  $^4$He confined in nanoporous Gelsil glass is a unique, strongly correlated
Bose system exhibiting quantum phase transition (QPT) by controlling pressure.
Previous studies revealed that the QPT occurs with four - dimensional (4D) XY
criticality, which appears in the zero-temperature limit of the superfluid
density. However, the $P-T$ phase diagram also suggested that 4D XY nature
appears at finite temperatures. Here, we have determined the critical exponent
of the superfluid density of $^4$He in two Gelsil samples that have pore
diameter to be about 3 nm, using a newly developed mechanical resonator
technique. The critical exponent $\zeta$ in the powerlaw fitting $\rho_{\mathrm
s} \propto \left| 1 - T/T_{\mathrm c} \right| ^{\zeta}$, where $T_{\mathrm c}$
is the superfluid transition temperature, was found to be 1.0 $\pm$ 0.1 for all
pressures realized in this experiment, 0.1 $<$ $P$ $<$ 2.4 MPa. This value of
$\zeta$ gives a decisive evidence that the finite-temperature superfluid
transition belongs to 4D XY universality class. The emergence of the 4D XY
criticality is explained by the existence of many nanoscale superfluid
droplets, the so called localized Bose - Einstein condensates (LBECs), above
$T_{\mathrm c}$. Due to the large energy cost for $^4$He atoms to move between
the LBECs, the phase of the LBEC order parameters fluctuates not only in
spatial (3D) but imaginary time ($+1$D) dimensions, resulting in the 4D XY
criticality by a temperature near $T_{\mathrm c}$, which is determined by the
finite size of the system in the imaginary time dimension. Below $T_{\mathrm
c}$, macroscopic superfluidity grows in the nanopores of Gelsil by the
alignment of the phases of the LBEC order parameters. An excess dissipation
peak observed below $T_{\mathrm c}$ is well explained by this phase matching
process.
