Density Functionals in the Presence of Magnetic Field

In this paper density functionals for Coulomb systems subjected to electric and magnetic fields are developed. The density functionals depend on the particle density, $\rho$, and paramagnetic current density, $j^p$. This approach is motivated by an adapted version of the Vignale and Rasolt formulation of Current Density Functional Theory (CDFT), which establishes a one-to-one correspondence between the non-degenerate ground-state and the particle and paramagnetic current density. Definition of $N$-representable density pairs $(\rho,j^p)$ is given and it is proven that the set of $v$-representable densities constitutes a proper subset of the set of $N$-representable densities. For a Levy-Lieb type functional $Q(\rho,j^p)$, it is demonstrated that (i) it is a proper extension of the universal Hohenberg-Kohn functional, $F_{HK}(\rho,j^p)$, to $N$-representable densities, (ii) there exists a wavefunction $\psi_0$ such that $Q(\rho,j^p)=(\psi_0,H_0\psi_0)_{L^2}$, where $H_0$ is the Hamiltonian without external potential terms, and (iii) it is not convex. Furthermore, a convex and universal functional $F(\rho,j^p)$ is studied and proven to be equal the convex envelope of $Q(\rho,j^p)$. For both $Q$ and $F$, we give upper and lower bounds.