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Journal of Low Temperature Physics

, Volume 197, Issue 3–4, pp 167–186 | Cite as

Mass Flux Experiments in Solid \(^4\)He: Some History, Recent Work and the Current Status

  • R. B. HallockEmail author
Article
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Abstract

Measurements that document a number of the characteristics of \(^{4}\hbox {He}\) mass flux through a cell filled with solid \(^{4}\hbox {He}\) carried out at the University of Massachusetts and elsewhere are reviewed. The mass flux is found to be a universal function of temperature, to have nonlinear behavior as a function of the driving chemical potential as might be expected for a Luttinger liquid and the flux can be extinguished at a concentration-dependent temperature by the presence of \(^{3}\hbox {He}\) . Strong evidence indicates that the flux is carried by the cores of dislocations present in the solid. Some contact with the history of quantum fluids and relevant early work in solid helium is made as is contact with theory related to the notion of a supersolid and mass flux in \(^{4}\hbox {He}\) .

Keywords

Solid helium Supersolid Quantum solids Phase transition 

1 Introduction

This is intended to be a short review of some of some of the experimental work that documents mass flux through solid \(^{4}\hbox {He}\) for this fiftieth anniversary issue of the Journal of Low Temperature Physics. There will be an emphasis on work done at the University of Massachusetts, but work by others in the community that has helped us all to converge toward a much better understanding of solid helium and some of its rather unique properties will also be included. But, of course, in a brief review it is not possible to cover everything or contributions by everyone. Our initial stimulation to study solid helium originated with the torsional oscillator results of Kim and Chan [1, 2]. Their experiments, which showed us unexpected behavior, represent a discovery of new phenomena and were originally thought to confirm the existence of a “supersolid” state of matter. Alas, as was subsequently shown, those experiments actually discovered something else.

Also provided here is a brief review of the history in which the notion of a supersolid phase emerged, as is mention of some of the early work to search for this potentially new phase of matter. We will also include here reference to a few theoretical works that have emerged to be especially relevant. This brief review includes material from and expands upon an earlier summary of some of our work [3]. A summary prepared for a wider audience is also available [4].

Phase transitions, especially exotic ones, have been of interest to the physics community for as long as there has been a physics community. For low temperature physicists, a key milestone was the liquefaction of helium first done by Kammerlingh Onnes [5] in 1908 in Leiden. This led rather promptly to the unexpected discovery in 1911 of the vanishing electrical resistance of multiply distilled mercury [6] when solid mercury was cooled below 4.2 K by pumping on liquid helium. Other superconductors were promptly discovered [7].

Following the initial intense interest in superconductivity and the continued availability of liquid \(^{4}\hbox {He}\) as a cooling agent, the liquid itself began to be studied in more detail. In 1924, Onnes and Boks [8] published clear evidence of a density maximum in liquid helium near 2.2 K. By the early 1930s, it became clear that the specific heat of liquid \(^{4}\hbox {He}\) had an interesting and very sharp maximum [9] also near 2.2 K. In 1938, there emerged reports of really unusual properties of the liquid when it was cooled below about 2.2 K under its own vapor pressure. Those experiments by Kapitza [10] and separately by Allen and Misener [11, 12] showed that liquid helium demonstrated a remarkable transition in flow properties when cooled below what became known as the Lambda point (so-called because of the shape of the specific heat of liquid helium when plotted against temperature). Kapitza even suggested that helium might enter a special state—a “superfluid.” Work by Allen and Jones [13] was the first to see a remarkable phenomenon in liquid helium, the fountain effect, which results when an imposed temperature difference between two reservoirs separated by a superleak creates a pressure difference and can drive a mass flux from one reservoir to the other. Key theoretical insights by Tisza [14], the Londons [15, 16] and Landau [17] provided some understanding and a basis for future work on superfluidity. For a detailed discussion of the early years of the discovery of superfluidity in \(^{4}\hbox {He}\) , with an emphasis on who did what and when, see the article by Balibar [18].

Helium has demonstrated a number of exceptional characteristics. In addition to the phase transition that leads to bulk superfluid behavior at low temperature, studies of the thin helium film have led to confirmation that liquid helium has a Kosterlitz–Thouless [19] phase transition. A variety of studies by Reppy and co-workers [20, 21, 22], using a high-sensitivity torsional oscillator [23], and others, by the use of other techniques [24] including third sound [25], have documented in detail the near textbook-like two-dimensional behavior that is predicted by theory [26, 27].

2 Background

Liquid \(^{4}\hbox {He}\) was recognized as special, and early efforts were made to try to solidify it. Kammerlingh Onnes attempted to achieve this goal by lowering the temperature of the liquid, but was not able to create solid because helium does not solidify under its own vapor pressure. Keesom [28] successfully solidified helium in 1926 in a narrow brass tube by the application of high pressure at several temperatures between 4.2 and 1.1 K. In doing so, he determined the general shape of the melting curve and noted that helium was unlikely to have a solid–liquid–gas triple point. He even indicated, correctly, that the \(^{4}\hbox {He}\) melting curve had a tendency to be nearly independent of temperature at low temperatures.

Given the remarkable behavior of liquid \(^{4}\hbox {He}\) , it was natural that theorists would eventually ask whether the solid might behave in some quite unexpected and unusual way. In 1969, Andreev and Lifshitz [29] suggested that under certain specific conditions it might be possible for a crystal with defects and with large zero point motions to demonstrate properties of superfluidity. In 1970, Chester [30] suggested, by the use of what he termed “almost rigorous arguments,” that quantum crystals with a finite number of vacancies might include a Bose–Einstein condensate. Leggett [31], also in 1970, explored whether solid helium could show superfluid-like properties. In doing so, he suggested that if there were a Bose–Einstein condensation, then there should be a non-classical rotational inertia (NCRI) which would be visible to experimentalists by the use of rotation techniques. This suggestion was analogous to the suggestion by Landau [17] that led to the Andronikashvili [32, 33] stacked-disk torsional oscillator measurements that first demonstrated temperature-dependent mass decoupling in bulk liquid helium, and ultimately helium films [20]. Leggett even proposed what he termed to be an “extremely tentative” superfluid fraction in the solid of order \(\sim 3 \times 10^{-4}\).

Guyer [34] considered the work of Chester [30], and Leggett [31] and also of Andreev and Lifshitz [29] in 1971 and concluded that ground-state vacancies likely do not exist and thus cannot be considered as evidence that would lead to the possibility of Bose–Einstein-type condensation in solid helium. He also concluded that a Bose–Einstein condensation due to cooperative tunneling of pairs of particles would only occur below 0.1 mK. He provided a constraint on any possible superfluid-like density in solid helium of \(\sim 1 \times 10^{-6}\).

The term “supersolid” seems to have first appeared in the theoretical work of Matsuda and Tsuneto [35] (1970) (who concluded that bulk \(^{4}\hbox {He}\) solid was unlikely to become a supersolid) and independently in work on a cell model of Bose–Einstein condensation in solid helium due to Mullin [36] (1971) (whose cell model was considered to be an explicit example of the more general approach taken by Andreev and Lifshitz [29]). Suffice it to say that theoretical work at the time was quite pessimistic about potential experimental realizations of superfluid-like behavior in solid helium at reasonably achievable temperatures. None the less, the theoretical work that suggested this interesting new state of matter attracted the attention of experimentalists.

Some of the earliest experimental work relevant to the possibility of a supersolid was the search for vacancies in solid helium by Andreev et al. [37] By Looking for any motion of a solid platinum–cobalt polished ball suspended magnetically in solid helium, they found no detectable motion and set an upper limit of 0.1 percent on the concentration of zero point vacancies at the lowest temperature that they could achieve, 0.5 K. Another early work was that of Suzuki [38], who studied plastic flow by applying force to a steel ball. The focus was on the motion of dislocations. A related experiment was that due to Tsymbalenko [39], which also had a focus on dislocation motion. None of this work resulted in evidence for behavior that might be directly related to supersolidity.

