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Reconstructed Fermi surface and quantum oscillation of doped resonating valence bond state with incommensurate charge order in underdoped cuprates

Recent experiments have revealed incommensurate charge density wave (CDW) in the pseudogap regime in underdoped cuprates, e.g. YBa$_2$Cu$_3$O$_{6+δ}$ and HgBa$_2$CuO$_{4+δ}$. However, its relationship with the pseudogap is still controversial. In this work, we take a phenomenological synthesis of the doped resonating valence bond (RVB) state and the CDW order. Starting from the Yang-Rice-Zhang Green's function ansatz for the doped RVB state [Phys. Rev. B {\bf 73}, 174501 (2006)], in which the Fermi surface is partially truncated into four nodal hole-like Fermi pockets by the antinodal RVB gap, we show that the CDW order at the wavevectors connecting the tips of the Fermi arcs (the hotspots) induces Fermi surface reconstruction, giving rise to an electron-like Fermi pocket ($α$ orbit) and a new hole-like Fermi pocket ($β$ orbit). The $α$ orbit is formed by joining the Fermi arcs at the hotspots and it dominates the quantum oscillation Fourier spectrum, while the $β$ orbit is formed by joining the outer patches of the original hole pockets, which has vanishingly small spectral weight. The areas enclosed by these orbits are extracted from the density of states oscillation in magnetic field and quantitatively agree with the experiments.