#  $K$-theory classification of Wannier localizability and detachable
topological boundary states

Ken Shiozaki<EMAIL_ADDRESS>Center for Gravitational
Physics and Quantum Information, Yukawa Institute for Theoretical Physics,
Kyoto University, Kyoto 606-8502, Japan  Daichi Nakamura
<EMAIL_ADDRESS>Institute for Solid State Physics,
University of Tokyo, Kashiwa, Chiba 277-8581, Japan  Kenji Shimomura  Center
for Gravitational Physics and Quantum Information, Yukawa Institute for
Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan  Masatoshi Sato
Center for Gravitational Physics and Quantum Information, Yukawa Institute for
Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan  Kohei Kawabata
<EMAIL_ADDRESS>Institute for Solid State Physics, University of
Tokyo, Kashiwa, Chiba 277-8581, Japan

###### Abstract

A hallmark of certain topology, including the Chern number, is the obstruction
to constructing exponentially localized Wannier functions in the bulk bands.
Conversely, other types of topology do not necessarily impose Wannier
obstructions. Remarkably, such Wannier-localizable topological insulators can
host boundary states that are detachable from the bulk bands. In our
accompanying Letter (D. Nakamura et al., arXiv:2407.09458), we demonstrate
that non-Hermitian topology underlies detachable boundary states in Hermitian
topological insulators and superconductors, thereby establishing their tenfold
classification based on internal symmetry. Here, using $K$-theory, we
elucidate the relationship between Wannier localizability and detachability of
topological boundary states. From the boundary perspective, we classify
intrinsic and extrinsic non-Hermitian topology, corresponding to nondetachable
and detachable topological boundary states, respectively. From the bulk
perspective, on the other hand, we classify Wannier localizability through the
homomorphisms of topological phases from the tenfold Altland-Zirnbauer
symmetry classes to the threefold Wigner-Dyson symmetry classes. Notably,
these two approaches from the boundary and bulk perspectives lead to the same
classification. We clarify this agreement and develop a unified understanding
of the bulk-boundary correspondence on the basis of $K$-theory.

††preprint: YITP-24-88

##  I Introduction

Topological phases of matter lay a foundation in modern condensed matter
physics [1, 2]. Specifically, topological phases of band insulators and gapped
Bogoliubov-de Gennes superconductors are generally classified by the
fundamental tenfold internal symmetry classification based on time reversal,
particle-hole transformation, and chiral transformation [3, 4, *Ryu-10, 6, 7].
A hallmark of the topologically nontrivial bulk bands is the emergence of
anomalous boundary states. Furthermore, certain topology, such as the Chern
number and the time-reversal-invariant $\mathbb{Z}_{2}$ topological invariant,
imposes obstructions to constructing exponentially localized Wannier functions
in the bulk bands [8, 9, 10, 11, 12]. The Wannier localizability also refines
definition of topological crystalline phases [13]. By contrast, other types of
topology do not necessarily impose the Wannier obstructions. Examples include
one-dimensional topological insulators [14, 15], higher-dimensional
topological superconductors [16], Hopf insulators [17, 18], and three-
dimensional topological insulators protected by chiral symmetry [19].
Remarkably, these Wannier-localizable topological insulators do not
necessitate spectral flow of the concomitant boundary states. As opposed to
the conventional intuition of the bulk-boundary correspondence, such
topological boundary states are detachable from the bulk bands [18, 19].

In our accompanying Letter [20], we elucidate that non-Hermitian topology
underlies Wannier localizability and detachable boundary states in Hermitian
topological insulators and superconductors, thereby establishing their tenfold
classification (Table 1). This connection is based on the identification of
topological boundary states as non-Hermitian topology [21, 22, 23, 24, 25,
26]. In general, complex-valued spectra of non-Hermitian systems allow two
types of complex-energy gaps: point and line gaps [27, *Kawabata-19, 29]. In
the presence of a point (line) gap, complex-energy bands are defined not to
cross a reference point (line) in the complex-energy plane. Non-Hermitian
systems with point gaps are continuously deformable to unitary systems and
thus can be intrinsic to non-Hermitian systems [30, 31]. By contrast, non-
Hermitian systems with real (imaginary) line gaps are continuously deformable
to Hermitian (anti-Hermitian) systems and hence have Hermitian counterparts.
We show that while intrinsic non-Hermitian topology prohibits the detachment
of topological boundary states, extrinsic non-Hermitian topology permits it
and further leads to anti-Hermitian topology of the detached boundary states.
Based on this connection, we also establish the tenfold classification of
Wannier localizability and detachable topological boundary states, as well as
topology of the detached boundary states.

Table 1: Wannier localizability of the $d$-dimensional bulk and detachability of $\left(d-1\right)$-dimensional boundary states in topological insulators and superconductors. The tenfold Altland-Zirnbauer symmetry classes consist of time-reversal symmetry (TRS), particle-hole symmetry (PHS), and chiral symmetry (CS). The entries with “$\checkmark$” (“$\times$”) accompany the Wannier nonlocalizable (localizable) topological phases in the bulk and exhibit nondetachable (detachable) boundary states with intrinsic (extrinsic) non-Hermitian topology. The entries with “$\checkmark/\times$” specify the Wannier nonlocalizable and localizable topological phases for the odd and even numbers of topological invariants, respectively. Class |  TRS |  PHS |  CS |  $d=1$ |  $d=2$ |  $d=3$ |  $d=4$ |  $d=5$ |  $d=6$ |  $d=7$ |  $d=8$
---|---|---|---|---|---|---|---|---|---|---|---
A | $0$ | $0$ | $0$ | $0$ | $\mathbb{Z}^{\checkmark}$ | $0$ | $\mathbb{Z}^{\checkmark}$ | $0$ | $\mathbb{Z}^{\checkmark}$ | $0$ | $\mathbb{Z}^{\checkmark}$
AIII | $0$ | $0$ | $1$ | $\mathbb{Z}^{\times}$ | $0$ | $\mathbb{Z}^{\times}$ | $0$ | $\mathbb{Z}^{\times}$ | $0$ | $\mathbb{Z}^{\times}$ | $0$
AI | $+1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $2\mathbb{Z}^{\checkmark}$ | $0$ | $\mathbb{Z}_{2}^{\checkmark}$ | $\mathbb{Z}_{2}^{\checkmark}$ | $\mathbb{Z}^{\checkmark}$
BDI | $+1$ | $+1$ | $1$ | $\mathbb{Z}^{\times}$ | $0$ | $0$ | $0$ | $2\mathbb{Z}^{\times}$ | $0$ | $\mathbb{Z}_{2}^{\times}$ | $\mathbb{Z}_{2}^{\times}$
D | $0$ | $+1$ | $0$ | $\mathbb{Z}_{2}^{\times}$ | $\mathbb{Z}^{\checkmark}$ | $0$ | $0$ | $0$ | $2\mathbb{Z}^{\checkmark}$ | $0$ | $\mathbb{Z}_{2}^{\times}$
DIII | $-1$ | $+1$ | $1$ | $\mathbb{Z}_{2}^{\times}$ | $\mathbb{Z}_{2}^{\checkmark}$ | $\mathbb{Z}^{\checkmark/\times}$ | $0$ | $0$ | $0$ | $2\mathbb{Z}^{\times}$ | $0$
AII | $-1$ | $0$ | $0$ | $0$ | $\mathbb{Z}_{2}^{\checkmark}$ | $\mathbb{Z}_{2}^{\checkmark}$ | $\mathbb{Z}^{\checkmark}$ | $0$ | $0$ | $0$ | $2\mathbb{Z}^{\checkmark}$
CII | $-1$ | $-1$ | $1$ | $2\mathbb{Z}^{\times}$ | $0$ | $\mathbb{Z}_{2}^{\times}$ | $\mathbb{Z}_{2}^{\times}$ | $\mathbb{Z}^{\times}$ | $0$ | $0$ | $0$
C | $0$ | $-1$ | $0$ | $0$ | $2\mathbb{Z}^{\checkmark}$ | $0$ | $\mathbb{Z}_{2}^{\times}$ | $\mathbb{Z}_{2}^{\times}$ | $\mathbb{Z}^{\checkmark}$ | $0$ | $0$
CI | $+1$ | $-1$ | $1$ | $0$ | $0$ | $2\mathbb{Z}^{\times}$ | $0$ | $\mathbb{Z}_{2}^{\times}$ | $\mathbb{Z}_{2}^{\checkmark}$ | $\mathbb{Z}^{\checkmark/\times}$ | $0$

In the present work, we develop the $K$-theory classification, further
elucidating the relationship between Wannier localizability and detachability
of topological boundary states. On the one hand, we classify intrinsic and
extrinsic non-Hermitian topology, corresponding to nondetachable and
detachable boundary states in Hermitian topological materials, respectively.
On the other hand, we classify Wannier localizability of bulk bands based on
the homomorphisms from topological phases in the tenfold Altland-Zirnbauer
(AZ) symmetry classes to those in the threefold Wigner-Dyson (WD) classes.
Notably, these two approaches from the boundary and bulk perspectives result
in the same classification. We clarify this agreement and develop a unified
understanding of the bulk-boundary correspondence on the basis of $K$-theory.

This work is organized as follows. In Sec. II, we provide an overview of this
work before the detailed discussions. In Sec. III, we present a chiral-
symmetric Hermitian lattice model with a flat and detached surface state in
three dimensions. In Sec. IV, we summarize intrinsic and extrinsic non-
Hermitian topology, corresponding to nondetachable and detachable boundary
states in Hermitian topological materials, respectively. In Sec. V, we develop
the classification of Wannier localizability based on the homomorphisms from
the AZ classification to the WD classification. Crucially, these two
approaches from the boundary and bulk perspectives lead to the same
classification (Table 1). In Sec. VI, we clarify this relationship, using
$K$-theory. In Sec. VII, we conclude this work.

##  II Overview

Using $K$-theory, we clarify the relationship between the detachability of
topological boundary states and Wannier localizability. In Sec. III, we
provide a prototypical chiral-symmetric model in three dimensions with a flat
and detached surface state before the general classification. For a gapped
$d$-dimensional Hermitian system, we consider the bulk Hamiltonian
$H({\bm{k}})$ with the periodic boundary conditions and the boundary
Hamiltonian $\hat{H}({\bm{k}}_{\parallel})$ with the semi-infinite boundary
conditions, which are classified by the degree-$0$ $K$-group
${}^{\phi}K_{G}^{(\tau,c)+0}(T^{d})$ and degree-($+1$) $K$-group
${}^{\phi}K_{G}^{(\tau,c)+1}(T^{d-1})$ [32, 33, 34, 35, 36, 37]. Here,
${\bm{k}}$ and ${\bm{k}}_{\parallel}$ are the $d$-dimensional wave vector in
the bulk and $\left(d-1\right)$-dimensional wave vector at the boundary,
respectively. Notably, while the Bloch Hamiltonian $H({\bm{k}})$ is a finite-
dimensional matrix, the boundary Hamiltonian $\hat{H}({\bm{k}}_{\parallel})$
is an infinite-dimensional operator. The bulk-boundary correspondence is
denoted by

$\begin{array}[]{rccc}f^{\rm
BBC}_{G}:&{}^{\phi}K_{G}^{(\tau,c)+0}(T^{d})&\longrightarrow&{}^{\phi}K_{G}^{(\tau,c)+1}(T^{d-1})\\\
&\rotatebox{90.0}{$\in$}&&\rotatebox{90.0}{$\in$}\\\
&[H({\bm{k}})]&\longmapsto&[\hat{H}({\bm{k}}_{\parallel})]\end{array}$ (1)

Here, $T^{d}$ and $T^{d-1}$ are respectively the bulk and boundary Brillouin
zone tori, $G$ is the symmetry group that is compatible with the boundary and
possibly includes crystalline symmetry, and $c:G\to{\mathbb{Z}}_{2}$ is the
homomorphism specifying whether $g\in G$ is symmetry or antisymmetry such as
chiral symmetry and particle-hole symmetry.

