  Topological insulators have been studied intensively over the last decades.
Among these materials, three-dimensional (3D) zirconium pentatelluride
(ZrTe$_5$) stands out as one of the most intriguing for both theoretical and
experimental studies because of its diverse range of distinct topological
phases. In this work, we employ density functional theory to study the
electronic structure and quantum oscillations exhibited by various topological
phases of 3D bulk ZrTe$_5$. We have discovered that by analyzing combined
patterns in band structures, Fermi surfaces, and Shubnikov-de Haas (SdH)
oscillations we can determine the corresponding topological phase without
relying on the conventional calculation of topological invariants or boundary
state contributions. This approach facilitates the identification of
topological phases in ZrTe$_5$ directly from experimental quantum oscillation
measurements. Using this method, we have analyzed the entire process of
topological phase transition, revealing changes in the topology of the Fermi
pockets and validating the shapes deduced from the experimental data for the
topological phases.
