  We study the Ruderman-Kittel-Kasuya-Yosida interaction between two magnetic
impurities connected to the edges of zigzag-terminated MoS$_{2}$ flakes. When
the impurities lie on the edges of the flake, the effective exchange
interaction exhibits sizable noncollinear Dzyaloshinskii-Moriya character that
competes with a strong Ising coupling. We analyze the characteristic decay
exponent for doping levels inside the band gap of the infinite layer,
corresponding to edge states of the flake at the Fermi level. The
characteristic exponents show sub-two-dimensional (sub-2D) behavior for these
band fillings, with decays much slower than quadratic. The Ising interaction
has effectively one-dimensional (1D) long range, while the noncollinear
component that grows for short impurity separation becomes comparable in
magnitude. The resulting tunable exchange interaction on these systems opens
the way for the study of interesting phases of impurity arrays with long-range
stable helical order.
