Abstract
A hallmark of symmetryprotected topological phases are topological boundary states, which are immune to perturbations that respect the protecting symmetry. It is commonly believed that any perturbation that destroys such a topological phase simultaneously destroys the boundary states. However, by introducing and exploring a weaker subsymmetry requirement on perturbations, we find that the nature of boundary state protection is in fact more complex. Here we demonstrate that the boundary states are protected by only the subsymmetry, using Su–Schrieffer–Heeger and breathing kagome lattice models, even though the overall topological invariant and the associated topological phase can be destroyed by subsymmetrypreserving perturbations. By precisely controlling symmetry breaking in photonic lattices, we experimentally demonstrate such subsymmetry protection of topological states. Furthermore, we introduce a longrange hopping symmetry in breathing kagome lattices, which resolves a debate on the higherorder topological nature of their corner states. Our results apply beyond photonics and could be used to explore the properties of symmetryprotected topological phases in the absence of full symmetry in different physical contexts.
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Main
Symmetryprotected topological (SPT) phases of matter are ubiquitous in nature and exist on versatile platforms including condensedmatter physics, ultracold atomic gases and photonics^{1,2}. Topological insulators (TIs) induced by spin–orbit coupling, which are protected by timereversal symmetry, are a paradigm for SPT phases of matter^{1,2,3}. In topological crystalline insulators, a crystalline point group symmetry protects topological metallic boundary states^{1,4}.
Imagine an SPT phase with a topological invariant characterizing the bulk states and the associated symmetryprotected boundary states. Any perturbation that respects the protecting symmetries will not destroy these boundary states or change the topological invariant without closing the gap between bands^{1,2}. However, as pictured in Fig. 1, the situation can be more complex: there are perturbations that preserve the topological invariant but oppose the existence of boundary states^{5,6} and, vice versa, there are perturbations that leave the boundary states unhurt while destroying the topological invariant^{7}. Here we explore the underlying physics for the latter scenario by using the concept of subsymmetry (SubSy), where the symmetry equation, involving the Hamiltonian and a symmetry operator, does not hold in the whole Hilbert space, but only in its subspace. For the prototypical onedimensional (1D) Su–Schrieffer–Heeger (SSH) and twodimensional (2D) breathing kagome lattice (BKL) models, the SubSys arise from their chiral symmetries, which restrict the possibilities of coupling between sublattices (rigorously defined below). Here the SubSy means that the symmetry equation holds on only one sublattice.
We demonstrate, theoretically and experimentally, that the boundary states in the SSH and BKL models are protected by their pertinent SubSy. Any SubSypreserving perturbation will leave its corresponding boundary state eigenvalue at zero, even though the bulk topological invariant is lost. We utilize zigzag and twisted lattices, and ‘bridge’ waveguides to experimentally introduce SubSybreaking and SubSypreserving perturbations in a controlled manner and thereby demonstrate SubSyprotected topological states. In the case of nonnegligible longrange hopping (that is, nonnegligible coupling between distant lattice sites) in BKLs, we find that SubSy and an additional longrange hopping symmetry are sufficient to protect the corner states. Our experiments are performed in photonic structures, which have been established as a fertile platform for exploring novel topological phenomena^{8,9,10}. The main message from our findings is summarized in Fig. 1.
The SSH lattice illustrated in Fig. 2a represents a typical 1D topological model, originally used to describe polyacetylene^{11}. It has subsequently been experimentally realized on versatile platforms including photonics and nanophotonics^{12,13,14,15}, plasmonics^{16} and quantum optics^{17} and in the context of parity–time symmetry and nonlinear nonHermitian phenomena^{18,19}.
The SSH lattice is composed of A and B sublattices (Fig. 2a), with the Hamiltonian \(H_{\mathrm{SSH}} ={\varSigma_n} {(t_1 b_n^{\dagger} a_n + t_2a_{n + 1}^{\dagger} b_n + {\mathrm{h.c.}})}\), where a_{n} is the annihilation operator at an A sublattice site in the nth unit cell, with an analogous definition for b_{n}, while t_{1} and t_{2} are the intracell and intercell coupling strengths, respectively. Its topological phase is protected by the chiral symmetry
where \({{\varSigma }}_z = P_{\mathrm{A}}  P_{\mathrm{B}}\) and P_{A} (P_{B}) denotes the projection operator on the A (B) sublattice (Methods). The system has a trivial phase for t_{1} > t_{2} and a topologically nontrivial phase for t_{1} < t_{2}, with the latter being characterized by the Zak phase of π and two topologically protected edge states at zero energy (Fig. 2b). The amplitudes of the left edge state \(\left {{A_{\mathrm{L}}}} \right\rangle\) are nonzero solely on the A sublattice, that is, \({{P_{\mathrm{A}}}}\left {A_{\mathrm{L}}} \right\rangle = \left {{A_{\mathrm{L}}}} \right\rangle\), and \(P_{\mathrm{B}}\left {{A_{\mathrm{L}}}} \right\rangle = 0\), and analogously for the right edge state \(\left {{B_{\mathrm{R}}}} \right\rangle\) (Fig. 2c).
