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

Abstract

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 ${E}_{\mathrm{xc}}$[${n}_{\ensuremath{\uparrow}}$,${n}_{\ensuremath{\downarrow}}$,${j}_{p\ensuremath{\uparrow}}$,${j}_{p\ensuremath{\downarrow}}$] ${n}_{m}$T (\ensuremath{\downarrow})r) is the \ensuremath{\uparrow} (\ensuremath{\downarrow}) component of the density and ${\mathrm{j}}_{\mathit{p}\mathrm{\ensuremath{\uparrow}}\phantom{\rule{0ex}{0ex}}(\mathrm{\ensuremath{\downarrow}})}$(r) is the \ensuremath{\uparrow} (\ensuremath{\downarrow}) component of the ``paramagnetic'' current density] is actually a functional of ${n}_{\ensuremath{\uparrow}}$(r), ${n}_{\ensuremath{\downarrow}}$(r), ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\uparrow}}}$(r)\ensuremath{\equiv}\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}${\mathrm{j}}_{\mathrm{p}\mathrm{\ensuremath{\uparrow}}}$(r)/${\mathit{n}}_{\mathrm{\ensuremath{\uparrow}}}$(r), and ${\ensuremath{\nu}}_{\mathrm{\ensuremath{\downarrow}}}$(r)\ensuremath{\equiv}\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}${\mathrm{j}}_{\mathrm{p}\mathrm{\ensuremath{\downarrow}}}$(r)/${\mathit{n}}_{\mathrm{\ensuremath{\downarrow}}}$(r). An explicit form of ${E}_{\mathrm{xc}}$, which is local in ${\ensuremath{\nu}}_{\ensuremath{\uparrow}}$(r) and ${\ensuremath{\nu}}_{\ensuremath{\downarrow}}$(r), is derived from linear-response theory. The generalizations to finite-temperature ensembles and to magnetic fields of arbitrarily varying directions are presented.