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 $-t^{\mathrm{\prime }}\mathrm{/}t\simeq 0.15$, as appropriate for the La${}_{2-x}$Sr${}_x$CuO${}_4$ family, support the formation of diagonal (vertical) stripe phases in the doping regime $x=1\mathrm{/}16$ ($x=1\mathrm{/}8$), respectively. In contrast, based on the fact that a larger value $-t^{\mathrm{\prime }}\mathrm{/}t=0.3$ 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.