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Quantized vortex state in hcp solid 4He

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.


Introduction
Quantized vortices are the essence of superfluidity 16 and superconductivity (superfluidity of electrons). They originate in the coherence of the wave function describing the system over macroscopic length scales much larger than the vortex core sizes. In the "classical" superfluids, the macroscopic coherence has been provided by the Bose-Einstein Condensation (BEC) of the constituent Bose particles 17 , and that of Cooper pairs 18 , in the case of Fermion systems. The superfluidnormal fluid transition has been discussed since the classical discussion by Feynmann 19 as a phase transition by the destruction of the macroscopic coherence by excitations of quantum vortex loops, and it is discussed much more concretely by a simpler vortex rings model by Williams 20 . More recently, since the discovery of the cuprate high T c super-conductors, HTSCs 15 , people realized that we do have the quantized vortex state, which is characterized by the complex T −H phase diagram of the new superconductors, namely, a VF state and different types of vortex solid states have been realized. All the "new type of superconductors" possess a common feature, namely the unique vortex state 21 . It is also true for the newly found FeAs based superconductors 22 . Early discussions for the vortex state in the cuprates argued for the importance of the 2D character of the CuO 2 plane Cooper pairs for the cuprate HTSCs 23 . This kind of appearance of the vortex state in the new superconductors, the strongly correlated electron systems, is often discussed nowadays in connection to the 2D pancake-type vortices 24 , but is never discussed for any mass superfluidity nor for the physics of solid He with a single exception 25 , though solid He is actually a candidate for being one of the most strongly correlated systems.
Ideal 2D superfluidity and the physics of 2D vortices had been discussed for the superfluidity of the 2D 4 He films by Berezinskii 26 , Kosterlitz and Thouless 27 and lead to the idea of topological BKT phase transition, where real BEC is absent, but 2D quantized vortices and local condensate were implicitly supposed. Its unique features, the 2D density linear transition temperature T KT and the universal jump 28 in the superfluid density are clearly experimentally confirmed 29,30 . A detailed discussion of the dynamics of the KT systems was given by Ambegaokar, Halperin, Nelson, and Siggia 31 . KT superfluidity of the CuO 2 plane electron Cooper pairs and Josephson coupling between the layers are the essence of the HTSC vortex physics. A unique vortex state, first discussed for the cuprate HTSCs supposes thermal excitations 23 of low dimensional quantized vortices, which have low enough energy and high entropy in high fields.
Readers are recommended to refer to a longer paper by the present author on the vortex physics in hcp 4 He (I), which includes a discussion on the essence of superfluidity and the vortex physics in 3D He film systems, where detection of 3D vortex lines penetration by torsional oscillator (TO) technique is also described 32 .

"Possible supersolidity" and Quantized vortex physics in hcp 4 He
The discussion of possible superfluidity in the quantum solid, the SS state, had historically assumed a non-negligible amount of vacancies, interstitials, and other possible excitations in the quantum solid 11,12,13 , because of the zero point motions in such systems and the possible Bose Einstein condensation BEC of these quasi particles or excitations. It was believed to certainly occur at some sufficiently low temperature. In 1970 no other possible mechanism for superfluidity in the solid was known. The BKT mechanism 27,28,29,31 for a 2D system was first introduced in 1972 and the lack of knowledge of other mechanisms like those possible in 1 D systems lead to this situation. It was later when a 1D dislocation network was considered as a candidate explanation for a possible SS state 33,34 .
New development started when Kim and Chan reported their new experimental results using torsional oscillator TO technique, 1,2 which raised the question whether or not they finally found the long sought, non-classical rotational inertia, NCRI 13 of the SS state.There have been frequent international workshops since 2006 when other groups started confirming results, yet the system remains still mysterious even after so many years of work by an increasing number of groups. By now several reviews 35,36,37,38 have appeared since the restart by Kim and Chan 1,2 . The main problem is that nobody knows what a SS state should look like. There are plenty of interesting aspects of the solid as well in the quantum solid He. First of all, Day and Beamish 39 reported an increase of the shear modulus towards lower temperatures near 200 mK, almost the same temperature range as TO anomalies had been reported. Substantial change of the properties by a minute amount of 3 He is also reported since the beginning 1 . A more detailed discussion of these peculiar behaviors is left for later in the discussion chapter. In the present chapter discussion is concentrated on the problem of whether quantized vortices are present in the system or not and if there is enough evidence supporting supersolidity and related phenomena in solid He.
