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

Topological electronic materials are new quantum states of matter hosting novel linear responses in the bulk and anomalous gapless states at the boundary, and are for scientific and applied reasons under intensive research in physics and in materials sciences. The detection for such materials has so far been hindered by the level of complication involved in the calculation of the so-called topological invariants, and is hence considered a specialized task that requires both e...

Topology in momentum space is the main characteristics of the ground states of a system at zero temperature, the quantum vacua. The gaplessness of fermions in bulk, on the surface or inside the vortex core is protected by topology. Irrespective of the deformation of the parameters of the microscopic theory, the energy spectrum of these fermions remains strictly gapless. This solves the main hierarchy problem in particle physics. The quantum vacuum of Standard Model is one of ...

This monograph offers an overview on the topological invariants in fermionic topological insulators from the complex classes. Tools from K-theory and non-commutative geometry are used to define bulk and boundary invariants, to establish the bulk-boundary correspondence and to link the invariants to physical observables.

Supersymmetry (SUSY) proposed as an elementary symmetry for physics beyond the Standard Model has found important applications in various areas outside high-energy physics. Here, we systematically implement supersymmetric quantum mechanics -- exhibiting fundamental SUSY properties in the simple setting of quantum mechanics -- into a wide range of topological semimetals, where the broken translational symmetry, e.g., by a magnetic field, is effectively captured by a SUSY poten...

Topological insulators are new states of quantum matter which can not be adiabatically connected to conventional insulators and semiconductors. They are characterized by a full insulating gap in the bulk and gapless edge or surface states which are protected by time-reversal symmetry. These topological materials have been theoretically predicted and experimentally observed in a variety of systems, including HgTe quantum wells, BiSb alloys, and Bi$_2$Te$_3$ and Bi$_2$Se$_3$ cr...

In this Book Chapter (invited) we briefly review the basic concepts defining topological insulators and focus on elaborating on the key experimental results that revealed and established their symmetry protected (SPT) topological nature. We then present key experimental results that demonstrate magnetism, Kondo insulation, mirror chirality or topological crystalline order and superconductivity in spin-orbit topological insulator settings and how these new (bulk insulating) ph...

The pseudogap state of high temperature superconductors is a profound mystery. It has tantalizing evidence of a number of broken symmetry states, not necessarily conventional charge and spin density waves. Here we explore a class of more exotic density wave states characterized by topological properties observed in recently discovered topological insulators. We suggest that these rich topological density wave states deserve closer attention in not only high temperature superc...

This article is meant as a gentle introduction to the "topological terms" that often play a decisive role in effective theories describing topological quantum effects in condensed matter systems. We first take up several prominent examples, mainly from the area of quantum magnetism and superfluids/superconductors. We then briefly discuss how these ideas are now finding incarnations in the studies of symmetry-protected topological phases, which are in a sense the generalizatio...

We discuss the thermal (or gravitational) responses in topological superconductors and in topological phases in general. Such thermal responses (as well as electromagnetic responses for conserved charge) provide a definition of topological insulators and superconductors beyond the single-particle picture. In two-dimensional topological phases, the Str\v{e}da formula for the electric Hall conductivity is generalized to the thermal Hall conductivity. Applying this formula to th...

Bloch theory describes the electronic states in crystals whose energies are distributed as bands over the Brillouin zone. The electronic states corresponding to a (few) isolated energy band(s) thus constitute a vector bundle. The topological properties of these vector bundles provide new characteristics of the corresponding electronic phases. We review some of these properties in the case of (topological) insulators and semi-metals.