The concept of SubSy focuses on perturbations that break the chiral symmetry but preserve a less strict SubSy requirement. This provides a theoretical framework and generalizes the partial chiral symmetrybreaking case proposed previously in ref. ^{7}. There are two SubSys in the SSH model, the ASubSy and the BSubSy, which are defined by
The most general perturbation in the couplings is of the form H_{AB} + H_{AA} + H_{BB}, which implies that the hopping parameter between any two lattice sites (irrespective of distance) can be changed without any restrictions. Here \(H_{\mathrm{AB}} = \mathop {\varSigma}\nolimits_{m,n} {\left( {s_{ab}^{m,n}a_m^{\dagger} b_n + {\mathrm{h.c.}}} \right)}\) denotes couplings between the A and B sublattice sites (A–B coupling), where \(s_{ab}^{m,n}\) are the individual coupling strengths, with an analogous definition for the A–A (H_{AA}) and B–B (H_{BB}) couplings (see Methods for details).
Without loss of generality, we consider ASubSypreserving perturbations, which are of the form H′ = H_{AB} + H_{BB}; they are more restrictive than general perturbations but less restrictive than chiral symmetrypreserving perturbations (H_{AB}). Perturbations involving the A–A coupling (H_{AA}) break the ASubSy. We emphasize that ASubSypreserving perturbations can be periodic (that is, respecting the lattice symmetry) or local (for example, perturbing only one coupling between two lattice sites) or even feature disorder.
One of our key results is that any such perturbation, if it respects the ASubSy, will not destroy the left edge zeroenergy state \(\left {{A_{\mathrm{L}}}} \right\rangle\) (and fully analogously for the BSubSy), as illustrated in Fig. 2a–c. The theoretical argument for this statement is made possible by the formulation of SubSy via projection operators in equation (2): because H_{AB} preserves the chiral symmetry, the edge states are protected under such perturbations until the gap closes. B–B perturbations do not affect \(\left {{A_{\mathrm{L}}}} \right\rangle\) because H_{BB}P_{A} = 0, which leads to \(H_{\mathrm{BB}}\left {{A_{\mathrm{L}}}} \right\rangle = H_{\mathrm{BB}}P_{\mathrm{A}}\left {{A_{\mathrm{L}}}} \right\rangle = 0\). Thus, \(\left {{A_{\mathrm{L}}}} \right\rangle\) is protected under ASubSypreserving perturbations H′ = H_{AB} + H_{BB}. However, the right edge zeroenergy state \(\left {{B_{\mathrm{R}}}} \right\rangle\) is not protected because the H_{BB} component affects this state. Moreover, H_{BB} perturbations generally break both the chiral symmetry and the Zak phase quantization.
SubSy protection of edge states is illustrated in Fig. 2b,c, which show the spectra and the eigenmode structure for the case of a single randomly chosen ASubSypreserving perturbation. The energy of the perturbed left edge mode \(\left {{A}_{\mathrm{L}}^\prime } \right\rangle\) is intact, but that of the right edge mode as well as the whole spectrum is altered by ASubSypreserving perturbations (Fig. 2b). The perturbed mode \(\left {{A}_{\mathrm{L}}^\prime } \right\rangle\) resides solely on the A sublattice, that is, \(\left {\left\langle {{{A}}_{\mathrm{L}}^\prime P_{\mathrm{A}}{{A}}_{\mathrm{L}}^\prime } \right\rangle } \right^2 = 1\); however, its structure can differ from the unperturbed mode (Fig. 2c). Detailed numerical analysis confirms that SubSy requirement is essential for protecting the edge states (Supplementary Information).
