Kitaev Honeycomb Research Articles

Dynamics of a Two-Dimensional Quantum Spin-Orbital Liquid: Spectroscopic Signatures of Fermionic Magnons.

We provide an exact study of dynamical correlations for the quantum spin-orbital liquid phases of an SU(2)-symmetric Kitaev honeycomb lattice model. We show that the spin dynamics in this Kugel-Khomskii type model is exactly the density-density correlation function of S=1 fermionic magnons, which could be probed in resonant inelastic x-ray scattering experiments. We predict the characteristic signatures of spin-orbital fractionalization in inelastic scattering experiments and compare them to the ones of the spin-anisotropic Kitaev honeycomb spin liquid. In particular, the resonant inelastic x-ray scattering response shows a characteristic momentum dependence directly related to the dispersion of fermionic excitations. The neutron scattering cross section displays a mixed response of fermionic magnons as well as spin-orbital excitations. The latter has a bandwidth of broad excitations and a vison gap that is three times larger than that of the spin-1=2 Kitaev model.

Raman Spectroscopic Probe of the Magnetic Specific Heat in Quantum Magnets

Quasielastic Raman scattering arising from spin energy fluctuations is used to extract the magnetic specific heat. To test the applicability of a Raman scattering technique for probing magnetic specific heat, we choose three different classes of quantum magnets: (i) a quasi-1D s = 1/2 antiferromagnetic chain CuSe2O5, (ii) a 3D Kitaev honeycomb magnet γ-Li2IrO3, and (iii) a coupled-spin tetrahedral system Cu2Te2O5Cl2. Despite distinct dimensionality, spin topology, and spin-exchange type, the three materials show a pronounced quasielastic response commonly, enabling the determination of the magnetic contributions to the specific heat. Our work demonstrates that a Raman spectroscopic method is useful, complementing the conventional thermodynamic method, in deriving the magnetic specific heat in quantum magnets.

Field stability of Majorana spin liquids in antiferromagnetic Kitaev models

Magnetic fields can give rise to a plethora of phenomena in Kitaev spin systems, such as the formation of non-trivial spin liquids in two and three spatial dimensions. For the original honeycomb Kitaev model, it has recently been observed that the sign of the bond-directional exchange is of crucial relevance for the field-induced physics, with antiferromagnetic couplings giving rise to an intermediate spin liquid regime between the low-field gapped Kitaev spin liquid and the high-field polarized state, which is not present in the ferromagnetically coupled model. Here, by employing a Majorana mean-field approach for a magnetic field pointing along the [001] direction, we present a systematic study of field-induced spin liquid phases for a variety of two and three-dimensional lattice geometries. We find that antiferromagnetic couplings generically lead to (i) spin liquid phases that are considerably more stable in field than those for ferromagnetic couplings, and (ii) an intermediate spin liquid phase which arises from a change in the topology of the Majorana band structure. Close inspection of the mean-field parameters reveal that the intermediate phase occurs due to a field-driven sign change in an 'effective' $z$-bond energy parameter. Our results clearly demonstrate the richness of the Majorana physics of the antiferromagnetic Kitaev models, in comparison to their ferromagnetic counterparts.

Field-driven gapless spin liquid in the spin-1 Kitaev honeycomb model

Recent proposals for spin-1 Kitaev materials, such as honeycomb Ni oxides with heavy elements of Bi and Sb, have shown that these compounds naturally give rise to antiferromagnetic (AFM) Kitaev couplings. Conceptual interest in such AFM Kitaev systems has been sparked by the observation of a transition to a gapless $U(1)$ spin liquid at intermediate field strengths in the AFM spin-1/2 Kitaev model. However, all hitherto known spin-1/2 Kitaev materials exhibit ferromagnetic bond-directional exchanges. Here we discuss the physics of the spin-1 Kitaev model in a magnetic field and show, by extensive numerical analysis, that for AFM couplings it exhibits an extended gapless quantum spin liquid at intermediate field strengths. The close analogy to its spin-1/2 counterpart suggests that this gapless spin liquid is a $U(1)$ spin liquid with a neutral Fermi surface, that gives rise to enhanced thermal transport signatures.