Two of the most notable of the early experiments with a more direct approach to explore the possibility of supersolidity were carried out at Bell Laboratories, one by Greywall [40] and one by Bishop et al. [41] The 1977 report by Greywall described an experiment in which two chambers filled with solid helium at different pressures were connected by a narrow solid-filled capillary. The experiments were carried out in an effort to see whether any pressure difference measured between the two chambers would relax with time. No evidence for pressure relaxation between the two chambers was seen over the pressure range studied (25.8–49.2 bar) at temperatures down to 30 mK. As a result, a limit on the possible value of \((\rho _s/\rho )V\) was determined to be \(< 2.5 \times 10^{-9}\) cm/s, where \(\rho _s/\rho \) is the super fraction and V is the flow velocity. Shortly thereafter, Bishop et al. [41] carried out an experiment with the specific goal of searching for the NCRI that Leggett had suggested might be present. In their work, a torsion oscillator was used down to 20 mK to search for a period shift that might indicate a fraction of the solid helium in the oscillator was not following the motion of the oscillator. No evidence for a non-classical moment of inertia was found to an accuracy of a few parts per million. Period changes were observed, but these were attributed to the behavior of dislocations and the phase separation of a small amount of \(^{3}\hbox {He}\) mixed with the solid \(^{4}\hbox {He}\) . Following the report of these results of Greywall [40] and Bishop et al. [41], most of the quantum fluids and solids community seemed to lose interest in the notion of a supersolid.

As mentioned in the 2013 review by Chan et al. [42], the negative results did not completely eliminate efforts to seek supersolidity. Bonfait et al. [43] used a capacitive technique to search for flow between two superfluid regions separated by a region of solid helium. In two separate measurements, one of which was designed to ensure that many dislocations were present in the solid, no evidence for the flow from one superfluid region to the other through solid helium was found down to the low temperature limit of 4 mK. The constraint found by Bonfait et al. [43] in 1989 was \((\rho _s/\rho )V\)\(< 1.8 \times 10^{-8}\) cm/s. This result extended the lower limit in temperature and added further evidence for the absence of measurable supersolidity in a reasonably accessible temperature range. The desire to purposely try to include dislocations had its origin in the suggestion by Shevchenko [44] in 1987 that superfluidity might be present in dislocation cores. The mechanism suggested by Shevchenko was quite different from that which Chester had suggested: quasi-one-dimensional behavior versus bulk three-dimensional behavior that relied on ground-state vacancies. As reviewed by Meisel [45], there were additional efforts, but none provided evidence that there might be a transition to a supersolid state of any kind.

But, in studies of the acoustic properties of solid \(^{4}\hbox {He}\) at frequencies in the range 9–45 MHz in the presence of small (tens of ppm) quantities of \(^{3}\hbox {He}\) , Ho et al. [46] found something unexpected. They reported in 1997 that the velocity and attenuation of sound showed an unusual temperature dependence; the velocity increased sharply (by a few parts in \(10^4\)) with falling temperature near 165 mK in a narrow temperature range, while the attenuation peaked. Ho et al. [46] interpreted their results to indicate the presence of a second-order phase transition and suggested that this might be related to the presence of a Bose condensed state in the solid (at temperatures above 165 mK). Subsequent work by Goodkind [47] with acoustic waves and heat pulses lent support to this interpretation. In later simultaneous torsional oscillator and ultrasound studies [48] with well helium, the sharp shifts in sound velocity were not seen.

3 Rekindled Interest in Solid Helium

The experiments by Ho et al. [46] appear to have helped to motivate Kim and Chan [1, 2] to begin a series of measurements on solid \(^{4}\hbox {He}\) by the use of a torsional oscillator. Their measurements provided results reported in 2004 that were interpreted as consistent with a transition to supersolid-like behavior below a characteristic temperature. The period of the oscillator shifted (Fig. 1) below \(\approx 250\) mK as might be expected if a fraction of the mass were to decouple from the oscillator. A number of careful control experiments were carried out to rule out known potential alternate interpretations.

The general results of the experiments of Kim and Chan [1, 2] were promptly confirmed [49], with additional confirmation at a number of laboratories around the world [50, 51, 52, 53, 54, 55] with similar results: a clear signature of an oscillator period shift that begins in the vicinity of 250 mK, with a magnitude that increases with decreasing temperature and flattens at the lowest temperatures as might be expected for a superfluid density such as seen in bulk \(^{4}\hbox {He}\) . But there was a roughly two orders of magnitude variance in the relative size of the period shift among the early results from laboratory to laboratory. Not everyone in the community was convinced of the presence of a supersolid. For example, one alternate explanation was that the torsional oscillator signatures seen could be explained by a low-temperature quench of topological defects into glass-like behavior [56, 57]. But these experiments greatly stimulated the quantum fluids and solids community and promptly attracted substantial theoretical interest from a variety of directions [58, 59, 60, 61, 62, 63, 64, 65]. In particular, Anderson et al. [58] thought that ground-state vacancies existed, but Prokofev and Svistunov [59] concluded that bulk \(^{4}\hbox {He}\) cannot have a supersolid state of the sort proposed by Chester. This was due to their conclusion that any ground-state vacancies would be purged from the solid, and a supersolid state would not exist in a commensurate solid. The absence of such vacancies was generally consistent with what Guyer [34] had previously concluded.
Fig. 1

Data from Kim and Chan [1, 2] [left] that show the deduced NCRIF (fraction) from the torsional oscillator at 26 bar for various levels of the rim velocity of the oscillator. Measurements of the shear modulus of solid helium due to Day and Beamish [66] showed a change in the shear modulus in the same temperature range. When the size of the shifts in the shear modulus is normalized, the shear modulus has a temperature dependence remarkably like that seen for the period shift in the torsional oscillator data. [right] Scaling the two, as shown here, was an early indicator that there must be a very direct relationship between the temperature dependence of the torsional oscillator period shift and that of the shear modulus. The two sets of NCRI data (left vs. right), which show different maximum NCRI percents are from different torsion oscillator experiments (Color figure online)

It was well known in the community that dislocations were present in solid helium. Indeed, some of the observations of Bishop et al. [41] were explained by the presence and behavior of such dislocations. Beamish [67] expressed the thought that dislocations might create effects in torsional oscillator measurements similar to those seen by the community and that these could be erroneously interpreted as mass decoupling. In an effort to explore the influence of dislocations and structural effects, Day and Beamish [66] carried out a direct measurement of the shear modulus in solid \(^{4}\hbox {He}\) by the use of piezoelectric crystals. When normalized, the temperature dependence of the shear modulus showed a remarkable similarity to the period shift seen in the torsional oscillator work (Fig. 1). John Beamish will report in detail on his group’s structural studies and dislocation motion in a separate contribution to this volume [68], and so this will not be discussed in substantial detail here.

The presence of unexpected behavior in the shear modulus was interpreted to be partially due to the ability of \(^{3}\hbox {He}\) to pin dislocations. Indeed, subsequent work by Haziot et al. [69] showed that in the absence of \(^{3}\hbox {He}\) (i.e., through the use of isotopically pure \(^{4}\hbox {He}\) ), as described elsewhere in this issue [68], the plasticity of the solid helium was “giant” since the dislocations were no longer pinned at low temperatures. Work to study the plastic flow by use of pressure-driven flow through a membrane with 6–8 \(\mu \) diameter pores was carried out by the Ukraine group [70] with the result that to the upper limit of their ability to apply a pressure difference across the membrane the flux of helium through the porous membrane was temperature independent below about 800 mK, i.e., there was no evidence of supersolid behavior in those experiments carried out by the group in Ukraine.

