# Universal non-Hermitian skin effect in two and higher dimensions

Kai Zhang  Beijing National Laboratory for Condensed Matter Physics and
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
University of Chinese Academy of Sciences, Beijing 100049, China  Zhesen Yang
<EMAIL_ADDRESS>Kavli Institute for Theoretical Sciences, Chinese Academy of
Sciences, Beijing 100190, China  Chen Fang<EMAIL_ADDRESS>Beijing National
Laboratory for Condensed Matter Physics and Institute of Physics, Chinese
Academy of Sciences, Beijing 100190, China  Kavli Institute for Theoretical
Sciences, Chinese Academy of Sciences, Beijing 100190, China  Songshan Lake
Materials Laboratory, Dongguan, Guangdong 523808, China

###### Abstract

Skin effect, experimentally discovered in one dimension, describes the
physical phenomenon that on an open chain, an extensive number of eigenstates
of a non-Hermitian hamiltonian are localized at the end(s) of the chain. Here
in two and higher dimensions, we establish a theorem that the skin effect
exists, if and only if periodic-boundary spectrum of the hamiltonian covers a
finite area on the complex plane. This theorem establishes the universality of
the effect, because the above condition is satisfied in almost every generic
non-Hermitian hamiltonian, and, unlike in one dimension, is compatible with
all spatial symmetries. We propose two new types of skin effect in two and
higher dimensions: the corner-skin effect where all eigenstates are localized
at one corner of the system, and the geometry-dependent-skin effect where skin
modes disappear for systems of a particular shape, but appear on generic
polygons. An immediate corollary of our theorem is that any non-Hermitian
system having exceptional points (lines) in two (three) dimensions exhibits
skin effect, making this phenomenon accessible to experiments in photonic
crystals, Weyl semimetals, and Kondo insulators.

## Introduction

The study of non-Hermitian hamiltonians, which can be regarded as the
effective description of dissipative processes, can be traced back to the
investigation of alpha decay, where real and imaginary parts of the complex
energy are related to the experimentally observed energy level and decay rate
Gamow (1928). When a lattice system is coupled with environments and has
dissipations, e.g. photonic crystals having radiational loss Feng  _et al._
(2017); El-Ganainy  _et al._ (2018); Ozawa  _et al._ (2019) and electronic
systems having finite quasiparticle lifetime Kozii and Fu (2017); Shen and Fu
(2018); Papaj  _et al._ (2019); Nagai  _et al._ (2020), the non-Hermitian band
theory becomes a conceptually simple and efficient approach Leykam  _et al._
(2017); Xu  _et al._ (2017); Shen  _et al._ (2018); Bergholtz  _et al._
(2021); Gong  _et al._ (2018); Lee  _et al._ (2019a); Ashida  _et al._ (2020);
Kawabata  _et al._ (2019a).

Skin effect Yao and Wang (2018); Yao  _et al._ (2018); Kunst  _et al._ (2018);
Martinez Alvarez  _et al._ (2018); Lee and Thomale (2019); Yokomizo and
Murakami (2019); Zhang  _et al._ (2020); Okuma  _et al._ (2020); Yang  _et
al._ (2020a); Yi and Yang (2020); Li  _et al._ (2020a); Borgnia  _et al._
(2020); Longhi (2019, 2020); Li  _et al._ (2020b), a phenomenon unique to the
non-Hermitian band theory, refers to the localization of eigenstates at the
boundary, the number of which scales with the volume of the system. For
example, in one dimension, all eigenstates of a non-Hermitian hamiltonian can
be localized at the ends of a chain Yao and Wang (2018). This suggests the
failure of Bloch’s theorem Xiong (2018); Yang (2020), which states that
eigenstates in the bulk are modulated plane waves. As Bloch’s theorem plays a
fundamental role in the development of condensed-matter physics Ashcroft and
Mermin (1976), the emergence of skin effect indicates a new and possibly
revolutionary direction. Especially, the skin effect has been experimentally
observed in one-dimensional classical systems Xiao  _et al._ (2020a, b);
Helbig  _et al._ (2020); Weidemann  _et al._ (2020); Brandenbourger  _et al._
(2019); Ghatak  _et al._ (2020); Wang  _et al._ (2021), inspiring further
studies on their higher dimensional generalizations Yao  _et al._ (2018); Lee
_et al._ (2019b); Ezawa (2019a, b); Hofmann  _et al._ (2020); Liu  _et al._
(2019); Yoshida  _et al._ (2020); Kawabata  _et al._ (2020); Scheibner  _et
al._ (2020); Zhang  _et al._ (2019a); Luo and Zhang (2019); Edvardsson  _et
al._ (2019); Ma and Hughes (2020); Okugawa  _et al._ (2020); Palacios  _et
al._ (2020); Song  _et al._ (2020); Fu  _et al._ (2021); Yu  _et al._ (2021);
Song  _et al._ (2021). However, a general theory for the higher-dimensional
skin effect has not been established.

Apart form the skin effect, another focus topic in non-Hermitian band systems
is the exceptional point (or line) Kato (2013); Miri and Alù (2019); Özdemir
_et al._ (2019); Lin  _et al._ (2011); Lee (2016); Molina and González (2018);
Carlström and Bergholtz (2018); Cerjan  _et al._ (2018); Kawabata  _et al._
(2019b); Moors  _et al._ (2019); Yang  _et al._ (2021); Xue  _et al._ (2020);
Zhen  _et al._ (2015); Zhou  _et al._ (2018); Zhang  _et al._ (2019b); Cerjan
_et al._ (2019); Yang and Hu (2019); Tang  _et al._ (2020); Denner  _et al._
(2020); Yang  _et al._ (2020b) that refers to stable point-type (or line-type)
non-Hermitian band degeneracy in the Brillouin zone. At the exceptional point,
not only eigenvalues but also eigenstates of the Bloch hamiltonian coalesce
Miri and Alù (2019). Many a novel phenomenon related to exceptional points has
been predicted and observed Dembowski  _et al._ (2001, 2003); Regensburger
_et al._ (2012); Feng  _et al._ (2013); Wiersig (2014); Gao  _et al._ (2015);
Xu  _et al._ (2016); Doppler  _et al._ (2016); Liu  _et al._ (2016); Hodaei
_et al._ (2017); Chen  _et al._ (2017); Yoon  _et al._ (2018); Tang  _et al._
(2020), such as the emergence of bulk-Fermi arc terminated at the exceptional
points Kozii and Fu (2017); Zhou  _et al._ (2018). Since the bulk-boundary
correspondence plays a central role in the development of topological phases
Hasan and Kane (2010), it is natural to ask if there exists a non-Hermitian
bulk-boundary correspondence in bands having exceptional points, analogous to
the surface Fermi arc in the Weyl semimetals in the Hermitian counterpart
Armitage  _et al._ (2018).

Figure 1:  The theorem of universal skin effect. (a) represents the Brillouin
zone. (b)(d) shows that when the spectral area of $\mathcal{H}(\mathbf{k})$ is
nonzero, the system on generic geometries must have universal skin effect.
(c)(e) shows that when the spectral area of $\mathcal{H}(\mathbf{k})$ is zero,
or forming one or several arcs on the complex plane, there is no skin effect
under any geometry.

In this paper, we establish a theorem that reveals a universal bulk-boundary
correspondence in two and higher dimensional non-Hermitian bands, as shown in
Fig. 1. The “bulk” refers to the area of the spectrum of the hamiltonian on
the complex plane with periodic boundary condition, and “boundary” the
presence (absence) of the skin effect for open-boundary system of an arbitrary
shape. The theorem states that the skin effect appears if and only if the
spectral area is nonzero. This skin effect is “universal” for three reasons:
(i) a randomly generated local non-Hermitian hamiltonian has the skin effect
with probability one; (ii) the skin effect is, unlike in one dimension,
compatible with all spatial symmetries and time-reversal symmetry; and (iii)
it does not require any special geometry of the open-boundary system. We also
propose two manifestations, restricted to two and higher dimensions, of the
universal skin effect, i.e., the corner-skin effect and the geometry-dependent
skin effect.

A surprising corollary of our theorem is that all stable exceptional points
Shen  _et al._ (2018); Kawabata  _et al._ (2019b); Yang  _et al._ (2021) imply
the presence of skin effect. Because exceptional points have been either
observed or proposed in meta-materials as well as in condensed matter, this
corollary makes skin effect observable in known systems. We predict the
geometry-dependent skin effect in the two-dimensional photonic crystal studied
in Ref. Zhou  _et al._ (2018), and propose to observe this effect in the
anomalous dynamics of wave packets.

## Theorem: an equivalence between spectral area and skin effect

For generic one-dimensional non-Hermitian systems, the correspondence between
the spectral shape and the skin effect has been derived Zhang  _et al._
(2020); Okuma  _et al._ (2020), i.e., when the Bloch spectrum is a loop-type
(an arc-type), the skin effect appears (disappears).

Generalizing the correspondence to two dimensions, we note two main
differences. One difference is in the periodic-boundary spectrum,
$E_{i}(\mathbf{k})$, where $i$ is the band index and $\mathbf{k}$ the crystal
momentum in the first Brillouin zone (BZ). Generally speaking,
$E_{i}(\mathbf{k})$ is a mapping from the $d$-dimensional torus to the complex
plane, $\mathbb{C}$. When $d=1$, the image of $E_{i}(k)$ forms a loop; but
when $d>1$, the image is generically a continuum on $\mathbb{C}$, denoted by
$E_{i}(\rm{BZ})$. The area covered by $E_{i}(\rm{BZ})$ on the complex plane is
called the spectral area, denoted by $A_{i}$. Another difference is in the
variety of open-boundary condition. There is only one geometry for an open
system in one dimension, i.e., an open chain; but there are an infinite number
of geometries in two dimensions such as triangle, rectangle and pentagon.

Now we are ready to state the theorem of universal skin effect: in the
thermodynamic limit, the skin effect is present in a hamiltonian having open
boundary of arbitrary geometry, if the spectral area is nonzero ($A_{i}\neq
0$); vice versa, the skin effect is absent for all possible geometries, if the
spectral area is zero ($A_{i}=0$). As the periodic-boundary hamiltonian
describes the dynamics in the bulk, the theorem relates a bulk property
(spectral area) to a boundary one (existence of skin modes). Fig. 1 shows some
schematic examples. In the Supplemental Material Sec. I, a complete proof of
the theorem has been provided.

The above theorem has implied the universality of skin effect in two and
higher dimensions. As $E_{i}(\rm{BZ})$ is the image of the $d\geq
2$-dimensional torus on the complex plane, it takes fine tuning of parameters
to make $A_{i}=0$ for every $i$. In fact, for single-band hamiltonian, we can
prove that $A=0$ if and only if $\mathcal{H}(\mathbf{k})=P[h(\mathbf{k})]$,
where $h(\mathbf{k})$ is a Hermitian hamiltonian and $P$ is a polynomial. In
other words, a randomly generated non-Hermitian hamiltonian
$\mathcal{H}(\mathbf{k})$ has skin effect: the first meaning of universality.
In previous studies, other types of skin effect, such as the line-skin and the
high-order-skin effect, in two and higher dimensions have been proposed Lee
_et al._ (2019b); Kawabata  _et al._ (2020). These types all require the open-
boundary system take a special geometry (usually a rectangle) and are hence
considered special and non-generic. The skin effect when $A_{i}\neq 0$ assumes
a completely generic geometry of boundary: the second meaning of universality.
The third meaning of universality lies in the fact that, unlike in one
dimension, higher-dimensional skin effect is compatible with all spatial
symmetries. A standing wave explanation for the above theorem is provided in
the Supplemental Material Sec. II.

Figure 2:  Two manifestations of skin effect. One is the CSE (a)-(d), the
other is the GDSE (e)-(h). In in (a)(b)(e)(f), the light blue regions
represent the spectrum under periodic boundary, and the red points represent
the eigenvalues under different open-boundary geometries. The spatial
distributions of eigenstates $W(\bm{x})$ are plotted in (c)(d)(g)(h). The GDSE
disappears under square geometry (open boundary 1) in (g), and reappears under
triangle geometry (open boundary 2) in (h).

## The corner-skin and the geometry-dependent-skin effect

While the theorem shows that the skin effect is universal, it does not specify
what skin modes look like in higher dimensions. Here, we report two types of
the universal skin effect, the corner-skin effect (CSE) and the geometry-
dependent skin effect (GDSE). Note that the CSE we mentioned in this paper has
the nature of non-reciprocity, similar to the one-dimensional skin effect.

