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[Huyen Thanh Phan](https://orcid.org/0000-0002-1031-6127), [Katsunori Wakabayashi](https://orcid.org/0000-0002-9147-9939)

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[Non-Hermitian corner skin effect in a two-dimensional photonic crystal](https://mdr.nims.go.jp/datasets/a791a3f0-f735-4f7e-819b-702115a1aef5)

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Non-Hermitian corner skin effect in a two-dimensional photonic crystalHuyen Thanh Phan1, 2, ∗ and Katsunori Wakabayashi1, 3, 4, †1Department of Nanotechnology for Sustainable Energy, School of Science and Technology,Kwansei Gakuin University, Gakuen Uegahara 1, Sanda 669-1330, Japan2Center for Materials Innovation and Technology, VinUniversity, Hanoi 100000, Vietnam3Center for Spintronics Research Network (CSRN), Osaka University, Toyonaka 560-8531, Japan4National Institute for Materials Science (NIMS), Namiki 1-1, Tsukuba 305-0044, Japan(Dated: April 7, 2026)We numerically study topological effects of electromagnetic (EM) waves in a two-dimensional (2D) non-Hermitian photonic crystal (PhC) composed of lossy magneto-optical materials. In this system, not only the EMwavefunctions but also the complex eigenfrequencies exhibit nontrivial topological properties. We demonstratethat the non-Hermitian skin effect, protected by point gaps in the complex eigenfrequency spectrum, emerges atboth the edges and corners of truncated structures. This phenomenon has no counterpart in Hermitian systems.In addition, we identify non-Hermitian topological edge states originating from the nontrivial topology of thebulk bands. While most previous studies of non-Hermitian topology have focused on tight-binding models, ourwork addresses a continuous photonic system, providing a more realistic platform and offering a concrete routetoward experimental realization of non-Hermitian effects.I. INTRODUCTIONTopological material is a class of quantum materials whosephysical properties are characterized by topological invari-ants. Prominent examples include topological insulators [1–6], topological photonic crystals (PhCs) [7–12] and topolog-ical acoustic materials [13–16]. Topology provides a math-ematical framework for classifying lattice systems accordingto whether their wavefunctions can be continuously deformedinto one another. This framework is formalized by topologicalband theory [17], which has played a central role in the studyof topological materials over the past decades. According totopological band theory, topological materials are insulatingin the bulk while supporting conducting states at their bound-aries. These boundary states are protected by the nontrivialtopology of the bulk wavefunctions [18–22]. This theory wasoriginally developed for Hermitian systems. Recently, it hasbeen extended to non-Hermitian systems [23–26], where gainand loss naturally arise due to coupling to the environment.Photonic systems are particularly attractive platforms for re-alizing non-Hermitian physics, as gain and loss can be readilyengineered in PhCs [27–29].There are several differences between topological band the-ory for Hermitian and non-Hermitian photonic systems. InHermitian systems, the emergence of band gap in eigenfre-quency spectrum is very important to classify topologicalproperties [17]. On the other hand, the definition of band gapis generalized to point gap, line gap, etc., in non-Hermitiansystems [23, 25, 26, 28]. Studies of non-Hermitian topologi-cal band theory have revealed the non-Hermitian skin effect,in which a macroscopic number of eigenstates in the finitesystem become localized at its boundaries [30–35]. Thesestates are associated with point gaps in the complex eigen-frequency spectrum and are protected by the nontrivial wind-ing of complex eigenfrequencies in the complex plane. This∗ phanthanhhuyenpth@gmail.com† waka@kwansei.ac.jpphenomenon is unique to non-Hermitian systems and has noHermitian counterpart. On the other hand, similar to Hermi-tian systems, topological properties of bulk wavefunctions innon-Hermitian systems are characterized by complex Berryphase [36–41], which can be quantized or non-quantized de-pending on the symmetries of systems [37, 42, 43].To date, the non-Hermitian skin effect is widely studiedin both tight-binding models and PhCs. However, most ofthem focus on the first-order skin