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Unlike a typical superfluid, the non-classical rotational inertia fraction (NCRIF) has been found to be partially hysteretic8. The low-temperature (T<∼20 mK) value of NCRIF for a sample was smaller when cooled down in a high-velocity field (HVF), versus that obtained at low speeds. On reducing the velocity at the lowest temperature, the NCRIF recovered the unsuppressed value. However, subsequent increases in the velocity did not result in any suppression of the NCRIF up to at least 800 μm s−1. This hysteresis was observed only below about 40 mK. Slow warming and cooling cycles (all below 65 mK) following the above type of velocity sweeps have revealed that the NCRIF can be multivalued at the lowest temperatures9. The two research groups involved have described these findings in terms of the pinning of low-temperature excitations (such as dislocations and/or vortices) in solid 4He.

Here we examine the phenomena systematically by measuring the dynamic response of a torsional oscillator containing a bulk 4He sample at a pressure of 40 bar. The sample was grown using the blocked capillary method. The torsional oscillator has an open volume of 1.92 cm3 in a cylindrical form, with a 13 mm diameter and a height of 14.5 mm. The surface-area to volume (S/V) ratio is 4.46 cm−1. The empty cell has a resonant period of 1.328 ms and a mechanical quality factor of 2×106 at 4.2 K.

Two procedures were used to reach a fixed temperature and rim velocity, both of which represent a different history. The first is a typical method from previous studies. After setting the driving voltage at 500 mK, the torsional oscillator was cooled to the base temperature of 23 mK. After equilibration at the lowest temperature, a slow warming scan was carried out. We will refer to this cooling procedure and the corresponding NCRIF as HVF cooling and HVF NCRIF, respectively. As the rim velocity vR of the torsional oscillator is temperature dependent for a constant driving voltage, we consistently denote each data set by the value of vR recorded at 23 mK.

The second procedure was as follows. At 500 mK, the driving voltage was set to a low value such that vR at 23 mK is less than 33 μm s−1. The sample was then cooled down to a particular target temperature. The time evolution of the period and that of the amplitude were subsequently measured as the driving voltage was increased discretely in several steps. When the driving voltage is stepped up or down there is an abrupt jump—dominated by an instrumental artefact—in the resonant period, followed by slow relaxation. Following each relaxation process, we acquire a value for the NCRIF at the target temperature for each respective rim velocity (see Fig. 1). Owing to the low-velocity field (LVF) cooling to each target temperature, we label each of these traces as LVF NCRIF.

Figure 1: Period change in a torsional oscillator versus temperature and conversion into NCRIF.
figure 1

a, Two different experimental procedures used in the experiment. The first is HVF cooling (see text). Data are obtained from a subsequent warming scan (solid blue and black curves). The second process involves cooling with vR≤33 μm s−1 (solid red curve). The rim velocity is increased discretely after the target temperature is reached (solid circles). The inset shows the time evolution of the period on increasing vR. Colour coded arrows mark Tp and Td at two different velocities. b, NCRIF measured with two different techniques. The dashed curves are guides for the eye.

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The maximum value of LVF NCRIF observed was 0.03%, which saturated at this value for T<∼35 mK while cooling with vR=33 μm s−1. This is one of the smallest reported values so far. Although a quantitatively accurate correlation is not well established, torsional oscillators with small S/V ratios tend to display smaller NCRIF values. This may in part be due to large defect densities near the walls of the cell17, and also the inefficiency to quickly drain away the latent heat of crystallization so as not to generate thermally induced strains18.

For HVF cooling, NCRIF at 23 mK is suppressed and projected to disappear at ∼3 mm s−1. Above a specific point that we call the depinning temperature Td, the HVF and LVF NCRIF values show the same temperature dependence all the way up to the onset temperature To. Below Td, where the two begin to differ, LVF NCRIF is always greater than HVF NCRIF. Moreover, the NCRIF recovers from the HVF value to its unsuppressed value (red curve in Fig. 1b) when the velocity is reduced to 33 μm s−1. This demonstrates that, below Td, the NCRIF depends on both the temperature and velocity histories of the sample.

We must emphasize that Td is not simply the temperature at which a difference between HVF and LVF NCRIF emerges, but also the point where the dynamic response of the torsional oscillator changes. Above Td, the resonant period decays exponentially on a relatively short timescale (see Fig. 2g–j). In contrast, decay takes place much more slowly below Td (see Fig. 2c–f). Interestingly, at even lower T, below what we define as the pinning temperature Tp, long-timescale characteristics once again disappear (see Fig. 2a and b). We note that Tp also represents the (velocity dependent) temperature above which LVF NCRIF is suppressed. As Tp is below 23 mK for vR>547 μm s−1, we extrapolated the low-T data to estimate the suppression temperature for high velocities (see Fig. 1a and Supplementary Information).

Figure 2: Dynamic response of a torsional oscillator.
figure 2

a–i, Time evolution of the resonant period, with respect to the equilibrium value, after vR is increased from 144 to 319 μm s−1 at various temperatures. The red dashed curves are fits to data using P(t)=P0+Ae−t/τ+B ln(1+t/t0) for c–f and P(t)=P0+Ae−t/τ for the rest. j, Zoom-in of a, d and h to emphasize the change in the relaxation process. The 25 mK data (a) have been inverted to fit in the figure. k,l, Time evolution in response to various rim velocity changes at 30 mK (k) and 50 mK (l).

