Mixed permutation symmetry quantum phase transitions of critical three-level atom models

We define the concept of Mixed Symmetry Quantum Phase Transition (MSQPT), considering each permutation symmetry sector \muμ of an identical particles system, as singularities in the properties of the lowest-energy state into each \muμ when shifting a Hamiltonian control parameter \lambdaλ. A three-level Lipkin-Meshkov-Glick (LMG) model is chosen to typify our construction. Firstly, we analyze the finite number NN of particles case, proving the presence of MSQPT precursors. Then, in the thermodynamic limit N\to∞N→∞, we calculate the lowest-energy density inside each sector \muμ, augmenting the control parameter space by \muμ, and showing a phase diagram with four different quantum phases.