Abstract

We study mode coupling of an edge state with bulk states in a coupled waveguide chain in the presence of gain and losses. In a low non-Hermiticity regime, it is found that the edge state associated with the interface between two topologically different waveguide chains is well isolated from the bulk states, sharing the exact same features as the well-known Su-Schrieffer-Heeger Model. As the non-Hermiticity increases, the two bands of the overall waveguide chain merge together, which leads to the overlap in the band diagram between the edge state and states of lower bands. We find that before the PT-symmetry breaking, the edge state is strongly coupled to bulk states of lower bands, evident by the anti-crossing feature in the band diagram. The strong mode hybridization is verified by a non-Hermitian coupled mode theory developed from reaction conservation, and is further examined using phenomenological models. We envisage that the strong coupling between the edge state and the bulk state may be useful in expanding our understanding in topological photonics in non-Hermitian condition, as well as in applications such as mode conversion between edge and bulk states.

© 2017 Optical Society of America

1. Introduction

Edge states emerge at the interfaces formed by two topologically distinct bulk materials [1–4]. And the number of the edge states is determined by the topological invariant for each individual band associated with different bulk materials. The link between the number of the edge states and the topological properties of the bulk materials, i.e., topological invariant, is coined as bulk-edge correspondence, which is discovered in mathematics and found further applications in quantum field theory, condense matter physics, as well as optics [5–9]. Thank to such theorem, a number of models and interesting phenomena, such as the Su-Schrieffer-Heeger (SSH) model for polyacetylene [10,11], quantum Hall effect (TKNN invariant) [12,13], Jackiw-Rebbi model [14,15], topological insulator [16,17] and many others, can be explained in a unified but concise manner.

The connection between the existence of the edge states and bulk properties has been investigated extensively in electronics, acoustics, and optics [18–23]. Very much attention is drawn to designing particular structures that fulfill certain requirements, to favor or forbid the existence of edge states. Ideally, those edge states are ’isolated’ such that they don’t interact with bulk states, as well as other edge states with different propagation direction (time reversal partner) or different polarization, which eventually leads to many interesting features, such as one-way transportation, back-scattering immunity, or optical diode [22–26]. All the aforementioned features about the edge states apply for the Hermitian bulk system without any imperfections.

Practically, one edge state may be possibly coupled to bulk modes or the edge state on the opposite edge. Indeed, a ’magnetic’ scatter can give rise to scattering of the spin edge state in topological insulator, making it couple to the oppositely propagating state. In non-Hermitian settings, it is found that much more is involved. As an example, the system with parity-time (PT) symmetry is one of typical types of non-Hermitian systems. In optics, PT-symmetric system needs the refractive index with the space distribution n(x) = n*(−x) (* respects complex conjugation). In such system, the eigenvalue spectrum is still real with smaller non-Hermiticity, while with large enough non-Hermiticity, the real spectrum bifurcates into complex plane. In recent years, there are several papers about PT-symmetric systems with edge states and their properties. Zhao and co-authors found that robust light states exist due to the PT-symmetric transition in non-Hermitian optical materials [27]. Malzard and co-authors found that topologically protected edge states exist in non-Hermitian system, while disappear in the corresponding Hermitian limit [28]. Interestingly, Tony Lee pointed out that in one-dimensional non-Hermitian lattice system, the topological invariant becomes fractional, which may lead to further modification of bulk edge correspondence in general non-Hermitian settings [29].

In this paper, we show that at low non-Hermitian regime, the edge states of a dimerized waveguide chain exist regardless of whether the overall structure respect PT-symmetry globally or not. We further show that the edge and bulk states can be strongly coupled, yielding anti-crossing feature as the non-Hermiticity increases. The mode hybridization between the edge and bulk states is due to the presence of non-Hermiticity, leading the merging of lower and upper bands, that inevitably gives rise to the overlapping of the edge state and bulk band states in the band diagram. Due to the presence of the strong coupling between the bulk states and edge states, those states may be adiabatically converted into each other in a judiciously designed PT-symmetric waveguide chain. Our results are further verified by a non-Hermitian coupled mode theory developed before [30].

The paper is organized as follows. In Section 2, we analyze the properties bulk states in dimerized PT-symmetric waveguide chains. In section 3, we study the edge state in waveguide chain with and without global PT-symmetry. It is found that the edge state exists in lower non-Hermiticity regime, and interacts strongly with the bulk states in high non-Hermiticity regime. In section 4, we study the strong modal hybridization between the edge state and the bulk states using a non-Hermitian coupled equation, as well as using phenomenological model. Lastly, Section 5 concludes the paper.

2. Bulk properties of the dimerized PT-symmetric waveguide chains

We examine the properties of the bulk states of dimerized PT-symmetric waveguide chains, in analogy to the SSH model, as shown in Fig. 1. In contrast to the SSH model, we replace the carbon atoms with single-mode waveguides. The width and height of the single-mode waveguide are given by w = 0.3λ0 and h = 0.1λ0, λ0 being vacuum wavelength, from which one obtain the effective refractive index of the single-mode waveguide neff0=1.51. To form a dimerized waveguide lattice, each unit cell consists of two identical single-mode waveguides, where d is the distance between the two waveguides, and D is the lattice constant. Depending on d > D/2 or d < D/2, The interaction of waveguides inside each unit cell can be taken as long bond or short bond in analogy to the SSH model. We study two different coupled waveguide lattices in Fig. 1, denoted by waveguide lattice A and B respectively. The waveguide lattice A and B share all the same geometrical parameters, except that the periodicity D differs. Specifically, the periodicity D of lattice A equals to 1.2λ0, while the periodicity D of lattice B equals to 1.3λ0. In both A and B structures, d = 0.8λ0. The relative permittivities of two waveguides are given by r=r0±iΔ to respect PT symmetry, indicated by red (gain) and blue (loss) rectangles shown in the insert of Figs. 1(b) and 1(e). The band structures as well as the modes of periodic waveguide chains are calculated using a commercial software package, COMSOL MULTIPHYSICS 5.2 [31].

 

Fig. 1 Bulk properties of the dimerized PT-symmetric waveguide lattices A and B. Each unit cell contains two identical single-mode waveguides with width and height denoted by w and h. The working wavelength is λ0, and the relative permittivity of single-mode waveguide core is r0=10, embedded in air cladding. (a)–(b)/(d)–(e) show the real part and imaginary part of eigenvalues of band structure with four values of Δ for waveguide lattice A/B. (c)/(f) shows the width of the projected band Δneff and the band-center as function of non-Hermiticity Δ.