Our discussion started with solid He, but now we limit ourselves to the hcp phase of the solid 4 He, because of the following development: The shear modulus increases below around 200 mK for hcp 4 He 39 . A similar change was also found 40 in hcp 3 He, but the TO response was found only in hcp 4 He.

Quantum Vortex State in hcp 4 He
Although enough details of the microscopic mechanism responsible for what is occurring in hcp 4 He are still not available to explain all the all the experimental observations, the existence of quantized circulations or quantized vortices can be shown by experimental evidence and it should be regarded as fundamental to determine the direction of the research activities. In this chapter experimental discoveries are discussed which are supporting quantized vortices in the VF state in hcp solid 4 He, a unique state among all the known superfluids. It is, however, quite common to discuss the VF state 3,4 in the so called "new type of superconductors" 41,42 , though there have not been many microscopic arguments from the standpoint of quantized vortices in the new type of superconductors. Anderson 25 has recently proposed an explanation which may cover a large portion of various experimental observations starting from the VF state picture.
The next subsection of this chapter starts with experimental discoveries of the peculiar TO drive velocity V ac dependence observed in hcp solid 4 He (Fig. 1), which turned out to indicate the unusual behavior of the VF state and that of onset of the VF state. Subsection 2.3 deals with vortex dynamics in the VF state, utilizing the standard theoretical method of handling the tangled vortices in superfluid turbulence 6 . In chapter 3 the recent discovery of the transition from the VF state into the real SS state 7,8,9 is reported and its properties are discussed in terms of the SS density ρ ss and the critical velocity V c .
2.2 Discovery of the onset temperature T o of the Vortex Fluid(VF) State in hcp 4 He and its unique properties Actually the vortex phase diagram of hcp 4 He was first experimentally proposed by reporting a definite onset temperature by Penzev et al. 5 , after Anderson's suggestion 3,4 and by describing unique features of the VF state, which appeared to be characterized by thermally excited vortex fluctuations 6 . These thermally excited fluctuations cause the unique drive amplitude dependence of the TO response 5 . Fig 1 shows a detailed study of hcp 4 He at T above the dissipation peak at T p . The V ac at the outer edge of the cylindrical sample influences both Period P shift ∆ P/P load and energy dissipation in the sample δ quite a lot, but only below an "onset temperature" T o . As noted before, although this strong V ac dependence was reported even in the original work by Kim and Chan 1 nobody discussed quantitatively its dependence, but refer to it as a sort of "critical velocity" phenomenon. This speed, on the order of 10µm/s, needs to be discussed as a characteristic speed 5 to describe system size fluctuations instead of a "critical velocity" in the later section on vortex dynamics in the VF state. The VF state may not have a critical velocity. In the "onset" paper 5 the authors discussed T o as the onset temperature of the VF state, below which quantized vortices are thermally excited and that NLRS exhibits a unique temperature dependence, which is different from any of the order parameters below a phase transition. Furthermore, they observed rather clear log(V ac ) linear behavior 43,5 (see also Fig 8) over a decade of V ac for the entire T range well above the dissipation peak at T p . This dependence is argued to be responsible either for appearance of the quantized vortex lines penetrating the sample, or polarizing the vortex rings in the VF state, in both cases, thereby reducing the disorder of the thermally excited VF state. This observation is unique compared with all other TO or other oscillator experimental results for known superfluid transitions, KT transition of 2D films 29,30 , bulk He liquids in Vycor 44 , and also in the artificial 3D superfluid made of 4 He monolayer superfluid films 45,46,32 , where energy dissipation ∆ Q −1 is larger when the oscillation amplitude is driven larger. Their intuitive discussion is also proven in terms of vortex dynamics in the randomly fluctuating vortex tangle 6 . The transition into the SS state 7,8 will be discussed, which is characterized by a unique order parameter and the critical velocity in the next chapter 3.

Vortex dynamics in the VF state and possible disappearance of the thermal excitations
The VF state has been discussed with regard to the H-T phase diagram of the new type of superconductors since discovery of the cuprate HTSCs, yet, there has not been a quantitative discussion from the quantized vortex dynamics viewpoint. There are, however, arguments developed for the vortex dynamics in the superfluid turbulence, which often deals with physics at T =0K, where no thermal excitations exist. The superfluid turbulence state is excited by various methods, but it is excited by some external flow velocity exceeding a characteristic critical velocity.
