Necklace beams carrying fractional angular momentum in fractional systems with a saturable nonlinearity

•The fractional system supports necklace beams with fractional angular momentum.•The expansion of necklace can be slow down by the decrease of Lévy index.•Energy transfers between pearls of necklace with fractional angular momentum.•Necklaces in fractional systems survive for a very long distance. We report the propagation dynamics of necklace beams in the framework of nonlinear fractional Schrödinger equation. The fractional diffraction can be partially compensated by a saturable nonlinearity and their combination leads to the quasi-stable propagation of necklace beams with integer or fractional angular momentum. The expansion of necklace can be slowed down remarkably by the decrease of Lévy index. There is an energy exchange between the neighboring pearls of necklace with fractional angular momentum, in sharp contrast to the necklace with integer or half-integer angular momentum. We, thus, put forward the first example of necklace-ring beams carrying fractional angular momentum in a fractional configuration.