# What is topological degeneracy in condensed matter physics?

1. What is topological degeneracy in strongly correlated systems such as FQH?

2. What is the difference between topological degeneracy and ordinary degeneracy?

3. Why is topological degeneracy important for the non-Abelian statistics?

(1) What is topological degeneracy in strongly correlated systems such as FQH?

Topological degeneracy is a phenomenon in quantum many-body physics, that the ground state of a gapped many-body system become degenerate in the large system size limit, and that such a degeneracy cannot be lifted by any local perturbations as long as the system size is large.

Topological degeneracy can be used as protected qubits which allows us to perform topological quantum computation. It is believed that the appearance of topological degeneracy implies the topological order (or long-range entanglements) in the ground state. Many-body states with topological degeneracy are described by topological quantum field theory at low energies.

Topological degeneracy was first introduced to physically define topological order. In two dimensional space, the topological degeneracy depends on the topology of space, and the topological degeneracy on high genus Riemann surfaces encode all information on the quantum dimensions and the fusion algebra of the quasiparticles. In particular, the topological degeneracy on torus is equal to the number of quasiparticles types.

The topological degeneracy also appears in the situation with trapped quasiparticles, where the topological degeneracy depends on the number and the type of the trapped quasiparticles. Braiding those quasiparticles leads to topologically protected non-Abelian geometric phase, which can be used to perform topologically protected quantum computation.

The topological degeneracy also appear in non-interacting fermion systems (such as p+ip superconductors) with trapped defects (such as vortices). In non-interacting fermion systems, there is only one type of topological degeneracy where number of the degenerate states is given by $2^{N_d/2}/2$, where $N_d$ is the number of the defects (such as the number of vortices). Such topological degeneracy is referred as "Majorana zero-mode" on the defects. In contrast, there are many types of topological degeneracy for interacting systems. A systematic description of topological degeneracy is given by tensor category (or monoidal category) theory.

(2) What is the difference between topological degeneracy and ordinary degeneracy?

Topological degeneracy, in general, is not exact degeneracy for finite systems. Topological degeneracy becomes exact in large system size limit. Ordinary degeneracy is usually exact degeneracy.

(3) Why is topological degeneracy important for the non-Abelian statistics?

Topologically protected non-Abelian geometric phases can only appear when there is a topological degeneracy. See How Non-abelian anyons arise in solid-state systems?