We map the boundary Hermitian Hamiltonian $\hat{H}({\bm{k}}_{\parallel})$ to a
point-gapped non-Hermitian Hamiltonian $H_{P}$ by

$H_{P}({\bm{k}}_{\parallel})\coloneqq\text{i}\exp\left[-\text{i}\pi\chi\,(\hat{H}({\bm{k}}_{\parallel}))\right],$
(2)

where $\chi(E)$ is a continuous function defined as

$\chi(E)\coloneqq\begin{cases}{\rm sgn}(E)&|E|>E_{\rm gap};\\\ E/E_{\rm
gap}&|E|\leq E_{\rm gap}\end{cases}$ (3)

with a bulk energy gap $E_{\rm gap}>0$. As shown in our accompanying Letter
[20], the detachment of $\hat{H}({\bm{k}}_{\parallel})$ corresponds to
imposing an imaginary line gap on $H_{P}({\bm{k}}_{\parallel})$, which is
further formulated by the anti-Hermitian condition for
$H_{P}({\bm{k}}_{\parallel})$ [i.e.,
$H_{P}^{{\dagger}}({\bm{k}}_{\parallel})=-H_{P}({\bm{k}}_{\parallel})$] [29].
In Sec. IV, we review intrinsic (extrinsic) point-gap topology irreducible
(reducible) to Hermitian or anti-Hermitian topology [30, 31]. Notably,
antisymmetries for $H_{P}({\bm{k}}_{\parallel})$ behave as symmetries for
$\text{i}H_{P}({\bm{k}}_{\parallel})$, and eventually, the imaginary-line-
gapped phases are classified by the degree-$0$ $K$-group with trivial $c$
[i.e., ${}^{\phi}K_{G}^{(\tau,c_{\rm triv})+0}(T^{d-1})$]. As demonstrated in
Sec. VI, if $G$ includes antisymmetry, we have the following commutative
diagram:

$\begin{CD}{}^{\phi}K_{G}^{(\tau,c_{\rm triv})+0}(T^{d-1})\\\ @V{}V{f_{\rm
L_{i}}^{\rm bdy}}V\\\ {}^{\phi}K_{G}^{(\tau,c)+0}(T^{d})@>{f_{G}^{\rm
BBC}}>{}>{}^{\phi}K_{G}^{(\tau,c)+1}(T^{d-1})\\\ @V{}V{f^{\rm
bulk}_{0}}V@V{}V{f^{\rm bdy}_{0}}V\\\
{}^{\phi}K_{G_{0}}^{\tau+0}(T^{d})@>{f_{G_{0}}^{\rm
BBC}}>{}>{}^{\phi}K_{G_{0}}^{\tau+1}(T^{d-1})\\\ \end{CD}$ (4)

Here, $G_{0}\subsetneq G$ is the subgroup excluding antisymmetries, the
$K$-groups ${}^{\phi}K_{G_{0}}^{\tau+0}(T^{d})$ and
${}^{\phi}K_{G_{0}}^{\tau+1}(T^{d-1})$ respectively classify the bulk and
boundary Hamiltonians protected only by $G_{0}$, and $f^{\rm bulk}_{0}$ and
$f^{\rm bdy}_{0}$ are defined by forgetting antisymmetries. The right vertical
line in Eq. (4) is exact, and $f^{\rm bdy}_{\rm L_{i}}$ maps imaginary-line-
gapped phases to point-gapped phases [30]. Saliently, the exactness ${\rm
im}\,f^{\rm bdy}_{\rm L_{i}}={\rm ker}\,f^{\rm bdy}_{0}$ indicates that the
boundary Hamiltonian $\hat{H}({\bm{k}}_{\parallel})$ is detachable and
exhibits no spectral flow ($[\hat{H}({\bm{k}}_{\parallel})]\in{\rm
im}\,f_{L_{i}}$) if and only if $\hat{H}({\bm{k}}_{\parallel})$ is gappable
($[\hat{H}({\bm{k}}_{\parallel})]\in{\rm ker}\,f_{0}$) when ignoring
antisymmetries.

Suppose that the bulk Hamiltonian $H({\bm{k}})$ exhibits spectral flow on the
boundary, expressed as $f^{\rm BBC}_{G}([H({\bm{k}})])\notin{\rm im}\,f^{\rm
bdy}_{L_{\rm i}}$. Then, the diagram in Eq. (4) implies that $H({\bm{k}})$
exhibits nontrivial topology in the $K$-group
${}^{\phi}K^{\tau+0}_{G_{0}}(T^{d})$, which obstructs the existence of
exponentially localized Wannier functions. This aligns with the fact that if a
real-space boundary hosts a spectral flow, either occupied or unoccupied Bloch
states are no longer Wannier localizable [19]. Such a boundary includes
cylindrical or spherical boundaries that can realize higher-order topological
boundary states. Furthermore, Table 1 is consistent with the symmetry-
forgetting homomorphisms $f^{\rm bdy}_{0}$ for the tenfold AZ symmetry
classes, as discussed in Sec. V.

##  III A model of detached and flat-band surface state

We begin with a prototypical model exhibiting detachable topological boundary
states. Specifically, we present a Hermitian lattice model with a flat and
detached surface state for three dimensions $d=3$ in class AIII. The
construction here can be straightforwardly generalized to any odd spatial
dimension $d=2n+1$ with the even winding numbers $W_{2n+1}\in 2\mathbb{Z}$.

###  III.1 Model

Consider the following four-band model in class AIII:

$\displaystyle H(\bm{h}(k_{x},k_{y}),k_{z})$ $\displaystyle=\left(\cos
k_{z}\right)\tau_{x}\otimes\sigma_{0}$ $\displaystyle\qquad+\left(\sin
k_{z}\right)\tau_{y}\otimes\bm{h}(k_{x},k_{y})\cdot\bm{\sigma},$ (5)

with

$\bm{h}(k_{x},k_{y})\coloneqq(\sin k_{x},\sin k_{y},m-\cos k_{x}-\cos k_{y}).$
(6)

This lattice model indeed respects chiral symmetry

$\Gamma
H(\bm{h}(k_{x},k_{y}),k_{z})\Gamma^{-1}=-H(\bm{h}(k_{x},k_{y}),k_{z}),$ (7)

with $\Gamma=\tau_{z}$, and hence belongs to class AIII [7]. By writing the
Hamiltonian as

$\displaystyle
H(\bm{h}(k_{x},k_{y}),k_{z})=\begin{pmatrix}&q^{\dagger}({\bm{k}})\\\
q({\bm{k}})\\\ \end{pmatrix},$ (8)

with

$q({\bm{k}})\coloneqq\cos k_{z}+\text{i}\left(\sin
k_{z}\right)\bm{h}(k_{x},k_{y})\cdot\bm{\sigma},$ (9)

the three-dimensional winding number is obtained as

$\displaystyle W_{3}$ $\displaystyle=\frac{1}{24\pi^{2}}\int_{T^{3}}{{\rm tr\
}}[q^{-1}dq]^{3}$ $\displaystyle=\left\\{\begin{array}[]{ll}0&(|m|>2);\\\
-2&(0<m<2);\\\ 2&(-2<m<0).\\\ \end{array}\right.$ (13)

This reflects twice the Chern number of the Hamiltonian
$\bm{h}({\bm{k}})\cdot\bm{\sigma}$ for two dimensions $d=2$ and class A.

Figure 1: Spectra of the model in Eq. (III.1) for the topological phase
($m=1$; $W_{3}=-2$) and trivial phase ($m=3$; $W_{3}=0$). The periodic
boundary conditions are imposed along the $y$ direction, by which the wave
number $k_{y}$ is introduced. The other boundary conditions are (i) the
periodic boundary conditions along the $x$ and $z$ directions, (ii) the
periodic boundary conditions along the $x$ direction and open boundary
conditions along the $z$ direction, (iii) the periodic boundary conditions
along the $z$ direction and open boundary conditions along the $x$ direction,
and (iv) the open boundary conditions along the $x$ and $z$ directions. The
system size along the $x$ and $z$ directions is taken as $L_{x}=L_{z}=21$.

###  III.2 Detached flat-band surface state

Viewing $\bm{h}$ as a parameter, we consider the following one-dimensional
lattice model along the $z$ direction:

$\displaystyle\hat{H}_{\rm
1D}(\bm{h})=\sum_{z\in\mathbb{Z}}\psi^{\dagger}_{z+1}\frac{\tau_{x}+\text{i}\tau_{y}\bm{h}\cdot\bm{\sigma}}{2}\psi_{z}+{\rm
H.c.},$ (14)

where $\psi_{z}$ is a four-component complex fermion operator at site $z$.
Under the semi-infinite boundary conditions where $z\leq 0$ is in the
topological insulator and $z>0$ is in the vacuum, for every $|\bm{h}|>0$, the
interface $z=0$ hosts two exponentially localized zero-energy states, given as

$\displaystyle\phi_{+}(z,\bm{h})$
$\displaystyle=u_{+}(\hat{h})\otimes\begin{pmatrix}0\\\ \eta_{-}(z;h)\\\
\end{pmatrix}_{\tau},$ (15) $\displaystyle\phi_{-}(z,\bm{h})$
$\displaystyle=u_{-}(\hat{h})\otimes\begin{pmatrix}\eta_{+}(z;h)\\\ 0\\\
\end{pmatrix}_{\tau},$ (16)

with $h\coloneqq|\bm{h}|$ and $\hat{h}\coloneqq\bm{h}/|\bm{h}|$. Here,
$u_{\pm}(\hat{h})\in\mathbb{C}^{2}$ is a normalized two-component vector in
the $\sigma$ space, satisfying

$\left(\bm{h}\cdot\bm{\sigma}\right)u_{\pm}(\hat{h})=\pm hu_{\pm}(\hat{h}),$
(17)

and $(0~{}\eta_{-}(z;h))_{\tau}^{T}$ and $(\eta_{+}(z;h)~{}0)_{\tau}^{T}$ are
respectively normalized vectors in the $\tau$ and $z$ spaces whose
localization lengths depend on $h$. Note that $\phi_{+}(z,\bm{h})$ and
$\phi_{-}(z,\bm{h})$ have chirality $\Gamma=1$ and $\Gamma=-1$, respectively.
With the polar coordinates of $\hat{h}$ as in $\hat{h}=(\theta_{h},\phi_{h})$,
$u_{\pm}(\hat{h})$ is explicitly given, up to a $U(1)$ phase, as

$\displaystyle u_{+}(\hat{h})$
$\displaystyle\propto\begin{pmatrix}\cos\left(\theta_{h}/2\right)\\\
e^{\text{i}\phi_{h}}\sin\left(\theta_{h}/2\right)\end{pmatrix}_{\sigma},$ (18)
$\displaystyle u_{-}(\hat{h})$
$\displaystyle\propto\begin{pmatrix}-e^{-\text{i}\phi_{h}}\sin\left(\theta_{h}/2\right)\\\
\cos\left(\theta_{h}/2\right)\\\ \end{pmatrix}_{\sigma}.$ (19)

Thus, we obtain the two-component flat detached surface state, irrespective of
the winding number $W_{3}$. However, its Chern number depends on that of
$\bm{h}({\bm{k}})\cdot\bm{\sigma}$, which further coincides with $W_{3}$.

