  The topological phase transition is theoretically studied in a metallic
single-wall carbon nanotube (SWNT) by applying a magnetic field $B$ parallel to
the tube. The $\mathbb{Z}$ topological invariant, winding number, is changed
discontinuously when a small band gap is closed at a critical value of $B$,
which can be observed as a change in the number of edge states owing to the
bulk-edge correspondence. This is confirmed by numerical calculations for
finite SWNTs of $\sim$ 1 $\mu$m length, using a one-dimensional lattice model
to effectively describe the mixing between $\sigma$ and $\pi$ orbitals and
spin-orbit interaction, which are relevant to the formation of the band gap in
metallic SWNTs.
