  We study the Ginzburg-Landau free energy functional for two coupled U(1)
charge order parameters describing two non-equivalent charge orders with wave
vector ${\bf Q}$ detected in X-ray and STM measurements of underdoped cuprates.
We do not rely on a mean-field analysis, but rather utilize a field-theoretical
technique suitable to study the interplay between vortex physics and discrete
symmetry breaking in two-dimensional systems with U(1) symmetry. Our
calculations support the idea that in the clean systems there are two
transitions: from a high temperature disordered state into a state with a
composite charge order which breaks time-reversal symmetry, but leaves U(1)
fields disordered, and then into a state with quasi long range order in the
U(1) fields.
