Topological Order, Invertible Phases, and Matter Diagnostics

Invertible phases have no nontrivial bulk quasiparticle sectors: stacking with an inverse yields a trivial phase. Intrinsic topological order instead has deconfined excitations with nonlocal braiding, long-range entanglement, and topology-dependent ground-state sectors. Both can have protected edges and quantized response, so Hall conductance alone does not distinguish them.

Required background. Topological field theory supplies metric-independent infrared descriptions; symmetry realization supplies the contrast with local order parameters; invertible response phases supplies the stacking notion.

Long-range entanglement and superselection sectors

A gapped topologically ordered system has quasiparticle types $a$ that cannot be created singly by a local operator. Local operators create total-trivial combinations, while widely separated particles retain a fusion channel detectable by braiding. The ground-state dimension depends on spatial topology; splittings at finite size are typically exponentially small in system size relative to the correlation length, not exactly zero.

For a simply connected region with boundary length $L$, the entanglement entropy often has

$$S(A)=\alpha L-\gamma+o(1), \qquad \gamma=\ln\mathcal D, \qquad \mathcal D=\sqrt{\sum_a d_a^2},$$

where $d_a$ is the quantum dimension. The area coefficient $\alpha$ is nonuniversal; the constant $\gamma$ is a topological prediction only after corners, finite correlation length, sector choice, and extrapolation are controlled Kitaev and Preskill 2006.

Invertible does not mean featureless

An integer Chern insulator is invertible but can have a quantized Hall response and a chiral boundary. Its closed-surface ground state is unique and it has no anyons beyond local electrons or holes. A fractional Hall fluid has a similar electromagnetic response but also fractionally charged sectors and genus-dependent degeneracy. Stacking a Laughlin state with an orientation-reversed copy cancels the Hall response yet generally leaves a nontrivial nonchiral topological order; cancellation of response is not cancellation of intrinsic order.

Short-range-entangled SPT phases are invertible only while the protecting symmetry is retained. Intrinsic order can exist without symmetry, although symmetry can enrich it by fractionalizing quantum numbers or permuting anyons. Spontaneous symmetry breaking is different again: its degenerate states are distinguished by a local order parameter and the degeneracy persists already on simple topology.

A diagnostic combination

A convincing intrinsic-order case combines:

• a stable many-body bulk gap or mobility gap;
• quasiparticle sectors with charge, fusion, or braiding data;
• topology or flux-insertion dependence consistent with those sectors;
• long-range-entanglement or modular information;
• exclusion of symmetry breaking, finite-size momentum multiplets, and boundary artifacts.

No finite torus calculation proves thermodynamic order by itself. A nearly degenerate manifold can arise from translation breaking, and a fitted entropy intercept can drift. The strongest conclusion matches all diagnostics to one candidate topological data set Wen 1990.

Exercise

An Abelian phase has four anyon types, all with quantum dimension one. Find its total quantum dimension and topological entanglement entropy.

Solution

$\mathcal D=\sqrt{4}=2$, so $\gamma=\ln2$. This does not by itself specify the mutual braiding or chiral central charge; different Abelian orders can share the same $\mathcal D$.

References

• Alexei Kitaev and John Preskill, “Topological Entanglement Entropy,” Physical Review Letters 96 (2006) 110404, doi:10.1103/PhysRevLett.96.110404.
• Xiao-Gang Wen, “Topological Orders in Rigid States,” International Journal of Modern Physics B 4 (1990) 239–271, doi:10.1142/S0217979290000139.