# Extrinsic topology of Floquet anomalous boundary states in quantum walks

Takumi Bessho<EMAIL_ADDRESS>Yukawa Institute for
Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan  Ken Mochizuki
<EMAIL_ADDRESS>Advanced Institute for Materials Research
(WPI-AIMR), Tohoku University, Sendai 980-8577, Japan  Hideaki Obuse
<EMAIL_ADDRESS>Department of Applied Physics, Hokkaido
University, Sapporo 060-8628, Japan  Institute of Industrial Science, The
University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa,Chiba 277-8574, Japan
Masatoshi Sato<EMAIL_ADDRESS>Yukawa Institute for Theoretical
Physics, Kyoto University, Kyoto 606-8502, Japan

###### Abstract

Bulk-boundary correspondence is a fundamental principle for topological phases
where bulk topology determines gapless boundary states. On the other hand, it
has been known that corner or hinge modes in higher order topological
insulators may appear due to "extrinsic" topology of the boundaries even when
the bulk topological numbers are trivial. In this paper, we find that Floquet
anomalous boundary states in quantum walks have similar extrinsic topological
natures. In contrast to higher order topological insulators, the extrinsic
topology in quantum walks is manifest even for first-order topological phases.
We present the topological table for extrinsic topology in quantum walks and
illustrate extrinsic natures of Floquet anomalous boundary states in concrete
examples.

##  I introduction

Recently, it has become clear that a class of gapless boundary states may
appear without bulk topological numbers because of topological properties of
boundaries. Such boundary states are called "extrinsic" Geier  _et al._
(2018). In particular, extrinsic topological phases are realized in higher-
order topological insulators with gapless corner and hinge modes Benalcazar
_et al._ (2017a, b, 2019); Song  _et al._ (2017); Langbehn  _et al._ (2017);
Schindler  _et al._ (2018); Ezawa (2018); Khalaf (2018); Matsugatani and
Watanabe (2018). For example, attaching a one-dimensional Su-Schrieffer-Heeger
chain Su  _et al._ (1979) onto an edge of a two-dimensional chiral-symmetric
topologically trivial insulator, we obtain extrinsic zero-energy corner
states. The zero-energy gapless modes are robust against continuous
deformation of the system unless the energy gap closes. The topological
invariant of the edge Su-Schrieffer-Heeger chain determines the types and the
numbers of these corner modes.

Floquet systems, of which Hamiltonians are periodic in time Kitagawa  _et al._
(2010a); Bukov  _et al._ (2015); Eckardt (2017); Oka and Kitamura (2019);
Jiang  _et al._ (2011); Carpentier  _et al._ (2015); Nathan and Rudner (2015);
Kolodrubetz  _et al._ (2018), and quantum walks, where periodically
multiplying unitary operators describe their dynamics Aharonov  _et al._
(1993); Kempe (2003); Kitagawa  _et al._ (2010b); Obuse and Kawakami (2011);
Kitagawa  _et al._ (2012); Gross  _et al._ (2012); Asbóth (2012); Asbóth and
Obuse (2013); Tarasinski  _et al._ (2014); Obuse  _et al._ (2015); Asboth and
Edge (2015); Cedzich  _et al._ (2016); Cardano  _et al._ (2016); Barkhofen
_et al._ (2017); Cedzich  _et al._ (2018); Chen  _et al._ (2018), have
attracted uprising interests due to topological phenomena intrinsic to non-
equilibrium systems. Both Floquet systems and quantum walks have the $2\pi/T$
energy periodicity because of the Bloch-Floquet theorem for time translation
symmetry with time period $T$ Sambe (1973). In contrast to conventional
topological insulators, the $2\pi/T$ periodicity leads to two high-symmetric
Fermi energy levels at $\epsilon=0$ and $\epsilon=\pi/T$ and defines gaps at
these energies. Gapless boundary states at $\epsilon=0$ and $\epsilon=\pi/T$
have been studied both in Floquet systems Rudner  _et al._ (2013); Kitagawa
_et al._ (2010a); Carpentier  _et al._ (2015); Jiang  _et al._ (2011) and
quantum walks Kitagawa  _et al._ (2010b, 2012); Asbóth and Obuse (2013). Due
to the shared properties above, quantum walks are often regarded as a kind of
Floquet systems in many previous literatures.

However, as we clarify below, there exists an essential difference between
Floquet systems and quantum walks. To see the difference, we compare quantum
walks with Floquet systems in more detail. In conventional Floquet systems,
the time-evolution operator $U(\bm{k},t_{1}\to t_{2})$ in the momentum
representation is given by a time-dependent microscopic Hamiltonian
$H(\bm{k},t)$,

$\displaystyle U(\bm{k},t_{1}\to
t_{2})=\mathcal{T}\exp\left[-i\int_{t_{1}}^{t_{2}}dtH(\bm{k},t)\right],$ (1)

where $\mathcal{T}$ is the time ordering operator. The one-cycle time-
evolution operator $U_{F}(\bm{k}):=U(\bm{k},0\to T)$ is called the Floquet
operator, where $T$ is the time period of the microscopic Hamiltonian
$H(\bm{k},t+T)=H(\bm{k},t)$. In quantum walks, however, the one-cycle time
evolution is given directly by a series of unitary operators $U_{j}(\bm{k})$,

$\displaystyle U_{\text{QW}}(\bm{k})=\prod_{j}U_{j}(\bm{k}).$ (2)

For both Floquet systems and quantum walks, we can define the effective
Hamiltonians $H_{F}$ and $H_{\text{QW}}$ as

$\displaystyle U_{F}=e^{-iH_{F}T},\quad U_{\text{QW}}=e^{-iH_{\text{QW}}},$
(3)

which describe stroboscopic dynamics of these systems. The quasi-energies of
these systems are defined as the eigenvalues of the effective Hamiltonians and
the periodicity $2\pi/T$ of the energy can be understood as the periodicity in
the phases of $U_{F}$ and $U_{\rm QW}$, where $T=1$ for quantum walks. When
$U_{j}(\bm{k})$’s are written by a microscopic Hamiltonian:
$U_{j}(\bm{k})=\mathcal{T}\exp\left[-i\int_{t_{j}}^{t_{j+1}}dtH(\bm{k},t)\right]$,
the quantum walk reduces to a Floquet system. However, this is not always
possible for general quantum walks [Fig. 1]. An illustrative example is a
quantum walk consisting only of the shift operator $U_{\text{QW}}(k)=S_{+}(k)$
Kempe (2003); Kitagawa  _et al._ (2010b),

$\displaystyle S_{+}(k)=\begin{pmatrix}e^{-ik}&0\\\ 0&1\end{pmatrix}.$ (4)

As one can check immediately, this model has a non-trivial winding number
$w_{1}[U_{\rm QW}(k)]=1$ with $w_{1}[U(k)]$ defined by

$\displaystyle
w_{1}[U(k)]=\int_{0}^{2\pi}\frac{dk}{2\pi}\mathrm{tr}[U(k)^{-1}i\partial_{k}U(k)].$
(5)

On the other hand, for any Floquet continuous time evolution $U(k,t_{1}\to
t_{2})$, the winding number becomes zero:

$\displaystyle w_{1}[U(k,t_{1}\to t_{2})]=w_{1}[U(k,0\to 0)]=0,$ (6)

as $U(k,t_{1}\to t_{2})$ is continuously deformed into $U(k,0\to 0)=1$.
Therefore, the quantum walk $U_{\text{QW}}(k)=S_{+}(k)$ cannot be written by a
Floquet continuous time evolution.

Figure 1:  Relation between quantum walks and Floquet systems. Any Floquet
operator $U_{F}$ can be regarded as a quantum walk $U_{\text{QW}}$. On the
other hand, some quantum walks $U_{\text{QW}}$ such as $S_{+}(k)$ in Eq. (4)
cannot be realized as a Floquet operator $U_{F}$.  Figure 2:  Lattice
structure and energy spectrum of the simple extrinsic topological model in Eq.
(13). (a) We attach the ($d-1$)-dimensional boundary onto the $d$-dimensional
bulk. In the simple mode, we consider $d=2$. We impose the periodic boundary
conditions (PBC) in the $y$ direction and the open boundary conditions (OBC)
in the $x$ direction. (b) The energy spectrum of Eq. (13). The blue (red) line
indicates the bulk (edge) energy spectrum.

In this paper, we argue that this difference enables extrinsic topological
phases in quantum walks. A simple example of the extrinsic topological phase
in a quantum walk is given as follows [Fig. 2]. First let us prepare a
topologically trivial bulk model in two dimensions (2D) with energy gaps at
$\epsilon=0$ and $\epsilon=\pi/T$ by using the coin operator $C(\theta)$,

$\displaystyle U_{\text{bulk}}(\bm{k})=C(\frac{\pi}{2}),\
C(\theta):=e^{-i\theta\sigma_{z}}.$ (7)

The real space representation of the bulk model is

$\displaystyle
U_{\text{bulk}}=\sum_{x=2}^{L_{x}}\sum_{y=1}^{L_{y}}-i\ket{\bm{r},+}\bra{\bm{r},+}+i\ket{\bm{r},-}\bra{\bm{r},-},$
(8)

where $\bm{r}$ represents the $(x,y)$ position, and $\pm$ indicate two
orthogonal internal states of the walker. The effective Hamiltonian $H_{\rm
QW}$ of this model is

$\displaystyle H_{\rm QW}=\frac{\pi}{2}\sigma_{z},$ (9)

which has the eigenvalues $\epsilon(\bm{k})=\pm\pi/2$. Thus, the model has
energy gaps both at $\epsilon=0,\pi$. Furthermore, as the Hamiltonian is a
constant, the bulk topological number is zero, so no gapless boundary mode is
expected.

However, we can obtain a gapless chiral edge mode by decorating the boundary
at $x=1$ of this model with a local unitary operator. For instance, we can
consider the following edge unitary operator

$\displaystyle U_{\text{edge}}(k_{y})=S_{+}(k_{y})C(\frac{\pi}{2}),$ (10)

where $k_{y}$ is the momentum along the edge. The effective Hamiltonian
$H_{\rm QW}^{\rm edge}$ of the edge unitary operator is

$\displaystyle H_{\rm QW}^{\rm edge}=\begin{pmatrix}k_{y}+\pi/2&0\\\
0&-\pi/2\end{pmatrix},$ (11)

and thus the edge unitary operator supports a chiral gapless mode with the
linear dispersion $\epsilon(k_{y})=k_{y}+\pi/2$. The real space representation
of the edge unitary operator is

$\displaystyle
U_{\text{edge}}=\sum_{y=1}^{L_{y}}-i\ket{1,y+1,+}\bra{1,y,+}+i\ket{1,y,-}\bra{1,y,-},$
(12)

where the periodic boundary condition is imposed in the $y$ direction. By
attaching this to the real space representation of the bulk model $U_{\rm
bulk}$ in Eq. (8) at $x=1$, we have a decorated bulk model,

$\displaystyle U_{\text{bulk}}\oplus U_{\text{edge}}.$ (13)

In spite of the trivial bulk topological number, the decorated model has a
chiral edge mode, as shown in Fig. 2 (b).

The extrinsic gapless chiral mode originates from the winding number of the
edge unitary operator. As explained above, in contrast to conventional Floquet
systems, a unitary operator in a quantum walk may have a nonzero winding
number, and the edge unitary operator $U_{\rm edge}(k_{y})$ in Eq. (10) has
$w_{1}[U_{\rm edge}(k_{y})]=1$. Following the theorem proved in Ref. Bessho
and Sato (2021), this implies the existence of gapless chiral modes both at
$\epsilon=0,\pi$ at the same time: The theorem in one-dimension (1D) indicates
the relation between the winding number of $U_{\rm edge}(k_{y})$ and the
gapless modes,

$\displaystyle\sum_{\epsilon_{p}(k_{p\alpha})=0}\mathrm{sgn}~{}v_{p\alpha}=\sum_{\epsilon_{p}(k_{p\alpha})=\pi}\mathrm{sgn}~{}v_{p\alpha}=w_{1}[U_{\text{edge}}(k_{y})],$
(14)

where $k_{p\alpha}$ is the $\alpha$-th gapless point of band $p$ of the edge
unitary operator defined by $\epsilon(k_{p\alpha})=0$ or $\pi$, and
$v_{p\alpha}=\left(\partial\epsilon_{p}/\partial
k_{y}\right)_{k_{y}=k_{p\alpha}}$ is the group velocity of the gapless mode at
$k_{p\alpha}$. Since ${\rm sgn}\ v_{p\alpha}$ measures the chirality of the
gapless mode, a nonzero winding number implies the existence of chiral gapless
modes both at $\epsilon=0,\pi$.

On the basis of the theorem in Ref. Bessho and Sato (2021), we classify
extrinsic topological phases in quantum walks in arbitrary dimensions.
Superficially, the classification coincides with that for gapless modes in
ordinary topological insulators and superconductors [Table 2]. This is because
the gapless boundary states in quantum walks have the same topological charges
as those in usual topological insulators and superconductors. However,
contrary to the conventional gapless modes, gapless modes in the extrinsic
topological phases appear in the absence of bulk topological numbers because
of non-trivial topology of boundary unitary operators. Furthermore, the
extrinsic gapless modes always appear in a pair at $\epsilon=0,\pi/T$. In even
(odd) dimensions, the net topological charge of extrinsic gapless modes at
$\epsilon=0$ is the same as (opposite to) the net topological charge of those
at $\epsilon=\pi/T$.

The extrinsic topological phase in quantum walks is closely related to the so-
called Floquet anomalous topological phase Roy and Harper (2017). In Floquet
systems, there are two types of topological phases: one is the conventional
topological phase defined by the effective Hamiltonian in Eq. (3), and the
other is the anomalous one determined from a refined dynamics of the
microscopic Hamiltonian in Eq. (1). These two types of topological phases are
needed to fully determine the boundary states both at $\epsilon=0$ and
$\epsilon=\pi/T$. In quantum walks, however, the microscopic Hamiltonian does
not always exist, and thus anomalous topological phase is not possible in
general. As a result, the bulk topology is insufficient to determine the
boundary gapless states. Instead, we find that boundary operators can be
topological, which enables us to fully control gapless states on the boundary.

We also discuss possible physical implementations of such extrinsic
topological phases. We numerically and analytically study the robustness of
extrinsic chiral gapless modes against disorders in the Anderson localization
Hamiltonian Anderson (1958). We also show that suitable modulations of
boundaries can eliminate gapless boundary states in the split step quantum
walk in one dimension Kitagawa  _et al._ (2010b, 2012); Obuse  _et al._ (2015)
and the five step model in two dimensions Rudner  _et al._ (2013), which
clearly illustrates the extrinsic nature of gapless modes in the quantum
walks.

This paper is organized as follows. In Sec. II, we present the classification
of extrinsic topological phases in quantum walks. In Sec. III, we examine
extrinsic topological phases in one-dimensional quantum walks in detail. In
Sec. IV, the relation between the topological classification of Floquet
systems and that of quantum walks is discussed. In Sec. V, we argue the bulk-
boundary correspondence for chiral-symmetric quantum walks in 1D. For chiral
symmetric quantum walks in 1D, it has been shown that the bulk topological
numbers fully determines the number of boundary zero modes Asbóth and Obuse
(2013), i.e. no extrinsic topological phase appears. On the other hand, our
classification indicates the presence of an extrinsic topological phase in
this case. We clarify that this difference comes from the difference in the
definition of chiral symmetry. Indeed, Ref. Asbóth and Obuse (2013) introduces
chiral symmetry in a specific manner, and we demonstrate that the extrinsic
topological phase becomes trivial under the special realization of chiral
symmetry. We also find that class CII quantum walks in 1D has a similar
property: Using an appropriate realization of symmetries, the bulk topological
numbers fully determine the number of boundary zero modes. In Sec. VI, we
present several physical implementations of extrinsic topological phases, and
examine their properties. Finally, we give the conclusion in Sec. VII

##  II Classification of extrinsic topology in quantum walks

Let us consider a general one-cycle time-evolution unitary operator $U_{\rm
QW}({\bm{k}})$ and the effective Hamiltonian $H_{\rm QW}({\bm{k}})$ defined by
$U_{\text{QW}}(\bm{k})=e^{-iH_{\text{QW}}({\bm{k}})}$ in $d$-dimensions with
$\bm{k}=(k_{1},k_{2},\ldots,k_{d})$. We first introduce the Altland-Zirnbauer
(AZ) symmetry classes Altland and Zirnbauer (1997). $H_{\text{QW}}(\bm{k})$
possibly satisfies time-reversal symmetry (TRS), particle-hole symmetry (PHS)
and/or chiral symmetry (CS):

$\displaystyle TH_{\text{QW}}(\bm{k})T^{-1}$
$\displaystyle=H_{\text{QW}}(-\bm{k}),$ (15) $\displaystyle
CH_{\text{QW}}(\bm{k})C^{-1}$ $\displaystyle=-H_{\text{QW}}(-\bm{k}),$ (16)
$\displaystyle\Gamma H_{\text{QW}}(\bm{k})\Gamma^{-1}$
$\displaystyle=-H_{\text{QW}}(\bm{k}).$ (17)

Here, $T$ and $C$ are anti-unitary operators with $T^{2}=\pm 1$ and $C^{2}=\pm
1$, and $\Gamma$ is a unitary operator with $\Gamma^{2}=1$. The AZ symmetry
classes are defined by the presence or absence of TRS, PHS and/or CS [Table
1]. In quantum walks, it is beneficial to rewrite the symmetries as those for
$U_{\text{QW}}(\bm{k})$:

$\displaystyle TU_{\text{QW}}(\bm{k})T^{-1}$
$\displaystyle=U_{\text{QW}}(-\bm{k})^{\dagger},$ (18) $\displaystyle
CU_{\text{QW}}(\bm{k})C^{-1}$ $\displaystyle=U_{\text{QW}}(-\bm{k}),$ (19)
$\displaystyle\Gamma U_{\text{QW}}(\bm{k})\Gamma^{-1}$
$\displaystyle=U_{\text{QW}}(\bm{k})^{\dagger}.$ (20)

When there are particle-hole and/or chiral symmetries, we have a symmetry
constraint $\epsilon_{n}=-\epsilon_{m}\ (\text{mod}\ 2\pi)$ for the energy
bands $n,m$ of $H_{\rm QW}$, and obtain high-symmetric energy gaps at
$\epsilon=0,\pi$. We assume that gaps are open at these levels for bulk bands
in the following [Fig. 3].