The VF state, on the other hand, is excited by thermal energy and it is considered that the thermal energy can exceed the critical velocity of the turbulent state for the vortex tangle in the VF state. As has been seen in the previous section, a peculiar feature of the VF state of hcp He is that some fluctuations can be reduced by the TO drive excitations 5 , namely larger drive velocity reduces the fluctuating signal more than smaller drive excitations, as can be seen in Fig 1 and Fig 4, for example. This is actively studied and there are references for it. Nemirovskii et al. 8,6 , consider quantized vortex element dynamics for the experimental data analysis of the TO responses of the VF state in hcp 4 He, using a phenomenological relaxation model; Angular momentum of the superfluid fraction appears only due to the presence of either aligned vortices (vortex array) or the polarized vortex tangle having nonzero total average polarization P =L<s ′ (ξ )> along the applied angular velocity Ω (axis z, the magnitude of Ω = V ac /R, where R is radius of the sample). Here L is the vortex line density (total length per unit volume), s(ξ ) is the vector line position as a function of label variable ξ and s ′ (ξ ) is the tangent vector. In the steady-state case there is a strictly fixed relation between the total polarization L<s ′ (ξ )> and applied angular velocity Ω , (2.1) They 8,6 pointed out that there are two possible mechanisms for relaxation-like polarization of the VF. The first is alignment of elements of the vortex lines due to interaction with the normal component (see Tsubota et al.(2004) 47 for detailed explanation). This interaction (mutual friction) is proportional to the local normal velocity, which in turn is proportional to the rim velocity V ac . Thus it is natural to suppose that polarization P of the vortex tangle due to alignment of filaments along Ω (t) occurs with typical inverse time τ −1 (V ac ) which is proportional to the rim velocity V ac . Let us illustrate according to Nemirovskii et al., with consideration of Tsubota et al.(2004) 47 . In the presence of mutual friction there is a torque acting on the line and the angle φ between axis z and the line element changes according to the equation dφ /dt = α(V ac /R) sinφ (α is the friction coefficient, dependent, in general, on T and pressure p). Except for a short transient, the solution to this equation can be described as a pure exponential ∼ exp(-t/τ 1 (V ac )), with the velocity dependent inverse time τ 1 −1 (V ac ) ∼ α(V ac )/R. Thus, it is concluded that during time-varying rotation or torsional oscillation vortex filaments tend to align along the angular velocity direction. However, there can be not enough pre-existing vortex lines in the tangle to involve all the superfluid part into the rotation to satisfy the relation (2.1), or on the contrary the initial vortex tangle can be excessively dense.
In such a case deficit (extra) vortices should penetrate into (leave from) the bulk of the sample. This penetration occurs in a diffusion-like manner 48 and leads to the relaxation-like saturation of the vortex line density L(t) Nemirovskii et al. 8,6 assume that this saturation occurs in an exponential manner with some characteristic inverse time τ −1 2 = β . Due to linearity of the diffusion process it is supposed that the coefficient β is velocity independent, but can be a function of T and P. Combining these two mechanisms it is assumed that the total polarization of the VF occurs in a relaxation-like manner with a pure exponential function φ (t ′ /τ) ∼ exp(t ′ /τ). The inverse time τ −1 of relaxation is just the sum of τ −1 1 (V ac ) and τ −1 2 : The parameters α, β , and ρ s (T )≡NLRS 0 (T )=∆ P/∆ P load at V ac →0, obtained using experimental data analysis for a 49 bar hcp 4 He sample, are given in Fig. 2. The ∆ Q −1 and NLRS at different T could be well reconstructed from these parameters and plotted in Fig. 3 of Nemirovskii et al. 6 . ρ s (T ) can be regarded as the VF state SS density and it will be contrasted with the real SS density ρ ss (T ) which will be discussed in chapter 3. Now let us consider what all these results of the measurements as well as the data analysis mean. An interesting observation of Fig. 2 α and especially β is that they are approaching zero when T approaches about 30 mK while ρ s is gradually increasing towards lower temperatures. That means the relaxation time τ is becoming infinite and vortices are becoming difficult to move, while ρ s is high. This situation reminds us of the dissipation peak appearing near the KT transition 31 , where it is interpreted that the thermally excited 2D vortices loose number density rapidly and the dissipation peak is localized near the transition at T KT , while the superfluid density remain towards T =0. The VF state may have lost the thermal activation below T =∼30 mK.
