The Topological Origin of Boundary Charges at Edges of One-Dimensional Crystals without Inversion Symmetry

Abstract

We report the edge states and non-zero boundary charges in one-dimensional photonic crystals (1D PhCs) without inversion symmetry. In contrast to common 1D systems, we show that edge states corresponding to non-zero boundary charges do exist in these asymmetric 1D PhCs even if we cannot obtain non-integral topological invariants. Moreover, an edge state could be observed in the interface between the PhC without inversion symmetry and the well-defined trivial PhC. Finally, the origin of the non-quantized boundary charges is unveiled by the non-central Wannier center. Not only exact solutions of photonic systems, but the above topological phenomena can also be found in the tight-binding models. This work proposes a way to study the 1D symmetries-broken systems and provides models to show the topological origin of boundary charges, which is suitable for both classic systems and quantum systems.

1. Introduction

The emergence of topological insulators has led to the gradual integration of the concept of topology into classical wave systems [1,2,3,4,5]. During the investigation of the topological properties of these systems, symmetries play a pivotal role in determining the overall behavior and characteristics of these systems [6]. For example, inversion symmetry ensures that the 1D Zak phase has only two quantized values of 0 or $\pi$ [7,8,9]. In addition, a 1D topological crystalline insulator (TCI) with inversion symmetry could exhibit quantized polarization [10,11,12,13,14]. One consequence of the quantization of the bulk polarization to $1/2$ in the nontrivial dipole phase is the appearance of $\pm 1/2$ charges at its edges, giving rise to the robust edge states. As for 1D systems without inversion symmetry, it is generally believed that the Zak phase is not well-defined due to the lack of symmetry [8,15]. As a result, these systems have often been considered topologically trivial, leading to a lack of thorough investigation into their topological properties and the presence of edge states. Surprisingly, despite being initially regarded as topologically trivial, the presence of edge states and non-zero boundary charges could still be observed in 1D PhC without inversion symmetry. Furthermore, if we gradually make the asymmetric model close to the symmetric model, boundary charges tend to be quantized. The successive evolution of non-quantized boundary charges in this process suggests that boundary charges arise from the breakdown of symmetry and have a topological origin. To further investigate the origin of non-quantized boundary charges, we calculate the Wannier center of 1D PhC without inversion symmetry, which unveils the underlying generation process of non-quantized boundary charges.

The aim of this work is to illustrate the topological origin of boundary charges for 1D crystals without inversion symmetry. To do so, we first construct 1D PhC with perfect inversion symmetry, i.e., layers A, B stacked along one direction. Next, by gradually changing the relative dielectric constant of part of layer B to make it be layer C, we break the inversion symmetry of the original model and split the degeneracy of the edge modes. Due to the symmetry breaking, boundary charges are no longer quantized but less than $1/2$. Initially, there is an inverse relationship between the increase in ${\epsilon }_C$ and the decrease in boundary charges. However, it is observed that when ${\epsilon }_C$ surpasses the value of the average relative dielectric constant of three layers being equal to layer A, boundary charges discontinue their downward trend and gradually ascend, ultimately approaching the value of $1/2$. Throughout this process, edge states consistently persist, appearing to undergo continuous evolution from the symmetric system. To investigate the topological origin of edge states and non-quantized boundary charges, we obtain the localized Wannier center of 1D PhC without inversion symmetry as a crucial parameter for analysis. Finally, these edge states could also be realized by the tight-binding model without inversion symmetry, which further illustrates the topological origin of boundary charges.

2. Models

Two kinds of models will be studied in this work, namely 1D layered PhC models with and without inversion symmetry. To explore the topological origin of boundary charges, we will study the changing trend of boundary charges in the process from a symmetric system to an asymmetric system in this section.