effect, where the local-ization is guaranteed by point gap of complex bulk eigen-frequencies. Studies on the second-order skin effect, wherethe localization emerges at corner of finite structures, are stilllimited. Some studies proposed the second-order skin effectbased on Benalcazar-Bernevig-Hughes model [44, 45], wherepositive/negative hoppings parameters are required for for-ward/backward directions. This model has not yet been re-alized experimentally. Besides, corner skin modes are numer-ically observed in a two-dimensional (2D) PhC with gain andloss under magnetic field [46, 47]. To experimentally realizenon-Hermitian effects, further materials design and theoreti-cal calculations are required.In this paper, we theoretically design a 2D non-Hermitianphotonic crystal made of magneto-optical materials that breakinversion, mirror, and time-reversal symmetries. The first-order non-Hermitian skin effect is numerically observed,which is protected by non-trivial winding of the complex bulkeigenfrequencies. Furthermore, non-Hermitian topologicaledge states are also obtained in the line gap between the firstand the second bands. We numerically calculate the complextopological phases of the bulk wavefunctions, which is consis-tent with the emergence of topological edge states. Complexeigenfrequencies of these topological edge states also formpoint gap in the complex plane. This point gap is responsi-ble for the emergence of the second-order non-Hermitian skineffect at the corner of finite structure. Our results provideessential insights into non-Hermitian topological states andrepresent a step toward the experimental realization of non-Hermitian effects.mailto:phanthanhhuyenpth@gmail.commailto:waka@kwansei.ac.jpII. THEORETICAL BACKGROUNDBefore studying non-Hermitian properties of a 2D PhC sys-tem, in this section, we give a brief review of theoretical back-ground. In general, eigenfrequencies of a non-Hermitian sys-tem are complex numbers. Assuming the lattice constant isa = 1, there are two possible cases of one-dimensional (1D)non-Hermitian band structure [30]. The first case is called re-ciprocal band structure, where ω(k)=ω(−k) as shown in Fig.1(a). This band structure becomes a curved line, which is of-ten called trivial loop in the complex plane as denoted in Fig.1(b). On the other hand, when ω(k) ̸= ω(−k), band structurewill form a closed loop (nontrivial loop) in the complex planeas shown in Figs. 1(c) and (d). The area inside the closed loopis called point gap. To analyze properties of the point gap, thewinding number is involved and defined as [30]w =12π∫π−π∂karg(ω(k)−ω0)dk, (1)where ω denotes the complex eigenfrequency of a given band,k is wave vector, ω0 is any frequency in the complex plane.- -dnab lacorpicernondnab lacorpicerpool laivirtnonpool laivirtbabacdcddcdcbaba(a)(b)(c)(d)FIG. 1. (a) Schematic of a 1D reciprocal complex band structure,where ω(k) = ω(−k). (b) The complex eigenfrequencies in (a) forma curved line (trivial loop) in complex plane. (c) Schematic of a 1Dnonreciprocal complex band structure, where ω(k) ̸= ω(−k). (d)The complex eigenfrequencies in (c) show a closed loop (nontrivialloop) in complex plane.In Eq. (1), the winding number w is the number of timesthat complex eigenfrequencies wind around point ω0 when kis varied in the first Brillouin zone (BZ). For any point in-side the point gap, the winding number is nonzero. Accordingto the above analysis, the necessary condition for obtaining anontrivial point gap in 1D system is ω(k) ̸= ω(−k). This cor-responds to the broken inversion symmetry in real space andbroken reciprocity [30]. For 2D systems, where the angularfrequencies are defined by ω(kx,ky), the integration in Eq. (1)is taken along one direction in k-space, the winding numberwill depend on the k-points in the other direction. For exam-ple, the ky-dependent winding number is defined as followw(ky) =12π∫π−π∂kx arg(ω(kx,ky)−ω0)dkx. (2)In this case, to achieve non-trivial loop, the necessary con-dition is ω(kx,ky) ̸= ω(−kx,ky). This means that the mirrorsymmetry with respect to kx = 0 axis and/or the reciprocity ofsystem are broken [30].It is mentioned in several studies that the non-zero windingnumber of the bulk band structures are responsible for highconcentration of EM waves at the edges of truncated struc-tures [23, 32, 33, 35]. In particular, if complex eigenfrequen-cies of infinite