Another interesting observation for the VF state of hcp 4 He is expressed in Fig. 3. The logV ac linear behavior at larger V ac values than about 40µm/s linear velocity has been already discussed, and now let us look at the situation when V ac approaches zero. The line through the data points in Fig. 3 is the Langevin func- with a = 0.0878±0.0011, and b = 0.0148±0.0004. Various other possible fits were tried, but they failed, for example, with x = 1/T . The origin of this interesting temperature dependence is not yet known. Actually this 1/T 2 dependence appears repeatedly in the following analysis as well. The Langevin function with x = 1/T is followed by the generalized Fig. 4 The log(V ac ) dependence of NLRS=∆ P/∆ P load of our solid 4 He sample at 49 bar at constant temperatures as given in the figure. Measurements were performed at each T starting at the largest drive which corresponds to 1,800µm/s and then to lower drive velocity, stepwise. After reaching the minimum drive velocity measurement at ∼27µm/s and after observing equilibrium, then upwards at each T for the remaining measurements. Appearance of the hysteretic behavior is found starting at 74 mK and lower temperatures. Limited data are shown for clarity.
susceptibility of an ensemble of classical dipoles with fixed number. It is not clear if the good fit is just accidental or not.
If one were to consider the 3D counterpart 3,4 of BKT theory 26,27 , one would need to consider the behavior of the ensemble of vortex rings, which are thermally excited in local domains, and a transition into the SS state would occur because of the coherence length growth towards zero temperature of the low dimensional subsystems, just as in the 3D superfluid made of He films 45,46 .
3 Discovery of the transition into the SS state from the VF state in hcp 4 He: Hysteresis found below a fixed temperature, T c When Anderson proposed the vortex fluid state to explain the reported experimental observation until that time, he pointed out that the real SS state transition should be located at much lower temperature 3,4 .
The occurrence of the unique hysteresis in the TO experiments with solid 4 He samples when the excitation velocity V ac was changed at low temperature had been reported first by Kojima's group, 49 and then by Chan's group, 50 , and then by Reppy's group. 51 They could not determine that the hysteresis is connected with a definite transition at a fixed temperature. Shimizu et al., 7 found for the first time that the hysteresis occurs below a characteristic temperature T c in solid He. They 8,7,9 propose from their detailed study of the hysteretic behavior that the hysteretic component of the NLRS, NLRS hys is actually additive to the NLRS quantity of the VF state. NLRS is found to continue to grow towards lower temperatures from the onset temperature T o as shown in Fig. 1. It has a unique, continuous temperature dependence towards T = 0 K as depicted in Fig. 3. The additive behavior of the hysteretic component can be also seen in Fig. 4. . The expected SS state should be characterized by a definite order parameter, supersolid density ρ ss , in addition to a critical velocity v c to destroy this state. These points are discussed in much more detail in the following section.

Hysteretic component of the TO responses and possible SS density
As pointed out above, the hysteretic components of the TO response are found to be additive to the response of the VF state. See Fig. ??. Shimizu et al.,7 as well as Kubota et al., 8,9 tried to analyze the hysteretic components separately from those for the VF state. Fig, 5 shows the dissipation change ∆ Q −1 hys (upper column) and the ∆ NLRS hys change across the hysterisis vs logV ac (lower column). The lower branch of the NLRS hys is named as the "equilibrium" state and the upper branch as the "excited" state branch. Fig, 5 shows the difference between the "excited" state and "equilibrium" state. The actual experimental method is described in detail in Kubota et al., 9 . Measurements are performed at a fixed temperature and after equilibrium is established first at high V ac of the TO and then going to lower excitations, step wise after reaching "equilibrium" or steady state at each step, to the lowest AC velocity on the order of 8 to 30µm/s.Then V ac is increased, also stepwise, up to the maximum speed 1,800µm/s. What is observed in Fig, 5 is a characteristic V ac value of about 40µm/s, above which hysteretic components develop. NLRS hys reaches a maximum value at 200 hys shows negative changes. This negative change indicates dissipation in the "excited" state is less than in the "equilibrium" state and implies that vortices are expelled from the sample. Shimizu et al. 7 , as well as Kubota et al. 8,9 , plotted the maximum of this NLRS hys (at V ac =200µm/s) as a function of temperature together with the NLRS extrapolated to V ac = 0 in Fig. 6. While NLRS of the VF state gradually appears as 1/T 2 below T o ∼500 mK, the hysteretic component NLRS hys (at V ac =200µm/s) starts to appear much more sharply below the temperature T c =75 mK and then changes slope around 60 mK. Furthermore, they 7,9 checked this NLRS hys (at V ac =200µm/s) in various ways and found an interesting feature that they could fit as (1-T /T c ) γ with γ=2/3 by choosing T c =56.7 mK. This is the critical behavior expected for a 3D superfluid transition described with 3D XY model. Using the Josephson's length relation 52,53,54 ξ = κ k B T c /ρ s , one can evaluate the coherence length for the SS state as plotted in Fig. 7, for the hcp He 49 bar sample as a function of reduced temperature, 1 − T /T c with T c =56.7 mK. The inset of the figure compares the Josephson's coherence length ξ for the VF state NLRS and for the SS state NLRS hys (at V ac =200µm/s). It is seen that the coherence length evaluated in the same manner differs for the VF state and for the SS state. The former is shorter for the same temperature. This may give some clue for the relation between the VF and the SS states, namely the SS state develops a macroscopic scale coherence, while the length of the VF state coherence is limited. From all of these trials, it is proposed that NLRS hys (at V ac =200µm/s) is actually the real SS density ρ ss . This proposal will be considered later once again in connection with vortex lines penetration phenomena under DC rotation. In connection to the above question of the relation between the VF state and the SS states and the transition between them, the details of the VF state NLRS behavior as a function of V ac will be checked once again in the next section.