Figure 3:  A typical energy spectrum of a quantum walk: The energy has $2\pi$
periodicity, so there exist two high-symmetric bulk energy gaps at
$\epsilon=0$ and $\pi$. Gapless boundary states may appear at both bulk gaps.
Here $a$ is a lattice constant.

Gapless boundary states appear at $\epsilon=0$ and $\epsilon=\pi$
individually. They are described by a gapless Dirac Hamiltonian:

$\displaystyle
H_{\text{QW}}(\bm{k}_{\parallel})=\sum_{j=1}^{d-1}k_{j}\gamma_{j}+\epsilon_{\text{Gap}}\hat{1},$
(21)

up to continuous deformations. Here, we take the open boundary condition in
the $x_{d}$ direction, $\bm{k}_{\parallel}=(k_{1},k_{2},\ldots,k_{d-1})$ is
the momentum along the boundary, and $\epsilon_{\text{Gap}}=0,\pi$ indicates
the energy gaps we consider. The gamma matrix $\gamma_{j}$ is Hermitian and
obeys $\\{\gamma_{i},\gamma_{j}\\}=2\delta_{ij}$. This boundary Hamiltonian is
taken to be compatible with AZ symmetries. The gapless Dirac Hamiltonian in
Eq. (21) has the same form as that for boundary states in conventional
topological insulators and superconductors in equilibrium Kobayashi  _et al._
(2014); Matsuura  _et al._ (2013); Chiu and Schnyder (2014); Chiu  _et al._
(2016). In the latter systems, from the bulk-boundary correspondence, the
topological classification of gapless boundary states in $(d-1)$-dimensions is
the same as that of insulators and superconductors in $d$-dimensions.
Therefore, the topological classification of $(d-1)$-dimensional boundary
states in quantum walks coincides with that of ordinary insulators and
superconductors in $d$-dimensions [Table. 1]. Note that the topological
numbers in Table 1 are doubled because each gap at $\epsilon=0,\pi$ may host
gapless states.

Table 1: Periodic table for $(d-1)$-dimensional gapless boundary states of $d$-dimensional quantum walks in AZ symmetry class. We assume bulk gaps at $\epsilon=0$ and $\pi$.  AZ class |  $T$ |  $C$ |  $\Gamma$ |  $d=1$ |  2 |  3
---|---|---|---|---|---|---
A | 0 | 0 | 0 | 0 | $\mathbb{Z}^{2}$ | 0
AIII | 0 | 0 | 1 | $\mathbb{Z}^{2}$ | 0 | $\mathbb{Z}^{2}$
AI | $+1$ | 0 | 0 | 0 | 0 | 0
BDI | $+1$ | $+1$ | 1 | $\mathbb{Z}^{2}$ | 0 | 0
D | 0 | $+1$ | 0 | $\mathbb{Z}^{2}_{2}$ | $\mathbb{Z}^{2}$ | 0
DIII | $-1$ | $+1$ | 1 | $\mathbb{Z}^{2}_{2}$ | $\mathbb{Z}^{2}_{2}$ | $\mathbb{Z}^{2}$
AII | $-1$ | 0 | 0 | 0 | $\mathbb{Z}^{2}_{2}$ | $\mathbb{Z}^{2}_{2}$
CII | $-1$ | $-1$ | 1 | $(2\mathbb{Z})^{2}$ | 0 | $\mathbb{Z}^{2}_{2}$
C | 0 | $-1$ | 0 | 0 | $(2\mathbb{Z})^{2}$ | 0
CI | $+1$ | $-1$ | 1 | 0 | 0 | $(2\mathbb{Z})^{2}$

The gapless boundary states in quantum walks have two different topological
origins. The first one is the bulk topology of the effective Hamiltonian
$H_{\text{QW}}(\bm{k})$. The two gaps $\epsilon=0$ and $\pi$ separate bulk
bands of $H_{\text{QW}}(\bm{k})$ into two, from which one can define
"occupied" and "empty" bands like conventional insulators. Therefore, in a
manner similar to usual topological insulators, $H_{\rm QW}({\bm{k}})$ can be
topological, which gives gapless boundary states at $\epsilon=0$. One of the
doubled topological numbers $\mathbb{Z}^{2}$, $\mathbb{Z}_{2}^{2}$ and
$(2\mathbb{Z})^{2}$ in Table 1 corresponds the boundary states determined from
the bulk topological invariant of $H_{\text{QW}}(\bm{k})$.

Another origin is the main subject of this paper, i.e., extrinsic topology of
boundary operators. Below, we discuss possible lattice terminations of
$d$-dimensional quantum walks, which is given by $(d-1)$-dimensional unitary
operators $U_{\text{BDQW}}(\bm{k}_{\parallel})$.

For this purpose, we employ a theory on gapless states of unitary operators
Bessho and Sato (2021). For a boundary unitary operator
$U_{\text{BDQW}}(\bm{k}_{\parallel})$ in a symmetry class specified by the
presence or absence of AZ symmetries, we have gapless states when the unitary
operator has a non-trivial topological number. The extended Nielsen-Ninomiya
theorem Bessho and Sato (2021) summarizes the explicit relation between
gapless states and the topological number for the $(d-1)$-dimensional unitary
operator $U_{\text{BDQW}}(\bm{k}_{\parallel})$:

$\displaystyle\sum_{\epsilon_{\alpha}=0}\nu_{\alpha}^{0}=(-1)^{d}\sum_{\epsilon_{\alpha}=\pi}\nu^{\pi}_{\alpha}=n,$
(22)

where $\nu_{\alpha}^{0,\pi}$ is the $\alpha$-th topological charge of gapless
states at $\epsilon=0,\pi$, and $n$ is the topological number of
$U_{\text{BDQW}}(\bm{k}_{\parallel})$. Equation (14) is an example of the
extended Nielsen-Ninomiya theorem for class A with $d=2$, and we will give the
explicit forms of $\nu_{\alpha}^{0,\pi}$ and $n$ for $d=1$ in Sec. III. In
general, we can define $n$ as follows: We introduce the doubled Hamiltonian
${\cal H}_{U}(\bm{k}_{\parallel})$ Higashikawa  _et al._ (2019):

$\displaystyle{\cal
H}_{U}(\bm{k}_{\parallel})=\begin{pmatrix}0&U_{\text{BDQW}}(\bm{k}_{\parallel})\\\
U_{\text{BDQW}}^{\dagger}(\bm{k}_{\parallel})&0\end{pmatrix},$ (23)

which is Hermitian, and has eigenvalues $\pm 1$ due to ${\cal
H}_{U}(\bm{k}_{\parallel})^{2}=\hat{1}$. It also obeys CS,

$\displaystyle\Sigma_{z}{\cal H}_{U}(\bm{k}_{\parallel})\Sigma_{z}=-{\cal
H}_{U}(\bm{k}_{\parallel}),\ \Sigma_{z}=\begin{pmatrix}\hat{1}&0\\\
0&-\hat{1}\end{pmatrix}.$ (24)

The AZ symmetries of $U_{\text{BDQW}}(\bm{k}_{\parallel})$, which should be
compatible with those for the bulk operator $U_{\rm QW}({\bm{k}})$ in Eqs.
(18)-(20), lead to

$\displaystyle\tilde{T}{\cal H}_{U}(\bm{k}_{\parallel})\tilde{T}^{-1}$
$\displaystyle={\cal H}_{U}(-\bm{k}_{\parallel}),\
\tilde{T}=\begin{pmatrix}0&T\\\ T&0\end{pmatrix},$ (25)
$\displaystyle\tilde{C}{\cal H}_{U}(\bm{k}_{\parallel})\tilde{C}^{-1}$
$\displaystyle={\cal H}_{U}(-\bm{k}_{\parallel}),\
\tilde{C}=\begin{pmatrix}C&0\\\ 0&C\end{pmatrix},$ (26)
$\displaystyle\tilde{\Gamma}{\cal
H}_{U}(\bm{k}_{\parallel})\tilde{\Gamma}^{-1}$ $\displaystyle={\cal
H}_{U}(\bm{k}_{\parallel}),\ \tilde{\Gamma}=\begin{pmatrix}0&\Gamma\\\
\Gamma&0\end{pmatrix}.$ (27)

Therefore, ${\cal H}_{U}(\bm{k}_{\parallel})$ can be regarded as a topological
insulator or superconductor with symmetries in Eq. (24) and Eqs. (25)-(27). As
shown in Appendix A, by using the standard Clifford algebra extension method,
Kitaev (2009); Morimoto and Furusaki (2013), we can perform the topological
classification of ${\cal H}_{U}(\bm{k}_{\parallel})$, which also provides a
topological classification of the boundary operator $U({\bm{k}}_{\parallel})$.
The resultant classification of $U_{\text{BDQW}}(\bm{k}_{\parallel})$ in
$(d-1)$-dimensions coincides with the classification of conventional
topological insulators and superconductors in $d$-dimensions [Table 2]. The
topological number of ${\cal H}_{U}({\bm{k}}_{\parallel})$ gives the
topological number $n$ in Eq. (22).

Table 2: Periodic table of extrinsic topological phases in $(d-1)$-dimensional boundaries of $d$-dimensional quantum walks. This table shows the presence or absence of the $\mathbb{Z}$ or $\mathbb{Z}_{2}$ topological invariant $n$ for $(d-1)$-dimensional boundary unitary operators.  AZ class |  $T$ |  $C$ |  $\Gamma$ |  $d=1$ |  2 |  3
---|---|---|---|---|---|---
A | 0 | 0 | 0 | 0 | $\mathbb{Z}$ | 0
AIII | 0 | 0 | 1 | $\mathbb{Z}$ | 0 | $\mathbb{Z}$
AI | $+1$ | 0 | 0 | 0 | 0 | 0
BDI | $+1$ | $+1$ | 1 | $\mathbb{Z}$ | 0 | 0
D | 0 | $+1$ | 0 | $\mathbb{Z}_{2}$ | $\mathbb{Z}$ | 0
DIII | $-1$ | $+1$ | 1 | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$ | $\mathbb{Z}$
AII | $-1$ | 0 | 0 | 0 | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}$
CII | $-1$ | $-1$ | 1 | $2\mathbb{Z}$ | 0 | $\mathbb{Z}_{2}$
C | 0 | $-1$ | 0 | 0 | $2\mathbb{Z}$ | 0
CI | $+1$ | $-1$ | 1 | 0 | 0 | $2\mathbb{Z}$

Attaching the boundary unitary operator with nonzero $n$ to a boundary of
quantum walks, we can change the number of gapless modes there, in accordance
with the extended Nielsen-Ninomiya theorem in Eq. (22): The number of boundary
states changes as

$\displaystyle\sum_{\epsilon_{\alpha}=0}\nu_{\alpha}^{0}\to$
$\displaystyle\sum_{\epsilon_{\alpha}=0}\nu_{\alpha}^{0}+n,$ (28)
$\displaystyle\sum_{\epsilon_{\alpha}=\pi}\nu_{\alpha}^{\pi}\to$
$\displaystyle\sum_{\epsilon_{\alpha}=\pi}\nu_{\alpha}^{\pi}+(-1)^{d}n.$ (29)

On the other hand, for even (odd) $d$, the difference (summation) of the
boundary states between $\epsilon=0$ and $\epsilon=\pi$ does not change, so it
is intrinsically determined by the bulk topological number $n_{\text{bulk}}$
of $H_{\rm QW}({\bm{k}})$.

$\displaystyle\sum_{\epsilon_{\alpha}=0}\nu_{\alpha}^{0}-(-1)^{d}\sum_{\epsilon_{\alpha}=\pi}\nu_{\alpha}^{\pi}=n_{\text{bulk}}.$
(30)

Thus, the bulk-boundary correspondence partially holds in quantum walks.

For classes A, AI and AII, we can choose the energy gaps arbitrarily, because
there is no symmetry constraint in the quasi-energy. If there are $l$ energy
gaps $\epsilon=\mu_{i}\ (i=1,\ldots,l)$, the topological classification of the
boundary states in Table 1 changes as $\mathbb{Z}^{2}\to\mathbb{Z}^{l}$,
$\mathbb{Z}_{2}^{2}\to\mathbb{Z}_{2}^{l}$,
$(2\mathbb{Z})^{2}\to(2\mathbb{Z})^{l}$. In these cases, the extended Nielsen-
Ninomiya theorem in Eq. (22) takes the form of

$\displaystyle\sum_{\epsilon_{\alpha}=\mu}\nu^{\mu}_{\alpha}=n,$ (31)

where $\nu^{\mu}_{\alpha}$ is the topological charge of $\alpha$-th gapless
state at $\epsilon=\mu$ Bessho and Sato (2021). We note that this formula is
compatible with Eq. (22) with odd $d$ since in class A, AI and AII, the
topological numbers $n$ of $U_{\text{BDQW}}(\bm{k}_{\parallel})$ is always $0$
or $\mathbb{Z}_{2}$ for odd $d$.

##  III Extrinsic boundary states of quantum walks in 1D

Topological phases of quantum walks have been studied mainly in 1D Kitagawa
_et al._ (2010b); Obuse and Kawakami (2011); Kitagawa  _et al._ (2012); Gross
_et al._ (2012); Asbóth (2012); Asbóth and Obuse (2013); Tarasinski  _et al._
(2014); Barkhofen  _et al._ (2017); Cedzich  _et al._ (2016); Cardano  _et
al._ (2016); Cedzich  _et al._ (2018). In this section, we study extrinsic
topological phases of quantum walks in 1D, i.e. zero-dimensional extrinsic
boundary states. According to the periodic table in Table 2, nontrivial
extrinsic topological phases appear for classes AIII, BDI, D, DIII and CII in
1D. We identify the topological numbers $n$ and $\nu^{0,\pi}$ for these
classes, and show that they obey the extended Nielsen-Ninomiya theorem in Eq.
(22).

###  III.1 class AIII

A boundary quantum walk operator $U_{\rm BDQW}$ in a class AIII quantum walk
in 1D is a unitary matrix obeying

$\displaystyle\Gamma
U_{\text{BDQW}}^{\dagger}\Gamma^{-1}=U_{\text{BDQW}},\quad\Gamma^{2}=1,$ (32)

with a unitary matrix $\Gamma$. This relation implies that
$U_{\text{BDQW}}\Gamma$ is Hermitian, and it has nonzero real eigenvalues
because of ${\rm det}[U_{\rm BDQW}\Gamma]\neq 0$. Therefore, we can define the
$\mathbb{Z}$ topological number $n$ in Eq. (22) as follows,

$\displaystyle
n=\frac{1}{2}\left[N_{+}(U_{\text{BDQW}}\Gamma)-N_{-}(U_{\text{BDQW}}\Gamma)\right],$
(33)

where $N_{+}(U_{\text{BDQW}}\Gamma)$ ($N_{-}(U_{\text{BDQW}}\Gamma)$) is the
number of positive (negative) eigenvalues of $U_{\text{BDQW}}\Gamma$. We can
also introduce the topological charge $\nu^{0,\pi}$ of gapless modes at
$\epsilon=0,\pi$ in the following manner. For a gapless mode $|u_{0}\rangle$
at $\epsilon=0$, we have

$\displaystyle U_{\rm BDQW}|u_{0}\rangle=|u_{0}\rangle,\quad U_{\rm
BDQW}^{\dagger}|u_{0}\rangle=|u_{0}\rangle,$ (34)

and CS implies

$\displaystyle U_{\rm BDQW}\Gamma|u_{0}\rangle=\Gamma|u_{0}\rangle.$ (35)

Thus, by taking a linear combination of $|u_{0}\rangle$ and
$\Gamma|u_{0}\rangle$, the gapless mode can be an eigenstate of $\Gamma$,

$\displaystyle\Gamma\ket{u_{0}}=\pm\ket{u_{0}}.$ (36)

Then, the eigenvalue of $\Gamma$ defines the topological number $\nu^{0}$ of
$|u_{0}\rangle$. In a similar manner, we can also define $\nu^{\pi}$ for a
gapless mode $|u_{\pi}\rangle$ at $\epsilon=\pi$. In summary, the topological
numbers $\nu^{0,\pi}$ are written as

$\displaystyle\nu^{0,\pi}=\bra{u_{0,\pi}}\Gamma\ket{u_{0,\pi}},$ (37)

with the normalization $\langle u_{0,\pi}|u_{0,\pi}\rangle=1$.

