Theory of Raman response in three-dimensional Kitaev spin liquids: Application to $\beta$- and $\gamma -{\mathrm{Li}}_2{\mathrm{IrO}}_3$ compounds

We calculate the Raman response for the Kitaev spin model on the $\mathcal{H}-0, \mathcal{H}-1$, and $\mathcal{H}-\infty$ harmonic honeycomb lattices. We identify several quantitative features in the Raman spectrum that are characteristic of the spin liquid phase. Unlike the dynamical structure factor, which probes both the Majorana spinons and flux excitations that emerge from spin fractionalization, the Raman spectrum in the Kitaev models directly probes a density of states of pairs of fractional, dispersing Majorana spinons. As a consequence, the Raman spectrum in all these models is gapless for sufficiently isotropic couplings, with a low-energy power law that results from the Fermi lines (or points) of the dispersing Majorana spinons. We show that the polarization dependence of the Raman spectrum contains crucial information about the symmetry of the ground state. We also discuss to what extent the features of the Raman response that we find reflect generic properties of the spin liquid phase, and comment on their possible relevance to $\alpha$-, $\beta$-, and $\gamma -{\mathrm{Li}}_2{\mathrm{IrO}}_3$ compounds.

Figure 1

An elementary plaquette of the honeycomb lattice with the bonds labeled by the components of spins that interact along them. The spin component shown at each vertex is the one that enters the product $W_P$ of Eq. (2), and corresponds to the color of the bond pointing out of the plaquette from that vertex.

Figure 2

The harmonic honeycomb ($\mathcal{H}-n$) lattices: $\mathcal{H}-0$ (left), $\mathcal{H}-1$ (center), and $\mathcal{H}-\infty$ (right). Here, $n$ counts the number of rows along the $c$ axis before the orientation of the honeycomb plane switches between the two nonparallel chains of $x$ and $y$ bonds. The $\mathcal{H}-0$ hyperhoneycomb lattice switches chains at every $c$ bond. The $\mathcal{H}-1$ has one set of $c$ bonds making rungs (gray) of ladders before a bridge ($c$ bonds in black) to the opposite ladder. The $\mathcal{H}-\infty$ never switches ladders. The ladders are labeled by red (orange) and blue (green) on $x$($y$) bonds. The primitive unit cell for $n<\infty$ contains $4n+4$ sites. The honeycomb lattice has only two sites in the unit cell. Our choice of unit cell is illustrated by the spheres, whose color alternates yellow and white indicating the odd and even sublattices, respectively. The excited bonds in the $\pi$ flux state are circled on the $\mathcal{H}-1$ lattice.

Figure 3

Raman intensities for the pure Kitaev model on the 2D honeycomb lattice computed for (a) the symmetric point $J^x=J^y=J^z$, (b) another gapless point $J^x=J^y=1.43J^z$, and (c) a gapped point $J^x=J^y=0.3J^z$. For each figure, the inset on the left is a plot of the one-particle DOS; the inset on the right is a plot of the derivatives of the Raman intensity $I^{\prime }\left(\omega \right)$.

Figure 4

The relative total spectral weight $SW=\int I\left(\omega \right)d\omega$ of the two representative Raman intensities as a function of the Kitaev exchange-coupling anisotropy for the 2D honeycomb lattice. The transition to the gapless phase occurs at $J^x/J^z=0.5$.

Figure 5

Raman intensities for the $\mathcal{H}-0$ lattice computed for (a) $J^x=J^y=J^z$ (gapless), (b) $J^x=J^y=1.43J^z$ (gapless), and (c) $J^x=J^y=0.3J^z$ (gapped). The spectrum looks qualitatively the same throughout the gapless phase, in contrast with the 2D case in which the symmetric coupling point has extra symmetry. The maxima and minima in $I\left(\omega \right)$ originate from van Hove singularities and band edges, respectively, for each band (see Fig. 6). The only two-particle combination that couples to the $B_{1g}$ channel is one from the lower band and one from the upper band causing the spectrum $I_{ab}$ to vanish unless $4J^z<\omega <4\sqrt{5}{J}^z$ [86].

Figure 6

The Majorana spinon spectrum on the $\mathcal{H}-0$ lattice plotted along high symmetry lines. The inset on the left shows the 2-DOS defined in Eq. (19). For details of the BZ see Appendix pp1-s3. The two complex fermions per unit cell make two bands, ${\varepsilon }_{\pm ,\mathbf{k}}$. This gives three distinct configurations for the two-particle states of interest in Raman scattering: two with both excitations in the same band, and one with one excitation per band. The 2-DOS is split into the contributions from each of the configurations by ${\rho }_{mn}\left(\omega \right)={\sum }_{\mathbf{k}}\delta (\omega -{\varepsilon }_{m,\mathbf{k}}-{\varepsilon }_{n,\mathbf{k}})$, where indices $m,n=\pm$. The maxima and minima in $I\left(\omega \right)$ appear as extrema in the spectrum of the appropriate band at some symmetry-distinguished point in the BZ.

Figure 7

Raman intensities for the $\mathcal{H}-1$ lattice in the zero-flux state at the isotropic point ($J^x=J^y=J^z$).

Figure 8

Raman intensities for the H-1 lattice $\pi$-flux state at the isotropic point ($J^x=J^y=J^z$) [72].

Figure 9

2-DOS for the Majorana spinons on the (a) $\mathcal{H}-0$, (b) $\mathcal{H}-1$, and (c) $\mathcal{H}-1\pi$ lattices. Up to polarization dependence these spectra determine the character of the Raman intensities. The number of maxima and minima grows linearly with the number of combinations of bands for two-particle states, which increases from left to right due to increasing unit cell size.

Figure 10

Relative spectral weights for Raman on the $\mathcal{H}-0$ lattice as a function of the bond-coupling anisotropy.