Interplay between Pair Density Wave and a Nested Fermi Surface

We show that spontaneous time-reversal-symmetry (TRS) breaking can naturally arise from the interplay between pair density wave (PDW) ordering at multiple momenta and nesting of Fermi surfaces (FSs). Concretely, we consider the PDW superconductivity on a hexagonal lattice with nested FS at $3/4$ electron filling, which is related to a recently discovered superconductor ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$. Because of nesting of the FSs, each momentum $\mathbf{k}$ on the FS has at least two counterparts $\ensuremath{-}\mathbf{k}\ifmmode\pm\else\textpm\fi{}{\mathbf{Q}}_{\ensuremath{\alpha}}$ ($\ensuremath{\alpha}=1$, 2, 3) on the FS to form finite momentum ($\ifmmode\pm\else\textpm\fi{}{\mathbf{Q}}_{\ensuremath{\alpha}}$) Cooper pairs, resulting in a TRS and inversion broken PDW state with stable Bogoliubov Fermi pockets. Various spectra, including (local) density of states, electron spectral function, and the effect of quasiparticle interference, have been investigated. The partial melting of the PDW will give rise to $4\ifmmode\times\else\texttimes\fi{}4$ and $(4/\sqrt{3})\ifmmode\times\else\texttimes\fi{}(4/\sqrt{3})$ charge density wave (CDW) orders, in addition to the $2\ifmmode\times\else\texttimes\fi{}2$ CDW. Possible implications to real materials such as ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$ and future experiments have been discussed further.