  We study a vortex chain in a thin film of a topological insulator with
proximity-induced superconductivity---a promising platform to realize Majorana
zero modes (MZMs)---by modeling it as a two-leg Majorana ladder. While each
pair of MZMs hybridizes through vortex tunneling, we hereby show that MZMs can
be stabilized on the ends of the ladder with the presence of tilted external
magnetic field and four-Majorana interaction. Furthermore, a fruitful phase
diagram is obtained by controlling the direction of magnetic field and the
thickness of the sample. We reveal many-body Majorana states and
interaction-induced topological phase transitions and also identify
trivial-superconducting and commensurate/incommensurate charge-density-wave
states in the phase diagram.
