# Bulk-boundary correspondence in point-gap topological phases

Daichi Nakamura<EMAIL_ADDRESS>Center for Gravitational
Physics and Quantum Information, Yukawa Institute for Theoretical Physics,
Kyoto University, Kyoto 606-8502, Japan  Takumi Bessho
<EMAIL_ADDRESS>Corporate Research and Development Center,
Toshiba Corporation, Kawasaki, Japan  Masatoshi Sato
<EMAIL_ADDRESS>Center for Gravitational Physics and Quantum
Information, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto
606-8502, Japan

###### Abstract

A striking feature of non-Hermitian systems is the presence of two different
types of topology. One generalizes Hermitian topological phases, and the other
is intrinsic to non-Hermitian systems, which are called line-gap topology and
point-gap topology, respectively. Whereas the bulk-boundary correspondence is
a fundamental principle in the former topology, its role in the latter has not
been clear yet. This paper establishes the bulk-boundary correspondence in the
point-gap topology in non-Hermitian systems. After revealing the requirement
for point-gap topology in the open boundary conditions, we clarify that the
bulk point-gap topology in open boundary conditions can be different from that
in periodic boundary conditions. On the basis of real space topological
invariants and the $K$-theory, we give a complete classification of the open
boundary point-gap topology with symmetry and show that the non-trivial open
boundary topology results in robust and exotic surface states.

Recently, non-Hermitian topological phases have attracted much attention
Rudner and Levitov (2009); Sato  _et al._ (2012); Esaki  _et al._ (2011); Hu
and Hughes (2011); Schomerus (2013); Malzard  _et al._ (2015); Lee (2016);
Leykam  _et al._ (2017); Xu  _et al._ (2017); Xiong (2018); Shen  _et al._
(2018); Kozii and Fu ; Takata and Notomi (2018); Martinez Alvarez  _et al._
(2018); Kawabata  _et al._ (2018a); Gong  _et al._ (2018); Kawabata  _et al._
(2019a); Yao and Wang (2018); Yao  _et al._ (2018); Kunst  _et al._ (2018);
Kawabata  _et al._ (2018b); McDonald  _et al._ (2018); Carlström and Bergholtz
(2018); Carlström  _et al._ (2019); Lee and Thomale (2019); Jin and Song
(2019); Budich  _et al._ (2019); Okugawa and Yokoyama (2019); Liu  _et al._
(2019); Yoshida  _et al._ (2019a); Zhou  _et al._ (2019); Lee  _et al._
(2019a); Kunst and Dwivedi (2019); Longhi (2019); Edvardsson  _et al._ (2019);
Kawabata  _et al._ (2019b); Zhou and Lee (2019); Herviou  _et al._ (2019a);
Zeng  _et al._ (2020); Hirsbrunner  _et al._ (2019); Zirnstein  _et al._
(2021); Pocock  _et al._ (2019); Kimura  _et al._ (2019); Borgnia  _et al._
(2020); Kawabata  _et al._ (2019c); Yokomizo and Murakami (2019); Song  _et
al._ (2019a); McClarty and Rau (2019); Okuma and Sato (2019); Song  _et al._
(2019b); Bergholtz and Budich (2019); Lee  _et al._ (2019b); Guo  _et al._
(2020); Yoshida  _et al._ (2019b); Rui  _et al._ (2019); Brzezicki and Hyart
(2019); Schomerus (2020); Imura and Takane (2019); Herviou  _et al._ (2019b);
Chang  _et al._ (2020); Zhang and Franz (2020); Zhang  _et al._ (2020); Yang
_et al._ (2020); Okuma  _et al._ (2020); Longhi (2020); Wang  _et al._ (2020);
Matsumoto  _et al._ (2020); Yoshida  _et al._ (2020a); Yokomizo and Murakami
(2020); Li  _et al._ (2020); Kawabata  _et al._ (2020a); Terrier and Kunst
(2020); Bessho and Sato (2021); Claes and Hughes (2021); Zirnstein and Rosenow
(2021); Denner  _et al._ (2021); Okugawa  _et al._ (2020); Kawabata  _et al._
(2020b); Fu  _et al._ (2021); Okuma and Sato (2021); Kawabata  _et al._
(2021); Yi and Yang (2020); Zhang  _et al._ (2022); Sun  _et al._ (2021);
Shiozaki and Ono (2021); Okugawa  _et al._ (2021); Vecsei  _et al._ (2021); Hu
_et al._ (2022); Ghosh and Nag (2022); Poli  _et al._ (2015); Zeuner  _et al._
(2015); Zhen  _et al._ (2015); Weimann  _et al._ (2017); Xiao  _et al._
(2017); St-Jean  _et al._ (2017); Parto  _et al._ (2018); Bahari  _et al._
(2017); Zhao  _et al._ (2018); Zhou  _et al._ (2018); Harari  _et al._ (2018);
Bandres  _et al._ (2018); Cerjan  _et al._ (2019); Zhao  _et al._ (2019);
Brandenbourger  _et al._ (2019); Ghatak  _et al._ (2020); Helbig  _et al._
(2020); Hofmann  _et al._ (2020); Xiao  _et al._ (2020); Weidemann  _et al._
(2020); Wang  _et al._ (2021a); Zhang  _et al._ (2021a, b); Zou  _et al._
(2021); Palacios  _et al._ (2021); Wang  _et al._ (2021b). Non-Hermitian
systems differ essentially from Hermitian ones: The complex-valued energy
spectra of non-Hermitian systems allow two types of the gap structure, i.e.,
line-gap and point-gap Kawabata  _et al._ (2019b). Whereas the line-gap is a
relatively straightforward generalization of a gap in Hermitian systems, the
point-gap is intrinsic in non-Hermitian systems. The multiple gap structures
enable corresponding topological phases of non-Hermitian systems, line-gap,
and point-gap topological phases Gong  _et al._ (2018); Kawabata  _et al._
(2019b). Both topological phases are indispensable to understanding non-
Hermitian topological phenomena.

A central character of topological phases is the bulk-boundary correspondence
(BBC): the bulk topology causes anomalous gapless boundary modes in the open
boundary conditions (OBCs). For example, the quantum Hall systems support
chiral edge modes from the nontrivial bulk Chern number Hatsugai (1993). The
exact quantization of the Hall conductance is a consequence of the
dissipationless current of the chiral edge modes.

Despite the importance, recent studies have shown that non-Hermiticity
obscures the BBC Xiong (2018); Yao and Wang (2018); Kunst  _et al._ (2018);
Lee and Thomale (2019); Helbig  _et al._ (2020); Weidemann  _et al._ (2020);
Borgnia  _et al._ (2020); Martinez Alvarez  _et al._ (2018); Shen  _et al._
(2018); Lee (2016); Yao  _et al._ (2018); Leykam  _et al._ (2017); Yokomizo
and Murakami (2019); Zirnstein  _et al._ (2021); Zirnstein and Rosenow (2021);
Jin and Song (2019); Herviou  _et al._ (2019a); Pocock  _et al._ (2019);
Brzezicki and Hyart (2019); Kawabata  _et al._ (2020a); Yang  _et al._ (2020).
A class of non-Hermitian systems shows completely different bulk spectra in
the OBCs than in the periodic boundary conditions (PBCs). Because of this
phenomenon-the non-Hermitian skin effect (NHSE) Yao and Wang (2018); Kunst
_et al._ (2018), the bulk energy gap in OBCs would be closed in PBCs.
Therefore, a bulk topological number in PBCs can be ill-defined even when a
gapless boundary mode exists in a gap in OBCs. Yao and Wang Yao and Wang
(2018) solved this problem in the case of line-gap topological phases. Using
the generalized Brillouin zone (GBZ) Yao and Wang (2018); Yao  _et al._
(2018); Yokomizo and Murakami (2019); Yang  _et al._ (2020); Kawabata  _et
al._ (2020a), they define a topological number in OBCs, and show that the new
bulk topological number correctly recovers the BBC. Furthermore, later, the
BBC in OBCs was also formulated in terms of real-space topological invariants,
enabling the application in the study of the BBC in higher dimensions and
higher-order line-gap topological phases Song  _et al._ (2019a); Tang  _et
al._ (2020); Liu  _et al._ (2020, 2021); Ghorashi  _et al._ (2021a, b).

Whereas the above prescription settled the BBC in line-gap topological phases,
the BBC in point-gap topological phases has not been clear yet. In one
dimension, point-gap topological numbers in PBCs result in NHSEs in OBCs, as a
result of the BBC Zhang  _et al._ (2020); Okuma  _et al._ (2020). However,
there remains uncertainty of the BBC in higher dimensions: It has been
suggested that topological surface states originate from the three-dimensional
winding number in PBCs Denner  _et al._ (2021); Vecsei  _et al._ (2021); Hu
_et al._ (2022), which is the point-gap topological number for general three-
dimensional systems. Nonetheless, these surface states can disappear without
changing the bulk topological number. Thus, the relation between the bulk
topology and the surface states is ambiguous.

In this paper, we establish the BBC in point-gapped topological phases.
Following the strategy learned from line-gap topological phases, our arguments
rely on topological numbers in OBCs. Remarkably, there appears an essentially
new feature intrinsic to point-gap topological phases. We find that a
particular class of non-Hermitian skin effects, which we dub in-gap skin
effects, ruins point-gap topological numbers in OBCs. As a result, the
topological classification in OBCs can be different from that in PBCs. Based
on this result, we resolve the uncertainty of the BBC in point-gap topological
phases, and show that non-trivial topological numbers in OBCs result in robust
and exotic surface states. Using the $K$-theory, we also give a complete
classification table for point-gap topological phases under OBCs in the
presence of symmetry.

Uncertainty of BBC in point-gap topological phases.– First, let us see the
fore-mentioned uncertainty of the BBC in point-gap topological phases. We
start with a model of exceptional topological insulators (ETIs) Denner  _et
al._ (2021),

$\displaystyle H_{\rm ETI}(\bm{k})$
$\displaystyle=\sin{k_{x}}\sigma_{x}+\sin{k_{z}}\sigma_{y}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{z}$
$\displaystyle-i\sin{k_{y}}\sigma_{0},$ (1)

where $\sigma_{i=x,y,z}$ are the Pauli matrices and $\sigma_{0}$ is the 2×2
identity matrix. As shown in Fig. 1(a), the ETI has a point gap at $E=0$ in
the complex energy plane, i.e., no complex spectrum crossing the reference
point $E=0$, under PBCs in all directions (full PBC). Therefore, we can define
the three-dimensional (3D) winding number over the 3D Brillouin zone (BZ) Gong
_et al._ (2018); Kawabata  _et al._ (2019b),

$\displaystyle w_{3}|_{\text{fullPBC}}$
$\displaystyle=-\dfrac{1}{24\pi^{2}}\int_{\text{
BZ}}\text{Tr}[(H-E)^{-1}d(H-E)]^{3},$ (2)

which takes $+1$ for $H=H_{\rm ETI}$ and $E=0$.

Interestingly, the ETI hosts surface states once we impose the OBC in the
$z$-direction (zOBC). See Fig. 1(b). Because the non-Hermitian term
$i\sin{k_{y}}\sigma_{0}$ in Eq. (1) only gives a complex shift of the energy
under the zOBC, the surface states of the ETI are equivalent to those of a
Weyl semimetal, $H_{\text{WSM}}(\bm{k})=H_{\rm
ETI}(\bm{k})+i\sin{k_{y}}\sigma_{0}=\sin{k_{x}}\sigma_{x}+\sin{k_{z}}\sigma_{y}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{z}$.
For a fixed $k_{y}$ with $-\pi/2<k_{y}<\pi/2$, the Weyl semimetal supports a
non-zero Chern number in the PBCs, and thus it has corresponding chiral edge
modes under the zOBC. By taking into account the complex energy shift from the
non-Hermitian term $i\sin{k_{y}}\sigma_{0}$, the chiral modes give surface
states filling the point-gapped region in Fig. 1(b).

Figure 1: The energy spectra of ETI in Eq. (1). (a) The full PBC spectrum
(gray). The point gap is open around $E=0$, with the nontrivial 3D winding
number $+1$. (b) The zOBC spectrum (blue) and the full PBC spectrum (gray) in
comparison. No NHSE occurs but surface states appear in the region where the
3D winding number takes the nontrivial value. The system size is
$L_{x}=L_{y}=100$ and $L_{z}=30$. (c) The yOBC spectrum (blue) and the full
PBC spectrum (gray) in comparison. NHSEs occur, and in-gap skin modes (orange)
appear. The system size in the $y$ direction is $L_{y}=30$ and the momentum
resolutions in both $k_{x,z}$-directions are taken as $\Delta
k_{x,z}=2\pi/100$.

These surface states have a topological number with the same value as the bulk
3D winding number: The effective Hamiltonian of the surface states around
$E=0$ takes the form of $h_{\rm surface}(k_{x},k_{y})=v_{x}k_{x}-ik_{y}$ as
the chiral edge mode of the Weyl semimetal has $v_{x}k_{x}$ with a real
positive constant $v_{x}>0$ and the non-Hermitian term of $H_{\rm ETI}$ gives
$-ik_{y}$. The 1D winding number $w_{1}$ on a circle $S_{1}$ around
$(k_{x},k_{y})=(0,0)$ on the surface BZ Denner  _et al._ (2021):

$\displaystyle w_{1}$
$\displaystyle=-\oint_{S^{1}}\dfrac{\text{d}{\bm{k}}}{2\pi i}\cdot h^{-1}_{\rm
surface}(k_{x},k_{y}){\bm{\nabla}}_{k}h_{\rm surface}(k_{x},k_{y})=1$ (3)

measures the topology of the surface states, of which value coincides with the
3D winding number.

The coincidence of the topological numbers suggests the BBC in point-gap
topological phases Denner  _et al._ (2021). However, there is an uncertainty
in this interpretation. In Fig. 1(c), we show the spectrum of the same model
under the OBC in the $y$-direction (yOBC). Whereas the bulk topological number
in Eq.(2) remains the same, no surface state covering the point-gapped region
appears. Instead, we have skin modes in the gap. Since the skin modes are
localized bulk modes 111skin modes are obtained using GBZ and thus bulk modes
Yao and Wang (2018), while surface states are not obtained using GBZ and thus
not bulk modes, and do not have 1D winding number Okuma  _et al._ (2020), the
simple BBC does not hold under the yOBC.

Point-gap topological number under OBCs and BBC. – The solution of the problem
is to use topological numbers in OBCs. Let us consider a 3D Hamiltonian
$\mathcal{H}(k_{x},k_{y})$ with momentum-space representation in the $x$\- and
$y$-directions and real-space representation in the $z$-direction under the
zOBC. Then, we construct the bulk Hamiltonian
$\mathcal{H}_{\text{bulk}}(k_{x},k_{y})$ by the projection of
$\mathcal{H}(k_{x},k_{y})$ onto the bulk Song  _et al._ (2019a). When the bulk
Hamiltonian has a point gap at $E$
($\text{det}[\mathcal{H}_{\text{bulk}}-E]\neq 0$), we can define the real-
space 3D winding number $w_{3}$ under the zOBC,

$\displaystyle w_{3}|_{\rm
zOBC}=-\dfrac{i}{12\pi}\int_{\text{BZ}}d^{2}\bm{k}\hskip
1.42262pt\mathcal{T}_{z}[\varepsilon^{ijk}Q_{i}Q_{j}Q_{k}],$ (4)

with
$Q_{i=x,y}=i(\mathcal{H}_{\text{bulk}}-E)^{-1}\partial_{k_{i}}(\mathcal{H}_{\text{bulk}}-E)$
and $Q_{z}=(\mathcal{H}_{\text{bulk}}-E)^{-1}[Z,\mathcal{H}_{\text{bulk}}-E]$,
where $Z$ is the position operator in the $z$-coordinate, and
$\mathcal{T}_{z}$ stands for the trace per unit length in the $z$-direction.
This is a non-Hermitian generalization of the real-space topological number in
Hermitian systems Mondragon-Shem  _et al._ (2014); Song and Prodan (2014);
Prodan and Schulz-Baldes (2016); Katsura and Koma (2018). For the full PBCs,
this quantity reproduces Eq. (2) with the identification

$\displaystyle\int\dfrac{dk_{z}}{2\pi}\text{Tr}[A(k_{z})]\leftrightarrow\mathcal{T}_{z}[\mathcal{A}],\quad
i\partial_{k_{z}}\leftrightarrow[Z,\cdot],$ (5)

where $A(k_{z})$ is a function of $k_{z}$ and $\mathcal{A}$ is the real-space
representation of $A(k_{z})$ Bellissard  _et al._ (1994); Kitaev (2006). Thus,
for the ETI in Eq. (1), the coincidence between $w_{3}|_{\rm fullPBC}$ with
$E=0$ in Eq. (2) and $w_{1}$ in Eq. (3) results in the correspondence between
$w_{3}|_{\rm zOBC}$ with $E=0$ in Eq. (4) and $w_{1}$ in Eq. (3). Since any
nontrivial 3D winding number under zOBC can be produced by stacking the ETIs
in Eq. (1) up to continuous deformations, we generally have the BBC

$\displaystyle w_{3}|_{\text{zOBC}}=\sum_{\bm{k}_{p}}w_{1}(\bm{k}_{p}),$ (6)

with

$\displaystyle w_{1}(\bm{k}_{p})=-\oint_{S^{1}_{p}}\dfrac{d\bm{k}}{2\pi
i}\cdot\text{Tr}[(h_{\text{surface}}-E)^{-1}\nabla(h_{\text{surface}}-E)],$
(7)

where $h_{\text{surface}}(k_{x},k_{y})$ is the surface effective Hamiltonian,
$\bm{k}_{p}$ is the Fermi point satisfying
$\text{det}[h_{\text{surface}}({\bm{k}}_{p})-E]=0$, and $S^{1}_{p}$ stands for
the counter clockwise circle around $\bm{k}=\bm{k}_{p}$ on the surface
Brillouin zone. As we shall show later, the bulk topological number under the
zOBC can be different from that under the yOBC. Therefore, the above BBC does
not require surface states under the yOBC.

The necessity of the OBC bulk topological number becomes obvious once we
consider another model. Figure 2(a) is the bulk spectra of the following model
under different boundary conditions:

$\displaystyle H(\bm{k})$
$\displaystyle=\sin{k_{x}}\sigma_{x}+2\sin{k_{z}}\sigma_{y}+2\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{z}$
$\displaystyle+\dfrac{3}{2}i(\sin{k_{y}}+\sin{k_{z}})\sigma_{0}.$ (8)

Because of NHSEs in the bulk spectrum, the OBC spectrum has a wider point-
gapped region than the PBC spectrum. Therefore, there is a region where only
the topological number under the zOBC is well-defined, as shown in Fig. 2. We
find that the BBC holds for the topological number under the zOBC, not for
that under the PBC.

