Weyl Spin Liquids

The fractionalization of quantum numbers in interacting quantum many-body systems is a central motif in condensed-matter physics with prominent examples including the fractionalization of the electron in quantum Hall liquids or the emergence of magnetic monopoles in spin-ice materials. Here, we discuss the fractionalization of magnetic moments in three-dimensional Kitaev models into Majorana fermions (and a ${\mathbb{Z}}_2$ gauge field) and their emergent collective behavior. We analytically demonstrate that the Majorana fermions form a Weyl superconductor for the Kitaev model on the recently synthesized hyperhoneycomb structure of $\beta \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ when applying a magnetic field. We characterize the topologically protected bulk and surface features of this state, which we dub a Weyl spin liquid, including thermodynamic and transport signatures.

Figure 1

The tricoordinated hyperhoneycomb lattice synthesized for $\beta \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$. ${\mathrm{IrO}}_6$ octahedra are indicated on the left. Green, red, and blue bonds correspond to an Ising-type interaction of ${\sigma }^x{\sigma }^x$, ${\sigma }^y{\sigma }^y$, and ${\sigma }^z{\sigma }^z$ type, respectively. The three-spin interaction ${\sigma }_j^x{\sigma }_k^y{\sigma }_l^z$ induces a next-nearest-neighbor hopping term between sites $j$ and $k$ in the effective Majorana model (3).

Figure 2

The energy dispersion of the hyperhoneycomb Kitaev model for various values of $\kappa$ (parametrizing the effective magnetic field) along certain high-symmetry lines indicated in the Brillouin zone on the right-hand side. The gray hexagon indicates the plane $k_x=-k_y$ on which the line of gapless mode (black line) is located.

Figure 3

In the presence of a finite magnetic field along the 111 direction the Fermi line of the hyperhoneycomb Kitaev model is gapped out except for two distinct Weyl points.

Figure 4

Plot of the Chern number of the effective Hamiltonian (3) for (a) $\kappa =0.05$ and (b) $\kappa =0.55$ restricted to the two-dimensional plane in reciprocal space defined by the three points $\mathbf{k}=(0,0,k_z)$, $\mathbf{k}+{\mathbf{q}}_2/2$, and $\mathbf{k}+{\mathbf{q}}_3/2$, where ${\mathbf{q}}_2$ and ${\mathbf{q}}_3$ are reciprocal lattice vectors (see the Supplemental Material [24]). The colored lines indicate the position of the respective (hexagonal) planes shown on the left.

Figure 5

Evolution of the (a) Weyl points and (b) corresponding Fermi arcs for increasing $\kappa$. The shading yellow-to-green indicates the evolution of negative chirality Weyl points for $\kappa =0\to \infty$, shading red-to-green the evolution of their particle-hole partners with positive chirality (see main text). The surface Brillouin zone for breaking translation invariance in the $a_1$ direction is indicated in (c).