Anatomy of skin modes and topology in non-Hermitian systems
AAnatomy of skin modes and topology in non-Hermitian systems
Ching Hua Lee
1, 2, ∗ and Ronny Thomale Institute of High Performance Computing, A*STAR, Singapore, 138632. Department of Physics, National University of Singapore, Singapore, 117542. Institute for Theoretical Physics and Astrophysics,University of W¨urzburg, Am Hubland, D-97074 W¨urzburg, Germany (Dated: May 7, 2019)A non-Hermitian system can exhibit extensive sensitivity of its complex energy spectrum to theimposed boundary conditions, which is beyond any known phenomenon from Hermitian systems.In addition to topologically protected boundary modes, macroscopically many “skin” boundarymodes may appear under open boundary conditions. We rigorously derive universal results forcharacterizing all avenues of boundary modes in non-Hermitian systems for arbitrary hopping range.For skin modes, we introduce how exact energies and decay lengths can be obtained by threadingan imaginary flux. Furthermore, for 1D topological boundary modes, we derive a new genericcriterion for their existence in non-Hermitian systems which, in contrast to previous formulations,does not require specific tailoring to the system at hand. Our approach is intimately based on thecomplex analytical properties of in-gap exceptional points, and gives a lower bound for the windingnumber related to the vorticity of the energy Riemann surface. It also reveals that the topologicallynontrivial phase is partitioned into subregimes where the boundary mode’s decay length dependsdifferently on complex momenta roots.
The avenue of topological phases has reshaped our per-spective on single-particle problems in condensed mat-ter [1–3]. Unlike interacting many-body problems whichare seldom exactly solvable, single-particle problems areoften regarded as conveniently analytically tractable,with quantum and classical realizations accessible onequal formal footing [4–13]. This view, however, un-derestimates the richness and intricacies derived fromthe parameter and phase space structure of the physi-cal system [14], as well as the added complexity impliedby investigations of boundary terminations [15], exter-nal driving [16], and open systems beyond the realm ofHermiticity [17].Non-Hermiticity from either inherent gain/loss or non-reciprocity is particularly interesting, exhibiting severalexciting new phenomena. For instance, complex energybands can develop branch cuts terminating at so-calledexceptional points [18–25] that can coalesce to form ex-ceptional rings [26–28], and bulk modes can morph intoboundary “skin” modes exhibiting an extensively largeboundary density of states [29, 30]. Non-Hermiticity pro-foundly affects topological localization in fascinating, yetpoorly understood ways. In a topologically non-trivialHermitian system, a boundary can only introduce a sub-extensive number of in-gap protected modes. The bulkmodes, being de-localized, remain largely undisturbed.In contrast, in a non-Hermitian system, the entire spec-trum of an arbitrary large system can be modified byintroducing a boundary, ostensibly violating the bulkboundary correspondence (BBC) [20, 29–33].As we shall elucidate, this seemingly counterintuitivesensitivity to boundary conditions is a consequence of thefundamental observation that non-reciprocal systems canbe driven into different regimes by local perturbations,each characterized by its distinct exceptional points and winding numbers. This is because non-reciprocity can lo-calize all eigenmodes at the boundaries, including thosewhich, for periodic boundary conditions, would have beenassigned extended bulk modes. There are two typesof non-Hermitian boundary eigenmodes: Extensive skinmodes which are adiabatically connected to Hermitianbulk modes through complex analytic continuation, andsub-extensive topological boundary modes, which are, aswe will show, protected by a universal non-Hermitiantopological winding number criterion.Recent attempts at characterizing these enigmatic non-Hermitian boundary modes have not always been con-clusive. Even after generalizing the Berry curvatureand Chern number to their biorthogonal non-Hermitiananalogs [23, 33–35], difficulties remain in choosing themost appropriate and efficient quantities and contoursfor capturing phase transitions [35]. While Refs. [29]and [32] have identified jumps in the biorthogonal po-larization as necessary conditions for topological phasetransitions, their sufficiency remains unclear beyond thesimplest models with nearest-neighbor hoppings. Sincenon-reciprocity fundamentally alters the non-Bloch en-ergy spectrum, the eigenmodes of generic models withmultiple non-reciprocal hopping ranges can only be un-derstood through a systematic analysis of their complex band structure. Quantitative predictions of the localiza-tion lengths and dispersions of skin modes are even moreelusive, with existing results restricted to numerical evi-dence or fine-tuned models where boundary modes can becalculated exactly [30, 32, 33]. Thus, the key outstandingquestions are: (i) What are not just necessary but also sufficient conditions for the skin effect in non-Hermitiansystems? (ii) How can one analytically characterize theenergies, density of states, and localization lengths of skinmodes? (iii) What is a universal criterion for topological a r X i v : . [ c ond - m a t . o t h e r] M a y boundary modes of 1D non-Hermitian systems that doesnot require specially tailored contours? In this work, weanswer these questions through a universal treatment ofboundary modes in non-Hermitian systems. Complex flux for characterizing skin modes –
Usually,open boundaries break translational invariance and pre-clude exact analytic characterization of the eigenmodes.For non-Hermitian skin modes, however, analytical re-sults exist via a mode pumping argument [36–42] with a complex flux φ . We propose to interpolate between peri-odic and open boundary conditions (PBCs and OBCs)by adiabatically reducing one of the boundary hop-pings to zero via this complex flux. As a minimalmodel to illustrate the idea, consider a generic 1D chainwith particle hopping of arbitrary range. In momen-tum space, it is represented by the Hamiltonian H = (cid:80) N R n = − N L (cid:80) k e ikn T n η † k η k , where T n is the hopping am-plitude across n unit cells, and η † k creates a particlewith quasi-momentum k . (Note that T n becomes matrix-valued as soon as there are multiple states per unitcell.) When the hoppings are non-reciprocal, T n (cid:54) = T T − n ( H (cid:54) = H T in real-space, see [43]), hence allowing theleft/right hopping ranges N L and N R to be not necessar-ily equal. To evolve from PBCs to OBCs, we first trans-form each hopping T n → T n e inφ through flux threading.Next, we perform a gauge transform H → V − HV with V = diag( e − iφ , e − iφ , ..., e − ilφ ), with l being the systemlength, to remove the complex phase from all but theboundary hoppings, which are consequently multipliedby e ∓ ilφ ∼ e ∓ l Im φ . Since skin modes are spatially local-ized, the divergent case e l | Im φ | can always be ignored bychoosing an appropriate sign [44] for φ . We are henceleft with boundary hoppings rescaled by e − l | Im φ | , whichcorresponds to perfect PBCs when φ = 0, and the OBClimit when φ → ∞ . Implementing φ threading by min-imal coupling k → k + iκ , this implies that the trans-lationally invariant analytic continuation of the originalHamiltonian, H κ ( k ) = H ( k + iκ ) , (1)has the same spectrum as the skin modes due to bound-ary hoppings suppressed by e − κl per unit hopping. Phys-ically, (S2) implies that all the original PBC bulk states must morph into left boundary modes with localizationlengths κ − under e − κl boundary hopping suppression.Furthermore, H κ ( k ) ∀ κ forms an equivalence class ofHamiltonians with identical OBC spectra [29, 45].Our approach allows us to understand why superfi-cially similar systems may still manifest markedly differ-ent non-Hermitian effects. We demonstrate this insightthrough the non-Hermitian Su-Schrieffer-Heeger (SSH)model [20, 29, 32, 46, 47]: H γ x ,γ y SSH ( k ) = (cid:18)
12 + cos k (cid:19) σ x + sin kσ y + iγ x σ x + iγ y σ y , (2) FIG. 1: a) Illustration of Eq. 2, with unbalanced intra-unit cell couplings T and balanced inter-unit cell cou-plings T ± . Spectra for H γ x , (b-d) and H ,γ y SSH (e-g)from Eq. 2. (d,g): Both systems exhibit topological zeromodes (black lines at Re[ E ] = 0) at small γ x or γ y , butonly H γ x , exhibits BBC. For H ,γ y SSH , all the OBC modes(black) differ from the PBC modes (red), not just fortopological modes. (b,c,e,f): Anatomy of PBC and OBCspectra in a topologically nontrivial ( γ x,y = 0 .