We numerically calculate the spectrum of this model, as shown in Fig. 1.
Consistent with the above discussion, flat topological surface states detached
from the bulk bands appear under the open boundary conditions along the $z$
direction. By contrast, the topological surface states are connected to the
bulk bands under the open boundary conditions along the $x$ direction. It is
also notable that flat surface states appear even in the topologically trivial
phase (i.e., $m=3$; $W_{3}=0$). However, these surface states do not exhibit
the nontrivial Chern number. Consequently, they are topologically unstable and
can be removed from zero energy, as shown in the next subsection.

###  III.3 Topological stability

Now, consider a generic perturbation to this flat-band surface state. The
effective $2\times 2$ Hamiltonian with a chiral-symmetry-preserving
perturbation $H_{V}(\bm{h})$ is

$\displaystyle H_{\rm eff}(\bm{h})=\int
dz\begin{pmatrix}\phi_{+}^{{\dagger}}(z,\bm{h})\\\
\phi_{-}^{{\dagger}}(z,\bm{h})\end{pmatrix}H_{V}(\bm{h})\begin{pmatrix}\phi_{+}(z,\bm{h})&\phi_{-}(z,\bm{h})\end{pmatrix}.$
(20)

Notably, while $H_{V}(\bm{h})$ is gauge-independent, $\phi_{\pm}(z,\bm{h})$’s
are gauge-dependent, and thus the effective Hamiltonian $H_{\rm eff}(\bm{h})$
is also gauge-dependent. Express $H_{V}(\bm{h})$ in the $\tau$ space as

$\displaystyle H_{V}(\bm{h})=\begin{pmatrix}&V^{\dagger}(\bm{h})\\\
V(\bm{h})&\\\ \end{pmatrix}_{\tau},$ (21)

and introduce the $2\times 2$ matrix

$\displaystyle v(\bm{h})=\int dz~{}\eta_{-}(z;h)^{*}V(\bm{h})\eta_{+}(z;h),$
(22)

then the effective Hamiltonian becomes

$\displaystyle H_{\rm
eff}(\bm{h})=\begin{pmatrix}&u_{+}^{\dagger}(\hat{h})v^{\dagger}(\bm{h})u_{-}(\hat{h})\\\
u_{-}^{\dagger}(\hat{h})v(\bm{h})u_{+}(\hat{h})&\\\ \end{pmatrix}.$ (23)

Thus, the spectrum of the surface states is

$\displaystyle\pm|z(\bm{h})|,\quad
z(\bm{h})=u_{-}^{\dagger}(\hat{h})v(\bm{h})u_{+}(\hat{h})\in\mathbb{C}.$ (24)

Note that the absolute value $|z(\bm{h})|$ is gauge-invariant. Now, we show
the following proposition:

Proposition.—There is no perturbation $v(\bm{h})$ such that $|z(\bm{h})|>0$
for all $\hat{h}\in S^{2}$.

Proof.—We rewrite $|z|^{2}$ in the following manner,

$\displaystyle|z|^{2}$ $\displaystyle={{\rm tr\
}}[vu_{+}u_{+}^{\dagger}v^{\dagger}u_{-}u_{-}^{\dagger}]$ $\displaystyle={{\rm
tr\
}}\left[v\frac{1+\hat{h}\cdot\bm{\sigma}}{2}v^{\dagger}\frac{1-\hat{h}\cdot\bm{\sigma}}{2}\right]$
$\displaystyle=\frac{1}{4}{{\rm tr\
}}[vv^{\dagger}+(v^{\dagger}v-vv^{\dagger})\hat{h}\cdot\bm{\sigma}-v\hat{h}\cdot\bm{\sigma}v^{\dagger}\hat{h}\cdot\bm{\sigma}].$
(25)

Using $v=v_{0}+\bm{v}\cdot\bm{\sigma}$, we have

$\displaystyle|z|^{2}=(\hat{h}\times\bm{v})\cdot(\hat{h}\times\bm{v}^{*})-\text{i}\hat{h}\cdot(\bm{v}^{*}\times\bm{v})$
(26)

from straightforward calculations. We decompose $\bm{v}$ into the component of
the polar coordinate
$\bm{v}=v_{h}\hat{h}+v_{\theta}\hat{\theta}+v_{\phi}\hat{\phi}$ and obtain

$\displaystyle|z|^{2}=|v_{\theta}-\text{i}v_{\phi}|^{2}.$ (27)

Further separating $\bm{v}$ into real and imaginary parts
$\bm{v}=\bm{v}_{1}+\text{i}\bm{v}_{2}$ leads to

$\displaystyle|z|^{2}=(v_{1\theta}+v_{2\phi})^{2}+(v_{2\theta}-v_{1\phi})^{2}.$
(28)

Introducing vectors on the tangent space $TS^{2}$,

$\displaystyle\bm{v}_{1}^{\perp}=\bm{v}_{1}-(\hat{h}\cdot\bm{v}_{1})\hat{h},\quad\bm{v}_{2}^{\perp}=\bm{v}_{2}-(\hat{h}\cdot\bm{v}_{2})\hat{h},$
(29)

we eventually obtain the following form:

$\displaystyle|z|^{2}=|\bm{v}_{1}^{\perp}-\hat{h}\times\bm{v}_{2}^{\perp}|^{2}.$
(30)

The vector $\bm{v}_{1}^{\perp}-\hat{h}\times\bm{v}_{2}^{\perp}$ is in the
tangent space $TS^{2}$. According to the Poincaré-Hopf theorem, there must be
two defects on $S^{2}$ where the tangent vector
$\bm{v}_{1}^{\perp}-\hat{h}\times\bm{v}_{2}^{\perp}$ vanishes. $\Box$

Alternative proof.—Note that $u_{+}(\hat{h})$ and $u_{-}(\hat{h})$ are
orthonormal bases of $\mathbb{C}^{2}$. The $2\times 2$ matrix $v(\bm{h})$ is
expanded locally in $S^{2}$ as

$\displaystyle v(\bm{h})$
$\displaystyle=\sum_{i,j\in\\{+,-\\}}v_{ij}(\bm{h})u_{i}(\hat{h})u_{j}^{\dagger}(\hat{h}),$
(31) $\displaystyle v_{ij}(\bm{h})$ $\displaystyle\coloneqq
u_{i}^{\dagger}(\hat{h})v(\bm{h})u_{j}(\hat{h}),$ (32)

so that $z(\bm{h})=v_{-+}(\bm{h})$. Note that

$\displaystyle u_{i}(\hat{h})u_{j}^{\dagger}(\hat{h})\in{\rm Mat}_{2\times
2}(\mathbb{C})\cong\mathbb{C}^{2}\otimes(\mathbb{C}^{2})^{*}$ (33)

defines a line bundle over $S^{2}$ with the fiber space ${\rm Mat}_{2\times
2}(\mathbb{C})$. The line bundle defined by
$u_{-}(\hat{h})u_{+}^{\dagger}(\hat{h})$ has the Chern number of 2. Assuming
$v_{-+}(\bm{h})\neq 0$ everywhere on $S^{2}$ means that there exists a global
section of the nontrivial line bundle
$u_{-}(\hat{h})u_{+}^{\dagger}(\hat{h})$, leading to a contradiction. $\Box$

A consequence of this statement is that if the detached surface state acquires
a uniform mass gap in the whole surface Brillouin zone $(k_{x},k_{y})\in
T^{2}$, the degree of the map $\hat{h}:T^{2}\to S^{2}$, which coincides with
the Chern number of the two-dimensional Hamiltonian
$\bm{h}(k_{x},k_{y})\cdot\bm{\sigma}$ in class A and thus the winding number
$W_{3}$ of the three-dimensional bulk, must be trivial.