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In Figs. 1(a)–1(c), the band structures of waveguide lattice A, i.e., the effective refractive index (denoted as neff onward for short) of these dimerized waveguide chains are calculated for four different Δ = 0, 0.25, 0.32, 0.4), from which we will discuss the properties of bulk state involution as non-Hermiticity increases. Firstly, at Δ = 0, the waveguide lattice is Hermitian and has real eigenvalues, i.e., Im(neff) = 0, evident by red triangles shown in Figs. 1(a) and 1(b). In this phase, the topological properties of the wave function associated with the lower and upper bands of the coupled waveguide lattice share the exact same feature as the SSH model. From full-wave simulation, one find the Zak phases of the lower and upper bands are 0 for short bond structure, while Zak phases are π for long bond structure, indicating the existence of distinct topological phases between two cases, i.e., the structure parameter d < D/2 and d > D/2. Secondly, at Δ = 0.25, the eigenvalues of the waveguide lattice, i.e., neff, remains real, thus one can expect that it shares similar topological features as Δ = 0. As the non-Hermiticity increases up to Δ = 0.32, one can find that the lower band and upper bands merge together at larger Bloch k-vector. Meanwhile, the imaginary part of neff splits. This is exactly the same as the PT-symmetry breaking phenomenon, with the parameter space being the Bloch k-vector, rather than being Δ which directly quantifies the non-Hermiticity. Lastly, it is found that at Δ = 0.4, the real parts of neff of the lower and upper bands overlap completely, while imaginary parts of neff are split throughout the Brillouin zone. From the viewpoint of the PT symmetry, We note that the aforementioned bulk state properties with different Δ can be classified as unbroken phase, half-broken phase, and completely broken phase as discussed in [27].

In Figs. 1(d)–1(f), the band structure of the waveguide lattice B is studied. The red triangles (blue circles) in Figs. 1(d) and 1(e) shows the two bands for Δ = 0 (Δ = 0.2), which corresponds to the unbroken phase (completely broken phase). Together with other values Δ (not shown here), we find that the band properties of the waveguide lattice share the similar features as that of Figs. 1(a) and 1(b), which depends on Δ and can be classified into three phases. However, there is few subtle differences in the band center and the width of the projected upper and lower bands of the two waveguide lattices, shown in Figs. 1(c) and 1(f). The left y-axis in panel (c)/(f) shows the band-width of the projected upper and lower bands in waveguide lattice A/B, while the right y-axis shows the band-center between upper and lower bands at kx = 0 in waveguide lattice A/B. In Figs. 1(c) and 1(f), one find that the changing of the band width of two bands have the same tendency. The two band width increase at first, and then suddenly reduce to a certain value with large enough Δ, and finally remain unchanged. As regards to the band center, one can realize that band-center denoted by neffc in waveguide lattice A, i.e., neff,Ac=1.52, is larger than that of waveguide lattice B, which approximately equals to the modal index of single waveguide( neff,Bc=neff0). The rationale behind the subtle difference between neff,Ac and neff,Bc, with respect to neff0, is the following: (1) the field distribution of upper band in waveguide lattice A is largely concentrated in a tiny gap with width of 0.1λ0, which gives rise to a larger effective modal index of the upper band. Meanwhile, the field distribution of the lower band is largely concentrated in the waveguide core medium. Thus, the overall effect of the field re-distribution gives rise to a up-ward shift of the band-center, as shown in Fig. 1(c); (2) as for waveguide lattice B, the fields of the lower and upper bands mostly remain inside the waveguide core, therefore the band-center coincides with neff0. The relatively large difference between neff,Ac and neff0, which turns out to be necessary to obtain the strong coupling between the edge state and the bulk states, will be discussed in the following section.

3. Edge state in a non-Hermitian waveguide chain

In Hermitian waveguide lattice structure with infinite periods, i.e., Δ = 0, there is a topological transition of the wavefunction for each band as the distance of the two waveguides d crosses D/2, which separates two distinct topological phases denoted by difference Zak phases, one is π and the other is 0. Interestingly, the mode properties of waveguide lattice structure with finite length, which will be referred as waveguide chains onward, crucially depends on bulk state properties of waveguide lattice. For a one-dimensional (1D) waveguide chain containing even number, i.e., 2N, single-mode waveguides, there are basically two options to group the single-mode waveguides into N unit cells, which can be either N short-bond unit cells or N long-bond unit cells. Due to topological reasons, there is no edge states associated with the waveguide chain that only contains short-bond unit cells. However, for a waveguide chain that only contains long-bond unit cells, there are two edge states localized at the two ends of the waveguide chain, due to the difference of the topological phases at the interface between the waveguide chain and air. When the chain is of finite length, two edge states interact with each other and split into an even mode and an odd mode. The splitting is quite small as both modes locate inside the bulk band gap region. As the length of waveguide chain becomes infinite, the splitting will go to zero. For a non-Hermitian long-bond waveguide chain, one can immediately realize that the two edges become PT-symmetric broken even though the perturbation of the non-Hermiticity Δ is very small, due to the very small difference of neff between the two edge states.

In this paper, we intend to explore how the non-Hermiticity impacts the edge state, thus we study the waveguide chain that contains odd number (4N + 1 and N is an integer) single mode waveguides in total to avoid the complication of the mode coupling between edge states. The 1D waveguide chains sketched in Figs. 2(a) and 2(b) can be considered as one waveguide chain with N short-bond unit cells, in combination with another waveguide chain that contains N + 1/2 unit cells, and N = 4 as an example. In Figs. 2(a) and 2(b), the coupled waveguide chain only supports one edge state, and preserves the local PT-symmetry at the meantime. The local PT-symmetry here refers to the fact that refractive indices n(x) in the unit cell obey the PT symmetric conditions, i.e., Re(n(x)) = Re(n(−x)) and Im(n(x)) = −Im(n(−x)), with respect to the mirror plane in the center of the unit cell. Specifically, the chains in Fig. 2 comprise two parts with seventeen waveguides. The left part ends with long bond composed of nine waveguides, and the right part starts with short bond composed of eight waveguides, thus the edge state appears in the ninth waveguides which can been seen as an interface. Globally, the coupled waveguide chains can be either PT symmetric or PT-symmetry broken, which are corresponding to the structures in Figs. 2(a) and 2(b), respectively. The structure on the left is PT symmetric which called GPT for short, and the other one on the right is local PT symmetric called LPT. Red (blue) rectangles stand for waveguides with gain (losses). The white one is the waveguide without gain and losses. In Fig. 2, the periodicity D of one unit cell is 1.2λ0, and the smaller space and larger space between two waveguides in one unit cell are dS = 0.4λ0 and dL = 0.8λ0, as same as the parameters of the unit cell in the inset of Fig. 1(b).