Suffice it to say that it is now believed by many in the community that the early torsional oscillator work clearly was measuring the temperature dependence of the structural effects and not supersolidity. As shown by Maris [71], the effect of these structural effects can be subtle and even include aspects associated with the level of rigidity of the torsional oscillator elements in contact with the solid helium. In work not directly published, but reported in a commentary in Nature [72], Reppy carried out a torsional oscillator experiment with a donut-shaped geometry, but with an open-channel crosslink. He concluded from the measurements that the signatures seen for solid \(^{4}\hbox {He}\) were not consistent with the presence of a superfluid-like state of matter. Evidence for supersolidity from torsional oscillator measurements is now thought to be extremely thin, with tight limits on any possible NCRI [73, 74, 75].

4 Flow Experiments

Searches for flow began soon after the work of Kim and Chan [1, 2] was reported. An early experiment was that of Day et al. [76] in which it was observed that there was no pressure-driven flow of solid helium through Vycor in the temperature and pressure range for which Kim and Chan had seen shifts in the period of their torsional oscillator. In subsequent work, Day and Beamish [77] used a diaphragm to attempt to cause a flow of solid helium through a large array of parallel glass capillaries of \(25 \mu \) diameter. No evidence for flow was seen, and an upper limit for \((\rho _s/\rho )V\) was determined to be \(< 1.2 \times 10^{-12}\) cm/s. An experiment by Sasaki et al. [78] was stimulating in that it did show flow, but it was promptly determined that the flow observed at the melting curve was due to grain boundaries. A discussion of this can be found in the review by Balibar and Caupin [79].

As this rapidly evolving field accelerated, my colleagues Boris Svistunov and Nikolay Prokofev suggested that we carry out a U-tube-type experiment with solid helium. The basic notion was that if one created a U-tube with solid in the lower part and superfluid in the upper part, such that the solid was on the melting curve, it might be possible to see mass migrate from one side to the other by adding mass to the liquid phase on one side. This, we later learned, was similar to the general approach taken by Bonfait et al. [43], mentioned earlier, who worked at very low temperatures. This simple direct approach seemed not possible for us to carry out since thermal conductivity issues with bulk samples on our experimental platform would likely be fatal and an experiment on the melting curve could lead to conceptual difficulties. For example, there might be a superfluid liquid pathway at grain boundaries [80], or in the incommensurate helium along the U-tube walls, that could bypass the solid in the U-tube. But, the general idea to try something quite different from the techniques employed in the many torsional oscillator experiments was intriguing.
Fig. 2

Schematic drawing [81] [left] and a more realistic schematic view [82] [right] of the experimental cell; not to scale. Two capillaries, 1 and 2, go to liquid reservoirs, R1 and R2, at the top ends of Vycor rods, V1 and V2, and a third capillary (for initial filling of the cell) enters from the center of the side of the cell. In our early work, the Vycor rods were epoxied into stainless steel tubes, but soon we eliminated the stainless steel tubes and simply coated the Vycor directly with epoxy, which reduced the heat flux to the cell. Two capacitance pressure gauges, C1 and C2, are located on each end of the cell. Pressures in the lines 1 and 2 are read by pressure gauges, P1 and P2, outside the cryostat. Each reservoir has a heater, H1, H2, which allows the temperatures of the reservoirs to be controlled, typically in the vicinity of 1.46 K. T1, T2 and TC are calibrated carbon resistance thermometers. The solid helium fills the horizontal cylindrical region of the cell, which has a diameter of 6.25 mm with a capacitor-to-capacitor distance of 44 mm. The Vycor rods, typically 1.4 mm diameter, are 25 mm apart, symmetric in the cell; but, at times these were replaced with Vycor rods of different diameters. There are also access ports (not shown) that are centered and perpendicular to the main axis that houses the solid helium, which were recently used to hold in place a barrier in the center of the cell to block part of the flow channel [83, 84] (Color figure online)

After further discussion, there emerged the conceptual idea illustrated in a schematic way in Fig. 2. The idea relies on the fact that superfluid helium in a micro-porous environment at a given temperature freezes at an elevated pressure [85, 86, 87, 88] compared with the freezing pressure for bulk \(^{4}\hbox {He}\) . Thus, it is possible to create an environment in which bulk solid helium, at pressures above the melting curve, can be in direct contact with superfluid helium (in the porous material). The basic question to be answered was whether atoms from one superfluid reservoir would pass into solid helium and as a result cause atoms to enter the other reservoir.

The concept shown in Fig. 2 works and resolves thermal conduction issues because the superfluid-filled micro-porous material Vycor has a very low thermal conductivity; in fact, more recent studies show that the thermal conductivity is anomalous [89]. The low thermal conductivity allows the reservoirs of superfluid at temperatures near 1.5 K to be available to allow application of pressure and temperature differences between the reservoirs on the two sides of the apparatus, while a much colder region of solid \(^{4}\hbox {He}\) is in close proximity. The apparatus was designed to have pressure gauges on each end of the region that would contain solid helium. This turned out to be a fortunate choice because of the resulting uncommon ability to determine the pressure in the solid-filled cell in situ in two different parts of the apparatus.

4.1 Injection of Atoms

In our initial work [81, 90], we simply added atoms to one of the reservoirs R1 or R2 (Fig. 2), by injection of \(^4\)He atoms via a capillary, which increased the pressure in that reservoir and we looked to see if the pressure in the other reservoir changed. As was the case with the work of Greywall [40], the goal was to see whether an imposed pressure difference would relax. Important differences between our approach and that of Greywall were (1) the presence of liquid reservoirs and (2) liquid reservoirs with bulk solid between them. If a pressure relaxation was present it would be clear that that could only happen if atoms somehow entered the lower pressure reservoir and there was only one way that could happen, i.e., by a transfer of atoms from one superfluid-filled reservoir to the other through the solid-filled cell, which could be held off the melting curve. A difference between our approach and that of Bonfait et al. [43] was the ability to study the solid off the melting curve.

Two important results from our experiments were promptly available as shown in Fig. 3. (1) With the addition of atoms to reservoir R1, there was a clear increase in the pressure in reservoir R2 and the pressure increase seen in R2 was linear in time and independent of the driving pressure difference. (2) Each in situ pressure gauge showed a pressure increase in the solid itself, which indicated that the density of solid \(^{4}\hbox {He}\) in the fixed-volume solid-filled cell was increasing while the flow took place. Thus, atoms were being added to the solid to increase its density at the same time as atoms passed through the solid to reach reservoir R2. We noted that a mass flux that appeared to be essentially constant in time and independent of the driving pressure head is consistent with the notion of a superflow at a limiting critical velocity [81].
Fig. 3

[Right] An early example [81] of flux measurements versus time for a sample with nominal well concentration of \(^{3}\hbox {He}\) of \(\sim 200\) ppb [81]. After the growth of solid helium from the melting curve at fixed temperature, typically 350 mK, and stabilization of the solid in the cell at about 26.5 bar, helium atoms were added to line 1, which caused an increase in the pressure in reservoir 1. In the case shown here, atoms continued to be added for about 60 min. What was promptly evident was an increase in the pressure in reservoir 2, accompanied by increases in the pressure in the cell as recorded by the in situ capacitive pressure gauges, C1 and C2. [left] Phase diagram of solid helium to show where flow was observed and where no flow was seen in our early mass injection experiments [91]. The solid line on the phase diagram is the melting curve for the bulk solid (Color figure online)

Fig. 4

Recent data from Chan’s group [92] for the flow versus no-flow boundary are quite consistent with our results, Fig. 3 [left], but his group sees evidence for flow in thinner samples that extends to somewhat higher temperatures than we do. The symbols here represent individually grown solids and also solids that have had mass added to increase the pressure and thus the density of the solid. Generally for our work, flow becomes unmeasurable above about \(\approx \) 600 mK, whereas Chan’s group sees flow in thinner solid samples up to \(\approx \) 1 K. The inset here shows the geometry used by Chan’s group, where the solid helium was initially a thin \(8 \mu \) thick layer. As will be described later, more recent work by Chan’s group [93] has studied a variety of different configurations in the region of solid (Color figure online)

In Fig. 3 [right] is shown a phase diagram for solid \(^{4}\hbox {He}\) above 25 bar that resulted from these mass injection experiments. Regions at lower pressures and lower temperatures showed finite mass flux, whereas at pressures above \(\sim 29\) bar or temperatures above \(\sim 600\) mK we did not see any evidence for mass flow. Recent work by Shin and Chan [92] (Fig. 4) to study solid \(^{4}\hbox {He}\) , also with the solid between reservoirs of superfluid contained in porous material, finds similar behavior: a distinct low-pressure, low-temperature region of the \(^{4}\hbox {He}\) solid phase diagram that shows evidence of the flow.

