Nonchiral edge states and cascading of exceptional points in the nonreciprocal Haldane model

We study a variant of the Haldane honeycomb model that has non-reciprocal hoppings between the next-nearest neighbours. The system on a torus hosts time-reversal symmetry protected exceptional rings(ER) in the spectrum. The ERs act as Berry-curvature flux tubes i.e the Berry curvature is non-zero only inside the ERs. The system on a cylinder having zig-zag boundaries (and transverse momentum $k_x$) hosts edge-states that have zero group velocity at $k_x=π$ and are therefore `non-chiral'. The edge states undergo a bifurcation transition at an exceptional point(EP)in the BZ and delocalise into the bulk. As the non-reciprocity is increased, the bulk states that are approaching each other are converted into pairs of EPs due to non-Hermiticity. As the non-reciprocity is further increased, there is a `Russian doll'-like nested proliferation of pairs of EPs, leading to an EP-cascade. The proliferation of EPs takes place only at specific values of the non-hermiticity parameter, leading to a step-like structure in the EP-pair density when plotted as a function of non-Hermiticity. Further, using wave packet dynamics, we find a tunable regime where the non-chiral edge states can be dynamically stabilised for large timescales. The `self-acceleration' term in the equations of motion tends to diffuse the wave packets into the bulk, thus making them `ephemeral edge states'. But we find that for small non-hermiticity, the edge localisation is stabilised until late times for sufficiently wider wave packets. Thus, we have brought forth an intriguing phenomenology of the exceptional phase of the non-reciprocal Haldane model, which may bear direct relevance for systems such as disordered Kitaev honeycomb model, wherein such ERs have been predicted.