Effect of the next-nearest neighbor hopping on the stability and band structure of the incommensurate phases in the cuprates

Using a spin-rotation invariant version of the slave-boson approach we investigate the relative stability and band structure of various incommensurate phases in the cuprates. Our findings obtained in the Hubbard model with next-nearest neighbor hopping $-t'/t\simeq 0.15$, as appropriate for the La$_{2-x}$Sr$_x$CuO$_4$ family, support the formation of diagonal (vertical) stripe phases in the doping regime $x=1/16$ ($x=1/8$), respectively. In contrast, based on the fact that a larger value $-t'/t=0.3$ expected for YBa$_2$Cu$_3$O$_{6+\delta}$ triggers a crossover to the diagonal (1,1) spiral phase at increasing doping, we argue that it might explain why the static charge order has been detected in YBa$_2$Cu$_3$O$_{6+\delta}$ only in the highly underdoped regime.