#  $\mathcal{PT}$-activated non-Hermitian skin modes

Zhoutao Lei  Guangdong Provincial Key Laboratory of Quantum Metrology and
Sensing $\&$ School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai
Campus), Zhuhai 519082, China  Ching Hua Lee<EMAIL_ADDRESS>Department of
Physics, National University of Singapore, Singapore 117551, Republic of
Singapore  Linhu Li<EMAIL_ADDRESS>Guangdong Provincial Key
Laboratory of Quantum Metrology and Sensing $\&$ School of Physics and
Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China

(August 28, 2024)

###### Abstract

Parity-time ($\mathcal{PT}$) symmetry is a cornerstone of non-Hermitian
physics as it ensures real energies for stable experimental realization of
non-Hermitian phenomena. In this work, we propose $\mathcal{PT}$ symmetry as a
paradigm for designing new families of higher-dimensional non-Hermitian states
with unique bulk, surface, hinge or corner dynamics. Through systematically
breaking or restoring $\mathcal{PT}$ symmetry in different sectors of a
system, we can selectively “activate” or manipulate the non-Hermitian skin
effect (NHSE) in both the bulk and topological boundary states. Some
fascinating new phenomena include the directional toggling of the NHSE, an
intrinsic hybrid skin-topological effect and the flow of boundary states
without chiral or dynamical pumping. Our results extend richly into 3D or
higher, with more sophisticated interplay with hybrid skin-topological
localizations and $\mathcal{CP}$ symmetry. Based on non-interacting lattices,
$\mathcal{PT}$-activated NHSE phenomena can be observed in various optical,
photonic, electric and quantum platforms that admit gain/loss and non-
reciprocity.

Introduction.– Parity-time ($\mathcal{PT}$) symmetry is a key ingredient in
contemporary non-Hermitian physics Bender and Boettcher (1998); Bender (2007);
Bergholtz  _et al._ (2021). As a special type of pseudo-Hermiticity
Mostafazadeh (2002), $\mathcal{PT}$ symmetry guarantees real eigenenergies for
$\mathcal{PT}$-unbroken eigenstates, enforcing a balance between gain and loss
and thus probability conservation Bender and Boettcher (1998); El-Ganainy  _et
al._ (2018); Feng  _et al._ (2017). As such, it provides an ubiquitous route
towards realizing exotic non-Hermitian phenomena in a stable manner, as
demonstrated in optical Regensburger  _et al._ (2012); Feng  _et al._ (2017);
Zhao and Feng (2018); Özdemir  _et al._ (2019), acoustic Zhu  _et al._ (2014);
Fleury  _et al._ (2015); Shi  _et al._ (2016); Shao  _et al._ (2020), circuit
Choi  _et al._ (2018); Wang  _et al._ (2020); Stegmaier  _et al._ (2021) and
atomic platforms Hang  _et al._ (2013); Zhang  _et al._ (2016); Peng  _et al._
(2016); Jahromi  _et al._ (2017); Li  _et al._ (2019a); Muniz  _et al._
(2019).

In this work, we found that beyond ensuring stability, $\mathcal{PT}$ symmetry
can also serve as a new paradigm for designing families of higher-dimensional
non-Hermitian lattices with novel bulk, surface, hinge or corner dynamics.
Previously, it was found that non-Hermitian lattices can possess robust
localized modes not just due to topological protection, but also from the non-
Hermitian skin effect (NHSE), where all eigenstates boundary accumulate
exponentially under open boundary conditions (OBCs) Lin  _et al._ (2023);
Zhang  _et al._ (2022a); Yao  _et al._ (2018); Song  _et al._ (2019a, b); Li
_et al._ (2019b); Jiang  _et al._ (2019); Longhi (2019a); Zeng  _et al._
(2020); Yokomizo and Murakami (2019); Lee and Thomale (2019); Yang  _et al._
(2020); Li  _et al._ (2020a); Yi and Yang (2020); Lee (2021); Lu  _et al._
(2021); Borgnia  _et al._ (2020); Zhang  _et al._ (2020); Okuma  _et al._
(2020); Li  _et al._ (2021a); Guo  _et al._ (2021); Li  _et al._ (2021b); Liu
_et al._ (2021a, b); Li  _et al._ (2021c); Cao  _et al._ (2022); Shen and Lee
(2022); Longhi (2022); Sun  _et al._ (2021); Bhargava  _et al._ (2021);
Schindler and Prem (2021); Panigrahi  _et al._ (2022); Lee  _et al._ (2019);
Li  _et al._ (2020b); Lee (2021); Li  _et al._ (2022); Tai and Lee (2022);
Yang  _et al._ (2022); Zhu and Gong (2022); Qin  _et al._ (2022); Zhang  _et
al._ (2022b); Qin  _et al._ ; Li and Lee (2022); Jiang and Lee ; Wang  _et
al._ (2022). In 2D or higher, topological localization and the NHSE can
conspire to yield hybrid skin-topological modes Lee  _et al._ (2019); Li  _et
al._ (2020b); Zou  _et al._ (2021); Li  _et al._ (2022); Zhu and Gong (2022);
Ou  _et al._ (2023); Zhang  _et al._ (2023). New intricate phenomenology
beyond 1D can also appear Petrides  _et al._ (2018); Lee  _et al._ (2018);
Zhang  _et al._ (2021); Okugawa  _et al._ (2020); Lee (2021), particularly
defect NHSE at dislocations and disclinations Sun  _et al._ (2021); Bhargava
_et al._ (2021); Schindler and Prem (2021); Panigrahi  _et al._ (2022),
universal and geometry-dependent NHSE Zhang  _et al._ (2022b) and reduced
effective dimensionality due to directional non-commutativity of the NHSE
Jiang and Lee .

Our approach is based on the central observation that NHSE always requires a
complex spectrum under periodic boundary conditions (PBCs)Okuma  _et al._
(2020); Zhang  _et al._ (2020); Lee and Thomale (2019); Li  _et al._ (2021b),
and hence broken $\mathcal{PT}$ symmetry. Hence we can generate new classes of
higher-dimensional NHSE phenomena by not breaking $\mathcal{PT}$ symmetry
globally, but by selectively “activating” or breaking it in different sectors
e.g. edges and surfaces. By systematically switching on/off various types of
NHSE in higher-dimensional lattices, we obtain new families of non-Hermitian
states beyond known higher-order topological, skin or hybrid states.

We start with 1D non-Hermitian topological models where $\mathcal{PT}$
symmetry can be independently controlled for bulk and edge states, and use
them as building blocks for higher dimensional lattices with selectively
activated bulk and boundary NHSE. This unveils exotic NHSE phenomena in 2D,
for instance (i) a directional NHSE toggling mechanism requiring a PBC
direction (unlike conventional NHSE which always requires OBCs), (ii) an
intrinsic hybrid skin-topological effect with NHSE-immune bulk independent of
lattice geometry Zhang  _et al._ (2022b) and (iii) flow of states from one
boundary to another without chiral or dynamical pumping. In 3D or higher,
$\mathcal{PT}$ symmetry can even conspire with $\mathcal{CP}$ symmetry and
hybrid higher-order topology to activate much richer arrays of non-Hermitian
phenomena.

Note that NHSE manifests as boundary accumulation and changes the
dimensionality of eigenstates. In the rest of this paper, we shall use the
terms of bulk/surface/edge/hinge states to denote eigenstates that are
adiabatically connected to them in the Hermitian limit to avoid confusion.

Corner NHSE from non-Bloch $\mathcal{PT}$ symmetry breaking.–

Figure 1:  Corner and edge modes activated by $\mathcal{PT}$ symmetry
breaking. (a,b): The OBC spectrum (gray) of $H_{\gamma,\rm 2D}$, regarded as a
1D model with fixed $k_{y}$, can be (a) complex or (b) real depending on
whether $\mathcal{PT}$ symmetry is restored by the NHSE. For reference, the
PBC spectrum (brown) is always complex. (c) Top: Full spectrum of
$H_{\gamma,\rm 2D}$ as a 2D model, with PBCs along the $y$ direction. Pink and
blue cross sections correspond to the spectra in (a) and (b). Bottom: The full
$x,y$-OBC spectrum (colored according to the fractal dimension FD) is
contained within the set of $y$-PBC eigenenergies summed over all $k_{y}$
(gray). Corner modes with FD $\approx 0$ (I,II triangles) are plotted in (d),
and exist at $\text{Im}(E)\neq 0$ where $\mathcal{PT}$ symmetry is broken. An
edge mode with FD $\approx 1$ (square) exist at $\text{Im}(E)=0$, as plotted
in (e). Parameters are $\gamma=1.4$, $t_{1}=0.1$, $t_{2}=0.1$, $v=1$, and
$u=0.7$.

