Fig. 1. Topological non-reciprocal wave network and its bulk band structure.

a, We consider a unitary scattering network made of three-port non-reciprocal elements, described by asymmetric unitary scattering matrices. b, Unit cell of the honeycomb lattice, highlighting the signals entering the non-reciprocal elements, their 120° rotational symmetry, and the reciprocal phase delay φ imparted by the links. The network is described by a unitary unit-cell scattering operator S(k) defining a Floquet unitary mapping with quasi-energy φ. c, Evolution of the Floquet band structure on increasing the level of reflection of the non-reciprocal elements from |R| = 0.16 (leftmost panel, with angular parameter values ξ = −η = 2.5π/12) to |R| = 0.51 (rightmost panel, ξ = −η = 3.5π/12). While the type 1 bandgaps do not change much, at |R| = 1/3 (centre panel, ξ = −η = π/4), the type 2 bandgap closes, symptomatic of a topological phase transition.