TQFTs in Nature
This chapter introduces the algebraic theory of anyons using unitary ribbon
fusion categories. It follows that quantum invariants of colored links are amplitudes
of anyon trajectories.
6.1. Emergence and anyons
TQFTs are very special quantum field theories. A physical Hamiltonian of
interacting electrons in real materials exhibits no topological symmetries. Then it
begs the question, is TQFT relevant to our real world? The answer is a resounding
yes; it is saved by the so-called emergence phenomenon. The idea is expressed well
by a line in an old Chinese poem:
Word for word it is: grass color far see close but not. It means that in early spring,
one sees the color of grass in a field from far away, yet no particular green spot can
be pointed to. TQFTs do exist in Nature as effective theories, though they are rare
and difficult to discover.
It is extremely challenging for experimental physicists to confirm the existence
of TQFTs in Nature. Physical systems whose low-energy effective theories are
TQFTs are called topological states or phases of matter. Elementary excitations
in topological phases of matter are particle-like, called quasiparticles to distinguish
them from fundamental particles such as the electron. But the distinction has be-
come less and less clear-cut, so very often we call them particles. In our discussion,
we will have a physical system of electrons or maybe some other particles in a plane.
We will also have quasiparticles in this system. To avoid confusion, we will call the
particles in the underlying system constituent particles or slave particles or some-
times just electrons, though they might be bosons or atoms, or even quasiparticles.
If we talk about a Hamiltonian, it is often the Hamiltonian for the constituent
While in classical mechanics the exchange of two identical particles does not
change the underlying state, quantum mechanics allows for more complex behav-
ior [LM]. In three-dimensional quantum systems the exchange of two identical
particles may result in a sign-change of the wave function which distinguishes
fermions from bosons. Two-dimensional quantum systems—such as electrons in
FQH liquids—can give rise to exotic particle statistics, where the exchange of two
identical (quasi)particles can in general be described by either abelian or non-
abelian statistics. In the former, the exchange of two particles gives rise to a com-
plex phase
where θ = 0,π correspond to the statistics of bosons and fermions
respectively, and θ = 0,π is referred to as the statistics of abelian anyons [Wi1].
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