Non-Bloch PT symmetry breaking: Universal threshold and dimensional surprise
NNon-Bloch PT symmetry breaking: Universal threshold and dimensional surprise
Fei Song, Hong-Yi Wang,
1, 2 and Zhong Wang ∗ Institute for Advanced Study, Tsinghua University, Beijing, 100084, China School of Physics, Peking University, Beijing, 100871, China
In the presence of non-Hermitian skin effect, non-Hermitian lattices generally have complex-valuedeigenenergies under periodic boundary condition, but they can have non-Bloch PT symmetry andtherefore completely real eigenenergies under open boundary condition. This novel PT symmetryand its breaking have been experimentally observed in one dimension. Here, we find that non-Bloch PT symmetry in two and higher dimensions exhibits drastically different behaviors comparedto its one-dimensional counterpart. Whereas Bloch PT breaking and one-dimensional non-BlochPT breaking generally have nonzero thresholds in the large-size limit, the threshold of two andhigher-dimensional non-Bloch PT breaking universally approaches zero as the system size increases.A product measure , namely the product of bare non-Hermiticity and system size, is introducedto quantify the PT breaking tendency. This product being small is required for the perturbationtheory to be valid, thus its growth with system size causes the breakdown of perturbation theory,which underlies the universal threshold. That the universal behaviors emerge only in two and higherdimensions indicates an unexpected interplay among PT symmetry, non-Hermitian skin effect, andspatial dimensionality. Our predictions can be confirmed on experimentally accessible platforms.
Introduction.–
In the standard quantum mechanics ofclosed systems, the Hamiltonians are always Hermitian.The time evolution of open systems is, however, oftengenerated by effective non-Hermitian Hamiltonians [1–3]. While complex-valued eigenenergies are natural con-sequences of gain and loss, a prominent class of non-Hermitian Hamiltonians can have purely real eigenener-gies when the non-Hermiticity is below a threshold [4, 5].They are known as parity-time (PT) symmetric Hamil-tonians, and the real-to-complex transition is called PTsymmetry breaking [6–8]. PT symmetry and its breakinghave many intriguing consequences, e.g., single-mode las-ing [9, 10], nonreciprocal transmission [11–13], and uni-directional invisibility [14, 15].Independent of the PT symmetry, non-Hermitiantopology has recently been attracting growing attention.Remarkably, the bulk-boundary correspondence is dras-tically modified by the non-Hermitian skin effect (NHSE)[16–20], namely that all the energy eigenstates are expo-nentially squeezed to the boundary. Owing to this failureof Bloch band picture, the edge states correspond to thenon-Bloch topological invariants defined in the general-ized Brillouin zone (GBZ) [16, 17, 21–37], which underliesthe non-Bloch band theory [16, 21, 24, 37]. Recent ex-periments have confirmed the novel bulk-boundary cor-respondence of non-Hermitian systems [38–43].Very recently, an intriguing interplay has been foundbetween PT symmetry and NHSE in one-dimensional(1D) systems [25, 26]. In the presence of NHSE, thenon-Bloch bands consisting of eigenstates under open-boundary condition (OBC) can have PT symmetry,though the Bloch bands under periodic-boundary condi-tion (PBC) cannot. Thus, NHSE becomes a mechanismof PT symmetry in periodic lattices that lies outside thefamiliar Bloch band framework. This NHSE-induced PTsymmetry, dubbed non-Bloch PT symmetry [25, 26], has been experimentally observed recently [44].In this paper, we find that non-Bloch PT symmetryin two and higher dimensions has drastically differentand unexpected behaviors compared to the 1D cases. Itturns out that the threshold of 2D non-Bloch PT sym-metry breaking universally approaches zero as the systemsize increases. Even for an infinitesimal non-Hermiticity,a large proportion of eigenenergies undergo the real-to-complex transitions as the system size increases. Thisfeature is in sharp contrast to the PT transitions of Blochbands, which generally have nonzero thresholds; evenwhen fine tuned to thresholdless points, an infinitesimalnon-Hermiticity can at most cause transitions of an in-finitesimal proportion of eigenenergies, regardless of thesystem size [45, 46]. Notably, non-Bloch PT breakingin 1D also has a size-independent (generally nonzero)threshold at large size. Thus, our finding reveals an un-expected interplay between non-Bloch bands and spatialdimensionality.
