Abstract
Robustness against disorder and defects is a pivotal advantage of topological systems^{1}, manifested by the absence of electronic backscattering in the quantumHall^{2} and spinHall effects^{3}, and by unidirectional waveguiding in their classical analogues^{4,5}. Twodimensional (2D) topological insulators^{4,5,6,7,8,9,10,11,12,13}, in particular, provide unprecedented opportunities in a variety of fields owing to their compact planar geometries, which are compatible with the fabrication technologies used in modern electronics and photonics. Among all 2D topological phases, Chern insulators^{14,15,16,17,18,19,20,21,22,23,24,25} are currently the most reliable designs owing to the genuine backscattering immunity of their nonreciprocal edge modes, brought via timereversal symmetry breaking. Yet such resistance to fabrication tolerances is limited to fluctuations of the same order of magnitude as their bandgap, limiting their resilience to small perturbations only. Here we investigate the robustness problem in a system where edge transmission can survive disorder levels with strengths arbitrarily larger than the bandgap—an anomalous nonreciprocal topological network. We explore the general conditions needed to obtain such an unusual effect in systems made of unitary threeport nonreciprocal scatterers connected by phase links, and establish the superior robustness of anomalous edge transmission modes over Chern ones to phaselink disorder of arbitrarily large values. We confirm experimentally the exceptional resilience of the anomalous phase, and demonstrate its operation in various arbitrarily shaped disordered multiport prototypes. Our results pave the way to efficient, arbitrary planar energy transport on 2D substrates for wave devices with full protection against large fabrication flaws or imperfections.
Main
Among the unique and counterintuitive attributes of topological systems, topological robustness^{1} against disorder and flaws is undoubtedly one of the most remarkable. This property shows substantial application potential by relaxing the tight constraints imparted by fabrication tolerances, and provides a way to route energy and information in a wide variety of 2D platforms^{4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27}, ranging from quantum electronics^{23} to classical photonic^{4,5} and phononic devices^{25,26,27}. Topological edge states were found in systems with broken timereversal symmetry, such as Chern insulators^{14,28}, and then extended to timereversal invariant scenarios, including the Z2 (ref. ^{3}) and other symmetryprotected schemes^{29}, simultaneously stimulating study of their classical analogues^{6,10,17}. So far, Chern topological edge modes^{14,15,16,17,18,19,20,21,22,23,24,25} undeniably represent the most reliable solution for pointtopoint energy guiding, as they provide truly unidirectional, backscatteringimmune wave transport at their boundaries. They have been reported in nonreciprocal artificial wave media, such as externally biased magnetophotonic crystals^{16} or mechanical systems^{13} with moving^{17,19,20,25} or timedependent^{8,24} elements. Albeit protected from the presence of local defects by the Chern number, the edge modes cannot survive the presence of distributed disorder of sufficiently large magnitude^{1,4,5,14}, especially when the average amplitude of frequency fluctuations gets larger than the bandgap size. This behaviour inherently confines the topological protection of Chern phases to small distributed disorder levels.
Here we demonstrate an anomalous nonreciprocal topological phase in which edge transmission is quantitatively stronger than for the Chern phase, surviving parametric fluctuations arbitrarily larger than the bandgap size. We find such anomalous robustness in unitary scattering networks made of interconnected nonreciprocal resonant scatterers coupled by nondirected phase links. We compare quantitatively the robustness of transmission through the anomalous and Chern channels to phaselink and scattering disorder, by statistical averaging over many disorder realizations. Our experiments at microwave frequencies confirm the superior resilience of the anomalous transmission channel over the Chern one. We apply our findings to the design of ideally robust networks with arbitrarily located ports and irregular shapes, including a perfect sixport circulator.
