In Pursuit of the Elusive Supersolid

The excitement following the initial report of supersolid behavior for ⁴He embedded in porous Vycor glass has been tempered by the realization that many of the early supersolid observations were contaminated by effects arising from an anomaly in the elastic properties of solid ⁴He. In an attempt to separate dynamic elastic effects from a true supersolid signal, we employed a torsional oscillator with two eigen-frequencies to study the ⁴He-Vycor system. We found that frequency-dependent elastic signals can entirely account for the observed period shift signals. Although, we conclude that supersolid does not exist for the ⁴He-Vycor case, the question of its presence in bulk samples remains open. In our current experiments we apply the two-frequency test to bulk samples of solid ⁴He. Again we find a frequency-dependent contribution arising from elastic effects; however, in some cases we also find a small frequency-independent contribution, which may indicate the existence of a remnant supersolid phase. Given the history of this subject such results must be treated with caution.     

Elementary Excitations in Solid and Liquid 4He at the Melting Pressure

Recent discovery of a nonclassical rotational inertia (NCRI) in solid ⁴He below 0.2 K by Kim and Chan has revived great interest in the problem of supersolidity and initiated intensive study on the properties of solid ⁴He. A direct proof that the onset of NCRI corresponds to the supersolid transition would be the observation of a corresponding drop of the entropy of solid ⁴He below the transition temperature. We have measured the melting pressure of ultrapure ⁴He in the temperature range from 0.01 to 0.45 K with several single crystals grown at different pressures and with the accuracy of 0.5 μbar. In addition, supplementary measurements of the pressure in liquid ⁴He at constant volume have been performed, which allowed us to eliminate the contribution of the temperature-dependent properties of the pressure gauge from the measured melting pressure data. With the correction to the temperature-dependent sensitivity of the pressure gauge, the variation of the melting pressure of ⁴He below 320 mK obeys the pure T ⁴ law due to phonons with the accuracy of 0.5 μbar, and no sign of the transition is seen (Todoshchenko et al. in JETP Lett. 85:454, 2007). This sets the upper limit of ∼5⋅10⁻⁸ R for a possible excess entropy in high-quality ⁴He crystals below 320 mK. At higher temperatures the contribution from rotons in the superfluid ⁴He has been observed. The thermal expansion coefficient of the superfluid ⁴He has been measured in the range from 0.01 to 0.7 K with the accuracy of ∼10⁻⁷ 1/K, or by two orders of magnitude better than in previous measurements. The roton contributions to the melting pressure and to the pressure in liquid at a constant volume are consistent and yield the value of 6.8 K for the roton gap, which is very close to the values obtained with other methods. As no contribution due to weakly interacting vacancies to the melting pressure of ⁴He has been observed, the lower limit of about 5.5 K for their activation energy can be set.      

Effect of Impurities on Supersolid Condensate: A Ginzburg–Landau Approach

Basing our arguments on a wavefunction that contains both positional and superfluid order, we propose a Ginzburg–Landau functional for a supersolid with the two order parameters necessary to describe such a phase: density n _B (r) and supersolid order parameter ψ _V . We argue that adding lighter. ³He atoms to a ⁴He supersolid produces attractive regions for vacancies, leading to patches of higher T _c. On the other hand, the supersolid stiffness decreases in this granular state with increased ³He disorder. Both effects are linear in ³He concentration.     

Observation of Non-Classical Rotational Inertia in Solid 4He Confined in Porous Gold

No Heading Recent torsional oscillator measurements on solid ⁴He confined in Vycor glass at 62 bars show supersolid response, an abrupt drop in rotational moment of inertia, at 175 mK¹. We have investigated the pore-size dependence of the supersolid behavior by confining solid ⁴He in a different porous host, porous gold, of a considerably larger pore diameter. When solid ⁴He in porous gold is cooled below 0.2 K a sharp drop in the resonant period is found. The supersolid response exhibits a strong dependence on the amplitude of oscillation.PACS numbers: 67.80–s. 67.80 Mg., 0.5.70.Fh, 05.30.Jp     

Interpreting Torsional Oscillator Measurements: Effect of Shear Modulus and Supersolidity

The torsional oscillator is the chief instrument for investigating supersolidity in solid ⁴He. These oscillators can be sensitive to the elastic properties of the solid helium, which show anomalies over the same range of temperature in which the supersolid phenomenon appears. In this report we present a detailed study of the influence of the elastic properties of the solid on the periods of torsional oscillators for the various designs that have been commonly employed in supersolid measurements. We show how to design an oscillator which measures supersolidity, and how to design one which predominantly measures elasticity. We describe the use of multiple frequency TOs for the separation of the elastic and supersolid phenomena.     

Search for Supersolidity in 4He in Low-Frequency Sound Experiments

We present results of the search for supersolid ⁴He using low-frequency, low-level mechanical excitation of a solid sample grown and cooled at fixed volume. We have observed low frequency non-linear resonances that constitute anomalous features. These features, which appear below ∼0.8 K, are absent in ³He. The frequency, the amplitude at which the nonlinearity sets in, and the upper temperature limit of existence of these resonances depend markedly on the sample history.     