 

Fig. 2 Edge state in coupled waveguide chains with global PT symmetry and local PT symmetry which consist of seventeen waveguides. (a) is the structure of PT symmetric waveguide chain with the condition n(x) = n*(−x), and x = 0 is the center of the structure. The larger space between two waveguides dL = 0.8λ0, the smaller one dS = 0.4λ0 and the periodicity D = 1.2λ0. (b) is the structure of waveguide chain with local PT symmetry and mirror symmetry. Local PT symmetry is in the unit cell with two waveguides and mirror symmetry means n(x) = n*(−x) with the center of the structure. (c) and (d) show the band dispersion of the structure in (a) and (b), respectively.

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From topological argument, one can expect that an edge state exists between the waveguide chains with different Zak phases. As shown in Figs. 2(c) and 2(d), it can be found that there are several similarities between the GPT and LPT structures. In low non-Hermiticity regime, i.e., Δ < 0.3, the edge state exists in GPT structure as well as in LPT structure. The existence of edge state at lower Hermiticity is ensured by sub-lattice symmetry in the waveguide chain, regardless of the global PT-symmetric property of the overall structure. Apparently, the edge state is robust in the presence of lower non-Hermiticity, i.e., Δ < 0.3, evident by the relatively large bandgap between the edge state and bulk states, as shown in Figs. 2(c) and 2(d). Apart from those similarities, there are still few distinctions in the band-diagram between GPT structure and LPT structure. Firstly, there is a clear transition point, namely, the exceptional point, where the PT-symmetric breaking occurs in GPT structure. In contrast, there is no such sharp transition point in LPT structure. The absence of exceptional points is due to the lack of the exact global PT-symmetry, which is different from the scenarios where the asymmetric coupling plays a role [32]. Secondly, the imaginary parts of neff of the edge state in GPT structure is exactly 0, the imaginary parts of neff of the LPT waveguide chain fluctuate around zero. Those distinctions indicate that the edge state of the GPT system is more robust than that of the LPT system, although the edge state can emerge in both structures.

We also note the fact that the edge state inevitably intersects with the bulk states from the lower bands for a number of times in Figs. 2(c) and 2(d). The rationale behind such an inevitable intersection is the following: Firstly, the edge state is located in the band gap and is rather isolated from the bulk states, with the effective modal index given by neff0. We also note that the effective modal index of the edge state is flat as non-Hermiticity increases; Secondly, as discussed in the previous section, the band-center of the waveguide lattice (D = 1.2λ0) is shifted upward ( neff,Ac>neff0), which is actually the location where the upper and lower bands merger together after the exceptional point; Lastly, one can immediately conclude that the edge state has to intersect with the lower bands, as non-Hermiticity increases. In Fig. 2(c), the edge state anti-crosses the lower band four times, which is equal to half of the number of pairs of waveguide chain, as well as half of the number of the modes in the lower band of the overall structure. In order to have a close look at the mode hybridization, i.e., the anti-crossing between the edge state and bulk states, we proceed to discuss a waveguide chain containing 5 guides in total using phenomenological model and non-Hermitian coupled mode formalism in the next section.

4. Anti-crossing of the edge state with bulk states

4.1. General coupled mode theory in waveguide chain

In this section, we study mode coupling between edge state and bulk states in a simplified structure that consists of five single-mode waveguides. The results in Figs. 3(a) and 3(b) are obtained from two independent approaches, blue circles obtained with COMSOL simulation, red stars obtained from non-Hermitian coupled mode equation denoted by GCMT [30]. Figure 3(a)(Figure 3(b)) show real (imaginary) part of effective modal index nneff as a function of non-Hermiticity. In such a simple structure, we can see clearly that there is a strong mode hybridization between the edge state and the bulk states from the lower bands, which is similar to Fig. 2(a) and will be re-examined using GCMT. Figure 3(c) sketches two options of the edge state intersecting with the bulk states: either crossing or anti-crossing as non-Hermiticity increases. From the full-wave simulation indicated by the black box in Fig. 3(a), one can see that the edge state couples to the bulk modes in non-trivial way, featuring as anti-crossing shown in Fig. 3(c2).

 

Fig. 3 Band structure and mode evolution sketch of waveguide chain with five waveguides. (a) and (b) show the real and imaginary part of band dispersion in coupled waveguide chains with global PT symmetry which consist of five waveguides, respectively. Blue circles represent the eigenvalues from COMSOL, and red stars represent the eigenvalues from GCMT. (c) The mode evolution in the mode crossing regime, i.e., the black rectangle in (a). (d) The values of k31, k32, k34, k35 and k45 with varying Δ.

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We proceed to discuss the non-trivial mode coupling between the edge state and bulk states using non-Hermitian coupled mode formalism. Based on GCMT in [30], we can get the coupling equation as follows,

jaj(ββ0,j)pij=jajkij,
where pij=z(e0,j×h0,i+)z(e0,i+×h0,j)dxdy, kij=k0Δ(x,y)e0,je0,i+dxdy, e0,i±(h0,i±) correspond to the electric (magnetic) field of the ith mode without perturbation in the transverse cross section for forward (+) and backward (−) propagation, and more details refer to [30]. 0 and i(j) represent no perturbation case and mode labels, respectively. aj is the coupled amplitude with the unperturbed field of the adjoint system. β0,j and β are eigenvalues of no perturbation case and perturbation case, respectively. It also can be rewritten in the form of eigenvalue problem. Based on Eq. (1), the coupled mode equation of five single-mode waveguides chain in our study can be explicitly written out as follows,
[h11h12h13h14h15h21h22h23h24h25h31h32h33h34h35h41h42h43h44h45h51h52h53h54h55][a1a2a3a4a5]=neffk0[p11p12p13p14p15p21p22p23p24p25p31p32p33p34p35p41p42p43p44p45p51p52p53p54p55][a1a2a3a5a5],
where hij = βj pijikij, i/j labels the five waveguide modes under study, and aj (βj) refers to the mode amplitude (propagation constant) of j mode for the perturbed system, neff is modal index of perturbation system. Specifically, i and j label the 5 modes, neff of which increases as i/j increases, as shown in Fig. 3(a). For instance, mode label i = 3 refers to the edge state.

