Topological invariants for Standard Model: from semi-metal to topological insulator

In this lecture for the Nobel symposium, we review previous research on a class of translational-invariant insulators without spin-orbit coupling. These may be realized in intrinsically spinless systems such as photonic crystals and ultra-cold atoms. Some of these insulators have no time-reversal symmetry as well, i.e., the relevant symmetries are purely crystalline. Nevertheless, topological phases exist which are distinguished by their robust surface modes. To describe thes...

Searching for topological insulators/superconductors is a central subject in recent condensed matter physics. As a theoretical aspect, various classification methods of symmetry-protected topological phases have been developed, where the topology of a gapped Hamiltonian is investigated from the viewpoint of its onsite/crystal symmetry. On the other hand, topological physics also appears in semimetals, whose gapless points can be characterized by topological invariants. Stimul...

These lecture notes were prepared for a mixed audience of students, postdocs and faculty from the Indian Institute of Technology Madras, India and neighboring institutions, particularly the Institute of Mathematical Sciences. I am not an expert on the subject and during the few years I spent working on the Quantum Hall effect, I had not fully appreciated that it was part of a family of topological insulators. It was a pleasure to dig a little deeper into this subject and to s...

This paper is a survey of the $\mathbb{Z}_2$-valued invariant of topological insulators used in condensed matter physics. The $\mathbb{Z}$-valued topological invariant, which was originally called the TKNN invariant in physics, has now been fully understood as the first Chern number. The $\mathbb{Z}_2$ invariant is more mysterious, we will explain its equivalent descriptions from different points of view and provide the relations between them. These invariants provide the cla...

The discovery of topological insulators has reformed modern materials science, promising to be a platform for tabletop relativistic physics, electronic transport without scattering, and stable quantum computation. Topological invariants are used to label distinct types of topological insulators. But it is not generally known how many or which invariants can exist in any given crystalline material. Using a new and efficient counting algorithm, we study the topological invarian...

This paper reviews several analytic tools for the field of topological insulators, developed with the aid of non-commutative calculus and geometry. The set of tools includes bulk topological invariants defined directly in the thermodynamic limit and in the presence of disorder, whose robustness is shown to have non-trivial physical consequences for the bulk states. The set of tools also includes a general relation between the current of an observable and its edge index, relat...

We present a pedagogical review of topological superconductivity and its consequences in spin-orbit coupled semiconductor/superconductor heterostructures. We start by reviewing the historical origins of the notions of Dirac and Majorana fermions in particle physics and discuss how lower dimensional versions of these emerge in one dimensional superconductors. Ultimately, we focus on Majorana zero-modes, which emerge at defects in the Majorana equation. We then review the defin...

To appear in Encyclopedia of Mathematical Physics, published by Elsevier in early 2006. Comments/corrections welcome. The article surveys topological aspects in gauge theories.

We give a self-contained and enriched review about topology properties in the rapidly growing field of topological states of matter (TSM). This review is mainly focus on the beautiful interplay of topology mathematics and condensed matter physics that issuing TSM. Fiber bundle theory is a powerful concept to describe the non-trivial topology properties underlying the physical system. So we briefly present some motivation of fiber bundle theory and following that several effec...

Topological quantum materials hold great promise for future technological applications. Their unique electronic properties, such as protected surface states and exotic quasiparticles, offer opportunities for designing novel electronic devices, spintronics, and quantum information processing. The origin of the interplay between various electronic orders in topological quantum materials, such as superconductivity and magnetism, remains unclear, particularly whether these electr...