To provide a first illustration of $\mathcal{PT}$-activated NHSE modes, we
consider a minimal 2D model $H_{\gamma,\rm 2D}$ whose robust corner/edge modes
are localized not from higher-order topology Benalcazar  _et al._ (2017a, b);
Langbehn  _et al._ (2017); Song  _et al._ (2017), but from the NHSE activated
by $\mathcal{PT}$ symmetry breaking. We construct $H_{\gamma,\rm 2D}$ by
extending the 1D non-Hermitian Su-Schrieffer-Heeger (nH-SSH) model to 2D with
a $k_{y}$-dependent coupling Su  _et al._ (1980); Lee (2016); Yao and Wang
(2018); Kunst  _et al._ (2018); Yin  _et al._ (2018); Lieu (2018)(also seen
Supplemental Materials sup ):

$\displaystyle H_{\gamma,\rm
2D}(k_{x},k_{y})=h_{\gamma}^{(0)}\,\sigma_{0}+h_{\gamma}^{(x)}\sigma_{x}+h_{\gamma}^{(y)}\sigma_{y}$
(1)

where $h_{\gamma}^{(0)}=t_{1}\cos k_{y}$, $h_{\gamma}^{(x)}=u+v\cos k_{x}$ and
$h_{\gamma}^{(y)}=v\sin k_{x}+i\gamma/2+it_{2}\sin k_{y}$, and
$\sigma_{\alpha=x,y,z}$ are the Pauli matrices acting on a pseudospin-1/2
space (e.g. two sublattices). Here $u,v$ and $t_{1},t_{2}$ are hopping
parameters along the $x$ and $y$ directions respectively.

First consider fixed $k_{y}$, when this Hamiltonian can be viewed as a 1D nH-
SSH model with $x$-direction NHSE Yao and Wang (2018) (with an extra real
energy shift of $t_{1}\cos k_{y}$). The non-Hermitian $\gamma$ term breaks the
$\mathcal{PT}$ symmetry and makes the PBC spectrum (brown) complex (Fig.
1(a,b)). Under $x$-OBCs, $\mathcal{PT}$ symmetry may continue to be broken as
in Fig. 1(a) for $k_{y}\in(0,\pi)$, if the OBC spectrum (gray) is also
complex. However, $\mathcal{PT}$ symmetry may be restored in another parameter
range $k_{y}\in[-\pi,0]$, when the OBC spectrum (gray) becomes real (Fig.
1(b)) Longhi (2019b); Xiao  _et al._ (2021); sup . This selective restoration
of $\mathcal{PT}$ symmetry is possible because the NHSE amplifies states in a
directed manner, and this amplification (and hence complex energies) may be
blocked by a boundary.

In the 2D context, Figs. 1(a,b) correspond to 1D slices [pink and blue planes
in Fig. 1(c)] of $H_{\gamma,\rm 2D}$ at different quasi-momenta $k_{y}$,
respectively with restored/broken non-Bloch $\mathcal{PT}$ symmetry. When OBCs
are also implemented in the $y$-direction, $k_{y}$ is no longer diagonal and
the spectrum with both $x,y$-OBCs [Fig. 1(c) Bottom] may differ from the
$x$-OBC, $y$-PBC spectrum (Top). In particular, $y$-NHSE can occur only when
the non-Bloch $\mathcal{PT}$ symmetry is already broken, since the NHSE
requires nontrivial spectral windingOkuma  _et al._ (2020); Zhang  _et al._
(2020); Lee and Thomale (2019); Li  _et al._ (2021b), which in turn requires
complex eigenenergies. Since $x$-NHSE acts on all eigenstates, $H_{\gamma,\rm
2D}$, there are no completely extended states; there are only edge or corner-
localized states under full $x,y$-OBCs, corresponding to single/double
direction NHSE from unbroken/broken non-Bloch $\mathcal{PT}$ symmetry, as
shown in Fig. 1(d) and (e) respectively. To quantify the effective
dimensionality of these eigenstates $\psi_{n,\mathbf{r}}$, we introduce the
fractal dimension

$\displaystyle{\rm
FD}=-\ln[\sum_{\mathbf{r}}|\psi_{n,\mathbf{r}}|^{4}]/\ln{\sqrt{N}},$ (2)

$N$ being the total number of sites. We have $\rm FD\approx 0,1$ or $2$ for
corner, edge and 2D bulk states. In Fig. 1(c) Bottom, ${\rm FD}\approx 1$ for
states with real eigenenergies, indicating their 1D edge-localized nature
[Fig. 1(e)]. In contrast, states with complex eigenenergies have FD closer to
$0$, corresponding to the corner localization in Fig. 1(d).

Directional toggling of NHSE and symmetry-driven corner modes.–

Figure 2:  Toggling between $x$-NHSE and $y$-NHSE through $x$-boundary tuning,
without any parameter adjustment. From (a) to (c): As we morph from $x$-OBC
(a) to $x$-PBC (c), the $y$-PBC (gray) and $y$-OBC (colored by fractal
dimension FD) spectra for $H^{\beta}_{\gamma,\rm 2D}$ transitions from purely
real to complex. Meanwhile, as activated by $\mathcal{PT}$ symmetry breaking,
the skin modes transition from $x$ to $y$-edge localized ($\overline{\rm FD}$
$\approx 1$), as plotted in the insets. Corner skin modes with $\overline{\rm
FD}$ $\approx 0.5$ appear in the intermediatete $\beta\approx 10$ regime
purely as a by-product of this $\mathcal{PT}$ activation mechanism. (d)
Average FD and its 1D projections as a function of $\beta$ under the $y$-OBC,
with the trough at $\beta\approx 10$ indicative of corner localization.
Parameters are $\gamma=1.4$, $t_{1}=0.1$, $t_{2}=0.1$, $v=1$, and $u=0.9$ in
all panels.

Noting that $\mathcal{PT}$ symmetry plays a crucial role in determining the
allowed NHSE pumping directions, we now show that it can be used to “activate”
the NHSE _exclusively_ in the $x$ or $y$ directions. We call this directional
NHSE toggling. Furthermore, at an intermediatete stage of the toggling,
localization exists briefly in both directions, leading to a novel mechanism
for _symmmetry-driven_ corner modes.

To demonstrate this intriguing directional toggling of the NHSE, we tune the
$\mathcal{PT}$ symmetry by varying the boundary hoppings. We interpolate
between $x$-PBCs and $x$-OBCs with a parameter $\beta$, such that
$H_{\gamma,\rm 2D}$ becomes

$H^{\beta}_{\gamma,\rm 2D}=H^{\rm OBC}_{\gamma,\rm
2D}+e^{-\beta}H_{1\leftrightarrow N_{x}}$ (3)

where $H^{\rm OBC}_{\gamma,\rm 2D}$ is the Hamiltonian under full $x,y$-OBCs,
and $H_{1\leftrightarrow N_{x}}$ denotes the hoppings between the first and
last unit cells along the $x$-direction sup .

We have $x$-OBCs and hence $x$-NHSE at $\beta\rightarrow\infty$, and if
parameters are chosen such that the non-Bloch $\mathcal{PT}$ symmetry is
unbroken for all states, no $y$-NHSE can occur, even though we also have
$y$-OBCs. (Corner NHSE shall occur otherwise, see Supplemental Materials sup
.) This gives the $x$-edge-localized modes of Fig. 2(a), with $\rm FD\approx
1$. By contrast, the other limit of $\beta=0$ recovers $x$-PBCs and leads to
$y$-NHSE, leading to $y$-localized states [Fig. 2(c)]. In both scenarios, the
$y$-OBCs remain unchanged yet, by adjusting the $x$-boundary hoppings, we can
“activate” the localization in the $y$-direction through $\mathcal{PT}$
symmetry.

Interestingly, with partial $x$-OBCs i.e. $\beta=10$, corner modes can appear
as an intermediate between $x$ and $y$-boundary localization [Fig. 2(b)].
These corner modes are evidently distinct from known higher-order skin,
topological or hybrid modes, which require proper full OBCs, and may in fact
harbor enigmatic scale-free properties inherited from 1D partial boundary
states Li  _et al._ (2021a); Guo  _et al._ (2021). We note that some
eigenstates always exist at real energies, and are thus always free from the
$y$-NHSE.

