Interplay between uni-directional and bi-directional charge orders in underdoped cuprates

We analyze the interplay between charge-density-wave (CDW) orders with axial momenta $(Q,0)$ and $(0,Q)$ ($\Delta_x$ and $\Delta_y$ respectively), detected in the underdoped cuprates. The CDW order in real space can be uni-directional (either $\Delta_x$ or $\Delta_y$ is non-zero) or bi-directional (both $\Delta_x$ and $\Delta_y$ are non-zero). To understand which of the two orders develop, we adopt the magnetic scenario, in which the CDW order appears due to spin-fluctuation exchange. We derive the Ginzburg-Landau action to the sixth order in $\Delta_x$ and $\Delta_y$ and argue that the CDW order is bi-directional at the onset but changes to uni-directional inside the CDW phase. This implies that, at a given temperature, CDW order is uni-directional at smaller dopings, but becomes bi-directional at larger dopings. These results are consistent with recent x-ray data on YBCO, which detected tendency towards bi-directional order at larger dopings. We also discuss for completeness the effect of yet unobserved intertwined pair-density-wave (PDW) order, which may appear along with CDW.