Dynamic Spin Correlations in the Honeycomb Lattice ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ Measured by Resonant Inelastic x-Ray Scattering

A Kitaev quantum spin liquid is a prime example of novel quantum magnetism of spin-orbit entangled pseudospin-$1/2$ moments in a honeycomb lattice. Most candidate materials such as ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ have many competing exchange interactions beyond the minimal Kitaev-Heisenberg model whose small variations in the strength of the interactions produce huge differences in low-energy dynamics. Our incomplete knowledge of dynamic spin correlations hampers identification of a minimal model and quantification of the proximity to the Kitaev quantum spin-liquid phase. Here, we report momentum- and energy-resolved magnetic excitation spectra in a honeycomb lattice ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ measured using a resonant inelastic x-ray scattering spectrometer capable of 12 meV resolution. Measured spectra at a low temperature show that the dynamic response lacks resolution-limited coherent spin waves in most parts of the Brillouin zone but has a discernible dispersion and spectral weight distribution within the energy window of 60 meV. A systematic investigation using the exact diagonalization method and direct comparison of high-resolution experimental spectra and theoretical simulations allow us to confine a parameter regime in which the extended Kitaev-Heisenberg model reasonably reproduces the main feature of the observed magnetic excitations. Hidden Kitaev quantum spin-liquid and Heisenberg phases found in the complex parameter space are used as references to propose the picture of renormalized magnons as explaining the incoherent nature of magnetic excitations. Magnetic excitation spectra are taken at elevated temperatures to follow the temperature evolution of the resonant inelastic x-ray scattering dynamic response in the paramagnetic state. Whereas the low-energy excitation progressively diminishes as the zigzag order disappears, the broad high-energy excitation maintains its spectral weight up to a much higher temperature of 160 K. We suggest that the dominant nearest-neighbor interactions keep short-range correlations up to quite high temperatures with a specific short-range dynamics which has a possible connection to a proximate spin-liquid phase.

Popular Summary

Quantum spin liquids (QSLs) are elusive states of matter where electron spins are unable to align even at zero temperature. Theory predicts that QSLs can arise in materials whose atoms are arranged in a special type of honeycomb-lattice structure. The compound ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ is one such material, however, measurements of how its spins interact with one another have proven difficult because standard probes for doing so are not well suited. To circumvent this hurdle, we turn to a different type of probe and successfully measure magnetic excitation spectra in a sample of ${\mathrm{Na}}_2{\mathrm{IrO}}_3$, thus providing information on spin correlations that can be compared to theoretical calculations.

Such measurements typically rely on a technique called inelastic neutron scattering, however, iridium compounds such as ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ absorb neutrons instead of scattering them. The only alternative tool is resonant inelastic x-ray scattering (RIXS), however, this approach does not offer sufficient energy resolution to compare observations with theory.

We overcome this barrier by replacing the silicon crystal that is conventionally used in the RIXS analyzer with a quartz crystal. On a high-quality sample of ${\mathrm{Na}}_2{\mathrm{IrO}}_3$, this alteration leads to an unprecedented energy resolution of 12 meV—a significant advancement over previous RIXS measurements. Our experimental and theoretical works show that a renormalized magnon dominates the magnetic response at low temperature. At high temperature, short-range dynamics, possibly connected to a proximate QSL, emerges.

Future experiments could combine the upgraded RIXS spectrometer used in our study with x-ray polarization analysis to directly probe fractional excitation—a key marker of QSLs—in ${\mathrm{Na}}_2{\mathrm{IrO}}_3$.

