Inherent non-Hermitian Chern insulators with PT-symmetry and synthetic translation dimension

Ke-Xin Sun; Jian-Wen Dong; Wen-Jie Chen

doi:10.1364/PRJ.561829

1. INTRODUCTION

In the past decade, the interplay between band topology and non-Hermiticity has attracted widespread interest in the non-Hermitian extension of topological concepts [1 –9], such as optical sensing [10 –12], the optical funnel [13], and topological lasers [14 –17]. Non-Hermiticity is universal in realistic systems because they are always coupled to their surrounding environment to some extent. Meanwhile, the introduction of non-Hermiticity provides more degrees of freedom, making the system more flexible and enriching the ways of optical field manipulation. Different from Hermitian systems, non-Hermitian systems have complex eigenvalue spectra and non-orthogonal eigen wavefunctions, enabling exotic band structures that have no Hermitian counterpart, such as nontrivial topology of eigenvalues [6 –9]. This gives rise to many unique and interesting physical phenomena in non-Hermitian systems, including parity-time (PT) symmetry phase transitions and exceptional points in energy spectra [18 –23]; the non-Hermitian skin effect within point band gaps [13,24 –32]; the non-Hermitian edge states within line band gaps [16,30 –37], etc. Among them, the combination of topological edge states and optical gain/loss has inspired a series of studies, such as topological lasers utilizing the amplification characteristics of non-Hermitian topological edge states [16]. However, most of the previous works on non-Hermitian topological edge states focus on 1D systems, for their easier realization, such as a 1D plasmonic array [33,34] and the PT-symmetric Su–Schrieffer–Heeger (SSH) model [36]. Recently, 2D non-Hermitian systems have been experimentally realized, such as 2D PT-symmetric coupled resonator optical waveguide systems, showing the great potential of non-Hermitian topological systems in optical signal transmission [35].

In the meantime, the synthetic dimension [38,39] has attracted much attention in different branches of physics, providing an effective way to explore higher-dimensional topological physics in low-dimensional systems. Synthetic dimensions refer to additional degrees of freedom (by coupling a set of physical states to form a synthetic lattice [40 –44] or by treating external parameters of the Hamiltonian as additional dimensions [45 –54]) to supplement the apparent geometric dimensions. Although there are only three geometric dimensions in real physical space, introducing synthetic dimensions allows us to define and explore high-dimensional systems without being constrained by the number of real geometric dimensions. Good examples are the four-dimensional (4D) quantum Hall effect [43,55], 5D Weyl semimetal [56], and high-order topological insulator [57,58]. In principle, all the structural and material parameters of a crystal can be deemed as an external parameter of the system, and thus can be deemed as a synthetic dimension. Among them, the translation distance of a lattice can serve as a special pseudo momentum, which will lead to the translation of a Wannier center [52]. Combined with Bloch momentum $k$ , one can construct an intrinsically nontrivial 2D/4D parameter space [say, ( $k_{x}, Δ x$ )]. These inherent band topologies give rise to interface/edge/dislocation states, which deterministically exist in any crystal [53]. This theory provides us with a universal and convenient way to achieve topological states. Based on the theory proposed in a Hermitian system, several types of passive photonic devices have been realized, including topological rainbow concentrator [45,51] and on-chip optical filter [48]. However, these findings are achieved within the Hermitian limit. What if a real open system has loss or gain? Especially, when the system’s non-Hermiticity exceeds a critical value, part of the energy band enters the PT-broken phase. In this case, Berry curvatures and resultant Chern numbers are ill-defined. Further extending this theory to a non-Hermitian system can help us to understand relevant physics in open systems and may inspire translation-based active devices, such as topological lasers [14 –17] and non-Hermitian optical switches [59].

