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

Abstract

We study the role of zero-point quantum fluctuations in a range of magnetic states 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 the Heisenberg and Dzyaloshinskii-Moriya interactions. For the former, we extend our previous results on quantum exactness of classical Bogomolny-Prasad-Sommerfield (BPS) ground-state degeneracies to the general case 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 $\mathrm{M}$. A promising candidate for their experimental implementation is monolayer graphene with $\mathrm{N=4}$ corresponding to spin and valley degrees of freedom at the charge neutrality point with $\mathrm{M=2}$ filled Landau levels. We find that the vanishing of the zero-point fluctuations in taking the continuum limit occurs differently in the case of BPS states compared to the case of more general smooth textures, with the latter exhibiting more pronounced lattice effects. This motivates us to consider the vanishing of zero-point fluctuations in such near-ferromagnets more generally. We present a family of lattice spin models for which the vanishing of zero-point fluctuations is evident, and show that some spirals can be thought of as having nonzero but weak zero-point fluctuations on account of their closeness to this family. Between them, these instances provide concrete illustrations of how the Casimir energy, dependent on the full UV-structure of the theory, evolves as the continuum limit is taken.