Spin-glass magnetism confined to individual weakly interacting vortices is detected in two different families of high-transition-temperature (T_c) superconductors, but only in samples on the low-doping side of the low-temperature normal state metal-to-insulator crossover (MIC). Our findings unravel the mystery of the MIC, but more importantly identify the true location of the field-induced quantum phase transition (QPT) in the superconducting state. The non-uniform appearance of magnetism in the vortex state favours a surprisingly exotic phase diagram, in which spatially inhomogeneous competing order is stabilized at the QPT, and an `avoided quantum critical point (QCP) is realized at zero magnetic field.
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