Unveiling quantum effects in materials at low temperatures, such as quantum spin-liquid behavior, have been a central task in condensed matter physics. A two-dimensional kagome antiferromagnet can achieve a spin-liquid ground state due to strong quantum fluctuations and geometrical frustrations, but the topological nature of its ground state and low-energy spin fluctuationsare still controversial. Recently, we have successfully synthesized polycrystalline of Cu3Zn(OH)6FBr with the barlowite type crystal structure. Our specific heat measurements show that there is no phase transition down to 50 mK in Cu3Zn(OH)6FBr and hence provides a unique opportunity to investigate spin-liquid ground state in this compound down to 50 mK using muon spin rotation and relaxation measurements. We would also measure Cu4(OH)6FBr, which exhibits an antiferromagnetic ordering at 15 K for a comparison.