Universal non-Hermitian skin effect in two and higher dimensions
UUniversal non-Hermitian skin effect in two and higher dimensions
Kai Zhang,
1, 2
Zhesen Yang, ∗ and Chen Fang
1, 3, 4, † Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China Kavli Institute for Theoretical Sciences, Chinese Academy of Sciences, Beijing 100190, China Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
Skin effect, experimentally discovered in one dimension, describes the physical phenomenon thaton an open chain, an extensive number of eigenstates of a non-Hermitian hamiltonian are localizedat the end(s) of the chain. Here in two and higher dimensions, we establish a theorem that the skineffect exists, if and only if periodic-boundary spectrum of the hamiltonian covers a finite area on thecomplex plane. This theorem establishes the universality of the effect, because the above conditionis satisfied in almost every generic non-Hermitian hamiltonian, and, unlike in one dimension, iscompatible with all spatial symmetries. We propose two new types of skin effect in two and higherdimensions: the corner-skin effect where all eigenstates are localized at one corner of the system, andthe geometry-dependent-skin effect where skin modes disappear for systems of a particular shape,but appear on generic polygons. An immediate corollary of our theorem is that any non-Hermitiansystem having exceptional points (lines) in two (three) dimensions exhibits skin effect, making thisphenomenon accessible to experiments in photonic crystals, Weyl semimetals, and Kondo insulators.
Introduction
The study of non-Hermitian hamiltonians, which canbe regarded as the effective description of dissipative pro-cesses, can be traced back to the investigation of alphadecay, where real and imaginary parts of the complexenergy are related to the experimentally observed en-ergy level and decay rate [1]. When a lattice systemis coupled with environments and has dissipations, e.g.photonic crystals having radiational loss [2–6] and elec-tronic systems having finite quasiparticle lifetime [7–10],the non-Hermitian band theory becomes a conceptuallysimple and efficient approach [11–25].Skin effect [19–39], a phenomenon unique to the non-Hermitian band theory, refers to the localization of eigen-states at the boundary, the number of which scales withthe volume of the system. For example, in one dimen-sion, all eigenstates of a non-Hermitian hamiltonian canbe localized at the ends of a chain [19]. This suggests thefailure of Bloch’s theorem [40], which states that eigen-states in the bulk are modulated plane waves. As Bloch’stheorem plays a fundamental role in the development ofcondensed-matter physics [41–43], the emergence of skineffect indicates a new and possibly revolutionary direc-tion. Especially, the skin effect has been experimentallyobserved in one-dimensional classical systems [36–39, 44–46], inspiring further studies on their higher dimensionalgeneralizations [20, 29, 30, 47–61]. However, a generaltheory for the higher-dimensional skin effect has not beenestablished.Apart form the skin effect, another focus topic innon-Hermitian band systems is the exceptional point (or ∗ Electronic address: [email protected] † Electronic address: [email protected] line) [14, 15, 62–82] that refers to stable point-type (orline-type) non-Hermitian band degeneracy in the Bril-louin zone. At the exceptional point, not only eigenval-ues but also eigenstates of the Bloch hamiltonian coa-lesce [65]. Many a novel phenomenon related to excep-tional points has been predicted and observed [82–95],such as the emergence of bulk-Fermi arc terminated atthe exceptional points [7, 79]. Since the bulk-boundarycorrespondence plays a central role in the developmentof topological phases [96–98], it is natural to ask if thereexists a non-Hermitian bulk-boundary correspondence inbands having exceptional points, analogous to the sur-face Fermi arc in the Weyl semimetals in the Hermitiancounterpart [99, 100].In this paper, we establish a theorem that reveals a uni-versal bulk-boundary correspondence in two and higherdimensional non-Hermitian bands, as shown in Fig. 1.The “bulk” refers to the area of the spectrum of thehamiltonian on the complex plane with periodic bound-ary condition, and “boundary” the presence (absence) ofthe skin effect for open-boundary system of an arbitraryshape. The theorem states that the skin effect appearsif and only if the spectral area is nonzero. This skineffect is “universal” for three reasons: (i) a randomlygenerated local non-Hermitian hamiltonian has the skineffect with probability one; (ii) the skin effect is, unlikein one dimension, compatible with all spatial symmetriesand time-reversal symmetry [17, 34]; and (iii) it does notrequire any special geometry of the open-boundary sys-tem. We also propose two manifestations, restricted totwo and higher dimensions, of the universal skin effect,i.e., the corner-skin effect and the geometry-dependentskin effect.A surprising corollary of our theorem is that all sta-ble exceptional points [15, 71, 73] imply the presence ofskin effect. Because exceptional points have been eitherobserved or proposed in meta-materials as well as in con- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b ReEImE
