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Superior robustness of anomalous non-reciprocal topological edge states

Network topology

The Chern number does not fully account for the topology of unitary operators, such as the scattering matrix in equation (1). For unitary evolutions, the eigenvalue (quasi-energy) spectrum being defined on a circle, each (quasi-energy) band is now allowed to be connected to the next one by an edge state42. Because of the cyclicity of the spectrum, and because the Chern number of a band counts the number of edge states that merge into that band, it follows that the Chern numbers of each band vanish. Since all the gaps are filled by a chiral edge state, this regime is called anomalous.

Actually, the topology of unitaries, such as evolution operators or our scattering matrix, is better described by the homotopy group π3(U(N)) = , whose elements are the topological numbers

$${W}_{\psi }={(24{{\rm{\pi }}}^{2})}^{-1}\int {\rm{tr}}{({{V}_{\psi }}^{-1}{\rm{d}}{V}_{\psi })}^{3}.$$

(2)

The power 3 must be understood in the language of differential forms, and the integral runs over a 3-torus, spanned by the quasi-momentum k = (kx, ky) and time t (over a time period T). Time is not explicit in scattering networks. However, the cyclicity of the network makes possible a direct mapping with a Floquet (that is, T-periodic in time) evolution operator U(t,k), such that an interpolation parameter that formally plays the role of time can be introduced33. Finally, the operator Vψ is a periodized (in time) evolution operator. For Floquet systems, it reads as42.

$${V}_{\psi }(t,{\bf{k}})=U(t,{\bf{k}})\exp (it{H}_{{\rm{eff}}}({\bf{k}}))$$

(3)

with

$${H}_{{\rm{eff}}}({\bf{k}})=i/T{\mathrm{ln}}_{-\psi }U(t=T,{\bf{k}}),$$

(4)

where −ψ denotes the branch-cut of the logarithm. The procedure to define such an operator Vψ and thus the invariant Wψ for discrete-time evolutions (that is, when the dynamics is given by a succession of scattering events and where time therefore does not appear explicitly), as in our model, was developed in a previous detailed study33 (in particular in sections V.A. and V.B.).

Importantly, the branch-cut ψ must be chosen in a spectral gap of U(T, k), or S(k) in our case. For this reason, Wψ is said to be a gap invariant, and indeed directly gives the number of chiral edge states in a given quasi-energy gap ψ. In contrast, Chern numbers are band invariants. They are inferred from the eigenstates of Heff(k) expressed in equation (4) and thus cannot capture the full unitary evolution. Finally, the details for the calculation of the invariants Wψ in oriented kagome graphs can be found in Delplace et al.33. Their values for the band structures of Fig. 1c are 1,1,1,1,1,1 in the anomalous case and 1,0,1,1,0,1 for the Chern case. For completeness, we provide the bandgap map of the network together with the values of the homotopy invariant in Supplementary Fig. 8.

Simulations

The simulation method of arbitrary finite non-reciprocal honeycomb networks is based on the scattering matrix method. For a finite non-reciprocal network with Nr input/output ports, once we have the information of the scattering matrix of each non-reciprocal element and the distribution of the phase delays of the links, this method can provide (i) the scattering matrix SNr regarding the Nr port system, and (ii) the field map across the network knowing the excitations at the Nr ports (see details in Supplementary Information part II).

We exemplify this method by calculating the transmission between ‘Geneva’ and ‘Davos’ through the Switzerland-shaped network (the network used in Fig. 4 of the main text) as a function of φ, and compare the transmission results with the ribbon band structures (see Supplementary Fig. 2). We assume a uniform distribution for the phase delay φ and the same non-reciprocal elements (in anomalous or Chern phase) in the Switzerland-shaped network. When both anomalous and Chern phases fall in a topological bandgap, the transmission is near unity. When both phases fall in a bulk band, the transmission undergoes sharp variations with φ, depending on the excited bulk mode. Only the Chern phase exhibits bands of blocked transmission, owing to the trivial bandgaps.

Design

The non-reciprocal networks are designed and fabricated on 0.508 mm thick Rogers RT/duroid 5880 substrate (dielectric loss tanδ = 0.0009 at 10 GHz) with 35 μm thick copper on each side. Here, the non-reciprocal element is a surface mount microwave circulator (UIYSC9B55T6, UIY Co.), designed from a ‘Y’-shaped strip line on a printed circuit board48. The three ports are placed 120° apart from each other such that they are iso-spaced. The printed circuit board is sandwiched between two pieces of ferrite. Without magnetic fields, the ‘Y’-junction strip line supports two degenerate modes at ω0: right-handed and left-handed. To bias it, two magnets are fixed outside, providing the required magnetic field of 50 kA m−1 = 628 Oe, normal to the printed circuit board and polarizing the ferrite, therefore lifting the initial degeneracy, with chiral modes at ω+ and ω. In our experiment, we first measure an individual circulator and retrieve its scattering matrix S0. The measured reflection of an individual circulator is shown in Extended Data Fig. 4a, and sets the frequency bands for CI and AFI operations.