To quantify the transition between skin localizations along the two
directions, we present in Fig. 2(d) the average fractal dimension
$\overline{\rm FD}$ and its 1D projections ($x$-$\overline{\rm FD}$ and
$y$-$\overline{\rm FD}$) over all states as functions of $\beta$, where

$\displaystyle\alpha\text{-}{\rm
FD}=-\ln[\sum_{\alpha}|\sum_{\alpha^{\prime}}|\psi_{n,\mathbf{r}}|^{2}|^{2}]/\ln{\sqrt{N}}$
(4)

with $\alpha,\alpha^{\prime}\in(x,y)$ and $\alpha\neq\alpha^{\prime}$. Similar
to the FD, $\alpha$-$\overline{\rm FD}$ quantifies the locality along a single
direction, which is $1$ ($0$) for fully extended (localized) states along the
$\alpha$ direction. While $\beta=0$ and $\beta\gg 1$ gives $\overline{\rm
FD}\approx 1$, a trough of $\overline{\rm FD}$ exists between them, indicative
of corner localization during the transition between single-directional
$y$-NHSE under $x$-PBC ($\beta=0$) and $x$-NHSE under $x$-OBC
($\beta\rightarrow\infty$). Meanwhile, $x$-$\overline{\rm FD}$ decreases and
$y$-$\overline{\rm FD}$ increases (almost) monotonously with $\beta$,
reflecting the mutual exclusion of NHSE along $x$ and $y$ directions.

Edge-$\mathcal{PT}$-breaking and selective boundary NHSE.—

Figure 3:  Selective boundary NHSE and non-monotonic state dynamics. (a)
$H_{g,\rm 2D}$ hosts $\mathcal{PT}$-unbroken bulk (black) and
$\mathcal{PT}$-broken topological $x$-OBC/$y$-PBC edge states (gray). Under
full OBCs, the latter experiences $y$-NHSE and become corner-localized (red).
(b) $H^{\prime}_{g,\rm 2D}$ is deformed from $H_{g,\rm 2D}$ such that
$\mathcal{PT}$ symmetry is restored for the branch of edge states (green) with
$\text{Im}(E)>0$, which remains edge-localized, at the expense of the bulk
states (black), which is thus $y$-NHSE localized. (c,d) depicts the dynamical
evolution of a center-localized initial state (black dot) due to $H_{g,\rm
2D}$ and $H^{\prime}_{g,\rm 2D}$ respectively, with the size of blue dots
indicating evolved state density. Qualitatively distinct stages occur before
and after encountering the upper boundary; in (d), the upper edge state
evolves into the left edge state, leading to non-monotonic $\langle y\rangle$.
Parameters are $u=0.2$, $g=0.3$, $t_{1}=0.3$ and $t^{\prime}_{2}=0.1$, with
system size $N_{x}=N_{y}=20$.

So far, we have only seen how non-Bloch $\mathcal{PT}$ symmetry breaking can
activate various NHSE channels for bulk states. We next discuss how that can
interplay with nontrivial topology, which separately gives $\mathcal{PT}$
symmetric topological edge modes. Specifically, we show that $\mathcal{PT}$
can be selectively broken for edge states only, leading to a new type of
boundary NHSE distinguished from the usual bulk NHSE. We introduce another 2D
model

$\displaystyle H_{g,\rm
2D}=\sum_{\alpha=0,x,y,z}{h}_{g}^{(\alpha)}(k_{x},k_{y})\sigma_{\alpha},$ (5)

where $h_{g}^{(0)}=t_{1}\cos k_{y}$, $h_{g}^{(x)}=u+v\cos k_{x}$,
$h_{g}^{(y)}=v\sin k_{x}$, and imaginary $h_{g}^{(z)}=i(g+t^{\prime}_{2}\sin
k_{y})$. This Hamiltonian satisfies the $\mathcal{PT}$ symmetry
$[\mathcal{PT},H_{g,\rm 2D}]=0$ with $\mathcal{PT}=\sigma_{x}\mathcal{K}$.
When $u^{2}+v^{2}-2uv>g+t_{2}^{\prime}$ (all parameters chosen to be
positive), all bulk states are $\mathcal{PT}$ unbroken and are free of any
type of NHSE. On the other hand, the effective edge Hamiltonian read Geier
_et al._ (2018); Trifunovic and Brouwer (2019); Lei  _et al._ (2022, 2023):

$H_{{\rm 2D,edge}}^{\pm}=\hat{P}_{\pm}H_{g,\rm 2D}\hat{P}_{\pm}=t_{1}\cos
k_{y}\pm i(g+t_{2}^{\prime}\sin k_{y})$

with $\hat{P}_{\pm}=(1\pm\sigma_{z})/2$ the projectors of edge states. It is
clear that $H_{{\rm 2D,edge}}^{\pm}$ holds no $\mathcal{PT}$ symmetry, and
gives a complex edge spectrum [gray dotted loop in the inset of Fig. 3(a)]
with nontrivial spectral winding for $k_{y}\in(-\pi,\pi]$. Consequently,
$y$-NHSE occurs for these $x$-boundary states under $y$-OBCs, accumulating on
opposite corners (red) depending on the sign of $\text{Im}(E)$ [Fig. 3(a)].
This mechanism represents a type of hybrid skin-topological effect, namely
that topological states are pushed to lower-dimensional boundaries by NHSE,
yet bulk states remain extended Lee  _et al._ (2019). Notably, the
$\mathcal{PT}$-protected bulk spectrum remains purely real (Fig. 3(a)) and
impervious to any NHSE throughout Indeed, this absence of bulk NHSE is
geometry-independent, representing an intrinsic hybrid skin-topological effect
sup .

Going further, we may also selectively recover/break $\mathcal{PT}$ symmetry
and hence turn off/on the NHSE for different branches of edge states.
Explicitly, by introducing an extra anti-Hermitian term $-i(t_{2}^{\prime}\sin
k_{y}+g)\sigma_{0}$:

$\displaystyle H_{g,\rm 2D}^{\prime}=H_{g,\rm 2D}-i(t_{2}^{\prime}\sin
k_{y}+g)\sigma_{0},$ (6)

one branch of edge eigenenergies becomes real, $H_{{\rm
2D,edge}}^{+}\rightarrow H_{{\rm 2D,edge}}^{{}^{\prime}+}=t_{1}\cos k_{y}$. As
seen in Fig. 3(b), these edge states remain left edge-localized (green) under
full OBCs because they cannot undergo additional NHSE, but those of the other
branch are $\mathcal{PT}$-broken and collapse into corner modes (red) due to
$y$-NHSE.

Anomalous $\mathcal{PT}$-activated state dynamics.— The selective activation
of these various forms of corner and edge NHSE entails a competition between
different NHSE channels, and interestingly gives rise to qualitative
transitions in different stages of the state dynamics, distinguished to that
of different bulk NHSE channels in $H_{\gamma,{\rm 2D}}$ sup . Consider an
initial state
$|\psi(0)\rangle=\left(|\uparrow,N_{x}/2,N_{y}/2\rangle+|\downarrow,N_{x}/2,N_{y}/2\rangle\right)/\sqrt{2}$
localized at the center of an $N_{x}\times N_{y}$ lattice. We dynamically
evolve it via $|\psi(t)\rangle=e^{-iHt}|\psi(0)\rangle$ and investigate the
evolution of its center-of-mass $\langle x\rangle$ and $\langle y\rangle$ in
Figs. 3(c) and (d), for $H=H_{g,{\rm 2D}}$ and $H^{\prime}_{g,{\rm 2D}}$ from
Eqs. (5) and (6) respectively.

For $H=H_{g,{\rm 2D}}$ (Fig. 3(c)), both $\langle x\rangle$ and $\langle
y\rangle$ remain roughly constant for a short time, since $|\psi(t)\rangle$ is
governed by NHSE-free bulk dynamics [see Fig. 3(a)] before it encounters a
boundary. After $t\approx 30$, the state arrives at the boundary and the
intrinsic hybrid skin-topological effect is activated. The dynamics are then
dominated by the corner states with $\text{Im}(E)>0$, where $\langle
x\rangle,\langle y\rangle\approx 1$.

More interestingly, $\langle x\rangle$ and $\langle y\rangle$ can evolve non-
monotonically in the dynamics governed by $H^{\prime}_{g,{\rm 2D}}$, which
possesses bulk $y$-NHSE and a skin-free branch of edge states (green in Fig.
3(b)). Due to bulk $y$-NHSE, the state is initially symmetrically pushed to
the top boundary, and $\langle y\rangle$ increases while $\langle x\rangle$
remains constant, as shown in Fig. 3(d). After encountering the top boundary
at $t\approx 50$, the dynamics become dominated by the left edge states with
$\text{Im}(E)>0$, and the $y$-NHSE is effectively deactivated. As such, the
edge state “flows” from the top to the left boundary, and $\langle
x\rangle\rightarrow 1$ and $\langle y\rangle\rightarrow N_{y}/2$. This
enigmatic behavior is possible because the left edge states are
$\mathcal{PT}$-protected and do not undergo any NHSE to become corner states.

