Interplay of non-symmorphic symmetry and spin-orbit coupling in hyperkagome spin liquids: Applications to Na${}_4$Ir${}_3$O${}_8$

Abstract

Na${}_4$Ir${}_3$O${}_8$ provides a material platform to study three-dimensional quantum spin liquids in the geometrically frustrated hyperkagome lattice of Ir${}^{4+}$ ions. In this work, we consider quantum spin liquids on hyperkagome lattice for generic spin models, focusing on the effects of anisotropic spin interactions. In particular, we classify possible ${\mathbb{Z}}_2$ and $U(1)$ spin liquid states, following the projective symmetry group analysis in the slave-fermion representation. There are only three distinct ${\mathbb{Z}}_2$ spin liquids, together with 2 different $U(1)$ spin liquids. The non-symmorphic space group symmetry of hyperkagome lattice plays a vital role in simplifying the classification, forbidding "$\pi$-flux" or "staggered-flux" phases in contrast to symmorphic space groups. We further prove that both $U(1)$ states and one $Z_2$ state among all 3 are symmetry-protected gapless spin liquids, robust against any symmetry-preserving perturbations. Motivated by the "spin-freezing" behavior recently observed in Na${}_4$Ir${}_3$O${}_8$ at low temperatures, we further investigate the nearest-neighbor spin model with dominant Heisenberg interaction subject to all possible anisotropic perturbations from spin-orbit couplings. We found a $U(1)$ spin liquid ground state with spinon fermi surfaces is energetically favored over $Z_2$ states. Among all spin-orbit coupling terms, we show that only Dzyaloshinskii-Moriya (DM) interaction can induce spin anisotropy in the ground state when perturbing from the isotropic Heisenberg limit. Our work paves the way for a systematic study of quantum spin liquids in various materials with a hyperkagome crystal structure.

27 pages, 9 figures, reference updated

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• Projective symmetry group classifications of quantum spin liquids on the simple cubic, body centered cubic, and face centered cubic lattices
• Topological spinon bands and vison excitations in spin-orbit coupled quantum spin liquids