The hamiltonian of the example for CSE is

$\begin{split}\mathcal{H}(\mathbf{k})=&[5(\cos{k_{x}}+\cos{2k_{x}})-i(\sin{k_{x}}+3\sin{2k_{x}})\\\
&+5\cos{k_{y}}+i\sin{k_{y}}]/2,\end{split}$ (1)

of which the spectral area under square geometry and triangle geometry is
shown in Fig. 2 (a)(b) with light blue color. Because of the nonzero spectral
area, the theorem tells us that the hamiltonian must have the universal skin
effect. This is verified in Fig. 2 (c)(d), where the spatial distributions of
all the eigenstates

$W(\bm{x})=\frac{1}{N}\sum_{n}|\psi_{n}(\bm{x})|^{2}$ (2)

under different open boundaries are plotted. Here $\psi_{n}(x)$ is a
normalized eigenstate and $N$ is the number of eigenstates. It is found that
the wave functions are always localized at the corner of the boundary in Fig.
2(c), even if the open-boundary geometry is changed in Fig. 2(d). We also plot
the corresponding eigenvalue spectra under different open boundaries, as shown
in Fig. 2 (a)(b) with red color. One can notice that the spectral areas under
periodic and open boundaries do not equal. This kind of skin effect is called
CSE, which can be explained by a nonzero current functional

$J_{\alpha}[n]=\sum_{i}\oint_{\rm{BZ}}{dk^{d}}n(E_{i},E^{\ast}_{i})\partial_{k_{\alpha}}{E}_{i}(\mathbf{k})$
(3)

under the periodic-boundary condition, where $n(E,E^{\ast})$ is any smooth
function Zhang  _et al._ (2020). The current functional is shown to vanish in
two and three dimensions under point groups $C_{i}$, $D_{2,3,4,6}$,
$C_{2h,3h,4h,6h}$, $D_{2d,3d,2h,3h,4h,6h}$, $T$, $T_{d,h}$, $O$ and $O_{h}$.
Therefore, the CSE is only compatible with point groups $C_{m}$ and
$C_{2,3,4,6,2v,3v,4v,6v}$ (see details in the Supplemental Material Sec. III).

Figure 3:  Two Dirac points (a) of a two-dimensional photonic crystal model
are split into four exceptional points (b) upon adding non-Hermitian term,
such as radiational loss. Correspondingly, the evolution of Gaussian wave
packet with initial velocity at the center of a diamond geometry for each ten
time intervals is shown in (c) with $\gamma=0$ (Hermitian) and (d) with
$\gamma=1/4$ (GDSE).

The hamiltonian of the example for GDSE reads

$\mathcal{H}(\mathbf{k})=2\cos{k_{x}}+2i\cos{k_{y}}.$ (4)

Since the spectral area is nonzero, our theorem tells us that the system must
have skin effect for generic geometry, such as a random polygon. However, an
interesting phenomenon in this example is that the skin effect disappears
under the square geometry due to the existence of two mirror symmetries shown
in Fig. 2 (g). Once we choose other types of boundaries where mirror
symmetries are broken, the skin effect reappears as shown in Fig. 2 (h). Since
the emergence of the skin effect and the position of localization depend on
the geometry, it is called the GDSE. Besides the distribution of the
eigenstates, another feature of the GDSE is that area of the open-boundary
spectrum seems to be the same as $A_{i}$. However, the corresponding density
of states is dependent by the choice of geometry as shown in Fig. 2 (e)(f). We
conjecture this is a universal phenomenon for the GDSE. In the Supplemental
Material Sec. III, we have provided some additional examples to illustrate
this new type of skin effect, and shown that the increase in the number of
skin modes is proportional to the increase in the system volume. For GDSE,
there is at least one spatial geometry such that skin modes vanish, and as
such is mutually exclusive with CSE. Additionally, GDSE is compatible with all
point groups, in contrast to CSE.

## Corollary: skin effect from exceptional points

An immediate corollary of our theorem is that all lattice hamiltonians having
stable exceptional points have universal skin effect, connecting two unique
phenomena in the non-Hermitian band theory. Consider a stable exceptional
point $\mathbf{k}_{0}$ in two dimensions. Due to the branch point structure of
exceptional point, the dispersion around $\bm{k_{0}}$ can be expressed as
$E_{\pm}(\mathbf{k})=\pm
c_{0}\sqrt{q_{x}+c_{1}q_{y}}+O(|\mathbf{k}-\mathbf{k}_{0}|)$, where
$q_{i=x,y}$ denotes a small derivation from exceptional point in $x$ or $y$
direction, that is, $q_{i}=k_{i}-k_{0i}$. Here $c_{0},c_{1}$ are nonzero
complex numbers and $c_{1}\notin\mathbb{R}$. Suppose the range of the
expansion is $r_{0}$, then it is clear that
$A_{\pm}\geq|c_{0}|\pi{r_{0}}^{2}/2\neq 0$. By the theorem, the system must
have skin effect (see more details in the Supplementary Material Sec. IV).

Now we use the photonic crystal model that has been experimentally realized in
Ref. Zhou  _et al._ (2018) to demonstrate our corollary. The tight-binding
model hamiltonian with periodic boundary can be written as

$H(k)=\bm{d}(k)\cdot\bm{\sigma}-i\gamma/2(\sigma_{0}-\sigma_{z}),$ (5)

where $\bm{\sigma}=(\sigma_{0},\sigma_{x},\sigma_{y},\sigma_{z})$ is a vector
of the Pauli matrices and $\bm{d}(k)$ is a vector with four components, that
is,

$\begin{split}\bm{d}(k)=&\\{\mu_{0}-(t_{2}+t_{3})(\cos{k_{x}}+\cos{k_{y}}),\\\
&t_{1}[1-\cos{k_{x}}-\cos{k_{y}}+\cos{(k_{x}-k_{y})}],\\\
&t_{1}[\sin{k_{x}}-\sin{k_{y}}-\sin{(k_{x}-k_{y})}],\\\
&\mu_{z}+(t_{2}-t_{3})(\cos{k_{x}}-\cos{k_{y}})\\}.\end{split}$ (6)

The parameters are chosen as follows,
$(t_{1},t_{2},t_{3},\mu_{0},\mu_{z})=(0.4,-0.1,0.5,1.35,-0.02)$. As shown in
Fig. 3 (a), in the Hermitian limit, i.e. $\gamma=0$, the system has two Dirac
points along the $x$-axis. When external dissipation or radiational loss is
added, i.e., $\gamma\neq 0$, each Dirac point splits into two exceptional
points shown in Fig. 2 (b), connected by the bulk Fermi arc. According to our
theorem, the system must have the universal skin effect, more precisely, the
GDSE shown in the Supplementary Material Sec. IV. The skin effect disappears
under square geometry but reappears under diamond geometry shown in Fig. 3
(c)(d). In this case, the majority of the eigenstates are concentrated on the
four edges.

As mentioned in the previous discussion, the appearance of the skin effect can
be reflected in the dynamical properties. In order to show this, we simulate
the time evolution of the wave packets with initial velocity at the center of
the diamond geometry. Here the initial state is chosen to be Gaussian form
$|\psi_{0}\rangle=\mathcal{N}$exp[${-(x-x_{0})^{2}/10-(y-y_{0})^{2}/10-i2x-i2y}]$$(1,1)^{T}$,
where $\mathcal{N}$ is the normalization factor and $x_{0}=y_{0}=21$ is the
center coordinate of the diamond geometry. We plot the corresponding
normalized final states
$|\psi(t_{f})\rangle=\mathcal{N}(t_{f})e^{-i\mathcal{H}_{\rm{OBC}}t_{f}}|\psi_{0}\rangle$
for every ten time intervals, where $\mathcal{H}_{\rm{OBC}}$ represents the
open-boundary hamiltonian on the diamond geometry. As shown in Fig. 3 (c), in
the Hermitian case, the center of the wave packets obeys the reflection rule.
Since the initial velocity of the wave packet is perpendicular to the two
edges, the center of the wave packet just bounces between the two edges
regardless of the dispersion of the wave packet. However, in the non-Hermitian
case ($\gamma=1/4$) with GDSE, besides the normal reflection channel, the wave
packet will evolve into the upper left boundary as shown in Fig. 3 (d). This
anomalous dynamical behavior is a experimental signature of GDSE.

We also propose the realization for CSE in a three-dimensional system with
exceptional lines in the Supplemental Material Sec. IV. Experimentally, the
non-reciprocity of the CSE can be detected by two-point Green function
approach.

## Discussion

Our work has built a bridge between two distinct phenomena that only exist in
non-Hermitian systems, i.e., the exceptional points (lines) and the non-
Hermitian skin effect, by establishing the correspondence between bulk
(spectral area) and boundary (universal skin effect). We prove that the skin
effect be universal and compatible with all spatial symmetries and reciprocity
in two and higher dimensions. Due to the universality, it is expected that the
skin effect is observable in a wide range of platforms, such as photonic
crystals with natural radiational loss, acoustic metamaterials and circuit
networks with lossy components such as resistors. Beyond these classical
systems, the skin effect can also be realized in condensed matter, e.g., the
heavy-fermion material with finite quasiparticle lifetime and the Weyl-
exceptional-ring semimetal. The latter is realizable in Weyl semimetals made
from inverting bands that have disparate effective masses, such as d- and
f-bands.

One should be reminded, however, that the results in this paper assume the
coherent dynamics of the constituent degrees of freedom, which is unlikely the
case in macroscopic condensed-matter systems where the coherence length is
shorter than the system size. On the contrary, for the systems where the
system size and the coherent length are comparable, as in mesoscopic systems,
we believe that the universal skin effect has a significant contribution to
the transport properties, a subject for future exploration.

