“There are more stars in the universe than grains of sand on earth.” The same applies to the electrons in a single grain of sand. An over-arching task of condensed matter physics is to uncover the hidden mechanisms that organize these countless billions of electrons.

In electronic systems, both correlations and topology prove to be significant drivers in generating novel physical phenomena. In this thesis, we study how they interplay with each other in gapless systems. Two types of materials are on our focus: (1) the heavy fermion systems and (2) the geometry-induced flat band systems.

In the study of correlated topology in heavy fermion systems, we explore how the space group symmetry constrains the electronic topology of the emergent composite quasiparticles in both nonmagnetic and magnetic settings. Furthermore, we consider the topological nodes in the new setting of non-Fermi liquid phases, where the Landau quasiparticles are completely destroyed. We successfully identify gapless topological states that do not have free-electron counterparts and discuss their physical properties.

In the study of geometry-induced flat band systems, we have successfully connected them with heavy fermion metals. This link is supported by the identification of a single emergent compact molecular orbital which is capable of representing the flat band. This emergent molecular orbital, with a small kinetic energy, serves a role akin to the f-electrons in heavy fermion systems. From this perspective, we first demonstrate the emergence of flat bands that are pinned to the Fermi energy, leading to a new platform for correlation-induced topological semimetals devoid of f-electrons. We then uncover a continuous selective transition of the molecular orbitals and the associated quantum criticality. Recent experimental findings lend support to both of these propositions.

Finally, we investigate other forms of correlation effects in electronic systems. Three aspects are covered: unconventional superconductivity, quantum criticality and orbital-selective correlations.