Versatile Braiding of Non-Hermitian Topological Edge States

Among the most intriguing features of non-Hermitian (NH) systems is the ability of complex energies to form braids under parametric variation. Several braiding behaviors, including link and knot formation, have been observed in experiments on synthetic NH systems, such as looped optical fibers. The exact conditions for these phenomena remain unsettled, but existing demonstrations have involved long-range nonreciprocal hoppings, which are hard to implement on many experimental platforms. Here, we present a route to realizing complex energy braids using 1D NH Aubry-Andr\'e-Harper lattices. Under purely local gain and loss modulation, the eigenstates exhibit a variety of braiding behaviors, including unknots, Hopf links, trefoil knots, Solomon links and catenanes. We show how these are created by the interplay between non-Hermiticity and the lattice's bulk states and topological edge states. The transitions between different braids are marked by changes in the global Berry phase of the NH lattice.