To experimentally test such edgestate protection with respect to the SubSypreserving perturbations, we break the chiral symmetry in a controlled fashion. To this end, we introduce the appropriate A–A or B–B hopping by twisting the SSH lattice into the angled structure illustrated in Fig. 2d,e (left), which either breaks the ASubSy (Fig. 2e) or preserves it (Fig. 2d). The probing is performed by launching a focused beam into the leftmost waveguide on the A sublattice. In Fig. 2d (middle), the output intensity resides dominantly on the A sublattice, indicating that the left edge mode is topologically protected when the ASubSy is preserved. For a direct comparison, in Fig. 2e (middle), we show the intensity of the same excitation beam propagating through the ASubSybreaking lattice. The presence of light in the second waveguide, that is, on the B sublattice, indicates that it is no longer a topologically protected edge mode^{12,19}. Numerical simulations (Fig. 2d,e, right) agree with experimental results.
Perturbations in the twisted SSH lattices (Fig. 2d,e) are localized. To experimentally probe the robustness of edge states under periodic ASubSypreserving (Fig. 2f) or ASubSybreaking (Fig. 2g) perturbations, we fabricated two zigzag photonic SSH lattices. The zigzag lattices plotted in Fig. 2f,g (left) are oriented such that the bottom site belongs to the A sublattice. By exciting the bottom edge waveguide in the ASubSypreserving lattice, we observe protection of the edge mode as light populates solely the A sublattice, without coupling to the bulk (Fig. 2f, middle). An identical excitation in the ASubSybreaking lattice (Fig. 2g, middle) clearly indicates that the edge mode is no longer topologically protected as light leaks into the B sublattice. Numerical simulations for much longer propagation distances (Fig. 2f,g, right) corroborate our experimental results. We emphasize that the zigzag lattice in Fig. 2f (left) breaks both the inversion and chiral symmetries, yet the edge mode \(\left {{A_{\mathrm{L}}}} \right\rangle\) is protected by the ASubSy.
The kagome lattice is an inexhaustible golden vein of intriguing physics, attracting the broad interest of the scientific community. BKLs, illustrated in Fig. 3a, have been classified as higherorder topological insulators (HOTIs), where topologically protected corner states were observed^{20,21,22,23,24,25,26}. HOTIs are a new class of topological materials^{27}, found in condensedmatter, networks of resonators, photonic and acoustic systems^{20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39}. The corner states in the BKLs were initially considered as HOTI states protected by the generalized chiral symmetry and the C_{3} crystalline symmetry^{20}. However, it was later debated that they are not HOTI states^{40,41,42} because they are not protected by some specific longrange hopping perturbations obeying these symmetries^{40}. In our discussion of SubSyprotected corner BKL states, we clarify this debated issue.
BKLs are composed of three sublattices (A, B and C), featuring intracell and intercell hopping amplitudes t_{1} and t_{2}, respectively (Fig. 3a and Methods). The bulk polarizations are the topological invariants that characterize the topological phase: for t_{1} < t_{2}, the system is in the nontrivial phase with \(P_x = P_y = \frac{1}{3}\), whereas for t_{1} > t_{2}, the polarizations are zero^{20,21}. The BKL Hamiltonian H_{K} possesses C_{3} symmetry and the generalized chiral symmetry \({{\varSigma }}_3H_{\mathrm{K}}{{\varSigma }}_3^{  1} + {{\varSigma }}_3^2H_{\mathrm{K}}{{\varSigma }}_3^{  2} =  H_{\mathrm{K}}\) (ref. ^{20}). Here \({{\varSigma }}_3 = P_{\mathrm{A}} + {\mathrm{e}}^{{\mathrm{i}} \frac{{2\uppi }}{3}}P_{\mathrm{B}} + {\mathrm{e}}^{  {\mathrm{i}}\frac{{2\uppi }}{3}}P_{\mathrm{C}}\) is the symmetry operator, where \(P_i,i \in \left\{ {\mathrm{A,B,C}} \right\}\) are the projection operators. The generalized chiral symmetry yields three equations
defining three SubSys corresponding to the three sublattices.
