New theory of superfluid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$

It is proposed that regions of low and high mass density exist in liquid helium and moreover are spatially well-ordered in the superfluid state with the result that the ground-state wave function ${\Psi }_0$ is periodic. A ground-state energy 8.1% lower than that determined by Monte Carlo procedure for a pair-correlated wave function $f\left(r_{\mathrm{ij}}\right)$ by McMillan is obtained with ${\Psi }_0=\left[\Pi \stackrel{}{i>j}\left[f\right(r_{\mathrm{ij}}\left)\right]\right]\left[1+B \Sigma \stackrel{}{i>j}\frac{cos2k_0{r}_{\mathrm{ij}}}{k_0{r}_{\mathrm{ij}}}\right]$ with $B=0.054\mathrm{}\pi$ and $k_0=2\mathrm{}\frac{\pi }{6.8}$ ${\mathrm{\AA }}^{-1\mathrm{}}$. A mean-field theory of the $\lambda$ transition has been formulated. The nature of the superfluid state is a consequence of the spatial periodicity of the wave function and a simple physical picture of the roton is presented as well as a rigorous roton wave function.