Topological Order

In physics, topological order (or intrinsic topological order) is a new kind of order in zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined/described by robust ground state degeneracy and quantized non-Abelian geometric phases of degenerate ground states, just like superfluid order is defined/described by vanishing viscosity and quantized vorticity. Microscopically, topological order corresponds to pattern of long-range quantum entanglement, just like superfluid order corresponds to boson condensation. States with different topological orders (or different patterns of long range entanglements) cannot change into each other without a phase transition.

Topologically ordered states have some amazing properties, such as ground state degeneracy that cannot be lifted by any local perturbations but depends on the topology of space, quasiparticle fractional statistics and fractional charges, perfect conducting edge states even in presence of magnetic impurities, topological entanglement entropy, etc. Topological order is important in the study of several physical systems such as spin liquids, the quantum Hall effect, along with potential applications to fault-tolerant quantum computation.

We note that topological insulators and topological superconductors (beyond 1D) do not have the topological order as defined above (see discussion below).