Given the manner in which our solids were grown (typically at fixed temperature by mass injection, but at times by the blocked capillary technique with similar results), it was highly likely that they were, and typically are, disordered. Annealing can reduce the disorder. While ideal crystals were by this point thought to not show supersolid behavior, it was shown that an inhomogeneous solid could house defects that would be superfluid [94].

One of the subsequent ideas proposed to explain our observations of (1) mass flux at a limiting velocity and also (2) the simultaneous growth of the density of the solid was due to Soyler et al. [65]. Following the original idea of Shevchenko [44], it was proposed that what carried the flux was helium atom motion along the cores of edge dislocations in the solid. Indeed, simulations showed, as first proposed by Shevchenko [44], that there was a superfluid density along these dislocation cores, i.e., a quasi-one-dimensional pathway with a superfluid density. The growth of the solid would be a natural occurrence if some atoms did not make the traverse along dislocation cores from one reservoir to the other but instead created jogs on the dislocation cores. By this process, atoms would be added to the solid by the “superclimb” of a dislocation [65] and thus change the solid density in the fixed-volume cell. The growth of the solid by the addition of atoms because of its isochoric compressibility was called “the syringe effect.” This suggestion was not widely accepted initially, with a possible alternate model being that the flux was due to liquid-like channels [80] in the solid at interfaces where different crystal faces meet or where crystals meet the cell walls. But cessation of flow above about 600 mK suggested that such channels were likely not the cause of the observed mass flux.

4.2 Utilization of the Fountain Effect

Subsequent work at UMass evolved with an improved data collection technique that utilized an imposed fountain effect [95, 96] between reservoirs R1 and R2 instead of direct injection of atoms into one of the two reservoirs (Fig. 5). In these measurements, an alternation of a finite temperature difference is applied between the two reservoirs, which drives a mass flow and results in changes in the pressures P1 and P2. A measure of the magnitude of the flux is obtained by a measure of the slope of the time-dependent pressure difference between the two reservoirs, which is reported as millibars per second, mbar/s. This is calibrated to find a mass flux; the rate 0.1 mbar/s results in the passage of \(\approx 4.8 \times 10^{-8}\) g/s of \(^{4}\hbox {He}\) from one reservoir to the other through a distance of 25 mm of solid \(^{4}\hbox {He}\) between the two Vycor rods. Given our cross-sectional area of bulk solid \(^{4}\hbox {He}\) of 31.65 mm\(^2\) perpendicular to the flux direction, this results in a calibration for the flux per mm\(^2\) of cross-sectional area of 0.1 mbar/s \(\approx 1.5 \times 10^{-9}\) g/mm\(^2\)s. By contrast, in the work of Shin and Chan [93] at 100 mK with solid samples with \(\approx 2\) mm thickness of solid, a typical flux of \(\approx 2.5 \times 10^{-8}\) g/mm\(^2\)s is found, with a much higher flux of \(\approx 7.2 \times 10^{-8}\) g/mm\(^2\)s seen for the sample with \(8 \mu \) thickness [92]. In general, it has been shown that thinner samples result in higher values of the flux.
Fig. 5

An example of our use of the fountain effect technique to collect most of our data [97]. To collect data, the temperature of the two reservoirs is changed, typically by alternation (top), which results in a fountain effect chemical potential difference between them (as recorded by the pressure changes, middle). The temperature of the solid \(^{4}\hbox {He}\) is changed in steps as shown at the bottom section of the figure. This chemical potential difference drives the mass flux. A measure of this flux is deduced from the time-dependent slope of the derivative of the pressure difference versus time. The availability of pressure gauges and thermometers allows a determination of the chemical potential difference versus time that drives the mass flow. These data are for a 10.2 ppm \(^3\)He sample at \(P = 26.30\) bar. Note here the abruptness of the extinction of the flux at the lowest of the three temperatures shown. Flow is present at 99 mK, but it has been extinguished once cooled to 97.5 mK. For this sample, recovery of the flux after prompt warming of a few mK was time dependent and complete in about 600 s [97, 98] (Color figure online)

Fig. 6

Here, we show data [99] for a 19.5 ppm \(^3\)He sample at \(P = 26.40\) bar. The deduced mass flux related to the measured rate of change in the pressure difference shows a strong increase with falling temperature. But, the flux is extinguished at a \(^{3}\hbox {He}\) concentration-dependent temperature that we denote as \(T_d\). This extinction is due to the phase separation of \(^{3}\hbox {He}\) out of the solid solution mixture (Color figure online)

Our measurements showed that the flux was temperature dependent, F(T), and increased with decreasing temperature (Figs. 6, 7 left) and did so in what was a universal manner as seen, for example, when the F(T)/F(0.2 K) was determined from the data from a variety of measurements and plotted against temperature [97, 98] (Fig. 7, right). We note here that the absolute value of the measured mass flux depends on the specific sample and its history, but the general temperature dependence is the same for most of the samples studied.

The magnitude of the flux per mm\(^2\) seen in our experiments as well as those of Chan’s group depends on the particular sample, the temperature and a partial anneal of the sample typically causes a reduction in the flux. We interpret this to mean that some of the dislocations have been annealed away and are no longer present to conduct the flux. Also, sample preparation and manipulation can influence the magnitude of the flux. This we interpret as being due to a different number of dislocation cores being present in different configurations in different samples.
Fig. 7

[Left] The temperature dependence of the maximum value of flux for different samples with a sequence of different \(^{3}\hbox {He}\) impurity concentrations and sample histories [97]. Lines are guides to the eye. [right] Evidence for a universal temperature dependence [97]. Here, the data have all been normalized to 0.2 K and the (arbitrary) function \(F = F_0^*[1 - 1.21\exp (-E/T)]\) represented by the solid curve is seen to provide a reasonable characterization of the data in the temperature range of our studies (Color figure online)

For all samples with \(\approx 200\) ppb \(^{3}\hbox {He}\) well helium, at the lower range of temperatures we could reach, the flux reproducibly (and nearly reversibly) dropped dramatically [96, 97] at a characteristic low temperature, \(T_d\) (Figs. 6, 7). It was speculated that what was happening was that the flux was being blocked by \(^{3}\hbox {He}\) due to phase separation in the solid helium mixture and the likely subsequent condensation of impurity \(^{3}\hbox {He}\) on the dislocation cores (or on their intersections). So, in this picture, the presence of the \(^{3}\hbox {He}\) impeded the flow of \(^{4}\hbox {He}\) along the cores. Simulation by Corboz et al. [100] showed that this indeed could be the case, at least for the case of increasing numbers of \(^{3}\hbox {He}\) atoms added to the nucleus of screw dislocations. Changing the concentration of \(^{3}\hbox {He}\) in the solid mixture results in a shift in the temperature \(T_d\) at which the flux was extinguished (Fig. 7, left) The increase in the value of \(T_d\) with an increase in \(^{3}\hbox {He}\) concentration was along the lines that could be predicted theoretically [97, 98, 101] or measured [102, 103].