4) and triv-ial ( γ x,y = 1 .
2) regime. Light blue-magenta taperingcurves illustrate the evolution of PBC modes into OBCmodes as Im φ increases from 0 to ∞ . Pale backgroundclosed curves are contours of constant κ = Im k with in-tervals of 0 .
1. For H γ x , , the PBC and OBC spectracoincide along open arcs and no evolution occurs, excepttowards the isolated topological zero mode in (b). For H ,γ y SSH , the OBC skin modes (black) morph en-masse fromthe PBC modes (red). Bulk modes become localized skinmodes as soon as they lie along κ (cid:54) = 0 contours. FromEq. 7, skin modes (black) of (f) satisfy Re[ E ] = − γ y and 4 γ y > E ]) .with σ x , σ y denoting the Pauli matrices. The breaking ofnon-reciprocity relies exclusively on γ y , which becomestransparent from the T , ± hopping matrix representa-tion in Fig. 1a. As shown in Fig. 1, H γ x , and H ,γ y SSH possess qualitatively different behavior as PBCs are mor-phed into OBCs via imaginary flux ( κ = Im φ ) pumping. H γ x , (Fig. 1b-d) respects the usual BBC, with its OBC(black) and PBC (red) spectra coinciding except for iso-lated topological boundary modes. For H ,γ y SSH (Fig. 1e-g),however, almost the entire spectrum collapses onto theOBC modes (black) when evolving towards OBCs, i.e.one finds the non-Hermitian skin effect. They all becomeboundary modes because only modes on the PBC loci(red in Fig. 1c,d,f,g) have real Bloch momenta. In par-ticular, PBC bulk modes tend to evolve into the interiorof their PBC loci, and will not move (i.e. obey the usualBBC) only if already located along an open arc, as for γ y = 0. In the following, we shall explain and analyticallycharacterize such behavior. OBC constraints and skin mode solutions –
One may betempted to find skin modes simply by taking the κ → ∞ OBC limit in Eq. S2. This, however, would yield un-detectable modes with vanishing decay lengths. To cor-rectly find the skin modes of a Hamiltonian H ( z ), where z = e ik , k ∈ C , we construct an ansatz eigenmode ψ ( x )from the eigenenergy E subset of the Hilbert space: ψ ( x ) = (cid:88) µ c µ β xµ ϕ µ , (3) Eϕ µ = H ( β µ ) ϕ µ , (4)where c µ denots complex coefficients and the set of β µ sconsists of all the roots of the characteristic polynomialDet[ H ( z ) − E I ] = 0, E regarded as a fixed parameter.The bulk Hamiltonian specifies that Hψ ( x ) = (cid:88) − N L 2. From Fig. S2, its eigenmodes (blue)flow towards the interior of the PBC energy loci (red),stopping only if they collide with other modes (black).Since these collisions occur at a single value of κ , theymust be solutions where the β s with e − κ = | β | coin-cide (Eq. 7). Saliently, not all solutions of equal | β | correspond to skin modes - only those with largest | β | ,i.e., smallest | κ | will be passed by the spectral evolution,and hence exist as OBC eigenmodes. All these observa-tions hold for arbitrarily complicated Hamiltonians, re-flecting the generic ”contraction” property of imaginaryflux flows [44]. In particular, models with PBC spectraalready confined to lines or arcs, i.e., including all Her-mitian and reciprocal systems (where T n = T T − n ), areprecisely those without such flows, and hence skin effect. Criterion for non-Hermitian topological phases – Besidesthe continuum of skin boundary modes, there can alsoexist isolated “topologically protected” boundary zeromodes. The general criterion for their existence, however,must invariably differ in non-Hermitian systems fromthat of Hermitian models, since the skin effect introducesnew decay length scales which manifest as additional sin-gularities in the complex band structure. Below, we shallderive a novel topological criterion (Eq. 11) for the mostintensely studied class of particle-hole (PH) symmetric1D systems. It generalizes previously proposed invari-ants for non-Hermitian systems [29, 32, 35, 47], and isstraightforwardly applicable to models with arbitrarilycomplicated non-Hermitian hoppings. Consider the mostgeneric PH symmetric 2-component Hamiltonian givenby H PH [ { r a/b } ; { p a/b } ]( z ) = (cid:18) a ( z ) b ( z ) 0 (cid:19) = (cid:32) z r a (cid:81) p a i ( z − a i ) z √ a i z r b (cid:81) p b i ( z − b i ) z √ a i (cid:33) , (9)where z = e ik and a i , b i are the complex roots of Lau-rent polynomials a ( z ) , b ( z ), both of which can be rescaledwithout changing the topology. In terms of OBC con-straints (Eqs. 6), “topological” modes are special solu-tions where the boundary system described by Eqs. 6 isnot of full rank, such that the eigenmode weights c µ havenonzero solutions despite | β µ | (cid:54) = | β ν | for any pair µ, ν .Rewriting Eqs. 6 as a matrix equation M c = , this con-dition for a topological mode translates to Det M = 0. Asmeticulously derived in the supplement [44], this prob-lem can be reformulated as the fundamental principle: An isolated topological zero mode exists when the r a + r b largest β µ s do not contain r a members from { a , ..., a p a } and r b members from { b , ..., b p b } . These conditions onthe zeros and poles of the Hamiltonian can also be re-cast [44] in terms of the windings W g ( R ) = (cid:73) | z | = R d (log g ( z ))2 πi = Z g ( R ) − P g , (10) g = a, b , which counts the number of zeros Z g ( R ) minusthe number of poles P g encircled by a circle | z | = R of radius R ∈ R . Evidently, P g = p g − r g does notdepend on R , since the poles are always at z = 0. If R is chosen such that | z | = R excludes the r a largestroots of a ( z ), W a ( R a ) = ( p a − r a ) − P a = 0 when atopological mode exists. The same | z | = R , however, isnot allowed to simultaneously exclude r b roots of b ( z ), forthat would cause the r a + r b excluded, i.e., largest roots,to be partitioned into r a a i ’s and r b b i ’s. Hence when W a ( R ) = 0, we must have W b ( R ) < 0, or vice versa.Thus a topological boundary mode exists iff ∃ R ∈ (0 , ∞ ) such that W a ( R ) W b ( R ) < , (11)or, in terms of the energy surface vorticity and eigenmodewinding V ( R ) , W ( R ) = ( W a ( R ) ± W b ( R )) / ∃ R ∈ (0 , ∞ ) such that | V ( R ) | < | W ( R ) | . (12) (a) (b) FIG. 3: a) Phase diagram of Eq. 13 with γ = 1 . 