Non-quantized square-root topological insulators:
a realization in photonic Aharonov-Bohm cages

Taking the square-root of the Klein-Gordon Hamiltonian has opened up an entirely new realm of physics Peskin1995 : the resulting Dirac operator provided a relativistic quantum description of massive spin-$1/2$ fermionic particles, thus disclosing the fine-structure spectra of atoms or the anomalous Zeeman effect Peskin1995 . Interestingly, as it emerged much later, Dirac’s Hamiltonian has found numerous realizations in a completely different context: in condensed-matter physics it describes various complex materials, such as Graphene Wallace1947 ; CastroNeto2009 , transition-metal dichalcogenide monolayers Mak2010 ; wang2012electronics , 3D Weyl semimetals Armitage2018 , the Bogoliubov-de-Gennes quasiparticles dispersion in superconductors de2018superconductivity , and, more recently, topological insulators (TIs) hasan2010colloquium ; qi2011topological ; ozawa2018topological .

TIs – a new phase of matter – have to date seen a variety of manifestations, with prominent examples such as the 2D Klitzing1980 and 4D quantum Hall effects zhang2001four , 1D topological superconductors Kitaev2001 , 2D Kane2005 and 3D TIs Hsieh2008 , crystalline and quasi-crystalline TIs fu2011topological ; kraus2012topologicalpump , and higher-order TIs benalcazar2017electric . All available realizations of TIs, however, share a common feature: their spectral bands are attributed with a nonlocal topological index that is quantized hasan2010colloquium ; qi2011topological ; ozawa2018topological . Hence, whereas different realizations of TIs can vary locally, as long as their topological characterization persists, they will exhibit the same topological phenomena. In other words, the quantization of topological indices lies at the foundation of the characteristic robustness of bulk responses and associated boundary-phenomena in TIs. Currently, there is no known setting that allows for TIs with bands possessing non-quantized topological indices to present such robust phenomena.

In our work, we theoretically devise and experimentally demonstrate a new type of topological insulator that exhibits non-quantized bulk topology alongside robust boundary states. Within our paradigm, the bands of the system do not admit quantized topological invariants, but the system holds a spectral symmetry such that by taking the square of its Hamiltonian quantized topological indices emerge. Notably, whereas taking the square of the Dirac operator leads to the topologically trivial Klein-Gordon model, in our case, the degenerate bands of the squared model exhibit non-Abelian topology. These generate the topological characterization of the square-root model, as well as its corresponding boundary phenomena.

We demonstrate our general paradigm using a quasi-1D chain made of photonic Aharonov-Bohm cages. Optical settings prove to be ideally suited for realizing various topological phenomena ozawa2018topological , such as Floquet TIs rechtsman2013photonic , TIs on a silicon platform hafezi2013 , 2D kraus2012topologicalpump and 4D topological Hall physics zilberberg2018photonic , as well as non-hermitian topological physics weimann2017topologically . In this vein, we employ photonic waveguide lattices with effective negative hopping amplitudes for realizing a square-root topological insulator.

We consider a chain made of Aharonov-Bohm cages, i.e., a quasi-1D lattice composed of inter-connected plaquettes, see Fig. 1A. Each lattice site is coupled to its neighbours with hopping amplitude $t$, while each plaquette is threaded by a flux $\phi$. The momentum space Hamiltonian of this model is given by

The spectrum of $H(k)$ contains three bands: a central band that remains non-dispersive for all values of the flux $\phi$, and two additional bands that are spectrally symmetric around $E=0$. For $\phi=0$, the three bands cross, while for $\phi=\pi$ the spectrum is gapped with three flat bands at energies $\lambda_{i}\in\{-2t,0,2t\}$, see Fig. 1B. The latter case corresponds to the Aharonov-Bohm caging effect, where the particles become immobile due to destructive interference vidal1998aharonov ; longhi2014aharonov .

For each band, we can evaluate a 1D topological invariant – Zak’s winding phase, $\gamma_{i}=\int\limits_{\text{BZ}}dk\mathcal{A}_{i}(k)$, where $\mathcal{A}_{i}(k)=i\bra{v_{i}(k)}\partial_{k}\ket{v_{i}(k)}$ is the Berry connection of the $i^{\rm th}$ band, and $\ket{v_{i}(k)}$ is the corresponding eigenstate zak1989berry . The winding phase commonly takes on a binary quantized value of $\pi$ (or $0$) corresponding to encircling (or not-encircling) a singularity in quasi-momentum phase space Asboth2015 . Interestingly, we find that the zero-energy band exhibits a quantized winding $\gamma_{2}=\pi$, whereas the top and bottom bands display a non-quantized phase $\gamma_{1}=\gamma_{3}=-\pi/2$. The latter are indeed not quantized, as can be seen by their response to on-site potentials that continuously scramble $\gamma_{1}$ and $\gamma_{3}$ and can even make each of them vanish (see supplementary text for more details).

