Ring Exchanges and the Supersolid Phase ofHe4

Using path integral Monte Carlo simulations we calculate exchange frequencies in bulk hcp ${}^4\mathrm{H}\mathrm{e}$ as atoms undergo ring exchange. We fit the frequencies to a lattice model and examine whether such atoms could become a supersolid, that is, have a nonclassical rotational inertia. We find that the scaling with respect to the number of exchanging atoms is such that superfluid behavior will not be observed in a perfect ${}^4\mathrm{H}\mathrm{e}$ crystal.