Activating non-Hermitian skin modes by parity-time symmetry breaking

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 rich 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 phenomena include the directional toggling of the NHSE, and the flow of boundary states without chiral or dynamical pumping, developed from selective boundary NHSE. Our results extend richly into 3D or higher, with more sophisticated interplay with selective bulk and boundary NHSE and charge-parity ($\mathcal{CP}$) symmetry. Based on non-interacting lattices, $\mathcal{PT}$-activated NHSEs can be observed in various optical, photonic, electric and quantum platforms that admit gain/loss and non-reciprocity.

Recently, 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 accumulate exponentially to the boundary under open boundary conditions (OBCs) [23][24][25].In particular, NHSE always requires a complex spectrum under periodic boundary conditions (PBCs) that forms loops in the complex energy plane [26][27][28][29][30], and hence broken PT symmetry.However, the potential application of PT symmetry in manipulating NHSE still remains largely unexplored and a comprehensive framework is lacking.
In this work, we found that beyond ensuring stability, PT symmetry can also serve as a paradigm for designing rich families of higher-dimensional non-Hermitian lattices with unconventional bulk, surface, hinge or corner dynamics.Note that in our discussion we use "bulk/surface/edge/hinge" to describe the co-dimensionality of states in their adiabatically connected Hermitian limit, even if the NHSE has effectively dimensionally-reduced them.Taking the spectral feature of NHSE as a cornerstone, we can generate various classes of higher-dimensional NHSEs not by breaking 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 twodimensional (2D) lattices as examples, we not only obtain rich families of non-Hermitian skin states beyond known NHSEs (e.g.corner NHSE and hybrid skin-topological effect), but also provide a scheme to manipulate them as desired.In three dimension (3D) or higher, PT symmetry can even conspire with charge-parity (CP) symmetry and selective NHSE to activate much richer arrays of non-Hermitian skin phenomena, which may prove useful for designing an abundant variety of non-Hermitian optical devices that exploit NHSE [31][32][33][34].We emphasize that these intriguing models can be realized in classical system, such as electrical circuits [16,[35][36][37][38][39][40][41][42][43][44].Moreover, it may also be extended to the quantum systems such as cold atoms and superconducting circuit setups, which are described by Lindblad master equation, where short-time dynamics can be effectively described by non-Hermitian Hamiltonians with the effects of quantum jump ignored [45,46].

Results
Corner NHSE from generalized PT symmetry breaking.As a warm-up, we begin with a minimal 2D model H γ,2D whose robust corner/edge modes are localized not from higher-order topology, but from the NHSE activated by PT symmetry breaking.We construct H γ,2D by extending the 1D non-Hermitian Su-Schrieffer-Heeger (nH-SSH) model [47,48] to 2D with a k y -dependent coupling (also see Methods section): where h k y , and σ α=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-NHSE [25] (with an extra energy shift of t 1 cos k y ).The real-space Hamiltonian of this 1D model satisfies a generalized PT symmetry KH γ,2D (k y )K = H γ,2D (k y ), as x-OBC Hamiltonian H γ,2D contains only real matrix elements [49,50].Under x-OBCs, the spectrum remains complex for k y ∈ (0, π) Fig. 1a], but becomes real in k y ∈ [−π, 0] [Fig.1b], reflecting the broken and unbroken phases of the generalized PT symmetry, respectively.More generally, the real OBC spectrum can be associated to the so-called non-Bloch PT symmetry [51][52][53] (also see Methods section), with non-Bloch exceptional points emerging during transition of these two scenarios at k = 0 and π [51].
In the 2D context, Figs.1a,b correspond to 1D slices [pink and blue planes in Fig. 1c] of H γ,2D at different quasi-momenta k y , respectively with unbroken/broken generalized PT symmetry and different types of NHSE.When OBCs are also implemented in the y-direction, k y is no longer diagonal and the spectrum with both x, y-OBCs [Fig.1c Bottom] may differ from the x-OBC, y-PBC spectrum (Top).In particular, y-NHSE can occur only for the states with the generalized PT symmetry already broken, since the NHSE requires nontrivial spectral winding, which in turn requires complex eigenenergies.Since x-NHSE acts on all eigenstates of H γ,2D , there are no completely extended states, but only edge or corner-localized states under full x, y-OBCs, corresponding to single/double direction NHSE from unbroken/broken generalized PT symmetry, as shown in Fig. 1d and e respectively.To quantify the effective dimensionality of these eigenstates, we introduce the fractal dimension [54] N being the total number of sites and ψ n,r being the amplitude of n-th eigenstates at r-site.We have FD ≈ 0, 1 or 2 for corner, edge and 2D bulk states.In Fig. 1c Bottom, FD ≈ 1 for states with real eigenenergies, indicating their 1D edge-localized nature [Fig.1e].In contrast, states with complex eigenenergies have FD closer to 0, corresponding to the corner localization in Fig. 1d.
