigenvalues to keep the balance of states in different symmetry sectors in
each layer. These boundary states are gapless on only a sharp boundary because
the sharpness of the boundary constrains the energy of the surface states and
forces them to cross the insulating bulk gap. This intuitive picture will
become more transparent in the following example.

To illustrate all the concepts and conclusions discussed above, let us take
the $C_{4}$-symmetric two-band tight-binding model of the Hopf insulator
constructed by Moore, Ran, and Wen (MRW) [29]. The Hamiltonian of the MRW
model is constructed from the well-known Hopf map:

$\displaystyle z=(z_{1}+iz_{2},z_{3}+iz_{4})^{T},$ (3)
$\displaystyle\mathbf{d}=z^{{\dagger}}\bm{\sigma}z,\
\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z}),$ $\displaystyle
H_{\text{MRW}}(\mathbf{k})=\mathbf{d}\cdot\bm{\sigma},$

where $\sigma_{x},\sigma_{y},\sigma_{z}$ are Pauli matrices, and

$\displaystyle z_{1}=\sin k_{x},z_{2}=\sin k_{y},z_{3}=\sin k_{z},$ (4)
$\displaystyle z_{4}=u-\cos k_{x}-\cos k_{y}-\cos k_{z}.$

The Hamiltonian $H_{\text{MRW}}(\mathbf{k})$ has a $C_{4}$ rotation symmetry
along the $z$-axis with rotation operator
$C_{4z}=\exp\left(i\pi\sigma_{z}/4\right)$. For $u\neq\pm 1,\pm 3$,
$H_{MRW}(\mathbf{k})$ is gapped and always has $l_{v}=\exp(i\pi/4)$ and
$l_{c}=\exp(-i\pi/4)$ at both high symmetry points $\Gamma$ ($k_{x}=k_{y}=0$)
and $M$ ($k_{x}=k_{y}=\pi$). This satisfies the mutually disjoint Hilbert
space restriction discussed above. We find that $\gamma_{v}(\Gamma)$ and
$\gamma_{v}(M)$ always have a $2\pi$-quantized difference when $1<|u|<3$ where
the bulk Hopf invariant is $-1$ 222When $|u|<1$, the bulk Hopf number is $2$
and $\gamma_{v}(\Gamma)-\gamma_{v}(M)=0$. However, there is a $2\pi$-quantized
difference between $\gamma_{v}(\Gamma)$ and $\gamma_{v}(X)$ that is protected
by the two-fold rotation symmetry. [2], which indicates a non-vanishing RTP
along $M\to\Gamma\to M$ as shown in Fig. 1 (a).

Figure 1: The left panel of (a) shows the RTP (change of $\gamma_{v}/(2\pi)$)
along the dotted-line path shown in the right panel. We include plots for the
MRW model with $u=-3.5,-1.5,0,1.5$. The right panel of (a) shows $C_{4}$
rotation invariant points $\Gamma$ and $M$ in the rBZ. (b) Illustration
showing bulk states protruding into the surface (green (orange) color for
occupied (unoccupied) states). The red (blue) line represents surface states
localized on the top (bottom) surface with the compensating angular momentum
at $\Gamma$ and $M$. (c) shows an exact diagonalization calculation of the
gapless surface states of the MRW model at $u=1.5$ with 10 unit cells along
the open $z$-direction (with sharp boundary), and periodic boundaries in
$x,y$. The surface band localized on the top ($z=10$) and the bottom ($z=1$)
surface are highlighted by red and blue colors, respectively.

Let us now focus on the MRW model with $u=1.5$ where
$\gamma_{v}(\Gamma)-\gamma_{v}(M)=2\pi,$ and
$\gamma_{c}(\Gamma)-\gamma_{c}(M)=-2\pi$. This RTP implies that in the bulk,
the occupied (unoccupied) states at $\Gamma$ are one layer higher (lower) in
the $z$-direction than the occupied (unoccupied) states at $M$, as illustrated
by red (orange) lines in Fig. 1 (b). Thus, on the top surface near $z=N$ there
will be one extra occupied (unoccupied) state with $l=l_{v}$ ($l=l_{c}$)
protruding from the bulk to the surface at $\Gamma$ ($M$). If we count both
occupied and unoccupied states there should always be states with $l=l_{v}$
and $l_{c}$ at both $\Gamma$ and $M$ in each layer along the $z$-direction.
Thus, the extra state (protruding from the bulk) with $l=l_{v}$ ($l=l_{c}$) at
$\Gamma$ ($M$) in the surface layer should be “compensated” by a surface state
with $l=l_{c}$ at $\Gamma$ ($l=l_{v}$ at $M$).

While the previous argument is generally true, the energies of the
compensating surface states are not generically required to cross the energy
gap, i.e., the states localized on the surface could only have energies in the
bulk band regions. If we make a further assumption of a sharp boundary then we
can make more definitive statements about surface state energies. The
sharpness of the boundary guarantees that each diagonal block
$H_{l}(\bm{\Lambda})$ of the effective 1D tight-binding Hamiltonian at $M$ or
$\Gamma$ is a block Toeplitz matrix, i.e., the hopping matrix elements between
unit cells at $\mathbf{R}$ and $\mathbf{R}^{\prime}$ depend on only
$\mathbf{R}-\mathbf{R}^{\prime}$ and satisfy
$H_{l,\mathbf{R}\mathbf{R}^{\prime}}\equiv
H_{l,\mathbf{R}-\mathbf{R}^{\prime}}=(H_{l,\mathbf{R}^{\prime}-\mathbf{R}})^{{\dagger}}$.
The spectrum theorem of block Toeplitz matrices tells us that in the
thermodynamic limit, the spectrum of $H_{l}(\bm{\Lambda})$ is bounded by the
spectrum of its corresponding Bloch Hamiltonian $H_{l}(k_{z},\bm{\Lambda})$
(i.e., the corresponding Hamiltonian under periodic boundary conditions) [2].
Thus, the surface state at $\Gamma$ ($M$) in the $l=l_{c}$ ($l=l_{v}$) sector
must lie in the bulk unoccupied (occupied) band of the energy spectrum. This
then guarantees that the surface bands must cross the bulk gap as $k_{\perp}$
goes from $\Gamma$ to $M$ shown in Fig. 1 (c). The same arguments can also be
made for the bottom surface. In summary, if we relax the sharpness condition,
the state-compensation argument is still valid, but the surface states are no
longer guaranteed to have energies in the insulating gap unless additional
symmetries are imposed as we show below.

##  III Scattering Theory for Delicate Topology

###  III.1 Pumped charge and reflection phase

From the review in Sec. II, we see that the RTP over a loop in the rBZ
describes an adiabatic pump in an effective 1D system. During this pump,
charge is pumped in one direction during part of the cycle, and then returns
to its starting point at the end. To understand the relation between the RTP
and the phase of a reflection amplitude in a scattering theory, we will first
discuss the general correspondence between the reflection phase and the charge
pumped in an adiabatically driven 1D wire. For simplicity, in this section, we
denote the general adiabatic parameter as $\tau$, and assume the adiabatic
period is from $\tau=0$ to $\tau=T$.