Our theoretical results on BKLs are presented in Fig. 3. We consider a rhombic flake illustrated in Fig. 3a, which has one zeroenergy corner state \(H_{\mathrm{K}}\left {{A}_{\mathrm{cor}}} \right\rangle = 0\) residing on the A sublattice, \(P_{\mathrm{A}}\left {{{A}}_{\mathrm{cor}}} \right\rangle = \left {{A}_{\mathrm{cor}}} \right\rangle\). The bandgap structure of one such flake is shown in Fig. 3d. First, we consider perturbations between B–B, C–C and B–C sites, H′ = H_{BB} + H_{CC} + H_{BC}, which obey the ASubSy, yet breaking the generalized chiral symmetry. These perturbations obey H′P_{A} = 0, which implies \(H^\prime \left {{A}_{\mathrm{cor}}} \right\rangle = H^\prime P_{\mathrm{A}}\left {{A}_{\mathrm{cor}} } \right\rangle = 0\), that is, any such perturbation does not affect the corner state. This is illustrated in Fig. 3c that shows the bandgap structure, with the corner state indicated by red crosses, for a set of randomly chosen perturbations H′ of various magnitudes quantified by δ′. These perturbations are randomly chosen from a set that respect both ASubSy and the lattice symmetries (Methods). Interestingly, at some higher perturbation strengths, \(\left {A_{\mathrm{cor}}} \right\rangle\) can become a bound state in the continuum^{43}.
Next, we consider the ASubSypreserving perturbations between A–B and A–C sites: H″ = H_{AC} + H_{AB}. Such perturbations can affect the zeroenergy corner state \(\left {{A}_{\mathrm{cor}}} \right\rangle\), even in a setup preserving both the full generalized chiral symmetry and the C_{3} symmetry^{40}. Longrange hopping parameters in H″ between site A in the unit cell (m, n) and site B in the unit cell (m_{0}, n_{0}) are denoted with \(t_{ab}^{m,n;m_0,n_0}\) and equivalently for A–C coupling (Methods). We analytically find that the zeroenergy state, residing solely on the A sublattice, can exist only if the following two conditions hold: \(t_{ac}^{m,n;m_0,n_0} = t_{ab}^{m,n;m_0  1,n_0 + 1}\) and \(t_{ac}^{m,n;m_0,n_0} = t_{ab}^{m,n;m_0,n_0}\) (Supplementary Section 6). These conditions are trivially satisfied if the longrange hopping is zero, that is, when the tightbinding approximation holds. Otherwise, they are too strict and unphysical as illustrated in Fig. 3b, as all coupling strengths between lattice sites indicated with solid lines must be equal.
However, as the coupling strength is typically correlated with the distance, an inspection of the A–B and A–C links in Fig. 3b and our theoretical analysis (Supplementary Section 6) suggest that, when t_{1} < t_{2}, we should consider an approximate but more physical and less restrictive longrange hopping symmetry (LRHS):
Equation (4) implies that only those couplings indicated by the same colour in Fig. 3b must be equal. To test the protection of the corner state under the ASubSy and LRHS, we calculate the spectra for a set of randomly chosen H′ + H″ perturbations of different magnitudes quantified by δ′ and δ″, respectively; these perturbations also retain the lattice symmetry by construction (Fig. 3e and Methods). We see that the zeroenergy corner state remains in the gap and protected, until it is too close to the band at strong perturbations (this is a finitesize effect). The perturbed corner state is dominantly on the A sublattice as long as it is in the gap (Supplementary Section 7).
We experimentally test the protection of the corner state under the SubSy by implementing targeted nextnearestneighbour hopping, introduced by imprinting bridge waveguides in the rhombic lattice (Methods). As shown in Fig. 4a (left), the lattice with B–B and C–C bridges preserves the ASubSy, while the lattice in Fig. 4b (left) with A–A bridges breaks the ASubSy. For the lattice with a broken ASubSy, after excitation of the corner site on the A sublattice, there is light in the B and C waveguides nearest to the corner site (Fig. 4b, middle). This offers clear evidence that the corner mode is not protected anymore. On the contrary, for the lattice with a preserving ASubSy, light is present solely on the A sublattice (Fig. 4a, middle), exhibiting the characteristics of HOTI corner states in BKLs^{20,21,39}. This proves that the corner state, in this case, is protected against the B–B and C–C bridge perturbations. To underpin the experimental results, in Fig. 4a,b (right), we show results from numerical simulations obtained in realistic BKLs with parameters corresponding to those from the experiment, which display an excellent agreement. Longdistance simulations also validify that light remains localized at the corner without traversing through the bridges in Fig. 4a (left) due to topological protection but travels through the two bridges (even now they are further away) and spread into the bulk in Fig. 4b (left) (Supplementary Section 8).
