Symmetry Breaking on the Three-Dimensional Hyperkagome Lattice of ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$

We study the antiferromagnetic spin-$1/2$ Heisenberg model on the highly frustrated, three-dimensional, hyperkagome lattice of ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$ using a series expansion method. We propose a valence bond crystal with a 72 site unit cell as a ground state that supports many, very low lying, singlet excitations. Low energy spinons and triplons are confined to emergent lower-dimensional motifs. Here, and for analogous kagome and pyrochlore states, we suggest finite temperature signatures, including an Ising transition, in the magnetic specific heat due to a multistep breaking of discrete symmetries.

Figure 1

Lattice structure: (a) cubic unit cell of the hyperkagome lattice (thin bonds and small sites show the extra bonds and sites of the pyrochlore lattice). (b) The 2D kagome lattice. (c) A “decagon,” the smallest loop of the hyperkagome lattice, highlighting the structure of two intersecting planes by different coloring.

Figure 2

VBC structure: (a) details of the VBC ground state on a slab of the hyperkagome viewed along a [110] direction: dimers are red [medium-dark gray] and weak bonds blue [dark gray] except on the decagon-rods, which are entirely yellow [light gray]. The empty triangles are indicated by green [medium gray] spheres. The two-dimensional, kagomelike, projection of the state is shown in (b), highlighting the positions and orientations of the decagon rods (yellow [light gray]) and the $\Delta$ chains (red [medium-dark gray] vs green [medium gray]). The shaded region corresponds to one choice of a VBC unit cell. The positions of the backbones of the decagon-rods are marked by black and their bridging dimers are indicated in brown. (c) A decagon rod—the main building block of the VBC ground state. The decagon-rods are directed along a backbone (blue [dark gray]). The two degenerate dimer coverings of a $\Delta$ chain (red [medium-dark gray] vs green [medium gray]) are shown in (d).

Figure 3

Dispersion of the triplons to first order in $\lambda$, plotted as function of the momenta $k_a$ and $k_b$ in the decagon-rod plane spanned by the vectors $\mathbf{a}$ and $\mathbf{b}$ defined in Fig. 2. The dispersion is flat in the perpendicular $k_c$ direction for all the modes. The inert triplets at constant energy $J$ are not shown.