We study the low-energy spin fluctuations and superfluid density of a series of pure and Zn-substituted high-T c superconductors ͑HTS͒ using the muon spin relaxation and ac-susceptibility techniques. At a critical doping state, p c , we find ͑i͒ simultaneous abrupt changes in the magnetic spectrum and in the superconducting ground state and ͑ii͒ that the slowing down of spin fluctuations becomes singular at Tϭ0. These results provide experimental evidence for a quantum transition that separates the superconducting phase diagram of HTS into two distinct ground states. DOI: 10.1103/PhysRevB.66.064501 PACS number͑s͒: 74.72.Ϫh, 74.25.Ha, 75.40.Ϫs, 76.75.ϩi Quantum phase transitions occur at zero temperature at a critical electron density separating distinct ground states. Near a quantum critical point, electrons in metals are highly correlated and the diverging fluctuations may induce unconventional superconductivity. [1][2][3][4][5][6][7][8] For example, in certain heavy fermion compounds a ''bubble'' of superconductivity occurs around the quantum critical point at which itinerant antiferromagnetism is suppressed by applied pressure. 9 The search for an underlying quantum phase transition in high-T c superconductors ͑HTS͒ is motivated by the potential for quantum fluctuations to bind electronic carriers into superconducting Cooper pairs and also to cause the celebrated linear temperature dependence of their electrical resistivity. [1][2][3][4][5][6][7][8]10 HTS exhibit a common generic phase diagram in which the superconducting transition temperature, T c , rises to a maximum at an optimal doping of approximately 0.16 holes per planar copper atom and then falls to zero on the overdoped side. In addition the underdoped normal state exhibits correlations, which introduce a gap in the density of states that strongly affects all physical properties. There is no phase transition associated with the opening of this gap and so it is called a pseudogap. Analysis of specific heat data, for example, suggests that the pseudogap energy decreases with doping and falls to zero at a critical doping of p c Ӎ0.19, just beyond optimal doping, 10,11 a behavior rather analogous to the quantum-critical heavy-fermion materials. 9 Many fundamental physical quantities such as the superconducting condensation energy, 10,11 the superfluid density, 12,13 and the quasiparticle weight, 10,14 show abrupt changes as p→p c . While compelling in their totality, 10,11 none of the results can be considered as evidence of a quantum transition. In particular there is no evidence for an associated order parameter and slowing down of the relevant fluctuations. With this in mind we examined the evolution with doping of the low-energy spin fluctuation spectrum using muon spin relaxation ( SR) combined with low-field ac-susceptibility measurements of the superfluid density.The samples studied were: ͑i͒ La 2Ϫx Sr x Cu 1Ϫy Zn y O 4 ͑LSCO͒ (xϭ0.03-0.24 and yϭ0, 0.01, and 0.02͒. Samples were synthesized using solid-state reaction and where necessary follow...