Further detailed study of the logV ac linear dependence study of the VF state NLRS and transition into the SS state
Having seen that the hysteretric behavior starts at a finite temperature and the excited state shows a higher value of the NLRS than the equilibrium state, which seems to be a continuation of the VF state from higher temperatures to lower temperatures, it is questioned if the VF state itself may undergo some change at T c . Kubota et al. 9 studied in detail the V ac dependence of the NLRS in the "equilibrium" state. Fig. 8 shows a logV ac linear relation over a decade of V ac value for each of given T . It then changes slope and again follows a logV ac linear relation for each temperature studied between 50 mK and 300 mK. The slope of the logV ac linear relation for each temperature was plotted versus a variety of temperature functions and an interesting relation was found for the slope as well as the turning point for NLRS as seen in the plot in Fig. 9. It was found that the initial slope varies linearly with 1/T 2 dependence and then it makes a jump and continues to follow another linear relation, whereas the NLRS turning point value is also found to follow a linear relation over some 1/T 2 range and then jumps to another linear relation. The jump is happening at 1/T 2 = 0.00025 ± 0.0004K 2 , or T = 59 -69 mK, which coincides with the T c within the error.
Actually the logV ac linear dependence may be expected, as originally pointed out by Anderson 3 , if a tangled quantized vortex state is to be polarized by the excitation or to form linear vortex lines so this result may be related to the polarizability change in the VF state at T c in addition to the hysteretic component appearance discussed in section 3.1. There remains a question as to the origin of the peculiar 1/T 2 dependence. Anderson 55 pointed out that it may be related to a quantum phase transition, but it needs further study.  7,9 . It is observed that the logV ac linear relation is followed over some range of V ac , starting at a certain V ac value. Then some other logV ac linear like dependence appears at a higher V ac range for each of data at a T . Expanded details are discussed by Kubota et al. 9 , in its Fig.4.and 5. The logV ac linear relation was discussed for solid 4 He first by Anderson 3,4 as evidence of quantized vortex lines involvement in the physics of solid 4 He, and then was experimentally demonstrated by Penzev et al. 5 for polarizing a tangled state of the VF state. Fig. 9 The turning point values of NLRS as well as those of the velocity V ac are plotted vs 1/T 2 . The turning point seems to show considerable regularity. A transition was found at 1/T 2 = 0.00025 ± 0.00004 K 2 , or T = 59−69 mK, which coincides with other T c determinations.

Experimental evidence for the quantized vortex lines penetration into the SS state under DC rotation
Straight quantized vortex lines are introduced into a macroscopic coherent state of a superfluid in a cylindrical vessel under DC rotation above a certain angular velocity Ω c1 56 , and at large enough Ω >> Ω c1 , the number of vortices would be changing in proportion to the angular velocity Ω , n v ∝ Ω until it comes close to Ω c2 , which is usually far beyond experimentally achievable speed of rotation in the roughly centimeter sized vessels used for the bulk 4 He. One needs a much larger size for the vortex core than 10 µm to reach the condition beyond Ω c2 57 . The state beyond Ω c2 means that vortex cores overlap each other and destroy the superfluidity of the dimensionality defined for the vortex core. It may not necessarily destroy all the coherence of the system, but that of the dimension to which the vortex core is connected. In the present discussion of rotation in a 3D system, DC rotation beyond Ω c2 (3) would destroy the 3D superfluid state. This consideration becomes important when various dimensional subsystems are involved in a single superfluid system. For example, the 2D vortex core diameter is found to be 2.5 nm for the monolayer film superfluid 58 and there is an independent 3D vortex core size which is found to be determined by the controlled pore size of the porous glass substrates 54 . It had been proposed to detect vortex lines through the SS state of solid 4 He by TO technique, observing the fact that solid 4 He also shows evidence of thermal excitations, namely the dissipation peak, presumably vortex rings excitations 59 .