Figure 1

Magnetic excitation spectra in ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ along high-symmetry Brillouin zone directions taken at $T=7\mathrm{K}$. (a) Scattering geometry. Yellow arrows indicate incident and scattered x rays, which define the scattering plane (gray). Brown arrows indicate x-ray polarizations. Green arrows indicate the cubic axes $(x,y,z)$ with respect to the octahedra (all of them point above the paper plane). (b) One of the collinear zigzag patterns and the corresponding direction of the ordered moments. The left-facing arrows have an out-of-plane component pointing above the paper plane; i.e., the corresponding moment direction lies approximately between the $x$ and $y$ axes. (c) Two-dimensional reciprocal space diagram showing the measured path along the symmetry directions. The inner hexagon (blue dashed line) indicates the first Brillouin zone of the honeycomb lattice. (d) RIXS intensity map of magnetic excitations in ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ as functions of the wave vector and energy loss. (e) The intensity profiles integrated over [60, 105] (open squares) and [105, 135] meV (filled triangles) show that the high-energy spectral intensities are broadly peaked at the $\mathrm{\Gamma }$ point, extending up to 105 meV. (f) The intensity profile integrated over $[-30,60]\mathrm{meV}$ (filled squares) shows a distinctive distribution of the spectral weight along the $K\text{−}\mathrm{\Gamma }\text{−}Y\text{−}{K}^{\prime }\text{−}{\mathrm{\Gamma }}^{\prime }$ path. Passing through the $K$ point, the excitation intensity rapidly increases and then decreases, which is followed by a near-constant intensity along the $\mathrm{\Gamma }\text{−}Y\text{−}{K}^{\prime }$ path. Large intensities at the $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}^{\prime }$ wave vectors correspond to elastic scatterings. The zero-energy loss intensity at the $M$ point is the diffuse magnetic Bragg peak. In the used scattering geometry, the magnetic Bragg peak at the $Y$ point is suppressed.

Figure 2

High-resolution RIXS spectra recorded at $T=7\mathrm{K}$. (a) The measured energy resolution function of the quartz analyzer is plotted at the bottom. At $\mathbf{Q}=(006.75)$ ($\mathrm{\Gamma }$), an elastic scattering is followed by a broad excitation without any indication of a narrow excitation. At $\mathbf{Q}=(0.450.456.5)$ (near $M$), a soft low-energy excitation is unresolved with 12 meV energy resolution, and a quasielastic scattering is followed by a broad shoulder excitation. At $\mathbf{Q}=(016.5)$ ($Y$), a low-energy excitation is seen with a broad high-energy feature. At $\mathbf{Q}=(0.6706.6)$ ($K$), a narrow excitation is discovered at a low energy. (b),(c) The measured RIXS spectra at the $\mathrm{\Gamma }$ and $K$ wave vectors, respectively, are fit by the pseudo-Voigt function (elastic scattering) and damped harmonic oscillator (DHO) function convoluted by the 12 meV resolution function. (d) Temperature dependence of the RIXS excitation spectrum at the $K$ point. The low-energy peak shows a clear decrease at $T=70\mathrm{K}$ and becomes featureless at $T=150\mathrm{K}$, indicating that it is an excitation peak of the zigzag magnetic order.

Figure 3

Zigzag phase in the phase diagram of the extended Kitaev-Heisenberg model with the main interactions parametrized using the radial coordinate $\theta$ and the azimuth $\varphi$ as $J=A\sin \theta \cos \varphi$, $K=A\sin \theta \sin \varphi$, $\mathrm{\Gamma }=A\cos \theta$, and the smaller interactions kept constant: ${\mathrm{\Gamma }}^{\prime }=-0.1A$, $J_2=0.05A$, and $J_3=0.1A$. Here, $A$ is the overall energy scale, which is irrelevant for the phase diagram but determines the characteristic energies of the magnetic excitations. The color indicates the angle of the zigzag-ordered moments to the honeycomb plane. The data are obtained using the exact-diagonalization-based method in Ref. [51]. Small panels around the phase diagram show, for selected parameter points, the theoretical RIXS intensity maps calculated by exact diagonalization of the extended Kitaev-Heisenberg model on 24- and 32-site clusters (see Appendix pp2 for details). We assume the scattering geometry depicted in Fig. 1 and present a sum of $\pi -{\pi }^{\prime }$ and $\pi -{\sigma }^{\prime }$ scattering channels (imitating an unpolarized detection) plotted along the same path through the Brillouin zone as in Fig. 1. The energies are determined by taking the value $A=29\mathrm{meV}$ giving the best match between the $A2$ parameter point and the experimental data. Gaussian broadening with a FWHM of 25 meV is applied to the spectra. Finally, the violet lines show the pathways in the parameter space connecting the selected points $A2$ and $A3$ with the points of hidden symmetry utilized in Sec. 3d2. They should be understood as projections only, since the hidden Heisenberg point (hH) has ${\mathrm{\Gamma }}^{\prime }\approx -0.4A$, whereas the hidden Kitaev point (hK) has ${\mathrm{\Gamma }}^{\prime }\approx -0.3A$, and $J_{2,3}=0$ for both points.

