A quantum critical line bounds the high field metamagnetic transition surface in UTe$_2$
Quantum critical phenomena are widely studied across various materials families, from high temperature superconductors to magnetic insulators. They occur when a thermodynamic phase transition is suppressed to zero temperature as a function of some tuning parameter such as pressure or magnetic field. This generally yields a point of instability - a so-called quantum critical point - at which the phase transition is driven exclusively by quantum fluctuations. Here we show that the heavy fermion metamagnet UTe$_2$ possesses a quantum phase transition at extreme magnetic field strengths of over 70 T. Rather than terminating at one singular point, we find that the phase boundary is sensitive to magnetic field components in each of the three Cartesian axes of magnetic field space. This results in the three-dimensional transition surface being bounded by a continuous ring of quantum critical points, the locus of which forms an extended line of quantum criticality - a novel form of quantum critical phase boundary. Within this quantum critical line sits a magnetic field-induced superconducting state in a toroidal shape, which persists to fields over 70~T. We model our data by a phenomenological free energy expansion, and show how a three-dimensional quantum critical phase boundary - rather than a more conventional singular point of instability - anchors the remarkable high magnetic field phase landscape of UTe$_2$.