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Hole-doped high-temperature cuprate superconductors below optimum doping havesmall electron-like Fermi surfaces occupying a small fraction of the Brillouinzone. There is strong evidence that this is linked to charge density wave (CDW)order, which reconstructs the large hole-like Fermi surfaces predicted by bandstructure calculations . Recent experiments have revealed the structure of thetwo CDW components in the benchmark bilayer material YBa$_2$Cu$_3$O$_{7-x}$ inhigh field where quantum oscillation (QO) measurements are performed. We havecombined these results with a tight-binding description of the bands in anisolated bilayer to give a minimal model revealing the essential physics of thesituation. Here we show that this approach, combined with the effects ofspin-orbit interactions and the pseudogap, gives a good qualitative descriptionof the multiple frequencies seen in the QO observations in this material.Magnetic breakdown through weak CDW splitting of the bands will lead to afield-dependence of the QO spectrum and to the observed fourfold symmetry ofthe results in tilted fields.

physical review. b./physical review. b

Fermi Surface Reconstruction and Quantum Oscillations in Underdoped  YBa$_2$Cu$_3$O$_{7-x}$ Modeled in a Single Bilayer with Mirror Symmetry  Broken by Charge Density Waves