ReEImE k x k y a Brillouin zone b Spectral area c Spectral arc d Universal skin effect e No skin effect ℋ( k ) FIG. 1: The theorem of universal skin effect. (a) represents the Brillouin zone. (b)(d) shows that when the spectral area of H ( k ) is nonzero, the system on generic geometries must have universal skin effect. (c)(e) shows that when the spectral area of H ( k ) is zero, or forming one or several arcs on the complex plane, there is no skin effect under any geometry. densed matter, this corollary makes skin effect observablein known systems. We predict the geometry-dependentskin effect in the two-dimensional photonic crystal stud-ied in Ref. [79], and propose to observe this effect in theanomalous dynamics of wave packets. We predict theskin effect in a Weyl-exceptional-ring semimetal, realiz-able in Weyl semimetals made from inverting bands thathave disparate effective masses, such as the d - and the f -bands. Theorem: an equivalence between spectral area andskin effect
Before introducing the theorem itself, we first use anexample to illustrate the intuition that motivates thetheorem. Consider the following one-dimensional model H ( k ) = 2 cos k placed on a chain of length L . Under theperiodic boundary condition, the two Bloch waves e ik x and e − ik x have the same energy E ( k ) = 2 cos k . Whenthe system has open boundary condition, the Bloch wave e ik x will be reflected to e − ik x with a π -phase shift.Their linear superposition e ik x − e − ik x is an eigen-state with energy 2 cos k that satisfies the zero bound-ary condition at x = 0 , L , thus being an open-boundaryeigenstate. When the system is added a momentum-dependent dissipation, e.g. H ( k ) = 2 cos k + i sin k , E ( k ) becomes complex and forms an ellipse in the com-plex plane. In this case, the degeneracy is broken, e.g. E ( k ) (cid:54) = E ( − k ), which implies the open-boundary eigen-states are no longer the linear superposition of Blochwaves. This is the emergence of skin effect. The cor-respondence between the spectral shape and the skin ef-fect has been derived for generic one-dimensional non-Hermitian systems [23, 24], i.e. when the Bloch spec-trum is a loop-type (an arc-type), the skin effect appears(disappears). The underlying reason is that the degen-eracy of Bloch waves is split (preserved) in a loop-type (arc-type) spectrum.Generalizing the above discussion to two dimensions,we note two main differences. One difference is in theperiodic-boundary spectrum, E i ( k ), where i is the bandindex and k the crystal momentum in the first Brillouinzone (BZ). Generally speaking, E i ( k ) is a mapping fromthe d -dimensional torus to the complex plane, C . When d = 1, the image of E i ( k ) forms a loop; but when d > C , denoted by E i (BZ). The area covered by E i (BZ) on the complexplane is called the spectral area , denoted by A i . Anotherdifference is in the variety of open-boundary condition.There is only one geometry for an open system in onedimension, i.e., an open chain; but there are an infinitenumber of geometries in two dimensions such as triangle,rectangle and pentagon.Now we are ready to state the theorem of universalskin effect: in the thermodynamic limit, the skin effect ispresent in a hamiltonian having open boundary of arbi-trary geometry, if the spectral area is nonzero ( A i (cid:54) = 0);vice versa, the skin effect is absent for all possible ge-ometries, if the spectral area is zero ( A i = 0). As theperiodic-boundary hamiltonian describes the dynamicsin the bulk, the theorem relates a bulk property (spec-tral area) to a boundary one (existence of skin modes).Fig. 1 shows some schematic examples. In the Supple-mental Material Sec. I, a complete proof of the theoremhas been provided.The above theorem has implied the universality of skineffect in two and higher dimensions. As E i (BZ) is the im-age of the d > A i = 0 for ev-ery i . In fact, for single-band hamiltonian, we can provethat A = 0 if and only if H ( k ) = P [ h ( k )], where h ( k ) is aHermitian hamiltonian and P is a polynomial. In otherwords, a randomly generated non-Hermitian hamiltonian H ( k ) has skin effect: the first meaning of universality. Inprevious studies, other types of skin effect, such as the a Spectrum (L x =L y =60) c Wave function (L x =L y =60) L x L y Open boundary 1 L x L y Open boundary 2 d Wave function (L x =L y =60) b Spectrum (L x =L y =60) h Wave function (L x =L y =60) - � - � � � � - � - ���� ��� � � � f Spectrum (L x =L y =60) L x L y Open boundary 2 e Spectrum (L x =L y =60) - � - � � � � - � - ���� ��� � � � L x L y Open boundary 1 g Wave function (L x =L y =60) Corner-skin effect Geometry-dependent-skin effect
FIG. 2: Two manifestations of skin effect. One is the CSE (a)-(d), the other is the GDSE (e)-(h). In in (a)(b)(e)(f), thelight blue regions represent the spectrum under periodic boundary, and the red points represent the eigenvalues under differentopen-boundary geometries. The spatial distributions of eigenstates W ( x ) (Eq. (2)) are plotted in (c)(d)(g)(h). The GDSEdisappears under square geometry (open boundary 1) in (g), and reappears under triangle geometry (open boundary 2) in (h). line-skin and the high-order-skin effect, in two and higherdimensions have been proposed [52, 55]. These typesall require the open-boundary system take a special ge-ometry (usually a rectangle) and are hence consideredspecial and non-generic. The skin effect when A i (cid:54) = 0assumes a completely generic geometry of boundary: thesecond meaning of universality. The third meaning ofuniversality lies in the fact that, unlike in one dimension,higher-dimensional skin effect is compatible with all spa-tial symmetries.Now we provide a physical explanation for the abovetheorem. In order to simplify the discussion, we con-sider the above single-band model H ( k ) = H ( k ) + i Γ( k )with a non-zero spectral area, the real and imaginaryparts of which are functionally independent. For agiven eigenvalue E of the Bloch hamiltonian, by solv-ing H ( k ) = Re E and Γ( k ) = Im E , one can obtain afinite set of pre-images of E , i.e, K ( E ) = { k , ..., k m } ,which includes all Bloch waves having energy E . Nowsuppose that one of the Bloch waves k i ∈ K ( E ) is inci-dent on the boundary, depending on the normal directionof the boundary, the corresponding momentum of the re-flected wave can be arbitrary. However, the number ofelements of K ( E ) is finite, and as such cannot supportso many reflection channels. This failure of reflectionat a generic boundary implies the failure in forming anopen boundary eigenstate from Bloch waves. However,if E i ( k ) collapses into an arc, the number of the corre-sponding solution of H ( k ) = E is infinite. It means thatthere are infinite reflection channels to satisfy the openboundary of any shape, and an open boundary eigenstatemay be formed from superimposing all channels. The corner-skin and the geometry-dependent-skineffect