periodic systems form point gap in the com-plex plane, those of their truncated structures, which have atleast one open boundary, will be within the point gap. Theirfield distribution are concentrated at the open boundaries anddecay to the bulk region. These phenomena are called non-Hermitian skin effect, which is unique in non-Hermitian sys-tems without Hermitian counterpart. To elucidate the origin ofthis effect, note that when the complex eigenfrequencies ex-hibit point gaps (Fig. 1(d)), a single real eigenfrequency cor-responds to two distinct imaginary eigenfrequencies, whichin turn correspond to two k values with opposite signs. Thismeans that, at the open boundaries, the incoming waves andreflected waves have different decay rates. When the reflectedwaves decay more slowly than the incoming waves, the skineffect emerges because residual energy accumulates at theboundary. On the other hand, when no point gap is present(Fig. 1(c)), each real eigenfrequency is associated with a sin-gle imaginary eigenfrequency. This complex eigenfrequencythen maps to two opposite k values, resulting in identical de-cay rates for the incoming and reflected waves and thus noskin effect is observed.III. THE FIRST-ORDER NON-HERMITIAN SKIN EFFECTAND TOPOLOGICAL EDGE STATESIn this section, based on the theoretical background in sec-tion II, we will design a 2D non-Hermitian PhC, which ex-hibits the non-Hermitian skin effect at both 1D edges and 0Dcorners. Our PhC structure is made of magneto-optical mate-rials, where non-symmetric blocks are arranged periodicallyin air medium. Figure 2(a) is the schematic of a square unitcell with lattice constant is a0. Inversion symmetry and mir-ror symmetry with respect to x and y direction are broken inthis PhC. We assume that the magnetization of materials isparallel to the z direction. Therefore, its relative permittivityis [48, 49]ε =4.2−0.6i −5i 05i 4.2−0.6i 00 0 1 , (3)In Eq. (3), the diagonal terms represent the actual complexrelative permittivity of the materials, while the off-diagonalterms denote the strength of the nonreciprocity. We note herethat, the main results are robust over the broad parameterrange, as long as the point gap is preserved.To numerically determine photonic band structure andtopological properties of this non-Hermitian PhC, we use thefinite-element method from COMSOL Multiphysics and theextended plane-wave expansion method [50]. Throughout thisPMCPMCperiodic boundary conditionperiodic boundary condition20 unit cells(a)(b)(c)(d)Chern number = 1-xy(e)FIG. 2. (a)Schematic of one unit cell of investigated 2D non-Hermitian PhC. (b) k-dependent dispersion of real part of eigenfre-quencies, showing a complete band gap between the first and the sec-ond bands. (c) Schematic of a super cell for ribbon structure, whichis periodic along y direction and finite in x direction. (d) Photonicband structure for the structure in (c). Topological edge states (greencolor) are numerically observed above the first bulk band. (e) Fieldprofile of the topological edge states marked by colored stars in (d).paper, the transverse electric (TE) waves, where electric fieldis parallel to the xy plane and the magnetic field is perpendic-ular to xy plane, will be considered. The dependence of thereal part of eigenfrequencies on wave vectors k is shown inFig. 2(b). There is a band gap between the first and the secondlowest bands. In Fig. 2(c), we show a super cell consisting of20 and 1 unit cell in x and y directions, respectively. In orderto examine ribbon structure, the periodic boundary conditionis applied to the upper and lower boundaries, the perfect mag-netic conductor (PMC) boundary condition is imposed on theleft and right boundaries. Photonic band structure for the rib-bon is shown in Fig. 2(d) as the ky-dependence of the real partof eigenfrequencies. Since the permittivity tensor is asym-metric, system becomes nonreciprocal, leading to the non-equivalence of two boundaries on the left and right sides ofthe ribbon structure. Therefore, two separated edge states col-ored in green are numerically observed above the first bulkband, where one state is the edge states on the left side andthe other one is the edge state on the right side as shown inFig. 