Session L33: Kitaev Physics in Honeycomb Iridates

Evidence for coexisting magnetic order in frustrated three-dimensional honeycomb iridates Li$_{2}$IrO$_{3}$

The search for unconventional magnetism has found a fertile hunting ground in 5d iridium oxide (iridate) materials. The competition between coulomb, spin-orbit, and crystal field energy scales in honeycomb iridates leads to a quantum magnetic system with localized spin-1/2 moments communicating through spin-anisotropic Kitaev exchange interactions. Although early and ongoing work has focused on layered two-dimensional honeycomb compounds such as Na$_{2}$IrO$_{3}$ and a 4d analog, RuCl$_{3}$, recently discovered polytypes of Li$_{2}$IrO$_{3}$ take on three-dimensional honeycomb structures. Bulk thermodynamic studies, as well as recent resonant x-ray diffraction and absorption spectroscopy experiments, have uncovered a rich phase diagram for these three-dimensional honeycomb iridates. Low temperature incommensurate and commensurate magnetic orders can be stabilized by tuning the applied magnetic field, displaying a delicate coexistence that signals highly frustrated magnetism.   

Vacancies in a 3D-Kitaev model on hyper-honeycomb lattice

We study the properties of isolated single and pairs of vacancies in an exactly solvable Kitaev model on a three dimensional hyper-honeycomb lattice. We show that each vacancy in the lattice is associated with a low energy spin like degree of freedom, similar to the case of previously studied honeycomb model. We calculate the contribution from these vacancy spin-moments to the low field magnetization response to a $z$-directed field. Isolated vacancies in the gapped phase act as free spins. In the gapless phase, these spins interact with the surrounding spin-liquid suppressing the low-field magnetization to $\frac{1}{\sqrt{\ln[1/h_z]}}$. Pair of vacancies have a sublattice-dependent, anisotropic, spin-liquid mediated interaction with each other. In the gapless phase, interaction between vacancies in the same (opposite) sublattice enhances (suppresses) the low-field magnetization, indicating a ferromagnetic (anti-ferromagnetic) nature. We also show that, unlike vacancies in the honeycomb lattice, the vacancies here do not bind a flux at low-energies.   

Spin-Peierls instability of three-dimensional Kitaev spin liquids with Majorana Fermi surface

The Kitaev honeycomb model is one of the paradigmatic examples of a frustrated spin system exhibiting a quantum spin liquid ground state. The emergent low-energy degrees of freedom are Majorana fermions that can form various different (semi-)metallic states. Three-dimensional variants of this model can, in particular, harbor gapless quantum spin liquids with a Majorana Fermi surface. In this talk, we discuss Fermi surface instabilities arising from additional spin exchange terms (such as a Heisenberg coupling), which induce interactions between the emergent Majorana fermion degrees of freedom. We show that independent of the details of the interactions, the Majorana Fermi surface is always unstable. Generically, the system spontaneously dimerizes at exponentially small temperatures and forms a quantum spin liquid with nodal lines. Depending on the microscopic details, further symmetries of the system may be broken at this transition. These spin-Peierls instabilities of a 3D spin liquid are closely related to BCS instabilities of fermions\footnote{M. Hermanns, S. Trebst, A. Rosch, PRL 115, 117205 (2015).}.   

Kitaev physics in three dimensional honeycomb iridates

It has been realized that Mott insulators with strong spin-orbit coupling may allow strongly bond-dependent exchange interactions between local moments. Such interactions may lead to magnetic frustration and possible quantum spin liquid phases. This is in contrast to usual frustrated magnets, where the magnetic frustration comes from the geometry of the underlying lattice structure. Hence, it offers a new avenue to generate exotic phases of matter. Recently, both two-dimensional ($\alpha$-Li$_2$IrO$_3$) and three-dimensional honeycomb iridates ($\beta$-Li$_2$IrO$_3$ and $\gamma$-Li$_2$IrO$_3$) have been discovered and it has been suggested that the magnetic exchange interactions contain the so-called Kitaev interaction, which depends on bond directions. In particular, the local moments of Ir ions in $\beta$-Li$_2$IrO$_3$ and $\gamma$-Li$_2$IrO$_3$ reside on the three-dimensional hyper-honeycomb and stripy-honeycomb lattices. The Kitaev model is exactly solvable on these lattices as well as the two-dimensional honeycomb lattice and the ground state is a quantum spin liquid with gapless excitations. We discuss recent progress in theoretical understanding of magnetic exchange interactions, possible presence of quantum spin liquid phases, and unusual magnetic order in $\beta$-Li$_2$IrO$_3$ and $\gamma$-Li$_2$IrO$_3$. These theoretical results are used to make connections to recent experimental data.   