As shown in Fig. 2 (a), to study the scattering problem in a 1D geometry, we
connect two reservoirs to a 1D wire; one on the left (L) and another on the
right (R). Fig. 2 (a) shows incoming and outgoing waves where the
wavefunctions in the reservoirs take the form
$\psi^{L}=Ae^{ik(x-x_{L})}+Be^{-ik(x-x_{L})}$ and
$\psi^{R}=Ce^{ik(x-x_{R})}+De^{-ik(x-x_{R})}$, where superscripts $L$ and $R$
label the left and right reservoirs, and $x_{L}$ and $x_{R}$ are the reference
points where the reflection and the transmission occur with the left and right
reservoirs respectively. The incoming channels $A$ and $D$ are related to the
outgoing channels $B$ and $C$ through the scattering matrix $S$:

$S\begin{pmatrix}A\\\ D\end{pmatrix}\equiv\begin{pmatrix}r&t^{\prime}\\\
t&r^{\prime}\end{pmatrix}\begin{pmatrix}A\\\
D\end{pmatrix}=\begin{pmatrix}B\\\ C\end{pmatrix},$ (5)

where $r$ ($r^{\prime}$) and $t$ ($t^{\prime}$) are the reflection and
transmission matrices of the left (right) interfaces. If the input waves have
an energy that falls in a bulk energy gap of the wire in the central region,
then there will be no bulk extended eigenstates that facilitate the
transmission across the gapped wire. In this case, any transmission would come
from only the overlap between possible boundary states that are exponentially
localized on the two boundaries. However, this process is exponentially
suppressed as a function of the size of the central wire region[30, 24]. In
such a scenario we therefore expect that in the thermodynamic limit the
reflection matrices $r$ and $r^{\prime}$, which respectively describe the
reflection of input waves from the left and right interfaces, exponentially
approach unit modulus, while the transmission matrices $t$ and $t^{\prime}$,
which respectively describe the transmission of input waves from the left and
right interfaces, are exponentially suppressed. Generally, the wavefunctions
in both reservoirs can be written as
$\psi^{L}=A(e^{ik(x-x_{L})}+re^{-ik(x-x_{L})})+t^{\prime}De^{-ik(x-x_{L})}$
and $\psi^{R}=D(r^{\prime}e^{ik(x-x_{R})}+e^{-ik(x-x_{R})})+tAe^{ik(x-x_{R})}$
with $k\geqslant 0$. As argued above, $t$ and $t^{\prime}$ go to zero in the
thermodynamic limit if the input waves have energy that falls in a bulk energy
gap of the wire. In this scenario the wavefunctions in both reservoirs become
$\psi^{L}=A(e^{ik(x-x_{L})}+re^{-ik(x-x_{L})})$ and
$\psi^{R}=D(r^{\prime}e^{ik(x-x_{R})}+e^{-ik(x-x_{R})}),$ where $r$ and
$r^{\prime}$ are both unit modulus.

Figure 2: (a)A schematic illustration showing the setup of the scattering
process in a 1D geometry. (b) An illustration showing the energy spectra of
the left and right reservoirs and the central 1D wire. The orange horizontal
line denotes the Fermi level $E_{F}$. The red dots represent charges, and the
dotted red arrow indicates a pump of charges from the left reservoir to the
right reservoir.

Let us now consider the reservoirs to be closed (finite but large, i.e.,
$|L_{x}|\rightarrow+\infty$) boxes, i.e., any wavefunction in the left or
right reservoirs satisfies $\psi^{R}(x=L_{x})=\psi^{L}(x=-L_{x})=0$. If we
express the reflection for input waves with an energy in the bulk gap of the
1D wire as $r=e^{-i\phi}$ ($r^{\prime}=e^{-i\phi^{\prime}}$), the boundary
conditions at $\pm L_{x}$ dictate that $k$ takes discrete values for a given
$\phi$:

$\displaystyle
k^{L}_{n}=\frac{2n+1}{2(L_{x}+x_{L})}\pi+\frac{\phi}{2(L_{x}+x_{L})},$ (6)
$\displaystyle
k^{R}_{n}=\frac{2n+1}{2(L_{x}-x_{R})}\pi+\frac{\phi^{\prime}}{2(L_{x}-x_{R})}.$

With this set up we will first review the scattering problem when the 1D wire
represents a Thouless pump [20] with Chern number one. During such a pumping
process we imagine tuning the wire adiabatically over a period (i.e., from
$\tau=0$ to $\tau=T$), such that one charge is pumped from the left reservoir
to the right reservoir as indicated by the red dotted arrows in Fig. 2 (b).
Since the wire returns to its initial state at $\tau=T,$ the reflection phases
$\phi$ and $\phi^{\prime}$ can change only by integer multiples of $2\pi$ if
they change continuously during this process. Thus $r$ should return to the
same value it had at $\tau=0$. Indeed, previous work on the scattering
formalism of the Thouless pump with unit Chern number has shown that if $\phi$
($\phi^{\prime}$) changes continuously, then it must have a non-vanishing
change of $2\pi$ ($-2\pi$) during the period [24, 30, 31, 32].

Let us try to understand the microscopic mechanism of this process. From Eq.
(6) we can identify that the change of $\phi$ $(\phi^{\prime})$ generates a
spectral flow such that $k^{L}_{n}\rightarrow k^{L}_{n+1}$
($k^{R}_{n+1}\rightarrow k^{R}_{n}$). Hence, the left (right) reservoir loses
(gains) one charge near the Fermi level, which must come through the
assistance of the gapless boundary states of the central wire that appear as
we tune $\tau$. The boundary states accommodate the charge transfer indicated
in Fig. 2 (b). Therefore, a $2\pi$ winding of $\phi$ is linked to the charge
transfer and can be observed assuming $\phi$ changes continuously when we
adiabatically pump the wire.

With the RTP in mind, let us consider a scenario where we know one charge has
been pumped from the left reservoir to the right reservoir (with the
assistance of gapless boundary states) over an incomplete part of the
adiabatic period. Since the period is incomplete, the wire has not yet
returned to its initial state, and one cannot assert that $\phi$ and
$\phi^{\prime}$ must have changed by a multiple of $2\pi$. However, the left
(right) reservoir still loses (gains) one charge, and hence $\phi$
($\phi^{\prime}$) must exhibit some change during the incomplete cycle such
that one charge enters the wire from the left reservoir, and one charge leaves
the wire to enter the right reservoir. We will illustrate in the following
that if one charge is continuously pumped from the left reservoir to the right
reservoir, then $\phi$ ($\phi^{\prime}$) crosses $(2n+1)\pi$ from below
(above) once for some value of $n$ (i.e., $r$ ($r^{\prime}$) must cross $-1$
counterclockwise (clockwise) once on the unit circle). Indeed, this behavior
will be a signature of the RTP.

To see this explicitly let us first consider the left reservoir. We recall
that the only valid $k$ are positive because we have identified $e^{ikx}$ as
input waves from the left. Hence, the allowed $k$ at $\tau=0$ are labeled by
the set of $n$ such that $2k^{L}_{n}(L_{x}+x_{L})=(2n+1)\pi+\phi\geq 0.$ From
this we see that if $\phi$ increases from a value less than $-(2n_{0}+1)\pi$
(for some $n_{0}$) to a value greater than $-(2n_{0}+1)\pi$ as we tune $\tau$,
then $2k_{n_{0}}^{L}(L_{x}+x_{L})=(2n_{0}+1)\pi+\phi$ becomes positive, and
thus one more allowed state appears at the bottom of the energy spectrum. In
the thermodynamic limit, the number of states under the Fermi level in the
reservoir is uniquely determined by the Fermi level $\mu$, i.e.,
$N(E\leq\mu)=\int_{0}^{\mu}\rho(E)dE$, where $\rho(E)$ is the density of
states for free particles, and the Fermi level $\mu$ is fixed to lie in the
bulk gap of the wire over the whole adiabatic process. Hence, one extra state
at the bottom of the energy spectrum implies that one occupied state at the
Fermi level is pushed beyond the Fermi level, and thus enters the wire from
the left reservoir.

Now let us consider the right reservoir. If a state from the wire enters the
right reservoir, we need $\phi^{\prime}$ to decrease from a value greater than
$-(2n_{0}^{\prime}+1)\pi$ to a value less than $-(2n_{0}^{\prime}+1)\pi$ (for
some $n_{0}^{\prime}$), which will remove a state with wave vector
$k_{n_{0}^{\prime}}^{R}$ at the bottom of the spectrum and thus make room at
the Fermi level for the electron coming from the wire. In conclusion, we have
shown that if one charge is continuously pumped from the left reservoir to the
right reservoir, then $\phi$ ($\phi^{\prime}$) crosses $(2n+1)\pi$
($(2n^{\prime}+1)\pi$) from below (above) once for some value of $n$
($n^{\prime}$), i.e., $r$ ($r^{\prime}$) must cross $-1$ counterclockwise
(clockwise) once on the unit circle.

