Magnetic breakdown refers to the tunneling of electrons between distinct orbitals beyond a critical magnetic field. In the past, studies on magnetic breakdown have largely focused on the semiclassical regime, by detecting the interference difference frequencies (usually designated as β − α) in the Shubnikov-de Haas or de Haas-van Alphen oscillations. Recently, researchers derived quantization rules under magnetic breakdown through matching WKB functions across regions of strong quantum interference. A host of topological solids were found to inevitably undergo magnetic breakdown and exhibit a quasi-random spectrum.

In this thesis I report the observation of magnetic breakdown in square net layered charge-density-wave (CDW) topological material GdTe_{3} under hydrostatic pressure. The high mobilities of electrons in symmetry-protected pockets lead to sharp quantum oscillations in the magnetoresistance (MR) of GdTe_{3}. The applied pressure causes FS reconstruction and the appearance of a second CDW. For a narrow pressure region from 16.8 kbar to 21.6 kbar, we observe Shubnikov-de Haas oscillations that appear chaotic and quasi-random. The FFT spectrum of the quantum oscillations in the chaotic region is found to be “continuous”.

In the rest of the thesis I explore three other topological chalcogenides: CeSb_{x}Te_{2−x−δ}, ZrTe_{5}, and Pb_{1−x}Sn_{x}Se/Pb_{1−x}Sn_{x}Te. CeSb_{x}Te_{2−x−δ} exhibits Dirac fermions which are protected by non-symmorphic symmetries that can survive under CDW. Here we study the magnetic and transport properties in several CeSb_{x}Te_{2−x−δ} compounds. Researchers have been exploring whether ZrTe_{5} is a Weyl semimetal or a topological insulator. We find that the properties of ZrTe_{5} vary with the growing conditions. In one batch of ZrTe_{5} we observe negative MR and anomalous Hall effect which are signs of a Weyl semimetallic phase and in another we see 2D quantum oscillations and RT/MR behaviors under pressure that suggest a TI phase. In the last part of the thesis I report the thermoelectric properties and quantum phase transitions in topological crystalline insulators Pb_{1−x}Sn_{x}Se/Pb_{1−x}Sn_{x}Te.