The effect of the presence of \(^{3}\hbox {He}\) on \(T_d\) has been independently confirmed by Cheng et al. [104], who used a variation of our approach. In their case, instead of a superfluid–solid–superfluid sample, they used a solid–superfluid–solid sample. They observed that increasing the pressure in one of the solid-filled chambers by the motion of a diaphragm resulted in an increase in the pressure of the other, a result of mass transport through their solid–liquid junctions; mass moved from one solid-filled region to the other. In their work, they also observed (Fig. 8) that additions of \(^{3}\hbox {He}\) caused a concentration-dependent shift in \(T_d\). They interpreted their results to suggest that the \(T_d\) reduction in flux was likely due to the presence of \(^{3}\hbox {He}\) at superfluid–solid interfaces. Our more recent experiments support our earlier conclusion [105] that this interface is not likely the bottleneck [83, 84]. The Beamish group [106] also observed that for much purer \(^{4}\hbox {He}\) (5 ppt \(^{3}\hbox {He}\) ), the flux continued to rise to their low temperature limit of 20 mK, Fig. 8, a result that confirmed again that the \(^{3}\hbox {He}\) was the source of the large flux reduction.
Fig. 8

The temperature dependence of the flow rates (normalized to 0.2 K) seen in the solid–superfluid–solid experiments of Cheng et al. [106]. The inset shows the non-normalized flow rates deduced from their pressure gauge (in bar/min). For this work, various concentrations of \(^{3}\hbox {He}\) were studied. These experiments showed an upper limit for the temperature at which flux could be measured of \(\approx 600\) mK, consistent with our results. The general temperature dependence was also similar to what our group has seen (Color figure online)

One of the goals for our future work is to carry the flux measurements to much lower temperatures with isotopically pure \(^{4}\hbox {He}\) to see the very low temperature behavior. If the flux is a measure of the quasi-one-dimensional superfluid density, then the temperature dependence might be expected to flatten and the flux approaches the low-temperature limit with zero slope. An experiment has been designed to explore the flux behavior down to the vicinity of 2 mK. Consideration of the behavior of the isochoric compressibility [65] in the limit of low temperature indicates [107] that the syringe effect will vanish [108, 109]. Another goal will be to complete work with a cell that has been constructed to allow flux measurements in the presence of in situ mechanical compression or expansion of the solid in the cell.

4.3 Luttinger-Like Behavior

Another interesting result we found was that the flux, F(T), is nonlinear in the driving chemical potential difference, \(\Delta \mu \). Our experiments monitor the temperature of the solid \(^{4}\hbox {He}\) and the pressures and temperatures of the reservoirs R1 and R2 as a function of time. So, when we, for example, apply a fixed temperature difference between the two reservoirs, we determine the full chemical potential difference, \(\Delta \mu \), that drives the temperature-dependent flux, F(T) as a function of time as the reservoir pressures change with time with the solid helium in the cell at a fixed temperature.
$$\begin{aligned} \Delta \mu = \int {\frac{\mathrm{d}P}{\rho }}-\int {S \mathrm{d}T}, \end{aligned}$$
(1)
We then deduce how the flux depends on that driving chemical potential difference. The experiments [82, 99] showed that \(F = A (\Delta \mu )^b\) (Fig. 9, right). b was found to be essentially independent of temperature over the range studied \((0.15\,{\hbox {K}}< T < 0.5\,{\hbox {K}}), b \sim 0.3\), but dependent on pressure (Fig. 10). This nonlinearity suggests that the behavior is as that of a Luttinger liquid, as might be expected were the flux to take place in a quasi-one-dimensional context [110].

We note here that the presence of the \(^{3}\hbox {He}\) while in solid mixture solution with the \(^{4}\hbox {He}\) does not seem to have any influence on the mass flux. So, when flux is present, neither the temperature dependence of the flux nor the Luttinger-like behavior is modified by the presence of the \(^{3}\hbox {He}\) . So, for example, as shown in Fig. 10 [right] the presence of the \(^{3}\hbox {He}\) does not appear to influence the pressure dependence of the exponent b. As shown in Fig. 7 [right], the presence of \(^{3}\hbox {He}\) for \(T > T_d\) does not influence the temperature dependence of the flux.

Under the assumption that Luttinger-like behavior is present, it is possible to obtain the Luttinger parameter, g, from the exponent b. To do so requires assumptions about the strength of the interactions among the \(^{4}\hbox {He}\) atoms on the dislocation cores. If one assumes that phase slips are introduced by random and independent scattering sites, then one finds [111] that \(g = [(1/b) + 1]/2\). g as deduced from our pressure-dependent determinations of b is shown in Fig. 11 [left]. It has been shown that an increase in pressure ultimately results in a region of the phase diagram that no longer supports mass flux. The decreasing values of g with an increase in pressure are consistent with this evolution.
Fig. 9

[Left] An example of flux measurements [82] with a 10.2 ppm concentration of \(^{3}\hbox {He}\) for several temperatures. Alternation of the temperatures of reservoirs R1 and R2 creates the chemical potential difference, \(\Delta \mu \), that drives the flux. The upper section shows the temperatures imposed on the reservoirs, which cause the pressures to change (middle) as the temperature of the solid is changed with time in a stepwise fashion (bottom). [right] Data for the measured flux versus the measured chemical potential difference are reasonably well fit [99] by \(F = A(\Delta \mu )^b\) and a determination of the coefficient A and the exponent b can be made, Fig. 10. Here are shown data for nominal well helium with a \(^{3}\hbox {He}\) concentration of \(\sim \) 200 ppb (Color figure online)

Fig. 10

[Left] Values of the fit parameters A and b for three pressures [112] for a sample with well helium of nominal \(^{3}\hbox {He}\) concentration. [right] Pressure dependence of the fit parameter [99] b as a function of the pressure above the melting curve, \(\delta P = P - Pmc\), where Pmc is the melting curve pressure, for several different \(^{3}\hbox {He}\) concentrations (Color figure online)

A number of the experimental results seen in the UMass experiments have been confirmed and extended by the use of variations of the UMass approach. We have already mentioned the solid–superfluid–solid approach taken by Cheng et al. [104]. As also mentioned earlier, work by Shin et al. [92] has found superfluid-like mass flux through an 8 \(\mu \) thickness of solid \(^{4}\hbox {He}\) held between two superfluid-filled Vycor rods. More recent work by Shin and Chan [93], with a superfluid–solid–superfluid configuration related to ours, has found such mass flux in thicker solid samples and confirmed the general nonlinear behavior, through use of \(F = a(\delta T)^b\), where \(\delta T\) is the difference in temperature between their two reservoirs. Their work shows evidence for significant temperature dependence for the exponent b over an expanded temperature range to higher temperature, \((0.03\,{\hbox {K}}< T < 0.9\,{\hbox {K}})\) (Fig. 11, right).

At the time of our study of the flux versus \(\Delta \mu \), \(F = a(\Delta \mu )^b\), we also explored the behavior through the use of \(F = C(\delta T)^k\). We found [113] that with this approach, the behavior was again nonlinear, with the temperature dependence in C, and with little temperature dependence to the exponent k. We found that at 25.7 bar and 0.3 K where \(b \approx 0.3, k\) was a bit larger, \(k \approx 0.5\). Our work and that of Chan’s group have led to growing acceptance of the idea that dislocation cores are the critical aspect of the solid that allows the observed mass flux and growth of solid density and that these behave as quasi-one-dimensional pathways.
Fig. 11

[Left] Values of the Luttinger parameter [112] g determined form our measurements of b by use of the relation \(g = [(1/b) + 1]/2\). [right] Measurements by Shin and Chan [93] of the exponent of the nonlinear behavior of the measured flux as determined by fits to the protocol \(F = a(\delta T)^b\). Note that in the case of these experiments, the independent variable is \(\delta T\), which differs from our choice of \(\Delta \mu \). Their data, over a wider temperature range than ours, show temperature dependence (Color figure online)

