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The quantized vortex state appearing in the recently discovered new states in hcp 4He since their discovery is discussed. Special attention is given to evidence for the vortex state as the vortex fluid (VF) state and its transition into the supersolid (SS) state. Its features are described. The historical expla- nations for the SS state in quantum solids such as solid 4He were based on the idea of Bose Einstein Condensation (BEC) of the imperfections such as va- cancies, interstitials and other possible excitations in the quantum solids which are expected because of the large zero-point motions. The SS state was proposed as a new state of matter in which real space ordering of the lattice structure of the solid coexists with the momentum space ordering of superfluidity. A new type of superconductors, since the discovery of the cuprate high Tc superconductors, HTSCs, has been shown to share a feature with the vortex state, involving the VF and vortex solid states. The high Tcs of these materials are being discussed in connection to the large fluctuations associated with some other phase transitions like the antiferromagnetic transition in addition to that of the low dimensionality. The supersolidity in the hcp solid 4He, in contrast to the new superconductors which have multiple degrees of freedom of the Cooper pairs with spin as well as angular momentum freedom, has a unique feature of possessing possibly only the momentum fluctuations and vortex ring excitations associated with the possible low dimensional fluctuations of the subsystem(s). The high onset temperature of the VF state can be understood by considering thermally excited low D quantized vortices and it may be necessary to seek low dimensional sub-systems in hcp He which are hosts for vortices., Comment: 20 pages, 12 figures, review

A phenomenological model is developed to explain a new set of detailed torsional oscillator data for hcp 4He. The model is based on Anderson's idea of the vortex fluid (vortex tangle) in solid 4He. Utilizing a well-studied treatment of dynamics of quantized vortices we describe how the "local superfluid component" is involved in rotation (torsional oscillation) via a polaried vortices tangle. The polarization in the tangle appears both due to aligning the remnant or thermal vortices and due to penetration of additional vortices into volume. Both are supposed to occur in a relaxation manner, and the inverse full relaxation time tau^-1 is the sum of them. One of them is found to change linearly with respect to the rim velocity Vac. The developed approach explains the behavior of both NLRS and Delta Q^-1 seen in the experiment. We reproduce not only the unique Vac dependence, but also obtain new information about the vortices tangle, for example, a divergence in tau at extrapolated T ~ 30 mK., Comment: 4 pages, 3 figures

A detailed torsional oscillator (TO) study on a stable solid 4He sample at 49 bar with To = 0.5K, is reported to T below the dissipation peak at Tp . We find hysteretic behavior starting below Tc = about 75 mK, in period shift, as well as in dissipation, with changes of AC excitation amplitude Vac . The derived difference of non-linear rotational susceptibility DeltaNLRS(T)hys across the hysteresis loop under a systematic condition is analyzed as a function of Vac and T . We propose that DeltaN LRS(T)hys is proportional to non- classical rortational inertia fraction, NCRIF, and is actually the supersolid density rho_ss of the 3D supersolid state below Tc . rho_ss changes linearly with T down to about 60 mK and then increases much more steeply, and approaching a finite value towards T = 0. We find an AC velocity of 40 micrometer/s beyond which the hysteresis starts at T < Tc and a critical AC velocity, about 1 cm/s, above which rho_ss is completely destroyed. We discuss also the coherence length of the supersolid., Comment: 4 pages, 5 figures

Detailed studies of AC velocity Vac and temperature dependence of torsional oscillator responses of solid 4He are reported. A characteristic onset temperature T0 about 0.5 K is found, below which a significant Vac dependent change occurs in the energy dissipation for the sample at 32 bar. A Vac dependence of the non-classical rotational inertia fraction, NCRIF also appears below about T0. This value of T0 excludes the possible explanation of supersolid by liquid superfluidity in grain boundaries or other liquid related origins. The log(Vac) linear dependence was found in NCRIF. Furthermore, this linear slope changes in proportion to 1/T 2 for 40 < Vac < 400 micro-m/s, then crosses over to about 1/T for larger Vac. We discuss properties of the vortex fluid state proposed by Anderson above Tc, below T0., Comment: 4pages,4 figures