Wavefront dislocations reveal the topology of quasi-1D photonic insulators
OObservation of wavefront dislocation as bulk evidence of topological transitionin a 1D microwave photonic insulator
Cl´ement Dutreix, ∗ Matthieu Bellec, Pierre Delplace, and Fabrice Mortessagne Universit´e de Bordeaux, France and CNRS, LOMA, UMR 5798, F-33400 Talence, France Universit´e Cˆote d’Azur, CNRS, Institut de Physique de Nice, Nice, France Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France
From amphidromic points of tides to singular optics,wave phenomena commonly involve topology. In particu-lar, the phase singularities of classical wave fields can leadto wavefront dislocations in the interference patterns. Inquantum mechanics, the U(1) gauge invariance also en-ables manifestations of the wave function phase singulari-ties in the context of topological phases of matter. If theirtopological properties are defined from the wave functionsdelocalised in the bulk, their main experimental manifes-tation is the existence of gapless excitations localised at theboundaries. Here we show wavefront dislocation as directevidence of the phase singularity of the delocalised wavefunctions and observe it through standing-wave interfer-ence in a 1D microwave photonic insulator. By bridgingthe bulk topology of insulators to a ubiquitous wave phe-nomenon, we open a promising route to investigate thetopological phases near topological defects in real-spaceinterference.
Topological insulating phases are described by integer-valued numbers (topological indices) that characterize thephase singularities of the bulk wave functions. A discretechange of the topological index requires the spectral gapto close. Such a topological transition is also associatedwith a phase singularity of the eigenmodes. By includingthe parameter that controls the spectral gap, a topologicaltransition of a D dimensional system is then described bya singular point in a D + + System
In 1D, the band topology of insulators may become nontrivialin the presence of chiral symmetry. For lattices with transla-tional invariance, this concerns a class of Bloch Hamiltoniansthat are bipartite H ( k ) = (cid:32) h ( k ) h † ( k ) 0 (cid:33) , (1)where k is the dimensionless 1D quasi-momentum. Anilluminating illustration of Hamiltonian (1) is found in thecelebrated Su-Schrie ff er-Heeger (SSH) model [2]. Firstintroduced to describe conducting electrons in polyacetylene,it is involved in the physics of various chiral systems [3–8].Here, we focus on an experimental realisation in a microwavephotonic insulator. The system consists of a dimerised latticeof dielectric resonators in a microwave cavity (Figs. 1a,b).The resonators couple each other through the evanescent fieldof the fundamental transverse electric (TE) modes and realisea tight-binding model [9]. We denote t and t the staggeredcoupling strengths between the sublattices A and B, so that h ( k ) = t + t e − ik for the choice of unit cell in Fig. 1b. TheSSH model is known to display a transition between twotopologically distinct insulators when varying the couplingratio t / t . Their band topology relies on the quantisation ofthe geometrical Zak phase of the Bloch wave function in the1D Brillouin zone (BZ) [10]. This quantisation is charac-terised by the winding number w = (cid:72) BZ ∇ k Arg[ h ( k )] dk / π ,which leads to w > = w < = − t ≷ t . Topology from localised boundary modes
Before presenting new evidence of the topological transition,let us recall that in experiments, the band topology is mainlyevidenced through the appearance / disappearance of midgapmodes localised at the lattice boundaries, by virtue of thecelebrated bulk-boundary correspondence. Here, the bulk-boundary correspondence predicts the existence of N A = − w ≷ bound states with sublattice polarization A at the leftmostedge of the crystal in Fig. 1b (See Supplementary Informa-tion). The situation at the rightmost edge is symmetric byspace inversion. Figures 1c,d report the observation of thesemidgap boundary modes. It shows the measured density of a r X i v : . [ c ond - m a t . m e s - h a ll ] J un c Frequency (GHz) D O S f min f max Lower band t /t =1.2 d D O S f min Frequency (GHz) f max Edge modesLower band
LDOS (a.u.) Dimer index m
01 22 mode A
Edge modes: t /t =0.8 ba mode B FIG. 1. 1D microwave photonic insulator a , Lattice of dielectric res-onators inside the microwave cavity made of two metallic plates. Thetop plate, which is partially shown, is movable in the plane (whitearrows), so that an antenna going through and connected to a vecto-rial network analyser (VNA) enables the generation and resolutionof the microwave signal both spectrally and spatially (LDOS). b , Il-lustration of the SSH lattice (top) and picture of its realisation withdielectric resonators (bottom). c , DOS of a 44-resonator SSH latticefor t / t = . ω > =
0) showing two bands separated by a frequencygap. d , DOS of a 44-resonator SSH lattice for t / t = . ω < = − m , whereas the B sitesare in between two consecutive ticks. states (DOS) of the TE waves in a lattice of 44 microwaveresonators. For t / t = .