Microstrip lines serve as phase delay links, with a width of 1.65 mm, corresponding to a standard 50 ohm characteristic impedance. The phase delay φ induced by a microstrip line with length L operating at frequency f is expressed as \(\phi =(2\pi Lf{\varepsilon }_{{\rm{e}}{\rm{f}}{\rm{f}}}^{1/2})/c\), where \({\varepsilon }_{{\rm{e}}{\rm{f}}{\rm{f}}}\) is the effective permittivity of the microstrip line, and can be determined by an empirical formula49. Taking into account the frequency dispersion of the lines and circulators, we construct a more practical topological bandgap map, shown in Extended Data Fig. 4b, as a function of the effective length of the microstrip lines L and the operating frequency f. With the aid of the map, we select L1 = 26.5 mm and L2 = 37.5 mm, which produce the conditions φ = π/8 and φ = π/2, respectively, in the simulations (Fig. 3a, Extended Data Figs. 5a, 6a). As exhibited in Extended Data Fig. 4c, the fabricated networks show the microstrip lines of L1 (blue dashed region) and L2 (red dashed region).

Measurements

The scattering parameters and field maps of three fabricated networks (network 1, network 2 and the Switzerland-shaped network) are measured by a vector network analyser (VNA; ZNB20, R&S), as demonstrated in Extended Data Fig. 8. For the scattering parameter measurements (Extended Data Fig. 4), as the networks are multiport, we connect the two ports of the VNA to two ports of the measured network, with the other network ports perfectly matched with 50-ohm terminations (no reflection). For the longer-range transport measurement shown in Extended Data Fig. 7, we connect ports 1 and 4 to the two VNA ports, while letting ports 2 and 3 be open (full reflection) and perfectly matching ports 5 and 6. For the field map measurements, we connect the signal input port of the measured network to VNA port 1, while perfectly matching the other ports of the network. We manually probe the field at the middle of the microstrip lines by using a coaxial probe, which is connected to VNA port 2, as shown in Extended Data Fig. 8b.

Validation of the model assumptions

The model is the one of a unitary scattering network, namely, lossless scatterers connected by links imparting phase delays. Microstrip transmission lines are known to behave as pure phase delays in this frequency range, since the propagation losses over so short distances are negligible (we indeed measured them to be 0.0167 dB cm−1). We are therefore left with checking that Supplementary equations (1)–(3) (see details in Supplementary Information part II) are a good model for the scatterers.

We start by checking the validity of the assumptions behind Supplementary equations (1)–(3), namely, that the scatterers have three-fold rotational symmetry (C3 symmetry), and that they are unitary. To do this, we measured the scattering matrix SM of our scatterers. We start with checking C3 symmetry, which implies that S12 = S23 = S31, as well as S11 = S22 = S33. Extended Data Fig. 3a plots the moduli and arguments of all these quantities in the considered frequency range. From these plots, we see that although some small deviations from C3 symmetry are observed in the reflection coefficients, they correspond to fluctuations of reflection below −20 dB. We conclude that C3 symmetry is a valid assumption.

Next, we check unitarity. Extended Data Fig. 3b plots the eigenvalues of the measured scattering matrix versus frequency, in the complex plane. We can see that they are always very close to the unit circle, meaning that unitarity is also a very reasonable assumption. This is expected since we used a substrate with a small loss tangent of 10−4 and circulators with low insertion losses of 0.2 dB. Absorption is therefore not expected to alter the prediction of the unitary theory, but simply to add an exponential decay which shows itself especially for large samples. For example, while long range transport from Geneva to Davos in the circulator network of Fig. 4b is associated with 20 dB of signal attenuation, the presence of the edge mode predicted by the unitary theory is not affected (see Extended Data Fig. 7).

Now, we estimate the error that we make by modelling the real matrix SM with Supplementary equations (1)–(3). To do this, we find the C3-symmetric unitary scattering matrix SU that is the closest to SM. We get SU by rescaling the eigenvalues of SM to make them exactly unitary, keeping their arguments. We then determine the parameters ξ and η of SU, which we plot against frequency in Extended Data Fig. 3c. We then define an S-parameter error metric as

$$\varepsilon ={\left[\frac{1}{3}({|{S}_{11}^{M}-{S}_{11}^{U}|}^{2}+{|{S}_{12}^{M}-{S}_{12}^{U}|}^{2}+{|{S}_{21}^{M}-{S}_{21}^{U}|}^{2})\right]}^{1/2}.$$

(5)

This quantity represents the error that we make by using Supplementary equations (1)–(3). It is plotted in Extended Data Fig. 3d. We see that this error is below 5% at all frequencies, which unambiguously validates the relevance of Supplementary equation (3).

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