Toward Kitaev's sixteenfold way in a honeycomb lattice model

Kitaev's sixteenfold way is a classification of exotic topological orders in which $\mathbb{Z}_2$ gauge theory is coupled to Majorana fermions of Chern number $C$. The $16$ distinct topological orders within this class, depending on $C \, \mathrm{mod} \, 16$, possess a rich variety of Abelian and non-Abelian anyons. We realize more than half of Kitaev's sixteenfold way, corresponding to Chern numbers $0$, $\pm 1$, $\pm 2$, $\pm 3$, $\pm 4$, and $\pm 8$, in an exactly solvable generalization of the Kitaev honeycomb model. For each topological order, we explicitly identify the anyonic excitations and confirm their topological properties. In doing so, we observe that the interplay between lattice symmetry and anyon permutation symmetry may lead to a "weak supersymmetry" in the anyon spectrum. The topological orders in our honeycomb lattice model could be directly relevant for honeycomb Kitaev materials, such as $\alpha$-RuCl$_3$, and would be distinguishable by their specific quantized values of the thermal Hall conductivity.

Raman Scattering Polarization and Single Spinon Identification in Two-Dimensional Kitaev Quantum Spin Liquids

Perfect deporalization of the Loudon-Fleury inelastic visible-light scattering in the Kitaev honeycomb model is well known. Though it happens in Heisenberg Kagome and triangular antiferromagnets as well, yet we prove it to be of geometric origin rather than peculiar to quantum spin liquids. A Kitaev spin liquid in the square planar geometry indeed exhibits polarized Raman spectra containing defferent symmetry species, each brought by symmetry-compatible spinon geminate excitations, i.e. arising from symmetry-compatible direct-product representations made of double-valued irreducible representations of mediating-spinon-belonging gauged $\bm{k}$-point symmetry groups. We combine a standard point-symmetry-group analysis of the Raman vertex in the real space and an elaborate projective-symmetry-group analysis of Raman-scattering-mediating Majorana spinons in the reciprocal space to identify emergent spinons singly.

Field evolution of low-energy excitations in the hyperhoneycomb magnet β−Li2IrO3

$^7$Li nuclear magnetic resonance (NMR) and terahertz (THz) spectroscopies are used to probe magnetic excitations and their field dependence in the hyperhoneycomb Kitaev magnet $\beta$-Li$_2$IrO$_3$. Spin-lattice relaxation rate ($1/T_1$) measured down to 100\,mK indicates gapless nature of the excitations at low fields (below $H_c\simeq 2.8$\,T), in contrast to the gapped magnon excitations found in the honeycomb Kitaev magnet $\alpha$-RuCl$_3$ at zero applied magnetic field. At higher temperatures in $\beta$-Li$_2$IrO$_3$, $1/T_1$ passes through a broad maximum without any clear anomaly at the N\'eel temperature $T_N\simeq 38$\,K, suggesting the abundance of low-energy excitations that are indeed observed as two peaks in the THz spectra, both correspond to zone-center magnon excitations. At higher fields (above $H_c$), an excitation gap opens, and a re-distribution of the THz spectral weight is observed without any indication of an excitation continuum, in contrast to $\alpha$-RuCl$_3$ where an excitation continuum was reported.