## References

  * Gamow (1928) G. Gamow, “Zur Quantentheorie des Atomkernes,” Zeitschrift für Physik 51, 204–212 (1928).
  * Feng  _et al._ (2017) Liang Feng, Ramy El-Ganainy, and Li Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nature Photonics 11, 752–762 (2017).
  * El-Ganainy  _et al._ (2018) Ramy El-Ganainy, Konstantinos G. Makris, Mercedeh Khajavikhan, Ziad H. Musslimani, Stefan Rotter, and Demetrios N. Christodoulides, “Non-Hermitian physics and PT symmetry,” Nature Physics 14, 11–19 (2018).
  * Ozawa  _et al._ (2019) Tomoki Ozawa, Hannah M. Price, Alberto Amo, Nathan Goldman, Mohammad Hafezi, Ling Lu, Mikael C. Rechtsman, David Schuster, Jonathan Simon, Oded Zilberberg, and Iacopo Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
  * Kozii and Fu (2017) Vladyslav Kozii and Liang Fu, “Non-Hermitian Topological Theory of Finite-Lifetime Quasiparticles: Prediction of Bulk Fermi Arc Due to Exceptional Point,” arXiv:1708.05841 (2017).
  * Shen and Fu (2018) Huitao Shen and Liang Fu, “Quantum Oscillation from In-Gap States and a Non-Hermitian Landau Level Problem,” Phys. Rev. Lett. 121, 026403 (2018).
  * Papaj  _et al._ (2019) Michał Papaj, Hiroki Isobe, and Liang Fu, “Nodal arc of disordered Dirac fermions and non-Hermitian band theory,” Phys. Rev. B 99, 201107 (2019).
  * Nagai  _et al._ (2020) Yuki Nagai, Yang Qi, Hiroki Isobe, Vladyslav Kozii, and Liang Fu, “DMFT Reveals the Non-Hermitian Topology and Fermi Arcs in Heavy-Fermion Systems,” Phys. Rev. Lett. 125, 227204 (2020).
  * Leykam  _et al._ (2017) Daniel Leykam, Konstantin Y. Bliokh, Chunli Huang, Y. D. Chong, and Franco Nori, “Edge Modes, Degeneracies, and Topological Numbers in Non-Hermitian Systems,” Phys. Rev. Lett. 118, 040401 (2017).
  * Xu  _et al._ (2017) Yong Xu, Sheng-Tao Wang, and L.-M. Duan, “Weyl Exceptional Rings in a Three-Dimensional Dissipative Cold Atomic Gas,” Phys. Rev. Lett. 118, 045701 (2017).
  * Shen  _et al._ (2018) Huitao Shen, Bo Zhen, and Liang Fu, “Topological Band Theory for Non-Hermitian Hamiltonians,” Phys. Rev. Lett. 120, 146402 (2018).
  * Bergholtz  _et al._ (2021) Emil J. Bergholtz, Jan Carl Budich, and Flore K. Kunst, “Exceptional topology of non-Hermitian systems,” Rev. Mod. Phys. 93, 015005 (2021).
  * Gong  _et al._ (2018) Zongping Gong, Yuto Ashida, Kohei Kawabata, Kazuaki Takasan, Sho Higashikawa, and Masahito Ueda, “Topological Phases of Non-Hermitian Systems,” Phys. Rev. X 8, 031079 (2018).
  * Lee  _et al._ (2019a) Jong Yeon Lee, Junyeong Ahn, Hengyun Zhou, and Ashvin Vishwanath, “Topological Correspondence between Hermitian and Non-Hermitian Systems: Anomalous Dynamics,” Phys. Rev. Lett. 123, 206404 (2019a).
  * Ashida  _et al._ (2020) Yuto Ashida, Zongping Gong, and Masahito Ueda, “Non-Hermitian Physics,” arXiv:2006.01837 (2020).
  * Kawabata  _et al._ (2019a) Kohei Kawabata, Ken Shiozaki, Masahito Ueda, and Masatoshi Sato, “Symmetry and Topology in Non-Hermitian Physics,” Phys. Rev. X 9, 041015 (2019a).
  * Yao and Wang (2018) Shunyu Yao and Zhong Wang, “Edge States and Topological Invariants of Non-Hermitian Systems,” Phys. Rev. Lett. 121, 086803 (2018).
  * Yao  _et al._ (2018) Shunyu Yao, Fei Song, and Zhong Wang, “Non-Hermitian Chern Bands,” Phys. Rev. Lett. 121, 136802 (2018).
  * Kunst  _et al._ (2018) Flore K. Kunst, Elisabet Edvardsson, Jan Carl Budich, and Emil J. Bergholtz, “Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems,” Phys. Rev. Lett. 121, 026808 (2018).
  * Martinez Alvarez  _et al._ (2018) V. M. Martinez Alvarez, J. E. Barrios Vargas, and L. E. F. Foa Torres, “Non-Hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points,” Phys. Rev. B 97, 121401(R) (2018).
  * Lee and Thomale (2019) Ching Hua Lee and Ronny Thomale, “Anatomy of skin modes and topology in non-Hermitian systems,” Phys. Rev. B 99, 201103(R) (2019).
  * Yokomizo and Murakami (2019) Kazuki Yokomizo and Shuichi Murakami, “Non-Bloch Band Theory of Non-Hermitian Systems,” Phys. Rev. Lett. 123, 066404 (2019).
  * Zhang  _et al._ (2020) Kai Zhang, Zhesen Yang, and Chen Fang, “Correspondence between Winding Numbers and Skin Modes in Non-Hermitian Systems,” Phys. Rev. Lett. 125, 126402 (2020).
  * Okuma  _et al._ (2020) Nobuyuki Okuma, Kohei Kawabata, Ken Shiozaki, and Masatoshi Sato, “Topological Origin of Non-Hermitian Skin Effects,” Phys. Rev. Lett. 124, 086801 (2020).
  * Yang  _et al._ (2020a) Zhesen Yang, Kai Zhang, Chen Fang, and Jiangping Hu, “Non-Hermitian Bulk-Boundary Correspondence and Auxiliary Generalized Brillouin Zone Theory,” Phys. Rev. Lett. 125, 226402 (2020a).
  * Yi and Yang (2020) Yifei Yi and Zhesen Yang, “Non-Hermitian Skin Modes Induced by On-Site Dissipations and Chiral Tunneling Effect,” Phys. Rev. Lett. 125, 186802 (2020).
  * Li  _et al._ (2020a) Linhu Li, Ching Hua Lee, and Jiangbin Gong, “Topological Switch for Non-Hermitian Skin Effect in Cold-Atom Systems with Loss,” Phys. Rev. Lett. 124, 250402 (2020a).
  * Borgnia  _et al._ (2020) Dan S. Borgnia, Alex Jura Kruchkov, and Robert-Jan Slager, “Non-Hermitian Boundary Modes and Topology,” Phys. Rev. Lett. 124, 056802 (2020).
  * Longhi (2019) Stefano Longhi, “Probing non-Hermitian skin effect and non-Bloch phase transitions,” Phys. Rev. Research 1, 023013 (2019).
  * Longhi (2020) S. Longhi, “Non-Bloch-Band Collapse and Chiral Zener Tunneling,” Phys. Rev. Lett. 124, 066602 (2020).
  * Li  _et al._ (2020b) Linhu Li, Ching Hua Lee, Sen Mu, and Jiangbin Gong, “Critical non-Hermitian skin effect,” Nature Communications 11, 5491 (2020b).
  * Xiong (2018) Ye Xiong, “Why does bulk boundary correspondence fail in some non-hermitian topological models,” Journal of Physics Communications 2, 035043 (2018).
  * Yang (2020) Zhesen Yang, “Non-perturbative Breakdown of Bloch’s Theorem and Hermitian Skin Effects,” arXiv:2012.03333 (2020).
  * Ashcroft and Mermin (1976) N. W. Ashcroft and N. D. Mermin,  _Solid State Physics_ (Holt-Saunders, 1976).
  * Xiao  _et al._ (2020a) Lei Xiao, Tianshu Deng, Kunkun Wang, Gaoyan Zhu, Zhong Wang, Wei Yi, and Peng Xue, “Non-Hermitian bulk–boundary correspondence in quantum dynamics,” Nature Physics 16, 761–766 (2020a).
  * Xiao  _et al._ (2020b) Lei Xiao, Tianshu Deng, Kunkun Wang, Zhong Wang, Wei Yi, and Peng Xue, “Observation of non-Bloch parity-time symmetry and exceptional points,” (2020b), arXiv:2009.07288 .
  * Helbig  _et al._ (2020) T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany, T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Szameit, M. Greiter, and R. Thomale, “Generalized bulk–boundary correspondence in non-Hermitian topolectrical circuits,” Nature Physics 16, 747–750 (2020).
  * Weidemann  _et al._ (2020) Sebastian Weidemann, Mark Kremer, Tobias Helbig, Tobias Hofmann, Alexander Stegmaier, Martin Greiter, Ronny Thomale, and Alexander Szameit, “Topological funneling of light,” Science 368, 311–314 (2020).
  * Brandenbourger  _et al._ (2019) Martin Brandenbourger, Xander Locsin, Edan Lerner, and Corentin Coulais, “Non-reciprocal robotic metamaterials,” Nature Communications 10, 4608 (2019).
  * Ghatak  _et al._ (2020) Ananya Ghatak, Martin Brandenbourger, Jasper van Wezel, and Corentin Coulais, “Observation of non-hermitian topology and its bulk–edge correspondence in an active mechanical metamaterial,” Proceedings of the National Academy of Sciences 117, 29561–29568 (2020).
  * Wang  _et al._ (2021) Kai Wang, Avik Dutt, Ki Youl Yang, Casey C. Wojcik, Jelena Vučković, and Shanhui Fan, “Generating arbitrary topological windings of a non-Hermitian band,” Science 371, 1240–1245 (2021).
  * Lee  _et al._ (2019b) Ching Hua Lee, Linhu Li, and Jiangbin Gong, “Hybrid Higher-Order Skin-Topological Modes in Nonreciprocal Systems,” Phys. Rev. Lett. 123, 016805 (2019b).
  * Ezawa (2019a) Motohiko Ezawa, “Non-Hermitian boundary and interface states in nonreciprocal higher-order topological metals and electrical circuits,” Phys. Rev. B 99, 121411 (2019a).
  * Ezawa (2019b) Motohiko Ezawa, “Non-Hermitian higher-order topological states in nonreciprocal and reciprocal systems with their electric-circuit realization,” Phys. Rev. B 99, 201411 (2019b).
  * Hofmann  _et al._ (2020) Tobias Hofmann, Tobias Helbig, Frank Schindler, Nora Salgo, Marta Brzezińska, Martin Greiter, Tobias Kiessling, David Wolf, Achim Vollhardt, Anton Kabaši, Ching Hua Lee, Ante Bilušić, Ronny Thomale, and Titus Neupert, “Reciprocal skin effect and its realization in a topolectrical circuit,” Phys. Rev. Research 2, 023265 (2020).
  * Liu  _et al._ (2019) Tao Liu, Yu-Ran Zhang, Qing Ai, Zongping Gong, Kohei Kawabata, Masahito Ueda, and Franco Nori, “Second-Order Topological Phases in Non-Hermitian Systems,” Phys. Rev. Lett. 122, 076801 (2019).
  * Yoshida  _et al._ (2020) Tsuneya Yoshida, Tomonari Mizoguchi, and Yasuhiro Hatsugai, “Mirror skin effect and its electric circuit simulation,” Phys. Rev. Research 2, 022062(R) (2020).
  * Kawabata  _et al._ (2020) Kohei Kawabata, Masatoshi Sato, and Ken Shiozaki, “Higher-order non-Hermitian skin effect,” Phys. Rev. B 102, 205118 (2020).
  * Scheibner  _et al._ (2020) Colin Scheibner, William T. M. Irvine, and Vincenzo Vitelli, “Non-Hermitian Band Topology and Skin Modes in Active Elastic Media,” Phys. Rev. Lett. 125, 118001 (2020).
  * Zhang  _et al._ (2019a) Zhiwang Zhang, María Rosendo López, Ying Cheng, Xiaojun Liu, and Johan Christensen, “Non-Hermitian Sonic Second-Order Topological Insulator,” Phys. Rev. Lett. 122, 195501 (2019a).
  * Luo and Zhang (2019) Xi-Wang Luo and Chuanwei Zhang, “Higher-Order Topological Corner States Induced by Gain and Loss,” Phys. Rev. Lett. 123, 073601 (2019).
  * Edvardsson  _et al._ (2019) Elisabet Edvardsson, Flore K. Kunst, and Emil J. Bergholtz, “Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence,” Phys. Rev. B 99, 081302 (2019).
  * Ma and Hughes (2020) Yuhao Ma and Taylor L. Hughes, “The Quantum Skin Hall Effect,” (2020), arXiv:2008.02284 .
  * Okugawa  _et al._ (2020) Ryo Okugawa, Ryo Takahashi, and Kazuki Yokomizo, “Second-order topological non-Hermitian skin effects,” Phys. Rev. B 102, 241202 (2020).
  * Palacios  _et al._ (2020) Lucas S. Palacios, Serguei Tchoumakov, Maria Guix, Ignasio Pagonabarraga, Samuel Sánchez, and Adolfo G. Grushin, “Guided accumulation of active particles by topological design of a second-order skin effect,” (2020), arXiv:2012.14496 .
  * Song  _et al._ (2020) Yiling Song, Weiwei Liu, Lingzhi Zheng, Yicong Zhang, Bing Wang, and Peixiang Lu, “Two-dimensional non-Hermitian Skin Effect in a Synthetic Photonic Lattice,” Phys. Rev. Applied 14, 064076 (2020).
  * Fu  _et al._ (2021) Yongxu Fu, Jihan Hu, and Shaolong Wan, “Non-hermitian second-order skin and topological modes,” Phys. Rev. B 103, 045420 (2021).
  * Yu  _et al._ (2021) Yang Yu, Minwoo Jung, and Gennady Shvets, “Zero-energy corner states in a non-Hermitian quadrupole insulator,” Phys. Rev. B 103, L041102 (2021).
  * Song  _et al._ (2021) Fei Song, Hong-Yi Wang, and Zhong Wang, “Non-Bloch PT symmetry breaking: Universal threshold and dimensional surprise,” arXiv:2102.02230 (2021).
  * Kato (2013) Tosio Kato,  _Perturbation theory for linear operators_ , Vol. 132 (Springer Science & Business Media, 2013).
  * Miri and Alù (2019) Mohammad-Ali Miri and Andrea Alù, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
  * Özdemir  _et al._ (2019) Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, “Parity–time symmetry and exceptional points in photonics,” Nature Materials 18, 783–798 (2019).
  * Lin  _et al._ (2011) Zin Lin, Hamidreza Ramezani, Toni Eichelkraut, Tsampikos Kottos, Hui Cao, and Demetrios N. Christodoulides, “Unidirectional Invisibility Induced by $\mathcal{P}\mathcal{T}$-Symmetric Periodic Structures,” Phys. Rev. Lett. 106, 213901 (2011).
  * Lee (2016) Tony E. Lee, “Anomalous Edge State in a Non-Hermitian Lattice,” Phys. Rev. Lett. 116, 133903 (2016).
  * Molina and González (2018) Rafael A. Molina and José González, “Surface and 3D Quantum Hall Effects from Engineering of Exceptional Points in Nodal-Line Semimetals,” Phys. Rev. Lett. 120, 146601 (2018).
  * Carlström and Bergholtz (2018) Johan Carlström and Emil J. Bergholtz, “Exceptional links and twisted fermi ribbons in non-hermitian systems,” Phys. Rev. A 98, 042114 (2018).
  * Cerjan  _et al._ (2018) Alexander Cerjan, Meng Xiao, Luqi Yuan, and Shanhui Fan, “Effects of non-Hermitian perturbations on Weyl Hamiltonians with arbitrary topological charges,” Phys. Rev. B 97, 075128 (2018).
  * Kawabata  _et al._ (2019b) Kohei Kawabata, Takumi Bessho, and Masatoshi Sato, “Classification of Exceptional Points and Non-Hermitian Topological Semimetals,” Phys. Rev. Lett. 123, 066405 (2019b).
  * Moors  _et al._ (2019) Kristof Moors, Alexander A. Zyuzin, Alexander Yu. Zyuzin, Rakesh P. Tiwari, and Thomas L. Schmidt, “Disorder-driven exceptional lines and Fermi ribbons in tilted nodal-line semimetals,” Phys. Rev. B 99, 041116 (2019).
  * Yang  _et al._ (2021) Zhesen Yang, A. P. Schnyder, Jiangping Hu, and Ching-Kai Chiu, “Fermion Doubling Theorems in Two-Dimensional Non-Hermitian Systems for Fermi Points and Exceptional Points,” Phys. Rev. Lett. 126, 086401 (2021).
  * Xue  _et al._ (2020) Haoran Xue, Qiang Wang, Baile Zhang, and Y. D. Chong, “Non-Hermitian Dirac Cones,” Phys. Rev. Lett. 124, 236403 (2020).
  * Zhen  _et al._ (2015) Bo Zhen, Chia Wei Hsu, Yuichi Igarashi, Ling Lu, Ido Kaminer, Adi Pick, Song-Liang Chua, John D. Joannopoulos, and Marin Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525, 354–358 (2015).
  * Zhou  _et al._ (2018) Hengyun Zhou, Chao Peng, Yoseob Yoon, Chia Wei Hsu, Keith A. Nelson, Liang Fu, John D. Joannopoulos, Marin Soljačić, and Bo Zhen, “Observation of bulk Fermi arc and polarization half charge from paired exceptional points,” Science 359, 1009–1012 (2018).
  * Zhang  _et al._ (2019b) Xufeng Zhang, Kun Ding, Xianjing Zhou, Jing Xu, and Dafei Jin, “Experimental Observation of an Exceptional Surface in Synthetic Dimensions with Magnon Polaritons,” Phys. Rev. Lett. 123, 237202 (2019b).
  * Cerjan  _et al._ (2019) Alexander Cerjan, Sheng Huang, Mohan Wang, Kevin P. Chen, Yidong Chong, and Mikael C. Rechtsman, “Experimental realization of a Weyl exceptional ring,” Nature Photonics 13, 623–628 (2019).
  * Yang and Hu (2019) Zhesen Yang and Jiangping Hu, “Non-Hermitian Hopf-link exceptional line semimetals,” Phys. Rev. B 99, 081102 (2019).
  * Tang  _et al._ (2020) Weiyuan Tang, Xue Jiang, Kun Ding, Yi-Xin Xiao, Zhao-Qing Zhang, C. T. Chan, and Guancong Ma, “Exceptional nexus with a hybrid topological invariant,” Science 370, 1077–1080 (2020).
  * Denner  _et al._ (2020) M. Michael Denner, Anastasiia Skurativska, Frank Schindler, Mark H. Fischer, Ronny Thomale, Tomáš Bzdušek, and Titus Neupert, “Exceptional Topological Insulators,” (2020), arXiv:2008.01090 .
  * Yang  _et al._ (2020b) Zhesen Yang, Ching-Kai Chiu, Chen Fang, and Jiangping Hu, “Jones Polynomial and Knot Transitions in Hermitian and non-Hermitian Topological Semimetals,” Phys. Rev. Lett. 124, 186402 (2020b).
  * Dembowski  _et al._ (2001) C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, “Experimental Observation of the Topological Structure of Exceptional Points,” Phys. Rev. Lett. 86, 787–790 (2001).
  * Dembowski  _et al._ (2003) C. Dembowski, B. Dietz, H.-D. Gräf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, “Observation of a Chiral State in a Microwave Cavity,” Phys. Rev. Lett. 90, 034101 (2003).
  * Regensburger  _et al._ (2012) Alois Regensburger, Christoph Bersch, Mohammad-Ali Miri, Georgy Onishchukov, Demetrios N. Christodoulides, and Ulf Peschel, “Parity–time synthetic photonic lattices,” Nature 488, 167–171 (2012).
  * Feng  _et al._ (2013) Liang Feng, Ye-Long Xu, William S. Fegadolli, Ming-Hui Lu, JoséE. B. Oliveira, Vilson R. Almeida, Yan-Feng Chen, and Axel Scherer, “Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies,” Nature Materials 12, 108–113 (2013).
  * Wiersig (2014) Jan Wiersig, “Enhancing the Sensitivity of Frequency and Energy Splitting Detection by Using Exceptional Points: Application to Microcavity Sensors for Single-Particle Detection,” Phys. Rev. Lett. 112, 203901 (2014).
  * Gao  _et al._ (2015) T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Höfling, Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott, R. G. Dall, and E. A. Ostrovskaya, “Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard,” Nature 526, 554–558 (2015).
  * Xu  _et al._ (2016) H. Xu, D. Mason, Luyao Jiang, and J. G. E. Harris, “Topological energy transfer in an optomechanical system with exceptional points,” Nature 537, 80–83 (2016).
  * Doppler  _et al._ (2016) Jörg Doppler, Alexei A. Mailybaev, Julian Böhm, Ulrich Kuhl, Adrian Girschik, Florian Libisch, Thomas J. Milburn, Peter Rabl, Nimrod Moiseyev, and Stefan Rotter, “Dynamically encircling an exceptional point for asymmetric mode switching,” Nature 537, 76–79 (2016).
  * Liu  _et al._ (2016) Zhong-Peng Liu, Jing Zhang, Şahin Kaya Özdemir, Bo Peng, Hui Jing, Xin-You Lü, Chun-Wen Li, Lan Yang, Franco Nori, and Yu-xi Liu, “Metrology with $\mathcal{PT}$-Symmetric Cavities: Enhanced Sensitivity near the $\mathcal{PT}$-Phase Transition,” Phys. Rev. Lett. 117, 110802 (2016).
  * Hodaei  _et al._ (2017) Hossein Hodaei, Absar U. Hassan, Steffen Wittek, Hipolito Garcia-Gracia, Ramy El-Ganainy, Demetrios N. Christodoulides, and Mercedeh Khajavikhan, “Enhanced sensitivity at higher-order exceptional points,” Nature 548, 187–191 (2017).
  * Chen  _et al._ (2017) Weijian Chen, Şahin Kaya Özdemir, Guangming Zhao, Jan Wiersig, and Lan Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
  * Yoon  _et al._ (2018) Jae Woong Yoon, Youngsun Choi, Choloong Hahn, Gunpyo Kim, Seok Ho Song, Ki-Yeon Yang, Jeong Yub Lee, Yongsung Kim, Chang Seung Lee, Jai Kwang Shin, Hong-Seok Lee, and Pierre Berini, “Time-asymmetric loop around an exceptional point over the full optical communications band,” Nature 562, 86–90 (2018).
  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
  * Armitage  _et al._ (2018) N. P. Armitage, E. J. Mele, and Ashvin Vishwanath, “Weyl and Dirac semimetals in three-dimensional solids,” Rev. Mod. Phys. 90, 015001 (2018).

