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# Photonic topological insulator with broken time-reversal symmetry

Edited by Zhi-Xun Shen, Stanford University, Stanford, CA, and approved March 30, 2016 (received for review December 24, 2015)

## Significance

Topological insulators are first discovered in electronic systems. A key factor is the Kramers doublet for the spin-1/2 electrons under fermionic time-reversal symmetry *T*_{p} (

## Abstract

A topological insulator is a material with an insulating interior but time-reversal symmetry-protected conducting edge states. Since its prediction and discovery almost a decade ago, such a symmetry-protected topological phase has been explored beyond electronic systems in the realm of photonics. Electrons are spin-1/2 particles, whereas photons are spin-1 particles. The distinct spin difference between these two kinds of particles means that their corresponding symmetry is fundamentally different. It is well understood that an electronic topological insulator is protected by the electron’s spin-1/2 (fermionic) time-reversal symmetry *T*_{p} (*T*_{p} rather than by the bosonic time-reversal symmetry *T*_{b}. This remarkable finding is expected to pave a new path to understanding the symmetry protection mechanism for topological phases of other fundamental particles and to searching for novel implementations for topological insulators.

- photonic topological insulator
- piezoelectric/piezomagnetic superlattice
- photonic crystal
- polariton
- time-reversal symmetry

Topological description of electronic phase has now become a new paradigm in the classification of condensed matters (1⇓⇓⇓⇓⇓⇓⇓⇓–10). Electronic topological insulators (TIs) are time-reversal symmetry (TRS) protected topological phase, exhibiting gapless edge/surface states in their bulk bandgap due to strong spin−orbit coupling (11, 12). This intriguing classification of the topologically protected phase has been applied to study other systems as well, for example, photonic systems. The photonic analog of the integer quantum Hall effect has been proposed and extensively studied with a single transverse electric (TE) polarization state (13, 14) or magnetic field (TM) polarization state (15, 16) in gyrotropic photonic crystals with broken TRS in the presence of an external magnetic bias; the fractional quantum Hall effect has also been investigated in a correlated photonic system (17). To date, to design photonic TIs (PTIs) with TRS, several models have been reported (18⇓⇓⇓⇓⇓⇓⇓–26) using a pair of degenerate photonic states, e.g., hybrid TE+TM/TE − TM states (21, 22, 24) or pseudospins represented by clockwise/counterclockwise helical energy flow states (19, 20, 23), where the degenerate states “see” opposite effective magnetic gauge fields. Additionally, the photonic Floquet TIs have also been explored with a helix photonic structure under a broken spatial inversion symmetry (18). However, the robustness of these topological phases has not been fully investigated and validated against a comprehensive set of TRS invariant impurities (18⇓⇓⇓⇓⇓⇓⇓⇓–27). According to the topological theory developed for condensed matters, having a Kramers doublet is the key to constructing a TI. It is well known that the spin quantum number of photons is different from that of electrons. Hence, they belong to two different classes of particles, boson and fermion, and have different TRS operators: *z* and *y* components, respectively, of a Pauli matrix, and *K* is the complex conjugation. Kramers degeneracy theorem states that there cannot exist any Kramers doublet for photons under

In this work, we explore this mysterious symmetry protection issue in a Tellegen photonic crystal medium (29), where the Tellegen magnetoelectric coupling can be realized in piezoelectric (PE) and piezomagnetic (PM) superlattice constituents (30, 31). The presence of Tellegen magnetoelectric coupling enables a photonic pseudospin−orbital coupling effect, which breaks the bosonic TRS but, at the same time, creates a fermionic-like pseudo TRS due to the electromagnetic duality. Consequently, two pseudospin states that are helical states represented by left and right circular polarizations (or any two desired modes/polarization states) can be carefully designed and matched to form Kramers degeneracy under the fermionic-like pseudo TRS. In addition, we show that a pair of degenerate gapless edge states for the Kramers doublet, i.e., left and right circular polarizations, exists in the bulk bandgap of the Tellegen photonic crystal. Such edge states exhibit pseudospin-dependent transportation, which characterizes a type of PTI with broken bosonic TRS in the presence of inherent magnetoelectric coupling (32). Our further analysis shows that this PTI is protected by the fermionic-like pseudo TRS

