  Pseudogap formation is an ubiquitous phenomena in strongly-correlated
superconductors, for example cuprates, heavy-fermion superconductors, and iron
pnictides. As the system is cooled, an energy gap opens in the excitation
spectrum before entering the superconducting phase. The origin of formation and
the relevancy to the superconductivity remains unclear, which is the most
challenging problem in condensed matter physics. Here, using the cuprate as a
model, we demonstrate that the formation of pseudogap is due to a massive gauge
interaction between electrons, where the mass of the gauge boson, determining
the interaction length scale, is the consequence of the remnant
antiferromagnetic fluctuation inherited from the parent compounds. Extracting
from experimental data, we predict that there is a quantum phase transition
belonging to the 2D XY universality class at the critical doping where
pseudogap transition vanishes.
