  A central quantity of importance for ultracold atoms is contact, which
measures two-body correlations at short distances in dilute systems. It appears
in universal relations among thermodynamic quantities, such as large momentum
tails, energy, and dynamic structure factors, through the renowned Tan
relations. However, a conceptual question remains open as to whether or not
contact can signify phase transitions that are insensitive to short-range
physics. Here we show that, near a continuous classical or quantum phase
transition, contact exhibits a variety of critical behaviors, including scaling
laws and critical exponents that are uniquely determined by the universality
class of the phase transition and a constant contact per particle. We also use
a prototypical exactly solvable model to demonstrate these critical behaviors
in one-dimensional strongly interacting fermions. Our work establishes an
intrinsic connection between the universality of dilute many-body systems and
universal critical phenomena near a phase transition.
