  Charge-density wave (CDW) modulations in underdoped high-temperature cuprate
superconductors remain a central puzzle in condensed matter physics. However,
despite a substantial experimental verification of this ubiquitous phase in a
large class of high $T_{\mathrm{c}}$ cuprates, a complete theoretical
explanation of this phase is still missing. Here, we build upon our recent
proposal that the CDW in underdoped cuprates (Y- and Bi- based compounds)
emerges from a unique cooperation of the B$_{1g}$ bond-buckling phonon with
strong electronic correlations. We assume a static mean-field lattice
distortion with B$_{1g}$ symmetry, regardless of its origin, with a
commensurate wave vector $\mathbf{q}^*=(2\pi/3,0)/(0,2\pi/3)$. We show that
such a phonon-induced CDW (both uni- and bi-axial) reconstructs the Fermi
surface, leading to electron and hole pockets, with relevant quantum
oscillation frequencies in close consistency with the experiments. Furthermore,
a systematic analysis of the symmetry of the intra-unit-cell charge modulations
on the copper-oxygen planes is provided. We find that the atomic charge
modulation on the CuO$_2$ unit cell is predominantly of $s$-wave character --
in support of the recent experimental observation.
