${\mathbb{Z}}_L$ symmetry breaking in SU(N) Fermi-Hubbard dots at zero and finite temperature

We address the SU(N) Fermi-Hubbard model on a chain, with $N$ the number of degenerate orbitals, or colors, for each fermion. In the limit of both large number of colors $N$ and particles, and small number of sites $L\ge 2$, the model is proved to undergo a ${\mathbb{Z}}_L$ symmetry breaking for attractive local interaction amplitude $U$. Using a combination of Exact Diagonalization with full SU(N) symmetry, generalized L-levels Holstein-Primakoff transformation, Hartree-Fock method and large-N saddle point approximation of the partition function, we extend the results obtained in [PRA 111, L020201 (2025)] to $L\ge 3$ and finite temperature $T>0$. In particular, we show that at $T=0$ for $U<U_c\sim -1\mathrm{/}N$, the ground state is L-fold degenerate, while for positive temperatures, the critical temperature is both proportional to $N$ and $U$, i.e. $T_c\propto -UN$, making this phase transition particularly suitable for large-N fermions.