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The superconducting ground state in a large number of two-dimensional (2d)systems can be created and destroyed through quantum phase transitions (QPTs)driven by non-thermal parameters such as the carrier density or magnetic field.The microscopic mechanism of QPTs has not been established in any 2dsuperconductor, in part due to an emergent scale-invariance near the criticalpoint, which conceals the specific processes driving the transitions. In thiswork, we find that the pair-breaking mechanism causing the suppression of theCooper pair density gives a unifyingly consistent description ofmagnetic-field-driven QPTs in amorphous MoGe, Pb and TaN films, as well as inquasi-2d high-temperature superconductor La$_{1.92}$Sr$_{0.08}$CuO$_{4}$. Thisdiscovery was facilitated by the development of a novel theoretical approach,one which goes beyond the standard determination of critical exponents andallows for the extraction of a microscopic seeding length scale of thetransitions. Remarkably, for the materials studied, and also for MoGenanowires, this scale matches the superconducting coherence length. Further,this approach has been successfully applied to many other complex,non-superconducting systems.

Microscopic scale of pair-breaking quantum phase transitions in superconducting films, nanowires and La$_{1.92}$Sr$_{0.08}$CuO$_{4}$