Diagonal charge modulation in the insulating La$_{2-x}$Sr$_x$CuO$_4$

We show that, due to the Dzyaloshinskii-Moriya and the XY anisotropies, the disordered diagonal spin spiral in the insulating La$_{2-x}$Sr$_x$CuO$_4$ generates a diagonal charge density wave (CDW) with the wave vector twice of that of the spin spiral. The amplitude of the CDW depends on values of the anisotropies, the doping level, and on the density of states at the chemical potential. Based on available experimental data we estimate that for 4% doping the amplitude of CDW is about 1/10 of the doping level. We believe that this mechanism explains the CDW observed recently in Zn-codoped detwinned La$_{2-x}$Sr$_x$CuO$_4$.