A nonreciprocal scattering network
Consider the nonreciprocal unitary scattering network of Fig. 1a, which consists of general threeport nonreciprocal scatterers connected by bidirectional links in a honeycomb periodic structure. The scattering elements exhibit threefold (C_{3}) rotational symmetry, while the links impart a phase delay of φ, as represented in the zoomedin view of the unit cell (Fig. 1b). The scattering process is described by a unitary 3 × 3 asymmetric scattering matrix S_{0} whose general parametrization involves only two angles, ξ and η, in the interval (−π/2, π/2) (see Supplementary Information and Extended Data Fig. 1). The wave propagation in the infinite network can be described by a Bloch eigenproblem, which considers the scattering at the nodes, described by a 6 × 6 unitary matrix S(k), and also involves the bidirectional phase delay φ induced by the links:
So far, topological unitary scattering wave networks^{6,30,31,32,33,34} have only been implemented in reciprocal systems^{7,35,36,37} exploiting two timereversed subspaces that are never genuinely decoupled. On the contrary, our nonreciprocal scattering network is formally analogous to a rigorously oriented kagome graph (see Supplementary Information), described by a unitary matrix^{33} S(k), which can be mapped^{38} onto the Floquet eigenproblem of a periodically driven lattice^{39,40,41,42,43,44,45}, with the angle variable φ taking the role of the quasienergy. Therefore, we can truly benefit from both the advantages of nonreciprocity^{46}, and the potentially richer topological physics of Floquet systems^{44}.
Chern and anomalous phases
We used the model of equation (1) to explore the parameters influencing potential topological phase transitions in the network. We found the individual reflection coefficient R of the nonreciprocal scatterers to be the main ‘control knob’ for the closing of the quasienergy bandgaps. The evolution of the bulk band structure with increasing values of R is shown in Fig. 1c. Our semianalytical model shows a systematic closing of two of the bandgaps at R = 1/3 (denoted type 2, in red) while the others (type 1, in blue) do not change much. This suggests that topological phase transitions may be controlled by the individual scatterer reflectance.
To confirm this intuition, we probe the existence of edge modes for each of these situations by numerically calculating the band structure of a ribbon terminated by fullreflection boundary conditions at top and bottom. As depicted in Fig. 2a, both the low and highreflection cases (respectively the leftmost and rightmost panels) exhibit chiral edge modes located at the walls either at the top (red line) or bottom (blue line), with profiles represented in Fig. 2b. The main difference is that the lowreflection case has edge modes in every quasienergy bandgap, whereas at high reflection, they are found only in bandgaps of type 1. This lowR behaviour is the hallmark of anomalous Floquet insulators^{33,35,42,45} (AFI), which possess topological edge states despite the Chern number of all surrounding bands being zero. In contrast, the highreflection case corresponds to the Chern insulator (CI). We map out in Fig. 2c the complete topological phase diagram for every possible realization of the scattering matrix S_{0}, represented by the angle parameters ξ and η. The CI and AFI regimes are shaded in red and blue, respectively. To connect this phase diagram with physically meaningful quantities, we plot it twice in the same parameter space, together with contour lines depicting the reflectance (Fig. 2c, left) and nonreciprocal isolation (Fig. 2c, right). Remarkably, the phase diagram unambiguously demonstrates the coincidence between the 1/3 reflection contours with the topological phase transition. Its centre corresponds to a semimetallic phase, with all bandgaps closed, whereas the green point is the perfect circulator case with R = 0 and infinite isolation, for which the bulk bands are flat and the edge modes are dispersionless (see Extended Data Fig. 2). Such a critical condition corresponds to a phase rotation symmetric point^{33}, which can only occur in the anomalous (or trivial) phases.
