User talk:00Ranjeet0/sandbox

THE QUANTUM DIMENSION Abelian and non-Abelian anyons Anyons are emergent quasiparticles in two dimensional systems that have exchange statistics which are neither fermionic nor bosonic. A system that contains anyonic quasiparticles has a ground state that is separated by a gap from the rest of the spectrum. We can move the quasiparticles around adiabatically, and as long as the energy we put in the system is lower than the gap we won't excite it and it will remain in the ground state. This is partly why we say the system is topologically protected by the gap.

The simpler case is when the system contains Abelian anyons, in which case the ground state is non-degenerate (i.e. one dimensional). When two quasiparticles are adiabatically exchanged we know the system cannot leave the ground state, so the only thing that can happen is that the ground state wavefunction is multiplied by a phase eiθ. If these were just fermions or bosons than we would have θ=π or θ=0 respectively, but for anyons θ can have other values.

The more interesting case is non-Abelian anyons where the ground state is degenerate (so it is in fact a ground space). In this case the exchange of quasiparticles can have a more complicated effect on the ground space than just a phase, most generally such an exchange applies a unitary matrix U on the ground space (the name 'non-Abelian' comes from the fact that these matrices do not in general commute with each other).

The quantum dimension So we know that the ground space of a system with non-Abelian anyons is degenerate, but what can we say about its dimension? We expect that the more quasiparticles we have in the system, the larger the dimension will be. Indeed it turns out that for M quasiparticles, the dimension of the ground space for large M is roughly ∼dM−2a where da is a number that depends on a - the type of the quasiparticles in the system. This scaling law is reminiscent of the scaling of the dimension of a tensor product of multiple Hilbert spaces of dimension da, and for this reason da is called the quantum dimension of a quasiparticle of type a. You can think of it as the asymptotic degeneracy per particle. For Abelian anyons we have a one-dimensional ground space no matter how many quasiparticles are in the system, so for them da=1.

Although we used the analogy to a tensor product of Hilbert spaces, note that in that case the dimension of each Hilbert space is an integer, while the quantum dimension is in general not an integer. This is an important property of non-Abelian anyons that differentiates them from just a set of particles with local Hilbert spaces - the ground space of non-Abelian anyons is highly nonlocal.

More details on anyons and the quantum dimension can be found in the review paper cited above. The quantum dimension can be generalized to other systems with topological properties, maintaining the same intuitive meaning of asymptotic degeneracy per particle. It is in general very hard to calculate the quantum dimension, and there is only a handful of papers that do