Table 2: Homomorphisms $f_{r},f_{i}$ from line-gap to point-gap topology for
the tenfold Altland-Zirnbauer† classification.
$\begin{array}[]{ccccccccccccc}\hline\cr\hline\cr\mbox{Class}&\mbox{Gap}&d=0&d=1&d=2&d=3&d=4&d=5&d=6&d=7\\\
\hline\cr{\rm A}&{\rm L}\to{\rm P}&{\mathbb{Z}}\to
0&0\to{\mathbb{Z}}&{\mathbb{Z}}\to 0&0\to{\mathbb{Z}}&{\mathbb{Z}}\to
0&0\to{\mathbb{Z}}&{\mathbb{Z}}\to 0&0\to{\mathbb{Z}}\\\ &&&&&&&&&\\\
\hline\cr{\rm AIII}&{\rm L_{r}}\to{\rm P}&0\to{\mathbb{Z}}&{\mathbb{Z}}\to
0&0\to{\mathbb{Z}}&{\mathbb{Z}}\to 0&0\to{\mathbb{Z}}&{\mathbb{Z}}\to
0&0\to{\mathbb{Z}}&{\mathbb{Z}}\to 0\\\ &&&&&&&&&\\\ &{\rm L_{i}}\to{\rm
P}&{\mathbb{Z}}\oplus{\mathbb{Z}}\to{\mathbb{Z}}&0\to
0&{\mathbb{Z}}\oplus{\mathbb{Z}}\to{\mathbb{Z}}&0\to
0&{\mathbb{Z}}\oplus{\mathbb{Z}}\to{\mathbb{Z}}&0\to
0&{\mathbb{Z}}\oplus{\mathbb{Z}}\to{\mathbb{Z}}&0\to 0\\\ &&(n,m)\mapsto
n-m&&(n,m)\mapsto n-m&&(n,m)\mapsto n-m&&(n,m)\mapsto n-m&\\\ \hline\cr{\rm
AI^{\dagger}}&{\rm L}\to{\rm P}&{\mathbb{Z}}\to 0&0\to 0&0\to 0&0\to
2{\mathbb{Z}}&2{\mathbb{Z}}\to
0&0\to{\mathbb{Z}}_{2}&{\mathbb{Z}}_{2}\to{\mathbb{Z}}_{2}&{\mathbb{Z}}_{2}\to{\mathbb{Z}}\\\
&&&&&&&&n\mapsto 0&\\\ \hline\cr{\rm BDI^{\dagger}}&{\rm L_{r}}\to{\rm
P}&{\mathbb{Z}}_{2}\to{\mathbb{Z}}&{\mathbb{Z}}\to 0&0\to 0&0\to 0&0\to
2{\mathbb{Z}}&2{\mathbb{Z}}\to
0&0\to{\mathbb{Z}}_{2}&{\mathbb{Z}}_{2}\to{\mathbb{Z}}_{2}\\\
&&&&&&&&&n\mapsto 0\\\ &{\rm L_{i}}\to{\rm
P}&{\mathbb{Z}}\oplus{\mathbb{Z}}\to{\mathbb{Z}}&0\to 0&0\to 0&0\to
0&2{\mathbb{Z}}\oplus 2{\mathbb{Z}}\to 2{\mathbb{Z}}&0\to
0&{\mathbb{Z}}_{2}\oplus{\mathbb{Z}}_{2}\to{\mathbb{Z}}_{2}&{\mathbb{Z}}_{2}\oplus{\mathbb{Z}}_{2}\to{\mathbb{Z}}_{2}\\\
&&(n,m)\mapsto n-m&&&&(n,m)\mapsto n-m&&(n,m)\mapsto n+m&(n,m)\mapsto n+m\\\
\hline\cr{\rm D^{\dagger}}&{\rm L_{r}}\to{\rm
P}&{\mathbb{Z}}_{2}\to{\mathbb{Z}}_{2}&{\mathbb{Z}}_{2}\to{\mathbb{Z}}&{\mathbb{Z}}\to
0&0\to 0&0\to 0&0\to 2{\mathbb{Z}}&2{\mathbb{Z}}\to 0&0\to{\mathbb{Z}}_{2}\\\
&&n\mapsto 0&&&&&&&\\\ &{\rm L_{i}}\to{\rm
P}&{\mathbb{Z}}\to{\mathbb{Z}}_{2}&0\to{\mathbb{Z}}&0\to 0&0\to
0&2{\mathbb{Z}}\to 0&0\to 2{\mathbb{Z}}&{\mathbb{Z}}_{2}\to
0&{\mathbb{Z}}_{2}\to{\mathbb{Z}}_{2}\\\ &&n\mapsto n&&&&&&&n\mapsto n\\\
\hline\cr{\rm DIII^{\dagger}}&{\rm L_{r}}\to{\rm
P}&0\to{\mathbb{Z}}_{2}&{\mathbb{Z}}_{2}\to{\mathbb{Z}}_{2}&{\mathbb{Z}}_{2}\to{\mathbb{Z}}&{\mathbb{Z}}\to
0&0\to 0&0\to 0&0\to 2{\mathbb{Z}}&2{\mathbb{Z}}\to 0\\\ &&&n\mapsto
0&&&&&&\\\ &{\rm L_{i}}\to{\rm
P}&{\mathbb{Z}}\to{\mathbb{Z}}_{2}&0\to{\mathbb{Z}}_{2}&{\mathbb{Z}}\to{\mathbb{Z}}&0\to
0&{\mathbb{Z}}\to 0&0\to 0&{\mathbb{Z}}\to 2{\mathbb{Z}}&0\to 0\\\ &&n\mapsto
n&&n\mapsto 2n&&&&n\mapsto n&\\\ \hline\cr{\rm AII^{\dagger}}&{\rm L}\to{\rm
P}&2{\mathbb{Z}}\to
0&0\to{\mathbb{Z}}_{2}&{\mathbb{Z}}_{2}\to{\mathbb{Z}}_{2}&{\mathbb{Z}}_{2}\to{\mathbb{Z}}&{\mathbb{Z}}\to
0&0\to 0&0\to 0&0\to 2{\mathbb{Z}}\\\ &&&&n\mapsto 0&&&&&\\\ \hline\cr{\rm
CII^{\dagger}}&{\rm L_{r}}\to{\rm P}&0\to 2{\mathbb{Z}}&2{\mathbb{Z}}\to
0&0\to{\mathbb{Z}}_{2}&{\mathbb{Z}}_{2}\to{\mathbb{Z}}_{2}&{\mathbb{Z}}_{2}\to{\mathbb{Z}}&{\mathbb{Z}}\to
0&0\to 0&0\to 0\\\ &&&&&n\mapsto 0&&&&\\\ &{\rm L_{i}}\to{\rm
P}&2{\mathbb{Z}}\oplus 2{\mathbb{Z}}\to 2{\mathbb{Z}}&0\to
0&{\mathbb{Z}}_{2}\oplus{\mathbb{Z}}_{2}\to{\mathbb{Z}}_{2}&{\mathbb{Z}}_{2}\oplus{\mathbb{Z}}_{2}\to{\mathbb{Z}}_{2}&{\mathbb{Z}}\oplus{\mathbb{Z}}\to{\mathbb{Z}}&0\to
0&0\to 0&0\to 0\\\ &&(n,m)\mapsto n-m&&(n,m)\mapsto n+m&(n,m)\mapsto
n+m&(n,m)\mapsto n-m&&\\\ \hline\cr{\rm C^{\dagger}}&{\rm L_{r}}\to{\rm
P}&0\to 0&0\to 2{\mathbb{Z}}&2{\mathbb{Z}}\to
0&0\to{\mathbb{Z}}_{2}&{\mathbb{Z}}_{2}\to{\mathbb{Z}}_{2}&{\mathbb{Z}}_{2}\to{\mathbb{Z}}&{\mathbb{Z}}\to
0&0\to 0\\\ &&&&&&n\mapsto 0&&&\\\ &{\rm L_{i}}\to{\rm P}&2{\mathbb{Z}}\to
0&0\to 2{\mathbb{Z}}&{\mathbb{Z}}_{2}\to
0&{\mathbb{Z}}_{2}\to{\mathbb{Z}}_{2}&{\mathbb{Z}}\to{\mathbb{Z}}_{2}&0\to{\mathbb{Z}}&0\to
0&0\to 0\\\ &&&&&n\mapsto n&n\mapsto n&&&\\\ \hline\cr{\rm CI^{\dagger}}&{\rm
L_{r}}\to{\rm P}&0\to 0&0\to 0&0\to 2{\mathbb{Z}}&2{\mathbb{Z}}\to
0&0\to{\mathbb{Z}}_{2}&{\mathbb{Z}}_{2}\to{\mathbb{Z}}_{2}&{\mathbb{Z}}_{2}\to{\mathbb{Z}}&{\mathbb{Z}}\to
0\\\ &&&&&&&n\mapsto 0&&\\\ &{\rm L_{i}}\to{\rm P}&{\mathbb{Z}}\to 0&0\to
0&{\mathbb{Z}}\to 2{\mathbb{Z}}&0\to
0&{\mathbb{Z}}\to{\mathbb{Z}}_{2}&0\to{\mathbb{Z}}_{2}&{\mathbb{Z}}\to{\mathbb{Z}}&0\to
0\\\ &&&&n\mapsto n&&n\mapsto n&&n\mapsto 2n&\\\
\hline\cr\hline\cr\end{array}$

##  IV Intrinsic and extrinsic non-Hermitian topology

As also discussed above, while intrinsic non-Hermitian topology prohibits the
detachment of topological boundary states, extrinsic non-Hermitian topology
permits it and further leads to anti-Hermitian topology of the detached
boundary states [20]. In this section, we discuss the classification of
intrinsic and extrinsic non-Hermitian topology, as shown in Table 2 (see also
Sec. SIX in the Supplemental Material of Ref. [30], as well as Ref. [31]).

###  IV.1 General formalism

In the presence of a point (line) gap, complex-energy bands are defined not to
cross a reference point (line) in the complex-energy plane [29]. Even if a
point gap is open, a line gap is not necessarily open. However, a point gap is
always open when a line gap is open by placing a reference point on the
reference line. Thus, we can introduce a map from line-gapped topological
phases to point-gapped topological phases for each spatial dimension and
symmetry class. Given a $d$-dimensional non-Hermitian Hamiltonian
$H\left(\bm{k}\right)$, we consider the Hermitized Hamiltonian

$\tilde{H}\left(\bm{k}\right)\coloneqq\begin{pmatrix}0&H\left(\bm{k}\right)\\\
H^{{\dagger}}\left(\bm{k}\right)&0\end{pmatrix}_{\sigma},$ (34)

which respects chiral symmetry

$\Gamma H\left(\bm{k}\right)\Gamma^{-1}=-H\left(\bm{k}\right)$ (35)

with $\Gamma=\sigma_{z}$ by construction. When the non-Hermitian Hamiltonian
$H\left(\bm{k}\right)$ is point-gapped, the Hermitized Hamiltonian
$\tilde{H}\left(\bm{k}\right)$ is also gapped, and vice versa. Thus, the
topological classification of $H\left(\bm{k}\right)$ coincides with that of
$\tilde{H}\left(\bm{k}\right)$, which is denoted by $K_{\rm P}$. By contrast,
the presence of a line gap imposes an additional constraint on
$\tilde{H}\left(\bm{k}\right)$. In fact, a non-Hermitian Hamiltonian
$H\left(\bm{k}\right)$ with a real or an imaginary line gap is continuously
deformable to a Hermitian or an anti-Hermitian Hamiltonian, respectively,
while keeping the line gap and relevant symmetry [29]. Notably, Hermiticity or
anti-Hermiticity of $H\left(\bm{k}\right)$ imposes another chiral symmetry
with $\Gamma_{\rm r}=\sigma_{y}$ or $\Gamma_{\rm i}=\sigma_{x}$ on the
Hermitized Hamiltonian $\tilde{H}\left(\bm{k}\right)$, respectively. This
additional chiral symmetry leads to the different topological classification
described by $K_{\rm L_{r}}$ or $K_{\rm L_{i}}$ in the presence of a real or
an imaginary line gap (and a point gap as well). Forgetting $\Gamma_{\rm r}$
($\Gamma_{\rm i}$) defines a homomorphism $f_{\rm r}:K_{\rm L_{r}}\to K_{\rm
P}$ ($f_{\rm i}:K_{\rm L_{i}}\to K_{\rm P}$) from $K_{\rm L_{r}}$ ($K_{\rm
L_{i}}$) to $K_{\rm P}$. Owing to the dimensional isomorphism of $K$-theory
[32], it is sufficient to compute $f_{\rm r}$ or $f_{\rm i}$ in zero dimension
to obtain those in arbitrary dimensions. Table 2 summarizes these
homomorphisms in the AZ† symmetry classes, which is relevant to boundary
states in Hermitian topological insulators and superconductors. The
homomorphisms in all the 38-fold symmetry classes are found in Refs. [30, 31].

If a point-gapped non-Hermitian Hamiltonian $H\left(\bm{k}\right)$ is included
in the image of either homomorphism $f_{\rm r}$ or $f_{\rm i}$, it is
continuously deformable to a Hermitian or an anti-Hermitian Hamiltonian,
implying that the topological nature is also attributed to the conventional
Hermitian one. Conversely, if $H\left(\bm{k}\right)$ is not included in the
image of $f_{\rm r}$ or $f_{\rm i}$, its topological nature is intrinsic to
non-Hermitian systems. Such intrinsic non-Hermitian topology is captured by
the quotient group $K_{\rm P}/\left(\mathrm{Im}\,f_{\rm
r}\cup\mathrm{Im}\,f_{\rm i}\right)$, which is also summarized in Table 2.

###  IV.2 Examples

As a prime example of intrinsic non-Hermitian topology, we study class A,
i.e., non-Hermitian systems without any internal symmetry. In class A, while
line-gap topology (or equivalently, Hermitian topology) is nontrivial only in
even spatial dimensions, point-gap topology is nontrivial only in odd spatial
dimensions [29]. Consequently, point-gap topology in class A is always
intrinsic to non-Hermitian systems. This intrinsic point-gap topology is the
origin of the non-Hermitian skin effect [38, 39, 40, 41, 42, 30] and
exceptional points [43, 44, 45, 46]. As discussed in Ref. [20], it also
underlies the spectral flow and nondetachability of chiral edge states in
Chern insulators.

