Strong magnetic frustration and anti-site disorder causing spin-glass behavior in honeycomb Li2RhO3

With large spin-orbit coupling, the $t_{2g}^5$ electron configuration in $d$-metal oxides is prone to highly anisotropic exchange interactions and exotic magnetic properties. In $5d^5$ iridates, given the existing variety of crystal structures, the magnetic anisotropy can be tuned from antisymmetric to symmetric Kitaev-type, with interaction strengths that outsize the isotropic terms. By many-body electronic-structure calculations we here address the nature of the magnetic exchange and the intriguing spin-glass behavior of Li$_2$RhO$_3$, a $4d^5$ honeycomb oxide. For pristine crystals without Rh-Li site inversion, we predict a dimerized ground state as in the isostructural $5d^5$ iridate Li$_2$IrO$_3$, with triplet spin dimers effectively placed on a frustrated triangular lattice. With Rh-Li anti-site disorder, we explain the observed spin-glass phase as a superposition of different, nearly degenerate symmetry-broken configurations.


Effective spin Hamiltonian
The effective magnetic Hamiltonian for two adjacent Rh sites is most conveniently written in a local reference frame {X,Y,Z} with the Rh-Rh bond along the X axis and Z perpendicular to the Rh 2 O 2 plaquette. For C 2h point-group symmetry, it reads Due to the inversion center, the antisymmetric anisotropy does not show up here. Further, for the symmetric anisotropic term, only one offdiagonal element is nonzero in C 2h . A straightforward diagonalization of H i j yields the following eigenvalues and eigenfunctions: Here Φ S is the total spin singlet and Φ 1−3 are combinations of the three triplet components. The latter are degenerate in the plain Heisenberg model. The above diagonalization procedure is equivalent with a rotation of the coordinate system {X,Y,Z} around X by an angle α to a new frame {X ,Y ,Z } in which the symmetric anisotropic exchange matrix is diagonal. 15 {X ,Y ,Z } are also referred to as principal axes and the angle α is given by In C 2h symmetry, the Φ S , Φ 1 , Φ 2 and Φ 3 spin-orbit wave functions transform according to the A g , B u , B u and A u irreducible representations, respectively. Since states Φ 1 and Φ 2 belong to the same irreducible representation B u , they are in general admixed, i.e., in the reference frame {X,Y,Z} the corresponding eigenfunctions should be written as The mixing parameter ξ = sin α is given by and is explicitly obtained from the QC data. For Rh-Rh links along the crystallographic b axis (labeled B1) the symmetry of a block of two edge-sharing octahedra is C 2h while for the other Rh-Rh links (labeled B2/B3) the Rh-O bonds on the Rh-O 2 -Rh plaquette have different lengths and the symmetry is lowered to C i . 16 Since the QC calculations were actually performed in C 1 symmetry, to determine the nature of each of the lowest four spin-orbit states, we explicitly computed the dipole and quadrupole transition matrix elements within that manifold. Standard selection rules and the nonzero dipole and quadrupole matrix elements in the QC outputs then clearly indicate which state is which. We also carried out the transformation of the spin-orbit wave functions from the usual This allows the study of Φ 1 -Φ 2 mixing due to the offdiagonal Γ yz and Γ zx couplings. With such an analysis, we find that in Li 2 RhO 3 the weight of Φ 1 in Ψ 1 (and of Φ 2 in Ψ 2 ) is 73% for links B1 and 94% for B2/B3.
Using Eqs. (2), (3) and (5) we obtain the effective coupling parameters of (1) as: In the local Kitaev reference system {x,y,z}, that is rotated from the reference frame {X,Y,Z} by 45 o about the Z = z axis (see Fig.1 and Refs. 5, 17), the Hamiltonian given in expression (1) above is transformed into equation (1) of the main text. For the latter, the exchange interaction parameters are given by : 5

Lattice spin model, notations and ab initio effective couplings
For each type of Rh-Rh link, we used two different local axes frames, namely 3} labels the type of bond. 5 Let us choose our global frame to be {x, y, z} = {x 1 , y 1 , z 1 }. The relation between the local axes {X b , Y b , Z b } and the global frame is then the following (see Fig. 1): We note that the system is invariant under two-fold rotations around the vector X 1 . This is why we must choose X 3 to be the rotated version of X 2 , i.e., C 2 · X 2 = X 3 . If we choose the opposite direction for X 3 , then we must change the sign of the coupling C.
Let us now write down the NN terms of the Hamiltonian for the three types of bonds b = 1, 2, 3. In the reference frame . For simplicity, we replace in the following J b=1 by J, J b=2,3 by J and similarly for the remaining parameters. Eqs. (9) and (8) then yield , , With the conventions introduced in Fig. 1, the spin Hamiltonian now reads where R = na + mb labels the unit cells and j ∈ {1, 2} labels the sublattice index.
The numerical values of the above coupling parameters, as found by ab initio multireference configuration-interaction (MRCI) calculations, are (in units of meV) : For completeness, we also provide the coupling parameters as found by spin-orbit CASSCF calculations (in units of meV) :