This conclusion can be generalized to multi-channel cases with $n$ pumped
charges from the left reservoir to the right reservoir, where $r$ is generally
a matrix. For an input wave with energy in the bulk gap of the 1D wire,
eigenvalues of the reflection matrix $r$ must cross $-1$ counterclockwise
(clockwise) $n$ times when $n$ charges have been pumped from the left (right)
reservoir to the right (left) reservoir 333The mathematical proof of the
correspondence between a Thouless pump and the reflection phase winding in
Ref. 30 implies the same conclusion.. Similar conclusions can also be drawn
for the eigenvalues of $r^{\prime}$. Though our argument relies on the
occupation and spectrum of the states in the reservoirs, the above conclusions
drawn for the reflection phase can be tested by looking at the reflection of
an input wave at a given energy in the bulk gap of only the wire without
considering states at any other energy [see the calculation of reflection
phase in Appendix A].

###  III.2 Probing the delicate TI by reflection phase on a sharp boundary

We can apply the above conclusions to the case of an RTP. We anticipate that
an RTP can be detected by counting the net number of crossings of the
eigenvalues of $r$ with $e^{i\pi}=-1$ during the first half of the adiabatic
cycle from $\tau=0$ to $\tau=T/2$. However, this conclusion can be reached
only if we input waves on a sharp boundary where there are gapless boundary
modes (see Fig. 1 (c)) that assist the charge exchange between the reservoirs
and the quantum wire. We need to also be sure to count the crossings with
direction. Namely, if an eigenvalue crosses from below (counterclockwise in
the complex plane) we count it as a positive crossing, otherwise it is a
negative crossing. As discussed in Sec III.1, the net crossing over a half
period for an insulator with sharp boundary is fixed by the number of charges
pumped from the left reservoir to the right reservoir, i.e., the value of the
RTP. Hence, the net number of crossings of $\phi$ with $\pi$ cannot be removed
by a continuous deformation of the Hamiltonian that preserves the bulk energy
gap, the symmetry, and the sharp boundary. The existence of such protected
crossings of the reflection phase distinguishes a non-trivial RTP from a
trivial band insulator. For example, a trivial insulator that has a momentum-
independent Hamiltonian will exhibit a $\phi$ fixed to $0$ during the whole
period, as illustrated in Fig. 3 (a). This distinction is made more apparent
when we have an RTP where $n>1.$ In this case the reflection phase has $n>1$
net crossings with $\pi$ over a half period. In Fig. 3 (b) we plot $\phi$
versus $\tau$ for an example with $n=2.$ The figure clearly shows a non-
contractible structure as $\tau$ goes from $0$ to $T/2$, i.e., $\phi$ first
winds from $0$ to $2\pi$, and then keeps increasing enough to intersect $\pi$
again, but does not finish the second winding.

Figure 3: (a) shows the crossing of the reflection phase $\phi$ with $\pi$. As
discussed in the main text, the reflection phase $\phi$ of a delicate TI with
RTP cannot be continuously deformed into the trivial case where $\phi=0$ for
all $\tau$ while maintaining the bulk gap, symmetry, and sharp boundary. (b)
illustrates the noncontractible structure of $\phi$ for a delicate TI with an
RTP of two. The $x$-axis in (a) and (b) represent the adiabatic parameter, and
$T$ is the period of the adiabatic cycle. (c) shows phases of two eigenvalues
of the reflection matrix for the MRW model with $u=1.5$. (d) shows the
returning winding of $\operatorname{arg}(\operatorname{det}r)$ for the MRW
model with $u=-3.5,-1.5,0.5,1.5$.

More explicitly, let us demonstrate that one can observe a returning winding
in the phase of $\operatorname{det}r$ for two-band delicate TIs protected by
rotation/mirror symmetry. In order to have delicate topology we want the two
basis orbitals in each unit cell to have different symmetry eigenvalues. In a
basis where the rotation/ mirror operator is diagonal we want the
rotation/mirror generator to be $\sigma_{z}$. Then, at high symmetry points
$\bm{\Lambda}$ in the rBZ, $H(k_{\parallel},\bm{\Lambda})$ is proportional to
$\sigma_{z}$, which has an emergent chiral symmetry $\mathcal{C}=\sigma_{x}$.

The emergent chiral symmetry will enforce the reflection matrix $r$ to be
unitarily similar to $r^{{\dagger}}$ [23, 24]. Thus, one can find a basis of
incoming and outgoing states such that all eigenvalues of $r$ are real (i.e.,
$\pm 1$). In such a basis, previous work on the reflection matrices of chiral
symmetric topological phases shows that the absolute value of the chiral
winding number is equal to the number of $-1$ eigenvalues of $r$ [23, 24].
Since $H(k_{\parallel},\bm{\Lambda})\propto\sigma_{z}$ always has chiral
winding number zero, we can conclude that all eigenvalues of $r$ at
rotation/mirror invariant $\bm{\Lambda}$ in the rBZ must be $1$ in the basis
where all eigenvalues of $r$ are real. Hence, the phase $\phi$ of each
eigenvalue can only continuously change by integer multiples of $2\pi$ over an
adiabatic path connecting a pair of rotation/mirror invariant $\bm{\Lambda}$.
With this constraint, if the reflection phase changes continuously and crosses
$\pm\pi$ $m$ times when required by a nonzero bulk RTP with value $m$, it is
straightforward to see that the phase of $\operatorname{det}r$,
$\operatorname{arg}(\operatorname{det}r)$ is basis-independent, and has a net
winding of $2m\pi$ over half a period. In the second half period, there will
be a corresponding returning winding. Thus, for any 2-band delicate TIs,
$\operatorname{arg}(\operatorname{det}r)$ is expected to show a returning
winding that replicates the RTP.

As an example, let us consider the 2-band MRW model with $u=1.5$ and $10$ unit
cells along the $z$-direction. We identify $\tau=0,T$ with the $M$-point in
the rBZ, and identify $\tau=T/2$ with the $\Gamma$-point in the rBZ. Thus, the
adiabatic pump from $\tau=0$ to $\tau=T$ is now identified with the dotted
line indicated in Fig. 1 (a). From $M$ (i.e., $\tau=0$) to $\Gamma$ (i.e.,
$\tau=T/2$), one charge is transferred from the left reservoir to the quantum
wire with the assistance of gapless boundary states (see Fig. 1 (c)) localized
on the boundary at $z=1.$ Additionally, one charge is transferred from the
quantum wire to the right reservoir with the assistance of gapless boundary
states (see Fig. 1 (c)) localized on the boundary at $z=10$. To confirm our
predictions we numerically calculate the two-channel reflection matrix for a
grid of values on the $M-\Gamma-M$ line, and plot the phases of its two
eigenvalues in Fig. 3 (c). As expected, we see one of the two phases,
$\phi_{1}$, crosses $\pi$ once in the first half period, and once in the
second half-period which captures the bulk RTP. We note that at $\Gamma$,
$\phi_{1}$ and $\phi_{2}$ are not real because the basis we use for input and
output waves in the numerical calculation is not the basis that makes
$r=r^{{\dagger}}$. However, $\phi_{1}=-\phi_{2}\ \operatorname{mod}\ 2\pi$,
which is consistent with the fact that $r$ is unitarily similar to
$r^{{\dagger}}$ for a chiral symmetric 1D chain. Furthermore, we plot
$\operatorname{arg}(\operatorname{det}r)$ over the $M-\Gamma-M$ line in the
rBZ in Fig. 3 (d). We see that the returning winding of
$\operatorname{arg}(\operatorname{det}r)$ exactly duplicates the Berry phase
calculations of the RTPs shown in Fig. 1 (a). The details about calculations
of the reflection matrix for 1D chains are given in Appendix A.