Twisted Kitaev bilayers and the moiré Ising model

In recent years, there have been numerous examples of twisted bilayer systems that host remarkable physical properties that are not found in their untwisted counterparts. Motivated by this, we study the properties of twisted bilayers of the Kitaev honeycomb model in the Abelian spin liquid phase. We show that for strong, short-ranged, interlayer interactions, a super-lattice of non-Abelian defects forms in the twisted bilayer system. These non-Abelian defects are wormhole-like genons that allow anyons from one layer to tunnel to the other layer. We find that when a magnetic field is applied to the system, the low energy dynamics of the twisted bilayer system can be mapped onto four quantum Ising models arising from the degrees of freedom localized on the genon defects. At small twist angles, the Ising models are in a trivial paramagnetic phase, and at large twist angles, they are in a ferromagnetic phase.

Geometric description of the Kitaev honeycomb lattice model

It is widely accepted that topological superconductors can only have an effective interpretation in terms of curved geometry rather than gauge fields due to their charge neutrality. This approach is commonly employed in order to investigate their properties, such as the behaviour of their energy currents. Nevertheless, it is not known how accurately curved geometry can describe actual microscopic models. Here, we demonstrate that the low-energy properties of the Kitaev honeycomb lattice model, a topological superconductor that supports localised Majorana zero modes at its vortex excitations, are faithfully described in terms of Riemann-Cartan geometry. In particular, we show analytically that the continuum limit of the model is given in terms of the Majorana version of the Dirac Hamiltonian coupled to both curvature and torsion. We numerically establish the accuracy of the geometric description for a wide variety of couplings of the microscopic model. Our work opens up the opportunity to accurately predict dynamical properties of the Kitaev model from its effective geometric description.

Magnetic field induced spin liquids in S=1 Kitaev honeycomb model

We investigate the ground state properties of the spin S=1 Kitaev honeycomb model under a magnetic field based on the density matrix renormalization group (DMRG) calculation. With the time reversal symmetry breaking due to the magnetic field, a gapped Kitaev spin liquid is identified for both ferromagnetic (FM) and antiferromagnetic (AFM) Kitaev couplings. The topological nature of such Kitaev spin liquid is manifested by the nearly quantized Wilson loop, degeneracy in the entanglement spectra and existence of edge modes. While the FM Kitaev spin liquid is destroyed by a weaker magnetic field $H_*^\text{FM}$, the AFM one demonstrates a robustness up to an order of magnitude larger critical field $H_*^\text{AFM}$. Moreover, an intermediate nonmagnetic phase appears only for the AFM case at larger fields, $H_*^\text{AFM} < H < H_{**}^\text{AFM}$, before the transition to a high-field polarized paramagnet. The stability of the Kitaev spin liquid against the Heisenberg interactions is also examined. Our findings may further inspire the investigation of recently proposed S=1 Kitaev materials.

Probing the continuum scattering and magnetic collapse in single-crystalline α−Li2IrO3 by Raman spectroscopy

Raman spectroscopy has been applied to investigate the lattice and magnetic excitations of the honeycomb Kitaev material $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{L}{\mathrm{i}}_{2}\mathrm{Ir}{\mathrm{O}}_{3}$. The temperature-dependent Raman spectra give evidence for an unusual magnetic excitation, typified by a broad continuum. The integrated intensity of this Raman continuum does not follow the bosonic statistics expected for the conventional insulating magnets, which indicates the emergence of itinerant Majorana fermions from spin fractionalization in $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{L}{\mathrm{i}}_{2}\mathrm{Ir}{\mathrm{O}}_{3}$. In addition, the Raman results under pressures resolve one critical pressure at ${P}_{c}=4.1\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$ with structural and magnetic phase transitions, signaled by the in-plane Ir-Ir bond dimerization and magnetic collapse, and the system turns into a correlated band insulator above ${P}_{c}$.