In this paper, we construct a synthetic Chern insulator by introducing a global translation deformation into a 1D non-Hermitian photonic crystal (PC). We theoretically investigate the 2D band structure of this insulator. Counter-intuitively, a frequency band gap is opened by system’s non-Hermiticity between the second and third bands. Its gap Chern number is calculated to be $- 2$ by integrating composite Berry curvatures, for the existence of a PT-broken regime. And two chiral edge states exist for this synthetic 2D crystal. Through rigorous derivation, we find that the inherent band topology associated with translation deformation persists in a non-Hermitian system. The Chern number of each band must be $- 1$ regardless of the structural parameters of the crystal [52,53] [an intrinsic property of the ( $k_{x}, Δ x$ ) space]. Our results imply that these translation-induced topological edge states can be extended from Hermitian systems to non-Hermitian systems, even when PT-phase transition occurs. To provide deeper insight, we derive and analyze the reflection phase and surface admittance of 1D crystal in 2D synthetic space using a plane wave expansion method. A $2 n π$ phase winding can always be found in the $n$ th band gap. This property is ubiquitous for any 1D crystal and can be understood through the eigen fields (standing wave modes) of band edge modes. From this special perspective, we present a clear physical picture of the topological edge states generated by translation deformation, which applies well for both Hermitian and non-Hermitian systems, indicating that this mechanism is independent of the system’s Hermiticity.

2. TOPOLOGICAL PROPERTIES OF SYNTHETIC SPACE: GAP CHERN NUMBER AND EDGE STATES

Consider a 1D PT-symmetric photonic crystal consisting of gain medium G and loss medium L [see Fig. 1(a)], with structural parameters: lattice constant $a$ , $ε_{L} = 2 + 0.5 i$ , $ε_{G} = 2 - 0.5 i$ , and $d_{L} = d_{G} = 0.5 a$ . Figures 1(b) and 1(c) calculate its complex band structure (real and imaginary parts) using a characteristic matrix method [60]. The lowest two bands merge near the Brillouin zone boundary, generating two exceptional points (EPs, red circles) and a PT-broken phase (black lines) in between. Its bulk eigenstates $| Ψ_{n} (k_{x}) ⟩$ are labeled by Bloch wave vector $k_{x} \in [- π / a, π / a]$ and band number $n$ . By continuously translating the PC, we introduce a new dimension, i.e., the translational parameter $Δ x$ [see Fig. 1(a)]. The cyclic parameter $Δ x$ and the Bloch wave vectors $k_{x}$ together form a 2D parametric space ( $k_{x}, Δ x$ ), which can be regarded as a 2D Brillouin zone ( $- π / a \leq k_{x} \leq π / a, 0 \leq Δ x \leq 1$ ) of the 1D PC. Because changing $Δ x$ will not truly change the bulk crystal, the $k_{x}$ dispersions for different values of $Δ x$ are identical. Thus, a 2D band structure about ( $k_{x}, Δ x$ ) can be obtained by simply extending the $k_{x}$ dispersion [Fig. 1(b)] along $Δ x$ -direction, forming a cylindrical surface that is flat along $Δ x$ -direction [Fig. 1(d)]. And the original exceptional points trace out two exceptional lines [red lines in Fig. 1(d)].

Figure 1.1D PT-symmetric photonic crystal with spatial translation. (a) One-dimensional photonic crystal considered as a 2D system by introducing a synthetic translation dimension $Δ x$ . (b), (c) Real and imaginary parts of complex band structure for a photonic crystal with $ε_{L} = 2 + 0.5 i, ε_{G} = 2 - 0.5 i, d_{L} = 0.5 a$ , and $d_{G} = 0.5 a$ . The blue and black curves highlight the PT-exact phase and the PT-broken phase, respectively. A frequency band gap (gray box) exists between the second and third bands. (d) 2D band structure in synthetic momentum space ( $k_{x}, Δ x$ ). (e) Composite Berry curvature in 2D periodic momentum space ( $k_{x}, Δ x$ ). The gap Chern number is $- 2$ .