Detecting Semimetal Surface Modes in Kitaev Spin Liquids

Raman scattering is a useful probe for “Kitaev-type" spin liquids because it couples only to the dispersing (and potentially gapless) fermionic degrees of freedom in these systems. I will discuss Raman scattering in Kitaev spin liquids on the 3D hyperhoneycomb (\mathcal{H}-0) lattice, where these fermionic degrees of freedom realize topologically non-trivial band structures with protected gapless surface states. I will describe Raman signatures both of bulk 3D samples, and thin-film samples of these materials, where the resonant Raman response can detect the protected surface modes.   

Large Band Gap of alpha-RuCl$_{\mathrm{3}}$ Probed by Photoemission and Inverse Photoemission Spectroscopy

The Kitaev honeycomb lattice model has attracted great attention because of its possibility to stabilize a quantum spin liquid ground state. Recently, it was proposed that alpha-RuCl$_{\mathrm{3}}$ is its material realization and the first 4$d$ relativistic Mott insulator from an optical spectrum and LDA$+U+$SO calculations. Here, we present photoemission and inverse photoemission spectra of alpha-RuCl$_{\mathrm{3}}$. The observed band gap is about 1.8 eV, which suggests that the previously assigned optical gap of 0.3 eV is misinterpreted, and that the strong peak at about 1.2 eV in the optical spectrum may be associated with an actual optical gap. Assuming a strong excitonic effect of 0.6 eV in the optical spectrum, all the structures except for the peak at 0.3 eV are consistent with our electronic spectra. When compared with LDA$+U+$SO calculations, the value of $U$ should be considerably larger than the previous one, which implies that the spin-orbit coupling is not a necessary ingredient for the insulating mechanism of alpha-RuCl$_{\mathrm{3}}$. We also present angle-resolved photoemission spectra to be compared with LDA$+U+$SO and LDA$+$DMFT calculations.   

Ab-initio study on crystal structure of $\alpha$-RuCl$_3$

$\alpha$-RuCl$_3$ was recently proposed as a candidate system for materialization of Kitaev model, but precise structural information of the compound has remained elusive. For the clarification of the full three-dimensional crystal structure of $\alpha$-RuCl$_3$, we performed ab-initio electronic structure calculations including effects of spin-orbit coupling (SOC) and electron correlations. We found that SOC prevents dimerization between Ru atoms, and keeps the system close to honeycomb lattice. The ground state crystal structure has monoclinic $C2/m$-type layer stacking, but trigonal $P3_112$-and orthorhombic $Cmc2_1$-type stacking orders are comparable to the $C2/m$ structure in energy, so that stacking faults can be easily introduced. The electronic structure and the $j_{\rm eff}$=1/2 pseudospin exchange interactions and possible magnetic states in $\alpha$-RuCl$_3$ will be presented.   

Nanoscale structural and electronic characterization of $\alpha $-RuCl$_{\mathrm{3}}$ layered compound

The exceptional interplay of spin-orbit effects, Coulomb interaction, and electron--lattice coupling is expected to produce an elaborate phase space of $\alpha $-RuCl$_{\mathrm{3}}$ layered compound, which to date remains largely unexplored. Here we employ a combination of scanning transmission electron microscopy (STEM) and scanning tunneling microscopy (STM) for detailed evaluation of the system's microscopic structural and electronic orders with a sub-nanometer precision. The STM and STEM measurements are further supported by neutron scattering, X-Ray diffraction, density functional theory (DFT), and multivariate statistical analysis. Our results show a trigonal distortion of Cl octahedral ligand cage along the $C_{\mathrm{3}}$ symmetry axes in each RuCl$_{\mathrm{3}}$ layer. The lattice distortion is limited mainly to the Cl subsystem leaving the Ru honeycomb lattice nearly intact. The STM topographic and spectroscopic characterization reveals an intra unit cell electronic symmetry breaking in a spin-orbit coupled Mott insulating phase on the Cl-terminated surface of $\alpha $-RuCl$_{\mathrm{3}}$. The associated long-range charge order (CO) pattern is linked to a surface component of Cl cage distortion. We finally discuss a fine structure of CO and its potential relation to variations of average unit cell geometries found in multivariate analysis of STEM data.   

