Yang, Lei2023-08-042023-08-042023-08-042023-08-01http://hdl.handle.net/10012/19652Topology and symmetry have become one of the backbones of modern condensed matter physics. These concepts play a large role in determining the possible effects of interactions and entanglement in both gapped and gapless systems. Gapped systems possess a well-developed description via topological quantum field theory (TQFT) that has given rise to many exciting concepts such as topological orders. However gapless systems are far less well-understood in the context of topology as they cannot be described by a simple TQFT due to the presence of local degrees of freedom at low energy. In this thesis I will explore these concepts in the framework of topological response in gapless systems with a focus on 3+1d Weyl and Dirac semimetallic systems. I develop a theory of unquantized topological response, as opposed to the usual quantized response of gapped systems, and explore the effects of strong interaction in the presence of these terms. I show that the associated unquantized topological quantities arise from crystalline symmetries such as discrete translations and rotations. Inspired by the topological crystalline quantity of momentum, I also develop a general theorem involving just discrete translation symmetry that can distinguish long-range entangled states from short-range entangled states. Such a statement can be seen as a generalisation of the well-known Lieb-Schultz-Mattis theorems and many are shown to be consequences from the pure translation theorem that I develop here.entopological semimetalsphases of mattermetalsLSM theoremsquantum phases of matterquantum entanglementweyl semimetaldirac semimetalquantum anomalytopological responsestrong interactionsvortex condensationquantum informationcrystalline defectstranslation symmetryrotation symmetrydislocationsdisclinationsLuttingers theoremsymmetry-protected topological phasesSPTSREtopological insulatorstopological quantum field theorychiral anomalycrystalline momentumTQFTLandau levelsstrongly-correlated electronscondensed mattertopological phases of mattertopology and symmetryDoctoral Thesis