When $n$ is non-zero, we have gapless modes according to the extended Nielsen-
Ninomiya theorem in Eq. (22). To check the theorem, we consider a general
$2\times 2$ unitary matrix in class AIII,

$\displaystyle
U_{\text{BDQW}}=a_{0}\sigma_{0}+ia_{1}\sigma_{1}+ia_{2}\sigma_{2}+a_{3}\sigma_{3},\quad\Gamma=\sigma_{3},$
(38)

where $a_{\mu}$ are real parameters. From the unitarity condition
$U_{\text{BDQW}}U_{\text{BDQW}}^{\dagger}=1$, we obtain three possible cases:

$\displaystyle\left\\{\begin{array}[]{l}(\text{i})\ a_{3}=0,\
a_{0}^{2}+a_{1}^{2}+a_{3}^{2}=1,\\\ (\text{ii})\ a_{3}=1,\\\ (\text{iii})\
a_{3}=-1.\end{array}\right.$ (42)

The eigenvalues of $U_{\text{BDQW}}$ in each case are given by

$\displaystyle\left\\{\begin{array}[]{l}(\text{i})\ \lambda_{\pm}=a_{0}\pm
i\sqrt{a_{1}^{2}+a_{2}^{2}},\\\ (\text{ii}),(\text{iii})\ \lambda_{\pm}=\pm
1.\end{array}\right.$ (45)

The corresponding eigenstates are

$\displaystyle\left\\{\begin{array}[]{l}(\text{i})\
\ket{u_{\pm}}=\frac{1}{\sqrt{2(a_{1}^{2}+a_{2}^{2})}}\begin{pmatrix}\pm
i\sqrt{a_{1}^{2}+a_{2}^{2}}\\\ ia_{1}-a_{2}\end{pmatrix},\\\ (\text{ii})\
\ket{u_{+}}=(1,0)^{\text{T}},\ \ket{u_{-}}=(0,1)^{\text{T}},\\\ (\text{iii})\
\ket{u_{+}}=(0,1)^{\text{T}},\
\ket{u_{-}}=(1,0)^{\text{T}}.\end{array}\right.$ (49)

which satisfy

$\displaystyle\nu_{\pm}=\bra{u_{\pm}}\Gamma\ket{u_{\pm}}=\left\\{\begin{array}[]{ll}0&\text{for}\
(\text{i}),\\\ \pm 1&\text{for}\ (\text{ii}),\\\ \mp 1&\text{for}\
(\text{iii}),\end{array}\right.$ (53)

Therefore, we obtain

$\displaystyle\sum\nu^{0}=-\sum\nu^{\pi}=\left\\{\begin{array}[]{ll}0&\text{for}\
(\text{i}),\\\ +1&\text{for}\ (\text{ii}),\\\ -1&\text{for}\
(\text{iii}).\end{array}\right.$ (57)

Now we compare this result with the topological number $n$. The Hermitian
matrix $U_{\text{BDQW}}\Gamma$ is given by

$\displaystyle
U_{\text{BDQW}}\Gamma=\left\\{\begin{array}[]{ll}a_{0}\sigma_{3}+a_{1}\sigma_{2}-a_{2}\sigma_{1}&\text{for}\
(\text{i}),\\\ +\sigma_{0}&\text{for}\ (\text{ii}),\\\ -\sigma_{0}&\text{for}\
(\text{iii}),\\\ \end{array}\right.$ (61)

of which eigenvalues are

$\displaystyle\mathcal{E}=\left\\{\begin{array}[]{ll}\pm 1&\text{for}\
(\text{i}),\\\ +1&\text{for}\ (\text{ii}),\\\ -1&\text{for}\
(\text{iii}).\end{array}\right.$ (65)

Thus, $n$ in Eq. (33) is evaluated as

$\displaystyle n=\left\\{\begin{array}[]{ll}0&\text{for}\ (\text{i}),\\\
+1&\text{for}\ (\text{ii}),\\\ -1&\text{for}\ (\text{iii}).\end{array}\right.$
(69)

Equations. (57) and (69) reproduce the extended Nielsen-Ninomiya theorem in
Eq. (22).

###  III.2 class BDI

In class BDI, a boundary unitary operator obeys TRS and PHS,

$\displaystyle TU_{\text{BDQW}}T^{-1}=U_{\text{BDQW}}^{\dagger},$
$\displaystyle CU_{\text{BDQW}}C^{-1}=U_{\text{BDQW}},$ (70)

where $T$ and $C$ are anti-unitary operators with $CT=TC$ and $T^{2}=C^{2}=1$.
By combining TRS with PHS, the boundary operator also has CS,

$\displaystyle\Gamma
U_{\text{BDQW}}^{\dagger}\Gamma^{-1}=U_{\text{BDQW}},\quad\Gamma=TC.$ (71)

Using CS, the topological number $n$ of $U_{\rm BDQW}$ and the topological
charge $\nu^{0,\pi}$ of gapless modes at $\epsilon=0,\pi$ are defined in the
same manner as in class AIII.

The theorem in Eq. (22) can be checked in a manner similar to class AIII. A
general $2\times 2$ unitary matrix in class BDI is given by

$\displaystyle U_{\rm BDQW}=a_{0}\sigma_{0}+ia_{1}\sigma_{1}+a_{3}\sigma_{3},$
$\displaystyle T=K,\quad C=\sigma_{3}K,$ (72)

where $a_{\mu}$ are real parameters, and $K$ is the complex conjugation
operator. From $U_{\text{BDQW}}U_{\text{BDQW}}^{\dagger}=1$, we obtain three
possible cases:

$\displaystyle\left\\{\begin{array}[]{l}(\text{i})\ a_{3}=0,\
a_{0}^{2}+a_{1}^{2}=1,\\\ (\text{ii})\ a_{3}=1,\\\ (\text{iii})\
a_{3}=-1.\end{array}\right.$ (76)

Then, $U_{\text{BDQW}}$ in Eq. (72) obeys the theorem since it is a special
case of Eq. (38) with $a_{2}=0$.

###  III.3 class D

For class D, a boundary operator $U_{\text{BDQW}}$ satisfies

$\displaystyle CU_{\text{BDQW}}C^{-1}=U_{\text{BDQW}},\quad C^{2}=1,$ (77)

with an anti-unitary operator $C$. This relation implies that ${\rm
det}(U_{\text{BDQW}})$ is real, and thus it takes $\pm 1$. Therefore, we can
define the $\mathbb{Z}_{2}$ topological invariant $n$ of $U_{\rm BDQW}$ by

$\displaystyle(-1)^{n}={\rm det}(U_{\text{BDQW}}).$ (78)

On the other hand, the presence or absence of a gapless state at
$\epsilon=0,\pi$ defines the $\mathbb{Z}_{2}$ invariant $\nu^{0,\pi}$ for the
gapless state. Note that an even number of gapless states trivializes
$\nu^{0,\pi}$.

A general $2\times 2$ unitary matrix in class D is given by

$\displaystyle
U_{\text{BDQW}}=a_{0}\sigma_{0}+a_{1}\sigma_{1}+ia_{2}\sigma_{2}+a_{3}\sigma_{3},\quad
C=K,$ (79)

Then, the unitarity of $U_{\text{BDQW}}$ leads to two possible cases:

$\displaystyle\left\\{\begin{array}[]{l}(\text{i})\ a_{0}=a_{2}=0,\
a_{1}^{2}+a_{3}^{2}=1,\\\ (\text{ii})\ a_{1}=a_{3}=1,\
a_{0}^{2}+a_{2}^{2}=1,\end{array}\right.$ (82)

where the eigenvalues of $U_{\text{BDQW}}$ are

$\displaystyle\left\\{\begin{array}[]{l}(\text{i})\ \lambda_{\pm}=\pm 1,\\\
(\text{ii})\ \lambda_{\pm}=a_{0}\pm ia_{2}.\end{array}\right.$ (85)

Thus, only for (i), a single gapless mode exists at $\epsilon=0,\pi$. From
this, we obtain

$\displaystyle\sum\nu^{0}=-\sum\nu^{\pi}=\left\\{\begin{array}[]{ll}1&\text{for}\
(\text{i}),\\\ 0&\text{for}\ (\text{ii}).\end{array}\right.$ (88)

On the other hand, a direct calculation shows

$\displaystyle\det(U_{\text{BDQW}})=a_{0}^{2}+a_{2}^{2}-a_{1}^{2}-a_{2}^{2},$
(89)

and thus, the $\mathbb{Z}_{2}$ invariant $n$ in Eq. (78) is

$\displaystyle n=\left\\{\begin{array}[]{ll}1&\text{for}\ (\text{i}),\\\
0&\text{for}\ (\text{ii}).\end{array}\right.$ (92)

The topological numbers in Eqs. (88) and (92) satisfy the extended Nielsen-
Ninomiya theorem in Eq. (22).

###  III.4 class DIII

For a boundary unitary operator $U_{\text{BDQW}}$ in class DIII, we have

$\displaystyle TU_{\text{BDQW}}T^{-1}=U_{\text{BDQW}}^{\dagger},$
$\displaystyle CU_{\text{BDQW}}C^{-1}=U_{\text{BDQW}},$ (93)

where $T$ and $C$ are anti-unitary operators with $T^{2}=-1$, $C^{2}=1$ and
$CT=TC$. For convenience, we decompose $T$ and $C$ into the unitary parts
${\cal T}$ and ${\cal C}$ and the complex conjugation operator $K$:

$\displaystyle T=\mathcal{T}K,\quad C=\mathcal{C}K.$ (94)

Then, the matrix $U_{\text{BDQW}}\mathcal{T}$ is found to be antisymmetric, so
we can introduce the Pfaffian ${\rm Pf}(U_{\text{BDQW}}\mathcal{T})$. We can
also show

$\displaystyle[{\rm Pf}(U_{\text{BDQW}}\mathcal{T})]^{*}={\rm
det}(\mathcal{C}^{*}){\rm Pf}(U_{\text{BDQW}}\mathcal{T}),$ (95)

and thus, we have ${\rm Pf}(U_{\text{BDQW}}\mathcal{T})=\pm 1$ in the basis
with ${\rm det}(\mathcal{C}^{*})=1$. Then, the sign of the Pfaffian defines
the $\mathbb{Z}_{2}$ invariant $n$ for $U_{\rm BDQW}$:

$\displaystyle(-1)^{n}=-{\rm Pf}(U_{\text{BDQW}}\mathcal{T}).$ (96)

On the other hand, for gapless states at $\epsilon=0,\pi$, the presence or
absence of a Kramers pair of gapless states defines the $\mathbb{Z}_{2}$
invariant $\nu^{0,\pi}$. Note that any eigenstate of $U_{\rm BDQW}$ doubly
degenerates due to the Kramers theorem for TRS.

To obtain a non-trivial example, we need at least a $4\times 4$ unitary matrix
in this class. Let us consider a general $4\times 4$ unitary matrix in class
DIII,

$\displaystyle
U_{\text{BDQW}}=a_{00}\tau_{0}\sigma_{0}+a_{10}\tau_{1}\sigma_{0}+a_{30}\tau_{3}\sigma_{0}$
$\displaystyle+ia_{21}\tau_{2}\sigma_{1}+ia_{22}\tau_{2}\sigma_{2}+a_{23}\tau_{2}\sigma_{3},$
(97)

where TRS and PHS are given by $T=\tau_{0}\sigma_{2}K$ and
$C=\tau_{0}\sigma_{1}K$. From $U_{\text{BDQW}}U_{\text{BDQW}}^{\dagger}=1$, we
obtain two possibilities,

$\displaystyle\left\\{\begin{array}[]{l}(\text{i})\ a_{00}=a_{21}=a_{22}=0,\
a_{30}^{2}+a_{10}^{2}+a_{23}^{2}=1,\\\ (\text{ii})\ a_{30}=a_{10}=a_{23}=0,\
a_{00}^{2}+a_{21}^{2}+a_{22}^{2}=1.\end{array}\right.$ (100)

The eigenvalues of $U_{\rm BDQW}$ with Kramers degeneracy are given by

$\displaystyle\left\\{\begin{array}[]{l}(\text{i})\ \lambda_{\pm}=\pm 1,\\\
(\text{ii})\ \lambda_{\pm}=a_{00}\pm
i\sqrt{a_{21}^{2}+a_{22}^{2}},\end{array}\right.$ (103)

and thus, the system supports a single Kramers pair of gapless states at
$\epsilon=0,\pi$ only for (i). Therefore, we obtain

$\displaystyle\sum\nu^{0}=-\sum\nu^{\pi}=\left\\{\begin{array}[]{ll}1&\text{for}\
(\text{i}),\\\ 0&\text{for}\ (\text{ii}).\end{array}\right.$ (106)

On the other hand, $\mathrm{Pf}(U_{\text{QW}}\mathcal{T})$ becomes

$\displaystyle\mathrm{Pf}(U_{\text{BDQW}}\mathcal{T})=a_{30}^{2}+a_{10}^{2}+a_{23}^{2}-a_{00}^{2}-a_{21}^{2}-a_{22}^{2},$
(107)

so $n$ in Eq. (96) is evaluated as

$\displaystyle n=\left\\{\begin{array}[]{ll}1&\text{for}\ (\text{i}),\\\
0&\text{for}\ (\text{ii}),\end{array}\right.$ (110)

which obeys the theorem in Eq. (22).

###  III.5 class CII

Finally, we examine boundary unitary operators for class CII quantum walks in
1D. The boundary unitary operator obeys

$\displaystyle TU_{\text{BDQW}}T^{-1}=U_{\text{BDQW}}^{\dagger},$
$\displaystyle CU_{\text{BDQW}}C^{-1}=U_{\text{BDQW}},$ (111)

where $T$ and $C$ are anti-unitary operators with $T^{2}=-1$, $C^{2}=-1$ and
$CT=TC$. Combining TRS with PHS, we also have CS,

$\displaystyle\Gamma
U_{\text{BDQW}}^{\dagger}\Gamma^{-1}=U_{\text{BDQW}},\quad\Gamma=TC.$ (112)

Using CS, we can introduce the topological numbers $n$ and $\nu^{0,\pi}$ in
Eqs. (33) and (37) in the same manner as those in class AIII. However, in
contrast to class AIII, because of additional CS and TRS in Eq. (111), these
topological numbers only take even integers. First, the Hermitian matrix
$U_{\text{BDQW}}\Gamma$ has its own TRS defined by $C$

$\displaystyle C[U_{\text{BDQW}}\Gamma]C^{-1}=U_{\text{BDQW}}\Gamma,$ (113)

which results in the Kramers degeneracy for eigenstates of
$U_{\text{BDQW}}\Gamma$. Therefore,

$\displaystyle
n=\frac{1}{2}\left[N_{+}(U_{\text{BDQW}}\Gamma)-N_{-}(U_{\text{BDQW}}\Gamma)\right],$
(114)

in Eq. (33) becomes a $2\mathbb{Z}$ topological invariant. Furthermore,
gapless modes at $\epsilon=0,\pi$ also form Kramers pairs due to the original
TRS by $T$. The Kramers pair has a common eigenvalue of $\Gamma$ since $T$
commutes with $\Gamma$, so the total charges $\sum\nu^{0,\pi}$ of gapless
modes with $\nu^{0,\pi}$ defined by Eq. (37) also become even integers.

A general $4\times 4$ unitary matrix in class CII is given by

$\displaystyle U_{\text{BDQW}}$
$\displaystyle=a_{00}\tau_{0}\sigma_{0}+ia_{10}\tau_{1}\sigma_{0}+a_{30}\tau_{3}\sigma_{0}$
$\displaystyle+ia_{21}\tau_{2}\sigma_{1}+ia_{22}\tau_{2}\sigma_{2}+ia_{23}\tau_{2}\sigma_{3},$
(115)

with $T=\tau_{0}\sigma_{2}K$ and $C=\tau_{3}\sigma_{2}K$. The unitarity
condition $U_{\text{BDQW}}U_{\text{BDQW}}^{\dagger}=1$ leads to the following
three possibilities:

$\displaystyle\left\\{\begin{array}[]{l}(\text{i})\ a_{30}=0,\
a_{00}^{2}+a_{10}^{2}+a_{21}^{2}+a_{22}^{2}+a_{23}^{2}=1,\\\ (\text{ii})\
a_{30}=1,\ a_{00}=a_{10}=a_{21}=a_{22}=a_{23}=0,\\\ (\text{iii})\ a_{30}=-1,\
a_{00}=a_{10}=a_{21}=a_{22}=a_{23}=0.\end{array}\right.$ (119)

The eigenvalues of $U_{\rm BDQW}$ with Kramers degeneracy are

$\displaystyle\left\\{\begin{array}[]{l}(\text{i})\ \lambda_{\pm}=a_{00}\pm
i\sqrt{a_{10}^{2}+a_{21}^{2}+a_{22}^{2}+a_{23}^{2}},\\\ (\text{ii})\
\lambda_{\pm}=\pm 1,\\\ (\text{iii})\ \lambda_{\pm}=\pm 1.\end{array}\right.$
(123)

and thus the boundary operator supports gapless states at $\epsilon=0,\pi$ for
(ii) and (iii). We can show that each Kramers pair satisfies

$\displaystyle\nu_{\pm}=\bra{u_{\pm}}\Gamma\ket{u_{\pm}}=\left\\{\begin{array}[]{ll}0&\text{for}\
(\text{i}),\\\ \mp 1&\text{for}\ (\text{ii}),\\\ \pm 1&\text{for}\
(\text{iii}),\end{array}\right.$ (127)

and thus, we obtain

$\displaystyle\sum\nu^{0}=-\sum\nu^{\pi}=\left\\{\begin{array}[]{ll}0&\text{for}\
(\text{i}),\\\ -2&\text{for}\ (\text{ii}),\\\ +2&\text{for}\
(\text{iii}).\end{array}\right.$ (131)

On the other hand, the Hermitian matrix $U_{\text{BDQW}}\Gamma$ is given by

$\displaystyle U_{\text{BDQW}}\Gamma=$ $\displaystyle-
a_{00}\tau_{3}\sigma_{0}-a_{10}\tau_{2}\sigma_{0}-a_{30}\tau_{0}\sigma_{0}$
$\displaystyle+a_{21}\tau_{1}\sigma_{1}+a_{22}\tau_{1}\sigma_{2}+a_{23}\tau_{1}\sigma_{3}$
(132)

of which eigenvalues are

$\displaystyle\mathcal{E}=-a_{30}\pm\sqrt{a_{00}^{2}+a_{10}^{2}+a_{21}^{2}+a_{22}^{2}+a_{23}^{2}}$
(133)

Thus, by taking into account the Kramers degeneracy, $n$ in Eq. (33) is
evaluated as

$\displaystyle n=\left\\{\begin{array}[]{ll}0&\text{for}\ (\text{i}),\\\
-2&\text{for}\ (\text{ii}),\\\ +2&\text{for}\ (\text{iii}).\end{array}\right.$
(137)

The results in Eqs. (131) and (137) satisfy the extended Nielsen-Ninomiya
theorem in Eq. (22).

