Interacting topological order and ground-state degeneracy

Put one spin-half degree of freedom on every edge of a square lattice. A star operator multiplies x Pauli matrices on the edges incident to a vertex; a plaquette multiplies z matrices around a face. Each operator squares to identity. A star and plaquette either share zero edges or two, so the two anticommutation signs cancel. All Hamiltonian terms commute, allowing simultaneous eigenstates labelled by their local constraint values.

Positive couplings favour every star and plaquette eigenvalue plus one. Flipping one term's eigenvalue raises its energy by twice that term's coupling. On a closed surface, global product constraints require the corresponding defects to occur with appropriate even total parity. Open strings create defect pairs at their endpoints, while transporting a defect around a noncontractible cycle can change a global sector without leaving local defects behind. This connects local excitations with nonlocal ground-space operations.

On a connected closed orientable surface, the product of every star is identity because each edge occurs twice. The same holds for the plaquettes. Thus there are two relations among the stabilizer constraints. With one qubit per edge, each independent commuting constraint halves the unconstrained Hilbert-space dimension. Combining this count with the Euler characteristic gives two logical qubits per handle, and four ground states on a torus.

These sectors are distinguished by noncontractible loop operators rather than a local order parameter. A loop and a dual loop crossing once have anticommuting logical actions. Local operators cannot distinguish ideal sectors in a large code without accessing a winding structure. Under sufficiently weak local perturbations within the gapped phase, finite-size splittings are suppressed with system size; exact degeneracy of the commuting model should not be demanded from every finite microscopic realization.

The toric code has intrinsic topological order: its anyon sectors and closed-surface degeneracy persist without requiring an external symmetry to forbid a local order parameter. A conventional symmetry-protected phase can instead have a unique closed-system ground state and lose its distinction when the protecting symmetry is relaxed. Band Chern numbers and Z₂ indices classify specific one-particle settings; interacting many-body phases can require different invariants, and some free-fermion classifications change when interactions are admitted.

A gapped many-body phase and its local indistinguishability assumptions must be established before interpreting near-degenerate numerical levels as topological sectors. Disorder and finite geometry can produce unrelated low-energy states. Nor does zero-temperature degeneracy alone prove passive thermal error correction: the elementary two-dimensional model permits creating a defect pair at finite cost and separating it without an energy growing with path length. Practical memories therefore need a temperature, noise, dynamics, and active-control analysis in addition to an elegant ground-state count.