Figure 4

(a) Experimental RIXS intensity map for $T=7\mathrm{K}$ with the positions of the intensity maxima for the individual wave vectors indicated as open circles. (b) Theoretical RIXS intensity map calculated for the parameter values corresponding to the $A2$ point in Fig. 3 and the energy scale $A=29\mathrm{meV}$, which gives $J\approx 12\mathrm{meV}$, $K\approx -24\mathrm{meV}$, $\mathrm{\Gamma }\approx 11\mathrm{meV}$, ${\mathrm{\Gamma }}^{\prime }\approx -3\mathrm{meV}$, $J_2\approx 1.5\mathrm{meV}$, and $J_3\approx 3\mathrm{meV}$. The scattering geometry depicted in Fig. 1 is assumed, and a sum of $\pi -{\pi }^{\prime }$ and $\pi -{\sigma }^{\prime }$ scattering channels is taken, corresponding to an unpolarized detection. The spectra are broadened in energy by Gaussians with a FWHM of 25 meV to imitate the experimental resolution as in (a). (c) The same as in (b) but for the parameter values corresponding to the $A3$ point in Fig. 3 and the energy scale of $A=24\mathrm{meV}$: $J\approx 10\mathrm{meV}$, $K\approx -15\mathrm{meV}$, $\mathrm{\Gamma }\approx 16\mathrm{meV}$, ${\mathrm{\Gamma }}^{\prime }=-2.4\mathrm{meV}$, $J_2=1.2\mathrm{meV}$, and $J_3=2.4\mathrm{meV}$. (d) Energy-integrated RIXS response (total spectral weight) corresponding to (a)–(c). The experimental data, obtained by integrating in the $[-30,60]\mathrm{meV}$ range, are arbitrarily scaled with respect to theoretical ones. (e) RIXS intensity profiles at selected wave vectors. High-resolution experimental data (gray dots) are compared to theoretical spectra for the above parameters (red and blue lines). The theoretical spectra are broadened by the experimental resolution function of the high-resolution setup.

Figure 5

(a) RIXS intensity maps for the parameter point $A2$ and the same energy scale as in Fig. 4. The left and right are obtained by exact diagonalization and the linear spin-wave approach, respectively. To better resolve the finer features, Gaussian broadening with a small FWHM of only 2 meV is used in both cases. In the case of LSW, the spectra are averaged by equally employing all three possible zigzag pattern directions. This procedure leads to a map that could be directly compared to the ED one, since the cluster ground state used in ED contains all zigzag pattern in an equal-weight superposition. (b) The same for the parameter point $A3$ and the energy scale as in Fig. 4. (c) Energy-integrated RIXS response (total spectral weight) corresponding to (a). Data for all the clusters are presented, showing negligible finite-size effects on the spectral weight. The curve obtained by the linear spin-wave approach (right) is shown also on the left by a dashed line. (d) Total spectral weight for the data in (b).

