A note on inhomogeneous ground states at large global charge

In this note we search for the ground state of the D = 3 Wilson-Fisher conformal O(4) model, at nonzero values of the two independent charge densities rho(1,2), on the torus spatial slice. Using an effective theory valid on scales longer than the scale defined by the charge density, we show that the ground-state configuration is inhomogeneous for generic ratios rho(1)/rho(2). This result confirms, within the context of a well-defined effective theory, a recent no-go result of [1]. We also show that any spatially periodic ground state solutions have an energetic preference towards longer periods, within some range of rho(1)/rho(2) containing a neighborhood of zero. This suggests that the scale of variation of the ground state solution in finite volume will be the infrared scale, and that the use of the effective theory at large charge in finite volume is self-consistent. Note added: the statements in this paper are true for arbitrary ratio of rho(1)/rho(2), which we proved after we uploaded this paper. See [2].