ABSTRACT Quadrupole topological phases, exhibiting protected boundary states that are themselves topological insulators of lower dimensions, have recently been of great interest. Extensions

of these ideas from current tight binding models to continuum theories for realistic materials require the identification of quantized invariants describing the bulk quadrupole order. Here

we identify the analog of quadrupole order in Maxwell’s equations for a gyromagnetic photonic crystal (PhC) through a double-band-inversion process. The quadrupole moment is quantized by the

simultaneous presence of crystalline symmetry and broken time-reversal symmetry, which is confirmed using three independent methods: analysis of symmetry eigenvalues, numerical calculations

QUADRUPOLE TOPOLOGICAL INSULATOR USING ORBITAL-INDUCED SYNTHETIC FLUX Article Open access 03 November 2022 REAL HIGHER-ORDER WEYL PHOTONIC CRYSTAL Article Open access 20 October 2023

BULK-LOCAL-DENSITY-OF-STATE CORRESPONDENCE IN TOPOLOGICAL INSULATORS Article Open access 14 November 2023 INTRODUCTION Symmetries play a pivotal role in understanding and classifying various

topological phases of matter1,2,3,4. In periodic media, systems with different symmetries can admit different classifications characterized by quantized topological invariants. Furthermore,

interface states in symmetry-protected topological (SPT) phases can only robustly exist, between two topologically distinct regions, when required bulk symmetries are preserved at the

boundaries. In the simplest example of SPT phases, a one-dimensional (1D) Su-Schrieffer-Heeger (SSH) model, the topological invariant—Zak phase5—is only quantized when inversion symmetry is

preserved, leading to edge states that are protected by non-zero bulk polarizations. Recently, the concept of polarization, or bulk dipole moment in crystals, has been generalized to include

multipole moments, such as quadrupole and octupole moments, leading to the discovery of higher-order topological insulators (HOTIs)6,7,8,9,10. In particular, these HOTIs have vanishing

dipole densities but non-zero higher-order multiple moments, which can be quantized by certain crystalline symmetries, such as reflection and rotation. In contrast to conventional

topological insulators (TIs) that support gapless boundary states, HOTIs exhibit protected boundaries that are, themselves, TIs in lower dimensions. Recently, quadrupole TIs have been

demonstrated in a number of classical systems, ranging from microwave11 and optics12 to acoustics13 and circuits14,15. Most experiments are analyzed as lattice models, following the

tight-binding approximation. In these lattice models, the adopted symmetry required to quantize the quadrupole moment is reflection or fourfold rotation (_C_4) with threaded _π_-flux, and

hence these systems are time-reversal invariant. Unfortunately, most realistic systems with subwavelength features, such as photonic crystals, cannot be modeled discretely but require a

different continuum approach. For example, the practical implementations of many lattice models no long preserve quantized bulk quadrupole moments in the continuum theory. In addition, most

experimental works solely used the existence of in-gap corner states as the measure of bulk quadrupole topology. The validity relies on the presence of additional chiral symmetry in the

lattice model, which is often not preserved in the continuum theory. Here, we find solutions to continuous Maxwell's equations in gyromagnetic photonic crystals that are the

electrodynamic analogs to quadrupole topological phases. The proposed topological PhCs have quantized bulk quadrupole moments, which are protected by the simultaneous presence of crystalline

symmetries and broken time-reversal symmetry (_T_)16. In particular, we show it is essential to break time-reversal symmetry—to open the energy gap—while preserving crystalline

symmetries—to quantize bulk dipole and quadrupole moments. In addition, we note the existence of corner states is neither sufficient nor necessary condition for quadrupole phases. Instead,

we validate the bulk quadrupole nature through analyzing the Wannier band polarization and its manifestations at boundaries as quantized edge polarizations and fractional corner charges. All

calculations are based on realistic parameters that are readily available in the microwave regime17,18. RESULTS QUADRUPOLE PHASE TRANSITION THROUGH BAND INVERSION We start by presenting the

topological phase transition between a trivial (Fig. 1b) and a quadrupole two-dimensional (2D) PhC (Fig. 1c). The 2D PhC consists of gyromagnetic rods in air and is homogeneous along the

out-of-plane (_z_) direction. Experimentally, this boundary condition can also be realized using metals17,18. The gapless transition point is achieved in a 2 × 2 super-cell structure with