While the theorem shows that the skin effect is uni-versal, it does not specify what skin modes look like inhigher dimensions. Here, we report two types of the uni-versal skin effect, the corner- skin effect (CSE) and thegeometry-dependent skin effect (GDSE).The hamiltonian of the example for CSE is H ( k ) =[5(cos k x + cos 2 k x ) − i (sin k x + 3 sin 2 k x )+ 5 cos k y + i sin k y ] / , (1)of which the spectral area under square geometry andtriangle geometry is shown in Fig. 2 (a)(b) with light bluecolor. Because of the nonzero spectral area, the theoremtells us that the hamiltonian must have the universal skineffect. This is verified in Fig. 2 (c)(d), where the spatialdistributions of all the eigenstates W ( x ) = 1 N (cid:88) n | ψ n ( x ) | (2)under different open boundaries are plotted. Here ψ n ( x )is a normalized eigenstate and N is the number of eigen-states. It is found that the wave functions are alwayslocalized at the corner of the boundary in Fig. 2(c), evenif the open-boundary geometry is changed in Fig. 2(d).We also plot the corresponding eigenvalue spectra underdifferent open boundaries, as shown in Fig. 2 (a)(b) withred color. One can notice that the spectral areas underperiodic and open boundaries do not equal. This kindof skin effect is called CSE, which can be explained by anonzero current functional J α [ n ] = (cid:88) i (cid:73) BZ dk d n ( E i , E ∗ i ) ∂ k α E i ( k ) (3)under the periodic-boundary condition (discussed in theSupplemental Material Sec. II), where n ( E, E ∗ ) is anysmooth function [23]. The current functional is shown tovanish in two and three dimensions under point groups C i , D , , , , C h, h, h, h , D d, d, h, h, h, h , T , T d,h , O and O h . Therefore, the CSE is only compatible with pointgroups C m and C , , , , v, v, v, v .The hamiltonian of the example for GDSE reads H ( k ) = 2 cos k x + 2 i sin k y . (4)Since the spectral area is nonzero, our theorem tells usthat the system must have skin effect for generic ge-ometry, such as a random polygon. However, an in-teresting phenomenon in this example is that the skineffect disappears under the square geometry shown inFig. 2 (g). Once we chose other types of boundariesshown in Fig. 2 (h), the skin effect reappears. Since theemergence of the skin effect and the position of localiza-tion depend on the geometry, it is called the GDSE. Be-sides the distribution of the eigenstates, another featureof the GDSE is that area of the open-boundary spectrumseems to be the same as A i . However, the correspondingdensity of states is dependent by the choice of geometryas shown in Fig. 2 (e)(f). We conjecture this is a universalphenomenon for the GDSE. In the Supplemental Mate-rial Sec. II, we have provided some additional examplesto illustrate this new type of skin effect.For GDSE, there is at least one spatial geometry suchthat skin modes vanish, and as such is mutually exclusivewith CSE. In other words, GDSE has the current func-tional J α [ n ] = 0, which is guaranteed by point groups C i , D , , , , C h, h, h, h , D d, d, h, h, h, h , T , T d,h , O and O h . Therefore, GDSE is compatible with all these pointgroups, in contrast to CSE. Corollary: skin effect from exceptional points
An immediate corollary of our theorem is that all lat-tice hamiltonians having stable exceptional points haveuniversal skin effect, connecting two unique phenomenain the non-Hermitian band theory. Consider a stableexceptional point k in two dimensions. Due to thebranch point structure of exceptional point, the disper-sion around k can be expressed as E ± ( k ) = ± c (cid:112) q x + iq y + O ( | k − k | ) , (5)where q i = x,y denotes a small derivation from exceptionalpoint in x or y direction, that is, q i = k i − k i . Here c is a nonzero complex number. Suppose the range of theexpansion is r , then it is clear that A ± ≥ | c | πr / (cid:54) = 0.By the theorem, the system must have skin effect. Now we use the photonic crystal model that has beenexperimentally realized in Ref. [79] to demonstrate ourcorollary. The tight-binding model hamiltonian with pe-riodic boundary can be written as H ( k ) = d ( k ) · σ − iγ/ σ − σ z ) , (6)where σ = ( σ , σ x , σ y , σ z ) is a vector of the Pauli ma-trices and d ( k ) is a vector with four components, thatis, d ( k ) =[ µ − ( t + t )(cos k x + cos k y ) ,µ z + ( t − t )(cos k x − cos k y ) ,t (1 − cos k x − cos k y + cos k x − k y ) ,t (sin k x − sin k y − sin k x − k y )] . (7)The parameters are chosen as follows, ( t , t , t , µ , µ z ) =(0 . , − . , . , . , − . γ = 0, the system has two Diracpoints along the x -axis. When external dissipation orradiational loss is added, i.e., γ (cid:54) = 0, each Dirac pointsplits into two exceptional points shown in Fig. 2 (b),connected by the bulk Fermi arc. According to our theo-rem, the system must have the universal skin