Dynamical Degeneracy Splitting and Directional Invisibility in Non-Hermitian Systems

Abstract

In this paper, we propose the concept called dynamical degeneracy splitting to characterize the anisotropic decay behaviors in non-Hermitian systems. We show that when the system has dynamical degeneracy splitting, it will exhibit (i) anomalous scattering in the bulk and (ii) the non-Hermitian skin effect under the open boundary condition of a generic geometry. As an application, we propose directional invisibility in terms of wave packet dynamics to probe the geometry-dependent skin effect in higher dimensions. Our work provides a feasible way to detect the non-Hermitian skin effect in experiments.

Full text

Dynamical Degeneracy Splitting and Directional Invisibility in Non-Hermitian
Systems
Kai Zhang,1Chen Fang,1, 2, 3, ∗and Zhesen Yang3, †
1Beijing National Laboratory for Condensed Matter Physics,
and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
2Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
3Kavli Institute for Theoretical Sciences, Chinese Academy of Sciences, Beijing 100190, China
In this paper, we introduce the concept of dynamical degeneracy splitting to describe the
anisotropic decay behaviors in non-Hermitian systems. We demonstrate that systems with dynam-
ical degeneracy splitting exhibit two distinctive features: (i) the system shows frequency-resolved
non-Hermitian skin effect; (ii) Green’s function exhibits anomalous at given frequency, leading to
uneven broadening in spectral function and anomalous scattering. As an application, we propose
directional invisibility based on wave packet dynamics to investigate the geometry-dependent skin
effect in higher dimensions. Our work elucidates a faithful correspondence between non-Hermitian
skin effect and Green’s function, offering a guiding principle for exploration of novel physical phe-
nomena emerging from this effect.
Introduction.— Non-Hermitian Hamiltonians [1–6],
which can effectively capture the dynamics of the sys-
tem that is coupled to external environments, have been
implemented in a wide variety of systems [7–18]. With-
out the constraint of Hermiticity, the eigenvalues of the
Hamiltonian can be complex, leading to many intriguing
phenomena in non-Hermitian systems. One such phe-
nomenon is the non-Hermitian skin effect (NHSE) [19–
63], which refers to that extensive eigenstates of system-
size order are concentrated on the open boundaries; and
that the energy spectrum is highly sensitive to the change
of boundary conditions. In one dimension, the NHSE
can be well understood within the framework of the non-
Bloch band theory [19,20,23,35]. Extending to higher
dimensions, a bunch of new appearances in NHSE have
been discovered [27,39,56,57] and the generalization of
the non-Bloch theory has also been attempted [21,64–
66]. However, up to date, many questions still need to
be solved. One representative is the recently discovered
geometry-dependent skin effect [57], where the appear-
ance of NHSE clearly depends on the geometric shapes
of the open boundary. How to interpret and find a guid-
ing principle to detect such kind of phenomena in higher
dimensions? This is one motivation for this work.
Another motivation comes from the following consid-
erations. Previous studies have suggested that the spec-
tral area of the Bloch spectrum can faithfully predict
the presence of higher-dimensional NHSE [57]; however,
this criterion losses the information of band structures
around a given frequency ω∈R[67], which is primarily
concerned in spectroscopic measurements [68]. Moreover,
the criterion fails to relate NHSE to realistic physical ob-
servations. For example, in condensed matter physics,
the transport and response properties of the system are
mainly determined by the excitations near the Fermi en-
∗cfang@iphy.ac.cn
†yangzs@ucas.ac.cn
ergy; thus, only NHSE appearing near the Fermi energy
does matter, which cannot be inferred from the spectral-
area criterion. These considerations inspire us to find
afrequency criterion for the manifestation of NHSE,
thereby delivering NHSE to broader physical scenarios.
In this paper, we demonstrate that the dynamical de-
generacy splitting plays the role of frequency criterion
for the appearance of NHSE. From the perspective of
bulk-boundary correspondence, the dynamical degener-
acy splitting reflects the bulk characteristic of the NHSE
and is independent of boundary conditions, which implies
that the dynamical degeneracy splitting maybe more in-
trinsic than the NHSE itself. As illustrated in Fig. 1(a),
the dynamical degeneracy splitting not only helps us to
understand the physical origin of NHSE, but also reveal a
deep relation to the Green’s function. Given this connec-
tion, we establish the scattering theory in non-Hermitian
systems, and reveal that when the dynamical degener-
acy splitting occurs, the scattered waves will be damped
away from impurities. Applying the scattering theory to
geometry-dependent skin effect, we propose a new phe-
nomenon unique to non-Hermitian systems called direc-
tional invisibility, which refers to that the reflected com-
ponents of the incident wave packet are visible when the
impurity line is in several spatial directions but invisible
in the remaining directions. As an application, direc-
tional invisibility can serve as an experimentally feasi-
ble method to directly detect the existence of geometry-
dependent skin effect without the need for open boundary
conditions.
Dynamical degeneracy splitting and frequency-resolved
NHSE.— Now we start with the non-Hermitian Bloch
band Eµ(k) to explain what dynamical degeneracy split-
ting is. For a given excitation frequency ω∈R(the real
frequency is assumed throughout this paper), the equal-
frequency contour K(ω) can be defined as
K(ω∈R) = K1(ω∈R)∪... ∪Km(ω∈R),(1)
where Kµ(ω∈R) = {k∈BZ|Re Eµ(k) = ω}. When
ωis chosen to be the chemical potential in electronic

2
(f)
(b)
(e)
0
0.2
0.4
0.6
0.8
1.0
(d)
(c)
0
0.2
0.4
0.6
0.8
1.0
Frequency-resolved
NHSE
Dynamical degeneracy
splitting
Green’s
function
Scattering theory
Spectral
function
Directional invisibility
BBC
(a)
FIG. 1. (a) Schematic of relation between dynamical degeneracy splitting (DDS), frequency-resolved NHSE, and the physical
consequences via Green’s function. (b)-(f ) show the bulk-boundary correspondence between dynamical degeneracy splitting and
frequency-resolved NHSE. For the Bloch spectrum in (b), there is no dynamical degeneracy splitting at excitation frequency ω1
in (c), where the color bar corresponds to imaginary energy ImE(k) on the equal-frequency contour (the dashed yellow lines).
Correspondingly, NHSE is absent at this frequency in (e). Dynamical degeneracy splitting occurs at ω2as shown in (d), and
consequently, NHSE appears at frequency ω2in (f).
systems, K(ω) is nothing but the renormalized Fermi
surface. An example is illustrated in Fig. 1(b)-(d). When
ω=ω1/2, the corresponding equal-frequency contour is
plotted in Fig. 1(c/d) by yellow dashing lines. Physically,
each point on the equal-frequency contour in Fig. 1(c)
corresponds to an excited mode at frequency ω1, and the
corresponding group velocity of this mode in real space
is along the normal direction at that point on the equal-
frequency contour [69].
In the Hermitian case, all excited modes at frequency
ωare degenerate since Im Eµ(k) = 0 for all k. Under
a generic open boundary geometry, the corresponding
eigenstate with energy ωare constructed by the linear
superposition of these k∈K(ω). However, once the
non-Hermitian term is introduced, the imaginary part
Im Eµ(k) will broaden the equal-frequency contour in
complex ways, which in general will split this degener-
acy as shown in Fig. 1(d). As a result, even under the
same open boundary geometry, the original linear super-
position of Bloch waves is no longer to be the eigenstate,
which implies the emergence of NHSE at frequency ω[69].
We refer to the above type of degeneracy splitting as
dynamic degeneracy splitting. This phenomenon results
from differences in the lifetimes of equal-frequency exci-
tation modes, has a dynamical consequence and corre-
sponds to the frequency-resolved NHSE.
We use the example HNH(k) = cos kx+cos ky+i[(1/2−
cos kx−cos ky) cos kx]−i9/16 to demonstrate this point.
As shown in Fig. 1(b), the dynamical degeneracy split-
ting occurs at ω2=−1/2 but not ω1= 1/2. It implies
that these eigenstates ψi∈ω2(ω1)(r) with eigenvalues sat-
isfying Re Ei,OBC =ω2(ω1) will (not) show NHSE at this
frequency. In Fig. 1(e)(f ), we plot Pω(r) = Pi∈ω|ψi(r)|2
on the diamond geometry with lattice size Lx=Ly= 80.
It is shown that Pω1(r) is extensive on the entire lattice in
(e), but Pω2(r) shows localization behavior in (f), which
demonstrates the correspondence between the dynamical
degeneracy splitting and frequency-resolved NHSE.
Dynamical degeneracy splitting in Green’s function.—
Apart from the relation to NHSE, dynamical degener-
acy splitting is also associated with Green’s function as
illustrated in Fig. 1(a). One consequence from dynami-
cal degeneracy splitting is the uneven broadening in the
spectral function at the excitation frequency ω. For a
given non-Hermitian Hamiltonian HNH(k), one can cal-
culate the spectral function A(ω, k) = −Im Tr[1/(ω+
iη −HNH(k))] to characterize the dynamical degeneracy
splitting, which can be measured directly, for example, by
the Angle-resolved photoemission spectroscopy. There-
fore, one can identify the dynamical degeneracy splitting
from the experimental side by observing the nonuniform
broadening of equal-frequency contour under a given ex-
citation frequency. Applying this result to condensed
matter physics, we further propose that quasiparticle in-
terference become anomalous as discussed in [69]. This
will be another experimental signature for the existence
of dynamical degeneracy splitting.
Next, we will demonstrate that anomalous scattering
behavior is another consequence of dynamical degeneracy
splitting via Green’s function. The anomalous scattering
here refers to the phenomenon that a defect can scatter
the propagating plane waves to exponentially damped
waves away from the scatterer.
Anomalous scattering theory.— We first establish a
general scattering theory in a two-dimensional non-
Hermitian system with single band. It is straightforward
to extend our discussion to general situations [69]. The
full Hamiltonian can be expressed as, H=HNH(k) + V,