Robustness comparison
From the band structures of Fig. 2a, we can already intuitively expect the AFI edge transmission to be much more robust than the CI one to quasienergy fluctuations, even those much larger than the bandgap size. Indeed, the AFI phase occurs in the ballistic regime, in which reflections at nodes are low, yielding relatively flat (slow) bulk bands and large bandgaps. An abrupt jump of φ within the lattice is very likely to land in a bandgap, which necessarily carries an edge mode. Conversely, in the CI phase, the probability of an edge mode being destroyed by fluctuations larger than the bandgap is much higher, owing to the increased width of the bulk bands^{33} and the occurrence of trivial bandgaps. As an example of such a situation, let us consider the transport properties of edge modes in a finite nonreciprocal network with an abrupt quasienergy jump in the middle (Fig. 3a, right). As a reference, we also include the case of a uniform sample (Fig. 3a, left). The two hexagonalshaped networks have three input/output ports, as shown in the top row of Fig. 3a. Network 1 (N1) consists of uniformly distributed phase links φ = π/8, while for network 2 (N2), a quasienergy jump is introduced by changing all phase links in the bottom part to π/2. With numerical simulations, we then compare the propagation of the anomalous and Chern edge modes, when exciting from port 1. The anomalous phase finds itself in topological bandgaps at both φ = π/8 and π/2 (Fig. 2a, left), whereas the Chern phase possesses a nontrivial bandgap only at φ = π/8 (Fig. 2a, right). As shown in Fig. 3a, the anomalous edge mode crosses the interface completely unperturbed. In stark contrast, the Chern edge mode is unable to transmit to port 2 in the presence of the interface, and all the energy is guided to port 3.
We validate experimentally this fundamental distinction between the anomalous and Chern phases by designing a nonreciprocal network operating at microwave frequencies. The scatterers are ferrite circulators connected with microstrip lines. Our experimental design, which takes into account both the frequency dispersion of the scatterers and delay lines, finds itself in the anomalous and Chern phases at 4.9 GHz and 3.6 GHz, respectively. Modification of the phase delays of the links is induced by changing the total lengths of the microstrip lines with serpentine paths. As shown in Fig. 3b, the measured field amplitude profiles confirm the resilience of the anomalous edge mode to the phase jump, in perfect agreement with the numerical predictions. Further evidence is provided by the measured changes in scattering parameters and field maps upon exciting ports 2 and 3 (Extended Data Figs. 3, 4d, e, 5 and 6).
The resilience of the anomalous edge transport in these interface scenarios, involving two periodic networks, raises the question of its generalization to nonperiodic quasienergy perturbations. To answer quantitatively, we consider the same hexagonal network as in the left of Fig. 3a, and impose sitedependent disorder on the phase links, with fluctuations of strength δ_{φ} randomly drawn with uniform probability in the interval π/8 + [−δ_{φ}/2, δ_{φ}/2]. We then numerically extract the transmission from ports 1 to 2 for 1,000 realizations of disorder, and plot its magnitude versus δ_{φ} in the top panel of Fig. 3c. The solid lines represent the ensemble average, and the dashed lines are the first and last quartiles (Q1 and Q3). In the clean limit (δ_{φ} = 0), both AFI and CI phases show high transmission, since the edge states exist in both cases and are unperturbed. We now turn on the disorder, up to the maximal possible strength, which corresponds to randomly drawn values in the entire 2π quasienergy range, much larger than the bandgap size of both AFI and CI phases (roughly π/4). Upon increasing δ_{φ}, the average transmission in the Chern case quickly drops to low values. Remarkably, the AFI transmission shows a markedly different behaviour, remaining near 90% even when δ_{φ} reaches 2π (fully random case). Note that this exceptional robustness does not require the critical condition R = 0 to be reached, since the figure is generated for R = 16%. Such statistically stable transmission constitutes solid evidence of the superiority of anomalous nonreciprocal topological networks, which survive phase disorder levels arbitrarily larger than their bandgap size. We also consider the other possible source of disorder, namely the scattering matrices of the nodes, which we pick randomly within the Chern or anomalous phases, fixing φ = π/8. The transmission statistics are shown in the bottom panel of Fig. 3c. We see that the anomalous transmission can tolerate 100% disorder in the choice of scattering matrices, whereas the Chern one falls after 25%. The reason for this surprising behaviour is that in a disordered Chern phase (random R > 1/3), transmission is mediated by both bulk and edge modes, but is blocked by trivial gaps, whereas in the anomalous case (random R < 1/3), those trivial gaps are absent (see Supplementary Fig. 8). This shows that the superior robustness of the anomalous phase is not restricted to phaselink disorder, but also to the other possible source of disorder: fluctuations of the scattering matrices.