four square rods (Fig. 1a). All rods are identical in shape and are of the same gyromagnetic material of Yttrium Iron Garnet (YIG) with isotropic dielectric permittivity of _ϵ_ = 15_ϵ_0 and

inplane permeability _μ_ = 14_μ_0. To break time-reversal symmetry, an external magnetic field is applied along the out-of-plane direction (_z_), which induces complex-valued off-diagonal

terms in the permeability tensor of YIG19: $$\mathop{\mu }\limits^{=}=\left[\begin{array}{ccc}\mu &i\kappa &0\\ -i\kappa &\mu &0\\ 0&0&{\mu }_{0}\end{array}\right]$$

(1) This gyromagnetic response, \({\mu }_{xy}={\mu }_{yx}^{* }=i\kappa\), breaks _T_ but preserves _C_4 and _M__x_(_y_)_T_. Here _M__x_(_y_) is mirror reflection that transforms _x_(_y_) to

−_x_(_y_). At the phase transition, all rods are placed at _a_/2 away from their neighbors; the corresponding band structure for TM modes (_E__z_, _H__x_, _H__y_) has twofold (fourfold)

degeneracies at the center (corner) of the folded Brilluion zone Γ (M). Both degeneracies are lifted when the four rods are simultaneously displaced inward (_d_ < 0, Fig. 1b) or outward

(_d_ > 0, Fig. 1c) along the diagonal lines of the unit cell. On either side of the transition point, the band structure is fully gapped owing to _T_-breaking (shaded in yellow) and

supports two topologically distinct phases determined by the displacement _d_: for the choice of cell in Fig. 1, inward displacements with negative _d_ give rise to trivial phases, whereas

outward displacements with positive _d_ correspond to quadrupole phases. Next, we present our calculations of the quadrupole topological invariant _q__x__y_, using three different

approaches, and demonstrate the phase transition between _q__x__y_ = 0 and _q__x__y_ = 1/2 by displacing the dielectric rods. We start by evaluating _q__x__y_ using the _C_4 eigenvalues at

high-symmetry K points of all bands below the gap6,16: $${e}^{i2\pi {q}_{xy}}={r}_{4}^{+}(\Gamma ){r}_{4}^{+* }({\rm{M}})={r}_{4}^{-}(\Gamma ){r}_{4}^{-* }({\rm{M}}).$$ (2) Here,

\({r}_{4}^{+}\) (\({r}_{4}^{-}\)) is the _C_4 eigenvalue of a mode with _C_2 eigenvalue _r_2 = +1(−1); naturally, \({r}_{4}^{+}={\pm}\!{1}\) and \({r}_{4}^{-}={\pm}\!{i}\). Accordingly, a

quadrupole topological phase transition happens when two pairs of bands switch their _C_4 eigenvalues at the same time—a process we call “double-band-inversion”. Specifically, the

double-band inversion happens in our system when _d_ changes from negative to positive: through this process, the TM mode at M with a phase winding of +2_π_ in the _E__z_ mode profile (_r_4 

= _i_, labeled as green) switches position with the one with the winding of −2_π_ (_r_4 = −_i_, label in red); meanwhile, the two modes with _r_4 = ±1 at M also switch positions. On the

other hand, all _C_4 eigenvalues at Γ remain unchanged. Using Eq. (2), we identify PhCs with _d_  > 0 to be topologically non-trivial, with bulk quadrupole invariant _q__x__y_ = 1/2, and

PhCs with _d_ < 0 to be topologically trivial with _q__x__y_ = 0. A more-comprehensive topological phase diagram of our system is shown in Fig. 1f, determined by displacement _d_ and

strength of gyromagnetic response _κ_. Both quadrupole topological insulators (green) and trivial insulators (orange) are identified, with the super-cell structure (_d_ = 0) being the

transition in between the two phases (_y_ axis). Interestingly, Chern insulating phases (purple) are observed at large displacements, consistent with the broken time-reversal symmetry19,20.