3D generalizations of $\mathcal{PT}$-activated NHSE.—

Figure 4:  NHSE selectively activated by $\mathcal{PT}$ or $\mathcal{CP}$
symmetry breaking in 3D. In most panels [except for (c) with $x$-PBC], colored
dots represent the bulk, surface or hinge spectra under full OBCs, and gray
dots represent the $xy$-OBC/$z$-PBC spectra. Insets display the average
spatial distribution of eigenstates of the respective color. (a,b,c), (d),
(e), and (f,g) pertain to $H_{\gamma,\text{3D}}$, $H_{g,\text{3D}}$,
$H^{\prime}_{g,\text{3D}}$, and $H^{\prime}_{\text{3D}}$ respectively. In
(a,b,c), the topologically $(001)$ and $(00\bar{1})$-surface localized states
(blue) experience corner pumping from $\mathcal{CP}$-breaking (b) or $x$ to
$y$-NHSE directional toggling (c). In (d,e), the hinge-localized states (red)
can have $\mathcal{CP}$ symmetry selectively broken (unbroken), respectively
resulting in corner (red) or edge (green) localization. In (f,g), on-demand
switching of 1st and 2nd-order 3D bulk NHSE (black surface and edges) through
$\mathcal{PT}$ breaking leaves other states (green,blue) invariant. In (a) to
(c), parameters are $v=1$, $u^{\prime}=0.6$, $v^{\prime}=3$ and
$t_{1}=t_{2}=0.1$, with $u=0.6$ and $\gamma=0.4$ for (a) and (c); $u=0.5$ and
$\gamma=1.4$ for (b). In (d) and (e), parameters are $u=0.2$, $v=1$,
$u^{\prime}=0.4$, $v^{\prime}=2$, $g=0.3$, $t_{1}=0.3$ and $t_{2}=0.1$. In (f)
and (g), parameters are $u^{\prime}=0.2$, $v^{\prime}=1$, $v_{2}=3$,
$\gamma=1.4$ and $t_{1}=t_{2}=0.1$, with $u=0.9$ for (f) and $u=0.5$ for (g).

The various avenues for $\mathcal{PT}$-activated NHSE effects discussed above
takes on even richer possibilities in higher dimensions. Below, we briefly
demonstrate how they acquire new interpretations as surface and hinge-
selective NHSE in 3D. Specifically, we consider a 3D Hamiltonian of the form

$\displaystyle H_{\gamma/g,\rm 3D}(\mathbf{k})=H_{\gamma/g,0}(k_{x},k_{y})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ +(u^{\prime}+v^{\prime}\cos
k_{z})\sigma_{0}\tau_{x}+v^{\prime}\sin k_{z}\sigma_{0}\tau_{y}$ (7)

with $\tau_{x,y,z}$ another set of Pauli matrices.
$H_{\gamma/g,0}(k_{x},k_{y})$ is inherited from the previous
$\mathcal{PT}$-breaking and directional toggling NHSE models
$H_{\gamma/g,\rm{2D}}$:

$\displaystyle H_{\gamma,0}$ $\displaystyle=$ $\displaystyle i\left[(u+v\cos
k_{x})\sigma_{x}+v\sin k_{x}\sigma_{y}+t_{1}\cos
k_{y}\sigma_{0}\right]\tau_{z}$ (8) $\displaystyle-(\gamma/2+t_{2}\sin
k_{y})\sigma_{y}\tau_{0},$ $\displaystyle H_{g,0}$ $\displaystyle=$
$\displaystyle iH_{g,\rm 2D}\tau_{z}.$ (9)

Under $z$-OBCs, $\tau_{0}$ and $\tau_{z}$ terms serve as overall and
$(001)$-$(00\bar{1})$-surface energy offsets respectively, and the surfaces
states obey the effective Hamiltonian $H_{\pm,{\rm surface}}=\hat{P}_{\pm,{\rm
3D}}H_{\gamma/g,\rm 3D}\hat{P}_{\pm,{\rm 3D}}$ with $\hat{P}_{\pm,{\rm
3D}}=(1\pm\sigma_{0}\tau_{z})/2$ $z$-surface state projectors Geier  _et al._
(2018); Trifunovic and Brouwer (2019); Lei  _et al._ (2022, 2023). Note that
the choices $H_{\gamma/g,0}$ are not unique, and the explicit forms in Eqs.
(8) and (9) are chosen to restore $\mathcal{PT}$ symmetry

$\displaystyle[\sigma_{x}\tau_{x}\mathcal{K},H_{\gamma/g,\rm
3D}(\mathbf{k})]=0.$ (10)

Also, an additional charge-parity ($\mathcal{CP}$) symmetry emerges for the
surface states Gong  _et al._ (2018); sup , which offers another avenue for
eliminating the NHSE by ensuring purely imaginary eigenenergies.

Our numerical results for these 3D models are shown in Fig. 4, where
eigenmodes are marked with different colors according to their corresponding
sectors and boundary conditions. In Fig. 4(a,b,c) for $H_{\gamma,\rm 3D}$, the
$(001)$ and ($00\bar{1}$) surface states essentially recapitulate the
symmetry-selected and directional toggling NHSE effects of Figs. 1 and 2.
Going from Fig. 4(a) to (b), these surface states (blue) become corner-
localized due to non-Bloch $\mathcal{CP}$-breaking, analogous to the corner
localization Fig. 1(d) due to $\mathcal{PT}$-breaking, while bulk (black) and
hinge (green) states remain unchanged. Likewise, directional toggling from
$x$-NHSE to $y$-NHSE localization occurs when the $x$-OBC of Fig. 4(a) is
replaced by the $x$-PBC of Fig. 4(c).

Similarly, the surface (blue) and hinge (red) states on $(001)$ and
($00\bar{1}$) surfaces of $H_{g,\rm 3D}$ in Fig. 4(d) are in direct
correspondence with the bulk (black) and boundary (red) states of $H_{g,{\rm
2D}}$ in Fig. 3(a). Meanwhile, bulk eigenenergies remain real as our choice of
parameters maintains unbroken $\mathcal{PT}$ symmetry in Eq. (7). In analogy
with $H^{\prime}_{g,{\rm 2D}}$ in Fig. 3(b), a $\mathcal{CP}$ symmetry can be
recovered in one branch of hinge states on each of $(001)$ and ($00\bar{1}$)
surfaces by introducing an extra Hermitian term to the Hamiltonian, i.e.
$H^{\prime}_{g,\rm 3D}(\mathbf{k})=H_{g,\rm
3D}(\mathbf{k})+(t_{2}^{\prime}\sin k_{y}+g)\sigma_{0}\tau_{0}$, which removes
the NHSE on these hinge states [Fig. 3(b)].

Alternatively, a 3D Hamiltonian with only bulk-NHSE can be designed such that
its bulk (PBC) states are not $\mathcal{PT}$-symmetric, but that non-Bloch
$\mathcal{PT}$-symmetry can be recovered in certain parameter regimes, for
instance $H^{\prime}_{\rm
3D}(\mathbf{k})=H_{\gamma,\rm{2D}}(k_{z},k_{y})\otimes\tau_{0}+\sigma_{z}\otimes
H_{\rm SSH}(k_{x})$, with

$\displaystyle H_{{\rm SSH}}(k_{x})$ $\displaystyle=$ $\displaystyle(u‘+v’\cos
k_{x})\tau_{x}+v‘\sin k_{x}\tau_{y}$ (11)

the Hermitian SSH model and $H_{\gamma,\rm{2D}}(k_{z},k_{y})$ from Eq. (1).
Consequently, bulk states accumulate to a 2D surface (or 1D hinges) when the
non-Bloch $\mathcal{PT}$ symmetry is unbroken (or broken), as shown in Fig.
4(f,g). In contrast to $H_{\gamma/g,\text{3D}}$, conventional surface (blue)
or hinge (green) states are immune from the NHSE and remain extended.

