Abstract
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.
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Introduction
Wavefront dislocations are a fundamental and ubiquitous wave phenomenon that originates from an indetermination of the phase of a wavefield, where the amplitude of the wave vanishes. Since the seminal work of John Nye and Michael Berry in 1974^{1}, it was realised that such topological defects could emerge in any wavefield irrespectively of its physical nature or its dispersion relation. Wavefront dislocations have been observed from the physics of fluids, sound, electromagnetism to oceanic tides and astronomy, and even led to the birth of a whole research field known as singular optics^{2,3,4,5,6,7}. In quantum mechanics, wavefront dislocations have been predicted in connection to the Aharonov–Bohm effect^{8,9}, and they have been observed recently in the electron density of graphene as a manifestation of the wavefunction Berry phase^{10,11}. In parallel, topology also spread in condensed matter physics, giving rise to the field of topological phases of matter^{12,13} and its various classical analogues such as topological photonics^{14}. In this context, topological properties are defined from singularities of the wave functions delocalised in the bulk of the material. Still, apart from the pioneering example of the quantised electric conductivity in the quantum Hall phase, their experimental manifestations are mainly indirect, through the existence of gapless excitations localised at the boundaries of the system. Despite this common underlying key role of phase singularities, topological phases and singular waves have remained two distinct fields^{15}. Here we reconcile them by showing a wavefront dislocation as a direct evidence of the phase singularity of the delocalised wave functions and observe it through standingwave interference in 1D microwave photonic insulators. By bridging the bulk topology of insulators to a ubiquitous wave phenomenon, we open a promising route to investigate quantum and classical topological systems through realspace interference patterns.
Topological insulating systems are associated with integervalued numbers (topological indices) that characterise the phase singularities of the bulk wave functions. A change of the topological index requires the spectral gap to close. Such a topological transition is also associated with a phase singularity of the eigenmodes. By including the parameter that controls the spectral gap, a topological transition of a D dimensional system is then described by a singular point in a D + 1 dimensional parameter space. The 1D case is particularly interesting as it allows us to describe topological transitions with vortices appearing in 2D parameter space. Vortices in real space are known to induce wavefront dislocations onto an incident propagating wave^{8}. Similarly, we reveal here that the vortex of the topological transition involves a quite analogue phenomenon in parameter space. When a defect or an edge is included to a topological insulator, a defectinduced interference pattern of bulk wave functions emerges and abruptly changes at the singular band crossing point, then giving rise to a wavefront dislocation in the D + 1 parameter space. We find that this wavefront dislocation is accessible experimentally through the local density of states (LDOS) and demonstrate its existence in 1D microwave photonic insulators. Moreover, we show that the quantised charge of the vortex, which corresponds to the variation of the number of interference fringes at the transition, consistently coincides with the variation in the number of topologicallyprotected boundary modes inside the spectral gap. This leads us to the experimental demonstration of the pillar of the topological phases of matter, the bulkboundary correspondence.
Results
Realisation of the 1D photonic insulator
In 1D, the band topology of insulators may become nontrivial in the presence of chiral symmetry. For lattices with translational invariance, this concerns a class of Bloch Hamiltonians that are bipartite
where k is the dimensionless 1D quasimomentum. An illuminating illustration of Hamiltonian (1) is found in the celebrated Su–Schrieffer–Heeger (SSH) model^{16}. First introduced to describe conducting electrons in polyacetylene, it is involved in the physics of various chiral systems^{17,18,19,20,21,22}. Here, we focus on an experimental realisation in a microwave photonic insulator. The system consists of a dimerised lattice of dielectric resonators in a microwave cavity (see Fig. 1a, b). Each cylindrical resonator is made of ZrSnTiO ceramics (radius r = 3 mm, height h = 5 mm, with an index of refraction n_{r} ≈ 6) and supports a fundamental transverseelectric (TE_{1}) mode of bare frequency of 7.435 GHz. This mode spreads out evanescently, so that the coupling strength can be controlled by adjusting the separation distance between the resonators^{23}. The lattice consists in two coupled sublattices A and B with staggered coupling strengths t_{1} and t_{2}, so that h(k) = t_{1} + t_{2}e^{−ik} for the choice of unit cell in Fig. 1b. The corresponding resonator separations are denoted d_{1} and d_{2}. In our experiments, the coupling strengths t_{1,2} can be typically adjusted from 10 to 115 MHz which corresponds to separations d_{1,2} of 16 mm and 7 mm, respectively.
