Quantum spin liquid with a Majorana Fermi surface on the three-dimensional hyperoctagon lattice

Motivated by the recent synthesis of $\beta$-Li${}_2$IrO${}_3$—a spin-orbit entangled $j=1/2$ Mott insulator with a three-dimensional lattice structure of the Ir${}^{4+}$ ions—we consider generalizations of the Kitaev model believed to capture some of the microscopic interactions between the iridium moments on various trivalent lattice structures in three spatial dimensions. Of particular interest is the so-called hyperoctagon lattice—the premedial lattice of the hyperkagome lattice, for which the ground state is a gapless quantum spin liquid where the gapless Majorana modes form an extended two-dimensional Majorana Fermi surface. We demonstrate that this Majorana Fermi surface is inherently protected by lattice symmetries and discuss possible instabilities. We thus provide the first example of an analytically tractable microscopic model of interacting SU(2) spin-$\frac{1}{2}$ degrees of freedom in three spatial dimensions that harbors a spin liquid with a two-dimensional spinon Fermi surface.

Figure 1

Illustration of the hyperoctagon lattice, a trivalent structure which contains two elementary motifs—a spiraling octagonal helix and a counter-spiraling square helix.

Figure 2

Relationship between various three-dimensional lattices. The hyperoctagon lattice is the premedial lattice of the hyperkagome lattice, which can be obtained from the pyrochlore lattice via depletion of 1/4 of the triangles. The premedial lattice of the pyrochlore is the diamond lattice, which can be depleted by 1/4 of its bonds to obtain the hyperoctagon lattice.

Figure 3

Illustration of (a) the honeycomb and (b) the hyperhoneycomb lattice.

Figure 4

Visualization of the (screw-) rotation symmetries of the hyperoctagon lattice, which belongs to the cubic space group I$4_{1}32$ (no. 214). (a) View along ($0,1,0$) as an example of a 90 degree screw-rotation symmetry. (b) View along ($1,1,-1$) as an example of a 120 degree rotation symmetry. (c) View along ($0,1,1$) as an example of a 180 degree rotation symmetry.

Figure 5

Structure of the edge-sharing O${}_6$ octahedra around the central Ir atoms. (a) View along the ($1,0,0$) direction. (b) View along the ($1,1,1$) direction.

Figure 6

Illustration of the different couplings of the Kitaev model on the hyperoctagon lattice. Green bonds correspond to $xx$ couplings, red bonds to $yy$ couplings, and blue bonds to $zz$ couplings.

Figure 7

The six distinct loops of the hyperoctagon lattice. Each loop contains ten bonds with two coupling types contributing four bonds and one coupling type contributing only two bonds. Note that the six loops realize all possible combinations reflecting the symmetries of the hyperoctagon lattice.

Figure 8

Two examples of elementary loops. The sites are numbered to facilitate the discussion in the main text.

Figure 9

The minimal closed surface of the hyperoctagon lattice is spanned by three neighboring loops (with the encompassed volume vanishing).

Figure 10

Illustration of the Majorana fermion representation of the spin degrees of freedom.

Figure 11

Phase diagram of the Kitaev model on the hyperoctagon lattice. The gapless spin-liquid phase with a Majorana Fermi surface extends in the triangle around the point of isotropic coupling $J_x=J_y=J_z$ (indicated by a point). Around the three corners of the phase diagram where one of the couplings dominates extends a gapped spin-liquid phase, which is separated from the gapless phase via a line of (continuous) phase transitions indicated by the yellow line.

Figure 12

Evolution of the Majorana Fermi surface with varying coupling parameters $J_x$ and $J_y=J_z=(1-J_x)/2$. As the phase transition to the gapped spin liquid is approached with increasing $J_z$ (upper row) the Majorana Fermi surface shrinks to a single point at momenta ${\mathbf{Q}}_1=(\pi ,\pi ,\pi )$ and ${\mathbf{Q}}_2=(\pi ,\pi ,-\pi )$. Approaching the decoupling point for $J_x\to 0$ (lower row) the Majorana Fermi surface flattens to two planes.