Universal threshold.–
We consider a simple 2D non-Hermitian lattice with hoppings shown in Fig. 1(a),whose Bloch Hamiltonian is H ( k ) = ( t − γ ) e ik x + ( t + γ ) e − ik x + ( t − γ ) e ik y +( t + γ ) e − ik y + s ( e ik x + e − ik x )( e ik y + e − ik y ) , (1)where k = ( k x , k y ) and t, γ, s are real parameters. As asingle-band model, its PBC eigenenergies are just H ( k ),which are generally complex-valued. The OBC eigenen-ergies are entirely different because of the NHSE. Forthe simplest case s = 0, all the eigenstates are lo-calized at a corner of the system, taking the form of ψ ( x, y ) ∼ ( β x ) x ( β y ) y with | β x | = | β y | = (cid:113) t + γt − γ . Thisis readily seen by a similarity transformation akin tothat used for the non-Hermitian SSH model[16]. Anequivalent statement is that the GBZ is the 2D torus a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b - - I m E - H b L Re E L = Γ= - - I m E - H c L Re E L = Γ= - - I m E - H d L Re E L = Γ= FIG. 1. Energy spectrums of the single-band model Eq. (1)for L × L squares. (a) Absolute values of imaginary partsof all eigenenergies for L = 40 with varying γ . The insetshows the hoppings in Hamiltonian. (b,c,d) Eigenenergiesfor L = 10 , ,
70, respectively, with fixed γ = 0 .
1. Otherparameters are fixed as t = 1 and s = 0 . { ( β x , β y ) : | β x | = | β y | = (cid:113) t + γt − γ } . Although analyticalformula is unavailable when s (cid:54) = 0, the NHSE is stillexpected.The real-space Hamiltonian has a generalized PT sym-metry K H K = H , where K stands for complex conju-gation, as the hoppings are all real-valued[47]. Accord-ingly, the OBC eigenenergies are real-valued for suffi-ciently small γ . As usual, PT breaking transition occursat some threshold. An example is shown in Fig. 1(a),where the threshold γ is close to 0 .
1. This is reminiscentof 1D non-Bloch PT transitions [25, 26, 44].The surprise comes when we study the size depen-dence. Taking γ = 0 . L = 10 [Fig. 1(b)]. However, at L = 40 thePT symmetry is broken, and at L = 70 even more propor-tion of complex-valued eigenenergies appear [Fig. 1(c,d)].Similarly, taking other values of γ , no matter how small,we always see PT breaking by increasing L . This meansthat the threshold approaches zero as size increases. Thisbehavior drastically differs from Bloch (NHSE-free) PTbreaking and non-Bloch PT breaking in 1D [25, 26, 44].In both cases, the threshold generally converges to anonzero constant as size increases.To have a more complete picture, we calculate thecomplex proportion P = N c /N with varying γ and L , where N c and N denotes the number of complex-valued eigenenergies and all eigenenergies, respectively[Fig. 2(a)]. While PT transition generally occurs as γ in- creases for any fixed L , it occurs at smaller γ for larger L .To see the robustness of this trend, we add weak randomonsite disorder to the Hamiltonian H (cid:48) = H + (cid:88) x,y w ( x, y ) | x, y (cid:105)(cid:104) x, y | (2)with w ( x, y ) uniformly distributed in [ − W/ , W/ H ( k ) = ( m + t cos k x + t cos k y ) σ z + ∆ σ y + iγσ x , (3)where σ x,y,z are the Pauli matrices and all parame-ters are real-valued. The eigenvalues are E ± ( k ) = ± (cid:112) ( m + t cos k x + t cos k y ) + ∆ − γ . This BlochHamiltonian also has a generalized PT-symmetry AH ( k ) A = H ( k ) where A = σ z K and A = 1[47].Apparently, PT symmetry is unbroken when γ < min[ (cid:112) ( m + t cos k x + t cos k y ) + ∆ ]. Without NHSE,the OBC energies are similar to the PBC (Bloch) ener-gies, and exhibit a nonzero PT threshold that is almostindependent of L [Fig. 2(c)]. Non-perturbative mechanism and non-Hermiticity-size product.–
