Observables in Inhomogeneous Ground States at Large Global Charge

TL;DR

This paper extends the analysis of the ground state in the $O(4)$ model at large charges, showing inhomogeneity for generic charge ratios, computing the energy and correlators, and revealing violations of cluster decomposition.

Contribution

It proves inhomogeneity of the ground state for generic charge ratios and computes the associated energy and two-point functions in the $O(4)$ model.

Findings

Ground state is inhomogeneous for generic charge ratios.

Ground state energy scales as $( ho_1+ ho_2)^{3/2}$.

Two-point functions violate cluster decomposition at large volume.

Abstract

As a sequel to previous work, we extend the study of the ground state configuration of the $D=3$, Wilson-Fisher conformal $O(4)$ model. In this work, we prove that for generic ratios of two charge densities, $\rho_1/\rho_2$, the ground-state configuration is inhomogeneous and that the inhomogeneity expresses itself towards longer spatial periods. This is the direct extension of the similar statements we previously made for $\rho_1/\rho_2\ll 1$. We also compute, at fixed set of charges, $\rho_1,\, \rho_2$, the ground state energy and the two-point function(s) associated with this inhomogeneous configuration on the torus. The ground state energy was found to scale $(\rho_1+\rho_2)^{3/2}$, as dictated by dimensional analysis and similarly to the case of the $O(2)$ model. Unlike the case of the $O(2)$ model, the ground also strongly violates cluster decomposition in the large-volume, fixed-density limit, with a two-point function that is negative definite at antipodal points of the torus at leading order at large charge.