Figure 2:  The energy spectra (top) and the 3D winding numbers (bottom) of the
model in Eq. (8) under different boundary conditions. The magenta lines in the
top figures represent $E=-0.5+i{\rm Im}E$ with $-4<{\rm Im}E<4$. (a)(top) The
bulk spectra under the full PBC (gray) and the zOBC (turquoise). The zOBC
spectrum is calculated by using the non-Bloch theory in Ref.Yokomizo and
Murakami (2019). The point-gapless region under the zOBC is wider than that
under the full PBCs. (bottom) The 3D winding number under the full PBC along
the magenta line in the top figure. The gray shadings represent point-gapless
regions. (b)(top) The energy spectrum under the zOBC. The blue modes are
surface states which are calculated with the system size
$L_{x}=L_{y}=100,L_{z}=20$. Surface states appear even in the regions where
$w_{3}|_{\text{fullPBC}}$ is ill-defined (inside the red circle). (bottom) The
real-space 3D winding number under the zOBC along the magenta line in the top
figure. The turquoise segments represent point-gapless regions.

In-gap skin effects and absence of surface states.– Now we show that the BBC
in Eq. (6) also explains the absence of surface states in Fig. 1(c) under the
yOBC. A key observation is the presence of in-gap skin modes. In a manner
similar to Eq. (4), we can introduce the 3D winding number under the yOBC, but
the in-gap bulk skin modes make the topological number ill-defined.
Consequently, the BBC under the yOBC does not require any surface states.

Let us see how this happens in detail. First, we note that the in-gap skin
modes originate from modes with $(k_{x},k_{z})=(0,0)$, where the Hamiltonian
in Eq. (1) becomes

$\displaystyle H_{\rm
ETI}(k_{x}=0,k_{y},k_{z}=0)=-\cos{k_{y}}\sigma_{z}-i\sin{k_{y}}\sigma_{0}.$
(9)

This 1D Hamiltonian gives the complex spectra $\mp e^{\pm ik_{y}}$ in the
eigensector of $\sigma_{z}=\pm 1$, which have the energy winding numbers $\pm
1$ along the $k_{y}$ direction, as illustrated in Fig. 3(a). Thus, from the
general theory of NHSE Okuma  _et al._ (2020), we have the skin modes inside
the point gap when one imposes the yOBC 222Note that the isotropic structure
of $\mp e^{\pm ik_{y}}$ in the complex energy plane causes highly degenerated
in-gap skin modes at $E=0$..

At first glance, the in-gap skin modes appear to be isolated from the other
bulk modes, but this is not the case. For a finite $L_{y}$, the Hamiltonian
under the yOBC is a continuous function with respect to $k_{x}$ and $k_{z}$,
so is its eigenvalues. Therefore, there must be ordinary bulk modes nearby the
skin modes. Figure 3(c) shows the energy spectrum of Eq. (1) under the yOBC,
with a high momentum resolution. The bulk modes around the in-gap skin modes
are now evident. Importantly, the point-gapped region disappears due to these
bulk modes. Therefore, we do not have a well-defined 3D winding number and
surface states under the yOBC.

The disappearance of surface modes can be regard as a result of a topological
phase transition under continuous change of the boundary conditions.
Decreasing the hopping terms between the $y=1$ sites and $y=L_{y}$ sites, we
can smoothly change the boundary condition from the full PBC to the yOBC.
According to the deformation, the modes at $(k_{x},k_{z})=(0,0)$ shrink to the
in-gap skin modes as shown in Fig. 3(b). Finally, the originally point-gapped
region is fully covered by bulk modes under the yOBC. We also show the change
of the 3D winding number throughout the topological phase transition for the
model in Eq. (8) in the Supplemental Material 333See Supplemental Material,
which includes Refs. Prodan  _et al._ (2010); Katsura and Koma (2016); Akagi
_et al._ (2017); Yang  _et al._ (2021), for a demonstration of how the real-
space 3D winding number changes with different boundary conditions, a proof of
the BBC for point-gap topological phases in all 38-fold symmetry classes, and
a connection between intrinsic point-gap topological phases in the OBC and a
single exceptional point..

Figure 3: The changes of the spectra of ETI in Eq. (1) from under the full PBC
to the yOBC. $L_{y}$ is the same as that in Fig. 1(c) but the momentum
resolution around $(k_{x},k_{z})=(0,0)$ is much finer. The orange modes
represent the modes with $k_{x}=k_{z}=0$. (a) The full PBC spectrum. A point
gap is open in the region containing $E=0$ with the nontrivial 3D winding
number $+1$. The modes at $k_{x}=k_{z}=0$ form a loop and have a nontrivial 1D
winding number in each eigensector of $\sigma_{z}=\pm 1$. (b) The spectrum
under a boundary condition between the full PBC and the yOBC. Here the hopping
amplitude between the $y=1$ sites and the $y=L_{y}$ sites is $10^{-6}$. The
point-gapped region with the non-zero 3D winding number shrinks. (c) The yOBC
spectrum. The region including $E=0$ is completely closed by the in-gap skin
effect of the modes at $k_{x}=k_{z}=0$. Comparing with Fig. 1(c), we can see
that the point-gapped region is completely collapsed by the modes near
$k_{x}=k_{z}=0$.

BBC of point-gap topology with symmetry.– The above arguments can also apply
to point-gapped systems under symmetries. Namely, when a symmetry-protected
point-gap topological number under the OBC is non-zero, then the corresponding
boundary states appear.

It should be noted here that possible point-gap topological numbers under OBCs
can be different from those under PBCs. The disagreement stems from the
property that some point-gap topological numbers in PBCs are always
accompanied by in-gap skin modes, which spoil the corresponding topological
numbers in OBCs.

Whereas we have considered the full general symmetries in the Supplemental
Material, we focus here on a particular class of symmetries, which we call AZ†
symmetry. The AZ† symmetries are a non-Hermitian generalization of the
Altland-Zirnbauer (AZ) symmetry Kawabata  _et al._ (2019b): It consists of
non-Hermitian versions of time-reversal symmetry (TRS†), ${\cal
C}H^{T}({\bm{k}}){\cal C}^{-1}=H(-{\bm{k}})$, particle-hole symmetry (PHS†),
${\cal T}H^{*}({\bm{k}}){\cal T}^{-1}=-H(-{\bm{k}})$, and chiral symmetry (CS)
$\Gamma H^{\dagger}({\bm{k}})\Gamma^{-1}=-H({\bm{k}})$, where ${\cal C}$,
${\cal T}$, and $\Gamma$ are unitary operators. The AZ† symmetry naturally
arises in the non-Hermitian Hamiltonian of the retarded Green function, and
thus it governs non-Hermitian topological phases in materials Yoshida  _et
al._ (2020b). The presence and/or absence of these symmetries define ten
symmetry classes, and the topological classification in these classes under
the full PBC has been known Kawabata  _et al._ (2019b).

In Table 1, we show how the point-gap topological classification under the
PBCs changes under OBCs: In one dimension, all the point-gap topological
numbers in the AZ† classes become trivial under the OBC, because their non-
trivial values in the PBC always result in in-gap skin modes under the OBC.
Actually, all the $\mathbb{Z}$ indices in 1D AZ† classes reduce to the 1D
winding number Kawabata  _et al._ (2019b), which gives in-gap skin modes under
the OBC Zhang  _et al._ (2020); Okuma  _et al._ (2020). Furthermore, the
$\mathbb{Z}_{2}$ topological number in 1D classes AII† and DIII† under the PBC
causes symmetry-protected skin modes inside the point gap under the OBC Okuma
_et al._ (2020). Therefore, no 1D point-gap topological number survives under
the OBC.

The reduction of topological numbers in Table 1 in two and three dimensions
occurs as a result of the dimensional reduction Qi  _et al._ (2008). First, we
focus on class AII†. From a Hamiltonian $H({\bm{k}})$ with a point gap at $E$
in class AII†, we can obtain a topologically equivalent gapped Hermitian
Hamiltonian $\tilde{H}({\bm{k}})$:

$\displaystyle\tilde{H}({\bm{k}})=\begin{pmatrix}0&H({\bm{k}})-E\\\
H^{\dagger}({\bm{k}})-E^{*}&0\end{pmatrix},$ (10)

which belongs to class DIII as it has an additional CS
$\Sigma\tilde{H}({\bm{k}})\Sigma^{-1}=-\tilde{H}({\bm{k}})$
($\Sigma=\sigma_{z}\otimes 1$) together with TRS $\tilde{\cal
C}\tilde{H}^{*}({\bm{k}})\tilde{\cal C}^{-1}=\tilde{H}(-{\bm{k}})$
$(\tilde{C}=\sigma_{x}\otimes{\cal C})$ Kawabata  _et al._ (2019b). Using the
dimensional reduction in class DIII Qi  _et al._ (2008), one can show that the
parity of the 3D $\mathbb{Z}$ index for $H(k_{x},k_{y},k_{z})$ equals to the
product of the 2D $\mathbb{Z}_{2}$ indices of
$H(k_{x},k_{y},k_{z}^{\text{0}})$ with $k_{z}^{0}=0$ and $\pi$, and similarly,
the 2D $\mathbb{Z}_{2}$ index of $H(k_{x},k_{y},k_{z}^{\text{0}})$ equals to
the product of the 1D $\mathbb{Z}_{2}$ indices of
$H(k_{x},k_{y}^{0},k_{z}^{\text{0}})$ with $k_{y}^{0}=0$ and $\pi$. Thus, for
class AII† under full PBCs, a non-trivial 2D $\mathbb{Z}_{2}$ index or an odd
parity of the 3D $\mathbb{Z}$ index at $E$ yield a non-zero 1D
$\mathbb{Z}_{2}$ index at $E$ along a high symmetric line in the BZ.
Therefore, they always accompany symmetry-protected in-gap skin modes Okuma
_et al._ (2020), trivializing the corresponding topological numbers in OBCs.
As a result, only the even part of the $\mathbb{Z}$ index in three dimensions
survives in the OBC. We can also show the reduction $\mathbb{Z}\to
2\mathbb{Z}$ in 2D class DIII†, using a similar dimensional reduction. On the
basis of the $K$-theory, we prove the BBC for point-gap topological phases
under the OBC in all 38-fold symmetry classes in non-Hermitian systems,
including AZ† ones, in the Supplemental Material Note (3).

Table 1: Classification of point-gap topological phases. For topological
numbers with arrows, the left specifies topological numbers under PBCs and the
right specifies those under OBCs. For topological numbers without arrows, the
classification under OBCs coincides with that under PBCs. We consider the
$\mathrm{AZ}^{{\dagger}}$ symmetry classes with the spatial dimension $d=1,2$
and $3$.

Symmetry class TRS† PHS† CS $d=1$ $d=2$ $d=3$ A $0$ $0$ $0$ $\mathbb{Z}\to 0$
$0$ $\mathbb{Z}$ AIII $0$ $0$ $1$ $0$ $\mathbb{Z}$ $0$ $\text{AI}^{{\dagger}}$
$+1$ $0$ $0$ $0$ $0$ $2\mathbb{Z}$ $\text{BDI}^{{\dagger}}$ $+1$ $+1$ $1$ $0$
$0$ $0$ $\text{D}^{{\dagger}}$ $0$ $+1$ $0$ $\mathbb{Z}\to 0$ $0$ $0$
$\text{DIII}^{{\dagger}}$ $-1$ $+1$ $1$ $\mathbb{Z}_{2}\to 0$ $\mathbb{Z}\to
2\mathbb{Z}$ $0$ $\text{AII}^{{\dagger}}$ $-1$ $0$ $0$ $\mathbb{Z}_{2}\to 0$
$\mathbb{Z}_{2}\to 0$ $\mathbb{Z}\to 2\mathbb{Z}$ $\text{CII}^{{\dagger}}$
$-1$ $-1$ $1$ $0$ $\mathbb{Z}_{2}$ $\mathbb{Z}_{2}$ $\text{C}^{{\dagger}}$ $0$
$-1$ $0$ $2\mathbb{Z}\to 0$ $0$ $\mathbb{Z}_{2}$ $\text{CI}^{{\dagger}}$ $+1$
$-1$ $1$ $0$ $2\mathbb{Z}$ $0$

Remarkably, we can predict novel topological phase transitions intrinsic to
non-Hermitian systems, using the reduction of the point-gap topological
numbers in the presence of symmetry: The symmetry-protected in-gap skin modes
in 3D class AII† systems may disappear suddenly once one breaks TRS† by an
infinitesimal perturbation. The disappearance of the in-gap skin modes allows
the well-defined 3D winding number under the OBC, and thus we have an abrupt
transmutation from the topologically trivial state with the in-gap skin modes
to a topologically non-trivial one with surface states. Infinitesimal
instability Okuma and Sato (2019) intrinsic to non-Hermitian systems induces
this topological phase transition, and thus it never happens in Hermitian
systems. We also find that boundary states in intrinsic point-gap topological
phases may support a single exceptional point Note (3), which is also unique
to non-Hermitian systems.

Conclusions.– In this paper, we establish the BBC for the point-gap topology
in non-Hermitian systems. We demonstrate that real-space topological numbers
under OBCs result in robust and exotic surface states. We find that the bulk
point-gap topology in OBCs can be different from that in PBCs, and give a
complete classification of the OBC point-gap topology in the presence of
symmetry. Our finding reveals a novel universal property of non-Hermitian
topological phases of matters.

We are grateful to Nobuyuki Okuma, Ken Shiozaki, Shuhei Ohyama, Yusuke Nakai,
and Hiroto Oka for valuable discussions. This work was supported by JST CREST
Grant No.JPMJCR19T2, the establishment of university fellowships towards the
creation of science technology innovation, Grant No.JPMJFS2123, and KAKENHI
Grant No.JP20H00131. This work was done while Takumi Bessho was at Yukawa
Institute for Theoretical Physics, Kyoto University.

Appendix on the BBC for point-gap topological phases in general symmetry
classes.– Our argument is applicable to any symmetry classes in non-Hermitian
systems. We summarize here our results for point-gap topological phases in
general symmetry classes in non-Hermitian systems. For the details of the
derivation, see Supplemental Material Note (3).

In addition to AZ† symmetries discussed in the main text, non-Hermitian
systems allow the original AZ symmetries defined by the following equations:

$\displaystyle{\cal T}H^{*}({\bm{k}}){\cal T}^{-1}$
$\displaystyle=H(-{\bm{k}}),\quad{\cal T}{\cal T}^{*}=\pm 1,$
$\displaystyle{\cal C}H^{T}({\bm{k}}){\cal C}^{-1}$
$\displaystyle=-H(-{\bm{k}}),\quad{\cal C}{\cal C}^{*}=\pm 1,$ (11)

where ${\cal T}$ and ${\cal C}$ are unitary operators corresponding to time-
reversal symmetry (TRS) and particle-hole symmetry (PHS), respectively.
Furthermore, as an additional general symmetry, one can also introduce
sublattice symmetry (SLS),

$\displaystyle{\cal S}H(\bm{k}){\cal S}^{-1}=-H(\bm{k}),\quad{\cal S}^{2}=1,$
(12)

or pseudo-Hermiticity (pH)

$\displaystyle\eta H^{\dagger}({\bm{k}})\eta^{-1}=H({\bm{k}}),\quad\eta^{2}=1$
(13)

with unitary operators ${\cal S}$ and $\eta$. The presence and/or absence of
these symmetries define symmetry classes intrinsic to non-Hermitian systems
Kawabata  _et al._ (2019b).

Tables S9 and S10 summarize our results on point-gap topological phases in
general symmetry classes. Here, the presence or absence of AZ or AZ†
symmetries define AZ or AZ† symmetry classes in Table S2. Moreover, each AZ or
AZ† class can host SLS (pH) additionally, where the subindex $+(-)$ of ${\cal
S}/\eta$ in Tables S9 and S10 specifies the commutation (anti-commutation)
relation between SLS/pH and AZ or AZ† symmetries. For an AZ (AZ†) class having
both TRS (TRS†) and PHS (PHS†), ${\cal S}$ or $\eta$ has a double subindex,
where the first index specifies the commutation or anticommutation relation
between SLS and TRS (TRS†), and the second one specifies those between SLS and
PHS (PHS†), respectively. Tables S9 and S10 show how the point-gap topological
classification under the PBCs changes under OBCs. For the topological numbers
with arrows, the classification under the PBCs shown on the left changes to
that under OBCs on the right, while the topological numbers without arrows
remain the same under both boundary conditions. We also find that the BBC
holds for the point-gap topological classification under OBCs: Topologically
protected boundary states appear when the bulk point-gap topological numbers
under OBCs are non-trivial.