3
where Vis the scattering potential. Now consider an in-
cident wave ϕi(r) (or an excitation) with momentum ki
propagating on the lattice and hits the scatterer. Then,
the scattered waves ϕs(r) can be captured by the follow-
ing integral equation [69],
ψ(r) = ϕi(r) + ϕs(r)
=ϕki(r) + Zdr′G+
0(E(ki); r,r′)V(r′)ψ(r′),(2)
where ϕki(r) = ⟨r|ϕki⟩=eikirrepresents the inci-
dent wave, and ϕs(r) comprises reflected and transmit-
ted waves. Here, V(r) = ⟨r|V|r⟩is the scattering po-
tential function, and G+
0(E(ki); r,r′) = ⟨r|[E(ki) + iη −
HNH(k)]−1|r′⟩with η→0+is the retarded Green’s func-
tion. The integral equation Eq. (2) tells us that: (i) after
introducing the scattering potential V, the eigenstate of
the full Hamiltonian H, i.e. ψ(r), can be decomposed
into the incident and scattered waves with the same en-
ergy E(ki) [69]; (ii) the anomalous behavior of the scat-
tering process comes from the anomalous property of the
retarded Green’s function in non-Hermitian systems.
Now we use an impurity line, labeled by Lθ, as an
example to demonstrate the anomalous scattering. As
shown in Fig. 2(a)(b), we specify the impurity line lying
on the position r⊥= 0 and along the θ-direction. For the
left (right) side of the impurity line Lθ, the corresponding
region is denoted by r⊥<0 (r⊥>0). Therefore, the
(b)
(a)
extended
localized
FIG. 2. The illustrations of conventional scattering in (a)
and anomalous scattering in (b). Here, the model Hamilto-
nian reads HNH(k) = 2 sin kxcos ky−2 cos kx+i(cos kx−1),
and the impurity line Lθis along θ= 3π/4 direction. The
dark and light blue dots represents parts of poles calculated
in Eq.(4). As η→0, the corresponding poles evolve from
dark to light blue dots, as indicated by arrows.
impurity-line scattering potential function reads
V(r) = λδ(r⊥= 0).(3)
Note that the translation symmetry along rθdirection is
preserved in the scattering process. Therefore, we sub-
stitute Eq.(3) into Eq.(2) and take the Fourier transform
from rθto kθ, and finally obtain the solution of scattering
wave [69] as
ϕs(kθ
i, r⊥) = λψ(kθ
i,0) 








X
|zin|<1
C(zin)zr⊥
in , r⊥>0;
X
|zout|>1−C(zout )zr⊥
out, r⊥<0,
(4)
where kθ
i=ki·eθis the θ-component of the incident
momentum; the coefficient ψ(kθ
i,0) is a constant for a
given kθ
i; C(zin/out) equals 2πi times the residue of the
function [z[E(ki)+iη −HNH(kθ
i, z)]]−1at the pole zin/out
inside/outside the |z|= 1 curve, as shown in Fig. 2(a)(b).
Now we show that when the dynamical degeneracy
splitting occurs, the reflected wave will become localized.
Without loss of generality, we assume the incident plane
wave ki= (kθ
i, k⊥
i) with energy E(ki) comes from r⊥<0
region. There are two cases of scattered waves. In case
(i), there are at least two poles that touch the |z|= 1
simultaneously from the inner and outer sides respec-
tively when η→0+, and one example is illustrated in
Fig. 2(a); In case (ii), there is only one pole touches
the |z|= 1 curve from the inside as η→0+, as shown
in Fig. 2(b). The more details are present in [69]. It
can be seen from Eq. (4) that there will be two domi-
nant propagating modes survive at infinity for case (i),
namely the transmitted wave in the region r⊥>0 and
the reflected wave in the region r⊥<0, as illustrated in
Fig. 2(a). In case (ii), we have one dominant transmitted
wave in the r⊥>0 region, while in the r⊥<0 region the
dominant reflected wave is a spatially localized wave as
η→0+. Therefore, the scattering process is anomalous,
as illustrated in Fig. 2(b).
Directional invisibility.— Now we show that for the
geometry-dependent skin effect (GDSE), the correspond-
ing scattering process exhibits directional invisibility.
The Bloch Hamiltonian of GDSE model [57] reads
H(k) = X
i=0,x,y,z
di(k)σi−iγ
2(σ0−σz),(5)
where di(k) is real function of kand σirepresents the
Pauli matrix. The only non-Hermitian parameter γ >
0 is used to describe the dissipative system. Specifi-
cally, {d0, dx, dy, dz}={µ0+t0(cos kx+ cos ky), t[1 −
cos kx−cos ky+ cos(kx−ky)], t[sin kx−sin ky−sin(kx−
ky)], µz+tz(cos kx−cos ky)}. We plot the spectral func-
tion A(ω0,k) in Fig. 3(a)(b). It shows the uneven broad-
ening on the equal-frequency contour (the gray curve
representing equal-frequency contour), which is a defi-
nite signature of the occurrence of dynamical degeneracy

4
2.5
5.0
7.5
10.0
12.5
15.0
(a)
(b)
(c1) (c2) (c3) (c4) (c5)
(d1) (d2) (d3) (d4) (d5)
FIG. 3. Directional invisibility in Hamiltonian Eq.(5) with the parameters (µ0, µz, t0, t, tz, γ) = (1.35,−0.05,−0.4,0.4,−0.6,1).
(a-b) show the spectral function A(ω, k) with ω= 3/2, of which the intensity corresponds to the opacity as shown in the color
bar. Here, Lθrepresents the impurity line, and kidenote the incident wave and ks(k′
s) indicates the scattered wave. The
incident wave packet with the momentum center at kihits the oblique impurity line in (c1)-(c5) and vertical impurity line in
(d1)-(d5), where the impurity strength λ= 1.
splitting. According to the established scattering theory,
the anomalous scattering will occur.
We assume an incident plane wave has ki= (kx
i, ky
i) =
(π/2,0) and hits the impurity line Lθwith a rightward
velocity in real space. Note that kilies on the equal-
frequency contour, i.e., K(ω) with ω= 3/2, as shown
in Fig. 3(a)(b). The impurity line Lθpreserves the mo-
mentum along this direction, which means that the scat-
terer Lθrelates kiwith ksand k′
sin the way illustrated
in Fig. 3(a) and (b), respectively. In Fig. 3(a), due to
the larger broadening at ksthan that at ki, the re-
flected wave is damped exponentially away from the im-
purity line, which means that anomalous scattering oc-
curs as discussed in case (ii). The band dispersion of the
Hamiltonian in Eq.(5) is mirror symmetric under Mx:
(kx, ky)→(−kx, ky) and My: (kx, ky)→(kx,−ky).
This means k′
s=Mxkihas the equal broadening with
ki, as shown in Fig. 3(b). Therefore, the conventional
scattering discussed in the case (i) occurs for the ver-
tical impurity line that scatters the incident plane wave
to another propagating plane wave k′
s. This phenomenon
that the visibility of reflected waves depends on the direc-
tion of impurity line is dubbed directional invisibility and
unique to higher-dimensional non-Hermitian systems.
To probe the directional invisibility in this example,
the incident wave is chosen as a Gaussian wave packet
with momentum center at kifor the scattering simu-
lation, as shown in Fig. 3(c)(d). The time evolution
of wave packet follows |ψ(t)⟩=N(t)e−iHt |ϕ0⟩, where
His the full Hamiltonian consisting of the free Hamil-
tonian in Eq.(5) and impurity-line scattering potential
V(r) = λσ0δ(r⊥= 0), and N(t) is the normalization fac-
tor at every time. The incident Gaussian wave packet
has the form ϕ0(r) = exp[−(r−r0)2/σ2+ik0
ir](1,1)T.
In Fig. 3(c)(d), the parameters are set as (x0, y0, σ) =
(14,20,4), and the lattice size is Lx=Ly= 40. It can be
observed that the Gaussian wave packet is almost com-
pletely transmitted through the oblique impurity line Lθ
without evident reflected waves, as shown in Fig. 3(c1)-
(c5). However, parts of the wave packet are reflected
by the vertical impurity line Lθas a propagating wave,
shown in Fig. 3(d1)-(d5). Therefore, the wave-packet
scattering simulation in Fig. 3(c)(d) verifies the direc-
tional invisibility.
The role of symmetry.— Now we discuss the role of
symmetry, which reveal the correspondence between di-
rectional invisibility and GDSE. For HNH(k), all sym-
metries preserving the complex energy form the scatter-
ing group of HNH(k), labeled by Gs, which includes, for
example, the reciprocity ¯
T[25] and point-group symme-
tries, such as rotation, inversion, and mirror symmetry
M. Now suppose that there is an incident wave with mo-
mentum ki. Under the action of Gs,kiwill be mapped
to a set of other points on the BZ with the same energy,
that is, Eµ(ki) = Eµ(gski) with gs∈Gs. Note that ki
and gskidetermines a direction crossing them, and we
label the impurity line perpendicular to this direction by
Lgski. Now we state the conclusion: for the incident wave
ki, the scattering process for the impurity line Lgskiis
conventional; while for all other directions, the scattering
process is not protected by symmetry gsand is generally
anomalous. It should be noted that if gs=Mor ¯
T M,
Lgskiis exactly parallel or perpendicular to the mirror
line, respectively, and is independent of ki. We label such
a impurity line as Lgs. It means that for the impurity
line Lgs, the scattering process for all possible incident
states is conventional. For example, if HNH(k) preserves
Mxsymmetry, then LMxis along the y-direction, and

5
the conventional scattering occurs on the impurity line
LMxfor all possible incident waves.
Now we relate it to the GDSE. For GDSE, if there
is an edge parallel to the impurity line Lgs, then the
edge shows conventional scattering for all ki∈BZ, cor-
respondingly, the open boundary eigenstates cannot be
localized at that edge. Based on this principle, one can
find that if the Hamiltonian has Mand/or ¯
T M sym-
metry, then open boundary eigenstates can no longer be
localized at the edges parallel to LMand/or L¯
T M under
any shape of open boundary geometry.
Conclusions and Discussions.— In summary, we in-
troduce the concept of dynamical degeneracy splitting
to characterize the nonuniform decay behavior of ex-
cited modes at a given frequency. On the one hand, the
dynamical degeneracy splitting predicts the frequency-
resolved NHSE; on the other hand, it associates with
the anomaly in Green’s function at a specified frequency,
leading to uneven broadening in spectral function and
anomalous scattering. As an application, we propose a
type of anomalous scattering, directional invisibility, as
an experimental indicator of the existence of GDSE.
This work essentially provides a frequency criterion
that can further help us understand and define NHSE in
the system beyond conventional band theory. For exam-
ple, in the electronic system with self-energy corrections,
the retarded Green function has the form G(ω, k) =
[ω− Heff (ω, k)]−1, where the effective Hamiltonian de-
pends on the frequency ω. The dynamical degeneracy
splitting can still be well-defined, correspondingly, the
concept of NHSE can be extended in such systems, which
laid the foundation for further studies.
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[69] See Supplemental Material for (i) Equal frequency con-
tour and dynamical degeneracy splitting; (ii) Scattering
theory in the non-Hermitian tight-binding Hamiltonian;
(iii) Analysis of the time evolution of scattering waves; (iv)
The relation between directional invisibility and geometry-
dependent skin effect; (v) The robustness of anomalous
scattering.