2, where the bulk winding numberis w > =
0, the sublattice structure produces two frequencybands of 22 modes each. In constrast, when t / t = .
8, the1D winding number switches to w < = − localised at boundariesis commonly considered as the hallmark of topologicaltransitions, as reported in mechanical, acoustic, photonic,microwave, cold-atomic, and electronic systems [11–18].Nevertheless, the band topology is defined from the delo-calised waves beyond the excitation gap. Now we presentdirect evidence of the topological transition through LDOSmeasurements of the delocalised waves. LDOS interference of delocalised waves
The delocalised waves correspond to resonance frequenciesin the two bands f ± . Figures 2a-d represent the sublattice-resolved LDOS ρ A , B of the delocalised waves of the lowerband f − probed in the two topological regimes. Only the left-most half of the crystal is shown, for the second half is inver- sion symmetric (see Supplementary Information). The LDOSmaps consist of standing-wave interference patterns due to thelattice boundaries. We focus in particular on the number N A ( B ) of constructive-interference fringes in the LDOS of sublat-tice A ( B ). For the sublattice A in Figs. 2a,b, we observe that N A changes identically on each site m through the topologicaltransition. For instance, there are six constructive-interferencefringes on site m = t / t = .
2, whereas there arefive of them when reducing the coupling ratio to t / t = . N A = m for t / t = . N A = m − t / t = .
8. In contrast, we do not ob-serve such a change on sublattice B, where there are always N B = m constructive-interference fringes per site, regardlessof the topological phase (Figs. 2c,d).To explain this striking feature in the LDOS maps near theedge, we focus on a semi-infinite SSH chain and describe theedge as an infinite potential barrier. Backscattering of the de-localised waves on the edge then leads to the LDOS [19] ρ A ( m , k ) ∝ + cos(2 km + δ A ( k ) + π ) (2) ρ B ( m , k ) ∝ + cos(2 km + π ) , (3)which reproduces very well the experimental LDOS maps inFig. 2 (see Supplementary Information). The wavelength ofthe oscillations on both sublattices relates to the backscatter-ing wave-vector 2 k . It varies with the frequency at which weprobe the cavity through the dispersion relation f − ( k ). Thewavelength of the oscillations is then a spectral measurementand does not imply the topology of the frequency band. Forinstance, the oscillations in ρ B only depend on the backscatter-ing wave-vector and give rise to similar interference patternsin the two topological regimes, as observed experimentally inFigs. 2c,d. In contrast, the oscillations in ρ A imply the addi-tional phase shift δ A ( k ) = h ( k )].The phase shift δ A further leads to dramatic modifications inthe LDOS interference patterns. Let us focus, for instance, onthe LDOS maxima in Eq. (2). They correspond to the specificwavefronts 2 n A π = km + δ A + π , where n A is an integer.When summing over the lower frequency band, we find thatthe number of constructive-interference fringes verifies N A = (cid:90) f max f min ∂ n A ∂ f − d f − = (cid:90) π dk π (cid:32) m + ∂δ A ∂ k (cid:33) = m + w . (4)This sum rule shows that the scattering phase shift δ A relatesan observable quantity ( N A ) of the delocalised waves to theirtopological winding w . In particular, the number of interfer-ence fringes depends on the site index m , but the topologicalinvariant shifts the interference fringes identically on all sites.Remarkably, although the winding number w depends on thechoice of unit cell [20, 21], this arbitrary choice is howeverreabsorbed in the labelling of the dimers m , such that theirsum yields the observable quantity N A . The sum rule then ex-plains the uniform change of N A observed in the LDOS mapsin Figs 2a,b for the winding numbers w > = w < = − 𝝆 A ( a . u . )
101 10Dimer index mf min f max F r e q u e n c y a f min f max F r e q u e n c y 𝝆 B ( a . u . ) f min f max F r e q u e n c y 𝝆 B ( a . u . )
101 10 f min f max F r e q u e n c y 𝝆 A ( a . u . ) bc Dimer index m Dimer index m Dimer index m t /t =0.8t /t =0.8t /t =1.2t /t =1.2 d FIG. 2.
