The Lieb-Mattis theorem orders the lowest-energy states of total spin $s$ of a system of $P$ interacting fermions. We generalize these predictions to fermionic mixtures of $P$ particles with more than $N=2$ spinor components or species in the thermodynamic limit $P\to \infty$. The lowest-energy state inside each permutation symmetry sector $h$, arising in the $P$-fold tensor product decomposition, is well approximated by a $\mathrm{U}\left(N\right)$ coherent (quasiclassical variational) state, especially in the limit $P\to \infty$. In particular, the ground state of the system belongs the most symmetric (dominant Young tableau $h_0$) configuration. We exemplify our construction with the $N=3$ level Lipkin-Meshkov-Glick model, with a previous motivation on pairing correlations and $\mathrm{U}\left(N\right)$-invariant quantum Hall ferromagnets. In the limit $P\to \infty$, each lowest-energy state within each permutation symmetry sector $h$ undergoes a quantum phase transition for a critical value ${\lambda }_c\left(h\right)$ of the exchange coupling constant $\lambda$, depending on $h$. This generalizes standard quantum phase transitions and their phase diagrams corresponding to the ground state belonging to the most symmetric sector $h_0$.

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• Phase diagrams
• Quantum phase transitions
• Quantum statistical mechanics
• Multicomponent systems
• Quantum many-body systems
• Group theory
• Heisenberg model
• Many-body techniques
• Phase space methods