Green's functions of and emission into discrete anisotropic and hyperbolic baths

In this work, we study wave propagation in generic Hermitian local periodic baths, and investigate the effects of anisotropy and quasi-breaking of periodicity on resonant emission into the band of the bath. We asymptotically decompose the Green's function into long-range travelling waves composed of all wavevectors (near-)resonant at the emitter frequency, and rapidly decaying evanescent waves. Our approximation then converges exponentially with increasing source-receiver separation ${\rho}$ when resonant wavepackets with group velocity parallel to ${\rho}$ exist. In hyperbolic media this condition may not be satisfied, and we find that the exponential decay length of oscillating evanescent waves locally around caustics generally depends as a power law with exponent 3/2 on the angle made between ${\rho}$ and the caustic. For ${\rho}$ beyond the caustic we observe that the Green's function can become almost imaginary, which results in exclusively incoherent emitter-emitter interactions and allows the simulation of purely dissipative systems with short-range interactions. Here the interaction length is tunable via the separation vector of the emitters. We finally probe the hyperbolic dispersion beyond the previous regimes by applying an artificial gauge field on the lattice. We find that emission resonant with the corresponding open orbits in the Brillouin zone is quasi-one dimensional, in contrast to an isotropic environment. The quasi-1D emission is further topologically protected against local and global lattice perturbations and periodically refocussing, offering a robust bi-directional transport of excitations in higher-dimensional media.