• Photonics Research
  • Vol. 11, Issue 2, 225 (2023)
Yingying Zhang1、†, Shiqiang Xia1、†,*, Xingdong Zhao1、5, Lu Qin1, Xuejing Feng1, Wenrong Qi1, Yajing Jiang1, Hai Lu1, Daohong Song2, Liqin Tang2、6, Zunlue Zhu1、7, Wuming Liu3, and Yufang Liu1、4
Author Affiliations
  • 1School of Physics, Henan Normal University, Xinxiang 453007, China
  • 2MOE Key Laboratory of Weak-Light Nonlinear Photonics, TEDA Applied Physics Institute and School of Physics, Nankai University, Tianjin 300457, China
  • 3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
  • 4Institute of Physics, Henan Academy of Sciences, Zhengzhou 450046, China
  • 5e-mail: phyzhxd@gmail.com
  • 6e-mail: tanya@nankai.edu.cn
  • 7e-mail: zl-zhu@htu.edu.cn
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    DOI: 10.1364/PRJ.478167 Cite this Article Set citation alerts
    Yingying Zhang, Shiqiang Xia, Xingdong Zhao, Lu Qin, Xuejing Feng, Wenrong Qi, Yajing Jiang, Hai Lu, Daohong Song, Liqin Tang, Zunlue Zhu, Wuming Liu, Yufang Liu. Symmetry-protected third-order exceptional points in staggered flatband rhombic lattices[J]. Photonics Research, 2023, 11(2): 225 Copy Citation Text show less

    Abstract

    Higher-order exceptional points (EPs), which appear as multifold degeneracies in the spectra of non-Hermitian systems, are garnering extensive attention in various multidisciplinary fields. However, constructing higher-order EPs still remains a challenge due to the strict requirement of the system symmetries. Here we demonstrate that higher-order EPs can be judiciously fabricated in parity–time (PT)-symmetric staggered rhombic lattices by introducing not only on-site gain/loss but also non-Hermitian couplings. Zero-energy flatbands persist and symmetry-protected third-order EPs (EP3s) arise in these systems owing to the non-Hermitian chiral/sublattice symmetry, but distinct phase transitions and propagation dynamics occur. Specifically, the EP3 arises at the Brillouin zone (BZ) boundary in the presence of on-site gain/loss. The single-site excitations display an exponential power increase in the PT-broken phase. Meanwhile, a nearly flatband sustains when a small lattice perturbation is applied. For the lattices with non-Hermitian couplings, however, the EP3 appears at the BZ center. Quite remarkably, our analysis unveils a dynamical delocalization-localization transition for the excitation of the dispersive bands and a quartic power increase beyond the EP3. Our scheme provides a new platform toward the investigation of the higher-order EPs and can be further extended to the study of topological phase transitions or nonlinear processes associated with higher-order EPs.

    1. INTRODUCTION

    Exceptional points (EPs), singularities at which eigenvalues and eigenvectors simultaneously coalesce, have attracted considerable interest in non-Hermitian physics [17]. For instance, EPs in the parity–time (PT)-symmetric optical arrangements [8,9] have provided an excellent platform for exploring intriguing phenomena such as PT-symmetric lasers [10,11] and topological phase transitions [1214]. In particular, recently, there has been a great deal of interest in proposing various methods for constructing higher-order (greater than second-order) EPs. The eigenvalue shift near the higher-order EPs follows the ϵ1/N (N is the order of the EPs) power law of the external perturbation ϵ, therefore leading to higher sensitivity of resonant optical structures to small external disturbances [1518]. Another interesting thing is that optical systems with PT symmetry can show additional symmetries, such as time-reversal symmetry and chiral symmetry (CS), making the higher-order EPs much more abundant [1921]. In fact, these symmetries ramify and show new features in non-Hermitian systems due to the fact that the Hamiltonian HH. A crucial example is CS which satisfies CHC1=H. The CS coincides with sublattice symmetry (SLS) (SHS1=H) in the presence of Hermiticity. Obviously, they are not equivalent for non-Hermitian Hamiltonians [2225].