We are now ready to discuss our results with the focus on the diagram in Fig. 1. In the 1D SSH lattice, the left edge mode is protected by ASubSy (encircled with a red line), while BSubSypreserving perturbations (encircled with a green line) do not affect the right edge mode. At the overlap region, one has the full chiral symmetry and the SPT phase. However, it has been shown that perturbations that respect the inversion symmetry (encircled with a grey line) protect the topological invariant, that is, the Zak phase, even if the full chiral symmetry is broken^{5,6}.
Recently, it was argued that the SSH model is a poor TI^{44,45}; more specifically, it is a band (Dirac) insulator featuring zero modes at a domain wall between two dimerizations arising from the Jackiw–Rebbi mechanism. Indeed, at low energy, in the longwavelength limit, the tightbinding SSH model can be described by an effective 1D Dirac equation, where the Jackiw–Rebbi mechanism gives rise to a topological defect mode at zero energy^{44,45}. Although we agree with this interpretation, the standardly used arguments for interpreting the SSH model as an SPT phase are holding (see, for example, refs. ^{2,46} and references therein and Supplementary Section 9). We emphasize that the intent of this paper is to accurately classify perturbations that destroy or protect the boundary states, where we use the SSH model as one of the examples to illustrate the suitability of the SubSy concept towards this goal.
The scenario in which BKLs are involved is more complex. First, we consider BKLs where longrange hopping is negligible, which is physically common when hopping is generated with evanescent coupling. The corner state on the A sublattice is robust with respect to ASubSypreserving perturbations and analogously for the corner states on other sublattices (their existence depends on the shape of the BKL flake). The topological invariant is quantized due to the C_{3} symmetry^{38}. Thus, for a triangular flake of the BKL with C_{3} and generalized chiral symmetry, one can classify perturbations with respect to symmetries in accordance with Fig. 1 with an additional CSubSy (not shown) and the grey encircled region corresponding to C_{3} symmetry. In this model, the BKL corner states are HOTI states.
When the longrange hopping beyond the neighbouring unit cells becomes appreciable, the protection of the corner state under ASubSy and the LRHS can be interpreted as being inherited from the underlying Hamiltonian H_{K} . This interpretation is underpinned by the calculation of the fractional corner anomaly (FCA)^{25} shown in Fig. 3f for an ensemble of randomly chosen ASubSy and LRHSpreserving perturbations. It is a clear signature of nontrivial topology and the existence of the corner state.
In conclusion, we have demonstrated SubSyprotected boundary states of SPT phases by employing perturbations that break the original topological invariants. Although the SubSy concept here arises from the chiral symmetries, we envision its applicability for other protecting symmetries as well. For the BKLs with nonnegligible longrange hopping, we have unveiled a previously undiscovered LRHS that is essential for protection of the corner states, providing a basis for understanding their HOTI characteristics. We have used the 1D SSH and the BKL models to demonstrate our main findings. However, with appropriately defined SubSys, our findings can be applied to other systems, such as the 2D SSH lattice. More generally, our results extend beyond photonics to condensedmatter and cold atom systems, where many intriguing phenomena are mediated by the interplay of symmetry and topology. For example, a periodic zigzag SSHlike photonic lattice with A–A and B–B nextnearestneighbour coupling (such as those in Fig. 2f,g (left)) can be engineered with Rydberg atoms^{47}. Even though perturbations respecting or breaking SubSy are artificially engineered in our work, we nevertheless expect that such perturbations could naturally appear in a number of existing materials, including polymers or other organic and inorganic structures.
Methods
Projection operators
The projection operator P_{A} is constructed by requiring that the amplitude of \(P_{\mathrm{A}}\left \psi \right\rangle\) is identical to the amplitude of a given state \(\left \psi \right\rangle\) on any A sublattice site and zero on any other sublattices. The other projection operators (P_{B}, P_{C}) are constructed fully analogously.
SSH lattice
The chiral symmetry of the SSH lattice in equation (1) implies that for every eigenstate \(\left e \right\rangle\) satisfying \(H_{\mathrm{SSH}}\left e \right\rangle = \beta \left e \right\rangle\), there is another eigenstate \({{\varSigma }}_z\left e \right\rangle\) with eigenvalue −β. This ensures that any perturbation of the Hamiltonian that preserves the chiral symmetry does not destroy the topologically protected edge states unless the gap closes and the system undergoes a topological phase transition to a trivial phase (for example, see ref. ^{2} and references therein).