The Glassy Response of Solid 4He to Torsional Oscillations

We calculated the glassy response of solid ⁴He to torsional oscillations assuming a phenomenological glass model. Making only a few assumptions about the distribution of glassy relaxation times in a small subsystem of otherwise rigid solid ⁴He, we can account for the magnitude of the observed period shift and concomitant dissipation peak in several torsion oscillator experiments. The implications of the glass model for solid ⁴He are threefold: (1) The dynamics of solid ⁴He is governed by glassy relaxation processes. (2) The distribution of relaxation times varies significantly between different torsion oscillator experiments. (3) The mechanical response of a torsion oscillator does not require a supersolid component to account for the observed anomaly at low temperatures, though we cannot rule out its existence.     

Absence of Slow Sound Modes in Supersolid 4He

Torsional oscillator experiments on solid ⁴He have been interpreted as showing mass decoupling similar to what one observes in a superfluid. Within the context of a two-component model for the supersolid one would expect the appearance of a second, slow acoustic mode. We have searched for this mode using an acoustic resonance technique. We have used porous membranes in bulk solid ⁴He analogous to a second sound experiment in the superfluid. We also investigated solid helium in Vycor using piezoelectrically driven titanium diaphragms (analogous to a fourth sound experiment in the liquid). Our measurements have shown no indication of an additional sound mode in the kHz range.     

Flow and Supersolidity in Helium

Recent torsional oscillator measurements showed evidence of non-classical rotational inertia in solid helium at temperatures below 200 mK and generated a great deal of interest in a possible supersolid state. The nature and properties of such a state are still unclear, although experiments involving ³He impurities and crystal annealing may provide clues. It would be very interesting to know whether supersolids share any of the other unusual properties of superfluids: superleaks, persistent currents, second sound and quantized vortices. We have studied the response of solid helium to pressure differences, in order to look for unusual flow properties that might be associated with supersolidity. Our measurements involved both helium confined in the nanometer pores of Vycor glass and bulk solid helium, at temperatures as low as 30 mK. Near melting, solid helium flows very easily but we did not see any evidence of superflow at low temperatures. If helium does become a supersolid at low temperatures, then its response to pressure gradients must be very different from that of liquid helium. We describe these and other experiments and discuss the role that defects may play in the low temperature behavior of solid helium.     

Observation of Non-Classical Rotational Inertia in Bulk Solid 4He

In recent torsional oscillator experiments by Kim and Chan (KC), a decrease of rotational inertia has been observed in solid ⁴He in porous materials (Kim, E., Chan, M.H.W. in Nature 427:225, 2004; J. Low Temp. Phys. 138:859, 2005) and in a bulk annular channel (Kim, E., Chan, M.H.W. in Science 305:1941, 2004). This observation strongly suggests the existence of “non-classical rotational inertia” (NCRI), i.e. superflow, in solid ⁴He. In order to study such a possible “supersolid” phase, we perform torsional oscillator experiments for cylindrical solid ⁴He samples. We have observed decreases in rotational inertia below 200 mK for two solid samples (pressures P=4.1 and 3.0 MPa). The observed NCRI fraction at 70 mK is 0.14%, which is about 1/3 of the fraction observed in the annulus by KC. Our observation is the first experimental confirmation of the possible supersolid finding by KC.     

On the Stability of Small Vacancy Clusters in Solid 4He

We study numerically properties of multiple vacancies in solid ⁴He at zero temperature. Up to four vacancies were introduced into the solid through incommensuration between the number of available lattice sites and the actual number of atoms. Vacancy-vacancy correlation function increases at very short distances indicating effective vacancy attraction between vacancies located on nearby lattice sites. The decay of the pair correlation function at large distances puts an upper bound on the absolute value of the binding energy varying from 4 mK at melting density to 150 mK at the highest considered density $\rho=32.2~\mbox{nm$^{-3}$}$ , and no lower bound; either the four-vacancy clusters are unbound, or are bound too weakly for the temperatures of the supersolid experiments.     

Elastic Properties of Solid 4He

The shear modulus of solid ⁴He increases below 200 mK, with the same dependence on temperature, amplitude and ³He concentration as the frequency changes recently seen in torsional oscillator (TO) experiments. These have been interpreted as mass decoupling in a supersolid but the shear modulus behavior has a natural explanation in terms of dislocations. This paper summarizes early ultrasonic and elastic experiments which established the basic properties of dislocations in solid helium. It then describes the results of our experiments on the low temperature shear modulus of solid helium. The modulus changes can be explained in terms of dislocations which are mobile above 200 mK but are pinned by ³He impurities at low temperature. The changes we observe when we anneal or stress our crystals confirm that defects are involved. They also make it clear that the shear modulus measured at the lowest temperatures is the intrinsic value—it is the high temperature modulus which is reduced by defects. By measuring the shear modulus at different frequencies, we show that the amplitude dependence depends on stress in the crystal, rather than reflecting a superfluid-like critical velocity. The shear modulus changes shift to lower temperatures as the frequency decreases, showing that they arise from a crossover in a thermally activated relaxation process rather than from a true phase transition. The activation energy for this process is about 0.7 K but a wide distribution of energies is needed to fit the broad crossover. Although the shear modulus behavior can be explained in terms of dislocations, it is clearly related to the TO behavior. However, we made measurements on hcp ³He which show essentially the same modulus stiffening but there is no corresponding TO anomaly. This implies that the TO frequency changes are not simply due to mechanical stiffening of the oscillator—they only occur in the Bose solid. We conclude by pointing out some of the open questions involving the elastic and TO behavior of solid helium.     