2(e). Origin of these two edge states will be studied inthe next section.In Fig. 3, we extract the complex eigenfrequencies of bothinfinite and ribbon structures at several fixed values of kyand plot them in the same complex plane. For the infi-nite structure, which is periodic in both the x and y direc-tions, the eigenfrequencies (black dots) at one fixed ky andkx ∈ [−π/a0,π/a0] form closed loops. On the other hand,the eigenfrequencies of the ribbon structure, subject to a PMCboundary condition on the left and right boundaries and peri-odic boundary conditions along the y direction, are indicatedby red plus signs, which form arcs. It is evident that the eigen-frequency spectrum of the ribbon structure lies predominantlywithin the region enclosed by that of the infinite structure. Asdiscussed in section II, any point located inside the point gapis associated with a nonzero winding number. Therefore, anyeigenfrequencies of semi-infinite structure that lie within theloops will possess a nonzero winding number. These statesexhibit skin effect, where EM waves are highly localized atthe boundaries of the ribbon structures as shown in the bot-tom panels of Fig. 3. The skin effect can be described withinnon-Bloch framework by complexifying the wave vectors; seeAppendix A for details. From the topological point of view,the non-Hermitian skin effect is topologically protected bynonzero winding number. This skin effect may emerge at theleft or right boundaries as shown in the field profile in Fig. 3,which depends on the sign of winding numbers. For exam-ple, at a0ky = −0.5π , the winding number for both the firstand second bands is 1, skin effect emerges at the left bound-ary. Meanwhile, the winding number for both the first andsecond bands at a0ky = 0.5π is −1, skin effect occurs at theright boundary. The winding number of point-gap is gener-ally ky-dependent except only some certain intervals such asky = 0,±π/a0. Their sign flip is reflected by the sign changeof the complex geometrical phase as shown in the next sectionand in Fig. 4(a).In Fig. 3, we present the complex eigenfrequency spec-tra under periodic boundary conditions along the y-direction.Identical spectra are obtained when periodic boundary con-ditions are applied along the x-direction. There are, how-ever, differences in the spatial localization of the skin effect.Specifically, if the skin effect appears at the left (right) bound-aries when the system is truncated along the y-direction, thecorresponding localization shifts to the upper (lower) bound-aries when the system is truncated along the x-direction.We note here that the skin effect in non-Hermitian PhCsis the effect where a macroscopic number of bulk eigenstatesbecome localized and decay exponentially at the boundariesunder open boundary conditions. The eigenfrequencies of theskin modes are within the bulk regions. In addition with theskin modes, there is also other edge states with the eigenfre-quencies in the gap regions as shown in Fig. 2(d). These edgestates may originate from nontrivial topology of the bulk asthe non-Hermitian counterpart of the Hermitian topologicaledge states. To confirm this statement, we will numericallycalculate topological phase of the first bulk band.IV. NON-HERMITIAN GEOMETRICAL PHASETopological phase of 2D systems is characterized by theZak phase [18] or the Wilson loop [51], which is the geomet-rical phase acquired by eigenstate when it is transported alonga path of BZ. In the Hermitian band theory, the Berry connec-tion is defined asAnk = i⟨unk|∂k|unk⟩, (4)where unk is the periodic part of the Bloch wave functionsHn(r) = unk(r)eik·r, n is band index, k = (kx,ky). In Eq. (4),the bra ⟨unk| and the ket |unk⟩ are the complex conjugatex: periodic,  y: periodicx: PMC,         y: periodicx: periodic,  y: periodicx: PMC,         y: periodicx: periodic,  y: periodicx: PMC,         y: periodic0 Max-MaxFIG. 3. Photonic band structure for the infinite structure (black) and ribbon structure (red) at a0ky =−0.5π (left panel), a0ky = 0 (middle panel)and a0ky = 0.5π (right panel). The green arrows denote the winding direction (clockwise or anti-clockwise) when kx is varied from −π/a0 toπ/a0. Number 1 and 2 indicate band indices.The lower panels are field profile of the states labeled by green stars in the corresponding bandstructures. The states enclosed by the point gap exhibit skin effect, where EM waves are highly localized at the boundaries of structures.of each other. Also, they are the left and right eigenvectorsof the eigenvalue equation, respectively. If systems are non-Hermitian, the Eq. (4) becomes ambiguous since the left andright eigenvectors of the eigenvalue equation are distinct [28].In non-Hermitian systems, the Berry connection is definedas [28, 52]Aαβnk = i⟨uαnk|∂k|uβnk⟩, (5)where α,β ∈ L,R denote left (L) or right (R) eigenvectors.Since ⟨uαnk| and |uαnk⟩ are not the complex conjugate of eachother in general, the Berry connection becomes complex.Therefore, the total Berry phase (geometrical phase) is alsocomplex, which is calculated by integrating the Berry connec-tion over a closed path (L) of the first BZ.