These authors contributed equally to this work. S1: Nanoskyrmion lattice solution to the Heisenberg modelSince the spin-spiral state is the exact solution of the classical Heisenberg model for a periodic arrangement of atoms, it is an excellent starting point to expand the solution due to further but smaller interactions such as the four-spin interaction. Therefore, we consider magnetic structures obtained from the superposition of spin spirals. These lead in general to two-dimensional magnetic structures.In order to be in accordance with the experimentally observed magnetic unit cell, the superposition or multi-Q state must include the Q-vectors determined experimentally from the Fourier transform of the spin-polarized STM image (Fig. 1c). In general, it is impossible to find a superposition state for two arbitrary but symmetry equivalent Q-vectors � and � in the 2D Brillouin zone, if we require the magnetic moment to be constant at all lattice sites, i.e. |� , in order to prevent a large loss of

A topologically ordered material is characterized by a rare quantum organization of electrons that evades the conventional spontaneously broken symmetry-based classification of condensed matter. Exotic spin-transport phenomena, such as the dissipationless quantum spin Hall effect, have been speculated to originate from a topological order whose identification requires a spin-sensitive measurement, which does not exist to this date in any system. Using Mott polarimetry, we probed the spin degrees of freedom and demonstrated that topological quantum numbers are completely determined from spin texture-imaging measurements. Applying this method to Sb and Bi(1-x)Sb(x), we identified the origin of its topological order and unusual chiral properties. These results taken together constitute the first observation of surface electrons collectively carrying a topological quantum Berry's phase and definite spin chirality, which are the key electronic properties component for realizing topological quantum computing bits with intrinsic spin Hall-like topological phenomena.