    The study of EPs has also been extended to flatband systems. A flatband is dispersionless throughout the Brillouin zone (BZ), and it provides a unique setting for exploring anomalous magnetic phases and strongly correlated states of matter [2629]. In flatband systems, diffraction is suppressed due to the destructive interference of the Bloch wave functions, resulting in typically compact localized states (CLSs) in real space [3037]. The most direct way to form a PT-symmetric flatband system is to introduce on-site gain/loss to the lattices. Recently, the PT-symmetric flatband photonic waveguide arrays with tailored on-site gain/loss distributions have been experimentally created using a laser-written technique [38,39]. On the other hand, the non-Hermitian couplings describe a situation where the mode amplitude undergoes gain/loss while hopping between sites [4042]. It has been shown that the PT-symmetric frustrated lattices with fine-tuned non-Hermitian couplings can also give rise to flatbands as well as EPs [43]. More importantly, recent progress has demonstrated that even the higher-order EPs can exist in the PT-symmetric flatband lattices [20,21,44]. Nevertheless, due to the band touching (vanishing bandgap) between flat and dispersive bands, so far, realizations of the nonzero-valued higher-order EPs have been limited to flatband systems with non-Hermitian couplings [4548]. Moreover, the distinct features of the two aspects of non-Hermiticity (on-site gain/loss and non-Hermitian couplings) as well as the dynamical behaviors near the phase transitions, also remain elusive.

    In this work, we demonstrate that a higher-order EP (EP3) can occur in staggered rhombic lattices by introducing the diagonal PT symmetry (with on-site gain/loss) and the off-diagonal PT symmetry (with non-Hermitian couplings) to the system. These PT-symmetric lattices possess both the non-Hermitian CS and SLS, thus leading to the presence of zero-energy flatbands and nonzero EP3s. However, the EP3 and the propagation dynamics of the lattices, unexpectedly, show strikingly different properties. For the diagonal PT-symmetric lattices, an EP3 arises at the BZ boundary. We find that the single-site excitations of different sublattices display identical exponential power increases in the PT-broken phase. By contrast, an EP3 is generated in the BZ center for the off-diagonal PT-symmetric lattices. Intriguing propagation dynamics, such as a dynamical delocalization-localization transition for the excitation of dispersive bands and a quartic power increase beyond the EP3, are observed. For both diagonal and off-diagonal PT-symmetric systems, the eigenvalue separations exhibit an ϵ1/3 dependence in the vicinity of the EP3. In particular, we show that a nearly flatband sustains for the diagonal PT-symmetric lattices in a small range of perturbations.

    2. DIAGONAL PT-SYMMETRIC FLATBAND RHOMBIC LATTICES

    Symmetry-protected flatband and EP3. We start from introducing the diagonal PT-symmetric staggered photonic rhombic lattices with on-site gain/loss to elucidate the lattice symmetry, the formation of flatband and the EP3. Figure 1(a) shows the lattice structure with each unit cell consisting of an edge site A with gain γ, an edge site C with loss γ, and a central neutral site B. The sublattice B is fourfold connected with the nearest neighbors, while the sublattices A and C have twofold connections with surrounding sites. The staggered couplings are tL=t(1+g) and tR=t(1g), where tL(tR) is the coupling on the left (right) side of the site B. Note that tL(tR) is real and represents conserved (Hermitian) coupling. Experimentally, such non-Hermitian lattices can be obtained using the optically induced method or the femtosecond laser-writing technique [14,38,39]. The couplings depend on the spacing of the neighboring waveguides; thus a staggered arrangement of the distance between the waveguides generates staggered couplings. Meanwhile, the need for optical amplification can be avoided by shifting the respective lattice sites from γ, 0, and γ to 0, γ, and 2γ [49,50]. As a consequence, the non-Hermitian environments can be experimentally introduced only by inscribing the waveguides with tailored loss distributions [14,39]. In the tight binding model, the Hamiltonian can be written as H=n(tLbnan+tRbnan+1+tLbncn+tRbncn+1+H.c.)+iγ(anancncn),where an (an), bn (bn), and cn (cn) are the creation (annihilation) operators in the n-th unit cell on the A, B and C sites, respectively. Transforming H into momentum space as H=kψkHkψk with ψk=(ak,bk,ck)T, one can get the momentum space Hamiltonian, Hk=(iγtL+tReik0tL+tReik0tL+tReik0tL+tReikiγ).

    (a) Schematic of the diagonal PT-symmetric photonic rhombic lattices formed by waveguide arrays without non-Hermitian coupling. The unit cell consists of three sites: two edge sites with gain γ (A, red) or loss−γ (C, blue), and one central site with a neutral imaginary part (B, gray). The staggered coupling coefficients are tL=t(1+g) and tR=t(1−g), where 0≤g<1 and t is the coupling coefficient of uniform lattices (g=0). For simplicity, we set the lattice period l=1. (b), (c) Calculated real (b) and imaginary (c) parts of the spectrum as a function of γ/t for g=0.2. The red circles represent the position of the EP3 and the corresponding critical value of phase transition is γc/t=0.56. (d) Field distribution of a flatband CLS, which occupies two unit cells. The characters show the corresponding amplitudes.