Essentially, Eq. (2) gives the mode hybridization in the full system under a small perturbation from Δ, marked as red stars in Figs. 3(a) and 3(b). As can be seen, the neff obtained from GCMT agrees well with those from the full-wave simulation. There also exists an edge state in band dispersion, which anti-crosses with bulk states. Evident in Eq. (2), it is the term kij that reflects the interaction among the relevant states. In Fig. 3(a), one can see that the edge state couples strongly with the two states from the lower bands at Δ ≥ 0.3, where the coupling coefficients k31 and k32 become significant. Indeed, Fig. 3(d) shows the calculated values of k31, k32, k34, k35 and k45 as functions of Δ, where k45 may be taken as a reference. It shows that the values of k3j are zero at small Δ. At Δ ≥ 0.3, the values of k3j increases with Δ. In particular, k31 and k32 are much larger than k34 and k35, which indicates that the edge state is coupled strongly to the lowest two bands, in consistency with results in Fig. 3(a).

4.2. Phenomenological model with anti-crossing

The emergence of edge state in Fig. 3 is mainly due to left waveguide chain with a long band end as indicated by the dashed rectangle in the insert of Fig. 3(b), thus we can choose such chain with 3 waveguides to make the coupled system further simpler. In Figs. 4(a) and 4(b), we obtain the band dispersion of three waveguides with COMSOL, marked as blue circles, in which there exist three bands and the edge state is coupled with bulk state in the form of anti-crossing. Similar to Fig. 3(c), Fig. 4.1(c) sketches two kinds of mode evolution of the intersection between two modes. In Fig. 4.1(c), (1) shows that two modes cross each other without coupling, and (2) shows that they are coupled with each other yielding anti-crossing, which is the same as the edge state coupled with the bulk state of lower band in Fig. 3(a). Therefore, the coupling of the three waveguides can be separated into two different mechanisms: one is the coupling between the supermodes induced by the perturbation from the local PT-symmetric condition, the other is the coupling between the edge states with the supermodes with smaller modal index. Phenomenologically, the Hamiltonian of those three waveguides system can been written as

H=[βoddiV10iV1βeveniV20iV2βsingle],
where βodd and βeven are eigenvalues of the odd and even modes of two waveguides with short bond and without gain and losses respectively, βodd = 1.57, βeven = 1.47, while βsingle is eigenvalue of the mode in single-mode waveguide without gain and losses, βsingle = 1.51. V1 and V2 are the mode coupling parameters which relate to the value of Δ. Since the edge state has no reaction with bulk state out of the region Δ ∈ [0.29, 0.34], the value of V2 is negligibly small and can be ignored, while inside of this region we can choose a reference point and linearly fit the trend of V2 as a function of Δ. Thus by parameter fitting, we can obtain V1 = 0.157Δ, V2={0.068Δ+0.023+(0.043+0.012)iΔ[0.29,0.34],0Δ[0.29,0.34]. Plugging these parameter in Eq. (3), we obtain mode coupling of three waveguides system as sketched in the inset on the top of Fig. 4.1(b), which are plotted in Figs. 4(a) and 4(b) marked as red dots. And the results are consistent with those from COMSOL. From Eq. (3), it is shown directly that the edge state is coupled with bulk state in the lower band, but there is no direct interaction with the bulk state in the upper band.

 

Fig. 4 Band structure and the sketch of mode evolution of waveguide chain with three waveguides. (a/b) shows the real (imaginary) part of band dispersion in coupled waveguide chains. Blue circles represent the eigenvalues from full-wave simulation, and Red dots represent results from phenomenological models. (c) The mode evolution of two mode intersection.

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5. Conclusion

We give a detailed examination of bulk state in a dimerized PT symmetric waveguide lattice with infinite periods, as well as the properties of the edge state in a finite dimerized PT symmetric waveguide chain in non Hermitian settings. In the Hermitian limit, we find that the bulk states have real eigenvalues, and the topological invariant associated with each band admits either 0 or π, which resembles SSH model. Consequently, the edge state can be formed at the interface, where two topologically different waveguide chains are combined. Further, we find that the edge state can exist in both global PT-symmetric and local PT-symmetric structures. The existence of edge state is ensured by the sub-lattice PT-symmetry, though the edge state in global PT-symmetric structure is more robust. As the non-Hermiticity increases, we find that the edge state can strongly couple to the bulk states from the lower band. It is due to the particular geometric configuration of waveguide lattice that shifts the band center in the band diagram upwards. Consequently, the shifts lead to inevitable crossing between the edge state and the lower bands. We further examine the strong coupling of the edge state to lower band states using non-Hermitian coupled mode theory, as well as phenomenological model. Indeed, both approaches indicate a significant interaction between the edge state and the bulk states, which leads to a strong hybridization between the edge state and the bulk states. We envisage that the strong hybridization may be useful for mode conversion between bulk and edge states in non-Hermitian settings.

Funding

National Natural Science Foundation of China (NSFC) (Grant No. 61405067, 61405066 and 61275201); Foundation for Innovative Research Groups of the Natural Science Foundation (NSF) of Hubei Province (Grant No. 2014CFA004); Research Grants Council (RGC), University Grants Committee (UGC), Hong Kong (AoE/P -02/12).

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24. P. Wang, L. Lu, and K. Bertoldi, “Topological phononic crystals with one-way elastic edge waves,” Phys. Rev. Lett. 115, 104302 (2015). [CrossRef]   [PubMed]  

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26. K. X. Liu, L. F. Shen, and S. He, “One-way edge mode in a gyromagnetic photonic crystal slab,” Opt. Lett. 37(19), 4110–4112 (2012). [CrossRef]   [PubMed]  

27. H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in non-Hermitian optical materials,” Sci. Rep. 5, 17022 (2015). [CrossRef]   [PubMed]  

28. S. Maizard, C. Poli, and H. Schomerus, “Topological protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity-time symmetry,” Phys. Rev. Lett. 115, 200402 (2015). [CrossRef]  

29. T. E. Lee, “Anomalous edge state in a non-Hermitian lattice,” Phys. Rev. Lett. 116, 133903 (2016). [CrossRef]   [PubMed]  

30. J. Xu and Y. Chen, “General coupled mode theory in non-Hermitian waveguides,” Opt. Express 23(17), 22619–22627 (2015). [CrossRef]   [PubMed]  

31. COMSOL MULTIPHYSICS 5.2, (2015). COMSOL Multiphysics: a finite element analysis, solver and simulation software for various physics and engineering application, especially coupled phenomena, or multiphysics. URL http://www.comsol.com/.