γαβ =∮LAαβnk dL, (6)In our study, we create a ribbon structure, which is periodicin y direction and finite in x direction. The isolated edge statesin the band gap are paralleled to the y direction. Therefore,we will calculate the Berry phase in y direction (often-calledthe Zak phase) as followγLRn (ky) =∫π/a0−π/a0i⟨uLnk|∂kx |uRnk⟩dkx, (7)Applying the numerical calculation method and the gaugesmoothing process described briefly in Appendix B and de-tailedly in ref. [42], we obtained the complex geometricalphase in y direction for the first lowest band as shown inFig. 4(a). The red and blue dot lines indicate the real andimaginary part of geometrical phase, respectively. It is the factthat the first point (a0ky =−π) and the last point (a0ky = π) ofthe first BZ are equivalent, the value π and −π of the geomet-rical phase are also equivalent. Therefore, the coordinate sys-tem in Fig. 4(a) is equivalent to a torus as shown in Fig. 4(b)realimaggeometrical phasephase(a) (b)(c)FIG. 4. (a) Complex geometrical phase in y direction of the first bandof investigated 2D non-Hermitian PhC. Red dots indicate real part ofgeometrical phase, blue dots denote imaginary part of geometricalphase. (b) The coordinate system in (a) is equivalent to a torus underperiodic boundary condition. (c) The real and imaginary part of thecomplex geometrical phase are represented by red and blue lines onthe torus.under periodic boundary condition for the Bloch states. Wedefine the winding of geometrical phase aswn =12π∮ky∂kyγLRn (ky)dky, (8)By using Eq. (8), we obtained the winding number for thereal and imaginary part of complex geometrical phase is 1and 0, respectively. We describe the winding properties of thecomplex geometrical phase on the torus as shown in Fig. 4(c).While the real part (red line) winds once along the torus,the imaginary part (blue line) changes smoothly without anywinding. Since the Chern number is consistent with this wind-ing number [53, 54], the Chern number for the first lowestband of our investigated PhC is 1. This is responsible for theemergence of edge states above the first bulk band shown inFig. 2(d).V. CORNER SKIN EFFECTIn this section, we numerically study the second-order skineffect. A finite structure which consist of 20× 20 unit cellswith PMC boundary condition in both x and y directions areinvestigated as shown in Fig. 5(a). For the ribbon structurein Fig. 2(c), its photonic dispersion are indicated by the blackdots in Fig. 5(b). In the inset, we enlarge the region that isclosed to the first band. There is a crossing point of the topo-logical edge states in this region. It is easily seen that thetopological edge states form point gap in the complex plane.The photonic dispersion of finite structure in Fig. 5(a) isplotted as red plus sign in Fig. 5(b). The states within thepoint gap exhibit corner modes, where EM waves are highlylocalized at the corners and decay exponentially. It can also beunderstood that all of the topological edge states of the semi-infinite structure are collapsed to corner skin modes when thestructure is truncated in both directions. It is evident that thetop (bottom) and left (right) boundaries share the same geo-metrical structure. Although PMC boundary conditions areimposed on all boundaries of the finite structure, the point gaparises from topological edge states associated with two dis-tinct boundary geometries. Therefore, corner modes are onlyobserved at the top right and bottom left corners as shownin the first four panels of Fig. 5(c). At the frequency ofthe crossing point, the extended topological edge states hap-pen as shown in the last panel of Fig. 5(c). This exceptionarises because at the crossing point, one state correspondsto the positive-k mode localized at the right (bottom) bound-ary, while