4.4 More Recent Work

Work of this general nature continues here and at other laboratories. In our most recent work [83, 84], we have shown that a change in the area of the macroscopic Vycor surface in contact with the solid reduces the flux, but not in direct proportion to the change in the area of the exposed Vycor. This helps to support the view that the behavior of the flux is not related in a fundamental way to the interface where the solid helium meets the liquid helium in the Vycor. That interface matters, but we believe that it is not a key factor limiting the flux or influencing its temperature dependence.
Fig. 12

Schematic diagram of the apparatus changes: original on left and with two changes made sequentially on right. First, the Vycor was shortened, then for separate experiments the block was inserted [83, 84]. The inserted plug blocked 83% of the cross-sectional area of the open cell along the axis between C1 and C2

Adapted from Fig. 1 in Ref. [81] (Color figure online)

In another experiment [83, 84], we inserted a rigidly fixed partial block (Fig. 12) in the solid helium cross section—to reduce the available pathway available to the flux. The flux was reduced, but not in direct proportion to the area of the horizontal cross section of the cell that was blocked; we blocked 83 percent of the cross-sectional area, with a cylindrical block of length 12.6 mm centered in the cell that resulted in an annular region around it filled with solid, and observed that his caused a flux reduction that averaged (over multiple samples) close to a factor of three, not five. Again, there was some sample to sample variation. It is possible that the dislocation pathways in the annual region have different constraints than was the case in the more open geometry, We interpret this result to confirm our belief that the flux is carried through the presence of dislocations in the macroscopic cross section of the solid and is not due to the surface layer of atoms along the cell walls or liquid channels. Had the flux been transported by the layer of atoms that are incommensurate at the cell walls, then the presence of the macroscopic block in the cell would have had essentially no effect on the flux.

This conclusion is much the same as the one also reached recently by Shin and Chan [93] during their quite extensive set of measurements with several different solid samples in a number of configurations. As mentioned, the phase diagram boundary that separates regions of flow from regions of non-flow in their work, Fig. 4, is similar to that which we found, Fig. 3 [right], but they see flow up to higher temperatures than we do. This may be due to thinner solid helium samples in their case. Their configuration has allowed several “thicknesses” of solid helium and other sample changes. The maximum flux they measure does decrease with increasing thickness of solid helium. In one of their experiments, they pre-filled the sample cell with aerogel and then studied the solid-filled sample to see whether the flux previously seen in an open geometry would remain. It did not. They interpret this to mean that the presence of the aerogel prevented the formation of a typical dislocation network from providing connectivity across the sample. This adds further support to the belief that it is the cores of dislocations that carry the flux.

Recent work by Cheng and Beamish [114], using techniques similar to their earlier work [106] on \(^{4}\hbox {He}\) , reports flow measurements in bcc \(^{3}\hbox {He}\) . The flux seen for solid \(^{3}\hbox {He}\) seems to be of a different character than the flux seen in the case of \(^{4}\hbox {He}\) in their earlier work. For example, while the flux in the case of \(^{4}\hbox {He}\) increases with a decrease in temperature, the flux seen for bcc \(^{3}\hbox {He}\) decreased monotonically at low temperature and approached a constant value at the lowest temperatures. The thought is that that this represents evidence for the presence of a quantum mechanism at work.

5 Conclusion

Mass flux measurements through a cell filled with solid \(^{4}\hbox {He}\) have now been made in several laboratories. Measurements under various conditions and with various concentrations of \(^{3}\hbox {He}\) present have shown the solid mixture system to be complex and interesting. Here, we have attempted to review some of the evidence that mass can indeed be transported through solid helium and the evidence strongly suggests that this mass flux is carried by the cores of dislocations, dislocations that are created with the sample solidification or subsequently through sample manipulation. The presence of Bosonic Luttinger-like behavior for the flux has been confirmed. What seems now clear is that while solid \(^{4}\hbox {He}\) shows little or no evidence for the presence of supersolidity of the type proposed nearly 50 years ago, it provides us with strong evidence for an entirely different type of superflow behavior reminiscent of what was initially proposed by Shevchenko [44]. Clearly, further work is needed to more completely unravel the full story of this fascinating quantum solid.

Notes

Acknowledgements

I very much appreciate the opportunity to contribute to this anniversary volume of the Journal of Low Temperature Physics. Our work at the University of Massachusetts Amherst has been done in collaboration with M. Ray, Ye. Vekhov and more recently with V. Rubanskyi. Without them, much of what has been reported here from UMass would not exist. We have benefitted from many discussions with numerous colleagues in the quantum solids community, particularly S. Balibar, J. Beamish, M.H.W. Chan, A. Kuklov, J.D. Reppy and, especially, our local colleagues W. Mullin, N. Prokofev, and B.V. Svistunov. We appreciate the permission to use the figures provided by Z. Cheng and J. Beamish and by J. Shin and M.H.W. Chan. Our most recent work was supported primarily through funds provided by NSF DMR 1205217 and DMR 1602616. Modest additional support was also available from funds provided by the Research Trust Fund administered by the University of Massachusetts Amherst.

