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The discovery of the charge-density-wave formation in the high-$T_c$ cupratesuperconductors has activated intensive theoretical studies for the pseudogapstates. However, the microscopic origin of the charge-density-wave state hasbeen unknown so far since the many-body effects beyond the mean-field-levelapproximations, called the vertex corrections, are essential. Toward solvingthis problem, we employ the recently developed functional renormalization-groupmethod, by which we can calculate the higher-order vertex corrections in asystematic and unbiased way with high numerical accuracy. We discover thecritical development of the $p$-orbital-density-wave ($p$-ODW) instability inthe strong-spin-fluctuation region. The obtained $p$-ODW state possesses thekey characteristics of the charge ordering pattern in Bi- and Y-basedsuperconductors, such as the wave vector parallel to the nearest Cu-Cudirection, and the $d$-symmetry form factor with the antiphase correlationbetween $p_x$ and $p_y$ orbitals in the same unit cell. In addition, from theobservation of the beautiful scaling relation between the spin susceptibilityand the $p$-ODW susceptibility, we conclude that the main driving force of thedensity wave is the Aslamazov-Larkin vertex correction that becomes verysingular near the magnetic quantum-critical point.

$p$-orbital density wave with $d$ symmetry in high-$T_c$ cuprate superconductors predicted by the renormalization-group + constrained RPA theory