Nodal Manifolds Bounded by Exceptional Points on Non-Hermitian Honeycomb Lattices and Electrical-Circuit Realizations
NNodal Manifolds Bounded by Exceptional Points on Non-Hermitian HoneycombLattices and Electrical-Circuit Realizations
Kaifa Luo, Jiajin Feng, Y. X. Zhao,
3, 4, ∗ and Rui Yu † School of Physics and Technology, Wuhan University, Wuhan 430072, China School of Physics, Sun Yat-sen University, Guangzhou 510275, China National Laboratory of Solid State Microstructures and department of Physics, Nanjing University, Nanjing, 210093, China Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
Topological semimetals feature a diversity of nodal manifolds including nodal points, various nodallines and surfaces, and recently novel quantum states in non-Hermitian systems have been arousingwidespread research interests. In contrast to Hermitian systems whose bulk nodal points must formclosed manifolds, it is fascinating to find that for non-Hermitian systems exotic nodal manifolds canbe bounded by exceptional points in the bulk band structure. Such exceptional points, at whichenergy bands coalesce with band conservation violated, are iconic for non-Hermitian systems. In thiswork, we show that a variety of nodal lines and drumheads with exceptional boundary can be realizedon 2D and 3D honeycomb lattices through natural and physically feasible non-Hermitian processes.The bulk nodal Fermi-arc and drumhead states, although is analogous to, but should be essentiallydistinguished from the surface counterpart of Weyl and nodal-line semimetals, respectively, for whichsurface nodal-manifold bands eventually sink into bulk bands. Then we rigorously examine the bulk-boundary correspondence of these exotic states with open boundary condition, and find that theseexotic bulk states are thereby undermined, showing the essential importance of periodic boundarycondition for the existence of these exotic states. As periodic boundary condition is non-realistic forreal materials, we furthermore propose a practically feasible electrical-circuit simulation, with non-Hermitian devices implemented by ordinary operational amplifiers, to emulate these extraordinarystates.
I. INTRODUCTION
Recently novel quantum states of non-Hermitian sys-tems have been a rapidly expanding field, accelerat-ingly attracting attention from the previously unrelatedfields, such as topological phases of quantum matter[1–20], many-particle physics [21–24], cold atoms [25–27], and the traditional field of quantum optics [28–36] with renewed interests. Maybe the most iconic fea-ture of non-Hermitian physics is the existence of excep-tional points [37] in parameter space, at which unitary ormore general similarity transformations cannot convertthe Hamiltonian under consideration into a completelydiagonal form, but optimally into upper-triangular blockseach with equal diagonal entries, namely a Jordan normalform [5]. Therefore, for a band theory a number of energybands coalesce at an exceptional point in the Brillouinzone (BZ), where accordingly energy-band conservationis violated. On the other hand, the recently enhancedinterest in non-Hermitian physics partially evolved fromtopological phases of quantum matter, where topologi-cal semimetals as a central topic feature nodal manifoldsin the BZ including degenerate nodal lines [38–50] andsurfaces [51–56]. Due to band conservation of Hermi-tian theory, such nodal manifolds are always closed andaccordingly have no boundary. Now considering nodalmanifolds in non-Hermitian systems, one may expect anexotic quantum state solely for non-Hermitian system,namely, that nodal manifold can terminate on a bound-ary consisting of exceptional points [57–60], and indeedrecently the bulk Fermi arc, which is an open nodal ended with two exceptional points [21, 22], have been realized innon-Hermitian photonic crystals with much attention at-tracted [61]. In this article we show that a variety of opennodal manifolds with exceptional boundaries, includingvarious Fermi arcs and particularly drumheads, namelyopen surfaces, can be realized in the bulk band structuresof 2D and 3D honeycomb lattices through natural andphysically feasible non-Hermitian processes. Our mod-els are quite simple with only nearest-neighbor hoppingsincluded, and may be understood as non-Hermitian the-ories of graphene and graphite.It is also interesting to compare the bulk nodal Fermiarcs and drumhead states with the boundary Fermi arcsand drumhead states of Weyl and nodal-line semimet-als, respectively. Although for both cases they are openmanifolds, Hermitian systems