The physics of quantum many-body systems is very rich and gives rise to remarkable collective states of matter that have no counterpart in their classical analogs. Some examples of such quantum states of matter include superfluids, superconductors, Mott insulators, topological quantum liquids, topological insulators, etc. Since these phenomena often arise from strong correlations and/or multiple energy scales the collective behavior of the elementary degrees of freedom (electrons or atoms) cannot be effectively described in terms of non-interacting entities. As a consequence, it is often very hard to connect the emergent collective behavior of interacting many-body systems to a microscopic understanding.

Topological insulators

A topological insulator is a system or material with certain symmetries (e.g., time-reversal symmetry or cristalline symmetries) and non-trivial topological order. The electronic band structure in the bulk of a non-interacting TI resembles an ordinary band insulator, where the Fermi level is between the valence and conduction bands, such that there is no electric conduction through the bulk. This behavior is just like a normal insulator like, for example, ceramics.

On the surface of a topological insulator, however, there are special states that lie within the bulk band gap between valence and conduction bands and that allow for electric conduction on the surfaces. In contrast to ordinary band insulators which can also have surface states, the surface states of a non-interacting topological insulator are said to be symmetry protected as they appear entirely due to time-reversal symmetry (or another symmetry) and the band structure of the material.

The spin-momentum locking in the topological insulator also gives rise to helical Dirac fermions on the surfaces of three-dimensional topological insulators which behave like massless relativistic particles. In a magnetic field, the surface states of a topological insulator are different from those in the so-called quantum Hall effect, because they are symmetry protected (i.e., not topological), while the surface states in the quantum Hall effect are topological (i.e., robust against any local perturbations that can break all the symmetries). Combining these two properties leads to interesting effects that, among other things, manifest in non-trivial networks of one-dimensional edge channels conducting electric current which can not be easily created in usual two-dimensional electron gases.