Koji Kobayashi

Topological properties are sometimes emergent or enforced by the breaking of translational invariance. Here, in this chapter we discuss dimensional crossover of topological properties in thin films of topological insulators (TI) and Weyl semimetals, electronic properties on the surface of TI nanoparticles and TI nanowires as a constrained electronic system. To discuss the effects of disorder is another highlight of this chapter. We cast on the unusual robustness of Dirac and Weyl semimetal phases against disorder, then the discussion is turned to a novel type of quantum criticality emergent from this unusual robustness, leading us to formulate the scaling theory of semimetal-metal transition. The concept of topological matter dose not fade under circumstances of absent translational invariance; it is on the contrary, emergent or enforced under such circumstances.

Topological properties are sometimes emergent or enforced by the breaking of translational invariance. Here, in this chapter we discuss dimensional crossover of topological properties in thin films of topological insulators (TI) and Weyl semimetals, electronic properties on the surface of TI nanoparticles and TI nanowires as a constrained electronic system. To discuss the effects of disorder is another highlight of this chapter. We cast on the unusual robustness of Dirac and Weyl semimetal phases against disorder, then the discussion is turned to a novel type of quantum criticality emergent from this unusual robustness, leading us to formulate the scaling theory of semimetal-metal transition. The concept of topological matter dose not fade under circumstances of absent translational invariance; it is on the contrary, emergent or enforced under such circumstances.