Directional toggling of NHSE and symmetrydriven corner modes.Noting that PT symmetry plays a crucial role in determining the allowed NHSE 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 intermediate stage of the toggling, localization occurs briefly in both directions, leading to symmmetry-driven corner modes.
To demonstrate this intriguing directional toggling of the NHSE, we tune the PT symmetry by varying the boundary hoppings.We interpolate between x-PBCs and x-OBCs with a parameter β, such that H γ,2D becomes where H OBC γ,2D is the Hamiltonian under full x, y-OBCs, and H 1↔Nx denotes the hoppings between the first and last unit cells along the x-direction (see Methods section).
We have x-OBCs and hence x-NHSE at β → ∞, and if parameters are chosen such that the generalized 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 Supplementary Note 1A.)This gives transition from purely real to complex.Meanwhile, as activated by PT symmetry breaking, the skin modes transition from x to y-edge localized (FD ≈ 1 with FD the average FD over all states), as plotted in the insets.Corner skin modes with FD ≈ 0.5 appear in the intermediate β ≈ 10 regime purely as a by-product of this PT activation mechanism.d Average FD and its 1D projections as a function of β under the y-OBC, with the trough at β ≈ 10 indicative of corner localization.Parameters are γ = 1.4,t1 = 0.1, t2 = 0.1, v = 1, and u = 0.9 in all panels.the x-edge-localized modes of Fig. 2a, with FD ≈ 1.By contrast, the other limit of β = 0 recovers x-PBCs that eliminate x-NHSE, but leads to y-NHSE with y-localized states [Fig.2c].In both scenarios, the y-OBCs remain unchanged, yet by adjusting the x-boundary hoppings, we can activate the localization in y-direction through PT symmetry breaking.
Interestingly, with partial x-OBCs i.e. β = 10, corner modes can appear as an intermediate between x and yboundary localization [Fig.2b].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 [55,56].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. 2d the average fractal dimension FD and its 1D projections (x-FD and y-FD) over all states as functions of β, where Edge-PT -breaking and selective boundary NHSE.
So far, we have only seen how generalized PT symmetry breaking can activate various NHSE channels for bulk states.We next discuss how that can interplay with nontrivial topology, which separately gives PT -asymmetric topological edge modes.Specifically, we show that PT can be selectively broken for edge states only, leading to a type of boundary NHSE distinguished from the usual bulk NHSE.We introduce another 2D model where h (0) g = v sin k x , and imaginary h 2 (all parameters chosen to be positive), all bulk states are PT unbroken and are free of any type of NHSE.On the other hand, edge states along x-boundaries will appear for all values of k y when their associating Wilson loop spectra take nontrivial values [57], and the effective edge Hamiltonian read [58][59][60][61]: with P± = (1 ± σ z )/2 the projectors of edge states.It is clear that H ± 2D,edge holds no PT symmetry, and gives a complex edge spectrum [gray dotted loop in the inset of Fig. 3a] with nontrivial spectral winding for k y ∈ (−π, π].Consequently, y-NHSE occurs for these x-boundary states under y-OBCs, accumulating on opposite corners (red) depending on the sign of Im(E) [Fig.3a].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 [64].In addition, the PTprotected bulk spectrum remains purely real [Fig.3a] and impervious to any NHSE throughout.Indeed, this absence of bulk NHSE is geometry-independent [62,63], representing an intrinsic hybrid skin-topological effect (Supplementary Note 2).
Going further, we may also selectively recover/break 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 sin k y + g)σ 0 : one branch of edge eigenenergies becomes real, As seen in Fig. 3b, these edge states remain left edge-localized (green) under full OBCs because they cannot undergo additional NHSE, but those of the other branch are PT -broken and collapse into corner modes (red) due to y-NHSE.