On the other hand, as an exemplary class of extrinsic non-Hermitian topology,
let us study class AIII, i.e., non-Hermitian systems that respect chiral
symmetry

$\Gamma H^{{\dagger}}\Gamma^{-1}=-H.$ (36)

Here, $\Gamma$ is a unitary matrix $\Gamma$ satisfying
$\Gamma\Gamma^{{\dagger}}=\Gamma^{{\dagger}}\Gamma=\Gamma^{2}=1$. In class
AIII, point-gap topology is nontrivial and leads to the $\mathbb{Z}$
classification in even spatial dimensions; real-line-gap topology results in
the $\mathbb{Z}$ classification in odd spatial dimensions, whereas imaginary-
line-gap topology the $\mathbb{Z}\oplus\mathbb{Z}$ classification in even
spatial dimensions. Consequently, the $\mathbb{Z}$ point-gap topology can be
incorporated in the $\mathbb{Z}\oplus\mathbb{Z}$ imaginary-line-gap topology.

To clarify the relationship between point-gap and imaginary-line-gap topology,
we investigate flattened chiral-symmetric non-Hermitian systems in zero
dimension. In the presence of an imaginary line gap, non-Hermitian systems can
be flattened into anti-Hermitian systems (i.e., $H^{{\dagger}}=-H$) while
preserving chiral symmetry and an imaginary line gap. Then, chiral symmetry
behaves as $\mathbb{Z}_{2}$ unitary antisymmetry for anti-Hermitian systems
$H$ (or equivalently, $\mathbb{Z}_{2}$ unitary symmetry for Hermitian systems
$H$). As a result, the $K$-group $K_{\rm L_{i}}$ for the
$\mathbb{Z}\oplus\mathbb{Z}$ topological classification is generated by

$\displaystyle\left(1,0\right)\in\mathbb{Z}\oplus\mathbb{Z}:\quad\left[\Gamma=+1,H_{0}=\text{i},H_{1}=-\text{i}\right],$
(37)
$\displaystyle\left(0,1\right)\in\mathbb{Z}\oplus\mathbb{Z}:\quad\left[\Gamma=-1,H_{0}=\text{i},H_{1}=-\text{i}\right],$
(38)

where $H_{0}$ and $H_{1}$ are topologically nontrivial and trivial flattened
systems in terms of an imaginary line gap. In the presence of a point gap, by
contrast, topology is captured by Hermitian matrices $\text{i}H\Gamma$ (see,
for example, Sec. SVII of the Supplemental Material in Ref. [44]). Thus, the
$K$-group $K_{\rm P}$ for the $\mathbb{Z}$ topological classification is
generated by

$1\in\mathbb{Z}:\quad\left[\text{i}\Gamma H=-1,\text{i}\Gamma H=+1\right].$
(39)

Comparing the $\mathbb{Z}\oplus\mathbb{Z}$ classification in Eqs. (37) and
(38) with the $\mathbb{Z}$ classification in Eq. (39), we have a homomorphism

$\begin{array}[]{rccc}f_{\rm i}:&\mathbb{Z}\oplus\mathbb{Z}~{}\left(K_{\rm
L_{i}}\right)&\longrightarrow&\mathbb{Z}~{}\left(K_{\rm P}\right)\\\
&\rotatebox{90.0}{$\in$}&&\rotatebox{90.0}{$\in$}\\\
&\left(n,m\right)&\longmapsto&n-m\end{array}$ (40)

Therefore, the $\mathbb{Z}$ point-gap topology is continuously deformable to a
part of the $\mathbb{Z}\oplus\mathbb{Z}$ imaginary-line-gap topology. This
extrinsic nature of point-gap topology leads to the detachment of topological
boundary states in class AIII, as discussed in Ref. [20].

Notably, the zero-dimensional analysis is generally extended to arbitrary
spatial dimensions, ensured by the dimensional isomorphism of $K$-theory [32].
In a similar manner, the homomorphisms from line-gap topology to point-gap
topology are identified in the tenfold AZ† symmetry classes, as summarized in
Table 2.

Table 3: Topological invariants for the tenfold Altland-Zirnbauer classes. There exist four types of invariants: For the $\mathbb{Z}$ indices, “$W_{d}$” is the $d$-dimensional winding number and “$\mathrm{Ch}_{d/2}$” is the $d/2$-th Chern number, which are defined in odd and even $d$ spatial dimensions, respectively. For the $\mathbb{Z}_{2}$ indices, “$\mathrm{FK}_{d/2}$” denotes the Fu-Kane invariant in even $d$ dimensions, and “$\mathrm{CS}_{d}~{}(\widetilde{\mathrm{CS}}_{d})$” represents the Chern-Simons invariant in symmetry classes without (with) chiral symmetry in odd $d$ dimensions. The entries “$0$” indicate the absence of nontrivial topological phases. Class |  TRS |  PHS |  CS |  $d=1$ |  $d=2$ |  $d=3$ |  $d=4$ |  $d=5$ |  $d=6$ |  $d=7$ |  $d=8$
---|---|---|---|---|---|---|---|---|---|---|---
A | $0$ | $0$ | $0$ | $0$ |  $\mathrm{Ch}_{1}\in\mathbb{Z}^{\checkmark}$ | $0$ |  $\mathrm{Ch}_{2}\in\mathbb{Z}^{\checkmark}$ | $0$ |  $\mathrm{Ch}_{3}\in\mathbb{Z}^{\checkmark}$ | $0$ |  $\mathrm{Ch}_{4}\in\mathbb{Z}^{\checkmark}$
AIII | $0$ | $0$ | $1$ |  $W_{1}\in\mathbb{Z}^{\times}$ | $0$ |  $W_{3}\in\mathbb{Z}^{\times}$ | $0$ |  $W_{5}\in\mathbb{Z}^{\times}$ | $0$ |  $W_{7}\in\mathbb{Z}^{\times}$ | $0$
AI | $+1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\mathrm{Ch}_{2}\in 2\mathbb{Z}^{\checkmark}$ | $0$ | $\mathrm{FK}_{3}\in\mathbb{Z}_{2}^{\checkmark}$ | $\mathrm{CS}_{7}\in\mathbb{Z}_{2}^{\checkmark}$ | $\mathrm{Ch}_{4}\in\mathbb{Z}^{\checkmark}$
BDI | $+1$ | $+1$ | $1$ | $W_{1}\in\mathbb{Z}^{\times}$ | $0$ | $0$ | $0$ | $W_{5}\in 2\mathbb{Z}^{\times}$ | $0$ | $\widetilde{\mathrm{CS}}_{7}\in\mathbb{Z}_{2}^{\times}$ | $\mathrm{FK}_{4}\in\mathbb{Z}_{2}^{\times}$
D | $0$ | $+1$ | $0$ | $\mathrm{CS}_{1}\in\mathbb{Z}_{2}^{\times}$ | $\mathrm{Ch}_{1}\in\mathbb{Z}^{\checkmark}$ | $0$ | $0$ | $0$ | $\mathrm{Ch}_{3}\in 2\mathbb{Z}^{\checkmark}$ | $0$ | $\mathrm{FK}_{4}\in\mathbb{Z}_{2}^{\times}$
DIII | $-1$ | $+1$ | $1$ | $\widetilde{\mathrm{CS}}_{1}\in\mathbb{Z}_{2}^{\times}$ | $\mathrm{FK}_{1}\in\mathbb{Z}_{2}^{\checkmark}$ | $W_{3}\in\mathbb{Z}^{\checkmark/\times}$ | $0$ | $0$ | $0$ | $W_{7}\in 2\mathbb{Z}^{\times}$ | $0$
AII | $-1$ | $0$ | $0$ | $0$ | $\mathrm{FK}_{1}\in\mathbb{Z}_{2}^{\checkmark}$ | $\mathrm{CS}_{3}\in\mathbb{Z}_{2}^{\checkmark}$ | $\mathrm{Ch}_{2}\in\mathbb{Z}^{\checkmark}$ | $0$ | $0$ | $0$ | $\mathrm{Ch}_{4}\in 2\mathbb{Z}^{\checkmark}$
CII | $-1$ | $-1$ | $1$ | $W_{1}\in 2\mathbb{Z}^{\times}$ | $0$ | $\widetilde{\mathrm{CS}}_{3}\in\mathbb{Z}_{2}^{\times}$ | $\mathrm{FK}_{2}\in\mathbb{Z}_{2}^{\times}$ | $W_{5}\in\mathbb{Z}^{\times}$ | $0$ | $0$ | $0$
C | $0$ | $-1$ | $0$ | $0$ | $\mathrm{Ch}_{1}\in 2\mathbb{Z}^{\checkmark}$ | $0$ | $\mathrm{FK}_{2}\in\mathbb{Z}_{2}^{\times}$ | $\mathrm{CS}_{5}\in\mathbb{Z}_{2}^{\times}$ | $\mathrm{Ch}_{3}\in\mathbb{Z}^{\checkmark}$ | $0$ | $0$
CI | $+1$ | $-1$ | $1$ | $0$ | $0$ | $W_{3}\in 2\mathbb{Z}^{\times}$ | $0$ | $\widetilde{\mathrm{CS}}_{5}\in\mathbb{Z}_{2}^{\times}$ | $\mathrm{FK}_{3}\in\mathbb{Z}_{2}^{\checkmark}$ | $W_{7}\in\mathbb{Z}^{\checkmark/\times}$ | $0$
Table 4: Homomorphisms from the tenfold Altland-Zirnbauer (AZ) symmetry classification to the threefold Wigner-Dyson (WD) classification. While the tenfold AZ classes are specified by time-reversal symmetry (TRS), particle-hole symmetry (PHS), and chiral symmetry (CS), the threefold WD classes are based solely on TRS. For the entries “$\mathbb{Z}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}$”, “$2\mathbb{Z}^{\checkmark}\xrightarrow{\cong}2\mathbb{Z}$”, and “$\mathbb{Z}_{2}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}_{2}$”, the original topological phases in the AZ classes are isomorphic to those in the corresponding WD classes. For the entries “$\mathbb{Z}_{2}^{\times}\xrightarrow{0}\mathbb{Z}_{2}$” and “$\mathbb{Z}_{2}^{\times}\xrightarrow{0}\mathbb{Z}$”, the original topological phases in the AZ classes vanish in the corresponding WD classes. For the entries “$2\mathbb{Z}^{\checkmark}\xrightarrow{2}\mathbb{Z}$”, the topological invariants double. For the entries $\mathbb{Z}^{\checkmark/\times}\xrightarrow{\bmod 2}\mathbb{Z}_{2}$, the topological phases with the odd (even) number of topological invariants survive (vanish).  Class |  TRS |  PHS |  CS |  $d=1$ |  $d=2$ |  $d=3$ |  $d=4$ |  $d=5$ |  $d=6$ |  $d=7$ |  $d=8$
---|---|---|---|---|---|---|---|---|---|---|---
A $\rightarrow$ A | $0$ | $0$ | $0$ | $0\rightarrow 0$ |  $\mathbb{Z}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}$ | $0\rightarrow 0$ |  $\mathbb{Z}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}$ | $0\rightarrow 0$ |  $\mathbb{Z}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}$ | $0\rightarrow 0$ |  $\mathbb{Z}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}$
AIII $\rightarrow$ A | $0$ | $0$ | $1$ |  $\mathbb{Z}^{\times}\rightarrow 0$ | $0\rightarrow\mathbb{Z}$ |  $\mathbb{Z}^{\times}\rightarrow 0$ | $0\rightarrow\mathbb{Z}$ |  $\mathbb{Z}^{\times}\rightarrow 0$ | $0\rightarrow\mathbb{Z}$ |  $\mathbb{Z}^{\times}\rightarrow 0$ | $0\rightarrow\mathbb{Z}$
AI $\rightarrow$ AI | $+1$ | $0$ | $0$ | $0\rightarrow 0$ | $0\rightarrow 0$ | $0\rightarrow 0$ | $2\mathbb{Z}^{\checkmark}\xrightarrow{\cong}2\mathbb{Z}$ | $0\rightarrow 0$ | $\mathbb{Z}_{2}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}_{2}$ | $\mathbb{Z}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}$
BDI $\rightarrow$ AI | $+1$ | $+1$ | $1$ | $\mathbb{Z}^{\times}\rightarrow 0$ | $0\rightarrow 0$ | $0\rightarrow 0$ | $0\rightarrow 2\mathbb{Z}$ | $2\mathbb{Z}^{\times}\rightarrow 0$ | $0\rightarrow\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}^{\times}\xrightarrow{0}\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}^{\times}\xrightarrow{0}\mathbb{Z}$
D $\rightarrow$ A | $0$ | $+1$ | $0$ | $\mathbb{Z}_{2}^{\times}\rightarrow 0$ | $\mathbb{Z}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}$ | $0\rightarrow 0$ | $0\rightarrow\mathbb{Z}$ | $0\rightarrow 0$ | $2\mathbb{Z}^{\checkmark}\xrightarrow{2}\mathbb{Z}$ | $0\rightarrow 0$ | $\mathbb{Z}_{2}^{\times}\xrightarrow{0}\mathbb{Z}$
DIII $\rightarrow$ AII | $-1$ | $+1$ | $1$ | $\mathbb{Z}_{2}^{\times}\rightarrow 0$ | $\mathbb{Z}_{2}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}_{2}$ | $\mathbb{Z}^{\checkmark/\times}\xrightarrow{\bmod 2}\mathbb{Z}_{2}$ | $0\rightarrow\mathbb{Z}$ | $0\rightarrow 0$ | $0\rightarrow 0$ | $2\mathbb{Z}^{\times}\rightarrow 0$ | $0\rightarrow 2\mathbb{Z}$
AII $\rightarrow$ AII | $-1$ | $0$ | $0$ | $0\to 0$ | $\mathbb{Z}_{2}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}_{2}$ | $\mathbb{Z}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $2\mathbb{Z}^{\checkmark}\xrightarrow{\cong}2\mathbb{Z}$
CII $\rightarrow$ AII | $-1$ | $-1$ | $1$ | $2\mathbb{Z}^{\times}\to 0$ | $0\to\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}^{\times}\xrightarrow{0}\mathbb{Z}_{2}$ | $\mathbb{Z}_{2}^{\times}\xrightarrow{0}\mathbb{Z}$ | $\mathbb{Z}^{\times}\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 2\mathbb{Z}$
C $\rightarrow$ A | $0$ | $-1$ | $0$ | $0\to 0$ | $2\mathbb{Z}^{\checkmark}\xrightarrow{2}\mathbb{Z}$ | $0\to 0$ | $\mathbb{Z}_{2}^{\times}\xrightarrow{0}\mathbb{Z}$ | $\mathbb{Z}_{2}^{\times}\to 0$ | $\mathbb{Z}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}$ | $0\to 0$ | $0\to\mathbb{Z}$
CI $\rightarrow$ AI | $+1$ | $-1$ | $1$ | $0\to 0$ | $0\to 0$ | $2\mathbb{Z}^{\times}\to 0$ | $0\to 2\mathbb{Z}$ | $\mathbb{Z}_{2}^{\times}\to 0$ | $\mathbb{Z}_{2}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}_{2}$ | $\mathbb{Z}^{\checkmark/\times}\xrightarrow{\bmod 2}\mathbb{Z}_{2}$ | $0\to\mathbb{Z}$

