Off-diagonal Exchange Research Articles

Hidden Plaquette Order in a Classical Spin Liquid Stabilized by Strong Off-Diagonal Exchange.

We report a new classical spin liquid in which the collective flux degrees of freedom break the translation symmetry of the honeycomb lattice. This exotic phase exists in the frustrated spin-orbit magnets where a dominant off-diagonal exchange, the so-called Γ term, results in a macroscopic ground-state degeneracy at the classical level. We demonstrate that the system undergoes a phase transition driven by thermal order by disorder at a critical temperature T_{c}≈0.04|Γ|. This transition reduces the emergent spherical spin symmetry to a cubic one: spins point predominantly toward the cubic axes, yet seem to remain disordered at T<T_{c}. Importantly, we show that the phase transition corresponds to a hidden plaquette ordering of hexagonal fluxes, which explicitly breaks the cubic symmetry, a scenario that is confirmed by our extensive MonteCarlo simulations. We further compute the dynamical structure factors of the spin-liquid phase and reveal unusual dynamical properties of the hexagonal flux parameters.

Impact of off-diagonal exchange interactions on the Kitaev spin-liquid state of α−RuCl3

Motivated by the recent experimental observation of the half-quantized thermal Hall conductivity for the candidate material of the Kitaev spin liquid, $\alpha$-RuCl$_3$[ Y. Kasahara et al., Nature 559, 227 (2018)], we investigate effects of non-Kitaev exchange interactions, which exist in the real material, on the gapped chiral spin liquid state realized in the case with an applied magnetic field. It is found that off-diagonal exchange interactions enhance significantly the mass gap of Majorana fermions. This result provides a possible explanation for robust quantization of the thermal Hall conductivity observed in the above-mentioned experiment. Furthermore, we demonstrate that if the Kitaev spin liquid state and the zigzag antiferromagnetic order coexist around the border of these two phases, off-diagonal exchange interactions can induce Fermi surfaces of Majorana fermions, leading to a state similar to the U(1) spin liquid.

Hierarchy of Exchange Interactions in the Triangular-Lattice Spin Liquid YbMgGaO4

The spin-1/2 triangular lattice antiferromagnet YbMgGaO$_{4}$ has attracted recent attention as a quantum spin-liquid candidate with the possible presence of off-diagonal anisotropic exchange interactions induced by spin-orbit coupling. Whether a quantum spin-liquid is stabilized or not depends on the interplay of various exchange interactions with chemical disorder that is inherent to the layered structure of the compound. We combine time-domain terahertz spectroscopy and inelastic neutron scattering measurements in the field polarized state of YbMgGaO$_{4}$ to obtain better microscopic insights on its exchange interactions. Terahertz spectroscopy in this fashion functions as high-field electron spin resonance and probes the spin-wave excitations at the Brillouin zone center, ideally complementing neutron scattering. A global spin-wave fit to all our spectroscopic data at fields over 4T, informed by the analysis of the terahertz spectroscopy linewidths, yields stringent constraints on $g$-factors and exchange interactions. Our results paint YbMgGaO$_{4}$ as an easy-plane XXZ antiferromagnet with the combined and necessary presence of sub-leading next-nearest neighbor and weak anisotropic off-diagonal nearest-neighbor interactions. Moreover, the obtained $g$-factors are substantially different from previous reports. This works establishes the hierarchy of exchange interactions in YbMgGaO$_{4}$ from high-field data alone and thus strongly constrains possible mechanisms responsible for the observed spin-liquid phenomenology.

Breakdown of Magnetic Order in the Pressurized Kitaev Iridate β-Li_{2}IrO_{3}.

Temperature-pressure phase diagram of the Kitaev hyperhoneycomb iridate β-Li_{2}IrO_{3} is explored using magnetization, thermal expansion, magnetostriction, and muon spin rotation measurements, as well as single-crystal x-ray diffraction under pressure and abinitio calculations. The Néel temperature of β-Li_{2}IrO_{3} increases with the slope of 0.9 K/GPa upon initial compression, but the reduction in the polarization field H_{c} reflects a growing instability of the incommensurate order. At 1.4GPa, the ordered state breaks down upon a first-order transition, giving way to a new ground state marked by the coexistence of dynamically correlated and frozen spins. This partial freezing in the absence of any conspicuous structural defects may indicate the classical nature of the resulting pressure-induced spin liquid, an observation paralleled to the increase in the nearest-neighbor off-diagonal exchange Γ under pressure.