preserve band number andtherefore the open manifolds of boundary band structurenecessarily sink into and connect with the bulk energybands. But maybe more profoundly the boundary of aHermitian system is not an independent system, and inthis sense it might bear certain connections with non-Hermitian physics that is essentially devoted to open sys-tems.Recently it is noticed that the physical property ofnon-Hermitian systems can be radically dependent onboundary conditions [9, 62–67]. For instance, the spec-trum of the Su-Schrieffer-Heeger (SSH) model with smallanti-Hermitian nearest-neighbor hoppings is complex un-der periodic boundary conditions, but is purely real un-der open boundary conditions. As the representation bythe BZ actually corresponds to periodic boundary condi- a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t tions, we proceed to study the bulk-boundary correspon-dence of our non-Hermitian honeycomb-lattice modelswith open boundary conditions, and find that the non-Hermitian tight-binding models are equivalent to Hermi-tian ones by similarity transformations. This shows thatthe exotic quantum states of nodal manifolds bounded byexceptional points can be undermined by open bound-ary conditions, and periodic boundary conditions aretherefore essential for their existence. To circumvent thedilemma that periodic boundary conditions are not real-istic for ordinary physical systems, such as real materials,photonic crystals, phononic crystals and cold atoms, andalso inspired by that novel band theory has broader ap-plications beyond electronic systems, we present a simu-lation of the non-Hermitian tight-binding models on hon-eycomb lattices through faithfully designating electricalcircuits, for which periodic boundary conditions are ob-viously realizable [68–70]. Particularly non-Hermitiandevices, emulating non-Hermitian terms, can be easilyimplemented via a standard application of a commonoperational amplifier in a voltage follower configuration.Furthermore, the feasibility of the particular design is en-sured by the fact that each unit cell only consists of a fewcapacitors, inductors and operational amplifiers.The article is organized as follows. Section II briefs the2D and 3D non-Hermitian lattice structures, which areused in the following discussions. In Sec. III we show theexceptional points and bulk Fermi-arc states terminatedat exceptional end-points in the 2D structure. In Sec. IVwe investigate the exceptional lines and bulk drumheadsstates with exceptional edges in the 3D structure. InSec. V we compare the band dispersions between periodicboundary conditions and open boundary conditions, andshow that the periodic boundary conditions are essentialfor the above bulk states. Finally Sec. VI presents thedesigned non-Hermitian electrical-circuit lattices, whichare easy to achieve periodic boundary conditions in 2Dand 3D cases, and can realize bulk quantum states ofnodal manifolds bounded by exceptional points. II. 2D AND 3D NON-HERMITIANHONEYCOMB LATTICES
Honeycomb lattice plays an important role in con-structing models of novel topological quantum states.For instance, electrons on 2D honeycomb lattices mayhave the Dirac-type energy dispersions, which havearoused tremendous research interests for topologicalphases. From the topological point of view, the mass-less Dirac point usually corresponds to criticality of phasetransition between two topologically distinct phases. No-tably both the quantum spin Hall states [71] and thequantum anomalous Hall state [72] were first proposedin the 2D honeycomb lattice as pioneering models oftopological insulators, which is in retrospect based on the Dirac criticality. It is also a good starting point tolook for nodal-line and Weyl semimetal semimetal phaseson 3D honeycomb lattices formed by stacking 2D honey-comb lattices along the vertical dimension [73]. As afore-mentioned honeycomb-lattice models are all Hermitian,the dissipative (gain/loss) and nonreciprocal effects arenot taken into consideration. In this work, we demon-strate that, in the non-Hermitian regime, honeycomb lat-tices are a cornerstone as well for seeking novel quantumstates, which essentially depend on non-Hermiticity. Asshown in Fig. 1, both 2D and 3D honeycomb latticesconsist of sublattices A and B, and the unit cell is indi-cated in the pink-dashed box. We assume the hoppingprocesses within each unit cell are asymmetric for sub-lattices A and B, resulting in the non-Hermitian terms,while the hoppings between unit cells, which are are sym-metric, lead to the corresponding Hermitian terms.
