Remarks on nuclear matter: how an $\omega_0$ condensate can spike the speed of sound, and a model of $Z(3)$ baryons

I make two comments about nuclear matter. First, I consider the effects of a coupling between the $O(4)$ chiral field, $\vec{\phi}$, and the $\omega_\mu$ meson, $\sim + \, \omega_\mu^2 \, \vec{\phi}^{\, 2}$; for any net baryon density, a condensate for $\omega_0$ is unavoidably generated. I assume that with increasing density, a decrease of the chiral condensate and the effective $\omega_0$ mass gives a stiff equation of state (EoS). In order to match that onto a soft EoS for quarkyonic matter, I consider an $O(N)$ field at large $N$, where at nonzero temperature quantum fluctuations disorder any putative pion "condensate" into a quantum pion liquid (Q$\pi$L) (arXiv:2005.10259). In this paper I show that the Q$\pi$L persists at zero temperature. If valid qualitatively at $N=4$, the $\omega_0$ mass goes up sharply and suppresses the $\omega_0$ condensate. This could generate a spike in the speed of sound at high density, which is of relevance to neutron stars. Second, I propose a toy model of a $Z(3)$ gauge theory with three flavors of fermions, where $Z(3)$ vortices confine fermions into baryons. In $1+1$ dimensions this model can be studied numerically with present techniques, using either classical or quantum computers.