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Abstract |
We study the role of zero-point quantum fluctuations in magnetic states which on the classical level are close to spin-aligned ferromagnets. These include Skyrmion textures which arise in the context of non-zero topological charge solutions of non-linear sigma-models, and topologically-trivial spirals generated by a competition of Heisenberg and Dzyaloshinskii-Moriya interactions. We show that the degeneracy of the Bogomolny-Prasad-Sommerfield (BPS) manifold is not lifted by quantum fluctuations in the case of general non-linear sigma-models with the target space given by Kähler manifolds presenting a physically-important example of the case of Grassmanian manifold relevant to quantum Hall effect in graphene. Further, we show that the phenomenon of vanishing zero-point motion can appear more generally in slowly-twisted almost ferromagnets. From a broader perspective, beyond the implications to exotic magnets, we suggest that this work provides two interesting angles on long-standing interesting issues in statistical physics and field theory. One is the existence of undressed states generally. The other item is the behaviour of ‘non-universal’ quantities, i.e. those involving information from the lattice scale, in the ‘universal’ continuum limit. Here, the Casimir energy of zero-point fluctuations vanishes in the continuum limit, but is nonzero for any lattice discretisation. This Casimir energy, however, does play a physical role, e.g. in the lifting of ground-state degeneracies in a process known as quantum order by disorder. Our results obtained for non-linear sigma models without supersymmetry about the absence of zero-point fluctuations in BPS manifolds may either be a feature entirely unrelated to the more familiar instances arising in relativistic field theory from the cancellation of fluctuations in bosonic and fermionic sectors, or they may be more pedestrian and perhaps intuitively accessible instances of the same physics. |