On the Existence of Supersolid 4He Monolayer Films

Extensive Monte Carlo simulations of ⁴He monolayer films adsorbed on weak substrates have been carried out, aimed at ascertaining the possible occurrence of a quasi-two-dimensional supersolid phase. Only crystalline films not registered with underlying substrates are considered. Numerical results yield strong evidence that ⁴He will not form a supersolid film on any substrate strong enough to stabilize a crystalline layer. On weaker substrates, continuous growth of a liquid film takes place.     

Specific Heat of Solid Helium

No Heading Recent torsional oscillator measurements of solid ⁴He in Vycor glass suggest the existence of a supersolid state of matter below 175mK. We have obtained preliminary data of the specific heat of solid helium down to 80mK. No sharp feature at the onset temperature is observable to within 1%. We are currently working towards extending our measurement to lower temperatures.PACS numbers: 67.40.–w, 67.80.–s, 67.80.Mg.     

Study of Solid 4He in Two Dimensions

Defects are believed to play a fundamental role in the supersolid state of ⁴He. We report on studies by exact Quantum Monte Carlo (QMC) simulations at zero temperature of the properties of solid ⁴He in presence of many vacancies, up to 30 in two dimensions (2D). In all studied cases the crystalline order is stable at least as long as the concentration of vacancies is below?2.5?%. In the 2D system for a small number, n _v , of vacancies such defects can be identified in the crystalline lattice and are strongly correlated with an attractive interaction. On the contrary when n _v ?10 vacancies in the relaxed system disappear and in their place one finds dislocations and a revival of the Bose-Einstein condensation. Thus, should zero-point motion defects be present in solid ⁴He, such defects would be dislocations and not vacancies, at least in 2D. In order to avoid using periodic boundary conditions we have studied the exact ground state of solid ⁴He confined in a circular region by an external potential. We find that defects tend to be localized in an interfacial region of width of about 15 ?. Our computation allows to put as upper bound limit to zero-point defects the concentration 3×10^?3 in the 2D system close to melting density.     

A Novel Design of a Low Temperature Preamplifier for Pulsed NMR Experiments of Dilute 3He in Solid 4He

Recent experimental studies of solid ⁴He indicate a strong correlation between the crystal defects and the onset of a possible supersolid state. We use pulsed NMR techniques to explore the quantum dynamics of the ³He impurities in the solid ⁴He in order to examine certain theoretical models that describe how the disordered states are related to supersolidity. Because of the very small signal-to-noise ratio at low ³He concentration and the long spin-lattice relaxation time (T ₁), it is essential to significantly enhance the NMR sensitivity to be able to carry out the experiments. Here we present the design of a novel low temperature preamplifier which is built with a low noise pseudomorphic HEMT transistor that is embedded into a cross-coil NMR probe. With a low power dissipation of about 0.7 mW, the preamplifier is capable of providing a power gain of 30 dB. By deploying the preamplifier near the NMR coil below 4 K, the noise temperature of the receiver is reduced to approximately 1 K. This preamplifier design also has the potential to be adapted into a low temperature amplifier with both input and output impedance at 50 Ω or a low temperature oscillator.     

Supersolidity and Superfluidity of Grain Boundaries

We have looked for dc mass transport through solid ⁴He in a simple experiment with two communicating vessels filled with solid ⁴He in equilibrium with liquid ⁴He. Through good quality crystals, we have observed no mass transport, in contradiction with the hypothesis of a Bose–Einstein condensation of vacancies. Through crystals containing grain boundaries, we have found superfluid flow along these grain boundaries. We discuss these results in the context of other experiments on supersolidity.     

Quantum lattice gas and the existence of a supersolid

We discuss, within mean field theory, the possible phase diagrams of a quantum lattice gas model (as considered by Matsuda and Tsuneto atT=0) with particular reference to the existence of a supersolid phase, displaying both long-range diagonal (or crystalline) order and long-range off-diagonal order (as characteristic of a superfluid). For parameter ranges which provide a reasonable representation of the known phase diagram of ⁴ He, we show that a supersolid phase may appear between the usual solid and superfluid phases. Such a phase might not, however, extend down to absolute zero.     

Influence of Deformation on the Formation of a Glassy Phase of 4He in the Supersolid Region

A new series of experiments has been performed to find out the conditions for the formation of a glass phase in solid ⁴He in the region where an anomalous behavior attributed to the supersolid effect was observed previously. A special two-chamber cell was used to deform the sample in situ. The contribution of the glass phase was identified by analyzing the measured temperature dependence of the sample pressure under different experimental conditions. It is found that the contribution of the glass phase increased sharply after deformation of the sample and practically disappears after its annealing.