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We study the role of zero-point quantum fluctuations in a range of magneticstates which on the classical level are close to spin-aligned ferromagnets.These include Skyrmion textures characterized by non-zero topological charge,and topologically-trivial spirals arising from the competition of theHeisenberg and Dzyaloshinskii-Moriya interactions. For the former, we extendour previous results on quantum exactness of classicalBogomolny-Prasad-Sommerfield (BPS) ground-state degeneracies to the generalcase of K\"ahler manifolds, with a specific example of Grassmann manifolds$\mathrm{Gr(M,N)}$. These are relevant to quantum Hall ferromagnets with$\mathrm{N}$ internal states and integer filling factor $\mathrm{M}$. Apromising candidate for their experimental implementation is monolayer graphenewith $\mathrm{N=4}$ corresponding to spin and valley degrees of freedom at thecharge neutrality point with $\mathrm{M=2}$ filled Landau levels. We find thatthe vanishing of the zero-point fluctuations in taking the continuum limitoccurs differently in the case of BPS states compared to the case of moregeneral smooth textures, with the latter exhibiting more pronounced latticeeffects. This motivates us to consider the vanishing of zero-point fluctuationsin such near-ferromagnets more generally. We present a family of lattice spinmodels for which the vanishing of zero-point fluctuations is evident, and showthat some spirals can be thought of as having nonzero but weak zero-pointfluctuations on account of their closeness to this family. Between them, theseinstances provide concrete illustrations of how the Casimir energy, dependenton the full UV-structure of the theory, evolves as the continuum limit istaken.

Zero point fluctuations for magnetic spirals and Skyrmions, and the fate of the Casimir energy in the continuum limit