2. Differ-ent colors represent regimes with topological mode de-cay rate − (log | β | ) − determined by β = a , a , b or b respectively. b) Illustration of how the ordering of − log | β | solutions determine the phase along the dashedline ( t = 0 . 05) of a), with β = a , a , b and b solutionscolored red, light red, dark green and light green. Fromcriterion 11, topological modes occur when no greenish(redish) curve falls between two redish (greenish) curves,with corresponding regimes colored as in a).Criterion 11 or S18 is a main result of this work, im-plying that to have topological modes, we need to find one value of R = e − κ such that W a ( R ) , W b ( R ) are ofopposite signs. Based on the insight that the OBC spec-trum remains invariant under imaginary flux pumping, itdoes not rely on any specially tailored contour [29]. Asformulated in Eq. S18, it expresses vorticity as a lowerbound for eigenmode winding in the topological phase.For instance, when the energy surface contains a branchcut ( V ( R ) = 1 / / 2, not 0 as in Hermitian cases.To illustrate Eqs. 11 and S18, we apply it to a generalnearest neighbor (nn) hopping model which is alreadybeyond the models previously studied in the literature H PHnn ( z ) = (cid:18) t − γ + z + t /zt + γ + 1 /z + t z (cid:19) . (13)Its phase diagram (Fig. 3a) contains a topological regionpartitioned into four subregions, depending on whetherthe zero mode decay length − (log | β | ) − is given by theroots a , = ( − t − γ ± (cid:112) ( t + γ ) − t ) / (2 t ), or b , =( γ − t ± (cid:112) ( t − γ ) − t ) / 2. The decisive β µ is the( r a + r b +1)th largest one [44] - not the one correspondingto the imaginary gap (largest β µ ), which controls thehopping decays [48, 49], as illustrated in Fig. 3b. For t =0 in (13), criterion 11 reduces to previous formulationsof a topological criterion [29, 32, 45] | t − γ | < a = ∞ , a = − t + γ , b = γ − t and b = 0.The fundamental advancement implied by criterion 11lies in its logical sufficiency, convenience of use and gen-eral applicability to all two-component PH-symmetricFIG. 4: Application of criterion 11 to a more complicatedinstance of (9) with p a = p b = 4 and r a = 3, r b = 2,which is completely topologically characterized by theroots of their a ( z ) and b ( z ) (purple and orange dots).Non-contractible contours in the purple region ( W a > q a +1 = p a − r a +1 = 2 purple roots, whilecontours in the orange region ( W b < 0) enclose fewerthan q b + 1 = p b − r b + 1 = 2 orange roots. In a)/b), thepresence/absence of a zero mode corresponds to the pres-ence/absence of an overlap region where W a > W b < W a W b < W a > W b < r a larger zeros of a ( z ) and less than r b smaller zeros of b ( z ). Discussion – We have provided a rigorous treatment ofboundary modes in non-Hermitian systems. We demon-strate how the skin modes can be characterized throughan imaginary flux threading argument, and developeda winding number criterion for 1D topological bound-ary modes in PH-symmetric models. Our criterion W a ( R ) W b ( R ) < entire complex band struc-ture, and, in the Hermitian case, reduces to the state-ment of nontrivial winding W a ( R ) > 0, where W b ( R ) = − W a ( R ) and R = 1. Our framework reveals the intuitionbehind the extreme sensitivity of non-Hermitian systemto its boundary: even in a large system, a small reduc-tion in the boundary hopping ∼ e − κl can be equivalentto a large change in κ for the entire system. Acknowledgements – We thank Zhong Wang, XiaoZhang, Xiong Ye, Tobias Helbig and Tobias Hoffmanfor helpful comments. R.T. is supported by the Eu-ropean Research Council through ERC-StG-Thomale-TOPOLECTRICS-336012, by DFG-SFB 1170 (projectB04), and by DFG-EXC 2471 ”ct.qmat”. ∗ [email protected]; [email protected][1] F. D. M. Haldane, Phys. Rev. Lett. , 2015 (1988).[2] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010).[3] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011).[4] G. Liang and Y. Chong, Phys. Rev. Lett. , 203904(2013).[5] W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou,J.-W. Dong, and C. T. Chan, Nature communications ,5782 (2014).[6] L. M. Nash, D. Kleckner, A. Read, V. Vitelli, A. M.Turner, and W. T. Irvine, Proceedings of the NationalAcademy of Sciences , 14495 (2015).[7] S. H. Mousavi, A. B. Khanikaev, and Z. Wang, NatureCommunications , 8682 (2015).[8] W. Hu, J. C. Pillay, K. Wu, M. Pasek, P. P. Shum, andY. Chong, Physical Review X , 011012 (2015).[9] X. Zhang, Y. Chen, Y. Wang, J. Y. Lin, N. C. Hu, B.Guan, and C. H. Lee, arXiv preprint arXiv:1710.08385(2017).[10] C. H. Lee, G. Li, G. Jin, Y. Liu, and X. Zhang, PhysicalReview B , 085110 (2018).[11] M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, andI. Bloch, Nature , 55 (2018).[12] T. Helbig, T. Hofmann, C. H. Lee, R. Thomale, S. Imhof,L. W. Molenkamp, and T. Kiessling, Physical Review B , 161114 (2019).[13] Y. Wang, L.-J. Lang, C. H. Lee, B. Zhang, and Y. Chong,Nature communications , 1102 (2019).[14] M. V. Berry, Proceedings of the Royal Society of Lon-don A: Mathematical, Physical and Engineering Sciences , 45 (1984).