Table 2: AZ and AZ† symmetry classes for non-Hermitian Hamiltonians. Here, $``0"$ denotes the absence of symmetries, while $``\pm 1"$ indicates the presence of each symmetry, dependent on whether its operator squares to $\pm 1$. Sym. class |  | TRS | PHS | CS | TRS† | PHS†
---|---|---|---|---|---|---
Complex AZ | A | $0$ | $0$ | $0$ | $0$ | $0$
| AIII | $0$ | $0$ | $1$ | $0$ | $0$
Real AZ | AI | $+1$ | $0$ | $0$ | $0$ | $0$
| BDI | $+1$ | $+1$ | $1$ | 0 | 0
| D | $0$ | $+1$ | $0$ | 0 | 0
| DIII | $-1$ | $+1$ | $1$ | $0$ | 0
| AII | $-1$ | $0$ | $0$ | 0 | $0$
| CII | $-1$ | $-1$ | $1$ | $0$ | 0
| C | $0$ | $-1$ | $0$ | $0$ | $0$
| CI | $+1$ | $-1$ | $1$ | $0$ | $0$
Real AZ† | $\text{AI}^{\dagger}$ | $0$ | $0$ | $0$ | $+1$ | $0$
| $\text{BDI}^{\dagger}$ | $0$ | $0$ | $1$ | $+1$ | $+1$
| $\text{D}^{\dagger}$ | $0$ | $0$ | $0$ | 0 | $+1$
| $\text{DIII}^{\dagger}$ | $0$ | $0$ | $1$ | $-1$ | $+1$
| $\text{AII}^{\dagger}$ | $0$ | $0$ | $0$ | $-1$ | $0$
| $\text{CII}^{\dagger}$ | $0$ | $0$ | $1$ | $-1$ | $-1$
| $\text{C}^{\dagger}$ | $0$ | $0$ | $0$ | $0$ | $-1$
| $\text{CI}^{\dagger}$ | $0$ | $0$ | $1$ | $+1$ | $-1$

AZ class Add. sym. $d=1$ $d=2$ $d=3$ A - $\mathbb{Z}\to 0$ $0$ $\mathbb{Z}$
AIII - $0$ $\mathbb{Z}$ $0$ A $\mathcal{S}$ ${\begin{aligned}
&\mathbb{Z}\oplus\mathbb{Z}\\\ \to&\mathbb{Z}[1,-1]\end{aligned}}$ $0$
$\mathbb{Z}\oplus\mathbb{Z}$ AIII $\mathcal{S}_{-},\eta_{-}$ $0$
$\mathbb{Z}\oplus\mathbb{Z}$ $0$ A $\eta$ $0$ $\mathbb{Z}$ $0$ AIII
$\mathcal{S}_{+},\eta_{+}$ $\mathbb{Z}$ $0$ $\mathbb{Z}$ AI - $\mathbb{Z}\to
0$ $0$ $0$ BDI - $\mathbb{Z}_{2}$ $\mathbb{Z}$ $0$ D - $\mathbb{Z}_{2}$
$\mathbb{Z}_{2}$ $\mathbb{Z}$ DIII - $0$ $\mathbb{Z}_{2}$ $\mathbb{Z}_{2}$ AII
- $2\mathbb{Z}\to 0$ $0$ $\mathbb{Z}_{2}$ CII - $0$ 2$\mathbb{Z}$ $0$ C - $0$
$0$ 2$\mathbb{Z}$ CI - $0$ $0$ $0$ AI $\mathcal{S}_{+}$ ${\begin{aligned}
&\mathbb{Z}\oplus\mathbb{Z}\\\ \to&\mathbb{Z}[1,-1]\end{aligned}}$ $0$ $0$ BDI
$\mathcal{S}_{+-},\eta_{-+}$
${\begin{gathered}\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\\\
\to\mathbb{Z}_{2}[1,1]\hskip 14.22636pt\end{gathered}}$
${\begin{gathered}\mathbb{Z}\oplus\mathbb{Z}\\\
\to\mathbb{Z}[2,0]\oplus\mathbb{Z}[1,-1]\hskip 4.55244pt\end{gathered}}$ $0$ D
$\mathcal{S}_{-}$ ${\begin{gathered}\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\\\
\to\mathbb{Z}_{2}[1,1]\hskip 14.22636pt\end{gathered}}$
${\begin{gathered}\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\\\
\to\mathbb{Z}_{2}[1,1]\hskip 14.22636pt\end{gathered}}$
${\begin{gathered}\mathbb{Z}\oplus\mathbb{Z}\\\
\to\mathbb{Z}[2,0]\oplus\mathbb{Z}[1,-1]\hskip 4.55244pt\end{gathered}}$ DIII
$\mathcal{S}_{+-},\eta_{+-}$ $0$ $\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$
$\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ AII $\mathcal{S}_{+}$
${\begin{gathered}2\mathbb{Z}\oplus 2\mathbb{Z}\\\ \to 2\mathbb{Z}[1,-1]\hskip
8.53581pt\end{gathered}}$ $0$ $\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ CII
$\mathcal{S}_{+-},\eta_{-+}$ $0$ $2\mathbb{Z}\oplus 2\mathbb{Z}$ $0$ C
$\mathcal{S}_{-}$ $0$ $0$ $2\mathbb{Z}\oplus 2\mathbb{Z}$ CI
$\mathcal{S}_{+-},\eta_{+-}$ $0$ $0$ $0$ AI $\mathcal{S}_{-}$ $\mathbb{Z}\to
0$ $0$ $\mathbb{Z}$ BDI $\mathcal{S}_{-+},\eta_{+-}$ $0$ $\mathbb{Z}$ $0$ D
$\mathcal{S}_{+}$ $\mathbb{Z}$ $0$ $\mathbb{Z}$ DIII
$\mathcal{S}_{-+},\eta_{-+}$ $0$ $\mathbb{Z}$ $0$ AII $\mathcal{S}_{-}$
$\mathbb{Z}\to 0$ $0$ $\mathbb{Z}$ CII $\mathcal{S}_{-+},\eta_{+-}$ $0$
$\mathbb{Z}$ $0$ C $\mathcal{S}_{+}$ $\mathbb{Z}\to 2\mathbb{Z}$ $0$
$\mathbb{Z}$ CI $\mathcal{S}_{-+},\eta_{-+}$ $0$ $\mathbb{Z}$ $0$ AI
$\eta_{+}$ $0$ $0$ $0$ BDI $\mathcal{S}_{++},\eta_{++}$ $\mathbb{Z}$ $0$ $0$ D
$\eta_{+}$ $\mathbb{Z}_{2}$ $\mathbb{Z}$ $0$ DIII $\mathcal{S}_{--},\eta_{++}$
$\mathbb{Z}_{2}$ $\mathbb{Z}_{2}$ $\mathbb{Z}$ AII $\eta_{+}$ $0$
$\mathbb{Z}_{2}$ $\mathbb{Z}_{2}$ CII $\mathcal{S}_{++},\eta_{++}$
2$\mathbb{Z}$ $0$ $\mathbb{Z}_{2}$ C $\eta_{+}$ $0$ 2$\mathbb{Z}$ $0$ CI
$\mathcal{S}_{--},\eta_{++}$ $0$ $0$ 2$\mathbb{Z}$ AI $\eta_{-}$
$\mathbb{Z}_{2}\to 0$ $\mathbb{Z}\to 2\mathbb{Z}$ $0$ BDI
$\mathcal{S}_{--},\eta_{--}$ $\mathbb{Z}_{2}$ $\mathbb{Z}_{2}$ $\mathbb{Z}$ D
$\eta_{-}$ $0$ $\mathbb{Z}_{2}$ $\mathbb{Z}_{2}$ DIII
$\mathcal{S}_{++},\eta_{--}$ 2$\mathbb{Z}$ $0$ $\mathbb{Z}_{2}$ AII $\eta_{-}$
$0$ 2$\mathbb{Z}$ $0$ CII $\mathcal{S}_{--},\eta_{--}$ $0$ $0$ 2$\mathbb{Z}$ C
$\eta_{-}$ $0$ $0$ $0$ CI $\mathcal{S}_{++},\eta_{--}$ $\mathbb{Z}\to
2\mathbb{Z}$ $0$ $0$

Table 3: Classification of point-gap topological phases in the AZ classes
without or with SLS or pH. The subscript of ${\cal S}_{\pm}/\eta_{\pm}$
specifies the commutation (+) or anti-commutation (-) relation to TRS or PHS.
For ${\cal S}_{\pm\pm}/\eta_{\pm\pm}$, the first subscript specifies the
relation to TRS and the second specifies the relation to PHS. For the
topological numbers with arrows, the classification under OBCs changes from
that under PBCs, where the left specifies the classification under PBCs and
the right specifies that under OBCs. The topological number $\mathbb{Z}[i,j]$
($\mathbb{Z}_{2}[i,j]$) under OBCs indicates the Abelian group $\mathbb{Z}$
($\mathbb{Z}_{2}$) generated by the element
$(i,j)\in\mathbb{Z}\oplus\mathbb{Z}$
[$(i,j)\in\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$] under PBCs. For the topological
numbers without arrows, the classification under OBCs coincides with that
under PBCs.

$\text{AZ}^{\dagger}$ class Add. sym. $d=1$ $d=2$ $d=3$
$\text{AI}^{{\dagger}}$ - $0$ $0$ 2$\mathbb{Z}$ $\text{BDI}^{{\dagger}}$ - $0$
$0$ $0$ $\text{D}^{{\dagger}}$ - $\mathbb{Z}\to 0$ $0$ $0$
$\text{DIII}^{{\dagger}}$ - $\mathbb{Z}_{2}\to 0$ $\mathbb{Z}\to 2\mathbb{Z}$
$0$ $\text{AII}^{{\dagger}}$ - $\mathbb{Z}_{2}\to 0$ $\mathbb{Z}_{2}\to 0$
$\mathbb{Z}\to 2\mathbb{Z}$ $\text{CII}^{{\dagger}}$ - $0$ $\mathbb{Z}_{2}$
$\mathbb{Z}_{2}$ $\text{C}^{{\dagger}}$ - $2\mathbb{Z}\to 0$ $0$
$\mathbb{Z}_{2}$ $\text{CI}^{{\dagger}}$ - $0$ $2\mathbb{Z}$ $0$
$\text{AI}^{{\dagger}}$ $\mathcal{S}_{+}$ $\mathbb{Z}$ $0$ $\mathbb{Z}$
$\text{BDI}^{{\dagger}}$ $\mathcal{S}_{+-},\eta_{-+}$ $0$ $\mathbb{Z}$ $0$
$\text{D}^{{\dagger}}$ $\mathcal{S}_{-}$ $\mathbb{Z}\to 0$ $0$ $\mathbb{Z}$
$\text{DIII}^{{\dagger}}$ $\mathcal{S}_{+-},\eta_{+-}$ $0$ $\mathbb{Z}$ $0$
$\text{AII}^{{\dagger}}$ $\mathcal{S}_{+}$ $\mathbb{Z}\to 2\mathbb{Z}$ $0$
$\mathbb{Z}$ $\text{CII}^{{\dagger}}$ $\mathcal{S}_{+-},\eta_{-+}$ $0$
$\mathbb{Z}$ $0$ $\text{C}^{{\dagger}}$ $\mathcal{S}_{-}$ $\mathbb{Z}\to 0$
$0$ $\mathbb{Z}$ $\text{CI}^{{\dagger}}$ $\mathcal{S}_{+-},\eta_{+-}$ $0$
$\mathbb{Z}$ $0$ $\text{AI}^{{\dagger}}$ $\mathcal{S}_{-}$ $0$ $0$
$2\mathbb{Z}\oplus 2\mathbb{Z}$ $\text{BDI}^{{\dagger}}$
$\mathcal{S}_{-+},\eta_{+-}$ $0$ $0$ $0$ $\text{D}^{{\dagger}}$
$\mathcal{S}_{+}$ ${\begin{aligned} &\mathbb{Z}\oplus\mathbb{Z}\\\
\to&\mathbb{Z}[1,-1]\end{aligned}}$ $0$ $0$ $\text{DIII}^{{\dagger}}$
$\mathcal{S}_{-+},\eta_{-+}$
${\begin{gathered}\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\\\
\to\mathbb{Z}_{2}[1,1]\hskip 14.22636pt\end{gathered}}$
${\begin{gathered}\mathbb{Z}\oplus\mathbb{Z}\\\
\to\mathbb{Z}[2,0]\oplus\mathbb{Z}[1,-1]\hskip 4.55244pt\end{gathered}}$ $0$
$\text{AII}^{{\dagger}}$ $\mathcal{S}_{-}$
${\begin{gathered}\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\\\
\to\mathbb{Z}_{2}[1,1]\hskip 14.22636pt\end{gathered}}$
${\begin{gathered}\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\\\
\to\mathbb{Z}_{2}[1,1]\hskip 14.22636pt\end{gathered}}$
${\begin{gathered}\mathbb{Z}\oplus\mathbb{Z}\\\
\to\mathbb{Z}[2,0]\oplus\mathbb{Z}[1,-1]\hskip 4.55244pt\end{gathered}}$
$\text{CII}^{{\dagger}}$ $\mathcal{S}_{-+},\eta_{+-}$ $0$
$\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ $\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$
$\text{C}^{{\dagger}}$ $\mathcal{S}_{+}$ ${\begin{gathered}2\mathbb{Z}\oplus
2\mathbb{Z}\\\ \to 2\mathbb{Z}[1,-1]\hskip 8.53581pt\end{gathered}}$ $0$
$\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ $\text{CI}^{{\dagger}}$
$\mathcal{S}_{-+},\eta_{-+}$ $0$ $2\mathbb{Z}\oplus 2\mathbb{Z}$ $0$
$\text{AI}^{{\dagger}}$ $\eta_{+}$ $\mathbb{Z}_{2}$ $\mathbb{Z}$ $0$
$\text{BDI}^{{\dagger}}$ $\mathcal{S}_{++},\eta_{++}$ $\mathbb{Z}$ $0$ $0$
$\text{D}^{{\dagger}}$ $\eta_{+}$ $0$ $0$ $0$ $\text{DIII}^{{\dagger}}$
$\mathcal{S}_{--},\eta_{++}$ $\mathbb{Z}_{2}$ $\mathbb{Z}_{2}$ $\mathbb{Z}$
$\text{AII}^{{\dagger}}$ $\eta_{+}$ $0$ 2$\mathbb{Z}$ $0$
$\text{CII}^{{\dagger}}$ $\mathcal{S}_{++},\eta_{++}$ 2$\mathbb{Z}$ $0$
$\mathbb{Z}_{2}$ $\text{C}^{{\dagger}}$ $\eta_{+}$ $0$ $\mathbb{Z}_{2}$
$\mathbb{Z}_{2}$ $\text{CI}^{{\dagger}}$ $\mathcal{S}_{--},\eta_{++}$ $0$ $0$
2$\mathbb{Z}$ $\text{AI}^{{\dagger}}$ $\eta_{-}$ $0$ $\mathbb{Z}_{2}$
$\mathbb{Z}_{2}$ $\text{BDI}^{{\dagger}}$ $\mathcal{S}_{--},\eta_{--}$ $0$ $0$
2$\mathbb{Z}$ $\text{D}^{{\dagger}}$ $\eta_{-}$ $\mathbb{Z}_{2}\to 0$
$\mathbb{Z}\to 2\mathbb{Z}$ $0$ $\text{DIII}^{{\dagger}}$
$\mathcal{S}_{++},\eta_{--}$ $\mathbb{Z}\to 2\mathbb{Z}$ $0$ $0$
$\text{AII}^{{\dagger}}$ $\eta_{-}$ $0$ $0$ $0$ $\text{CII}^{{\dagger}}$
$\mathcal{S}_{--},\eta_{--}$ $\mathbb{Z}_{2}$ $\mathbb{Z}_{2}$ $\mathbb{Z}$
$\text{C}^{{\dagger}}$ $\eta_{-}$ $0$ 2$\mathbb{Z}$ $0$
$\text{CI}^{{\dagger}}$ $\mathcal{S}_{++},\eta_{--}$ 2$\mathbb{Z}$ $0$
$\mathbb{Z}_{2}$

Table 4: Classification of point-gap topological phases in the real AZ†
classes without or with SLS or pH. The subscript of ${\cal
S}_{\pm}/\eta_{\pm}$ specifies the commutation (+) or anti-commutation (-)
relation to TRS† or PHS†. For ${\cal S}_{\pm\pm}/\eta_{\pm\pm}$, the first
subscript specifies the relation to TRS† and the second specifies the relation
to PHS†. For the topological numbers with arrows, the classification under
OBCs changes from that under PBCs, where the left specifies the classification
under PBCs and the right specifies that under OBCs. The topological number
$\mathbb{Z}[i,j]$ ($\mathbb{Z}_{2}[i,j]$) under OBCs indicates the Abelian
group $\mathbb{Z}$ ($\mathbb{Z}_{2}$) generated by the element
$(i,j)\in\mathbb{Z}\oplus\mathbb{Z}$
[$(i,j)\in\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$] under PBCs. For the topological
numbers without arrows, the classification under OBCs coincides with that
under PBCs.

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Supplemental Material

##  S1 From PBC to yOBC in Eq. (8)

We here demonstrate that the real-space 3D winding number detects a
topological phase transition under continuous change of the boundary
condition. Let us start with the Hamiltonian in Eq. (8) of the main text:

$\displaystyle
H(\bm{k})=\sin{k_{x}}\sigma_{x}+2\sin{k_{z}}\sigma_{y}+2\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{z}+\dfrac{3}{2}i(\sin{k_{y}}+\sin{k_{z}})\sigma_{0}.$
(S1)

While this model has the non-trivial 3D winding number $-1$ in a region
including $E=0$ under the full PBCs, in-gap skin modes arise under the yOBC,
instead of surface states, as shown in Fig. S1. In the following, we will
reveal that the absence of surface states can be understood as a topological
phase transition under a continuous deformation of the boundary condition by
using the real-space 3D winding number.

Figure S1: The yOBC spectrum of the model in Eq. (S1). The system size in the
$y$ direction is $L_{y}=21$ and the momentum resolution in the
$k_{x,z}$-directions is taken as $\Delta k_{x,z}=2\pi/100$. No surface states
appear, and instead, the skin modes appear.

First, we note that the in-gap skin modes originate from modes with
$(k_{x},k_{z})=(0,0)$, where the Hamiltonian in Eq. (S1) becomes

$\displaystyle
H(k_{x}=0,k_{y},k_{z}=0)=-2\cos{k_{y}}\sigma_{z}+\dfrac{3}{2}i\sin{k_{y}}\sigma_{0}.$
(S2)

This Hamiltonian gives elliptical complex spectra in the eigensector of
$\sigma_{z}=\pm 1$, which have the energy winding numbers $\mp 1$ along the
$k_{y}$ direction. According to the general theory of NHSE Zhang  _et al._
(2020); Okuma  _et al._ (2020), the energy winding results in skin modes in
Fig. S1.

In Fig. S2, we show how the complex spectrum and the real-space 3D winding
number changes under the continuous deformation of the boundary conditions.
Here we define the real-space 3D winding number as follows. Taking the real-
space representation of $H({\bm{k}})$ in the $y$-direction and performing the
projection to the bulk, we have the bulk Hamiltonian ${\cal H}_{\rm
bulk}(k_{x},k_{z})$. Then, the real-space 3D winding number at $E$ is

$\displaystyle w_{3}=-\frac{i}{12\pi}\int_{BZ}d^{2}{\bm{k}}{\cal
T}_{y}[\varepsilon^{ijk}Q_{i}Q_{j}Q_{k}]$ (S3)

with $Q_{i=x,z}=i({\cal H}_{\rm bulk}-E)^{-1}\partial_{k_{i}}({\cal H}_{\rm
bulk}-E)$ and $Q_{y}=({\cal H}_{\rm bulk}-E)^{-1}[Y,{\cal H}_{\rm bulk}-E]$,
where $Y$ is the position operator of the $y$-coordinate, and ${\cal T}_{y}$
stands for the trace per unit length in the $y$-direction. Decreasing the
hopping amplitudes between the $y=1$ sites and $y=L_{y}$ sites, we can change
the boundary conditions from the full PBC to the yOBC smoothly. During the
process, the point-gapped region becomes smaller because of the shrinking of
the modes with $(k_{x},k_{z})=(0,0)$ in the complex energy plane. Accordingly,
we can confirm that the real-space 3D winding number changes from the well-
defined value $-1$ to ill-defined ones in the original point-gapped region
around $E=0$, as illustrated in Fig. S2. Therefore, we conclude that no
surface states appear under the yOBC in Eq. (S1) as a result of the
topological phase transition.

