Lattice-layer entanglement in Bernal-stacked bilayer graphene

The complete lattice-layer entanglement structure of Bernal-stacked bilayer graphene is obtained for the quantum system described by a tight-binding Hamiltonian which includes mass and bias voltage terms. Through a suitable correspondence with the parity-spin $SU\left(2\right)\otimes SU\left(2\right)$ structure of a Dirac Hamiltonian, when it brings up tensor and pseudovector external field interactions, the lattice-layer degrees of freedom can be mapped into such a parity-spin two-qubit basis which supports the interpretation of the bilayer graphene eigenstates as entangled ones in a lattice-layer basis. The Dirac Hamiltonian mapping structure simply provides the tools for the manipulation of the corresponding eigenstates and eigenenergies of the Bernal-stacked graphene quantum system. The quantum correlational content is then quantified by means of quantum concurrence, in order to have the influence of mass and bias voltage terms quantified, and in order to identify the role of the trigonal warping of energy in the intrinsic entanglement. Our results show that while the mass term actively suppresses the intrinsic quantum entanglement of bilayer eigenstates, the bias voltage term spreads the entanglement in the Brillouin zone around the Dirac points. In addition, the interlayer coupling modifies the symmetry of the lattice-layer quantum concurrence around a given Dirac point. It produces some distortion on the quantum entanglement profile which follows the same pattern of the isoenergy line distortion in the Bernal-stacked bilayer graphene.

Figure 1

(Left) Top view of the geometrical configuration of the AB (Bernal) stacking. Half of the atoms of the upper layer (joined by dotted lines) are exactly above half of the atoms of the lower layer (joined by dashed lines). Sites that are placed exactly above a site of the lower layer are called dimer sites ($A1$ and $B2$), while sites that are localized above the center of the other honeycomb are called nondimer sites ($B1$ and $A2$). (Right) Schematic representation of the hopping amplitudes of the tight binding model for the bilayer graphene. The parameter $t$ describes the hopping between next neighbors in the same layer; $t_{\perp }$ is the hopping from a nondimer site to its nearest nondimer site; $t_3$ is the hopping from a dimer site to its nearest dimer site, and $t_4$ is the hopping from a dimer to the nearest nondimer site.

Figure 2

(Left) Energy bands for the branch $s=1,t/t_{\perp }=8.29,t_3/t_{\perp }=0.99$, and for $m/t_{\perp }=\mathrm{\Lambda }/t_{\perp }=0$. The energy bands touch the corners of the first Brillouin zone (Dirac points). Near such points the energy profile approaches a parabola in $k$ and the dispersion relation reproduces a massive Dirac equation. (Right) Zoom around the Dirac point. The isoenergy lines are distorted due to the $t_3$ hopping—the trigonal warping effect. The constant energy lines have a $\frac{2\pi }{3}$ symmetry around these points due to the $cos(3\varphi \left(\mathit{k}\right))$ term in (20). In the absence of the $t_3$ hopping, the isoenergy lines around a Dirac point are perfectly symmetric.

Figure 3

Auxiliary plot for the energy bands for the branch $s=1$ as function of $k_x$ for $k_y=\frac{2\pi }{3\sqrt{3}a},t/t_{\perp }=8.29,t_3/t_{\perp }=0.99$, and for $m/t_{\perp }=\mathrm{\Lambda }/t_{\perp }=0$ (solid lines) and $m/t_{\perp }=\mathrm{\Lambda }/t_{\perp }=1$ (dashed lines). For $m=\mathrm{\Lambda }=0$ the bands $n=0$ and $n=1$ touch the Dirac point. Nonzero values of $m$ and $\mathrm{\Lambda }$ open an energy gap between the valence and conduction band, deforming the linear dispersion (for $m=\mathrm{\Lambda }=0$) around the Dirac point into a hyperbolic one.

Figure 4

(Left) Concurrence in $\mathit{k}$ space for $s=1,m=0,\mathrm{\Lambda }=0$ and the same set of parameters adopted in Fig. 2. (Right) When both bias voltage and mass parameters are missed, i.e., for the AB Hamiltonian Eq. (4), the lattice-layer quantum correlations are maximized for states near the Dirac points, which depicts concurrence around ${\mathit{K}}_{+}$.

Figure 5

Concurrence in the $\mathit{k}$ space for $s=1$ around ${\mathit{K}}_{+}$, for the same set of parameters adopted in Fig. 4, with $m/t_{\perp }=0$ and $\mathrm{\Lambda }/t_{\perp }=1$ (first plot), $\mathrm{\Lambda }/t_{\perp }=5$ (second plot), and $\mathrm{\Lambda }/t_{\perp }=10$ (third plot). The inclusion of the bias voltage Hamiltonian (6) spreads lattice-layer entanglement around the $k$ space. Considering the angular coordinate on the $k_x-k_y$ plane, maximally entangled states correspond to the red region around the Dirac point.

Figure 6

Concurrence in the $\mathit{k}$ space for $s=1$ around ${\mathit{K}}_{+}$, for the same set of parameters adopted in Fig. 4, for $\mathrm{\Lambda }/t_{\perp }=0$, with $m/t_{\perp }=0.1$ (first plot), $m/t_{\perp }=0.5$ (second plot), and $m/t_{\perp }=1$ (third plot). The corresponding mass term (5) destroys the quantum entanglement. For $m/t_{\perp }\to \infty$, the entanglement vanishes in the whole first Brillouin zone.

Figure 7

Entanglement profile comparison for $\mathit{k}$ in the first Brillouin zone. The plots are for $(\mathrm{\Lambda }/t_{\perp },m/t_{\perp })=(0,0)$ (first plot), $(\mathrm{\Lambda }/t_{\perp },m/t_{\perp })=(10,0)$ (second plot), and $(\mathrm{\Lambda }/t_{\perp },m/t_{\perp })=(0,1)$ (third plot); other parameters are fixed with the same values adopted in Fig. 4.

Figure 8

Contour plot of the concurrence near a Dirac point for $t_3/t_{\perp }=0.997$ (first row), $t_3/t_{\perp }=5$ (second row) and $t_3/t_{\perp }=10$ (third row), for $m/t_{\perp }=\mathrm{\Lambda }/t_{\perp }=0$ (first column), $m/t_{\perp }=0$ and $\mathrm{\Lambda }/t_{\perp }=1$ (second column), and $m/t_{\perp }=1$ and $\mathrm{\Lambda }/t_{\perp }=0$ (third column). Additional parameters are in correspondence with the previous plots. Black lines correspond to isoenergy lines while green lines correspond to isoconcurrence lines. Concurrence also exhibits a $\frac{2\pi }{3}$ polar symmetry such that the distortion of the entanglement has a behavior similar to those exhibited by the eigenenergies due to the $t_3$ hopping parameter.