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 particular, we focus on the Nambu–Goldstone modes associated with the spontaneous breakdown of the flavor symmetries. In the strong coupling regime for the interactions, we prove the following: (i) For the spatial dimension ν ≥ 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 ν ≥ 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 appear gapless excitations above the sector of the infinite-volume ground states. These are nothing but the Nambu–Goldstone modes associated with the spontaneous breakdown of the global rotational symmetry of flavors. (v) In particular, we establish that the number of the linearly independent Nambu–Goldstone modes is equal to the number of the broken symmetry generators on the Hilbert space constructed from a certain symmetry-breaking infinite-volume ground state.
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As is well known, in the Euclidean formalism, the reflection positivity is needed to make the corresponding Hamiltonian self-adjoint.
The generators Q(a) are ill-defined in the infinite-volume limit. However, we will always use them in finite-volume systems. Therefore, mathematical problems in the infinite-volume systems do not arise concerning the generators.
In order to prove the equivalence between V and , consider transformations, Vt≔ exp[itA] and , with the real parameter t. It is enough to show the equivalence between VtΨ(x) and . By differentiating with respect to t, one can obtain the two differential equations with the same form. They have also the same initial value at t = 0. Therefore, the uniquness of their solutions implies the desired result at t = 1.