To understand the mechanism underlyingFig. 1, we consider the overlap between a pair of eigen-states: η ( n, m ) = |(cid:104) ψ n | ψ m (cid:105)| (cid:112) (cid:104) ψ n | ψ n (cid:105)(cid:104) ψ m | ψ m (cid:105) , (4)where H | ψ n (cid:105) = E n | ψ n (cid:105) , and similar for m . We order allthe real eigenvalues as E ≤ E ≤ · · · ≤ E N R . Whenan adjacent pair ( i, i + 1) undergo the real-to-complextransition, the two eigenvectors become parallel to eachother and we have η ( i, i +1) = 1 [6]. Therefore, a measureof the transition tendency is the mean value of overlaps¯ η = 1 N R − N R − (cid:88) i =1 η ( i, i + 1) . (5)When ¯ η is closer to 1, the real-to-complex transitionstend to occur more frequently as γ increases. The nu-merical ¯ η for model Eq. (1) is shown in Fig. 2(d,e), whichexhibits very similar trend as the corresponding complex-value proportion P in Fig. 2(a,b). Thus, η ( n, m ) and ¯ η contain useful information of the transition.Now we exploit this information to understand the in-triguing size-dependent behaviors. It should be explainedwhy a very small γ can induce η ( n, m ) = 1 for many ofthe ( n, m )’s, given that the reference point γ = 0 guar-antees η ( n, m ) = 0 ( n (cid:54) = m ). We express the real-spaceHamiltonian as H = H + iγV , where H and V are bothHermitian. We treat the non-Hermitian term iγV as aperturbation, so that the eigenstates read | ψ n (cid:105) = | ψ (0) n (cid:105) + iγ (cid:88) l (cid:54) = n | ψ (0) l (cid:105)(cid:104) ψ (0) l | V | ψ (0) n (cid:105) E (0) n − E (0) l + O ( γ ) , (6) FIG. 2. Complex eigenenergies proportion P and wavefunc-tion overlap ¯ η for L × L squares. (a) P for the single-band model [Eq. (1)], with t = 1 and s = 0 .
3. (b)The same as (a) except that onsite disorder is added with W = 0 .
05, and the data is the average from ten disorderconfigurations. (c) P for the NHSE-free model [Eq. (3)],with m = 0 . , t = 0 . , ∆ = 0. Numerically, eigenenergieswith imaginary part | Im( E ) | > − are regarded as com-plex. (d),(e),(f) The wavefunction overlap ¯ η corresponding to(a),(b),(c), respectively. The dashed line represents γL = 1in (b)(e) and γ = 0 . where {| ψ (0) n (cid:105)} are the unperturbed eigenstates spanningan orthonormal basis. By definition in Eq. (4), we have η ( n, m ) ≈ γ | (cid:104) ψ (0) n | V | ψ (0) m (cid:105) E n − E m | ≈ |(cid:104) ψ (0) n | ψ m (cid:105)| . (7)For the simpler case of PBC, n and m has definitewavevector k , k (cid:48) , respectively, and η ( n, m ) is propor-tional to |(cid:104) u (0) n ( k ) | u m ( k (cid:48) ) (cid:105)| δ k , k (cid:48) , where | u (cid:105) stands for theBloch wavefunction. Thus, η ( n, m ) = 0 when k (cid:54) = k (cid:48) ,and η ( n, m ) ∼ γ when k = k (cid:48) . This perturbation pic-ture breaks down only at Bloch-band-gap closing, when E (0) n ( k ) − E (0) m ( k ) = 0 and η ( n, m ) could be large ac-cording to Eq. (7). Requiring multiple bands, this con-ventional scenario is irrelevant to our single-band model.Consequently, a small γ generally cannot drive η ( n, m )to 1 and cause PT