effect, moreprecisely, the GDSE. The skin effect disappears undersquare geometry but reappears under diamond geometryshown in Fig. 3 (c)(d). In this case, the majority of theeigenstates are concentrated on the four edges.As mentioned in the previous discussion, the ap-pearance of the skin effect can be reflected in thedynamical properties. In order to show this, we simulatethe time evolution of the wave packets with initialvelocity at the center of the diamond geometry. Herethe initial state is chosen to be Gaussian form | ψ (cid:105) = N exp[ − ( x − x ) / − ( y − y ) / − i x − i y ](1 , T ,where N is the normalization factor and x = y = 21is the center coordinate of the diamond geometry.We plot the corresponding normalized final states | ψ ( t f ) (cid:105) = N ( t f ) e − i H OBC t f | ψ (cid:105) for every ten timeintervals, where H OBC represents the open-boundaryhamiltonian on the diamond geometry. As shown inFig. 3 (c), in the Hermitian case, the center of the wavepackets obeys the reflection rule. Since the initial veloc-ity of the wave packet is perpendicular to the two edges,the center of the wave packet just bounces betweenthe two edges regardless of the dispersion of the wavepacket. However, in the non-Hermitian case ( γ = 1 / H ( k ) = [ d r ( k ) + iδ d i ( k )] · σ , (8)where d r ( k ) and d i ( k ) are vectors with four components, b Exceptional points BZ k x k y a Dirac points BZ k x k y t=0 t=10 t=20 t=30 t=40 t=50 Non-Hermitian case de Weyl points f Exceptional lines (1,-1,0)(-1,1,0) (1,-1, π ) (1,-1,0)(-1,1,0) (1,-1, π )k x k y ReE h Non-reciprocal response - - × - ω P ( ω ) - - × - c Hermitian case t=0 t=10 t=20 t=30 t=40 t=50 g Corner skin modes (L=16) yxz / i o P io ( ω ) P oi ( ω ) FIG. 3: Two Dirac points (a) of a two-dimensional photonic crystal model are split into four exceptional points (b) uponadding non-Hermitian term, such as radiational loss. Correspondingly, the evolution of Gaussian wave packet with initialvelocity at the center of a diamond geometry for each ten time intervals is shown in (c) with γ = 0 (Hermitian) and (d) with γ = 1 / W ( x ), is plotted in (g). The modulussquare of the propagator from i to o P io ( ω ) and that from o to i P oi ( ω ), as functions of ω , are plotted with red color and darkcyan color in (h), respectively. that is, d r ( k ) = (0 , sin k x , sin k y , − cos k x − cos k y + sin k z ) , d i ( k ) = ( −√ , k z , − cos k z , cos k z ) . (9)The Hermitian part d r · σ is a Weyl semimetal with twoWeyl points. One Weyl point with +1 topological charge[red cone in Fig 3 (e)] is at (0 , , − , , π ).Upon turning on the non-Hermitian term ( δ (cid:54) = 0), theWeyl points evolve into two exceptional rings as shownin Fig. 3 (f). According to our theorem, the system withexceptional lines must have the universal skin effect. Spe-cially, the system described in Eq. (8) always has CSE asshown in Fig. 3 (g) with δ = 1 /
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I. THE PROOF OF THE THEOREM
In this section, we will prove the following theorem appeared in the main text:
Theorem : In the thermodynamic limit, the skin effect is present in a hamiltonian having open boundary ofarbitrary shape, if the periodic-boundary spectral area of the same hamiltonian is nonzero; vice versa, the skineffect is absent for all possible shapes of open boundary, if the spectral area is zero.Here the spectral area refers to the area of the region covered by the periodic boundary spectrum on the complexplane. In the following contents, we will first show some numerical verifications of the theorem, and then prove thetheorem in two dimensional systems, and finally extend the proof to three-dimensional cases.
A. Some numerical examples of the theorem
In this subsection, we provide some numerical examples of the theorem. In order to simplify the discussion, weconsider the following single-band model H ( k ) = (cid:88) i,j t ij β − ix β − jy , β x/y = e ik x/y , (S1)where t ij represents the hoping parameter that an electron or particle hopes from site ( m, n ) to site ( m, n ) + ( i, j ).The two rows in Fig. S1 represent two different models. In the first one (the first row), the hoping parameters areshown in Fig. S1 (a), and the periodic boundary spectrum, i.e. E ( k ) = H ( k ) with k belonging to the Brillouin zone(BZ), is shown in Fig. S1 (b). Here the color strength represents the cover times of E ∈ E ( k ) when k sweeps overthe whole BZ. One can notice that the spectral area of the first model is nonzero. As a result, the open boundary - - - - (cid:1) (cid:1)
400 0 (cid:1) (cid:1) (cid:2)(cid:1) (cid:1) (cid:1) (cid:1) (cid:2) + (cid:1) (cid:1) (cid:2)(cid:1) (cid:1) (cid:1) (cid:1) (cid:2)(cid:1) (cid:1)
20 00 (cid:1) (cid:1) a b c de f g h - (cid:1) - (cid:2) - (cid:3) (cid:4) (cid:3) (cid:2) - (cid:5)(cid:4) - (cid:6)(cid:4)(cid:6)(cid:5)(cid:4) (cid:1)(cid:2)(cid:3) (cid:1) (cid:2) (cid:3) (cid:4)(cid:5)(cid:6) (cid:7)(cid:8)(cid:2)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13) (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:5)(cid:2)(cid:3) (cid:5)(cid:2)(cid:4) (cid:6)(cid:2)(cid:3)(cid:3)(cid:5)(cid:7)(cid:8) (cid:1)(cid:2)(cid:3) (cid:1) (cid:2) (cid:3) (cid:4)(cid:5)(cid:6) (cid:7)(cid:8)(cid:2)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13) FIG. S1: Some numerical examples of the theorem. (a-d) show an example having skin effect, and (e-h) show an examplewithout skin effect. (a) and (e) show the corresponding hoping parameters of the hamiltonian shown in Eq. S1. (b) and (f)show the periodic boundary spectrum. (c-d) and (g-h) show the distribution of W ( x ) in Eq. S2 under different open boundarygeometries. Spectral AreaSpectral Winding Universal skin a Triangular relation b Spectral winding
