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April 24th 2019

Topological states in condensed matter form an important research theme in modern physics. Topology is the branch of mathematics that studies shapes and spaces. Objects in the same topological class can be reshaped into each other by bending and stretching. However, some objects can only be reshaped into each other if you cut them open somewhere, reshape them and then stick them back together again. Topology can be applied to classify knots: which knots can be reshaped into each other? And which knots cannot be unravelled?

Electron knot that cannot be unravelled
This mathematics really comes to life in the quantum world in special insulating crystals. The wavelengths of electrons in these crystals together form a knot that cannot be unravelled. Because of this, the electrons exhibit collective quantum behaviour, even though they cannot move individually. That makes the material a so-called topological insulator

Although there are odd examples of topological crystals, these were until recently mainly discovered by trial and error. The researchers from this FOM programme are therefore trying to unravel which characteristics a topological crystal must have.

From search to quantum computer
The researchers discovered that the lattice symmetry of the crystal is vital for the mathematical description of the knot of wave functions and with that for the stability of the knot as well. Thanks to this discovery the researchers can now initiate a systematic search for new topological insulators. 

This search is also vitally important for the realisation of a quantum computer. Due to the laws of quantum mechanics the bits in a quantum computer can be in many states at once. Quantum computers therefore have an unparalleled processing power. However, the largest barrier to making a quantum computer is the fragile nature of the quantum states – the special characteristics of the bits disappear if they come into contact with the 'normal outside world'. Topological insulators are interesting in this respect because with their topological characteristics they can protect quantum information.

For further information please see the news release from Leiden University.