An experimental tool to realize vortex lines penetration is rotation of the vessel. The ISSP high speed rotation cryostat(ISSP-HSRC), with which vortex penetration as well as rejection phenomena are being studied in the SS state of hcp 4 He, is described in Yagi et al. 60 . It also reviews how vortex lines in different superfluids have been observed and discusses features of the ISSP-HSRC in detail.
Since the first report by Kim and Chan, one of the most specific features of the solid He TO responses is the sensitivity to minute AC speed, V ac on the order of 10µm/s 1,2 . It is the linear edge speed which seems to be important and not amplitude or acceleration according to Aoki, Graves, and Kojima 49 . The sensitivity to the linear velocity of the TO responses is also reproduced in our measurements (Fig. 1, Fig. 4, and so on). Based on these observations it is essential to minimize the variations in the rotational speed, in addition to the direct vibrations of the TO itself, for the successful TO study of solid 4 He. This is easily seen by taking a 10 mm inner diameter torsion bob in which sample He is contained and DC rotational speed of 1 radian/sec with 10 −3 stability. This situation alone would already cause variations on the order of 6.3µm/s linear velocity in addition to the usual V ac . This size of AC velocity change would already cause significant modification of the TO responses. One needs to be concerned about actual stability of the rotational speed apart from the average drive velocity accuracy. If the sample diameter were larger then the linear velocity effect would be further enlarged and would easily cause misleading TO results under DC rotation. There is some idea of the constancy of ISSP rotating cryostats DC angular speed since they have been performing both dilution refrigerator temperature TO experiments on solid He and nuclear demagnetization T range experiments on superfluid 3 He 61,62 . They record heat leak under DC rotation on the order of 10 nW for the superfluid 3 He experiments under world record rotational speeds up to 12 radian/sec with our first rotating frame. 63,64 and cryostat 64 .

Vortex lines penetration experiments in hcp 4 He
With a specially designed rotating cryostat which does not have a 1K pot for the purpose of condensation of circulating 3 He for the dilution refrigerator, experiments under DC rotation have been performed for quite some time.  46,65,66,63,64,60 . NLRS(upper frame (a)) is plotted against T for three AC drive linear velocities at the cell edge under DC rotational velocities indicated in revolution/sec (rps). Details of the actual experiments are described elsewhere 67 . The SS density or NLRS does not change under DC rotation under the present experimental conditions. Evidence of the vortex lines penetration through the sample is given (lower frame (b)) by the change in the energy dissipation under DC rotation at constant velocities as given in the figure below a certain temperature, actually below 75 mK, exactly below T c , derived from the start of hysteretic behavior 7,9 . Detailed Ω as well as T dependence analysis will be discussed in the text.
Let us consider how vortex lines penetrating a superfluid can be studied experimentally. We proposed a method for solid 4 He to detect the vortex lines penetration events 59 by TO technique, detecting extra energy dissipation caused by superflow around the vortex core and the interaction with thermal excitations in the system, similar to the experimental method used for the 3D He film system 46 . From the first evidence for the penetration by vortex lines as presented at the workshop in Trieste 68 , Fig.'s 10(a), (b) show NLRS and ∆ Q −1 under DC rotation with given rotational speed for three different V ac 's. While NLRS follows the already discussed V ac dependence 5 it does not change as DC rotational speed changes within the experimental error (see Fig. 10(a)), ∆ Q −1 data in Fig. 10(b) show a significant change below T ∼75 mK according to the DC angular velocity Ω change for each of three different AC excitation velocities V ac 's. The Ω dependent change in the ∆ Q −1 does not seem to depend so much on V ac . Fig. 11 shows the almost linear dependence of ∆ Q −1 for each of temperature T . This behavior certainly supports the picture of the vortex lines in the SS state of hcp 4 He. Let us check if this is right by comparing the expected T dependence of such extra energy dissipation. Following the same analysis as for the 3D He film superfluid 46 , the expected extra energy dissipation caused by the interaction between the superflow around each of the vortex lines and the thermal excitations in the system would be expressed as follows: under DC rotation for the given T s. Ω linear dependence is observed for each T until ≈ 0.9 rad/s rotational speed. The T dependence of the slope in this figure may represent T dependence of the SS density. See text below and the next section. Further detailed analysis according to our experiences 46,59 is in progress as well.
