In the underdoped pseudogap regime of cuprate superconductors, the normal state is commonly probed by applying a magnetic field (H). However, the nature of the H-induced resistive state has been the subject of a long-term debate, and clear evidence for a zero-temperature (T = 0) H-tuned superconductor-insulator transition (SIT) has proved elusive. Here we report magnetoresistance measurements in underdoped La 2−x Sr x CuO 4 , providing striking evidence for quantum critical behavior of the resistivity -the signature of a H-driven SIT. The transition is not direct: it is accompanied by the emergence of an intermediate state, which is a superconductor only at T = 0. Our finding of a two-stage H-driven SIT goes beyond the conventional scenario in which a single quantum critical point separates the superconductor and the insulator in the presence of a perpendicular H. Similar two-stage H-driven SIT, in which both disorder and quantum phase fluctuations play an important role, may also be expected in other copper-oxide high-temperature superconductors.The SIT is an example of a quantum phase transition (QPT): a continuous phase transition that occurs at T = 0, controlled by some parameter of the Hamiltonian of the system, such as doping or the external magnetic field 1 . A QPT can affect the behavior of the system up to surprisingly high temperatures. In fact, many unusual properties of various strongly correlated materials have been attributed to the proximity of quantum critical points (QCPs). An experimental signature of a QPT at nonzero T is the observation of scaling behavior with relevant parameters in describing the data. Although the SIT has been studied extensively 2 , even in conventional superconductors many questions remain about the perpendicular-field-driven SIT in two-dimensional (2D) or quasi-2D systems 3 . In high-T c cuprates (T c -transition temperature), which have a quasi-2D nature, early magnetoresistance (MR) experiments showed the suppression of superconductivity with high H, revealing the insulating behavior 4-6 and hinting at an underlying H-field-tuned SIT 7 . However,

The nature of the superconducting transition in highly underdoped thick films of La2−xSrxCuO4 (x = 0.07 and 0.08) has been investigated using the in-plane transport measurements. The contribution of superconducting fluctuations to the conductivity in zero magnetic field, or paraconductivity, was determined from the magnetoresistance measured in fields applied perpendicular to the CuO2 planes. Both the temperature dependence of the paraconductivity above the transition and the nonlinear current-voltage (I − V ) characteristics measured across it, exhibit the main signatures of the Berezinskii-Kosterlitz-Thouless (BKT) transition. The quantitative comparison of the superfluid stiffness, extracted from the I − V data, with the renormalization-group results for the BKT theory, reveals a large value of the vortex-core energy. This finding is confirmed by the analysis of the paraconductivity obtained using different methods. The results strongly suggest that the characteristic energy scale controlling the BKT behavior in this layered system corresponds to the superfluid stiffness of a few layers.