## Models

Previous studies on polarization degeneracy-based PTIs mainly focus on a pair of linearly polarized states, e.g., TE/TM states (22) and TE+TM/TE − TM states (21, 24) as shown in a Poincaré sphere (Fig. 1*A*, *Left*). There also exists another important pair of polarization states, left circular polarization (LCP: positive helix)/right circular polarization (RCP: negative helix). These two states are commonly used to mimic the electronic spin states (Fig. 1*A*, *Right*), due to their opposite helicity−orbital couplings. Herein, they are referred as *x* component of a Pauli matrix) should also be taken into account during the time-reversal operation because of the magnetoelectric interaction

Clearly, the key to realizing a photonic Kramers doublet in our system is to implement the required magnetoelectric coupling. Here we construct a quasi-2D superlattice photonic crystal (SLPC) consisting of arrayed meta-atoms (34, 35): superlattices with alternating PE/PM layers with the periodicity at a deep subwavelength scale along the *z* direction (Fig. 1*B*). To realize the pseudospin degenerate states, the point groups of PE and PM materials are chosen to be 622 and 6 mm (or 422 and 4 mm) with nonzero PE coefficient _{3}−CoFe_{2}O_{4} superlattice (30, 31) (*SI Appendix*, *Part A*). The effective bianisotropic constitutive relation for these deep-subwavelength superlattices can be described as*SI Appendix*, *Part A*). The elastic coefficient is *ρ* is the material density. In a lossless case (*A* and *B*). The *C*. In contrast to conventional isolators, such an isolator exhibits a clear pseudospin-dependent transmission manifested by the opposite transmission direction of the LCP and RCP states. This unique one-way transmission property can be further leveraged to construct PTIs.

## Results and Discussion

### Gapless Edge State.

In this work, the thickness of each PE and PM layer is set to be *d* = 500 nm; the transverse vibration frequency is *a* = 9 mm. The material property PE/PM superlattice inside the SLPC at frequency of *A*) clearly shows band crossing at **M** point of the Brillouin zone for the second and third bulk bands. However, after introducing pseudospin−orbital coupling, e.g., **M** point is lifted, creating a bandgap for the bulk states with their normalized frequency ranging from 0.55 to 0.61 (corresponding to wavelengths from 1.82*a* to 1.64*a*, i.e., 16.38 mm to 14.76 mm). In the bandgap, there exist gapless helical edge states for LCP and RCP states (Fig. 3*B*), which exhibit pseudospin-dependent edge state transportation. Because LCP and RCP states experience opposite effective gauge fields, their power flux transportation behavior is completely reversed, i.e., counterclockwise for LCP and clockwise for RCP (Fig. 3 *C*–*F*).

### Robustness of Pseudospin-Dependent Transportation.

The robustness for the edge states transportation is studied in various SLPC configurations (Fig. 4). Without any defects, the pseudospin-dependent transportation is obvious (Fig. 4*A*), because the LCP (RCP) source only excites one-way clockwise (counterclockwise) light transportation (*SI Appendix*, *Part B*). This one-way robust transportation is then verified and confirmed against three different types of geometric defects, an L-shaped slab obstacle, a cavity obstacle, and a strongly disordered domain inside the cavity as shown in Fig. 4 *B*–*D*. Note that the impact of propagation loss can be safely omitted in the above analysis (*SI Appendix*, *Part C*), because the frequency of operation is far away from the resonance condition of the superlattice (as shown in Fig. 2*A*).

Next, the robustness of such photonic pseudospin-dependent transportation is further checked against a comprehensive set of impurities with optical properties satisfying all possible combination of *i*) a uniaxial dielectric impurity that is *ii*) a Tellegen impurity that is neither *iii*) a chiral impurity that is *iv*) a chiral impurity that is both *i* and *ii*, the LCP state can be backscattered into the RCP state by impurities (Fig. 5 *A* and *B*, *Left*). Such findings imply that the pseudospin-dependent transportation is not robust if impurities’ *A* and *B*, *Right*), which indicates the emergence of a bandgap from an initially gapless band structure. In case *iii*, the eigen equation of the chiral medium can be treated as a superposition of two orthogonal LCP and RCP eigen states with different Bloch wave vectors. Although the impurity cannot backscatter the RCP state (Fig. 5*C*, *Left*), the LCP and RCP edge states decouple, and both states exhibit an independent integer photonic quantum Hall effect (Fig. 5*C*, *Right*). In case *iv*, the RCP state can be excited at the impurity site as shown in Fig. 5*D*, *Left*. However, the excited RCP state is localized in the vicinity of the impurity site and no RCP state is backscattered because *D*, *Right*, the edge states remain gapless, and such a system still supports pseudospin-dependent transportation. The above analysis strongly indicates that LCP and RCP eigen states can only be considered as a pair of pseudospin states under the fermionic-like pseudo TRS **3** commutes with