Anomalous PT -activated state dynamics.The selective activation of these various forms of corner and edge NHSE entails a competition between different NHSE channels, and gives rise to qualitative transitions in different stages of the state dynamics, distinguished to that of different bulk NHSE channels in H γ,2D (Supplementary Note 1B).Consider an initial state |ψ(0 √ 2 localized at the center of an N x × N y lattice.We dynamically evolve it via |ψ(t)⟩ = e −iHt |ψ(0)⟩ and investigate the evolution of its center-of-mass ⟨x⟩ and ⟨y⟩ in Figs.3c and d, for H = H g,2D and H ′ g,2D from Eqs. ( 5) and ( 6) respectively.For H = H g,2D [Fig.3c], both ⟨x⟩ and ⟨y⟩ remain roughly constant for a short time, since |ψ(t)⟩ is governed by NHSE-free bulk dynamics [see Fig. 3a] before it encounters a boundary.After t ≈ 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 Im(E) > 0, where ⟨x⟩, ⟨y⟩ ≈ 1.
A unusual observation is that ⟨x⟩, ⟨y⟩ can evolve non-monotonically in the dynamics governed by H ′ g,2D .Due to bulk y-NHSE, the state is initially symmetrically pushed to the top boundary, and ⟨y⟩ increases while ⟨x⟩ remains constant [Fig.3d].After spreading to the left and right boundaries at t ≈ 50, the dynamics become dominated by the left edge states with Im(E) > 0, and the y-NHSE is effectively deactivated.As such, the edge state evolves from the top to the left boundary, and ⟨x⟩ → 1 and ⟨y⟩ → N y /2.This enigmatic behavior is possible because the left edge states are PT -protected and do not undergo any NHSE to become corner states.
3D generalizations of PT -activated NHSE.The various avenues for PT -activated NHSE effects discussed above takes on even richer possibilities in higher dimensions.Below, we briefly demonstrate how they acquire extra interpretations as surface and hinge-selective NHSE in 3D.Specifically, we consider a 3D Hamiltonian of the form with τ x,y,z another set of Pauli matrices.H γ/g,0 (k x , k y ) is inherited from the previous PT -breaking and directional toggling NHSE models H γ/g,2D : which contain only diagonal terms of τ 0,z .Since k z enters only off-diagonal terms of τ x,y , the surfaces states under z-OBCs obey the effective Hamiltonian H ±,surface = P±,3D H γ/g,3D P±,3D with P±,3D = (1 ± σ 0 τ z )/2 z-surface state projectors [58][59][60][61].Note that the choices H γ/g,0 are not unique, and the explicit forms in Eq. ( 8) are chosen to restore PT symmetry Also, an additional CP symmetry emerges for the surface states [65] (also see Methods section), which offers another avenue for eliminating the NHSE by ensuring purely imaginary eigenenergies.In Fig. 4, eigenmodes are marked with different colors according to their corresponding sectors and boundary conditions.In Fig. 4a,b,c for H γ,3D , the (001) and (00 1) surface states essentially recapitulate the symmetryselected and directional toggling NHSE effects of Figs. 1  and 2. Going from Fig. 4a to b, these surface states (blue) become corner-localized due to generalized CP-breaking, analogous to the corner localization Fig. 1d due to PTbreaking, 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. 4a is replaced by the x-PBC of Fig. 4c.
Similarly, the surface (blue) and hinge (red) states on (001) and (00 1) surfaces of H g,3D in Fig. 4d are in direct correspondence with the bulk (black) and boundary (red) states of H g,2D in Fig. 3a.Meanwhile, bulk eigenenergies remain real as our choice of parameters maintains unbroken PT symmetry in Eq. ( 7).In analogy with H ′ g,2D in Fig. 3b, a CP symmetry can be recovered in one branch of hinge states on each of (001) and (00 1) surfaces by introducing an extra Hermitian term to the Hamiltonian, i.e.H ′ g,3D (k) = H g,3D (k) + (t ′ 2 sin k y + g)σ 0 τ 0 , which removes the NHSE on these hinge states [Fig.4e].
Alternatively, a 3D Hamiltonian with only bulk-NHSE can be designed such that its bulk (PBC) states are not PT -symmetric, but that generalized PT -symmetry can be recovered in certain parameter regimes, for instance the Hermitian SSH model and H γ,2D (k z , k y ) from Eq. (1).Consequently, bulk states accumulate to a 2D surface (or 1D hinges) when the generalized PT symmetry is unbroken (or broken), as shown in Fig. 4f,g.In contrast to H γ/g,3D , conventional surface (blue) or hinge (green) states are immune from the NHSE and remain extended.Thereby, our work provides a systemic framework for constructing and exploring various types of PT activated bulk and boundary skin effect, which also can be applicable in the study of non-Hermitian gapless phases [66,67].

