  We study the mechanism of decay of a topological (winding-number) excitation
due to finite-size effects in a two-dimensional valence-bond solid state,
realized in an $S=1/2$ spin model ($J$-$Q$ model) and studied using projector
Monte Carlo simulations in the valence bond basis. A topological excitation
with winding number $|W|>0$ contains domain walls, which are unstable due to
the emergence of long valence bonds in the wave function, unlike in effective
descriptions with the quantum dimer model. We find that the life time of the
winding number in imaginary time diverges as a power of the system length $L$.
The energy can be computed within this time (i.e., it converges toward a
"quasi-eigenvalue" before the winding number decays) and agrees for large $L$
with the domain-wall energy computed in an open lattice with boundary
modifications enforcing a domain wall. Constructing a simplified two-state
model and using the imaginary-time behavior from the simulations as input, we
find that the real-time decay rate out of the initial winding sector is
exponentially small in $L$. Thus, the winding number rapidly becomes a
well-defined conserved quantum number for large systems, supporting the
conclusions reached by computing the energy quasi-eigenvalues. Including
Heisenberg exchange interactions which brings the system to a quantum-critical
point separating the valence-bond solid from an antiferromagnetic ground state
(the putative "deconfined" quantum-critical point), we can also converge the
domain wall energy here and find that it decays as a power-law of the system
size. Thus, the winding number is an emergent quantum number also at the
critical point, with all winding number sectors becoming degenerate in the
thermodynamic limit. This supports the description of the critical point in
terms of a U(1) gauge-field theory.
