Topological order

Remarks

Clause 2 says that no local observable distinguishes the ground states, so no local order parameter separates them. Definitions in the literature vary; some also require that the ground states cannot be connected to a product state by a short local unitary circuit.

Example: Kitaev's toric code on a square lattice of linear size $L$ on a surface of genus $g$, with one spin on each edge. Its Hamiltonian is $H_L = -\sum_v A_v - \sum_p B_p$. Here $A_v$ is the product of $\sigma^x$ on the edges at a vertex $v$, and $B_p$ is the product of $\sigma^z$ on the edges of a face $p$. All these terms commute and have eigenvalues $\pm 1$, so the gap is at least $2$, and the ground space has dimension $4^g$. An operator that changes one ground state into another must act along a loop that winds around the surface, so its support has diameter of order $L$.

Non-example: the classical Ising Hamiltonian $-\sum \sigma^z_x \sigma^z_y$ on a torus has the two ground states with all spins up or all down. The single-site operator $\sigma^z_x$ takes the value $+1$ on one and $-1$ on the other, so clause 2 fails, and the degeneracy $2$ does not depend on the surface.