  Since the 1950s, topological solitons have been used to describe elementary
particles[1-3] and particle-like field configurations[4-13] that appear in
almost all branches of physics ranging from subatomic to cosmological
scales[3,14-16]. In this context, formation and annihilation of particles
corresponds to topological phase transitions, whose actual process and
mechanism remain elusive. Here, we show that annihilation of an isolated
skyrmion in helimagnets as the applied magnetic field increases initiates at
its center point by a local magnetization reversal. At the critical field, a
soft mode of the spin-wave excitation appears, which vanishes the local
magnetization modulus as well as the soft-mode-modulated emergent elastic
stiffness[17] at the center, which makes it an emergent magnetic monopole[6].
This softened point vanishes the energy barrier of reversing the local
magnetization at the center, which inevitably annihilates the skyrmion. On this
basis, we propose a general critical condition for topological phase
transitions of field solutions with a soft mode: topological phase transitions
initiate at those spatial points where the soft-mode-modulated emergent
stiffness matrix losses its positive-definiteness.