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Such a 2D PC is a Chern insulator, as the band Chern number in the synthetic momentum space ( $k_{x}, Δ x$ ) must be $- 1$ , no matter the specific parameters of the photonic crystal. Before proving this, it is important to note that our system is non-Hermitian; its left eigenstates $| Ψ_{n}^{L} (k_{x}, Δ x) ⟩$ and right eigenstates $| Ψ_{n}^{R} (k_{x}, Δ x) ⟩$ are generally unrelated. The Berry curvature and Berry connection calculated using left/right eigenstates will be different. One can define four types of Berry curvatures: $A_{n}^{α β} (k_{x}, Δ x) = i ⟨ u_{n}^{α} (k_{x}, Δ x) | \nabla_{k_{x}, Δ x} | u_{n}^{β} (k_{x}, Δ x) ⟩ .$ (1)Here $α, β = L, R$ . $| u_{n}^{α / β} (k_{x}, Δ x) ⟩$ is the normalized eigenstate. But it does not affect the resulting Chern number [7]. We take $A_{n}^{L R}$ as an example for analysis. In fact, the 1D PCs with different $Δ x$ can be connected through translations, and so are their wavefunctions. So we have $| E_{n}^{R} (k_{x}, Δ x, x) ⟩ = | E_{n, 0}^{R} (k_{x}, x - Δ x) ⟩$ , $⟨ E_{0}^{L} (k_{x}, Δ x, x) | = ⟨ E_{n, 0}^{L} (k_{x}, x - Δ x) |$ , where $| E_{n, 0}^{R} (k_{x}, x) ⟩$ and $⟨ E_{n, 0}^{L} (k_{x}, x) |$ are the eigen electric fields of the 1D PC with $Δ x = 0$ . The Berry connection is given by $A_{n}^{L R} (k_{x}, Δ x) = i ⟨ u_{n}^{L} (k_{x}, Δ x) | \nabla_{k_{x}, Δ x} | u_{n}^{R} (k_{x}, Δ x) ⟩ = i ⟨ u_{n, 0}^{L} (k_{x}, x - Δ x) | \nabla_{k_{x}, Δ x} | u_{n, 0}^{R} (k_{x}, x - Δ x) ⟩ .$ (2)Since $u_{n, 0}^{L / R} (k_{x}, x)$ is a periodic function in the unit cell, the shift $Δ x$ would not affect the integral in Eq. (2) and $A_{n}^{L R} (k_{x}, Δ x)$ is $Δ x$ -independent, and so is Berry curvature $B_{n}^{L R} (k_{x}, Δ x) = \nabla_{k_{x}, Δ x} \times A_{n}^{L R} (k_{x}, Δ x)$ . The $A_{k_{x}}$ component does not contribute to Berry curvature and subsequently the band Chern number $C$ : $C_{n}^{L R} = \frac{1}{2 π} \oint_{\partial B Z} A_{n}^{L R} (k_{x}, Δ x) \cdot d (k_{x}, Δ x) = \int_{a}^{0} A_{n, Δ x}^{L R} (- \frac{π}{a}, Δ x) d Δ x + \int_{0}^{a} A_{n, Δ x}^{L R} (\frac{π}{a}, Δ x) d Δ x .$ (3)Note that the eigen fields for $k_{x} = \pm \frac{π}{a}$ are identical, i.e., $| E_{0}^{R} (- \frac{π}{a}, x) ⟩ = | e^{- i \frac{π}{a} x} u_{0}^{R} (- \frac{π}{a}, x) ⟩ = | E_{0}^{R} (\frac{π}{a}, x) ⟩ = | e^{i \frac{π}{a} x} u_{0}^{R} (\frac{π}{a}, x) ⟩ .$ (4)We have $| u_{0}^{R} (\frac{π}{a}, x) ⟩ = | e^{- i \frac{2 π}{a} x} u_{0}^{R} (- \frac{π}{a}, x) ⟩$ . In a similar way, $⟨ u_{0}^{L} (\frac{π}{a}, x) | = ⟨ u_{0}^{L} (- \frac{π}{a}, x) e^{- i \frac{2 π}{a} x} |$ . Then $A_{n, Δ x}^{L R} (\frac{π}{a}, Δ x) = i ⟨ u_{n, 0}^{L} (- \frac{π}{a}, x - Δ x) e^{- i \frac{2 π}{a} (x - Δ x)} | \frac{\partial}{\partial Δ x} | e^{- i \frac{2 π}{a} (x - Δ x)} u_{n, 0}^{R} (- \frac{π}{a}, x - Δ x) ⟩ = i ⟨ u_{n, 0}^{L} (- \frac{π}{a}, x - Δ x) | \frac{\partial}{\partial Δ x} | u_{n, 0}^{R} (- \frac{π}{a}, x - Δ x) ⟩ + i (i \frac{2 π}{a}) ⟨ u_{n, 0}^{L} (- \frac{π}{a}, x - Δ x) | u_{n, 0}^{R} (- \frac{π}{a}, x - Δ x) ⟩ = A_{n, Δ x}^{L R} (- \frac{π}{a}, Δ x) - \frac{2 π}{a},$ (5)with $⟨ u_{n, 0}^{L} (- \frac{π}{a}, x - Δ x) | u_{n, 0}^{R} (- \frac{π}{a}, x - Δ x) ⟩ = 1$ . Plugging into Eq. (3), we arrive at $C \equiv - 1$ . It should be noted that in our system, the lowest two bands merge at EPs, where both eigenvectors and eigenvalues are degenerate. Thus, in the PT-broken phase [Fig. 1(c)], the two complex bands cannot be definitively attributed to band 1 or band 2. The definitions of bands 1 and 2 are not unique, and consequently, neither are their individual Chern numbers. We calculate the composite Chern number [61] by integrating composite Berry curvature, which mixes the contributions from the lowest two bands. Figure 1(e) plots the calculated composite Berry curvature (we use the system’s right eigenstates that can be directly obtained in COMSOL simulation; for more details see Appendix A). As we mentioned before, composite Berry curvature is $Δ x$ -independent. This independence is not affected by the specific parameters of PC. The resulting composite Chern number is $- 2$ . We emphasize again that this nontrivial topology stems from this 2D parameter space itself (translation of the wavefunction and its Wannier orbit), which is its inherent topological property; therefore it does not depend on structural details of the crystal.

