Topological Order - The Discovery and Characterization of Topological Order

The Discovery and Characterization of Topological Order

However, since late 1980s, it has become gradually apparent that Landau symmetry-breaking theory may not describe all possible orders. In an attempt to explain high temperature superconductivity the chiral spin state was introduced. At first, physicists still wanted to use Landau symmetry-breaking theory to describe the chiral spin state. They identified the chiral spin state as a state that breaks the time reversal and parity symmetries, but not the spin rotation symmetry. This should be the end of story according to Landau's symmetry breaking description of orders. However, it was quickly realized that there are many different chiral spin states that have exactly the same symmetry, so symmetry alone was not enough to characterize different chiral spin states. This means that the chiral spin states contain a new kind of order that is beyond the usual symmetry description. The proposed, new kind of order was named "topological order". (The name "topological order" is motivated by the low energy effective theory of the chiral spin states which is a topological quantum field theory (TQFT)). New quantum numbers, such as ground state degeneracy and the non-Abelian geometric phase of degenerate ground states, were introduced to characterize/define the different topological orders in chiral spin states. Recently, it was shown that topological orders can also be characterized by topological entropy.

But experiments soon indicated that chiral spin states do not describe high-temperature superconductors, and the theory of topological order became a theory with no experimental realization. However, the similarity between chiral spin states and quantum Hall states allows one to use the theory of topological order to describe different quantum Hall states. Just like chiral spin states, different quantum Hall states all have the same symmetry and are beyond the Landau symmetry-breaking description. One finds that the different orders in different quantum Hall states can indeed be described by topological orders, so the topological order does have experimental realizations.

Fractional quantum Hall (FQH) states were discovered in 1982 before the introduction of the concept of topological order. But FQH states are not the first experimentally discovered topologically ordered states, as superconductors, having Z2 topological order, were discovered earlier, in 1911.

Although topologically ordered states usually appear in strongly interacting boson/fermion systems, a simple kind of topological order can also appear in free fermion systems. This kind of topological order correspond in integral quantum Hall state, which can be characterized by the Chern number of the filled energy band if we consider integral quantum Hall state on a lattice. Theoretical calculations have proposed that such Chern number can be measured for a free fermion system experimentally. It is also well known that such a Chern number can be measured (maybe indirectly) by edge states.