Copper metaborate CuB$_2$O$_{4}$ was studied by muon spin relaxation measurements in order to clarify its static and dynamic magnetic properties. The time spectra of muon spin depolarization suggest that the local fields at the muon site contain both static and fluctuating components in all ordered phases down to 0.3 K. In the weak ferromagnetic phase (20 K~$>T>$~9.3 K), the static component is dominant. On the other hand, upon cooling the fluctuating component becomes dominant in the incommensurate helix phase (9.3K > T > 1.4K). The dynamical fluctuations of the local fields persist down to 0.3K, where a new incommensurate phase (T < 1.4K) is expected to appear. This result suggests that spins fluctuate even at T to 0. We propose two possible origins of the remnant dynamical spin fluctuations: frustration of the exchange interactions and the dynamic behavior of the soliton lattice.
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