Figure S2: The changes of the spectrum (top) and the real-space 3D winding
number (bottom) of the model in Eq. (S1) from under the full PBC to the yOBC.
$L_{y}$ is the same as that in Fig. S1, but the momentum resolution around
$(k_{x},k_{z})=(0,0)$ is much finer in the energy spectrum. The orange circles
in the top figures represent the modes with $(k_{x},k_{z})=(0,0)$. In the
bottom figures, the 3D winding numbers are calculated at $E=0+i{\rm Im}E$ with
$-4<{\rm Im}E<4$. The purple shaded regions in the bottom figures correspond
to point-gapless regions in the top figures. (a) The full PBC. A point gap is
open in the region containing $E=0$ with the nontrivial 3D winding number
$-1$. The orange modes at $(k_{x},k_{z})=(0,0)$ form a loop and have a
nontrivial 1D winding number in each eigensector of $\sigma_{z}=\pm 1$. (b) A
boundary condition between the full PBC and the yOBC. The hopping amplitude
between the $y=1$ sites and the $y=L_{y}$ sites is $0.00005$. The point gapped
region containing $E=0$ shrinks, but the 3D winding number is $-1$ in the
point gapped region. (c) The yOBC. The point-gapped region including $E=0$ is
completely closed by the in-gap skin effect with the orange modes near
$(k_{x},k_{z})=(0,0)$. The in-gap skin effect makes the 3D winding number ill-
defined.

##  S2 Proof of the BBC for point-gap topological phases in AZ† symmetry
classes

In this section, we give a proof of the BBC for point-gap topological phases
with AZ† symmetry. Below, we assume that the spatial dimension $d$ is $d=2,3$,
but the generalization to higher dimensions is straightforward.

Our basic strategy of the proof is to use the additivity of topological phases
in the framework of $K$-theory. Since we can add the topological number of
$H_{1}$ to that of $H_{2}$ by stacking $H_{1}$ and $H_{2}$ as

$\displaystyle\begin{pmatrix}H_{1}&0\\\ 0&H_{2}\end{pmatrix},$ (S4)

we can generate any topological phases by considering a proper stacks of a
model with the minimal topological number. From $K$-theory, we also know that
models with the same topological numbers can deform into each other without
gap-closing, up to addition of topologically trivial Hamiltonians. Thus, once
we can show the BBC for a model with the minimal topological number, the BBC
holds generally.

We can construct a model with the minimal topological number, which we call
the minimal model, as follows. Except for 3D class CII†, we consider the
following form of Hamiltonian,

$\displaystyle H_{\text{minimal}}(\bm{k})$
$\displaystyle=H_{\text{SM}}(\bm{k})+i\sin{k_{d}}\Gamma_{d},\quad
H_{\text{SM}}(\bm{k})=\sum_{i=1}^{d-1}\sin{k_{i}}\Gamma_{i}+\left[(d-1)-\sum_{i=1}^{d}\cos{k_{i}}\right]\Gamma_{0},$
(S5) $\displaystyle E_{\text{minimal}}(\bm{k})$
$\displaystyle=\pm\sqrt{\sum_{i=1}^{d-1}\left(\sin
k_{i}\right)^{2}+\left[(d-1)-\sum_{i=1}^{d}\cos k_{i}\right]^{2}}+s\,i\sin
k_{d},$ (S6)

where $d\geq 2$ is the spatial dimension, $\Gamma_{i=0,1,2,\dots,d-1}$ are
Gamma matrices that anti-commute with each other, and $\Gamma_{d}$ is a
matrice that commutes with all $\Gamma_{i=0,1,2,\dots,d-1}$, and
$\Gamma_{\mu}^{2}=1$ ($\mu=0,1,\dots,d$). The sign factor $s$ in Eq. (S6)
takes $s=+1$ ($s=\pm 1$) for $\Gamma_{d}=\hat{1}$ ($\Gamma_{d}\neq\hat{1}$).
Note that $H_{\text{SM}}(\bm{k})$ is the Hamiltonian of a Hermitian Weyl/Dirac
semimetal, and $H_{\rm minimal}$ is straightforward generalization of Eq.
($1$) in the main text. Since $H_{\text{minimal}}(\bm{k})$ commutes with
$i\sin{k_{d}}\Gamma_{d}$, the non-Hermitian term $i\sin{k_{d}}\Gamma_{d}$ only
gives a complex energy shift if we impose the PBC in the $x_{d}$-direction.
Therefore, skin effects do not occur when we keep the PBC in the
$x_{d}$-direction.

The point-gap topological number of $H_{\rm minimal}({\bm{k}})$ under the full
PBC is given by the topological number of the doubled Hermitian Hamiltonian
Kawabata  _et al._ (2019b),

$\displaystyle\tilde{H}({\bm{k}})=\begin{pmatrix}0&H_{\rm
minimal}({\bm{k}})\\\ H^{\dagger}_{\rm
minimal}({\bm{k}})&0\end{pmatrix}=\sum_{i=1}^{d}\sin
k_{i}\tilde{\Gamma}_{i}+\left[(d-1)-\sum_{i=1}^{d}\cos
k_{i}\right]\tilde{\Gamma}_{0}.$ (S7)

Here $\tilde{\Gamma}_{\mu=0,\dots,d}$ are the Gamma matrices in the doubled
space,

$\displaystyle\tilde{\Gamma}_{i=0,\dots,d-1}=\begin{pmatrix}0&\Gamma_{i}\\\
\Gamma_{i}&0\end{pmatrix},\quad\tilde{\Gamma}_{d}=\begin{pmatrix}0&i\Gamma_{d}\\\
-i\Gamma_{d}&0\end{pmatrix},$ (S8)

which satisfy
$\\{\tilde{\Gamma}_{\mu},\tilde{\Gamma}_{\nu}\\}=\delta_{\mu,\nu}$. We can
easily evaluate the topological number of Eq. (S7), namely the point-gap
topological number of $H_{\rm minimal}({\bm{k}})$, because Eq. (S7) is the
standard massive Dirac Hamiltonian for topological insulators. The obtained
point-gap topological number coincides with that under the OBC in other than
the $x_{d}$-direction, say the OBC in the $x_{1}$-direction (x1OBC), as no
skin effects occur. One can also show that this model has surface states under
the x1OBC at the same time. Since the Weyl/Dirac semimetal model
$H_{\text{SM}}$ has Fermi arc surface states between two Weyl/Dirac points,
$H_{\text{minimal}}(\bm{k})$ also has surface states under the x1OBC. As a
result, we have the BBC between the surface states and the bulk point-gap
topological number.

For 3D class CII†, the minimal model has the form

$\displaystyle H_{\rm minimal}({\bm{k}})=H_{\rm SM}({\bm{k}})+i\sin
k_{2}\Gamma_{2}+i\sin k_{3}\Gamma_{3},\quad H_{\rm SM}({\bm{k}})=\sin
k_{1}\Gamma_{1}+\left[2-\sum_{i=1}^{3}\cos k_{i}\right]\Gamma_{0},$ (S9)

where $\Gamma_{\mu}$ satisfy

$\displaystyle\\{\Gamma_{0},\Gamma_{1}\\}=\\{\Gamma_{2},\Gamma_{3}\\}=[\Gamma_{i=0,1},\Gamma_{j=2,3}]=0,\quad\Gamma_{\mu=0,1,2,3}^{2}=1.$
(S10)

The double Hamiltonian of this model also has the form of the standard massive
Dirac Hamiltonian, and thus we can easily evaluate the point-gap topological
number. We also find that $H_{\rm SM}$ has $E=0$ surface states, which give
surface states with ${\rm Re}E=0$ of $H_{\rm minimal}$. Thus, we have the BBC
again.

In Table S1, we summarize possible point-gap topological phases under the OBC
for AZ† classes with the spatial dimension $d=1,2,3$. Below, we present the
minimal model for each point-gap topological phase in Table S1. In the
following, $\tau_{i=x,y,z}$ and $\sigma_{i=x,y,z}$ represent the Pauli
matrices, and $\tau_{0}$ and $\sigma_{0}$ are the 2×2 identity matrix.

Table S1: Classification of point-gap topological phases under the OBC. We consider the $\mathrm{AZ}^{{\dagger}}$ symmetry classes with the spatial dimension $d=1,2$ and $3$. The section numbers for the minimal models are shown for each topological number. Symmetry class |  TRS† |  PHS† |  CS |  $d=1$ |  $d=2$ |  $d=3$
---|---|---|---|---|---|---
A | $0$ | $0$ | $0$ | 0 | $0$ |  $\mathbb{Z}$ [Sec.S2.1]
AIII | $0$ | $0$ | $1$ | $0$ |  $\mathbb{Z}$ [Sec.S2.2] | $0$
$\text{AI}^{{\dagger}}$ | $+1$ | $0$ | $0$ | $0$ | $0$ | 2$\mathbb{Z}$ [Sec.S2.3]
$\text{BDI}^{{\dagger}}$ | $+1$ | $+1$ | $1$ | $0$ | $0$ | $0$
$\text{D}^{{\dagger}}$ | $0$ | $+1$ | $0$ | $0$ | $0$ | $0$
$\text{DIII}^{{\dagger}}$ | $-1$ | $+1$ | $1$ | $0$ |  $2\mathbb{Z}$ [Sec.S2.4] | $0$
$\text{AII}^{{\dagger}}$ | $-1$ | $0$ | $0$ | 0 | 0 |  $2\mathbb{Z}$ [Sec.S2.5]
$\text{CII}^{{\dagger}}$ | $-1$ | $-1$ | $1$ | $0$ |  $\mathbb{Z}_{2}$ [Sec.S2.6] |  $\mathbb{Z}_{2}$ [Sec.S2.6]
$\text{C}^{{\dagger}}$ | $0$ | $-1$ | $0$ | 0 | $0$ |  $\mathbb{Z}_{2}$ [Sec.S2.7]
$\text{CI}^{{\dagger}}$ | $+1$ | $-1$ | $1$ | $0$ | 2$\mathbb{Z}$ [Sec.S2.8] | $0$

###  S2.1 class A

A non-Hermitian system in class A has a nontrivial point-gap topological phase
in $d=3$. The minimal model is

$\displaystyle H(\bm{k})$
$\displaystyle=\sin{k_{x}}\sigma_{x}+\sin{k_{y}}\sigma_{y}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{z}+i\sin{k_{z}}\sigma_{0},$
(S11)

and the point-gap topological number under the full PBC is the 3D winding
number in Eq. (2), which takes $+1$ for Eq. (S11). As shown in the main text,
the corresponding point-gap topological number under the xOBC is

$\displaystyle w_{3}|_{\rm
xOBC}=-\dfrac{i}{12\pi}\int_{\text{BZ}}d^{2}\bm{k}\hskip
1.42262pt\mathcal{T}_{x}[\varepsilon^{ijk}Q_{i}Q_{j}Q_{k}],$ (S12)

with $Q_{x}=\mathcal{H}_{\text{bulk}}^{-1}[X,\mathcal{H}_{\text{bulk}}]$ and
$Q_{i=y,z}=i\mathcal{H}_{\text{bulk}}^{-1}\partial_{k_{i}}\mathcal{H}_{\text{bulk}}$,
where $X$ is the position operator in the $x$-coordinate, $\mathcal{T}_{x}$
stands for the trace per unit length in the $x$-direction, and
$\mathcal{H}_{\text{bulk}}(k_{y},k_{z})$ is the 3D bulk Hamiltonian with
momentum-space representation in the $y$\- and $z$-directions and real-space
representation in the $x$-direction.

###  S2.2 class AIII

A non-Hermitian system in class AIII has a nontrivial point-gap topological
phase in $d=2$. The minimal model is

$\displaystyle H(\bm{k})$
$\displaystyle=\sin{k_{x}}\sigma_{x}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\sigma_{y}+i\sin{k_{y}}\sigma_{0},$
(S13)

which hosts CS

$\displaystyle\Gamma
H^{{\dagger}}(\bm{k})\Gamma^{-1}=-H(\bm{k}),\quad\Gamma=\sigma_{z}.$ (S14)

The point-gap topological number under the full PBC is the first Chern number
of the Hermitian matrix $iH(\bm{k})\Gamma$ Kawabata  _et al._ (2019b):

$\displaystyle iH(\bm{k})\Gamma$
$\displaystyle=\sin{k_{x}}\sigma_{y}-\sin{k_{y}}\sigma_{z}-\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\sigma_{x},$
(S15)

which takes $+1$ for Eq. (S13). The corresponding point-gap topological number
under the xOBC is

$\displaystyle Ch_{1}|_{\rm xOBC}=2\pi
i\mathcal{T}_{xy}\left(P_{\alpha}[[X,P_{\alpha}],[Y,P_{\alpha}]]\right),$
(S16)

where $P_{\alpha}$ is the bulk-band projection operator of $i{\cal H}\Gamma$
in a band $\alpha$, $X$ is the position operator in the $x$-coordinate, $Y$ is
the position operator in the $y$-coordinate, $\mathcal{T}_{xy}$ stands for the
trace per unit area in the $xy$-plane, and $i{\cal H}\Gamma$ is the bulk
Hamiltonian of the real-space representation of $iH(\bm{k})\Gamma$ Bellissard
_et al._ (1994); Kitaev (2006); Prodan  _et al._ (2010); Katsura and Koma
(2018); Song  _et al._ (2019a).

###  S2.3 class $\text{AI}^{{\dagger}}$

In class $\text{AI}^{{\dagger}}$, we have a nontrivial point-gap topological
phase in $d=3$. The required symmetry is

$\displaystyle{\cal C}H^{T}({\bm{k}}){\cal C}^{-1}=H(-{\bm{k}}),\quad{\cal
C}{\cal C}^{*}=+1,\quad{\cal C}=\tau_{x}\sigma_{x},$ (S17)

and the minimal model is

$\displaystyle H(\bm{k})$
$\displaystyle=\left[\sin{k_{x}}\tau_{x}+\sin{k_{y}}\tau_{y}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\tau_{z}+i\sin{k_{z}}\tau_{0}\right]\sigma_{z}.$
(S18)

The point-gap topological number in 3D class $\text{AI}^{{\dagger}}$ is the
same as that in 3D class A. Since Eq. (S18) is a stacking of the class A model
in Eq. (S11) to the eigensector of $\sigma_{z}=\pm 1$, the 3D winding number
takes $+2\in 2$$\mathbb{Z}$ for Eq. (S18).

###  S2.4 class $\text{DIII}^{{\dagger}}$

In class $\text{DIII}^{{\dagger}}$, there exists a nontrivial point-gap
topological phase in $d=2$. The minimal model is

$\displaystyle H(\bm{k})$
$\displaystyle=\sin{k_{x}}\tau_{x}\sigma_{x}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\tau_{x}\sigma_{y}-i\sin{k_{y}}\tau_{y}\sigma_{z},$
(S19)

which has TRS†, PHS† and their combination CS

$\displaystyle{\cal C}H^{T}({\bm{k}}){\cal C}^{-1}$
$\displaystyle=H(-{\bm{k}}),\quad{\cal C}{\cal C}^{*}=-1,\quad{\cal
C}=i\tau_{y}\sigma_{0},$ $\displaystyle{\cal T}H^{*}({\bm{k}}){\cal T}^{-1}$
$\displaystyle=-H(-{\bm{k}}),\quad{\cal T}{\cal T}^{*}=+1,\quad{\cal
T}=\tau_{0}\sigma_{0},$ (S20) $\displaystyle\Gamma
H^{\dagger}({\bm{k}})\Gamma^{-1}$
$\displaystyle=-H({\bm{k}}),\quad\Gamma=i{\cal C}{\cal T}^{*}.$

The point-gap topological number in 2D class $\text{DIII}^{{\dagger}}$ is the
same as that in 2D class AIII. $iH(\bm{k})\Gamma$ for Eq. (S19) is

$\displaystyle iH(\bm{k})\Gamma$
$\displaystyle=\sin{k_{x}}\tau_{z}\sigma_{x}-\sin{k_{y}}\tau_{0}\sigma_{z}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\tau_{z}\sigma_{y},$
(S21)

which is a stacking of Chern insulators to the eigensector of $\tau_{z}=\pm
1$. Thus, the first Chern number takes $+2\in 2$$\mathbb{Z}$ for Eq. (S19).

###  S2.5 class $\text{AII}^{{\dagger}}$

In class $\text{AII}^{{\dagger}}$, there exists a nontrivial point-gap
topological phase in $d=3$. The minimal model is

$\displaystyle H(\bm{k})$
$\displaystyle=\tau_{y}\left[\sin{k_{x}}\sigma_{z}+\sin{k_{y}}\sigma_{x}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{y}+i\sin{k_{z}}\sigma_{0}\right],$
(S22)

which has TRS†,

$\displaystyle{\cal C}H^{T}({\bm{k}}){\cal C}^{-1}=H(-{\bm{k}}),\quad{\cal
C}{\cal C}^{*}=-1,\quad{\cal C}=i\tau_{y}\sigma_{0}.$ (S23)

The point-gap topological number in 3D class $\text{AII}^{{\dagger}}$ is the
same as that in 3D class A. Since Eq. (S22) is a stacking of the minimal model
of 3D class A to the eigensector of $\tau_{y}=\pm 1$, the 3D winding number
takes $+2\in 2$$\mathbb{Z}$ for Eq. (S22).