The SSH model is known to display a transition between two topologically distinct insulators when varying the coupling ratio t_{1}/t_{2}. Its spectrum exhibits two bands given by f_{±}(k) = ±∣h(k)∣ and whose topology relies on the quantisation of the geometrical Zak phase of the Bloch wave function in the 1D Brillouin zone (BZ)^{24}. This quantisation is characterised by the winding number \(w={\oint }_{{\rm{BZ}}}{\nabla }_{k}{\rm{Arg}}[h(k)]dk/2\pi\), which leads to w_{>} = 0 and w_{<} = −1 in the two insulating regimes t_{1} ≷ t_{2}.
Topology from localised boundary modes
Before shedding new light on the topological transition, let us recall that in experiments, the band topology is mainly evidenced through the appearance/disappearance of midgap modes localised at the lattice boundaries, by virtue of the famous bulkboundary correspondence^{25}. Here, the bulkboundary correspondence predicts the existence of \({{\mathcal{N}}}_{A}={w}_{\gtrless }\) (\({{\mathcal{N}}}_{B}={w}_{\gtrless }\)) bound states with sublattice polarisation A (B) at the leftmost (rightmost) edge of the crystal in Fig. 1b (see Supplementary Note 3). We report the observation of these midgap boundary modes in Fig. 1c, d. It shows the measured density of states (DOS) of the TE_{1} waves in a lattice of 44 microwave resonators (see Supplementary Note 1). For t_{1}/t_{2} = 1.2, where the bulk winding number is w_{>} = 0, the sublattice structure produces two frequency bands of 22 modes each. In contrast, when t_{1}/t_{2} = 0.8, the 1D winding number switches to w_{<} = −1 and we observe two modes pinned in the gap. We then confirm that they are sublattice polarised and spatially localised at the two ends of the crystal by resolving their LDOS (inset of Fig. 1d).
The observation of midgap modes localised at boundaries is commonly considered as the hallmark of topological transitions, as reported in mechanical, acoustic, photonic, microwave, coldatomic and electronic systems^{26,27,28,29,30,31,32,33}. Nevertheless, the band topology is defined from the delocalised waves beyond the excitation gap. Now we present direct evidence of the topological transition through LDOS measurements of the delocalised waves.
LDOS interferences of delocalised waves
The delocalised waves correspond to resonance frequencies in the two bands f_{±}. Figures 2a–d represents the sublatticeresolved LDOS ρ_{A,B} of the delocalised waves of the lower band f_{−} probed in the two topological regimes. Only the leftmost half of the photonic crystal is shown, for the second half is inversion symmetric (see Supplementary Note 2). The LDOS maps consist of standingwave interference patterns due to the lattice boundaries. We focus in particular on the number N_{A(B)} of constructiveinterference fringes in the LDOS of sublattice A(B). For the sublattice A in Fig. 2a, b, we observe that N_{A} changes identically on each site m through the topological transition. For instance, there are six constructiveinterference fringes onsite m = 6 when t_{1}/t_{2} = 1.2, whereas there are five of them when reducing the coupling ratio to t_{1}/t_{2} = 0.8. More generally, it shows that N_{A} = m for t_{1}/t_{2} = 1.2 and N_{A} = m − 1 for t_{1}/t_{2} = 0.8. In contrast, we do not observe such a change on sublattice B, where there are always N_{B} = m constructiveinterference fringes per site, regardless of the topological phase (Fig. 2c, d).