Perturbations of the SSH model corresponding to A–B coupling are formally defined as \(H_{\mathrm{AB}} = \mathop {\varSigma}\nolimits_{m,n} {\left( {s_{ab}^{m,n}a_m^{\dagger} b_n + {\mathrm{h.c.}}} \right)}\), where a_{m} is the annihilation operator at an A sublattice site in the mth unit cell, and analogously for b_{n}, while \(s_{ab}^{m,n}\) is the strength of the coupling. Similarly, \(H_{\mathrm{BB}} = \mathop {\varSigma}\nolimits_{m,n} {\left( {s_{bb}^{m,n}b_m^{\dagger} b_n + {\mathrm{h.c.}}} \right)}\), where m ≠ n, and analogously for H_{AA}.
Breathing kagome lattice
The BKL Hamiltonian is given by
where \(a_{m,n}\) is the annihilation operator at an A sublattice site in the unit cell labelled with (m, n) indices and analogously for \(b_{m,n}\) and \(c_{m,n}\). All perturbations between sublattices A and B beyond the hopping corresponding to t_{1} and t_{2} can be described by
and analogously for H_{AC} and H_{BC}. The B–B hopping perturbations are described by
and analogously for H_{AA} and H_{CC}.
To construct perturbations, the hopping amplitudes between the unit cells \((m,n) \to (m \pm i,n \pm j)\), \(i,j =  3, \ldots ,3\), are perturbed with strength \({\mathrm{rand}}\delta ^\prime t_2  t_1\) for H′ = H_{BC} + H_{BB} + H_{CC} and \({\mathrm{rand}} \delta ^{\prime\prime} t_2  t_1\) for H″ = H_{AC} + H_{AB}. Here rand is a random number between 0 and 1 chosen with respect to uniform probability distribution. The nearest couplings t_{1} and t_{2} are not perturbed. The strength of the perturbations is given relative to the size of the gap, which is given by \(t_2  t_1\). All perturbations retain the lattice symmetry. In Fig. 3c, for each magnitude of the perturbation δ′, we calculate and plot an ensemble of 70 spectra for randomly chosen H′. An equivalent procedure is used for Fig. 3e, where parameter δ″ is now varied, and δ′ = 0.05 is kept fixed. Every H″ respects the lattice symmetry, the ASubSy, and the LRHS. The projections \(\left\left\langle {{A}_{\mathrm{cor}}^\prime {{{\mathrm{}}}}{{A}}_{\mathrm{cor}}} \right \rangle\right^2\) and \(\left\left \langle {{A}_{\mathrm{cor}}^\prime {{{\mathrm{}}}}P_{\mathrm{A}}{{A}}_{\mathrm{cor}}^\prime } \right \rangle \right^2\) between the unperturbed \(\left {{A}_{\mathrm{cor}}} \right\rangle\) and the perturbed \(\left {{A}_{\mathrm{cor}}^\prime } \right\rangle\) corner states are exactly unity for any H′ = H_{BB} + H_{CC} + H_{BC}. However, for H′ + H″, \(\left\left\langle {A_{\mathrm{cor}}^\prime {{{\mathrm{}}}}A_{\mathrm{cor}}} \right \rangle \right^2 < 1\) (Supplementary Section 7).
The FCA is calculated as (ρ − 2 σ) mod 1 following ref. ^{25}; the red and black circles in Fig. 3f represent the mode density of the corner unit cell ρ and the average mode density of edge unit cells σ (with edges that intersect at the corner), respectively. The mode density is calculated as the local density of states integrated over all states above the bandgap in the propagation constant spectrum (shown in Fig. 3e).