NESTED WILSON LOOP AND WANNIER REPRESENTATION To confirm the quadrupole topology of our PhC, we explicit show that its dipole moments are zero (_p__x_ = _p__y_ = 0), but its quadrupole

moment is non-zero (_q__x__y_ = 1/2). To this end, we perform two separate sets of calculations using two different methods: (1) the nested Wilson loop formulation6; (2) and the expectation

value of the exponentiated quadrupole operator21,22, and show they reach the same conclusions. Here, we present the nested Wilson loop calculations, leaving the second method in

Supplementary Note 3. We start by computing the band structure and mode profiles (_E__z_) of the TM modes for a particular displacement of dielectric rods _d_ = _a_/4 − _w_/2 (Fig. 2a, same

as in Fig. 1c). Using these as input, we compute the Wannier bands _ν__x_,_y_ for the two lowest-energy bands (Supplementary Note 1). The results (Fig. 2a) show that the Wannier bands are

related as \(\{{\nu }_{x}^{j}({k}_{y})\}\to \{-{\nu }_{x}^{j}(-{k}_{y})\}\) mod 1 owing to the presence of _C_2 symmetry (Supplementary Note 2). Therefore, the bulk polarization _p__x_,

which is simply the integral of the two Wannier bands over the 1D B.Z., vanishes: \({p}_{x}={\sum }_{n = {\pm} }{\int}_{{\rm{B}}.{\rm{Z}}.}d{k}_{y}\,{\nu }_{x}^{n}({k}_{y})=0\). A similar

argument can also be applied to bulk polarization _p__y_, proving the bulk dipole moments are zero: _p__x_ = _p__y_ = 0. Meanwhile, owing to the existence of a gap in the Wannier spectrum,

the upper and lower Wannier bands can be separated into two sectors, labeled as \({\nu }_{x}^{\pm }\) (\({\nu }_{y}^{\pm }\)), respectively. To confirm the quadrupole topology, we compute

the polarizations of the Wannier bands within one sector, \({p}_{y}^{{\nu }_{x}^{-}}\) and \({p}_{x}^{{\nu }_{y}^{-}}\), using the nested Wilson loop formulation6,23. As shown in Fig. 2b,

both Wannier band polarizations are quantized to be 1/2—due to the simultaneous preservation of _M__x__T_ and _M__y__T_ symmetries (Supplementary Note 2)—which further confirms our PhC has a

non-zero bulk quadrupole moment: \({q}_{xy}=2{p}_{x}^{{\nu }_{y}^{-}}{p}_{y}^{{\nu }_{x}^{-}}=1/2\). To compare, we repeat the same set of calculations for a different unit cell with a

negative displacement of _d_ = −(_a_/4 − _w_/2) (Fig. 2c). As shown, the Wannier bands are also gapped and sum to zero, meaning the bulk dipoles _p__x_,_y_ remain zero. However, the nested

Wilson loops, \({p}_{y}^{{\nu }_{x}^{-}}\) and \({p}_{x}^{{\nu }_{y}^{-}}\), are also zero, leading to a trivial quadrupole moment _q__x__y_ = 0. This correspondence between positive

(negative) displacements leading to non-trivial (trivial) quadrupole phases is consistent with our preceding conclusions based on _C_4 symmetry eigenvalues. Importantly, the non-trivial and

trivial PhCs in Fig. 2 can be simply related to each other, by shifting the choice of the unit cell center by _a_/2. This observation leads to an intuitive understanding of the difference in

quadrupole moment between the two phases as discussed in Supplementary Note 5. BULK-EDGE CORRESPONDENCE OF WANNIER BANDS Next, we present the physical consequences of quadrupole topological

PhCs at interfaces, originating from the bulk-edge correspondence of the Wannier bands. Following classical electromagnetism, a non-zero bulk quadrupole moment in a finite system is

manifested as edge polarizations at its boundaries. Here, we study the 1D interfaces between quadrupole (trivial) PhCs and perfect electric conductors (PECs). We find that the non-trivial

quadrupole topology indeed leads to a quantized edge polarization along the interface, which is absent for a trivial PhC. Specifically, we consider a strip of 20 unit cells of the quadrupole

(trivial) PhC design, which satisfies periodic boundary condition in the _x_—direction and closed boundary condition in the _y_—direction, owing to the two PECs on top and bottom as shown

in Fig. 3a, b. The energy dispersions _ω_(_k__x_) of the quadrupole and trivial strips (Fig. 3c, d) share the same bulk energy spectra (gray areas) but have different edge dispersions (gray

solid lines), owing to their different edge terminations. The Wannier centers _ν__x_ are calculated for the two different strips based on their energy dispersions and Bloch mode profiles