Experimental LDOS maps : a , LDOS of the lower bandbetween frequencies f min and f max resolved on the first 10 sites ofsublattices A for t / t = . b , Same as a for t / t = . c - d Sameas a - b for sublattice B. topology of the delocalised waves in the 1D microwave insu-lator.The phase shifts of wave functions also play a central role inscattering physics, because they relate to the number of (vir-tual) bound states at a potential barrier. Fundamental theo-rems, such as Levinson theorem or Friedel sum rule, showthat, for given wave functions, the number of (virtual) boundstates change with the depth of the barrier [22–24]. Similarlyhere, the bulk-boundary correspondence can be rephrased asa relation between the scattering phase shift δ A and the num-ber N A of bound states localised at the potential barrier of theedge. Since δ A ( π ) − δ A (0) = π w ( Eq. 4)), we readily find δ A (0) − δ A ( π ) = π N A . (5)The important di ff erence, however, is that the number ofbound states here changes with the topological transition,whereas the strength of the potential barrier remains thesame. This change results from an intrinsic property of thedelocalised waves and the potential barrier at the edge onlyacts as a natural interferometer that reveals their topologythrough the scattering phase shift. If the phase shift vari-ation has been measured through N A in the LDOS mapsof sublattice A (Figs. 2a,b), we have also resolved the N A midgap bound states localised at the edge (inset of Fig. 1d).Thus, both sides of Eq. (5) are observable independently, andour measurements also bring evidence of the bulk-boundarycorrespondence. This demonstrates an e ffi cient method to testthis pillar of gapped topological phases through the LDOS inthe experiments. Wavefront dislocations in the LDOS
Now we show that the change of N A observed in the LDOSmaps arises as a ubiquitous wave phenomenon and is the sig-nature of a wavefront dislocation in the LDOS. Topological defects in waves rely on generic assumptions that do not in-volve the wave equation, and so they are ubiquitously involvedin branches of physics as various as electromagnetism, optics,acoustics, fluid physics, astrophysics, and condensed matterphysics [1, 25–33]. The wavefront dislocations are associ-ated with the topological phase singularities of wavefields ina space at least 2D.Here, the microwave photonic insulator is 1D and its topo-logical transition relies on a spectral band crossing in the 1DBZ (Fig. 3a). Nevertheless, the topological transition is drivenby the coupling ratio t / t . Thus, it is fully characterised in a2D space associated with the parameter s = ( k , t / t ). In thisparameter space, the spectral bands are f ± ( s ) = ±| h ( s ) | and theeigenstates can be chosen as | u ± ( s ) (cid:105) ∝ | A (cid:105) ± e i θ ( s ) | B (cid:105) , where θ ( s ) = Arg[ h ( s )]. The zeroes of h ( s ) are points where i) thespectral band gap closes and ii) the eigenstate phase θ ( s ) be-comes ill-defined. This phase singularity in 2D is nothing buta vortex that constrains the surrounding phase texture to wind.The vortex winding is then quantified by a topological index W s , such that (cid:72) C ∇ s θ · ds = π W s along a closed circuit C en-closing the phase singularity. In the SSH model, s = ( π,
1) isthe only point where h ( s ) vanishes (Fig. 3b). This leads to thesingularity charge W s = C in Fig. 3c.The phase singularity in the 2D parameter space is thesource of a wavefront dislocation of strength 2 W s in theLDOS. We can evidence the dislocation by following the evo-lution of the LDOS interference patterns through the topolog- A r g [ h ] 𝜋 𝜋 𝜋𝜋𝜋𝜋 k k k f ± ( k ) t /t =1.2t /t =1.0t /t =0.8 | h | 𝜋 t /t t /t 𝜋 k 𝜋 k C s s ab c FIG. 3.
