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I make two comments about nuclear matter. First, I consider the effects of acoupling between the $O(4)$ chiral field, $\vec{\phi}$, and the $\omega_\mu$meson, $\sim + \, \omega_\mu^2 \, \vec{\phi}^{\, 2}$; for any net baryondensity, a condensate for $\omega_0$ is unavoidably generated. I assume thatwith 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 thatonto 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 thispaper I show that the Q$\pi$L persists at zero temperature. If validqualitatively 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 athigh density, which is of relevance to neutron stars. Second, I propose a toymodel 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 bestudied numerically with present techniques, using either classical or quantumcomputers.

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