Photoemission experiments have shown that Bi$_2$Se$_3$ is a topological insulator. By controlled doping, we have obtained crystals of Bi$_2$Se$_3$ with non-metallic conduction. At low temperatures, we uncover a novel type of magnetofingerprint signal which involves the spin degrees of freedom. Given the mm-sized crystals, the observed amplitude is 200-500$times$ larger than expected from universal conductance fluctuations. The results point to very long phase breaking lengths in an unusual conductance channel in these non-metallic samples. We discuss the nature of the in-gap conducting states and their relation to the topological surface states.
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