Bose-Einstein condensation in solidHe4

We have computed the one-body density matrix ${\rho }_1$ in solid ${}^4\mathrm{He}$ at $T=0\mathrm{K}$ using the shadow wave-function (SWF) variational technique. The accuracy of the SWF has been tested with an exact projector method. We find that off-diagonal long-range order is present in ${\rho }_1$ for a perfect hcp and bcc solid ${}^4\mathrm{He}$ for a range of densities above the melting one, at least up to 54 bars. This is first microscopic indication that Bose Einstein condensation (BEC) is present in perfect solid ${}^4\mathrm{He}$. At melting, the condensate fraction in the hcp solid is $5\times {10}^{-6}$ and it decreases by increasing the density. The key process giving rise to BEC is the formation of vacancy interstitial pairs. We also present values for Leggett’s upper bound on the superfluid fraction deduced from the exact local density.