32. Z. Z. Liu, Q. Zhang, X. L. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016). [CrossRef]   [PubMed]  

References

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  1. C. L. Kane and E. J. Mele, “Z2 topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95(14), 205414 (2005).
  2. M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
    [Crossref]
  3. X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys 83(4), 175–179 (2011).
    [Crossref]
  4. S. Q. Shen, “Topological insulators: Dirac equation in condensed matters,” Springer Series in Solid-State Sciences174 (2012).
    [Crossref]
  5. S. Ryu and Y. Hatsugai, “Topological origin of zero-energy edge states in particle-hole symmetric systems,” Phys. Rev. Lett. 89(7), 077002 (2002).
    [Crossref] [PubMed]
  6. R. Mong and V. Shivamoggi, “Edge states and the bulk-boundary correspondence in Dirac Hamiltonians,” Phys. Rev. B 83, 125109 (2011).
    [Crossref]
  7. M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).
  8. C. W. Ling, M. Xiao, C. T. Chan, S. F. Yu, and K. H. Fung, “Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles,” Opt. Express 23, 2021–2031 (2015).
    [Crossref] [PubMed]
  9. X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
    [Crossref]
  10. W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
    [Crossref]
  11. W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22(4), 2099–2111 (1980).
    [Crossref]
  12. D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Den Nijs, “Quantized Hall conductance in a two-Dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405 (1982).
    [Crossref]
  13. Y. Ando, “Topological insulator materials,” J. Phys. Soc. Jpn. 82(10), 102001 (2013).
    [Crossref]
  14. S. Ho, F. Lin, and X. Wen, “Majorana zero-modes and topological phases of multi-flavored Jackiw-Rebbi model,” J. High Energy Phys. 210707412 (2012).
  15. D. G. Angelakis, P. Das, and C. Noh, “Probing the topological properties of the Jackiw-Rebbi model with light,” Sci. Rep. 4, 6110 (2014).
    [Crossref] [PubMed]
  16. A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
    [Crossref] [PubMed]
  17. W. J. Chen, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 67821 (2014).
  18. Y. Hatsugai, “Chern number and edge states in the integer quantum Hall effect,” Phys. Rev. Lett. 71(22), 3697–3700 (1993).
    [Crossref] [PubMed]
  19. P. Delplace, D. Ullmo, and G. Montambaux, “Zak phase and the existence of edge states in graphene,” Phys. Rev. B 84, 195452 (2011).
    [Crossref]
  20. D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
    [Crossref]
  21. X. F. Zhu, Y. G. Peng, X. Y. Yu, H. Jia, M. Bao, Y. X. Shen, and D. G. Zhao, “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices,” arXiv:1508.06243 (2015).
  22. R. Fleury, A. B. Khanikaev, and A. Al, “Floquet topological insulators for sound,” Nat. Commun. 7, 11744 (2016).
    [Crossref] [PubMed]
  23. C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).
  24. P. Wang, L. Lu, and K. Bertoldi, “Topological phononic crystals with one-way elastic edge waves,” Phys. Rev. Lett. 115, 104302 (2015).
    [Crossref] [PubMed]
  25. Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljai, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 145–150 (2008).
    [Crossref]
  26. K. X. Liu, L. F. Shen, and S. He, “One-way edge mode in a gyromagnetic photonic crystal slab,” Opt. Lett. 37(19), 4110–4112 (2012).
    [Crossref] [PubMed]
  27. H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in non-Hermitian optical materials,” Sci. Rep. 5, 17022 (2015).
    [Crossref] [PubMed]
  28. S. Maizard, C. Poli, and H. Schomerus, “Topological protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity-time symmetry,” Phys. Rev. Lett. 115, 200402 (2015).
    [Crossref]
  29. T. E. Lee, “Anomalous edge state in a non-Hermitian lattice,” Phys. Rev. Lett. 116, 133903 (2016).
    [Crossref] [PubMed]
  30. J. Xu and Y. Chen, “General coupled mode theory in non-Hermitian waveguides,” Opt. Express 23(17), 22619–22627 (2015).
    [Crossref] [PubMed]
  31. COMSOL MULTIPHYSICS 5.2, (2015). COMSOL Multiphysics: a finite element analysis, solver and simulation software for various physics and engineering application, especially coupled phenomena, or multiphysics. URL http://www.comsol.com/ .
  32. Z. Z. Liu, Q. Zhang, X. L. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
    [Crossref] [PubMed]

2016 (3)

R. Fleury, A. B. Khanikaev, and A. Al, “Floquet topological insulators for sound,” Nat. Commun. 7, 11744 (2016).
[Crossref] [PubMed]

T. E. Lee, “Anomalous edge state in a non-Hermitian lattice,” Phys. Rev. Lett. 116, 133903 (2016).
[Crossref] [PubMed]

Z. Z. Liu, Q. Zhang, X. L. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref] [PubMed]

2015 (6)

J. Xu and Y. Chen, “General coupled mode theory in non-Hermitian waveguides,” Opt. Express 23(17), 22619–22627 (2015).
[Crossref] [PubMed]

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

P. Wang, L. Lu, and K. Bertoldi, “Topological phononic crystals with one-way elastic edge waves,” Phys. Rev. Lett. 115, 104302 (2015).
[Crossref] [PubMed]

H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in non-Hermitian optical materials,” Sci. Rep. 5, 17022 (2015).
[Crossref] [PubMed]

S. Maizard, C. Poli, and H. Schomerus, “Topological protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity-time symmetry,” Phys. Rev. Lett. 115, 200402 (2015).
[Crossref]

C. W. Ling, M. Xiao, C. T. Chan, S. F. Yu, and K. H. Fung, “Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles,” Opt. Express 23, 2021–2031 (2015).
[Crossref] [PubMed]

2014 (5)

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).

D. G. Angelakis, P. Das, and C. Noh, “Probing the topological properties of the Jackiw-Rebbi model with light,” Sci. Rep. 4, 6110 (2014).
[Crossref] [PubMed]

W. J. Chen, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 67821 (2014).

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

2013 (1)

Y. Ando, “Topological insulator materials,” J. Phys. Soc. Jpn. 82(10), 102001 (2013).
[Crossref]

2012 (3)

S. Ho, F. Lin, and X. Wen, “Majorana zero-modes and topological phases of multi-flavored Jackiw-Rebbi model,” J. High Energy Phys. 210707412 (2012).