Probing Spin Excitations Using Magneto-Raman Spectroscopy

The presence of a 2D quantum spin liquid state was recently suggested for the spin-orbit coupled Mott insulator $\alpha$-RuCl$_3$ with a honeycomb lattice.[Phys. Rev. 90, 041112 (2014)] Optical spectroscopy, Raman scattering, specific heat as well as magnetic susceptibility measurements on $\alpha$-RuCl$_3$ identified elementary excitations due to electronic correlations and spin-orbit coupling.[arXiv:1503.07593, Phys. Rev. Letters 114, 147201 (2015), and Phys. Rev. 91, 144420 (2015)] These observations appear to be consistent with theoretical expectations for Heisenberg-Kitaev model for QSL.[Phys. Rev. 91, 241110 (2015)] The underlying mechanism for the unconventional magnetism in $\alpha$-RuCl$_3$ was further investigated by probing the effect of external magnetic field on the Raman spectroscopic signatures. Raman scattering experiments were performed at temperatures down to 5 K and magnetic fields up to 10 T. The intensity of strongest A$_{1g}$ phonon was found to decrease with increasing magnetic field strength suggesting the presence of strong magnetic interactions. The experimental observations and its implications will be presented.   

Phase diagram and quantum order by disorder in the Kitaev $K_1$-$K_2$ honeycomb magnet

We show that the topological Kitaev spin liquid on the honeycomb lattice is extremely fragile against the second neighbor Kitaev coupling $K_2$, which has been recently identified as the dominant perturbation away from the nearest neighbor model in iridate Na$_2$IrO$_3$, and may also play a role in $\alpha$-RuCl$_3$. This coupling explains naturally the zig-zag ordering and the special entanglement between real and spin space observed recently in Na$_2$IrO$_3$. The minimal $K_1$-$K_2$ model that we present here holds in addition the unique property that the classical and quantum phase diagrams and their respective order-by-disorder mechanisms are qualitatively different due to their fundamentally different symmetry structure.   

High Pressure Transport and Structural Study on the Honeycomb Lattice Iridates $A_2$IrO$_3$ ($A =$ Na, Li)

The honeycomb lattice iridates $A_2$IrO$_3$ ($A =$ Na, Li) have been predicted and shown to exhibit novel magnetic properties which suggest that these materials could realize bond-directional Kitaev-like magnetic exchange interactions [1 - 5]. We will present high pressure (P) electrical transport ($P \leq 80$~GPa) and powder X-ray diffraction ($P \leq 40$~GPa) on these materials. The PXRD data for both materials show a structural transition around $P \sim 5$~GPa. The transport data show a dramatic reduction of the charge gap for Na$_2$IrO$_3$. References: [1] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev.Lett. {\bf 105}, 027204 (2010). [2] Y. Singh and P. Gegenwart, Phys. Rev. B {\bf 82}, 064412 (2010). [3] Y. Singh et al., Phys. Rev. Lett. {\bf 108}, 127203 (2012). [3] F. Ye et al., Phys. Rev. B {\bf 85}, 180403 (2012) [5] S. H. Chun et al. Nature Phys. {\bf 11}, 462 (2015).   

Order by disorder in Kitaev-Heisenberg models on the honeycomb lattice

Recent diffuse magnetic x-ray scattering data in Na2IrO3 [1] clearly determined the spin orientation in this zigzag state and showed that, unexpectedly, it is along  the 44.3 degrees direction  with respect to a axis, which is approximately half way in between the cubic x and y axes.  This experiment provides an important check of the validity of any  model proposed to described the  magnetic properties  of Na2IrO3 as the model should correctly predict not only the type of the magnetic order but also  its orientation in space. We propose that order by disorder mechanism in quantum J1-K1-J2-K2-J3 model [2] gives the experimentally observed direction along cubic face diagonals. Our findings are based on both the calculation of the contribution  of thermal fluctuations of quantum spins into free energy   obtained by Hubbard-Stratonovich transformation and  the zero-point correction to the ground state energy due to quantum spin fluctuations obtained by the spin-wave expansion at zero temperature. [1] S. H. Chun et al, Nature Physics10,1038 (2015).? [2] Y. Sizyuk, C. Price, P. Woelfle, and N. B. Perkins,? Phys. Rev. B  90, 155126 (2014).