Current- and spin-density-functional theory for inhomogeneous electronic systems in strong magnetic fields

We formulate the current- and spin-density-functional theory for electronic systems in arbitrarily strong magnetic fields. A set of single-particle self-consistent equations which determine, in addition to the ground-state energy, the density, the spin density, the current density, and the spin-current density, is derived and is proved to be gauge invariant and to satisfy various physical requirements, including the continuity equation. For a magnetic field of constant direction in space, we prove that the exchange-correlation energy functional $E_{\mathrm{xc}}$[$n_{\uparrow }$,$n_{\downarrow }$,$j_{p\uparrow }$,$j_{p\downarrow }$] $n_m$T (↓)r) is the ↑ (↓) component of the density and ${\mathrm{j}}_{\mathit{p}\mathrm{\uparrow }\left(\mathrm{\downarrow }\right)}$(r) is the ↑ (↓) component of the ‘‘paramagnetic’’ current density] is actually a functional of $n_{\uparrow }$(r), $n_{\downarrow }$(r), ${\nu }_{\mathrm{\uparrow }}$(r)≡∇×${\mathrm{j}}_{\mathrm{p}\mathrm{\uparrow }}$(r)/${\mathit{n}}_{\mathrm{\uparrow }}$(r), and ${\nu }_{\mathrm{\downarrow }}$(r)≡∇×${\mathrm{j}}_{\mathrm{p}\mathrm{\downarrow }}$(r)/${\mathit{n}}_{\mathrm{\downarrow }}$(r). An explicit form of $E_{\mathrm{xc}}$, which is local in ${\nu }_{\uparrow }$(r) and ${\nu }_{\downarrow }$(r), is derived from linear-response theory. The generalizations to finite-temperature ensembles and to magnetic fields of arbitrarily varying directions are presented.