The nontrivial topology in the parametric space ( $k_{x}, Δ x$ ) naturally leads to edge states by joining the translated PC and the perfect electric conductor (PEC) together, as illustrated in Fig. 2(a). Figure 2(b) shows the edge spectrum as a function of translation $Δ x$ . As expected by the gap Chern number of $- 2$ , there are two gapless edge bands traversing the gap. This indicates that bulk-edge correspondence is still valid in a non-Hermitian system. Especially, the eigenfrequency of the edge band [orange lines in Fig. 2(b)] in this non-Hermitian crystal deviates from the real plane and becomes a complex number. Both edge bands have a positive slope versus $Δ x$ . Note that the slope here is not the group velocity but the parameter dependence of the edge mode. It does not mean a unidirectional transport state. Figure 2(c) plots the electric field profile for $Δ x = 0.9 a$ [magenta star in Fig. 2(b)], which is localized near the photonic crystal boundary.

Figure 2.Edge states between translated photonic crystal and perfect electric conductor. (a) Photonic crystal under the perfect electric conductor (PEC) boundary condition with different $Δ x$ . (b) Edge spectrum as a function of translation $Δ x$ . Two gapless edge bands with complex frequencies are plotted in yellow. The blue dashed lines are their projections onto the real frequency plane. Black dotted lines along $ω_{i}$ direction are projection lines. As expected by the gap Chern number of $- 2$ , the edge bands traverse the band gap twice. The gray areas represent the bulk bands. (c) Electric field distribution of edge state for $Δ x = 0.9 a$ , corresponding to the magenta star in (b).

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Figures 3(a)–3(c) calculate the edge spectrum near the EP frequency. Interestingly, an edge band (yellow solid line) exists even when the two bands are degenerate at EPs. Actually this edge band is topologically protected by the inherent Chern number in ( $k_{x}, Δ x$ ) space. To see this, we slightly break the PT symmetry by changing the permittivity of the gain/lossy layer. A complete band gap opens between the lowest two bands, near the EP [see Figs. 3(d)–3(f)]. The edge band persists, which connects the two band edges with complex frequencies [see Fig. 3(e)]. These two complex frequencies [ $0.351 + 0.029 i$ and $0.352 - 0.024 i (2 π c / a)$ ] correspond to the eigenfrequencies of bulk states at the Brillouin zone boundary in Fig. 1(c). This edge band seems to traverse the imaginary gap opened in the PT-broken phase. From the top view in Fig. 3(f), the slope of this gapless edge band is positive, consistent with the bulk Chern number. Therefore it is a chiral edge band between two complex bulk bands.

Figure 3.Chiral edge states between band 1 and band 2. (a) Edge spectrum for the lowest two bands. (b), (c) Side view and top view of the band structure in (a). Red line in (a) and red circle in (b) correspond to the EPs in bulk band structure. (d) Edge spectrum for a PC without PT symmetry. Here $ε_{L} = 2 + 0.45 i, ε_{G} = 2 - 0.5 i$ . (e), (f) Side view and top view of the band structure in (f). In these figures, the translucent gray/blue plane denotes the real/complex bulk band in PT-exact/broken phase. Yellow solid line denotes the gapless edge band while yellow dashed line is its projection onto the complex frequency plane.