ReE I m E - � � � - ��� � � k x k y � π - π � π - π � � FIG. S2: (a) shows the equivalence relation between spectral area, spectral winding and the universal skin effect. Eachequivalence relation is sufficient and necessary. (b) illustrates the spectral winding for hamiltonian Eq. S12. Here the lightblue region represents the periodic boundary spectrum, and the paths on BZ corresponds to the spectral loops (or arcs) on thecomplex plane with the same color. eigenstates show the localization behaviors, namely, the emergence of skin effect, as shown in Fig. S1 (c-d). In orderto illustrate the localization properties, W ( x ) = 1 N (cid:88) n | ψ n ( x ) | (S2)is plotted in Fig. S1 (c-d), , where ψ n ( x ) is the n th normalized eigenstate of the open boundary hamiltonian, and N is the number of open boundary eigenstates. For the second example, since the spectral area is zero, as shown inFig. S1 (f), there is no skin effect. Indeed, W ( x ) shown in Fig. S1 (g-h) are extended.In the following subsection, we will prove the theorem in two-dimensional systems. Our strategy of the proof isillustrated in Fig. S2(a). The equivalence relation between “spectral area” and “the universal skin effect” is linked by“spectral winding” (see the following discussion). B. The proof of the theorem in two-dimensions
1. Spectral area and spectral winding
We begin with a two-dimensional single-band non-Hermitian model with periodic boundary in both x and y direc-tions H ( k ) = u ( k ) + iv ( k ) , (S3)where u and v being real functions about k = ( k x , k y ). For any k r ∈ BZ , one can define the following winding number ν ( k r ) = (cid:73) Γ k r d k πi · ∇ k log det[ H ( k ) − E r ] , k r ∈ BZ , (S4)where Γ k r represents the infinitesimal counterclockwise loop enclosing k r . Here E r = H ( k r ) is the base energy point,which is shown in Fig. S2 (b) with red point. We note that for different k r , the base energy point is different. Thistopological invariant describes the spectral winding on the complex plane. As shown in Fig. S2 (b), if ν ( k r ) is nonzero,the image of Γ k r , i.e. H ( k ∈ Γ k r ), forms a closed loop that encloses E r .The first step of our proof is to show that1. if there are some k r points in the BZ with nonzero topological charge, the spectral area must be nonzero;2. if all the k r points on the BZ have zero topological charge, the spectral area must be zero.The above two statement can be represented by the equivalence relation between “spectral winding” and “spectralarea” shown in Fig. S2 (a).Based on the definition of winding number, the statement 1 is obvious. Therefore, we only need to prove thestatement 2. In order to show this, we expand the hamiltonian at the point k r ≡ ( k rx , k ry ) as follows H ( k ) − H ( k r ) ≈ ∂ x H ( k r ) q x + ∂ y H ( k r ) q y , (S5)0where q x , q y are the displacements k − k r in x, y directions respectively. For a generic k r point, the first derivative of H should not vanish, i.e., the coefficients of q x and q y in Eq. (S5) cannot be zero at the same time. This is becausein order to have ∂ x H ( k r ) = ∂ y H ( k r ) = 0, there are four real independent equations that must be satisfied. However,in two-dimension, there are only two free parameters ( k x , k y ), which cannot satisfy the above four equations.In order to calculate th topological charge, according to Eq. S3, one can define C ( k r ) = (cid:18) ∂ x u ( k r ) ∂ y u ( k r ) ∂ x v ( k r ) ∂ y v ( k r ) (cid:19) , (S6)where ∂ x/y = ∂/∂ k x/y . It can be shown that when det[ C ( k r )] (cid:54) = 0, ν ( k r ) is proportional to the sign of det[ C ( k r )].Therefore, a necessary condition for the zero charge for all the k r ∈ BZ isdet[ C ( k r )] = ∂ x u ( k r ) ∂ y v ( k r ) − ∂ y u ( k r ) ∂ x v ( k r ) = 0 . (S7)A theorem (the corollary of theorem 13.2) in Ref. [1] tells us that if C ( k ) (cid:54) = 0 and det[ C ( k )] = 0 for an open set k ∈ S ⊂ BZ , then, u ( k ) and v ( k ) have a functional dependent relation in the open set S . Applying the theorem to theentire BZ (except for some isolated points at which the first derivative vanishes), one can reexpressed the hamiltonianas H ( k ) = P [ h ( k )] (S8)where h ( k ) is a real and periodic function of k , and P is a complex polynomial of h ( k ). Since h ( k ) is a real periodicfunction, its image must be an arc on the real axis, e.g. h ( k ) ∈ [ h , h ], where h / are real numbers. Therefore, theimage of P [ h ( k )] is also an arc on the complex plane. This completes the proof of the statement 2 for single-bandmodels.Generalizing the above discussion to the multi-band case, for each k r ∈ BZ , the topological charge defined for the m -th band is ν m ( k r ) = (cid:73) Γ k r d k πi · ∇ k log det[ H ( k ) − E m ( k r )] , = (cid:88) n (cid:73) Γ k r d k πi · ∇ k log[ E n ( k ) − E m ( k r )] , (S9)where E m ( k r ) is the energy of the m -th band whose momentum is k r . In the second step, we have used det[ H ( k ) − E m ( k r )] = (cid:81) n [ E n ( k ) − E m ( k r )]. If k r is not the degeneracy point, only n = m term in the summation hascontributions to the winding number. Therefore, the winding number becomes ν m ( k r ) = (cid:73) Γ k r d k πi · ∇ k log[ E m ( k ) − E m ( k r )] . (S10)Using the similar argument in the single-band case, one can conclude that locally, the the real and imaginary parts of E m ( k ) are functional dependent. As a result, the spectral of E m ( k ) must be an arc. The above conclusion applies forevery band. Finally, we have proved that if all the k r points on the BZ having zero topological charge, the spectralarea must be zero.
