Non-Loudon-Fleury Raman scattering in spin-orbit coupled Mott insulators

We revisit the theory of magnetic Raman scattering in Mott insulators with strong spin-orbit coupling, with a major focus on Kitaev materials. We show that Kitaev materials with bond-anisotropic interactions are generally expected to show both one- and two-magnon responses. It is further shown that, in order to obtain the correct leading contributions to the Raman vertex operator $\mathcal{R}$, one must take into account the precise, photon-assisted microscopic hopping processes of the electrons and that, in systems with multiple hopping paths, $\mathcal{R}$ contains terms beyond those appearing in the traditional Loudon-Fleury theory. Most saliently, a numerical implementation of the revised formalism to the case of the three-dimensional hyperhoneycomb Kitaev material $\beta \text{−}{\mathrm{Li}}_2\mathrm{Ir}{\mathrm{O}}_3$ reveals that the non-Loudon-Fleury scattering terms actually dominate the Raman intensity. In addition, they induce a qualitative modification of the polarization dependence, including, e.g., the emergence of a sharp one-magnon peak at low energies, which is not expected in the traditional Loudon-Fleury theory. This peak is shown to arise from microscopic photon-assisted tunneling processes that are of similar type with the ones leading to the symmetric off-diagonal interaction $\mathrm{\Gamma }$ (known to be present in many Kitaev materials), but take the form of a bond-directional magnetic dipole term in the Raman vertex. These results are expected to apply across all Kitaev materials and mark a drastic change of paradigm for the understanding of Raman scattering in materials with strong spin-orbit coupling and multiple exchange paths.

Figure 1

The square plaquette that is relevant for the superexchange processes between two magnetic ions sharing a $z$ bond in $\mathrm{A}{}_2\mathrm{Ir}\mathrm{O}{}_3$ compounds (e.g., $\beta \text{−}\mathrm{Li}{}_2\mathrm{Ir}\mathrm{O}{}_3$). The same plaquette provides the superexchange processes in $\alpha \text{−}\mathrm{Ru}\mathrm{Cl}{}_3$ with substitution Ir $\to$ Ru, O $\to$ Cl.

Figure 2

Direct hopping ($a\text{−}b$ and $b\text{−}a$).

Figure 3

The eight different oxygen-mediated hopping paths connecting $\mathrm{Ir}{}_1$ and $\mathrm{Ir}{}_2$ in the plaquette of Fig. 1.

Figure 4

The eight different mixed hopping paths connecting $\mathrm{Ir}{}_1$ and $\mathrm{Ir}{}_2$ in the basic plaquette of Fig. 1.

Figure 5

Sketch of a hyperhoneycomb lattice of $\beta \text{−}{\text{Li}}_2{\text{IrO}}_3$. The orthorhombic unit cell is defined by the crystallographic axes $\{\widehat{\mathbf{a}},\widehat{\mathbf{b}},\widehat{\mathbf{c}}\}$ related to the Cartesian axes $\{\widehat{\mathbf{x}},\widehat{\mathbf{y}},\widehat{\mathbf{z}}\}$ appearing in the spin Hamiltonian by the following relations: $\widehat{\mathbf{x}}=(\widehat{\mathbf{a}}+\widehat{\mathbf{c}})/\sqrt{2},\widehat{\mathbf{y}}=(\widehat{\mathbf{c}}-\widehat{\mathbf{a}})/\sqrt{2},\widehat{\mathbf{z}}=-\widehat{\mathbf{b}}.$ The five NN bonds of the $J\text{−}K\text{−}\mathrm{\Gamma }$ model are marked in red for $\mathit{d}\in \{x,x^{\prime }\}$, green for $\mathit{d}\in \{y,y^{\prime }\}$, and blue for $\mathit{d}\in \left\{z\right\}$. Each octahedral denotes to the ${\text{IrO}}_6$ cage.

Figure 6

Linear spin wave spectrum along a high symmetry path in the Brillouin zone of the orthorhombic unit cell (inset).

Figure 7

One-magnon (a)–(b) and two-magnon (c)–(d) Raman intensities, computed with [(b) and (d)] and without [(a) and (c)] taking into account the non-Loudon-Fleury scattering terms, see main text. Lines with different colors correspond to different polarization channels (with the color scheme being consistent across all the panels). The inset of (b) shows the Raman response without the non-Loudon-Fleury, magnetic-dipole term $\propto h_{\mathrm{\Gamma }}^{\left(3\right)}$. The inset of (d) shows the two-magnon density of states ${\rho }_2$ of Eq. (52).

Figure 8

Direct hopping from one-hole state on $\mathrm{Ir}{}_1$ and $\mathrm{Ir}{}_2$ to the intermediate two-hole states (denoted in Table 4) on $\mathrm{Ir}{}_2$ from one of four possible $|{\psi }_n\rangle$ ground states: (a) $|{\psi }_1\rangle \equiv |+1/2;+1/2\rangle$, (b) $|{\psi }_4\rangle \equiv |-1/2;-1/2\rangle$, (c) $|{\psi }_3\rangle \equiv |+1/2;-1/2\rangle$, and (d) $|{\psi }_2\rangle \equiv |-1/2;+1/2\rangle$.

Figure 9

Oxygen mediated hopping from one-hole state on $\mathrm{Ir}{}_1$ and $\mathrm{Ir}{}_2$ to the intermediate two-hole states (denoted in Table 4) on $\mathrm{Ir}{}_2$ from one of four possible $|{\psi }_n\rangle$ ground states. (b)–(e) represent the hopping via the upper path (a) and (g)–(j) represent the hopping via the lower path (f).