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To interpret the gapless edge bands in the frequency band gap, we calculate the reflection phase of translated PC. Figure 4(a) shows the schematics of reflection from a semi-infinite crystal with different $Δ x$ . In Fig. 4(b), we calculate the reflection phase inside the band gap between the second band and third band. One finds that, for each frequency inside the gap, reflection phase winds up by $4 π$ from $Δ x = 0$ to $Δ x = a$ . For instance, Fig. 4(c) plots the reflection phase at the lower band edge [ $0.691 (2 π c / a)$ ], mid-gap [ $0.7 (2 π c / a)$ ], and upper band edge frequency [ $0.713 (2 π c / a)$ ]. They have the same trend of change, but with different initial values at $Δ x = 0$ . Remarkably, the condition for an edge state between PEC and translated PC should be $ϕ_{P E C} (Δ x, ω) + ϕ_{t P C} (Δ x, ω) = 2 π n,$ (6)where $ϕ_{t P C} (Δ x, ω)$ is the reflection phases from translated PC. Due to the $4 π$ winding of $ϕ_{t P C} (Δ x, ω)$ , there must be two suitable values of $Δ x$ satisfying the resonance condition in Eq. (6). Thus there would be two edge modes for each frequency inside the gap, resulting in two edge bands traversing the gap (Fig. 2). In fact, this $4 π$ winding is the property of the crystal surface and related to the evanescent decay mode (surface impedance) of the crystal, which is guaranteed by the deterministic Chern number in ( $k_{x}, Δ x$ ) space.

Figure 4. $4 π$ reflection phase winding of the translated photonic crystal. (a) Reflection phase of translated photonic crystals with different $Δ x$ . (b) Reflection phase $ϕ_{r}$ in the second band gap. From $Δ x = 0$ to $Δ x = a$ , the phase winds up by $4 π$ . (c) Reflection phase at lower band edge [ $ω = 0.691 (2 π c / a)$ ], mid-gap [ $ω = 0.7 (2 π c / a)$ ], upper band edge frequency [ $ω = 0.713 (2 π c / a)$ ].

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3. INTERPRETATION OF REFLECTION PHASE WINDING

From the above analysis about the inherent Chern number in ( $k_{x}, Δ x$ ) space, this $4 π$ phase winding in the second band gap should be a universal property for any 1D periodic structure. In this section, we try to understand the universality of this inherent topology using a three-plane-wave expansion.

Without loss of generality, we consider a binary 1D PC with $Δ x = 0$ . The permittivities and thicknesses of two layer components are $ε_{A} = ε_{0} + ε_{Δ}$ , $ε_{B} = ε_{0} - ε_{Δ}$ , $d_{A}$ , and $d_{B} = a - d_{A}$ . The electric field in the PC satisfies the following Helmholtz equation [62]: $(\frac{d^{2}}{d x^{2}} + {(\frac{ω}{c})}^{2} ε (x)) E (x) = 0,$ (7)where $E_{k} (x) = u_{k} (x) e^{i k x}$ , with $u_{k} (x) = \sum_{n = - \infty}^{\infty} α_{n k} e^{i n 2 π x / a}$ . Substituting this expansion into Eq. (3), we arrive at $\sum_{n = - \infty}^{\infty} k^{2} {\bar{ε}}_{0, n^{'} n} (k) α_{n k} = \sum_{n = - \infty}^{\infty} {(\frac{ω}{c})}^{2} ({\bar{ε}}_{0, n^{'} n} (k) + {\bar{ε}}_{Δ, n^{'} n} (k)) α_{n k},$ (8)in which ${\bar{ε}}_{0 / Δ, n^{'} n} (k) = \int d x e^{- i k_{n^{'}} x} (x) ε_{0 / Δ} (x) e^{i k_{n} x} (x)$ . As we are concerned with the eigen modes near the second band gap, three dominant $k$ components with coefficients of $α_{- 1}, α_{0}$ , and $α_{1}$ are retained for simplicity. Equation (4) can be rewritten in a matrix form as $(\begin{matrix} {(- 2 π + k)}^{2} ε_{0} & 0 & 0 \\ 0 & k^{2} ε_{0} & 0 \\ 0 & 0 & {(2 π + k)}^{2} ε_{0} \end{matrix}) (\begin{matrix} α_{- 1} \\ α_{0} \\ α_{1} \end{matrix}) = {(\frac{ω}{c})}^{2} (\begin{matrix} ε_{0} + ε_{Δ} (2 d_{A} - 1) & - \frac{2 ε_{Δ} \sin (π d_{A})}{π} & \frac{ε_{Δ} \sin (2 π d_{A})}{π} \\ - \frac{2 ε_{Δ} \sin (π d_{A})}{π} & ε_{0} + ε_{Δ} (2 d_{A} - 1) & - \frac{2 ε_{Δ} \sin (π d_{A})}{π} \\ \frac{ε_{Δ} \sin (2 π d_{A})}{π} & - \frac{2 ε_{Δ} \sin (π d_{A})}{π} & ε_{0} + ε_{Δ} (2 d_{A} - 1) \end{matrix}) (\begin{matrix} α_{- 1} \\ α_{0} \\ α_{1} \end{matrix}) .$ (9)