Supplemental Material for “Universal non-Hermitian skin effect in two and
higher dimensions”

##  I The proof of the theorem

In this section, we will prove the following theorem appeared in the main
text:

  * Theorem: In the thermodynamic limit, the skin effect is present in a hamiltonian having open boundary of arbitrary shape, if the periodic-boundary spectral area of the same hamiltonian is nonzero; vice versa, the skin effect is absent for all possible shapes of open boundary, if the spectral area is zero.

Here the spectral area refers to the area of the region covered by the
periodic boundary spectrum on the complex plane. In the following contents, we
will first show some numerical verifications of the theorem, and then prove
the theorem in two dimensional systems, and finally extend the proof to three-
dimensional cases.

###  I.1 Some numerical examples of the theorem

Figure S1:  Some numerical examples of the theorem. (a-d) show an example
having skin effect, and (e-h) show an example without skin effect. (a) and (e)
show the corresponding hoping parameters of the hamiltonian shown in Eq. (S1).
(b) and (f) show the periodic boundary spectrum. (c-d) and (g-h) show the
distribution of $W(x)$ in Eq. (S2) under different open boundary geometries.

In this subsection, we provide some numerical examples of the theorem. In
order to simplify the discussion, we consider the following single-band model

$\mathcal{H}(\mathbf{k})=\sum_{i,j}t_{ij}\beta_{x}^{-i}\beta_{y}^{-j},\qquad\beta_{x/y}=e^{ik_{x/y}},$
(S1)

where $t_{ij}$ represents the hoping parameter that an electron or particle
hopes from site $(m,n)$ to site $(m,n)+(i,j)$.

The two rows in Fig. S1 represent two different models. In the first one, the
hoping parameters are shown in Fig. S1 (a), and the periodic boundary
spectrum, i.e. $E(\mathbf{k})=\mathcal{H}(\mathbf{k})$ with $\mathbf{k}$
belonging to the Brillouin zone (BZ), is shown in Fig. S1 (b). Here the color
strength represents the cover times of $E_{0}\in E(\mathbf{k})$ when
$\mathbf{k}$ sweeps over the whole BZ. One can notice that the spectral area
of the first model is nonzero. As a result, the open boundary eigenstates show
the localization behaviors, namely, the emergence of skin effect, as shown in
Fig. S1 (c-d). In order to illustrate the localization properties,

$W(x)=\frac{1}{N}\sum_{n}|\psi_{n}(x)|^{2}$ (S2)

is plotted in Fig. S1 (c-d), , where $\psi_{n}(x)$ is the $n$-th normalized
eigenstate of the open boundary hamiltonian, and $N$ is the number of open
boundary eigenstates. For the second example, since the spectral area is zero,
as shown in Fig. S1 (f), there is no skin effect. Indeed, $W(x)$ shown in Fig.
S1 (g-h) are extended.

In the following subsection, we will prove the theorem in two-dimensional
systems. Our strategy of the proof is illustrated in Fig. S2(a). The
equivalence relation between “spectral area” and “the universal skin effect”
is linked by “spectral winding” (see the following discussion).

###  I.2 The proof of the theorem in two-dimensions

####  I.2.1 Spectral area and spectral winding

We begin with a two-dimensional single-band non-Hermitian model with periodic
boundary in both $x$ and $y$ directions

$\mathcal{H}(\mathbf{k})=u(\mathbf{k})+iv(\mathbf{k}),$ (S3)

where $u$ and $v$ are real functions about $\mathbf{k}=(k_{x},k_{y})$. For any
$\mathbf{k}_{r}\in\rm{BZ}$, one can define the following winding number

$\nu(\mathbf{k}_{r})=\oint_{\Gamma_{\mathbf{k}_{r}}}\frac{d\mathbf{k}}{2\pi
i}\cdot\nabla_{\mathbf{k}}\log\det[\mathcal{H}(\mathbf{k})-E_{r}],\,\,\,\,\mathbf{k}_{r}\in\rm{BZ},$
(S4)

where $\Gamma_{\mathbf{k}_{r}}$ represents the infinitesimal counterclockwise
loop enclosing $\mathbf{k}_{r}$. Here $E_{r}=\mathcal{H}(\mathbf{k}_{r})$
represents the reference energy, which is shown in Fig. S2 (b) with red point.
We note that for different $\mathbf{k}_{r}$, the reference energy is
different. This topological invariant describes the spectral winding on the
complex plane. As shown in Fig. S2 (b), if $\nu(\mathbf{k}_{r})$ is nonzero,
the image of $\Gamma_{\mathbf{k}_{r}}$, i.e.
$\mathcal{H}(\Gamma_{\mathbf{k}_{r}})$, forms a closed loop that encloses
$E_{r}$.

Figure S2:  (a) shows the equivalence relation between spectral area, spectral
winding and the universal skin effect. Each equivalence relation is sufficient
and necessary. (b) illustrates the spectral winding for hamiltonian Eq. S13.
Here the light blue region represents the periodic boundary spectrum, and the
paths on BZ corresponds to the spectral loops (or arcs) on the complex plane
with the same color.

First we prove the following two statements,

  * 1.

if there are some $\mathbf{k}_{r}$ points in the BZ with nonzero topological
charge, the spectral area must be nonzero;

  * 2.

if all the $\mathbf{k}_{r}$ points on the BZ have zero topological charge, the
spectral area must be zero.

The above two statement can be represented by the equivalence relation between
“spectral winding” and “spectral area” shown in Fig. S2 (a).

Based on the definition of winding number, the statement 1 is obvious.
Therefore, we only need to prove the statement 2. In order to show this, we
expand the hamiltonian at the point
$\mathbf{k}_{r}\equiv(k_{x}^{r},k_{y}^{r})$ as follows

$\mathcal{H}(\mathbf{k})-\mathcal{H}(\mathbf{k}_{r})\approx\partial_{x}\mathcal{H}(\mathbf{k}_{r})q_{x}+\partial_{y}\mathcal{H}(\mathbf{k}_{r})q_{y},$
(S5)

where $q_{x},q_{y}$ are the displacements $\mathbf{k}-\mathbf{k}_{r}$ in $x,y$
directions respectively. For a generic $\mathbf{k}_{r}$ point, the first
derivative of $\mathcal{H}$ does not vanish, i.e., the coefficients of $q_{x}$
and $q_{y}$ in Eq. (S5) cannot be zero at the same time. The reason is that in
order to make
$\partial_{x}\mathcal{H}(\mathbf{k}_{r})=\partial_{y}\mathcal{H}(\mathbf{k}_{r})=0$,
four independent real conditions,
$\operatorname{Re}[\partial_{x}\mathcal{H}(\mathbf{k}_{r})]=\operatorname{Im}[\partial_{x}\mathcal{H}(\mathbf{k}_{r})]=\operatorname{Re}[\partial_{y}\mathcal{H}(\mathbf{k}_{r})]=\operatorname{Im}[\partial_{y}\mathcal{H}(\mathbf{k}_{r})]=0$,
need to be satisfied. However, in two dimensions, there are only two free
parameters $(k_{x},k_{y})$, which cannot satisfy the above four equations
generally.