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

K. X. Liu, L. F. Shen, and S. He, “One-way edge mode in a gyromagnetic photonic crystal slab,” Opt. Lett. 37(19), 4110–4112 (2012).
[Crossref] [PubMed]

2011 (3)

P. Delplace, D. Ullmo, and G. Montambaux, “Zak phase and the existence of edge states in graphene,” Phys. Rev. B 84, 195452 (2011).
[Crossref]

R. Mong and V. Shivamoggi, “Edge states and the bulk-boundary correspondence in Dirac Hamiltonians,” Phys. Rev. B 83, 125109 (2011).
[Crossref]

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys 83(4), 175–179 (2011).
[Crossref]

2010 (1)

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

2008 (1)

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljai, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 145–150 (2008).
[Crossref]

2005 (1)

C. L. Kane and E. J. Mele, “Z2 topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95(14), 205414 (2005).

2002 (1)

S. Ryu and Y. Hatsugai, “Topological origin of zero-energy edge states in particle-hole symmetric systems,” Phys. Rev. Lett. 89(7), 077002 (2002).
[Crossref] [PubMed]

1993 (1)

Y. Hatsugai, “Chern number and edge states in the integer quantum Hall effect,” Phys. Rev. Lett. 71(22), 3697–3700 (1993).
[Crossref] [PubMed]

1982 (1)

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Den Nijs, “Quantized Hall conductance in a two-Dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405 (1982).
[Crossref]

1980 (1)

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22(4), 2099–2111 (1980).
[Crossref]

1979 (1)

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

Al, A.

R. Fleury, A. B. Khanikaev, and A. Al, “Floquet topological insulators for sound,” Nat. Commun. 7, 11744 (2016).
[Crossref] [PubMed]

Ando, Y.

Y. Ando, “Topological insulator materials,” J. Phys. Soc. Jpn. 82(10), 102001 (2013).
[Crossref]

Angelakis, D. G.

D. G. Angelakis, P. Das, and C. Noh, “Probing the topological properties of the Jackiw-Rebbi model with light,” Sci. Rep. 4, 6110 (2014).
[Crossref] [PubMed]

Bandres, M. A.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Bao, M.

X. F. Zhu, Y. G. Peng, X. Y. Yu, H. Jia, M. Bao, Y. X. Shen, and D. G. Zhao, “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices,” arXiv:1508.06243 (2015).

Bertoldi, K.

P. Wang, L. Lu, and K. Bertoldi, “Topological phononic crystals with one-way elastic edge waves,” Phys. Rev. Lett. 115, 104302 (2015).
[Crossref] [PubMed]

Chan, C. T.

C. W. Ling, M. Xiao, C. T. Chan, S. F. Yu, and K. H. Fung, “Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles,” Opt. Express 23, 2021–2031 (2015).
[Crossref] [PubMed]

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

Chen, W. J.

W. J. Chen, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 67821 (2014).

Chen, Y.

Chen, Y. B.

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

Chen, Y. F.

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

Chong, Y.

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljai, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 145–150 (2008).
[Crossref]

Das, P.

D. G. Angelakis, P. Das, and C. Noh, “Probing the topological properties of the Jackiw-Rebbi model with light,” Sci. Rep. 4, 6110 (2014).
[Crossref] [PubMed]

Delplace, P.

P. Delplace, D. Ullmo, and G. Montambaux, “Zak phase and the existence of edge states in graphene,” Phys. Rev. B 84, 195452 (2011).
[Crossref]

Den Nijs, M.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Den Nijs, “Quantized Hall conductance in a two-Dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405 (1982).
[Crossref]

Feng, L.

H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in non-Hermitian optical materials,” Sci. Rep. 5, 17022 (2015).
[Crossref] [PubMed]

Fleury, R.

R. Fleury, A. B. Khanikaev, and A. Al, “Floquet topological insulators for sound,” Nat. Commun. 7, 11744 (2016).
[Crossref] [PubMed]

Fung, K. H.

Ge, H.

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

Guzman-Silva, D.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Hasan, M. Z.

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

Hatsugai, Y.

S. Ryu and Y. Hatsugai, “Topological origin of zero-energy edge states in particle-hole symmetric systems,” Phys. Rev. Lett. 89(7), 077002 (2002).
[Crossref] [PubMed]

Y. Hatsugai, “Chern number and edge states in the integer quantum Hall effect,” Phys. Rev. Lett. 71(22), 3697–3700 (1993).
[Crossref] [PubMed]

He, C.

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

He, S.

Heeger, A. J.

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22(4), 2099–2111 (1980).
[Crossref]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

Ho, S.

S. Ho, F. Lin, and X. Wen, “Majorana zero-modes and topological phases of multi-flavored Jackiw-Rebbi model,” J. High Energy Phys. 210707412 (2012).

Huang, X. Q.

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

Jia, H.

X. F. Zhu, Y. G. Peng, X. Y. Yu, H. Jia, M. Bao, Y. X. Shen, and D. G. Zhao, “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices,” arXiv:1508.06243 (2015).

Joannopoulos, J.

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljai, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 145–150 (2008).
[Crossref]

Kane, C. L.

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

C. L. Kane and E. J. Mele, “Z2 topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95(14), 205414 (2005).

Kargarian, M.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Khanikaev, A. B.

R. Fleury, A. B. Khanikaev, and A. Al, “Floquet topological insulators for sound,” Nat. Commun. 7, 11744 (2016).
[Crossref] [PubMed]

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Kohmoto, M.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Den Nijs, “Quantized Hall conductance in a two-Dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405 (1982).
[Crossref]

Lee, T. E.

T. E. Lee, “Anomalous edge state in a non-Hermitian lattice,” Phys. Rev. Lett. 116, 133903 (2016).
[Crossref] [PubMed]

Lin, F.

S. Ho, F. Lin, and X. Wen, “Majorana zero-modes and topological phases of multi-flavored Jackiw-Rebbi model,” J. High Energy Phys. 210707412 (2012).

Ling, C. W.

Liu, K. X.

Liu, X. L.

Z. Z. Liu, Q. Zhang, X. L. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref] [PubMed]

Liu, X. P.

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

Liu, Z. Z.

Z. Z. Liu, Q. Zhang, X. L. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref] [PubMed]

Longhi, S.

H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in non-Hermitian optical materials,” Sci. Rep. 5, 17022 (2015).
[Crossref] [PubMed]

Lu, L.

P. Wang, L. Lu, and K. Bertoldi, “Topological phononic crystals with one-way elastic edge waves,” Phys. Rev. Lett. 115, 104302 (2015).
[Crossref] [PubMed]

Lu, M. H.

