The fate of chiral symmetry in two-flavor matrix adjoint QC$_2$D

Abstract: In the matrix model of two-flavor adjoint QC$_2$D, we study the low-lying states and their properties in the intermediate-to-strong (Yang-Mills) coupling regime. The model has a classical $SU(2)_R$ chiral symmetry and the eigenstates of the Hamiltonian can be organized in its irreps. We construct the energy eigenstates in presence of a chiral chemical potential $c$ using the variational techniques. We find that when $c=0$, the ground state is always a $SU(2)_R$ singlet, irrespective of the coupling strength $g$. However, as $g$ is tuned from intermediate to strong coupling, the system under goes a crossover from a unique to a doubly degenerate ground state. The degeneracy in the strong coupling regime spontaneously breaks the axial $\mathbb{Z}_8$, while preserving the chiral symmetry. When $c \neq 0$, we find that there can be level crossings which correspond to quantum phase transitions (QPTs). Depending on the ground state, there are three possible phases. The $SU(2)_R$ symmetry is spontaneously broken in only one of these phases, and this phase can only emerge for intermediate $g$ with moderate values of $c$. In the $g-c$ plane this phase corresponds to a narrow window, outside which $SU(2)_R$ is always preserved.