To explain this striking feature in the LDOS maps near the edge, we focus on a semiinfinite SSH chain and model the edge as an infinite potential barrier. Backscattering of the delocalised waves on the edge then leads to the LDOS^{34}
which reproduces very well the experimental LDOS maps in Fig. 2 (see Supplementary Note 3). The oscillating terms in the righthand sides describe the LDOS fluctuations induced by the edge. The wavelength of the oscillations on both sublattices relates to the backscattering wavevector 2k. Such 2kwavevector oscillations are often referred to as (frequencyresolved) Friedel oscillations, with reference to the charge density oscillations that screen charged impurities in metals^{35}. It varies with the frequency at which we probe the cavity through the dispersion relation f_{−}(k). The wavelength of the oscillations is then a spectral measurement and does not imply the topology of the frequency band. For instance, the oscillations in ρ_{B} only depend on the backscattering wavevector and give rise to similar interference patterns in the two topological regimes, as observed experimentally in Fig. 2c, d. In contrast, the oscillations in ρ_{A} imply the additional phase shift \({\delta }_{A}(k)=2{\rm{Arg}}[h(k)]\) (see ref. ^{34}).
The phase shift δ_{A} further leads to dramatic modifications in the LDOS interference patterns. In particular, the number of constructiveinterference fringes N_{A} is given by the variation of the phase φ_{A} = 2km + δ_{A} + π in Eq. (2) over the lower frequency band. It reads
This sum rule shows that the scattering phase shift δ_{A} relates an observable quantity of the delocalised waves, N_{A}, to their topological winding w. In particular, the number of interference fringes depends on the site index m, but the topological invariant shifts the interference fringes identically on all sites. Remarkably, if the winding number w depends on the choice of unit cell^{36,37}, this arbitrary choice is however included in the labelling of the dimers m, such that their sum yields the observable quantity N_{A}. The sum rule then explains the uniform change of N_{A} observed in the LDOS maps in Fig. 2a, b for the winding numbers w_{>} = 0 and w_{<} = −1. Therefore, the LDOS maps reveal direct evidence of the band topology of the delocalised waves in the 1D microwave insulator.
The phase shifts of wave functions also play a central role in scattering physics, because they relate to the number of (virtual) bound states at a potential barrier. Fundamental theorems, such as Levinson theorem or Friedel sum rule, show that, for given wave functions, the number of (virtual) bound states change with the depth of the barrier^{35,38,39}. Similarly here, the bulkboundary correspondence can be rephrased as a relation between the scattering phase shift δ_{A} and the number \({{\mathcal{N}}}_{A}\) of bound states localised at the potential barrier of the edge. Since δ_{A}(π) − δ_{A}(0) = 2πw (see Eq. (4)), we readily find
We stress that the number of bound states here changes with the topological transition, whereas the strength of the potential barrier remains the same, in sharp contrast with usual defects bound states. This change results from an intrinsic property of the delocalised waves and the potential barrier at the edge only acts as a natural interferometer that reveals their topology through the scattering phase shift. If the phase shift variation has been measured through N_{A} in the LDOS maps of sublattice A (Fig. 2a, b), we have also resolved the \({{\mathcal{N}}}_{A}\) midgap bound states localised at the edge (inset of Fig. 1d). Thus, both sides of Eq. (5) are observable independently, and our measurements also bring evidence of the bulkboundary correspondence. This demonstrates an efficient method to test this key concept of gapped topological systems through the LDOS in the experiments.
Wavefront dislocations in the LDOS
Now we show that the change of N_{A} observed in the LDOS maps arises as a ubiquitous wave phenomenon and is the signature of a wavefront dislocation in the LDOS. Topological defects in waves rely on generic assumptions that do not involve the wave equation, and so they are ubiquitously involved in branches of physics as various as electromagnetism, optics, acoustics, fluid physics, astrophysics, and condensed matter physics^{2,3,4,5,6,7,8,10,11,40,41}. The wavefront dislocations are associated with the topological phase singularities of wavefields in a space of at least dimension 2.
Here, the microwave photonic insulator is 1D and its topological transition relies on a spectral band crossing in the 1D BZ (Fig. 3a). Nevertheless, the topological transition is driven by the coupling ratio t_{1}/t_{2}. Thus, it is fully characterised in a 2D space associated with the parameter s = (t_{1}/t_{2}, k). In this parameter space, the spectral bands are f_{±}(s) = ±∣h(s)∣ and the eigenstates can be chosen as \(\left{u}_{\pm }({\bf{s}})\right\rangle \propto \leftA\right\rangle \pm {e}^{i\theta ({\bf{s}})}\leftB\right\rangle\), where \(\theta ({\bf{s}})={\rm{Arg}}[h({\bf{s}})]\). The zeroes of h(s) are points where i) the spectral band gap closes and ii) the eigenstate phase θ(s) becomes illdefined. This phase singularity in 2D is nothing but a vortex that constrains the surrounding phase texture to wind. The vortex winding is then quantified by a topological index W_{s}, such that ∮_{C}∇_{s}θ ⋅ ds = 2πW_{s} along a closed circuit C enclosing the phase singularity. In the SSH model, s_{0} = (1, π) is the only point where h(s) vanishes (Fig. 3b). This leads to the singularity charge W_{s} = 1 for the counterclockwise circuit C in Fig. 3c.
