Signatures of thermally excited vortices in a superconductor with competing orders

Experimental evidence for the existence of a fluctuating charge-density wave order in the pseudogap regime of YBa$_2$Cu$_3$O$_{6+x}$ has renewed interest in its interplay with superconductivity. Here, we consider the problem within a nonlinear sigma model, which was recently proposed to describe the apparent competition between the two order parameters. In particular, we use a saddle-point approximation to calculate the properties of superconducting vortex excitations within such a model. In addition, we analytically calculate a collection of experimentally observable quantities, which probe both the superconducting and charge-density wave fluctuations, and identify expected signatures of thermally excited vortices.