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

Macdonald, A. H.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Maizard, S.

S. Maizard, C. Poli, and H. Schomerus, “Topological protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity-time symmetry,” Phys. Rev. Lett. 115, 200402 (2015).
[Crossref]

Mejia-Cortes, C.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Mele, E. J.

C. L. Kane and E. J. Mele, “Z2 topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95(14), 205414 (2005).

Mong, R.

R. Mong and V. Shivamoggi, “Edge states and the bulk-boundary correspondence in Dirac Hamiltonians,” Phys. Rev. B 83, 125109 (2011).
[Crossref]

Montambaux, G.

P. Delplace, D. Ullmo, and G. Montambaux, “Zak phase and the existence of edge states in graphene,” Phys. Rev. B 84, 195452 (2011).
[Crossref]

Mousavi, S. H.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Ni, X.

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

Nightingale, M. P.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Den Nijs, “Quantized Hall conductance in a two-Dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405 (1982).
[Crossref]

Noh, C.

D. G. Angelakis, P. Das, and C. Noh, “Probing the topological properties of the Jackiw-Rebbi model with light,” Sci. Rep. 4, 6110 (2014).
[Crossref] [PubMed]

Nolte, S.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Peng, Y. G.

X. F. Zhu, Y. G. Peng, X. Y. Yu, H. Jia, M. Bao, Y. X. Shen, and D. G. Zhao, “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices,” arXiv:1508.06243 (2015).

Poli, C.

S. Maizard, C. Poli, and H. Schomerus, “Topological protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity-time symmetry,” Phys. Rev. Lett. 115, 200402 (2015).
[Crossref]

Qi, X. L.

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys 83(4), 175–179 (2011).
[Crossref]

Rechtsman, M. c.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Ryu, S.

S. Ryu and Y. Hatsugai, “Topological origin of zero-energy edge states in particle-hole symmetric systems,” Phys. Rev. Lett. 89(7), 077002 (2002).
[Crossref] [PubMed]

Schomerus, H.

S. Maizard, C. Poli, and H. Schomerus, “Topological protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity-time symmetry,” Phys. Rev. Lett. 115, 200402 (2015).
[Crossref]

Schrieffer, J. R.

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22(4), 2099–2111 (1980).
[Crossref]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

Segev, M.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Shen, L. F.

Shen, S. Q.

S. Q. Shen, “Topological insulators: Dirac equation in condensed matters,” Springer Series in Solid-State Sciences174 (2012).
[Crossref]

Shen, Y. X.

X. F. Zhu, Y. G. Peng, X. Y. Yu, H. Jia, M. Bao, Y. X. Shen, and D. G. Zhao, “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices,” arXiv:1508.06243 (2015).

Shivamoggi, V.

R. Mong and V. Shivamoggi, “Edge states and the bulk-boundary correspondence in Dirac Hamiltonians,” Phys. Rev. B 83, 125109 (2011).
[Crossref]

Shvets, G.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Soljai, M.

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljai, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 145–150 (2008).
[Crossref]

Su, W. P.

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22(4), 2099–2111 (1980).
[Crossref]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

Sun, X. C.

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

Szameit, A.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Thouless, D. J.

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Den Nijs, “Quantized Hall conductance in a two-Dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405 (1982).
[Crossref]

Tse, W. K.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Ullmo, D.

P. Delplace, D. Ullmo, and G. Montambaux, “Zak phase and the existence of edge states in graphene,” Phys. Rev. B 84, 195452 (2011).
[Crossref]

Vicencio, R. A.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Wang, P.

P. Wang, L. Lu, and K. Bertoldi, “Topological phononic crystals with one-way elastic edge waves,” Phys. Rev. Lett. 115, 104302 (2015).
[Crossref] [PubMed]

Wang, Z.

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljai, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 145–150 (2008).
[Crossref]

Weimann, S.

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Wen, X.

S. Ho, F. Lin, and X. Wen, “Majorana zero-modes and topological phases of multi-flavored Jackiw-Rebbi model,” J. High Energy Phys. 210707412 (2012).

Xiao, J. J.

Z. Z. Liu, Q. Zhang, X. L. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref] [PubMed]

Xiao, M.

C. W. Ling, M. Xiao, C. T. Chan, S. F. Yu, and K. H. Fung, “Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles,” Opt. Express 23, 2021–2031 (2015).
[Crossref] [PubMed]

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

Xu, J.

Yao, Y.

Z. Z. Liu, Q. Zhang, X. L. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref] [PubMed]

Yu, S. F.

Yu, X. Y.

X. F. Zhu, Y. G. Peng, X. Y. Yu, H. Jia, M. Bao, Y. X. Shen, and D. G. Zhao, “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices,” arXiv:1508.06243 (2015).

Zhang, Q.

Z. Z. Liu, Q. Zhang, X. L. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref] [PubMed]

Zhang, S. C.

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys 83(4), 175–179 (2011).
[Crossref]

Zhang, Z. Q.

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

Zhao, D. G.

X. F. Zhu, Y. G. Peng, X. Y. Yu, H. Jia, M. Bao, Y. X. Shen, and D. G. Zhao, “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices,” arXiv:1508.06243 (2015).

Zhao, H.

H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in non-Hermitian optical materials,” Sci. Rep. 5, 17022 (2015).
[Crossref] [PubMed]

Zhu, X. F.

X. F. Zhu, Y. G. Peng, X. Y. Yu, H. Jia, M. Bao, Y. X. Shen, and D. G. Zhao, “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices,” arXiv:1508.06243 (2015).

J. High Energy Phys. (1)

S. Ho, F. Lin, and X. Wen, “Majorana zero-modes and topological phases of multi-flavored Jackiw-Rebbi model,” J. High Energy Phys. 210707412 (2012).

J. Phys. Soc. Jpn. (1)

Y. Ando, “Topological insulator materials,” J. Phys. Soc. Jpn. 82(10), 102001 (2013).
[Crossref]

Nat. Commun. (2)

W. J. Chen, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 67821 (2014).

R. Fleury, A. B. Khanikaev, and A. Al, “Floquet topological insulators for sound,” Nat. Commun. 7, 11744 (2016).
[Crossref] [PubMed]

Nat. Mater. (1)

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2012).
[Crossref] [PubMed]

Nat. Phys. (1)

C. He, X. Ni, H. Ge, X. C. Sun, Y. B. Chen, M. H. Lu, X. P. Liu, and Y. F. Chen, “Acoustic topological insulator and robust one-way sound transport,” Nat. Phys. 15123273 (2015).

