Electrons dance themselves in knots

“More is different”. This is how Philip Anderson, former head of the theory group at the Cavendish, summarised the intrigue of condensed matter physics, the study of phenomena that emerge when a large number of particles are brought together. Unexpected behaviours often appear in complex systems that were not predicted ahead of time based on the properties of the
constituents alone. Perhaps the most famous example is superconductivity, where a material conducts electricity without loss of energy.

A powerful mathematical tool for understanding condensed matter systems is the concept of topology: the study of the properties of a system that remain unchanged by smooth transformations like bending, stretching or twisting. The number holes in an object, for example, makes a donut and a coffee cup with a handle identical from a topological point of view, even if trying to drink your coffee from a donut is a bad idea. We can make this exact from an algebraic viewpoint by integrating socalled ‘surface curvatures’ over objects to give ‘topological numbers’ that relate to the number of holes.

While topology is an active field of mathematics, these concepts come to life in the electron systems that we explore in the laboratory. In such topological systems, the electronic wave functions weave together to form knots
that cannot be smoothly disentangled or deformed into each other. These ‘topological phases’ have distinct and extremely robust physical properties controlled by these topological numbers, such as novel quantum excitations
and electromagnetic couplings. What’s more they hold promise for applications ranging from advanced electronics to quantum computing, representing a holy
grail in modern physics.

Specific mathematical rules have been developed that lay out the types of knots and associated topological patterns that can emerge in a crystalline material, but the past few years have seen the discovery of new topological phases that follow novel rules. In these systems, electrons perform a precise dance to create emergent particles that can be moved around each other, or ‘braided’, storing topological information in a distinctive manner.

We discovered that these phenomena come into their full glory when the system undergoes a change. By suddenly altering the parameters and then releasing the system, we can induce specific knotted patterns that
can then be measured. Most recently, we fired periodic laser pulses at the electronic phases and found that we could not only induce braiding,
but also produce anomalous phases, characterised by topological numbers that can’t exist in static settings.

Perhaps most excitingly, we found that these phases are highly resilient against external perturbations, making them promising candidates for robust information storage and processing. They are now actively being
investigated by the quantum simulation groups here at the Cavendish and in labs around the world, making it an interesting time for this nascent field.

R.-J. Slager, A. Bouhon, F.N. Ünal, 'Non-Abelian Floquet braiding and anomalous Dirac string phase in periodically driven systems', Nature Communications 15, 1144 (2024)