In order to calculate the topological charge of a generic $\mathbf{k}_{r}$
point, according to Eq. (S3), one can define

$C(\mathbf{k}_{r})=\begin{pmatrix}\partial_{x}u(\mathbf{k}_{r})&\partial_{y}u(\mathbf{k}_{r})\\\
\partial_{x}v(\mathbf{k}_{r})&\partial_{y}v(\mathbf{k}_{r})\end{pmatrix},$
(S6)

where the notation $\partial_{x/y}$ refers to $\partial/\partial_{k_{x/y}}$.
When $\det[C(\mathbf{k}_{r})]\neq 0$, the topological charge of
$\mathbf{k}_{r}$ is the sign of the determinant of $C(\mathbf{k}_{r})$,
expressed as

$\nu(\mathbf{k}_{r})=\mathrm{sgn}[\det[C(\mathbf{k}_{r})]].$ (S7)

Therefore, a sufficient and necessary condition for the zero charge for each
$\mathbf{k}\in\rm{BZ}$ is

$\det[C(\mathbf{k})]=\partial_{x}u(\mathbf{k})\partial_{y}v(\mathbf{k})-\partial_{y}u(\mathbf{k})\partial_{x}v(\mathbf{k})=0.$
(S8)

A theorem (the corollary of theorem 13.2) in Ref. Loomis and Sternberg (1968)
tells us that if $C(\mathbf{k})\neq 0$ and $\det[C(\mathbf{k})]=0$, for an
open set $S$, then, $u(\mathbf{k})$ and $v(\mathbf{k})$ have a functional
dependent relation with $\mathbf{k}\in S\subset\rm{BZ}$. Applying the theorem
to the entire BZ (except for some isolated points at which the first
derivative vanishes), one can reexpress the single-band hamiltonian that
satisfies Eq. (S8) as

$\mathcal{H}(\mathbf{k})=P[h(\mathbf{k})],$ (S9)

where $h(\mathbf{k})$ is a real and periodic function of $\mathbf{k}$, and $P$
is a complex polynomial of $h$. Since $h(\mathbf{k})$ is a real periodic
function, its image must be an arc on the real axis, e.g.
$h(\mathbf{k})\in[h_{1},h_{2}]$, where $h_{1/2}$ are real numbers. Therefore,
the image of $P[h(\mathbf{k})]$ is also an arc on the complex plane. This
completes the proof of the statement 2 for single-band models.

Generalizing the above discussion to the multi-band case, for each
$\mathbf{k}_{r}\in\rm{BZ}$, the topological charge defined for the $m$-th band
is

$\displaystyle\nu_{m}(\mathbf{k}_{r})$
$\displaystyle=\oint_{\Gamma_{\mathbf{k}_{r}}}\frac{d\mathbf{k}}{2\pi
i}\cdot\nabla_{\mathbf{k}}\log\det[\mathcal{H}(\mathbf{k})-E_{m}(\mathbf{k}_{r})]$
(S10)
$\displaystyle=\sum_{n}\oint_{\Gamma_{\mathbf{k}_{r}}}\frac{d\mathbf{k}}{2\pi
i}\cdot\nabla_{\mathbf{k}}\log[E_{n}(\mathbf{k})-E_{m}(\mathbf{k}_{r})],$

where $E_{m}(\mathbf{k}_{r})$ is the energy of the $m$-th band with the
momentum $\mathbf{k}_{r}$. For the second equal sign in Eq. (S10), we have
used
$\det[\mathcal{H}(\mathbf{k})-E_{m}(\mathbf{k}_{r})]=\prod_{n}[E_{n}(\mathbf{k})-E_{m}(\mathbf{k}_{r})]$.
For Eq. (S10), if $\mathbf{k}_{r}$ is not the degeneracy point, only $n=m$
term in the summation has contributions to the topological charge. Therefore,
the Eq. (S10) further becomes

$\nu_{m}(\mathbf{k}_{r})=\oint_{\Gamma_{\mathbf{k}_{r}}}\frac{d\mathbf{k}}{2\pi
i}\cdot\nabla_{\mathbf{k}}\log[E_{m}(\mathbf{k})-E_{m}(\mathbf{k}_{r})].$
(S11)

Using the similar approaches in the single-band case, one can conclude that,
the the real and imaginary parts of $E_{m}(\mathbf{k})$ are locally functional
dependent on the neighbourhood of $\mathbf{k}\in\rm{BZ}$. As a result, the
spectrum of $E_{m}(\mathbf{k})$ must be an arc. The above conclusion applies
for each band. So far, we have proved that if each $\mathbf{k}$ point on the
BZ has zero topological charge, the spectral area must be zero.

####  I.2.2 Spectral winding and the universal skin effect

In the above contents, we have proved the equivalence relation between
“spectral winding” and “spectral area”. Here, we will prove the equivalence
condition between “spectral winding” and “universal skin effect”, as shown in
Fig. S2 (a).

We first notice that the BZ in two-dimensional systems can be covered by a set
of straight lines of any slope, labeled as $\\{L_{s}\\}$. Here, the subscript
$s$ indicates the slope of the set $\\{L_{s}\\}$, and $L_{s}$ represents a
generic straight line belonging to $\\{L_{s}\\}$. For example, if we fix
$k_{y}(k_{x})$ and change $k_{x}(k_{y})$ from $0$ to $2\pi$, we get a
horizontal (vertical) straight line with the slope being $0$ ($\infty$) on BZ,
and the set of all horizontal or vertical straight lines ($\\{L_{0}\\}$ or
$\\{L_{\infty}\\}$) covers the entire BZ. Particularly, an inclined straight
line goes out from one side of BZ and again enters from another side as shown
in Fig. S2(b). Since the straight lines on the BZ are periodic, one can define
the spectral winding number for each straight lines with respect to the
prescribed reference energy $E_{r}$.

$\nu(L_{s},E_{r})=\oint_{L_{s}}\frac{d\mathbf{k}}{2\pi
i}\cdot\nabla_{\mathbf{k}}\log\det[\mathcal{H}(\mathbf{k})-E_{r}].$ (S12)

Obviously, if all the $\mathbf{k}$ points on the BZ have zero topological
charge, $\nu(L_{s},E_{r})$ must be zero for arbitrary $L_{s}$ and $E_{r}$.
Otherwise, one can always find some $L_{s}$, such that $\nu(L_{s},E_{r})$ is
nonzero. Next we briefly prove the latter statement. Assuming that each
straight line in $\\{L_{0}\\}$ and $\\{L_{\infty}\\}$ has zero spectral
winding, and there is a $k_{r}$ point carrying nonzero topological charge on
the $\rm{BZ}$. For a generic inclined straight line, one can always find
corresponding horizontal and vertical straight lines, such that together with
the inclined straight line to form a closed path enclosing $\mathbf{k}_{r}$ on
BZ. Therefore, the closed path has nonzero winding number with respect to
$E_{r}$. Due to the zero spectral winding of the horizontal and vertical
straight lines as we assumed, hence a generic inclined straight line must have
nonzero spectral winding number.

Let’s use an example to show this. Consider the following hamiltonian

$\mathcal{H}(k_{x},k_{y})=2\cos{k_{x}}+2i\sin{k_{y}}.$ (S13)

As shown in Fig. S2, the spectrum of the hamiltonian along each horizontal or
vertical straight line (gray lines) on BZ has zero winding number with respect
to any reference energy on the spectral area (lightblue square region).
However, once we choose the straight line with darker cyan color, these three
straight lines (two gray lines with the darker cyan line) together form a
closed path that encloses $\mathbf{k}_{r}$. Therefore, the closed path has
nonzero spectral winding number with respect to $E_{r}$. Due to the zero
spectral winding of two gray lines, the darker cyan straight line must have
nonzero winding number for $E_{r}$ as illustrated in Fig. S2(b).

If the system has no skin effect under a specific parallelepiped open
boundary, the winding number $\nu_{m}(L_{m},E_{r})$ along each straight lines
that are perpendicular to the boundary cut directions should be zero. (This is
a conjecture that we cannot exactly proved currently. However, we believe this
statement is true as all the numerical results we obtained obey this
conclusion. Furthermore, it is also a natural generalization of the one-
dimensional results Zhang  _et al._ (2020); Okuma  _et al._ (2020) and has
been mentioned or applied in some recent works Yokomizo and Murakami (2019);
Denner  _et al._ (2020)) More generally, if the system has no skin effect
under any parallelepiped open boundary, then the spectral windings of all
straight lines on BZ are required to be zero, which is satisfied when the
spectral area of the system is zero. Therefore, nonzero spectral area means
that there must be skin effect under certain open boundaries, namely, the
existence of the universal skin effect.

###  I.3 The proof of the theorem in three-dimensions

In this section, we extend the above proof of two-dimensional systems into
three dimensions. We obtain the similar conclusion that nonzero spectral area
is equivalent to the existence of the universal skin effect.

Consider a general three-dimensional single-band tight-binding hamiltonian,
which consists of real- and imaginary-part functions

$\mathcal{H}(k_{x},k_{y},k_{z})=u(k_{x},k_{y},k_{z})+iv(k_{x},k_{y},k_{z}).$
(S14)

We choose a generic $\mathbf{k}_{r}$ point and use its energy
$\mathcal{H}(\mathbf{k}_{r})$ as the reference energy. For a given reference
energy $E_{r}$, we can obtain a one-dimensional curve in the there-dimensional
BZ by solving the following two real equations,

$\begin{split}u(k_{x},k_{y},k_{z})=\operatorname{Re}(E_{r});\\\
v(k_{x},k_{y},k_{z})=\operatorname{Im}(E_{r}).\end{split}$ (S15)

Each equation determines a surface, and the intersection of two surfaces is
one-dimensional curve in there-dimensional BZ. The tangent direction of the
curve at $\mathbf{k}_{r}$ is perpendicular to the normal vector of the two
surfaces at this $\mathbf{k}_{r}$ point. The tangent vector at
$\mathbf{k}_{r}$ is expressed as

$\bm{T}_{\mathbf{k}_{r}}=\bm{\nabla}u(\mathbf{k}_{r})\times\bm{\nabla}v(\mathbf{k}_{r}),$
(S16)

where $\bm{\nabla}u(\mathbf{k}_{r})$ represents the gradient of $u$. We choose
the local coordinate system ($R^{3}$ space) with $\mathbf{k}_{r}$ as the
origin, and the gradient is reexpressed as

$\bm{\nabla}u(\mathbf{k}_{r})=\partial_{x}u(\mathbf{k}_{r})q_{x}+\partial_{y}u(\mathbf{k}_{r})q_{y}+\partial_{z}u(\mathbf{k}_{r})q_{z},$
(S17)

where
$q_{i}\equiv(\frac{\mathbf{k}-\bm{k_{0}}}{|\mathbf{k}-\bm{k_{0}}|})_{i}$,
$\mathbf{k}$ and $\bm{k_{0}}$ represent two vectors in the global coordinate
system.

Next we expand the hamiltonian into Taylor series around the origin of the
local coordinate system,

$\mathcal{H}(\mathbf{k})-\mathcal{H}(\mathbf{k}_{r})=\sum_{i=x,,y,z}\partial_{i}\mathcal{H}(\mathbf{k}_{r})q_{i}+o(|\bm{q}|),$
(S18)

where the subscription $i$ represents the partial differential to $x,y,z$. And
$q_{i}$ represents the deviation of $k_{i}$ from $k_{r,i}$ and the last term
is the infinitesimal of higher order of $|\bm{q}|$. Obvious, the zero winding
condition requires

$\begin{split}\partial_{x}u(\mathbf{k}_{r})\partial_{y}v(\mathbf{k}_{r})-\partial_{x}v(\mathbf{k}_{r})\partial_{y}u(\mathbf{k}_{r})=0;\\\
\partial_{x}v(\mathbf{k}_{r})\partial_{z}u(\mathbf{k}_{r})-\partial_{x}u(\mathbf{k}_{r})\partial_{z}v(\mathbf{k}_{r})=0;\\\
\partial_{y}u(\mathbf{k}_{r})\partial_{z}v(\mathbf{k}_{r})-\partial_{y}v(\mathbf{k}_{r})\partial_{z}u(\mathbf{k}_{r})=0,\end{split}$
(S19)

or equivalently,

$\bm{T}_{k_{r}}=\bm{0}.$ (S20)

Next, we prove that if all $\mathbf{k}$ points in three-dimensional BZ satisfy
$\bm{T}_{k}=\bm{0}$, then the entire 3D periodic-boundary spectrum must be an
arc in the complex plane. We define a two-tuple function
$W(\mathbf{k}):=[u(\mathbf{k})\;\;\;v(\mathbf{k})]^{t}$ with three variables,
the exterior derivative of the vector-valued function is expressed as

$dW=\begin{pmatrix}\partial_{x}u(\mathbf{k})&\partial_{y}u(\mathbf{k})&\partial_{z}u(\mathbf{k})\\\
\partial_{x}v(\mathbf{k})&\partial_{y}v(\mathbf{k})&\partial_{z}v(\mathbf{k})\end{pmatrix}.$
(S21)

Eq. (S20) implies that the rank of $dW$ less than 2 (the number of components
of $W$). To be precise, there are the following cases. (i.) Both the gradients
of $u$ and $v$ are not zero vector, and they are linearly dependent on each
other. (ii.) One of the gradients of $u$ and $v$ is zero vector. (iii.) Both
the gradients of $u$ and $v$ are zero vector. In all these cases, we can
obtain the final conclusion that $u$ and $v$ are linearly functional dependent
on each other. Therefore, the spectrum must be arcs on the complex plane.