7
Supplemental Material for “Dynamical Degeneracy Splitting and Directional
Invisibility in Non-Hermitian Systems”
CONTENTS
S-1. Equal frequency contour and dynamical degeneracy splitting 7
A. General formula of equal frequency contour in non-Hermitian systems 7
B. An example in two-band model 8
C. The singularities and intersections on equal-frequency contour 9
D. The relationship between dynamical degeneracy splitting and non-Hermitian skin effect 10
E. Quasiparticle interference and dynamical degeneracy splitting 11
S-2. Scattering theory in non-Hermitian tight-binding model 12
A. The general scattering theory 12
B. The derivation of Eq.(4) in the main text 13
C. Eq.(4) in the multi-band and multiple pole cases 14
D. Two cases of scattering waves 15
E. The relation between dynamical degeneracy splitting and pole’s behavior 16
S-3. Scattered waves in time evolution 17
S-4. Directional invisibility and geometry-dependent skin effect 19
A. The directional invisibility in single-band model 19
B. The dissipative reflected and transmitted intensity 20
C. The normalization of spectral function in dissipative system 21
S-5. The robustness of anomalous scattering 21
A. Anomalous scattering on general scattering potential 21
B. The numerical simulation of anomalous scattering on finite step potential and open boundary 23
C. The robustness of directional invisibility 23
References 24
S-1. EQUAL FREQUENCY CONTOUR AND DYNAMICAL DEGENERACY SPLITTING
In this section, we will give a general method for obtaining the equal-frequency contour in non-Hermitian systems.
As an example, we use a two-band model to demonstrate the equal-frequency contour and dynamical degeneracy
splitting. Then, we discuss the cases where singularities and intersections exist on the equal-frequency contour. In
the last part, we give the relation between dynamical degeneracy splitting and the emergence of non-Hermitian skin
effect.
A. General formula of equal frequency contour in non-Hermitian systems
We assume a (non-Hermitian) Bloch Hamiltonian H(k) with mbands in ddimensions, of which the characteristic
equation can be expressed as
f(ω, ImE , k) = det[H(k)−(ω+iImE)Im]=0,(S1)
where E=ω+iImEis the complex energy with the excitation frequency ωand line width ImE, and ω, ImE∈R.
Imrepresents the m×midentity matrix. In general, f(ω, ImE, k) is a complex function, hence, Eq.(S1) imposes
two real constraints, fr(ω , ImE, k) = fi(ω, ImE , k) = 0. Here, the subscript r/i represents the real/imaginary part of
f(ω, ImE , k), respectively. By definition, equal-frequency contour is only determined by ωand is regardless of ImE.
Therefore, we can give an analytic traceable formula to equal-frequency contour. We define
F(ω, k) := Res[fr(ω, ImE , k), fi(ω, ImE, k),ImE].(S2)

8
(a)
(b)
(d)
(c)
K(ω1)
0
10
20
30
40
50
0
10
20
30
40
K(ω0)
FIG. S1. Non-Hermitian Bloch spectrum (a), equal-frequency contour and dynamical degeneracy splitting (b-d) for the
Hamiltonian Eq.(S7) with system parameters {µ0, µz, t0, t, tz, γ}={1.35,−0.05,−0.4,0.4,−0.6,1}. The red and blue surface
in (b) represent fr(ω1,ImE, k) = 0 and fi(ω1,ImE, k) = 0, respectively. The gray solid lines in (c) and (d) represent the
equal-frequency contours at ω0and ω1, respectively. The spectral function in Eq.(S6) is plotted in (b)(c) with η= 1/50, and
the intensity of spectral function corresponds to the opacity as shown in the color bar.
Here, Res[fr, fi,ImE] represents resultant [1] between frand firegarding variable ImE. After doing this, variable ImE
is eliminated and a real function F(ω, k) is obtained. Finally, for a generic ω0, we can obtain the (d−1)-dimensional
equal-frequency contour expressed as,
K(ω0) = {k∈BZ|F(ω0,k)=0}.(S3)
Here the expression of equal-frequency contour hides the information of the energy band index, which is slightly
different from the definition in the main text. Note that for each k∈BZ, the energy band index is well defined.
Therefore, we can obtain the expression of equal-frequency contour in the main text,
K(ω∈R) = K1(ω∈R)∪... ∪Km(ω∈R),(S4)
where Kµ(ω∈R) = {k∈BZ|Re Eµ(k) = ω}, and Eµ(k) represents the µ-th band of the Bloch Hamiltonian H(k).
Here, we discuss the physical meaning of equal-frequency contour with dynamical degeneracy splitting. As a general
assumption, equal-frequency contour is always formed by simple closed curves without singularities or intersections.
It means that ∇kRe Eµ(k)|k∈Kµ(ω)= 0 for µ= 1, ..., m, which is satisfied for a generic frequency ω. For a given
frequency ω0, each point on the equal-frequency contour corresponds to an excited mode, and the group velocity in
real space is along the normal direction at that point, which can be calculated by
v(k)=(vx, vy) = −∂kxF(ω , k)
∂ωF(ω, k),−∂kyF(ω, k)
∂ωF(ω, k)ω=ω0
,(S5)
where F(ω, k) in defined in Eq.(S2). When dynamical degeneracy splitting occurs, the excited modes on Kµ(ω)
obtains different decay rates and will evolve as ∼e−ImEµ(k)t. From the perspective of spectral information, the
imaginary part Im Eµ(k) will broaden the equal-frequency contour. This can be usually reflected in spectral function,
A(ω, k) = −ImTr[(ω+iη − H(k))−1]/N, (S6)
where η→0+and Nis the total number of energy bands, therefore, it is detectable in experiments.
B. An example in two-band model
As an example, we consider the two-band Bloch Hamiltonian in Eq.(5) in the main text, that is,
H(k) = X
i=0,x,y,z
di(k)σi−iγ
2(σ0−σz),(S7)
where {d0, dx, dy, dz}={µ0+t0(cos kx+ cos ky), t[1−cos kx−cos ky+cos(kx−ky)], t[sin kx−sin ky−sin(kx−ky)], µz+
tz(cos kx−cos ky)}. This model demonstrates the bulk Fermi arc [2] in photonic crystal experiment. Theoretically,
this model has the geometry-dependent skin effect [3], i.e., non-Hermitian skin effect appears under open boundary
conditions of generic geometries (e.g., the diamond geometry), but disappears under square geometry.

9
The Bloch spectrum of Hamiltonian Eq.(S7) is shown in Fig. S1(a). We choose ω0and ω1as the excitation
frequency, as indicated by the dashed lines. When ω=ω0, we can get two surfaces in (kx, ky,ImE) space, namely
fr(ω1,ImE, k) = 0 and fi(ω1,ImE, k) = 0, as shown by the red and blue surfaces in Fig. S1(b), respectively. The
intersection of these two surfaces is shown by the red line in Fig. S1(b), and its projection onto kplane is exactly the
equal-frequency contour curve in (c). Actually, according to the formula in Eq.(S3), we can directly obtain the equal-
frequency contours K(ω0) and K(ω1), as plotted by the gray solid lines in Fig. S1(c) and (d), respectively. It can
be seen from Fig. S1(b) that the momenta on the equal-frequency contour have different imaginary energies ImE(k),
which means that dynamical degeneracy splitting occurs. It can be reflected in the spectral function A(ω, k). We
plot the spectral function A(ω0,k) and A(ω1,k) in Fig. S1(c) and Fig. S1(d), respectively. The intensity of spectral
function corresponds to the opacity of red color. It demonstrates the occurrence of dynamical degeneracy splitting in
this two-band model.
C. The singularities and intersections on equal-frequency contour
In this part, we first demonstrate that equal-frequency contour is always formed by simple closed curves without
singularities or intersections. Then, we discuss the case where exceptional points exist on the equal-frequency contour
and give an example to show it.
As we defined in Eq.(S2), the algebraic equation that determines the equal-frequency contour is
F(ω, k) := Res [fr(ω, Im E , k), fi(ω, Im E, k),Im E] = 0,
which is a real function with variables kand ω. Mathematically, singularities or intersections on the equal-frequency
contour must satisfy
F(ω, k) = ∇kF(ω, k)=0.(S8)
In 2D cases, Eq.(S8) reduces to F(ω, k) = ∂kxF(ω, k) = ∂kyF(ω , k) = 0. Since we have three real variables ω , kx, ky,
and three real equations, the dimension of corresponding solutions is zero-dimension, which is nothing but a set of
points. The ωin the solution determines the corresponding equal-frequency contour with singularities or intersections.
Since the solution is a set of points, for a generic excitation frequency ω0, it is expected that it does not include
singularities or intersections. Therefore, we assume that equal-frequency contour is always formed by simple closed
curves without singularities or intersections.
In general, the non-Hermitian band degeneracy is determined by the following equations
f(E, k) = ∂Ef(E , k)=0.(S9)
Here E∈Cand f(E, k) is a complex equation with two real components. In general, the solution of the above
equation determines the exceptional points in the Bloch bands.
For a given equal-frequency contour band that satisfies Eq.(S8)F(ω, k) = 0, the corresponding band degeneracy is
determined by the following equations
F(ω, k) = ∂ωF(ω, k)=0.(S10)
For 2D cases, we have three real variables and two real equations. Therefore, the corresponding solution is general a
1D line in the parameter space. This solution exactly determines the Fermi arc in the BZ. In general, the end point
of the Fermi arc is the exceptional point. Therefore, the solution of Eq.(S9) in general belongs to Eq.(S10). Actually,
we can find the common solutions between Eq.(S8) and Eq.(S10), which are determined by F(ω, k) = ∂ωF(ω, k) =
∇kF(ω, k) = 0. These points are singularities of F(ω, k).
Here, we take a two-band example that has equal-frequency contour with exceptional points and Fermi arcs. The
Hamiltonian of this example reads
H(k) = (2 −2 cos kx−2 cos ky)σ0+ cos kxcos kyσx−(cos kx+ cos ky)σz−iγ (σ0−σz),(S11)
where σx,y represent Pauli matrices and σ0is the identity matrix. The last term is the non-Hermitian term, meaning
that the second orbital has a decay rate γ. First, let γbe zero, the Hamiltonian reduces to Hermitian and has the
band structure plotted in Fig. S2(a), where the blue (red) surface indicates the first (second) energy band and the
green plane refers to the energy plane E= 2. The corresponding equal-frequency contour at ω0= 2 is shown in
Fig. S2(b), where the blue (red) curve corresponds to the band with the same color. Note that on the equal-frequency
contour at ω0, there are four degeneracies indicated by green dots in Fig. S2(b).

