# A Note on Inhomogeneous Ground States at Large Global Charge

## TL;DR

This paper investigates the ground state configurations of the 3D Wilson-Fisher $O(4)$ model at nonzero charge densities, revealing inhomogeneous states and their energetic preferences, confirming recent theoretical no-go results.

## Contribution

It demonstrates that the ground state is inhomogeneous for generic charge ratios and analyzes the energetic favorability of spatially periodic solutions within an effective theory framework.

## Key findings

- Ground state is inhomogeneous for generic charge ratios.
- Spatially periodic solutions favor longer periods.
- Effective theory remains self-consistent at large charge.

## Abstract

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 independent charge densities $\rho_{1,2}$. 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 Alvarez-Gaume' et al. 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.