A 3D BZ can be divided into a series of plane systems, each plane corresponds
to a two-dimensional subsystem. If the spectral area of a three-dimensional
system is nonzero, then for each reference energy on the spectral area, its
preimage (1D ring) has nonzero topological charge. Equivalently, the two-
dimensional subsystem, of which the BZ (2D plane) has intersections with the
ring, also has nonzero topological charge for the intersecting $k$ points.
Hence, the 2D subsystem has nonzero spectral area, and has the universal skin
effect. Correspondingly, we come to the same conclusion in 3D systems that
nonzero spectral area signifies the existence of the universal skin effect.

##  II A physical explanation for the theorem

Figure S3:  (a) shows the periodic-boundary spectrum of Eq. (S25) with gray
color, and the pre-images of $E_{0}=1+i$ (red point in (a)) are the four red
points in (b). The periodic-boundary spectrum of Eq. (S26) is the gray line in
(c), and $\mathbf{k}(E_{0}=1+i)$ is plotted by the red lines in (d).

Here, we use some examples to illustrate the intuition that motivates the
theorem. Consider the following one-dimensional model

$\mathcal{H}_{0}(k)=2\cos k$ (S22)

placed on a chain of length $L$. Under the periodic boundary condition, the
two Bloch waves $e^{ik_{0}x}$ and $e^{-ik_{0}x}$ have the same energy
$E(k_{0})=2\cos{k_{0}}$. When the system has open boundary condition, the
Bloch wave $e^{ik_{0}x}$ will be reflected to $e^{-ik_{0}x}$ with a
$\pi$-phase shift. Their linear superposition $e^{ik_{0}x}-e^{-ik_{0}x}$ is an
eigenstate with energy $2\cos{k_{0}}$ that satisfies the zero boundary
condition at $x=0,L$, thus being an open-boundary eigenstate. When the system
is added a momentum-dependent dissipation,

$\mathcal{H}(k)=2\cos k+i\sin k,$ (S23)

the spectrum $E(k)$ becomes complex and forms an ellipse in the complex plane.
In this case, the degeneracy is broken, e.g. $E(k)\neq{}E(-k)$, which implies
the open-boundary eigenstates are no longer the linear superposition of the
extended Bloch waves. This implies the emergence of skin effect.

Extend the above arguments to two dimensions, and we can provide a physical
explanation for the theorem proved in the section I.

Formally, we consider a single-band model

$\mathcal{H}(\mathbf{k})=\mathcal{H}_{0}(\mathbf{k})+i\Gamma(\mathbf{k}).$
(S24)

When the real and imaginary parts of which are functionally independent, the
hamiltonian will have a non-zero spectral area. For a given eigenvalue $E_{0}$
of the Bloch hamiltonian, by solving
$\mathcal{H}_{0}(\mathbf{k})=\operatorname{Re}E_{0}$ and
$\Gamma(\mathbf{k})=\operatorname{Im}E_{0}$, one can obtain a finite set of
pre-images of $E_{0}$, i.e,
$\mathbf{K}({E_{0}})=\\{\mathbf{k}_{1},...,\mathbf{k}_{m}\\}$, which includes
all Bloch waves having energy $E_{0}$. Now suppose that one of the Bloch waves
$\mathbf{k}_{i}\in\mathbf{K}({E_{0}})$ is incident on the boundary, depending
on the normal direction of the boundary, the corresponding momentum of the
reflected wave can be arbitrary. However, the number of elements of
$\mathbf{K}({E_{0}})$ is finite, and as such cannot support so many reflection
channels. This failure of reflection mechanism at a generic boundary means the
failure in forming an open boundary eigenstate from Bloch waves, which implies
the emergence of skin effect under a generic open-boundary geometry. However,
the spectrum collapses into an arc (zero spectral area) if the real and
imaginary parts of the hamiltonian are functionally dependent, and the number
of the corresponding solutions of $\mathcal{H}(\mathbf{k})=E_{0}$ is infinite.
It means that there are infinite reflection channels to satisfy the open
boundary of any shape, and an open boundary eigenstate can be formed from
superimposing all Bloch-wave channels.

Concretely, we choose two examples to demonstrate the above arguments. The
first example is

$\mathcal{H}(\mathbf{k})=2\cos{k_{x}}+2i\cos{k_{y}},$ (S25)

of which the spectral area is nonzero shown in Fig. S3(a). For a given
eigenvalue $E_{0}=1+i$, by solving $2\cos{k_{x}}=1$ and $2\cos{k_{y}}=1$, we
can obtain a finite set of pre-images of $E_{0}$, that is,
$\mathbf{K}(E_{0})=\\{k_{1},k_{2},k_{3},k_{4}\\}$ [red points in Fig. S3(b)].
The finite solutions of $\mathcal{H}(\mathbf{k})=E_{0}$ cannot support so many
reflection channels, that is to say, cannot form an open-boundary eigenstate
on a generic geometry by superimposing these Bloch waves specified by
$k_{i=1,2,3,4}$. Therefore, the hamiltonian Eq. (S25) has skin effect under
open-boundary geometry of a generic shape. The second example reads

$\mathcal{H}(\mathbf{k})=2\cos{(k_{x}+k_{y})}+2i\cos{(k_{x}+k_{y})},$ (S26)

the periodic-boundary spectrum of which is an arc [the gray line in Fig.
S3(c)]. The set of pre-images of $E_{0}=1+i$ has infinite elements [the red
lines in Fig. S3(d)], which means that there are infinite ways of
superimposing these Bloch waves to satisfy the open boundary condition of any
shape. Therefore, the hamiltonian Eq. (S26) has no skin effect under any open-
boundary geometry.

##  III Corner-skin effect and Geometry-dependent-skin effect

In this section, we will provide some examples to demonstrate the
characteristics of the two manifestations of the universal skin effect. We
discuss the role of symmetry on the universal skin effect. We define the
current functional to explain the appearance of corner-skin effect. In
addition, we numerically verified that geometry-dependent-skin effect obeys
the volume law, which is a significant feature to distinguish skin modes form
conventional boundary states.

###  III.1 Symmetry and the universal skin effect

Figure S4:  The distribution of $W(x)$ for Hamiltonian Eq. (S27) with
different parameters and on different geometries. The system size is
$L_{x}=L_{y}=25$. The probability density is proportional to the opacity of
the red color. (a) shows the corner-skin effect with $t_{1}=t_{2}=1,w=0$; (b)
shows line skin with $t_{1}=1,t_{2}=w=0$; (c) has no skin effect with
$t_{1}=t_{2}=0,w=1$, and geometry-dependent-skin effect appears in (d) under
triangle and diamond geometries.

We have proved that if the system has nonzero spectral area, the universal
skin effect will occur under a general open boundary condition. According to
the symmetry restriction, the universal skin effect has two manifestations,
that is, corner-skin effect and geometry-dependent-skin effect. Consider a
system with nonzero spectral area, if its Hamiltonian has no any symmetry, the
skin modes are localized at one or several vertices on an open geometry of any
shape, and the number of modes are proportional to the volume of the system.
If the Hamiltonian has certain spatial symmetries, such as mirror symmetry,
the corner-skin effect will be forbidden. The reason is that the corner-skin
effect has the nature of non-reciprocity, which is incompatible with mirror
symmetry. However, if we change the open boundary geometry such that the
symmetry is broken on the boundary, then geometry-dependent-skin modes will
appear, which is a unique but universal phenomenon in higher-dimensional
systems. We take a concrete example to demonstrate the role of symmetry on the
skin effect. Consider a tight-binding model, of which the periodic-boundary
Hamiltonian reads

$\begin{split}H(k_{x},k_{y})&=h_{0}(k_{x},k_{y})+ih_{1}(k_{x},k_{y})\\\
&=\sin{k_{x}}\sigma_{x}+\sin{k_{y}}\sigma_{y}+(2-\cos{k_{x}}-\cos{k_{y}})\sigma_{z}+i[t_{1}\sin{k_{x}}+t_{2}\sin{k_{y}}+w(\cos{k_{x}}-\cos{k_{y}})]\sigma_{z},\end{split}$
(S27)

where $h_{0}$ and $h_{1}$ are the Hermitian and non-Hermitian parts,
respectively. The Hermitian part is in a gapless phase with a Dirac point at
$k_{x}=k_{y}=0$, and has inversion symmetry
$\sigma_{z}h_{0}(k_{x},k_{y})\sigma_{z}=h_{0}(-k_{x},-k_{y})$.

If we only add the $w$ non-Hermitian term ($t_{1}=t_{2}=0;w=1$), the combined
mirror and non-Hermitian time-reversal symmetry $M_{x}T=A_{t}$ and
$M_{y}T=\sigma_{z}A_{t}$ are preserved ($A_{t}$ representing transpose
operator),

$\begin{split}(M_{x}T)H(k_{x},k_{y})(M_{x}T)^{-1}=H(k_{x},-k_{y});\\\
(M_{y}T)H(k_{x},k_{y})(M_{y}T)^{-1}=H(-k_{x},k_{y}).\end{split}$ (S28)

There is no skin effect under open boundary with square geometry as shown in
Fig. S4(c). However, if we cut the square lattice into triangles and diamond
lattices, the skin effect will retrieve on the boundaries that destroy the two
symmetries, which is shown in Fig. S4(d).

If we only turn on $t_{1}$ non-Hermitian term ($t_{1}=1;t_{2}=w=0$), the
Hamiltonian preserves $M_{x}T$ symmetry but destroys $M_{y}T$ symmetry. In
this case, the system has skin modes along $x$ direction as shown in Fig.
S4(b), the number of the skin modes are proportional to the volume of the
system. If we add $t_{1}$ and $t_{2}$ non-Hermitian terms
($t_{1}=t_{2}=1;w=0$), the Hamiltonian destroys the two symmetries, and skin
modes will be concentrated on the corner as shown in Fig. S4(a).

###  III.2 Current functional

In this section, we first simply introduce the concept of current functional,
then discuss the restrictions of point groups in two and three dimensions on
the current functional, a quantity faithfully depicting the appearance of
corner-skin effect, which demonstrates that corner-skin effect is only
compatible with point groups $C_{m}$ and $C_{2,3,4,6,2v,3v,4v,6v}$.

####  III.2.1 An introduction to the current functional

We define the current functional to depict the corner-skin effect in $d$
dimensions.

$J_{\alpha}[n]=\sum_{i}J_{i,\alpha}[n]=\sum_{i}\oint_{\rm{BZ}}{dk^{d}}n(E_{i},E^{\ast}_{i})\partial_{k_{\alpha}}{E}_{i}(\mathbf{k}),$
(S29)

where $i$ is band index, and $k_{\alpha}$ is a vector, expressed as
$k_{\alpha}=\sum_{i}k_{i}\mathbf{e}_{i}$ in $d$-dimensional momentum space
with unit vector basis $\mathbf{e}_{i}$. Here $n(E,E^{\ast})$ is a
distribution function depending on $E$ and $E^{\ast}$, but does not depend on
$k$ explicitly, such as the Bose distribution
$n(E,E^{\ast})=(e^{\operatorname{Re}E(k)/k_{B}T}-1)^{-1}$. If there exists a
$n(E,E^{\ast})$ such that the current functional is nonzero for any $\alpha$,
then the system must have the corner-skin effect. If for any possible
$n(E,E^{\ast})$ and $\alpha$, the current functional is zero, then the system
has no corner-skin effect.

For example, we take a single-band tight-binding model [Eq. (1) in the main
text] as

$\mathcal{H}(\mathbf{k})=[5(\cos{k_{x}}+\cos{2k_{x}})-i(\sin{k_{x}}+3\sin{2k_{x}})+5\cos{k_{y}}+i\sin{k_{y}}]/2,$
(S30)

and $n(E,E^{\ast})$ as $\operatorname{Im}{[\mathcal{H}(\mathbf{k})]}$. In this
case, $J_{x}$ is equal to $25\pi^{2}/2$ and $J_{y}$ is equal to $-5\pi^{2}/2$.
Hence, the system has corner-skin effect. Another example of tight-binding
model [Eq. (3) in the main text] reads

$\mathcal{H}(\mathbf{k})=2\cos{k_{x}}+2i\cos{k_{y}},$ (S31)

of which the current functional $J_{x}=J_{y}=0$ regardless of the choose of
$n(E,E^{\ast})$. Therefore, the hamiltonian has no corner-skin effect
(although hosts the geometry-dependent-skin effect due to the existence of two
mirror symmetries). Meanwhile, the example also reminds us that certain
symmetries may prohibit the corner-skin effect. Next, we will systematically
analyze the interplay of point-group symmetries and the corner-skin effect.

####  III.2.2 Corner-skin effect under point groups

Here we consider the hamiltonian with only spatial symmetries (without any
anti-unitary symmetry such as non-Hermitian time-reversal symmetry), and
investigate the restrictions of the point-group symmetries on the current
functional, and further conclude that corner-skin effect is only compatible
with point groups $C_{m}$ and $C_{2,3,4,6,2v,3v,4v,6v}$.

