What Is Topological Order

Most phases of matter are organized by symmetry breaking: a magnet picks a direction, a crystal picks a lattice, and Landau's theory classifies the phase by a local order parameter that distinguishes it from the symmetric phase. Topological order is a fundamentally different kind of organization. It describes gapped quantum phases that are not distinguished by any local order parameter, yet are still sharply different from a trivial product state. The differences live in global, nonlocal properties of the ground-state wavefunction.

Three fingerprints of topological order

A gapped many-body system is said to be topologically ordered when its ground space exhibits the following hallmarks, none of which a conventional (e.g. symmetry-broken) phase possesses.

Topology-dependent ground-state degeneracy. Placed on a closed surface of genus $g$ (a sphere has $g=0$ , a torus $g=1$ ), the system has a ground-state degeneracy that depends on $g$ and on nothing local. For the toric code the degeneracy is $4^g$ ; on a torus it is exactly $4$ . No local perturbation can split this degeneracy, because the degenerate states are locally indistinguishable.
Long-range entanglement. The ground state cannot be transformed into a product state by any finite-depth local unitary circuit. This is captured quantitatively by the topological entanglement entropy. For a region $A$ with smooth boundary, the entanglement entropy obeys
$S(A) = \alpha\,|\partial A| - \gamma + \cdots,$
where $\alpha|\partial A|$ is the non-universal "area law" piece and the universal subleading constant $\gamma = \ln \mathcal{D}$ is set by the total quantum dimension $\mathcal{D}$ of the anyon theory. A trivial phase has $\gamma = 0$ ; topological order means $\gamma > 0$ .
Anyonic excitations. The elementary excitations above the ground state are point-like quasiparticles with fractional exchange statistics — anyons — rather than the bosons or fermions of ordinary matter. Braiding one anyon around another applies a nontrivial unitary to the ground space.

Why "topological"

The defining data are insensitive to smooth local deformations. The degeneracy depends only on the genus of the surface, not its shape; the statistics of a braiding process depend only on the topology of the worldlines (how many times one particle encircled another), not on the geometric details of the path; and $\gamma$ is invariant under any finite-depth local unitary. This robustness is precisely what makes topological order attractive for quantum computation: information stored in the degenerate, locally indistinguishable ground space is protected from local noise.

The exactly solvable anchor: the toric code

Kitaev's toric code is the canonical example and the thread running through this module. It is a model of spins on the edges of a lattice with mutually commuting stabilizer-like terms, a unique gapped phase, and four locally indistinguishable ground states on a torus. Its excitations come in two species, $e$ and $m$ , that are individually bosons but acquire a phase of $-1$ when one is braided around the other — the simplest possible mutual anyonic statistics. We will use it as a concrete laboratory for the abstract ideas of anyons, fusion, and braiding developed in this module.

What to take away

Topological order is a gapped phase with no local order parameter, identified by genus-dependent ground-state degeneracy, a universal long-range-entanglement constant $\gamma=\ln\mathcal{D}$ , and anyonic excitations. These global properties — not any local quantity — are what protect quantum information, and they are the foundation for topological quantum computation.