Discussions
PT symmetry ensures real spectra, thus eliminating the prospect for spectral winding.This fundamental observation leads to the paradigm of PT -activated NHSE, where rich families of NHSE-related phenomena can be designed by selectively symmetry breaking in bulk or boundary subspaces.In 2D, we discussed the directional NHSE toggling and PT -mediated corner modes, as well as selective edge-PT breaking and restoring that causes non-monotonic transfer of edge states.In 3D or higher, PT activation leads to far more varied phenomenology by interplaying with the already rich classes of boundary and bulk NHSE.This paradigm of symmetry-controlled NHSE can be extended to include other symmetries which forbid spectral winding, such as non-Bloch and generalized PT symmetries, CP symmetry, or pseudo-Hermiticity.It also provides a versatile scheme for constructing models with different types of NHSE on different sectors of the systems.We also note that the conditions for yielding a real OBC spectrum with complex PBC spectrum can be geometrically understood with an electrostatics analogy of the NHSE problem [84], which may provide further possibilities for activating/deactivating NHSE based on our method.

Methods
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 [47,48], described by the Hamiltonian: where For the other non-Hermitian scenario of Eq. ( 11) with g = 0 but γ ̸ = 0, the 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 [26][27][28].Yet its OBC bulk spectrum become real in Fig. 5c due to the recovery of a generalized symmetry PT symmetry KH γ K = H γ [49,50] or a non-Bloch PT symmetry [51][52][53], or more generally, a pseudo-Hermiticity [4].Explicitly, the OBC bulk spectrum can be described by a non-Bloch Hamiltonian Hγ (k 25,29,86].That is, this non-Hermitian Hamiltonian can be transformed into a Hamiltonian without NHSE through a similarity transformation [25]: ψ † j,A/B → e −jκ ψ † j,A/B , where ψ † j,A/B is the annihilate operator for j-th cell with sublattices A/B.In particular, a Hermitian SSH model will be obtained directly through this transformation in the non-Bloch PT unbroken phase.A different route of transforming non-Hermitian Hamiltonians into Hermitian ones can be found in Ref. [87].This Hamiltonian Hγ (k x ) satisfies a non-Bloch PT symmetry, [ηK, Hγ ] = 0 with η a unitary operator, which is unbroken for all non-Bloch eigenstates when |u| > |γ/2| (as discussed later).In contrast, here the topological edge states always remain PT -symmetric and possess real (zero) eigenenergies, as a nonzero γ does not enter their effective Hamiltonian: H ± edge = P± H γ P± = 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 [88].Similarly, edge states of 2D generalization Hamiltonian H γ,2D with selective bulk NHSE, namely those near zero energy in Fig. 1c and Fig. 2a of the main text, are seen to also possess real eigenergies with their distributing dimensionality unchanged by NHSE.
Non-Bloch PT symmetry.While the models of H γ in the last section and H γ,2D satisfy a generalized PT symmetry, it is the non-Bloch PT symmetry that guarantees real OBC spectrum for more generic non-Hermitian Hamiltonians.Here, we discuss the non-Bloch PT symmetry of the non-Hermitian SSH model with asymmetric hoppings When γ = 0, this Hamiltonian satisfies the conventional PT symmetry [σ x K, H γ (k x )] = 0 with K the complex conjugate operator.Its corresponding non-Bloch Hamiltonian reads [25,29,86] with with W ± = (u ± γ/2)/(u ∓ γ/2).The quantity κ describes the inverse localization length of skin modes under OBCs, which can be obtained by requiring non-Bloch eigenenergies to form degenerate pairs for different quasimomenta (so that their linear combination can satisfy OBCs).W ± as well as all factors defined in Eq. ( 14) are real when |u| > |γ/2|.In this regime, we can define non-Bloch PT symmetry.To identify the non-Bloch PT symmetry, we rewrite the non-Bloch Hamiltonian as with with η = σ ′ x a parameter-dependent unitary operator and (ηK) 2 = 1, indicating the non-Bloch PT symmetry analogous to the conventional one.