##  IV Classification of Floquet systems v.s. quantum walks

In this section, we compare the topological classification of quantum walks
and that of Floquet systems. Let us start with a brief review of the
topological classification of Floquet topological insulators and
superconductors Roy and Harper (2017). We consider a general time periodic
Hamiltonian $H(\bm{k},t+T)=H(\bm{k},t)$ and the time-evolution operator

$\displaystyle
U(\bm{k},t)=\mathcal{T}\exp\left[-i\int_{0}^{t}dtH(\bm{k},t)\right].$ (138)

From the one-cycle time evolution $U_{F}(\bm{k})=U(\bm{k},T)$, we define the
effective Hamiltonian by $U_{F}(\bm{k})=e^{-iH_{F}(\bm{k})T}$. The fundamental
symmetries, TRS, PHS and CS in Floquet systems are defined in the microscopic
Hamiltonian:

$\displaystyle TH(\bm{k},t)T^{-1}$ $\displaystyle=H(-\bm{k},-t),$ (139)
$\displaystyle CH(\bm{k},t)C^{-1}$ $\displaystyle=-H(-\bm{k},t),$ (140)
$\displaystyle\Gamma H(\bm{k},t)\Gamma^{-1}$ $\displaystyle=-H(\bm{k},-t).$
(141)

Here, $T$ and $C$ are anti-unitary operators with $T^{2}=\pm 1$ and $C^{2}=\pm
1$, and $\Gamma$ is a unitary operator with $\Gamma^{2}=1$. These symmetries
lead to TRS, PHS and CS for the effective Hamiltonian $H_{F}(\bm{k})$:

$\displaystyle TH_{F}(\bm{k})T^{-1}$ $\displaystyle=H_{F}(-\bm{k}),$ (142)
$\displaystyle CH_{F}(\bm{k})C^{-1}$ $\displaystyle=-H_{F}(-\bm{k}),$ (143)
$\displaystyle\Gamma H_{F}(\bm{k})\Gamma^{-1}$ $\displaystyle=-H_{F}(\bm{k}).$
(144)

For convenience, we also rewrite the symmetries as those for time-evolution
operator $U(\bm{k},t)$:

$\displaystyle TU(\bm{k},t)T^{-1}=U(-\bm{k},-t),$ (145) $\displaystyle
CU(\bm{k},t)C^{-1}=U(-\bm{k},t),$ (146) $\displaystyle\Gamma
U(\bm{k},t)\Gamma^{-1}=U(\bm{k},-t).$ (147)

Instead of the microscopic Hamiltonian $H(\bm{k},t)$, we classify the time-
evolution operator $U(\bm{k},t)$ because it has the same information as
$H({\bm{k}},t)$. We decompose $U(\bm{k},t)$ into two parts:

$\displaystyle C(\bm{k},t)=e^{-iH_{F}(\bm{k})t},\quad
L(\bm{k},t)=U(\bm{k},t)C(\bm{k},t)^{-1},$ (148)

where $L(\bm{k},t)$ is periodic in $t$. We call $C(\bm{k},t)$ and
$L(\bm{k},t)$ as the constant time evolution and the loop unitary,
respectively. When $H_{F}(\bm{k})$ is gapped at $\epsilon=0,\ \pi/T$ and has
the branch cut at $\epsilon=\pi/T$, this decomposition is shown to be unique
up to homotopy equivalence Roy and Harper (2017). Thus the topological
classification of $U(\bm{k},t)$ reduces to those of the constant time
evolution $C(\bm{k},t)$ and the loop unitary $L(\bm{k},t)$.

The topological classification of $C(\bm{k},t)$ is the same as that of
$H_{F}(\bm{k})$, and thus the same as that of ordinary topological insulators
and superconductors. This coincides with the classification of
$H_{\text{QW}}(\bm{k})$ in the previous section.

On the other hand, the loop unitary $L({\bm{k}},t)$ realizes the Floquet
anomalous topological phase intrinsic to dynamical systems Rudner  _et al._
(2013). Interestingly, for a very different reason, the topological
classification of $L(\bm{k},t)$ also coincides with that of ordinary
topological insulators and superconductors as shown below: One can classify
$L({\bm{k}},t)$ by using the doubled Hamiltonian ${\cal H}_{L}(\bm{k},t)$,

$\displaystyle{\cal H}_{L}(\bm{k},t)=\begin{pmatrix}0&L(\bm{k},t)\\\
L^{\dagger}(\bm{k},t)&0\end{pmatrix},$ (149)

which is Hermitian, gapped due to ${\cal H}_{L}(\bm{k},t)^{2}=\hat{1}$,
periodic both in ${\bm{k}}$ and $t$, and obeys CS

$\displaystyle\Sigma_{z}{\cal H}_{L}(\bm{k},t)\Sigma_{z}=-{\cal
H}_{L}(\bm{k},t),\ \Sigma_{z}=\begin{pmatrix}\hat{1}&0\\\
0&-\hat{1}\end{pmatrix}.$ (150)

Since $L(\bm{k},t)$ obeys the same symmetries as $U(\bm{k},t)$ in Eqs.
(145)-(147), ${\cal H}_{L}(\bm{k},t)$ may have

$\displaystyle\tilde{T}{\cal H}_{L}(\bm{k},t)\tilde{T}^{-1}$
$\displaystyle={\cal H}_{L}(-\bm{k},-t),\ \tilde{T}=\begin{pmatrix}T&0\\\
0&T\end{pmatrix},$ (151) $\displaystyle\tilde{C}{\cal
H}_{L}(\bm{k},t)\tilde{C}^{-1}$ $\displaystyle={\cal H}_{L}(-\bm{k},t),\
\tilde{C}=\begin{pmatrix}C&0\\\ 0&C\end{pmatrix},$ (152)
$\displaystyle\tilde{\Gamma}{\cal H}_{L}(\bm{k},t)\tilde{\Gamma}^{-1}$
$\displaystyle={\cal H}_{L}(\bm{k},-t),\
\tilde{\Gamma}=\begin{pmatrix}\Gamma&0\\\ 0&\Gamma\end{pmatrix}.$ (153)

Therefore, by regarding $t$ as a space direction and then ${\cal
H}_{L}(\bm{k},t)$ as a $(d+1)$-dimensional topological insulator with proper
symmetries defined above, we can classify $L({\bm{k}},t)$. As shown in
Appendix B, one can perform the classification by using the Clifford algebra
extension method Kitaev (2009); Morimoto and Furusaki (2013), and find that
the classification coincides with that of ordinary topological insulators and
superconductors in $d$-dimensions. Thus, the periodic table of $L(\bm{k},t)$
is the same as that of extrinsic topological phases in quantum walks [Table
2].

Therefore, combining the classifications of $C({\bm{k}},t)$ and
$L({\bm{k}},t)$, we find that the topological periodic table of $U(\bm{k},t)$
agrees with that for quantum walks in Table 1. Furthermore, the bulk-boundary
correspondence in Floquet systems is summarized as Roy and Harper (2017)

$\displaystyle\sum_{\epsilon_{\alpha}=0}\nu_{\alpha}^{0}=n_{\text{C}}+n_{\text{L}},$
(154)
$\displaystyle\sum_{\epsilon_{\alpha}=\pi/T}\nu_{\alpha}^{\pi}=(-1)^{d}n_{\text{L}},$
(155)

where $n_{C}$ is the topological invariant of $C(\bm{k},t)$ or equivalently
$H_{F}(\bm{k})$, and $n_{L}$ is the topological invariant of $L(\bm{k},t)$,
and $\nu_{\alpha}^{0,\pi}$ is the topological charge of $\alpha$-th gapless
states at $\epsilon=0$ or $\pi/T$. This formula corresponds to Eq. (29) with
Eq. (30) in quantum walks.

In summary, extrinsic boundary states determined by the boundary topology of
quantum walks correspond to boundary states determined by $L(\bm{k},t)$ of
Floquet systems. In other words, the extrinsic boundary states in quantum
walks correspond to the Floquet anomalous boundary states Rudner  _et al._
(2013). Note that no well-defined loop unitary exists in quantum walks due to
the absence of the microscopic Hamiltonian $H(\bm{k},t)$: Whereas one can
introduce $L({\bm{k}},t)$ for a quantum walk in a specific manner Asboth and
Edge (2015), it is not unique. For instance, the same one-cycle time-evolution
operator for the quantum walk can be obtained by the effective Hamiltonian
$H({\bm{k}},t)=H_{\rm QW}(\bm{k})$, for which $L({\bm{k}},t)=\hat{1}$. As a
result, we can construct a microscopic Hamiltonian with $n_{\text{L}}=0$ and
$n_{\text{L}}$ cannot be uniquely determined from time-evolution operators for
any quantum walks. This observation is also consistent with the extrinsic
nature of Floquet anomalous boundary states in quantum walks.

##  V Bulk-Boundary correspondence in 1D chiral-symmetric quantum walks

For 1D chiral symmetric quantum walks, it has been known that the bulk-
boundary correspondence holds Asbóth and Obuse (2013) with a different
definition of CS from ours. In this section, we explain why the bulk
topological numbers fully determine gapless boundary states in their
definition of CS, and also discuss another possibility of a similar bulk-
boundary correspondence in other symmetry classes.

We first review a specific realization of CS introduced in Ref. Asbóth and
Obuse (2013). To define CS, Asbóth and Obuse decomposed the time-evolution
unitary operator of a quantum walk into two parts

$\displaystyle U_{\text{QW}}=U_{2}U_{1},$ (156)

where $U_{1}$ and $U_{2}$ may consist of multiple unitary operators. Then,
they introduced the decomposed CS as Asbóth and Obuse (2013),

$\displaystyle\Gamma U_{1}\Gamma^{-1}=U_{2}^{\dagger},\quad\Gamma^{2}=1,$
(157)

with a unitary operator $\Gamma$, which leads to the original CS in Eq. (20).
Note that Floquet systems with CS in Eq. (141) naturally realize Eq. (157) by
regarding $U_{1}$ and $U_{2}$ as $U_{1}=U(0\to T/2)$ and $U_{2}=U(T/2\to T)$.

Under this special realization of CS, we can show that the bulk-boundary
correspondence holds Asbóth and Obuse (2013); Mochizuki  _et al._ (2020). For
this purpose, one can take the basis where $\Gamma$ and $U_{1}$ are given by

$\displaystyle\Gamma=\begin{pmatrix}\hat{1}&0\\\ 0&-\hat{1}\end{pmatrix},\
U_{1}=\begin{pmatrix}a&b\\\ c&d\end{pmatrix},$ (158)

and show that if the band gap at $\epsilon=0$ ($\epsilon=\pi$) is open, $b$
and $c$ ($a$ and $d$) have the well-defined 1D winding number $w_{1}[b]$ and
$w_{1}[c]$ ($w_{1}[a]$ and $w_{1}[d]$) Mochizuki  _et al._ (2020). Then, we
have the relation Asbóth and Obuse (2013),

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle\sum_{\epsilon_{\alpha}=0}\nu^{0}_{\alpha}=\frac{w_{1}[a]+w_{1}[d]}{2},\\\
\displaystyle\sum_{\epsilon_{\alpha}=\pi}\nu^{\pi}_{\alpha}=\frac{w_{1}[b]+w_{1}[c]}{2},\end{array}\right.$
(161)

where $\nu_{\alpha}^{0,\pi}$ is the topological charge of boundary zero modes
defined by Eq. (37). Therefore, the net topological charges of boundary modes
at $\epsilon=0$ and $\pi$ are determined by the bulk topological numbers. In
other words, no extrinsic boundary modes are possible in this case.

We can easily show why extrinsic boundary modes are prohibited under CS in Eq.
(157). From the decomposed CS, $U_{\rm QW}\Gamma$ is recast into

$\displaystyle
U_{\text{QW}}\Gamma=U_{2}U_{1}\Gamma=U_{2}(\Gamma^{-1}U_{2}^{\dagger}\Gamma)\Gamma=U_{2}\Gamma
U_{2}^{\dagger},$ (162)

which takes the form of unitary transformation of $\Gamma$ by $U_{2}$.
Therefore, any boundary operator $U_{\rm BDQW}$ decomposed into two parts
$U^{\prime}_{2}U^{\prime}_{1}$ by CS also has the same form,

$\displaystyle U_{\rm BDQW}\Gamma=U^{\prime}_{2}\Gamma
U^{\prime}_{2}{}^{\dagger}.$ (163)

From this, we can show that the zero-dimensional (0D) $\mathbb{Z}$ topological
number defined by Eq. (33) becomes zero: Because of the above relation,
$U_{\rm BDQW}\Gamma$ has the same eigenvalues as $\Gamma$, so we have

$\displaystyle
n=\frac{1}{2}[N_{+}(U_{\text{BDQW}}\Gamma)-N_{-}(U_{\text{BDQW}}\Gamma)]=0.$
(164)

Therefore, the boundary unitary operator gives no additional gapless state in
0D boundaries. Thus, in 1D quantum walks with decomposed CS Eq. (157), the
bulk topological numbers fully determine the numbers of boundary modes at
$\epsilon=0,\pi$.

Whereas the decomposed CS in Eq. (157) prohibits the extrinsic zero modes at
boundaries of 1D systems, it may allow extrinsic topological phases in other
dimensions. For instance, extrinsic zero modes of 2D boundaries are not
prohibited by the decomposed CS in Eq. (157). The topological invariant for a
2D boundary operator $U_{\rm BDQW}(k_{x},k_{y})$ with CS is the Chern number
of $U_{\rm BDQW}(k_{x},k_{y})\Gamma$. Equation (163) merely implies that
$U^{\prime}_{2}$ diagonalizes $U_{\rm BDQW}\Gamma$, so we can realize any
Chern number by choosing a proper $U^{\prime}_{2}$. Consequently, we can add
arbitrary numbers of 2D extrinsic boundary states to three-dimensional quantum
walks under the decomposed CS in Eq. (157).

One may ask a question if there is any other symmetry class that can recover
the bulk-boundary correspondence with an appropriate definition of symmetries.
The answer is yes. We find that 1D quantum walks in class CII also have the
same property. To see this, we again decompose the time-evolution unitary
operator of a quantum walk into two parts,

$\displaystyle U_{\text{QW}}=U_{2}U_{1}.$ (165)

Then, we consider decomposed TRS and PHS, which give the original TRS and PHS
for $U_{\rm QW}$ in Eq. (18) and (19),

$\displaystyle TU_{1}(k)T^{-1}$ $\displaystyle=U_{2}^{\dagger}(-k),$ (166)
$\displaystyle CU_{1}(k)C^{-1}$ $\displaystyle=U_{1}(-k),\
CU_{2}(k)C^{-1}=U_{2}(-k).$ (167)

Here $T$ and $C$ are anti-unitary operators with $CT=TC$ and $T^{2}=C^{2}=-1$.
Combining TRS with PHS, we also have decomposed CS,

$\displaystyle\Gamma U_{1}\Gamma^{-1}=U_{2}^{\dagger},\quad\Gamma=TC.$ (168)

Using the decomposed CS, we can again take the basis in Eq. (158) and prove
the bulk-boundary correspondence in Eq. (161). Note that the decomposed PHS in
Eq. (167) leads to two-fold degeneracy similar to the Kramers doublet, and
thus topological numbers on both sides of Eq. (161) take only even integers.
(See Appendix C.) In a manner similar to the above, we can also show that 0D
extrinsic boundary states are prohibited by Eq. (168). The 0D $2\mathbb{Z}$
topological invariant in Eq. (114) is always zero by the decomposed CS.

##  VI Physical implementations

In this section, we present three possible physical implementations of the
extrinsic topological phases of quantum walks.

###  VI.1 2D disordered systems with extrinsic edge states

In this section, we examine robustness of extrinsic edge modes against
disorders, which is an analog of robustness of quantum Hall edge states
against impurity scatterings Halperin (1982). We consider a single-band model
with an extrinsic edge mode in quantum walks. The edge mode is robust against
impurities, and shows a directed position displacement along the edge
characterized by the topological winding number Thouless (1983); Kitagawa  _et
al._ (2010a); Titum  _et al._ (2016); Nakagawa  _et al._ (2020); Cardano  _et
al._ (2016); Fedorova  _et al._ (2020).

