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

Composite charge order in the pseudogap region of the cuprates