##  V From Altland-Zirnbauer classification to Wigner-Dyson classification

While topology protected by no internal symmetry or time-reversal symmetry,
including the Chern number, should impose Wannier obstructions, topology
protected by chiral or particle-hole symmetry should be exempt from such
Wannier obstructions [19]. Consequently, studying how essential chiral or
particle-hole symmetry for given topological invariants, we identify the
presence or absence of the Wannier obstructions, leading to their tenfold
classification summarized in Table 1. To clarify it, we explicitly provide
homomorphisms from topological phases in the tenfold AZ symmetry classes to
those in the threefold WD classes (i.e., classes A, AI, and AII).

Let us first introduce the topological invariants for the AZ classification.
As summarized in Table 3, each nontrivial $d$-dimensional topological phase is
characterized by one of
$\\{W_{d},\mathrm{Ch}_{d/2},\mathrm{FK}_{d/2},\mathrm{CS}_{d}~{}(\widetilde{\mathrm{CS}}_{d})\\}$,
which are the $d$-dimensional winding number, the $d/2$-th Chern number, the
Fu-Kane invariant, and the Chern-Simons invariant in symmetry classes without
(with) chiral symmetry, respectively [7]. The homomorphisms are given in Table
4, based on the connections among these topological invariants, i.e., how the
$d$-dimensional topological invariants in the AZ classes relate to those in
the WD classes if one virtually forgets chiral symmetry and/or particle-hole
symmetry. To derive these homomorphisms, we divide all the entries in Table 4
into the following seven categories. We identify the homomorphism for each
case below.

###  V.1 $\mathbb{Z}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}$,
$2\mathbb{Z}^{\checkmark}\xrightarrow{\cong}2\mathbb{Z}$,
$\mathbb{Z}_{2}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}_{2}$

Let us focus on class DIII in two dimensions to explain how to see and derive
Table 4. Since class DIII respects spinful time-reversal symmetry, it reduces
to class AII if one ignores both chiral symmetry and particle-hole symmetry.
See the first four columns of Table 3 and Table 4. The entries
“$\mathbb{Z}_{2}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}_{2}$” show how the
topological phase in class DIII, the left side of the arrow (i.e.,
“$\mathbb{Z}_{2}^{\checkmark}$”), is mapped to that in class AII, the right
side of the arrow (i.e., “$\mathbb{Z}_{2}$”). Since, for two dimensions, both
the $\mathbb{Z}_{2}$ invariants coincide and are given by the Fu-Kane
invariant $\mathrm{FK}_{1}$ (see Table 3) [47], we conclude that the
nontrivial topological phase in two-dimensional class DIII is isomorphically
mapped to that in two-dimensional class AII. We use the symbol “$\cong$” in
“$\mathbb{Z}_{2}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}_{2}$” to emphasize
that this homomorphism is indeed an (group) isomorphism. In a similar manner,
we have an isomorphism for each of
“$\mathbb{Z}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}$”,
“$2\mathbb{Z}^{\checkmark}\xrightarrow{\cong}2\mathbb{Z}$”, and
“$\mathbb{Z}_{2}^{\checkmark}\xrightarrow{\cong}\mathbb{Z}_{2}$” in Table 4.

###  V.2 $2\mathbb{Z}^{\checkmark}\xrightarrow{2}\mathbb{Z}$

The entries “$2\mathbb{Z}^{\checkmark}\xrightarrow{2}\mathbb{Z}$” appear in
two-dimensional class C and six-dimensional class D. These two classes reduce
to class A when one forgets particle-hole symmetry. In two dimensions, the
first Chern number characterizes both classes A and C, where the presence of
particle-hole symmetry makes it even integers. Thus, we have a map from the
topological phases in two-dimensional class C to the even parts of those in
two-dimensional class A. This homomorphism is denoted by
“$2\mathbb{Z}^{\checkmark}\xrightarrow{2}\mathbb{Z}$”. Similarly, one can show
this kind of map for six-dimensional class D.

###  V.3 $\mathbb{Z}^{\checkmark/\times}\xrightarrow{\bmod 2}\mathbb{Z}_{2}$

In three dimensions, we have the entry
“$\mathbb{Z}^{\checkmark/\times}\xrightarrow{\bmod 2}\mathbb{Z}_{2}$” for
class DIII. Since the corresponding WD symmetry class is class AII, what we
should know is the connection between the three-dimensional winding number
$W_{3}$ and the Chern-Simons invariant $\mathrm{CS}_{3}$ (see Table 3).
Notably, $W_{d}$ relates to $\mathrm{CS}_{d}$ through
$(-1)^{W_{d}}=(-1)^{\mathrm{CS}_{d}}$ for general odd $d$ spatial dimensions
[7]. Therefore, only the even parts of topological phases in three-dimensional
class DIII are sent to the nontrivial phase in three-dimensional class AII.
This homomorphism is written as
“$\mathbb{Z}^{\checkmark/\times}\xrightarrow{\bmod 2}\mathbb{Z}_{2}$”. One can
also prove the presence of such a “mod 2” map for seven-dimensional class CI
in the same manner.

###  V.4 $\mathbb{Z}_{2}^{\times}\xrightarrow{0}\mathbb{Z}$

Because we consider homomorphisms that preserve the group structure, only the
possible map from the $\mathbb{Z}_{2}$ classification to the $\mathbb{Z}$
classification is a zero map. In fact, all elements of the $\mathbb{Z}_{2}$
topological phases are sent to a trivial one in the $\mathbb{Z}$ topological
phases. We use the symbol “$\mathbb{Z}_{2}^{\times}\xrightarrow{0}\mathbb{Z}$”
to represent this zero-map homomorphism. For instance, in four dimensions, the
Fu-Kane invariant $\mathrm{FK}_{2}$ for class CII does not result in the
nontrivial values of the second Chern number $\mathrm{Ch}_{2}$ for class AII
if one ignores both chiral symmetry and particle-hole symmetry.

###  V.5 $\mathbb{Z}_{2}^{\times}\xrightarrow{0}\mathbb{Z}_{2}$

The entries “$\mathbb{Z}_{2}^{\times}\xrightarrow{0}\mathbb{Z}_{2}$” show that
all the topological phases in the original AZ classes are mapped to a trivial
phase in the corresponding WD classes. We show this using the dimensional
reduction of topological phases [48]. Let us consider three-dimensional class
CII as an example. This class is characterized by the Chern-Simons invariant
$\widetilde{\mathrm{CS}_{3}}$, while the corresponding WD class (i.e., class
AII) is characterized by the other type of Chern-Simons invariant
$\mathrm{CS}_{3}$. The dimensional reduction connects the $\mathbb{Z}_{2}$
descendants to their parent $\mathbb{Z}$ topology in each complex symmetry
class. For the present case,
$(-1)^{\widetilde{\mathrm{CS}_{3}}}=(-1)^{\mathrm{FK}_{2}}=(-1)^{W_{5}}$ holds
for class CII in $d=3,4,5$. Similarly, we have
$(-1)^{\mathrm{FK}_{1}}=(-1)^{\mathrm{CS}_{3}}=(-1)^{\mathrm{Ch}_{2}}$ for
class AII in $d=2,3,4$; see Table 3. As stated in the previous case,
$\mathrm{FK}_{2}$ in four-dimensional class CII always leads to a trivial
value of $\mathrm{Ch}_{2}$ in four-dimensional class AII. Combining this with
the dimensional reduction in $d=3,4$, we conclude that
$\widetilde{\mathrm{CS}_{3}}$ only relates to trivial $\mathrm{CS}_{3}$.
Similarly, one can prove this kind of zero map for other classes.

###  V.6 $\mathbb{Z}^{\times}\rightarrow 0$, $2\mathbb{Z}^{\times}\rightarrow
0$, $\mathbb{Z}_{2}^{\times}\rightarrow 0$

Since the corresponding WD classes exhibit topologically trivial phases, the
homomorphisms are just zero maps. Here, we do not use the symbol
“$\xrightarrow{0}$” like the previous two cases as the homomorphisms are
clearly zero maps.