10
(c) (d)(a) (b)
FIG. S2. (a) The band structure and (b) equal-frequency contour at ω= 2 of Hamiltonian Eq.(S11) when γ= 0. The green
plane in (a) refers to E= 0 plane and the green dots in (b) corresponds to the band degeneracies. (c) The band structure
and (d) equal-frequency contour at ω= 2 of Hamiltonian Eq.(S11) when γ= 1/4. The green dots and blend-color arcs in (d)
represent exceptional points and bulk Fermi arcs, respectively.
When adding the loss term (γ= 1/4), the real energy bands of the non-Hermitian Hamiltonian H(k) are plotted in
Fig. S2(c), and the corresponding equal-frequency contour at ω0= 2 is shown in Fig. S2(d). It shows that the original
degeneracies split into eight exceptional points (the green dots) that are connected by four pieces of bulk Fermi arcs
(in blend color). On the Fermi arcs, two real energy bands degenerate but the imaginary parts are different. The
group velocity at exceptional points cannot be defined, similar to the intersections. Note that in this example, only
the equal-frequency contour at ω= 2 has exceptional points. When we excite at the frequency of Fermi arcs, there
are always two modes are excited with velocities, vµ=1,2= (∂kxRe Eµ, ∂kyRe Eµ), and these two excited modes must
have different decay rates.
D. The relationship between dynamical degeneracy splitting and non-Hermitian skin effect
Here, we discuss the relation between dynamical degeneracy splitting and non-Hermitian skin effect by the following
logic: (i) The appearance of dynamical degeneracy splitting; (ii) The original linear superposition of Bloch waves is
no longer to be the eigenstate of HOBC; (iii) If E0is the eigenvalue of HOBC, the corresponding eigenstate must
contain some non-Bloch wave components. The appearance of non-Bloch waves in the OBC eigenstate implies the
emergence of non-Hermitian skin effect; (iv) If E0is not the eigenvalue of HOBC, the mismatch between PBC and
OBC spectrum also indicates the emergence of non-Hermitian skin effect [4,5]. Therefore, no matter whether E0is
the OBC eigenvalue or not, the non-Hermitian skin effect always appears.
Now we explain it in more details. In order to simplify the discussion, we consider a single band model with the
Hamiltonian
H(k) = h0(k) + iλΓ0(k),
where h0(k) and Γ0(k) are real functions. For a given excitation energy ω0∈R, the corresponding equal-frequency
contour K(ω0) is defined as
K(ω0) = {k∈BZ|ω0= Re H(k)}.
And the set of pre-images of energy E0in BZ is defined as
Q(E0) = {k∈BZ|E0=H(k)}.
These k(plane waves) in the set Q(E0) can be linearly superimposed into the wavefunction with energy E0under
some particular OBC geometry (for example, square geometry), that is,
ψ0(r) = X
k∈Q(E0)
ckeikr ,(S12)
where these independent coefficients ckcan be determined by specific open boundary conditions in a 2D lattice.
Note that the wavefunction ψ0(r) compose of plane waves is always extended wavefunction. By definition, the equal-
frequency contour K(ω0∈R) is always 1D line in BZ, while Q(E0) is not always 1D line.

11
We consider the non-Hermitian case with λ= 0. Owing to the term iλΓ0(k), dynamical degeneracy splitting
may occur, and the equal-frequency contour will split in term of imaginary energy. Correspondingly, the degenerate
eigenvalue ω0in Hermitian limit splits into ω0+iλΓ0(k) with k∈K(ω0). Taking one of the eigenvalues as an
example, E0=ω0+iγ0. The corresponding pre-images in BZ Q(E0) include some (order-1) kpoints. However, for
a two-dimensional lattice, there are order-Ldifferent open boundary conditions (Lrepresents the system length); for
example, there are order-Ledges in different angles. Therefore, we need order-Lindependent coefficients cksuch
that the extended wavefunction ψ0(r) with energy E0=ω0+iγ0can always be superimposed under all order-L
possible open boundary conditions. Otherwise, there must be some open boundary (geometries) conditions under
which the wave function ψ0(r) with E0is a localized skin mode instead of an extended wave. Finally, we conclude
that Q(ω0+iγ0) having some (order-1) kpoints is insufficient to form extended wavefunction ψ0(r) in Eq.(S12)
under all order-Ldifferent open boundary (geometries) conditions. Therefore, for a wave function ψ0with energy
E0=ω0+iγ under generic open boundary geometry, if we assume it to be an extended wavefunction, the number
of plane waves kto form the extended wavefunction will in general be order-L in two dimensions. In the case of
dynamical degeneracy splitting, the set Q(ω0+iγ0) is not sufficient to form the Bloch wave under a generic OBC
geometry. It implies that if ω0+iγ0is the OBC eigenvalue, the corresponding OBC eigenstate must contain some
non-Bloch components with Im k= 0, which means the appearance of non-Hermitian skin effect. If ω0+iγ0is not
the OBC eigenvalue, the mismatch between PBC and OBC spectrum also indicates the emergence of non-Hermitian
skin effect. Therefore, no matter whether E0=ω0+iγ0is the OBC eigenvalue or nor, the non-Hermitian skin effect
always appears.
E. Quasiparticle interference and dynamical degeneracy splitting
In condensed matter physics, the interference pattern of quasiparticles around a single impurity observed through
scanning tunneling spectroscopy has been widely used for electronic structure characterization of unconventional
states, for example, the topological surface states. Consider a point impurity in the system, the local density of states
is no longer uniform due to the breakdown of translation symmetry. The quasiparticle interference pattern can be
ω0
-6-3 0 3 6
-5
-2
1
ReE
ImE
Spectrum
ω0
-6-3 0 3 6
1
-1
0
ReE
ImE
Spectrum
0
0.2
0.4
0.6
0.8
1.0
0
0.2
0.4
0.6
0.8
1.0
ω0
-6-3 0 3 6
1
-1
0
ReE
ImE
Spectrum
ω0
-6-3 0 3 6
1
-1
0
ReE
ImE
Spectrum
ω0
-6-3 0 3 6
1
-1
0
ReE
ImE
Spectrum
ω0
-6-3 0 3 6
1
-1
0
ReE
ImE
Spectrum
2000
4000
6000
8000
0
1000
2000
3000
4000
5000
(a)
(b)
(c)
(d)
(e)
(f)
FIG. S3. (a)-(c) and (d)-(f) show the spectrum, equal-frequency contour weighted by different imaginary energies, and
quasiparticle interference pattern for the Hamiltonian in Eq.(S14) with γ= 0 and γ= 1, respectively. In (b)(e) the yellow
dashed circle represents equal-frequency contour at ω0=−3, and the color bar corresponds to the Im E(k) on the equal-
frequency contour. In (c)(f ) the color bar corresponds to the quantity ρ(q, ω0) calculated in Eq.(S13) .

12
calculated by the Fourier transform of the local density of states. It can be derived as
ρ(q, ω) = 1
Nπ X
k
Tr[G(k,k+q, ω )−G∗(k+q,k, ω)],(S13)
where G(k,k′, ω) represents the retarded Green’s function including the impurity potential. The position of high-
intensity peaks in ρ(q, ω) generically depends on the geometric features of the equal-frequency contour at frequency ω.
As we pointed out in our manuscript, the dynamical degeneracy splitting leads to the anomalous scattering behaviors.
Therefore, we can expect dynamical degeneracy splitting will significantly affect the quasiparticle interference pattern,
which can be directly detected in scanning tunneling spectroscopy experiments.
Next, we use a simple example to demonstrate it. The Hamiltonian of the simple model can be written as
HNH(k) = −2 cos kx−2 cos ky−2iγ(1 −cos ky),(S14)
where γcontrols the non-Hermitian part that leads to the dynamical degeneracy splitting. We can clearly observe in
Fig. S3(a)(b) that when γ= 0, there is no dynamical degeneracy splitting at ω0=−3, and consequently, the highest
intensity peaks in the quasiparticle interference pattern in Fig. S3(c) reveal the dominant scattering channels on the
equal-frequency contour at ω0, that is, q= 2kcorresponding to the back scattering in this simple example. The
secondary intensity peaks looks like petal shape inside the dashed black circle in Fig. S3(c) and corresponds to the
scattering channels from kto keiθ (θ= 0, π) on the equal-frequency contour in Fig. S3(b) .
As shown in Fig. S3(d)(e), when γ= 1, dynamical degeneracy splitting occurs at ω0=−3. Consequently, the
quasiparticle interference pattern in Fig. S3(f) significantly different from that in Fig. S3(c). We conclude the following
main points from the differences in their quasiparticle interference patterns. First, the highest peaks Fig. S3(f) reveal
that the dominant scattering channels occur between the momenta with longest lifetimes on the equal-frequency
contour shown in Fig. S3(e); Second, the scattering channels from kto keiθ (θ= 0, π) in most directions θare
suppressed by dynamical degeneracy splitting while in several directions are reserved as we discussed in the anomalous
scattering in the main text; correspondingly, the petal shape inside the black dashed circle in Fig. S3(c) disappears in
Fig. S3(f). Therefore, quasiparticle interference pattern reflects the geometric shape and the dynamical degeneracy
splitting on the equal-frequency contour.
S-2. SCATTERING THEORY IN NON-HERMITIAN TIGHT-BINDING MODEL
In this section, we first give the general form of the scattering equation. Then, we derive Eq.(4) of the main text
and extend it into multi-band and multiple-pole cases. After this, we demonstrate that there are two cases for the
scattered waves. Finally, we discuss the relation between dynamical degeneracy splitting and the pole’s behavior
shown in Fig. 2 of the main text.
A. The general scattering theory
We begin with a general non-Hermitian Hamiltonian in ddimensions,
H=H0+V=X
rX
l
q
X
α,β=1
tl,αβ|r, α⟩⟨r+l, β|+X
r
q
X
α,β=1 Vαβ (r)|r, α⟩⟨r, β |,(S15)
where H0is the free Hamiltonian and Vrepresents the scattering potential. Here r= (r1, r2, . . . , rd) represents the
lattice coordinate in ddimensions, and l= (l1, l2, . . . , ld) indicates the hopping displacement with the largest distance
|lmax|.αand βlabel different degrees of freedom per unit. Note that the free Hamiltonian has translation symmetry,
which can be written in momentum space as,
H0=X
k∈BZ X
l
q
X
α,β=1
tl,αβeikl|k, α⟩⟨k, β |=X
k∈BZ H0(k)|k⟩⟨k|,(S16)
where H0(k) is a q×qnon-Hermitian Bloch Hamiltonian. Then, we define the right and left eigenstates [6] as
H0(k)|uR
µ(k)⟩=Eµ(k)|uR
µ(k)⟩;
⟨uL
µ(k)|H0(k) = Eµ(k)⟨uL
µ(k)|.(S17)