The phase singularity in the 2D parameter space is the source of a wavefront dislocation of strength 2W_{s} in the LDOS. We can evidence the dislocation by following the evolution of the LDOS interference patterns through the topological transition. Figure 4a shows the predicted LDOS evolution on a given site of sublattice A (m = 2). It exhibits an edge dislocation with two constructiveinterference fringes emerging from the core in s_{0}. Wavefront dislocations are known to occur as the phase singularity of a complex scalar field whose real (or imaginary) part represents a physical quantity^{1}. Here, the physical quantity is the LDOS fluctuations defined as \({{\Delta }}{\rho }_{A}={\rm{Im}}{{\Delta }}{G}_{A}/\pi\) (oscillating term in Eq. (2)), so that the complex scalar field is the scattering Green function ΔG_{A} that describes the delocalised waves backscattering on the edge of the microwave insulator
where φ_{A}(s) = 2km + δ_{A}(s) + π (see Supplementary Note 3). The scattering phase shift \({\delta }_{A}({\bf{s}})=2{\rm{Arg}}[h({\bf{s}})]\) maps the phase singularity of the eigenstates into the phase of the scattering Green function. The latter effectively describes a plane wave (e^{i2km}) passing through a vortex (\({e}^{i{\delta }_{A}({\bf{s}})}\)) in the 2D parameter space. This effective vortex perturbs the surrounding phase of the wave in such a way that, for the counterclockwise Burgers circuit C in Fig. 4a, the phase accumulated by the scattering Green function satisfies
The phase variation is 2πquantised because ΔG_{A} must be single valued to describe observable LDOS fluctuations along the circuit C. Thus, the number of additional interference fringes required to fulfil the phase variation along the Burgers circuit C is W_{d}. In analogy with Burgers’ vectors whose length provides the dislocation strength for atomic planes in solids, W_{d} is the strength of the wavefront dislocation. Since W_{s} = 1 in the SSH model, there are W_{d} = 2 additional interference fringes emerging from the dislocation core, as shown in Fig. 4a. It is worth stressing that although the expression of the phase φ_{A} depends on the choice of the unit cell, its variation over C does not and is observable.
To confirm this prediction, we measure the LDOS on the site m = 2 of sublattice A for 20 values of the coupling ratio t_{1}/t_{2} between 0.2 and 2.0. Since the LDOS is resolved as a function of the frequency f_{−}(k) instead of the wave vector k in our experiments, we do not expect W_{d} but W_{d}/2 interference fringes emerging from the dislocation core (Fig. 4b). Figure 4c experimentally confirms that the number N_{A} of constructiveinterference fringes changes from one to two at the dislocation core. This is also in agreement with the LDOS change on site m = 2 shown in Fig. 2a, b. This observation reveals the wavefront dislocation at the topological transition that causes the uniform change in the number of fringes N_{A} in the LDOS interference pattern.
The number of interference fringes in the LDOS can also reveal the bulk topology of 1D insulators with winding numbers larger than ∣w∣ = 1. To demonstrate this beyond the SSH insulator, we consider the (quasi)1D microwave lattice depicted in Fig. 5a. By varying the coupling ratio t_{1}/t_{2}, one can experience two distinct topological regimes associated with the winding numbers w = 0 and w = −2 (see Supplementary Note 4). Remarkably, we show theoretically that the scattering phase shift in the LDOS also leads to the sum rule (4) on sublattice A_{1} for m ≥ 2, where \({N}_{{A}_{1}}=m+w\) (see Supplementary Note 4). It is further confirmed experimentally by our LDOS measurements reported in Figs. 5b,c, where we observe that \({N}_{{A}_{1}}\) shifts by two units between w = 0 and w = −2. This change in the wavefronts of the interference field is direct evidence of a dislocation of charge W_{d} = 4 associated with the topological transition (see Supplementary Note 4). Moreover, the simultaneous resolution of the corresponding number of midgap edge modes on sublattice A_{1} allows us to demonstrate experimentally the bulkedge correspondence in that microwave insulator (inset in Fig. 5d).
