k -point, it is meaningful to assume some perturbation, such as alloying with Bi [27] that does not significantly alter the spin splitting [Fig. 5], that pushes the bulk va- lence H and conduction L bands completely below and above E$_{F}$ respectively without changing its Z$_{2}$ class. Un- der such an operation, it is clear that the spin polarized surface bands must traverse E$_{F}$ an odd number of times between ¯ Γ and ¯ M, consistent with the 1;(111) topological classification of Sb. This conclusion can also be reached by noticing that the spin-split pair of surface bands that emerge from ¯ Γ do not recombine at ¯ M, indicative of a “partner switching” [9] characteristic of topological insu- lators. 

In conclusion, we have mapped the spin structure of the surface bands of Sb(111) and shown that the purely surface bands located in the projected bulk gap are spin 

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