13
Here we assume the Bloch Hamiltonian has no spinful reciprocity [7] to enforce the band degeneracy, namely Eν(k) =
E˜ν(−k).
Next, we establish the scattering theory in the d-dimensional non-Hermitian lattice. Once the scattering process
occurs far away from the boundary, the Bloch Hamiltonian is a good starting point. Consider an incident wave |ϕi⟩
as an eigenstate with energy E0of the free Hamiltonian H0. The local scattering potential Vis a perturbation to the
continuum spectra of scattering states in the thermodynamic limit. Therefore, we have two eigenequations,
H0|ϕi⟩=E0|ϕi⟩; (S18)
(H0+V)|ψ⟩=E0|ψ⟩,|ψ⟩=|ϕi⟩+|ϕs⟩.(S19)
Here |ψ⟩represents the eigenstate of the full Hamiltonian H, which comprises the incident wave |ϕi⟩and scattering
wave |ϕs⟩. Subtracting Eq.(S18) from Eq.(S19), we can get (E0−H0)(|ψ⟩−|ϕi⟩) = V|ψ⟩. Further, we can obtain the
scattering equation,
|ψ⟩=|ϕi⟩+G+
0(E0)V|ψ⟩,(S20)
where G+
0(E0)=[E0−H0+iη]−1with η→0+. Finally, we can write the scattering equation in real space, i.e.,
ψα(r) = ϕi,α(r) + ϕs,α (r)
=ϕi,α(r) + X
βσ Zdr′G+
0,αβ(E0;r,r′)Vβσ (r′)ψσ(r′),(S21)
where ϕs,α(r) = ⟨r, α|ϕs⟩is the α-th component of scattered wave at r-th lattice site. It is clear that the characteristics
of scattered wave are encoded in the retarded Green function. Note that E0is an eigenvalue of Hamiltonian H0, hence
E0∈ {E|E=Eµ(k),∀µ, k∈BZ}. Without loss of generality, we assume E0=Eν(ki). The retarded Green function
can be further written as
G+
0,αβ(Eν(ki); r,r′) = ZBZ
dkeik(r−r′)⟨α|[Eν(ki) + iη − H(k)]−1|β⟩=
q
X
µ=1 ZBZ
dkeik(r−r′)Pµ,αβ(k)
Eν(ki) + iη −Eµ(k),
(S22)
where Pµ(k) = |uR
µ(k)⟩⟨uL
µ(k)|is the projection operator on µ-th non-Hermitian Bloch band, and |uR/L
µ(k)⟩is the
right/left eigenstate of Bloch Hamiltonian H0(k) defined in Eq.(S17).
B. The derivation of Eq.(4) in the main text
Here, we consider the scattering potential in d-dimensional non-Hermitian system with impurities of codimension-1.
It can be expressed as
Vβσ (r) = λ δβσ δ(r⊥= 0),(S23)
where λis the strength of impurities. We specify these codimension-1 impurities lying on the r⊥= 0, that is point, line
or plane in d= 1,2 or 3 dimensions, respectively. Therefore, the full Hamiltonian has still the translation symmetry
along remaining d−1 directions. Correspondingly, the scattering equation can be written in (k∥, r⊥) representation
with k∥= (k1, k2, . . . , kd−1). We can always expressed the momentum of the incident wave as ki= (k∥
i, k⊥).
Combining Eq.(S21) with Eq.(S22), one can finally obtain the scattered wave with codimension-1 impurities as
ϕs,α(k∥
i, r⊥) = λ
q
X
µ,σ=1 Zπ
−π
dk⊥eik⊥r⊥Pµ,ασ(k∥
i, eik⊥)
Eν(ki) + iη −Eµ(k∥
i, eik⊥)ψσ(k∥
i,0).(S24)
Now we consider the q= 1 case in d= 2 dimensions, which gives the same properties of scattered wave as multi-
band (q > 1) cases, but in a more compact form. In this case, the codimension-1 scattering potential in Eq.(S23)
becomes an impurity line lying on the r⊥= 0 and along the θ-direction, which is denoted as Lθ. For the left (right)
side of the impurity line Lθ, the corresponding region is denoted by r⊥<0 (r⊥>0), as shown in Fig. S4(a). We
define z:= eik⊥, then the scattered wave becomes
ϕs(kθ
i, r⊥) = λ ψ(kθ
i,0) I|z|=1
dz
z
zr⊥
E(ki) + iη − H0(kθ
i, z),(S25)

14
where kθ
i=ki·eθis the θ-component of the incident momentum and eθis the unit vector along θdirection in kspace.
The coefficient ψ(kθ
i,0) is a constant for a given kθ
i, which can be solved by Eq.(S25). Note that for given incident
wave kiand impurity line Lθ,E(ki) + iη − H0(kθ
i, z) in Eq.(S25) is a complex function of variable z. Therefore, we
can define
g(E(ki) + iη, kθ
i, z) = E(ki) + iη − H0(kθ
i, z) = (E(ki) + iη)zm−Pm+n(kθ
i, z)
zm(S26)
with an m-order pole at the origin and m+nsimple zeros located in the complex zplane. We mark those zeros inside
|z|= 1 curve as {zin}and those outside |z|= 1 as {zout }. Physically, the scattered wave is required to converge at
infinity (|r⊥| → ∞) and be continuous at the impurity line (r⊥= 0). Taking these into account, the scattered wave
in Eq.(S25) can be finally obtained as
ϕs(kθ
i, r⊥) = λψ(kθ
i,0) 








X
|zin|<1
C(zin)zr⊥
in , r⊥>0;
X
|zout|>1−C(zout )zr⊥
out, r⊥<0,
(S27)
where C(zin/out) equals 2πi times the residue of the function [z g(E(ki) + iη, kθ
i, z)]−1at the pole zin/out inside/outside
the |z|= 1 curve. So far, we have the Eq.(4) in the main text.
From Eq.(S27) we can see that the behavior of scattered waves is dominated by those poles zin/out when |r⊥|→∞,
which is the same as that in q > 1 cases. Next, we will discuss the form of Eq.(4) in the multi-band and multiple pole
cases.
C. Eq.(4) in the multi-band and multiple pole cases
We begin with the Eq.(S21). The α-component of the scattered wave can be expressed as ϕs,α(r) = ⟨r, α|ϕs⟩, where
αrepresents the internal degree of freedom of the unit cell. For simplicity, we can take the basis transformation from
this natural basis to Bloch band basis, that is,
ϕs,α(r) = ⟨r, α|ϕs⟩=⟨r| ⊗
q
X
µ=1⟨α|uR
µ⟩⟨uL
µ|ϕs⟩=
q
X
µ=1
Sαµϕs,µ (r),(S28)
where ϕs,µ(r) = ⟨r|⊗⟨uL
µ|ϕs⟩, and |uR
µ⟩is the right wave vector on the µ-th band and satisfies the bi-orthogonality
and normalization. Once the µ-th band component of the scattered wave is obtained, the scattered wave in natural
basis can be obtained by the above basis transformation.
The ϕs,µ(r) can be expressed as
ϕs,µ(r) =
q
X
ν,ξ=1 Zdr′G+
0,µν (E0;r,r′)Vνξ (r′)ψξ(r′),(S29)
with E0the energy of the incident wave. Correspondingly, the matrix element of the Green’s operator becomes
G+
0,µν (E0;r,r′) = ZBZ
dkeik(r−r′)⟨uL
µ|[E0+iη − H(k)]−1|uR
ν⟩
=ZBZ
dkeik(r−r′)δµν
E0+iη −Eµ(k).
(S30)
In general, the codimension-1 scattering potential can be written as
V(r) =
q
X
ν,ξ=1
λVνξ δ(r⊥= 0)|uR
ν⟩⟨uL
ξ|,(S31)
Combining Eq.(S30) and Eq.(S31), we can obtain the the µ-th component scattered wave as
ϕs,µ(k∥
i, r⊥) = λ
q
X
ξ=1 Zdz
zzr⊥Vµξ
E0+iη −Eµ(k∥
i, z)ψξ(k∥
i,0).(S32)