Two-magnon bound states in the Kitaev model in a [111] field

It is now well established that the Kitaev honeycomb model in a magnetic field along the $[111]$-direction harbors an intermediate gapless quantum spin liquid (QSL) phase sandwiched between a gapped non-abelian QSL at low fields $H< H_{c1}$ and a partially polarized phase at high fields $H> H_{c2}$. Here, we analyze the low field and high field phases and phase transitions in terms of single- and two-magnon excitations using exact diagonalization (ED) and density matrix renormalization group (DMRG) methods. We find that the energy to create a bound state of two-magnons $\Delta_p$ becomes lower than the energy to create a single spin flip $\Delta_s$ near $H_{c2}$. In the entire Kitaev spin liquid $\Delta_p<\Delta_s$ and both gaps vanish at $H_{c2}$. We make testable predictions for magnon pairing that could be observable in Raman scattering measurements on Kitaev QSL candidate materials.

Magnon bound states versus anyonic Majorana excitations in the Kitaev honeycomb magnet \u03b1-RuCl3

The pure Kitaev honeycomb model harbors a quantum spin liquid in zero magnetic fields, while applying finite magnetic fields induces a topological spin liquid with non-Abelian anyonic excitations. This latter phase has been much sought after in Kitaev candidate materials, such as α-RuCl3. Currently, two competing scenarios exist for the intermediate field phase of this compound (B = 7 − 10 T), based on experimental as well as theoretical results: (i) conventional multiparticle magnetic excitations of integer quantum number vs. (ii) Majorana fermionic excitations of possibly non-Abelian nature with a fractional quantum number. To discriminate between these scenarios a detailed investigation of excitations over a wide field-temperature phase diagram is essential. Here, we present Raman spectroscopic data revealing low-energy quasiparticles emerging out of a continuum of fractionalized excitations at intermediate fields, which are contrasted by conventional spin-wave excitations. The temperature evolution of these quasiparticles suggests the formation of bound states out of fractionalized excitations.

Impact of disorder on dynamics and ordering in the honeycomb-lattice iridate Na2IrO3

Kitaev's honeycomb spin-liquid model and its proposed realization in materials such as $\alpha$-RuCl$_3$, Li$_2$IrO$_3$ and Na$_2$IrO$_3$ continue to present open questions about how the dynamics of a spin-liquid are modified in the presence of non-Kitaev interactions as well as the presence of inhomogeneities. Here we use $^{23}$Na nuclear magnetic resonance to probe both static and dynamical magnetic properties in single crystal Na$_2$IrO$_3$. We find that the NMR shift follows the bulk susceptibility above 30 K but deviates from it below; moreover below $T_N$ the spectra show a broad distribution of internal magnetic fields. Both of these results provide evidence for inequivalent magnetic sites at low temperature, suggesting inhomogeneities are important for the magnetism. The spin-lattice relaxation rate is isotropic and diverges at $T_N$, suggesting that the Kitaev cubic axes may control the critical quantum spin fluctuations. In the ordered state, we observe gapless excitations, which may arise from site substitution, emergent defects from milder disorder, or possibly be associated with nearby quantum paramagnetic states distinct from the Kitaev spin liquid.

Randomly Hopping Majorana Fermions in the Diluted Kitaev System α-Ru_{0.8}Ir_{0.2}Cl_{3}.

dc and ac magnetic susceptibility, magnetization, specific heat, and Raman scattering measurements are combined to probe low-lying spin excitations in α-Ru_{1-x}Ir_{x}Cl_{3} (x≈0.2), which realizes a disordered spin liquid. At intermediate energies (ℏω>3 meV), Raman spectroscopy evidences linearly ω-dependent Majorana-like excitations, obeying Fermi statistics. This points to robustness of a Kitaev paramagnetic state under spin vacancies. At low energies below 3meV, we observe power-law dependences and quantum-critical-like scalings of the thermodynamic quantities, implying the presence of a weakly divergent low-energy density of states. This scaling phenomenology is interpreted in terms of the random hoppings of Majorana fermions. Our results demonstrate an emergent hierarchy of spin excitations in a diluted Kitaev honeycomb system subject to spin vacancies and bond randomness.

