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The helimagnetic compounds MnSi$_{1-x}$Ge$_{x}$ show the three-dimensionalmultiple-$q$ order as referred to as spin-hedgehog-anti-hedgehog (SHAH)lattice. Two representative forms of SHAH are cubic-3$q$ lattice with $q \|\langle100\rangle$ and tetrahedral-4$q$ lattice with $q \| \langle111\rangle$,which show up typically for $x=1.0-~0.8$ and for $x=0.6$, respectively. Here,we have investigated the spin fluctuations in the MnSi$_{1-x}$Ge$_{x}$polycrystalline samples with $x=0.6$ and $0.8$ by using the time-of-flight(TOF) neutron inelastic scattering and MIEZE-type neutron spin echo techniquesto elucidate the microscopic origin of the unconventional Hall effect in theSHAH lattice states. This research is motivated by the observation of a signchange in the unconventional Hall resistivity as a function of temperature [Y.Fujishiro et al., Nat. Comm. $\textbf{10}$, 1059 (2019)]. The present resultsreveal the correspondences between the temperature ranges where the positiveHall resistivity and spin fluctuations are observed. These results agree wellwith the theoretical model of the conduction electrons scattered by thefluctuating spin clusters with a non-zero average of sign-biased scalar spinchirality as a mechanism of the positive Hall resistivity [H. Ishizuka and N.Nagaosa, Sci. Adv. $\textbf{4}$, eaap9962 (2018)].

physical review. b./physical review. b

Direct observations of spin fluctuations in spin-hedgehog-anti-hedgehog  lattice states in MnSi$_{1-x}$Ge$_x$ ($x=0.6$ and $0.8$) at zero magnetic  field