Let us consider the following single-band tight-binding model with random
onsite potentials in 2D, which is typically used for the study of the Anderson
localization:

$\displaystyle H_{A}=\sum_{x,y}J\ket{x+1,y}\bra{x,y}+J\ket{x,y+1}\bra{x,y}$
$\displaystyle+h.c.+\delta_{x,y}\ket{x,y}\bra{x,y},$ (169)

where $J$ is the hopping amplitude and $\delta_{x,y}\in[-W,W]$ is a random
potential uniformly distributed within $[-W,W]$. Here $J$ and $\delta_{x,y}$
are real, so the above Hamiltonian belongs to class AI and is topologically
trivial. For class AI in 2D, it is known that all the eigenstates are
localized for any $W$ if the system is large enough Anderson (1958). Below, we
impose the periodic boundary condition in the $x$ direction, and the open
boundary condition in the $y$ direction.

The one-cycle time evolution by the Hamiltonian in Eq. (VI.1) is described by

$\displaystyle U_{A}=e^{-iH_{A}T}.$ (170)

with $T=1$. The wavefunction at time step $t$ is derived by multiplying the
state by $U_{A}^{t}$. Figure 4 shows (a) the wave packet dynamics at time step
$t$ starting at an edge of the system, (b) the density of states (DOS)
histogram, and (c) a typical eigenstate of $U_{A}$. The Anderson localization
occurs for $W/J=5$ in Fig. 4. After long time steps, the wave packet on the
edge shows a localization with almost the same radius as the typical
eigenstate, and does not diffuse into the bulk.

Figure 4: (a) Dynamics of a wave packet, (b) DOS and (c) a typical eigenstate
of the Anderson model in Eq. (VI.1). The parameters are $J=0.2,W=1,L_{x}=25$
and $L_{y}=15$. The initial state of the wave packet is $\ket{x=10,y=1}$. We
consider the strongly localized regime $W\gg J$, and thus all the eigenstates
are localized. (a) We see that a wave packet starting at an edge does not
diffuse into the bulk and shows a localization with almost the same radius as
a bulk eigenstate. (b) The width of the energy band is broader than $2J$ due
to the random potential $\delta_{x,y}\in[-W,W]$. (c) A typical eigenstate
shows the Anderson localization.  Figure 5: (a) Dynamics of a wave packet, (b)
DOS and (c) delocalized eigenstates of the Anderson model with the extrinsic
edge mode in Eq. (172). The parameters are $J=0.2$, $W=1$, $L_{x}=25$,
$L_{y}=15$ and $T=1$. The color scale is different between (a) and (c). The
initial state of the wave packet is $\ket{x=10,y=1}$. Since $W\gg J$, all bulk
eigenstates are localized. (a) The wave packet propagates to the
$+x$-direction. (b) The DOS around $\epsilon=\pi$ indicates the presence of an
anomalous chiral edge mode. (c) Delocalized eigenstates with eigenenergies
$\epsilon=3.12$ and $\epsilon=0.06$ correspond to the anomalous chiral edge
mode.

We next introduce an extrinsic chiral edge state onto the boundary at $y=1$.
We multiply $U_{A}$ by the unitary operator $A$,

$\displaystyle
A(k_{x})=U_{\text{edge}}(k_{x})\otimes\ket{y=1}\bra{y=1}+\sum_{y=2}^{L_{y}}1\otimes\ket{y}\bra{y}.$
(171)

where the edge unitary operator is $U_{\text{edge}}(k_{x})=e^{-ik_{x}}$, and
consider the decorated time-evolution operator

$\displaystyle U_{A}^{\prime}=AU_{A}.$ (172)

In the above sections, we have considered attaching completely decoupled
boundary unitary operators. In this section, however, we consider attaching
boundary unitary operators coupled with the bulk. As discussed in Sec. I,
$U_{\text{edge}}(k_{x})$ provides a chiral edge mode with $\epsilon=k_{x}$,
which should be robust against disorders.

Figure 5 shows (a) the wave packet dynamics starting at the decorated edge
with $y=1$, (b) the DOS histogram, and (c) a typical eigenstate profile of
$U_{A}^{\prime}$. The wave packet propagates to the $+x$-direction, as we
expected. Thus, the boundary operator $A$ induces a robust edge state
analogous to a quantum Hall edge state. We also find that the extended state
along the edge survives even when its energy is overlapped with that of the
original bulk band. This is because of the Anderson localization of the bulk
states, which suppresses the mixing with the edge state. In Appendix F, we
also show the extrinsic chiral edge state is robust even in the presence of
random phases along the edge at $y=1$.

In general, we can characterize the directed wave packet movement due to
extrinsic chiral modes by the winding number in Eq. (5). To show this, we
consider time-evolution by $\hat{U}$ in 1D for a while, then apply the result
to the extrinsic edge modes in 2D.

For a quantum walk in 1D, we introduce the polarization at $x$ as

$\displaystyle P_{x}=\sum_{\alpha}\bra{x,\alpha}\hat{x}\ket{x,\alpha},$ (173)

where $\ket{x,\alpha}$ represents a state at position $x$ with internal
degrees of freedom $\alpha$ such as spin, orbital and so on. After the one-
cycle time evolution by $\hat{U}$, the polarization becomes

$\displaystyle
P_{x}(T)=\sum_{\alpha}\bra{x,\alpha}\hat{U}^{\dagger}\hat{x}\hat{U}\ket{x,\alpha}.$
(174)

If $\hat{U}$ has translation symmetry, one can show that the position
displacement after one-cycle evolution is given by the winding number Thouless
(1983); Kitagawa  _et al._ (2010a); Titum  _et al._ (2016); Nakagawa  _et al._
(2020); Cardano  _et al._ (2016); Fedorova  _et al._ (2020)

$\displaystyle P_{x}(T)-P_{x}=w_{1}[U(k)],$ (175)

where $U(k)$ is the momentum space representation of $\hat{U}$. The proof is
as follows: From the Fourier transformation, the polarization in Eq. (174) is
rewritten as

$\displaystyle P_{x}(T)$
$\displaystyle=\sum_{\alpha}\bra{x}\bra{\alpha}\hat{U}^{\dagger}\hat{x}\hat{U}\ket{x}\ket{\alpha}$
$\displaystyle=\sum_{\alpha}\left[\frac{1}{\sqrt{L}}\sum_{k}e^{ikx}\bra{k}\right]\bra{\alpha}\hat{U}^{\dagger}\left[\sum_{x^{\prime}}x^{\prime}\ket{x^{\prime}}\bra{x^{\prime}}\right]$
$\displaystyle\quad\times\hat{U}\left[\frac{1}{\sqrt{L}}\sum_{k^{\prime}}e^{-ik^{\prime}x}\ket{k^{\prime}}\right]\ket{\alpha}$
$\displaystyle=\frac{1}{L^{2}}\sum_{\alpha,k,k^{\prime},x^{\prime}}e^{ikx}\bra{\alpha}U^{\dagger}(k)e^{-ikx^{\prime}}[-i\partial_{k^{\prime}}e^{ik^{\prime}x^{\prime}}]$
$\displaystyle\quad\times U(k^{\prime})e^{-ik^{\prime}x}\ket{\alpha},$ (176)

where $U(k)$ acts on the space of internal degrees of freedom $\alpha$. Using
the partial integration and the relation

$\displaystyle\frac{1}{L}\sum_{x^{\prime}}e^{i(k^{\prime}-k)x^{\prime}}=\delta_{k,k^{\prime}},$
(177)

we obtain

$\displaystyle P_{x}(T)$
$\displaystyle=\frac{1}{L}\sum_{\alpha,k}e^{ikx}\bra{\alpha}U^{\dagger}(k)i\partial_{k}[U(k)e^{-ikx}]\ket{\alpha}$
$\displaystyle=\frac{1}{L}\sum_{\alpha,k}\bra{\alpha}U^{\dagger}(k)[i\partial_{k}U(k)]\ket{\alpha}+\sum_{\alpha}x.$
(178)

The first term in the right hand side in the above is the winding number in
Eq. (5)

$\displaystyle\frac{1}{L}\sum_{\alpha,k}\bra{\alpha}U^{\dagger}(k)[i\partial_{k}U(k)]\ket{\alpha}$
$\displaystyle=\int_{0}^{2\pi}\frac{dk}{2\pi}\mathrm{tr}[U^{\dagger}(k)i\partial_{k}U(k)]=w_{1}[U(k)],$
(179)

while the second term is the polarization $P_{x}$

$\displaystyle\sum_{\alpha}x=\sum_{\alpha}\bra{x,\alpha}\hat{x}\ket{x,\alpha}=P_{x}.$
(180)

Therefore, we obtain $P_{x}(T)=w_{1}[U(k)]+P_{x}$, and thus the formula in Eq.
(175).

Let us check this formula in Eq. (175) for a simple example
$U_{\text{QW}}=S_{+}R(\theta)$ with

$\displaystyle S_{+}$ $\displaystyle=\begin{pmatrix}e^{-ik}&0\\\
0&1\end{pmatrix},\ R(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix},$ (181)

which has $w_{1}[U_{\text{QW}}(k)]=1$. Consider the initial states

$\displaystyle\ket{x,\uparrow}=\begin{pmatrix}\ket{x}\\\
0\end{pmatrix},\ket{x,\downarrow}=\begin{pmatrix}0\\\ \ket{x}\end{pmatrix},$
(182)

with the polarization

$\displaystyle P_{x}$
$\displaystyle=\bra{x,\uparrow}\hat{x}\ket{x,\uparrow}+\bra{x,\downarrow}\hat{x}\ket{x,\downarrow}$
$\displaystyle=2x.$ (183)

The states after one cycle are

$\displaystyle U_{\text{QW}}\begin{pmatrix}\ket{x}\\\
0\end{pmatrix}=\begin{pmatrix}\cos\theta\ket{x+1}\\\
\sin\theta\end{pmatrix}\ket{x},$ (184) $\displaystyle
U_{\text{QW}}\begin{pmatrix}0\\\
\ket{x}\end{pmatrix}=\begin{pmatrix}-\sin\theta\ket{x+1}\\\
\cos\theta\ket{x}\end{pmatrix},$ (185)

and thus, we have

$\displaystyle P_{x}(T)$
$\displaystyle=\bra{x,\uparrow}U_{\text{QW}}^{\dagger}\hat{x}U_{\text{QW}}\ket{x,\uparrow}$
$\displaystyle\quad+\bra{x,\downarrow}U_{\text{QW}}^{\dagger}\hat{x}U_{\text{QW}}\ket{x,\downarrow}$
$\displaystyle=2x+1=P_{x}+1.$ (186)

Therefore, we obtain $P_{x}(T)-P_{x}=w_{1}[U_{\text{QW}}]=1$, which coincides
with the formula in Eq. (175)

We can generalize the above relation between the polarization and the winding
number even in the presence of disorders. For this purpose, we consider the
flux inserted unitary operator $\hat{U}(\Phi)$, where the hopping terms in
$\hat{U}$ are modified by the uniform gauge potential $A_{x}=\Phi/L$ as
$\ket{x+q}\bra{x}\to e^{-i(\Phi/L)q}\ket{x+q}\bra{x}$ Niu  _et al._ (1985).
Then, we introduce its winding number Gong  _et al._ (2018)

$\displaystyle
w_{1}[\hat{U}(\Phi)]=\int_{0}^{2\pi}\frac{d\Phi}{2\pi}\mathrm{tr}[\hat{U}^{\dagger}(\Phi)i\partial_{\Phi}\hat{U}(\Phi)].$
(187)

We note that the flux inserted unitary operator $\hat{U}(\Phi)$ is periodic in
$\Phi$ with the period $2\pi$ up to the large gauge transformation
$\hat{U}_{G}=e^{-\frac{2\pi i}{L}\hat{x}}$,
$\hat{U}(\Phi+2\pi)=\hat{U}_{G}\hat{U}(\Phi)\hat{U}_{G}^{\dagger}.$ Thus, the
winding number takes an integer. In a manner similar to Eq. (175), we can
prove

$\displaystyle P(T)-P=w_{1}[\hat{U}(\Phi)],$ (188)

where $P(T)-P$ is an averaged version of the position displacement
$P_{x}(T)-P_{x}$ after one-cycle evolution. See Appendix D for details.

Now, we extend the above results to extrinsic edge modes in 2D. Since an
extrinsic edge mode is localized at an edge of the system, say at $y=1$, we
consider the polarization at $y=1$,

$\displaystyle
P_{x}|_{y=1}=\sum_{\alpha}\bra{x,y=1,\alpha}\hat{x}\ket{x,y=1,\alpha}.$ (189)

Then, the polarization after one cycle time evolution is

$\displaystyle
P_{x}(T)|_{y=1}=\sum_{\alpha}\bra{x,y=1,\alpha}\hat{U}^{\dagger}\hat{x}\hat{U}\ket{x,y=1,\alpha},$
(190)

where $\hat{U}$ is the time-evolution operator in 2D. As shown in Appendix E,
if the system has translation symmetry, the formula in Eq. (175) can be
generalized as

$\displaystyle P_{x}(T)|_{y=1}-P_{x}|_{y=1}=w_{P}[U(k_{x})],$ (191)

where $w_{P}$ is the projected winding number defined by

$\displaystyle w_{P}[U(k_{x})]$
$\displaystyle=\int_{0}^{2\pi}\frac{dk_{x}}{2\pi}\mathrm{tr}_{y,\alpha}\left[\hat{P}_{\rm
edge}U^{\dagger}(k_{x})i\partial_{k_{x}}U(k_{x})\right],$ (192)

with the projection operator $\hat{P}_{\rm edge}=\ket{y=1}\bra{y=1}$ at the
edge. Similarly, if the system has disorders, we have a generalization of Eq.
(188) with the projected winding number for Eq. (187).

We remark that the projected winding numbers are not quantized in general
because edge modes at $y=1$ can diffuse into the bulk. However, if bulk states
are gapped or localized, edge modes rarely diffuse into the bulk, so the
quantization of the projected winding numbers is almost recovered. Under such
situations, an extrinsic chiral mode induces a directed movement of wave
packets at the edge since it has a non-trivial winding number. For instance,
our model in 2D has a well-localized bulk state as shown in Fig. 4 (c), and
thus the above mechanism explains the wave packet dynamics in Fig. 5 (a). Here
note that the argument here does not necessarily require a bulk gap. In Fig.
6, we show the wave packet dynamics in the model of Eq. (172) with $W=10,J=1$,
where the bulk band covers the whole energy range from $-\pi$ to $\pi$. Even
in this case, we observe that wave packets propagate to the $+x$-direction and
there exists delocalized eigenstates along the edge.

Figure 6: In case of no band gap: (a) Dynamics of a wave packet, (b) DOS and
(c) a delocalized eigenstate of the Anderson model with the extrinsic edge
mode in Eq. (172). The parameters are $J=1$, $W=10$, $L_{x}=25$, $L_{y}=15$
and $T=1$. The color scale is different between (a) and (c). The initial state
of the wave packet is $\ket{x=10,y=1}$. Due to $W\gg J$, all bulk eigenstates
are localized. Whereas no band gap exists due to the strong random potential,
the extended edge state survives and enables a directed wave packet motion.
Figure 7: In case of noisy environment (spatial and temporal disorder):
Dynamics of a wave packet (a) for the Anderson model in Eq. (193), and (b)
that with the extrinsic edge mode in Eq. (194), where the random potentials
change at each time step. The parameters are $J=0.2$, $W=1$, $L_{x}=25$,
$L_{y}=15$ and $T=1$. (a) For the Anderson model without the extrinsic edge
mode, a wave packet starting at the edge diffuses into the bulk. (b) For the
Anderson model with the extrinsic edge mode, the wave packet propagates to the
$+x$-direction during the time scale shorter than diffusion.

Finally, we consider a noisy environment where the random potential in Eq.
(VI.1) fluctuates in each time step Konno (2004); Joye and Merkli (2010); Joye
(2011); Evensky  _et al._ (1990); Obuse and Kawakami (2011). Our numerical
simulations show that the directed wave packet dynamics due to the extrinsic
chiral edge mode survives during the time scale shorter than diffusion. In
Fig. 7, we compare the $t$ steps dynamics without the edge mode,

$\displaystyle U_{\rm w/o}=\prod_{s=1}^{t}e^{-iH_{A}(s)T},$ (193)

to that with the edge mode,

$\displaystyle U_{\rm w/e}=\prod_{s=1}^{t}A\cdot e^{-iH_{A}(s)T}.$ (194)

Here $H_{A}(s)$ at time step $s$ has the same form as in Eq. (VI.1) but the
random potential $\delta_{x,y}$ changes at each step $s$. As seen in Fig. 7
(a), the time dependent randomness leads to diffusion. The details of the
diffusive behavior is given in the Appendix G. In the case with the extrinsic
chiral edge mode [Fig. 7 (b)], on the other hand, we find again a directed
wave packet displacement due to the nontrivial winding number $w_{1}[U_{\rm
w/e}(\Phi)]=N$. Such robustness in a noisy environment would enable extrinsic
modes to realize fault-tolerance quantum devices.

###  VI.2 class AIII the split step quantum walk in 1D: cancelation of the
edge mode

Boundary modes of chiral-symmetric quantum walks in 1D have been
experimentally observed as a localization of dynamics Kitagawa  _et al._
(2012); Barkhofen  _et al._ (2017). The extrinsic topological phase, however,
can cancel the boundary states, leading to a dynamics with delocalized
behaviors.

