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The Kitaev model is an exactly-soluble quantum spin model, whose ground stateprovides a canonical example of a quantum spin liquid. Spin excitations fromthe ground state are fractionalized into emergent matter fermions and $Z_2$fluxes. The $Z_2$ flux excitation is pointlike in two dimensions, while itcomprises a closed loop in three dimensions because of the local constraint foreach closed volume. In addition, the fluxes obey global constraints involving(semi)macroscopic number of fluxes. We here investigate such global constraintsin the Kitaev model on a three-dimensional lattice composed of nine-siteelementary loops, dubbed the hypernonagon lattice, whose ground state is achiral spin liquid. We consider two different anisotropic limits of thehypernonagon Kitaev model where the low-energy effective models are describedsolely by the $Z_2$ fluxes. We show that there are two kinds of globalconstraints in the model defined on a three-dimensional torus, namely, surfaceand volume constraints: the surface constraint is imposed on the even-oddparity of the total number of fluxes threading a two-dimensional slice of thesystem, while the volume constraint is for the even-odd parity of the number ofthe fluxes through specific plaquettes whose total number is proportional tothe system volume. In the two anisotropic limits, therefore, the elementaryexcitation of $Z_2$ fluxes occurs in a pair of closed loops so as to satisfyboth two global constraints as well as the local constraints.

Global constraints on $Z_2$ fluxes in two different anisotropic limits of a hypernonagon Kitaev model