Wavefront dislocations reveal the topology of quasi-1D photonic insulators

Phase singularities appear ubiquitously in wavefields, regardless of the wave equation. Such topological defects can lead to wavefront dislocations, as observed in a humongous number of classical wave experiments. Phase singularities of wave functions are also at the heart of the topological classification of the gapped phases of matter. Despite identical singular features, topological insulators and topological defects in waves remain two distinct fields. Realising 1D microwave insulators, we experimentally observe a wavefront dislocation – a 2D phase singularity – in the local density of states when the systems undergo a topological phase transition. We show theoretically that the change in the number of interference fringes at the transition reveals the topological index that characterises the band topology in the insulator.


Supplementary Note 1. MICROWAVE REALISATION OF TIGHT-BINDING SYSTEMS
The experimental setup is designed to realise a microwave system that is well approximated by a nearest-neighbour tightbinding description [1]. The sites of the lattice are occupied by dielectric microwave resonators with a cylindrical shape and made of ZrSnTiO ceramics (Temex-Ceramics, E8000 series: 5 mm height, 3 mm radius and a refractive index n r ≈ 6) sandwiched between two metallic plates at a distance h = 16 mm. Each resonator supports a fundamental TE mode of bare frequency of 7.435 GHz, which corresponds to the on-site energy of atoms in a tight-binding model. As the resonance frequency is below the cut-off frequency of the first TE mode defined by the two plates the adjacent resonators are coupled through evanescent wave components leading to an approximately exponential decay of the coupling strength t with the distance between the resonators [1]. The system is excited via a loop antenna fixed in the movable top plate thus allowing to scan spatially the magnetic field B z , which is the only magnetic field component for this mode. From the reflection measurements performed by a vector network analyzer (ZVA 24 from Rohde & Schwarz) the local density of states can be extracted (for details see Ref. [1]) and finally, by integrating over space, the density of states. In all these experiments, we face an intrinsic on-site disorder of ∼ 0.15% in the values of the bare frequency.

Supplementary Note 2. EXPERIMENTAL LDOS MAPS
In the manuscript, we focus specifically on the LDOS fluctuations induced by the leftmost edge of the SSH chain (see Manuscript Fig. 1b). Here, we present in Supplementary Fig. 1 the LDOS of the whole SSH chain. It shows that the discussions in the manuscript also hold for the rightmost edge, thus exemplifying space inversion symmetry. For instance, we observe that the shift in the number of interference fringes on sublattice A close to the leftmost edge also occurs on sublattice B close to the rightmost edge (compare e.g. the inversion symmetric fringes in ρ A (m = 6) and ρ B (m = 17) in Supplementary Fig. 1 The winding number w introduced in the manuscript characterises the bulk topology of the insulator in the 1D Brillouin zone. It can be written equivalently as where z = e ik and C is the unit circle centred on the origin of the complex plane. We can evaluated this integral through Cauchy's argument principle where r and p are respectively the numbers of zeros and poles of the function h(z) inside the unit circle. Since h(z) = (t 1 z + t 2 )z −1 (cf. Eq. (1) in the manuscript), z = 0 is the single pole inside C and so p = 1. The number of zeros inside C depends on the values of the coupling strengths t 1 and t 2 . Given the expression of h(z), we find either r = 1 when t 1 > t 2 , or r = 0 when t 1 < t 2 .
Thus, the winding number can take two different values This characterises two distinct topological regimes for the wave functions delocalised through the bulk of the insulator, as introduced in the manuscript.

Midgap boundary modes
The chiral symmetry enforces the midgap modes to be sublattice polarised at zero energy. Assuming nearest-neighbour coupling, the wave functions of such modes satisfy independent recursive equations where A m (resp. B m ) is the wave-function weight in unit cell m on sublattice A (resp. B). These recursive relations respectively have the characteristic equations h(1/z) = 0 and h(z) = 0 .
Thus, the number of evanescent wave-functions on sublattice A (resp. B) is given by the number of zeros of h(1/z) (resp. h(z)) inside the unit circle C. The polynomial h is of first order and has a single zero. So we find that there is 1 − r (resp. r) midgap boundary mode on sublattice A (resp. B). Nevertheless, the open boundary condition at the leftmost edge is B 0 = 0 (see Supplementary Fig. 2), which further prohibits the boundary mode on sublattice B. This demonstrates the bulk-boundary correspondence discussed in the manuscript at the leftmost edge: there is N A = 1 − r ≡ −w (from Supplementary Eq. (3)) boundary mode polarised on sublattice A.

B A
Semi-infinite crystal m=0 Supplementary Fig. 2. Modelisation of the edge: Within a nearest-neighbour tight-binding description, the edge is regarded as a vacancy breaking the translation invariance of the infinite crystal. We then focus on the semi-infinite crystal associated with positive dimer indices (m > 0).