New J. Phys. (1)

D. Guzman-Silva, C. Mejia-Cortes, M. A. Bandres, M. c. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, “Experimental observation of bulk and edge transport in photonic Lieb lattices,” New J. Phys. 16(6), 063061 (2014).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (4)

P. Delplace, D. Ullmo, and G. Montambaux, “Zak phase and the existence of edge states in graphene,” Phys. Rev. B 84, 195452 (2011).
[Crossref]

X. Q. Huang, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

R. Mong and V. Shivamoggi, “Edge states and the bulk-boundary correspondence in Dirac Hamiltonians,” Phys. Rev. B 83, 125109 (2011).
[Crossref]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22(4), 2099–2111 (1980).
[Crossref]

Phys. Rev. Lett. (9)

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Den Nijs, “Quantized Hall conductance in a two-Dimensional periodic potential,” Phys. Rev. Lett. 49(6), 405 (1982).
[Crossref]

S. Ryu and Y. Hatsugai, “Topological origin of zero-energy edge states in particle-hole symmetric systems,” Phys. Rev. Lett. 89(7), 077002 (2002).
[Crossref] [PubMed]

Y. Hatsugai, “Chern number and edge states in the integer quantum Hall effect,” Phys. Rev. Lett. 71(22), 3697–3700 (1993).
[Crossref] [PubMed]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42(25), 1698–1701 (1979).
[Crossref]

C. L. Kane and E. J. Mele, “Z2 topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95(14), 205414 (2005).

P. Wang, L. Lu, and K. Bertoldi, “Topological phononic crystals with one-way elastic edge waves,” Phys. Rev. Lett. 115, 104302 (2015).
[Crossref] [PubMed]

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljai, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100(1), 145–150 (2008).
[Crossref]

S. Maizard, C. Poli, and H. Schomerus, “Topological protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity-time symmetry,” Phys. Rev. Lett. 115, 200402 (2015).
[Crossref]

T. E. Lee, “Anomalous edge state in a non-Hermitian lattice,” Phys. Rev. Lett. 116, 133903 (2016).
[Crossref] [PubMed]

Phys. Rev. X (1)

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).

Rev. Mod. Phys (1)

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys 83(4), 175–179 (2011).
[Crossref]

Rev. Mod. Phys. (1)

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

Sci. Rep. (3)

D. G. Angelakis, P. Das, and C. Noh, “Probing the topological properties of the Jackiw-Rebbi model with light,” Sci. Rep. 4, 6110 (2014).
[Crossref] [PubMed]

H. Zhao, S. Longhi, and L. Feng, “Robust light state by quantum phase transition in non-Hermitian optical materials,” Sci. Rep. 5, 17022 (2015).
[Crossref] [PubMed]

Z. Z. Liu, Q. Zhang, X. L. Liu, Y. Yao, and J. J. Xiao, “Absence of exceptional points in square waveguide arrays with apparently balanced gain and loss,” Sci. Rep. 6, 22711 (2016).
[Crossref] [PubMed]

Other (3)

COMSOL MULTIPHYSICS 5.2, (2015). COMSOL Multiphysics: a finite element analysis, solver and simulation software for various physics and engineering application, especially coupled phenomena, or multiphysics. URL http://www.comsol.com/ .

X. F. Zhu, Y. G. Peng, X. Y. Yu, H. Jia, M. Bao, Y. X. Shen, and D. G. Zhao, “Topologically protected acoustic helical edge states and interface states in strongly coupled metamaterial ring lattices,” arXiv:1508.06243 (2015).

S. Q. Shen, “Topological insulators: Dirac equation in condensed matters,” Springer Series in Solid-State Sciences174 (2012).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 Bulk properties of the dimerized PT-symmetric waveguide lattices A and B. Each unit cell contains two identical single-mode waveguides with width and height denoted by w and h. The working wavelength is λ0, and the relative permittivity of single-mode waveguide core is r 0 = 10, embedded in air cladding. (a)–(b)/(d)–(e) show the real part and imaginary part of eigenvalues of band structure with four values of Δ for waveguide lattice A/B. (c)/(f) shows the width of the projected band Δneff and the band-center as function of non-Hermiticity Δ.
Fig. 2
Fig. 2 Edge state in coupled waveguide chains with global PT symmetry and local PT symmetry which consist of seventeen waveguides. (a) is the structure of PT symmetric waveguide chain with the condition n(x) = n*(−x), and x = 0 is the center of the structure. The larger space between two waveguides dL = 0.8λ0, the smaller one dS = 0.4λ0 and the periodicity D = 1.2λ0. (b) is the structure of waveguide chain with local PT symmetry and mirror symmetry. Local PT symmetry is in the unit cell with two waveguides and mirror symmetry means n(x) = n*(−x) with the center of the structure. (c) and (d) show the band dispersion of the structure in (a) and (b), respectively.
Fig. 3
Fig. 3 Band structure and mode evolution sketch of waveguide chain with five waveguides. (a) and (b) show the real and imaginary part of band dispersion in coupled waveguide chains with global PT symmetry which consist of five waveguides, respectively. Blue circles represent the eigenvalues from COMSOL, and red stars represent the eigenvalues from GCMT. (c) The mode evolution in the mode crossing regime, i.e., the black rectangle in (a). (d) The values of k31, k32, k34, k35 and k45 with varying Δ.
Fig. 4
Fig. 4 Band structure and the sketch of mode evolution of waveguide chain with three waveguides. (a/b) shows the real (imaginary) part of band dispersion in coupled waveguide chains. Blue circles represent the eigenvalues from full-wave simulation, and Red dots represent results from phenomenological models. (c) The mode evolution of two mode intersection.

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

j a j ( β β 0 , j ) p i j = j a j k i j ,
[ h 11 h 12 h 13 h 14 h 15 h 21 h 22 h 23 h 24 h 25 h 31 h 32 h 33 h 34 h 35 h 41 h 42 h 43 h 44 h 45 h 51 h 52 h 53 h 54 h 55 ] [ a 1 a 2 a 3 a 4 a 5 ] = n eff k 0 [ p 11 p 12 p 13 p 14 p 15 p 21 p 22 p 23 p 24 p 25 p 31 p 32 p 33 p 34 p 35 p 41 p 42 p 43 p 44 p 45 p 51 p 52 p 53 p 54 p 55 ] [ a 1 a 2 a 3 a 5 a 5 ] ,
H = [ β odd i V 1 0 i V 1 β even i V 2 0 i V 2 β single ] ,

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