###  III.3 Beyond sharp boundaries

If we relax the sharpness requirement of the boundaries, the boundary modes
can be gapped out by adding local potentials on the boundary. In such a
scenario there will no longer be charge exchanges between the reservoirs and
the quantum wire. For example, by adding local potentials on the boundary that
decrease (increase) the energy of the red (blue) color boundary band in Fig. 1
one can gap out the surface states as shown in Fig. 4 (a) 444Details about the
local potential can be found in Appendix B.. Since the surface states no
longer cross the Fermi level (of the quantum wire or the reservoirs), there
cannot be charge exchange between the reservoir and the quantum wire, and thus
the reflection phase will not cross $\pi$ anymore as we confirm in Fig. 4 (b).
Instead, the pumped charge during the first half-period accumulates on the
boundary at $z=10$ and polarizes the system along the $z$-direction, as shown
in Fig. 4 (c)

Figure 4: (a) Gapped surface states of the MRW model when the boundary is
decorated with a local potential. The MRW model has 10 unit cells along the
$z$-direction and $u=1.5$. (b) shows the corresponding reflection phases that
no longer cross $\pi$ since the boundary states are gapped. (c) shows the
change of the charge distribution during the first half of the adiabatic cycle
at $M$ (left) and $\Gamma$ (right). At $M$, each unit cell has one charge and
the combined charge in the 3 unit cells nearest each boundary is $Q=3$. At
$\Gamma$, by looking at the total charge $Q$ on the three unit cells closest
to the bottom boundary at $z=1$ (the top boundary at $z=10$), one can observe
that there is exactly one less (more) charge on the bottom (top) boundary
comparing to that at $M$.

The charge build-up mechanism that breaks the connection between the RTP and
the reflection phase winding gives us a key hint about how to restore it.
Indeed, to guarantee the nontrivial behavior of the reflection matrix
discussed in Sec. III.2, we just need to make sure that the charge exchange
between the reservoirs and the quantum wire is always allowed and that the
system cannot polarize. In general, extra symmetries can forbid the ground
state of the quantum wire at $\tau=0$ and $\tau=T/2$ from having a polarized
charge distribution like that in Fig. 4 (c). As a result, if we impose such
symmetries the charge exchange between the reservoirs and the quantum wire
would be guaranteed when there is a nonzero RTP in the quantum wire.

To implement the above requirement on the charge distribution, one can impose
symmetries on the 1D Hamiltonians at the high symmetry points in the rBZ,
i.e., impose symmetries at $\tau=0$ and $\tau=T/2$ in the adiabatic pump
language. One choice is the requirement of chiral or particle-hole symmetry at
high symmetry points in the BZ. In fact, for an insulator with a unity-valued
RTP and chiral symmetry at $\tau=0$ and $\tau=T/2$ we know: (i) the unoccupied
subspace has an opposite RTP to the occupied subspace, i.e., if in the
occupied subspace one charge accumulates on the right boundary, then in the
unoccupied subspace, one charge should accumulate on the left, and (ii) in
each unit cell, the chiral symmetry enforces the charge distribution of the
occupied states to be the same as that of the unoccupied states. It is
straightforward to see that (i) and (ii) can be simultaneously satisfied only
if there is no charge accumulation on the boundary. Then, the pumped charges
during the first half-period must go into the reservoir with the help of
gapless boundary modes. One can also straightforwardly argue that particle-
hole symmetry has the same effect.

Figure 5: The three columns of sub-figures represent calculations for three
different boundary terminations that preserve mirror symmetry at the high
symmetry points of the MRW model. The calculations are for $u=1.5$, and there
are 10 unit cells along the $z$-direction. (a), (b), (c) show the charge
distribution of the occupied ground state at ${\textbf{k}}=\Gamma$. Each unit
cell has two sites/orbitals, and the summed charge density $Q$ in the last
three unit cells near each of $z=1,10$ are marked by the gray, dotted boxes.
In (a)/(b), we see one more/less charge on _both_ boundaries compared to the
cases where the boundary is sharp as in (c). The second row (d), (e), (f)
shows gapless surface states localized near boundaries at $z=1$ (blue) and
$z=10$ (red). The third row and the fourth row show the nontrivial crossings
with $\pi$ of the reflection phases. The reflection phase $\phi$ at the $z=1$
boundary is shown in (g), (h), (i). The reflection phase $\phi^{\prime}$ at
the $z=10$ boundary are shown in (j),(k),(l).

Another, perhaps more realistic choice, is to require an additional mirror
symmetry along the direction of the symmetry axis. With such a choice, the
charge distribution in the quantum wire at high symmetry points (i.e.,
$\tau=0$ and $\tau=T/2$) must be mirror symmetric along the 1D chain, and thus
guarantees that the system will not be polarized and that there will be charge
exchange between the reservoirs and a quantum wire with a nonzero RTP. As will
be discussed below, the existence of charge exchange between the reservoirs
and the quantum wire has an additional important consequence that gapless
modes survive on the boundary of the quantum wire despite the boundary not
being sharp.

Let us illustrate this idea for the MRW model at $u=1.5$ with 10 unit cells
along the $z$-direction. Consider local perturbations on the boundary that are
much smaller than the gap at $M,$ but comparable with the gap at $\Gamma$
555Details about the local potentials we use can be found in Appendix B.. Then
at $M$, we expect the charge distribution of the ground state to always be
uniform as in the case without the perturbations. On the path from $M$ to
$\Gamma$, the bulk, unity-valued RTP indicates that one charge is pumped from
the boundary at $z=1$ to the boundary at $z=10$ in the wire. If the mirror
symmetry is preserved during this process, then at $\Gamma$ there are three
possible cases.

(i) For the first scenario, two charges enter the reservoirs through the
boundary at $z=10$. As shown in Fig. 5 (a), this leads to a mirror symmetric
charge distribution at $\Gamma$ such that each boundary has one less charge
compared to that at $M$ (which has a charge distribution identical to that in
Fig. 5(c)). This also implies that there are two branches of surface bands
near $z=10$ that cross the Fermi-level between $M$ and $\Gamma$ such that
there are more states occupied at $M$ than $\Gamma$ (see Fig. 5(d)). The
spectral flow from the two branches of surface states indicates that there are
two charges entering the right reservoir. According to the theory discussed in
Sec III.1, the two additional charges in the right reservoir after a half
period indicate that the reflection phase near the $z=10$ boundary,
$\phi^{\prime}$, would cross $\pi$ twice from above as shown in Fig. 5(j).
However, the reflection phase near the $z=1$ boundary, $\phi$, does not cross
$\pi$ as observed in Fig. 5(g).

(ii) In a second scenario two charges enter the wire through the boundary at
$z=1$, as shown in Fig. 5(b). This leads to a mirror symmetric charge
distribution at $\Gamma$ such that each boundary has one more charge compared
with that at $M$. Similarly, this also implies that from $M$ to $\Gamma$ there
are two surface bands localized near $z=1$ that cross the Fermi level such
that there are less occupied states at $M$ than are at $\Gamma$ (see Fig.
5(e)). According to the theory discussed in Sec III.1, the reflection phase at
the $z=1$ boundary, $\phi$, should cross $\pi$ twice from below (as shown in
Fig. 5(h)) to indicate the left reservoir having two fewer charges. In
comparison, the reflection phase near the $z=10$ boundary, $\phi^{\prime}$,
does not cross $\pi$ as observed in Fig. 5(k).

(iii) For the third scenario, one charge enters the wire through the boundary
at $z=1$, and one leaves the wire through the boundary at $z=10$, which is
similar to the case when the boundary is sharp (see Fig. 5(c)). Here there is
just one branch of edge states at $z=1$ and one branch at $z=10$ that cross
the Fermi level between $M$ and $\Gamma$ (see Fig. 5(f)). Each of the
reflection phases $\phi,\phi^{\prime}$ will exhibit a single crossing with
$\pi$ (see Fig. 5(i)(l) respectively).

