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Broken-symmetry $ν=0$ quantum Hall states in bilayer graphene: Landau level mixing and dynamical screening

For bilayer graphene in a magnetic field at the neutral point, we derive and solve a full set of gap equations including all Landau levels and taking into account the dynamically screened Coulomb interaction. There are two types of the solutions for the filling factor $ν=0$: (i) a spin-polarized type solution, which is the ground state at small values of perpendicular electric field $E_{\perp}$, and (ii) a layer-polarized solution, which is the ground state at large values of $E_{\perp}$. The critical value of $E_{\perp}$ that determines the transition point is a linear function of the magnetic field, i.e., $E_{\perp,{\rm cr}}=E_{\perp}^{\rm off}+a B$, where $E_{\perp}^{\rm off}$ is the offset electric field and $a$ is the slope. The offset electric field and energy gaps substantially increase with the inclusion of dynamical screening compared to the case of static screening. The obtained values for the offset and the energy gaps are comparable with experimental ones. The interaction with dynamical screening can be strong enough for reordering the levels in the quasiparticle spectrum (the $n=2$ Landau level sinks below the $n=0$ and $n=1$ ones).