Diamagnetism and density wave order in the pseudogap regime of YBa$_2$Cu$_3$O$_{6+x}$

Clear experimental evidence of charge density wave correlations competing with superconducting order in YBCO have thrust their relationship with the pseudogap regime into the spotlight. To aid in characterizing the pseudogap regime, we propose a dimensionless ratio of the diamagnetic susceptibility to the correlation length of the charge density wave correlations. Using Monte Carlo simulations, we compute this ratio on the classical model of Hayward et. al. (Science 343, 1336 (2014)), which describes angular fluctuations of a multicomponent order, capturing both superconducting and density wave correlations. We compare our results with available data on YBa$_2$Cu$_3$O$_{6+x}$, and propose experiments to clarify the value of this dimensionless ratio using existing samples and techniques.