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In this note we search for the ground state, in infinite volume, of the $D=3$Wilson-Fisher conformal $O(4)$ model, at nonzero values of the two independentcharge densities $\rho_{1,2}$. Using an effective theory valid on scales longerthan the scale defined by the charge density, we show that the ground-stateconfiguration is inhomogeneous for generic ratios $\rho_1 / \rho_2$. Thisresult confirms, within the context of a well-defined effective theory, arecent no-go result of Alvarez-Gaume' et al. We also show that any spatiallyperiodic ground state solutions have an energetic preference towards longerperiods, within some range of $\rho_1 / \rho_2$ containing a neighborhood ofzero. This suggests that the scale of variation of the ground state solution infinite volume will be the infrared scale, and that the use of the effectivetheory at large charge in finite volume is self-consistent.

A Note on Inhomogeneous Ground States at Large Global Charge