2. Spectral winding and the universal skin effect
In the above subsubsection, we have proved the equivalence relation between “spectral winding” and “spectralarea”. In this subsubsection, we will prove the equivalence condition between “spectral winding” and “universal skineffect”, as shown in Fig. S2 (a).We first notice that the BZ in two-dimensional systems can be carpeted by straight lines with different slopes, whichis labeled by L m . For example, if we fix k x ( k y ) and change k y ( k x ) from 0 to 2 π , we get a vertical (horizontal) straightline on BZ, and all horizontal or vertical straight lines cover the entire BZ. Particularly, an inclined straight line goesout from one side of BZ and again enters from another side as shown in Fig. S2(b). Since the straight lines on theBZ are periodic, one can define the spectral winding number for each straight lines with respect to the prescribedreference energy E r . ν m ( L m , E r ) = (cid:73) L m d k πi · ∇ k log det[ H ( k ) − E r ] . (S11)1Obviously, if the all the k points on the BZ have zero topological charge, ν m ( L m , E r ) must be zero for arbitrary L m and E r . Otherwise, one can always find some L m , such that ν m ( L m , E r ) is nonzero. Let’s use an example to showthis. Consider the following hamiltonian H ( k x , k y ) = 2 cos k x + 2 i sin k y . (S12)As shown in Fig. S2, the spectrum of the hamiltonian along each horizontal or vertical straight line (gray lines) on BZhas zero winding number with respect to any reference energy on the spectral area (lightblue square region). However,once we choose the straight line with darker cyan color, the corresponding spectral has a nonzero winding. If thesystem has no skin effect under a specific parallelepiped open boundary, the winding number ν m ( L m , E r ) along eachstraight lines that are perpendicular to the boundary cut directions should be zero. (This is a conjecture that wecannot exactly proved currently. However, we believe this statement is true as all the numerical results we obtainedobeys this conclusion. Furthermore, it is also a natural generalization of the one-dimensional results [3 ? ] and hasbeen mentioned or applied in some recent works [4, 5]) More generally, if the system has no skin effect under anyparallelepiped open boundary, then the spectral windings of all straight lines on BZ are required to be zero, whichis satisfied when the spectral area of the system is zero. Therefore, nonzero spectral area means that there must beskin effect under certain open boundaries, namely, the existence of the universal skin effect. C. The proof of the theorem in three-dimensions
In this section, we extend the above proof of two-dimensional systems into three dimensions. We obtain the similarconclusion that nonzero spectral area is equivalent to the existence of the universal skin effect.Consider a general three-dimensional single-band tight-binding hamiltonian, which consists of real- and imaginary-part functions H ( k x , k y , k z ) = u ( k x , k y , k z ) + iv ( k x , k y , k z ) . (S13)We choose a generic k r point and use its energy H ( k r ) as the reference energy. For a given reference energy E r , wecan obtain a one-dimensional curve in the there-dimensional BZ by solving the following two real equations, u ( k x , k y , k z ) = Re( E r ); v ( k x , k y , k z ) = Im( E r ) . (S14)Each equation determines a surface, and the intersection of two surfaces is one-dimensional curve in there-dimensionalBZ. The tangent direction of the curve at k r is perpendicular to the normal vector of the two surfaces at this k r point. The tangent vector at k r is expressed as T k r = ∇ u ( k r ) × ∇ v ( k r ) , (S15)where ∇ u ( k r ) represents the gradient of u . We choose the local coordinate system ( R space) with k r as the origin,and the gradient is reexpressed as ∇ u ( k r ) = ∂ x u ( k r ) q x + ∂ y u ( k r ) q y + ∂ z u ( k r ) q z , (S16)where q i ≡ ( k − k | k − k | ) i , k and k represent two vectors in the global coordinate system.Next we expand the hamiltonian into Taylor series around the origin of the local coordinate system, H ( k ) − H ( k r ) = (cid:88) i = x,,y,z ∂ i H ( k r ) q i + o ( | q | ) , (S17)where the subscription i represents the partial differential to x, y, z . And q i represents the deviation of k i from k r,i and the last term is the infinitesimal of higher order of | q | . Obvious, the zero winding condition requires ∂ x u ( k r ) ∂ y v ( k r ) − ∂ x v ( k r ) ∂ y u ( k r ) = 0; ∂ x v ( k r ) ∂ z u ( k r ) − ∂ x u ( k r ) ∂ z v ( k r ) = 0; ∂ y u ( k r ) ∂ z v ( k r ) − ∂ y v ( k r ) ∂ z u ( k r ) = 0 , (S18)or equivalently, T k r = . (S19)2 a L y L x L x L x L x Corner skin b Line skin c No skin d Geometry dependent skin
FIG. S3: The distribution of W ( x ) for Hamiltonian Eq. (S21) with different parameters and on different geometries. Thesystem size is L x = L y = 25. The probability density is proportional to the opacity of the red color. (a) shows the corner-skineffect with t = t = 1 , w = 0; (b) shows line skin with t = 1 , t = w = 0; (c) has no skin effect with t = t = 0 , w = 1, andgeometry-dependent-skin effect appears in (d) under triangle and diamond geometries. Next, we prove that if all k points in three-dimensional BZ satisfy T k = , then the entire 3D periodic-boundaryspectrum must be an arc in the complex plane. We define a two-tuple function W ( k ) := [ u ( k ) v ( k )] t with threevariables, the exterior derivative of the vector-valued function is expressed as dW = (cid:18) ∂ x u ( k ) ∂ y u ( k ) ∂ z u ( k ) ∂ x v ( k ) ∂ y v ( k ) ∂ z v ( k ) (cid:19) . (S20)Eq. (S19) implies that the rank of dW less than 2 (the number of components of W ). To be precise, there are thefollowing cases. (i.) Both the gradients of u and v are not zero vector, and they are linearly dependent on each other.(ii.) One of the gradients of u and v is zero vector. (iii.) Both the gradients of u and v are zero vector. In all thesecases, we can obtain the final conclusion that u and v are linearly functional dependent on each other. Therefore, thespectrum must be arcs on the complex plane.A 3D BZ can be divided into a series of plane systems, each plane corresponds to a two-dimensional subsystem.If the spectral area of a three-dimensional system is nonzero, then for each reference energy on the spectral area, itspreimage (1D ring) has nonzero topological charge. Equivalently, the two-dimensional subsystem, of which the BZ(2D plane) has intersections with the ring, also has nonzero topological charge for the intersecting k points. Hence,the 2D subsystem has nonzero spectral area, and has the universal skin effect. Correspondingly, we come to the sameconclusion in 3D systems that nonzero spectral area signifies the existence of the universal skin effect. II. CORNER-SKIN EFFECT AND GEOMETRY-DEPENDENT-SKIN EFFECT
In this section, we will provide some examples to demonstrate the characteristics of the two manifestations of theuniversal skin effect. We discuss the role of symmetry on the universal skin effect. We define the current functional toexplain the appearance of corner-skin effect. In addition, we numerically verified that geometry-dependent-skin effectobeys the volume law, which is a significant feature to distinguish skin modes form conventional boundary states.
