Superconductivity and quantum phase transitions in dense QCD$_3$

We present a new perspective on thermal and quantum phase transitions (QPT) in $(2+1)$-dimensional quantum chromodynamics based on symmetries, topology, and quantum dynamical structure of the baryon ground state in the large $N_c$ limit for quarks in the two-index antisymmetric representation. The intermediate and high density regimes are modeled through effective four-fermion interactions, which include attractive scalar and diquark terms, with a term to account for vector meson repulsion at high densities. We address beyond-mean-field phenomena by constructing the quantum field theory for fluctuations in internal degrees of freedom of the baryon ground state using a Madelung decomposition. A key QPT occurs at the meson-diquark transition driven by the ratio of baryon chemical potential to quark mass, $\mu_B/m$, or to an external applied magnetic field, $\mu_B/|eB|$. Properties of the system at ${\mu^c_B} \equiv m, |eB|$ stem from diverging quantum fluctuations of a continuous field $\gamma$ that modulates the spin-space angular positions of baryon potential minima, identified with discrete chiral $\mathbb{Z}_2^\sigma$, $\mathbb{Z}_2^{LR} \times \mathbb{Z}_2^{LR}$, and $\mathbb{Z}_4^\chi$ symmetries for meson, diquark, and asymptotically free regimes. Remarkably, competition between $\mu_B$ and $m$ (or $B$), destroys superconductivity via a \emph{quantum} Berezinskii-Kosterlitz-Thouless (BKT) phase transition at ${\mu^c_B}$. Moreover, the large $\gamma$ fluctuations at ${\mu^c_B }$ behave as an additional momentum scale, resulting in an effective $(3+1)$d conformal critical theory there. We use these insights to elucidate the nature of \emph{holographic} BKT transitions, from the field theory side, a result which has remained elusive to date. We derive the QCD phase diagram for $\mu_B$ above the baryon mass and find good agreement with results obtained by other methods.