We validate the resilience of the anomalous transmission by performing experiments on irregularly shaped disordered networks. First, we demonstrate the use of anomalous phases in a practical scenario, where an anomalous nonreciprocal topological network is used to create a robust sixport circulator with arbitrary shape and port locations. The prototype is shaped like Switzerland, and we place six ports at the locations of six boundary cities (Fig. 4a). We aim at connecting each city to its clockwise closest neighbour, with strong nonreciprocal isolation to any other city. A picture of the fabricated prototype is shown in Fig. 4b. We sequentially excite each input of this sixport nonreciprocal network, and report the measured experimental field maps in the AFI band (Fig. 4c). Despite the presence of finite fabrication tolerances, such as the inaccuracy in the surface mounting process of the elements, and shrinking effects due to the employed reflow oven method, and regardless of the tortuous shape of the border, we observe the expected clockwise nonreciprocal circulation of the energy, consistent with simulations (Extended Data Fig. 7c). Such robustness is also observed in longerrange transmission tests between ports 1 and 4 (Extended Data Fig. 7a and b). Second, we provide an experimental validation of the superiority of the anomalous transmission in the presence of fully random phase delays. We built five different prototypes, one of them shown in Fig. 4d, with phase fluctuations in a 2π range implemented via serpentine links. The measured field maps in the AFI and CI phases show that only the anomalous channel survives such strong distributed perturbations, consistent with our statistical studies.
Conclusion and outlook
We envision that such anomalous wave platforms may be used in a new generation of multipleinput multipleoutput devices, capable of reaching an unprecedented level of robustness. Since individual reflection is the sole ‘control knob’ for the transition from the CI to the AFI phase, one could foresee very practical ways to reconfigure a domain wall between the two phases—for example, by simply changing the matching of the scatterers—without the need for flipping a magnetic field. Our tabletop experiment, compatible with standard printed circuit board microwave technologies and offtheshelf surface mount components, provides genuine nonreciprocity and large robustness, not only to local defects, but also to distributed imperfections. This opens an avenue to a new generation of wave systems^{47} that can provide reconfigurable pointtopoint unidirectional energy guiding, with arbitrary control over the imparted phase delays and full immunity against backscattering. Finally, exploration of the interplay between anomalous nonreciprocal networks and nonHermitian perturbations (such as radiation losses occurring when coupling the edge mode to the freespace continuum) represents a promising future opportunity for topologically controlled radiation patterns in applications such as multiple beam antennas for 5G communications.
Methods
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 (quasienergy) spectrum being defined on a circle, each (quasienergy) band is now allowed to be connected to the next one by an edge state^{42}. 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 π_{3}(U(N)) = ℤ, whose elements are the topological numbers
The power 3 must be understood in the language of differential forms, and the integral runs over a 3torus, spanned by the quasimomentum k = (k_{x}, k_{y}) 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, Tperiodic in time) evolution operator U(t,k), such that an interpolation parameter that formally plays the role of time can be introduced^{33}. Finally, the operator V_{ψ} is a periodized (in time) evolution operator. For Floquet systems, it reads as^{42}.
with
where −ψ denotes the branchcut of the logarithm. The procedure to define such an operator V_{ψ} and thus the invariant W_{ψ} for discretetime 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 study^{33} (in particular in sections V.A. and V.B.).