breaking. For an OBC system withoutNHSE, the energy spectrum is asymptotically the sameas in the PBC case for large size, so the same conclusion holds true.Crucially, this perturbation approach breaks down forOBC systems with NHSE. In this case, the eigenstate | ψ m (cid:105) ∼ exp( κ x x + κ y y ) | ψ (0) m (cid:105) ≈ (1 + κ x x + κ y y )) | ψ (0) m (cid:105) ,where κ x,y is of order γ (normalization of | ψ m (cid:105) is unim-portant because it only has higher-order contributions tothe following results). Therefore, it follows from Eq. (7)that η ( n, m ) ≈ |(cid:104) ψ (0) n | ( κ x x + κ y y ) | ψ (0) m (cid:105)| . Since x, y takevalues in { , , · · · , L } , we expect that (cid:104) ψ (0) n | x | ψ (0) m (cid:105) and (cid:104) ψ (0) n | y | ψ (0) m (cid:105) can reach the order of L for certain ( n, m ).Hence, η ( n, m ) ∼ γL. (8)The key feature is the presence of the L factor, whichwould be absent without NHSE. Regardless of how small γ is, it cannot be treated as a perturbation as size in-creases to L ∼ /γ , otherwise we would have the appar-ently wrong result η ( n, m ) >
1. While Eq. (8) is not sup-posed to be quantitatively precise, it offers a qualitativeunderstanding. It means η ( n, m ) ∼ γ ∼ /L ,and therefore η ( n, m ) = 1 is possible, enabling the PTbreaking. Numerically, we indeed see that γ ∼ /L is avisible characteristic scale of PT breaking [see the dashedline in Fig. 2(b)(e)]. Intuitively, Eq. (8) suggests the non-Hermiticity-size product γL as a measure of the effectivenon-Hermiticity strength. Thus, the weakness of non-Hermiticity requires γL being small, in addition to theusual requirement of γ/t being small.Now we turn to the NHSE-free model Eq. (3), for whichthe Bloch band theory applies and the OBC energies areasymptotically the same as the PBC ones. For PBC, E i and E i +1 generally correspond to different k , so thatthe overlap η ( i, i + 1) vanishes. Thus, Eq. (5) is not agood measure of PT breaking tendency in the absenceof NHSE. Instead, we define ¯ η = (cid:80) N R + i =1 η ( i, i (cid:48) ) /N R + forthe model Eq. (3), where E i (cid:48) = − E i < N R + is thenumber of positive real eigenenergies. For PBC, the op-posite eigenenergies E + ( k ) and E − ( k ) = − E + ( k ) sharethe same k , so that their wavefunction overlap can benonzero as non-Hermiticity is turned on. Since the Blochband theory applies, this definition is expected to remaininformative under OBC. One can see this from Fig. 2(f),in which the trend of ¯ η is similar to that of P in Fig. 2(c).We emphasize that for Bloch bands, modes with differ-ent k cannot be coupled by non-Hermitian perturbationswithout breaking the translational symmetry. Hence, PTbreaking can only occur in multi-band systems wherenon-Hermitian terms can couple different bands with thesame k . Without NHSE, taking OBC would not alterthis conclusion as the PBC and OBC energy bands areidentical. In contrast, the non-Bloch PT breaking canoccur even for single-band models, which indicates itsdifferent origin. For non-Bloch bands, energetically ad-jacent eigenstates from the same band are driven by theNHSE to be more parallel to each other as size increases, FIG. 3. Complex eigenenergies proportion P for the 1D modeland 3D cubic model (see text). (a) P for H on open chainswith length L . t = 1, s = 0 .
15. (b) P for H on L × L × L cubes. t = 1, s = 0 .