At the same time, the AB-cages chain with open boundary conditions exhibits two localized boundary states at energies $\pm\sqrt{2}t$ on each side of the system, see Fig. 1C. These boundary states are robust against on-site potentials that do not close the bulk bands, see supplementary text. Commonly, robust topological boundary states appear in a gap that lies above bands that add up to a quantized topological index hasan2010colloquium ; qi2011topological ; ozawa2018topological . The robust appearance of such boundary states is a surprising occurrence for our case where non-quantized bulk windings arise.

The elusive topological aspects of the model are revealed by taking the square of the Hamiltonian 6,

where $m_{0}=2t^{2}$, $m_{1}(k)=2t^{2}\cos(k)$ and $m_{2}(k)=2t^{2}\sin(k)$. The squared Hamiltonian is block diagonal with a single band $\ket{w_{1}}$ at energy $\Lambda_{1}=4t^{2}$ that is decoupled from the other two states. The latter form a topologically non-trivial 1D model that is mappable to the Su-Schrieffer-Heeger (SSH) model by a rotation of the $2\times 2$ block by $e^{i\sigma_{y}\frac{\pi}{4}}$, i.e., the second block is equivalent to a SSH chain with $0$ intra-cell coupling, $2t^{2}$ inter-cell coupling, and a constant $2t^{2}$ energy shift, see supplementary text. The two bands, $\ket{w_{2}}$ and $\ket{w_{3}}$, of the effective SSH sub-model are found at energies $\Lambda_{2}=0$ and $\Lambda_{3}=4t^{2}$, respectively.

Importantly, the bands $\ket{w_{1}}$ and $\ket{w_{3}}$ form a degenerate subspace at energy $\Lambda_{1}=\Lambda_{3}=4t^{2}$, thus barring our ability to evaluate an Abelian Zak phase for either of these bands. Instead, the bands of the squared Hamiltonian can be assigned a non-Abelian topological phase using the Wilzcek-Zee formulation which generalizes Zak’s phase to multiband scenarios wilczek1984appearance , $\gamma=\int\limits_{k}^{k+2\pi}\text{Tr}\Big{(}\mathcal{A}(k)\Big{)}\mathrm{d}k$, where $\mathcal{A}(k)^{nm}=\langle v_{n}(k)|\partial_{k}|v_{m}(k)\rangle$, and $n,m$ run over the involved states. For our model, the Wilzcek-Zee phase of both the zero-energy band and the degenerate subspace is $\pi\,\text{ mod }\,2\pi$, corresponding to a nontrivial TI, see supplementary text. Consequently, the standard bulk-boundary correspondence of 1D TIs applies resta2007theory , and the squared Hamiltonian (12) maintains topological boundary modes in the middle of its gap, i.e., at energy $2t^{2}$. Note that unlike the standard SSH model and in similitude to the Haldane chain pollmann2010entanglement , at proper termination, two degenerate states appear on each boundary of our model, see supplementary text. Crucially, under the square-root operation these boundary states are mapped onto specific boundaries of $H$, giving rise to the topological boundary states found in the spectrum of $H(k)$, see supplementary text.

We implement the AB-cages chain (6) in photonic waveguide lattices fabricated using the femtosecond laser writing technique in bulk glass szameit2010discrete , see methods. The evolution of light propagating along the $z$-direction of an array of single-mode waveguides can be well-described in the paraxial approximation through a set of coupled mode equations $i\partial_{z}\psi=H\psi$. The wave function $\psi$ represents the excited optical wavepacket as a superposition of bound modes of the waveguides. The matrix $H$ has diagonal elements corresponding to the refractive indices of the waveguides and off-diagonal coupling elements being proportional to the overlap between the bound modes of neighboring waveguides. Thus, discrete Schrödinger equations can be simulated in waveguide arrays with the benefit that the time coordinate in the quantum regime is mapped onto a spatial propagation distance in the optical system. In other words, the propagation of an optical wavepacket through a waveguide system simulates the temporal dynamics of an electron in a potential landscape. Notably, using fluorescence microscopy we can directly image the light propagation along the device szameit2007quasi .