Inversion: Consider a system that only has inversion symmetry $I$, and each
band satisfies $E_{i}(\mathbf{k})=E_{i}(I\,\mathbf{k})=E_{i}(-\mathbf{k})$.
The current functional for $i$-th band can be expressed as

$J_{i,\alpha}[n]=\oint_{\rm{BZ}}{dk^{d}}n(E_{i},E^{\ast}_{i})\partial_{k_{\alpha}}{E}_{i}(\mathbf{k}),$
(S32)

which is invariant by replacing $\mathbf{k}$ with
$\mathbf{k^{\prime}}\equiv-\mathbf{k}$. After the transformation, Eq. (S32)
becomes

$\begin{split}J_{i,\alpha}[n]&=\oint_{\rm{BZ}^{\prime}}(-1)^{d}{dk^{\prime
d}}n(E_{i},E^{\ast}_{i})\partial_{-k^{\prime}_{\alpha}}{E}_{i}(-\mathbf{k}^{\prime})=(-1)^{d}\oint_{\rm{BZ}}(-1)^{d}{dk^{\prime
d}}n(E_{i},E^{\ast}_{i})\partial_{-k^{\prime}_{\alpha}}{E}_{i}(-\mathbf{k}^{\prime})\\\
&=-\oint_{\rm{BZ}}{dk^{\prime
d}}n(E_{i},E^{\ast}_{i})\partial_{k^{\prime}_{\alpha}}{E}_{i}(\mathbf{k}^{\prime})=-\oint_{\rm{BZ}}{dk^{d}}n(E_{i},E^{\ast}_{i})\partial_{k_{\alpha}}{E}_{i}(\mathbf{k})=-J_{i,\alpha}[n]=0.\end{split}$
(S33)

It means that if the hamiltonian has only inversion symmetry, the current
functional for each band must be zero regardless of the choose of
$n(E,E^{*})$. Equivalently, the corner-skin effect must vanish in the system
with inversion symmetry, and is incompatible with the point groups including
inversion symmetry, such as $C_{i,3i,2h,4h,6h}$, $D_{3d,2h,4h,6h}$, $T_{h}$
and $O_{h}$.

Rotation: Consider a system that is invariant under a point group including
rotation operator $R$, then $E_{i}(\mathbf{k})=E_{i}(R\,\mathbf{k})$. The Eq.
(S32) is also invariant under the transformation from $\mathbf{k}$ to
$\mathbf{k}^{\prime}\equiv R^{-1}\,\mathbf{k}$ =
$\sum_{ij}\mathbf{e}_{i}\rm{c}_{ij}k_{j}$, where $k_{j}$ is the $j$-th
component (along $\mathbf{e}_{j}$) of $\mathbf{k}$. After the transformation,
the current functional becomes

$\begin{split}J_{i,\alpha}[n]&=\oint_{\rm{BZ}^{\prime}}\det{[J_{\mathbf{k},\mathbf{k}^{\prime}}]}{dk^{\prime
d}}n(E_{i},E^{\ast}_{i})\partial_{Rk^{\prime}_{\alpha}}{E}_{i}(R\,\mathbf{k}^{\prime})=\oint_{\rm{BZ}}dk^{{}^{\prime}d}n(E,E^{\ast})\partial_{R\,k^{{}^{\prime}}_{\alpha}}E_{i}(\mathbf{k^{\prime}})\\\
&=\oint_{\rm{BZ}}dk^{d}n(E,E^{\ast})\partial_{R\,k_{\alpha}}E_{i}(\mathbf{k})=\oint_{\rm{BZ}}dk^{d}n(E,E^{\ast})\partial_{k_{\alpha}}E_{i}(\mathbf{k}),\end{split}$
(S34)

where $\det{[J_{\mathbf{k},\mathbf{k}^{\prime}}]}$ in the second term is the
determinant of the Jacobian $J_{\mathbf{k},\mathbf{k}^{\prime}}$ that measures
the change of differential volume element under different representations and
the sign of $\det{[J_{\mathbf{k},\mathbf{k}^{\prime}}]}$ is positive because
the rotational operator preserves orientation. One can always choose an
appropriate basis transformation such that
$\det{[J_{\mathbf{k},\mathbf{k}^{\prime}}]}=1$. In addition, since the
Brillouin zone has the same symmetry group as the hamiltonian and rotational
operator $R$ does not change the orientation, the integral region $\rm{BZ}$ is
invariant under the point group. For example, consider a rotation $R$ that
rotates $\pi/4$ along $\mathbf{e}_{z}$ axis, then $k_{x}=-k^{\prime}_{y}$ and
$k_{y}=k^{\prime}_{x}$. The Jacobian matrix can be written as

$J_{\mathbf{k},\mathbf{k}^{\prime}}=\begin{pmatrix}\partial_{x^{\prime}}k_{x}&\partial_{y^{\prime}}k_{x}\\\
\partial_{x^{\prime}}k_{y}&\partial_{y^{\prime}}k_{y}.\end{pmatrix}$ (S35)

In this case, the determinant of Jacobian
$\det{[J_{\mathbf{k},\mathbf{k}^{\prime}}]}$ is 1.

The last equation of Eq. (S34) requires

$Rk_{\alpha}=k_{\alpha},$ (S36)

which means that the direction of $k_{\alpha}$ is parallel to the rotational
axis of $R$, equivalently, the component of $k_{\alpha}$ perpendicular to the
rotational axis must be zero. For example, if $R$ is a rotation that rotates
$\theta$ along $\mathbf{e}_{z}$ axis, that is,

$R=\begin{pmatrix}\cos{\theta}&-\sin{\theta}&0\\\
\sin{\theta}&\cos{\theta}&0\\\ 0&0&1\end{pmatrix},$ (S37)

then Eq. (S36) restricts $k_{\alpha}$ as
$0\,\mathbf{e}_{x}+0\,\mathbf{e}_{y}+k_{z}\,\mathbf{e}_{z}$, along
$\mathbf{e}_{z}$ axis.

As a consequence, if a point group contains two or more rotations with non-
parallel rotational axes, the current functional for each band must be zero.
If the point group contains only one rotation, a nonzero current functional
for each band is allowed, thus corner-skin effect is compatible with the point
groups including only one rotation, such as $C_{2,3,4,6}$.

Mirror: In a similar way, a mirror symmetry requires

$Mk_{\alpha}=k_{\alpha}$ (S38)

if nonzero current functional for each band is allowed, which means the
corner-skin effect is compatible with the group containing only one mirror
symmetry, that is, $C_{m}$.

It is notable that an additional rotation symmetry is allowed when the
rotational axis lies in the mirror plane, and $k_{\alpha}$ is restricted as

$RMk_{\alpha}=k_{\alpha}.$ (S39)

Therefore, the corner-skin effect is also compatible with such point groups,
i.e., $C_{2v,3v,4v,6v}$.

In summary, we have explicitly shown that corner-skin effect does not exist
under most point groups because the current functional is restricted to zero
under such point groups, regardless of the choice of $n(E,E^{\ast})$, but is
allowed to appear under these point groups, i.e.,

$\\{C_{m},C_{2},C_{3},C_{4},C_{6},C_{2v},C_{3v},C_{4v},C_{6v}\\}.$ (S40)

###  III.3 Volume Law

Figure S5:  The norm squared of all wave functions of the Hamiltonian Eq.
(S13) on square lattice (a) and parallelogram lattice (b) is plotted. The
system size is chosen as $L_{x}=L_{y}=25$. The volume law is shown in (c), in
which blue line represents $N_{skin}\propto V$, gray line
$N_{skin}\propto\sqrt{V}$ and black line $N_{skin}\propto 1$.

We numerically show that the geometry-dependent-skin effect satisfies the
volume law, that is, the increase in the number of skin modes is proportional
to the increase in volume of the system,

$\delta N_{skin}\propto\delta V.$ (S41)

For example, If the shape of the open boundary is a parallelogram whose side-
lengths are $L_{x}$ and $L_{y}$ as shown in Fig. S5, then, $V=L_{x}L_{y}$. Our
criterion for judging a mode as a skin mode is to check whether ninety percent
of the probability density of this mode lies within the boundary we appointed.

Consider a tight-binding model with periodic-boundary Hamiltonian
$\mathcal{H}(k_{x},k_{y})=2\cos{k_{x}}+2i\sin{k_{y}}$, there is no skin effect
under square geometry Fig. S5(a), but the skin effect appears under
parallelogram geometry Fig. S5(b) due to the spectral area being nonzero,
which is geometry-dependent-skin effect. The distributions of $W(\bm{x})$
under different open boundaries are plotted. For parallelogram geometry, we
specify the thickness of the boundary to be the width of three unit cells, and
use black dashed lines to distinguish the boundary from the bulk. If ninety
percent of the probability density of a mode lies in the boundary, we count it
as a skin mode. We count the number of skin modes for different volumes
($\mathcal{L}_{x}\mathcal{L}_{y}$), and the fitting curve (blue curve in Fig.
S5(c)) shows that the two are in a linear relation, that is, $\delta
N_{skin}=0.52\delta V$. The volume law of geometry-dependent-skin effect has
been verified numerically.

##  IV Exceptional semimetals

In this section, we will prove the corollary of our theorem, that is all
stable exceptional semimetals imply the universal skin effect. We first review
the topological charge of non-Hermitian band degeneracies.

###  IV.1 Non-Hermitian band degeneracy

Consider a general $m$-band non-Hermitian Bloch Hamiltonian (with periodic
boundary condition),

$\mathcal{H}(\mathbf{k})=\sum_{s=1}^{m^{2}-1}\left[h_{s}^{r}(\mathbf{k})+ih_{s}^{i}(\mathbf{k})\right]\Gamma_{s},$
(S42)

where $\Gamma_{s}$ are the generators of Lei algebra $\mathfrak{su}(m)$ and
$h_{s}^{r}(\mathbf{k})$ and $h_{s}^{i}(\mathbf{k})$ are real functions of
$\mathbf{k}$. When $m=2,3,4$, $\Gamma_{s}$ are the Pauli, GellMann, and
$\gamma$ matrices, respectively. The eigenvalues of $\mathcal{H}(\mathbf{k})$
can be obtained by solving the following characteristic polynomial

$f_{E}(\mathbf{k})=\det[E-\mathcal{H}(\mathbf{k})]=\prod_{i=1}^{m}[E-E_{i}(\mathbf{k})],$
(S43)

where $E_{i}(\mathbf{k})$ is the $i$th eigenvalue of the non-Hermitian
Hamiltonian $\mathcal{H}(\mathbf{k})$. At the degeneracy point
$\mathbf{k}_{D}$, two bands must have the same energy, i.e.

$E_{i}(\mathbf{k}_{D})=E_{j}(\mathbf{k}_{D})$ (S44)

for some $i\neq j$. In Ref. Yang  _et al._ (2020, 2021), the authors have
shown that the above condition is equivalent to the vanishing of the
discriminant of $f_{E}(\mathbf{k})$, i.e.

$\operatorname{Disc}_{E}[\mathcal{H}](\mathbf{k}_{D})=0,$ (S45)

where

$\operatorname{Disc}_{E}[\mathcal{H}](\bm{k})=\prod_{i<j}\left[E_{i}(\bm{k})-E_{j}(\bm{k})\right]^{2}$
(S46)

is the discriminant of $f_{E}(\mathbf{k})$. Although the discriminant is
defined by the roots of $f_{E}(\mathbf{k})=0$, it can be computed directly
from the determinant of the Sylvester matrix of $f_{E}(\mathbf{k})$ and
$\partial_{E}f_{E}(\mathbf{k})$, which can be expressed by the coefficients of
$f_{E}(\mathbf{k})$. Now we show a concrete example of the discriminant
method.

  * Example: Consider a generic two-band model

$\mathcal{H}(\mathbf{k})=h_{0}(\mathbf{k})\sigma_{0}+h_{x}(\mathbf{k})\sigma_{x}+h_{y}(\mathbf{k})\sigma_{y}+h_{z}(\mathbf{k})\sigma_{z},$
(S47)

where $h_{\mu}(\mathbf{k})=h_{\mu}^{r}(\mathbf{k})+ih_{\mu}^{i}(\mathbf{k})$
are complex functions of $\mathbf{k}$. The characteristic polynomial of the
two-band model can be written as

$f_{E}(\mathbf{k})=E^{2}+b(\mathbf{k})E+c(\mathbf{k}),$ (S48)

where $b(\mathbf{k})=-2h_{0}(\mathbf{k})$ and
$c(\mathbf{k})=h_{0}^{2}(\mathbf{k})-h_{x}^{2}(\mathbf{k})-h_{y}^{2}(\mathbf{k})-h_{z}^{2}(\mathbf{k})$.
Computing the discriminant of polynomial (S48) with respect to the energy $E$,
we obtain the following condition for the existence of DPs

$\operatorname{Disc}_{E}[\mathcal{H}](\mathbf{k})=b^{2}(\mathbf{k})-4c(\mathbf{k})=4[h_{x}^{2}(\mathbf{k})+h_{y}^{2}(\mathbf{k})+h_{z}^{2}(\mathbf{k})]=0.$
(S49)

This condition can also be obtained from the energy spectrum, that is the two
bands
$E_{\pm}=h_{0}(\mathbf{k})\pm([h_{x}^{2}(\mathbf{k})+h_{y}^{2}(\mathbf{k})+h_{z}^{2}(\mathbf{k}))^{1/2}$
are degenerate, whenever the square root is vanishing.