[15] D. S. Sholl and J. A. Steckel, Density Functional Theory (Wiley, New Jersey, 2009).[16] M. Bukov, L. D’Alessio, and A. Polkovnikov, Advancesin Physics , 139 (2015).[17] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H.Musslimani, S. Rotter, and D. N. Christodoulides, Na-ture Physics , 11 (2018).[18] M. V. Berry, Czechoslovak journal of physics , 1039(2004).[19] W. D. Heiss, Journal of Physics A: Mathematical andTheoretical , 444016 (2012).[20] T. E. Lee, Phys. Rev. Lett. , 133903 (2016).[21] W. Hu, H. Wang, P. P. Shum, and Y. Chong, Phys. Rev.B , 184306 (2017).[22] V. Achilleos, G. Theocharis, O. Richoux, and V. Pag-neux, Phys. Rev. B , 144303 (2017).[23] H. Shen, B. Zhen, and L. Fu, Phys. Rev. Lett. ,146402 (2018).[24] Q. Zhong, M. Khajavikhan, D. Christodoulides, and R.El-Ganainy, arXiv preprint arXiv:1805.07620 (2018).[25] X.-L. Zhang, S. Wang, B. Hou, and C. Chan, PhysicalReview X , 021066 (2018).[26] B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A.Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljaˇci´c,Nature , 354 (2015).[27] J. Carlstr¨om and E. J. Bergholtz, arXiv preprintarXiv:1807.03330 (2018).[28] Z. Yang and J. Hu, Physical Review B , 081102 (2019). [29] S. Yao and Z. Wang, Phys. Rev. Lett. , 086803(2018).[30] V. M. Alvarez, J. B. Vargas, and L. F. Torres, Phys. Rev.B , 121401 (2018).[31] Y. Xiong, Journal of Physics Communications , 035043(2018).[32] F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J.Bergholtz, Phys. Rev. Lett. , 026808 (2018).[33] K. Kawabata, K. Shiozaki, and M. Ueda, Phys. Rev. B , 165148 (2018).[34] S. Yao, F. Song, and Z. Wang, Physical review letters , 136802 (2018).[35] Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Hi-gashikawa, and M. Ueda, Physical Review X , 031079(2018).[36] R. B. Laughlin, Phys. Rev. B , 5632 (1981).[37] Q. Niu, Phys. Rev. Lett. , 1812 (1990).[38] N. Hatano and D. R. Nelson, Phys. Rev. Lett. , 570(1996).[39] A. A. Soluyanov and D. Vanderbilt, Phys. Rev. B ,035108 (2011).[40] A. Alexandradinata, T. L. Hughes, and B. A. Bernevig,Phys. Rev. B , 195103 (2011).[41] C. H. Lee and P. Ye, Physical Review B , 085119(2015).[42] Y. Hatsugai and T. Fukui, Phys. Rev. B , 041102(2016).[43] S. Malzard, C. Poli, and H. Schomerus, Physical reviewletters , 200402 (2015).[44] Supplemental Materials .[45] L. Jin and Z. Song, Physical Review B , 081103 (2019).[46] S. Lieu, Physical Review B , 045106 (2018).[47] C. Yin, H. Jiang, L. Li, R. L¨u, and S. Chen, PhysicalReview A , 052115 (2018).[48] C. H. Lee, D. P. Arovas, and R. Thomale, Phys. Rev. B , 155155 (2016).[49] C. H. Lee, M. Claassen, and R. Thomale, Phys. Rev. B , 165150 (2017).[50] Equivalently, we can rephrase this as: Among the q a + q b smallest β µ ’s, we must not have q a of them belonging to { a , ..., a p a } and q b of them belonging to { b , ..., b p b } .[51] C. H. Lee, L. Li, and J. Gong, arXiv preprintarXiv:1810.11824 (2018).[52] C. H. Lee, G. Li, Y. Liu, T. Tai, R. Thomale, and X.Zhang, arXiv preprint arXiv:1812.02011 (2018). Supplemental Online Material for “Anatomy of skin modes and topology innon-Hermitian systems” Ching Hua Lee , and Ronny Thomale Institute of High Performance Computing, A*STAR, Singapore, 138632. Department of Physics, National University of Singapore, Singapore, 117542. University of W¨urzburg, Am Hubland, D-97074 W¨urzburg, Germany This supplementary contains the following material arranged by sections:1. Periodic-open boundary condition (PBC-OBC) evolution through imaginary flux - detailed derivations leading to keyresults Eqs. 1 and the discussion after Eq. 7 of the main text.2. Pedagogical derivation of our topological criterion from first principles (Eqs. 11 and 12 of the main text). PBC-OBC EVOLUTION THROUGH IMAGINARY FLUXImaginary flux threading argument and semi-OBCs We treat a generic lattice system as a collection of 1D chains perpendicular to the open boundary, with coordinatesof the other dimensions taken as external parameters. Consider a 1D chain described by a Hamiltonian H = N R (cid:88) n = − N L (cid:88) x ; γδ [ T n ] γδ η † x,γ η x + n,δ = N R (cid:88) n = − N L (cid:88) k (cid:88) γδ e ikn [ T n ] γδ η † k,γ η k,δ , (S1)such that hoppings across a displacement of n unit cells (i.e. sites) are given by the elements of the matrix T n in thesublattice (internal component) basis indexed by γ, δ . η † x,γ and η † k,γ are the creation operators of a γ -th sublatticestate at unit cell x and quasi-momentum k respectively. For brevity, we shall henceforth drop the sublattice indices.We assume reasonably local hoppings, so N L , N R ∼ O (1). Under periodic/open boundary conditions (PBCs/OBCs),the chain can be visualized as a ring with hoppings present/absent across its endpoints. Via Faraday’s law, we canthread flux through this ring by shifting the momentum k via minimal coupling k → k + φ , where ˙ φ is the rateof change of flux which equals the induced (ficticious) electromagnetic field. Equivalently, this flux multiplies eachhopping with a phase factor viz. T n → T n e inφ .To relate this flux pumping with the boundary conditions (BCs), one performs a gauge transformation H → V − HV with V = diag( e − iφ , e − iφ , ..., e − ilφ ), l being the system length. This removes the phase from all the hoppings exceptfor those across the endpoints, which acquire a phase of e ∓ ilφ . Through this, we have managed to re-express BCs onthe boundary hoppings in terms of translationally-invariant fluxes.We next construct an interpolation between PBCs and OBCs for studying how non-Hermitian skin modes