Figure 1. (a) 2D and (b) 3D honeycomb lattice. The dashedpink box indicates the unit cell. The Hopping parameters t g ± γ y inside a unit cell are asymmetric, leading to the non-Hermitian term. The interactions between unit cells on the2D plane are set as t . The inter-layer couplings are set as t A , t B and t between A-A, B-B and A-B sites, respectively. III. EXCEPTIONAL POINTS AND BULKFERMI-ARC STATES IN 2D NON-HERMITIANHONEYCOMB LATTICE
We begin with the 2D case, for which the tight-bindingHamiltonian is written as H ( k ) = d x ( k ) σ x + ( d y ( k ) + iγ y ) σ y , (1)where d x = t g + t (cos k · a + cos k · a ), d y = t (sin k · a +sin k · a ), a , = a (cid:0) / , ±√ / (cid:1) are the lattice vectorsand we set the atom-atom distance a = 1 hereafter. t g , t and γ y are hopping parameters as indicated in Fig. 1(a). σ x,y,z are the Pauli matrices while the term involving σ z vanishes under the assumption of chiral symmetry of thesystem. The energy dispersions are then calculated as E ± ( k ) = ± (cid:113) d x ( k ) + d y ( k ) − γ y + 2 id y ( k ) γ y , (2)which is generally complex for nonzero γ y . The excep-tional point appears if two bands coalesce, leading to d y ( k ) = 0 and d x ( k ) = ± γ y , (3)and can be combined into a single complex equation d x + id y = t g + t ( e i k · a + e i k · a ) = ± γ y . (4)By tuning the parameters, we get different numbers ofsolutions for Eq. (4), i.e., different number of exceptionalpoints in the BZ: 1) max[ | ( t g ± γ y ) /t | ] <
2, there arefour exceptional points in the first BZ (Fig. 2(a1)). 2)min[ | ( t g ± γ y ) /t | ] < < max[( t g ± γ y ) /t ], two exceptionalpoints appear (Fig. 2(b1)). 3) No exceptional point ex-ists when min[ | ( t g ± γ y ) /t | ] >
2. The band dispersionsthrough the exceptional points are shown in Fig. 2(a2,b2).
Figure 2. The solutions of Eq. (4) with (a) 4 and (b) 2exceptional points (red dots) in the first Brillouin zone. Theparameters are setting as t = 1 , t g = 1 .
5, and (a) γ y = 0 . γ y = 0 .
6. The real and imaginary part of the band dispersionsalong the line through the exceptional points are showing in(a2) and (b2).
Quite different from the Hermitian systems, a pairof exceptional points in the complex spectrum of non-Hermitian Hamiltonian will lead to an open-ended bulkstates, i.e., the so-called bulk Fermi-arc [21, 22, 61]. Asschematically plotted in Fig. 3(a), the bulk Fermi-arc,degenerate with real part of the eigenvalues while non-degenerate with the imaginary part, links a pair of ex-ceptional points. The number of bulk Fermi-arcs andexceptional points can be tuned by the parameter γ y asdiscussed above. Below, we demonstrate the existence ofbulk Fermi-arc in our model of Eq. (1). From the dis-persion expression Eq. (2), it is easy to find that if thefollowing expressionsRe (cid:0) E ± (cid:1) < (cid:0) E ± (cid:1) = 0 (5) are satisfied, the real parts of the dispersions are degener-ate while the imaginary parts are non-degenerate, whichare the solutions for the bulk Fermi-arc. SubstitutingEq. (2) into Eq. (5), we obtain d y ( k ) = 0 and d x ( k ) < γ y . (6)Comparing Eq. (3) with Eq. (6), it is obvious that theexceptional points are the boundaries of bulk Fermi-arcs. Solving Eq. (6), one obtains the explicit ranges ofbulk Fermi-arc in the BZ, which read 1) k x = 4 nπ/ | t g + 2 t cos( √ k y / | < | γ y | ; 2) k x = (4 n + 2) π/ | t g − t cos( √ k y / | < | γ y | ; and 3) k y = (2 n + 1) π/ √ | t g | < | γ y | . The configurations of the bulk Fermi arcs for γ y = 0 . . Figure 3. (a) The real part of the spectrum, which aredegenerate along a line, form the so-called bulk Fermi-arc. (b)and (c) are the configurations of the bulk Fermi-arc with theparameters the same as Fig. 2(a) and (b), respectively. Eachbulk Fermi-arc (green lines) is ended with two exceptionalpoints (red points).
