Electrically controllable magnetic order in the bilayer Hubbard model on honeycomb lattice: A determinant quantum Monte Carlo study

In this paper, we use the determinant quantum Monte Carlo method to study the effect of the electric field on the magnetic order in a bilayer Hubbard model on a honeycomb lattice, in which only the direct interlayer hopping energy is included. Our results qualitatively support the layered antiferromagnetic, spin-density wave ground state found in the mean-field theory at the charge neutrality point. The obtained magnetic moments, however, are much smaller than what are estimated in the mean-field theory. As the electric field increases, the magnetic order parameter rapidly decreases.

Figure 1

Bernal stacked bilayer graphene lattice and restricted antiferromagnetic (AFM) orders. A and B denote two sublattices. The subscripts 1 and 2 are layer indices. Intralayer hopping $t$ links two nearest-neighbor sites within the same layer, while interlayer hopping $t_{\perp }$ links two nearest-neighbor sites in opposite layers. The coordination number $z=3$ for the $A_1$ and $B_2$ sites, and $z=4$ for the $A_2$ and $B_1$ sites. (a) Restricted AFM order within $z=3$ sites. (b) Restricted AFM order within $z=4$ sites.

Figure 2

The dominant magnetization square $m^2$ is plotted as a function of $1/L^2$ for different values of $U$. Here we set $1/T=1.2L$ and $V=0$. (a) $t_{\perp }=1.0t$, (b) $t_{\perp }=0.2t$. The symbols are numerical results of DQMC simulations, and the lines are linear fitting. Insets: $S(q=0)=4m^2{L}^2$ is plotted as a function of $1/L^2$, from which it is clearly seen that $m(L\to \infty )=0$ at $U/t=2.0$.

Figure 3

The restricted magnetization square $m_z^2$ in the absence of an external electric field is plotted as a function of $1/L^2$. Here we set $U=4.0t$.

Figure 4

The dominant magnetization square $m^2$ is plotted as a function of $1/L^2$ in the presence of an external perpendicular electric field. Here we set $U=4.0t$. (a) $t_{\perp }=1.0t$, (b) $t_{\perp }=0.2t$. Insets: The electron occupation numbers in four sublattices.

Figure 5

The spin susceptibility $\chi$ as a function of $T$ in the presence of an external perpendicular electric field. Here we set $U=4.0t$. (a) $t_{\perp }=1.0$, (b) $t_{\perp }=0.2$.

Figure 6

(a) The mean-field results of magnetization $m$ and (b) charge number for $U=4.0t$ and $t_{\perp }=1.0t$ as a function of potential bias $V$. Here we use $U=4.0$. Inset: A zoom for the charge number at the range of $0<V<1.0t$. (c) The critical potential bias $V_c$ for various values of $U$ for $t_{\perp }=1.0t$.