  Recently quantum simulators have been constructed to investigate
experimentally the most prominent theoretical four-point many-body system
described by the Hubbard model. By varying the coupling strength of the
four-point interaction in relation to the kinetic term, one can analyze the
phase structure of the model. This intriguing fact leads us to investigate
whether similar Hamiltonians with four-point interactions can also be studied
as a function of their four-point coupling strength. In this paper, we
reexamine the Nambu-Jona-Lasinio model, regarding it generally beyond the
context of quantum chromodynamics. Essentially, it is a model in which
particle-antiparticle pairing leads to a BCS-like condensate, with the result
that chiral symmetry is broken dynamically in the strong-coupling regime. To
study the behavior of the system, it is necessary to move from this regime to a
hypothetical regime of weak coupling, altering the coupling strength of the
interaction arbitrarily. In order to do this, the gap equation must be regarded
as complex and its Riemann surface structure must be known. We do this and
obtain a continuous quantum phase transition characterized by the development
of a complex order parameter (the dynamically generated mass) from the second
sheet of the Riemann surface, as we move into the weak-coupling regime. The
power-law behavior of the order parameter in the vicinity of the phase
transition point is demonstrated to be independent of the choice of the
regularization scheme with the critical exponent as $\beta \approx 0.55$. At
the same time, the isovector pseudoscalar modes retain their feature as
Goldstone modes and still have zero mass, while the isoscalar scalar meson
follows the behavior of the order parameter and gains a width. Energetically,
this mode is not favored over the normal, uncondensed mode but would have to be
accessed through an excitation process.