Importantly, the branchcut ψ must be chosen in a spectral gap of U(T, k), or S(k) in our case. For this reason, W_{ψ} is said to be a gap invariant, and indeed directly gives the number of chiral edge states in a given quasienergy gap ψ. In contrast, Chern numbers are band invariants. They are inferred from the eigenstates of H_{eff}(k) expressed in equation (4) and thus cannot capture the full unitary evolution. Finally, the details for the calculation of the invariants W_{ψ} 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 nonreciprocal honeycomb networks is based on the scattering matrix method. For a finite nonreciprocal network with N_{r} input/output ports, once we have the information of the scattering matrix of each nonreciprocal element and the distribution of the phase delays of the links, this method can provide (i) the scattering matrix S_{Nr} regarding the N_{r} port system, and (ii) the field map across the network knowing the excitations at the N_{r} ports (see details in Supplementary Information part II).
We exemplify this method by calculating the transmission between ‘Geneva’ and ‘Davos’ through the Switzerlandshaped 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 nonreciprocal elements (in anomalous or Chern phase) in the Switzerlandshaped 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 nonreciprocal 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 nonreciprocal element is a surface mount microwave circulator (UIYSC9B55T6, UIY Co.), designed from a ‘Y’shaped strip line on a printed circuit board^{48}. The three ports are placed 120° apart from each other such that they are isospaced. The printed circuit board is sandwiched between two pieces of ferrite. Without magnetic fields, the ‘Y’junction strip line supports two degenerate modes at ω_{0}: righthanded and lefthanded. 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 S_{0}. 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 formula^{49}. 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 L_{1} = 26.5 mm and L_{2} = 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 L_{1} (blue dashed region) and L_{2} (red dashed region).
Measurements
The scattering parameters and field maps of three fabricated networks (network 1, network 2 and the Switzerlandshaped 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 50ohm terminations (no reflection). For the longerrange 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 threefold rotational symmetry (C3 symmetry), and that they are unitary. To do this, we measured the scattering matrix S^{M} of our scatterers. We start with checking C3 symmetry, which implies that S_{12} = S_{23} = S_{31}, as well as S_{11} = S_{22} = S_{33}. 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 S^{M} with Supplementary equations (1)–(3). To do this, we find the C3symmetric unitary scattering matrix S^{U} that is the closest to S^{M}. We get S^{U} by rescaling the eigenvalues of S^{M} to make them exactly unitary, keeping their arguments. We then determine the parameters ξ and η of S^{U}, which we plot against frequency in Extended Data Fig. 3c. We then define an Sparameter error metric as
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).
Data availability
The data that support the findings of this study are available at https://doi.org/10.5281/zenodo.5101825.
Code availability
The codes that support the findings of this study are available at https://doi.org/10.5281/zenodo.5101825.
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Acknowledgements
Z.Z. and R.F. acknowledge funding from the Swiss National Science Foundation under the Eccellenza award number 181232. P.D. acknowledges support from the IDEX Lyon Breakthrough programme ToRe (contract no.ANR16IDEX0005).
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Z.Z. performed the numerical simulations and experiments, under the supervision of R.F. P.D. and R.F. conceived the project. All authors participated in writing and revising the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Detailed schematic of the unit cell of the nonreciprocal network and signal labelling convention.
We define three state vectors: a >, b >, and c >, which represent scattering wave amplitudes propagating out, between and into the nonreciprocal elements, respectively. The total phase delay between two scatterers is φ.
Extended Data Fig. 2 Floquet band structures at two special points of the topological phase diagram.
a, b, Bulk band structures at the green (a) and centre (b) points of the phase diagram of Fig. 2c in the main text. The green point corresponds to a phaserotation symmetric network of perfect matched circulators, thus in AFI phase. The red centre point represents a network of reciprocal resonant scatterers, with all bandgaps closed. c, d, Ribbon band structures corresponding to panel a and b, respectively. The perfect circulator network features flat bulk band with dispersionless edge modes regardless of the value of the quasienergy φ, which can only occur in the AFI phase.