Let us study the split step quantum walk model in 1D Kitagawa  _et al._
(2010b, 2012); Obuse  _et al._ (2015):

$\displaystyle U_{\text{QW}}=U_{2}U_{1},$ $\displaystyle
U_{1}=R_{2}^{1/2}S_{-}R_{1}^{1/2},\ U_{2}=R_{1}^{1/2}S_{+}R_{2}^{1/2}.$ (195)

Here, $S_{+}$ and $S_{-}$ are the shift operators, and $R_{j}=R(\theta_{j})$
is the spin rotation coin operator, which are defined as follows:

$\displaystyle S_{+}(k)=\begin{pmatrix}e^{-ik}&0\\\ 0&1\end{pmatrix},\
S_{-}(k)=\begin{pmatrix}1&0\\\ 0&e^{ik}\end{pmatrix},$ (196) $\displaystyle
R(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}.$ (197)

This model has the decomposed CS in Eq. (157) with $\Gamma=\sigma_{x}$. After
performing a unitary transformation of the basis so that $\Gamma$ becomes
$\Gamma=\sigma_{z}$, we calculate the bulk topological numbers
$(w^{0},w^{\pi})$ defined by the right hand side of Eq. (161),

$\displaystyle w^{0}=\frac{w_{1}[a]+w_{1}[d]}{2},\quad
w^{\pi}=\frac{w_{1}[b]+w_{1}[c]}{2},$ (198)

for the split step quantum walk in Eq. (195). The obtained topological numbers
are summarized in the phase diagram in Fig. 8 (a).

Figure 8: (a) Topological phase diagram of the split-step quantum walk.
$(w^{0},w^{\pi})$ represent the bulk topological invariants given by Eq.
(198). The red triangle and star represent the parameters for the left chain,
$(\theta_{1}^{L},\theta_{2}^{L})=(0,\pi/4)$, and the right one,
$(\theta_{1}^{R},\theta_{2}^{R})=(0,-\pi/4)$, used in the numerical
simulations in Figs. 9 and 10. (b) The setup of the split-step quantum walk.
The left and right regions of the quantum walk are joined at two edges at
$x=1$ and $L$.  Figure 9: (a) Energy spectrum, (b) dynamics, and (c-f)
eigenstates of the split step walk in Eq. (195). The total system size is
$2N=20$. The parameters in the left half chain are
$(\theta_{1}^{L},\theta_{2}^{L})=(0,\pi/4)$, while those in the right half
chain are $(\theta_{1}^{R},\theta_{2}^{R})=(0,-\pi/4)$. The initial state is
$\ket{x=11,\downarrow}$. At $x=N+1$, we have a single $\epsilon=0$ mode and an
$\epsilon=\pi$ mode. The topological number of the $\epsilon=0$ mode is $-1$,
while that of the $\epsilon=\pi$ mode is $+1$.

To examine boundary modes of the system, we consider the loop configuration
shown in Fig. 8 (b). The loop consists of left and right chains with the same
length $N$, where $U_{\rm QW}$ in the left (right) chain has the coin
operators with the parameter $(\theta_{1}^{L},\theta_{2}^{L})=(0,\pi/4)$
($(\theta_{1}^{R},\theta_{2}^{R})=(0,-\pi/4)$). Boundary modes appear at the
interfaces between the left and right chains.

Figure 9 (a) shows eigenvalues $\lambda$ of the eigenequation
$U_{\text{QW}}\ket{\psi}=\lambda\ket{\psi}$ on the loop configuration, where
$\lambda=1$ ($\lambda=-1$) indicates the $\epsilon=0$ ($\epsilon=\pi$)
boundary modes. Both of the interfaces at $x=1,N+1$ host a single $\epsilon=0$
mode $\ket{\psi^{0}_{x=1,N+1}}$ and a $\epsilon=\pi$ one
$\ket{\psi^{\pi}_{x=1,N+1}}$ with the topological number
$\nu_{x=1,N+1}^{0,\pi}=\langle\psi^{0,\pi}_{x=1,N+1}|\Gamma|\psi_{x=1,N+1}^{0,\pi}\rangle$
shown in Figs. 9 (c)-(f). We note that the boundary gapless modes satisfy the
bulk-boundary correspondence:

$\displaystyle\nu_{x=1}^{0,\pi}=-\nu^{0,\pi}_{x=N+1}=w^{0,\pi}_{L}-w^{0,\pi}_{R},$
(199)

where $w^{0,\pi}_{R,L}$ is the bulk topological number in the right and left
chains. Here the original bulk-boundary correspondence in Eq. (161) is
slightly modified because we have considered interfaces between two
topologically non-trivial chains. Due to the existence of the boundary modes,
a wave packet initially localized at one of the interfaces remains localized
after time evolution [Fig. 9(b)].

As discussed in Sec. V, if one relaxes the decomposed CS in Eq. (157) by the
original CS in Eq. (20), we can change the number of boundary gapless modes by
using the extrinsic topological phase. To see this, we introduce the following
unitary operator $A$

$\displaystyle A=\sum_{x\neq
N}^{2N}\ket{x}\bra{x}\otimes\sigma_{0}+\ket{x=N}\bra{x=N}\otimes
U_{\text{BDQW}},$ (200)

where the boundary unitary operator $U_{\text{BDQW}}$ is taken to be
nontrivial

$\displaystyle U_{\text{BDQW}}=\sigma_{x}.$ (201)

We insert $A$ between $U_{1}$ and $U_{2}$ in Eq. (195),

$\displaystyle U_{\text{QW}}=U_{2}AU_{1}.$ (202)

Since $A$ satisfies $\Gamma A^{\dagger}\Gamma^{-1}=A$, $U_{\rm QW}$ in the
above has the original CS. We can show that $U_{\text{BDQW}}$ has nontrivial
extrinsic boundary states. From the Hermitian matrix
$U_{\text{BDQW}}\Gamma=\sigma_{0}$, we have a nontrivial topological invariant
in Eq. (33),

$\displaystyle
n=\frac{1}{2}[N_{+}(U_{\text{BDQW}}\Gamma)-N_{-}(U_{\text{BDQW}}\Gamma)]=1.$
(203)

Therefore, from the extended Nielsen-Ninomiya theorem in Eq. (22), we obtain
extrinsic boundary modes at $x=N$ with the energies $\epsilon=0,\pi$ and the
topological numbers $\nu^{0}=-\nu^{\pi}=1$. As shown below, these extrinsic
boundary modes may cancel the original edge modes at the interface at $x=N+1$.

The eigenenergies of $U_{\rm QW}$ in Eq. (202) is shown in Fig. 10 (a). While
the spectrum has $\epsilon=0$ and $\epsilon=\pi$ modes, they are localized
near the interface at $x=1$, as shown in Fig. 10 (c) and (d). No gapless edge
mode exists near the interface at $x=N+1$. As a result, in contrast to the
previous case, a wave packet initially localized at $x=N+1$ spreads after time
evolution [Fig. 10 (b)].

Figure 10: (a) Energy spectrum, (b) dynamics, and (c,d) eigenstates of the
decorated split step walk in Eq. (202). The total system size is $2N=20$. The
parameters in the left half chain is
$(\theta_{1}^{L},\theta_{2}^{L})=(0,\pi/4)$, while those in the right half
chain is $(\theta_{1}^{R},\theta_{2}^{R})=(0,-\pi/4)$. The initial state is
$\ket{x=11,\downarrow}$. No localized state at boundary $x=N+1$ is found both
in the spectrum and the dynamics.

###  VI.3 class A in 2D: cancelation of the chiral edge mode

Floquet topological phases may host chiral edge modes without the Chern number
Kitagawa  _et al._ (2010a); Rudner  _et al._ (2013); Asboth and Edge (2015).
The Floquet anomalous edge states originate from the bulk topological
invariant of the time-evolution operator $U(\bm{k},t)$ in Eq. (138), and have
been experimentally realized in photonic systems Maczewsky  _et al._ (2017);
Mukherjee  _et al._ (2017); Chen  _et al._ (2018). In this section, we
demonstrate that the Floquet anomalous edge states can be eliminated by using
the extrinsic topology of quantum walks in 2D.

We consider the model in 2D with Floquet anomalous edge states in Ref. Rudner
_et al._ (2013):

$\displaystyle H(t)=H_{j},\ t\in[(j-1)T/5,jT/5],$ (204)

with

$\displaystyle
H_{j=1,2,3,4}=Je^{i\bm{b}_{j}\cdot\bm{k}}\sigma_{+}+Je^{-i\bm{b}_{j}\cdot\bm{k}}\sigma_{-}+\delta_{AB}\sigma_{z},$
$\displaystyle H_{5}=\delta_{AB}\sigma_{z}.$ (205)

Here $\bm{b}_{1}=-\bm{b}_{3}=(a,0)$ and $\bm{b}_{2}=-\bm{b}_{4}=(0,a)$,
$\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$, and $\delta_{AB}$ is a real
parameter. The one-cycle time evolution of the model is given by

$\displaystyle U_{R}=\prod_{j=1}^{5}e^{-iH_{j}T/5}.$ (206)

The energy spectrum and the dynamics of anomalous edge modes in the model are
shown in Figs. 11 (a) and (c). This model has gapless chiral edge modes both
at the $\epsilon=0$ and $\epsilon=\pi$ gaps. The existence of the chiral edge
modes is easily understood for $JT/5=\pi/2$ and $\delta_{AB}=0$. In this case,
each time-evolution unitary operator becomes

$\displaystyle
e^{-iH_{j=1,2,3,4}T/5}=-i(e^{i\bm{b}_{j}\cdot\bm{k}}\sigma_{+}+e^{-i\bm{b}_{j}\cdot\bm{k}}\sigma_{-}),$
$\displaystyle e^{-iH_{5}T/5}=1,$ (207)

which makes the total time-evolution operator trivial $U_{R}=\hat{1}$, but
there exists a chiral mode at the boundary, as shown in Fig. 12. The chiral
edge mode remains even if we modify the parameters $\delta_{AB}$ and $J$
unless the energy gaps at $\epsilon=0,\pi$ are closed.

Figure 11: Energy spectrum and dynamics (a,c) without and (b,d) with the
unitary operator $A$ in Eq. (210). We use (a,c) the model with $U_{R}$ in Eq.
(206) and (b,d) that with $U^{\prime}_{R}$ in Eq. (209). The parameters are
$J/T=2.2\pi$, $\delta=1.3\pi$. For (a) and (b), the system size is $L_{y}=30$.
For (c) and (d), the system size is $L_{x}=25$ and $L_{y}=15$. The initial
state is $\ket{x=5,y=1,\downarrow}$. We take the periodic (open) boundary
condition in the $x$ ($y$) direction. (a) The model $U_{R}$ in Eq. (206) has
Floquet anomalous chiral edge modes. (c) Due to the existence of chiral edge
modes, a wave packet on the edge at $y=1$ propagates to the $+x$-direction.
(b) The decorated model $U^{\prime}_{R}$ in Eq. (209) exhibits no chiral edge
mode. (d) Due to the disappearance of chiral edge modes, only a diffusive
behaviour into the bulk is observed.  Figure 12: One-cycle dynamics of the
model Eq. (206) at $JT/5=\pi/2$ and $\delta_{AB}=0$. This model has a
bipartite lattice structure with spin up (red) and spin down (blue). After the
one-cycle time evolution, bulk states return to the same position, while a
spin up (down) edge state at $y=1\ (L_{y})$ propagates in the
$+(-)x$-direction.

The presence of the Floquet anomalous chiral edge state is ensured by the
three-dimensional winding number of the loop unitary $L({\bm{k}},t)$ in Eq.
(148),

$\displaystyle w_{3}[L]$ $\displaystyle=\frac{1}{8\pi^{2}}\int dk_{x}dk_{y}dt$
$\displaystyle\times\mathrm{tr}\left(L^{-1}\partial_{t}L[L^{-1}\partial_{k_{x}}L,L^{-1}\partial_{k_{y}}L]\right).$
(208)

However, as we already pointed out in Sec. IV, $L({\bm{k}},t)$ is well-defined
only when the microscopic Hamiltonian $H({\bm{k}},t)$ exists. If one allows a
general deformation of the time-evolution unitary operator, one can trivialize
$L({\bm{k}},t)$ without closing gaps at $\epsilon=0,\pi$. Therefore, in the
framework of quantum walks, where no microscopic Hamiltonian is assumed in
general, the Floquet anomalous edge mode is not protected by the bulk
topological number.

Actually, we can eliminate the anomalous edge mode in Fig. 11 (a) by
multiplying $U_{R}$ by $A$,

$\displaystyle U^{\prime}_{R}=AU_{R},$ (209) $\displaystyle
A(k_{x})=U_{\text{edge}}^{y=1}(k_{x})\otimes\ket{y=1}\bra{y=1}$
$\displaystyle+U_{\text{edge}}^{y=L_{y}}(k_{x})\otimes\ket{y=L_{y}}\bra{y=L_{y}}$
$\displaystyle+\sum_{y=2}^{L_{y}-1}\sigma_{0}\otimes\ket{y}\bra{y},$ (210)
$\displaystyle U_{\text{edge}}^{y=1}(k_{x})=\begin{pmatrix}e^{2ik_{x}}&0\\\
0&1\end{pmatrix},\ U_{\text{edge}}^{y=L_{y}}(k_{x})=\begin{pmatrix}1&0\\\
0&e^{-2ik_{x}}\end{pmatrix},$ (211)

where we impose the open boundary conditions on $U_{R}$ at $y=1,L_{y}$. As
shown in Figs. 11 (b) and (d), no gapless chiral edge mode exists in the
quasi-particle spectrum of $U_{R}^{\prime}$, and no directed wave packet
motion is observed on the edges.

The extrinsic topology of $U_{\text{edge}}^{y=1}$ and
$U_{\text{edge}}^{y=L_{y}}$ in $A$ explains the disappearance of the anomalous
edge modes. For these boundary unitary operators, the winding number in Eq.
(5) becomes nonzero:

$\displaystyle
w_{1}\left[U_{\text{edge}}^{y=1}(k_{x})\right]=-w_{1}\left[U_{\text{edge}}^{y=L_{y}}(k_{x})\right]=2.$
(212)

Thus, $U_{\text{edge}}^{y=1}$ and $U_{\text{edge}}^{y=L_{y}}$ have the
extrinsic topology phase at the boundaries. From the the extended Nielsen-
Ninomiya theorem in Eq. (22), $U_{\text{edge}}^{y=1}$ and
$U_{\text{edge}}^{y=L_{y}}$ provide an anomalous edge state on each boundary
that has the chirality opposite to that in the original model. As a result,
$U^{\prime}_{R}$ does not have no net topological number for the anomalous
edge modes, and thus no stable anomalous edge mode remains.

We emphasize again that the boundary unitary operator $A$ affects nontrivially
only on the edges, and it controls the presence and absence of anomalous
gapless edge states, which implies the extrinsic nature of the edge states.

##  VII Conclusion

In this work, we argue classification of topological phases in quantum walks.
Due to the discrete nature of quantum walk dynamics, the bulk topological
invariant is insufficient to determine the boundary states. The numbers of
boundary states depend on both the bulk topology and the boundary topology.
While the conventional topological insulators and superconductors in
equilibrium systems also may have similar extrinsic nature in higher-order
topological phases, quantum walks may support it even in the first-order
topological phases.

The extrinsic boundary states in quantum walks resemble to anomalous boundary
states in Floquet systems, but their topological origin is different. For
Floquet systems, the anomalous boundary states originate from the non-trivial
topology of the bulk continuous time-evolution operator, but for quantum
walks, the continuous time-evolution operator is not given.

In the previous work Asbóth and Obuse (2013), it was shown that the bulk-
boundary correspondence holds for class AIII systems in 1D with a decomposed
realization of CS. We explain how their definition assures the bulk-boundary
correspondence, and discuss a similar bulk-boundary correspondence in other
dimensions and symmetry classes. We find that class CII quantum walks in 1D
obey the bulk-boundary correspondence under decomposed TRS and PHS.

We also examine physical implementations of extrinsic topological phases in
quantum walks. We numerically show that the extrinsic boundary states induce
the charge pump which is robust against disorders along the edge of 2D
systems. We also present general arguments for the robustness of the charge
pump. Moreover, we show that using the extrinsic topology, one can eliminate
the pre-existing anomalous boundary states in the class AIII 1D split step
walk and a class A 2D model, respectively. We can change the types and the
numbers of gapless boundary states without changing the bulk, which implies
the breakdown of the bulk-boundary correspondence in quantum walks.

In this work, we have focused on quantum walks, but most of the arguments hold
for any other systems described by unitary operators such as cellular
automatons. Applying the present arguments to such systems could be
interesting. Another promising direction is higher-order boundary states in
quantum walks. While we have studied only the first-order boundary states in
this work, higher-order boundary states may have richer extrinsic topological
behaviors. Quantum walks also support unique symmetry that have no counterpart
in static systems such as time-glide symmetry Mochizuki  _et al._ (2020). Such
symmetry could produce extrinsic topological phases unique to dynamical
systems.

This work was supported by JST CREST Grant No. JPMJCR19T2, JST ERATO-FS Grant
No. JPMJER2105, and KAKENHI Grant Nos. JP21J11810, JP20H00131, JP20H01845,
JP18H01140, JP20H01828, and JP21H01005 from the JSPS.