As we can see, for all three cases, there are always gapless boundary states
(albeit with different configurations) and we observe the predicted non-
trivial behavior of the reflection phase crossings.

After the above discussions, we close this section by emphasizing that for a
system with boundaries that are not sharp, an imposed symmetry can help us
have an observable RTP. Indeed, using the data in the reflection phase and the
charge distribution, one can unambiguously determine the value of the RTP in
the wire: the crossing of the reflection phase with $\pi$ over a half period
provides the information about charge exchange between the wire and the
reservoirs. Together with the charge distribution at $\tau=0$ and $\tau=T/2$,
one can determine the number of charges pumped from one side of the wire to
the other side of the wire over the half period. As an example for scenario
(i) above, the reflection phase $\phi^{\prime}$ has two $\pi$-crossings which
implies that two states are transferred to the right reservoir. If there is no
pump then we would expect the charge near the right boundary to have two less
charges than the initial case, but we see in Fig. 5(a) that it has only one
less charge. No charges traveled into the left reservoir, but we see that the
left edge also has one less charge. Thus, we can conclude that one charge was
also pumped from left to right during this process, and hence the RTP has a
value of unity. Finally, we emphasize that the existence of gapless boundary
states in each scenario illustrates that imposing additional symmetry to a
delicate topological insulator can stabilize boundary modes away from a sharp
boundary limit.

Figure 6: (a) Illustration of a lattice model with the position space hoppings
and staggered onsite potentials of $H_{\text{2DRT}}$ at a fixed $k_{x}$. (b)
The returning winding of $\operatorname{arg}(\operatorname{det}r)$ for the 2D
RTP model with $m=0.5,1.5$. (c) illustration of waveguides in a photonic
crystal, where the disks with different colors represent waveguides with
different radii. $d_{0,1,2}$ and $r_{A,B}$ are tuned by a parameter $\theta$
that can be identified with $k_{x}$ in $H_{\text{2DRT}}$. (d) Spectra of full-
continuum waveguide chains with $\theta$ ranging from $0$ to $1$, which
corresponds to $k_{x}$ in the range $0$ to $\pi$. Note that all energy scales
are normalized with respect to the inter-cell hopping at $\theta=\pi/2$ in the
insulating chain.

##  IV Experimental proposal

In this section, we provide an experimental proposal to implement a delicate
TI in a photonic crystal. Furthermore, we demonstrate that the reflection
phase can be measured from the wavefunction of static states in reservoirs.
Since the hoppings in the MRW model are complicated to engineer in photonics,
we instead study a 2D mirror symmetric model that also has a bulk RTP, but is
more experimentally feasible. Let us consider a two-band, 2D Bloch
Hamiltonian:

$\displaystyle H_{\text{2DRTP}}(k_{x},k_{y})$
$\displaystyle=\sin(k_{x})(m+\cos(k_{y}))\sigma_{x}$ (7)
$\displaystyle+\sin(k_{x})\sin(k_{y})\sigma_{y}+\cos(k_{x})\sigma_{z},$

which has a mirror symmetry $m_{x}=\sigma_{z}$ along the $x$-direction, and a
time-reversal symmetry $\mathcal{T}=\sigma_{z}K$ where $K$ is the complex
conjugation operator. When $0<m<1$, this model has a $-2\pi$ Berry flux on the
$k_{x}>0$ half of the BZ, and thus has a $2\pi$ Berry flux on the $k_{x}<0$
half of the BZ to satisfy time-reversal symmetry. This configuration
guarantees $\gamma_{v}(k_{x=0})-\gamma_{v}(k_{x=\pi})=2\pi$ by Stokes Theorem.
Hence this model manifests a RTP [33].

Treating $k_{x}$ as an adiabatic parameter, $H_{\text{2DRTP}}$ at each fixed
$k_{x}$ is a Su-Schrieffer-Heeger(SSH)-type 1D chain along the $y$-direction
with a staggered potential $\pm\cos k_{x}$ (see Fig. 6 (a)). We numerically
calculate the single-channel reflection coefficient $r$ and extract
$\phi=\operatorname{arg}(r)$ for this 2D RTP model as shown in Fig. 6 (b). The
details of the single-channel calculations are discussed in Appendix A.

Treating the 2D model in Eq. (7) as a set of 1D chains along the $y$-direction
that are parameterized by different $k_{x}$ values, we can implement the model
using 1D photonic crystals composed of evanescently coupled waveguides. The
on-site potential and hopping strength depend on the radii of the waveguides
(e.g., $r_{A}$ and $r_{B}$ in Fig. 6 (c)), and the distance between the
adjacent waveguides (e.g., $d_{0}$, $d_{1}$, and $d_{2}$ in Fig. 6 (c)),
respectively. These quantities can be fully controlled by fabricating the
evanescently coupled waveguide array using 3D printing by two-photon
lithography[34, 35]. The diffraction of light through the waveguide array is
governed by the paraxial wave equation

$\displaystyle i\partial_{z}\psi(\textbf{r},z)=$
$\displaystyle\left[-\frac{1}{2k_{0}}\nabla^{2}_{\textbf{r}}-\frac{k_{0}\Delta
n(\textbf{r},z)}{n_{0}}\right]\psi(\textbf{r},z)$ $\displaystyle\equiv
H_{\textrm{cont}}\psi(\textbf{r},z),$ (8)

where ${\textbf{r}}=(x,y)$, $\psi(\textbf{r},z)$ is the envelope function of
the electric field
$\textbf{E}(\textbf{r},z)=\psi(\textbf{r},z)e^{i(k_{0}z-\omega t)}\hat{x}$,
$k_{0}=2\pi n_{0}/\lambda$ is the wave number within the medium, and $\Delta
n$ is the refractive index of the waveguide relative to the index of our
medium: $n_{0}$. In addition, $\lambda$ is the wavelength of laser light,
$\nabla^{2}_{\textbf{r}}$ is the Laplacian in the transverse $(x,y)$ plane,
$\omega=2\pi c/\lambda$, and $H_{\textrm{cont}}$ is the continuum Hamiltonian
for the propagation of light in the waveguide array. The eigenvalues and
eigenmodes of the waveguide array can be obtained by diagonalizing
$H_{\textrm{cont}}$, which is called a full-continuum simulation [36].

Using realistic experimental values, we can parameterize a family of photonic
waveguides by a variable $\theta$ via: $r_{A,B}(\theta)=1.5\pm
0.1\cos(\theta\pi)~{}\mu$m, $d_{0}=6.5~{}\mu$m,
$d_{1}(\theta)=9-2.85\sin(\theta\pi)~{}\mu$m,
$d_{2}(\theta)=8-3.0\sin(\theta\pi)~{}\mu$m. Through the full-continuum
simulation, we confirmed that we can tune $\theta,$ which controls the size
of, and the distance between, waveguides, so that as it is tuned from $0$ to
$1$ the system changes in the same manner that tuning $k_{x}\in[0,\pi]$
changes parameters of the tight-binding model in Eq. 7. The energy spectrum of
such a system derived from the continuum calculation is shown in Fig. 6 (d),
where we indeed observe the expected in-gap states in the first half period,
i.e., from $\theta=0$ to $\theta=1,$ which corresponds to $k_{x}=0$ to
$k_{x}=\pi$. Here we show the spectrum of only the first half period, because
with mirror symmetry $m_{x}$ we know that the spectrum of the second half
period at a $k_{x0}\in[\pi,2\pi]$ is the same as the spectrum at $2\pi-
k_{x0}\in[0,\pi]$.

To detect the reflection phase, we attach a 1D gapless chain
$H_{\text{gapless}}=\sum_{y}h_{0}(c^{{\dagger}}_{y+1}c_{y}+h.c.)$ to the left
of each 1D insulating chain as shown in Fig. 6 (c). The attached gapless chain
plays the role of a reservoir. For clarity, we label the central 1D insulating
chain with the range $1\leqslant y\leqslant L_{0}$, and label the gapless
chain with the range $y\leqslant 0$. In our tight-binding calculations, we
connect the gapless chain and the insulating chain between the sites $y=0$ and
$y=1$ using a hopping with strength $\sin^{2}k_{x}$. The effects of
alternative choices of the form of this hopping are discussed in Appendix A.