    Figure 1.(a) Schematic of the diagonal PT-symmetric photonic rhombic lattices formed by waveguide arrays without non-Hermitian coupling. The unit cell consists of three sites: two edge sites with gain γ (A, red) or lossγ (C, blue), and one central site with a neutral imaginary part (B, gray). The staggered coupling coefficients are tL=t(1+g) and tR=t(1g), where 0g<1 and t is the coupling coefficient of uniform lattices (g=0). For simplicity, we set the lattice period l=1. (b), (c) Calculated real (b) and imaginary (c) parts of the spectrum as a function of γ/t for g=0.2. The red circles represent the position of the EP3 and the corresponding critical value of phase transition is γc/t=0.56. (d) Field distribution of a flatband CLS, which occupies two unit cells. The characters show the corresponding amplitudes.

    The diagonal terms of Hk describe the propagation constant and on-site gain/loss, whereas the off-diagonal terms describe the lattice couplings. Such a Hamiltonian possesses PT symmetry (PT)Hk(PT)1=Hk, and more importantly, a CS as well as a SLS satisfying CHkC1=Hk and SHkS1=Hk, where the unitary operators C and S are defined as C=(001010100),S=(100010001).

    By calculating the eigenvalues of Hk, one can easily get the three corresponding eigenvalues, E0(k)=0,E±1(k)=±γ2+4t2(1+cosk)+4t2g2(1cosk).

    In the Hermitian case (γ/t=0), the CS/SLS symmetry requires all the energy eigenvalues appear in pairs, namely, the three-band system must have a zero-energy flatband E0=0 and two dispersive bands with opposite energies E1=E1 [51,52]. Figures 1(b) and 1(c) show the spectrum as a function of γ/t for g=0.2. It can be clearly seen that the Hermitian spectrum in the limit γ/t=0 has three bands: a zero-energy flatband located between two symmetric dispersive bands [dotted lines in Fig. 1(b)]. Unlike in uniform lattices where all the bands touch each other at k=π [5355], here the band degeneracy is lifted and two symmetric gaps with gap width Egap=2γ2+8t2g2 emerge as a result of the staggered couplings. When the on-site gain/loss is introduced (γ/t0), the energy eigenvalues become complex. As mentioned above, the CS bifurcates into CS and SLS in the non-Hermitian case. However, we find both the CS and SLS are preserved. Meanwhile, they retain E1=E1 and non-Hermitian zero mode occurs for the central band. As a consquence, the zero-energy flatband survives and the spectrum stays symmetric about the imaginary and real energy axis of the complex energy plane. The bandgaps decrease by increasing γ/t. More importantly, all the three eigenvalues coalesce at E0=0 for the critical value γc/t=22g=0.56 [Figs. 1(b) and 1(c)]. Note that γc depends on the value of g, i.e., the relative strength of the staggered couplings. The three eigenvectors also coalesce at this specific degenerate point and have the form [ψnA,ψnB,ψnC,ψn+1A,ψn+1C]=[0.8,0.2i,0.8,1.2,1.2]. Accordingly, the spectrum is completely real for γ<γc, and a CS/SLS-protected EP3 emerges at γ=γc. For γ>γc, the system turns into the PT-broken phase. Figure 1(d) illustrates the irreducible CLS of the flatband. As demonstrated in previous studies, CLSs are localized in a single unit cell in Hermitian rhombic lattices [5658]. Here, in contrast, the CLS is confined in the five-site-cross configurations in two adjacent unit cells. Nonzero amplitudes appear in A and C sites and the on-site gain/loss remodulates the amplitude of the B site. The equal amplitude and opposite phase ensure the destructive interference between the two legs formed by the A and C sites.

    Propagation dynamics for single-site excitation. To verify our analysis, we simulate the light propagation dynamics for initial excitations localized to either an A or B site. A sufficiently large number of lattice sites (101 unit cells) are considered. The input is an(0) or bn(0)=(π/σ)e(n2/σ2)(σ=0.5), i.e., only the central single site is excited. Experimentally, a Gaussian beam is sent to the front facet of the sample as a probe beam, and the output pattern can be monitored at the back facet of the sample [14,34,39,53,54]. Figure 2 shows the evolution of the beam intensity P(z)=n|ψn(z)|2=n(|an(z)|2+|bn(z)|2+|cn(z)|2), where z is an effective propagation distance. For comparison, we first consider the result for the Hermitian case at γ/t=0. A suppression of diffraction is observed for the excitation of A site [Fig. 2(a)]. The input excites a superposition of dispersive bands and flatband states. The energy is mainly distributed in the flatband; therefore, most of the power remains confined to the initially excited site. When the system is operated below the PT threshold value (γ/t=0.5), part of the energy evolves into the dispersive bands, though most of the energy still stays localized [Fig. 2(a2)]. It should be noted that what is conserved here is the quasi-energy Q(z)=nψ(n,z)ψ(n,z) as opposed to the actual power P(z). The presence of the gain/loss results in exponential amplification [P(z)ez] of the total power when the non-Hermitian parameter is tuned to γ/t=0.6 [Fig. 2(a3)] (see Appendix A). A different scenario is found when a localized B site is excited. In Fig. 2(b1), the input generates a conventional discrete diffraction pattern with two ballistically expanding lobes for the Hermitian lattices. In this case, only the dispersive bands are excited, since no flatband energy is distributed in the B sublattice [cf. Fig. 1(d)]. Nevertheless, the dispersive bands and flatband states, including the EP3 states, are excited in the PT symmetry phase γ/t=0.5 [Fig. 2(b2)]. The resulting diffraction consists of a discrete pattern and conserved total power as required for a PT-symmetric lattice in the unbroken phase. Furthermore, there is a residual localized component due to the excitation of the flatband. With the further increase of γ/t, the energy is also obviously amplified when the system is operated in the PT symmetry broken phase for γ/t=0.6 [Fig. 2(b3)]. Interestingly, we find that the power growth is the same as that in Fig. 2(a3).

