Interplay between unidirectional and bidirectional charge-density-wave orders in underdoped cuprates

We analyze the interplay between charge-density-wave (CDW) orders with axial momenta (Q,0) and (0,Q) (Δx and Δy, respectively), detected in the underdoped cuprates. The CDW order in real space can be unidirectional (either Δx or Δy is nonzero) or bidirectional (both Δx and Δy are nonzero). To understand which of the two orders develop, we adopt the magnetic scenario, in which the CDW order appears due to spin-fluctuation exchange, and derive the Ginzburg-Landau action to the sixth order in Δx and Δy. We argue that at the mean-field level, the CDW order is bidirectional at the onset, with equal amplitudes of Δx and Δy, but changes to unidirectional inside the CDW phase. This implies that at a given temperature, CDW order is unidirectional at smaller dopings but becomes bidirectional at larger dopings. This is consistent with recent x-ray data on YBa2Cu3Oy, which detected tendency towards bidirectional order at larger dopings. We discuss the role of discrete symmetry breaking at a higher temperature for the interplay between bidirectional and unidirectional CDW orders and also discuss the role of pair-density-wave (PDW) order, which may appear along with CDW. We argue that PDW with the same momentum as CDW changes the structure of the bidirectional charge order by completely replacing either Δx or Δy CDW components by PDW. However, if a so-called Amperean PDW order, which pairs fermions with approximately the same momenta, is also present, both Δx and Δy remain nonzero in the bidirectional phase, albeit with nonequal amplitudes. This is again consistent with x-ray experiments, which at larger doping found nonequal Δx and Δy in every domain.