  A topological insulator (TI) nanowire (NW), where the core is insulating and
the 2D spin-helical Dirac fermion topological surface states (TSS) are
circumferentially quantized into a series of 1D sub-bands, promises novel
topological physics and applications. An axial magnetic flux ({\Phi}) through
the core drives periodic topological transitions in the surface sub-bands,
changing from being all doubly-degenerate with a gapped Dirac point (DP) at
integer (including zero) flux quanta ({\Phi}$_0$ = h/e, with h being the Planck
constant and e the electron charge), to contain a topologically-protected,
non-degenerate 1D spin helical mode with restored DP at half-integer flux
quanta. The resulting magnetoconductance is predicted to exhibit Aharonov-Bohm
oscillations (ABO) with maxima occurring alternatively at half-integer or
integer flux quanta (referred to as {\pi}-ABO or 0-ABO), depending periodically
on the Fermi wavevector (k$_F$, with period 2{\pi}/C, C being the NW
circumference). Here, we report a clear observation of such k$_F$-periodic
alternations between 0-ABO and {\pi}-ABO in Bi$_2$Te$_3$ TI nanoribbon (NR, a
rectangular cross sectional NW) field effect devices, which exhibit
quasi-ballistic transport over ~2 {\mu}m (as manifested in length-independent
conductance, exponential decaying ABO amplitude with increasing temperature
(T), and an 1/T dependence of the extracted phase coherence length). The
conductances as functions of the gate voltage at half and zero flux quanta also
exhibit clear, but anti-correlated oscillations periodic in k$_F$ (with period
2{\pi}/C, equivalently when C encloses an integer multiples of Fermi wavelength
2{\pi}/k$_F$), consistent with the circumferentially quantized surface
sub-bands. We also extract the minimal Fermi energy and momentum for TSS to
emerge out of the bulk valence band, in agreement with the known Bi$_2$Te$_3$
bandstructure.