Band degeneracy and phase singularity : a , Spectral bands f ± ( k ) = ±| h ( k ) | in the 1D BZ for various values of the parameter t / t .The band crossing for t / t = b , Energy band f + ( s ) = | h ( s ) | rep-resented in the 2D parameter space. The spectral degeneracy occursat point s = ( π, c , Eigenstate phase θ ( s ) = Arg[ h ( s )]in the 2D parameter space. It is singular at the spectral degeneracypoint s . F r e q u e n c y ( a . u . ) f min Coupling ratio t /t 𝜋 k -11 𝝆 A ( a . u . ) a 𝝆 A ( a . u . ) f max F r e q u e n c y ( a . u . ) f min bs s f max s 𝝆 A ( a . u . ) Coupling ratio t /t Coupling ratio t /t c C FIG. 4.
Wavefront dislocation in the LDOS : a , Theoretical LDOS ρ A in the 2D parameter space on site m = b , Theoretical LDOS ρ A in the 2D parameter space where the wavevector k is replaced by the lower band frequency f − ( k ). c , Experimental LDOS ρ A resolved on site m = ical transition. Figure 4a shows the predicted LDOS evolutionon a given site of sublattice A ( m = s . Wavefront dislocations are known to occuras the phase singularity of a complex scalar field whose real(or imaginary) part represents a physical quantity [1]. Here,the complex scalar field is the standing-wave Green functionthat describes the delocalised waves backscattering on the po-tential barrier at the edge of the microwave insulator G A ( m , s ) ∝ e i ϕ A ( s ) , (6)where ϕ A ( s ) = km + δ A ( s ) + π , and the physical quantity is theLDOS ρ A = − Im G A /π , as introduced in Eq (2). The scatter-ing phase shift δ A ( s ) = h ( s )] maps the phase singularityof the eigenstates into the phase of the standing-wave Greenfunction. Thus, the latter e ff ectively describes a plane wave( e i km ) passing through a vortex ( e i δ A ( s ) ) in the 2D parameterspace. This e ff ective vortex perturbs the surrounding phase ofthe wave in such a way that, for the counter-clockwise Burg-ers circuit C in Fig. 4a, the phase accumulated by the Greenfunction satisfies W d ≡ π (cid:73) C ∇ s ϕ A ( s ) · ds = W s (7)The phase variation is 2 π -quantised because G A must be singlevalued to describe the LDOS along the circuit C . Thus, thenumber of additional interference fringes required to fulfill thephase variation along the Burgers circuit C is W d . In analogywith Burgers’ vectors whose length provides the dislocationstrength for atomic planes in solids, W d is the strength of thewavefront dislocation. Since W s = W d = ϕ A depends on the choiceof the unit cell, its variation over C does not and is observable.To confirm this prediction, we measure the LDOS on thesite m = t / t between 0.2 and 2.0. Since the LDOS is resolvedas a function of the frequency f − ( k ) instead of the wavevector k in our experiments, we do not expect W d but W d / N A of constructive-interference fringes abruptly changes fromone to two at the dislocation core. This is also in agreementwith the LDOS change on site m = N A in the LDOS interference pattern. Conclusion
We have probed the band topology of a 1D photonic insulatorthrough the standing-wave interference pattern in the LDOSresulting from backscattering on a boundary. We have shownthat the uniform change in the number of interference fringesat the topological transition is a measurement of the disloca-tion strength and then of the eigenstate phase singularity. This2D phase singularity constrains the 1D winding numbers ofthe two nonequivalent SSH insulators as W s = w > − w < (seeFig. 3). Although there is a gauge choice in the definition ofthe 1D winding numbers, their di ff erence is gauge invariantand the uniform change in the number of interference fringescharacterises unambiguously the change of band topology atthe transition. Thus, the wavefront dislocation in the LDOS isan observable phenomenon that reveals the topological transi-tion in 1D insulators.The band topology of 1D insulators is also known to af-fect the electron response to external force fields through phe-nomena such as the electric polarisation and Bloch oscilla-tions [34–36]. Nevertheless, these phenomena are observablein very specific systems. Bloch oscillations, for instance, arehardly observable with electrons in solids, where impuritiesare usually detrimental to phase coherence, and so they leadto band topology measurements in cold atoms [5]. In contrast,the LDOS is an observable routinely resolved in various kindsof systems [11–18]. 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