###  V.7 $0\rightarrow 0$, $0\rightarrow\mathbb{Z}$, $0\rightarrow
2\mathbb{Z}$, $0\rightarrow\mathbb{Z}_{2}$

Since the original AZ classes are topologically trivial, the homomorphisms are
given by zero maps to retain the group structure. Here, we do not use the
symbol “$\xrightarrow{0}$” to express the zero maps.

##  VI $K$-theory perspective

In Sec. IV, we classified intrinsic and extrinsic non-Hermitian topology,
corresponding to nondetachable and detachable topological boundary states. In
Sec. V, on the other hand, we classified Wannier localizability of the bulk
bands based on the homomorphisms from the AZ classification to the WD
classification. These two results give the same classification. In this
section, we discuss why the two approaches coincide with each other from the
viewpoint of $K$-theory. The discussion in this section applies to any
magnetic space group symmetry compatible with planar boundary conditions.

###  VI.1 $K$-group

The degree $+1$ $K$-group $K^{1}$ is represented by self-adjoint Fredholm
operators whose essential spectrum contains both positive and negative values
[33]. Gapless Hamiltonians naturally represent such operators realized in
topological boundary states, meaning that there exists an energy gap $E_{\rm
gap}>0$ with a finite number of boundary-state spectra in $E\in[-E_{\rm
gap},E_{\rm gap}]$ and an infinite number of bulk spectra outside this range.

Let $\hat{H}({\bm{k}}_{\parallel})$ be the infinite-dimensional gapless
Hamiltonian on the boundary Brillouin zone $T^{d-1}$ (${\bm{k}}_{\parallel}\in
T^{d-1}$), $\mathcal{G}$ be the symmetry group compatible with the presence of
the boundary, $\Pi\cong\mathbb{Z}^{d-1}$ be the lattice translation subgroup,
and $G=\mathcal{G}/\Pi$ be their quotient group. The group $G$ acts on the
boundary Brillouin zone as $g:{\bm{k}}_{\parallel}\mapsto
g{\bm{k}}_{\parallel}$. We use the homomorphism $\phi,c:G\to\\{\pm 1\\}$ to
specify

$\begin{cases}\phi_{g}=1:\text{unitary~{}symmetry};\\\
\phi_{g}=-1:\text{antiunitary~{}symmetry},\end{cases}$ (41)

and

$\begin{cases}c_{g}=1:\text{conventional~{}symmetry};\\\
c_{g}=-1:\text{antisymmetry}.\end{cases}$ (42)

For matrices $M$, we also introduce the notation $M^{\phi_{g}}$ as $M$ for
$\phi_{g}=1$ and $M^{*}$ for $\phi_{g}=-1$. Antisymmetry is defined to change
the sign of Hamiltonians, prime examples of which include chiral symmetry and
particle-hole symmetry. We summarize the symmetry algebra among symmetry
operators in $G$ as

$\displaystyle
u_{g}(h{\bm{k}}_{\parallel})u_{h}({\bm{k}}_{\parallel})^{\phi_{g}}=\tau_{g,h}(gh{\bm{k}}_{\parallel})u_{gh}({\bm{k}}_{\parallel}),\quad
g,h\in G,$ (43)

where $u_{g}({\bm{k}}_{\parallel})$ is a unitary operator for $g\in G$. Here,
$\tau$ is a 2-cocycle $\tau\in Z_{\rm group}^{2}(G,C(T^{d-1},U(1)))$, where
$C(T^{d-1},U(1))$ is the group of $U(1)$-valued functions on $T^{d-1}$ with a
left $G$-action defined by
$(g.\tau)({\bm{k}}_{\parallel})=\tau(g^{-1}{\bm{k}}_{\parallel})$. The twisted
equivariant $K$-group ${}^{\phi}K_{G}^{(\tau,c)+1}(T^{d-1})$ [34, 37]
classifies topology of $(d-1)$-dimensional gapless Hamiltonians with the
following symmetry:

$\displaystyle
u_{g}({\bm{k}}_{\parallel})\hat{H}({\bm{k}}_{\parallel})^{\phi_{g}}u_{g}^{{\dagger}}({\bm{k}}_{\parallel})=c_{g}\hat{H}(g{\bm{k}}_{\parallel}),\quad
g\in G.$ (44)

###  VI.2 Gapless Hermitian Hamiltonians and point-gapped non-Hermitian
Hamiltonians

First, we relate gapless Hermitian Hamiltonians to point-gapped non-Hermitian
Hamiltonians. From the gapless Hermitian Hamiltonian
$\hat{H}({\bm{k}}_{\parallel})$, we obtain the point-gapped non-Hermitian
Hamiltonian $H_{P}({\bm{k}}_{\parallel})$ by Eq. (2), with a continuous odd
function $\chi(E)$ defined in Eq. (3). As a consequence of the symmetries of
the gapless Hermitian Hamiltonian $\hat{H}({\bm{k}}_{\parallel})$, the
obtained point-gapped Hamiltonian $H_{P}({\bm{k}}_{\parallel})$ satisfies the
following symmetries,

$\displaystyle
u_{g}({\bm{k}}_{\parallel})H_{P}({\bm{k}}_{\parallel})^{\phi_{g}}u_{g}^{\dagger}({\bm{k}}_{\parallel})=c_{g}\begin{cases}H_{P}(g{\bm{k}}_{\parallel})&(\phi_{g}c_{g}=1);\\\
H_{P}^{\dagger}(g{\bm{k}}_{\parallel})&(\phi_{g}c_{g}=-1).\\\ \end{cases}$
(45)

In other words, the point-gapped Hamiltonian shares the same unitary
symmetries with the original gapless Hamiltonian, while it realizes the other
symmetries in a different manner. In particular, it hosts the AZ† symmetries,
instead of the original AZ symmetries [29]. Importantly, this map ensures that
the $K$-group ${}^{\phi}K_{G}^{(\tau,c)+1}(T^{d-1})$ also classifies topology
of point-gapped Hamiltonians satisfying symmetry in Eq. (45).

We also use another equivalent relation between Hermitian and non-Hermitian
topology. For the point-gapped Hamiltonian in Eq. (2), we have the Hermitized
Hamiltonian [see also Eq. (34)]

$\displaystyle\tilde{H}({\bm{k}}_{\parallel})=\begin{pmatrix}0&H_{P}({\bm{k}}_{\parallel})\\\
H_{P}^{\dagger}({\bm{k}}_{\parallel})&0\end{pmatrix}_{\sigma},$ (46)

which is gapped because of $\tilde{H}({\bm{k}}_{\parallel})^{2}=1$. By design,
$\tilde{H}({\bm{k}}_{\parallel})$ respects chiral symmetry

$\displaystyle\sigma_{z}\tilde{H}({\bm{k}}_{\parallel})\sigma_{z}^{-1}=-\tilde{H}({\bm{k}}_{\parallel}).$
(47)

The symmetry constraints in Eq. (45) are mapped as

$\displaystyle\tilde{u}_{g}({\bm{k}}_{\parallel})\tilde{H}({\bm{k}}_{\parallel})^{\phi_{g}}\tilde{u}_{g}^{\dagger}({\bm{k}}_{\parallel})=c_{g}\tilde{H}(g{\bm{k}}_{\parallel}),$
(48)

with

$\tilde{u}_{g}({\bm{k}}_{\parallel})\coloneqq
u_{g}({\bm{k}}_{\parallel})\otimes(\sigma_{x})^{\frac{1-\phi_{g}c_{g}}{2}},\quad
g\in G.$ (49)

They satisfy the following algebraic relation with chiral symmetry in Eq.
(47):

$\displaystyle(\text{i}\sigma_{z})^{2}=-1,\quad\tilde{u}_{g}({\bm{k}}_{\parallel})(\text{i}\sigma_{z})^{\phi_{g}}=c_{g}(\text{i}\sigma_{z})\tilde{u}_{g}({\bm{k}}_{\parallel}).$
(50)

If chiral symmetry in Eq. (47) is absent, the gapped Hamiltonian with $G$
symmetry in Eq. (48) gives the degree $0$ $K$-group
${}^{\phi}K^{(\tau,c)+0}_{G}(T^{d-1})$. Additional chiral symmetry satisfying
the algebraic relation in Eq. (50) increases the degree of the $K$-group by
$+1$ [32], and thus, the Hermitian Hamiltonian
$\tilde{H}({\bm{k}}_{\parallel})$ gives the same $K$-group
${}^{\phi}K^{(\tau,c)+1}_{G}(T^{d-1})$ as the point-gapped Hamiltonian
$H_{P}({\bm{k}}_{\parallel})$.

###  VI.3 Imaginary line gap and detachable topological boundary states

In the presence of an imaginary line gap, the spectrum of
$H_{P}({\bm{k}}_{\parallel})$ is defined not to contain ${\rm Im}\,E=0$.
Additionally, imposing the imaginary line gap is equivalent to imposing the
anti-Hermitian condition

$H_{P}^{\dagger}({\bm{k}}_{\parallel})=-H_{P}({\bm{k}}_{\parallel})$ (51)

on $H_{P}({\bm{k}}_{\parallel})$ [29]. As discussed in Sec. IV, this is also
equivalent to imposing chiral symmetry with $\sigma_{x}$ on the gapped
Hermitized Hamiltonian $\tilde{H}({\bm{k}}_{\parallel})$ in addition to Eqs.
(47) and (48):

$\displaystyle\sigma_{x}\tilde{H}({\bm{k}}_{\parallel})\sigma_{x}^{-1}=-\tilde{H}({\bm{k}}_{\parallel}).$
(52)

Using this chiral operator $\sigma_{x}$, we construct a Hamiltonian
parameterized by the interval $I=[-1,1]$ as

$\displaystyle\tilde{H}^{\prime}({\bm{k}}_{\parallel},t)=\left(\sin\frac{\pi
t}{2}\right)\sigma_{x}+\left(\cos\frac{\pi
t}{2}\right)\tilde{H}({\bm{k}}_{\parallel}),\quad t\in I.$ (53)

It should be noted that the symmetry group $G$ acts on the additional
parameter space as $c_{g}t$,

$\displaystyle\tilde{u}_{g}({\bm{k}}_{\parallel})\tilde{H}^{\prime}({\bm{k}}_{\parallel},t)\tilde{u}_{g}^{\dagger}({\bm{k}}_{\parallel})=c_{g}\tilde{H}^{\prime}(g{\bm{k}}_{\parallel},c_{g}t),\quad
g\in G.$ (54)

Therefore, the Hamiltonian $\tilde{H}^{\prime}({\bm{k}}_{\parallel},t)$, and
thus the imaginary-line-gapped phase, are classified by the relative $K$-group
${}^{\phi}K^{(\tau,c)+1}_{G}(T^{d-1}\times I,T^{d-1}\times\partial I)$, where
$G$ acts on $T^{d-1}\times I$ as
$g:({\bm{k}}_{\parallel},t)\mapsto(g{\bm{k}}_{\parallel},c_{g}t)$, and
$\partial I=\\{-1,1\\}$ is the boundary of the interval. From the long exact
sequence for the pair $(T^{d-1}\times I,T^{d-1}\times\partial I)$, we obtain
the exact sequence