where ∆ Q −1 Ω is the extra energy dissipation under DC rotation, n v is the number of vortex lines penetrating the sample, and ρ ss (T ) is the SS density, which represents the macroscopic coherence over the whole sample as a function of T . Therefore the simplest idea of its T dependence comes from that of ρ ss (T ).

SS Density ρ ss (T )
It is discussed briefly how the value of the SS density ρ ss (T ) could be derived from at least two independent experimental quantities, d∆ Q −1 Ω /dΩ and NLRS hys in 4.1. Now, how does this appear as evidence for the real supersolidity in hcp solid 4 He? Fig. 12(a) on the left indicates the slope d∆ Q −1 Ω /dΩ of Fig. 11 as a function of T , whereas (b) on the right the hysteretic component NLRS h ys discussed in Chapter 3 is displayed. The former is obtained from the data under DC rotation and the other is data obtained without rotation. A significant resemblance between them is observed, namely, both quantities appear below the same temperature near 75 mK and furthermore, linear T dependence below 75 mK continues until some temperature slightly below 57 mK and then increases towards lower T .
It is recalled that the hysteretic component of NLRS has shown a critical behavior as seen in Fig. 7. The critical behavior of the SS density with the critical exponent 2/3 with T c = 56.7 mK, consistent with 3D XY model, suggests a second order phase transition on one hand, but the existence of the hysteresis implies a first order phase transition. Shimizuet al. 7 have discussed a possible weak first order phase transition as the type of transition from the VF into the SS state 9 with critical behavior. As has been seen in section 4.2 there is a hint of this kind of transition also in the logV ac linear dependence jump in the slope as a property of the VF state as in Fig. 8, 9. All these observations tell us that the transition between the VF and the SS states is a unique phase transition. Further study is needed. Please note, in Fig. 12, the left hand extra energy dissipation divided by Ω data are taken in the "equilibrium" state under DC rotation, whereas the right hand data of the hysteresis component of NLRS were observed in still condition in an additive manner to the NLRS data from higher T . What does this result mean? The "excited" state is discussed as expelling the external vortices in section 4.1 from the upper section of Fig. 5, where the hysteretic energy dissipation change ∆ Q −1 hys showed a negative change across the hysteretic path difference between the"excited" and "equilibrium" states. The result expressed on the right side of Fig. 12 indicates that the "excited" state is really a Landau-like state, where external vortices are expelled. The ρ ss is an additive quantity to the VF state NLRS.

The excited state and the equilibrium state
It has been seen that AC vortices are expelled from the "excited" state of hcp 4 He which shows Landau-like state behavior. The vortex lines have been observed to penetrate the "equilibrium" state sample under DC rotation. All these experimental facts show us that the ρ ss is present even in the "equilibrium" state, while the Landau-like state is excited in the hysteretic process below T c . Then there remains a question whether there is a real Ω c1 below which one should expect the real Landau state. Such a question should be answered by future experiments. One may be exciting some shielding current for the "excited" state and the NLRS is increased at the same time as external vortices are expelled.

Supporting experimental results by other groups
In this chapter we discuss the consistency of our results and their relation to the results by other groups. The following subjects are discussed, namely, the pressure dependence of the NLRS extrapolated to T = 0 K, the specific heat anomaly observed around 75 to 100 mK by Chan's group and the superflow experiments by Ray and Hallock next, and then TO experiments on "ultra pure" or single crystal samples with "onset" near 60 to 79 mK.
Pressure dependence of NLRS as T → 0K: One of the most important properties to check consistency with results by other groups is the absolute size of the NLRS extrapolated to T = 0 K NLRS 0 , since Rittner and Reppy 51 reported a huge change of NLRS 0 according to the sample cell geometry, namely to the ratio between surface area and the volume of the solid 4 He sample. The present author thinks that their findings might include some effects of viscoelasticity. In any case it is important to quote some characteristic property to identify the sample studied. Kubota group data, except for the earliest ones in a cell with slightly complicated geometry 69 , all show rather low NLRS 0 and are stable during repeated measurements over a period exceeding a year, as in Penzev et al. 5 .
