Quantum phases at high chemical potential in 2-flavor matrix-QC$_2$D

We investigate the matrix model of two-color two-flavor QCD (matrix-QCD$_{2,2}$) in regimes with large baryon ($\mu_{_B}$), isospin ($\mu_{_I}$), and/or chiral ($c$) chemical potentials. In these regimes, the Hamiltonian simplifies considerably, making it possible to investigate the ground state for intermediate and strong Yang-Mills coupling. By diagonalizing the Hamiltonian using the variational techniques, we show that in regimes where $\mu_{_B}$ and $c$ (or $\mu_{_B}$ and $\mu_{_I}$) dominate, tuning the remaining parameters leads to quantum phase transitions (QPTs). These transitions form a complex web of phases, each of which has a ground state uniquely labelled by baryon number $B$ and isospin $I$. Several of these phases are LOFF-like, characterized by a ground state carrying non-zero spin and hence spontaneously breaking rotational symmetry. These results are consistent with older effective field theory predictions by Splittorff-Son-Stephanov \cite{Splittorff:2000mm}. The fermionic content of these LOFF-like ground states consists of spin-1 di-(anti-) quarks which are analogous to Cooper pairs. We compute the spin-fraction carried by the quarks and find that it constitutes a significant portion -- in some cases nearly the entirety -- of the total spin.