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In Fig. 3 we have constructed a phase diagram in the vR–T plane on the basis of the observed Tp, Td and To. Identification of each phase was possible by analysing the relaxation dynamics of the torsional oscillator. In the red shaded region, the time evolution of the resonant period is exponential. The equilibrium time constant τ is of the order of a few hundred seconds near To. This is shorter than that of the empty cell, indicating that there is an extra dissipative process in the solid sample that provides damping to the torsional oscillator. We find that τ increases for T<100 mK, which is consistent with the literature10,19. An increase in τ, that is, a slowing down of the internal relaxation of 4He, can be explained by the progressive freezing out of excitations present in the solid10,20. The leading candidates for these excitations are dislocations21,22, vortices and glassy topological defects12.

Figure 3: Velocity–temperature phase diagram of solid 4He obtained by the second procedure described in the text.
figure 3

Above To (solid red circles), solid 4He is a normal Debye solid. Below To, a fluid of unpinned vortices forms. On a further decrease in temperature, down to T≈Td (solid yellow circles), the vortices become weakly pinned. The yellow curve is a fit of the functional form . Tp (filled blue circles) marks the true transitions to a superfluid-like vortex glass at various velocities. The red and blue curves are guides for the eye. The error bars mark uncertainty in transition temperatures determined by the method explained in Supplementary Information Section 2.

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In the yellow shaded region between Tp and Td, the same exponential decay manifested above Td is present with τ constantly increasing with decreasing temperatures. However, as mentioned earlier, there is a relaxation process that takes place on a different timescale. This behaviour has also been observed in previous works8,9. We follow the argument in ref. 8 and fit to a logarithmic time dependence, which is quite successful, as shown in Fig. 2c–f. The logarithmic behaviour could be caused by creep of weakly pinned excitations23,24. That is, if the pinned state is metastable in this region, excitations can escape one trapping site and creep to another. As the temperature is reduced, the thermal energy of the excitations will decrease accordingly, allowing them to be weakly pinned on local pinning sites. The pinning of originally free excitations through this mechanism is a continuous process; thus, Td would not correspond to a thermodynamic phase transition.

Although the type of excitations cannot be directly identified, the velocity dependence of Td can provide us with some insight. The positive slope of Td in vR–T phase space is at first glance counterintuitive. A higher rim velocity adds more kinetic energy to the system so that one would expect a higher probability of depinning (resulting in a negative slope of Td). However, if the excitations are vortices, the increasing velocity will result in their proliferation and ultimately will yield stronger vortex–vortex interactions. This would give an extra constraint to free-flowing vortices, hence leading to higher depinning temperatures at higher velocities. However, there is no conclusive evidence at this point.

Below Tp, the jump in the resonant period is followed by exponential relaxation, and is similar to that seen in the empty cell. This indicates that the excitations present in solid 4He are no longer metastably pinned, but rather, completely pinned. Such a process would be expected to result from a change in the pinning potential at Tp; that is, the potential is diverging such that excitations can no longer escape. This is different from a gradual crossover from an unpinned to a weakly pinned state, and could possibly be a true phase transition. This is also the region that presents the hysteresis originally observed previously8. It is argued that this could be analogous to the Meissner–Ochsenfeld effect in superconductors, implying that Tp is the temperature at which vortices freeze in both position and number. As Tp is velocity dependent, by inverting the relation, the temperature dependence of the velocity at which NCRIF suppression occurs can be obtained. This is the critical velocity vc(T) of the hidden Meissner–Ochsenfeld-like state. Determining vc(0) cannot be done accurately owing to the lack of data below 20 mK. Nevertheless, vc(0)>2 mm s−1, which is much closer to that of superfluid 4He than original estimates1,2.

Note that Tp≈35 mK in the zero-velocity limit coincides with the temperature at which NCRIF saturation is observed. Similar behaviour was found in ref. 8, in which they see saturation at ∼40 mK (see Fig. 2 in ref. 8). In ref. 10, NCRIF saturation is not achieved even down to 20 mK. Thus, the continuously increasing relaxation time with decreasing temperature that they observed is to be expected. Fig. 3A and B in ref. 10 show that the change in the frequency shift and dissipation both become smaller as T approaches 20 mK, indicating that the time evolution will eventually vanish at a lower temperature (this is also implied in Fig. 3C of their work). The relatively small Tp reported in ref. 10 (the point at which their observed relaxation will disappear) might be related to a rather large NCRIF (4.8%), as opposed to ours (0.03%) or that of ref. 8 (0.1%).

Another type of hysteresis observed in ref. 9 is seen by LVF cooling of the sample, followed by a single-step velocity increase. This will put the solid 4He in the yellow shaded region of our phase diagram. Temperature variation in this region then creates a mixed state between the weakly pinned and HVF-cooled states, resulting in a multivalued NCRIF. Comparison with previous measurements8,9,10,19 reveals that despite the variation in the exact values of velocity and temperature, the general features of the phase diagram seem to be universal.