    Intensity profile |ψ|2 of a single-site A (a1)–(a3) or B (b1)–(b3) excitation. (a1) The A sublattice excitation for Hermitian case (γ/t=0) exhibits a suppression of diffraction due to the excitation of flatband CLSs. (a2) The input remains conserved below the EP3 threshold (γ/t=0.5). (a3) The input leads to an exponentially growing total intensity in the PT-broken phase (γ/t=0.6). (b1) The B site excitation for Hermitian case (γ/t=0), showing a typically discrete diffraction pattern. (b2) The input excites both the flatband and dispersive bands, producing discrete diffraction as well as a residual localized component (γ/t=0.5). (b3) The input shows same exponential growth of total intensity as in (a3) (γ/t=0.6). The propagation length is z=20t−1. Note the different scales of the vertical axis in these panels, though the initial amplitudes of the excitation all equal 1.

    Figure 2.Intensity profile |ψ|2 of a single-site A (a1)–(a3) or B (b1)–(b3) excitation. (a1) The A sublattice excitation for Hermitian case (γ/t=0) exhibits a suppression of diffraction due to the excitation of flatband CLSs. (a2) The input remains conserved below the EP3 threshold (γ/t=0.5). (a3) The input leads to an exponentially growing total intensity in the PT-broken phase (γ/t=0.6). (b1) The B site excitation for Hermitian case (γ/t=0), showing a typically discrete diffraction pattern. (b2) The input excites both the flatband and dispersive bands, producing discrete diffraction as well as a residual localized component (γ/t=0.5). (b3) The input shows same exponential growth of total intensity as in (a3) (γ/t=0.6). The propagation length is z=20t1. Note the different scales of the vertical axis in these panels, though the initial amplitudes of the excitation all equal 1.

    (a), (b) Analytical (dashed lines) and numerical (solid lines) results for the (a) real and (b) imaginary parts of the spectrum as a function of the detuning ϵ/t in the vicinity of EP3. Red lines, E−1; green lines, E0; blue lines, E1. (c) Corresponding results for E0−E1=ΔE, exhibiting a cube-root behavior; (d) results from (c) on a logarithmic scale. The slope of about 1/3 confirms the cube-root response.

    Figure 3.(a), (b) Analytical (dashed lines) and numerical (solid lines) results for the (a) real and (b) imaginary parts of the spectrum as a function of the detuning ϵ/t in the vicinity of EP3. Red lines, E1; green lines, E0; blue lines, E1. (c) Corresponding results for E0E1=ΔE, exhibiting a cube-root behavior; (d) results from (c) on a logarithmic scale. The slope of about 1/3 confirms the cube-root response.

    Bifurcations of eigenvalues around the EP3. In practical implementations, the propagation of photons is subject to inevitable perturbations. As mentioned above, the eigenvalues bifurcation characteristics of the higher-order EPs can enhance the response of the optical structure to small disturbances. In the following, we demonstrate the eigenvalues evolution of our ternary PT-symmetric photonic structure around the EP3 and show how a small lattice perturbation affects the wave propagation dynamics.

