Fig. 3: Topological ‘phase diagram’ of the scattering matrix.

a Gain \({\mathcal{G}}(0)\) and reverse gain \(\bar{{\mathcal{G}}}(0)\), see Eqs. (5) and (6), respectively, for N = 10 (solid line), N = 15 (dashed), N = 20 (dotted), all for \(\theta =\frac{\pi }{2},\frac{3\pi }{2}\), and b general topological ‘phase diagram’ on resonance, ω = 0, with distinct winding numbers according to Eq. (11). We can associate a scattering matrix S(0) with each point in the diagram and we show some ∣S(0)∣² as inset with Λ =  2 and γ = 2Γ − κ in Eq. (9) to obtain impedance matching at the exceptional point (EP). Note in particular the color scales of the scattering matrices revealing the amplification, and the asymmetry of the matrix signifying non-reciprocity. Condition (15) yields the orange lobes in b and corresponds to winding numbers ν = ±1, whereas the rest is the trivial regime ν = 0. Directional amplification, i.e., \({\mathcal{G}}\;> \;1\), sets in as we move into a topologically non-trivial regime. For the parameters shown in a this occurs at \({\mathcal{C}}=1\). In this regime, the gain grows exponentially with N. At the EP the transmission in the reverse direction is completely suppressed, i.e., \(\bar{{\mathcal{G}}}=0\), and the upper (lower) triangle of S(0) is exactly zero. The system becomes unstable (gray overlay), when \(\mathop{\max }\nolimits_{m}{\rm{Re}}\ {\lambda }_{m}\;> \;0\), in which λ_m is the mth eigenvalue of the dynamic matrix M_obc(0), see Eqs. (3) and (47). \({\rm{Re}}\ {\lambda }_{m}=0\) coincides with the onset of the parametric instability and can be seen as divergence in the gain in a. Non-reciprocity also occurs outside of non-trivial topological regimes and is governed by the phase θ. Complete directionality is achieved at \(\theta =\frac{\pi }{2}\) for ν = −1 from left to right (\(\frac{3\pi }{2}\) for ν = +1 from right to left). While the gain only depends weakly on Λ, larger (smaller) Λ shifts the location of the EP to the right (left) and extends (shrinks) the stable regime. c The number of zeros inside the unit circle determines the winding number. On the boundary between trivial and non-trivial regimes, one of the zeros lies on the unit circle and hence \({\mathcal{G}}={\mathcal{O}}(1)\) independent of N, see (a) at \({\mathcal{C}}=1\). d Off-resonance, ω ≠ 0 shifts the two lobes corresponding to non-trivial topological regimes ν = ±1 towards each other. Where they overlap, we obtain a trivial regime.