References

  1. 1.
    E. Kim, M. Chan, Nature 427, 225 (2004)ADSGoogle Scholar
  2. 2.
    E. Kim, M. Chan, Science 305, 1941 (2004)ADSGoogle Scholar
  3. 3.
    R.B. Hallock, J. Low Temp. Phys. 180, 6 (2015)ADSGoogle Scholar
  4. 4.
    R.B. Hallock, Phys. Today 68, 30 (2015)ADSGoogle Scholar
  5. 5.
    H.K. Onnes, KNAW Proc. 11, 168 (1909)Google Scholar
  6. 6.
    H.K. Onnes, Commun. Phys. Lab Univ. Leiden Suppl. 29 (1911)Google Scholar
  7. 7.
    H. Onnes, KNAW Proc. 16, 113 (1913)Google Scholar
  8. 8.
    H.K. Onnes, J. Boks, Commun. Phys. Lab Univ. Leiden 170b (1924)Google Scholar
  9. 9.
    W. Keesom, K. Clusius, Leiden Comm. 219e; Proc. Sect. Sci. K. Ned. Acad. Wet. 35, 307 (1932). 219e (1932)Google Scholar
  10. 10.
    P. Kapitza, Nature 141, 74 (1938)ADSGoogle Scholar
  11. 11.
    J. Allen, A. Misener, Nature 141, 75 (1938)ADSGoogle Scholar
  12. 12.
    J. Allen, A. Misener, Nature 142, 643 (1938)ADSGoogle Scholar
  13. 13.
    J. Allen, H. Jones, Nature 141, 243 (1938)ADSGoogle Scholar
  14. 14.
    L. Tisza, Nature 141, 913 (1938)ADSGoogle Scholar
  15. 15.
    F. London, Nature 141, 643 (1938)ADSGoogle Scholar
  16. 16.
    H. London, Proc. R. Soc. A 171, 484 (1939)ADSGoogle Scholar
  17. 17.
    L. Landau, J. Phys. (USSR) 5, 71 (1941)Google Scholar
  18. 18.
    S. Balibar, J. Low Temp. Phys. 146, 441 (2007)ADSGoogle Scholar
  19. 19.
    J.M. Kosterlitz, D.J. Thouless, J. Phys. C 6, 1181 (1973)ADSGoogle Scholar
  20. 20.
    D.J. Bishop, J.D. Reppy, Phys. Rev. Lett. 40, 1727 (1978)ADSGoogle Scholar
  21. 21.
    D.J. Bishop, J.D. Reppy, Phys. Rev. B 22, 5171 (1980)ADSGoogle Scholar
  22. 22.
    M.D. Agnolet, G.J.D. Reppy, Phys. Rev. B 39, 8934 (1989)ADSGoogle Scholar
  23. 23.
    J. Berthold, D. Bishop, J.D. Reppy, Phys. Rev. Lett. 39, 348 (1977)ADSGoogle Scholar
  24. 24.
    J. Maps, R. Hallock, Phys. Rev. B 26, 3979 (1982)ADSGoogle Scholar
  25. 25.
    I. Rudnick, Phys. Rev. Lett. 40, 1454 (1978)ADSGoogle Scholar
  26. 26.
    D. Nelson, J. Kosterlitz, Phys. Rev. Lett. 39, 1201 (1977)ADSGoogle Scholar
  27. 27.
    D.N.V. Ambegaokar, B.I. Halperiin, E. Siggia, Phys. Rev. B 21, 1806 (1980)ADSGoogle Scholar
  28. 28.
    W.H. Keesom, Nature 118, 81 (1926)ADSGoogle Scholar
  29. 29.
    A. Andreev, I. Lifshitz, Sov. Phys. JETP 29, 1107 (1969)ADSGoogle Scholar
  30. 30.
    G.V. Chester, Phys. Rev. A 2(1), 256 (1970).  https://doi.org/10.1103/PhysRevA.2.256 ADSCrossRefGoogle Scholar
  31. 31.
    A.J. Leggett, Phys. Rev. Lett. 25(22), 1543 (1970).  https://doi.org/10.1103/PhysRevLett.25.1543 ADSCrossRefGoogle Scholar
  32. 32.
    E. Andronikashvili, ZH. Eksp. Theo. Fiz 16, 780 (1946)Google Scholar
  33. 33.
    E. Andronikashvili, J. Phys. (USSR) 10, 201 (1946)Google Scholar
  34. 34.
  35. 35.
    H. Matsuda, T.T. Tsuneto, Prog. Theor. Phys. 46, 411 (1970)Google Scholar
  36. 36.
    W.J. Mullin, Phys. Rev. Lett. 26(11), 611 (1971).  https://doi.org/10.1103/PhysRevLett.26.611 ADSCrossRefGoogle Scholar
  37. 37.
    A. Andreev, K. Keshishev, L. Mezhov-Deglin, A. Shal’nikov, Sov. Phys. JETP Lett. 9, 306 (1969)ADSGoogle Scholar
  38. 38.
    H. Suzuki, J. Phys. Soc. Jpn. 35, 1472 (1973)ADSGoogle Scholar
  39. 39.
    V.L. Tsymbalenko, Sov. Phys. JETP Lett. 23, 653 (1976)ADSGoogle Scholar
  40. 40.
    D.S. Greywall, Phys. Rev. B 16(3), 1291 (1977).  https://doi.org/10.1103/PhysRevB.16.1291 ADSCrossRefGoogle Scholar
  41. 41.
    D.J. Bishop, M.A. Paalanen, J.D. Reppy, Phys. Rev. B 24(5), 2844 (1981).  https://doi.org/10.1103/PhysRevB.24.2844 ADSCrossRefGoogle Scholar
  42. 42.
    M.H.W. Chan, R.B. Hallock, L. Reatto, J. Low Temp. Phys. 172, 317 (2013)ADSGoogle Scholar
  43. 43.
    G. Bonfait, H. Godfrin, B. Castaing, J. Phys. Fr. 50, 1997 (1989)Google Scholar
  44. 44.
    S.I. Shevchenko, Sov. J. Low Temp. Phys. 13(2), 61 (1987)Google Scholar
  45. 45.
    M. Meisel, Phys. B 178, 121 (1992)ADSGoogle Scholar
  46. 46.
    P. Ho, I. Bindloss, J. Goodkind, J. Low Temp. Phys. 109, 409 (1997).  https://doi.org/10.1007/s10909-005-0094-0 ADSCrossRefGoogle Scholar
  47. 47.
    J. Goodkind, Phys. Rev. Lett. 89(9), 095301 (2002).  https://doi.org/10.1103/PhysRevLett.89.095301 ADSCrossRefGoogle Scholar
  48. 48.
    B. Hein, J. Goodkind, I. Iwasa, H. Kojima, J. Low Temp. Phys. 171, 322 (2013)ADSGoogle Scholar
  49. 49.
    A.S.C. Rittner, J.D. Reppy, Phys. Rev. Lett. 97(16), 165301 (2006).  https://doi.org/10.1103/PhysRevLett.97.165301 ADSCrossRefGoogle Scholar
  50. 50.
    M. Kondo, S. Takada, Y. Shibayama, K. Shirahama, J. Low Temp. Phys. 148, 695 (2007).  https://doi.org/10.1007/s10909-007-9471-1 ADSCrossRefGoogle Scholar
  51. 51.
    Y. Aoki, J.C. Graves, H. Kojima, Phys. Rev. Lett. 99(1), 015301 (2007).  https://doi.org/10.1103/PhysRevLett.99.015301 ADSCrossRefGoogle Scholar
  52. 52.
    Y. Aoki, J.C. Graves, H. Kojima, J. Low Temp. Phys. 150, 252 (2008)ADSGoogle Scholar
  53. 53.
    A. Penzev, Y. Yasuta, M. Kubota, J. Low Temp. Phys. 148, 677 (2007).  https://doi.org/10.1007/s10909-007-9452-4 ADSCrossRefGoogle Scholar
  54. 54.
    B. Hunt, E. Pratt, V. Gadagkar, M. Yamashita, A.V. Balatsky, J.C. Davis, Science 324(5927), 632 (2009).  https://doi.org/10.1126/science.1169512 ADSCrossRefGoogle Scholar
  55. 55.
    R. Toda, P. Gumann, K. Kosaka, M. Kanemoto, W. Onoe, Y. Sasaki, Phys. Rev. B 81, 214515 (2010)ADSGoogle Scholar
  56. 56.
    Z. Nussinov, A.V. Balatsky, M.J. Graf, S.A. Trugman, Phys. Rev. B 76(1), 014530 (2007).  https://doi.org/10.1103/PhysRevB.76.014530 ADSCrossRefGoogle Scholar
  57. 57.
    A. Balatsky, M. Graf, Z. Nussinov, J.J. Su, J. Low Temp. Phys. 172(5–6), 388 (2013)ADSGoogle Scholar
  58. 58.
    P. Anderson, W. Brinkman, D.A. Huse, Science 310, 1164 (2005)ADSGoogle Scholar
  59. 59.
    N. Prokof’ev, B. Svistunov, Phys. Rev. Lett. 94(15), 155302 (2005).  https://doi.org/10.1103/PhysRevLett.94.155302 ADSCrossRefGoogle Scholar
  60. 60.
    A.T. Dorsey, P.M. Goldbart, J. Toner, Phys. Rev. Lett. 96(5), 055301 (2006).  https://doi.org/10.1103/PhysRevLett.96.055301 ADSCrossRefGoogle Scholar
  61. 61.