    Note that the perturbation could be introduced anywhere along the diagonal of the matrix Hk. For simplicity and without loss of generality, we suppose that the perturbation ϵ is imposed on the sublattice A. When fabricating the lattices using femtosecond or continuous-wave laser-writing technique [14,34,39,53,54], the slight difference of the relative laser power or “writing” time between A and other sublattices will result in such a perturbation. The energy bands can be analytically obtained through solving the cubic characteristic equation (see Appendix B). Figures 3(a) and 3(b) illustrate analytical and numerical solutions for the real and imaginary parts of the eigenvalues as a function of ϵ/t. The small perturbation ranges from 1010 to 101. The analytical solutions show a good agreement with numerical results. The perturbation drives the system out of the degenerate state, and three complex eigenvalues are obtained. To have a quantitative comparison, we calculate the numerical eigenvalue difference of the upper band in Fig. 3(a) for ϵ/t=0.03 and ϵ/t=0.06, which are 0.1 and 0.15, respectively. The difference between the lower two eigenvalues (E0 and E1) is plotted in Fig. 3(c). Obviously, the eigenvalues’ splitting increases along with the strength of the perturbation. By considering the logarithmic behavior of this curve, we find that the slope is about 1/3, which is exactly the characteristic of an EP3 [Fig. 3(d)]. Another intriguing feature is that both the real and imaginary parts of the central band change slightly, especially for ϵ/t<0.03. This indicates that an approximate flatband can be sustained in such a range of perturbations.

    The perturbation on the spectrum can also be reflected by light propagation dynamics. For ϵ/t=0.03, one can find a similar pattern for the excitation of B sublattice in Fig. 4(a), which is compared with that in Fig. 2(a2). An increase of the lattice perturbation (ϵ/t=0.06) results in an enhancement of wave intensity in Fig. 4(b). The input behaves differently for the excitation of the flatband CLSs. An interesting property of flatband is that, despite the nonzero couplings between lattice sites, propagation of flatband eigenstates is completely suppressed. Therefore, if we excite five sites according to Fig. 1(d), we can observe a completely localized state with z-invariant intensity. In Fig. 4(c), a CLS excitation can stay well localized in the presence of ϵ/t=0.03, confirming that the flatband can preserve in this case. Moreover, the input keeps nearly localized, even when the lattice detuning is increased up to ϵ/t=0.06, as shown in Fig. 4(d).

    (a), (b) Intensity profile for the B site excitation in the presence of small perturbations ϵ/t=0.03 and ϵ/t=0.06, respectively. For ϵ/t=0.03, a similar pattern compared with Fig. 2(b2) is obtained. (c), (d) Same as (a), (b) but for the excitation of a flatband CLS. The CLS can stay well localized despite the presence of the lattice perturbations.

    Figure 4.(a), (b) Intensity profile for the B site excitation in the presence of small perturbations ϵ/t=0.03 and ϵ/t=0.06, respectively. For ϵ/t=0.03, a similar pattern compared with Fig. 2(b2) is obtained. (c), (d) Same as (a), (b) but for the excitation of a flatband CLS. The CLS can stay well localized despite the presence of the lattice perturbations.

    3. OFF-DIAGONAL PT-SYMMETRIC FLATBAND RHOMBIC LATTICES

    Symmetry-protected flatband and EP3. The most intriguing peculiarity of our scheme is that the non-Hermiticity can also be introduced by employing the non-Hermitian couplings. Such a system also possesses an EP3 and a flatband but exhibits different features. As illustrated in Fig. 5(a), we maintain the geometric symmetry of the lattices in Fig. 1(a) and perform imaginary processing on the coupling parameters. TL=tL+iγ, TL*=tLiγ, TR=tR+iγ, and TR*=tRiγ are the complex-valued coupling coefficients on the left (right) side of the site B, respectively. These complex couplings can be regarded as the phase of a hopping amplitude [5961], which can be experimentally realized by positioning auxiliary waveguide between adjacent waveguides [54,62]. This approach allows the flip of the sign of the coupling amplitude and obtaining an effective negative coupling. Embedding such a negative hopping amplitude inside the structure can realize an artificial magnetic field of θ flux for the photons, i.e., when a particle goes around a plaquette (AnBnCnBn1An), it collects the negative sign, which is equivalent to an eiθ phase. In this case, the Hamiltonian can be written as H=n(TLbnan+TRbnan+1+TL*bncn+TR*bncn+1+H.c.).

    (a) Schematic of the off-diagonal PT-symmetric photonic rhombic lattices formed by waveguide arrays with non-Hermitian coupling. The complex-coupling coefficients are TL=tL+iγ, TL*=tL−iγ, TR=tR+iγ, and TR*=tR−iγ, respectively. Note that the real parts (tL and tR) are the same as in Fig. 1(a), which represent conserved staggered couplings. (b), (c) Calculated (b) real and (c) imaginary parts of the spectrum as a function of γ/t for g=0.2; the EP3 forms at k=0 and the critical value of the phase transition is γc=t. (d) Field distribution of the flatband CLS, which also occupies two unit cells but only distributes in A and C sites.

