Credit: Courtesy of Charles Kane
Topology is a branch of mathematics concerning the properties of geometric objects and their shapes. These properties are essentially invariant under continuous deformations such as stretching, twisting, etc. Entanglement on the other hand is purely a physical phenomenon wherein two particles can influence each other instantaneously irrespective of the spatial distance between them.
In new research published in the journal Physical Review X, Charles Kane, who is the Christopher H. Browne Distinguished Professor of Physics in U. Penn's School of Arts & Sciences established a conceptual duality between topology and entanglement along with his collaborators .
Consider a sphere and a donut. The difference between the two lies in the fact that a donut, which has a toroidal topology, is specified by a single hole whereas there are no holes in a sphere. In this sense, a coffee mug and donut are topologically similar since both of them are characterized by a common property and that is a single hole (see figure below). Building upon this feature, topology can be utilized as a universal scheme for describing quantum states. Quantum entanglement on the other hand is already a well-known procedure that quantum states use to characterize non-local correlations. What the researchers have found is a circumstance where these two ideas are tightly interwoven. In a sense, it is a connection between an abstract and a concrete phenomenon wherein topology comes under the former and entanglement under the latter.
Continuous deformation (a type of homeomorphism) turning a coffee mug into a torus shaped object. Source: Henry Segerman
The relationship was originally explored in the Fermi surface of copper which Kane initially realized as a topological surface. The Fermi surface of copper essentially possesses four holes, also called a genus in technical terminology. Kane and his collaborators derived a mathematical formula between the genus of the Fermi surface and a measure of quantum entanglement called mutual information which brings together individual correlations from different regions of space to meet at a single point.
This work displays a beautiful blend of mathematics and physics and also opens up a box of possibilities such as its potential application to understanding phases of matter, investigating the manifestation of quantum mechanics in a system with multiple degrees of freedom, and measuring topological genus among others.
 Pok Man Tam et al, Topological Multipartite Entanglement in a Fermi Liquid, Physical Review X (2022). DOI: 10.1103/PhysRevX.12.031022