Stable magnetic Lorentz-violating vacua in gauge-invariant nonlinear electrodynamics

We investigate gauge-invariant nonlinear electrodynamics in the Plebański first-order Hamiltonian formulation, taking the single-invariant potential V̂(P) V ^ ( P ) as the primary object. Our focus is on the existence of stable Lorentz-violating magnetic vacua. For three explicit two-parameter models – rational asymmetric, logarithmic, and exponential – we determine the regions of parameter space in which nontrivial constant electromagnetic vacua are compatible with an effective Hamiltonian bounded from below and a positive-semidefinite Hessian. In all three cases, physically admissible Lorentz-violating vacua are realized in the magnetic branch. We further discuss the electric branch and several additional one-parameter models, illustrating that Hamiltonian boundedness by itself does not ensure spontaneous Lorentz symmetry breaking. We also comment on how the symmetry-breaking conditions are related to known strong-field causality criteria.