Discussion.— $\mathcal{PT}$ symmetry ensures real spectra, thus eliminating
the prospect for spectral winding. This fundamental observation leads to the
new paradigm of $\mathcal{PT}$-activated NHSE, where new families of NHSE-
related phenomena can be designed by selectively symmetry breaking in bulk or
boundary subspaces. In 2D, we discussed novel directional NHSE toggling and
$\mathcal{PT}$-mediated corner modes, as well as selective edge-$\mathcal{PT}$
breaking and restoring that causes non-monotonic transfer of edge states. In
3D or higher, $\mathcal{PT}$ activation leads to far more varied phenomenology
by interplaying with the already rich classes of higher-order hybrid skin-
topological modes. This paradigm of symmetry-controlled NHSE can be extended
to include other symmetries which forbid spectral winding, such as
$\mathcal{CP}$ symmetry or pseudo-Hermiticity.

Since $\mathcal{PT}$-activated NHSE is fundamentally a phenomenon on a linear
lattice, it can be experimentally demonstrated on any metamaterial or
artificial lattice, whether classical or quantum, as long as $\mathcal{PT}$
symmetry can be broken through a combination of gain/loss and non-reciprocity.
Possible platforms include non-reciprocal lossy acoustic or photonic crystals
or waveguides Xiao  _et al._ (2020); Lin  _et al._ (2022); Xiao  _et al._
(2023); Wang  _et al._ (2023), ultracold atomic lattices Liang  _et al._
(2022), superconducting quantum circuit lattices Kollár  _et al._ (2019);
Smith  _et al._ (2019); Bassman  _et al._ (2021); Ippoliti  _et al._ (2021);
Koh  _et al._ (2022a, b); Zhang  _et al._ (2022c); Koh  _et al._ (2023);
Fleckenstein  _et al._ (2022); Frey and Rachel (2022); Chen  _et al._ (2022),
and in particular op-amp controlled electrical circuits which can be
constructed with great versatility Hofmann  _et al._ (2019); Li  _et al._
(2019c); Helbig  _et al._ (2020); Stegmaier  _et al._ (2021); Zou  _et al._
(2021); Shang  _et al._ (2022); Wu  _et al._ (2022); Lenggenhager  _et al._
(2022); Zhu  _et al._ (2023).

###### Contents

  1. I Non-Hermitian SSH model
  2. II Non-Bloch $\mathcal{PT}$ symmetry
  3. III The explicit form of Hamiltonian $H^{\beta}_{\gamma,\rm 2D}$
  4. IV Further results of $H_{\gamma,\rm 2D}$
    1. IV.1 Corner NHSE in the non-Bloch $\mathcal{PT}$ broken regime
    2. IV.2 Evolution process for $H_{\gamma,\rm 2D}$ with mutual exclusively or conventional NHSE
  5. V Geometry-independence of the intrinsic hybrid skin-topological effects
  6. VI Effective surface Hamiltonian for $3$D generalizations

Supplementary Materials

##  I Non-Hermitian SSH model

The models supporting various types of NHSE discussed in the main text are
mostly designed based on a non-Hermitian Su-Schrieffer-Heeger (SSH) model,
described by the Hamiltonian Lieu (2018):

$\displaystyle
H(k_{x})=\sum_{\alpha=x,y,z}{h}_{\alpha}(k_{x})\sigma_{\alpha},$ (S1)

where $h_{x}(k_{x})=u+v\cos k_{x}$, $h_{y}(k_{x})=v\sin k_{x}+i\gamma/2$,
$h_{z}=ig$, and $\sigma_{\alpha=x,y,z}$ is the Pauli matrix acting on a
pseudospin-1/2 space [e.g. two sublattices $|A\rangle$ and $|B\rangle$ as
shown in Fig. S1(a)]. $u$ and $v$ denote the Hermitian staggered hopping
amplitudes. The two non-Hermitian parameters $\gamma$ and $g$ describes
asymmetric hoppings and on-site gain and loss respectively, as illustrated in
Fig. S1(a). By separately tuning the two non-Hermitian parameters $g$ and
$\gamma$, we can break or restore $\mathcal{PT}$-symmetry selectively for bulk
or topological edge states in this model, as discussed below.

When $\gamma=0$, non-Hermiticity enters the Hamiltonian only through the on-
site gain and loss of $g$, and the resultant Hamiltonian

$H_{g}\equiv H(\gamma=0)$

holds a $\mathcal{PT}$ symmetry: $[\mathcal{PT},H_{g}]=0$ with
$\mathcal{PT}=\sigma_{x}\mathcal{K}$ and $\mathcal{PT}^{2}=+1$. A nonzero $g$
does not break the $\mathcal{PT}$ symmetry as the gain and loss are balanced
between sublattices. As shown in Fig. S1(b), in the $\mathcal{PT}$-unbroken
phase, all bulk states have real eigenenergies since they are also eigenstates
of $\mathcal{PT}$ symmetry operator Bender and Boettcher (1998); Bender
(2007). On the other hand, topological edge states for the SSH model are
sublattice polarized and hence $\mathcal{PT}$-broken. Instead of being
eigenstates of $\mathcal{PT}$ symmetry operator, these states are related to
each other through the symmetry, i.e.
$\mathcal{PT}|\psi_{+}\rangle\propto|\psi_{-}\rangle$, with
$|\psi_{+,-}\rangle$ the topological edge states localized at left and right
ends respectively. The sublattice-polarization of these edge states allow us
to write down their effective Hamiltonians through projectors
$\hat{P}_{\pm}=(1\pm\sigma_{z})/2$, i.e.

$H_{{\rm edge}}^{\pm}=\hat{P}_{\pm}H_{g}\hat{P}_{\pm}=\pm ig,$

which directly give their eigenenergies Geier  _et al._ (2018); Trifunovic and
Brouwer (2019); Lei  _et al._ (2022, 2023).

Figure S1: (color online). The non-Hermitian SSH model. (a) A sketch of the
model of Eq. (S1), with $\gamma$ ($g$) the asymmetric hopping (gain/loss)
parameter. (b) Typical PBC (gray) and OBC (other colors) spectra of $H_{g}$.
Under the OBCs, bulk states can be $\mathcal{PT}$-symmetric and have real
eigenenergies (black), but edge states are $\mathcal{PT}$-broken and have
imaginary eigenenergies $\pm ig$. Insets demonstrate the spatial distribution
of corresponding eigenstates. (c) Typical PBC and OBC spectra of $H_{\gamma}$,
with the same color marks as in (b). The PBC spectrum in (c) is
$\mathcal{PT}$-broken and possesses a nontrivial spectral winding, leading to
a boundary accumulation of all eigenstates (see insets). The OBC spectrum
restores a non-Bloch $\mathcal{PT}$ symmetry and becomes purely real.
Parameters are chosen to be $u=0.5$, $v=1$, and (b) $g=0.2$, (c) $\gamma=0.2$
for demonstration.

For the other non-Hermitian scenario of Eq. (S1) with $g=0$ but $\gamma\neq
0$,

$\displaystyle H_{\gamma}\equiv H(g=0)$ (S2)

the $\mathcal{PT}$ symmetry is broken, and the resultant complex bulk spectrum
is known to have nontrivial spectral winding topology under the PBC, a
signature of the emergence of NHSE under the OBC Borgnia  _et al._ (2020);
Zhang  _et al._ (2020); Okuma  _et al._ (2020). Yet its OBC bulk spectrum
become real in Fig. S1(c) due to the recovery of a non-Bloch $\mathcal{PT}$
symmetry Longhi (2019c); Xiao  _et al._ (2021) or a pseudo-Hermiticity
Mostafazadeh (2002). Explicitly, the OBC bulk spectrum can be described by a
non-Bloch Hamiltonian $\bar{H}_{\gamma}(k_{x})=H_{\gamma}(k_{x}+i\kappa)$ with
$\kappa=\ln\sqrt{|(u+\gamma/2)/(u-\gamma/2)|}$ Yao and Wang (2018); Yokomizo
and Murakami (2019); Lee and Thomale (2019). This Hamiltonian
$\bar{H}_{\gamma}(k_{x})$ satisfies a non-Bloch $\mathcal{PT}$ symmetry,

$[\eta\mathcal{K},\bar{H}_{\gamma}]=0$

with $\eta$ a $k$-dependent unitary operator, which is unbroken for all non-
Bloch eigenstates when $|u|>|\gamma/2|$ (as shown in Sec II of Supplement
Materials). In contrast, here the topological edge states always remain
$\mathcal{PT}$-symmetric and possess real (zero) eigenenergies, as a nonzero
$\gamma$ does not enter their effective Hamiltonian:

$H_{{\rm edge}}^{\pm}=\hat{P}_{\pm}H_{\gamma}\hat{P}_{\pm}=0.$

Note that the the localizing direction of these edge states may be altered by
the NHSE, yet their distributing dimensionality remains unchanged, namely they
are always 0D edge states except for some critical points in the parameter
space Zhu  _et al._ (2021). Similarly, edge states of $2$D generalization
Hamiltonian $H_{\gamma,\rm 2D}$ with selective bulk NHSE, namely those near
zero energy in Fig.1(c) and Fig.2(a) of the main text, are seen to also
possess real eigenergies with their distributing dimensionality unchanged by
NHSE.