Extended Data Fig. 3 Experimental validation of the model assumptions.
a, C3 symmetry holds when S_{12} = S_{23} = S_{31} as well as S_{11} = S_{22} = S_{33}, which is very well satisfied in the considered frequency range. b, Eigenvalues of the measured scattering matrix, with nearlyunitary behaviour over the entire experimental bandwidth. c, ξ and η parameters used to approximate the real scattering matrix with a C3symmetric unitary matrix. The red area is the Chern phase, and the blue area the anomalous one. d, Error in % made by approximating the real scattering matrix with equation (4) over the entire bandwidth.
Extended Data Fig. 4 Experimental network design and measured scattering parameters.
a, Measured reflection spectrum of an individual ferrite circulator. The blueshaded area represents the bandwidth of the anomalous phase, corresponding to low reflection (R < −9.5 dB = 20∙log_{10}(1/3)). By contrast, the redshaded area shows the Chern phase with high reflection (R > −9.5 dB). Topological phase transitions happen at around 3.9 GHz and 7 GHz. b, Topological bandgap map predicted from the individual scattering data, when varying the length of the microstrip connections and the operating frequency. The blue and red regions correspond to bandgaps with and without topological edge modes, respectively. The white regions represent bulk bands. c, Design details of the experimental networks probed in Fig. 3b of the main text. Network 1 (N1) has a uniform length distribution of microstrip lines with L = L1. For network 2 (N2), we introduce an interface and replace the bottom part with lines of different length L2. d, Measured amplitudes of the scattering parameters S_{21} (left), S_{31} (middle) and S_{22} (right) in the Chernphase frequency band (green dashed box in panel b). e, Measured scattering parameters in the anomalousphase frequency band (yellow dashed box in panel b).
Extended Data Fig. 5 Numerical and experimental field maps for excitation at port 2.
a, Numerical predictions for excitation at port 2 for the same system as in Fig. 3 of the main text. While the anomalous phase supports transmission to port 3 regardless of the phase link distribution, the Chern phase possesses a trivial bandgap at φ = π/2, and reflects all the energy incident from port 2, see bottom right plot (the field distribution exhibits exponential decay). b, Corresponding experimental data.
Extended Data Fig. 6 Numerical and experimental field maps for excitation at port 3.
a, Numerical predictions for excitation at port 3 for the same system as in Fig. 3 of the main text. Both the anomalous and Chern phases fall in topological bandgap at φ = π/2, leading to transmission to port 1. b, Corresponding experimental data.
Extended Data Fig. 7 Additional field maps for the anomalous topological Switzerlandshaped network.
Extended Data Fig. 8 Experimental setups for scattering parameter and field distribution measurements.
a, The setup consists of a vector network analyser (VNA) and three microwave nonreciprocal networks: the Switzerlandshaped network (left), N1 (middle), and N2 (right). b, Field map measurement with a coaxial probe for measuring fields on the microstrip lines.
Extended Data Fig. 9 Experimental validation of anomalous phase disorder robustness in four other prototypes with distinct disorder realizations.
a, Pictures of the prototypes, having the same irregular shape but different phase delay distributions implemented by varying the geometry of the serpentine links. b, Measured field maps in the AFI phase. c, Measured field maps in the CI phase.
Supplementary information
Supplementary Information
This file contains six Supplementary Discussion sections I–VI, Supplementary Figures 1–11 and Supplementary References. The six Supplementary Discussion Sections include: I. Theoretical modelling; II. Simulation method of arbitrary finite nonreciprocal networks; III. Examples of disorder realisations for phaselink delays and scattering matrices; IV. Discussion of the measured amplitudes of the scattering parameters in Extended Data Fig. 4; V. Mapping of the honeycomb lattice to an oriented graph and VI. Robustness comparisons with other Chern phases.
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Zhang, Z., Delplace, P. & Fleury, R. Superior robustness of anomalous nonreciprocal topological edge states. Nature 598, 293–297 (2021). https://doi.org/10.1038/s41586021038687
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DOI: https://doi.org/10.1038/s41586021038687
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