Fate of Vacancy-Induced Supersolidity in ${}^4\mathrm{He}$

The supersolid state of matter, exhibiting nondissipative flow in solids, has been elusive for 35 years. The recent discovery of a nonclassical moment of inertia in solid ${}^4\mathrm{He}$ by Kim and Chan provided the first experimental evidence, although the interpretation in terms of supersolidity of the ideal crystal phase remains a subject to debate. Using quantum Monte Carlo methods we investigate the long-standing question of vacancy-induced superflow and find that vacancies in a ${}^4\mathrm{He}$ crystal phase separate instead of forming a supersolid. On the other hand, nonequilibrium vacancies relaxing on defects of polycrystalline samples could provide an explanation for the experimental observations.

Figure 1

A sketch of the different scenarios for vacancy states in the ground state of solid helium at the melting density. Supersolidity can occur at finite vacancy densities $n_V$, according to the Andreev-Lifshitz-Chester (ALC) or Dai-Ma-Zhang–(DMZ-) Anderson-Brinkman-Huse (ABH) scenarios. ALC assume an energy gain already for a single vacancy (negative initial slope), while DMZ/ABH postulate a ground state with a low density of strongly correlated vacancies, without making any prediction for the single-vacancy behavior. For the scenarios with gapped vacancies (positive slope ${\Delta }_V$), a (meta)stable gas of out-of-equilibrium vacancies will be formed if the energy curve bends up (the upper dashed line) due to repulsion between the vacancies. The curve labeled “unstable” represents the situation supported in this Letter: an attractive vacancy gas is unstable and phase separates into the commensurate insulating crystal (with density $n_S$) and the liquid (with density $n_L<n_S$).

Figure 2

Single-particle Green function $G(\mathbf{k}=0,\tau )$ computed for hcp ${}^4\mathrm{He}$ at the melting density $n_0=0.0287{\AA }^{-3}$ and $T=0.2\mathrm{K}$. Symbols refer to numerical data; solid lines are fits to the long-time exponential decay. The given numerical values are the interstitial (${\Delta }_I=22.8\pm 0.7\mathrm{K}$) and the vacancy (${\Delta }_V=13.0\pm 0.5\mathrm{K}$) activation energies, inferred from the slopes of $G$. The inset shows the vacancy-interstitial gap $E_{\mathrm{gap}}={\Delta }_I+{\Delta }_V$ for different system sizes.

Figure 3

Chemical potential thresholds for creating a single interstitial (upper curve, ${\mu }_I$) and vacancy (lower curve, ${\mu }_V$) at different densities. The middle curve, ${\mu }_{\mathrm{hcp}}$ is the chemical potential of the hcp solid deduced from the compressibility data [30] and the value of ${\mu }_{\mathrm{hcp}}={\mu }_0$ at melting density. The gaps ${\Delta }_{V(I)}$ for activating a vacancy (or interstitial) are given by the difference in chemical potentials, ${\mu }_{\mathrm{hcp}}-{\mu }_V$ (or ${\mu }_I-{\mu }_{\mathrm{hcp}}$).

Figure 4

The vacancy-vacancy correlation function $\nu (r)$ as a function of the distance $r$ between the vacancies shows that three vacancies easily cluster and form a tight bound state. The inset shows a typical snapshot of a layer of atomic positions averaged over the time interval [0, $\beta$] (filled black circles), where we see that the three vacancies (triangles) have a tendency to cluster in layers.