To set up the scattering problem let us consider an incoming wave
$\psi_{\text{in}}(y)=e^{iky}$ ($y=0,-1,-2,\ldots$) from the gapless chain
directed toward the insulating chain with energy $E=0$ (i.e., the in-gap Fermi
level of the insulating chain), where $y$ is the site coordinate, and $k>0$ is
the wave vector. Note that for the gapless chain we use, an input wave with
$E=0$ has a wave vector $k=\pi/2$ because the dispersion of the gapless chain
is $E(k)=2h_{0}\cos k$. After a reflection by the insulating chain, there will
be an outgoing wave with wave function $\psi_{\text{out}}(y)=e^{-iky+\phi}$ as
discussed above. Then, the wavefunction of the state with energy $E=0$ in the
gapless reservoir chain is $\psi(y)=(e^{iky}+e^{-iky+\phi})/\sqrt{2}$. Thus,
up to a normalization factor the intensity is given by

$|\psi(y)|^{2}=1+\cos(2ky-\phi),$ (9)

which contains information about $\phi$ and can be measured in the experiment.

Figure 7: (a) A tight-binding calculation of the intensity of the states with
energy closest to zero at $y=0$ for $k_{x}\in[0,\pi]$. (b) A corresponding
full-continuum waveguide simulation, where $\theta$ plays the role of $k_{x}$
and parameterizes the size of, and the distance between, waveguides. The blue
circle identifies that there is a point where $|\psi(0)|^{2}=0$ and thus where
$\phi=\pi$.

More specifically, we now study the intensity for a gapless chain attached to
the family of 1D insulating chains that correspond to $k_{x}\in[0,\pi]$. To
extract the information about $\phi$ from Eq. (9), in Fig. 7 (a) we plot the
intensity at $y=0$ for $k_{x}\in[0,\pi],$ which should be proportional to
$1+\cos\phi$ (recall $k=\pi/2$). We note that more details about the intensity
over the whole gapless chain can be found in Appendix C. We find that the plot
in Fig. 7 (a) is approximately captured by $0.01(1+\cos\phi)$ from which we
can observe that $\phi$ indeed winds by $2\pi$ when $k_{x}$ changes from $0$
to $\pi$. A corresponding full-continuum simulation for coupled waveguides is
shown in Fig. 7 (b), which qualitatively shows the same results.
Interestingly, even though the reflection phase in the full-continuum
simulation does not precisely match the tight-binding results because it does
not exactly wind by $2\pi$ when $\theta$ changes from $0$ to $1$666This is
ascribed to the subtlety of the connection between the gapless chain and the
insulating chains discussed in Appendix A.2 and C., one can still see that
$\phi$ crosses $\pi$ once (i.e., $|\psi(0)|^{2}$ touches zero as indicated by
the blue circle) when $k_{x}$ changes from $0$ to $\pi$, as expected from the
analysis in Sec III.2.

In an experimental context the eigenstate closest to mid-gap, and the
corresponding reflection phase, can be accessed and/or observed by placing an
auxiliary waveguide sufficiently far away from the system [37]. The auxiliary
waveguide should be identical to the waveguides in the gapless region. Such a
waveguide can act as an external drive to inject light into the main system
without significant perturbations of the intrinsic eigenstates of the main
system. This results in the selective excitation of the eigenstates near
$E=0$, for which we can immediately apply our theoretical analysis above.

##  V Conclusions and remarks

In conclusion, we developed a scattering theory to understand delicate TIs.
Furthermore, we proposed a mechanism to stabilize surface states of delicate
TIs on non-sharp boundaries. To test our predictions we proposed a photonic
experiment to measure the reflection phase of delicate TIs. Since a nonzero
RTP indicates an intrinsically polarized system that breaks inversion
symmetry, such materials potentially have large optical effects like the
photovoltaic effect discussed recently in Ref. 33. Thus, it is important and
interesting to search for such kinds of materials in solid state and
metamaterial platforms. The scattering theory discussed in this work can be
used as an important indicator for the search. We hope that in the future, the
experimental implementation of delicate TIs and the measurement of their
reflection phase can be done not only in photonic systems, but also in
acoustic systems, electric circuits, and even quantum materials.

## Acknowledgements

P. Z. thanks Yidong Chen, Tomáš Bzdušek, Aris Alexandradinata, Sachin Vaidya,
and Xiaoqi Sun for insightful discussions. P. Z. and T. L. H. thank the US
Office of Naval Research (ONR) Multidisciplinary University Research
Initiative (MURI) grant N00014-20- 1-2325 on Robust Photonic Materials with
High-Order Topological Protection for support.

##  Appendix A Calculation of the reflection phase in a 1D chain

The setup of the scattering problem on a 1D chain is shown in Fig. 2 (a). We
label the positions of the lattice sites of the target 1D system in the range
$[1,N]$, and connect its two boundaries to reservoirs. For simplicity, we take
the reservoir to be a simple 1D gapless chain with Hamiltonian
$H_{\text{gapless}}=\sum_{n}\left[(h_{0}c_{n+1}^{{\dagger}}c_{n}+h.c.)-\mu
c_{n}^{{\dagger}}c_{n}\right].$ The spectrum is given by
$E=-\mu+2h_{0}\cos(k)$, where $\mu$ is the chemical potential of the whole
system, i.e., the target system and the reservoirs have the same chemical
potential $\mu$. For simplicity, we choose the lattice constant to be $1$ and
$\mu=0$.

To calculate the scattering matrix $S$ in Eq. (5), we define a transfer matrix
$T$ such that

$T\begin{pmatrix}B\\\ A\end{pmatrix}=\begin{pmatrix}T_{11}&T_{12}\\\
T_{21}&T_{22}\end{pmatrix}\begin{pmatrix}B\\\
A\end{pmatrix}=\begin{pmatrix}D\\\ C\end{pmatrix}.$ (10)

Then,

$\displaystyle r=-T_{11}^{-1}T_{12},\ t=-T_{21}T_{11}^{-1}T_{12}+T_{22},$ (11)
$\displaystyle r^{\prime}=T_{21}T_{11}^{-1},\ t^{\prime}=T_{11}^{-1}.$

For the central region we consider a general short-range 1D tight-binding
model with Hamiltonian:

$H(s)=\sum_{n=1}^{N-1}(c^{{\dagger}}_{n+1}h_{n+1,n}(s)c_{n}+\text{h.c.})+h_{n}^{\prime}(s)c^{{\dagger}}_{n}c_{n},$
(12)

where $h_{n+1,n}$ and $h^{\prime}_{n}$ are generically matrices, and $s$ is a
periodic parameter that will be used when we consider 2D systems as pumps of a
1D chain.

