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We review our study of critical phenomena in superfluid $^4$He confined in nanoporous glasses. $^4$He in nanoporous media is an ideal ground to survey the quantum phase transition of bosons. In the present work, critical phenomena were examined using a newly developed hydrodynamic mechanical resonator. The critical exponent of superfluid density $\zeta$ was found to be 1.0, in contrast to 0.67 in bulk $^4$He. We also demonstrate that the superfluid density is proportional to $|P - P_{\mathrm c}|^{\zeta_p}$ with $\zeta_p = 1$ at any finite temperatures. These are the decisive evidences for the 4D XY criticality, which should have been observed only at 0 K, at finite temperatures. We propose a mechanism of the quantum criticality at finite temperatures in terms of phase alignment among the nanoscale localized Bose condensates (LBECs) in nanopores. The proposed mechanism are discussed in the consideration of the correlation length compared with the quantum effect.

$^4$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.

$^4$He confined in nanoporous media is a model Bose system that exhibits quantum phase transition (QPT) by varying pressure. We have precisely determined the critical exponent of the superfluid density of $^4$He in porous Gelsil glasses with pore size of 3.0 nm using the Helmholtz resonator technique. The critical exponent $\zeta$ of the superfluid density was found to be 1.0 $\pm$ 0.1 for the pressure range 0.1 < P < 2.4 MPa. This value provides decisive evidence that the finite-temperature superfluid transition belongs to the four-dimensional (4D) XY universality class, in contrast to the classical 3D XY one in bulk liquid 4He, in which $\zeta$ = 0.67. The quantum critical behavior at a finite temperature is understood by strong phase fluctuation in local Bose-Einstein condensates above the superfluid transition temperature. $^4$He in nanoporous media is a unique example in which quantum criticality emerges not only at 0 K but at finite temperatures.

We report measurements of elastic moduli of hcp solid $^4$He down to 15 mK when the samples are rotated unidirectionally. Recent investigations have revealed that the elastic behavior of solid $^4$He is dominated by gliding of dislocations and pinning of them by $^3$He impurities, which move in the solid like Bloch waves (impuritons). Motivated by the recent controversy of torsional oscillator studies, we have preformed direct measurements of shear and Young's moduli of annular solid $^4$He using pairs of quarter-circle shape piezoelectric transducers (PZTs) while the whole apparatus is rotated with angular velocity $\Omega$ up to 4 rad/s. We have found that shear modulus $\mu$ is suppressed by rotation below 80 mK, when shear strain applied by PZT exceeds a critical value, above which $\mu$ decreases because the shear strain unbinds dislocations from $^3$He impurities. The rotation - induced decrement of $\mu$ at $\Omega = 4$ rad/s is about 14.7 (12.3) % of the total change of temperature dependent $\mu$ for solid samples of pressure 3.6 (5.4) MPa. The decrements indicate that the probability of pinning of $^3$He on dislocation segment, $G$, decreases by several orders of magnitude. We propose that the motion of $^3$He impuritons under rotation becomes strongly anisotropic by the Coriolis force, resulting a decrease in $G$ for dislocation lines aligning parallel to the rotation axis.

We study the odd-frequency Cooper pairs formed near the surface of superfluid 3He. The odd-frequency pair amplitude is closely related to the local density of states in the low energy limit. We derive a formula relating explicitly the two quantities. This formula holds for arbitrary boundary condition at the surface. We also present some numerical results on the surface odd-frequency pair amplitude in superfluid 3He-B. Those analytical and numerical results allow one to interpret the midgap surface density of states, observed recently by transverse acoustic impedance measurements on superfluid 3He-B, as the manifestation of the surface odd-frequency state.

The superfluid $^3$He B phase, one of the oldest unconventional fermionic condensates experimentally realized, is recently predicted to support Majorana fermion surface states. Majorana fermion, which is characterized by the equivalence of particle and antiparticle, has a linear dispersion relation referred to as the Majorana cone. We measured the transverse acoustic impedance $Z$ of the superfluid$^3$He B phase changing its boundary condition and found a growth of peak in $Z$ on a higher specularity wall. Our theoretical analysis indicates that the variation of $Z$ is induced by the formation of the cone-like dispersion relation and thus confirms the important feature of the Majorana fermion in the specular limit.