5. For (b), onsite disorder is added akinto Eq. (2), but only on boundary sites with W = 0 .
7. Thedata is the average from six disorder configurations. which enables the PT symmetry breaking. Since the en-tire band is involved, P can be of order unity even fora small γ ∼ /L . This sharply differs from the BlochPT breaking where, irrespective of the size, P is at mostproportional to γ [45, 46]. Dimensional surprise.–
Perhaps the most unexpectedaspect of our finding is the dependence on spatial dimen-sions. In the mechanism outlined above, NHSE appearsto be the only crux of the matter, irrespective of thedimension. However, it is known for 1D non-Bloch PTbreaking that the threshold does not approach zero assize increases [25, 26, 44]. For example, we consider a 1DHamiltonian H ( k ) = ( t − γ ) e ik + ( t + γ ) e ik + 2 s cos 2 k under OBC. As shown in Fig. 3(a), the threshold ap-proaches a nonzero constant as the size increases. Similarbehaviors are known for the non-Hermitian SSH model[16, 48].The puzzling difference between 1D and 2D can be un-derstood as follows. Eq. (8) means that η ( n, m ) ∼ γ ∼ /L , but it does not enforce η ( n, m ) = 1. Itis possible that the largest η ( n, m ) approaches 1 as sizeincreases, but never equal to 1. It turns out that it isindeed the case for 1D non-Bloch PT symmetry. Thestability of real eigenenergies as size increases is closelyrelated to a recently proved theorem stating that when L → ∞ the OBC eigenenergies form lines or arcs enclos-ing no area in the complex energy plane, while the Bloch H ( k ) eigenenergies form loops [49, 50]. A line can beentirely on the real axis, meaning that PT symmetry canbe exact at large size. Since this theorem stems from the1D GBZ properties that are not generalizable to higherdimensions [49], we believe that the 2D physics here ismore generic and intrinsic to non-Bloch PT symmetry.To confirm this understanding, we study a 3Dmodel H ( k ) = (cid:80) i = x,y,z [( t − γ ) e ik i + ( t + γ ) e − ik i ] +8 s cos k x cos k y cos k z . From Fig. 3(b), we see that thethreshold decreases towards 0 as size increases, which issimilar to 2D rather than 1D. Thus, the non-Bloch PTsymmetry in spatial dimensions higher than one shares FIG. 4. The complex eigenenergies proportion of non-Hermitian Chern bands. m = 1 .
4. (a) Square geometry withside length L . (b) Disk geometry with diameter D . similar universal features. Note that we exclude fine-tuned cases that can be similarity-transformed to a Her-mitian model, such as the s = 0 case of Eq. (1). At thefine-tuned points, the behavior of η ( n, m ) resembles thatof 1D. Non-Hermitian Chern bands.–
To further demonstratethe generality of the phenomenon, we consider a non-Hermitian Chern model in 2D [17, 51]: H ( k ) =(sin k x + iγ ) σ z + sin k y σ y +( m + cos k x + cos k y ) σ x . (9)The non-Bloch PT symmetry at small size has been no-ticed before [17], but its breaking at larger size was nottouched. Here, we calculate the size dependence of P for the square geometry, and find that P increases assize increases [Fig. 4(a)]. This trend is even stronger forthe disk geometry [Fig. 4(b)], though different geome-tries share qualitatively similar behavior. This is rea-sonable as the square geometry is more “regular” thanthe disk, leading to the suppression of certain matrix el-ements (cid:104) ψ (0) n | V | ψ (0) m (cid:105) [see Eq. (7)]. In fact, after addingweak disorder as in Eq. (2) to the square to break thesymmetry, we find that P is significantly increased, re-sembling that of the disk geometry.Finally, we emphasize that Eq. (9) only involves on-site gain/loss, meaning that nonreciprocal hopping isnot a necessary ingredient to induce the described phe-nomenon. Conclusions.–
We have uncovered generic behaviors ofnon-Bloch PT symmetry in spatial dimensions higherthan one. In the presence of NHSE, we find that theproduct of bare non-Hermiticity and system size is a mea-sure of the effective non-Hermiticity, meaning that evena weak non-Hermiticity becomes effectively strong as sizeincreases. This non-perturbative effect causes the asymp-totic vanishing of PT threshold, which is a universal prop-erty of non-Bloch PT breaking in two and higher dimen-sions. Notably, 1D systems evade the above physics, sug-gesting rich and unexpected interplay between non-Blochphysics and spatial dimensionality. Our theory is testableon several experimental platforms. For example, the non-reciprocal model Eq. (1) can be realized in topolectricalcircuits [38], and the onsite-non-Hermiticity model Eq.(9) can be constructed in coupled ring resonators withgain/loss [52, 53].