15
localized
(a) (d)(c)
E1
E0
E2
0 3
-1
0
ReE
ImE
(b)
FIG. S4. (a) The schematic of the anomalous scattering process. (b) The path in momentum space preserves the component
along θdirection and transverses the incident wave ki, which is denoted by |z|= 1. (c) The Bloch spectrum of H(k) in Eq.(S7)
as ktransverses the path in (b). (d) the flow of zeros zin Eq.(S26) when the reference energy changes from E1to E2, here
only the zeros in the red annular region are shown.
This equation shows that the general scattered potential couples the different band components to form the scattered
wave. Note that Vµξ here is the matrix representation under the Bloch basis. For simplicity, the scattering potential
is also chosen to be
V(r) =
q
X
ν,ξ=1
λ δνξ δ(r⊥= 0)|uR
ν⟩⟨uL
ξ|,(S33)
which always has the diagonal matrix form under any basis transformation. Then, Eq.(S32) becomes
ϕs,µ(k∥
i, r⊥) = λ ψµ(k∥
i,0) Zdz
z
zr⊥
E0+iη −Eµ(k∥
i, z),(S34)
which reduces to the one-band form as shown in Eq.(S25) of the supplementary materials, and the similar form of
Eq.(4) in the main text can be obtained with the band index
ϕs,µ(kθ
i, r⊥) = λψµ(kθ
i,0) 








X
|zin|<1
Cµ(zin)zr⊥
in , r⊥>0;
X
|zout|>1−Cµ(zout )zr⊥
out, r⊥<0,(S35)
where Cµ(z) corresponds to the residue for the µ-th band.
Next, we discuss the form of Eq.(4) in the cases having multiple pole. In this case, the only different thing is the
calculation of the coefficient C(zi). We need to take the multiplicity of each pole ziunder consideration. Consider
the one-band system with the n-order pole zi, then the coefficient C(zi) in Eq.(4) of the main text can be calculated
as
C(zi)=2πi Res(F(z), zi) = 2πi
n!lim
z→zi
dn−1
dzn−1[(z−zi)nF(z)] (S36)
with F(z) = [zg(E0+iη, k∥
i, z)]−1, and g(E0+iη, k∥
i, z) is given in Eq.(S26) of the supplementary materials.
D. Two cases of scattering waves
Now we demonstrate that there are only two cases of scattered wave as discussed in the main text. It means that
there are two types of behaviors of the poles {zin/out}in Eq.(S27). In case (i), there are two poles that touch the
|z|= 1 curve simultaneously from the inner and outer sides respectively when η→0+; and in case (ii), there is only
one pole touches the |z|= 1 curve from the inside as η→0+.
Without loss of generality, we assume the incident plane wave ki= (kθ
i, k⊥
i) with energy Eν(ki) comes from r⊥<0
region as show in Fig. S4(a). Correspondingly, the transmitted wave is in the r⊥>0 region and the reflected wave

16
is in the r⊥<0 region. From Eq.(S27) one can see that when |r⊥| → ∞, the transmitted wave is dominated by the
maximum zmax
in ∈ {zin}(having the maximal amplitude |zmax
in |), and the reflected wave is mainly controlled by the
minimum zmin
out ∈ {zout}(having the minimal amplitude |zmin
out |). Also, as can be seen from Eq.(S25), when η→0+
there is always a pole z=eik⊥
itouching |z|= 1 curve. Next, we show that this pole always approaches |z|= 1 curve
from the inside, not from the outside.
Generally, for the scattering process shown in Fig. (S4)(a), the momentum kθalong the impurity line Lθis preserved.
When we give the momentum kiof the incident wave, the path in kspace that is perpendicular to the θdirection
is determined. This path traverses the momentum ki, as shown in Fig. (S4)(b). Here we use the Hamiltonian in
Eq.(S7) as an example, and choose the incident wave to be ki= (kx
i, ky
i)=(π/2,0) with the energy E0= 3/2. As k
transverses the path in Fig. (S4)(b), a loop-shape spectrum on the complex energy plane can be obtained as plotted
in Fig. (S4)(c). Note that the incident wave has the rightward velocity, which requires vx(ki)>0 defined in Eq.(S5).
Now we can define the Bloch spectral winding number regarding the reference energy E0+iη [4,5] as
w(E0+iη) = 1
2πI|z|=1
d
dz arg [g(E0+iη, kθ
i, z)]dz =Nzeros −Npoles ,(S37)
where Nzeros refers to the number of zeros of g(E0+iη, kθ
i, z) inside |z|= 1 and Npoles is the order of the pole at the
origin. The spectral winding number is ill-defined when η= 0. As shown in Fig. S4(c), the ηis tuned from η < 0 to
η > 0 (correspondingly, the reference energy goes from E1→E0→E2).
Note that we always choose the rightward incident wave as shown in Fig. S4(a). Correspondingly, as kincreases,
the Bloch spectrum always passes through E0from right to left in the complex energy plane, as shown in Fig. S4(c).
Therefore, the spectral winding number always increases by 1 when the reference energy E0+iη runs from the region
below E0to the region above E0, namely w(E0+ 0+)−w(E0+ 0−) = 1. For example, in Fig. S4(c), w(E1) = −1
and w(E2) = 0. Meanwhile, in this process Npoles is always invariant, thus the number of zeros inside |z|= 1 curve
Nzeros increases +1. Equivalently, there is one pole in Eq.(S27) that moves into |z|= 1 curve as the reference energy
changes from E1to E2, as shown in Fig. S4(d). Therefore, for the rightward incident wave, there is always one pole
approaches to |z|= 1 curve from inside when η→0+. Finally, we can obtain only the two cases of scattered wave as
discussed in the main text.
E. The relation between dynamical degeneracy splitting and pole’s behavior
Here, we demonstrate that when dynamical degeneracy splitting occurs, there are at least two poles in Eq.(S26)
reach |z|= 1; when dynamical degeneracy splitting does not occur, only one pole approaches |z|= 1, as illustrated in
Fig. 2 of the main text.
For simplicity, one-band case is considered below. Assuming that the impurity line is along the θdirection (labeled
as Lθ) and the incident wave has (complex) energy E0=ω0+iΓ0. Here, we clarify that when dynamical degeneracy
splitting does not occur at ω0, for each Lθ, there are at least two poles going to |z|= 1 as η→0+; when dynamical
degeneracy splitting occurs at ω0, there exist Lθsuch that only one pole approaches |z|= 1 when η→0+.
When dynamical degeneracy splitting does not occur at ω0, as shown in Fig. 1(a)(b) of the main text, all momenta
on the equal-frequency contour at ω0have the same real part ω0(by the definition of equal-frequency contour in
Eq.(1)) and the same imaginary part Γ0. Therefore, it means in this case that the solution of E0−H(k) = 0 is
exactly the equal-frequency contour at ω0. Note that all momenta on the equal-frequency contour are real number.
Once incident wave and scattering impurity line Lθare given, the kθ
iand E0are determined, where kθ
irepresents
the θ-component of the incident wave momentum ki. When dynamical degeneracy splitting does not occur at ω0, for
each kθ
i, one can always find at least two real momenta, ki= (kθ
i, k⊥
i) and ks= (kθ
i, k⊥
s) that satisfy E0−H(ki) = 0
and E0−H(ks) = 0. Note that kimust belong to the solutions of E0−H(k) = 0 because E0is the energy of incident
wave ki.
Here, we look at the poles of
lim
η→0+z[E0+iη −H(kθ
i, z)]−1=zm−1
E0zm−Pm+n(kθ
i, z),(S38)
where Pm+n(kθ
i, z) is a polynomial of zfor given kθ
i. The poles are exactly the solutions of E0−H(kθ
i, z) = 0 for
given kθ
iand E0. Therefore, when dynamical degeneracy splitting does not occur at ω0, for each kθ
i, we have at least
two solutions, zi=eik⊥
iand zs=eik⊥
sthat satisfy |zi|=|zs|= 1 because k⊥
iand k⊥
sare real. In other words, when
dynamical degeneracy splitting does not occur at ω0, for each kθ
i, there are at least two poles approaching |z|= 1 as
η→0+.

17
When dynamical degeneracy splitting occurs at ω0, different momenta on the equal-frequency contour have the same
real part ω0but different imaginary parts. Therefore, there exist some kθ
isuch that the solutions of E0−H(k) = 0
have only one real momentum kiand other complex momenta, for example, ks= (kθ
i, k⊥
s) with kθ
i∈Rbut k⊥
s∈C.
Correspondingly, the poles of Eq.(S38) include one zi=eik⊥
ion the |z|= 1 unit circle and other zs=eik⊥
swith
|zs| = 1. Therefore, when dynamical degeneracy splitting occurs at ω0, there exist kθ
isuch that only one pole
approaches |z|= 1 when η→0+.
S-3. SCATTERED WAVES IN TIME EVOLUTION
In this section, we will study the time evolution of incident waves and discuss under what conditions the stationary
scattering equation (Eq.(2) in the main text) in non-Hermitian systems can describe the scattered waves well.
The total Hamiltonian comprises the unperturbed Hamiltonian H0and scattering potential λV with the strength
λ, that is H=H0+λV . We define their eigenequations as follows:
H|ψR
n⟩=En|ψR
n⟩, H†|ψL
n⟩=E∗
n|ψL
n⟩;
H0|ϕR
n⟩=ϵn|ϕR
n⟩, H†
0|ϕL
n⟩=ϵ∗
n|ϕL
n⟩,(S39)
where the superscript Rand Lrepresent the right and left eigenvectors, respectively. They satisfy the bi-orthogonality,
namely, ⟨ψL
n|ψR
m⟩=⟨ϕL
n|ϕR
m⟩=δmn. The incident wave (initial state) is assumed to be an eigenstate |ϕ0⟩of H0with
the energy ϵ0. After encountering the scattering potential, the time evolution of the scattered wave can be defined as:
|ϕs(t)⟩=U(t)|ϕ0⟩ − U0(t)|ϕ0⟩,(S40)
where the time evolution operators reads U(t) = e−iHt and U0(t) = e−iH0t. Under spectral representation, the above
equation can be further expressed as
|ϕs(t)⟩=X
m
e−iEmt|ψR
m⟩⟨ψL
m|ϕ0⟩ − e−iE0t|ϕ0⟩
=e−iE0t[⟨ψL
0|ϕ0⟩|ψR
0⟩−|ϕ0⟩+X
m=0
e−i(Em−E0)t⟨ψL
m|ϕ0⟩|ψR
m⟩].(S41)
Here, ⟨ψL
0|is the left eigenvector of Hwith the same complex energy of the initial state |ϕ0⟩, that is E0=ϵ0. From this
one can see that when (i) ⟨ψL
0|ϕ0⟩ ≈ 1 and (ii) Pm=0 e−i(Em−E0)t⟨ψL
m|ϕ0⟩|ψR
m⟩ ≃ 0, the scattered wave in Eq.(S41)
reduces to
|ϕs(t)⟩ ≈ e−iE0t(|ψR
0⟩−|ϕR
0⟩),(S42)
which is exactly the time evolution form given by the Lippmann-Schwinger equation. Therefore, if the approximations
(i) and (ii) hold, the Lippmann-Schwinger equation in Eq.(2) of the main text gives the scattered waves for the non-
Hermitian Hamiltonian.
Next, we discuss the approximation conditions under which Eq.(S42) holds. We divide the discussion by the
following two cases:
•The incident wave has the largest imaginary energy (the lowest decay rate).
In this case, Im ϵm<Im ϵ0for all ϵm=ϵ0, where ϵ0is the energy of the incident wave. Note that in the scattering
process of a the impurity line, the momentum k∥
0of the incident wave, which is parallel to the impurity line, is
conserved. Therefore, the largest imaginary energy here refers to the spectrum of the reduced 1D Hamiltonian
H(k∥
0, k⊥), instead of the whole spectrum of H(k).
In what follows, we will use the perturbation theory to calculate En,⟨ψL
0|, and ⟨ψL
m|.
Eigenvalue En: With the first-order correction, the n-th eigenvalue of Hcan be approximated to
En≈ϵn+⟨ϕL
n|λV |ϕR
n⟩=ϵn+OλV0
N,(S43)
where |ϕR
n⟩and ⟨ϕL
n|are the extended waves (plane waves) and satisfy normalization condition ⟨ϕL
n|ϕR
n⟩= 1,
thus each element ⟨r|ϕR
n⟩has the order of 1/√Nwith Nrepresenting the system size. Given that λV is a local