A. Symmetry and the universal skin effect
We have proved that if the system has nonzero spectral area, the universal skin effect will occur under a generalopen boundary condition. According to the symmetry restriction, the universal skin effect has two manifestations,that is, corner-skin effect and geometry-dependent-skin effect. Consider a system with nonzero spectral area, if itsHamiltonian has no any symmetry, the skin modes are localized at one or several vertices on an open geometry of anyshape, and the number of modes are proportional to the volume of the system. If the Hamiltonian has certain spatialsymmetries, such as mirror symmetry, the corner-skin effect will be forbidden. The reason is that the corner-skineffect has the nature of non-reciprocity, which is incompatible with mirror symmetry. However, if we change the openboundary geometry such that the symmetry is broken on the boundary, then geometry-dependent-skin modes willappear, which is a unique but universal phenomenon in higher-dimensional systems. We take a concrete example todemonstrate the role of symmetry on the skin effect. Consider a tight-binding model, of which the periodic-boundary3Hamiltonian reads H ( k x , k y ) = h ( k x , k y ) + ih ( k x , k y )= sin k x σ x + sin k y σ y + (2 − cos k x − cos k y ) σ z + i [ t sin k x + t sin k y + w (cos k x − cos k y )] σ z , (S21)where h and h are the Hermitian and non-Hermitian parts, respectively. The Hermitian part is in a gapless phasewith a Dirac point at k x = k y = 0, and has inversion symmetry σ z h ( k x , k y ) σ z = h ( − k x , − k y ).If we only add the w non-Hermitian term ( t = t = 0; w = 1), the combined mirror and non-Hermitian time-reversalsymmetry M x T = A t and M y T = σ z A t are preserved ( A t representing transpose operator),( M x T ) H ( k x , k y )( M x T ) − = H ( k x , − k y );( M y T ) H ( k x , k y )( M y T ) − = H ( − k x , k y ) . (S22)There is no skin effect under open boundary with square geometry as shown in Fig. S3(c). However, if we cut thesquare lattice into triangles and diamond lattices, the skin effect will retrieve on the boundaries that destroy the twosymmetries, which is shown in Fig. S3(d).If we only turn on t non-Hermitian term ( t = 1; t = w = 0), the Hamiltonian preserves M x T symmetry butdestroys M y T symmetry. In this case, the system has skin modes along x direction as shown in Fig. S3(b), thenumber of the skin modes are proportional to the volume of the system. If we add t and t non-Hermitian terms( t = t = 1; w = 0), the Hamiltonian destroys the two symmetries, and skin modes will be concentrated on the corneras shown in Fig. S3(a). B. Current functional
We define the current functional to depict the corner-skin effect in d dimensions. J α [ n ] = (cid:88) i (cid:73) BZ dk d n ( E i , E ∗ i ) ∂ k α E i ( k ) , (S23)where i is band index, α labeling the components of k , and d is the dimension of the system. Here n ( E, E ∗ ) is adistribution function depending on E and E ∗ , but does not depend on k explicitly, such as the Bose distribution n ( E, E ∗ ) = ( e Re E ( k ) /k B T − − . If there exists a n ( E, E ∗ ) such that the current functional is nonzero for any α ,then the system must have the corner-skin effect. For example, in two dimensions, α = x, y , as long as there exists a n ( E, E ∗ ) such that J x [ n ] (cid:54) = 0 and J y [ n ] (cid:54) = 0, the system has corner-skin effect. Additionally, if J x [ n ] (cid:54) = 0 and J y [ n ] = 0for any smooth function n ( E, E ∗ ), then the system has a line skin effect along x direction. If J x [ n ] = J y [ n ] = 0 forany smooth function n ( E, E ∗ ), then the system has no skin effect. From this, we can see that the appearance of thecorner-skin effect is generic if the system has no symmetry.If d = 1, the Eq. (S23) reduces to J [ n ] = (cid:80) i (cid:82) π n ( E i , E ∗ i ) ∂ k E i ( k ) dk , a similar form is obtained in Ref. [2]. In onedimension, if there is no skin effect under open boundary condition, the periodic-boundary spectrum for each band isan arc on the complex plane. The current functional can be written as J [ n ] = (cid:72) L i, BZ n ( E i , E ∗ i ) dE i , where the integralpath L i, BZ refers to the image of i -th band on BZ (an arc on the complex plane), therefore, the current functionalmust be zero (the integral path encloses zero area). C. Volume Law
We numerically show that the geometry-dependent-skin effect satisfies the volume law, that is, the increase in thenumber of skin modes is proportional to the increase in volume of the system, δN skin ∝ δV. (S24)For example, If the shape of the open boundary is a parallelogram whose side-lengths are L x and L y as shown inFig. S4, then, V = L x L y . Our criterion for judging a mode as a skin mode is to check whether ninety percent of theprobability density of this mode lies within the boundary we appointed.Consider a tight-binding model with periodic-boundary Hamiltonian H ( k x , k y ) = 2 cos k x + 2 i sin k y , there is noskin effect under square geometry Fig. S4(a), but the skin effect appears under parallelogram geometry Fig. S4(b)due to the spectral area being nonzero, which is geometry-dependent-skin effect. The distributions of W ( x ) under4 a Square b Parallelogram c Volume Law L x L y B ounda r y B ounda r y L x ∝ � ∝ � � / � ∝ � �
200 700 12000300600 V N sk i n FIG. S4: The norm squared of all wave functions of the Hamiltonian Eq. (S12) on square lattice (a) and parallelogram lattice(b) is plotted. The system size is chosen as L x = L y = 25. The volume law is shown in (c), in which blue line represents N skin ∝ V , gray line N skin ∝ √ V and black line N skin ∝ different open boundaries are plotted. For parallelogram geometry, we specify the thickness of the boundary to bethe width of three unit cells, and use black dashed lines to distinguish the boundary from the bulk. If ninety percentof the probability density of a mode lies in the boundary, we count it as a skin mode. We count the number of skinmodes for different volumes ( L x L y ), and the fitting curve (blue curve in Fig. S4(c)) shows that the two are in a linearrelation, that is, δN skin = 0 . δV . The volume law of geometry-dependent-skin effect has been verified numerically. III. EXCEPTIONAL SEMIMETALS
In this section, we will prove the corollary of our theorem, that is all stable exceptional points imply the universalskin effect. We first review the topological charge of non-Hermitian band degeneracies.
