Topological lattices lit at the corners


Higher-order topological states that are robust against certain classes of disorder and pinned to lattice corners are now observed in photonics platforms.

The discovery that some material systems acquire their properties from certain global topological features has opened a whole new world in condensed-matter physics. A central concept in topological band theory is the bulk–edge correspondence, according to which the interface between two topologically distinct lattices exhibits robust defect states1. Perhaps the simplest model that displays this character is the celebrated one-dimensional (1D) Su–Schrieffer–Heeger (SSH) Hamiltonian that can support localized, immobile 0D edge states. Another important class of topological arrangements is topological insulators — 2D networks that exhibit a bandgap in the bulk but support 1D propagating edge states.

While these concepts were originally developed in the context of condensed-matter physics, they were later mapped to other physical platforms such as photonics, mechanics, microwave and acoustics. In particular, photonics is a unique platform that offers flexibility in design and implementation as well as a large degree of control over linear and nonlinear interactions and processes. In this regard, photonic SSH lattices and topological insulators have been widely studied, and their potential utility in laser systems, nonlinear optics and quantum photonics has been demonstrated2,3.

Recently, the notion of higher-order topological structures was theoretically introduced4 and experimentally demonstrated using electronic5, microwave6 and mechanical systems7. The key feature of these arrangements is their ability to support localized 0D topological states pinned at the corners of 2D lattices (higher-dimensional structures can also be considered). As described in refs. 4,5,6,7, this behaviour can be understood by invoking the connection between topological band theory and electric polarization of periodic crystals that exhibit electric quadruple moments.

Now, writing in Nature Photonics, two independent teams — El Hassan and colleagues8 and Mittal and colleagues9 — present experimental realizations of photonic topological corner states in two different 2D photonic structures.

The origin of the localized states in these works can be understood by a simple model based on the SSH Hamiltonian. As shown in Fig. 1a, the 1D SSH lattice consists of identical sites with alternating weak and strong hopping coefficients. As a result, the unit cell is made of two sites. By expressing the Hamiltonian in a basis where the first and second elements of the unit cells are grouped separately, the Hamiltonian takes the form (after shifting the on-site energies to zero):

$$H = \left( \begin{array}{cc} 0 & J \\ J^{\ast} & 0 \end{array} \right)$$

where J is the hopping matrix. In other words, there is no coupling between similar elements in different unit cells, a feature termed sublattice symmetry. This, in turn, gives rise to chiral symmetry described by the relation \(\chi H\chi ^{ - 1} = - H\), with

$$\chi = \left(\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array} \right)$$

which holds irrespective of the details of J. The spectrum of H is thus symmetric around the zero-energy level. For a finite SSH array consisting of an odd number of sites, the symmetry of the spectrum can be satisfied only if the eigenvalue of one eigenstate is always pinned to zero, regardless of the hopping matrix. It is this particular property that provides the robustness against off-diagonal disorder (for example, deviations from the ideal coupling strengths between the sites). Additionally, invoking topological band theory together with the notion of bulk–edge correspondence1 asserts that this zero state is localized at the side terminated by the weak bond. Figure 1b depicts a 2D version of the SSH lattice. In this case, the unit cell is composed of four sites, and similar to the 1D case the identical sites in each cell can be grouped together to express the Hamiltonian in the form

$$H = \left( \begin{array}{cccc}0 & J & 0 & {J}\\{J^{\ast}} & 0 & J & 0 \\0 & {J^{\ast}} & 0 & J\\{J^{\ast}} & 0 & {J^{\ast}} & {0}\end{array} \right)$$

Again, this Hamiltonian displays a sublattice symmetry \(\chi H\chi ^{ - 1} = - H\), with

$$\chi = \left( \begin{array}{cccc}1 & 0 & 0 & {0}\\0 & -1 & 0 & 0 \\0 & 0 & 1 & 0\\0 & 0 & 0 & {-1}\end{array} \right)$$

which explains the similarity between its spectral features and those of the 1D structure. By inspecting Fig. 1 it is not difficult to see that the edge state in Fig. 1a will map into a corner state in Fig. 1b. As such, the emergence of corner states is rooted in the judicious coupling of SSH chains and is not due to some local effects originating from disorder. These corner states are two dimensions lower than the bulk modes and one dimension lower than the edge modes, suggesting that they are the boundary modes of higher-order topological systems and are topologically protected at the boundary of the boundaries.

Fig. 1: Topologically protected edge and corner states.

An SSH array consists of identical sites with alternating weak (dashed arrows) and strong (solid arrows) coupling. Properly setting coupling coefficients allows topologically trivial or non-trivial phases to be created, including localized edge and corner states. a, An infinite 1D SSH lattice supports a localized edge state when the lattice is terminated at the weak coupling link. b, Each unit cell in a 2D SSH lattice consists of four weakly coupled sites where each site is strongly coupled to the sites of a neighbouring cell. Engineering the intercell and intracell couplings allows a localized corner state to be created when the lattice is terminated at the weak links in both directions. The haloes in a and b depict (not to scale) the light intensity of the localized states at the corresponding site. The works of El Hassan and colleagues and Mittal and colleagues are variants of this simple construction.