Discussion
We have probed the band topology of 1D photonic insulators through the standingwave interference pattern in the LDOS resulting from backscattering on a boundary. We have shown that the uniform change in the number of interference fringes at the topological transition is a measurement of the dislocation strength and then of the eigenstate phase singularity. This 2D phase singularity constrains the 1D winding numbers of the two nonequivalent insulators as W_{s} = w_{>} − w_{<} (see Fig. 3). Although there is a gauge choice in the definition of the 1D winding numbers, their difference is gauge invariant and the uniform change in the number of interference fringes characterises unambiguously the change of band topology at the transition. Thus, the wavefront dislocation in the LDOS is an observable phenomenon that reveals the topological transition in 1D insulators. We also emphasise that this direct characterisation of the bandstructure topology in real space relies here on the monotonic feature of the dispersion relation—see e.g. Eq. (4). For nonmonotonic dispersion relations, there exist several scattering wavevectors at same energy. However, these are usually well resolved from the LDOS in Fourier space, where the topological scattering phase can be resolved too^{34,40}. Such a Fourier analysis is what enabled recent experiments to extract realspace wavefront dislocations as manifestation of topological semimetals with nonmonotonic dispersion relations^{10,11}.
The band topology of 1D insulators is also known to affect the electron response to external force fields through phenomena such as the electric polarisation and Bloch oscillations^{42,43,44}. Nevertheless, these phenomena are observable in very specific systems. Bloch oscillations, for instance, are hardly observable with electrons in solids, where impurities are usually detrimental to phase coherence, and so they lead to band topology measurements in cold atoms^{19} or coupled electronic circuits^{45}. The concept of mean chiral displacement has been also used in photonic or cold atoms experiments to extract topological invariants from the bulk^{46,47,48}. Our approach lies on an universal observable, the LDOS, which is routinely resolved in various kinds of systems^{26,27,28,29,30,31,32,33}. Thus, we expect that topological defects in realspace LDOS interference can reveal the band topology in experiments involving propagating waves of very different natures.
In addition to the band topology through wavefront dislocations, the LDOS also leads to the resolution of midgap modes localised at boundaries. This enabled us to test of the bulkboundary correspondence through a single observable and, thus, a single experiment. This efficient approach could then shed light into breakdowns of the bulkboundary correspondence, as recently reported in systems where the number of bound states may no longer be provided by the bulk topological invariant^{22,49}.
Methods
Experimental realisation of the quasi1D photonic insulators: Each dielectric microwave resonator (see Fig. 1a, b) is made of ZrSnTiO ceramics (radius r = 3 mm, height h = 5 mm, with an index of refraction n_{r} ≈ 6) and supports a fundamental transverseelectric mode TE_{1} of bare frequency ν_{0} = 7.435 GHz. This mode spreads out evanescently, so that the coupling strength can be controlled by adjusting the separation distance between the resonators^{23}. As shown in Fig. 1b, the lattice consists of two coupled sublattices A and B with staggered coupling strengths t_{1} and t_{2}. The corresponding resonator separations are denoted d_{1} and d_{2}. In our experiments, the coupling strengths t_{1,2} can be typically adjusted from 10 to 115 MHz which corresponds to separations d_{1,2} of 16 mm and 7 mm, respectively.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The codes that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
M.B. and F.M. acknowledge fruitful discussions with Ulrich Kuhl. C.D. acknowledges the support of Idex Bordeaux (Maesim Risky project 2019 of the LAPHIA Programme).
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M.B. and F.M. performed the experiments, while P.L.D. and C.D. provided theoretical support.
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Dutreix, C., Bellec, M., Delplace, P. et al. Wavefront dislocations reveal the topology of quasi1D photonic insulators. Nat Commun 12, 3571 (2021). https://doi.org/10.1038/s4146702123790w
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DOI: https://doi.org/10.1038/s4146702123790w
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