IV. EXCEPTIONAL LINES, BULK DRUMHEADSTATES IN 3D NON-HERMITIAN HONEYCOMBLATTICES
Inspired by the existence of bulk Fermi-arc terminatedat the exceptional points in 2D BZ, for the 3D hon-eycomb lattice, due to the increasing of spatial dimen-sionality, we expect to obtain lines of exceptional pointsand drumhead-like bulk states bounded by this excep-tional lines. The tight-binding Hamiltonian for the lat-tice model given in Fig. 1(b) is H ( k ) = d x ( k ) σ x + ( d y ( k ) + iγ y ) σ y + d z ( k ) σ z , (7)where d x ( k ) = t g + t (cid:80) j =1 cos k · a j , d y ( k ) = t (cid:80) j =1 sin k · a j , d z ( k ) = t AB cos k · a + µ AB , t AB ≡ t A − t B , and µ AB ≡ ( µ A − µ B ) / A and B .In the following, we investigate this model for two cases.We first consider a simplified case, where d z = 0, i.e.,setting t AB = µ AB = 0. Then the exceptional points arethe solutions of the following equations d x ( k ) + id y ( k ) = t g + h ( k ) = ± γ y (8)where h ( k ) ≡ t (cid:80) j =1 e i k · a j . If γ y = 0, the solutions ofEq. (8) form a nodal-ring in momentum space (shown inFig. 4(a)) for appropriate values of t g and t as discussedin [73]. For a nonzero γ y , there are three types of solu-tions. 1) max[ | ( t g ± γ y ) /t | ] <
3, the solutions of Eq. (8)form two exceptional rings in the 3D BZ, as shown inFig. 4(b). 2) min[ | ( t g ± γ y ) /t | ] < < max[ | ( t g ± γ y ) /t | ],a single exceptional ring exists, as shown in Fig. 4(c).3) min[ | ( t g ± γ y ) /t | ] >
3, no exceptional point solutionsexist for Eq. (8).Substituting Eq. (7) into Eq. (5), we obtain( t g + h ( k )) < γ y . (9)The solutions of Eq. (9) determine the range of the de-sired drumhead states. Comparing Eq. (8) with Eq. (9),we obtain that the exceptional rings are the boundaryof the bulk drumhead states. The configuration of bulkdrumhead states is dependent on the parameters as dis-cussed below Eq. (8). By numerically solving Eq. (9), wefind two types of bulk drumhead states. The first typeis a drumhead with a hole, bounded by two exceptionallines (Fig. 5(a)). The second type is a whole drumheadbounded by one exceptional line (Fig. 5(b)). These bulkdrumhead states are essentially different from the drum-head surface states in the Hermitian nodal-line semimet-als. For the latter, the degenerate points form nodal-ringsin the 3D bulk BZ, and due to the bulk-boundary cor-respondence, lead to the drumhead boundary states onthe 2D surface BZ, whose edges eventually sink into andconnect with the bulk nodal-line states. While for the3D non-Hermitian system, the drumhead states are bulkstates bounded by the exceptional lines, with eigenvaluedegenerate for the real part but splitted for the imaginarypart.Now we discuss the more general case with d z (cid:54) = 0in Eq. (7). For Hermitian system with γ y = 0, Weylpoints can be realized in this 3D honeycomb lattice if d x ( k ) = d y ( k ) = d z ( k ) = 0 are satisfied [73]. Fornon-Hermitian system with γ y (cid:54) = 0, the configurationof drumhead states with exceptional edges is enrichedcompared to the Hermitian case and the non-Hermitiancase of d z = 0. In parallel to the discussions in previoussections, the exceptional points and the bulk drumheadstates are determined by the following equation and in- equation respectively. d y ( k ) = 0 , d x ( k ) + d z ( k ) = γ y , (10) d y ( k ) = 0 , d x ( k ) + d z ( k ) < γ y . (11)Setting µ AB = 0 . γ y = 0 . t AB = 1, we obtaintwo drumhead states bounded by two exceptional ringsas shown in Fig. 6. V. THE BULK-BOUNDARYCORRESPONDENCE
In the previous two sections, we discussed the bandstructures of the bulk states, where the periodic bound-ary conditions were actually implicitly presumed forthe Fourier transforms can be applied to produce theBZ. For non-Hermitian system, the bulk energy spectra