We emphasis here that the system is non-Bloch PT symmetric only when |u| > |γ/2|, as Eq. ( 17) is not satisfied when |u| < |γ/2|.In the non-Bloch PT symmetric regime with |u| > |γ/2|, eigenenergies of H γ (k x + iκ), i.e. with , always take real values, since we have In other words, the system always falls in the non-Bloch PT unbroken phase in the symmetric regime of |u| > |γ/2|.On the other hand, when |u| < |γ/2|, the system no longer holds the symmetry of Eq. ( 17), and the eigenenergies E γ generally become complex.An exception is when k x ∈ {0, π}, where the eigenenergies reduce to when |u| < |γ/2|, which are also real provided The explicit form of Hamiltonian H β γ,2D .For analytic tractability, we give the explicit form of H β γ,2D in Eq. ( 3).According to this definition, the Hamiltonian H β γ,2D contains two parts: the Hamiltonian H γ,2D in Eq. ( 3) under full OBCs, and its hopping between boundaries along x-direction.The first part reads where ψj,k = [â j,k , bj,k ] T , with âj,k ( bj,k ) being the annihilate operator for the pseudospin-↑ (↓) [sublattices A (B)] on the (j, k)-th unit cell.Similarly, the hoppings between boundaries along xdirection H 1↔Nx and y-direction H 1↔Ny are Thereby, the Hamiltonian under full PBCs can be expressed by Through the Fourier transformation, the momentum space Hamiltonian H γ,2D (k x , k y ) in Eq. ( 3) can be obtained.In Eq. ( 3), an extra factor e −β is introduced to describe the partial OBCs aong x direction.
Effective surface Hamiltonian for 3D generalizations.The Hamiltonians H γ/g,3D and H ′ γ,3D are chosen to support surface states on (001) and (00 1) surfaces.Corresponding projectors of these surface states are P±,3D = (1 ± σ 0 τ z )/2.Thereby, the effective surface Hamiltonian of H γ,3D can be given by: Comparing to Eq. ( 3), we find the effective Hamiltonian H ± γ,surf can be considered as ±iH γ,2D (the signs of γ and t 2 are flipped for H − γ,surf ).As discussed in before, H γ,2D holds a non-Bloch PT symmetry.Correspondingly, H ± γ,surf holds a non-Bloch CP symmetry, due to the additional imaginary factor ±i. Therefore, the surface states of H γ,3D can support various NHSE channels activated by CP symmetry breaking, as discussed in Fig. 4a  to c.
Therefore, H ± g,surf satisfies a CP symmetry [CP, H ± g,surf ] + = 0 with CP = σ x K.And the surface (hinge) states of H g,3D displays similar NHSE phenomenon as the bulk( and edge) states of H g,2D , as shown in Fig. 4d.
The effective surface Hamiltonian of H ′ g,3D is given by where the CP symmetry is broken and then surface states suffer from NHSE.However, one branch of boundary states of H ± ′ g,surf recovers CP symmetry and is free from NHSE, analogous with the discussion of H ′ g,2D in Eq. ( 5).In this 3D model, CP symmetry is recover for distinct branches of boundary states in the (001) and (00 1) surfaces because of the ∓ sign in the square bracket of the Eq.( 27).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. 4e (green).
Their effective surface Hamiltonian can be obtained as which satisfies a PT symmetry [PT , H ± ′ 3D,surf ] = 0 with PT = τ x K. Actually, H ± ′ 3D,surf is Hermitian and without NHSE.Correspondingly, the dimensionality of surface and hinge states of H ′ 3D is unchanged by NHSE, despite that the localizing direction of hinge states (green) are altered by the z-NHSE, as shown in Figs.4f and 4g.Meanwhile, the bulk states can show distinct localization with various types of NHSE, as shown in Fig. 4f and 4g.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. S4 (c) and (d).

Figure 1 .
Figure 1.Corner and edge modes activated by paritytime (PT ) symmetry breaking.a,b: The spectrum under open boundary conditions (OBCs) (gray) of Hγ,2D with fixed ky can be a complex or b real depending on whether PT symmetry is restored by the non-Hermitian skin effect.For reference, the spectrum under periodic boundary conditions (PBCs) (brown) is always complex.c Full spectrum of Hγ,2D as a 2D model, with PBCs along the y direction.Pink and blue cross sections correspond to the spectra in a and b (also indicated by gray arrows).The full x, y-OBC spectrum (colored according to the fractal dimension FD) is contained within the set of y-PBC eigenenergies, as shown by the lower part of the panel (gray).Corner modes with their fractal dimension FD ≈ 0 (I,II, triangles) are plotted in d, and exist at Im(E) ̸ = 0 where PT symmetry is broken.An edge mode with FD ≈ 1 (III, square) exists at Im(E) = 0, as plotted in e. Parameters are γ = 1.4,t1 = 0.1, t2 = 0.1, v = 1, and u = 0.7.

