  Critical universality classes are ubiquitous in nature. The standard scenario
is that the two-point correlation algebraically decreases with the distance $r$
as $g(r) \asymp r^{2-d-\eta}$ ($d$ is spatial dimensionality). Recently, an
unusual logarithmic universality was put forward to describe the extraordinary
surface transition of three-dimensional O($N$) critical systems, which has long
been theoretically debated. In this logarithmic universality, $g(r)$ decays in
a power of logarithmic distance as $g(r) \asymp [{\rm ln}(r)]^{-\eta}$,
dramatically different from the standard scenario. We explore the
three-dimensional $XY$ model by extensive Monte Carlo simulations, and provide
strong evidence for the emergence of logarithmic universality. Moreover, on the
basis of our simulations, we propose that the finite-size scaling of $g(r,L)$
is described by an ansatz simultaneously containing the $r$-dependent term
$g(r) \asymp [{\rm ln}(r)]^{-\eta'}$ as well as an $r$-independent plateau at
large distance whose height decays logarithmically with $L$ as $g(r) \asymp
[{\rm ln}(L)]^{-\eta}$. These results expand our understanding of critical
universality classes.
