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4D-XY Superfluid Transition and Dissipation in $^4$He Confined in Nanoporous Media

$^4$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 $ζ$ in the powerlaw fitting $ρ_{\mathrm s} \propto \left| 1 - T/T_{\mathrm c} \right| ^ζ$, 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 $ζ$ 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.