The experiment by El Hassan and colleagues employs a variant of the above model based on 2D kagome lattices with modulated coupling coefficients implemented in optical waveguide arrays inscribed in silica glass by laser-writing techniques. The authors studied two different terminations of the lattice (corresponding to rhombic and triangular geometries) that support distinct corner states. Light is coupled directly to the lattice corner and its discrete diffraction pattern is collected at the output. The authors indeed confirm that for the correct set of parameters (that is, the ratio between the two hopping parameters or coupling strengths shown in Fig. 1), light remains localized at the corners of the lattice without significant coupling to the bulk. But when the condition for the topological phase is violated, the input light diffracts and spreads all over the lattice structure.

On the other hand, Mittal and colleagues start with a photonic topological insulator realized in an integrated silicon photonics platform consisting of microring resonator arrays (‘site resonators’) coupled through ‘link resonators’. Tuning these link resonators to provide positive or negative coupling between site resonators creates the required synthetic gauge flux (no external magnetic field is required) to realize topological edge states. Furthermore, by varying the coupling coefficients in a fashion similar to that shown in Fig. 1b, the authors create an effective quadrupole moment and observe that the otherwise travelling edge state collapses to a pinned corner mode when the lattice is terminated at the weak bond. When the arrangement is altered to terminate the array at the strong bond, the optical wave penetrates into the bulk elements. The authors experimentally confirm these predictions by spectroscopic measurements and direct imaging.

Both studies have demonstrated through rigorous experiments that the corner states in their respective structures are boundary modes of the second-order topological phase and that they are robust against certain types of disorder and imperfection. For example, to show that the corner states observed are a result of the topological phase, both teams fabricated samples with trivial phase and revealed that these samples do not support corner states. Furthermore, El Hassan and colleagues deformed a single edge of the triangular kagome lattice into the topologically trivial regime while keeping the rest in the non-trivial regime, showing that light spreads along the trivial edge when injected at the corner of the trivial side but is pinned at the corner when it is injected at the corner of the non-trivial side. Both teams introduced defects and deformations into their respective structures (that is, Mittal and colleagues broke the periodicity of the structure and El Hassan and colleagues removed waveguides or shifted the positions of some waveguides while keeping the rest in their original positions) and observed that corner states survive against some of these defects and deformations but not against some others. Furthermore, Mittal and colleagues compared the corner modes and topological protection in the quadrupole system with a system with zero quadrupole moment, demonstrating that although both the quadrupole and the zero-quadrupole systems support corner states, the corner states in the latter are not completely isolated from the bulk and thus they are not topologically protected (they are prone to disorder and deformations and fabrication errors).

Although it remains to be seen how the reported observations and intriguing results will lead to novel devices and technological advances, they open the door for further future investigations that aim at using topological band theory to achieve better control over light trapping, propagation and generation as well as over the interaction of light with matter. Towards such scientific and technological goals, there are still many issues that need to be addressed. On the experimental front, the topological states studied so far in tight-binding optical systems are robust against off-diagonal disorder (that is, deviations in the coupling strength between the sites) but not to on-diagonal perturbations (that is, deviations in resonant frequencies and propagation constants of each cavity or waveguide forming the structure), which can severely hinder their topological protection. For instance, in arrays made of optical cavities, the coupling between any two cavities is at best two or three orders of magnitudes less than the resonant frequency of the individual cavity. Thus, a minute shift in the resonant frequency can have adverse effects on the topological protection. A similar argument also applies to waveguide arrays. In addition, despite increasing theoretical and experimental interest, the effect of light–matter interactions on topological protection is still not well explored. For instance, in topological laser set-ups, what is the role of nonlinear gain saturation and amplitude–phase modulation on the phase locking of the individual microcavities and how do they affect the global topological protections? Similarly, how will nonlinearities affect the corner states and their robustness against disorder and defects?

A research direction that will particularly benefit from the current studies is the interplay between topology and non-Hermiticity10. In 1D set-ups, it was shown that employing non-Hermitian parameters (gain and loss in optics) as extra degrees of freedom can be used to engineer additional symmetries that exist in condensed matter with no analogy in Hermitian optics. This includes, for example, particle–hole symmetry11. It will be interesting to explore both theoretically and experimentally whether these results can extend naturally to higher-order topological structures or perhaps whether new symmetries will arise or disappear due to the reduced dimensionality of the topological state with respect to the host lattice.

Furthermore, it will be interesting to explore if topological protection can be combined with other strategies borrowed from non-Hermitian physics12,13 to achieve more robustness and/or altogether new optical functionalities. For instance, recent work on state flipping in non-Hermitian systems by adiabatically encircling exceptional points (a peculiar type of degeneracy where the eigenvalues and eigenfunctions associated with different eigenstates coalesce) suggests that this process exhibits robustness, despite the fact that the system does not exhibit topological features. Can one combine these two approaches to achieve spatiotemporal robustness?

The studies by El Hassan and colleagues and Mittal and colleagues are interesting additions to the intriguing journey into the realm of topological physics and photonics. These and previous studies are just scratching the surface, with further counterintuitive phenomena waiting beneath. Given the current status of the field, there is no doubt that future studies will uncover many key intriguing results and, with a bit of luck, also some surprises that are not intuitively anticipated.


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Correspondence to Şahin K. Özdemir or Ramy El-Ganainy.

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Özdemir, Ş.K., El-Ganainy, R. Topological lattices lit at the corners. Nat. Photonics 13, 660–662 (2019). https://doi.org/10.1038/s41566-019-0523-2

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