the other corresponds to the negative-k mode local-ized at the top (left) boundary. Consequently, at the corner,the incoming and reflected waves have identical decay rates,thereby canceling the skin effect.To distinguish from the topological corner states in Hermi-tian systems, these corner states are called corner skin states,which are protected by the point gap of the investigated pho-tonic system. This means that whenever the point gap emergesamong topological edge states, the corner skin effect happens.While the number of topological corner states is limited anddepends on the number of corners in each specific structure,the number of corner skin modes is much larger than that oftopological corner state and depends on the system size. Weconclude that the emergence of point gap in the edge-statespectrum is the condition for the corner skin effect, as doubletruncation causes these edge states to collapse and accumulateat the corner.VI. SUMMARY AND CONCLUSIONSIn summary, we have theoretically designed a 2D non-Hermitian photonic crystal and numerically investigated theskin effect and topological states of electromagnetic waves.Owing to the absence of inversion and mirror symmetries aswell as broken reciprocity, complex eigenfrequencies exhibitpoint gaps in the complex plane, which protect the emergenceof the non-Hermitian skin effect at the edges of truncatedstructures.Furthermore, we demonstrated that topological edge statesarising from a nonzero Chern number also form point gapsin their complex eigenfrequency spectra. These point gapsgive rise to the second-order non-Hermitian skin effect, whereelectromagnetic waves become localized at the corners of fi-nite structures. Unlike conventional topological edge or cor-ner states, the skin modes appear over broad frequency rangeand scale with system size. Our results provide a realisticphotonic platform for exploring higher-order non-Hermitiantopology and may stimulate future experimental studies.ACKNOWLEDGEMENTSThis work was supported by JSPS KAKENHI (Grants No.JP25K01609, No. JP22H05473, and No. JP21H01019), JSTCREST (Grant No. JPMJCR19T1). K.W. acknowledges thefinancial support for Basic Science Research Projects (GrantNo. 2401203) from the Sumitomo Foundation.DATA AVAILABILITYThe data used and analyzed during the current study avail-able from corresponding authors on reasonable request.APPENDIX A: USING COMPLEX WAVE VECTORS TODESCRIBE SKIN MODESIn this Appendix, we present a practical method for obtain-ing complex wave vectors (a non-Bloch implementation) in2D continuous photonic systems. EM waves propagating inPhC systems are mathematically described by the Bloch bandtheory, which includes a periodic term and a plane wave termwith real wave vectors Hn(r) = unk(r)eik·r. Following Blochband theory, the band structure of a finite large system canbe reproduced by using one unit cell with periodic bound-ary condition. However, in a truncated non-Hermitian sys-tem, a macroscopic number of bulk states are not extendedthroughout the entire structure. They become localized atthe boundaries and exponentially decay. Moreover, it is eas-ily seen in Fig. 3 that the eigenfrequencies of one unit cellwith periodic boundary condition (black) and the eigenfre-quencies of the structure with two PMC boundaries (red) aredistinct. Therefore, the conventional Bloch band theory withreal wave vectors k is not suitable to describe the skin modes.To ensure the mathematical formulation accurately capturesthe physics, the localization and decay properties of skin ef-fect will be expressed using complex wave vectors k̃ insteadof real ones. This framework corresponds to the non-Blochband theory [55, 56].In our investigated ribbon structure, the skin effect emergesat the two boundaries paralleled to y direction. EM waves de-cay in x direction and extend in y direction. Therefore, weassume that kx is complex k̃x = krex + ikimx and ky is real asnormal. We note here that both krex and ky take values from(a) (b)(c)x: PMC,  y: periodicx: PMC,  y: PMCFIG. 