##  Appendix A Topological classification of
$U_{\text{BDQW}}(\bm{k}_{\parallel})$

As explained in the main text, we can classify
$U_{\text{BDQW}}(\bm{k}_{\parallel})$ by using ${\cal
H}_{U}({\bm{k}}_{\parallel})$ with symmetry in Eqs. (24)-(27). In this
section, we perform the topological classification of ${\cal
H}_{U}(\bm{k}_{\parallel})$ by the extension of Clifford algebras Morimoto and
Furusaki (2013); Kitaev (2009).

We first review the Clifford algebra. The Clifford algebra is a ring in
mathematics, which has addition and multiplication. The complex Clifford
algebra $Cl_{n}$ has $n$ generators $\\{e_{1},\ldots,e_{n}\\}$ satisfying

$\displaystyle\\{e_{i},e_{j}\\}=\delta_{i,j},$ (213)

and the linear combination of the products $e_{1}^{p_{1}}e_{2}^{p_{2}}\cdots
e_{n}^{p_{n}}$ form a $2^{n}$-dimensional complex vector space. The real
Clifford algebra $Cl_{p,q}$ has $p+q$ generators
$\\{e_{1},\ldots,e_{p};e_{p+1},\ldots,e_{p+q}\\}$ satisfying

$\displaystyle\\{e_{i}$ $\displaystyle,e_{j}\\}=0\ \text{for}\ i\neq j,$ (214)
$\displaystyle e_{i}^{2}$ $\displaystyle=\begin{cases}-1&\text{for}\ 1\leq
i\leq p,\\\ 1&\text{for}\ p+1\leq i\leq p+q,\end{cases}$ (215)

and their products form a $2^{p+q}$-dimensional real vector space. For
instance, generators of $Cl_{2}$ are given by the Pauli matrices:

$\displaystyle\\{e_{1},e_{2}\\}=\\{\sigma_{x},\sigma_{y}\\},$ (216)

of which products provide the basis of the $2^{2}$-dimensional complex vector
space,

$\displaystyle 1,\sigma_{x},\sigma_{y},i\sigma_{z}.$ (217)

The space coincides with that of $2\times 2$ matrices $\mathbb{C}(2)$, and
thus we obtain the isomorphism $Cl_{2}\simeq\mathbb{C}(2)$.

We can obtain $Cl_{n+1}$ by adding a generator $e_{0}$ to a given
representation of $Cl_{n}$. The problem to identify all possible
representations of $e_{0}$ is called the extension problem of $Cl_{n}\to
Cl_{n+1}$, and the space of the representations is called the classifying
space $C_{n}$. In Table 3, we summarize the classifying space $C_{n}$ and the
number of connected parts of $C_{n}$, $\pi_{0}(C_{n})$. $C_{n}$ has the Bott
periodicity $C_{n+2}=C_{n}$.

We also have a similar extension problem for real Clifford algebras. For a
given representation of $Cl_{p,q}$, we add a generator $e_{0}$ that satisfies
$e_{0}^{2}=-1$ ($e_{0}^{2}=1$), then obtain the real Clifford algebra
$Cl_{p+1,q}$ ($Cl_{p,q+1}$). The extension problem $Cl_{p,q}\to Cl_{p+1,q}$
($Cl_{p,q}\to Cl_{p,q+1}$) defines the classifying space $R_{p+2-q}$
($R_{q-p}$). Table 4 summarizes the classifying space $R_{q}$ and the number
of connected parts of $R_{q}$, $\pi_{0}(R_{q})$. $R_{q}$ has the Bott
periodicity $R_{q+8}=R_{q}$.

Table 3: Classifying space $C_{n}$.  $n\mod 2$ |  Classifying space $C_{n}$ |  $\pi_{0}(C_{n})$
---|---|---
0 | $[U(k+m)/(U(k)\times U(m))]\times\mathbb{Z}$ | $\mathbb{Z}$
1 | $U(k)$ | 0
Table 4: Classifying space $R_{q}$.  $q\mod 2$ |  Classifying space $R_{q}$ |  $\pi_{0}(R_{q})$
---|---|---
0 | $[O(k+m)/(O(k)\times O(m))]\times\mathbb{Z}$ | $\mathbb{Z}$
1 | $O(k)$ | $\mathbb{Z}_{2}$
2 | $O(2k)/U(k)$ | $\mathbb{Z}_{2}$
3 | $U(2k)/Sp(k)$ | 0
4 | $[Sp(k+m)/(Sp(k)\times Sp(m))]\times\mathbb{Z}$ | $\mathbb{Z}$
5 | $Sp(k)$ | 0
6 | $Sp(k)/U(k)$ | 0
7 | $U(k)/O(k)$ | 0
Table 5: Clifford algebra extensions and classifying spaces for ${\cal H}_{U}(\bm{k}_{\parallel})$.  AZ class |  $T$ |  $C$ |  $\Gamma$ |  Generator |  Extension |  Classifying space
---|---|---|---|---|---|---
A | 0 | 0 | 0 | $\\{\gamma_{1},\ldots,\gamma_{d-1},\gamma_{0},\Sigma_{z}\\}$ | $Cl_{d}\to Cl_{d+1}$ | $C_{d}$
AIII | 0 | 0 | 1 | $\\{\gamma_{1},\ldots,\gamma_{d-1},\gamma_{0},\Sigma_{z},J\Sigma_{z}\tilde{\Gamma}\\}$ | $Cl_{d+1}\to Cl_{d+2}$ | $C_{d+1}$
AI | $+1$ | 0 | 0 | $\\{J\gamma_{0};\gamma_{1},\ldots,\gamma_{d-1},\Sigma_{z},\tilde{T},J\tilde{T}\\}$ | $Cl_{0,d+2}\to Cl_{1,d+2}$ | $R_{-d}$
BDI | $+1$ | $+1$ | 1 | $\\{J\gamma_{0},\tilde{\Gamma}\Sigma_{z};\gamma_{1},\ldots,\gamma_{d-1},\Sigma_{z},\tilde{T},J\tilde{T}\\}$ | $Cl_{1,d+2}\to Cl_{2,d+1}$ | $R_{1-d}$
D | 0 | $+1$ | 0 | $\\{J\gamma_{0},J\Sigma_{z};\gamma_{1},\ldots,\gamma_{d-1},\tilde{C},J\tilde{C}\\}$ | $Cl_{1,d+1}\to Cl_{2,d+1}$ | $R_{2-d}$
DIII | $-1$ | $+1$ | 1 | $\\{J\gamma_{0},\tilde{T},J\tilde{T};\gamma_{1},\ldots,\gamma_{d-1},\Sigma_{z},\tilde{\Gamma}\Sigma_{z}\\}$ | $Cl_{2,d+1}\to Cl_{3,d+1}$ | $R_{3-d}$
AII | $-1$ | 0 | 0 | $\\{J\gamma_{0},\tilde{T},J\tilde{T};\gamma_{1},\ldots,\gamma_{d-1},\Sigma_{z}\\}$ | $Cl_{2,d}\to Cl_{3,d}$ | $R_{4-d}$
CII | $-1$ | $-1$ | 1 | $\\{J\gamma_{0},\tilde{T},J\tilde{T},\tilde{\Gamma}\Sigma_{z};\gamma_{1},\ldots,\gamma_{d-1},\Sigma_{z}\\}$ | $Cl_{3,d}\to Cl_{4,d}$ | $R_{5-d}$
C | 0 | $-1$ | 0 | $\\{J\gamma_{0},J\Sigma_{z},\tilde{C},J\tilde{C};\gamma_{1},\ldots,\gamma_{d-1}\\}$ | $Cl_{3,d-1}\to Cl_{4,d-1}$ | $R_{6-d}$
CI | $+1$ | $-1$ | 1 | $\\{J\gamma_{0};\gamma_{1},\ldots,\gamma_{d-1},\Sigma_{z},\tilde{T},J\tilde{T},\tilde{\Gamma}\Sigma_{z}\\}$ | $Cl_{0,d+3}\to Cl_{1,d+3}$ | $R_{7-d}$
Table 6: Clifford algebra extensions and classifying spaces for ${\cal H}_{L}(\bm{k},t)$.  AZ class |  $T$ |  $C$ |  $\Gamma$ |  Generator |  Extension |  Classifying space
---|---|---|---|---|---|---
A | 0 | 0 | 0 | $\\{\gamma_{1},\ldots,\gamma_{d},\gamma_{t},\gamma_{0},\Sigma_{z}\\}$ | $Cl_{d+2}\to Cl_{d+3}$ | $C_{d}$
AIII | 0 | 0 | 1 | $\\{\gamma_{1},\ldots,\gamma_{d},\gamma_{t},\gamma_{0},\Sigma_{z},J\gamma_{t}\tilde{\Gamma}\\}$ | $Cl_{d+3}\to Cl_{d+4}$ | $C_{d+1}$
AI | $+1$ | 0 | 0 | $\\{J\gamma_{0},J\Sigma_{z};\gamma_{1},\ldots,\gamma_{d},\gamma_{t},\tilde{T},J\tilde{T}\\}$ | $Cl_{1,d+3}\to Cl_{2,d+3}$ | $R_{-d}$
BDI | $+1$ | $+1$ | 1 | $\\{J\gamma_{0},J\Sigma_{z},\gamma_{t}\tilde{\Gamma};\gamma_{1},\ldots,\gamma_{d},\gamma_{t},\tilde{T},J\tilde{T}\\}$ | $Cl_{2,d+3}\to Cl_{3,d+3}$ | $R_{1-d}$
D | 0 | $+1$ | 0 | $\\{J\gamma_{0},J\gamma_{t},J\Sigma_{z};\gamma_{1},\ldots,\gamma_{d},\tilde{C},J\tilde{C}\\}$ | $Cl_{2,d+2}\to Cl_{3,d+2}$ | $R_{2-d}$
DIII | $-1$ | $+1$ | 1 | $\\{J\gamma_{0},J\Sigma_{z},\tilde{T},J\tilde{T};\gamma_{1},\ldots,\gamma_{d},\gamma_{t},\gamma_{t}\tilde{\Gamma}\\}$ | $Cl_{2,d+2}\to Cl_{3,d+2}$ | $R_{2-d}$
AII | $-1$ | 0 | 0 | $\\{J\gamma_{0},J\Sigma_{z},\tilde{T},J\tilde{T};\gamma_{1},\ldots,\gamma_{d},\gamma_{t}\\}$ | $Cl_{3,d+1}\to Cl_{4,d+1}$ | $R_{4-d}$
CII | $-1$ | $-1$ | 1 | $\\{J\gamma_{0},J\Sigma_{z},\gamma_{t}\tilde{\Gamma},\tilde{T},J\tilde{T};\gamma_{1},\ldots,\gamma_{d},\gamma_{t}\\}$ | $Cl_{4,d+1}\to Cl_{5,d+1}$ | $R_{5-d}$
C | 0 | $-1$ | 0 | $\\{J\gamma_{0},J\gamma_{t},J\Sigma_{z},\tilde{C},J\tilde{C};\gamma_{1},\ldots,\gamma_{d}\\}$ | $Cl_{4,d}\to Cl_{5,d}$ | $R_{6-d}$
CI | $+1$ | $-1$ | 1 | $\\{J\gamma_{0},J\Sigma_{z};\gamma_{1},\ldots,\gamma_{d},\gamma_{t},\gamma_{t}\tilde{\Gamma},\tilde{T},J\tilde{T}\\}$ | $Cl_{1,d+4}\to Cl_{2,d+4}$ | $R_{7-d}$

We perform the topological classification of ${\cal
H}_{U}(\bm{k}_{\parallel})$ as the extension problem of Clifford algebras.
Since it is enough to classify the system near topological phase transitions,
we consider ${\cal H}_{U}({\bm{k}}_{\parallel})$ in the form of the Dirac
Hamiltonian

$\displaystyle{\cal
H}_{U}(\bm{k}_{\parallel})=\sum_{j=1}^{d-1}k_{j}\gamma_{j}+\gamma_{0},$ (218)

where $\gamma_{j}$ and $\gamma_{0}$ are the gamma matrices that satisfy the
relation $\\{\gamma_{\mu},\gamma_{\nu}\\}=\delta_{\mu,\nu}$
($\mu,\nu=0,1,\ldots,d-1$). Then, the gamma matrices $\gamma_{\mu}$, symmetry
operations for ${\cal H}_{U}({\bm{k}}_{\parallel})$ in Eqs. (24)-(27), and the
imaginary unit $J:=i$ if ${\cal H}_{U}({\bm{k}}_{\parallel})$ has anti-unitary
symmetry, form the Clifford algebra in Table 5. Possible mass terms
$\gamma_{0}$ in the Dirac Hamiltonian provide possible topological phases, and
thus the classification reduces to the extension problem of the Clifford
algebra without $\gamma_{0}$ to that with $\gamma_{0}$. For each symmetry
class, we summarize the extension and the classifying space in Table 5. Using
Tables 3 and 4, we can specify the connected component $\pi_{0}$ of the
classification space, which gives the topological numbers in Table 2 in the
main text.

##  Appendix B Topological classification of $L(\bm{k},t)$

In a manner similar to Appendix A, we can perform the topological
classification of $L({\bm{k}},t)$ by using ${\cal H}_{L}(\bm{k},t)$ under
symmetry in Eqs. (150)-(153). In Table 6, we summarize the Clifford algebra,
the extension by adding the mass term $\gamma_{0}$, and the classifying space
for the "Hamiltonian"

$\displaystyle{\cal
H}_{L}(\bm{k},t)=\sum_{j=1}^{d}k_{j}\gamma_{j}+t\gamma_{t}+\gamma_{0},$ (219)

for each symmetry class with symmetry in Eqs. (150)-(153). Note that the
obtained classifying spaces are the same as those for ${\cal
H}_{U}({\bm{k}}_{\parallel})$ in Table 5. Therefore, the topological
classification of $L(\bm{k},t)$ coincides with that of
$U({\bm{k}}_{\parallel})$.

##  Appendix C The winding number $w_{1}[a]$, $w_{1}[b]$, $w_{1}[c]$,
$w_{1}[d]$ for class CII in 1D

In this section, we show that the winding number $w_{1}[a]$ in Sec. V takes an
even integer for class CII in 1D with decomposed TRS and PHS in Eqs. (166) and
(167). In a similar manner, one can show that $w_{1}[b]$ $w_{1}[c]$, and
$w_{1}[d]$ are also even integers.

For this purpose, we take the basis where CS and PHS are given by

$\displaystyle\Gamma=\begin{pmatrix}\hat{1}&0\\\ 0&-\hat{1}\end{pmatrix},\quad
C=\begin{pmatrix}\sigma_{2}&0\\\ 0&-\sigma_{2}\end{pmatrix}K.$ (220)

Then, PHS in Eq. (167) leads to

$\displaystyle\begin{pmatrix}\sigma_{2}a^{*}(k)\sigma_{2}&-\sigma_{2}b^{*}(k)\sigma_{2}\\\
-\sigma_{2}c^{*}(k)\sigma_{2}&\sigma_{2}d^{*}(k)\sigma_{2}\end{pmatrix}=\begin{pmatrix}a(-k)&b(-k)\\\
c(-k)&d(-k)\end{pmatrix}.$ (221)

When the system has a gap at $\epsilon=0$, it holds that $\det[a(k)]\neq 0$
Mochizuki  _et al._ (2020), and thus one can define the winding number
$w_{1}[a]$. To show that $w_{1}[a]$ is an even integer, we first deform $a(k)$
into a unitary matrix. During the deformation, we can retain the condition
$\det[a(k)]\neq 0$, and so $w_{1}[a]$ does not change. After the
unitarization, $a(k)$ is diagonalizable. Because of
$\sigma_{2}a^{*}(k)\sigma_{2}=a(-k)$ in Eq. (221), any eigenstate of $a(k)$
has a Kramers partner, and thus $a(k)$ is written as

$\displaystyle a(k)=$
$\displaystyle\sum_{n}\big{[}\lambda_{n}(k)\ket{\psi_{n}(k)}\bra{\psi_{n}(k)}$
$\displaystyle+\lambda_{n}^{*}(-k)\sigma_{2}\ket{\psi_{n}^{*}(-k)}\bra{\psi_{n}^{*}(-k)}\sigma_{2}\big{]},$
(222)

where $|\psi_{n}(k)\rangle$ is an eigenstate of $a(k)$ with the eigenvalue
$\lambda_{n}(k)$. Then, we have

$\displaystyle\mathrm{tr}[a(k)^{\dagger}\partial_{k}a(k)]$
$\displaystyle=\partial_{k}\log\det[a(k)]$
$\displaystyle=\partial_{k}\log\left[\prod_{n}\lambda_{n}(k)\lambda_{n}^{*}(-k)\right]$
$\displaystyle=\partial_{k}\left[\sum_{n}\log\lambda_{n}(k)+\sum_{n}\log\lambda_{n}^{*}(-k)\right],$
(223)

which leads to

$\displaystyle w_{1}[a]$ $\displaystyle=\int_{0}^{2\pi
i}\frac{dk}{2\pi}\mathrm{tr}[a(k)^{\dagger}\partial_{k}a(k)]$
$\displaystyle=\int_{0}^{2\pi
i}\frac{dk}{2\pi}\partial_{k}\left[\sum_{n}\log\lambda_{n}(k)+\sum_{n}\log\lambda_{n}^{*}(-k)\right]$
$\displaystyle=2\int_{0}^{2\pi
i}\frac{dk}{2\pi}\partial_{k}\sum_{n}\log\lambda_{n}(k).$ (224)

Because the integral

$\displaystyle\int_{0}^{2\pi
i}\frac{dk}{2\pi}\partial_{k}\sum_{n}\log\lambda_{n}(k)$ (225)

is an integer due to the periodicity of $a(k)$ in $k$, $w_{1}[a]$ is an even
integer.

