Theory of Quantum Oscillations in Cuprate Superconductors

Cuprate supercoductors show pseudogap state below its transition temperature $T^*$. The pseudogap state is distinct from superconducting state(critical temperature $T_c < T^*$). Two states may coexist and superconductivity developes from the pseudogap state which can be explained by commensurate and incommensurate $d$-density wave ($d$DW) orders. Quantum oscillations help us to understand the pseudogap state. Hall coefficient, magnetization, conductance and specific heat oscillate as a function of external field in $Y Ba_2 Cu_3 O_{6+\\delta}$ (YBCO), $Nd_{2-x}Ce_xCuO_4$ (NCCO) and other cuprates. The oscillation frequency $F$ is proportional to the area of closed-orbits in reconstructed Fermi surface. High magnetic field for quantum oscillation experiment suppresses superconductivity. Therefore, quantum oscillations measure reconstructed Fermi surface of pseudogap state.Electron-doped cuprate superconductor NCCO shows only hole pocket frequency peak. Experimental results can be explained by period-2 $d$DW order and white-noise disorder. Disorder in the system removes electron pocket frequency and adjusts the amount of magnetic breakdown effect corresponding to very high frequency. Period-8 density wave order can explain quantum oscillations in the pseudogap state of hole-doped $YBCO$. Only electron pocket frequency is observed. Period-8 $d$DW order generates small but many hole pockets in the reconstructed Fermi surface. Small hole pocket frequency is too slow to be observed.