Acknowledgements.–
This work is supported by NSFCunder Grant No. 11674189. ∗ [email protected][1] Yuto Ashida, Zongping Gong, and MasahitoUeda, “Non-Hermitian Physics,” arXiv e-prints ,arXiv:2006.01837 (2020), arXiv:2006.01837 [cond-mat.mes-hall].[2] Emil J. Bergholtz, Jan Carl Budich, and Flore K.Kunst, “Exceptional Topology of Non-Hermitian Sys-tems,” arXiv e-prints , arXiv:1912.10048 (2019),arXiv:1912.10048 [cond-mat.mes-hall].[3] Ingrid Rotter, “A non-hermitian hamilton operator andthe physics of open quantum systems,” Journal of PhysicsA: Mathematical and Theoretical , 153001 (2009).[4] Carl M. Bender and Stefan Boettcher, “Real spectra innon-hermitian hamiltonians having pt symmetry,” Phys.Rev. Lett. , 5243–5246 (1998).[5] Carl M Bender, “Making sense of non-hermitian hamil-tonians,” Reports on Progress in Physics , 947 (2007).[6] S¸. K. ¨Ozdemir, S. Rotter, F. Nori, and L. Yang, “Parity-time symmetry and exceptional points in photonics,” Na-ture Materials , 783–798 (2019).[7] Ramy El-Ganainy, Konstantinos G Makris, MercedehKhajavikhan, Ziad H Musslimani, Stefan Rotter, andDemetrios N Christodoulides, “Non-hermitian physicsand pt symmetry,” Nature Physics , 11 (2018).[8] Mohammad-Ali Miri and Andrea Al`u, “Exceptionalpoints in optics and photonics,” Science (2019),10.1126/science.aar7709.[9] Liang Feng, Zi Jing Wong, Ren-Min Ma, Yuan Wang,and Xiang Zhang, “Single-mode laser by parity-time sym-metry breaking,” Science , 972–975 (2014).[10] Hossein Hodaei, Mohammad-Ali Miri, Matthias Hein-rich, Demetrios N Christodoulides, and Mercedeh Kha-javikhan, “Parity-time–symmetric microring lasers,” Sci-ence , 975–978 (2014).[11] Liang Feng, Maurice Ayache, Jingqing Huang, Ye-LongXu, Ming-Hui Lu, Yan-Feng Chen, Yeshaiahu Fainman,and Axel Scherer, “Nonreciprocal light propagation in asilicon photonic circuit,” Science , 729–733 (2011).[12] Christian E R¨uter, Konstantinos G Makris, RamyEl-Ganainy, Demetrios N Christodoulides, MordechaiSegev, and Detlef Kip, “Observation of parity–time sym-metry in optics,” Nature physics , 192 (2010).[13] Bo Peng, S¸ahin Kaya ¨Ozdemir, Fuchuan Lei, FarazMonifi, Mariagiovanna Gianfreda, Gui Lu Long, Shan-hui Fan, Franco Nori, Carl M Bender, and LanYang, “Parity–time-symmetric whispering-gallery micro-cavities,” Nature Physics , 394 (2014).[14] Zin Lin, Hamidreza Ramezani, Toni Eichelkraut,Tsampikos Kottos, Hui Cao, and Demetrios N.Christodoulides, “Unidirectional invisibility induced by PT -symmetric periodic structures,” Phys. Rev. Lett. , 213901 (2011).[15] Alois Regensburger, Christoph Bersch, Mohammad-Ali Miri, Georgy Onishchukov, Demetrios N Christodoulides,and Ulf Peschel, “Parity–time synthetic photonic lat-tices,” Nature , 167 (2012).[16] Shunyu Yao and Zhong Wang, “Edge states and topo-logical invariants of non-hermitian systems,” Phys. Rev.Lett. , 086803 (2018).[17] Shunyu Yao, Fei Song, and Zhong Wang, “Non-hermitian chern bands,” Phys. Rev. Lett. , 136802(2018).[18] Flore K. Kunst, Elisabet Edvardsson, Jan Carl Budich,and Emil J. Bergholtz, “Biorthogonal bulk-boundary cor-respondence in non-hermitian systems,” Phys. Rev. Lett. , 026808 (2018).