Figure 4. Fixing t g = 2 . , t = 1 and tuning the value ofparameter γ y , one obtain (a1) Ring shape bands degeneratepoints for γ y = 0. In this case the system is Hermitian, theenergy of bands are real as show in (b2). (b1) The exceptionalpoints form two rings in the BZ for γ y = 0 . max [ | ( t g ± γ y ) /t | ] <
3. The complex bands is square-rootnearby the exceptional points as shown in (b2). (c1) and(c2) For γ y = 0 .
6, which satisfy min [ | ( t g ± γ y ) /t | ] < 6. The edge states (red lines) ap-pear and connect a pair of nodal points for the non-Hermitian2D honeycomb lattice with zigzag edges. ing the band-crossing points (blue stars) now present.The above results indicate that the proposed excep-tional points and the bulk Fermi-arc states can only existin a system with periodic boundary conditions. This re-quirement clearly brings difficulty to realize these statesexperimentally, for the periodic boundary conditions arenot easy, if not impossible, to implement in commonlyused experimental systems, such as real materials, pho-tonic crystals, phononic crystals, and cold atoms.To solve this problem, we propose to simulate thesestates in electrical-circuit lattices, for which the periodicboundary conditions are quite easy to be implemented ifwe connect the head with the tail, showing a significantadvantage compared with other realization scenarios. Inthe following section, we detail how to design the non-Hermitian honeycomb lattice to realize the novel statesdiscussed above. VI. NON-HERMITIAN ELECTRICAL CIRCUITLATTICE Recently, there has been growing interest in realizingtopological phases by electrical circuits, including thetime-reversal-invariant topological insulators [68, 69, 74],3D Weyl semimetals [70, 73, 75], 1D topological insu-lators [76] and the higher-order topological insulators[77–79]. In this section, we construct a 2D electrical-circuit lattice, consisting of capacitors, inductors and op- Figure 8. (a) The elementary circuit cell that gives the non-Hermitian effect. The operational amplifier’s inputs consistof a non-inverting input (+) with voltage V + and an invertinginput ( − ) with voltage V − . The output voltage of the opera-tional amplifier is denoted as V out . Connecting the invertinginput ( − ) and the output, the operational amplifier is used asvoltage follower, which gives V + = V out but no current flowsinto the non-inverting input (+). C and C are capacitors.The nodes A and B are connected to ground through parallelconnected capacitor and inductor C G , C , L . erational amplifiers, as an experimental setup to realizethe bulk Fermi-arc states bounded by exceptional pointsdiscussed in Sec. III. The 3D electrical-circuit lattice forthe bulk drumhead states with exceptional edge statesdiscussed in Sec. IV can be designated in a similar way.We first elaborate how to construct the elementary cir-cuit cell, which corresponds to the non-Hermitian term,as shown in Fig. 8. The key idea is to utilize operationalamplifiers, which are standard components in electricalcircuits, to emulate gain and loss, the characteristics ofnon-hermiticity. Hence, let us begin with some basics ofoperational amplifier. The differential inputs of the op-erational amplifier are characterized by a non-invertinginput (+) with voltage V + and an inverting input ( − )with voltage V − . Ideally the operational amplifier am-plifies the difference in voltage between the two inputs.The output voltage of the operational amplifier V out isgiven by the equation V out = A ( V + − V − ), where A is theopen-loop gain of the amplifier that is very high for anideal amplifier. Connecting the inverting input ( − ) andthe output, leading to V − = V out , the amplifier is usedas a voltage follower, for that V out = A ( V + − V out ) ⇒ V out = AA +1 V + ≈ V + . For an ideal operational amplifier,there is no voltage across its inputs. Therefore the inputterminals V + and V − behave like a short circuit. Butthis kind of short is virtual, different from a real one,and draws no current because of the infinite impedancebetween the two inputs. With these properties and ac-cording to Kirchhoff’s current law, we get the following equations I A = jω ( C + C )( v B − v A )+ jωC G (0 − v A )+ 1 jωL (0 − v A ) , (12) I B = jωC ( v A − v B )+ jω ( C + C G )(0 − v B )+ 1 jωL (0 − v B ) , (13)where ω is the frequency of voltage and j ≡ √− 1. Con-sidering the current conservation, namely, that the sum-mation of the inflow and outflow currents at every nodeis zero, these equations can be simplified, and then recastinto the matrix form, (cid:20) ( C + C + C G ) − C − C − C ( C + C + C G ) (cid:21) (cid:20) v A v B (cid:21) = 1 ω L (cid:20) v A v B (cid:21) . (14)The two-by-two matrix on the right hand of Eq. (14) isclearly non-Hermitian because it is real but not symmet-ric. Hence, a non-Hermitian device has been constructedby using conventional operational amplifiers, and repeat-ing this elementary non-Hermitian cell, we can build the2D non-Hermitian honeycomb lattices. Consequently,the desired electrical-circuit lattice can be constructedas illustrated in Fig. 9, which is made of the elementarynon-Hermitian cells and capacitors C . Figure 9. A 2D circuit lattice consist of operational amplifiersand capacitors. The dashed pink box indicates the elementarynon-Hermitian unit cells. The capacitors C connect the unitcell, forming a honeycomb-type lattice. Now periodic boundary conditions can be readily im-posed on the 2D electric-circuit lattice by accordinglyconnecting components on the left (upper) edge to thoseon the right (lower) edge. And the Fourier transformscan be performed, so that the Kirchhoff equations canbe expressed into an eigenvalue-like equation for the sta-tionary systems H ( k ) V = 1 ω L V, (15)where H ( k ) = C s σ + d x ( k ) σ x + ( d y ( k ) + iγ y ) σ y , (16)and d x ( k ) = − ( C + C − C (cos k · a + cos k · a ) ,d y ( k ) = − C (sin k · a + sin k · a ) ,γ y = − C . (17)Here, C s = C + 2 C + C + C G , and V = [ v A ( k ) , v B ( k )] T is the Bloch-like states for the potential distributions onthe A and B nodes. a , are the basis vectors of the2D lattice as shown in Fig. (9). The details of deriva-tion of Eqs. (15-17) are given in Appendix C. Compar-ing Eq. (17) with Eq. (1), we find that the parame-ters in these two equations can be related as t = − C , t g = − ( C + C / 2) and γ y = − C / 2. Therefore, by tun-ing capacitors C , , , one can realize the nodal points andbulk Fermi arc states in the 2D non-Hermitian electrical-circuit honeycomb lattice. The 3D non-Hermitian hon-eycomb lattice to simulate the nodal drumhead with ex-ceptional edges can be constructed by the same methodas well. VII. CONCLUSION In this work we investigated possible exotic non-hermitian quantum states on 2D and 3D honeycomb lat-tices models with only nearest-neighbor hoppings beingconsidered. More specifically, the bulk Fermi-arc statesconnecting the exceptional points, and the bulk drum-head states bounded by the exceptional lines were foundin 2D and 3D cases, respectively. By investigating thebulk-boundary correspondence of these models with openboundary conditions, we observed that the above exoticstates are undermined, indicating the periodic bound-ary conditions are essential for the existence of these ex-ceptional points and open nodal manifolds. Since peri-odic conditions are actually