arise.For that, we have to first introduce the semi-open boundary condition (semi-OBC), which has the boundary hoppingsvanish in one direction but not the other. This is necessary because an imaginary flux component will always producea rescaling factor O ( e ± l Im φ ) that diverges with l at one of the boundaries. Without loss of generality, we set hoppings T n< | R from the right to the left boundary to zero, but preserve their reciprocal hoppings T n> | L . As φ becomescomplex, T n> | L will be rescaled by a factor of e − l Im φ . When Im φ = 0, we have perfect PBC in one direction; asIm φ → ∞ , we approach the OBC limit. Had the non-reciprocity be directed in the opposite direction, an identicalarguments holds with left and right sides switched, and φ ↔ φ ∗ .Hence, to find the spectrum of H ( k ) under the semi-OBC of T n> | R = 0 and T n> | L rescaled by a factor e − κl ,which tends to the exact OBC when κl → ∞ , we can perform the analytic continuation k → k + iκ . In other words,we can simply diagonalize the translationally invariant analytic continuation of the original Hamiltonian (Eq. 1 ofthe main text): H κ ( k ) = H ( k + iκ ) , (S2)which possesses an identical spectrum as the semi-OBC system. Physically, (S2) implies that all the original PBC bulkstates must morph into left boundary modes with localization lengths κ − under e − κl boundary hopping suppression.Furthermore, H κ ( k ) ∀ κ forms an equivalence class of Hamiltonians with identical OBC spectra. Such macroscopiccondensation of modes onto one edge does not happen in Hermitian systems because semi-OBCs, being non-reciprocal,2destroy hermiticity, and as such is a physically unrealistic proxy for OBC. But for the skin modes, OBCs and (correctlychosen) semi-OBCs are essentially equivalent, since the BCs are only consequential at the boundary where the skinmode is localized. Henceforth, we shall no longer distinguish OBCs from semi-OBCs. Geometric argument for when skin mode evolution stops (Eq. 7 and subsequent arguments of the main text) To intuitively understand why the eigenmodes should converge along exceptional points or arcs in the OBC limit, weconsider their spectral flow upon threading of the real part of a flux: φ = Re φ + i Im φ → (Re φ + 2 π/l ) + i Im φ . Thiscorresponds to multiplying the boundary hopping by a suppression factor together with a phase: e − l Im φ → e − l Im φ e πi .Since e πi = 1, this real flux evolution must map the full set of eigenvalues onto itself after a 2 π period, i.e. it canonly permute the eigenmodes.However, even this permutation should be trivial in the exact OBC limit of Im φ → ∞ , since in this limit theboundary hopping disappears, and there will be no more boundary hopping to be rotated! As such, we intuitivelyexpect the spectrum to contract into smaller and smaller loops when approaching the OBC limit (Fig. S2a), haltingwhen the loops degenerate into arcs or isolated “phenomenal” exceptional points [30, 31] which exist only underOBCs and not PBCs. Hamiltonians which do not host skin modes are precisely those whose PBC spectra already arelocated along an arc. This includes all Hermitian systems, with spectra confined to the real line, as well as reciprocal non-Hermitian systems, whose symmetric hoppings ( T n = T T − n ) force the PBC spectrum to retrace itself. (Note thatup to now, those are the models that have predominantely been realized in experimental setups.)3 Examples Here we present more detailed results on the non-reciprocal SSH model ( γ x = 0 , γ y (cid:54) = 0 from Eq. 2 of the maintext). For convenience, we have defined z = e ik and σ ± = ( σ x ± iσ y ) / H γ y SSH = (cid:18) 12 + cos k (cid:19) σ x + sin k σ y + iγ y σ y = (cid:18) 12 + z − + γ y (cid:19) σ + + (cid:18) 12 + z − γ y (cid:19) σ − . (S3)FIG. S1: (a-f) The H γ y SSH (Eq. S3) spectrum for γ y = 0 , . , . , . , . . 7. As in the main text, the red curverepresents the PBC spectrum and the blue/magenta tapering lines represent the PBC-OBC evolution trajectories ofincreasing κ , which collides to form the OBC spectrum (black). The pale background curves are contours of constant κ with intervals of 0 . 1. When γ y = 0 (a), we have the Hermitian SSH model, whose PBC and OBC spectra coincideexcept for the zero mode (black dot at E = 0). As γ y increases, the PBC bands (red) broaden into ellipses (b).Before γ y exceeds 0 . 5, the OBC limit can still be gauge transformed into that of the Hermitian SSH model[29], andits spectrum (black) is hence confined to the real line. Beyond γ y = 0 . 5, the OBC spectrum also extends into theimaginary direction (c), finally annihilating with the zero mode and reopening in the perpendicular direction (d). ThisOBC topological phase transition occurs when the PBC spectrum is merged as a single loop (has nontrivial vorticity,i.e. is 4 π -periodic). Finally, at γ y = 1 . 