20 Figure 2 .
Figure 2. Toggling between non-Hermitian skin effect along x and y directions through x-boundary tuning.From a to c: As we morph the boundary conditions along x direction from open boundary conditions (OBCs) a to periodic boundary conditions (PBCs) c, the y-PBC (gray) and y-OBC (colored by fractal dimension FD) spectra for H β γ,2D with α, α ′ ∈ (x, y) and α ̸ = α ′ .Similar to the FD, α-FD quantifies the locality along a single direction, which is 1 (0) for fully extended (localized) states along the α direction.While β = 0 and β ≫ 1 gives FD ≈ 1, a trough of FD exists between them, indicative of corner localization during the transition between single-directional y-NHSE under x-PBC (β = 0) and x-NHSE under x-OBC (β → ∞).Meanwhile, x-FD decreases and y-FD increases (almost) monotonously with β, reflecting the mutual exclusion of NHSE along x and y directions.

Figure 3 .
Figure 3. Selective boundary non-Hermitian skin effect (NHSE) and non-monotonic state dynamics.a Hg,2D hosts bulk states with unbroken parity-time (PT ) symmetry (black) and PT -broken topological edge states under open/periodic boundary conditions (OBCs/PBCs) along x/y direction (gray).Under full OBCs, the latter experiences y-NHSE and become corner-localized (red).b H ′ g,2D is deformed from Hg,2D such that PT symmetry is restored for the branch of edge states (green) with Im(E) > 0, which remains edge-localized; and broken for the bulk states (black), which are thus edge-localized due to y-NHSE.c,d depict the dynamical evolution of a center-localized initial state (black dot) due to Hg,2D and H ′ g,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 ⟨y⟩.Parameters are u = 0.2, g = 0.3, t1 = 0.3 and t ′ 2 = 0.1, with system size Nx = Ny = 20.
and σ α=x,y,z is the Pauli matrix acting on a pseudospin-1/2 space [e.g.two sublattices |A⟩ and |B⟩ as shown in Fig.5a].u and v denote the Hermitian staggered hopping amplitudes.The two non-Hermitian parameters γ and g describes asymmetric hoppings and imaginary on-site potential, which correspond to to nonlocal and local dissipation[85], respectively, as illustrated in Fig.5a.By separately tuning the two non-Hermitian parameters g and γ, we can break or restore PT -symmetry selectively for bulk or topological edge states in this model, as discussed below.When γ = 0, non-Hermiticity enters the Hamiltonian only through the on-site gain and loss of g, and the resultant Hamiltonian H g ≡ H(γ = 0) holds a PT symmetry: [PT , H g ] = 0 with PT = σ x K and PT 2 = +1.A nonzero g does not break the PT symmetry as the gain and loss are balanced between sublattices.As shown in Fig. 5b, in the PT -unbroken phase, all bulk states have real eigenenergies since they are also eigenstates of PT symmetry operator [1, 2].On the other hand, topological edge states for the SSH model are sublattice polarized and hence PT -broken.Instead of being eigenstates of PT symmetry operator, these states are related to each other through the symmetry, i.e.PT |ψ + ⟩ ∝ |ψ − ⟩, with |ψ +,− ⟩ 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 P± = (1 ± σ z )/2, i.e.H ± edge = P± H g P± = ±ig, which directly give their eigenenergies [58-61].

Figure 5 .
Figure 5.The non-Hermitian Su-Schrieffer-Heeger model.a A sketch of the model of Eq. (11), with γ (g) the asymmetric hopping (gain/loss) parameter.b Typical spectra of Hg under periodic (gray) and open (other colors) boundary conditions (PBCs and OBCs).Under the OBCs, bulk states can be parity-time-symmetric (PT symmetric) and have real eigenenergies (black), but edge states are PTbroken and have imaginary eigenenergies ±ig (red and blue).Insets demonstrate the spatial distribution of corresponding eigenstates.c Typical PBC and OBC spectra of Hγ, with the same color marks as in b.The PBC spectrum in c is PTbroken and possesses a nontrivial spectral winding, leading to a boundary accumulation of all eigenstates (see insets).The OBC spectrum restores a non Bloch (or a generalized) PT symmetry and becomes purely real.Parameters are chosen to be u = 0.5, v = 1, and b g = 0.2, c γ = 0.2 for demonstration.

Figure S4 .
Figure S4.Geometry-dependence of conventional hybrid skin-topological effect of H (geo) g,2D .(a) Energy spectrum of Hamiltonian H (geo) g,2D in Eq. (1) 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, g = 0.3, t1 = 0.3 and t ′ 2 = 0.1 for all panels.