Figure 6

(a) Evolution of the theoretical $\mathrm{\Gamma }$-point ($\mathit{q}=0$) RIXS intensity on a line through the parameter space going from the hidden Heisenberg point $(J,K,\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime })\approx (-0.1,-0.6,0.8,-0.4)A$ to the point $A2$ of Fig. 3. The line lies completely in the zigzag phase. Finite-size effects can be estimated by comparing data for various clusters shown in Appendix pp2. The average is taken over all of them (i.e., 24a, 32a, and 32b1 to 32b3). The spectra are presented in units of the energy scale $A$ and only a tiny broadening is used to resolve the structure of the continuum. (b) The same for a line connecting the point $A2$ and the hidden Kitaev point $(J,K,\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime })\approx (0.6,-0.5,0.6,-0.3)A$. Apart from the last point, the ground state remains zigzag ordered. The dotted line indicates scaled exact solution in the AFM Kitaev limit [63]. (c,d) Same as panels (a,b) but for the $A3$ point of Fig. 3. (e) Two-magnon density of states for selected points of panels (c) and (d) calculated using the linear spin-wave dispersions (gray lines). Single zigzag pattern of Fig. 1 was assumed, its Bragg spots correspond to the $Y$ points.

Figure 7

Temperature evolution of magnetic excitation spectra. (a) $T=7\mathrm{K}$. (b) $T=70\mathrm{K}$. The magnetic Bragg peak at the $M$ disappears, indicating disappearing zigzag correlations. Spectral intensities fill in between $\mathrm{\Gamma }$ and all first Brillouin zone symmetric points ($M$, $K$, $K^{\prime }$, and $Y$), connecting the high-energy feature at the $\mathrm{\Gamma }$ to the low-energy features at the $M$, $K$, and $Y$. This trend becomes more pronounced at (c) $T=120\mathrm{K}$ and (d) $T=160\mathrm{K}$. (e) $T=200\mathrm{K}$. Spectral intensities move toward lower energy. (f) $T=280\mathrm{K}$. The RIXS intensity over the whole Brillouin zone substantially diminishes.

Figure 8

(a) Clusters used in the exact diagonalization calculations: fully symmetric hexagonal 24- and 32-site clusters and a rectangular 32-site cluster in three possible orientations with respect to the honeycomb lattice. (b) Wave vectors compatible with periodic boundary conditions applied to the individual clusters. They are shown in one quadrant of the first Brillouin zone of the honeycomb lattice (dashed black line) and that of the completed triangular lattice (solid black line). High-symmetry points and the path along which the measured or simulated data are presented (gray line) are indicated [cf. Fig. 1].

Figure 9

(a) Four-sublattice transformation used in the text. At the individual sublattices, the spin axes are rotated by 180° around the cubic axes $x$, $y$, and $z$ or left intact. In the momentum space (right), the spin components are shifted by the zigzag ordering vectors. (b) Model parameter values along the lines used in Fig. 6. The values are expressed using the original $xyz$-coordinate system for the spins. The full lines and symbols correspond to Figs. 6 and 6, and dashed lines and open symbols to Figs. 6 and 6. The figure is organized like Fig. 6 with $A2$ or $A3$ parameters being in the middle and the evolution toward the hidden Heisenberg (Kitaev) point corresponding to the left (right) direction from the middle. Only nearest-neighbor interaction parameters $JK\mathrm{\Gamma }{\mathrm{\Gamma }}^{\prime }$ are presented; $J_{2,3}$ linearly vanish when approaching the points of hidden symmetry. The marked parameter points correspond to the spectra shown in Fig. 6. (c) The same lines through the parameter space but expressed using the $x^{\prime }{y}^{\prime }{z}^{\prime }$-coordinate system rotated by 180° around the axis perpendicular to the honeycomb plane. In both $xyz$ and $x^{\prime }{y}^{\prime }{z}^{\prime }$ frames depicted by the insets, the axes point above the paper plane.