5, the PBC gap also closes (e) before re-opening as the trivial phase (f).Next, we further study a more complicated next nearest neighbor hopping model (Eq. 8 of the main text): H nnn ( z ) = 94 σ x − z σ − + 3 (cid:18) − z − z (cid:19) σ + , (S4)as well as a possible extension with third-nearest unit cell hoppings:˜ H ( z ) = 94 σ x + (cid:18) z − z (cid:19) σ − + 3 (cid:18) − z − z (cid:19) σ + , (S5)In these models, the higher powers of z enable more complicated twists and turns in the PBC loop, although theirOBC spectrum generally consist of relative straight sections (Fig. S2).4 (cid:45) (cid:45) (cid:45) (cid:64) E (cid:68) (cid:45) (cid:45) (cid:45) (cid:64) E (cid:68) (a) (cid:45) (cid:64) E (cid:68) (cid:45) (cid:45) (cid:64) E (cid:68) (b) FIG. S2: a) Illustration eigenmodes of H min flowing into each other under the threading of a real flux Re φ → Re φ +2 π/l (see above geometric argument), tracing successively smaller loops as the boundary hopping e − lκ diminisheswith increasing κ (Shown are κ = 0 , . , . κ , each eigenvalue will flow into itself. b) PBC(red), OBC (black) and their interpolation trajectories (blue/magenta) for the extended minimal model ˜ H min , whichexhibit more convoluted loops which interpolate into more OBC branches hinged on by OBC exceptional points. II. DERIVATION OF THE CRITERION FOR NON-HERMITIAN PARTICLE-HOLE(PH) SYMMETRICTOPOLOGICAL ZERO MODEGeneral treatment of the open boundary condition In this section, we detail, from first principles, the detailed derivation of the topological criterion of particle-holesymmetric topological modes given by Eqs. 11 and 12 of the main text, as well as an equivalent formulation in termsof poles and zeros of the Hamiltonian. For a generic N-component Hamiltonian H ( z ) where z = e ik , any openboundary condition (OBC) eigenmode ψ at eigenenergy E can always be expanded in the Hilbert subspace of modes ϕ µ that satisfy H ( β µ ) ϕ µ = Eϕ µ , (S6)where β µ is a root of the characteristic polynomial Det[ H ( β ) − E I ] = 0. Fourier transforming into real space, theOBC eigenmode ψ can be written as ψ ( x ) = (cid:88) µ c µ β xµ ϕ µ (S7)where the coefficients c µ are chosen such that Hψ ( x ) satisfies the OBC condition, i.e. vanishes outside an interval x ∈ [0 , l ], where l is the system length. Although there can be many more µ ’s than the number of bands in H , thebasis spanned by β xµ ϕ µ is not necessarily overcomplete: This is because each β xµ ϕ µ for each different x should be takenas a different basis mode. Although this sounds like an additional stringent requirement on ψ ( x ), the OBC allows forcertain spatially decaying solution modes that are prohibited under periodic boundary conditions (PBCs). Supposethat H ( z ) contains hoppings of up to N L unit cells to the left, and up to N R unit cells to the right, i.e. H ( z ) = (cid:88) − N L To make the above treatment more concrete, we now specialize to finding zero modes ( E = 0 eigenenergies), andconsider Hamiltonians of the form H ( z ) = (cid:18) a ( z ) b ( z ) 0 (cid:19) = (cid:18) z − q a (cid:81) p a i ( z − a i ) z − q b (cid:81) p b i ( z − b i ) 0 (cid:19) (S10)where z = e ik and a i , b i are the p a , p b complex roots of a ( z ) , b ( z ) respectively. Here N L = max {| q a | , | q b |} and N R = max { r a , r b } , where r a = p a − q a and r b = p b − q b . We have set the overall constants of a ( z ) and b ( z ) to unity,since their only effect is to rescale the energy trivially.The particle-hole symmetry of H ( z ) allows for considerable simplification of Eq. S9. At E = ± (cid:112) a ( z ) b ( z ) = 0,either a ( z ) or b ( z ) vanishes, and the µ max = p a + p b roots of Det H ( β ) = 0 are precisely the a i and b i ’s. Since E = 0is an exceptional point in this case, each β µ correspond to only one normalized eigenvector ϕ µ , which must be of theform (1 , T or (0 , T , depending on whether β µ ∈ { b i } or β µ ∈ { a i } respectively. Substituting these into Eq. S9 at6 E ≈ M ∼ A , A , ... A ,p a ... 00 0 ... B , B , ... B ,p b A , A , ... A ,p a ... 00 0 ... B , B , ... B ,p b ... ... ... ... ... ... A N L , A N L , ... A N L ,p a ... 00 0 ... B N L , B N L , ... B N L ,p b A (cid:48) , a l A (cid:48) , a l ... A (cid:48) ,p a a lp a ... 00 0 ... B (cid:48) , b l B (cid:48) , b l ... B (cid:48) ,p b b lp b A (cid:48) , a l A (cid:48) , a l ... A (cid:48) ,p a a lp a ... 00 0 ... B (cid:48) , b l B (cid:48) , b l ... B (cid:48) ,p b b lp b ... ... ... ... ... ... A N R , a l A (cid:48) N R , a l ... A (cid:48) N R ,p a a lp a ... 00 0 ... B (cid:48) N R , b l B (cid:48) N R , b l ... B (cid:48) N R ,p b b lp b + O ( E ) (S11)In the above, we have separated all the entries from Eq. S9 into constant scalars A i,j , B i,j , A (cid:48) i,j and B (cid:48) i,j which donot depend on the system size l , as well as factors a li and b li which decreases exponentially with l . In general, thereare q a , q b , r a , r b nonzero rows of the A i,j , B i,j , A (cid:48) i,j and B (cid:48) i,j s respectively, adding up to µ max = p a + p b constraints for µ max unknown c µ coefficients. The O ( E ) = O (cid:16)(cid:112) a ( z ) b ( z ) | z ≈ β µ (cid:17) correction arises from the small corrections fromthe T n ϕ µ ’s at E slightly away from zero, which also decreases as a power of e − l . Out[783]= (cid:200) Ψ (cid:72) x (cid:76)(cid:200) (cid:45) (cid:45) (cid:200) Ψ (cid:72) x (cid:76)(cid:200) (a) Out[729]= (cid:200) Ψ (cid:72) x (cid:76)(cid:200) (cid:45) (cid:45) (cid:200) Ψ (cid:72) x (cid:76)(cid:200) (b) FIG. S3: Two possible solutions to Eq. S9, which approximately lie in the kernel of the matrix M . The Hamiltonianis given by Eq. 14 of the main text, with parameters t = 1 , t = 0 . 05 and γ = 1 . essential step in the derivation of thetopological criterion; other details of the entries of M are inconsequential. From it, one can extract conditions on the | a i | , | b i | ’s such that the edge mode condition Det M = 0 is consistent with the scaling behavior; these conditions canthen be recast in terms of winding numbers.In the expansion of Det M , one necessarily have l ( r a + r b )-degree monomials of the formconst. × β l β l ...