  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 novel topological order whose identification
requires a spin sensitive measurement. Using Spin-resolved-ARPES, we probe the
spin degrees of freedom and demonstrate that topological quantum numbers are
uniquely determined from spin-texture Berry Phase imaging measurements.
Applying this method to pure antimony (Sb) and Bi-Sb, we identify the origin of
its novel Topological Order and the negative value of the mirror Chern number.
These results taken together constitute the first observation of surface
electrons collectively carrying a topological Berry's phase and definite mirror
Chern chirality in pure Antimony (Sb) which are the key electronic properties
for realizing topological quantum computing via the interface Majorana fermion
framework. This paper contains the details of the above mentioned previously
reported (Science \textbf{323}, 919 (2009)) results.