Backscattering of the delocalised waves on the edge
The Bloch Hamiltonian of the SSH model is bipartite and reads in the sublattice basis (A, B). The single-particle Green function of the system with translation invariance satisfies where the integers m and n label the dimers, and I 2 is the 2 × 2 identity matrix. We describe the edge as a vacancy and model it by a large on-site potential V B δ m,0 on sublattice B (see Supplementary Fig. 2). We focus on the standing-wave interference it induces on dimers m > 0. Summing all the scattering orders in the potential V B leads to the T -matrix where The sublattice components of the Green function that describes the backscattering of the delocalised Bloch waves on the edge satisfy where 0 < k < π and C(k) = −ωt 2 /(2t 1 sin(k)) is a positive coefficient for the lower band (ω < 0). Backscattering on the leftmost edge involves the additional phase shift δ A (k) = 2Arg[h(k)] on sublattice A. The LDOS verifies ρ = −Im G/π, so its space behaviour consists of where we have introduced the vector s = (t 1 /t 2 , k). These sublattice-polarised expressions of the LDOS are the ones introduced in the manuscript. Supplementary Figure 3 shows that the space behaviour of the LDOS in Supplementary Eq. (10) describes well the one observed in the 1D photonic insulator in our experiment.
The wavefront dislocation discussed in the manuscript arises from the LDOS fluctuations ∆ρ A = −Im ∆G AA /π, as a phase singularity of the scattering Green function This is the expression of the scattering Green function we introduce in the manuscript to explain the origin the LDOS dislocation.

Supplementary Note 4. GENERALISATION TO LARGER WINDING NUMBERS
Here, we extend beyond the SSH model the relevance of LDOS topological defects as new manifestation of the nontrivial bulk of 1D insulators. We first present how to experience 1D insulators with winding numbers larger than one through a nearestneighbour tight-binding model. We derive a bulk-boundary correspondence showing that the winding number still provides the number of midgap edge states. We then describe the elastic backscattering of the delocalised waves on the edge of the system and reach similar conclusions as for the SSH insulator. The number of interference fringes in the LDOS still exhibits a wavefront dislocation whose strength reveals the larger winding number of the band structure. We finally realise such a system in our microwave experiment, which demonstrates that our approach is further relevant to measure topological indices larger than one. One possibility to experience larger winding numbers in a 1D insulator is to consider additional coupling processes between distant neighbours in the SSH chain [2,3]. However, this is not possible in our microwave platform. The dielectric resonators couple each other through their evanescent fundamental TE modes, which mainly couples nearest-neighbour resonators. Now we show that this is nonetheless possible to experience larger winding numbers within a nearest-neighbour tight-binding description, as represented in Fig. 4 and Fig. 5 in the main manuscript. In the sublattice basis (A 1 , A 2 , B 1 , B 2 ), the bipartite Bloch Hamiltonian reads and h 1 (k) = t 1 + t 2 e −ik , h 2 (k) = t 3 + t 4 e −ik , while the dimensionless momentum verifies −π ≤ k ≤ π and t i > 0. There are four frequency bands associated with the dispersion relations ± f 1 (k) and ± f 2 (k). They satisfy where ∆(k) = (|h 1 (k)| 2 + |h 2 (k)| 2 + t 2 ⊥ ) 2 − 4|h 1 (k)| 2 |h 2 (k)| 2 . The bulk gap between the frequency bands f 1 and f 2 cannot close because ∆(k) > 0 for t ⊥ 0. Only can the gap close at zero frequency when ± f 1 = 0. This occurs at k = π for t 1 = t 2 or t 3 = t 4 . When t 1 t 2 and t 3 t 4 , the Bloch Hamiltonian H(k) defines a mapping from the 1D Brillouin zone S 1 to the space of gapped Hamiltonians, which reduces to S 1 under chiral symmetry. The first homotopy group of spheres π 1 (S 1 ) = Z then characterises the band structure topology. We find that the topological index is a winding number w satisfying where z = e ik and r n (p n ) denotes the number of zeros (poles) of h n inside the unit circle C. Therefore, the topological index of the two coupled SSH chains is nothing but the sum of the individual winding numbers of each uncoupled SSH chain. From our previous study of the SSH chain, we readily identify three distinct topological regimes: w = 0 when t 1 > t 2 and t 3 > t 4 (15) w = −1 when t 1 < t 2 and t 3 > t 4 or t 1 > t 2 and t 3 < t 4 w = −2 when t 1 < t 2 and t 3 < t 4 .
Note that this analysis can be generalised straightforwardly to the case of N coupled SSH chains following the pattern in Supplementary Fig. 4, which then allows arbitrary winding numbers up to w = N.

