Topology is the study of shape, in particular the properties that are preserved when a shape is squeezed, stretched and battered but not torn or ripped.
In the past, topology was little more than an amusing diversion for mathematicians doodling about the difference between donuts and dumplings.
But that is beginning to change. In recent years, physicists have begun to use topology to explain some of the most important puzzles at the frontiers of physics.
For example, certain quantum particles cannot form pairs but do form triplets called Efimov states.That’s curious—surely the bonds that allow three particles to bond together should also allow two to become linked?
Actually, no and topology explains why. The reason is that the mathematical connection between these quantum particles takes the form of a Borromean ring: three circles intertwined in such a way that cutting one releases the other two. Only three rings can be connected in this way, not two. Voila!
But this kind of topological curiosity is merely the tip of the iceberg if Xiao-Gang Wen at the Perimeter Institute for Theoretical Physics, in Waterloo, Canada, is to be believed.
Today, Wen combines topology, symmetry and quantum mechanics in a new theory that predicts the existence of new states of matter, unifies various puzzling phenomena in solid state physics and allows the creation artificial vacuums populated with artificial photons and electrons.
So where to start? Wen begins by explaining the fundamental role of symmetry in the basic states of matter such as liquids and solids. A symmetry is a property that remains invariant under a transformation of some kind.
So in a liquid, for example, atoms are randomly distributed and so the liquid looks the same if it is displaced in any direction by any distance. Physicists say it has a continuous translation symmetry.
However, when a liquid freezes, the atoms become locked into a crystal lattice and a different symmetry applies. In this case, the lattice only appears the same if it is displaced along the crystal axis by a specific distance. So the material now has discrete translation symmetry and the original symmetry is broken.
In other words, when the material undergoes a phase change, it also undergoes a change in symmetry, a process that physicists call symmetry breaking.
But in addition to the four ordinary phases of matter—liquid, solid, gas and plasma,—physicists have discovered many quantum phases of matter such as superconductivity, superfluidity.and so on.
These phases are also the result of symmetry breaking but symmetry alone cannot explain what’s going on.
So physicists have turned to topology to help. It turns out that the mathematics of quantum mechanics has topological properties that, when combined with symmetry, explain how these phases form.
This kind of work has led to the discovery of additional phases of matter such as topological conductors and insulators,
The important point here is that the properties of these systems are guaranteed not by the ordinary laws of physics but by the topological properties of quantum mechanics, just like the Borromean rings that explain the Efimov states described earlier.
Xiao-Gang Wen’s approach is to explore the properties of matter when the topological links between particles become much more general and complex. He generalises these links, thinking of them as strings that can connect many particles together. In fact, he considers the way many strings can form net-like structures that have their own emergent properties.
So what kind of emergent properties do these string-nets have? It turns out that string-nets are not so different from the ordinary matter. String nets can support waves which Xiao-Gang Wen says are formally equivalent to photons.
That makes string nets a kind of “quantum ether” through which electromagnetic waves travel. That’s a big claim.
Wen also says that various properties of string nets are equivalent to fundamental particles such as electrons. And that it may be possible to derive the properties of other particles too. That’s another big idea.
Of course, no theory is worth more than bag of beans unless it makes testable predictions about the universe.
Wen says that his theory has significant implications for the states of matter that existed soon after the Big Bang but doesn’t develop the idea into specific predictions.
Presumably, the same ought to be true of other extreme astrophysical phenomenon. For example, it’d be interesting to see what conditions this kind of approach places on the nature of black holes.
Wen also says that it ought to be possible to manipulate the topological properties of materials to create artificial vacuums complete with artificial photons and artificial particles like electrons. In other words, topology is the key to creating entirely new worlds in the lab.
Clearly, Wen’s ideas will take some digesting. And the implications he discusses need to firmed up into specific experimental predictions.
But it’s not the first time we’ve come across the notion that topology plays a more fundamental role in the universe than anyone imagined. We explored a similar idea a couple of years ago.
Physicists have known for many decades that symmetry plays a powerful role in the laws of physics. In fact, it’s fair to say that symmetry has changed the way we think about the universe.
It’s just possible that adding topology to the mix could be equally revolutionary.
Ref: arxiv.org/abs/1210.1281: Topological Order: From Long-Range Eentangled Quantum Matter To An Unification Of Light And Electrons
Copyright Technology Review 2012.