Massless Dirac-Weyl fermions in a ${\mathcal{T}}_3$ optical lattice

We propose an experimental setup for the observation of quasirelativistic massless fermions. It is based on a ${\mathcal{T}}_3$ optical lattice, realized by three pairs of counterpropagating lasers, filled with fermionic cold atoms. We show that in the long wavelength approximation the ${\mathcal{T}}_3$ Hamiltonian generalizes the Dirac-Weyl Hamiltonian for the honeycomb lattice, however, with a larger value of the pseudospin $S=1$. In addition to the Dirac cones, the spectrum includes a dispersionless branch of localized states producing a finite jump in the atomic density. Furthermore, implications for the Landau levels are discussed.

Figure 1

(Color online) (a) The ${\mathcal{T}}_3$ lattice. It is characterized by translation vectors ${\mathbf{v}}_1=\left(3/2;-\sqrt{3}/2\right){\ell }_0$ and ${\mathbf{v}}_2=\left(3/2;\sqrt{3}/2\right){\ell }_0$, with lattice constant ${\ell }_0$. Open circles mark the two sublattices A and B, forming a HCL. Solid circles mark the hub sites H forming a (larger) triangular lattice. (b) Contour plot of $E_{+}\left(\mathbf{k}\right)$ in units of the hopping energy $t$, cf. Eq. (3b). The dashed hexagon defines the first Brillouin zone, $\mathbf{K}=2\pi {\ell }_0^{-1}\left(1/3;-\sqrt{3}/9\right)$ and ${\mathbf{K}}^{\prime }=2\pi {\ell }_0^{-1}\left(1/3;\sqrt{3}/9\right)$ are two nonequivalent Dirac points.

Figure 2

(Color online) (a) Distribution of the laser field intensity for generating a ${\mathcal{T}}_3$ lattice. (b) Cut of the field intensity along the $y$ axis.

Figure 3

(Color online) The number of atoms $n$ per unit cell at zero temperature as a function of $\mu /t$, ignoring physical spin and valley degeneracies, where $t$ is the nearest-neighbor hopping amplitude, cf. Eq. (2). Inset: density of states, $dn/d\mu$, versus $\mu /t$.