In this talk, we study a lattice Nambu–Jona-Lasinio model with SU(2) and SU(3) flavor symmetries of staggered fermions
in the Kogut–Susskind Hamiltonian formalism.
This type of four-fermion interactions has been widely used for describing low-energy behaviors of strongly interacting quarks as an effective model.
In the strong coupling regime for the interactions, we prove the following:
(i) For the spatial dimension $d\geqq 5$, the SU(3) model shows a long-range order at sufficiently low temperatures.
(ii) In the case of the SU(2) symmetry, there appears a long-range order in the spatial dimension $d\geqq 3$ at sufficiently low temperatures.
(iii) These results hold in the ground states as well.
(iv) In general, if a long-range order emerges in this type of models, then there exists a gapless excitation above an infinite-volume ground state. This is nothing but the Nambu–Goldstone mode associated with the spontaneous breakdown of the global rotational symmetry of flavors.
(v) It is also established that the number of Nambu–Goldstone modes is equal to the number of broken symmetry generators.