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

Study of the Weyl and Dirac topological materials (topological semimetals, insulators, superfluids and superconductors) opens the route for the investigation of the topological quantum vacua of relativistic fields. The symmetric phase of the Standard Model (SM), where both electroweak and chiral symmetry are not broken, represents the topological semimetal. The vacua of the SM (and its extensions) in the phases with broken Electroweak symmetry represent the topological insula...

We discuss some aspects of topological invariants that classify topological states of matter with emphasis on topological insulators. The main aspect addressed is if there are only two topological phases to Bloch Hamiltonian that are time reversal invariant or if there are more phases that has different topological invariants. From a mathematical point of view may exist more topological phases of matter as a subclass of one well established phase.

Topological semimetals and metals have emerged as a new frontier in the field of quantum materials. Novel macroscopic quantum phenomena they exhibit are not only of fundamental interest, but may hold some potential for technological applications.

Topological Structures in the Standard Model at high $T$ are discussed.

The momentum-space topological invariants, which characterize the ground state of the Standard Model, are continuous functions of two parameters, generated by the hypercharge and by the weak charge. These invariants provide the absence of the mass of the elementary fermionic particles in the symmetric phase above the electroweak transition (the mass protection). All the invariants become zero in the broken symmetry phase, as a result all the elementary fermions become massive...

Topological media are gapped or gapless fermionic systems, whose properties are protected by topology, and thus are robust to deformations of parameters of the system and generic. We discuss the class of gapless topological media, which contains the quantum vacuum of Standard Model in its symmetric phase, and condensed matter systems with zeroes in the energy spectrum, which form Fermi surfaces, Weyl and Dirac points, Dirac lines, Khodel-Shaginyan flat bands, etc. Some zeroes...

The theory of topological insulators and superconductors has mostly focused on non-interacting and gapped systems. This review article discusses topological phases that are either gapless or interacting. We discuss recent progress in identifying gapless systems with stable topological properties (such as novel surface states), using Weyl semimetals as an illustration. We then review recent progress in describing topological phases of interacting gapped systems. We explain how...

An old branch of mathematics, Topology, has opened the road to the discovery of new phases of matter. A hidden topology in the energy spectrum is the key for novel conducting/insulating properties of topological matter.

In this article, we will give a brief introduction to the topological insulators. We will briefly review some of the recent progresses, from both theoretical and experimental sides. In particular, we will emphasize the recent progresses achieved in China.

Many quantum condensed matter systems are strongly correlated and strongly interacting fermionic systems, which cannot be treated perturbatively. However, physics which emerges in the low-energy corner does not depend on the complicated details of the system and is relatively simple. It is determined by the nodes in the fermionic spectrum, which are protected by topology in momentum space (in some cases, in combination with the vacuum symmetry). Close to the nodes the behavio...