###  S2.6 class $\text{CII}^{{\dagger}}$

In class $\text{CII}^{{\dagger}}$, there exist nontrivial point-gap
topological phases in $d=2,3$. We impose the following AZ† symmetry,

$\displaystyle{\cal C}H^{T}({\bm{k}}){\cal C}^{-1}$
$\displaystyle=H(-{\bm{k}}),\quad{\cal C}{\cal C}^{*}=-1,\quad{\cal
C}=i\tau_{y}\sigma_{0},$ $\displaystyle{\cal T}H^{*}({\bm{k}}){\cal T}^{-1}$
$\displaystyle=-H(-{\bm{k}}),\quad{\cal T}{\cal T}^{*}=-1,\quad{\cal
T}=i\tau_{y}\sigma_{z},$ (S24) $\displaystyle\Gamma
H^{\dagger}({\bm{k}})\Gamma^{-1}$
$\displaystyle=-H({\bm{k}}),\quad\Gamma={\cal C}{\cal T}^{*}.$

For $d=2$, the minimal model is

$\displaystyle H(\bm{k})$ $\displaystyle=\sin
k_{x}\tau_{x}\sigma_{x}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\tau_{x}\sigma_{y}+i\sin
k_{y}\tau_{z}\sigma_{z}.$ (S25)

The point-gap topological number under the full PBCs is the Kane-Male
invariant for 2D class AII Hermitian matrix $iH(\bm{k})\Gamma$ Kawabata  _et
al._ (2019b), which takes $1\in$ $\mathbb{Z}_{2}$ for Eq. (S25). The
corresponding point-gap topological number under the xOBC is

$\displaystyle\eta|_{\rm xOBC}$ $\displaystyle=\text{dim
ker}(A-1)~{}\text{mod}~{}2$ (S26)

with

$\displaystyle A=P_{F}-D^{{\dagger}}_{\bm{a}}P_{F}D_{\bm{a}},\quad
D_{\bm{a}}(\bm{r})=\dfrac{\hat{x}+i\hat{y}-(a_{x}+ia_{y})}{|\hat{x}+i\hat{y}-(a_{x}+ia_{y})|}.$

Here $P_{F}$ is the projection operator of $i{\cal H}\Gamma$ onto the bulk
states below the Fermi energy, where $i{\cal H}\Gamma$ is the real-space
representation of $iH(\bm{k})\Gamma$,
$\bm{r}=(\hat{x},\hat{y})\in\mathbb{Z}^{2}$ represents the position operator
in the $xy$-plane, and
$\bm{a}=(a_{x},a_{y})\in\mathbb{R}^{2}\backslash\mathbb{Z}^{2}$ is a two
dimensional vector in the $xy$-plane Katsura and Koma (2016, 2018); Akagi  _et
al._ (2017).

For $d=3$, the minimal model is

$\displaystyle H(\bm{k})$ $\displaystyle=\sin
k_{x}\tau_{x}\sigma_{x}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\tau_{x}\sigma_{y}+i\sin
k_{y}\tau_{x}\sigma_{0}+i\sin k_{z}\tau_{z}\sigma_{z}.$ (S27)

The point-gap topological number under the full PBC is the $\mathbb{Z}_{2}$
invariant of the class AII Hermitian matrix $iH(\bm{k})\Gamma$ Kawabata  _et
al._ (2019b), which takes $1\in$ $\mathbb{Z}_{2}$ for Eq. (S27). The
corresponding point-gap topological number under the xOBC is the 3D version of
Eq. (S26) Katsura and Koma (2018); Akagi  _et al._ (2017).

###  S2.7 class $\text{C}^{{\dagger}}$

In class $\text{C}^{{\dagger}}$, there exists a nontrivial point-gap
topological phase in $d=3$. The minimal model is

$\displaystyle H(\bm{k})$ $\displaystyle=\sin k_{x}\tau_{x}\sigma_{x}+\sin
k_{y}\tau_{x}\sigma_{y}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\tau_{0}\sigma_{z}+i\sin
k_{z}\tau_{z}\sigma_{z},$ (S28)

which has the following PHS†,

$\displaystyle{\cal T}H^{*}({\bm{k}}){\cal T}^{-1}$
$\displaystyle=-H(-{\bm{k}}),\quad{\cal T}{\cal T}^{*}=-1,\quad{\cal
T}=i\tau_{z}\sigma_{y}.$ (S29)

The point-gap topological number under the full PBCs is the $\mathbb{Z}_{2}$
invariant of the Hermitian Hamiltonian

$\displaystyle\begin{pmatrix}0&H({\bm{k}})\\\
H^{\dagger}({\bm{k}})&0\end{pmatrix}$ (S30)

in class CII Kawabata  _et al._ (2019b), which takes $1\in$ $\mathbb{Z}_{2}$
for Eq. (S28). The corresponding point-gap topological number under the xOBC
is its real-space representation given in Ref. Katsura and Koma (2018).

###  S2.8 class $\text{CI}^{{\dagger}}$

In class $\text{CI}^{{\dagger}}$, there exists a nontrivial point-gap
topological phase in $d=2$. The minimal model is

$\displaystyle H(\bm{k})$ $\displaystyle=\sin
k_{x}\tau_{y}\sigma_{x}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\tau_{x}\sigma_{x}+i\sin
k_{y}\tau_{z}\sigma_{y},$ (S31)

which obeys

$\displaystyle{\cal C}H^{T}({\bm{k}}){\cal C}^{-1}$
$\displaystyle=H(-{\bm{k}}),\quad{\cal C}{\cal C}^{*}=+1,\quad~{}{\cal
C}=\tau_{0}\sigma_{0},$ $\displaystyle{\cal T}H^{*}({\bm{k}}){\cal T}^{-1}$
$\displaystyle=-H(-{\bm{k}}),\quad{\cal T}{\cal T}^{*}=-1,\quad~{}{\cal
T}=i\tau_{0}\sigma_{y},$ (S32) $\displaystyle\Gamma
H^{\dagger}({\bm{k}})\Gamma$ $\displaystyle=-H({\bm{k}}),\quad\Gamma=i{\cal
C}{\cal T}^{*}.$

The point-gap topological number under the full PBC is the same as that in 2D
class AIII. Since $iH(\bm{k})\Gamma$ for Eq. (S31) is a stacking of two Chern
insulators to the eigensectors of $\sigma_{z}=\pm 1$:

$\displaystyle iH(\bm{k})\Gamma$
$\displaystyle=\sin{k_{x}}\tau_{y}\sigma_{z}+\sin{k_{y}}\tau_{z}\sigma_{0}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\tau_{x}\sigma_{z},$
(S33)

the first Chern number takes $+2\in 2$$\mathbb{Z}$ for Eq. (S31).

##  S3 Bulk-boundary correspondence in 2D class AIII

We here explain more details of the BBC in point-gap topological phases for 2D
class AIII. We start with the model in Eq. (S13):

$\displaystyle H_{\rm AIII}(\bm{k})$
$\displaystyle=\sin{k_{x}}\sigma_{x}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\sigma_{y}+i\sin{k_{y}}\sigma_{0}.$
(S34)

As explained in Sec.S2.2, this model has the first Chern number $+1$ of
$iH_{\rm AIII}\Gamma$ under the xOBC. Since $H_{\rm AIII}(\bm{k})$ commutes
with $i\sin{k_{y}}\sigma_{0}$, the surface state of this model with a fixed
$k_{y}$ is that of the Su-Schrieffer–Heeger (SSH) model,
$H_{\text{SSH}}(\bm{k})=H_{\rm
AIII}(\bm{k})-i\sin{k_{y}}\sigma_{0}=\sin{k_{x}}\sigma_{x}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\sigma_{y}$.
For $-\pi/2<k_{y}<\pi/2$, the SSH model supports the 1D winding number $+1$,
and thus it has corresponding zero modes $\ket{\psi_{+}(k_{y})}$ with
chirality, $\Gamma\ket{\psi_{+}(k_{y})}=+1\ket{\psi_{+}(k_{y})}$, under the
xOBC. By taking into account the complex energy shift from the non-Hermitian
term $i\sin{k_{y}}\sigma_{0}$, the zero modes give surface states in the
point-gapped region, as shown in Fig. S3(b).

Figure S3: The energy spectra of the model in Eq. (S34). The system sizes are
$L_{x}=L_{y}=100$. (a) The full PBC spectrum. A point gap is open in the
region containing $E=0$, with the nontrivial first Chern number $+1$. No NHSE
occurs and the spectra coincide with the bulk spectra under the xOBC. (b) The
xOBC spectrum. Surface states appear along $\mathrm{Re}E=0$ where the first
Chern number takes +1. Note that the first Chern number is well-defined only
on the line $\mathrm{Re}E=0$ in class AIII, since the system has chiral
symmetry only when the reference energy of the point gap is purely imaginary.

These surface states have a topological number with the same value as the bulk
first Chern number. The effective Hamiltonian of the surface states around
$E=0$ takes the form of $h_{\rm
surface}(k_{y})=ik_{y}\ket{\psi_{+}(k_{y})}\bra{\psi_{+}(k_{y})}$ as the zero
mode of the SSH model shifted by the non-Hermitian term of $H_{\rm AIII}$,
$ik_{y}$. Then, we can define the occupation number ${\cal N}(k_{y})$ (the
number of negative eigenvalues) for $ih_{\rm surface}(k_{y})\Gamma$ on the
surface BZ. Since $\Gamma\ket{\psi_{+}(k_{y})}=+1\ket{\psi_{+}(k_{y})}$,
$ih_{\rm surface}(k_{y})\Gamma$ is equal to
$-k_{y}\ket{\psi_{+}(k_{y})}\bra{\psi_{+}(k_{y})}$ and we get

$\displaystyle{\cal N}(k_{y})$ $\displaystyle=\begin{cases}1&\text{for
$k_{y}>0$}\\\ 0&\text{for $k_{y}<0$}\end{cases}.$ (S35)

The difference of the occupation number $\nu_{\rm AIII}$:

$\displaystyle\nu_{\rm AIII}$ $\displaystyle={\cal N}(k_{y}>0)-{\cal
N}(k_{y}<0)=+1$ (S36)

protects the surface energy $E=0$ at $k_{y}=0$ and thus measures the topology
of the surface states, of which value coincides with the bulk first Chern
number. Consequently, we have the BBC under the xOBC for Eq. (S34):

$\displaystyle Ch_{1}|_{\rm xOBC}=\nu_{\rm AIII}.$ (S37)

In a general case, we can get the BBC by stacking the model in Eq. (S34) up to
continuous deformations. The BBC for other symmetry classes also can be proved
in a similar manner.

##  S4 The BBC for point-gap topological phases in 38-fold symmetry classes

According to a general theory in Ref.Kawabata  _et al._ (2019b), there are
38-fold symmetry classes in non-Hermitian systems. In this section, we extend
our arguments to all the 38-fold symmetry classes.

First, we briefly explain the 38-fold symmetry classes. In addition to AZ†
symmetries discussed in the main text, non-Hermitian systems may host the
original AZ symmetries defined by the following equations:

$\displaystyle{\cal T}H^{*}({\bm{k}}){\cal T}^{-1}$
$\displaystyle=H(-{\bm{k}}),\quad{\cal T}{\cal T}^{*}=\pm 1,$
$\displaystyle{\cal C}H^{T}({\bm{k}}){\cal C}^{-1}$
$\displaystyle=-H(-{\bm{k}}),\quad{\cal C}{\cal C}^{*}=\pm 1,$ (S38)
$\displaystyle\Gamma H^{\dagger}({\bm{k}})\Gamma^{-1}$
$\displaystyle=-H({\bm{k}}),\quad\Gamma^{2}=1,$

where ${\cal T},{\cal C}$ and $\Gamma$ are unitary operators corresponding to
time-reversal symmetry (TRS), particle-hole symmetry (PHS), and chiral
symmetry (CS), respectively. Furthermore, one can introduce sublattice
symmetry (SLS),

$\displaystyle{\cal S}H(\bm{k}){\cal S}^{-1}=-H(\bm{k}),\quad{\cal S}^{2}=1,$
(S39)

with a unitary operator ${\cal S}$, which is distinct from CS in non-Hermitian
systems. The presence and/or absence of these symmetries give the 38-fold
independent symmetry classes, which is a natural generalization of the
Hermitian 10-fold AZ symmetry classes to non-Hermitian systems Kawabata  _et
al._ (2019b).

To specify the 38-fold symmetry classes, we introduce the convention used in
Ref.Kawabata  _et al._ (2019b) as follows: (i) The AZ and AZ† symmetries
define 18-fold symmetry classes in Table S2. (ii) Furthermore, each of them
can host SLS additionally, which defines AZ class + ${\cal S}$ and AZ† class +
${\cal S}$. We also introduce the subindex $+(-)$ of ${\cal S}$ specifying the
commutation (anti-commutation) relation between SLS and AZ or AZ† symmetries.
For an AZ (AZ†) class having both TRS (TRS†) and PHS (PHS†), ${\cal S}$ has a
double subindex, where the first index specifies the commutation or
anticommutation relation between SLS and TRS (TRS†), and the second one
specifies those between SLS and PHS (PHS†), respectively.

Table S2: AZ and AZ† symmetry classes for non-Hermitian Hamiltonians. Sym. class |  | TRS | PHS | CS | TRS† | PHS†
---|---|---|---|---|---|---
Complex AZ | A | $0$ | $0$ | $0$ | $0$ | $0$
| AIII | $0$ | $0$ | $1$ | $0$ | $0$
Real AZ | AI | $+1$ | $0$ | $0$ | $0$ | $0$
| BDI | $+1$ | $+1$ | $1$ | 0 | 0
| D | $0$ | $+1$ | $0$ | 0 | 0
| DIII | $-1$ | $+1$ | $1$ | $0$ | 0
| AII | $-1$ | $0$ | $0$ | 0 | $0$
| CII | $-1$ | $-1$ | $1$ | $0$ | 0
| C | $0$ | $-1$ | $0$ | $0$ | $0$
| CI | $+1$ | $-1$ | $1$ | $0$ | $0$
Real AZ† | $\text{AI}^{\dagger}$ | $0$ | $0$ | $0$ | +1 | $0$
| $\text{BDI}^{\dagger}$ | $0$ | $0$ | $1$ | +1 | +1
| $\text{D}^{\dagger}$ | $0$ | $0$ | $0$ | 0 | +1
| $\text{DIII}^{\dagger}$ | $0$ | $0$ | $1$ | -1 | +1
| $\text{AII}^{\dagger}$ | $0$ | $0$ | $0$ | -1 | $0$
| $\text{CII}^{\dagger}$ | $0$ | $0$ | $1$ | -1 | -1
| $\text{C}^{\dagger}$ | $0$ | $0$ | $0$ | $0$ | $-1$
| $\text{CI}^{\dagger}$ | $0$ | $0$ | $1$ | $+1$ | $-1$

Let us now identify the independent 38-fold symmetry classes. As was shown in
Ref.Kawabata  _et al._ (2019b), we can divide the 38-fold symmetry classes
according to the number $N\leq 3$ of their generators TRS, PHS, TRS† and PHS†.
(i) For $N$=0, we have 5 classes, which are classes A, AIII, A + ${\cal S}$,
AIII + ${\cal S}_{+}$, and AIII + ${\cal S}_{-}$. (ii) For $N=1$, we have 6
classes, AI, AII, AI†, AII†, D, and C. (iii) For $N=2$, there are 15 classes,
BDI, DIII, CI, CII, BDI†, DIII†, CI†, CII†, AI + ${\cal S}_{\pm}$, AII +
${\cal S}_{+}$, D + ${\cal S}_{\pm}$, and C + ${\cal S}_{\pm}$. (iv) For
$N=3$, we have 12 classes, BDI + ${\cal S}_{\pm\pm}$, DIII + ${\cal
S}_{+\pm}$, CI + ${\cal S}_{\pm\pm}$, and CII + ${\cal S}_{+\pm}$. Therefore,
we have, in total, 5+6+15+12=38 classes. Here we have several remarks: First,
AZ† \+ ${\cal S}$ classes do not appear above because the combination between
TRS† (PHS†) and SLS gives PHS (TRS) so they are not independent of AZ + ${\cal
S}$ classes. Second, AII + ${\cal S}_{-}$ coincides with AI + ${\cal S}_{-}$
by multiplying $i$ to the Hamiltonian and replacing the original TRS with the
combination of TRS and SLS. In a similar manner, DIII + ${\cal S}_{-\pm}$ and
CII + ${\cal S}_{-\pm}$ are equal to BDI + ${\cal S}_{-\pm}$ and CI + ${\cal
S}_{-\pm}$, respectively. Thus, the above list does not include them.

Among these 38-fold symmetry classes, we have proved the BBC in point-gap
topological phases for 10-fold AZ† classes in Sec.S2. The AZ† symmetry classes
correspond to A, AIII, AI†, BDI†, AI, DIII†, AII†, CII†, AII, and CI† in the
above 38 classes since one can identify D† and C† with AI and AII by
multiplying the imaginary unit $i$ to the Hamiltonian. We argue below the BBC
in point-gap topological phases for the remaining 28 symmetry classes.

###  S4.1 Classes without SLS

In the remaining 28 classes, 6 classes (BDI, D, DIII, CII, C, and CI) in Table
S3 do not support SLS. We first argue these classes in $d=1,2,3$. In a manner
similar to Sec.S2, on the basis of $K$-theory, we can prove the BBC by
constructing a model with a minimal topological number. Below we present such
minimal models. Except for $d=1,2$ in class BDI and $d=1$ in class D, the
minimal model has the form of Eq.(S5), and for $d=1,2$ in class BDI and $d=1$
in class D, the minimal model is either anti-Hermitian or Hermitian.
Therefore, all the minimal models can avoid skin effects, so have the common
value of the point-gap topological numbers between OBC and PBC. Furthermore,
using the conventional BBC for Hermitian systems, each of these models has a
boundary state, so we have the BBC.

AZ class |  $d=1$ |  $d=2$ |  $d=3$
---|---|---|---
BDI |  $\mathbb{Z}_{2}$ [Sec.S4.1.1] |  $\mathbb{Z}$ [Sec.S4.1.1] | $0$
D |  $\mathbb{Z}_{2}$ [Sec.S4.1.2] |  $\mathbb{Z}_{2}$ [Sec.S4.1.2] |  $\mathbb{Z}$ [Sec.S4.1.2]
DIII | $0$ |  $\mathbb{Z}_{2}$ [Sec.S4.1.3] |  $\mathbb{Z}_{2}$ [Sec.S4.1.3]
CII | $0$ | 2$\mathbb{Z}$ [Sec.S4.1.4] | $0$
C | $0$ | $0$ | 2$\mathbb{Z}$ [Sec.S4.1.5]
CI | $0$ | $0$ | $0$
Table S3: Point-gap topological phases in 6 classes without SLS. These classes
do not show the skin effects, and thus the classification under the OBCs
coincides with that under the PBC. The section numbers for the proof of the
classification under OBCs and the BBC are shown for each topological number.

