Ordered states of undoped AB bilayer graphene: bias induced cascade of transitions

Abstract

Using mean-field theory, we determine the electronic phase diagram of undoped AB-stacked bilayer graphene in the presence of a transverse electric field. In addition to multiple competing electronic instabilities characterized by excitonic order parameters, our framework incorporates the long-range Coulomb energy associated with interlayer polarization. This long-range interaction plays a crucial role, as it significantly influences both the structure and the relative energies of the competing ordered states. We derive a set of self-consistency equations and solve them both numerically and analytically. Our findings reveal that, as the bias field is varied, the bilayer undergoes a cascade of first-order transitions between several ordered insulating phases for which order-parameter structures are explicitly identified. Some of these phases are characterized by two inequivalent single-particle gaps, whose magnitudes depend on the valley and spin quantum numbers. Field-driven transitions are accompanied by discontinuous and non-monotonic variations of the single-electron gap. We relate our results to Hartree-Fock numerical calculations and to experimental research, including observations of fractional metallic phases that emerge upon doping the bilayer system.

Figures (5)

• Figure 1: Function $y = x \ln (1/|x|)$. Formally, the derivative of the function diverges at $x = 0$ as $\sim \ln (1/|x|)$. Yet, due to weakness of this divergence, vertical tangent is essentially unobservable on the plot.
• Figure 2: The phase diagram of the system at $V>0$ and the dependence of the order parameters ${\sl x}_m=D_m/\Delta_0$ on the applied voltage (red lines are the solutions ${\sl x}_+$ and blue ${\sl x}_-$). The voltage $V^*\approx 2.3$ corresponds to the transition between the "antiferromagnetic states" with $(n_+=2,n_-=2)$ and $(n_+=3,n_-=1)$. When $V>V_C\approx 8.8$ the "antiferromagnetic orders" disappear and the gap in the spectrum is due to the layers polarization only.
• Figure 3: The system energies as functions of the applied voltage. For $V \geqslant 0$, only three ground states are possible, see the legend. The plotted energy is the sum of the first and the second terms in definition (\ref{['bar_E']}). The third term in Eq. (\ref{['bar_E']}), while contributing a large offset to the total energy, does not influence relative stability of phases. It is therefore omitted for clarity. For $V<V^*$, the solid (red) curve is the lowest. Thus, the solution $n_\pm = 2$ is the ground state. As $V$ increases, a transition into $n_+ = 3$, $n_- = 1$ type, followed by the second transition into the FE state $n_+ = 4$, $n_- = 0$, occurs. Note that in the interval $V^* < V < V_{\rm C}$ the solution $n_\pm = 2$ disappears, while the FE solution emerges. When $V$ exceeds $V_{\rm C}$ sufficiently, the solution $n_+ = 3$, $n_- = 1$ disappears, leaving the FE solution the only one at elevated $V$.
• Figure 4: Dependence of dimensionless single-electron gap $\min_m |{\sl x}_m|$ (red solid line) versus dimensionless bias $V$. In the interval $0 < V < V_{\rm C}$, both $n_\pm > 0$, thus, positive and negative ${\sl x}_m$'s may have unequal $|{\sl x}_m|$, see Fig. \ref{['Figure_OP']}. For such values of $V$, dashed (blue) line shows $\max_m |{\sl x}_m|$. The gap is a non-monotonic function of $V$, with discontinuities at $V = V^*, V_{\rm C}$. At $V = 2 \Lambda$, the gap has local maximum ${\sl x} = 1$, see Sec. \ref{['subsection_Vstar']}. Dash-dotted (green) line is $\Delta = e\Phi/2$, or, in dimensionless units, ${\sl x}=\lambda_{\textrm{FE}}V$. It represents bias-induced gap in the model with no interaction. We see that interaction effects screen the bare field significantly.
• Figure 5: Numerically estimated dimensionless polarization $\bar{P}$ versus the dimensionless bias voltage $V$. The plot for $\bar{P}$ demonstrates discontinuities at the transition points $V=V^*$ and $V=V_{\rm C}$. Between the transitions, the graph is virtually indistinguishable from linear function. This behavior is in agreement with Eq. (\ref{['polarization_linear']}), attesting the quality of approximate solution (\ref{['self_cons_linear_solution']}).