##  II Non-Bloch $\mathcal{PT}$ symmetry

In this section, we discuss the non-Bloch $\mathcal{PT}$ symmetry of the non-
Hermitian SSH model with asymmetric hoppings $H_{\gamma}(k_{x})$ of Eq.(S2).
Its corresponding non-Bloch Hamiltonian reads Yao and Wang (2018); Yokomizo
and Murakami (2019); Lee and Thomale (2019)

$\displaystyle\bar{H}_{\gamma}(k_{x})=H_{\gamma}(k_{x}+i\kappa)=(A+iB)\sigma_{x}+(C+iD)\sigma_{y},$
(S3)

with

$\displaystyle A$ $\displaystyle=$ $\displaystyle u+v\frac{\cos
k_{x}}{2}\left(\sqrt{\bigg{|}\frac{u-\gamma/2}{u+\gamma/2}\bigg{|}}+\sqrt{\bigg{|}\frac{u+\gamma/2}{u-\gamma/2}\bigg{|}}\right),\leavevmode\nobreak\
B=v\frac{\sin
k_{x}}{2}\left(\sqrt{\bigg{|}\frac{u-\gamma/2}{u+\gamma/2}\bigg{|}}-\sqrt{\bigg{|}\frac{u+\gamma/2}{u-\gamma/2}\bigg{|}}\right),$
$\displaystyle C$ $\displaystyle=$ $\displaystyle v\frac{\sin
k_{x}}{2}\left(\sqrt{\bigg{|}\frac{u-\gamma/2}{u+\gamma/2}\bigg{|}}+\sqrt{\bigg{|}\frac{u+\gamma/2}{u-\gamma/2}\bigg{|}}\right),\leavevmode\nobreak\
D=-v\frac{\cos
k_{x}}{2}\left(\sqrt{\bigg{|}\frac{u-\gamma/2}{u+\gamma/2}\bigg{|}}-\sqrt{\bigg{|}\frac{u+\gamma/2}{u-\gamma/2}\bigg{|}}\right)+\frac{\gamma}{2}.$

To identify the non-Bloch $\mathcal{PT}$ symmetry, we rewrite the non-Bloch
Hamiltonian as

$\displaystyle
H_{\gamma}(k_{x}+i\kappa)=h_{x}^{\prime}\sigma_{x}^{\prime}+h_{y}^{\prime}\sigma_{y}^{\prime}$
(S5)

with

$\displaystyle
h_{x}^{\prime}=\frac{(A+iB)C-(C+iD)iB}{\sqrt{C^{2}-B^{2}}},\leavevmode\nobreak\
\leavevmode\nobreak\
h_{y}^{\prime}=\frac{(A+iB)iB+(C+iD)C}{\sqrt{C^{2}-B^{2}}},$
$\displaystyle\sigma_{x}^{\prime}=\frac{C}{\sqrt{C^{2}-B^{2}}}\sigma_{x}-\frac{iB}{\sqrt{C^{2}-B^{2}}}\sigma_{y},\leavevmode\nobreak\
\leavevmode\nobreak\
\sigma_{y}^{\prime}=\frac{iB}{\sqrt{C^{2}-B^{2}}}\sigma_{x}+\frac{C}{\sqrt{C^{2}-B^{2}}}\sigma_{y}.$
(S6)

It is straightforward to check that
$h_{x}^{\prime},h_{y}^{\prime}\in\mathbb{R}$,
$(\sigma_{x}^{\prime})^{2}=(\sigma_{i}^{\prime})^{2}=\sigma_{0}$, and
$\sigma_{x}^{\prime}\sigma_{y}^{\prime}\sigma_{x}^{\prime}=-\sigma_{y}^{\prime}$.
Therefore we obtain

$\displaystyle[\eta\mathcal{K},H_{\gamma}(k_{x}+i\kappa)]=0$ (S7)

with $\eta=\sigma_{x}^{\prime}$ a unitary operator and
$(\eta\mathcal{K})^{2}=1$, indicating the non-Bloch $\mathcal{PT}$ symmetry.
Note that $B$ and $C$ are $k_{x}$-dependent coefficients, thus $\eta$ also
varies with $k_{x}$, represents a special form of pseudo-Hermiticity
Mostafazadeh (2002). $\eta$ is ill-defined when $|u|=|\gamma/2|$, which is
also the the boundary between non-Bloch $\mathcal{PT}$ unbroken and broken
phases. Specifically, the eigenenergies of $H_{\gamma}(k_{x}+i\kappa)$ are
given by

$\displaystyle E_{\gamma}=\pm\sqrt{A^{2}+C^{2}-B^{2}-D^{2}-2i(AB+CD)},$ (S8)

and the unbroken phase requires $AB+CD=0$ and $A^{2}+C^{2}-B^{2}-D^{2}\geq 0$
for $\forall k_{x}\in(-\pi,\pi]$. The first condition gives

$\displaystyle AB+CD=v\frac{\sin k_{x}}{2}\left({\rm sgn}[u+\gamma/2]-{\rm
sgn}[u-\gamma/2]\right)\sqrt{|u^{2}-\gamma^{2}/4|}=0,$ (S9)

which is satisfied either when $|u|>|\gamma/2|$, or when
$k_{x}\in\\{0,\pi\\}$. The second condition gives

$\displaystyle A^{2}+C^{2}-B^{2}-D^{2}=u^{2}+v^{2}-\gamma^{2}/4+v\cos
k_{x}\left({\rm sgn}[u+\gamma/2]+{\rm
sgn}[u-\gamma/2]\right)\sqrt{|u^{2}-\gamma^{2}/4|}\geqslant 0.$ (S10)

When $|u|>|\gamma/2|$, it reduces to $u^{2}+v^{2}-\gamma^{2}/4+2v\cos
k_{x}\sqrt{u^{2}-\gamma^{2}/4}\geq(\sqrt{u^{2}-\gamma^{2}/4}-v)^{2}\geq 0$ and
is always satisfied. In the non-Bloch $\mathcal{PT}$ broken phase with
$|u|<|\gamma/2|$, this condition may also hold as long as
$u^{2}+v^{2}\geqslant\gamma^{2}/4$. Combining these results, the non-Bloch
$\mathcal{PT}$ is unbroken when $|u|>|\gamma/2|$ with all $E_{\gamma}$ being
real, partially broken when $u^{2}+v^{2}>\gamma^{2}/4>u^{2}$ where only
$E_{\gamma}(k_{x}\in\\{0,\pi\\})$ are real, and fully broken where all bulk
states acquire imaginary eigenenergies when $\gamma^{2}/4>u^{2}+v^{2}$.

##  III The explicit form of Hamiltonian $H^{\beta}_{\gamma,\rm 2D}$

In this section, we give the explicit form of $H^{\beta}_{\gamma,\rm 2D}$ in
Eq.(3) of the main text. According to this definition, the Hamiltonian
$H^{\beta}_{\gamma,\rm 2D}$ contains two parts: the Hamiltonian $H_{\gamma,\rm
2D}$ in Eq.(1) of the main text under full OBCs, and its hopping between
boundaries along $x$-direction. The first part reads

$\displaystyle H^{\rm OBC}_{\gamma,\rm 2D}=$
$\displaystyle\frac{v}{2}\sum_{j}^{N_{x}-1}\sum_{k}^{N_{y}}[\hat{\psi}^{{\dagger}}_{j,k}(\sigma_{x}-i\sigma_{y})\hat{\psi}_{j+1,k}+{\rm
H.c.}]$ (S11)
$\displaystyle+\frac{1}{2}\sum_{j}^{N_{x}}\sum_{k}^{N_{y}-1}[t_{1}(\hat{\psi}^{{\dagger}}_{j,k}\hat{\psi}_{j,k+1}+{\rm
H.c.})-it_{2}(i\hat{\psi}^{{\dagger}}_{j,k}\sigma_{y}\hat{\psi}_{j,k+1}+{\rm
H.c.})]$
$\displaystyle+\sum_{j}^{N_{x}}\sum_{k}^{N_{y}}\hat{\psi}^{{\dagger}}_{j,k}(u\sigma_{x}+\frac{i\gamma}{2}\sigma_{y})\hat{\psi}_{j,k},$

where $\hat{\psi}_{j,k}=[\hat{a}_{j,k},\hat{b}_{j,k}]^{T}$, with
$\hat{a}_{j,k}$ ($\hat{b}_{j,k}$) being the annihilate operator for the
pseudospin-$\uparrow$ ($\downarrow$) [sublattices $A$ ($B$)] on the $(j,k)$-th
unit cell.