####  S4.1.1 class BDI

In class BDI, there exist nontrivial point-gap topological phases in $d=1,2$
Kawabata  _et al._ (2019b). For $d=1$, the minimal model is

$\displaystyle H(k_{x})$ $\displaystyle=i\left(\sin
k_{x}\sigma_{x}-\cos{k_{x}}\sigma_{y}\right),$ (S40)

which has TRS and PHS of class BDI with ${\cal T}=\sigma_{0}$ and ${\cal
C}=\sigma_{0}$. This model has a non-zero $\mathbb{Z}_{2}$ point-gap
topological number of BDI class in $d=1$, which is defined by $(-1)^{\nu}={\rm
sgn}({\rm Pf}[H(\pi){\cal C}]/{\rm Pf}[H(0){\cal C}])$ Kawabata  _et al._
(2019b). Furthermore, as Eq.(S40) is anti-Hermitian so does not show skin
effects, this model has the same non-zero point-gap topological number under
OBC, which is defined by

$\displaystyle\eta|_{\rm OBC}$ $\displaystyle=\text{dim
ker}(A-1)~{}\text{mod.}~{}2$ (S41)

with

$\displaystyle A=P_{F}-D_{a_{x}}P_{F}D_{a_{x}},\quad
D_{a_{x}}(\hat{x})=\dfrac{\hat{x}-a_{x}}{|\hat{x}-a_{x}|}.$

Here $P_{F}$ is the projection operator of the Hertmian Hamltonian $i{\cal
H}{\cal T}{\cal C}^{*}$ onto the its negative energy states, where ${\cal H}$
is the real-space representation of $H(k_{x})$, $\hat{x}\in\mathbb{Z}$ is the
position operator in the real space lattice $x$, and
$a_{x}\in\mathbb{R}\backslash\mathbb{Z}$ Katsura and Koma (2018). One can
easily check that this model has a zero energy boundary state under OBC, so
the BBC holds.

For $d=2$, the minimal model is

$\displaystyle H(\bm{k})$ $\displaystyle=i\left[\sin k_{x}\sigma_{x}+\sin
k_{y}\sigma_{z}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\sigma_{y}\right],$
(S42)

with ${\cal T}=\sigma_{0}$ and ${\cal C}=\sigma_{0}$. The point-gap
topological number in 2D class BDI is the first Chern number of the Hermitian
Hamiltonian $iH(\bm{k}){\cal T}{\cal C}^{*}$. For Eq. (S42), we have

$\displaystyle iH(\bm{k})\Gamma$ $\displaystyle=-\sin k_{x}\sigma_{x}-\sin
k_{y}\sigma_{z}-\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\sigma_{y},$ (S43)

which gives the first Chern number $+1\in$$\mathbb{Z}$. This model does not
show skin effects because of the anti-Hermiticity and shows a chiral edge
state closing the point gap at $E=0$ under OBCs. Thus we have the BBC.

####  S4.1.2 class D

In class D, there exist nontrivial point-gap topological phases in $d=1,2$ and
$3$. For $d=1$, the minimal model is

$\displaystyle H(k_{x})$ $\displaystyle=\sin
k_{x}\sigma_{x}+\cos{k_{x}}\sigma_{y},$ (S44)

which has PHS of class D with ${\cal C}=\sigma_{0}$. The point-gap topological
number is given by

$\displaystyle(-1)^{\nu_{1}[H]}=\text{sgn}\left\\{\dfrac{\operatorname{Pf}[H(\pi)\mathcal{C}]}{\operatorname{Pf}[H(0)\mathcal{C}]}~{}\times~{}\text{exp}\left[-\dfrac{1}{2}\int_{k_{x}=0}^{k_{x}=\pi}\operatorname{\it{d}}\operatorname{log}\operatorname{det}[H(k_{x})\mathcal{C}]\right]\right\\},$
(S45)

which becomes non-trivial for Eq.(S44). This model is Hermitian and thus does
not show skin effects. Therefore, this model supports the same non-trivial
point-gap topological number under OBC. Because the minimal model has a
boundary zero mode under OBC, we have the BBC.

For $d=2$, the minimal model is

$\displaystyle H(\bm{k})$ $\displaystyle=\sin
k_{x}\sigma_{x}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\sigma_{y}+i\sin
k_{y}\sigma_{0},\quad{\cal C}=\sigma_{0}.$ (S46)

This model has the same form as Eq.(S5), and thus we can prove the BBC in a
similar manner. The $\mathbb{Z}_{2}$ point-gap topological number is defined
by Kawabata  _et al._ (2019b)

$\displaystyle(-1)^{\nu_{2}[H]}=\prod_{\text{X=I,II}}\text{sgn}\left\\{\dfrac{\operatorname{Pf}[H(\bm{k}_{\text{X}+})\mathcal{C}]}{\operatorname{Pf}[H(\bm{k}_{\text{X}-})\mathcal{C}]}~{}\times~{}\text{exp}\left[-\dfrac{1}{2}\int_{\bm{k}=\bm{k}_{\text{X}-}}^{\bm{k}=\bm{k}_{\text{X}+}}\operatorname{\it{d}}\operatorname{log}\operatorname{det}[H(\bm{k})\mathcal{C}]\right]\right\\},$
(S47)

where $(\bm{k}_{\text{I}+},\bm{k}_{\text{I}-})$ and
$(\bm{k}_{\text{II}+},\bm{k}_{\text{II}-})$ are two pairs of particle-hole
symmetric momenta.

For $d=3$, the minimal model is

$\displaystyle H(\bm{k})$
$\displaystyle=\sin{k_{x}}\sigma_{x}+\sin{k_{y}}\sigma_{z}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{y}-i\sin{k_{z}}\sigma_{0},\quad{\cal
C}=\sigma_{0},$ (S48)

which also has the form of Eq.(S5). The point-gap topological number in 3D
class D coincides with the winding number for 3D class A, which takes $+1\in$
$\mathbb{Z}$ for Eq. (S48).

####  S4.1.3 class DIII

In class DIII, there exist nontrivial point-gap topological phases in $d=2,3$.
For $d=2$, the minimal model is

$\displaystyle H(\bm{k})$
$\displaystyle=\sin{k_{x}}\tau_{x}\sigma_{x}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\tau_{x}\sigma_{y}+i\sin{k_{y}}\tau_{0}\sigma_{0},\quad{\cal
T}=i\tau_{y}\sigma_{0},\quad{\cal C}=\tau_{0}\sigma_{0}.$ (S49)

The point-gap topological number is the Kane-Male $\mathbb{Z}_{2}$ invariant
for 2D class AII Hermitian matrix $iH(\bm{k}){\cal T}{\cal C}^{*}$ Kawabata
_et al._ (2019b), which is non-trivial for Eq. (S49). For $d=3$, the minimal
model is

$\displaystyle H(\bm{k})$
$\displaystyle=\sin{k_{x}}\tau_{x}\sigma_{x}+\sin{k_{y}}\tau_{x}\sigma_{z}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\tau_{x}\sigma_{y}+i\sin{k_{z}}\tau_{0}\sigma_{0},\quad{\cal
T}=i\tau_{y}\sigma_{0},\quad{\cal C}=\tau_{0}\sigma_{0}.$ (S50)

The point-gap topological number under the full PBC is the $\mathbb{Z}_{2}$
invariant of the class AII Hermitian matrix $iH(\bm{k}){\cal T}{\cal C}^{*}$
Kawabata  _et al._ (2019b), which is non-trivial for Eq. (S50). These models
have the form of Eq.(S5), and thus we have the BBC.

####  S4.1.4 class CII

In class CII, there exist nontrivial point-gap topological phases in $d=2$.
For $d=2$, the minimal model is

$\displaystyle H(\bm{k})$
$\displaystyle=\tau_{0}\left[\sin{k_{x}}\sigma_{x}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\sigma_{y}+i\sin{k_{y}}\sigma_{0}\right],$
(S51)

with ${\cal T}=i\tau_{y}\sigma_{z}$ and ${\cal C}=i\tau_{y}\sigma_{0}$. The
point-gap topological number is the first Chern number of $iH({\bm{k}}){\cal
T}{\cal C}^{*}$. Since the above model is a stacking of Eq. (S13), this model
has the first Chern number $+2\in 2$$\mathbb{Z}$ and the BBC holds.

####  S4.1.5 class C

In class C, there exists a non-trivial point-gap topological phase in $d=3$.
The minimal model is

$\displaystyle H(\bm{k})$
$\displaystyle=\tau_{0}\left[\sin{k_{x}}\sigma_{x}+\sin{k_{y}}\sigma_{z}+\left(1-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{y}-i\sin{k_{z}}\sigma_{0}\right],\quad{\cal
C}=i\tau_{y}\sigma_{0}.$ (S52)

The point-gap topological number is equal to the 3D winding number in $d=3$
class A. Since Eq. (S52) is a stacking of Eq.(S11), the 3D winding number for
Eq. (S52) takes $+2\in 2$$\mathbb{Z}$, and the BBC holds.

####  S4.1.6 class CI

The point gap topological phase is trivial for class CI in $d=1,2,3$.

###  S4.2 Classes with SLS

So far, we have argued the BBC for point-gap topological phases in 16 classes,
all of which do not support SLS. Below we discuss the remaining 22 classes
supporting SLS in Eq.(S39). See Table S4. Without loss of generality, we
choose ${\cal S}$ as $\mu_{z}\otimes\hat{1}$ such that $H(\bm{k})$ takes the
form of

$\displaystyle H(\bm{k})=\begin{pmatrix}0&h_{+}({\bm{k}})\\\
h_{-}({\bm{k}})&0\end{pmatrix},$ (S53)

where $\mu_{i=x,y,z}$ are the Pauli matrices, $\mu_{0}$ is the 2×2 identity
matrix and $\hat{1}$ is an identity matrix acting on $h_{\pm}(\bm{k})$.

In a manner similar to the above, we prove the BBC by constructing models with
minimal point-gap topological numbers. To show the existence of boundary
states for these models under OBCs, we use the following lemmas:

###### Lemma 1

Suppose $h_{\pm}$ is diagonalizable and $[h_{+},h_{-}]=0$, and let
$|\phi_{n}\rangle$ be an eigenvector diagonalizing $h_{+}$ and $h_{-}$
simultaneously,

$\displaystyle h_{\pm}|\phi_{n}\rangle=E^{\pm}_{n}|\phi_{n}\rangle.$ (S54)

Then, we obtain the eigenvectors and eigenenergies of $H$ from those of
$h_{\pm}$,

$\displaystyle H\begin{pmatrix}c_{n}^{\pm}\sqrt{E^{+}_{n}}\ket{\phi_{n}}\\\
\pm
c_{n}^{\pm}\sqrt{E^{-}_{n}}\ket{\phi_{n}}\end{pmatrix}=\pm\sqrt{E^{+}_{n}E^{-}_{n}}\begin{pmatrix}c_{n}^{\pm}\sqrt{E^{+}_{n}}\ket{\phi_{n}}\\\
\pm c_{n}^{\pm}\sqrt{E^{-}_{n}}\ket{\phi_{n}}\end{pmatrix},$ (S55)

where $c_{n}^{\pm}$ is a constant.

###### Lemma 2

Suppose $h_{-}h_{+}$ is diagonalizable and $h_{-}$ has its inverse
$h_{-}^{-1}$, and let $|\psi_{n}\rangle$ be an eigenvector of $h_{-}h_{+}$,

$\displaystyle h_{-}h_{+}|\psi_{n}\rangle=\lambda_{n}|\psi_{n}\rangle.$ (S56)

Then, we obtain the eigenvectors and eigenenergies of $H$ as,

$\displaystyle H\begin{pmatrix}\sqrt{\lambda_{n}}h_{-}^{-1}|\psi_{n}\rangle\\\
\pm|\psi_{n}\rangle\end{pmatrix}=\pm\sqrt{\lambda_{n}}\begin{pmatrix}\sqrt{\lambda_{n}}h_{-}^{-1}|\psi_{n}\rangle\\\
\pm|\psi_{n}\rangle\end{pmatrix}.$ (S57)

These lemmas imply that if $h_{\pm}$ ($h_{-}h_{+}$) in the former (latter)
lemma does not show skin effects and supports a boundary state, then $H$ also
does. Here it should be noted that the commutation relation $[h_{-},h_{+}]=0$
and the invertibility $h_{-}^{-1}$ may depend on the boundary condition. For
instance, whereas $e^{ik_{x}}$ and $e^{-ik_{x}}$ commute with each other,
their matrix representations under the OBC do not. The invertibility of
$e^{ik_{x}}$ also depends on the boundary condition because $e^{ik_{x}}$ has a
zero mode under the OBC.

For a system satisfying the assumption for Lemma 2, we also have the following
corollary.

###### Corollary 1

If $h_{+}$ has a (right) zero eigenvalue, $h_{+}|\psi_{0}\rangle=0$, then we
have

$\displaystyle H|\Psi_{0}^{1}\rangle=0,\quad
H|\Psi_{0}^{2}\rangle=|\Psi_{0}^{1}\rangle,$ (S58)

with

$\displaystyle|\Psi_{0}^{1}\rangle=\begin{pmatrix}0\\\
|\psi_{0}\rangle\end{pmatrix},\quad|\Psi_{0}^{2}\rangle=\begin{pmatrix}h_{-}^{-1}|\psi_{0}\rangle\\\
0\end{pmatrix}.$ (S59)

The equation in Corollary implies that $H$ has a $2\times 2$ Jordan block with
zero eigenenergy

$\displaystyle H|\Psi_{0}^{i}\rangle=\sum_{j=1,2}|\Psi_{0}^{j}\rangle
J_{ji},\quad J=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix},$ (S60)

and thus it has an exceptional point. We use this result in Sec.S6.

Once we choose the basis in Eq.(S53), we only need to consider how AZ
symmetries act on $h_{\pm}$. For complex AZ + ${\cal S}$ classes, we have 3
different situations:

  1. a.

$H$ does not have any AZ symmetries, which corresponds to A + ${\cal S}$. In
this case, $h_{\pm}$ does not have any AZ symmetries.

  2. b.

CS commutes with SLS, i.e. AIII + ${\cal S}_{+}$. Without loss of generality,
we take

$\displaystyle\Gamma=\mu_{0}\otimes\gamma$ (S61)

with $\gamma$ a unitary operator acting on $h_{\pm}$, which leads to

$\displaystyle\gamma
h^{\dagger}_{\pm}({\bm{k}})\gamma^{-1}=-h_{\mp}({\bm{k}}),\quad\gamma^{2}=1.$
(S62)

Therefore, $h_{\pm}$ does not have CS, but they are not independent.

  3. c.

CS anti-commutes with SLS, i.e. AIII + ${\cal S}_{-}$. Without loss of
generality, we take

$\displaystyle\Gamma=\mu_{x}\otimes\gamma$ (S63)

with $\gamma$ a unitary operator acting on $h_{\pm}$, which leads to

$\displaystyle\gamma
h^{\dagger}_{\pm}({\bm{k}})\gamma^{-1}=-h_{\pm}({\bm{k}}),\quad\gamma^{2}=1.$
(S64)

Therefore, $h_{\pm}$ has its own CS, and they are independent.

For real AZ + ${\cal S}$ classes, we have 4 different situations:

  1. d.

TRS (if exists) commutes with SLS, while PHS (if exists) anti-commutes with
SLS: We can take generally

$\displaystyle{\cal T}=\mu_{0}\otimes T,\quad{\cal C}=\mu_{x}\otimes C,$ (S65)

where $T$ and $C$ are unitary operators acting on $h_{\pm}(\bm{k})$. In this
basis, TRS and PHS in Eq.(S4) reduce to

$\displaystyle Th^{*}_{\pm}({\bm{k}})T^{-1}$
$\displaystyle=h_{\pm}(-{\bm{k}}),\quad TT^{*}=\pm 1,$ $\displaystyle
Ch^{T}_{\pm}({\bm{k}})C^{-1}$ $\displaystyle=-h_{\pm}(-{\bm{k}}),\quad
CC^{*}=\pm 1.$ (S66)

Therefore, $h_{+}(\bm{k})$ and $h_{-}(\bm{k})$ are independent and belong to
the same real AZ class as $H(\bm{k})$. We have 8 such classes, AI + ${\cal
S}_{+}$, BDI + ${\cal S}_{+-}$, D + ${\cal S}_{-}$, DIII + ${\cal S}_{+-}$,
AII + ${\cal S}_{+}$, CII + ${\cal S}_{+-}$, C + ${\cal S}_{-}$, and CI +
${\cal S}_{+-}$.

  2. e.

TRS (if exists) anti-commutes with SLS, but PHS (if exists) commutes with SLS:
Generally, we can take the basis,

$\displaystyle{\cal T}=\mu_{x}\otimes T,\quad{\cal C}=\mu_{0}\otimes C,$ (S67)

which leads to

$\displaystyle Th^{*}_{\pm}({\bm{k}})T^{-1}$
$\displaystyle=h_{\mp}(-{\bm{k}}),\quad TT^{*}=+1,$ $\displaystyle
Ch^{T}_{\pm}({\bm{k}})C^{-1}$ $\displaystyle=-h_{\mp}(-{\bm{k}}),\quad
CC^{*}=\pm 1.$ (S68)

Thus, $h_{\pm}$ supports neither TRS nor PHS, but it retains the combination
of TRS and PHS, i.e. CS. There are 5 such independent classes, AI + ${\cal
S}_{-}$, BDI + ${\cal S}_{-+}$, D + ${\cal S}_{+}$, C + ${\cal S}_{+}$, and CI
+ ${\cal S}_{-+}$.

  3. f.

$H$ has TRS and PHS both commuting with SLS: In this case, we can take the
basis

$\displaystyle{\cal T}=\mu_{0}\otimes T,\quad{\cal C}=\mu_{0}\otimes C,$ (S69)

from which we have

$\displaystyle Th^{*}_{\pm}({\bm{k}})T^{-1}$
$\displaystyle=h_{\pm}(-{\bm{k}}),\quad TT^{*}=\pm 1,$ $\displaystyle
Ch^{T}_{\pm}({\bm{k}})C^{-1}$ $\displaystyle=-h_{\mp}(-{\bm{k}}),\quad
CC^{*}=\pm 1.$ (S70)

Thus, $h_{\pm}$ may retain only TRS. (If $H$ does not have TRS, $h_{\pm}$ has
no symmetry.) We have 4 such independent classes: BDI + ${\cal S}_{++}$, DIII
+ ${\cal S}_{++}$, CII + ${\cal S}_{++}$, and CI + ${\cal S}_{++}$.

