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arXiv:0909.5553 · Physical Review B

Off-diagonal long-range order and supersolidity in a quantum solid with vacancies

Yu Shi, Yin Yang, Shao-Ming Fei

Published Nov 3, 2009• 0 citations
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Abstract

We consider a lattice of bosonic atoms, whose number $N$ may be smaller than the number of lattice sites $M$. We study the Hartree-Fock wave function built up from localized wave functions $w(\mathbf{r})$ of single atoms, with nearest-neighboring overlap. The zero-momentum particle number is expressed in terms of permanents of matrices. In one dimension, it is analytically calculated to be $\ensuremath{\alpha}N(M\ensuremath{-}N+1)/M$, with $\ensuremath{\alpha}={|\ensuremath{\int}w(\mathbf{r})d\ensuremath{\Omega}|}^{2}/[(1+2a)l]$, where $a$ is the nearest-neighboring overlap and $l$ is the lattice constant. $\ensuremath{\alpha}$ is on the order of 1. The result indicates that the condensate fraction is proportional to and of the same order of magnitude as that of the vacancy concentration, hence there is off-diagonal long-range order or Bose-Einstein condensation of atoms when the number of vacancies $M\ensuremath{-}N$ is a finite fraction of the number of the lattice sites $M$.

DOI
10.1103/physrevb.80.174503

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We have computed the one-body density matrix ${\ensuremath{\rho}}_{1}$ in solid 4He^{4}\mathrm{He}4He at T=0KT=0\phantom{\rule{0.3em}{0ex}}\mathrm{K}T=0K 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 ${\ensuremath{\rho}}_{1}$ for a perfect hcp and bcc solid 4He^{4}\mathrm{He}4He 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 4He^{4}\mathrm{He}4He. At melting, the condensate fraction in the hcp solid is $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ 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.
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We consider interacting Bose particles in an external potential. It is shown that a Bose-Einstein condensate is possible at finite temperatures that describes a super solid in three dimensions (3D) for a wide range of potentials in the absence of an external potential. However, for 2D, a self-organized super solid exists for finite temperatures provided the interaction between bosons is nonlocal and of infinitely long-range. It is interesting that in the absence of the latter type of potential and in the presence of a lattice potential, there is no Bose-Einstein condensate and so in such a case, a 2D super solid is not possible at finite temperatures. We also propose the correct Bloch form of the condensate wave function valid for finite temperatures, which may be used as the correct trial wave function.