Topological Spinon Semimetals and Gapless Boundary States in Three Dimensions

Recently, there has been much effort in understanding topological phases of matter with gapless bulk excitations, which are characterized by topological invariants and protected intrinsic boundary states. Here we show that topological semimetals of Majorana fermions arise in exactly solvable Kitaev spin models on a series of three-dimensional lattices. The ground states of these models are quantum spin liquids with gapless nodal spectra of bulk Majorana fermion excitations. It is shown that these phases are topologically stable as long as certain discrete symmetries are protected. The corresponding topological indices and the gapless boundary states are explicitly computed to support these results. In contrast to previous studies of noninteracting systems, the phases discussed in this work are novel examples of gapless topological phases in interacting spin systems.

Figure 1

Unit cell, lattice vectors, sublattices, and coordinate systems for the $\mathcal{H}-0$ and $\mathcal{H}-1$ lattices. The conventional (orthorhombic) unit cells are drawn, while the sublattices in the primitive unit cells are labeled from 1 to $4n+4$. The $x$, $y$, and $z$ bonds within the conventional unit cell as defined in the Kitaev model are colored in orange, blue, and turquoise, respectively.

Figure 2

Position of nodal rings, surface spectra, and winding numbers. In (a)–(c), the red lines show the location of the nodal rings of the indicated lattice, flux sector, and $\delta =J_z/J_x$. The red lines are the intersections of the yellow and turquoise surfaces, which are the $\mathrm{LHS}=1$ and $\mathrm{RHS}=1$ of Eqs. (5), (6), and (11). In (c), the first Brillouin zone spans the region $-\pi /2<k_1\le \pi /2$ due to doubling of the unit cell in the $\pi$-flux sector. (d)–(h) show the surface spectra along one-dimensional momentum cuts on the various lattices and flux sectors, while the insets within indicate the location of the momentum cuts and projection of the nodal ring on the surfaces indicated. The colors in the insets correspond to the winding numbers, where yellow, turquoise, and red are $\nu =0$, $\pm 1$, and $\pm 2$, respectively. When $\nu \ne 0$, as shown in (d)–(h), we find the presence of zero-energy surface flat bands with $|\nu |$-fold degeneracy, due to the bulk-boundary correspondence.