Using three-plane-wave approximation, band structures for the PT-symmetric crystal in Fig. 1 are calculated in Fig. 5(a). The lower and upper band edge modes of the second band gap can be obtained at the Brillouin zone center ( $k = 0$ ), whose eigenvalues and eigenvectors are $ω_{1} = \frac{4 π^{4} (2 d_{A} ε_{Δ} - ε_{Δ} + ε_{0}) C}{ε_{0}^{2} π^{2} - 2 (1 - 2 d_{A}) ε_{0} ε_{Δ} π^{2} - 8 ε_{Δ}^{2} \sin^{2} (π d_{A}) + π^{2} ε_{Δ}^{2} {(1 - 2 d_{A})}^{2} - π ε_{Δ} (ε_{Δ} - ε_{0} - 2 d_{A} ε_{Δ}) \sin (2 π d_{A})}, ω_{2} = - \frac{4 π^{3}}{ε_{Δ} π - ε_{0} π + ε_{Δ} \sin (2 π d_{A}) - 2 π ε_{Δ} d_{A}}, V_{1} = {(1, C, 1)}^{T}, V_{2} = {(- 1, 0, 1)}^{T} .$ (10)

$C = \frac{4 ε_{Δ} \sin (π d_{A})}{(ε_{0} - ε_{Δ} + 2 d_{A} ε_{Δ}) π}$ is a constant determined by the structural parameters of PC. Corresponding eigen electric fields are $E_{1} = 2 \cos (2 π x / a) + C, E_{2} = - 2 i \sin (2 π x / a) .$ (11)And magnetic fields can be obtained from $H_{y} (x) = \frac{1}{ω_{n} μ_{0}} k_{x} \times E_{z} (x)$ : $H_{1} = i D_{1} \sin (2 π x / a), H_{2} = D_{2} \cos (2 π x / a) .$ (12) $D_{1} = 4 π / (a ω_{1} μ_{0})$ and $D_{2} = 4 π / (a ω_{2} μ_{0})$ are the amplitudes of $H_{1}$ and $H_{2}$ , respectively. Not surprisingly, both band edge modes are typical standing wave modes, which are the superposition of two waves with the same frequency propagating in opposite directions, as a consequence of Bragg scattering. The nodes of $E$ fields coincide with the antinodes of $H$ fields and conversely, a remarkable feature of standing waves. At each point, the $E$ field and $H$ field have a phase difference of $π / 2$ . Note that these properties do not depend on the crystal’s specific structural parameters (dielectric constant, film thickness, and so on) or whether the system is Hermitian or not. For the second band gap, the plane wave components with $e^{- 2 π x / a}$ and $e^{2 π x / a}$ dominate and are mixed with each other due to Bragg scattering, leading to two standing wave modes with a period of $a / 2$ . And their eigenfrequencies’ splitting results in the band gap.