18
operator in the real space, therefore, we have the conclusion: ⟨ϕL
n|λV |ϕR
n⟩has the order of λV0/N, where V0
represents the characteristic energy of the scattering potential. It tells us that under the thermodynamic limit,
the continuum spectrum of Hwell approximates to that of H0.
Eigenstates ⟨ψL
0|, and ⟨ψL
m|: Here, the left eigenvector of Hcan be obtained with the first-order correction:
⟨ψL
n| ≈ ⟨ϕL
n|+X
m=n
⟨ϕL
n|λV |ϕR
m⟩
ϵm−ϵn⟨ϕL
m|,(S44)
where the minimal value of ϵm−ϵncan be approximated by Ebw /N with Ebw representing the bandwidth of
the system. Therefore,
⟨ψL
n|≈⟨ϕL
n|+X
m=n
OλV0
Ebw ⟨ϕL
m|.(S45)
Note that the perturbation condition requires that the off-diagonal term (λV0)/Ebw ≪1. Therefore, one can
obtain the following approximated bi-orthogonal normalization condition,
⟨ψL
m|ϕR
0⟩ ≈ δm0+OλV0
Ebw (S46)
Note that in the Eq.(S41), e−i(Em−E0)twill decay with time due to face that Im ϵm<Im ϵ0. Therefore, putting
Eq.(S46) into Eq.(S41), one can obtain
|ϕs(t)⟩ ≈ e−iE0t(|ψR
0⟩−|ϕR
0⟩),(S47)
which shows that the Lippmann-Schwinger equation in Eq.(2) can capture the scattered wave in the non-
Hermitian system.
Actually, we can show that the adiabatic theorem still holds for the eigenstates with largest imaginary energy,
which is another way to justify that Eq.(2) indeed gives the scattered states for non-Hermitian systems. We will
discuss the adiabatic theorem for the eigenstates with largest imaginary energy later.
•The incident wave has no the largest imaginary energy.
In this case, e−i(Em−E0)twill become larger and larger than 1 if Im ϵm>Im ϵ0. Therefore the approximation in
Eq.(S42) holds only when the evolution time t≪tc, where tcrepresents the characteristic time. It means that
for this case the Eq.(2) can capture the scattered waves with short-time evolution. This makes sense because
after a long time, the eigenstates with the largest imaginary energy will dominate the dynamics of the system.
Now we examine the characteristic time tc. From e−i(Em−E0)t⟨ψL
m|ϕ0⟩|ψR
m⟩, one can find that if
e−i(Em−E0)t⟨ψL
m|ϕ0⟩ ≪ 1,the Eq.(S42) still holds. Therefore, the characteristic time can be given by
e−i(Em−E0)tc⟨ψL
m|ϕ0⟩=e−i(Em−E0)tc⟨ϕL
m|λV |ϕR
0⟩
ϵ0−ϵm
= 1,(S48)
Based on the above perturbation analysis in Eq.(S43) and Eq.(S44), we know that
cn0:= ⟨ψL
n=0|ϕR
0⟩ ≈ ⟨ϕL
n|λV |ϕR
0⟩
ϵn−ϵ0
;c00 := ⟨ψL
0|ϕR
0⟩ ≈ 1.(S49)
Therefore, the scattered wave in Eq.(S41) can be approximated into
|ϕs(t)⟩ ≈ e−iE0t{|ψR
0⟩−|ϕR
0⟩+X
n=0
e−i(ϵn−ϵ0)tcn0|ψR
n⟩}.(S50)
The Eq.(S42) holds when |e−i(ϵn−ϵ0)tcn0| ≪ 1. Therefore, the characteristic time can be estimated by
e∆n0tcλ
N∆n0∼1; ⇒tc∼1
∆n0
ln N∆n0
λ,(S51)
where ∆n0:= |ϵn−ϵ0|represents the spectral distance between ϵnand ϵ0in the complex energy plane. There
are two different cases:

19
0
2.5
5.0
7.5
10.0
12.5
15.0
(a)
(b)
(c)
(d)
t=1t=5t=9t=13 t =17
t=1t=5t=9t=13 t =17
t=1t=5t=9t=13 t =17
t=1t=5t=9t=13 t =17
t=1t=5t=9t=13 t =17
FIG. S5. Directional invisibility in Hamiltonian Eq.(S54). (a-b) show the spectral function A(ω, k) with ω= ReH(ki), where
ki= (kx
i, ky
i) = (1,0). The intensity of spectral function corresponds to the opacity indicated in the color bar. The incident
wave packet with the momentum center at kihits the oblique impurity line in (c) and vertical impurity line in (d), where the
impurity strength λ= 3/2. The snapshots of wave packets at t= 1, , 5,9,13,17 are shown in the (c)(d).
(i) when ∆m0has the order of O(1) (the energies away from ϵ0), then Eq.(S42) holds when the evolution time
t≪tc∼ln N
λ; (S52)
(ii) when ∆m0has the order of 1/N, the nearest energies from the ϵ0has the gap δϵ ∼Ebw/N, where Ebw
represents the bandwidth of the system as discussed before. Therefore, the Eq.(S42) holds when the evolution
time
t≪tc∼N
Ebw
ln Ebw
λ,(S53)
where λ≪1 is required such that the perturbation analysis of the wave functions can be applied.
So far, we have clarify the applicable conditions under which the Lippmann-Schwinger equation in Eq.(2) can well
describe the scattered waves.
S-4. DIRECTIONAL INVISIBILITY AND GEOMETRY-DEPENDENT SKIN EFFECT
In this section, we use a single-band example having GDSE to show the directional invisibility in terms of wave
packet dynamics. Then, we show the transmitted and reflected intensity without time-dependent normalization factor
using the two-band example (Eq.(5) in the main text). Finally, we prove that the normalization of spectral function
still works for the dissipative non-Hermitian systems.
A. The directional invisibility in single-band model
The Hamiltonian of the model is
H(k) = −2(cos kx+ cos ky) + ig(cos ky−1) (S54)
with a non-Hermitian parameter g= 1. Obviously, the imaginary part is k-dependent, thus leading to dynamical
degeneracy splitting in this system. We plot the spectral function A(ω0,k) = −Im[G+
0(ω0,k)] in Fig. S5 (a)(b),
showing the uneven broadening of equal-frequency contour (the gray circle), which is a definite signature of the

20
occurrence of dynamical degeneracy splitting. According to the anomalous scattering theory, the anomalous scattering
will occur. We assume an incident plane wave has ki= (kx, ky) = (1,0) and hits the impurity line Lθwith a rightward
velocity in real space. Note that kilies on the equal-frequency contour K(ω0) with ω0= Re[H(ki)], as shown in
Fig. S5(a)(b). The impurity line Lθpreserves the momentum along this direction, which means the scatterer Lθ
relates k0
iwith ksand k′
sin the way illustrated in Fig. S5(a) and (b), respectively. Due to the larger broadening
at ks, it means that the reflected wave is a non-Bloch wave, damped away from the impurity line. However, if we
rotate the oblique impurity line into vertical as shown in Fig. S5(b), the ky-component will be preserved during the
scattering process. Because of the Hamiltonian respecting Mxsymmetry, k′
s=M−1
xkihas the equal broadening with
ki. Therefore, the vertical impurity line scatters the incident plane wave kito a propagating plane wave k′
s, then the
normal scattering occurs. Such a phenomenon unique to GDSE is dubbed directional invisibility.
We use the wave packet dynamics to probe the directional invisibility in this example. The incident wave is a
Gaussian wave packet with momentum center at ki= (kx
i, ky
i) = (1,0). The time evolution of wave packet follows
|ψ(t)⟩=N(t)e−iHt |ϕ0⟩, where His the full Hamiltonian consisting of the free Hamiltonian in Eq.(S54) and impurity-
line scattering potential V(r) = λδ(r⊥= 0), and N(t) is the normalization factor at every time. The incident
Gaussian wave packet has the form ϕ0(r) = exp[−(r−r0)2/σ2+ik0
ir]. In Fig. S5(c)(d), the parameters are set as
(x0, y0, σ) = (24,30,4), and the system size of the lattice is Lx=Ly= 60. It can be observed that the Gaussian wave
packet is almost completely transmitted through the oblique impurity line without any propagating reflected waves,
as shown in Fig. S5(c). However, parts of the wave packet are reflected by the vertical impurity line as a propagating
wave, as shown in Fig. S5(d). Therefore, the real space dynamics in Fig. S5(c)(d) shows the directional invisibility.
B. The dissipative reflected and transmitted intensity
Here, we plot the intensity of transmitted and reflected components of the Gaussian wave packet in Fig. S6(a). The
model Hamiltonian is taken as Eq.(5) in the main text. The wave function has not been renormalized after every
time step. In such a dissipative non-Hermitian Hamiltonian, the system waves always decay with time. Even so, it
clearly shows that the Gaussian wave packet is fully transmitted.
In our dissipative model, the Gaussian wave packet decays with time and follows the time-evolution equation:
|ψ(t)⟩=e−iHt |ϕ0⟩,(S55)
where ϕ0(r) is the Gaussian wave packet and has the same form as that in Fig. 3 of the main text. The total
probability of wave function at time tbecomes
N(t) = ⟨ψ(t)|ψ(t)⟩.(S56)
And the transmitted and reflected intensities at time tcan be defined as
T(t) = X
r∈ST⟨ψ(t)|r⟩⟨r|ψ(t)⟩;R(t) = X
r∈SR⟨ψ(t)|r⟩⟨r|ψ(t)⟩,(S57)
where STrepresents the region of transmitted wave, and SRindicates the reflected region. In our scattering setting
of the main text, r⊥>0 is ST, and r⊥<0 is RT. Obviously, when the scattered wave packet moves away from the
scattering section (the impurity line), T(t) + R(t) = N(t).
(a)
(b)
FIG. S6. (a) The total probability N(t), transmitted wave intensity T(t) and reflected wave intensity R(t) change with time.
Note that the wave function here is not renormalized. (b) The snapshots of renormalized wave packet at t= 0, t= 10, and
t= 20.

