Inducing topological order in a honeycomb lattice

We explore the possibility of inducing a topological insulator phase in a honeycomb lattice lacking spin-orbit interaction using a metallic (or Fermi gas) environment. The lattice and the metallic environment interact through a density-density interaction without particle tunneling, and integrating out the metallic environment produces a honeycomb sheet with in-plane oscillating long-ranged interactions. We find the ground state of the interacting system in a variational mean-field method and show that the Fermi wave vector $k_F$ of the metal determines which phase occurs in the honeycomb lattice sheet. This is analogous to the Ruderman-Kittel-Kasuya-Yosida (RKKY) mechanism in which the metal's $k_F$ determines the interaction profile as a function of the distance. Tuning $k_F$ and the interaction strength may lead to a variety of ordered phases, including a topological insulator and anomalous quantum-Hall states with complex next-nearest-neighbor hopping, as in the Haldane and the Kane-Mele model. We estimate the required range of parameters needed for the topological state and find that the Fermi vector of the metallic gate should be of the order of $3\pi /8a$ (with $a$ being the graphene lattice constant). The net coupling between the layers, which includes screening in the metal, should be of the order of the honeycomb lattice bandwidth. This configuration should be most easily realized in a cold-atoms setting with two interacting Fermionic species.

Figure 1

Phase diagram for graphene with induced long-ranged interaction of the form of Eq. (4). The $x$ axis is $2k_F$ of the metallic layer (measured in units of $1/a$ where $a=2.46\text{Å}$ is the graphene lattice constant), which governs the relative strength of interaction in different distances and the $y$ axis is the overall interaction strength, which is determined by the coupling $\alpha$. The on-site interaction parameter $U$ was scanned. Here we show (a) $U=0$, (b) $U=t$, and (c) $U=2t$ where $t$ is the hopping amplitude. The different symbols represent different phases with circle (black) semimetal, square (blue) charge-density wave, diamond (gray) spin-density wave, up triangle (purple) superconductor, down triangle (orange) Kekulé, open circle (red) anomalous Hall, open square (green) anomalous spin Hall.

Figure 2

An example of order-parameter magnitudes as a function of the metal's Fermi wavelength (measured in units of $1/a$ where $a=2.4\text{Å}$ is the graphene lattice constant). The data are given for a constant $U=0$ and $V_0=4t$. The curves represent the order-parameter magnitude in units of the bare hopping $t$ with square (blue) charge-density wave, circle (red) anomalous Hall, and triangle (purple) superconductivity.