In order to generate an effective AB-phase threading each cage, we use Peierls’ substitution and associate an effective phase to one of the hopping amplitudes, see Fig. 2. Engineering a hopping phase for photons is challenging due to the positive refractive index of each waveguide, and consequently the coupling between the waveguides is always real and positive. Nevertheless, by positioning an auxiliary waveguide with a well-tuned refractive index in between two waveguides keil2016universal , an effective negative coupling between the two original waveguides is generated, see Fig. 2A and the method section. Crucially, the auxiliary waveguide is engineered such that it does not contribute significantly to the dynamics of the system. Therefore, by placing one (or three) negative couplings in each plaquette of our waveguide structure [see Fig. 2B], an overall flux of $\pi$ within the plaquettes is created, resulting in the desired AB-caging effect, see Fig. 2C for an illustration of the device.

We start by establishing our ability to generate the AB-caging effect in the bulk of the chain, i.e., we probe the light dynamics in the bulk of the lattice in order to test the flatness of the bands, see Fig. 3. Exciting a single waveguide within a plaquette will excite all $k$-states of Bloch bands that overlap with this site. At the same time, if the bands are flat the light will stay bound to the injection point and will not disperse. We perform two experiments corresponding to two different injection sites within the unit cell, see Figs. 3A-C. Indeed, in both experiments, despite of some residual spreading, the propagating wavepacket remains confined to the injected unit cell. The experiments agree well with tight-binding simulations of the AB-chain (6).

Interestingly, from the experimental results shown in Fig. 3C, we can additionally detect the energy of the bands: launching light into a waveguide that connects two plaquettes (Fig. 3C) solely excites the two modes in the bands at $E=\pm 2t$, as the mode from the band at $E=0$ has no weight in this site, see Fig. 3D. The resulting beating pattern is, therefore, generated by two modes with a beating length $l_{b}$ that is connected to the energy difference $\Delta E$ of the participating modes by yariv1989quantum $l_{b}=\frac{\pi}{\Delta E}$. From the beating in Fig. 3C, we measure $l_{b}=0.9$cm, which corresponds to $\Delta E=\pm 3.4$ cm^-1. Taking into account the spectral symmetry of the model, the energy of the two bands are therefore measured to be at $E=\pm 1.7$ cm^-1 while the third band lies at $E=0$.

We now turn to demonstrate the existence of the topological boundary states in our square-root model. The amplitude distribution of the predicted boundary modes is shown in Figs. 4A and B. The two boundary states differ by a phase flip and appear at two different eigenenergies, cf. Fig. 1C. Hence, similarly to the bulk experiments above, light injected into the outermost waveguide simultaneously excites both topological boundary modes and the resulting light pattern exhibits a beating with a frequency corresponding to the difference between their eigenenergies, see Fig. 4C. Our experimental data agree well with tight-binding simulations shown in Fig. 4D. From the beating structure, we can determine the energy of the boundary modes $E_{e}$: we observe a beating with $l_{b}=1.3$cm and, hence, deduce that $E_{e}=\pm 1.2$ cm^-1. Comparing the observed energies in the bulk and in the boundary, we find that $\frac{E^{2}}{E_{e}^{2}}\sim 2$, in agreement with the model predictions (6). In other words, the observed boundary modes are indeed the topological boundary states that originate from the squared model (12).

We have predicted and demonstrated the physics of a non-quantized square-root topological insulator, using a photonic platform. Specifically, we show that such systems can exhibit bands with non-quantized topological invariants, but with robust in-gap boundary phenomena. Quantized bulk topology is revealed by mathematically squaring the system’s model, where a standard non-Abelian topological characterization is available. We establish this fascinating relationship in the bulk of the system and derive the respective bulk-edge correspondence. Using a chain of AB-cages, we have presented a minimal example of the non-quantized square-root topological insulator phase. We, therefore, expect that our work will stimulate a range of new theoretical and experimental studies exploring the implications and breadth of such topological phases. In this vein, our experimental results prompt various important questions: Can the square-root topological insulator phase be realized in ultracold atomic setups, where topological quantities can be observed via bulk wavepacket dynamics, rather than by detection of boundary states? What happens to this phase in the presence of interactions, whether in optical, atomic, or condensed matter systems? Is there an analogous square-root topological insulator in the quantum many-body regime and, if yes, will such a model exhibit exotic excitations on the boundary, e.g., parafermions fendley2012parafermionic . Are there other non-linear maps between Hamiltonians that admit such a description arkinstall2017topological ? The answers to these questions are now in experimental reach.

During the submission of our manuscript, we came across another recent realization of photonic Aharonov-Bohm cages Muk2018 .

Acknowledgments I.P. and O.Z. acknowledge financial support from the Swiss National Foundation. A.S. thanks the Deutsche Research Foundation (grants SZ 276/7-1, SZ 276/9-1, BL 574/13-1, SZ 276/19-1).