A. Non-Hermitian band degeneracy
Consider a general m -band non-Hermitian Bloch Hamiltonian (with periodic boundary condition), H ( k ) = m − (cid:88) s =1 (cid:2) h rs ( k ) + ih is ( k ) (cid:3) Γ s , (S25)where Γ s are the generators of Lei algebra su ( m ) and h rs ( k ) and h is ( k ) are real functions of k . When m = 2 , , s are the Pauli, Gell-Mann, and γ matrices, respectively. The eigenvalues of H ( k ) can be obtained by solving thefollowing characteristic polynomial f E ( k ) = det[ E − H ( k )] = m (cid:89) i =1 [ E − E i ( k )] , (S26)where E i ( k ) is the i th eigenvalue of the non-Hermitian Hamiltonian H ( k ). At the degeneracy point k D , two bandsmust have the same energy, i.e. E i ( k D ) = E j ( k D ) (S27)for some i (cid:54) = j . In Ref. [6, 7], the authors have shown that the above condition is equivalent to the vanishing of thediscriminant of f E ( k ), i.e. Disc E [ H ]( k D ) = 0 , (S28)where Disc E [ H ]( k ) = (cid:89) i In this subsection, we will review the topological charge of the non-Hermitian band degeneracies. Based on thediscriminant of the characteristic polynomial, one can define the topological charge of the degeneracy point k D , i.e. ν ( k D ) = i π (cid:73) Γ( k D ) d k · ∇ k ln Disc E [ H ]( k ) . (S34)where Γ ( k D ) is a loop encircling the degeneracy point k D . Since Disc E [ H ]( k ) is single valued, this invariant isquantized, which is called the discriminant number in Ref. [7]. PuttingDisc E [ H ]( k ) = (cid:89) i 2. The eigenvalues of H ( δ k ) are E ± ( δ k ) = ± (cid:112) δk x + iδk y . (S39)When k = k D , which is equivalent to δk x = δk y = 0, one can find E + ( δ k = 0) = E − ( δ k = 0) = 0. This means k D is a non-Hermitian degeneracy point. Now we choose Γ( k D ) = k D + δk r (cos θ, sin θ ), then, E ± ( δ k ) = ± δk / r e iθ/ , θ ∈ ( − π, π ] . (S40)One can find that E + ( δ k ) and E − ( δ k ) forms a spectral loop that encloses E + ( k D ) = E − ( k D ) = 0. The windingnumber ν ( k D ) describes this topological properties of degeneracy points.The topological charge ν ( k D ) can be used to classify the non-Hermitian degeneracies. However, the classificationis not complete. As a comparison with H ( δ k ), consider the following two low energy Hamiltonians, H ( δ k ) = ( δk x + iδk y ) σ + + ( δk x − iδk y ) σ − . (S41)Obvious, δ k = 0 is a degeneracy point. One can further prove that its topological charge is +1, which is equal to thecharge of δ k = 0 in H ( δ k ). However, these two degeneracy points have different properties. For example H ( δ k = 0) = σ + , H ( δ k = 0) = 0 . (S42)One can notice that H ( δ k = 0) is non-diagonal. This type of non-Hermitian degeneracy points are called exceptionalpoints. In Ref. [7], the authors have shown that only the exceptional points with ν ( k D ) = ± ± 1, the corresponding spectrumarea must be nonzero.7 C. The photonic crystal model In this subsection, we numerically calculate the spectrum and spatial distribution of the wave function, i.e. W ( x )in Eq. S2, for the photonic crystal model in the main text under different geometries.It shows that the skin effect disappears under square geometry in Fig. S5(b), and reappears under diamond geometryin Fig. S5(d), which is a characteristic signature of geometry-dependent-skin effect. Here we take the non-Hermitianparameter γ as 1 / 4. The spectrum under square geometry (red points in Fig. S5(a)) coincides with the spectrum ofperiodic boundary (light blue region in Fig. S5(a)(c)). We conjecture that the spectrum under diamond geometry (redpoints in Fig. S5(d)) will also coincide with the periodic-boundary spectrum as the system size increases. However, itclearly shows that the density of states under different geometries is completely different. The dependence of densityof states on the choice of boundary geometry is another significant feature of geometry-dependent-skin effect. [1] L. H. Loomis and S. Sternberg, Advanced calculus , (World Scientific, 1968).[2] K. Zhang, Z. Yang, and C. Fang, Phys. Rev. Lett. , 126402 (2020).[3] N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Phys. Rev. Lett. , 086801 (2020).[4] K. Yokomizo and S. Murakami, Phys. Rev. Lett. , 066404 (2019).[5] M. M. Denner, A. Skurativska, F. Schindler, M. H. Fischer, R. Thomale, T. Bzdusek, and T. Neupert, arXiv:2008.01090(2020).[6] Z. Yang, C.-K. Chiu, C. Fang, and J. Hu, Phys. Rev. Lett.124