Lieb-Mattis ordering theorem of electronic energy levels in the thermodynamic limit

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/species in the thermodynamic limit $P\to \mathrm{\infty }$. The lowest-energy state inside each permutation symmetry sector $h$, arising in the $P$-fold tensor product decomposition, is well approximated by a U$(N)$ coherent (quasi-classical, variational) state, specially in the limit $P\to \mathrm{\infty }$. 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 U$(N)$-invariant quantum Hall ferromagnets. In the limit $P\to \mathrm{\infty }$, each lowest-energy state within each permutation symmetry sector $h$ undergoes a quantum phase transition for a critical value ${\lambda }_c(h)$ 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 $h_0$.