Spin-orbit effects in Na$_4$Ir$_3$O$_8$, a hyper-kagom\'{e} lattice antiferromagnet

TL;DR

This paper investigates the effects of spin-orbit coupling in Na$_4$Ir$_3$O$_8$, revealing how different limits of spin-orbit strength influence magnetic interactions and the potential for quantum spin liquid behavior.

Contribution

It provides a semi-microscopic analysis of spin-orbit effects, considering both weak and strong limits, and explores their impact on magnetic frustration and spin-liquid states in Na$_4$Ir$_3$O$_8$.

Findings

Strong spin-orbit coupling leads to unfrustrated symmetric exchange anisotropy.

Weak spin-orbit coupling allows for Heisenberg and Dzyaloshinskii-Moriya interactions.

Quantum spin liquid may persist depending on the orientation of the DM vector.

Abstract

We consider spin-orbit coupling effects in Na$_4$Ir$_3$O$_8$, a material in which Ir$^{4+}$ spins form an hyper-kagom\'{e} lattice, a three-dimensional network of corner-sharing triangles. We argue that both low temperature thermodynamic measurements and the impurity susceptibility induced by dilute substitution of Ti for Ir are suggestive of significant spin-orbit effects. Because of uncertainties in the crystal-field parameters, we consider two limits in which the spin-orbit coupling is either weak or strong compared to the non-cubic atomic splittings. A semi-microscopic calculation of the exchange Hamiltonian confirms that indeed large antisymmetric Dzyaloshinskii-Moriya (DM) and/or symmetric exchange anisotropy may be present. In the strong spin-orbit limit, the Ir-O-Ir superexchange contribution consists of unfrustrated strong symmetric exchange anisotropy, and we suggest that spin-liquid behavior is unlikely. In the weak spin-orbit limit, and for strong spin-orbit and direct Ir-Ir exchange, the Hamiltonian consists of Heisenberg and DM interactions. The DM coupling is parametrized by a three component DM vector (which must be determined empirically). For a range of orientation of this vector, frustration is relieved and an ordered state occurs. For other orientations, even the classical ground states are very complex. We perform spin-wave and exact diagonalization calculations which suggest the persistence of a quantum spin liquid in the latter regime. Applications to Na$_4$Ir$_3$O$_8$ and broader implications are discussed.