Similarly, the hoppings between boundaries along $x$-direction
$H_{1\leftrightarrow N_{x}}$ and $y$-direction $H_{1\leftrightarrow N_{y}}$
are

$\displaystyle H_{1\leftrightarrow N_{x}}=$
$\displaystyle\frac{v}{2}\sum_{k}^{N_{y}}[\hat{\psi}^{{\dagger}}_{N_{x},k}(\sigma_{x}-i\sigma_{y})\hat{\psi}_{1,k}+{\rm
H.c.}],$ (S12) $\displaystyle H_{1\leftrightarrow N_{y}}=$
$\displaystyle\frac{1}{2}\sum_{j}^{N_{x}}[t_{1}(\hat{\psi}^{{\dagger}}_{j,N_{y}}\hat{\psi}_{j,1}+{\rm
H.c.})-it_{2}(i\hat{\psi}^{{\dagger}}_{j,N_{y}}\sigma_{y}\hat{\psi}_{j,1}+{\rm
H.c.})].$ (S13)

Thereby, the Hamiltonian under full PBCs can be expressed by

$\displaystyle H^{\rm PBC}_{\gamma,\rm 2D}=H^{\rm OBC}_{\gamma,\rm
2D}+H_{1\leftrightarrow N_{x}}+H_{1\leftrightarrow N_{y}}.$ (S14)

Through the Fourier transformation, the momentum space Hamiltonian
$H_{\gamma,\rm 2D}(k_{x},k_{y})$ in Eq.(1) of the main text can be obtained.
In Eq.(3) of the main text, an extra factor $e^{-\beta}$ is introduced to
describe the partial OBCs aong $x$ direction.

##  IV Further results of $H_{\gamma,\rm 2D}$

###  IV.1 Corner NHSE in the non-Bloch $\mathcal{PT}$ broken regime

In the main text, we have discussed the 2D Hamiltonian $H_{\gamma,\rm 2D}$ in
the non-Bloch $\mathcal{PT}$ symmetry unbroken (Fig.2 of the main text) and
intermediate regime (Fig.1 of the main text), where non-Bloch $\mathcal{PT}$
symmetry is unbroken for either all or some values of $k_{y}$. In this
section, we discuss another situation where the non-Bloch $\mathcal{PT}$
symmetry is partially broken [$u^{2}+v^{2}>(\gamma+2t_{2}\sin
k_{y})^{2}/4>u^{2}$] for all values of $k_{y}$.

Figure S2: (color online). The spectrum and distribution of a single
eigenstate of $H_{\gamma,\rm 2D}$ with corner NHSE. (a) $x$-OBC/$y$-PBC (gray)
and full OBCs (colored) spectra. Different colors indicate the FD of each
eigenstate. (b) and (c) the distributions of eigenstates under full OBCs
indexed by triangles and square in (a), respectively. In all panels, the
parameters are chosen as $\gamma=1.4$, $t_{1}=t_{2}=0.1$, $v=1$, and $u=0.5$,
and full OBCs.

In this regime, almost all bulk states become corner-skin states under the
full OBCs, as shown in Fig. S2(a) and S2(b). The only exceptions are a
$\mathcal{O}(L)$ number of states falling on the real axis of the complex
energy plane, which are extended along $y$ direction [Fig. S2(c)]. This is
because these states correspond to those with $k_{x}\in\\{0,\pi\\}$ of the 1D
non-Bloch Hamiltonian $\bar{H}_{\gamma}(k_{x})$, which remains non-Bloch
$\mathcal{PT}$ unbroken even in the non-Bloch $\mathcal{PT}$ broken phase (See
Sec. II of Supplement materials).

###  IV.2 Evolution process for $H_{\gamma,\rm 2D}$ with mutual exclusively or
conventional NHSE

In this section, we discuss the evolution process governed by Hamiltonian
$H_{\gamma,\rm 2D}$. As in the main text, here the initial state is also
chosen to be
$|\psi(0)\rangle=\left(|\uparrow,x_{0},y_{0}\rangle+|\downarrow,x_{0},y_{0}\rangle\right)/\sqrt{2}$
with $(x_{0},y_{0})=(N_{x}/2,N_{y}/2)$, and the state at time $t$ is given by
$|\psi(t)\rangle=e^{-iH_{\gamma,\rm 2D}t}|\psi(0)\rangle$.

We first discuss the regime $u<\gamma/2-t_{2}$ where the non-Bloch PT symmetry
is broken and almost all bulk state suffer from corner NHSE. Specifically,
bulk states with positive imaginary energy (which shall dominate the dynamics)
locate at the bottom left corner, as discussed in Sec IV.1 and shown in Fig.
S2(a) and S2(b). As a result, $\langle x\rangle$ and $\langle y\rangle$
decreases rapidly to $\sim 1$, as shown in Fig. S3(a). Figure S3(b) further
demonstrates the distribution of $|\psi(t)\rangle$ at different values of $t$.

Figure S3: (color online). The dynamical process for Hamiltonian
$H_{\gamma,\rm 2D}$ with various types of NHSE. (a), (c), and (e) The mean
position of states $\langle x\rangle$ and $\langle y\rangle$ during the
evolution process. Here the initial state is prepared in the center of the
$2$D lattices. (b), (d), and (f) The normalized distribution of the state
$|\psi(t)\rangle$ at different time. In (a) and (b), parameters are
$\gamma=1.4$, $t_{1}=t_{2}=0.1$, $v=1$, $u=0.5$, the same as in Fig. S2 where
corner NHSE occurs. (c) to (f) have the same parameters except for $u=0.9$,
the same as in Fig. 2 of the main text corresponding to non-Bloch
$\mathcal{PT}$ unbroken phase. The boundary conditions are chosen to be full
OBCs in (a) to (d), and $x$-PBC/$y$-OBC in (e) and (f). In all panels the
system’s size is $N_{x}=N_{y}=20$.

In contrast, bulk states of this model locates at left edge under full OBCs
when the non-Bloch $\mathcal{PT}$ symmetry is unbroken, as shown in Fig.2(a)
in the main text. Correspondingly, $\langle x\rangle$ decreases rapidly while
$\langle y\rangle$ does not show a clear increasing or decreasing tendency, as
shown in Fig. S3(c). Consistently, a state prepared deep in the bulk will
evolve to the left boundary and remain there in a long-time dynamics, as shown
in Fig. S3(d).

Finally, we demonstrate the dynamics for the same parameters as in Fig. S3(c)
and (d), but with $x$-PBC/$y$-OBC instead of full OBCs. As discussed in the
main text, a $y$-NHSE shall emerge in this scenario. Specifically, bulk states
with positive imaginary parts distribute along the bottom edge, as shown in
Fig.2(c) of the main text. Therefore, after replacing $x$-OBC with $x$-PBC,
$\langle y\rangle$ decreases rapidly while $\langle x\rangle$ oscillates
around $N_{x}/2$, as shown in Fig. S3(e). More explicitly, evolution of a
state behaves the same under the full OBCs and $x$-PBC/$y$-OBC before it
reaches a boundary of the system, as shown in Fig. S3(d) and Fig. S3(f) with
$t=10$. Afterward, different boundary conditions lead to distinct dynamical
behaviors, and $|\psi(t)\rangle$ remains in the bottom boundary in a long-time
dynamics for the system under $x$-PBC/$y$-OBC [Fig. S3(f)].

##  V Geometry-independence of the intrinsic hybrid skin-topological effects

In this section, we discuss the intrinsic hybrid skin-topological effect of
$H_{g,\rm 2D}$ in Eq.(5) of the main text with different geometries. As shown
in Fig. 3(a) of the main text, the bulk spectrum of $H_{g,\rm 2D}$ is given by
1D lines and covers zero area in the complex plane. As discussed in Ref. Zhang
_et al._ (2022b), this property reveals that the absence of bulk NHSE is
independent from the system’s geometry, hence we denote it as the intrinsic
hybrid skin-topological effect. To varify this, we demonstrate the complex
spectrum and spatial distribution of eigenstates of Hamiltonian $H_{g,\rm 2D}$
under triangle geometry in Fig. S4. As seen in Fig. S4(a), the bulk spectrum
(black) is still consistent with the $x$-OBC/$y$-PBC spectrum (gray). And as
further demonstrated in Fig. S4(b), bulk states are extended in the triangle
lattice and are free from NHSE, while edge states suffer from NHSE and are
located at corners.

