Broken symmetry states in bilayer graphene in electric and in-plane magnetic fields

Broken symmetry states in bilayer graphene in perpendicular electric $E_{\perp }$ and in-plane magnetic $B_{\parallel }$ fields are studied in the presence of the dynamically screened long-range Coulomb interaction and the symmetry-breaking contact four-fermion interactions. The integral gap equations are solved numerically, and it is shown that the momentum dependence of gaps is essential: It diminishes by an order of magnitude the gaps compared to the case of momentum-independent approximation, and the obtained gap magnitudes are found to agree well with existing experimental values. We derived a phase diagram of bilayer graphene at the neutrality point in the plane $(B_{\parallel },E_{\perp })$ showing that the (canted) layer antiferromagnetic (LAF) state remains a stable ground state of the system at large $B_{\parallel }$. On the other hand, while the LAF phase is realized at small values of $E_{\perp }$, the quantum valley Hall (QVH) phase is the ground state of the system at values $E_{\perp }>E_{cr}\left(B_{\parallel }\right)$, where a critical value $E_{cr}\left(B_{\parallel }\right)$ increases with in-plane magnetic field $B_{\left|\right|}$.

Figure 1

(a) Solutions of gap equation (21) for ${\delta }_0=0$ and different values of $g_{\delta }$. (b) The dependence of the band gap (solid line) and $2\delta \left(0\right)$ (dotted line) on the coupling constant $g_{\delta }$ shown on a logarithmic scale; the dashed line shows the band gap $2\delta$ obtained within the approximation $\delta =\text{const}$. (c) The energy spectrum of the LAF state with $g_{m_z}=0$. The value $\kappa =1$ is used.

Figure 2

The energy densities (a) and the gap parameters at $k=0$ (b) of the LAF and QVH states in external electric field for $g_{m_z}=0,g_m=\tilde{v}$. Panel (c) shows the dependence of the critical field $E_{\perp }^{\text{cr}}$ for the LAF-QVH phase transition [vertical dotted line in panels (a) and (b)] on the local interaction constants $g_{m_z}$ and $g_m$.

Figure 3

The energy densities (a) and gaps at $k=0$ (b) of the LAF and QSH states in external electric field for $g_{m_z}=0,g_{{\mathrm{\Delta }}_z}=-0.02$.

Figure 4

The gaps at $k=0$ of the LAF and QVH states as functions of external electric field for $B_{\parallel }=1$ T (a) and $B_{\parallel }=9$ T (b), of parallel magnetic field for $E_{\perp }=6$ mV/nm (c) and the phase diagram in electric and in-plane magnetic fields (d). Parameters used: $g_{m_z}=g_{\mu }=0,g_m=\tilde{v}$, and $\kappa =1$. The line types used in panels (a)–(c) are described in the inset of panel (a). In panel (d), the solid and dashed lines describe the first and second order phase transitions between the LAF and QVH states. The black point marks the critical point where the line of the first order phase transition terminates. The dotted line shows the boundary of the region of existence of the gapped QVH state and the dot-dashed line splits the LAF into regions where $m_z\left(k\right)$ or ${\mu }_x\left(k\right)$ dominates.

Figure 5

The energy densities of the LAF and QVH states at $\mathbf{B}=0$ (a), their corrections due to finite $B_{\perp }$ in the vicinity of the phase transition at $E_{\perp }=E_{\perp }^{\text{cr}}$ (marked by the vertical dotted line) (b), and the phase diagram in the $(B_{\perp },E_{\perp })$ plane (c) for $g_{\alpha \beta }=0$ and $\kappa =3$.