Experimental setup and methods
In our experiments, we establish the desired photonic lattices (either the 1D ‘twisted’ and zigzag SSH lattices shown in Fig. 2 or the twodimensional rhombic kagome lattice as shown in Fig. 4) by sitetosite writing of waveguides in a strontium–barium niobate (SBN:61) photorefractive crystal with a continuouswave laser^{19,39}. As illustrated in Extended Data Fig. 1, a lowpower laser beam featuring a 532 nm wavelength illuminates a spatial light modulator, which creates a quasinondiffracting writing beam with variable input positions onto the 20mmlong biased crystal. The latticewriting beam is ordinarily polarized, while the probe beam launched to the lattice edge is extraordinarily polarized. Because of the selffocusing nonlinearity and the photorefractive ‘memory’ effect^{19,39}, all waveguides are induced and remain intact during the subsequent probing processes. Compared with the femtosecond laserwriting method largely employed in glass materials^{9,26}, the photonic lattices in our crystal can be readily reconfigured from topological nontrivial to trivial structures simply by controlling the lattice spacing. After the multistep writing process (with a bias field of 130 kV m^{−1}) is completed, the whole lattice can be examined by sending a set of Gaussian beams into the crystal to probe the waveguides one by one, which leads to superimposed lattice structures shown in Figs. 2 and 4. To investigate the evolution of the topological states in this work, the probe beam used to excite the lattice edge/corner is set at a much weaker power of only about 20 nW, so it undergoes only linear propagation without nonlinear selfaction through the lattice. (We note that the probe power can be increased to locally change the index structure of the lattices—the ingredient used for nonlinear control of topological states as in our previous work^{19,39}.) The intensity patterns of the probe beam exiting the lattices (Figs. 2 and 4) are captured by an imaging lens paired to a chargecoupled device camera.
Data availability
Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
References
Chiu, C.K. et al. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).
Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).
Kane, C. L. & Mele, E. J. Z_{2} topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).
Fu, L. Topological crystalline insulators. Phys. Rev. Lett. 106, 106802 (2011).
Longhi, S. Probing onedimensional topological phases in waveguide lattices with broken chiral symmetry. Opt. Lett. 43, 4639–4642 (2018).
Jiao, Z. Q. et al. Experimentally detecting quantized Zak phases without chiral symmetry in photonic lattices. Phys. Rev. Lett. 127, 147401 (2021).
Poli, C. et al. Partial chiral symmetrybreaking as a route to spectrally isolated topological defect states in twodimensional artificial materials. 2D Mater. 4, 025008 (2017).
Wang, Z. et al. Observation of unidirectional backscatteringimmune topological electromagnetic states. Nature 461, 772–775 (2009).
Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).
Hafezi, M. et al. Imaging topological edge states in silicon photonics. Nat. Photon. 7, 1001–1005 (2013).
Su, W. P., Schrieffer, J. R. & Heeger, A. J. Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698–1701 (1979).
Malkova, N. et al. Observation of optical Shockleylike surface states in photonic superlattices. Opt. Lett. 34, 1633–1635 (2009).
Keil, R. et al. The random mass Dirac model and longrange correlations on an integrated optical platform. Nat. Commun. 4, 1368 (2013).
Xiao, M., Zhang, Z. Q. & Chan, C. T. Surface impedance and bulk band geometric phases in onedimensional systems. Phys. Rev. X 4, 021017 (2014).
Kruk, S. et al. Edge states and topological phase transitions in chains of dielectric nanoparticles. Small 13, 1603190 (2017).
Poddubny, A. et al. Topological Majorana states in zigzag chains of plasmonic nanoparticles. ACS Photon. 1, 101–105 (2014).
BlancoRedondo, A. et al. Topological protection of biphoton states. Science 362, 568–571 (2018).
Weimann, S. et al. Topologically protected bound states in photonic paritytimesymmetric crystals. Nat. Mater. 16, 433–438 (2017).
Xia, S. et al. Nonlinear tuning of PT symmetry and nonHermitian topological states. Science 372, 72–76 (2021).
Ni, X. et al. Observation of higherorder topological acoustic states protected by generalized chiral symmetry. Nat. Mater. 18, 113–120 (2019).
Xue, H. et al. Acoustic higherorder topological insulator on a Kagome lattice. Nat. Mater. 18, 108–112 (2019).
Li, M. et al. Higherorder topological states in photonic Kagome crystals with longrange interactions. Nat. Photon. 14, 89–94 (2019).
El Hassan, A. et al. Corner states of light in photonic waveguides. Nat. Photon. 13, 697–700 (2019).
Kempkes, S. N. et al. Robust zeroenergy modes in an electronic higherorder topological insulator. Nat. Mater. 18, 1292–1297 (2019).
Peterson Christopher, W. et al. A fractional corner anomaly reveals higherorder topology. Science 368, 1114–1118 (2020).
Kirsch, M. S. et al. Nonlinear secondorder photonic topological insulators. Nat. Phys. 17, 995–1000 (2021).
Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017).
Song, Z., Fang, Z. & Fang, C. (d2)dimensional edge states of rotation symmetry protected topological states. Phys. Rev. Lett. 119, 246402 (2017).