[19] Ching Hua Lee and Ronny Thomale, “Anatomy of skinmodes and topology in non-hermitian systems,” Phys.Rev. B , 201103 (2019).[20] V. M. Martinez Alvarez, J. E. Barrios Vargas, andL. E. F. Foa Torres, “Non-hermitian robust edge states inone dimension: Anomalous localization and eigenspacecondensation at exceptional points,” Phys. Rev. B ,121401 (2018).[21] Kazuki Yokomizo and Shuichi Murakami, “Non-blochband theory of non-hermitian systems,” Phys. Rev. Lett. , 066404 (2019).[22] Zhesen Yang, Kai Zhang, Chen Fang, and JiangpingHu, “Non-hermitian bulk-boundary correspondence andauxiliary generalized brillouin zone theory,” Phys. Rev.Lett. , 226402 (2020).[23] Tian-Shu Deng and Wei Yi, “Non-bloch topological in-variants in a non-hermitian domain wall system,” Phys.Rev. B , 035102 (2019).[24] Kohei Kawabata, Nobuyuki Okuma, and MasatoshiSato, “Non-bloch band theory of non-hermitian hamilto-nians in the symplectic class,” Phys. Rev. B , 195147(2020).[25] Stefano Longhi, “Non-Bloch PT symmetry breaking innon-Hermitian photonic quantum walks,” Optics Letters , 5804 (2019), arXiv:1909.06211 [physics.optics].[26] Stefano Longhi, “Probing non-hermitian skin effect andnon-bloch phase transitions,” Phys. Rev. Research ,023013 (2019).[27] Fei Song, Shunyu Yao, and Zhong Wang, “Non-hermitian skin effect and chiral damping in open quan-tum systems,” Phys. Rev. Lett. , 170401 (2019).[28] Fei Song, Shunyu Yao, and Zhong Wang, “Non-hermitian topological invariants in real space,” Phys.Rev. Lett. , 246801 (2019).[29] Yifei Yi and Zhesen Yang, “Non-hermitian skin modesinduced by on-site dissipations and chiral tunneling ef-fect,” Phys. Rev. Lett. , 186802 (2020).[30] Linhu Li, Ching Hua Lee, Sen Mu, and JiangbinGong, “Critical non-Hermitian skin effect,” Nature Com-munications , 5491 (2020), arXiv:2003.03039 [cond-mat.mes-hall].[31] S. Longhi, “Non-bloch-band collapse and chiral zenertunneling,” Phys. Rev. Lett. , 066602 (2020).[32] Tao Liu, Yu-Ran Zhang, Qing Ai, Zongping Gong,Kohei Kawabata, Masahito Ueda, and Franco Nori,“Second-order topological phases in non-hermitian sys-tems,” Phys. Rev. Lett. , 076801 (2019).[33] Ching Hua Lee, Linhu Li, Ronny Thomale, and JiangbinGong, “Unraveling non-hermitian pumping: Emergentspectral singularities and anomalous responses,” Phys.Rev. B , 085151 (2020). [34] Chun-Hui Liu, Kai Zhang, Zhesen Yang, and Shu Chen,“Helical damping and dynamical critical skin effect inopen quantum systems,” Phys. Rev. Research , 043167(2020).[35] Kazuki Yokomizo and Shuichi Murakami, “Non-Blochband theory in bosonic Bogoliubov-de Gennes sys-tems,” arXiv e-prints , arXiv:2012.00439 (2020),arXiv:2012.00439 [cond-mat.mes-hall].[36] Zhesen Yang, “Non-perturbative Breakdown of Bloch’sTheorem and Hermitian Skin Effects,” arXiv e-prints, arXiv:2012.03333 (2020), arXiv:2012.03333 [cond-mat.mes-hall].[37] Kazuki Yokomizo and Shuichi Murakami, “Non-Blochband theory and bulk-edge correspondence in non-Hermitian systems,” Progress of Theoretical and Exper-imental Physics , 12A102 (2020), arXiv:2009.04220[cond-mat.mes-hall].