2.1. The Boundary Charges with Inversion Symmetry

Firstly, we consider the 1D PhC unit cell with inversion symmetry, as shown in Figure 1a. The lattice constant of PhC is a, i.e., $\epsilon (x+a)=\epsilon \left(x\right)$, the dielectric constant and the thickness of two kinds of layers are ${\epsilon }_A=9$, $d_A=0.5$ μm and ${\epsilon }_B=4$, $d_B=1$ μm, respectively. A finite PhC which contains N cells with the boundary condition at both ends of PhC as perfect electric conductor (PEC) is considered, where N is chosen as 20 for PhC in this work. By using finite-element method (FEM) software COMSOL Multiphysics and the wave optics module, the eigen-frequencies of the finite PhC are shown in Figure 1b, in which the bulk states and edge states are marked by black and red dots, respectively. In the context of PEC boundary conditions, the choice of the electric field component in the z-direction ($E_z$ polarization) is made. Since the inversion symmetry is preserved in the system, the edge-localized modes are degenerated and the field distribution of these states is shown in Figure 1b. According to the theory of dipole moments in crystalline insulators [11], the topological charges, which need to be half-integral, will accumulate at the boundaries of the finite systems when the non-trivial edge polarization exists. As for 1D PhC, similar photonic edge states are protected by non-trivial topology charges too. Hence, we can describe the topological properties of the band-gap by using boundary charges. To calculate boundary charges of photonic systems, the integration of local density of states (LDOSs) over all bands below the gap is taken [16,17,18]. The charge of the j-th unit cell is calculated by the sum of the spatial distribution of electromagnetic field energy for the states below the gap

$$Q_j=\sum _{i=1}^N{\int }_{x_j}^{x_j+a}\mathit{d}x{\left|{\phi }_{i,j}\left(x\right)\right|}^2$$

(1)

where $|{\phi }_{\mathit{i},\mathit{j}}\left(x\right){|}^2$ is the intensity of the i-th wave function at the j-th cell for electronic systems, which should be replaced by $\epsilon \left(x\right)|E_{i,j}{\left(x\right)|}^2$ for photonic systems and $\epsilon \left(x\right)$ is the permittivity distribution. The calculation results are shown in Figure 1c, wherein boundary charges at the edges are quantized to $\pm 1/2$ relative to the background.

2.2. The Boundary Charges without Inversion Symmetry

In this section, we will discuss the model for 1D PhC without spatial inversion symmetry. It is generally believed that the Zak phase cannot take integer values due to the lack of symmetry. Thus, it is a difficult temptation to establish a bulk-edge correspondence in such system. As a consequence, the significance of topological properties and the existence of edge states are frequently overlooked or undervalued. In fact, the presence of the edge states can still be observed in such system. Our objective is to investigate the topological origin of these edge states when inversion symmetry is broken. To do so, we suppose that the relative dielectric constant of half of layer B undergoes a change and the new layer is called C layer. In Figure 2a, if the relative dielectric constants of the three layers are not equal, inversion symmetry is broken. The eigen-frequencies of the finite PhC with ${\epsilon }_A=9,{\epsilon }_B=4,{\epsilon }_C=3$ and $d_A=d_B=d_C=0.5$ μm is shown in Figure 2b. Remarkably, the presence of edge states remains evident even in the absence of inversion symmetry. Edge states are no longer degenerated but localized to one side, as shown in Figure 2b. We can also calculate the distribution of boundary charges of $ABC$ model according to Equation (1), just adding C layer to the calculation of unit cells. In this case of ${\epsilon }_C=3$ (Figure 2c), we conclude that boundary charges which are not equal to $1/2$ also correspond to edge states. Moreover, the value of boundary charges is also influenced by the number of cells. For small numbers of cells, the localization of the electric field at edges diminishes, which would result in a significant calculation error of boundary charges. As the number of cells increases sufficiently, the value of boundary charges tends to converge. After the number of cells reach the value of 20, boundary charges have already achieved convergence. Based on this, we have set the number of cells to 20 in the main text (see details in Appendix C).