Note that the $4 π$ phase winding is actually related to the surface admittance of the band edge mode (standing wave mode). To see this, we take the upper band edge frequency ( $ω_{2}$ ) as an example, and consider the reflection from a semi-infinite crystal with translation $Δ x$ [Figs. 5(b)–5(d)]. The reflection phase is determined by the surface admittance of the crystal, which is the ratio between surface electric and magnetic fields [circles in Figs. 5(b)–5(d)]. Surface electromagnetic fields ( $E_{s}$ and $H_{s}$ ) as a function of $Δ x$ are summarized in Fig. 5(e). This curve is actually the mode profile of the upper band edge mode, because the mode profiles in Figs. 5(b)–5(d) are only different by a translation. Similarly, $E_{s}$ and $H_{s}$ travel two complete periods from $Δ x = 0$ to $Δ x = a$ . And their ratio (surface admittance $η_{s} = H_{s} / E_{s}$ ) reads $η_{2 s} = i D_{2} \cdot \cot (\frac{2 π Δ x}{a}) / 2 .$ (13)It is a purely imaginary number, changing from $+ \infty$ to $- \infty$ two times, as $Δ x$ increases from zero to $a$ [Fig. 5(f)]. Finally, we obtain the reflection coefficient at the upper band edge frequency via $r = \frac{1 - η_{s}}{1 + η_{s}} .$ (14)Since $η_{s}$ is a purely imaginary number, the numerator and denominator are a complex conjugated pair with arguments of $\pm θ$ , and reflection phase $ϕ_{r} = \arg (r) .$ (15)As plotted in Fig. 5(g), the reflection coefficient’s trajectory on the complex plane is a unit circle. And as $Im (η_{s})$ changes from $- \infty$ to $+ \infty$ , $ϕ_{r}$ changes from $- π$ to $π$ . Therefore, as $Δ x$ changes from zero to $a$ , $r$ circulates twice around the origin and $ϕ_{r}$ winds up $4 π$ .

Figure 5.Interpretation of $4 π$ phase winding via the surface admittance of translated photonic crystals. (a) Band structure near the second band gap. Magenta star marks the band edge mode at $ω_{u} = 0.713 (2 π c / a)$ . (b)–(d) Total reflection of translated PCs with $Δ x = 0, 0.15 a, 0.3 a$ , when a plane wave with frequency of $ω_{u}$ impinges onto the surface. The solid and dashed lines plot the magnetic field (purely real) and the electric field (purely imaginary) of the bulk propagating mode [magenta star in (a)]. (e) Surface electric/magnetic field $E_{s} / H_{s}$ for different truncations. The three cases in (b)–(d) are highlighted by colored circles. (f) Surface admittance $η_{s} = H_{s} / E_{s}$ . This purely imaginary admittance changes from $- \infty$ to $+ \infty$ two times, as $Δ x$ increases. (g) Corresponding reflection coefficients $r = (1 - η_{s}) / (1 + η_{s})$ on the complex plane. As $Δ x$ increases from zero to $a$ , reflection phase winds up by $4 π$ .

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Interestingly, this is actually a direct consequence of the band edge mode (typically standing wave mode). Similar statements can be reached for the lower band edge mode (also a $4 π$ winding) and even the higher-frequency band edges (will be discussed later). As restricted by the two band edge frequencies, the reflection phases inside the gap should always have the same winding properties (shown in Fig. 4).

Likewise, the above discussion about reflection phase can be extended to higher-frequency band gaps. From the perspective of perturbation theory, the formation of the $n$ th band gap is due to the Bragg scattering and mixing between plane wave components of $e^{\pm 2 i n π x / a}$ . Take the fourth gap (between the fourth and fifth bands) for example. As shown in Figs. 6(a) and 6(b), there exists a frequency band gap between the fourth and fifth bands at the zone center, just like the case of the second gap. Here the band edge modes are standing waves with a period of $a / 4$ , although they are not so perfectly periodic due to other plane wave components. By comparing the eigen fields for the second and fourth gaps [Figs. 6(c) and 6(d)], we can clearly see their different periods ( $a$ and $a / 2$ ). These different periods ultimately lead to different reflection phase windings as a function of $Δ x$ . A period of $a / 2$ leads to a $4 π$ phase winding (Fig. 5), while a period of $a / 4$ leads to an $8 π$ phase winding. For a clearer display, we numerically calculate the reflection phase in the fourth band gap in Fig. 7. Figures 7(a) and 7(b) show surface electric and magnetic fields, respectively. As $Δ x$ changes, the dominant wave components [ $Im (E_{s})$ and $Re (H_{s})$ ] oscillate with a period of $a / 4$ . Similar to the results in Fig. 5, the surface admittance $η_{s}$ is almost an imaginary number with its imaginary part Im( $η_{s}$ ) changing from positive to negative four times [Fig. 7(c)], resulting in an $8 π$ phase winding. Since the eigen fields on higher-frequency bands are usually more complicated (involving more $k$ -components), surface admittance $η_{s}$ deviates from a purely imaginary number (different from the second gap) but its main feature sustains. Thus, reflection coefficient $r_{s}$ circulates four times on the complex plane [Fig. 7(d)], although its trajectory is not a unit circle, indicating a reflectivity $R = {| r |}^{2} \neq 1$ for the sake of gain/loss. A similar winding behavior is observed for the upper band edge mode of the fourth gap, which together constrains the reflection phase’s winding inside the fourth gap [Fig. 7(f)].