##  Appendix D Proof of Eq. (188)

In this section, we prove the formula in Eq. (188) in the main text. For a
rigorous argument, instead of the original quantum walk operator $\hat{U}$
having $L$ sites in the $x$-direction, we consider $N$ copies of $\hat{U}$
arranged in a line in the $x$-direction, which is denoted by $\hat{U}_{N}$.
Whereas $\hat{U}$ does not have translation symmetry in the presence of onsite
disorders, $\hat{U}_{N}$ has translation invariance with respect to $L$-sites
translations. We also assume the periodic boundary condition of $\hat{U}_{N}$
in the $x$-direction. Below we show that the position displacement after one-
cycle evolution by $\hat{U}_{N}$ is equal to the winding number of
$\hat{U}(\Phi)$:

$\displaystyle\sum_{x=1}^{L}\sum_{\alpha}\bra{x,\alpha}\hat{U}_{N}^{\dagger}\frac{\hat{x}}{L}\hat{U}_{N}\ket{x,\alpha}-\sum_{x=1}^{L}\sum_{\alpha}\bra{x,\alpha}\frac{\hat{x}}{L}\ket{x,\alpha}$
$\displaystyle=\int_{0}^{2\pi}\frac{d\Phi}{2\pi}\mathrm{tr}\left[\hat{U}^{\dagger}(\Phi)i\partial_{\Phi}\hat{U}(\Phi)\right].$
(226)

The right hand side of the above equation defines $P(T)-P$, i.e. an averaged
version of $P_{x}(T)-P_{x}$, and thus Eq. (D) is the exact description of Eq.
(188) in the main text.

To show the above equation, we introduce $\hat{U}_{N}(\Phi)$ by

$\displaystyle\hat{U}_{N}(\Phi)=e^{-i(\Phi/L)\hat{x}}\hat{U}_{N}e^{i(\Phi/L)\hat{x}},$
(227)

where $\Phi$ takes the discrete values of $\Phi=2\pi p/N$ with integers $p$.
For these values of $\Phi$, $\hat{U}_{N}(\Phi)$ keeps the same periodic
boundary condition as $\hat{U}_{N}$, and equals to the flux inserted
$\hat{U}_{N}$ by the uniform gauge potential $A_{x}=\Phi/L$. Below, we assume
the large $N$ limit, and treat $\Phi$ as a continuous real number.

The flux inserted operator obeys the Heisenberg equation

$\displaystyle
i\partial_{\Phi}\hat{U}_{N}(\Phi)=\left[\frac{\hat{x}}{L},\hat{U}_{N}\right].$
(228)

Therefore, we obtain

$\displaystyle\hat{U}_{N}^{\dagger}(\Phi)i\partial_{\Phi}\hat{U}_{N}(\Phi)=\hat{U}_{N}^{\dagger}(\Phi)\frac{\hat{x}}{L}\hat{U}_{N}(\Phi)-\frac{\hat{x}}{L},$
(229)

which leads to

$\displaystyle\sum_{x=1}^{L}\sum_{\alpha}\bra{x,\alpha}\hat{U}_{N}^{\dagger}(\Phi)i\partial_{\Phi}\hat{U}_{N}(\Phi)\ket{x,\alpha}$
$\displaystyle=\sum_{x=1}^{L}\sum_{\alpha}\bra{x,\alpha}\hat{U}_{N}^{\dagger}(\Phi)\frac{\hat{x}}{L}\hat{U}_{N}(\Phi)\ket{x,\alpha}$
$\displaystyle-\sum_{x=1}^{L}\sum_{\alpha}\bra{x,\alpha}\frac{\hat{x}}{L}\ket{x,\alpha}.$
(230)

From translation symmetry of $\hat{U}_{N}(\Phi)$ under $L$ sites translations
${\cal T}_{L}$,

$\displaystyle\mathcal{T}_{L}^{\dagger}\hat{U}_{N}(\Phi)\mathcal{T}_{L}=\hat{U}_{N}(\Phi),$
(231)

the left hand side of Eq. (230) becomes

$\displaystyle\sum_{x=1}^{L}\sum_{\alpha}\bra{x,\alpha}\hat{U}_{N}^{\dagger}(\Phi)i\partial_{\Phi}\hat{U}_{N}(\Phi)\ket{x,\alpha}$
$\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{x=1}^{L}\sum_{n=1}^{N}\sum_{\alpha}\bra{x,\alpha}\left(\mathcal{T}_{L}^{\dagger}\right)^{n}\hat{U}_{N}^{\dagger}(\Phi)i\partial_{\Phi}\hat{U}_{N}(\Phi)\left(\mathcal{T}_{L}\right)^{n}\ket{x,\alpha}$
$\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{x=1}^{NL}\sum_{\alpha}\bra{x,\alpha}\hat{U}_{N}^{\dagger}(\Phi)i\partial_{\Phi}\hat{U}_{N}(\Phi)\ket{x,\alpha}$
$\displaystyle=$
$\displaystyle\frac{1}{N}\mathrm{tr}_{N}\left[\hat{U}_{N}^{\dagger}(\Phi)i\partial_{\Phi}\hat{U}_{N}(\Phi)\right],$
(232)

where ${\rm tr}_{N}$ is the trace over the Hilbert space for $\hat{U}_{N}$. On
the other hand, the first term of the right hand side in Eq. (230) is recast
into

$\displaystyle\sum_{x=1}^{L}\sum_{\alpha}\bra{x,\alpha}\hat{U}_{N}^{\dagger}(\Phi)\frac{\hat{x}}{L}\hat{U}_{N}(\Phi)\ket{x,\alpha}$
$\displaystyle=$
$\displaystyle\sum_{x=1}^{L}\sum_{\alpha}\bra{x,\alpha}e^{-i(\Phi/L)\hat{x}}\hat{U}_{N}^{\dagger}e^{i(\Phi/L)\hat{x}}\frac{\hat{x}}{L}e^{-i(\Phi/L)\hat{x}}$
$\displaystyle\quad\quad\quad\times\hat{U}_{N}(\Phi)e^{i(\Phi/L)\hat{x}}\ket{x,\alpha}$
$\displaystyle=$
$\displaystyle\sum_{x=1}^{L}\sum_{\alpha}\bra{x,\alpha}\hat{U}_{N}^{\dagger}\frac{\hat{x}}{L}\hat{U}_{N}\ket{x,\alpha}.$
(233)

Therefore, Eq. (230) leads to

$\displaystyle\frac{1}{N}\int_{0}^{2\pi}\frac{d\Phi}{2\pi}\mathrm{tr}_{N}\left[\hat{U}_{N}^{\dagger}(\Phi)i\partial_{\Phi}\hat{U}_{N}(\Phi)\right]$
$\displaystyle=$
$\displaystyle\sum_{x=1}^{L}\sum_{\alpha}\bra{x,\alpha}\hat{U}_{N}^{\dagger}\frac{\hat{x}}{L}\hat{U}_{N}\ket{x,\alpha}$
$\displaystyle-\sum_{x=1}^{L}\sum_{\alpha}\bra{x,\alpha}\frac{\hat{x}}{L}\ket{x,\alpha}.$
(234)

Finally, it holds that 111Note that Eq. (235) can be easily shown for
$U(k)=e^{\pm ik}$. In general, any model can be continuously deformed into a
direct product of $U(k)=e^{\pm ik}$ up to addiction or subtraction of trivial
models, and thus Eq. (235) holds.

$\displaystyle\frac{1}{N}\int_{0}^{2\pi}\frac{d\Phi}{2\pi}\mathrm{tr}_{N}\left[\hat{U}_{N}^{\dagger}(\Phi)i\partial_{\Phi}\hat{U}_{N}(\Phi)\right]$
$\displaystyle=\int_{0}^{2\pi}\frac{d\Phi}{2\pi}\mathrm{tr}\left[\hat{U}^{\dagger}(\Phi)i\partial_{\Phi}\hat{U}(\Phi)\right],$
(235)

and thus we have Eq. (D).

##  Appendix E Proof of Eq. (191)

The polarization at $t=T$ is rewritten as

$\displaystyle
P_{x}(T)|_{y=1}=\sum_{\alpha}\bra{x,y=1,\alpha}\hat{U}^{\dagger}\hat{x}\hat{U}\ket{x,y=1,\alpha}$
$\displaystyle=\sum_{\alpha}\left[\frac{1}{\sqrt{L_{x}}}\sum_{k_{x}}e^{ik_{x}x}\bra{k_{x}}\right]\bra{y=1,\alpha}\hat{U}^{\dagger}\left[\sum_{x^{\prime}}x^{\prime}\ket{x^{\prime}}\bra{x^{\prime}}\right]$
$\displaystyle\quad\times\hat{U}\left[\frac{1}{\sqrt{L_{x}}}\sum_{k^{\prime}_{x}}e^{-ik^{\prime}_{x}x}\ket{k^{\prime}_{x}}\right]\ket{y=1,\alpha}$
$\displaystyle=\frac{1}{L_{x}^{2}}\sum_{\alpha,k_{x},k^{\prime}_{x},x^{\prime}}e^{ik_{x}x}\bra{y=1,\alpha}U^{\dagger}(k_{x})e^{-ik_{x}x^{\prime}}[-i\partial_{k^{\prime}_{x}}e^{ik^{\prime}_{x}x^{\prime}}]$
$\displaystyle\quad\times
U(k^{\prime}_{x})e^{-ik^{\prime}_{x}x}\ket{y=1,\alpha}$
$\displaystyle=\frac{1}{L_{x}}\sum_{\alpha,k_{x}}e^{ik_{x}x}\bra{y=1,\alpha}U^{\dagger}(k_{x})i\partial_{k_{x}}[U(k_{x})e^{-ik_{x}x}]\ket{y=1,\alpha}$
$\displaystyle=\frac{1}{L_{x}}\sum_{\alpha,k_{x}}\bra{y=1,\alpha}U^{\dagger}(k_{x})[i\partial_{k_{x}}U(k_{x})]\ket{y=1,\alpha}+\sum_{\alpha}x$
$\displaystyle=\int_{0}^{2\pi}\frac{dk_{x}}{2\pi}\mathrm{tr}_{y,\alpha}\left[\hat{P}_{\rm
edge}U^{\dagger}(k_{x})i\partial_{k_{x}}U(k_{x})\right]+P_{x}|_{y=1},$ (236)

with $\hat{P}_{\rm edge}=|y=1\rangle\langle y=1|$, which means Eq. (191).

To check Eq. (191), we consider the model with 2 sites in the $y$-direction,

$\displaystyle U(k_{x})$
$\displaystyle=e^{-ik_{x}}\cos\theta\ket{y=1}\bra{y=1}-e^{-ik_{x}}\sin\theta\ket{y=1}\bra{y=2}$
$\displaystyle+\sin\theta\ket{y=2}\bra{y=1}+\cos\theta\ket{y=2}\bra{y=2}.$
(237)

The model provides one-site displacement in the $x$-direction when a particle
is located at $y=1$ after one-cycle time evolution. If the particle starts at
$y=1$, only the first term in the right hand side of Eq. (237) gives the one-
site displacement. Thus, the particle is expected to move in the $x$-direction
at rate $\cos^{2}\theta$.

Equation (191) reproduces this result. Actually, from direct calculations

$\displaystyle\hat{P}_{\rm edge}U^{\dagger}(k_{x})$
$\displaystyle=e^{ik_{x}}\cos\theta\ket{y=1}\bra{y=1}$
$\displaystyle+\sin\theta\ket{y=1}\bra{y=2},$ $\displaystyle
i\partial_{k_{x}}U(k_{x})$
$\displaystyle=e^{-ik_{x}}\cos\theta\ket{y=1}\bra{y=1}$
$\displaystyle-e^{-ik_{x}}\sin\theta\ket{y=1}\bra{y=2},$ (238)

we have

$\displaystyle w_{P}[U(k_{x})]$
$\displaystyle=\int_{0}^{2\pi}\frac{\mathrm{d}k_{x}}{2\pi}\mathrm{tr}_{y,\alpha}\left[\hat{P}_{\rm
edge}U^{\dagger}(k_{x})i\partial_{k_{x}}U(k_{x})\right]$
$\displaystyle=\int_{0}^{2\pi}\frac{\mathrm{d}k_{x}}{2\pi}\mathrm{tr}_{y,\alpha}\left[\cos^{2}\theta\ket{y=1}\bra{y=1}\right]$
$\displaystyle=\cos^{2}\theta.$ (239)

##  Appendix F Effects of random phases of the boundary unitary operator in
the Anderson model

In this section, we numerically check that the extrinsic edge mode in Eq.
(172) is robust against random phases on the edge. For this purpose, we
consider $U_{A}^{\prime\prime}$ below, instead of $U_{A}^{\prime}$,

$\displaystyle U_{A}^{\prime\prime}$ $\displaystyle=A^{\prime}\cdot
e^{-iH_{A}T},$ $\displaystyle A^{\prime}$
$\displaystyle=\sum_{x=1}^{L_{x}}e^{-i\phi_{x}}\ket{x+1}\bra{x}\otimes\ket{y=1}\bra{y=1}$
$\displaystyle+\sum_{x=1}^{L_{x}}\sum_{y=2}^{L_{y}}\ket{x}\bra{x}\otimes\ket{y}\bra{y},$
(240)

where $\phi_{x}$ is uniformly distributed in a range $[0,2\pi]$. The resultant
dynamics of a wave packet starting at an edge of the system, the DOS
histogram, and a typical eigenstate profile are shown in Fig. 13. We can see
the directed wave packet motion along $y=1$ occurs even in this case.
Actually, this behavior is assured by the formulae Eqs. (188) and (191).

Figure 13: (a) Dynamics of a wave packet, (b) DOS and (c) a delocalized
eigenstate of the Anderson model with the disordered extrinsic edge mode in
Eq. (240). The parameters are $J=0.2$, $W=1$, $L_{x}=25$, $L_{y}=15$ and
$T=1$. The color scale is different between (a) and (c). The initial state of
the wave packet is $\ket{x=10,y=1}$. We take the periodic (open) boundary
condition in the $x$ ($y$) direction. Even in the presence of random phases on
the boundary unitary operator, the anomalous edge state survives, and enables
a directed wave packet motion.

##  Appendix G Diffusive behavior of the time-dependent Anderson model

Here we show diffusive behaviors of our time-dependent Anderson models in Eqs.
(193) and (194). For this purpose, we numerically examine $\rho(y,t)$ and
$\langle y^{2}\rangle$ defined by

$\displaystyle\rho(y,t)=\int\rho(x,y,t)dx,$ $\displaystyle\langle
y^{2}\rangle=\int y^{2}\rho(x,y,t)dxdy,$ (241)

Figure 14: Density distribution $\rho(y,t)$ at $t=100T$ and the mean squared
distance $\braket{y^{2}}$ of the time-dependent Anderson model in Eq. (193).
Parameters are $L_{x}=40$, $L_{y}=25$, $J=0.2$, $W=1$, and $T=1$. The initial
state of the wave packet is $\ket{x=20,y=1}$. The density distribution
$\rho(y,t)$ exhibits a Gaussian tail, and the mean squared distance
$\braket{y^{2}}$ shows a nearly linear behavior in time.

where $\rho(x,y,t)=|\psi(x,y,t)|^{2}$ is the density distribution. If
$\rho(x,y)$ obeys the diffusion equation

$\displaystyle\frac{\partial\rho}{\partial
t}=\frac{D}{2}\left(\frac{\partial^{2}\rho}{\partial
x^{2}}+\frac{\partial^{2}\rho}{\partial y^{2}}\right),$ (242)

with a diffusion constant $D$, then the typical solution takes the form of the
Gaussian distribution:

$\displaystyle\rho(x,y,t)=\frac{1}{2\pi\sigma_{t}^{2}}\exp\left[-\frac{x^{2}+y^{2}}{2\sigma_{t}^{2}}\right],\
\sigma_{t}=\sqrt{Dt},$ (243)

which leads to

$\displaystyle\rho(y,t)=\frac{1}{\sqrt{2\pi\sigma_{t}^{2}}}\exp\left[-\frac{y^{2}}{2\sigma_{t}^{2}}\right],\quad\braket{y^{2}}=Dt.$
(244)

We show our numerical results of $\rho(y,t=100T)$ and $\braket{y^{2}}$ for the
time-dependent Anderson models in Eqs. (193) and (194) in Figs. 14 and 15,
respectively. These numerical results display Gaussian tails for
$\rho(y,t=100T)$ and nearly linear behaviors in time for $\langle
y^{2}\rangle$, which are consistent with the diffusive behaviors in Eq. (244).
Similar results have been reported for models in Refs. Konno (2004); Joye and
Merkli (2010); Joye (2011); Evensky  _et al._ (1990).

Figure 15: Density distribution $\rho(y,t)$ at $t=100T$ and the mean squared
distance $\braket{y^{2}}$ of the time-dependent Anderson model with the
extrinsic edge mode in Eq. (194). Parameters are $L_{x}=40$, $L_{y}=25$,
$J=0.2$, $W=1$, and $T=1$. The initial state of the wave packet is
$\ket{x=20,y=1}$. Even in this case, the density distribution $\rho(y,t)$
exhibits a Gaussian tail, and the mean squared distance $\braket{y^{2}}$ shows
a nearly linear behavior in time.

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