β lr a + r b , (S12)where { β , ..., β r a + r b } ⊂ { a , ..., a p a , b , ..., b p b } . The monomials resulting from the expansion of the leading ordermatrix expression in Eq. S11 necessarily contain r a of the a li s and r b of the b li s. However, the monomials from the O ( E ) or higher order contributions can contain any number of the a li s and b li s, as long as there is a total of r a + r b of them. For a boundary mode to exist, the roots β µ need to be consistent with the fact that E is exponentiallydecaying in l , while satisfying Det M = 0. Below, we present two equivalent formulations for the criterion for satisfyingthe above requirements: Topological criterion: Decay length hierarchy formulation We order the roots β µ ∈ { a , ..., a p a , b , ..., b p b } of the characteristic polynomial Det[ H ( β ) − E I ] = 0 by | β | > | β | > | β | > ... ϕ µ ,which is given by L β µ = − (log | β µ | ) − . Whether a boundary mode can exist or not depends entirely on the largest r a + r b = p a + p b − q a − q b roots | β | , ..., | β r + r | : • For an isolated topological boundary mode to exist at E = 0 when l → ∞ , we must not , among the r a + r b largest β µ ’s, have r a of them belonging to { a , ..., a p a } and r b of them belonging to { b , ..., b p b } [50].If this criterion is violated, we will always find a monomial in the leading order contribution that contains all of the r a + r b largest β µ ’s. Since this is already the monomial with the largest possible magnitude, we will never be able tocancel it off with the subleading O ( E ) contribution to give Det M = 0 in the l → ∞ limit.Suppose that this criterion is respected. Let the largest roots be { a , ..., a r a + r b − j } (cid:83) { b , ...b j } , where j < r b . FromEq. S11 and the arguments following it, the leading order monomial in the O ( E ) contribution can only scale like( a ...a r a ) l ( b ...b r b ) l . However, the O ( E ) contribution generically contains every possible monomial, and will thus bedominated by ( a ...a r a + r b − j ) l ( b ...b j ) l . Hence E will scale like E ∼ ( a ...a r a ) l ( b ...b r b ) l ( a ...a r a + r b − j ) l ( b ...b j ) l = (cid:32) (cid:81) r b i = j +1 b i (cid:81) r a + r b − ji = r a +1 a i (cid:33) l (S13)which always converges to zero. In terms of decay lengths L a i and L b i ’s, Eq. S13 readslog | E | ∼ − l r b − j (cid:88) i =1 (cid:18) L b j + i − L a ra + i (cid:19) → −∞ (S14)Practically, we can directly obtain | a | , ..., | a p a | and | b | , ..., | b p b | from any given 2-band PH symmetric Hamiltonian.Of the (cid:0) p a + p b p a (cid:1) possible partitions of these ordered roots into the two a i and b i sets, the above criterion gives p a (cid:88) j (cid:54) = q a (cid:18) q a + q b j (cid:19)(cid:18) r a + r b p a − j (cid:19) = (cid:18) p a + p b p a (cid:19) − (cid:18) q a + q b q a (cid:19)(cid:18) r a + r b r a (cid:19) (S15)partitions that yield a boundary mode. Although all these partitions appears to give rise to the same topological zeromode, their decay lengths can differ. These results are also useful in the analysis of higher dimensional non-Hermitiansystems like higher-order topological lattices [51] and, more crucially, 3D non-Hermitian nodal metals [52]. Topological criterion: Winding number formulation The condition for the existence of isolated boundary modes Det M = 0 will now be recast into an equivalent butmore “topological” language (Eq. S18). We define the winding numbers W a ( R ) = 12 πi (cid:73) | z | = R d (log a ( z )) W b ( R ) = 12 πi (cid:73) | z | = R d (log b ( z )) (S16)which measure how many zeros minus the number of poles is encircled by each along the contour | z | = R . Evidently,the number of poles encircled by both do not depend on R , since a ( z ) and b ( z ) possess poles of order q a and q b at z = 0 respectively. As for the zeros, we first choose an R a such that | z | = R a excludes the r a largest roots of a ( z ).This gives W a ( R a ) = ( p a − r a ) − q a = 0. Now, in the previously formulated criterion for edge modes, the set of thelargest r a + r b roots of the characteristic polynomial cannot be the union of the largest r a roots of a ( z ) and the largest r b roots of b ( z ); in other words,min {| a r a | , | b r b |} / ∈ {| β | , ..., | β r a + r b |} ⇒ min {| a r a | , | b r b |} < | β r a + r b | , (S17)the above roots all ordered by magnitude. Hence | z | = R a must enclose at least one fewer root of a ( z ) than of b ( z ),or vice versa, i.e. if W a ( R = R a ) = 0, W b ( R = R a ) < 0, or vice versa. In a nutshell,8 • A topological boundary mode exists at E = 0 when l → ∞ iff ∃ R ∈ (0 , ∞ ) such that W a ( R ) W b ( R ) < W a ( R ) , W b ( R ) are defined in Eq. S16.Condition S18 is a new result that generalizes the topological criterion in Hermitian systems, where W b ( R ) = − W a ( R ).In the latter, it reduces to the usual criterion of W a ( R ) > R = 1.Eq. S18 can be expressed in terms of more familiar quantities [23]: The winding W ( R ) = W a ( R ) − W b ( R )2 (S19)of the eigenmode ∝ √ a ( z ) b ( z ) ( b ( z ) , a ( z )) T as z traces a circle of radius R around the origin, and the vorticity V ( R ) = W a ( R ) + W b ( R )2 (S20)which gives the winding on the energy Riemann surface along the same contour; half-integer V ( R ) signify a branchcut along a double-valued energy surface. It is trivial to show that V ( R ) − W ( R ) = W a ( R ) W b ( R ), so that W a ( R ) W b ( R ) < V ( R ) < W ( R ) or | V ( R ) | < | W ( R ) | (Eq. 12 of the main text). In Hermitiansystems, V ( R ) always vanishes, and | V ( R ) | < | W ( R ) | simply reduces to the usual condition of nonzero eigenmodewinding. DETAILED EXAMPLE: NON-HERMITIAN BOUNDARY MODE FROM NEAREST-NEIGHBOR (NN)HOPPINGS For pedagogical clarity, we provide the explicit mathematical details for a PH symmetric 2-component Hamiltonianof the form H ( z ) = (cid:18) a ( z ) b ( z ) 0 (cid:19) = (cid:18) z − a )( z − a ) /z ( z − b )( z − b ) /z (cid:19) (S21)with p a = p b = p = 2, q a = q b = q = 1 and µ max = p a + p b = 4 eigenmodes. Such Hamiltonians are simply enough tobe analytically treated, but still possess sufficient richness for realizing most representive non-Hermitian phenomena.In more familiar notation, it is proportional to a generalized SSH model with complex coefficients: H PHmin ( z ) = ( α + cos k + iα − sin k − α ) σ + + [ a i ↔ b i ] σ − (S22)where α ± = √ a a ± √ a a and α = (cid:113) a a + a a (remember that a , a , b , b can all be complex).For any finite system size l , we expect the energy E of a topological mode to be exponentially close to 0, such thattwo solutions β , β of the characteristic equation Det[ M − I E ] = 0 are approximately equation to the roots a , a of a ( z ). Likewise, β ≈ b and β ≈ b . Their corresponding eigenmodes can be arbitrarily normalized since the c µ coefficients can be rescaled at will, and we shall choose the following for convenience: ϕ = (cid:18) E ( a − b )( a − b ) (cid:19) ; ϕ = (cid:18) E ( a − b )( a − b ) (cid:19) ; ϕ = (cid:18) ( b − a )( b − a ) E (cid:19) ; ϕ = (cid:18) ( b − a )( b − a ) E (cid:19) (S23)Note that we have neglected the exponentially small differences between the β µ ’s and the roots of a ( z ) and b ( z ),except when they are of leading order (as in E ). The translation hopping matrices from H ( z ) are given by T = (cid:18) (cid:19) and T − = (cid:18) a a b b (cid:19) . (S24)With them, we can construct the M matrix representing the OBC constraints: M = a a ( a − b )( a − b ) a a ( a − b )( a − b ) a a E a a Eb b E b b E b b ( b − a )( b − a ) b b ( b − a )( b − a ) a l ( a − b )( a − b ) a l ( a − b )( a − b ) b l E b l Ea l E a l E b l ( b − a )( b − a ) b l ( b − a )( b − a ) (S25)9A boundary mode can exist if Det M = 0 can be satisfied. Explicitly, the latter is given by( a a b b ) − Det M = − ( a l − a l )( b l − b l ) (cid:89) i,j =1 ( a i − b j ) + E + E ( a l a l + b l b l )( a − a )( b − b )( a + a − b − b ) + E ( a l b l + a l b l )( a − b )( a − b )(( a − a ) + ( a − b ) ) − E ( a l b l + a l b l )( a − b )( a − b )(( a − b ) + ( a − b ) ) (S26)which is the explicit form of Eq. S11 with all the higher order terms in E written down. We see that although theleading order term does not contain monomials of the forms a l a l and b l b l , the subleading E contributions containsall (cid:0) (cid:1) = 6 types of monomials of degree 2 l .In the l → ∞ limit, only the monomials containing the largest two β µ ∈ { a , a , b , b } will dominate. Supposethat | a | > | a | are the two largest. Then, since a l a l is absent from the O ( E ) term, we can rearrange the dominantterms to obtainDet M = 0 ⇔ E | | a | > | a | > | b | > | b | ∼ ( a − l − a − l )( b l − b l ) (cid:81) i,j =1 ( a i − b j ) ( a − a )( b − b )( a + a − b − b ) ∼ (cid:18) b a (cid:19) l ( a − b )( a − b )( a − b )( a − b )( a − a )( b − b )( a + a − b − b ) (S27)Since | b | < | a | , this is consistent with the requirement that E ∼ (cid:16)(cid:113) b a (cid:17) l → j = 0 and r a = r b = 1, although we have also evaluated the coefficient of the exponent.Note that, if we had chosen say a , b to be the largest two roots, in violation with the boundary mode criterion,the dominant monomial a l b l would have appeared in the contributions at all orders of E , and E will have to tendtowards a finite value instead, i.e. not lead to a zero mode.In summary, the exceptional nature of the E = 0 in-gap point turns out to be key in expressing Det M = 0 as aconstraint on winding numbers. Exactly at E = 0, either a ( z ) or b ( z ) vanishes and the p a + p b roots β µ correspond tothe a i ’s or b i ’s, with corresponding eigenmodes (1 , T or (0 , T . For any finite system size l , however, the topologicalmode is displaced from zero by E ∼ e − l , and the eigenmodes will acquire O ( E ) corrections.For a solution Det M = 0 to exist, both sides of Eq. S26 must scale similarly with l . Since the RHS is alreadysuppressed by E , the LHS cannot contain the most slowly decaying terms. Specifically, the pairs a , a or b , b must be the two β µ ’s with largest magnitude, since they are absent in the LHS but not the RHS. This leads to the result ofEq. S27. Such constraints imposed by the scaling suppression from E also appear in generic cases, and is guaranteedby the defective eigenspace of H ( z ) at E = 0. Simplest case of the non-reciprocal SSH model In the special simplest case of the non-reciprocal SSH model (Eq. S3), H γ y SSH ( z ) = (cid:18) t + γ + z t − γ + z (cid:19) = (cid:32) t + γ ) ( z + t + γ ) z ( z − ( γ − t ))( z − z (cid:33) (S28)and, after discarding inconsquential scalar factors, we identify a = ∞ , a = − t + γ , b = γ − t and b = 0. Theboundary mode criterion states that the two largest roots must be either a , a or b , b . But since a and b arealready fixed, the only option is to have let them be a , a . Hence | a | > | b | , i.e. we need1 | t + γ | > | γ − t | ⇒ | t − γ | < γ − SSH model, in agreement with the literature[29, 32].In terms of the equivalent winding criterion Eq. S18, we have W a ( R ) = (cid:40) R > | γ + t | − R < | γ + t | W b ( R ) = (cid:40) R > | γ − t | R < | γ − t | To have W a ( R ) W b ( R ) < 0, the W a ( R ) = − W b ( R ) = 1 region. This is possible if thereexists R such that | γ − t | < R < | γ + t | , i.e. the same conclusion | t − γ | <<