Two-stage magnetic-field-tuned superconductor-insulator transition in underdoped La$_{2-x}$Sr$_{x}$CuO$_{4}$

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.