  Universality is a pillar of modern critical phenomena. The standard scenario
is that the two-point correlation algebraically decreases with the distance $r$
as $g(r) \sim r^{2-d-\eta}$, with $d$ the spatial dimension and $\eta$ the
anomalous dimension. Very recently, a logarithmic universality was proposed to
describe the extraordinary surface transition of O($N$) system. In this
logarithmic universality, $g(r)$ decays in a power of logarithmic distance as
$g(r) \sim ({\rm ln}r)^{-\hat{\eta}}$, dramatically different from the standard
scenario. We explore the three-dimensional $XY$ model by Monte Carlo
simulations, and provide strong evidence for the emergence of logarithmic
universality. Moreover, we propose that the finite-size scaling of $g(r,L)$ has
a two-distance behavior: simultaneously containing a large-distance plateau
whose height decays logarithmically with $L$ as $g(L) \sim ({\rm
ln}L)^{-\hat{\eta}'}$ as well as the $r$-dependent term $g(r) \sim ({\rm
ln}r)^{-\hat{\eta}}$, with ${\hat{\eta}'} \approx {\hat{\eta}}-1$. The critical
exponent $\hat{\eta}'$, characterizing the height of the plateau, obeys the
scaling relation $\hat{\eta}'=(N-1)/(2\pi \alpha)$ with the RG parameter
$\alpha$ of helicity modulus. Our picture can also explain the recent numerical
results of a Heisenberg system. The advances on logarithmic universality
significantly expand our understanding of critical universality.