(Fig. 3e, f). For the topological strip, two additional Wannier states (red circles) are found to emerge outside the Wannier bands (blue) at the middle of the gap (±0.5), which is protected

by the non-trivial bulk quadrupole moment. This can be understood in a similar way as the in-gap edge states in the 1D SSH model with non-trivial bulk polarization. The quantization of the

mid-gap Wannier states at ±0.5 is due to the additional _M__x__T_ symmetry in our system, which is retained even at the boundary of a finite strip as shown in Fig. 3a. (Supplementary Note 

2). In comparison, no additional Wannier states are found inside the Wannier gaps for the trivial strip, consistent with the lack of bulk quadrupole moment. Furthermore, the two mid-gap

Wannier states in the topological strip are spatially localized at the top and bottom edges. To demonstrate this, we further study the spatial distribution of Wannier states by calculating

the polarization density _p__x_ as a function of position along _y_ (Supplementary Note 6). In order to choose a definite sign of the polarization, we introduce an infinitesimal perturbation

to break the _C_2 symmetry of the semi-infinite strip. For the quadrupole PhC, as shown in Fig. 3g, there are non-zero polarization densities developed near the two edges at _y_ = 0 and _y_

 = 20_a_, and the edge polarizations are quantized to ±0.5. In comparison, neither edge nor bulk polarization are observed in the trivial PhC (Fig. 3h). FRACTIONAL CORNER CHARGES AND FILLING

ANOMALY Finally, we show the physical consequence of quadrupole PhCs as localized 0D corner states, which are the photonic analogs of states responsible for filling anomalies and fractional

charges in an electronic setting24. To this end, we solve the eigenstates in a finite 2D quadrupole PhC enclosed by PECs, with a thin air gap in between (Fig. 4a). The eigenstates are

labeled according to their energies, with the lowest-energy state labeled as 1. Aside from delocalized bulk states, four degenerate states—199 through 202—are found, with their energies

inside the bulk energy gap (Fig. 4b). Their mode profiles (_E__z_) confirm that these states are spatially localized at the four corners, with one example shown in Fig. 4c. Owing to the lack

of chiral symmetry in our system, which is generic in the continuum theory expanded around non-zero frequencies, the energy of the four corner states is not pinned at the center of the bulk

energy gap25; instead they can be shifted, even immersed into the bulk continuum, by modifying the refractive index at the corners. This renders the appearance of corner states less of an

essential signature of quadrupole phases10,11. In fact, corner states are also found in other higher-order topological phases with vanishing bulk quadrupole moments23,24,25,26,27,28,29,30.

Instead, here we illustrate the non-trivial quadrupole nature of our PhC using the filling anomaly, by counting the number of energy eigenstates below and above a given bulk energy gap10

(Fig. 4a). Specifically, even though trivial samples may support corner states, they originate from either the top or bottom band alone, leaving 2_N_2−4_n_ states in the bulk continuum (_n_

is an integer). On the other hand, for non-trivial PhCs, the number of states below and above the energy gap are both 2_N_2 − 2 = 198 (_N_ = 10 in our case). This indicates the four

degenerate corner states we observed are “contributed” by both the top and bottom bands together, and thus proves the non-trivial quadrupole topology of our design. As a consequence, for a

quadrupole PhC, quantized fractional charges appear at four corners of a finite sized system (Fig. 4d) when calculate the spatial distribution of the lowest 200 energy states, in a similar

vein as the calculations of charge density in electronic systems at “half-filling” (here, we have introduced an infinitesimal perturbation to break _C_4 symmetry in order to split the

fourfold degenerate corner states). Remarkably, these corner charges are shared by two convergent dipoles on the two perpendicular edges, as the magnitude of the corner charges is equal to

the edge polarizations. This further confirms that these corner charges originate from a non-zero bulk quadrupole moment. We point out that the observed fractional corner charges arise from

the fundamental difference in the counting of bulk states10, and was recently proposed in electronic systems by Benalcazar et al.24 as a filling anomaly: a mismatch between number of states

in an energy band and the number of electrons required for charge neutrality. DISCUSSION In summary, we present quadrupole topological photonic crystals with truly quantized invariants and

the physical consequences at material’s edges and corners. The proposed gyromagnetic PhCs can be readily realized in the microwave regime. Meanwhile, the coexistence of multiple topological

phases in our system, both quadrupole TIs and Chern insulators, provides a versatile platform to further demonstrate topological photonic circuits with protected elements immune to disorders

in various dimensions. Finally, our findings of inducing quadrupole phase transitions and quantizing quadrupole moments—through crystalline symmetries in conjunction with broken

time-reversal symmetry—can also be applied to other wave systems, including electrons, phonons, and polaritons. METHODS NUMERICAL SIMULATION OF MAXWELL’S EQUATION USING THE FINITE ELEMENT