$\displaystyle{}^{\phi}K^{(\tau,c)+1}_{G}(T^{d-1}\times
I,T^{d-1}\times\partial I)$ $\displaystyle\qquad\xrightarrow{f^{\rm bdy}_{\rm
L_{i}}}{}^{\phi}K^{(\tau,c)+1}_{G}(T^{d-1})$
$\displaystyle\qquad\xrightarrow{f^{\rm
bdy}_{0}}{}^{\phi}K^{(\tau,c)+1}_{G}(T^{d-1}\times\partial I).$ (55)

Here, we used the isomorphism

${}^{\phi}K^{(\tau,c)+1}_{G}(T^{d-1}\times
I)\cong{}^{\phi}K^{(\tau,c)+1}_{G}(T^{d-1})$ (56)

due to the contractibility of the interval $I$. This exactness implies

${\rm im}\ f^{\rm bdy}_{\rm L_{i}}={\rm ker}\ f^{\rm bdy}_{0}.$ (57)

Notably, the homomorphism $f^{\rm bdy}_{\rm L_{i}}$ coincides with the map
from imaginary-line-gapped phases to point-gapped phases discussed in Sec. IV
[30, 31]. If there are no group elements $g\in G$ with $c_{g}=-1$, the
$K$-group ${}^{\phi}K^{(\tau,c)+1}_{G}(T^{d-1}\times\partial I)$ reduces to
two copies defined on two independent points $\\{-1,1\\}=\partial I$, leading
to:

$\displaystyle{}^{\phi}K^{\tau+1}_{G}(T^{d-1}\times I,T^{d-1}\times\partial
I)$ $\displaystyle\qquad\xrightarrow{f^{\rm bdy}_{\rm
L_{i}}}{}^{\phi}K^{\tau+1}_{G}(T^{d-1})$
$\displaystyle\qquad\xhookrightarrow{f^{\rm
bdy}_{0}}{}^{\phi}K^{\tau+1}_{G}(T^{d-1})^{\oplus 2}.$ (58)

Here, we dropped $c$ owing to $c_{g}\equiv 1$ from the assumption. In this
case, since $f^{\rm bdy}_{0}$ sends $x$ to $(x,x)$, we have ${\rm ker}\,f^{\rm
bdy}_{0}=0$ and hence ${\rm im}\,f^{\rm bdy}_{L_{\rm i}}=0$, meaning that
there are no point-gapped phases representable by imaginary-line-gapped
phases. On the other hand, if there is a group element $g\in G$ with
$c_{g}=-1$, $g$ exchanges $t=1$ and $t=-1$, and hence merely relates the
$K$-group at $t=1$ with that at $t=-1$. Thus, we only need to consider the
$K$-group at $t=1$, excluding group elements with $c_{g}=1$. This
consideration gives the isomorphism

${}^{\phi}K^{(\tau,c)+1}_{G}(T^{d-1}\times\partial
I)\cong{}^{\phi}K^{\tau+1}_{G_{0}}(T^{d-1}),$ (59)

where $G_{0}\coloneqq{\rm ker}\ c=\\{g\in G\mid c_{g}=1\\}$ is the subgroup of
$G$ excluding antisymmetries. As a result, we obtain

$\displaystyle{}^{\phi}K^{(\tau,c)+1}_{G}(T^{d-1}\times
I,T^{d-1}\times\partial I)$ $\displaystyle\qquad\xrightarrow{f^{\rm bdy}_{\rm
L_{i}}}{}^{\phi}K^{(\tau,c)+1}_{G}(T^{d-1})$
$\displaystyle\qquad\xrightarrow{f^{\rm
bdy}_{0}}{}^{\phi}K^{\tau+1}_{G_{0}}(T^{d-1}).$ (60)

Here, $f^{\rm bdy}_{0}$ in Eq. (60) can be redefined by forgetting
antisymmetries.

To derive Eqs. (58) and (60), we assumed that the point-gapped Hamiltonians
$H_{P}(k_{\parallel})$ can be anti-Hermitian. Under the same assumption, we
can obtain another useful expression for the $K$-group. For the anti-Hermitian
point-gapped Hamiltonian $H_{P}({\bm{k}}_{\parallel})$ with Eq. (51), the
symmetry constraints in Eq. (45) are rewritten as

$\displaystyle
u_{g}({\bm{k}}_{\parallel})(\text{i}H_{P}({\bm{k}}_{\parallel}))^{\phi_{g}}u_{g}^{\dagger}({\bm{k}}_{\parallel})=\text{i}H_{P}(g{\bm{k}}_{\parallel}),\quad
g\in G,$ (61)

for the Hermitian Hamiltonian $\text{i}H_{P}({\bm{k}}_{\parallel})$.
Remarkably, antisymmetries with $c_{g}=-1$ behave as the corresponding
symmetries with $c_{g}=1$. Therefore, the imaginary-line-gapped Hamiltonian is
classified by the $K$-group ${}^{\phi}K_{G}^{(\tau,c_{\rm triv})+0}(T^{d-1})$
with the trivial homomorphism $c=c_{\rm triv}$, i.e., $c_{{\rm triv},g}=1$ for
all $g\in G$. This is nothing but the Thom isomorphism [37]

$\displaystyle{}^{\phi}K^{(\tau,c)+1}_{G}(T^{d-1}\times
I,T^{d-1}\times\partial I)\cong{}^{\phi}K_{G}^{(\tau,c_{\rm
triv})+0}(T^{d-1}).$ (62)

In summary, in the absence of antisymmetry ($c_{g}\equiv 1$), we have the
exact sequence

$\displaystyle{}^{\phi}K^{\tau+0}_{G}(T^{d-1})$
$\displaystyle\qquad\xrightarrow{{\rm im}\,f^{\rm bdy}_{\rm
L_{i}}=0}{}^{\phi}K^{\tau+1}_{G}(T^{d-1})$
$\displaystyle\qquad\xhookrightarrow{f^{\rm
bdy}_{0}}{}^{\phi}K^{\tau+1}_{G}(T^{d-1})^{\oplus 2}.$ (63)

The triviality of $f^{\rm bdy}_{\rm L_{i}}$ implies that all the nontrivial
gapless boundary Hamiltonians involve spectral flow. In the presence of
antisymmetries (${}^{\exists}g\in G,c_{g}=-1$), on the other hand, we have

$\displaystyle{}^{\phi}K^{(\tau,c_{\rm triv})+0}_{G}(T^{d-1})$
$\displaystyle\qquad\xrightarrow{f^{\rm bdy}_{\rm
L_{i}}}{}^{\phi}K^{(\tau,c)+1}_{G}(T^{d-1})$
$\displaystyle\qquad\xrightarrow{f^{\rm
bdy}_{0}}{}^{\phi}K^{\tau+1}_{G_{0}}(T^{d-1}).$ (64)

Since the boundary Hamiltonian $\hat{H}({\bm{k}}_{\parallel})$ hosts a
detachable boundary state when the corresponding point-gapped Hamiltonian may
have an imaginary line gap, this equation implies that
$\hat{H}({\bm{k}}_{\parallel})$ may host a detachable boundary state
($[\hat{H}({\bm{k}}_{\parallel})]\in{\rm im}\,f^{\rm bdy}_{\rm L_{i}}$) if and
only if one can open a gap in the boundary state
($[\hat{H}({\bm{k}}_{\parallel})]\in{\rm ker}\,f^{\rm bdy}_{0}$) in
$\hat{H}({\bm{k}}_{\parallel})$ by ignoring the antisymmetries.

###  VI.4 Wannier localizability

Finally, we clarify the relationship with Wannier localizability. It was
argued that if the $d$-dimensional bulk admits exponentially localized Wannier
states for both occupied and unoccupied states, no spectral flow arises in the
boundary Hamiltonian for arbitrary boundary geometry [19]. Therefore, the
existence of spectral flow in the boundary indicates that either occupied or
unoccupied states are not Wannier localizable. To see this in $K$-theory, let
us introduce the bulk-to-boundary map

$f^{\rm
BBC}_{G}:{}^{\phi}K_{G}^{(\tau,c)+0}(T^{d})\to{}^{\phi}K^{(\tau,c)+1}_{G}(T^{d-1}),$
(65)

which sends the bulk Hamiltonian $H({\bm{k}})$, classified by the degree-$0$
$K$-group ${}^{\phi}K^{(\tau,c)+0}_{G}(T^{d})$, to the boundary Hamiltonian
$\hat{H}({\bm{k}}_{\parallel})$. The bulk-to-boundary map in the absence of
antisymmetry

$f^{\rm
BBC}_{G_{0}}:{}^{\phi}K_{G_{0}}^{\tau+0}(T^{d})\to{}^{\phi}K^{\tau+1}_{G_{0}}(T^{d-1})$
(66)

is also defined, and forgetting antisymmetries defines the homomorphism

$f^{\rm
bulk}_{0}:{}^{\phi}K_{G}^{(\tau,c)+0}(T^{d})\to{}^{\phi}K_{G_{0}}^{\tau+0}(T^{d})$
(67)

for the bulk. Then, we obtain the commutative diagram in Eq. (4) in which the
right vertical line is exact. Given a bulk Hamiltonian
$[H({\bm{k}})]\in{}^{\phi}K_{G}^{(\tau,c)+0}(T^{d})$, the existence of
spectral flow at the boundary means

$f^{\rm BBC}_{G}([H({\bm{k}})])\notin{\rm im}\,f^{\rm bdy}_{L_{\rm i}}={\rm
ker}\,f^{\rm bdy}_{0}.$ (68)

The commutativity of the diagram in Eq. (4) implies

$f^{\rm BBC}_{G_{0}}\circ f^{\rm bulk}_{0}([H({\bm{k}})])\neq
0\in{}^{\phi}K_{G_{0}}^{\tau+1}(T^{d-1}).$ (69)

This further means that the bulk topological invariant without antisymmetry,
which detects the element of the $K$-group $K_{G}^{\tau+0}(T^{d})$ signaling
the boundary spectral flow, should provide the obstruction of Wannier
localizability.

##  VII Conclusion

In this work, we elucidate the relationship between Wannier localizability and
detachability of topological boundary states. From the boundary perspective,
we classify intrinsic and extrinsic non-Hermitian topology, underlying
nondetachable and detachable topological boundary states, respectively. From
the bulk perspective, by contrast, we classify Wannier localizability by the
homomorphisms of topological phases from the tenfold AZ symmetry classes to
the threefold WD symmetry classes. We clarify the agreement of the two
classification from the boundary and bulk perspectives on the basis of
$K$-theory. Our work, combined with our accompanying Letter [20], provides a
fundamental understanding of Wannier localizability and detachable boundary
states, as well as the bulk-boundary correspondence, in topological materials.

Note added.—After the completion of this work, we became aware of a recent
related work [49].

###### Acknowledgements.

We thank Akira Furusaki, Shingo Kobayashi, and Shinsei Ryu for helpful
discussion. We appreciate the long-term workshop “Recent Developments and
Challenges in Topological Phases” (YITP-T-24-03) held at Yukawa Institute for
Theoretical Physics (YITP), Kyoto University. K. Shiozaki, D.N., K. Shimomura,
and M.S. are supported by JST CREST Grant No. JPMJCR19T2. K. Shiozaki is
supported by JSPS KAKENHI Grant Nos. 22H05118 and 23H01097. K. Shimomura is
supported by JST SPRING, Grant No. JPMJSP2110. M.S. is supported by JSPS
KAKENHI Grant No. 24K00569. K.K. is supported by MEXT KAKENHI Grant-in-Aid for
Transformative Research Areas A “Extreme Universe” No. 24H00945.

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