A. Zee, Quantum Field Theory in a Nutshell · Problem V.7.1

The fundamental group $\Pi_0(S^0)$ counts the topologically distinct connected components of the vacuum manifold—in this case, two disconnected points representing the two ground states. Kinks are topological defects that interpolate between these vacua, and their charge $Q$ labels which component they connect; since there are only two components, a kink can only wind once between them, making $|Q| = 1$ the maximum. Any configuration with $|Q| \geq 2$ would require the field to "wrap around" the vacuum manifold multiple times, but a discrete two-point space has no such higher winding—the topology simply forbids it.

The fundamental group $\Pi_0(S^0)$ counts the topologically distinct connected components of the vacuum manifold—in this case, two disconnected points representing the two ground states. Kinks are topological defects that interpolate between these vacua, and their charge $Q$ labels which component they connect; since there are only two components, a kink can only wind once between them, making $|Q| = 1$ the maximum. Any configuration with $|Q| \geq 2$ would require the field to "wrap around" the vacuum manifold multiple times, but a discrete two-point space has no such higher winding—the topology simply forbids it.