METHOD The band structures and mode profiles are calculated using the Finite Element Method in COMSOL Multiphysics 5.4. Specifically, we compute the band structures and mode profiles in a 2D

geometry with periodic boundary conditions along all directions. The corresponding Wannier bands are calculated by using the Bloch mode profiles as input (Supplementary Note 1). DATA

AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. CODE AVAILABILITY The code that supports the plots within

this paper and other findings of this study is available from the corresponding authors upon reasonable request. REFERENCES * Chiu, C.-K., Teo, J. C. Y., Schnyder, A. P. & Shinsei, R.

Classification of topological quantum matter with symmetries. _Rev. Mod. Phys._ 88, 035005 (2016). Article  ADS  Google Scholar  * Kitaev, A. Periodic table for topological insulators and

superconductors. _AIP conference proceedings_ 1134, 22–30 (2009). * Ryu, S., Schnyder, A. P., Furusaki, A. & Ludwig, A. W. W. Topological insulators and superconductors: tenfold way and

dimensional hierarchy. _N. J. Phys._ 12, 065010 (2010). Article  Google Scholar  * Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. W. Classification of topological insulators and

superconductors in three spatial dimensions. _Phys. Rev. B_ 78, 195125 (2008). Article  ADS  Google Scholar  * Zak, J. Berry’s phase for energy bands in solids. _Phys. Rev. Lett._ 62, 2747

(1989). Article  ADS  CAS  Google Scholar  * Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Electric multipole moments, topological multipole moment pumping, and chiral hinge states

in crystalline insulators. _Phys. Rev. B_ 96, 245115 (2017). Article  ADS  Google Scholar  * Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators.

_Science_ 357, 61–66 (2017). Article  ADS  MathSciNet  CAS  Google Scholar  * Langbehn, J., Peng, Y., Trifunovic, L., von Oppen, F. & Brouwer, P. W. Reflection-symmetric second-order

topological insulators and superconductors. _Phys. Rev. Lett._ 119, 246401 (2017). Article  ADS  Google Scholar  * Schindler, F. et al. Higher-order topological insulators. _Sci. Adv._ 4,

eaat0346 (2018). Article  ADS  Google Scholar  * Song, Z., Fang, Z. & Fang, C. (d- 2)-dimensional edge states of rotation symmetry protected topological states. _Phys. Rev. Lett._ 119,

246402 (2017). Article  ADS  Google Scholar  * Peterson, C. W., Benalcazar, W. A., Hughes, T. L. & Bahl, G. A. quantized microwave quadrupole insulator with topologically protected

corner states. _Nature_ 555, 346–350 (2018). Article  ADS  CAS  Google Scholar  * Mittal, S. et al. Photonic quadrupole topological phases. _Nat. Photonics_ 13, 692–696 (2019). Article  ADS

  CAS  Google Scholar  * Serra-Garcia, M. et al. Observation of a phononic quadrupole topological insulator. _Nature_ 555, 342–345 (2018). Article  ADS  CAS  Google Scholar  * Imhof, S. et

al. Topolectrical-circuit realization of topological corner modes. _Nat. Phys._ 14, 925–929 (2018). Article  CAS  Google Scholar  * Serra-Garcia, M., Süsstrunk, R. & Huber, S. D.

Observation of quadrupole transitions and edge mode topology in an lc circuit network. _Phys. Rev. B_ 99, 020304 (2019). Article  ADS  CAS  Google Scholar  * Wieder, B. J. et al. Strong and

fragile topological dirac semimetals with higher-order fermi arcs. _Nat. Commun._ 11, 1–13 (2020). Article  Google Scholar  * Skirlo, S. A. et al. Experimental observation of large chern

numbers in photonic crystals. _Phys. Rev. Lett._ 115, 253901 (2015). Article  ADS  Google Scholar  * Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljačić, M. Observation of unidirectional

backscattering-immune topological electromagnetic states. _Nature_ 461, 772–775 (2009). Article  ADS  CAS  Google Scholar  * Wang, Z., Chong, Y. D., Joannopoulos, J. D. & Soljačić, M.

