Colloquium: Supersolids: What and where are they?

The ongoing experimental and theoretical effort aimed at understanding nonclassical rotational inertia in solid helium has sparked renewed interest in the supersolid phase of matter, its microscopic origin and character, and its experimental detection. The purpose of this Colloquium is to provide a general theoretical framework for the phenomenon of supersolidity, review some of the experimental evidence for solid ${}^4\mathrm{He}$, and discuss its possible interpretation in terms of physical effects underlain by extended defects (such as dislocations). Quantitative support to our theoretical scenarios by means of first-principle numerical simulations is provided. Alternate avenues for the observation of the supersolid phase, not involving helium but rather assemblies of ultracold atoms, are also discussed.

Figure 1

Schematic of the experimental torsional oscillator setup aimed at measuring nonclassical rotational inertia. The cell is filled with a sample of condensed matter, whose superfluid fraction affects the moment of inertia of the overall system.

Figure 2

Schematic of an experiment measuring mass flow through a supersolid. Superfluid liquid flows across the central section filled with supersolid, of the same substance. The setup as shown can work only at the melting pressure. In the case of ${}^4\mathrm{He}$, modifications involving Vycor segments and temperature gradients allow one to extend measurements to higher pressures. From 73.

Figure 3

One-body density matrix $n(r)$, computed by Monte Carlo simulation, of solid (hcp) near (density $\overline{\rho }=0.0292{\AA }^{-3}$) and away from (density $\overline{\rho }=0.0359{\AA }^{-3}$) the melting curve at a low temperature $T=0.2\mathrm{K}$. Results obtained at lower temperature (as low as 0.1 K) are not distinguishable from those shown here, within the statistical uncertainties of the calculation. The exponential decay of $n(r)$ signals no off-diagonal long-range order (i.e., Bose-Einstein condensation) in the system. From 17.

Figure 4

(a) Domain wall in the checkerboard solid. (b) When an atom is moved along the wall by one atomic distance, the wall position shifts in the perpendicular direction while the kink on the wall moves along the wall by two atomic distances.

Figure 5

The minimal phase defect in the network is obtained by imagining a vortex ring winding around the pipe and creating $\pm 2\pi$ phase windings in elementary plaquettes.

Figure 6

One-dimensional crystal of particles interacting via a soft-sphere potential, equal to some energy $V>0$ if the distance between two particles is less than their effective diameter $a$, and zero if it is greater. As the density equals $1/a$, (b) becomes energetically advantageous over (a), as its cost is $V/2$ per particle, as opposed to $V$ as in (a).

Figure 7

Snapshots from computer simulations of a system of bosons interacting via a power-law potential flattening off at short distances [see 25) for details]. Snapshots are taken at four different temperatures, decreasing from (a) to (d); each snapshot is taken at a temperature an order of magnitude lower than the immediately previous one. From 25.