unrealistic for conventionalsystems, such as real materials, photonic crystals andcold atoms in optical lattices, we therefore proposed theelectrical-circuit simulations, which have the advantageof easily achieving periodic boundary conditions, to real-ize the exotic states. Moreover, the constructed electricalcircuits in principle can be easily fabricated experimen-tally, since all components and their usage are conven-tional. ACKNOWLEDGMENTS This work was supported by the National KeyResearch and Development Program of China (No.2017YFA0304700, No.2017YFA0303402), the NationalNatural Science Foundation of China (No. 11674077, No.11874048). The numerical calculations in this work havebeen done on the supercomputing system in the Super-computing Center of Wuhan University. Appendix A: Edge states and skin effect in a strip of2D non-Hermitian honeycomb lattice Considering the strip of 2D honeycomb lattice withzigzag edge in the x -direction, the Hamiltonian can bewritten as H = N (cid:88) j =1 (cid:16) tc † jA,k y c jB,k y e − ik y a y + h.c. (cid:17) + N − (cid:88) j =1 (cid:0) ( t g − γ y ) c † ( j +1) B,k y c jA,k y + ( t g + γ y ) c † jA,k y c ( j +1) B,k y (cid:1) , (A1)where N is the number of unit cell in the x -direction.The band dispersions for Hamiltonian (A1) are shownin Fig. 7. It clear that the band-crossing points (bluestars) are not correspond to the exceptional points (reddots), and the number of gap-closing point can be notequal to the number of exceptional points. The bulkFermi-arc states connecting exceptional points E - E and E - E (red dots) disappear. But edge states ariseand connect a pair of the new gap closing points (bluestars), instead of connecting the projection of the ex-ceptional points. These anomalies can be resolved byusing the auxiliary Hamiltonian proposed in reference[67]. Taking a similarity transformation to H , we ob-tain ˜ H = P − HP , where P is a 2 N × N diagonalmatrix P = diag[1 , , α, α, · · · , α N − , α N − ] and α = (cid:112) | ( t g − γ y ) / ( t g + γ y ) | . The transformed Hamiltonian ˜ H has explicit form as˜ H = N (cid:88) j =1 tc † jA,k y c jB,k y e − ik y a y + N − (cid:88) j =1 ˜ t g c † ( j +1) B,k y c jA,k y + h.c.. (A2)After taking Fourier transform in the x direction, oneobtains ˜ H ( k ) = ˜ d x ( k ) σ x + ˜ d y ( k ) σ y , (A3)where ˜ d x = ˜ t g + t (cos k · a + cos k · a ), ˜ d y = t (sin k · a + sin k · a ), and ˜ t g = (cid:113) t g − γ y . Hence, the Eq. (A2)and (A3) is Hermitian if | t g | ≥ | g y | is satisfied. The gapclosing points calculated from ˜ H ( k ) are consistent withthe gap closing points of the strip structure as shownin Fig. 7. One can calculate the Berry phase φ ( k y ) for˜ H ( k x , k y ) with k y fixed. φ ( k y ) = π reveals that the edgestates exist on the boundary, while φ ( k y ) = 0 indicatesno edge states existing. Therefore, the bulk-boundarycorrespondence is recovered by using ˜ H ( k ).Now, we show the skin effect for the non-Hermitian2D honeycomb lattice with open-boundary conditions.Considering the similarity transformation ˜ H = P − HP .If ˜ H | ψ (cid:105) = λ | ψ (cid:105) , then P − HP | ψ (cid:105) = λ | ψ (cid:105) ⇒ HP | ψ (cid:105) = λP | ψ (cid:105) . Thus, if | ψ (cid:105) is an eigenvector of ˜ H with eigen-value λ , then P | ψ (cid:105) is an eigenvector of H with thesame eigenvalues. With periodical boundary condi-tions, all states in both Hermitian and non-Hermitian2D honeycomb lattice are Bloch waves, ensured bythe translational symmetry of the lattice. With openboundary conditions, the bulk states of ˜ H , ψ = (cid:0) ψ B,k y , ψ A,k y , · · · , ψ NB,k y , ψ NA,k y (cid:1) T , are still nearlyBloch-type, when the number of layers is large enough.However, this is not the same as in non-Hermitian cases.We can find that the wave function P ψ of H becomes P ψ nA,B = α n − ψ nA,B . As | α | (cid:54) = 1. Therefore we getthat P ψ is localized at one side of the effective 1D sys-tem (as shown in Fig. 10), dubbed as “non-Hermitianskin effect” [62, 67]. Figure 10. (a) The two edge states(red lines) is localized at theleft side as expected and degenerate for the chiral symmetrywith γ y = 0 . 