Reflection-free one-way edge modes in a gyromagnetic photonic crystal. _Phys. Rev. Lett._ 100, 013905 (2008). Article  ADS  Google Scholar  * Skirlo, S. A., Lu, L. & Soljačić, M.

Multimode one-way waveguides of large chern numbers. _Phys. Rev. Lett._ 113, 113904 (2014). Article  ADS  Google Scholar  * Kang, B., Shiozaki, K. & Cho, G. Y. Many-body order parameters

for multipoles in solids. _Phys. Rev. B_ 100, 245134 (2019). Article  ADS  CAS  Google Scholar  * Wheeler, W. A., Wagner, L. K. & Hughes, T. L. Many-body electric multipole operators in

extended systems. _Phys. Rev. B_ 100, 245135 (2019). Article  ADS  CAS  Google Scholar  * Xie, B.-Y. et al. Second-order photonic topological insulator with corner states. _Phys. Rev. B_

98, 205147 (2018). Article  ADS  CAS  Google Scholar  * Benalcazar, W. A., Li, T. & Hughes, T. L. Quantization of fractional corner charge in c n-symmetric higher-order topological

crystalline insulators. _Phys. Rev. B_ 99, 245151 (2019). Article  ADS  CAS  Google Scholar  * Noh, J. et al. Topological protection of photonic mid-gap defect modes. _Nat. Photonics_ 12,

408–415 (2018). Article  ADS  CAS  Google Scholar  * Chen, X.-D. et al. Direct observation of corner states in second-order topological photonic crystal slabs. _Phys. Rev. Lett._ 122, 233902

(2019). Article  ADS  CAS  Google Scholar  * Hassan, A. E. L. et al. Corner states of light in photonic waveguides. _Nat. Photonics_ 13, 697–700 (2019). Article  ADS  Google Scholar  * Ni,

X., Weiner, M., Alù, A. & Khanikaev, A. B. Observation of higher-order topological acoustic states protected by generalized chiral symmetry. _Nat. Mater._ 18, 113–120 (2019). Article 

ADS  CAS  Google Scholar  * Xue, H., Yang, Y., Gao, F., Chong, Y. & Zhang, B. Acoustic higher-order topological insulator on a kagome lattice. _Nat. Mater._ 18, 108–112 (2019). Article 

ADS  CAS  Google Scholar  * Zhang, X. et al. Second-order topology and multidimensional topological transitions in sonic crystals. _Nat. Phys._ 15, 582–588 (2019). Article  CAS  Google

Scholar  Download references ACKNOWLEDGEMENTS This work was partly supported by the National Science Foundation through the University of Pennsylvania Materials Research Science and

Engineering Center DMR-1720530 and the US Office of Naval Research (ONR) Multidisciplinary University Research Initiative (MURI) grant N00014-20-1-2325 on Robust Photonic Materials with

High-Order Topological Protection. L.H. was supported by the Air Force Office of Scientific Research under award number FA9550-18-1-0133. Work by Z.A. and E.M. on the symmetry analysis of

the quadrupole phase was supported by the Department of Energy, Office of Basic Energy Sciences under grant DE FG02 84ER45118. B.Z. was supported by the Army Research Office under award

contract W911NF-19-1-0087. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, 19104, USA Li He, Zachariah

Addison, Eugene J. Mele & Bo Zhen Authors * Li He View author publications You can also search for this author inPubMed Google Scholar * Zachariah Addison View author publications You

can also search for this author inPubMed Google Scholar * Eugene J. Mele View author publications You can also search for this author inPubMed Google Scholar * Bo Zhen View author

publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS L.H. and B.Z. conceived the idea. L.H. carried out numerical simulations. L.H., Z.A., E.M., and B.Z.

discussed and interpreted the results. L.H and B.Z. wrote the paper with contribution from all authors. B.Z. supervised the project. CORRESPONDING AUTHOR Correspondence to Bo Zhen. ETHICS

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