Langbehn, J. et al. Reflectionsymmetric secondorder topological insulators and superconductors. Phys. Rev. Lett. 119, 246401 (2017).
Schindler, F. et al. Higherorder topology in bismuth. Nat. Phys. 14, 918–924 (2018).
Noh, J. et al. Topological protection of photonic midgap defect modes. Nat. Photon. 12, 408–415 (2018).
Imhof, S. et al. Topolectricalcircuit realization of topological corner modes. Nat. Phys. 14, 925–929 (2018).
SerraGarcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342–345 (2018).
Mittal, S. et al. Photonic quadrupole topological phases. Nat. Photon. 13, 692–696 (2019).
Zhang, X. et al. Secondorder topology and multidimensional topological transitions in sonic crystals. Nat. Phys. 15, 582–588 (2019).
Xie, B. Y. et al. Visualization of higherorder topological insulating phases in twodimensional dielectric photonic crystals. Phys. Rev. Lett. 122, 233903 (2019).
Chen, X. D. et al. Direct observation of corner states in secondorder topological photonic crystal slabs. Phys. Rev. Lett. 122, 233902 (2019).
Benalcazar, W. A., Li, T. & Hughes, T. L. Quantization of fractional corner charge in C_{n}symmetric higherorder topological crystalline insulators. Phys. Rev. B 99, 245151 (2019).
Hu, Z. et al. Nonlinear control of photonic higherorder topological bound states in the continuum. Light. Sci. Appl. 10, 164 (2021).
Van Miert, G. & Ortix, C. On the topological immunity of corner states in twodimensional crystalline insulators. npj Quantum Mater. 5, 63 (2020).
Jung, M., Yu, Y. & Shvets, G. Exact higherorder bulk–boundary correspondence of cornerlocalized states. Phys. Rev. B 104, 195437 (2021).
Herrera, M. A. J. et al. Corner modes of the breathing kagome lattice: origin and robustness. Phys. Rev. B 105, 085411 (2022).
Hsu, C. W. et al. Bound states in the continuum. Nat. Rev. Mater. 1, 16048 (2016).
Fuchs, J.N. & Piéchon, F. Orbital embedding and topology of onedimensional twoband insulators. Phys. Rev. B 104, 235428 (2021).
Cayssol, J. & Fuchs, J. N. Topological and geometrical aspects of band theory. J. Phys. Mater. 4, 034007 (2021).
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).
de Leseleuc, S. et al. Observation of a symmetryprotected topological phase of interacting bosons with Rydberg atoms. Science 365, 775–780 (2019).
Acknowledgements
We acknowledge assistance from R. Cheng and Y. Wang. This research is supported by the National Key R&D Program of China under grant no. 2022YFA1404800, the National Natural Science Foundation (12134006, 12274242) and the QuantiXLie Center of Excellence, a project cofinanced by the Croatian Government and European Union through the European Regional Development Fund—the Competitiveness and Cohesion Operational Programme (grant KK.01.1.1.01.0004). D.B. acknowledges support from the 66 Postdoctoral Science Grant of China and the National Natural Science Foundation (12250410236). R.M. acknowledges support from NSERC and the CRC programme in Canada.
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Z.H., D.B. and X.W. realized the photonic lattices and performed the experiments. Z.W., X.W., D.J. and H.B. performed theoretical analysis and numerical simulations of the discrete models. D.B. and Z.H. performed numerical simulations of the continuous models. H.B., Z.C. and R.M. supervised the work. All the authors discussed the results and contributed to this work.
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Extended data
Extended Data Fig. 1 Schematic illustration of the experimental setup employed for writing and probing a photonic lattice in a photorefractive crystal.
CW: the continuouswave laser beam; SLM: spatial light modulator; BS: beam splitter; FM: Fourier mask; L: circular lens; SBN: strontium barium niobite crystal; M: mirror; λ/2: halfwavelength plate; CCD: chargecoupled device. The inset shows a laserwritten “bridged” kagome lattice used in the experimental work of Fig. 4. The bottom path is used as a reference beam for interference measurement when needed.
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Supplementary Figs. 1–7 and Discussion.
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Source Data Fig. 2
Numerically calculated data.
Source Data Fig. 3
Numerically calculated data.
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Wang, Z., Wang, X., Hu, Z. et al. Subsymmetryprotected topological states. Nat. Phys. 19, 992–998 (2023). https://doi.org/10.1038/s41567023020119
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DOI: https://doi.org/10.1038/s41567023020119
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