To calculate the transfer matrix, we look at the static
Schr$\ddot{\rm{o}}$dinger equation given by $H(s)$:

$H(s)\sum_{n}\psi_{n}\left|n\right\rangle=E\sum_{n}\psi_{n}\left|n\right\rangle,$
(13)

which leads to

$0=h_{n,n-1}(s)\psi_{n-1}+h^{{\dagger}}_{n+1,n}(s)\psi_{n+1}+(h_{n}^{\prime}(s)-E)\psi_{n}.$
(14)

Note that we focus on the scattering process for input waves at the Fermi
level, i.e., $E=\mu=0$. Then, the above equation can be rewritten as

$\begin{pmatrix}\psi_{n}\\\
\psi_{n+1}\end{pmatrix}=\mathcal{M}_{n}\begin{pmatrix}\psi_{n-1}\\\
\psi_{n}\end{pmatrix},$ (15)

where

$\mathcal{M}_{n}=\begin{pmatrix}0&1\\\
-(h^{{\dagger}}_{n+1,n}(s))^{-1}h_{n,n-1}(s)&-(h^{{\dagger}}_{n+1,n}(s))^{-1}h^{\prime}_{n}(s)\end{pmatrix}.$
(16)

Note that we assume the $h_{n+1,n}$ to be invertible. For non-invertible
$h_{n+1,n}$, one needs use the singular-value-decomposition (SVD) method as
discussed in Ref. 38. At the interfaces between the target system and the
reservoirs, we apply continuity of the eigenstates to find

$\begin{pmatrix}\psi_{0}\\\
\psi_{1}\end{pmatrix}=\Lambda\theta_{L}\begin{pmatrix}B\\\ A\end{pmatrix},\
\begin{pmatrix}\psi_{N}\\\
\psi_{N+1}\end{pmatrix}=\Lambda\theta^{-1}_{R}\begin{pmatrix}D\\\
C\end{pmatrix},$ (17)

where

$\Lambda=\begin{pmatrix}\mathbbm{1}&\mathbbm{1}\\\
e^{-ik}\mathbbm{1}&e^{ik}\mathbbm{1}\end{pmatrix},$ (18)

and

$\theta_{L}=\begin{pmatrix}e^{ikx_{L}}\mathbbm{1}&0\\\
0&e^{-ikx_{L}}\mathbbm{1}\end{pmatrix},\
\theta_{R}=\begin{pmatrix}e^{ik(N-x_{R})}\mathbbm{1}&0\\\
0&e^{-ik(N-x_{R})}\mathbbm{1}\end{pmatrix}.$ (19)

Comparing Eq. (10), (15), and (17), we can write the transfer matrix $T$ as

$T=\theta_{R}\Lambda^{-1}\left(\prod_{n=1}^{N}\mathcal{M}_{n}\right)\Lambda\theta_{L}.$
(20)

With $T$ calculated in Eq. (20), we apply Eq. (11) to get the reflection
coefficient and subsequently its phase.

In our calculation, input states with $E=0$ in the gapless chain with
dispersion $E=2h_{0}\cos k$ have momentum $k=\pi/2$. At the interface between
gapless chains and the target system, we have $h_{1,0}=h_{0}$ and
$h_{N+1,N}=h_{0}$. Since in Eq. (20), $\mathcal{M}_{n}$ matrices connect site
$0$ and site $N+1$, it is reasonable to regard $0$ and $N+1$ as the reflection
point and thus $x_{L}=0$ and $x_{R}=N+1$. This corresponds to

$\theta_{L}=\begin{pmatrix}\mathbbm{1}&0\\\ 0&\mathbbm{1}\end{pmatrix},\
\theta_{R}=\begin{pmatrix}-i\mathbbm{1}&0\\\ 0&i\mathbbm{1}\end{pmatrix}.$
(21)

###  A.1 Reflection matrix of the MRW model in Eq. (3)

For the 3D model in Eq. (3), we can read off the real space hopping (in the
$z$-direction) after a Fourier transformation of the Bloch Hamiltonian. This
model has just nearest-neighbor tunneling in the $z$ direction. For labeling
purposes, in this subsection $n$ refers to the unit cell label, with implicit
labeling of sites within a unit cell. As a result, the hopping amplitudes are
matrices, and they are the same for even and odd unit cells within the central
region, namely $h_{n+1,n}=h_{n,n-1}$. Using this convention we have the matrix
elements of $h_{n,n+1}$ as

$\displaystyle\left(h_{n,n+1}\right)_{11}=$ $\displaystyle
u-\cos(k_{x})-\cos(k_{y}),$ (22) $\displaystyle\left(h_{n,n+1}\right)_{12}=$
$\displaystyle 2\imath\sin(k_{x})-2\sin(k_{y}),$
$\displaystyle\left(h_{n,n+1}\right)_{21}=$ $\displaystyle 0,$
$\displaystyle\left(h_{n,n+1}\right)_{22}=$
$\displaystyle-u+\cos(k_{x})+\cos(k_{y}).$

The matrix elements of $h^{\prime}_{n}$ are

$\displaystyle\left(h^{\prime}_{n}\right)_{11}=$ $\displaystyle
2\cos(k_{x})(u-\cos(k_{y}))-\cos(2k_{x})$ (23)
$\displaystyle+2u\cos(k_{y})-\cos(2k_{y})-u^{2}-1,$
$\displaystyle\left(h^{\prime}_{n}\right)_{12}=$ $\displaystyle
2(-\sin(k_{y})+\imath\sin(k_{x}))(\cos(k_{x})+\cos(k_{y})-u),$
$\displaystyle\left(h^{\prime}_{n}\right)_{21}=$ $\displaystyle
2(\sin(k_{y})+\imath\sin(k_{x}))(-\cos(k_{x})-\cos(k_{y})+u),$
$\displaystyle\left(h^{\prime}_{n}\right)_{22}=$ $\displaystyle
2\cos(k_{x})(\cos(k_{y})-u)$
$\displaystyle+\cos(2k_{x})-2u\cos(k_{y})+\cos(2k_{y})+u^{2}+1.$

Notice that the above two matrices are both invertible. We can use Eq. (15) to
calculate the matrix $\mathcal{M}_{n}$, treating the hopping strength
$h_{n,n+1}$, $h^{\prime}_{n}$ as matrices. The rest follows.

###  A.2 Reflection matrix of the 2D model in Eq. (7)

For the 2D model in Eq. (7) the hoppings and onsite potentials are illustrated
in Fig. 6(a). If we use the same method as in Appendix A.1, we find the
hopping matrices are singular. We can either use the singular value
decomposition method to calculate the canonical $n$-channel transfer matrix
(where $n$ is the number of internal degrees of freedom in a unit cell) as is
discussed in Ref. 38, or we can make a simplification assuming the incoming
wave is a single channel. Since our model is an SSH-type 1D chain, the
transfer matrix of single channel incoming waves has the benefit of
simplicity.

Therefore, in this subsection, we will consider single channel incoming waves,
where we have $h_{n+1,n}(s=k_{x})$ equal to $m\sin k_{x}$ ($\sin k_{x}$) for
odd (even) $n$, and $h_{n}^{\prime}=(-1)^{n-1}\cos k_{x}$. Since we are
considering only single channel incoming waves, here $n$ label sites instead
of unit cells, and $h$’s are now complex numbers instead of matrices. In doing
so we lose information about the unit cell structure so we will need to
enforce the total number of sites to be an even number by hand so that we have
translation symmetry. Assuming the hoppings are all non-zero, in the target
system we can write the matrices

$\mathcal{M}_{2n+1}=\begin{pmatrix}0&1\\\
-\frac{1}{m}&-\frac{\cos(k_{x})}{m\sin(k_{x})}\end{pmatrix},\mathcal{M}_{2n}=\begin{pmatrix}0&1\\\
-m&\frac{\cos(k_{x})}{\sin(k_{x})}\end{pmatrix}.$ (24)

The simplicity of these matrices hides a subtlety when it comes to choosing
the form of the hopping strength between the reservoir and the central region.
Notice the transfer matrix here has a natural singularity whenever the hopping
vanishes, i.e., when $\sin(k_{x})=0$. This corresponds to a fully decoupled
case, where the intermediate matrices will contain zero matrices. Our transfer
matrix method will no longer work in this completely decoupled case. However,
we may consider the limit where $\sin(k_{x})\to 0$. In this limit the matrices
in Eq. (24) both take the form $\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}$ up to
an overall factor. This matrix form is maintained when taking products with
itself regardless of the number of sites in the central region.

Figure 8: The reflection phase for the model in Eq. (7) when connecting to the
metallic chain with a hopping comparable to the onsite potential.