From the above example, one can notice that the discriminant
$\operatorname{Disc}_{E}[\mathcal{H}](\mathbf{k})$ is a complex periodic
function of $\mathbf{k}$. Its vanishing is equivalent to the vanishing of the
real and imaginary parts, i.e.

$\operatorname{Re}\operatorname{Disc}_{E}[\mathcal{H}](\mathbf{k})=\operatorname{Im}\operatorname{Disc}_{E}[\mathcal{H}](\mathbf{k})=0.$
(S50)

The solution of the above equation are the non-Hermitian degeneracy points in
2D and lines in 3D.

Figure S6:  The periodic-boundary spectrum of the photonic crystal model is
shown in (a)(c) with light blue color. Under the square geometry with the
systems size $L_{x}=L_{y}=31$, the eigenvalues (red points) and the norm
squared of all wave functions are shown in (a) and (b), respectively. Under
the diamond geometry with the systems size $L_{x}=L_{y}=45$, the eigenvalues
(red points) and the norm squared of all wave functions are plotted in (c) and
(d), respectively.

###  IV.2 Topological charge of non-Hermitian band degeneracies

In this subsection, we will review the topological charge of the non-Hermitian
band degeneracies. Based on the discriminant of the characteristic polynomial,
one can define the topological charge of the degeneracy point
$\mathbf{k}_{D}$, i.e.

$\nu\left(\bm{k}_{D}\right)=\frac{1}{2\pi
i}\oint_{\Gamma\left(\bm{k}_{D}\right)}d\bm{k}\cdot\nabla_{\bm{k}}\ln\operatorname{Disc}_{E}[\mathcal{H}](\bm{k}).$
(S51)

where $\Gamma\left(\bm{k}_{D}\right)$ is a loop encircling the degeneracy
point $\mathbf{k}_{D}$. Since $\operatorname{Disc}_{E}[\mathcal{H}](\bm{k})$
is single valued, this invariant is quantized, which is called the
discriminant number in Ref. Yang  _et al._ (2021). Putting

$\operatorname{Disc}_{E}[\mathcal{H}](\bm{k})=\prod_{i<j}\left[E_{i}(\bm{k})-E_{j}(\bm{k})\right]^{2}$
(S52)

into $\nu(\mathbf{k}_{D})$, one can obtain

$\displaystyle\nu(\mathbf{k}_{D})$ $\displaystyle=\frac{1}{2\pi
i}\oint_{\Gamma(\mathbf{k}_{D})}d\mathbf{k}\cdot\nabla_{\mathbf{k}}\ln\prod_{1\leq
i<j\leq n}\left[E_{i}(\bm{k})-E_{j}(\bm{k})\right]^{2}$ (S53)
$\displaystyle=\frac{1}{2\pi i}\sum_{i\neq
j}\oint_{\Gamma(\mathbf{k}_{D})}d\mathbf{k}\cdot\nabla_{\mathbf{k}}\ln\left[E_{i}(\bm{k})-E_{j}(\bm{k})\right]$
$\displaystyle=\frac{1}{2\pi}\sum_{i\neq
j}\oint_{\Gamma(\mathbf{k}_{D})}d\mathbf{k}\cdot\nabla_{\mathbf{k}}\arg\left[E_{i}(\bm{k})-E_{j}(\bm{k})\right].$

Therefore, for a two-band system,

$\nu(\mathbf{k}_{D})=\frac{1}{2\pi}\oint_{\Gamma(\mathbf{k}_{D})}d\mathbf{k}\cdot\nabla_{\mathbf{k}}\arg\left[E_{+}(\bm{k})-E_{-}(\bm{k})\right]$
(S54)

which describes the winding of the complex energy between two bands. Now we
show a concrete example of the winding number.

  * Example: Consider the following low energy Hamiltonian around $\mathbf{k}_{D}$,

$\mathcal{H}_{1}(\delta\mathbf{k})=\sigma_{+}+(\delta k_{x}+i\delta
k_{y})\sigma_{-},$ (S55)

where $\delta\mathbf{k}=\mathbf{k}-\mathbf{k}_{D}$ and
$\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$. The eigenvalues of
$\mathcal{H}(\delta\mathbf{k})$ are

$E_{\pm}(\delta\mathbf{k})=\pm\sqrt{\delta k_{x}+i\delta k_{y}}.$ (S56)

When $\mathbf{k}=\mathbf{k}_{D}$, which is equivalent to $\delta k_{x}=\delta
k_{y}=0$, one can find
$E_{+}(\delta\mathbf{k}=0)=E_{-}(\delta\mathbf{k}=0)=0$. This means
$\mathbf{k}_{D}$ is a non-Hermitian degeneracy point. Now we choose
$\Gamma(\mathbf{k}_{D})=\mathbf{k}_{D}+\delta k_{r}(\cos\theta,\sin\theta)$,
then,

$E_{\pm}(\delta\mathbf{k})=\pm\delta
k_{r}^{1/2}e^{i\theta/2},\qquad\theta\in(-\pi,\pi].$ (S57)

One can find that $E_{+}(\delta\mathbf{k})$ and $E_{-}(\delta\mathbf{k})$
forms a spectral loop that encloses
$E_{+}(\mathbf{k}_{D})=E_{-}(\mathbf{k}_{D})=0$. The winding number
$\nu(\mathbf{k}_{D})$ describes this topological properties of degeneracy
points.

The topological charge $\nu(\mathbf{k}_{D})$ can be used to classify the non-
Hermitian degeneracies. However, the classification is not complete. As a
comparison with $\mathcal{H}_{1}(\delta\mathbf{k})$, consider the following
two low energy Hamiltonians,

$\mathcal{H}_{2}(\delta\mathbf{k})=(\delta k_{x}+i\delta
k_{y})^{2}\sigma_{+}+(\delta k_{x}-i\delta k_{y})\sigma_{-}.$ (S58)

Obvious, $\delta\mathbf{k}=0$ is a degeneracy point. One can further prove
that its topological charge is $+1$, which is equal to the charge of
$\delta\mathbf{k}=0$ in $\mathcal{H}_{1}(\delta\mathbf{k})$. However, these
two degeneracy points have different properties. For example

$\mathcal{H}_{1}(\delta\mathbf{k}=0)=\sigma_{+},\qquad\mathcal{H}_{2}(\delta\mathbf{k}=0)=0.$
(S59)

One can notice that $\mathcal{H}_{1}(\delta\mathbf{k}=0)$ is non-diagonal.
This type of non-Hermitian degeneracy points are called exceptional points. In
Ref. Yang  _et al._ (2021), the authors have shown that only the exceptional
points with $\nu(\mathbf{k}_{D})=\pm 1$ are robust in 2D. Any other non-
Hermitian band degeneracies are unstable against non-Hermitian perturbations.

Having clarifying the topological charge of non-Hermitian degeneracies, now we
can prove the corollary of our theorem. Since in 2D, the topological charge of
the stable exceptional points must be $\pm 1$, the corresponding spectrum area
must be nonzero.

###  IV.3 The photonic crystal model

In this subsection, we numerically calculate the spectrum and spatial
distribution of the wave function, i.e. $W(x)$ in Eq. S2, for the photonic
crystal model in the main text under different geometries.

It shows that the skin effect disappears under square geometry in Fig. S6(b),
and reappears under diamond geometry in Fig. S6(d), which is a characteristic
signature of geometry-dependent-skin effect. Here we take the non-Hermitian
parameter $\gamma$ as $1/4$. The spectrum under square geometry (red points in
Fig. S6(a)) coincides with the spectrum of periodic boundary (light blue
region in Fig. S6(a)(c)). We conjecture that the spectrum under diamond
geometry (red points in Fig. S6(d)) will also coincide with the periodic-
boundary spectrum as the system size increases. However, it clearly shows that
the density of states under different geometries is completely different. The
dependence of density of states on the choice of boundary geometry is another
significant feature of geometry-dependent-skin effect.

Figure S7:  Two Weyl points (a) of a three-dimensional Weyl semimetal are
expanded into two exceptional rings (b) after the addition of non-Hermitian
perturbations. The spatial distribution of eigenstate is plotted in (c). The
modulus square of the propagator from $i$ to $o$ $P_{io}(\omega)$ and that
from $o$ to $i$ $P_{oi}(\omega)$, as functions of $\omega$, are plotted with
red color and dark cyan color in (d), respectively.

###  IV.4 The corner-skin effect in a three-dimensional exceptional-line
semimetal

We propose the realization for corner-skin effect in a three-dimensional
system with exceptional lines. Consider a Weyl semimetal with non-Hermitian
term as a perturbation, of which the periodic-boundary hamiltonian reads

$H(k)=[\bm{d_{r}}(k)+i\delta\bm{d_{i}}(k)]\cdot\bm{\sigma},$ (S60)

where $\bm{d_{r}}(k)$ and $\bm{d_{i}}(k)$ are vectors with four components,
that is,

$\begin{split}&\bm{d_{r}}(k)=(0,\sin{k_{x}},\sin{k_{y}},2-\cos{k_{x}}-\cos{k_{y}}+\sin{k_{z}}),\\\
&\bm{d_{i}}(k)=(-\sqrt{5},1+\cos{k_{z}},1-\cos{k_{z}},\cos{k_{z}}).\end{split}$
(S61)

The Hermitian part $\bm{d_{r}}\cdot\bm{\sigma}$ is a Weyl semimetal with two
Weyl points. One Weyl point with $+1$ topological charge [red cone in Fig S7
(a)] is at $(0,0,0)$, and another with $-1$ topological charge [blue cone in
Fig S7 (a)] is at $(0,0,\pi)$. Upon turning on the non-Hermitian term
($\delta\neq 0$), the Weyl points evolve into two exceptional rings as shown
in Fig. S7 (b). According to our theorem, the system with exceptional lines
must have the universal skin effect. Specially, the system described in Eq.
(S60) always has corner-skin effect as shown in Fig. S7 (c) with $\delta=1/6$.

Due to the non-reciprocity of the corner-skin effect, we propose an
experimental approach of two-point Green function to detect the corner-skin
effect. We give a source at $i=(1,1,1)$ position, and probe it at
$o=(16,16,16)$ position in Fig. S7 (c). The modulus square of the propagator
from $i$ to $o$ is expressed as

$P_{io}(\omega)=\sum_{\alpha,\beta=1,2}|\langle
o,\beta|\frac{1}{\omega-\hat{H}}|i,\alpha\rangle|^{2},$ (S62)

where $\alpha,\beta$ label the orbitals of the unit cell. We calculate
$P_{io}(\omega)$ and plot it with red color in Fig. S7 (d). We do the same
process but interchange $i$ and $o$, and $P_{oi}(\omega)$ is plotted with dark
cyan color in Fig. S7 (d). A significant difference between $P_{io}(\omega)$
and $P_{oi}(\omega)$ demonstrates the non-reciprocity of corner-skin effect.

## References

  * Loomis and Sternberg (1968) Lynn Harold Loomis and Shlomo Sternberg,  _Advanced calculus_ (World Scientific, 1968).
  * Zhang  _et al._ (2020) Kai Zhang, Zhesen Yang, and Chen Fang, “Correspondence between Winding Numbers and Skin Modes in Non-Hermitian Systems,” Phys. Rev. Lett. 125, 126402 (2020).
  * Okuma  _et al._ (2020) Nobuyuki Okuma, Kohei Kawabata, Ken Shiozaki, and Masatoshi Sato, “Topological Origin of Non-Hermitian Skin Effects,” Phys. Rev. Lett. 124, 086801 (2020).
  * Yokomizo and Murakami (2019) Kazuki Yokomizo and Shuichi Murakami, “Non-Bloch Band Theory of Non-Hermitian Systems,” Phys. Rev. Lett. 123, 066404 (2019).
  * Denner  _et al._ (2020) M. Michael Denner, Anastasiia Skurativska, Frank Schindler, Mark H. Fischer, Ronny Thomale, Tomáš Bzdušek, and Titus Neupert, “Exceptional Topological Insulators,” (2020), arXiv:2008.01090 .
  * Yang  _et al._ (2020) Zhesen Yang, Ching-Kai Chiu, Chen Fang, and Jiangping Hu, “Jones Polynomial and Knot Transitions in Hermitian and non-Hermitian Topological Semimetals,” Phys. Rev. Lett. 124, 186402 (2020).
  * Yang  _et al._ (2021) Zhesen Yang, A. P. Schnyder, Jiangping Hu, and Ching-Kai Chiu, “Fermion Doubling Theorems in Two-Dimensional Non-Hermitian Systems for Fermi Points and Exceptional Points,” Phys. Rev. Lett. 126, 086401 (2021).