Figure 13

Band structure of the four principal bands of the Kitaev model along a path connecting the high-symmetry points $\Gamma =(0,0,0)$, $P=(\pi ,\pi ,\pi )$, $N=(\pi ,\pi ,0)$, and $H=(0,2\pi ,0)$ of the Brillouin zone depicted on the left. The dispersion of the elementary excitations at the Fermi energy is linear for all couplings $0<J_x<1/2$ and corresponding couplings $J_y=J_z=(1-J_x)/2$. At the transition to the gapped spin liquid, $J_x=1/2$, the dispersion becomes quadratic. At the point $J_x=0$ the Hamiltonian reduces to decoupled $J_y$-$J_z$ spirals, resulting in a flat spectrum between the high-symmetry points $P$ and $N$.

Figure 14

Table of the energy relationships that are imposed by various symmetries for the hyperoctagon lattice (left column), the honeycomb lattice (middle column), and the hyperhoneycomb lattice (right column).

Figure 15

Illustration of the different couplings in the Kitaev model, when enlarging the unit cell along the (a) ${\mathbf{a}}_3$ and (b) ${\mathbf{a}}_1$ direction. The arrow marks the broken translation vector. The unit cell of the model in (a) is compatible with a bipartite coloring of the lattice while the one in (b) is not.

Figure 16

Visualization of the surface vs lines of gapless modes. On the line $J_x=J_y$, the model exhibits a Majorana Fermi surface—centered around $(\pi ,\pi ,\pi )$ as shown in panel (a). Away from $J_x=J_y$, the surface reduces to a line, which lies in the $k_z=\pm \pi$ plane. Panels (b) and (c) show the behavior of the gapless line, when (b) increasing $J_x$ and (c) decreasing $J_x$, and setting $J_y=J_z=(1-J_x)/2$.

Figure 17

Sketch of the phase diagram of the model defined in Fig. 15. Along the line $J_x=J_y$, there is a Majorana Fermi surface, which is identical to the original model defined in Sec. 3a. Away from this line, the Majorana surface is partly gapped out and reduced to a closed line of Dirac cones.

Figure 18

Two possibilities of placing atoms on the interstitial sites in the IrO${}_3$ structures, indicated by the gray octahedra. Panel (a) shows the crystal structure by placing atoms of valency $+2$ on the sites of type (i) (see text). In panel (b), atoms with valency $+1$ are placed on sites of type (iii), which are generated by the representative $(0,0,0)$.

Figure 19

The hyperhoneycomb lattice. The four-site unit cell and translation vectors are indicated. Green bonds correspond to $xx$ couplings, red bonds to $yy$ coupling, and blue bonds to $zz$ coupling.

Figure 20

The medial lattice of the hyperhoneycomb lattice, which we dub orthorhombic kagome lattice. The lines of dark-shaded triangles run along the ($1,1,0$) direction, the light-shaded ones along the ($0,1,-1$) direction. The lattice has been slightly deformed for better illustration.

Figure 21

Another set of relationships between various three-dimensional lattices—see also Fig. 2 in comparison. The hyperhoneycomb lattice is the medial lattice of the so-called orthorhombic-kagome lattice depicted in Fig. 20. Like the hyper-kagome lattice the orthorhombic-kagome lattice can be obtained from the pyrochlore lattice via depletion of 1/4 of the triangles; see the inset on left. The premedial lattice of the pyrochlore is the diamond lattice, which can be depleted by 1/4 of its bonds to obtain the hyperhoneycomb lattice.

Figure 22

The minimal closed surface is spanned by four loops in the hyperhoneycomb lattice.

Figure 23

Gapless lines in the Kitaev model on the hyperhoneycomb lattice. Panel (a) shows the position of the gapless line in the Brillouin zone at the isotropic point $J_x=J_y=J_z$. The gapless line is located in the plane $k_x+k_y=0$, which is indicated in gray. The other panels show the behavior of the gapless line when (a) increasing $J_x$ and (b) decreasing $J_x$, while setting $J_y=J_z=(1-J_x)/2$. The extent of the Brillouin zone in the plane $k_x+k_y=0$ is indicated by the hexagon.