  An effective Proca Lagrangian action is used to address the vector
condensation Lorentz violation effects on the equation of state of the strongly
interacting fermions system. The interior quantum fluctuation effects are
incorporated as an external field approximation indirectly through a fictive
generalized Thomson Problem counterterm background. The general analytical
formulas for the $d$-dimensions thermodynamics are given near the unitary limit
region. In the non-relativistic limit for $d=3$, the universal dimensionless
coefficient $\xi ={4}/{9}$ and energy gap $\Delta/\epsilon_f ={5}/{18}$ are
reasonably consistent with the existed theoretical and experimental results. In
the unitary limit for $d=2$ and T=0, the universal coefficient can even
approach the extreme occasion $\xi=0$ corresponding to the infinite effective
fermion mass $m^*=\infty$ which can be mapped to the strongly coupled
two-dimensions electrons and is quite similar to the three-dimensions
Bose-Einstein Condensation of ideal boson gas. Instead, for $d=1$, the
universal coefficient $\xi$ is negative, implying the non-existence of phase
transition from superfluidity to normal state. The solutions manifest the
quantum Ising universal class characteristic of the strongly coupled unitary
fermions gas.
