We review the intriguing many-body physics resulting out of the interplay of a single, local impurity and the two-particle interaction in a one-dimensional Fermi system. Even if the underlying homogeneous correlated system is taken to be metallic, this interplay leads to an emergent quantum phase transition between metallic and insulating states. We show that the zero temperature critical point and the universal low-energy physics associated to it, is realized in two different models, the field theoretical local sine-Gordon model and spinless fermions on a lattice with nearest-neighbor hopping and two-particle interaction, as well as in an experimental setup consisting of a highly tunable quantum circuit. Despite the different high-energy physics of the three systems the universal low-energy scaling curves of the conductance as a function of temperature agree up to a very high precision without any free parameter. Overall this provides a convincing example of how emergent universality in complex systems originating from a common underlying quantum critical point establishes a bridge between different fields of physics. In our case between field theory, quantum many-body theory of correlated Fermi systems, and experimental circuit quantum electrodynamics.
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