Double-hump solitons in fractional dimensions with a 𝒫𝒯-symmetric potential.

We investigate the properties of double-hump solitons supported by the nonlinear Schrödinger equation featuring a combination of parity-time symmetry and fractional-order diffraction effect. Two classes of nonlinear states, i.e., out-of-phase and in-phase solitons are found. Each class contains two families of solitons originating from the same linear mode in both focusing and defocusing nonlinear Kerr media. The critical phase-transition point increases monotonously with increasing Lévy index. For strong gain and loss, out-of-phase solitons in focusing media are stable in a wide parameter window and are almost completely unstable in media with a defocusing nonlinearity. The stability of in-phase solitons is opposite to that of out-of-phase solitons. In-phase solitons in defocusing media are stable in their entire existence domains provided that the gain-loss strength is below a critical value. Meanwhile, the stability region shrinks with the decrease of Lévy index. We, thus, put forward the first example of spatial solitons in fractional dimensions with a parity-time symmetry.