    Figure 5.(a) Schematic of the off-diagonal PT-symmetric photonic rhombic lattices formed by waveguide arrays with non-Hermitian coupling. The complex-coupling coefficients are TL=tL+iγ, TL*=tLiγ, TR=tR+iγ, and TR*=tRiγ, respectively. Note that the real parts (tL and tR) are the same as in Fig. 1(a), which represent conserved staggered couplings. (b), (c) Calculated (b) real and (c) imaginary parts of the spectrum as a function of γ/t for g=0.2; the EP3 forms at k=0 and the critical value of the phase transition is γc=t. (d) Field distribution of the flatband CLS, which also occupies two unit cells but only distributes in A and C sites.

    The corresponding Hamiltonian in momentum space is Hk=(0TL+TReik0TL+TReik0TL*+TR*eik0TL*+TR*eik0),which satisfies the PT symmetry (PT)Hk(PT)1=Hk. Interestingly, one can find that such an arrangement also possesses CS and SLS. Here, the unitary operators C and S can be set as C=(100010001),S=(001010100).

    The energy eigenvalues of Hamiltonian [Eq. (6)] are E0(k)=0,E±1(k)=±2(t2γ2)(1+cosk)+g2t2(1cosk).

    For direct comparison of the two non-Hermitian arrangements, we also set g=0.2 and plot the real and imaginary parts of the spectrum [as shown in Figs. 5(b) and 5(c)]. Due to the presence of CS and SLS, the central flatband still survives, and the three bands distribute symmetrically. With the increase in γ/t, the bandgap tends to narrow and merge. In contrast, here, the PT symmetry-broken transition occurs at the center of the BZ (k=0), and the critical value is γc=t, i.e., is invariable and independent of the value of g. The eigenmodes of the bands also coalesce into a degenerated one, namely, a CS/SLS-protected EP3 is generated. Above the EP3, the spectrum becomes complex. In addition, the irreducible flatband CLS also occupies two unit cells, but the energy only distributes in A and C sites, as shown in Fig. 5(d).

    Propagation dynamics for single-site excitation. In Fig. 6, we illustrate the propagation dynamics of the system for the same initial excitations as in Fig. 2. Similar patterns as depicted in Figs. 2(a1) and 2(b1) can be obtained in the Hermitian limit γ/t=0. However, diverse propagation dynamics can be observed for γ/t0. The input experiences periodical oscillations for the A site excitation due to the unfolding of the nonorthogonal Floquet–Bloch modes in PT symmetry phase γ/t<1 [Figs. 6(a1) and 6(a2)]. The beat length, which is equal to L=2π/(δE) (δE is the difference between the propagation constants of two modes), increases as γ/t approaches the EP3 [9,63,64]. As can be found in Fig. 6(a2), the oscillation is enhanced and has a double period for γ/t=0.9. The beat length goes to infinity as the EP3 is approached. Furthermore, we observe a sharp transition in Fig. 6(a3). The input becomes localized and leads to a quartic power increase [P(z)z4] in the PT symmetry-broken phase (see Appendix A). The dynamics exhibited in Figs. 6(b1)–6(b3) is distinct from that in Figs. 6(a1)–6(a3). Transport is always ballistic in the PT-symmetry phase (γ/t<1) for the B site excitation. In this case, the input will never excite the flatband CLSs [cf. Fig. 5(d)], and thus no localized component appears. More saliently, the excitation displays a transition from ballistic wave packet spreading to dynamical localization, as the non-Hermitian parameter γ/t is increased to the value at the EP3. In Figs. 6(b1) and 6(b2), the excitation propagates bidirectionally along the lattice with the spreading speed νσ(z)/z, where the position operator σ2=nn2ψn2/nψn2. ν decreases as γ/t is increased and the wave packet spreading decreases faster for large γ/t (see Appendix C). When γ/t approaches the EP3, the diffraction is prevented. In fact, the largest velocity at which an excitation propagates along the lattice can also be described by the group velocity νg=Re{(dE/dk)k=π/2}. As the γ/t is increased, the energy spectrum in Fig. 5(b) undergoes a deformation, which changes the dispersion relation and results in a decrease of νg [65,66]. This fact can be used for a direct observation of the EP: the propagation constants approach when the non-Hermitian parameter is increased to the value at the EP3. Though the total intensity in Fig. 6(b3) is much smaller than that in Fig. 6(a3), we find that it also follows a quartic power increase in the PT symmetry-broken phase.

    Same as Fig. 2, but for the off-diagonal PT-symmetric lattices. (a1) The A sublattice excitation oscillates periodically below the EP3 threshold (γ/t=0.45). Most of the energy stays localized due to the excitation of the flatband. (a2) The quasi-energy is conserved, and the oscillation period is doubled for γ/t=0.9. (a3) The input leads to quartic total intensity growth once the system is operated in the PT symmetry-broken phase (γ/t=1.001). (b1) The B input only excites eigenstates of the dispersive bands and therefore experiences conventional diffraction. (b2) The spreading velocity diminishes with the increase of γ/t. (b3) The input becomes localized, and the total intensity also show a quartic increase in the PT symmetry-broken phase.