3, while the bulk states (gray lines in the inset)are also localized on the boundary. (b) With γ y = 0 . 6, all thebulk and edge states are localized more heavily. Below we give a more intuitive way to understand theskin effect. For the non-Hermitian originated from asym-metric hopping terms t g ± γ y , the particles have largerhopping probability in a specific direction. Although, thewave functions are Bloch type in the periodical bound-ary conditions, the particles accumulate to one side ofthe system in the open boundary conditions. The stateslargely deviate from the bulk Bloch type, therefore, thebreakdown of the correspondence between bulk excep-tional points with periodical boundary conditions andthe edge states with open boundary condition is not sur-prising. Figure 11. For the d z = 0 case with t g = 2 . t = 1,there are (a) two exceptional rings for γ y = 0 . γ y = 0 . k y . The gap-closing points are located at T and T (pink points indicated in (a) and (c), not at the exceptionalpoints. Appendix B: Surface states for slab structure of 3Dnon-Hermitian honeycomb lattice We consider slab geometry terminated in the x direc-tion of the 3D non-Hermitian honeycomb lattice. Theband structures of the slab are calculated as shown inFig. 11. The bulk exceptional lines (red color curves) areprojected to the surface BZ as shown in Fig. 11(a, c).The gap closing points for the slab band structures arelocated at T and T points instead of at the exceptionalpoints. The bulk drumhead states are damaged, with nocorresponding states on the surface. While new surfacestates (red color bands), connecting T and T points,emerge as shown in Fig. 11(b, d). Appendix C: Details of the derivation ofEqs. (15-17) in the main text The currents, which flow into nodes A and B in the celllocated at R = 0 of the circuit lattice (shown in Fig. 9),are given as I A (0) = jω ( C + C )[ v B (0) − v A (0)]+ jωC [ v B ( − a ) − v A (0)] + jωC [ v B ( − a ) − v A (0)]+ jωC G [0 − v A (0)] + 1 jωL [0 − v A (0)] , (C1) I B (0) = jωC [ v A (0) − v B (0)]+ jωC [ v A ( a ) − v B (0)] + jωC [ v A ( a ) − v B (0)]+ jω ( C + C G )[0 − v B (0)] + 1 jωL [0 − v B (0)] , (C2)where ω is the frequency for the sinusoidal signal, j ≡√− 1, the vectors a , a in the parentheses correspond-ing to lattice sites. The relations for the nodes currents I A,B ( R ) and the potential distributions v A,B ( R ) on thewhole lattice can be obtained with the same method.Kirchhoff’s law demands that I A ( R ) and I B ( R ) arezero. Therefore writing above equations into a matrixform, we get a tight-binding-like Hamiltonian equation . . . ... − C − C C s − ( C + C ) − C C s − C − C ... . . . ... v B ( − a ) v B ( − a ) v A (0) v B (0) v A ( a ) v A ( a )... = 1 ω L ... v B ( − a ) v B ( − a ) v A (0) v B (0) v A ( a ) v A ( a )... . (C3)The hopping terms can be extracted from the left matrixas listed below: H AA ( R = 0) = C s ≡ C +2 C + C + C G , H AB ( R = 0) = − ( C + C ), H AB ( R = − a ) = − C , H AB ( R = − a ) = − C , H BB ( R = 0) = C S , H BA ( R =0) = − C , H BA ( R = a ) − C , and H BA ( R = a ) = − C , where R is the lattice vector and H nm ( R ) are thetight-binding parameters between node n located at thehome unit cell and node m located at R . 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