It is noteworthy that the edge states consistently exist throughout the entire range of ${\epsilon }_C$ from 1 to 40. In Figure 3a, we show the evolution process of the 20th and 21st states, i.e., edge states as ${\epsilon }_C$ changes. Firstly, when ${\epsilon }_C$ changes to 4 (unit cell is “$ABB$”) and 9 (unit cell is “$ABA$”), the system recovers inversion symmetry and the two edge states are degenerated. Once the value of ${\epsilon }_C$ is not 4 or 9, these two states are separated. Specifically, as the value of ${\epsilon }_C$ approaches 4 or 9, the two edge states would gradually converge toward each other. However, these two states exhibit a trend of moving away initially and then approaching each other when ${\epsilon }_C$ is greater than 9. In addition, the field distribution of these edge states with ${\epsilon }_C=40$ shows that the localization of edge states remains robust for large values of ${\epsilon }_C$. We consider that the difference between ${\epsilon }_A$ and ${\epsilon }_B$ could be ignored when ${\epsilon }_C$ is very large. Therefore, the $ABC$ model gradually approaches the symmetric system and these two edge states are close to each other as ${\epsilon }_C$ increases. The value of ${\epsilon }_C=14$ serves as the critical turning point that distinguishes the state of separation from the state of proximity between these two edge states. We will provide an explanation for the phenomenon in the following. Secondly, there is a notable alteration in the field distribution before and after reaching the degeneracy point of the edge states. Taking ${\epsilon }_C=9$ as an example, the field distribution with ${\epsilon }_C=8.9$ and ${\epsilon }_C=9.1$ is shown in Figure 3c. Interestingly, the position of the field distribution is changed after ${\epsilon }_C$ varying from $8.9$ to $9.1$. In symmetric systems, it is widely acknowledged that the band-gap closing could trigger a transformation in the system’s topological properties. This process is linked with modifications in the field distribution of high-symmetry points. In 1D PhC without inversion symmetry, the degeneracy of edge states is also observed to coincide with changes in the distribution of field. However, it is important to note that this process does not induce any alteration in the topological properties of the energy band. The above results indicate that the edge states of asymmetric systems develop continuously from symmetric systems and have topological origin.

To further investigate the topological origin of edge states in the first gap, we also calculate the evolution of boundary charges as ${\epsilon }_C$ changes. However, in contrast to traditional system with inversion symmetry, the boundary charges of our system are not always equal to $1/2$. This abnormal phenomenon can be explained by the absence of crystalline symmetries, e.g., the inversion symmetry for 1D PhC. To confirm it, we calculate the boundary charges with ${\epsilon }_C$ increasing from 1 to 40 and the results are shown in Figure 3b. Boundary charges would increase to $1/2$ when the system is closer to inversion symmetry with ${\epsilon }_C$ changing. Especially, when ${\epsilon }_C$ changes to 4 (unit cell is “$ABB$”) and 9 (unit cell is “$ABA$”), the system recovers inversion symmetry, which makes the boundary charge be quantized. Here, we focus on the range of ${\epsilon }_C$ varying from 9 to 40. When ${\epsilon }_C$ is slightly greater than 9, the inversion symmetry of this system is broken. Therefore, boundary charges are no longer quantized but gradually decrease. But boundary charges tend to be $1/2$ again with ${\epsilon }_C$ increasing. That is because the system is more like a symmetrical system in the extreme case that ${\epsilon }_C$ is much larger than ${\epsilon }_A$ and ${\epsilon }_B$. The turning point from asymmetric system to symmetric system is the average dielectric constant of layer B and C being equal to that of layer A. Under the parameters of this model, this turning point is ${\epsilon }_C=14$. Interestingly, the topological singularity of the first band crosses the zero frequency to reach imaginary frequency at this time [19,20]. For the topologically nontrivial band-gap structure of periodic systems, topological singularity could be directly positioned by the zero-scattering condition, which means perfect transmission of period structure [8,9]. The evolution of topological singularities has been extensively researched because it is the origin of topological phase transition, which is signed by the gap closing–reopening process. However, in this case of topological singularity passing through the zero frequency into the imaginary frequency by changing material parameter, the topological properties of energy band will not change because this situation will not cause the band-gap to close [21,22]. This also illustrates why boundary charges are non-zero and edge states still exist during the process of ${\epsilon }_C$ undergoing changes from another perspective.

In this section, we delve into the continuous evolution process of both edge states and boundary charges with ${\epsilon }_C$ varying from 1 to 40. Based on the obtained results, it is evident that the edge states in asymmetric systems also correspond to non-zero boundary charges. Both edge states and boundary charges of 1D PhC without inversion symmetry have undergone continuous evolution originating from a symmetric system.