Note that, in our proposed PTI, a unity normalized impedance (**3** for details). For a large mismatch, two elliptical polarized eigen states

### Topological Values.

It should be noticed that each energy band in Fig. 3*A* is doubly degenerate for LCP and RCP states experiencing opposite gauge fields. Thus, although the total Chern number is zero in our broken TRS system, the spin Chern number for LCP and RCP is +1 and −1, respectively (38). Therefore, in such a system, the photonic quantum spin Hall effect (or PTI) rather than the photonic quantum Hall effect is expected. On the other hand, because the pseudo TRS *m*th occupied energy band at four *SI Appendix*, *Part D*).

## Conclusions

In summary, we study the topological property of a Tellegen photonic crystal, which has broken bosonic TRS due to Tellegen magnetoelectric coupling. Such a coupling leads to the interchange of electric field and magnetic field of an eigen state and thus gives rise to an artificial fermionic pseudo TRS

## Materials and Methods

### Theoretical Model.

A full description of the mathematic derivation and parameters of PE and PM superlattice to obtain the effect constitutive relation can be found in *SI Appendix*, *Part A*. It should be noticed that a key factor to realizing the pseudospin degenerate states for LCP and RCP is the point groups of PE and PM materials, which are chosen to be 622 and 6 mm in this work to realize off-diagonal magnetoelectric coupling. The loss of such a superlattice can be safely omitted because the operating frequency is far away from the resonance frequency. Furthermore, we also derived the theoretical model of SLPC via the tight-binding approximation approach as shown in *SI Appendix*, *Part D*.

### Numerical Method.

Numerical investigations in this work are conducted using a hybrid RF mode of commercial FEM software (COMSOL Multiphysics). The parameters used in numerical investigations are based on the effective constitutive relation of PE and PM superlattice.

## Acknowledgments

We thank Dr. P. Nayar for helpful discussion. The work was jointly supported by the National Basic Research Program of China (Grants 2012CB921503 and 2013CB632702) and the National Nature Science Foundation of China (Grants 11134006, 11474158, and 11404164). We also acknowledge support from Academic Program Development of Jiangsu Higher Education. Y.C. acknowledges support from a DARPA MESO project (187 N66001-11-1-4105). L.F. was funded by Department of Energy (DE-SC0014485) for analyzing the results of magnetoelectric coupling and photonic topological insulator.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: luminghui{at}nju.edu.cn or yfchen{at}nju.edu.cn.

Author contributions: C.H. and M.-H.L. designed research; C.H. performed research; C.H., X.-C.S., and M.-H.L. contributed new reagents/analytic tools; C.H., X.-C.S., X.-P.L., M.-H.L., Y.C., L.F., and Y.-F.C. analyzed data; and C.H., X.-C.S., X.-P.L., M.-H.L., L.F., and Y.-F.C. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1525502113/-/DCSupplemental.

Freely available online through the PNAS open access option.

## References

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Bernevig BA,
- Hughes TL,
- Zhang S-C

- ↵.
- König M, et al.

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Kapit E,
- Hafezi M,
- Simon SH

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Bliokh KY,
- Smirnova D,
- Nori F

- ↵
- ↵
- ↵.
- Süsstrunk R,
- Huber SD

- ↵.
- Sihvola AH,
- Lindell IV

- ↵
- ↵
- ↵.
- Kong JA

- ↵.
- Lakhtakia A

- ↵.
- Smith DR,
- Pendry JB,
- Wiltshire MCK

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Fleury R,
- Sounas DL,
- Sieck CF,
- Haberman MR,
- Alù A

- ↵.
- Kane CL,
- Lubensky TC

- ↵.
- Chen BG,
- Upadhyaya N,
- Vitelli V

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