Midgap boundary modes
Under chiral symmetry, the midgap modes are sublattice polarised at zero frequency. Assuming nearest-neighbour coupling, the wave functions of such modes satisfy independent recursive equations where A n (m) is the wave-function weight in unit cell m on sublattice A n . The second equation leads to the number of evanescent wave-functions on sublattice A 2 . Following our previous analysis of the SSH chain, it is given by the number of zeros of h 2 (1/z) inside the unit circle C. Thus, we find that there is N A 2 = 1 − r 2 = −w 2 midgap boundary mode on sublattice A 2 . From the knowledge of A 2 (m), we can then determine the number of evanescent solutions on sublattice A 1 satisfying Supplementary Eq. (16). The homogeneous solution A 1,h (m) is associated with the characteristic equation h 1 (1/z) = 0. Following our previous analysis of the SSH chain, it leads to 1 − r 1 = −w 1 evanescent modes. The peculiar solution is of the form A 1,p (m) ∝ A 2 (m) and so describes −w 2 extra evanescent modes on sublattice A 1 . In the end, the full solution is A 1 (m) = A 1,h (m) + A 1,p (m). Therefore, the number of independent evanescent modes localised at the leftmost edge on sublattice A 1 is On the other hand, the zero-frequency modes polarised on sublattice B n satisfy the recurrence relations where B n (m) is the wave-function weight in unit cell m on sublattice B n . A similar analysis as for the sublattices A 1,2 readily shows that there is no evanescent mode localised at the leftmost edge on sublattice B n satisfying the boundary condition B n (0) = 0. This proves the bulk-boundary correspondence: the number of zero-energy edge states polarised on sublattices A 1 is given by the topological invariant of the bulk band structure N A 1 = −w.

Effective third-nearest-neighbour tight-binding model
We can show that the low-frequency physics of our nearest-neighbour tight-binding microwave insulator realises an effective 1D insulator coupling resonators up to the third-nearest neighbours on sublattices A 1 and B 2 . Indeed, the Bloch Hamiltonian in Supplementary Eq. (12) leads to the set of secular equations

-t 1 t 3/ t ⊥ -(t 1 t 4+ t 2 t 3)/ t ⊥ -t 2 t 4/ t ⊥ a b
Effective couplings : Therefore, the wave functions are mainly polarised on sublattices A 1 and B 2 in the limit of low frequency f t ⊥ , and this two-sublattice picture captures entirely the band-structure topology of the Bloch Hamiltonian in Supplementary (12). As we show below, this has strong consequences for the elastic scattering problem in the two lowest frequency bands ± f 1 (k).

Backscattering on the leftmost edge
The non-interacting Green function in the sublattice basis (A 1 , where the integers m and n label the unit cells, and I 4 is the 4×4 identity matrix. We model the leftmost edge by two infinite onsite potentials V 1 = V 2 in unit cell m = 0, and we focus on the standing-wave interference on unit cells m ≥ 1 (see. Supplementary  Fig. 4). Summing up the scattering diagrams in all orders in the onsite potentials leads to a T -matrix that generically reads We aim to probe the winding number w and the bulk-boundary correspondence through the LDOS fluctuations on sublattice A 1 .
The standing-wave interference on sublattice A 1 are obtained from the scattering Green function To probe a larger winding number in the experiment, it is sufficient to focus on the situation where t 1 = t 3 and t 2 = t 4 . This allows us to realise two different topological regimes in Supplementary Eq. (15), that are, w = 0 and w = −2. The transition is then driven by the single parameter t 1 /t 2 . The bare Green matrix of such a system satisfies where h(k) = α + e −ik and we introduce α = t 1 /t 2 and β = t ⊥ /t 2 . So the frequency is in units of t 2 . The space behaviour of the LDOS fluctuations on sublattice A 1 requires the knowledge of the real-space components G A 1 B 1 and G A 1 B 2 of the non-interacting Green matrix. These are obtained as The poles provide the frequency bands ± f 1 (k) and ± f 2 (k) since Furthermore, we have shown above that the energy bands ± f 1 are always gapped from ± f 2 . Since only ± f 1 can vanish at zero frequency, the different topological regimes can be fully characterised from the two frequency bands ± f 1 . In the limit |h(k)| β, the frequency bands verify f 1 (k) |h(k)| and f 2 (k) β. For unit cells satisfying m ≥ 2, we find We have used the residue theorem for the complex variable z = e ik with respect to the unit circle C (see the appendix of Ref. [4] for more details). Importantly, the expressions above describe the elastic scattering for the frequency band − f 1 on unit cells m ≥ 2. Otherwise, for the unit cell m = 1, there are extra poles inside C at z = 0. These must be taken into account to suitably describe the LDOS on site m = 1.

LDOS fluctuations and wavefront dislocation
The expressions above imply that the scattering processes described by T B 2 B 2 are dominant in Supplementary Eq. (25). This leads to where ϕ A 1 = 2km + δ A 1 (k) + π and the scattering phase shift is The space modulations of the LDOS in unit cells m ≥ 2 then satisfy where we introduce the vector s = (t 1 /t 2 , k). This space behaviour of the LDOS is represented in Supplementary Fig. 6 and shows good agreements with the LDOS measured in the experiment on sites m ≥ 2.
The number of interference fringes N A 1 on sublattice A 1 is also given by a sum rule over the frequency band f − = − f 1

4.0
Coupling ratio t1/t2 The bulk-boundary correspondence further relates the variation of the scattering phase shift to the number of boundary modes as δ A 1 (0) − δ A 1 (π) = N A 1 . Furthermore, Supplementary Fig. 7 shows that the homogeneous change observed in N A 1 is direct evidence of a wavefront dislocation of strength W d = 4 at the topological transition whereas W d = 2 for the SSH chain. This demonstrates that wavefront dislocations can also reveal topological transitions involving winding numbers larger than 1.