    P. Anderson, Nat. Phys. 3, 160 (2007).  https://doi.org/10.1038/nphys539 CrossRefGoogle Scholar
  62. 62.
    L. Pollet, M. Boninsegni, A.B. Kuklov, N.V. Prokof’ev, B.V. Svistunov, M. Troyer, Phys. Rev. Lett. 98(13), 135301 (2007).  https://doi.org/10.1103/PhysRevLett.98.135301 ADSCrossRefGoogle Scholar
  63. 63.
    M. Boninsegni, A.B. Kuklov, L. Pollet, N.V. Prokof’ev, B.V. Svistunov, M. Troyer, Phys. Rev. Lett. 99(3), 035301 (2007).  https://doi.org/10.1103/PhysRevLett.99.035301 ADSCrossRefGoogle Scholar
  64. 64.
    P. Anderson, Phys. Rev. Lett. 100, 215301 (2008)ADSGoogle Scholar
  65. 65.
    Ş.G. Söyler, A.B. Kuklov, L. Pollet, N.V. Prokof’ev, B.V. Svistunov, Phys. Rev. Lett. 103(17), 175301 (2009)ADSGoogle Scholar
  66. 66.
    J. Day, J. Beamish, Nature 450, 853 (2007)ADSGoogle Scholar
  67. 67.
    J. Beamish, Private communicationGoogle Scholar
  68. 68.
    J. Beamish, This issueGoogle Scholar
  69. 69.
    A. Haziot, X. Rojas, A. Fefferman, J. Beamish, S. Balibar, Phys. Rev. Lett. 110, 035301 (2013)ADSGoogle Scholar
  70. 70.
    V. Zhuchkov, A. Lisunov, V. Maidanov, A. Neoneta, V. Rubanskyi, S. Rubets, E. Rudavskii, S. Smirnov, Low Temp. Phys. 41, 169 (2015)ADSGoogle Scholar
  71. 71.
    H.J. Maris, Phys. Rev. B 86, 020502 (2012)ADSGoogle Scholar
  72. 72.
    E.S. Reich, Nature 468, 748 (2010)ADSGoogle Scholar
  73. 73.
    D.Y. Kim, M.H.W. Chan, Phys. Rev. Lett. 109, 155301 (2012)ADSGoogle Scholar
  74. 74.
    J. Choi, J. Shin, E. Kim, Phys. Rev. B 92, 144505 (2015)ADSGoogle Scholar
  75. 75.
    A. Eyal, X. Mi, L. Talanov, A.V.J. Reppy, Proc. Natl. Acad. Sci. 113, 6335 (2017)Google Scholar
  76. 76.
    J. Day, T. Herman, J. Beamish, Phys. Rev. Lett. 95(3), 035301 (2005).  https://doi.org/10.1103/PhysRevLett.95.035301 ADSCrossRefGoogle Scholar
  77. 77.
    J. Day, J. Beamish, Phys. Rev. Lett. 96(10), 105304 (2006)ADSGoogle Scholar
  78. 78.
    S. Sasaki, R. Ishiguro, F. Caupin, H. Maris, S. Balibar, Science 313, 1098 (2006)ADSGoogle Scholar
  79. 79.
    S. Balibar, F. Caupin, J. Phys. Condens. Matter 20, 173201 (2008)ADSGoogle Scholar
  80. 80.
    S. Sasaki, F. Caupin, S. Balibar, Phys. Rev. Lett. 99(20), 205302 (2007).  https://doi.org/10.1103/PhysRevLett.99.205302 ADSCrossRefGoogle Scholar
  81. 81.
    M.W. Ray, R.B. Hallock, Phys. Rev. B 79(22), 224302 (2009).  https://doi.org/10.1103/PhysRevB.79.224302 ADSCrossRefGoogle Scholar
  82. 82.
    Y. Vekhov, R.B. Hallock, Phys. Rev. Lett. 109, 045303 (2012)ADSGoogle Scholar
  83. 83.
    V. Rubanskyi, R. Hallock, in APS March Meeting 2019, March 4–8, Boston, Massachusetts; V06.00004 (2019)Google Scholar
  84. 84.
    V. Rubanskyi, R. Hallock, in preparation (2019)Google Scholar
  85. 85.
    J.R. Beamish, A. Hikata, L. Tell, C. Elbaum, Phys. Rev. Lett. 50(6), 425 (1983).  https://doi.org/10.1103/PhysRevLett.50.425 ADSCrossRefGoogle Scholar
  86. 86.
    E. Adams, Y. Tang, K. Uhlig, G. Haas, J. Low Temp. Phys. 66, 85 (1987).  https://doi.org/10.1007/BF00681469 ADSCrossRefGoogle Scholar
  87. 87.
    C. Lie-zhao, D.F. Brewer, C. Girit, E.N. Smith, J.D. Reppy, Phys. Rev. B 33(1), 106 (1986).  https://doi.org/10.1103/PhysRevB.33.106 ADSCrossRefGoogle Scholar
  88. 88.
    L. Pollet, A.B. Kuklov, Phys. Rev. Lett. 113, 045301 (2014)ADSGoogle Scholar
  89. 89.
    Z.G. Cheng, M. Chan, New J. Phys. 15, 063030 (2013)ADSGoogle Scholar
  90. 90.
    M.W. Ray, R.B. Hallock, Phys. Rev. Lett. 100(23), 235301 (2008)ADSGoogle Scholar
  91. 91.
    M.W. Ray, R.B. Hallock, J. Low Temp. Phys. 158, 560 (2010)ADSGoogle Scholar
  92. 92.
    J. Shin, D. Kim, A. Haziot, M. Chan, Phys. Rev. Lett. 118, 325301 (2017)Google Scholar
  93. 93.
    J. Shin, M. Chan, Phys. Rev. B 99, 140502 (2019)ADSGoogle Scholar
  94. 94.
    L. Pollet, M. Boninsegni, A.B. Kuklov, N.V. Prokof’ev, B.V. Svistunov, M. Troyer, Phys. Rev. Lett. 101(9), 097202 (2008).  https://doi.org/10.1103/PhysRevLett.101.097202 ADSCrossRefGoogle Scholar
  95. 95.
    M.W. Ray, R.B. Hallock, Phys. Rev. B 82(1), 012502 (2010)ADSGoogle Scholar
  96. 96.
    M.W. Ray, R.B. Hallock, Phys. Rev. B 84(17), 144512 (2011)ADSGoogle Scholar
  97. 97.
    Y. Vekhov, W.J. Mullin, R.B. Hallock, Phys. Rev. Lett. 113, 035302 (2014)ADSGoogle Scholar
  98. 98.
    Y. Vekhov, W.J. Mullin, R.B. Hallock, Phys. Rev. Lett. 115, 019902 (2015)ADSGoogle Scholar
  99. 99.
    Y. Vekhov, R.B. Hallock, Phys. Rev. B 92, 104509 (2015)ADSGoogle Scholar
  100. 100.
    P. Corboz, L. Pollet, N.V. Prokof’ev, M. Troyer, Phys. Rev. Lett. 101, 155302 (2008)ADSGoogle Scholar
  101. 101.
    D.O. Edwards, S. Balibar, Phys. Rev. B 39(7), 4083 (1989)ADSGoogle Scholar
  102. 102.
    C. Huan, S. Kim, L. Yin, J. Xia, D. Candela, N. Sullivan, J. Low Temp. Phys. 162(3–4), 167 (2011)ADSGoogle Scholar
  103. 103.
    C. Huan, L. Yin, J. Xia, D. Candela, B. Cowan, N. Sullivan, Phys. Rev. B 95, 104107 (2017)ADSGoogle Scholar
  104. 104.
    Z. Cheng, J. Beamish, A. Fefferman, F. Souris, S. Balibar, Phys. Rev. Lett. 114, 165301 (2015)ADSGoogle Scholar
  105. 105.
    Y. Vekhov, R.B. Hallock, Phys. Rev. B 91, 180506(R) (2015)ADSGoogle Scholar
  106. 106.
    Z.G. Cheng, J. Beamish, Phys. Rev. Lett. 117, 025301 (2016)ADSGoogle Scholar
  107. 107.
    A. Kuklov, L. Pollet, B. Prokofev, N.V. Svistunov, Phys. Rev. B 90, 184508 (2014)ADSGoogle Scholar
  108. 108.
    A. Kuklov, Phys. Rev. B 92, 134504 (2015)ADSGoogle Scholar
  109. 109.
    M. Yarmolinsky, A. Kuklov, Phys. Rev. B 96, 024505 (2017)ADSGoogle Scholar
  110. 110.
    F.D.M. Haldane, Phys. Rev. Lett. 47(25), 1840 (1981)ADSGoogle Scholar
  111. 111.
    B.V. Svistunov, N.V. Prokof’ev, private communicationGoogle Scholar
  112. 112.
    Y. Vekhov, R.B. Hallock, Phys. Rev. B 90, 134511 (2014)ADSGoogle Scholar
  113. 113.
    Y. Vekhov, R. Hallock, to be publishedGoogle Scholar
  114. 114.
    Z.G. Cheng, J. Beamish, Phys. Rev. Lett. 121, 225304 (2018)ADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratory for Low Temperature Physics, Department of PhysicsUniversity of MassachusettsAmherstUSA

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