  The ability to tailor the hopping interactions between the constituent
elements of a physical system could enable the observation of unusual phenomena
that are otherwise inaccessible in standard settings. In this regard, a number
of recent theoretical studies have indicated that an asymmetry in either the
short- or long-range complex exchange constants can lead to counterintuitive
effects, for example, the possibility of a Kramer's degeneracy even in the
absence of spin 1/2 or the breakdown of the bulk-boundary correspondence. Here,
we show how such tailored asymmetric interactions can be realized in photonic
integrated platforms by exploiting non-Hermitian concepts, enabling a class of
topological behaviors induced by optical gain. As a demonstration, we implement
the Haldane model, a canonical lattice that relies on asymmetric long-range
hopping in order to exhibit quantum Hall behavior without a net external
magnetic flux. The topological response observed in this lattice is a result of
gain and vanishes in a passive but otherwise identical structure. Our findings
not only enable the realization of a wide class of non-trivial phenomena
associated with tailored interactions, but also opens up avenues to study the
role of gain and nonlinearity in topological systems in the presence of quantum
noise.
