Abstract
We study the Ginzburg-Landau free energy functional for two coupled $U(1)$ charge order parameters describing two nonequivalent charge orders with wave vector $\mathbf{Q}$ 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.
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