Determining topological order from infinite projected entangled pair states

We present a method of extracting information about the topological order from the ground state of a strongly correlated two-dimensional system computed with the infinite projected entangled pair state (iPEPS). For topologically ordered systems, the iPEPS wrapped on a torus becomes a superposition of degenerate, locally indistinguishable ground states. Projectors in the form of infinite matrix product operators (iMPO) onto states with well-defined anyon flux are used to compute topological $S$ and $T$ matrices (encoding mutual- and self-statistics of emergent anyons). The algorithm is shown to be robust against a perturbation driving string-net toric code across a phase transition to a ferromagnetic phase. Our approach provides accurate results near quantum phase transition, where the correlation length is prohibitively large for other numerical methods. Moreover, we used numerically optimized iPEPS describing the ground state of the Kitaev honeycomb model in the toric code phase and obtained topological data in excellent agreement with theoretical prediction.

Correlation effects on the magnetization process of the Kitaev model

By using the variational Monte Carlo method, we study the magnetization process of the Kitaev honeycomb model in a magnetic field. Our trial wavefunction is a generalized Bardeen-Cooper-Schrieffer wave function with the Jastrow correlation factor, which exactly describes the ground state of the Kitaev model at zero magnetic field using the Jordan-Wigner (JW) transformation. We find that two phase transitions occur for the antiferromagnetic Kitaev coupling, while only one phase transition occurs for the ferromagnetic Kitaev coupling. For the antiferromagnetic Kitaev coupling, we also find that the topology of the momentum distribution of the JW fermions changes at the transition point from the Kitaev spin liquid to an intermediate state. Our numerical results indicate that the intermediate state between the Kitaev spin liquid and the fully polarized phases stably exists in the bulk system on two dimensions for the antiferromagnetic Kitaev coupling against many-body correlations.

Topological mechanics from supersymmetry

In topological mechanics, the identification of a mechanical system's rigidity matrix with an electronic tight-binding model allows to infer topological properties of the mechanical system, such as the occurrence of `floppy' boundary modes, from the associated electronic band structure. Here we introduce an approach to systematically construct topological mechanical systems by an exact supersymmetry (SUSY) that relates the bosonic (mechanical) and fermionic (e.g. electronic) degrees of freedom. As examples we discuss mechanical analogues of the Kitaev honeycomb model and of a second-order topological insulator with floppy corner modes. Our SUSY construction naturally defines hitherto unexplored topological invariants for bosonic (mechanical) systems, such as bosonic Wilson loop operators that are formulated in terms of a SUSY-related fermionic Berry curvature.

Ab initio prediction of a two-dimensional variant of the iridate IrO2

We propose an insulating 2D phase of IrO$_{2}$, predicted by $\textit{ab initio}$ evolutionary algorithms. The predicted phase is a van der Waals crystal, in which Ir forms a triangular lattice, and is energetically competitive with the metastable spinel phase, observed experimentally. Electronic structure calculations show that the magnetic properties of this phase are highly nontrivial, with an almost perfect degeneracy of 120$^{\circ}$ $\textit{Neel}$ and $Y$-stripe orders, and unusually soft magnetic moments. The resulting behavior, which we term $\textit{easy plane anisotropy}$, is entirely different from what is realized in previously-explored Kitaev honeycomb lattices. Our results thus suggest that IrO$_{2}$ may be an ideal candidate to realize highly unusual magnetic properties.

Quenching the Kitaev honeycomb model

I studied the non-equilibrium response of an initial Néel state under time evolution with the Kitaev honeycomb model. With isotropic interactions (J_x = J_y = J_zJx=Jy=Jz) the system quickly loses its antiferromagnetic order and crosses over into a steady state valence bond solid, which can be inferred from the long-range dimer correlations. There is no signature of a dynamical phase transition. Upon including anisotropy (J_x = J_y \neq J_zJx=Jy≠Jz), an exponentially long prethermal regime appears with persistent magnetization oscillations whose period derives from an effective toric code.