Figure 6.Frequency band gap induced by Bragg reflection. (a) Band structure using the extended zone scheme. Red for homogeneous medium ( $ε = 2$ ), and blue lines for PC with $ε = 2 \pm 0.5 i$ . Different background colors represent different Brillouin zones with different orders. Frequency gaps open at Brillouin zone boundaries at $k = 2 N π / a$ ( $N$ is integer). The band gaps are highlighted with blue areas in the two subgraphs. (b) Corresponding folded band structure in reduced Brillouin zone. Due to the periodic modulation in $ε_{i}$ , two plane waves are mixed together forming the lower and upper band edge modes (standing wave modes). (c), (d) Corresponding eigen electromagnetic field in (b).

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Figure 7. $8 π$ reflection phase winding in the fourth band gap. (a) Surface electric field $E_{s}$ for different truncations at the lower band edge frequency $ω = 1.38 (2 π c / a)$ . The solid and dashed lines represent the real and imaginary parts of the electric field, respectively. (b) Corresponding surface magnetic field $H_{s}$ . (c) Surface admittance $η_{s}$ . There are four inflection points as $Δ x$ increases. (d) Corresponding reflection coefficients $r$ on the complex plane. (e) Reflection phase $ϕ_{r} = \arg (r)$ . As $Δ x$ increases from zero to $a$ , reflection phase winds up by $8 π$ . (f) Reflection phase in the entire band gap. White dashed line is the result of (e).

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Likewise, the band edge modes for the $n$ th band gap must have a period of $a / n$ ( $n$ is integer), which will result in a $2 n π$ phase winding along the $Δ x$ dimension. This quite well explains the intrinsic nontrivial topology in synthetic space ( $k_{x}, Δ x$ ) and related topological phenomena [52]. More importantly, our analysis points out that these effects are also applicable to a non-Hermitian system and not restricted to a Hermitian system.

4. CONCLUSION

In summary, we construct an inherent non-Hermitian Chern insulator in a 2D synthetic $k$ -space ( $k_{x}, Δ x$ ) by introducing a global translation deformation in a 1D PC. We rigorously prove that each band Chern number in this specific 2D space must be $- 1$ , even for a non-Hermitian system. This is verified by phase winding and edge dispersion. More importantly, this is always true [an intrinsic property of ( $k_{x}, Δ x$ ) space] regardless of the structural parameters or unit cell geometry of the crystal. The inherent reason is that the eigen modes for different values of $Δ x$ have a deterministic relation (they are related by a certain translation). To give deep insight into these ubiquitous phenomena, we derive and analyze the crystal’s reflection phase and surface admittance in 2D synthetic space. A $2 n π$ phase winding can always be found in the $n$ th band gap. This property can be understood via the profiles of their band edge modes, which are standing wave modes. From this special perspective, we give a clear physical picture on the mechanism generating topological edge states by translation deformation. These findings may pave the way for translation-based photonic devices.

Category: Nanophotonics and Photonic Crystals

Received: Mar. 17, 2025

Accepted: May. 21, 2025

Published Online: Jul. 31, 2025

The Author Email: Wen-Jie Chen (chenwj5@mail.sysu.edu.cn)

DOI:10.1364/PRJ.561829

CSTR:32188.14.PRJ.561829