Reduced Density Matrix Functional Theory for Bosons

Based on a generalization of Hohenberg-Kohn’s theorem, we propose a ground state theory for bosonic quantum systems. Since it involves the one-particle reduced density matrix $\gamma$ as a variable but still recovers quantum correlations in an exact way it is particularly well suited for the accurate description of Bose-Einstein condensates. As a proof of principle we study the building block of optical lattices. The solution of the underlying $v$-representability problem is found and its peculiar form identifies the constrained search formalism as the ideal starting point for constructing accurate functional approximations: The exact functionals $\mathcal{F}[\gamma ]$ for this $N$-boson Hubbard dimer and general Bogoliubov-approximated systems are determined. For Bose-Einstein condensates with $N_{\mathrm{BEC}}\approx N$ condensed bosons, the respective gradient forces are found to diverge, ${\nabla }_{\gamma }\mathcal{F}\propto 1/\sqrt{1-N_{\mathrm{BEC}}/N}$, providing a comprehensive explanation for the absence of complete condensation in nature.

Figure 1

Left: Illustration of the spherical representation (7) of the 1RDM $\gamma$. Right: Straight line $\gamma (\lambda )$ as constructed within the approach (10) at an angle $\phi$.

Figure 2

For the Bose-Hubbard dimer we plot (a) the pure functional ${\mathcal{F}}_N^{(p)}$ and (b) the ensemble functional ${\mathcal{F}}_N^{(e)}$ (both renormalized to [0, 1]) as functions of the diagonal ${\gamma }_{LL}$ and the off-diagonal entry ${\gamma }_{LR}$ of the 1RDM for the particle numbers $N=2,4,\infty$. In (c) the $v$-representable 1RDMs are shown in orange and the nonphysical ones in black (see text for more details).