Theory of excitonic order in kagome metals ScV$_6$Sn$_6$ and LuNb$_6$Sn$_6$

We argue that kagome metals can feature an excitonic condensate of unconventional nature. Studying the recently discovered variants ScV$_6$Sn$_6$ and LuNb$_6$Sn$_6$ we identify electron and hole pockets due to a pair of van Hove singularities (vHS) close to the Fermi level, with an approximate spectral particle-hole symmetry. A significant fraction of the Fermi level density of states away from the vHS is removed by the onset of high temperature charge density wave order, and makes the bands more two-dimensional, setting the stage for the formation of excitons. We develop a two-orbital minimal tight-binding model of these materials which captures these features along with the sublattice support of the wavefunctions, and find $s$- or $d$-wave excitons depending on interaction parameters -- the latter of which exhibits either charge nematicity or time-reversal symmetry breaking (TRSB) depending on strain, offering an explanation of recent STM and transport experiments. The presence of particle- and hole-type vHS, and the associated excitonic resonance, may be a common thread to understanding nematicity and TRSB in kagome metals.