  4. g.

$H$ has TRS and PHS both anti-commuting with SLS: Taking the basis

$\displaystyle{\cal T}=\mu_{x}\otimes T,\quad{\cal C}=\mu_{x}\otimes C,$ (S71)

we have

$\displaystyle Th^{*}_{\pm}({\bm{k}})T^{-1}$
$\displaystyle=h_{\mp}(-{\bm{k}}),\quad TT^{*}=\pm 1,$ $\displaystyle
Ch^{T}_{\pm}({\bm{k}})C^{-1}$ $\displaystyle=-h_{\pm}(-{\bm{k}}),\quad
CC^{*}=\pm 1,$ (S72)

so $h_{\pm}$ supports only PHS. We have such 2 independent classes: BDI +
${\cal S}_{--}$, and CI + ${\cal S}_{--}$.

Below, we construct the minimal models for these 22 classes.

Case |  | AZ class |  |  SLS |  |  $d=1$ |  $d=2$ |  $d=3$
---|---|---|---|---|---|---|---|---
a |  | A |  | $\mathcal{S}$ |  |  $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbb{Z}\oplus\mathbb{Z}\to\mathbb{Z}[1,-1]$ [Sec.S4.2.1] | $0$ |  $\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbb{Z}\oplus\mathbb{Z}$ [Sec.S4.2.1]
b |  | AIII |  | $\mathcal{S}_{+}$ |  |  $\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbb{Z}$ [Sec.S4.2.2] | $0$ |  $\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbb{Z}$ [Sec.S4.2.2]
c |  | AIII |  | $\mathcal{S}_{-}$ |  | $0$ |  $\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbb{Z}\oplus\mathbb{Z}$ [Sec.S4.2.3] | $0$
d |  | AI |  | $\mathcal{S}_{+}$ |  |  $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbb{Z}\oplus\mathbb{Z}\to\mathbb{Z}[1,-1]$ [Sec.S4.2.4] | $0$ | $0$
|  | BDI |  | $\mathcal{S}_{+-}$ |  |  $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\to\mathbb{Z}_{2}[1,1]$ [Sec.S4.2.5] |  $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbb{Z}\oplus\mathbb{Z}\to\mathbb{Z}[2,0]\oplus\mathbb{Z}[1,-1]$ [Sec.S4.2.5] | $0$
|  | D |  | $\mathcal{S}_{-}$ |  |  $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\to\mathbb{Z}_{2}[1,1]$ [Sec.S4.2.6] |  $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\to\mathbb{Z}_{2}[1,1]$ [Sec.S4.2.6] |  $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbb{Z}\oplus\mathbb{Z}\to\mathbb{Z}[2,0]\oplus\mathbb{Z}[1,-1]$ [Sec.S4.2.6]
|  | DIII |  | $\mathcal{S}_{+-}$ |  | $0$ |  $\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ [Sec.S4.2.7] |  $\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ [Sec.S4.2.7]
|  | AII |  | $\mathcal{S}_{+}$ |  |  $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2\mathbb{Z}\oplus 2\mathbb{Z}\to 2\mathbb{Z}[1,-1]$ [Sec.S4.2.8] | $0$ |  $\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ [Sec.S4.2.8]
|  | CII |  | $\mathcal{S}_{+-}$ |  | $0$ |  $\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}2\mathbb{Z}\oplus 2\mathbb{Z}$ [Sec.S4.2.9] | $0$
|  | C |  | $\mathcal{S}_{-}$ |  | $0$ | $0$ |  $\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}2\mathbb{Z}\oplus 2\mathbb{Z}$ [Sec.S4.2.10]
|  | CI |  | $\mathcal{S}_{+-}$ |  | $0$ | $0$ | $0$
e |  | AI |  | $\mathcal{S}_{-}$ |  |  $\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbb{Z}\to 0$ [Sec.S4.2.12] | $0$ |  $\mathbb{Z}$ [Sec.S4.2.12]
|  | BDI |  | $\mathcal{S}_{-+}$ |  | $0$ |  $\mathbb{Z}$ [Sec.S4.2.13] | $0$
|  | D |  | $\mathcal{S}_{+}$ |  |  $\mathbb{Z}$ [Sec.S4.2.14] | $0$ |  $\mathbb{Z}$ [Sec.S4.2.14]
|  | C |  | $\mathcal{S}_{+}$ |  |  $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbb{Z}\to 2\mathbb{Z}$ [Sec.S4.2.15] | $0$ |  $\mathbb{Z}$ [Sec.S4.2.15]
|  | CI |  | $\mathcal{S}_{-+}$ |  | $0$ |  $\mathbb{Z}$ [Sec.S4.2.16] | $0$
f |  | BDI |  | $\mathcal{S}_{++}$ |  |  $\mathbb{Z}$ [Sec.S4.2.17] | $0$ | $0$
|  | DIII |  | $\mathcal{S}_{++}$ |  | 2$\mathbb{Z}$ [Sec.S4.2.18] | $0$ |  $\mathbb{Z}_{2}$ [Sec.S4.2.18]
|  | CII |  | $\mathcal{S}_{++}$ |  | 2$\mathbb{Z}$ [Sec.S4.2.19] | $0$ |  $\mathbb{Z}_{2}$ [Sec.S4.2.19]
|  | CI |  | $\mathcal{S}_{++}$ |  |  $\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbb{Z}\to 2\mathbb{Z}$ [Sec.S4.2.20] | $0$ | $0$
g |  | BDI |  | $\mathcal{S}_{--}$ |  |  $\mathbb{Z}_{2}$ [Sec.S4.2.21] |  $\mathbb{Z}_{2}$ [Sec.S4.2.21] |  $\mathbb{Z}$ [Sec.S4.2.21]
|  | CI |  | $\mathcal{S}_{--}$ |  | $0$ | $0$ | 2$\mathbb{Z}$ [Sec.S4.2.22]
Table S4: Point-gap topological phases in 22 classes with SLS. For topological
numbers colored red or blue, the left specifies topological numbers under PBCs
and the right specifies those under OBCs. For topological numbers colored
green, the classification under OBCs coincides with those under PBCs. The
section numbers for the proof of the classification under OBCs and the BBC are
shown for each topological number.

####  S4.2.1 class A + ${\cal S}$ (case a.)

This class has nontrivial point-gap topological phases in $d=1,3$. For $d=1$,
the point-gap topological number under the PBC is a pair of the 1D winding
numbers for $h_{\pm}(k_{x})$,
$(w_{1}[h_{+}],w_{1}[h_{-}])\in\mathbb{Z}\oplus\mathbb{Z}$ Kawabata  _et al._
(2019b), where $w_{1}[h_{\pm}]$ is given by

$\displaystyle w_{1}[h_{\pm}]=\int_{0}^{2\pi}\dfrac{dk_{x}}{2\pi
i}~{}\text{Tr}[h_{\pm}^{-1}\partial_{k_{x}}h_{\pm}].$ (S73)

We also have the 1D winding number $w_{1}[H]$ of $H(k_{x})$,
$w_{1}[H]=w_{1}[h_{+}]+w_{1}[h_{-}]$, which gives skin effects. Therefore, we
have $w_{1}[h_{+}]+w_{1}[h_{-}]=0$ under the OBC Zhang  _et al._ (2020); Okuma
_et al._ (2020). As a result, under the OBC, the possible point-gap
topological number is $(w_{1}[h_{+}],-w_{1}[h_{+}])$ and the
$\mathbb{Z}\oplus\mathbb{Z}$ classification changes to the $\mathbb{Z}$ one.
Here we use the notation $\mathbb{Z}[1,-1]$ to represent the latter
$\mathbb{Z}$ classification because its generator is given by the element
$(1,-1)\in\mathbb{Z}\oplus\mathbb{Z}$ of the former. (We often use the same
notation below.) The minimal model of this phase is

$\displaystyle h_{+}(k_{x})=-i\text{e}^{ik_{x}},$ $\displaystyle
h_{-}(k_{x})=i\text{e}^{-ik_{x}}.$ (S74)

which has $(+1,-1)\in\mathbb{Z}\oplus\mathbb{Z}$. This model gives

$\displaystyle H(k_{x})=\sin k_{x}\mu_{x}+\cos k_{x}\mu_{y},\quad{\cal
S}=\mu_{z},$ (S75)

which supports a boundary zero mode under the OBC. Thus, we have the BBC.

For $d=3$, the classification of the point-gap topological phase is
$\mathbb{Z}\oplus\mathbb{Z}$, of which topological numbers are a pair of the
3D winding numbers of $h_{\pm}(\bm{k})$,
$(w_{3}[h_{+}],w_{3}[h_{-}])\in\mathbb{Z}\oplus\mathbb{Z}$ Kawabata  _et al._
(2019b). Thus, we have two generators, $(1,0)$ and $(0,1)$. The minimal model
for the generator $(1,0)$ is

$\displaystyle
h_{+}(\bm{k})=\sin{k_{x}}\sigma_{x}+\sin{k_{y}}\sigma_{y}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{z}+i\sin{k_{z}}\sigma_{0},$
$\displaystyle h_{-}(\bm{k})=\sigma_{0}.$ (S76)

We also have the minimal model for another generator
$(0,+1)\in\mathbb{Z}\oplus\mathbb{Z}$, by swapping $h_{+}(\bm{k})$ and
$h_{-}(\bm{k})$ in Eq. (S4.2.1). Under the xOBC, we have $[h_{+},h_{-}]=0$ and
either $h_{+}$ or $h_{-}$ supports boundary states. Thus we have the BBC.

####  S4.2.2 class AIII + ${\cal S}_{+}$ (case b.)

In this class, there exist nontrivial point-gap topological phases
characterized by the winding numbers of $h_{+}$ in $d=1,3$. For $d=1$, the
minimal model is

$\displaystyle h_{+}(k_{x})=\text{e}^{ik_{x}},$ $\displaystyle
h_{-}(k_{x})=-\gamma h^{\dagger}_{+}(k_{x})\gamma^{-1}=-\text{e}^{-ik_{x}},$
(S77)

where $\gamma=1$ and $w_{1}[h_{+}]=+1\in\mathbb{Z}$. From this, we have
$H(k_{x})=i(\sin k_{x}\mu_{x}+\cos k_{x}\mu_{y})$ with ${\cal S}=\mu_{z}$ and
$\Gamma=1$. This model has a boundary state under the OBC, and thus we have
the BBC.

For $d=3$, we have the minimal model with $\gamma=\sigma_{0}$ and
$w_{3}[h_{+}]=+1\in\mathbb{Z}$ as follows.

$\displaystyle
h_{+}(\bm{k})=\sin{k_{x}}\sigma_{x}+\sin{k_{y}}\sigma_{y}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{z}+i\sin{k_{z}}\sigma_{0},$
$\displaystyle h_{-}(\bm{k})=-\gamma
h^{\dagger}_{+}({\bm{k}})\gamma^{-1}=-\sin{k_{x}}\sigma_{x}-\sin{k_{y}}\sigma_{y}-\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{z}+i\sin{k_{z}}\sigma_{0}.$
(S78)

Note that $h_{+}({\bm{k}})$ concides with Eq.(S11) so $h_{\pm}$ has surface
states under the xOBC. Thus, from $[h_{+},h_{-}]=0$, we have the BBC.

####  S4.2.3 class AIII + ${\cal S}_{-}$ (case c.)

In this class, $h_{\pm}$ has its own CS so belongs to class AIII. Thus, we
have a $\mathbb{Z}\oplus\mathbb{Z}$ point-gap topological phase in $d=2$. A
pair of the first Chern numbers for $ih_{\pm}(\bm{k})\gamma$,
$(Ch_{1}[ih_{+}\gamma],Ch_{1}[ih_{-}\gamma])$$\in\mathbb{Z}\oplus\mathbb{Z}$,
is the point-gap topological numberKawabata  _et al._ (2019b). Using the model
in Eq. (S13), we obtain the minimal model with the topological number
$(+1,0)\in\mathbb{Z}\oplus\mathbb{Z}$ as

$\displaystyle
h_{+}(\bm{k})=\sin{k_{x}}\sigma_{x}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\sigma_{y}+i\sin{k_{y}}\sigma_{0},$
$\displaystyle h_{-}(\bm{k})=i\sigma_{0},\quad\gamma=\sigma_{z}.$ (S79)

We also have another minimal model with topological number
$(0,+1)\in\mathbb{Z}\oplus\mathbb{Z}$ by exchanging $h_{+}(\bm{k})$ and
$h_{-}(\bm{k})$ in Eq. (S4.2.3). In both cases, $h_{+}$ and $h_{-}$ commute,
and either $h_{-}$ or $h_{-}$ has a boundary state. Thus, the BBC holds.

####  S4.2.4 class AI + ${\cal S}_{+}$ (case d.)

In this class, $h_{\pm}$ belongs to class AI, and there is a
$\mathbb{Z}\oplus\mathbb{Z}$ point-gap topological phase in $d=1$. The point-
gap topological number under the PBC is a pair of the 1D winding number
$(w_{1}[h_{+}],w_{1}[h_{-}])$, where $w_{1}[h_{\pm}]$ is given by Eq.(S73). In
a manner similar to the $d=1$ class A + ${\cal S}$ case, one can avoid skin
effects only when $w_{1}[h_{+}]+w_{1}[h_{-}]=0$, so the point-gap topological
phase under the OBC is classified as $\mathbb{Z}[1,-1]$. The minimal model is

$\displaystyle h_{+}(k_{x})=\text{e}^{ik_{x}}\quad
h_{-}(k_{x})=\text{e}^{-ik_{x}},\quad T=1.$ (S80)

This model gives $H(k_{x})=\cos k_{x}\mu_{x}-\sin k_{x}\mu_{y}$ with ${\cal
T}=\mu_{0}$ and ${\cal S}=\mu_{z}$, which has a zero energy boundary state.
Thus we have the BBC.

####  S4.2.5 class BDI + ${\cal S}_{+-}$ (case d.)

In class $\text{BDI}+\mathcal{S}_{+-}$, $h_{\pm}$ belongs to class BDI, which
has nontrivial point-gap topological phases in $d=1,2$. For $d=1$, the point-
gap topological number under the PBC is a pair of the $\mathbb{Z}_{2}$
invariants for $h_{\pm}(k_{x})$,
$(\nu[h_{+}],\nu[h_{-}])\in\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ Kawabata  _et
al._ (2019b),

$\displaystyle(-1)^{\nu_{1}[h_{\pm}]}=\text{sgn}\left\\{\dfrac{\operatorname{Pf}[h_{\pm}(\pi)C]}{\operatorname{Pf}[h_{\pm}(0)C]}\right\\},$
(S81)

which coincides with Eq. (S41) in OBCs. As the Hamiltonian $H(k_{x})$ has TRS†
for class AII†,

$\displaystyle\mathcal{T}_{\text{AII}^{\dagger}}H^{\text{T}}(k_{x})\mathcal{T}_{\text{AII}^{\dagger}}^{-1}=H(-k_{x}),\quad\mathcal{T}_{\text{AII}^{\dagger}}\mathcal{T}_{\text{AII}^{\dagger}}^{*}=-1,$
(S82)

where $\mathcal{T}_{\text{AII}^{\dagger}}$ is given by the combination of
${\cal C}$ and ${\cal S}$, i.e.
$\mathcal{T}_{\text{AII}^{\dagger}}=\mathcal{SC}=i\mu_{y}\otimes C$, we also
have the $\mathbb{Z}_{2}$ number for $H(k_{x})$ in class AII†,

$\displaystyle(-1)^{\nu_{1}[H]}=\text{sgn}\left\\{\dfrac{\operatorname{Pf}[H(\pi)\mathcal{C}_{\text{AII}^{\dagger}}]}{\operatorname{Pf}[H(0)\mathcal{C}_{\text{AII}^{\dagger}}]}~{}\times~{}\text{exp}\left[-\dfrac{1}{2}\int_{k_{x}=0}^{k_{x}=\pi}\operatorname{\it{d}}\operatorname{log}\operatorname{det}\left[H(k_{x})\mathcal{C}_{\text{AII}^{\dagger}}\right]\right]\right\\},$
(S83)

which satisfies
$\nu_{1}[H]=\nu_{1}[h_{+}]+\nu_{1}[h_{-}]~{}(\text{mod.}~{}2)$. Since the
$\mathbb{Z}_{2}$ number results in the symmetry-skin effect Okuma  _et al._
(2020), we have the condition $\nu_{1}[h_{+}]+\nu_{1}[h_{-}]=0$ (mod.2) to
avoid it. Therefore, the point-gap topological phase under OBC is
$\mathbb{Z}_{2}[1,1]$. The minimal model is

$\displaystyle h_{+}(k_{x})=h_{-}(k_{x})=i\left(\sin
k_{x}\sigma_{x}-\cos{k_{x}}\sigma_{y}\right),$ (S84)

which shows the BBC.

For $d=2$, the point-gap topological phase is $\mathbb{Z}\oplus\mathbb{Z}$,
which topological number is given by a pair of the first Chern numbers,
$(Ch_{1}[ih_{+}\gamma],Ch_{1}[ih_{-}\gamma])$ Kawabata  _et al._ (2019b).
However, using the dimension reduction discussed in the main text, we have the
symmetry-protected skin effect when the first Chern number of $iH\Gamma$,
$Ch_{1}[iH\Gamma]=Ch_{1}[ih_{+}\gamma]+Ch_{1}[ih_{-}\gamma]$ is odd.
Therefore, the point-gap topological phase becomes
$\mathbb{Z}[1,-1]\oplus\mathbb{Z}[2,0]$ under OBCs.

Using Eq.(S42) for $h_{\pm}$, we can construct the minimal model with the
topological number $(1,-1)$ as

$\displaystyle h_{+}(\bm{k})=-h_{-}(\bm{k})=i\left[\sin k_{x}\sigma_{x}+\sin
k_{y}\sigma_{z}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\sigma_{y}\right],\quad
C=T=\sigma_{0}$ (S85)

In a similar manner, another minimal model with the topological number $(2,0)$
is given by

$\displaystyle h_{+}({\bm{k}})=i\tau_{0}\left[\sin k_{x}\sigma_{x}+\sin
k_{y}\sigma_{z}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\sigma_{y}\right],$
$\displaystyle h_{-}({\bm{k}})=i\tau_{y}\sigma_{0},\quad
T=C=\tau_{0}\sigma_{0}.$ (S86)

In both models, $h_{+}$ and $h_{-}$ commute with each other and they have
boundary states when their first Chern numbers are non-zero. Therefore, the
BBC holds.

