Weyl Points in Three-Dimensional Optical Lattices: Synthetic Magnetic Monopoles in Momentum Space

We show that a Hamiltonian with Weyl points can be realized for ultracold atoms using laser-assisted tunneling in three-dimensional optical lattices. Weyl points are synthetic magnetic monopoles that exhibit a robust, three-dimensional linear dispersion, identical to the energy-momentum relation for relativistic Weyl fermions, which are not yet discovered in particle physics. Weyl semimetals are a promising new avenue in condensed matter physics due to their unusual properties such as the topologically protected “Fermi arc” surface states. However, experiments on Weyl points are highly elusive. We show that this elusive goal is well within experimental reach with an extension of techniques recently used in ultracold gases.

Figure 1

Sketch of the 3D cubic lattice with phase engineered hopping along $x$ and $z$ directions, which possesses Weyl points in momentum space. Dashed (solid) lines depict hopping with acquired phase $\pi$ (0, respectively). (a) The $xy$ planes of the lattice are equivalent to the lattice of the Harper Hamiltonian for $\alpha =1/2$. Centers of inversion symmetry for this 2D lattice are denoted by orange crosses. Green triangles along the axes denote the tilted directions. (b) A pair of Raman lasers enabling laser-assisted tunneling is sketched with arrows. The 3D lattice can be visualized as alternating stacks of 2D lattices parallel to the $xz$ plane, which are shown in (c) and (d); the hopping between these planes (along $y$) is regular. The hopping along $z$ is alternating with phases 0 or $\pi$, depending on the position in the $xy$ plane [see (b), (c), and (d)], which breaks the inversion symmetry.

Figure 2

Sketch of the first Brillouin zone of the lattice depicted in Fig. 1, energy spectrum and Weyl points. (a) The positions of the Weyl points in the Brillouin zone and their chiralities are indicated with $+$ and $-$ signs. If a tunable $A\text{−}B$ sublattice energy offset is introduced, Weyl points move along dotted lines, and can annihilate at points denoted with stars (see text). (b) Energy spectrum in the $k_x=0$ plane [shaded plane in (a)] shows linear dispersion in the proximity of the Weyl points. The insets show the Berry curvature of two Weyl points, demonstrating that they are synthetic magnetic monopoles in momentum space.

Figure 3

Surface states and Weyl points. (a) A slab of finite width is cut from the 3D lattice along planes orthogonal to the $\widehat{x}-\widehat{y}$ direction; cross section in the $xy$ plane is sketched. The two sides of the slab are indicated with letters $L$ and $R$. (b) Energy spectrum of the slab. The two dispersion sheets of surface states (corresponding to the two surfaces of the slab) are denoted with $R$ (blue) and $L$ (orange). The intersections of the two sheets are Fermi arcs (denoted with dashed lines). The arcs connect Weyl points of opposite chiralities. The insets show examples of the profile of the Fermi arc surface states across the slab, as indicated by the green dashed box in (a).