  Long-range spin-spin interactions are known to generate non-equilibrium
dynamics which can squeeze the collective spin of a quantum spin ensemble in a
scalable manner, leading to states whose metrologically useful entanglement
grows with system size. Here we show theoretically that scalable squeezing can
be produced in 2d U(1)-symmetric systems even by short-range interactions, i.e.
interactions that at equilibrium do not lead to long-range order at finite
temperatures, but rather to an extended, Berezhinski-Kosterlitz-Thouless (BKT)
critical phase. If the initial state is a coherent spin state in the easy plane
of interactions, whose energy corresponds to a thermal state in the critical
BKT phase, the non-equilibrium dynamics exhibits critical slowing down,
corresponding to a power-law decay of the collective magnetization in time.
This slow decay protects scalable squeezing, whose scaling reveals in turn the
decay exponent of the magnetization. Our results open the path to realizing
massive entangled states of potential metrological interest in many relevant
platforms of quantum simulation and information processing -- such as Mott
insulators of ultracold atoms, or superconducting circuits -- characterized by
short-range interactions in planar geometries.
