Superfluidity of solid4He

Abstract

Using a procedure suggested by Leggett, an upper bound to the superfluid fraction in ground state solid 4 He slightly above the melting density is obtained numerically. The value obtained is 0.3±0.1. To judge the usefulness of this upper bound, we examine the conditions under which a symmetrized product of single-particle functions times a Jastrow function exhibits ODLRO, a necessary and sufficient condition for superfluid flow. It is found that ifU ij (U ij=ί φ i (x)φ j (x) dx, and φ i (x) is a single-particle wave function centered on the pointi) satisfies σ′i U ij>x, wherex varies from unity for long rangeU ij (i.e.,U ij decreases slowly enough asR i−Rj increases) to a value of 12/7 for nearest-neighbor overlap only in the hcp lattice, then there is ODLRO, but not otherwise. Therefore, if the accepted single-particle functions are the true ones, then there is no ODLRO in solid 4 He, since the overlap is too small. We have explored the possibility of adding a flat tail, of magnitude λ′(VN)−1/2 to the accepted single-particle functions. It is shown that if λ → 1 [λ 2 =(λ′) 2 +2(vNV −1)1/2, andv=(ί φi(x)dx)2], the system wave function becomes a pure Jastrow function, whereas if λ2−1≲−2×10 −1 , we have in effect the case where λ′=0; furthermore, there is ODLRO if λ 2 −1∼−2×10 −1 . It is also concluded that the superfluid fraction upper bound of 0.3±0.1 obtained here as well as one suggested by Leggett are not very useful. We have not attempted to establish if there is some value of λ satisfying the above inequality such that the ground-state energy is lower than the value it takes for λ′=0.