    Figure 6.Same as Fig. 2, but for the off-diagonal PT-symmetric lattices. (a1) The A sublattice excitation oscillates periodically below the EP3 threshold (γ/t=0.45). Most of the energy stays localized due to the excitation of the flatband. (a2) The quasi-energy is conserved, and the oscillation period is doubled for γ/t=0.9. (a3) The input leads to quartic total intensity growth once the system is operated in the PT symmetry-broken phase (γ/t=1.001). (b1) The B input only excites eigenstates of the dispersive bands and therefore experiences conventional diffraction. (b2) The spreading velocity diminishes with the increase of γ/t. (b3) The input becomes localized, and the total intensity also show a quartic increase in the PT symmetry-broken phase.

    Bifurcations of eigenvalues around the EP3. Similarly, we calculate the spectrum separation of the system in the presence of the external perturbations. The eigenvalues’ difference depicted in Fig. 7 also has a 1/3 power law of the external perturbation [Figs. 7(c) and 7(d)]. Note that the changes of the actual value of both the dispersive bands and flatband are greater compared with that in Fig. 3. For instance, the numerical eigenvalue difference in Fig. 7(a) for ϵ/t=0.03 and ϵ/t=0.06 is 0.54 and 0.69, respectively. In addition, we find that the imaginary part of the central band is identical to the upper band, indicating that the flatband transforms into a dispersive band [Fig. 7(b)].

    Same as Fig. 3, but for the off-diagonal PT-symmetric lattices.

    Figure 7.Same as Fig. 3, but for the off-diagonal PT-symmetric lattices.

    Figure 8 displays the corresponding propagation dynamics in the presence of the lattice perturbations. For B site excitation, the energy becomes obviously amplified for ϵ/t=0.06 due to the complex spectrum [Fig. 8(b)]. More importantly, one can clearly find that the CLSs no longer exist even when only a small perturbation is applied [Fig. 8(c)], and the pattern in Fig. 8(d) is similar to Fig. 8(b). These results reveal that the flatband is readily destroyed and evolves into a complex dispersive band as a result of the lattice perturbations.

    Same as Fig. 4, but for the off-diagonal PT-symmetric lattices.

    Figure 8.Same as Fig. 4, but for the off-diagonal PT-symmetric lattices.

    4. CONCLUSION

    In summary, we have demonstrated a feasible scheme to realize higher-order EPs (EP3s) in flatband rhombic lattices. By introducing on-site gain/loss and non-Hermitian couplings, we have established a diagonal PT-symmetric and an off-diagonal PT-symmetric rhombic lattices, respectively. Furthermore, we have made careful consideration of the distributions of the higher-order EPs and the dynamical behaviors around them. It has been shown that the systems possess both the non-Hermitian CS and SLS, thus leading to the presence of zero-energy flatbands and nonzero EP3. We have also observed varied propagation dynamics in these systems, such as sustained flatband CLSs, exponential and quartic power increase, as well as dynamical localization. These results provide insightful information about the underlying properties of the higher-order EPs in flatband systems. In future works, it will be interesting to investigate higher-order EPs in other flatband lattices. For example, one may realize the order of EPs greater than 3 in a one-dimensional Lieb lattice, which consist of five bands including a flatband [51]. Meanwhile, it will also be interesting to study a wide range of phenomena, such as the non-Hermitian topological phase transitions, anomalous edge states, and nonlinear optical processes in these systems [59,6770].

    APPENDIX A: THE ENERGY AMPLIFICATION IN THE PT SYMMETRY-BROKEN PHASE

    A localized single-site excitation in these PT-symmetric lattices can display exponential and quartic power increases in the PT symmetry-broken phase. Specifically speaking, for the diagonal PT-symmetric lattices, the power increases for the excitation of A and B sites are identical and follow an exponential amplification [P(z)=e0.39z] [green line in Fig. 9(a)]. For the off-diagonal PT-symmetric lattices, the power increase follows quartic amplifications [P(z)z4]. However, the energy amplification is greater for the excitation of the A site [Fig. 9(b)]. As can be seen in Fig. 9(d), the corresponding intensities P1/4 increase linearly, but the two lines have different slopes.

