Yong-Baek Kim: "Heisenberg Antiferromagnet on the Hyper-Kagome Lattice: Application to Na4Ir3O8"

Yong-Baek Kim discussed very interesting recent experiments on a new three dimensional antiferromagnetic compound [Okamoto et al., "Spin liquid state in S=1/2 hyper-kagome antiferromagnet Na4Ir3O8"] and recent theoretical work by his group directed at understanding the magnetic behavior of the material [Hopkinson, Isakov, Kee, and Kim, "Classical antiferromagnet on a hyperkagome lattice," PRL 99, 037201 (2007) and "Topological spin liquid on the hyper-kagome lattice Na4Ir3O8"].

Stoichiometry shows that the Ir ions are in a 4+ valence state and the five 5d electrons form a spin-1/2 state in the t2g level. The 3D lattice is like the better known pyrochlore lattice except for the fact that the Ir ions occupy only 3 of the 4 sites of each tetrahedron; the (spinless) Na ions occupy the 4th site. The Ir ions on the decimated pyrochlore lattice form a network of corner-sharing triangles that has been dubbed a "hyperkagome" lattice. As the unit cell contains 12 spins, the material doesn't have an odd number of spin-1/2's per unit cell (in contrast to several 2D candidate spin liquids that may support gapless spin excitations). Nevertheless, magnetic susceptibility measurements find a large Curie-Weiss temperature of -650K and no sign of ordering down to 2K. Due to the large nuclear charge of Iridium (Z=77) there could be a sizable spin-orbit interaction, but as the lattice has inversion symmetry, Dzyaloshinsky-Moriya interactions are apparently forbidden.

Kim's group first investigated the behavior of a classical Heisenberg antiferromagnet on the hyperkagome lattice. Assuming nearest-neighbor exchange, the Hamiltonian can be rewritten as a sum over all the triangles of:


(J/2) * (S_triangle)^2


where S_triangle is the sum of the three spins on each triangle. At zero temperature the ground state is specified simply by setting S_triangle = 0 on each triangle; as there are many choices of the spins that satisfy the constraint, there is a macroscopic degeneracy. Here Andrey Chubukov asked if the constraint could be implemented independently on each triangle, and Yong-Baek clarified that the triangles are not independent, but nevertheless the ground state does have macroscopic degeneracy.

The question then arises as to whether or not there is an order--by-disorder transition induced by classical thermal fluctuations at non-zero temperatures; Hopkinson et al. addressed this by classical Monte Carlo calculations and found a transition from a "cooperative paramagnet" at high temperatures to a spin-nematic phase at low temperatures (below 0.3 to 1.5 K for an exchange constant of about 300K). The rather unusual spin-nematic order parameter is given in terms of cross-products of pairs of spin operators on the triangles. Correlations are found numerically to become long-ranged in the ordered phase, and the change in the entropy is consistent with the formation of spin-nematic order.

Analysis of the corresponding quantum Heisenberg antiferromagnet by Read and Sachdev's bosonic Sp(N) method finds two other phases: An ordered co-planar state and a Z2 spin liquid with massive deconfined spinon excitations. Thus quantum fluctuations have a markedly different effect than classical thermal fluctuations. I noted that spin-nematic order would be difficult to find within the Sp(N) method, as the order parameter doesn't generalize to Sp(N) in a natural way.

John Mydosh suggested experimental investigation by neutron scattering would be interesting as it could detect a spin-ordered phase, and Yong-Baek agreed but pointed out that the common isotope of Iridium strongly absorbs neutrons. Amit Keran suggested that NMR measurements on the sodium atoms would be the next logical step, especially as lower temperatures can be reached.