[38] T Helbig, T Hofmann, S Imhof, M Abdelghany,T Kiessling, LW Molenkamp, CH Lee, A Szameit, M Gre-iter, and R Thomale, “Generalized bulk–boundary cor-respondence in non-hermitian topolectrical circuits,” Na-ture Physics , 747 (2020).[39] Lei Xiao, Tianshu Deng, Kunkun Wang, Gaoyan Zhu,Zhong Wang, Wei Yi, and Peng Xue, “Non-Hermitianbulk-boundary correspondence in quantum dynamics,”Nature Physics , 761 (2020), 1907.12566 [cond-mat.mes-hall].[40] Sebastian Weidemann, Mark Kremer, Tobias Helbig, To-bias Hofmann, Alexander Stegmaier, Martin Greiter,Ronny Thomale, and Alexander Szameit, “Topologicalfunneling of light,” Science , 311–314 (2020).[41] Tobias Hofmann, Tobias Helbig, Frank Schindler,Nora Salgo, Marta Brzezi´nska, Martin Greiter, TobiasKiessling, David Wolf, Achim Vollhardt, Anton Kabaˇsi,Ching Hua Lee, Ante Biluˇsi´c, Ronny Thomale, and Ti-tus Neupert, “Reciprocal skin effect and its realization ina topolectrical circuit,” Phys. Rev. Research , 023265(2020).[42] Ananya Ghatak, Martin Brandenbourger, Jasper vanWezel, and Corentin Coulais, “Observation of non-hermitian topology and its bulk–edge correspondence inan active mechanical metamaterial,” Proceedings of theNational Academy of Sciences , 29561–29568 (2020). [43] Lucas S. Palacios, Serguei Tchoumakov, Maria Guix, Ig-nasio Pagonabarraga, Samuel S´anchez, and Adolfo G.Grushin, “Guided accumulation of active particles bytopological design of a second-order skin effect,” arXive-prints , arXiv:2012.14496 (2020), arXiv:2012.14496[cond-mat.soft].[44] Lei Xiao, Tianshu Deng, Kunkun Wang, Zhong Wang,Wei Yi, and Peng Xue, “Observation of non-Blochparity-time symmetry and exceptional points,” arXive-prints , arXiv:2009.07288 (2020), arXiv:2009.07288[quant-ph].[45] I. V. Barashenkov, L. Baker, and N. V. Alexeeva,“ PT -symmetry breaking in a necklace of coupled opti-cal waveguides,” Phys. Rev. A , 033819 (2013).[46] Vladimir V. Konotop, Jianke Yang, and Dmitry A.Zezyulin, “Nonlinear waves in PT -symmetric systems,”Rev. Mod. Phys. , 035002 (2016).[47] Carl M Bender, M V Berry, and Aikaterini Mandilara,“Generalized PT symmetry and real spectra,” Journalof Physics A: Mathematical and General , L467–L471(2002).[48] Chuanhao Yin, Hui Jiang, Linhu Li, Rong L¨u, and ShuChen, “Geometrical meaning of winding number and itscharacterization of topological phases in one-dimensionalchiral non-hermitian systems,” Phys. Rev. A , 052115(2018).[49] Kai Zhang, Zhesen Yang, and Chen Fang, “Correspon-dence between winding numbers and skin modes in non-hermitian systems,” Phys. Rev. Lett. , 126402 (2020).[50] Nobuyuki Okuma, Kohei Kawabata, Ken Shiozaki, andMasatoshi Sato, “Topological origin of non-hermitianskin effects,” Phys. Rev. Lett. , 086801 (2020).[51] Huitao Shen, Bo Zhen, and Liang Fu, “Topological bandtheory for non-hermitian hamiltonians,” Phys. Rev. Lett. , 146402 (2018).[52] Mohammad Hafezi, Eugene A. Demler, Mikhail D.Lukin, and Jacob M. Taylor, “Robust optical delay lineswith topological protection,” Nature Physics , 907–912(2011), arXiv:1102.3256 [quant-ph].[53] Miguel A. Bandres, Steffen Wittek, Gal Harari, MidyaParto, Jinhan Ren, Mordechai Segev, Demetrios N.Christodoulides, and Mercedeh Khajavikhan, “Topo-logical insulator laser: Experiments,”359