    Energy evolution in the symmetry-broken phase of the (a) diagonal and (b)–(d) off-diagonal PT-symmetric lattices. (a) The power increase for the excitation of A and B sublattices is identical and follows an exponential amplification [P(z)=e0.39z]. (b)–(d) The power increases for the excitation of (b) A and (c) B sublattices follow quartic amplifications [P(z)∼z4], and the corresponding intensities P1/4 increase linearly. Note that the two lines in (d) have different slopes.

    Figure 9.Energy evolution in the symmetry-broken phase of the (a) diagonal and (b)–(d) off-diagonal PT-symmetric lattices. (a) The power increase for the excitation of A and B sublattices is identical and follows an exponential amplification [P(z)=e0.39z]. (b)–(d) The power increases for the excitation of (b) A and (c) B sublattices follow quartic amplifications [P(z)z4], and the corresponding intensities P1/4 increase linearly. Note that the two lines in (d) have different slopes.

    APPENDIX B: EIGENVALUE EVOLUTIONS OF THE PT-SYMMETRIC LATTICES

    Here, we provide an analytical study of the eigenvalue evolutions of our PT-symmetric structure in the vicinity of the EP3. For simplicity, we only discuss the case of perturbing the gain sublattice A. As a result, the perturbation terms only appear along the diagonal element of Hk. For the diagonal PT-symmetric lattices, the Hamiltonian Hk can be written as Hk=(iγ+ϵtL+tReik0tL+tReik0tL+tReik0tL+tReikiγ).

    The corresponding determinant of Hk equated to zero is |En+iγ+ϵtL+tReik0tL+tReikEntL+tReik0tL+tReikEniγ|=0.

    The energy bands can be analytically obtained through solving the cubic characteristic equation, which is simplified to 0=En3En2ϵEn(iγϵγ2+2A)+ϵA,where γc2A=0, because A=tL2+tR2+2tLtRcosk=0.16t2 at the EP3 (k=π,γc/t=22g). By means of the Newton–Puiseux series Enc1ϵ1/3+c2ϵ2/3 (c1 and c2 are complex constants), we can expand the above equation by perturbation, 0=(c13+A)ϵ3/3+(3c12c2iγc1)ϵ4/3+(3c1c22c12c2iγ)ϵ5/3+(c232c1c2)ϵ6/3c22ϵ7/3.

    Forcing the coefficients of the first two terms to be zero, we obtain three sets of values for the coefficients c1 and c2, corresponding to the three eigenvalues, E14253eiπ/3ϵ1/3+4253i23eiπ/3ϵ2/3,E04253eiπ/3ϵ1/3+4253i23eiπ/3ϵ2/3,E14253eiπ/3ϵ1/34253i23eiπ/3ϵ2/3.

    Similarly, we can get the results of the off-diagonal PT-symmetric system. The Hamiltonian Hk can be written as Hk=(ϵTL+TReik0TL+TReik0TL*+TR*eik0TL*+TR*eik0).

    The determinant of Hk equated to zero is |En+ϵTL+TReik0TL+TReikEnTL+TReik0TL+TReikEn|=0.

    Here, A=TL2+TR2+2TLTRcosk=8i and A*=TL*2+TR*2+2TL*TR*cosk=8i at the EP3 (k=0,γc/t=1). So, the determinant can be simplified to 0=En3En2ϵ+ϵA*.

    Using the Newtonian series expansion for Enc1ϵ1/3+c2ϵ2/3, one can get 0=(c138i)ϵ3/3+3c12c2ϵ4/3+(3c1c22c12c2iγ)ϵ5/3+(c232c1c2)ϵ6/3c22ϵ7/3.

    The bifurcations in the eigenvalues have the form E12ei5π/6ϵ1/3,E02eiπ/6ϵ1/3,E12iϵ1/3.

    APPENDIX C: THE SPREADING VELOCITY ν FOR THE B SITE EXCITATION OF THE OFF-DIAGONAL PT-SYMMETRIC LATTICES

    For the off-diagonal PT-symmetric lattices, the B site excitation displays a transition from the ballistic wave packet spreading to dynamical localization. As can be seen in Fig. 10(a), the diffraction is prevented when the non-Hermitian parameter γ/t approaches the value at the EP3. Meanwhile, the spreading speed ν decreases faster for large γ/t, as shown in Fig. 10(b).

    (a) Intensity profile |ψ|2 of a localized B sublattice excitation for γ/t=0.98. When γ/t approaches the EP3, the diffraction is prevented. (b) Behavior of the spreading velocity ν as a function of γ/t, computed for the largest propagation length z=20t−1; ν is expressed in units of t.

    Figure 10.(a) Intensity profile |ψ|2 of a localized B sublattice excitation for γ/t=0.98. When γ/t approaches the EP3, the diffraction is prevented. (b) Behavior of the spreading velocity ν as a function of γ/t, computed for the largest propagation length z=20t1; ν is expressed in units of t.

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