21
As shown in Fig. S6(a), the red solid curve represents N(t) and the blue solid curve (green dotted line) corresponds
to the transmitted (reflected) part. At the beginning, the wave packet is in the reflection region SR. With the
occurrence of scattering event, the transmitted component is greater than the reflected one, and eventually all the
wave packets are in the transmitted region (correspondingly, the red and blue curve coincide after t= 15). As a
comparison, we select the snapshots of wave packet at t= 0, t= 10, and t= 20 in Fig. S6(b), where the wave packet
has been renormalized as in the main text.
C. The normalization of spectral function in dissipative system
In this part, we will prove that in the dissipative non-Hermitian systems, the normalization of the spectral function
is still true for any given k, that is,
1
2πZdωA(ω, k)=1.(S58)
Here, the spectral function can be expressed as
A(ω, k) = −1
NImTr[1/(ω+iη −H(k))] = −1
NIm "N
X
µ=1
1
ω+iη −Eµ(k)#,(S59)
where ωis real excitation frequency and η= 0+, and µrepresents the band index and Nis the total number of energy
bands.
Note that in the dissipative systems, the imaginary part of energy satisfies ImEµ(k)≤0 for all µand k, which is
considered throughout our paper. Now we prove the universal normalization. We have
1
2πZ∞
−∞
dωA(ω, k) = −1
2πN Im
N
X
µ=1 Z∞
−∞
1
ω+iη −Eµ(k)dω. (S60)
Note that for any given band µand momentum k, the energy Eµ(k)−iη is below the real axis in the complex
energy plane due to Im[Eµ(k)−iη]<0. Therefore, we use the contour integral technique, and then Eq.(S60) can be
calculated as
1
2πZ∞
−∞
dωA(ω, k) = 1
2πN Im
N
X
µ=1 IΓ+
1
ω−[Eµ(k)−iη]dω =−1
2πN Im[N×(−2πi)] = 1.(S61)
Here, Γ+indicates the positively oriented integral contour that surrounds the lower half energy plane. Therefore, the
universal normalization 1
2πRdωA(ω, k) = 1 is always valid for different kin the dissipative non-Hermitian systems.
S-5. THE ROBUSTNESS OF ANOMALOUS SCATTERING
On the theoretical perspective, we demonstrate the relation between dynamical degeneracy splitting and anomalous
scattering on the general scattering potentials in the first part, which suggests the robustness of anomalous scattering
(including robustness of directional invisibility in GDSE). Then, we numerically show that directional invisibility
is robust against the changes in orbital components, and can be still observed with finite step potential and open
boundary.
A. Anomalous scattering on general scattering potential
We begin with the general scattering equation, that is,
|ψ⟩=|ϕi⟩+G+
0(E0)V|ψ⟩=|ϕi⟩+G+
0(E0)V|ϕi⟩+G+
0(E0)V G+
0(E0)V|ϕi⟩+··· ,(S62)
where |ϕi⟩represents incident wave vector with the energy E0in H0. For analytical convenience, we here consider
Vas a weak scattering potential and take the Born approximation, under the real space representation, then the

22
scattering equation can be expressed as
ψ(r) = ϕi(r) + ϕs(r)
≈ϕi(r) + Zdr′G+
0(E0;r,r′)V(r′)ϕi(r′),(S63)
where G+
0(E;r,r′) is the retarded free Green’s function and V(r′) has translation symmetry along θdirection in
real space and are local function along the perpendicular direction. Therefore, the scattering process preserves the
momentum along θdirection. After Fourier transformation, we have kθ-component scattered wave
ϕs(kθ, r⊥) = Zdr′
⊥G+
0(E0;kθ, r⊥−r′
⊥)V(kθ, r′
⊥)ϕi(kθ, r′
⊥),(S64)
which is very similar to the results for delta potential scattering potential but scattering potential is a local function
of r⊥in general cases.
The scattering potential is localized along r⊥direction, which means
V(kθ, r⊥) = 0,if r⊥/∈S;S:= [rL
⊥, rR
⊥],(S65)
where rL
⊥and rR
⊥represent the left and right boundary position in r⊥direction. Therefore, the integral over r′
⊥in
Eq.(S64) reduce to the sum over Sregion, that is,
ϕs(kθ, r⊥) = X
r′
⊥∈S
α(kθ, r′
⊥)G+
0(E0;kθ, r⊥−r′
⊥); α(kθ, r′
⊥) = V(kθ, r′
⊥)ϕi(kθ, r′
⊥),(S66)
Finally, we can expand the free Green’s function in k⊥space and obtain the similar results as Eq.(4), that is,
ϕs(kθ, r⊥) = X
r′
⊥∈S
α(kθ, r′
⊥)








X
|zin|<1
C(zin)zr⊥−r′
⊥
in , r⊥> rR
⊥;
X
|zout|>1−C(zout )zr⊥−r′
⊥
out , r⊥< rL
⊥,
(S67)
(a2) (a3) (a5)(a4)(a1)
(b2) (b3) (b5)(b4)
(b1)
(c2) (c3) (c5)(c4)
(c1)
FIG. S7. Directional invisibility in step potential(a)(b) and open boundary (c). The Hamiltonian adopts Eq.(5) in the main
text, and the parameters are set to be (µ0, µz, t0, t, tz, γ ) = (1.35,−0.05,−0.4,0.4,−0.6,1). In (a)(b), the gray region represent
the region where the step potential is nonzero (λ= 1). In (c), the incident wave packet hits the oblique open boundary and
gets stuck, which indicates the anomalous scattering on the open boundary.

23
(a2)
(a3)
(a5)
(a4)
(a1)
(b2)
(b3)
(b5)
(b4)
(b1)
FIG. S8. Directional invisibility for the incident Gaussian wave packet with (1, i)Torbital components.
where α(kθ, r′
⊥) is a finite constant for each fixed kθand r′
⊥, and C(z) has the same definition as Eq.(4), that is 2πi
times the residue of [z(E0+iη −H0(kθ, z))]−1. Note that when V(kθ, r⊥) takes the delta potential along , r⊥direction,
the Eq.(S67) will reduce to Eq.(4) in the main text.
From Eq.(S67) one can see that, when dynamical degeneracy splitting occurs, for some kθ, there is just one pole z
approaching |z|= 1, which belongs to case (ii) in the main text and results in anomalous scattering that the reflected
wave are exponentially damped away from the scattering potential section S. Therefore, we have demonstrated that
our results of the anomalous scattering based on Eq.(4) of the main text is robust against the perturbed scattering
potential and does not depend on the special form of V, just requires that dynamical degeneracy splitting occurs in
the free Hamiltonian.
B. The numerical simulation of anomalous scattering on finite step potential and open boundary
Here, we present some numerical results of wave-packet scattering process on finite step potential in Fig. S7(a)(b)
and open boundary in Fig. S7(c), which shows the similar result of anomalous scattering as in Fig. 3 of the main text.
The free Hamiltonian of this example adopts Eq.(5) in the main text. The scattering potential in Fig. S7(a)(b) is
the step potential having the form: λ(r)σ0, where λ(r) = 1 when ris in the gray region, otherwise λ(r) = 0. In
Fig. S7(c), the scattering potential is the open boundary, that is, infinite step potential.
According to figures 3(a)(b) in the main text, it can be seen that dynamical degeneracy splitting occurs at
ω= 3/2. Therefore, we adopt the same form of incident wave packet as that in the main text, that is, ϕ0(r) =
exp[−(r−r0)2/σ2+ik0
ir](1,1)T. In Fig. S7(a), we can see a relatively obvious reflected wave component, while
in Fig. S7(b), the incident wave almost all enters the gray area, similar to the results obtained in the Fig. 3 of the
main text. In Fig. S7(c), the incoming wave packet hits the open boundary and gets stuck. The numerical results
indicate that when dynamical degeneracy splitting occurs, the anomalous scattering based on Eq.(4) also applies to
step potential and open boundary.
C. The robustness of directional invisibility
We present the numerical results in Fig. S8 to verify that the directional invisibility is robust against such changes of
orbital degrees of freedom. We adopt the Hamiltonian in Eq.(5) in the main text, and use the same incident Gaussian
wave packet as that in Fig. 3 but its orbital components are changed as (1, i)T. The real space dynamics (from t= 1
to t= 21) for vertical impurity line and oblique impurity line are show in Fig. S8(a1)-(a5) and (b1)-(b5), respectively.

24
It clearly demonstrates the directional invisibility.
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