####  S4.2.6 class D + ${\cal S}_{-}$ (case d.)

In class $\text{D}+\mathcal{S}_{-}$, $h_{\pm}$ belongs to class D, which
realizes non-trivial point-gap topological phases in $d=1,2,3$. For $d=1$,
$h_{\pm}$ supports the $\mathbb{Z}_{2}$ point-gap topological number
$\nu_{1}[h_{\pm}]$ in Eq.(S45), and thus the classification under the PBC is
given by
$(\nu_{1}[h_{+}],\nu_{1}[h_{-}])\in\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$. Then,
as $H({\bm{k}})$ has TRS† of class AII†,

$\displaystyle{\cal T}_{\rm AII^{\dagger}}H^{T}({\bm{k}}){\cal T}_{\rm
AII^{\dagger}}^{-1}=H(-{\bm{k}})$ (S87)

with $\mathcal{T}_{\text{AII}^{\dagger}}=\mathcal{SC}$, we have the 1D
$\mathbb{Z}_{2}$ number $\nu_{1}[H]$ for the symmetry-protected skin effect,
so we have the condition $\nu_{1}[H]=\nu_{1}[h_{+}]+\nu_{1}[h_{-}]=0$ (mod.2)
to avoid the skin effects. As a result, the point-gap topological phase under
the OBC becomes $\mathbb{Z}_{2}[1,1]$. From Eq.(S44), the minimal model is

$\displaystyle h_{+}(k_{x})=h_{-}(k_{x})=\sin
k_{x}\sigma_{x}+\cos{k_{x}}\sigma_{y},\quad C=\sigma_{0},$ (S88)

which has $(1,1)\in\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$. Since it supports a
boundary state, we have the BBC.

For $d=2$, the point-gap topological number under the PBC is a pair of
$(\nu_{2}[h_{+}],\nu_{2}[h_{-}])$ in Eq. (S47) Kawabata  _et al._ (2019b).
Then, from TRS† in the above, the dimension reduction argument in the main
text indicates that there arises the symmetry-protected skin effect when
$\nu_{2}[h_{+}]+\nu_{2}[h_{-}]=1$ (mod.2). Hence, the topological
classification under OBCs becomes $\mathbb{Z}_{2}[1,1]$. The minimal model
with topological number $(1,1)$ is

$\displaystyle h_{+}(\bm{k})=h_{-}(\bm{k})=\sin
k_{x}\sigma_{x}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\sigma_{y}+i\sin
k_{y}\sigma_{0},\quad C=\sigma_{0},$ (S89)

which shows the BBC.

In $d=3$, a pair of the 3D winding numbers $(w_{3}[h_{+}],w_{3}[h_{-}])$
characterize the point-gap topological phase under the PBC Kawabata  _et al._
(2019b). In a manner similar to the above, we have the symmetry-protected skin
effect when $w_{3}[h_{+}]+w_{3}[h_{-}]$ is odd. Consequently, the
classification under OBCs becomes $\mathbb{Z}[1,-1]\oplus\mathbb{Z}[2,0]$. The
minimal model with topological number $(1,-1)$ is

$\displaystyle
h_{+}(\bm{k})=\sin{k_{x}}\sigma_{x}+\sin{k_{y}}\sigma_{z}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{y}-i\sin{k_{z}}\sigma_{0},$
$\displaystyle
h_{-}(\bm{k})=\sin{k_{x}}\sigma_{x}+\sin{k_{y}}\sigma_{z}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{y}+i\sin{k_{z}}\sigma_{0},\quad
C=\sigma_{0},$ (S90)

and the minimal model with topological number $(2,0)$ is

$\displaystyle
h_{+}(\bm{k})=\tau_{0}\left(\sin{k_{x}}\sigma_{x}+\sin{k_{y}}\sigma_{z}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{y}-i\sin{k_{z}}\sigma_{0}\right),$
$\displaystyle h_{-}(\bm{k})=\tau_{y}\sigma_{0},\quad C=\tau_{0}\sigma_{0},$
(S91)

both of which show the BBC.

####  S4.2.7 class DIII + ${\cal S}_{+-}$ (case d.)

The off-diagonal components $h_{\pm}$ belong to class DIII, so we have
nontrivial point-gap topological phases in $d=2,3$. Under the PBC, these
phases are characterized by a pair of the $\mathbb{Z}_{2}$ invariants in
$d=2,3$ class DIII Kawabata  _et al._ (2019b) for $h_{\pm}(\bm{k})$.

For $d=2$, the minimal model is

$\displaystyle
h_{+}(\bm{k})=\sin{k_{x}}\tau_{x}\sigma_{x}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\tau_{x}\sigma_{y}+i\sin{k_{y}}\tau_{0}\sigma_{0},$
$\displaystyle h_{-}(\bm{k})=i\tau_{y}\sigma_{z},\quad
T=i\tau_{y}\sigma_{0},\quad C=\tau_{0}\sigma_{0},$ (S92)

which takes $(1,0)\in\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$. We can also
construct another minimal model with topological number
$(0,1)\in\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ by exchanging $h_{+}(\bm{k})$ and
$h_{-}(\bm{k})$ in Eq. (S4.2.7). Since $h_{+}$ in Eq.(S4.2.7) has a boundary
state and commutes with $h_{-}$, we have the BBC.

Similarly, for $d=3$, the minimal model with topological number $(1,0)$ is

$\displaystyle
h_{+}(\bm{k})=\sin{k_{x}}\tau_{x}\sigma_{x}+\sin{k_{y}}\tau_{x}\sigma_{z}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\tau_{x}\sigma_{y}+i\sin{k_{z}}\tau_{0}\sigma_{0},$
$\displaystyle h_{-}(\bm{k})=i\tau_{y}\sigma_{0},\quad
T=i\tau_{y}\sigma_{0},\quad C=\tau_{0}\sigma_{0},$ (S93)

We also have another minimal model with topological number $(0,1)$ by swapping
$h_{+}(\bm{k})$ and $h_{-}(\bm{k})$ in Eq. (S4.2.7). Both models show the BBC
from Lemma 2.

####  S4.2.8 class AII + ${\cal S}_{+}$ (case d.)

Since $h_{\pm}$ belongs to class AII, there are nontrivial point-gap
topological phases in $d=1,3$. For $d=1$, the point-gap topological phase is
$2\mathbb{Z}\oplus 2\mathbb{Z}$ Kawabata  _et al._ (2019b), which topological
number is given by a pair of the 1D winding numbers $w_{1}[h_{\pm}]$. To avoid
skin effects, the 1D winding number $w_{1}[H]$ of $H$ should vanish, so we
have the condition $w_{1}[H]=w_{1}[h_{+}]+w_{1}[h_{-}]=0$. Thus, the
classification under the OBC is $2\mathbb{Z}[1,-1]$. The minimal model is

$\displaystyle h_{+}(k_{x})=i\text{e}^{ik_{x}}\sigma_{y},\quad
h_{-}(k_{x})=-i\text{e}^{-ik_{x}}\sigma_{y},\quad T=i\sigma_{y},$ (S94)

which has $(+2,-2)\in 2\mathbb{Z}\oplus 2\mathbb{Z}$ and shows the BBC.

For $d=3$, the point-gap topological number under the PBC is a pair of the
$\mathbb{Z}_{2}$ invariants in class AII Kawabata  _et al._ (2019b) for
$h_{\pm}(\bm{k})$. We have the minimal model with
$(1,0)\in\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$,

$\displaystyle h_{+}(k_{x})=i\left[\sin k_{x}\tau_{x}\sigma_{x}+\sin
k_{y}\tau_{x}\sigma_{y}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\tau_{0}\sigma_{z}\right]-\sin
k_{z}\tau_{z}\sigma_{z},$ $\displaystyle h_{-}(k_{x})=\tau_{0}\sigma_{0},\quad
T=i\tau_{z}\sigma_{y}.$ (S95)

We can obtain another minimal model with topological number $(0,1)$ by
exchanging $h_{+}(\bm{k})$ and $h_{-}(\bm{k})$ in Eq. (S4.2.8). From them, the
BBC holds.

####  S4.2.9 class CII + ${\cal S}_{+-}$ (case d.)

In class $\text{CII}+\mathcal{S}_{+-}$, $h_{\pm}$ belongs to class CII, so
there exist nontrivial point-gap topological phases in $d=2$. The point-gap
topological number is a pair of the first Chern numbers
$(Ch_{1}[ih_{+}TC^{*}],Ch_{1}[ih_{-}TC^{*}])\in 2\mathbb{Z}\oplus
2\mathbb{Z}$. The minimal model with topological number $(2,0)$ is

$\displaystyle
h_{+}(\bm{k})=\tau_{0}\left[\sin{k_{x}}\sigma_{x}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\sigma_{y}+i\sin{k_{y}}\sigma_{0}\right],$
$\displaystyle h_{-}(\bm{k})=i\tau_{y}\sigma_{0},\quad
T=i\tau_{y}\sigma_{z},\quad C=i\tau_{y}\sigma_{0},$ (S96)

and another one with topological number $(0,2)$ is obtained by switching
$h_{+}(\bm{k})$ and $h_{-}(\bm{k})$ in Eq. (S4.2.9). From them, we have the
BBC.

####  S4.2.10 class C + ${\cal S}_{-}$ (case d.)

$h_{\pm}$ belongs to class C, and we have nontrivial point-gap topological
phases in $d=3$. A pair of the 3D winding numbers
$(w_{3}[h_{+}],w_{3}[h_{-}])\in 2\mathbb{Z}\oplus 2\mathbb{Z}$ characterize
the point-gap topological phases under the PBC Kawabata  _et al._ (2019b). The
minimal model with topological number $(2.0)$ is

$\displaystyle
h_{+}(\bm{k})=\tau_{0}\left[\sin{k_{x}}\sigma_{x}+\sin{k_{y}}\sigma_{z}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{y}-i\sin{k_{z}}\sigma_{0}\right],$
(S97) $\displaystyle h_{-}(\bm{k})=i\tau_{y}\sigma_{0},\quad
C=i\tau_{y}\sigma_{0},$ (S98)

and we also have aother minimal model with $(0,+2)\in 2\mathbb{Z}\oplus
2\mathbb{Z}$ similarly. These models show the BBC.

####  S4.2.11 class CI + ${\cal S}_{+-}$ (case d.)

Since the point gap topological phases are trivial for class CI in $d=1,2,3$,
so are also for this class.

####  S4.2.12 class AI + ${\cal S}_{-}$ (case e.)

In this class, $h_{+}$ belongs to class A, and TRS exchanges $h_{+}$ and
$h_{-}$. Thus, we have nontrivial point-gap topological phases in $d=1,3$
under the PBC. For $d=1$, the corresponding point-gap topological number is
the 1D winding number $w_{1}[h_{+}]$ for $h_{+}(k_{x})$ Kawabata  _et al._
(2019b). We can also define the 1D winding number $w_{1}[H]$ for $H(k_{x})$,
which satisfies $w_{1}[H]=w_{1}[h_{+}]+w_{1}[h_{-}]$, where $w_{1}[h_{-}]$ is
the 1D winding number for $h_{-}$. Then, from
$h_{-}(k_{x})=Th^{*}_{+}(-k_{x})T^{-1}$, we have $w_{1}[h_{+}]=w_{1}[h_{-}]$.
Thus, to avoid skin effect, we need $w_{1}[h_{+}]=0$, which implies no point-
gap topological phase survives under the OBC.

For $d=3$, we have a nontrivial point-gap topological phase under the OBC. The
minimal model is

$\displaystyle
h_{+}(\bm{k})=\sin{k_{x}}\sigma_{x}+\sin{k_{y}}\sigma_{y}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{z}+i\sin{k_{z}}\sigma_{0},$
$\displaystyle
h_{-}(\bm{k})=Th^{*}_{+}(-{\bm{k}})T^{-1}=-\sin{k_{x}}\sigma_{x}-\sin{k_{y}}\sigma_{y}-\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{z}+i\sin{k_{z}}\sigma_{0},$
(S99)

where $T=\sigma_{x}$. As $h_{\pm}$ has a boundary state and $[h_{+},h_{-}]=0$,
the BBC holds.

####  S4.2.13 class BDI + ${\cal S}_{-+}$ (case e.)

In class $\text{BDI}+{\cal S}_{-+}$, $h_{+}(\bm{k})$ has CS, $\gamma
h^{\dagger}_{+}({\bm{k}})\gamma^{\dagger}=-h_{+}({\bm{k}})$ with
$\gamma=TC^{*}$, and TRS and PHS exchange $h_{+}$ and $h_{-}$. So $h_{+}$
belongs to class AIII. Thus, we have a non-trivial point-gap topological phase
in $d=2$. The Chern number for $ih_{+}(\bm{k})\gamma$ characterizes the point-
gap topological phaseKawabata  _et al._ (2019b). The minimal model is

$\displaystyle
h_{+}(\bm{k})=\sin{k_{x}}\sigma_{x}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\sigma_{y}+i\sin{k_{y}}\sigma_{0},$
$\displaystyle
h_{-}(\bm{k})=Th^{*}_{+}(-{\bm{k}})T^{-1}=-Ch^{\text{T}}_{+}(-{\bm{k}})C^{-1}=\sin{k_{x}}\sigma_{x}+\left(1-\sum_{i=x,y}\cos{k_{i}}\right)\sigma_{y}+i\sin{k_{y}}\sigma_{0},$
(S100)

with $T=\gamma=\sigma_{z},C=\sigma_{0}$. $h_{+}$ and $h_{-}$ in the minimal
model commute and they support boundary states. Thus, we have the BBC.

####  S4.2.14 class D + ${\cal S}_{+}$ (case e.)

In this class, $h_{+}(\bm{k})$ belongs to class A and PHS exchanges $h_{+}$
and $h_{-}$. The system has non-trivial topological phases in $d=1,3$. For
$d=1$, the corresponding point-gap topological number under the PBC is the 1D
winding number $w_{1}[h_{+}]\in\mathbb{Z}$ for $h_{+}(k_{x})$. We can also
consider the 1D winding number $w_{1}[H]$ for $H(k_{x})$,
$w_{1}[H]=w_{1}[h_{+}]+w_{1}[h_{-}]$, where $w_{1}[h_{-}]$ is the 1D winding
number for $h_{-}(k_{x})$. However, the relation
$h_{-}(k_{x})=-Ch^{\text{T}}_{+}(-k_{x})C^{-1}$ leads
$w_{1}[h_{+}]=-w_{1}[h_{-}]$, so $w_{1}[H]$ always vanishes. Thus, no skin
effect occurs. The minimal model with $w_{1}[h_{+}]=1$ is

$\displaystyle h_{+}(k_{x})=\text{e}^{ik_{x}},\quad
h_{-}(k_{x})=-Ch^{\text{T}}_{+}(-k_{x})C^{-1}=-\text{e}^{-ik_{x}},$ (S101)

where $C=1$. This model gives $H(k_{x})=i(\sin k_{x}\mu_{x}+\cos
k_{x}\mu_{y})$, which supports a boundary state. Thus the BBC holds.

In $d=3$, the point-gap topological number is the 3D winding number
$w_{3}[h_{+}]\in\mathbb{Z}$. The minimal model is

$\displaystyle
h_{+}(\bm{k})=\sin{k_{x}}\sigma_{x}+\sin{k_{y}}\sigma_{y}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{z}+i\sin{k_{z}}\sigma_{0},$
$\displaystyle
h_{-}(\bm{k})=-Ch^{\text{T}}_{+}(-{\bm{k}})C^{-1}=\sin{k_{x}}\sigma_{x}+\sin{k_{y}}\sigma_{y}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{z}+i\sin{k_{z}}\sigma_{0},$
(S102)

where $C=\sigma_{x}$ and $w_{3}[h_{+}]=+1$. This model shows the BBC.

####  S4.2.15 class C + ${\cal S}_{+}$ (case e.)

For this class, $h_{+}(\bm{k})$ belongs to class A, and PHS exchanges $h_{+}$
and $h_{-}$. Thus, point-gap topological phases under the PBC are $\mathbb{Z}$
in $d=1,3$. For $d=1$, the point-gap topological number is the 1D winding
number $w_{1}[h_{+}]$ for $h_{+}(k_{x})$. Because $H({\bm{k}})$ has TRS for
class AII†, ${\cal T}_{\rm AII^{\dagger}}H^{T}({\bm{k}}){\cal T}_{\rm
AII^{\dagger}}^{-1}=H(-{\bm{k}})$ with ${\cal T}_{\rm AII^{\dagger}}={\cal
S}{\cal C}$, we also have the 1D $\mathbb{Z}_{2}$ number $\nu_{1}[H]$
responsible for the symmetry-protected skin effect. From the straightforward
calculation, we can show that $\nu_{1}[H]=w_{1}[h_{+}]$ (mod.2), thus only the
even part of $w_{1}[h_{+}]$ survives under the OBC. Thus, the classification
under the OBC becomes $2\mathbb{Z}$. The minimal model with $w_{1}[h_{+}]=2$
is

$\displaystyle h_{+}(k_{x})=\text{e}^{ik_{x}}\sigma_{0},\quad
h_{-}(k_{x})=-Ch^{\text{T}}_{+}(-k_{x})C^{-1}=-\text{e}^{-ik_{x}}\sigma_{0},$
(S103)

with $C=i\sigma_{y}$. This model gives $H(k_{x})=i(\sin k_{x}\mu_{x}+\cos
k_{x}\mu_{y})\sigma_{0}$, which support a boundary state. Thus, we have the
BBC.

For $d=3$, the point-gap topological number is the 3D winding number
$w_{3}[h_{+}]\in\mathbb{Z}$ for $h_{+}$. Whereas $H({\bm{k}})$ has TRS for
class AII† in the above, the 3D winding number $w_{3}[H]$ of $H$ takes an even
number because $H({\bm{k}})$ has PHS for class C at the same time. Therefore,
no symmetry-protected skin effect occurs by the dimensional reduction. The
minimal model with $w_{3}[h_{+}]=1$ is

$\displaystyle
h_{+}(\bm{k})=\dfrac{\tau_{0}+\tau_{z}}{2}\left[\sin{k_{x}}\sigma_{x}+\sin{k_{y}}\sigma_{y}+\left(2-\sum_{i=x,y,z}\cos{k_{i}}\right)\sigma_{z}+i\sin{k_{z}}\sigma_{0}\right]+\dfrac{\tau_{0}-\tau_{z}}{2}\sigma_{0},$
$\displaystyle