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, $\overrightarrow{\varphi }$, and the ${\omega }_{\mu }$ meson, $\sim +{\overrightarrow{\varphi }}^2{\omega }_{\mu }^2$; 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 “condensates” into a quantum pion liquid ($\mathrm{Q}\pi \mathrm{L}$) [R. D. Pisarski et al., Phys. Rev. D 102, 016015 (2020)]. In this paper, I show that the $\mathrm{Q}\pi \mathrm{L}$ 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.