Spontaneous symmetry breaking, a spontaneous course of breaking the spatial symmetry (parity) of the system, is known to exist in many branches of physics, including condensed-matter physics, high-energy physics, nonlinear optics, and Bose-Einstein condensates. In recent years, the spontaneous symmetry breaking of solitons in nonlinear wave systems is broadly studied; understanding such a phenomenon in nonlinear fractional quantum mechanics with space fractional derivatives (the purely nonlinear fractional systems whose fundamental properties are governed by a nonlinear fractional Schrodinger equation), however, remains pending. Here, we survey symmetry breaking of solitons in fractional systems (with the fractional diffraction order being formulated by the Levy index alpha) of a nonlinear double-well structure and find several kinds of soliton families in the forms of symmetric and anti-symmetric soliton states as well as asymmetric states. Linear stability and dynamical properties of these soliton states are explored relying on linear-stability analysis and direct perturbed simulations, with which the existence and stability regions of all the soliton families in the respective physical parameter space are identified.