5. (a) Schematic of investigated finite structure, system size is 20 × 20 unit cells. PMC boundary condition is applied to all fourboundaries. (b) Photonic dispersion in the complex plane of the finite structure (red) in (a) and semi-infinite (black) in Fig. 2(c). The insetshows enlarged figure of the region enclosed by the blue dot line. The topological edge states form loop in complex plane and enclosesthe complex eigenfrequencies of the corner modes. (c) Field profile of the corner skin modes labeled by the stars in (b) and the extendedtopological edge states labeled by the magenta circle in (b).vary          until black loop become arc and identical to red line  (a)(b)x, y: periodic,x: PMC, y: periodic,x, y: periodic,      FIG. 6. (a) Schematic of eigenfrequencies of the infinite structure(black) and eigenfrequencies of semi-infinite structure (red). Fromleft to right panels, kimx is varied from 0 to the critical point, whereeigenfrequencies of two structures become identical. (b) Band struc-ture of infinite structure with real kx (black), semi-infinite structure(red) and infinite structure with complex k̃x.−π/a0 to π/a0 in the first BZ. To find kimx , the transfer ma-trix method or the tight binding approximation method can beused. These two methods work well for 1D systems [57] andkimx can be solved analytically. However, for 2D systems, thesetwo methods become complicated since there are several de-pendent parameters. As mentioned previously, we expect toreproduce the photonic band structure of truncated systemsusing one unit cell with periodic boundary condition. There-fore, here, we introduce a simulation method, which can beused to obtain kimx more easily than tight binding approxima-tion and transfer matrix method.In Fig. 6(a), the black and red dashed lines denote the eigen-frequencies of the infinite and ribbon structures, respectively.The red lines remain fixed throughout the procedure. For theinfinite structure, the eigenfrequencies are recalculated by em-ploying complex wave vectors k̃x. The imaginary componentkimx is systematically varied until the eigenfrequencies of theinfinite (black) and ribbon (red) structures become identical.The corresponding value of kimx is thereby identified as the de-sired one.Focusing on the lowest photonic band at a0ky = 0.5π asshown in Fig. 6(b), the point gap form by eigenfrequencies ofinfinite structure (black) is collapsed to an arc (green) whenthe imaginary component kimx is 0.048π/a0. This arc coin-cides with the eigenfrequencies of the ribbon structure (red).This means that the eigenfrequencies of skin modes are repro-duced by using one unit cell with periodic boundary condi-tion and complex wave vectors. The localization length of thewavefunctions are determined by the imaginary part of com-plex wave vectors, given by 1/|kimx |. In this case, at a0ky =0.5π , localization length is 1/|kimx | = 1/|0.048π/a0| ≈ 7a0.This value is consistent with the field profile shown in the leftpanel of Fig. 3.APPENDIX B: NUMERICAL GAUGE SMOOTHING FORTHE COMPLEX BERRY PHASEIn this appendix, we give a brief review of a numericalmethod to obtain the left and right eigenvectors ⟨uLnk| and|uRnk⟩ and in a suitably smooth gauge, enabling the calculationof complex Berry phases along a discretized kx in parameterspace [42]. This procedure is essential for the numerical eval-uation of Eq. (7). The properties of EM waves in photoniccrystals are analyzed using the following eigenvalue equation∇× 1ε(r)∇×H(r) =(ωc)2H(r), (9)here, we put Θ = ∇× 1ε(r)∇× is the operator. The left andright eigenvectors can be obtained by⟨unL(k)|Θ = ⟨unL(k)|(ωc)2, (10a)Θ|unR(k)⟩=(ωc)2|unR(k)⟩. (10b)In general, the left and right eigenvectors are computed in-dependently for each discrete k-point, and each carries an ar-bitrary global phase. According to ref. [58], the biorthogonalnormalization condition for the left and right eigenvectors ateach ki point in kx direction are⟨unL(ki)| →⟨unL(ki)|√⟨unL(ki)|unR(ki)⟩, (11a)|unR(ki)⟩ →|unR(ki)⟩√⟨unL(ki)|unR(ki)⟩. (11b)By normalizing left and right eigenvectors,⟨umL (ki)|unR(ki)⟩ = δmn is satisfied. To smooth the gaugein the 1st BZ, we choose a based point such as k1, thennormalized the eigenvectors according to Eq. (11). Thearbitrary global phases are adjusted by the following process⟨unL(ki)| → ⟨unL(ki)|e−iarg⟨unL(ki)|unR(ki−1)⟩, (12a)|unR(ki)⟩ → |unR(ki)⟩e−iarg⟨unL(ki−1)|unR(ki)⟩. (12b)After the above step, the gauge is basically smooth. How-ever, they are not single-valued in the 1st BZ. In particular,at the starting point k1 and ending point kN , eigenvectors arenot identical. Next, we adjust the phase difference betweenthe starting and ending point. 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