Role of mixed permutation symmetry sectors in the thermodynamic limit of critical three-level Lipkin-Meshkov-Glick atom models

We introduce the notion of Mixed Symmetry Quantum Phase Transition (MSQPT) as singularities in the transformation of the lowest-energy state properties of a system of identical particles inside each permutation symmetry sector $mu$, when some Hamiltonian control parameters $lambda$ are varied. We use a three-level Lipkin-Meshkov-Glick (LMG) model, with $U(3)$ dynamical symmetry, to exemplify our construction. After reviewing the construction of $U(3)$ unirreps using Young tableaux and Gelfand basis, we firstly study the case of a finite number $N$ of three-level atoms, showing that some precursors (fidelity-susceptibility, level population, etc.) of MSQPTs appear in all permutation symmetry sectors. Using coherent (quasi-classical) states of $U(3)$ as variational states, we compute the lowest-energy density for each sector $mu$ in the thermodynamic $Ntoinfty$ limit. Extending the control parameter space by $mu$, the phase diagram exhibits four distinct quantum phases in the $lambda$-$mu$ plane that coexist at a quadruple point. The ground state of the whole system belongs to the fully symmetric sector $mu=1$ and shows a four-fold degeneracy, due to the spontaneous breakdown of the parity symmetry of the Hamiltonian. The restoration of this discrete symmetry leads to the formation of four-component Schrodinger cat states.