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Signatures of Majorana fermion bound states in one-dimensional topologicalinsulator (TI) nanowires with proximity effect induced superconductivity arestudied. The phase diagram and energy spectra are calculated for single TInanowires and it is shown that the nanowires can be in the topologicalinvariant phases of winding numbers $W=0, \pm 1$, and $\pm 2$ corresponding tothe cases with zero, one and two pairs of Majorana fermions in the single TInanowires. It is also shown that the topological winding numbers, i.e., thenumbers of pairs of Majorana fermions in the TI nanowires can be extracted fromthe transport measurements of a Josephson junction device made from two TInanowires, while the sign in the winding numbers can be extracted using asuperconducting quantum interference device (SQUID) setup.

Majorana fermions in topological insulator nanowires: from single superconducting nanowires to Josephson junctions