Specific heat anomaly at 75 -100 mK: A phase transition of second kind with critical fluctuations would cause a sharp specific heat peak and that of first kind would produce a jump in the specific heat. There have been serious searches by Chan's group and they reported a definite specific heat peak at T ≈75-100 mK, independent of a small 3 He content 70,71 . One does not yet know the detailed connection to the transition from the VF state into the SS state, but the observation of the transition into SS state from the VF state is consistent with the specific heat peak reported by Lin et al. 70,71 .
Super flow experiments: While earlier attempts to observe DC super flow failed 72,73 , recent flow experiments utilizing the thermomechanical effect with liquid in porous Vycor to drive the bulk solid He mass flow by Ray and Hallock 74 seem to have some correspondence to our observation of the VF and the transition into the SS states 7,8,9 . Actually they claim to have observed mass flow below ∼600 mK and some drastic change below ∼70 mK and increase of the mobility below this T . Theoretical consideration by Aleinikava, Dedits, and Kuklov 75 describes a microscopic picture of glide and superclimb of dislocations to explain the above experiment.
Ultra pure, single crystal samples experiments: Another interesting observation, which was made for solid 4 He samples near 60mK, is reported by Clark, West and Chan 76 . They have grown large crystals with high purity and compared them to standard, pure 4 He samples with various qualities. The onset temperature for a number of samples with constant temperature or constant pressure crystallization collapse onto a single value of ∼79±5 mK and the low temperature part is well fitted with what they call the NCRI, which is proportional to (1-T /T c ) γ , where γ = 2/3 with T c = 0.06 K. It looks as if the purest (single crystal) samples with smallest amount of imperfections would not have the VF contribution, but only the SS part. The absolute size of their NLRS beyond 0.2 % is still much larger than our observation for NLRS or ρ ss (see Fig. 6). Nobody yet knows the relation between their results and the VF to SS transition. It is interesting to see a 3D XY critical behavior for their purest samples. It is important to ask the relative size of the VF state and the SS state contributions to the observable NLRS.

Summary and Discussion
Results for the VF state and the SS state in hcp 4 He are summarized as follows: 1]. The VF state was found below T o ∼ 500mK. The VF state is characterized by the dynamics of the vortex tangle and its cessation at 30mK. 2].The transition from the VF state into the SS state is marked by a) appearance of hysteresis below T c with evolution of the equilibrium state and the excited state, b) the VF state logV ac linear dependence change in the equilibrium state at T c . 3]. Vortex lines penetration in the equilibrium state under DC rotation has been detected by the extra energy dissipation signal under rotation. The extra dissipation was found to follow the expected relation (4.1) and its temperature dependence was found to coincide with the SS density ρ ss (T ), which was derived from the proposed quantity of the hysteretic component of the NLRS, NLRS hys . 4]. The excited state was discussed to be rejecting external vortices totally and this would lead to a consistent picture of a Landau-like state with 3]. 5]. The critical velocity V c (T =0) is evaluated to be about 1 cm/s from the logV ac linear dependence of the hysteretic component NLRS ac to the extrapolated value for NLRS ac =ρ ss =0. This is in reasonable agreement with the evaluation V c =h/(m 4 ξ 0 π) = 6-12 cm/s for ξ 0 =25-50nm.
The evidence of the VF state and its transition into the SS state has been shown. It requires some different mechanism than the BCS scenario. The VF triggering SS state appearance is happening, where the VF state is describable by a tangled vortex state model probably involving some thermally excited low dimensional subsystems. The macroscopic coherence appearance at low T well below T c should be related to the problem of the ground state discussed by Anderson 77 . The shear modulus observation 39,40 problem has been so far neglected, but it would be essential to construct a real microscopic description of the supersolidity and the vortex state in hcp 4 He. So far the vortex physics in solid 4 He has been discussed in the VF state and the SS state, in one of the simplest Bose quantum crystals. Before solid 4 He the vortex state had been discussed only among so called "new type of superconductors" 42 . Then, one may ask a question: is hcp 4 He a high temperature superfluid? It would be a fundamental question of the present day understanding of superfluidity, superconductivity and the vortex state, related to the next question. Are there further new supersolids awaiting discovery? After studying the case of hcp 4 He, one may ask oneself such a question. It would be a new challenge to find one in the second century following the first production of a superfluid about a century ago.