When connected to the reservoir, we have a choice to make. If we set the
gapless reservoir chain’s hopping strength to be a constant that is comparable
with the strength of the onsite potential at $k_{x}=0$ or $\pi$, then at the
interface when $\sin(k_{x})\to 0$ the matrix $\mathcal{M}$ is of the form
$\begin{pmatrix}0&0\\\ \pm 1&1\end{pmatrix}$ up to an overall factor. This
leads to a resultant reflection phase of $\pm\pi/2$. We numerically verify the
predicted number of crossings with $\pi$ of the reflection matrix eigenvalues
as shown in Fig. 8: When we go from $k_{x}=0$ to $k_{x}=\pi$, the reflection
phase varies from $+\pi/2$ to $-\pi/2$ continuously, and crosses $\pi$ once.
Still, there is a small discrepancy from the conclusion we drew from the
multi-channel case, which is that the values of the phase at $k_{x}=0,\pi$ are
not integer multiples of $2\pi$ as predicted in the multichannel case, they
are $\pm\pi/2$. However, if we go back to the two-channel case, where we have
one more channel connected to another sublattice in the boundary unit cell,
then at $k_{x}=0$ and $k_{x}=\pi$, the reflection matrices are just
$\begin{pmatrix}e^{-i\pi/2}&0\\\ 0&e^{i\pi/2}\end{pmatrix}$ and
$\begin{pmatrix}e^{i\pi/2}&0\\\ 0&e^{-i\pi/2}\end{pmatrix}$, and the
determinants are $1$ which is consistent with the argument in the main text.
As a consequence, if we consider only single-channel scattering and set the
hopping to a constant comparable with the strength of onsite potential at
$k_{x}=0,\pi$, we cannot see the returning winding number of the phase.

To fix this, we will prescribe a better way of setting the reservoir coupling
that is free of the aforementioned discrepancy. To accomplish this we can vary
the hopping strength between the gapless chain and the insulating chain in
accordance with the change of hopping strength in the insulating chain, i.e.,
we vary the gapless chain’s hopping strength as $\sin^{2}(k_{x})$. This smooth
choice of the reservoir’s hopping strength goes to zero simultaneously with
the insulating chain. At the interface when $\sin(k_{x})\to 0$ the matrix
$\mathcal{M}$ now takes the form $\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}$ up
to an overall factor. This will lead to a phase which is an integer multiple
of $2\pi$. In this case, we can nicely observe the returning winding even in a
single-channel setup.

The deviation from returning winding at $k_{x}=0,\pi$ can also be removed by
setting the hopping between the gapless chain and the insulating chain to a
small value compared to the onsite potential achievable in the insulating
chain. Doing so ensures the $\mathcal{M}$ matrices approximately retain the
form in the decoupled limit in the insulating chain, which again leads to the
observation of the returning winding as shown in the main text. In realistic
cases, the hopping between the gapless chain and the insulating chain can be
smaller than the onsite potential achievable in the insulating chain but might
not be much smaller. In this case we would expect to observe a reflection
phase between $0$ and $\pi/2$ at $k_{x}=0,\pi$, and an approximate returning
winding.

##  Appendix B Local potential on the boundary

Given the Hamiltonian in Appendix A.2, we add boundary potential
$V=v_{1}c^{{\dagger}}_{1}c_{1}+v_{N}c^{{\dagger}}_{N}c_{N}$. For the plots in
Fig. 4, we use $v_{1}=-v_{N}=1.2$ which breaks the chiral and mirror
symmetries at both $M$ and $\Gamma$. For the plots in Fig. 5, we use
$v_{1}=v_{N}$ which breaks the chiral symmetry but preserves the mirror
symmetry. In Fig. 5 (a), (b), and (c), we use $v_{1}=v_{N}=1.2,-1.2,0.5$,
respectively.

##  Appendix C Supplementary discussion about the experimental proposal

Figure 9: (a) shows the intensity of the states with energy closest to zero in
the 100 site gapless chain attached to insulating chains corresponding to
different $k_{x}\in[0,\pi]$. (b) shows the density for the ten sites closest
to the insulating chain. (c) shows a full-continuum waveguide simulation,
where $\theta$ parametrizes the size of and the distance between waveguides
and plays the role of $k_{x}$.

Here, we show that the intensity over the gapless chain is indeed captured by
Eq.(9) for our setup in the main text. Since both the gapless chain and the
insulating chain are finite (100 and 10 unit cells respectively), we have
eigenstates close to $E=0,$ but not exactly at $E=0.$ Additionally, the
associated wave vector is not exactly $k=\pi/2,$ but instead is
$k=\pi/2+\delta k$ with $\delta k\sim 2\pi/100\ll\pi/2$. Substituting
$k=\pi/2+\delta k$ into Eq. (9), we can derive the intensity to be
$|\psi(y)|^{2}\sim 1+\cos(\pi y+2\delta kn-\phi)=1+\cos(2\delta
ky-\phi)\cos\pi y$ up to a normalization factor. As shown in Fig. 9 (a), our
tight-binding calculations of intensity for eigenstates of the gapless chain
closest to $E=0$ are consistent with this result. Indeed at the five typical
$k_{x}$ values between $0$ and $\pi$ that we show, the intensity plots are
essentially the plots of $0.01(1+\cos((\pi+2\delta k)y-\phi)\cos\pi y)$ with
$(\delta k=\pi/100,\phi=0)$, $(\delta k=\pi/200,\phi=\pi/2)$, $(\delta
k=0,\phi=\pi)$, $(\delta k=3\pi/200,\phi=-\pi/2)$, and $(\delta
k=\pi/100,\phi=0).$ Crucially, the series of intensities indicate a $2\pi$
winding of $\phi$ as the effective $k_{x}$ goes from $0$ to $\pi$.

We show the results of the reverting process from $k_{x}=\pi$ to $k_{x}=2\pi$
in Fig. 10, which manifest a $-2\pi$ winding (returning winding) of the
reflection phase $\phi$ from $k_{x}=\pi$ to $k_{x}=2\pi$. We focus on $y=0$ in
the main text, but we can also extract information about $\phi$ by looking at
a part of the gapless chain with a length much shorter than the total length
of the gapless chain. If we choose say the ten sites closest to the insulating
chains (i.e., sites with coordinate labels $y\in[-9,0]$) in the gapless chain,
then the variation due to $\delta k$ can be ignored. Then, $\phi$ can be
derived by simply comparing our plots with the function $1+\cos(\pi y-\phi)$
because it has the largest oscillation when $\phi=0,\pi$, and gradually
becomes flat when $\phi$ goes to $\pm\pi/2$, as shown in Fig. 7 (b). A
corresponding full-continuum simulation for coupled waveguides is shown in
Fig. 7 (c), which qualitatively shows the same results.

Figure 10: (a) shows the intensity of the states with energy closest to zero
in the 100 site gapless chain attached to insulating chains corresponding to
different $k_{x}\in[\pi,2\pi]$. (b) shows the density for the ten sites
closest to the insulating chain.

Next, we discuss the subtlety of the distance between the gapless chain and
different insulating chains, which is related to the subtlety discussed in the
tight-binding calculation discussed in Appendix A, i.e., the reflection phase
we observe depends on how we connect the gapless chain and the insulating
chain. In the full-continuum simulation, we fix the distance between the
gapless chain and the insulating chain to be $d_{0}$, which is the distance
between adjacent waveguides in the gapless chain. Thus, the coupling between
the gapless chain and the insulating chain is the same as the coupling between
adjacent sites in the gapless chain. As shown in Fig. 11, the onsite potential
oscillation amplitude (i.e., the onsite potential proportional to $\sigma_{z}$
in the tight-binding Hamiltonian) is twice the coupling between the gapless
chain and the insulating chain (i.e., not that much larger than the coupling
between the gapless chain and the insulating chain). Thus, as discussed in
Appendix A.2, from $\theta=0$ to $\theta=\pi$, the reflection phase does not
exactly wind by $2\pi$, which is consistent with what we show in Fig . 7 (b)
and 9 (c).

Figure 11: The left panel shows the on-site potential with oscillation
amplitude $\sim 0.9$. The right panel shows the internal and external
couplings in the insulating chain and the couplings in the gapless chain. Note
that all energy scales are normalized with respect to the external coupling
strength in the insulating chain at $\theta=0.5$.

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