3. Discussion

In the previous section, we discuss the influence of symmetry breaking on the topological properties of the band-gap in photonic systems and reach the conclusion that edge states still exist even when the boundary charges are non-quantized. However, there is no in-depth discussion of the underlying mechanism and no comprehensive explanation to address the observed phenomenon that the boundary charges are always non-zero during the evolution process. In Appendix A, we obtain the relationship between boundary charges and Wannier center according to the modern theory of polarization [23,24]. For bulk dipole moment in 1D crystals, boundary charges could be described in terms of localized Wannier states and boundary charges are proportional to the Wannier center for the $ABC$ model of one occupied band. Thus, we could explore the topological origin of boundary charges based on the evolution trajectory of Wannier center.

We could calculate maximally localized Wannier function (MLWF) [25,26,27] to obtain the Wannier center of 1D PhC without inversion symmetry (see details in Appendix B). The choice of MLWF is motivated by its inherent property of being less influenced by symmetry, thereby enabling a more localized characterization of Wannier function. For an isolated band of symmetrical system, the MLWF is centered at either 0 or $a/2$, the two inversion centers in the unit cell. MLWF would be centered at 0 if ${\epsilon }_A$<${\epsilon }_B$ (Figure 4a) with the origin of the coordinate frame being fixed at the center of layer A, which makes the first gap be topologically trivial. Thus, by constructing a system with topologically non-trivial PC1 (the center of MLWF of the first band is at $a/2$) on one side and topologically trivial PC2 (the center of MLWF of the first band is at 0) on the other side, we should see an interface state inside the first gap. For our system without inversion symmetry, MLWF is no longer localized at 0 or $a/2$, which can be seen in Figure 4b. However, by constructing a system consisting of a slab of PC1 (with 10 unit cells, ${\epsilon }_A=9$, ${\epsilon }_B=4$, ${\epsilon }_C=6$ and $d_A=d_B=2d_C$) on one side and a slab of PC2 (with 10 unit cells, ${\epsilon }_A=4$, ${\epsilon }_B={\epsilon }_C=9$ and $d_A=d_B=d_C$) on the other side embedded in vacuum to obtain a transmission spectrum, Figure 4c clearly shows a resonance transmission in the first gap due to an interface state.

The above results prove that, even for an asymmetric system with non-integer topological invariants, the edge state also has topological origin. In order to identify whether there is a topological phase transition with ${\epsilon }_C$ varying from 1 to 40, the evolution track of the Wannier center of the first band is drawn in Figure 4d. The Wannier center is localized at $a/2$ with ${\epsilon }_A=9$, ${\epsilon }_B=4$, ${\epsilon }_C=4$ (${\epsilon }_A$ =9, ${\epsilon }_B=4$, ${\epsilon }_B=9$). But it could not move from $a/2$ to another inversion center, i.e., 0 or a due to a lack of band crossing, which means that the topological properties of the first band do not transform non-triviality to triviality. This also explains why there are always edge states with ${\epsilon }_c$ varying from 1 to 40 from another angle.

Different from common cases, the special property is that boundary charges are not equal to $1/2$. We can also explain the topological origin of boundary charges in a 1D asymmetric photonic system from the point of view of the change of the Wannier center. In the asymmetric system, the Wannier center exhibits the ability to be localized across the entire range of 0–$a/2$. In this case, the bulk polarization which arises from the displacement of the Wannier center with respect to the center of the lattice is no longer quantized. Therefore, boundary charges which are attributed to the edge polarization could not take quantized values. With the enhancement of the symmetry of the system, the Wannier center gradually approaches to 0 or $a/2$, and boundary charges are also close to $1/2$. If the Wannier center remains unable to transition from one inversion center to another, the topological properties of the energy band will not be reversed and boundary charges will not evolve to zero.

4. Conclusions

In this paper, we construct a 1D PhC model without inversion symmetry and discuss the physical origin of its edge states and boundary charges in detail. The results show that, even for those 1D systems without inversion symmetry, there are still edge states when their topological invariants are non-integer. These edge states are associated with non-zero boundary charges which are no longer quantized but less than $1/2$. At the same time, we explore the topological origin of its boundary charges from the movement of the Wannier center. Since the Wannier center can assume any value between 0 and $a/2$, it follows that boundary charges are non-quantized. Under the condition that the Wannier center does not transition between different inversion centers during parameter evolution, the existence of non-zero boundary charges remains constant. The discovery of these phenomena can serve as inspiration to expand this idea to encompass high-dimensional systems or other wave systems as well.