Fig. 2: Topological invariant for the dynamic matrix, and intuitive explanation for the gain.

Under PBC, the eigenvalues of the dynamic matrix M(ω), Eq. (7), describe a closed curve h(k) (red) in the complex plane—the generating function (10). This allows us to define the winding number ν of Eq. (11) counting the revolutions of h(k) around the origin. a On resonance, ω = 0, the non-local dissipation has to surpass the local dissipation to yield a non-trivial winding number, see Eq. (14). Otherwise, b ν is trivial. This competition between local and non-local contributions in the generating function is indicated by the purple and blue arrows. For θ = 0 or π, h(k) degenerates into a line, which is shown for θ = 0 in a and b as dashed lines. When θ = π the slope changes sign. In a \(\theta =\frac{\pi }{2}\) and in b θ = 0.5. c Under PBC with ν ≠ 0, excitations travel directionally around the ring and gain energy at each revolution causing instability. d Removing one link (OBC) leads to the accumulation of excitations at one end, which determines the end-to-end gain. e For reciprocal dynamics, removing the link only induces local changes and no gain.