In contrast, we can construct another Hamiltonian as followed,

$\displaystyle H^{(\rm geo)}_{g,\rm 2D}=H_{g}+t^{\prime}_{2}\cos
k_{y}\sigma_{z}+it_{1}\sin k_{y}\sigma_{0},$ (S15)

which is similar to Hamiltonian $H_{g,\rm 2D}$ but with an extra $\pi$ phase
to the hopping toward positive $y$ direction. This Hamiltonian also supports
hybrid skin-topological effect under square geometry, as demonstrated in Fig.
S5 (a) and S5 (b). However, its bulk spectrum covers a non-vanishing area in
the complex plane, suggesting the emergence of bulk NHSE on certain
geometries. Specifically, we find that the bulk states suffer from NHSE under
triangle geometry, localizing along the hypotenuse of the triangle, as
demonstrated in Fig. S5 (c) and (d).

Figure S4: (color online). Geometry-independence of intrinsic hybrid skin-
topological effect of $H_{g,\rm 2D}$. (a) Energy spectrum for Hamiltonian
$H_{g,\rm 2D}$ in Eq.(5) of the main text under triangle geometry. Black
circles correspond to bulk states and red (blue) squares correspond to
boundary states with positive (negative) imaginary energies. Gray circles show
$x$-OBC/$y$-PBC spectra for the same Hamiltonian. (b) Spatial distribution of
states under triangle geometry, marked by the same colors as in (a).
Parameters are $u=0.2$, $v=1$, $g=0.3$, $t_{1}=0.3$ and $t^{\prime}_{2}=0.1$
for all panels. Figure S5: (color online). Geometry-dependence of conventional
hybrid skin-topological effect of $H^{(\rm geo)}_{g,\rm 2D}$. (a) Energy
spectrum of Hamiltonian $H^{(\rm geo)}_{g,\rm 2D}$ in Eq. (S15) under square
geometry. Black circles corresponds to bulk states and red (blue) squares
corresponds to boundary states with positive (negative) imaginary energies.
Gray circles shows $x$-OBC/$y$-PBC spectrum for the same Hamiltonian. (b)
Spatial distribution for states under square geometry, marked by the same
colors as in (a). (c) and (d) the same as (a) and (b), but under triangle
geometry. Parameters are $u=0.2$, $v=1$, $g=0.3$, $t_{1}=0.3$ and
$t^{\prime}_{2}=0.1$ for all panels.

##  VI Effective surface Hamiltonian for $3$D generalizations

In this section, we provide the effective surface Hamiltonian of the $3$D
system discussed in the main text.

The Hamiltonians $H_{\gamma/g,\rm 3D}$ and $H^{\prime}_{g,\rm 3D}$ in the main
text are chosen to support surface states on $(001)$ and $(00\bar{1})$
surfaces. Corresponding projectors of these surface states are

$\hat{P}_{\pm,{\rm 3D}}=(1\pm\sigma_{0}\tau_{z})/2.$

Thereby, the effective surface Hamiltonian of $H_{\gamma,\rm 3D}$ can be given
by:

$\displaystyle H^{\pm}_{\gamma,\rm surf}$ $\displaystyle=$
$\displaystyle\hat{P}_{\pm,{\rm 3D}}H_{\gamma,\rm 3D}\hat{P}_{\pm,{\rm 3D}}$
(S16) $\displaystyle=$ $\displaystyle\pm i\\{t_{1}\cos
k_{y}\sigma_{0}+(u+v\cos k_{x})\sigma_{x}+[v\sin k_{x}\pm i(\gamma/2+t_{2}\sin
k_{y})]\sigma_{y}\\}.$

Comparing to Eq.(1) of the main text, we find the effective Hamiltonian
$H^{\pm}_{\gamma,\rm surf}$ can be considered as $\pm iH_{\gamma,\rm 2D}$ (the
signs of $\gamma$ and $t_{2}$ are flipped for $H^{-}_{\gamma,\rm surf}$). As
discussed in Sec II, $H_{\gamma,\rm 2D}$ holds a non-Bloch $\mathcal{PT}$
symmetry. Correspondingly, $H^{\pm}_{\gamma,\rm surf}$ holds a non-Bloch
$\mathcal{CP}$ symmetry, due to the additional imaginary factor $\pm i$.
Therefore, the surface states of $H_{\gamma,\rm 3D}$ can support various NHSE
channels activated by $\mathcal{CP}$ symmetry breaking, as discussed in the
main text with Fig. 4(a) to (c).

Similarly, the effective surface Hamiltonian of $H_{g,\rm 3D}$ reads

$\displaystyle H^{\pm}_{g,\rm surf}=\hat{P}_{\pm,{\rm 3D}}H_{g,\rm
3D}\hat{P}_{\pm,{\rm 3D}}=\pm iH_{g,\rm 2D}.$ (S17)

Therefore, $H^{\pm}_{g,\rm surf}$ satisfies a $\mathcal{CP}$ symmetry
$[\mathcal{CP},H^{\pm}_{g,\rm surf}]_{+}=0$ with
$\mathcal{CP}=\sigma_{x}\mathcal{K}$. And the surface (hinge) states of
$H_{g,\rm 3D}$ displays similar NHSE phenomenon as the bulk( and edge) states
of $H_{g,\rm 2D}$, as shown in Fig.4(d) of the main text.

The effective surface Hamiltonian of $H^{\prime}_{g,\rm 3D}$ is given by

$\displaystyle H^{\pm^{\prime}}_{g,\rm surf}=\hat{P}_{\pm,{\rm
3D}}H^{\prime}_{g,\rm 3D}\hat{P}_{\pm,{\rm 3D}}=\pm i[H_{g,\rm 2D}\mp
i(t_{2}^{\prime}\sin k_{y}+g)\sigma_{0}],$ (S18)

where the $\mathcal{CP}$ symmetry is broken and then surface states suffer
from NHSE. However, one branch of boundary states of $H^{\pm^{\prime}}_{g,\rm
surf}$ recovers $\mathcal{CP}$ symmetry and is free from NHSE, analogous with
the discussion of $H^{\prime}_{g,\rm 2D}$ in the main text. In this 3D model,
$\mathcal{CP}$ symmetry is recover for distinct branches of boundary states in
the $(001)$ and $(00\bar{1})$ surfaces because of the $\mp$ sign in the square
bracket of the Eq.(S18). As a result, these NHSE-free hinge states living in
these two surfaces are localized in opposite sides along $x$-direction, as
shown in Fig.4(e) (green) in the main text.

Finally, the projectors of $(001)$ and $(00\bar{1})$ surface states for the
Hamiltonian $H^{\prime}_{\rm 3D}$ in the main text are

$\hat{P}^{\prime}_{\pm,{\rm 3D}}=(1\pm\sigma_{z}\tau_{0})/2.$

Their effective surface Hamiltonian can be obtained as

$\displaystyle H^{\pm^{\prime}}_{3D,\rm surf}$ $\displaystyle=$
$\displaystyle\hat{P}^{\prime}_{\pm,{\rm 3D}}H^{\prime}_{\rm
3D}\hat{P}_{\pm,{\rm 3D}}$ (S19) $\displaystyle=$ $\displaystyle t_{1}\cos
k_{y}\tau_{0}\pm[(u^{\prime}+v^{\prime}\cos k_{x})\tau_{x}+v^{\prime}\sin
k_{x}\tau_{y}],$

which satisfies a $\mathcal{PT}$ symmetry
$[\mathcal{PT},H^{\pm^{\prime}}_{3D,\rm surf}]=0$ with
$\mathcal{PT}=\tau_{x}\mathcal{K}$. Actually, $H^{\pm^{\prime}}_{3D,\rm surf}$
is Hermitian and without NHSE. Correspondingly, the dimensionality of surface
and hinge states of $H^{\prime}_{\rm 3D}$ is unchanged by NHSE, despite that
the localizing direction of hinge states (green) are altered by the $z$-NHSE,
as shown in Figs.4(f) and 4(g) of the main text. Meanwhile, the bulk states
can show distinct localization with various types of NHSE, as shown in
Figs.4(f) and 4(g) of the main text.

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