Bilayer graphene with parallel magnetic field and twisting: Phases and phase transitions in a highly tunable Dirac system

The effective theory for bilayer graphene (BLG), subject to parallel/in-plane magnetic fields, is derived. With a sizable magnetic field the trigonal warping becomes irrelevant, and one ends up with two Dirac points in the vicinity of each valley in the low-energy limit, similar to the twisted BLG. Combining twisting and parallel field thus gives rise to a Dirac system with tunable Fermi velocity and cutoff. If the interactions are sufficiently strong, several fully gapped states can be realized in these systems, in addition to the ones in a pristine setup. Transformations of the order parameters under various symmetry operations are analyzed. The quantum critical behavior of various phase transitions driven by the twisting and the magnetic field is reported. The effects of an additional perpendicular field and possible ways to realize the new massive phases are highlighted.

Figure 1

Left: Splitting of the Dirac points (red and blue arrows) due to in-plane field (B) (here $K^{\prime }=-K$). $k_x,k_y$ are measured in unit $\hslash =a=1$. Right: A schematic variation of $B_C$ with $\theta$ (see left), with $v_F={10}^6\mathrm{m}/\mathrm{s}$, $t_{\perp }/t=10$, $v_F/v_2=30$, $d=3.5$ Å (Refs. [34, 37]).