Today I have the pleasure of chairing the last symposium of the year of the MRS chapter at UHasselt. During this invited lecture, Bart Sorée (Professor at UAntwerp and KULeuven, and alumnus of my own Alma Mater) will introduce us into the topic of topological insulators.

This topic became unexpectedly a hot topic as it is part of the 2016 Nobel Prize in Physics, awarded last Saturday.

This year’s Nobel prize in physics went to: David J. Thouless (1/2), F. Duncan M. Haldane (1/4) and J. Michael Kosterlitz (1/4) who received it

*“for theoretical discoveries of topological phase transitions and topological phases of matter.”*

*“for theoretical discoveries of topological phase transitions and topological phases of matter.”*

On the Nobel Prize website you can find this document which gives some background on this work and explains what it is. Beware that the explanation is rather technical and at an abstract level. They start with introducing the concept of an **order parameter**. You may have heard of this in the context of dynamical systems (as I did) or in the context of phase transitions. In the latter context, order parameters are generally **zero in one phase, and non-zero in the other**. In overly simplified terms, one could say an order parameter is a kind of hidden variable (not to be mistaken for a hidden variable in QM) which becomes visible upon symmetry breaking. An example to explain this concept.

__Example:__** Magnetization of a ferromagnet.**

__Example:__

In a ferromagnetic material, the atoms have what is called a spin (imagine it as a small magnetic needle pointing in a specific direction, or a small arrow). At high temperature these spins point randomly in all possible directions, leading to a net zero *magnetization* (the sum of all the small arrows just lets you run in circles going nowhere). This magnetization is the order parameter. At the high temperature, as there is no preferred direction, the system is invariant under rotation and translations (*i.e.* if you shift it a bit or you rotate it, or both you will not see a difference) When the temperature is lower, you will cross what is called a critical temperature. Below this temperature all spins will start to align themselves parallel, giving rise to a non-zero magnetization (if all arrows point in the same direction, their sum is a long arrow in that direction). At this point, the system has lost the rotational invariance (because all spins point in direction, you will know when someone rotated the system) and the symmetry is said to have broken.

Within the context of phase transitions, order parameters are often temperature dependent. In case of topological materials this is not the case. A **topological material** has a **topological order**, which means both phases are present at absolute zero (or the temperature you will never reach in any experiment no matter how hard you try) or maybe better without the presence of temperature (this is more the realm of computational materials science, calculations at 0 Kelvin actually mean without temperature as a parameter). So the order parameter in a topological material will not be temperature dependent.

__Topological insulators__

__Topological insulators__

To complicate things, topological insulators are materials which have a topological order which is not as the one defined above 😯 —yup why would we make it easy 🙄 . It gets even worse, a topological insulator is conducting.

OK, before you run away or loose what is remaining of your sanity. A topological insulator is an insulating material which has surface states which are conducting. In this it is not that different from many other “normal” insulators. What makes it different, is that these surface states are, what is called, **symmetry protected**. What does this mean?

In a topological insulator with 2 conducting surface states, one will be linked to spin up and one will be linked to spin down (remember the ferromagnetism story of before, now the small arrows belong to the separate electrons and exist only in 2 types: pointing up=spin up, and pointing down=spin down). Each of these surface states will be populated with electrons. One state with electrons having spin up, the other with electrons having spin down. Next, you need to know that these states also have a real-space path let the electrons run around the edge of material. Imagine them as one-way streets for the electrons. Due to symmetry the two states are mirror images of one-another. As such, if electrons in the up-spin state more left, then the ones in the down-spin state move right. We are almost there, no worries there is a clue. Now, where in a normal insulator with surface states the electrons can scatter (bounce and make a U-turn) this is not possible in a topological insulator. But there are roads in two directions you say? Yes, but these are restricted. And up-spin electron cannot be in the down-spin lane and vice versa. As a result, a current going in such a surface state will show extremely little scattering, as it would need to change the spin of the electron as well as it’s spatial motion. This is why it is called symmetry protected.

If there are more states, things get more complicated. But for everyone’s sanity, we will leave it at this. 😎