Magnetic structure and excitation spectrum of the hyperhoneycomb Kitaev magnet β−Li2IrO3

We present a theoretical study of the static and dynamical properties of the Kitaev magnet $\beta$-Li$_2$IrO$_3$ based on the minimal $J$-$K$-$\Gamma$ model. We argue that the incommensurate magnetic order of $\beta$-Li$_2$IrO$_3$ can be qualitatively understood in terms of a long-wavelength twisting of a nearby commensurate, period-3 state, that shares the same symmetry, propagation vector, counter-rotation of the two spin sublattices, and non-coplanarity. This period-3 state shows very different structure when either the Kitaev interaction $K$ or the off-diagonal exchange anisotropy $\Gamma$ is dominant, featuring in particular ferromagnetic or antiferromagnetic spin dimers, respectively. We compute the magnon excitation spectra and the dynamical spin structure factor for both structures. The results for the momentum and energy dependence of the neutron-scattering intensity provide distinctive qualitative and quantitative fingerprints of the $K$-dominant or $\Gamma$-dominant states and thus gives additional information on the relative strength of these interactions in $\beta$-Li$_2$IrO$_3$.

Cluster-Glass Phase in Pyrochlore XY Antiferromagnets with Quenched Disorder.

We study the impact of quenched disorder (random exchange couplings or site dilution) on easy-plane pyrochlore antiferromagnets. In the clean system, order by disorder selects a magnetically ordered state from a classically degenerate manifold. In the presence of randomness, however, different orders can be chosen locally depending on details of the disorder configuration. Using a combination of analytical considerations and classical MonteCarlo simulations, we argue that any long-range-ordered magnetic state is destroyed beyond a critical level of randomness where the system breaks into magnetic domains due to random exchange anisotropies, becoming, therefore, a glass of spin clusters, in accordance with the available experimental data. These random anisotropies originate from off-diagonal exchange couplings in the microscopic Hamiltonian, establishing their relevance to other magnets with strong spin-orbit coupling.

Topological superconductivity in the extended Kitaev-Heisenberg model

We study superconducting pairing in the doped Kitaev-Heisenberg model by taking into account the recently proposed symmetric off-diagonal exchange $\Gamma$. By performing a mean-field analysis, we classify all possible superconducting phases in terms of symmetry, explicitly taking into account effects of spin-orbit coupling. Solving the resulting gap equations self-consistently, we map out a phase diagram that involves several topologically nontrivial states. For $\Gamma<0$, we find a competition between a time-reversal symmetry breaking chiral phase with Chern number $\pm1$ and a time-reversal symmetric nematic phase that breaks the rotational symmetry of the lattice. On the other hand, for $\Gamma \geq 0$ we find a time-reversal symmetric phase that preserves all the lattice symmetries, thus yielding clearly distinguishable experimental signatures for all superconducting phases. Both of the time-reversal symmetric phases display a transition to a $\mathbb{Z}_2$ non-trivial phase at high doping levels. Finally, we also include a symmetry-allowed spin-orbit coupling kinetic energy and show that it destroys a tentative symmetry protected topological order at lower doping levels. However, it can be used to tune the time-reversal symmetric phases into a $\mathbb{Z}_2$ non-trivial phase even at lower doping.

Theoretical investigation of magnetic dynamics in α−RuCl3

We study the spin-wave excitations in $\alpha$-RuCl$_3$ by the spin-wave theory. Starting from the five-orbital Hubbard model and the perturbation theory, we derive an effective isospin-$1/2$ model in the large Hubbard ($U$) limit. Based on the energy-band structure calculated from the first-principle method, we find that the effective model can be further reduced to the $K$-$\Gamma$ model containing a ferromagnetic nearest-neighbor (NN) Kitaev interaction ($K$) and a NN off-diagonal exchange interaction ($\Gamma$). With the spin-wave theory, we find that the $K$-$\Gamma$ model can give magnetic excitations which is consistent with the recent neutron scattering experiments.

Magnetization processes of zigzag states on the honeycomb lattice: Identifying spin models for α−RuCl3 and Na2IrO3

We study the field-induced magnetization processes of extended Heisenberg-Kitaev models on the honeycomb lattice, taking into account off-diagonal and longer-range exchange interactions, using a combination of Monte-Carlo simulations, classical energy minimization, and spin-wave theory. We consider a number of different parameter sets, previously proposed to describe the magnetic behavior of $\alpha$-RuCl$_3$ and Na$_2$IrO$_3$ with their antiferromagnetic zigzag ground states. By classifying these parameter sets, we reveal the existence of three distinct mechanisms to stabilize zigzag states, which differ in the sign of the nearest-neighbor Kitaev interaction, the role of longer-range interactions, and the magnitude of the off-diagonal $\Gamma_1$ interaction. While experimentally hardly distinguishable at zero field, we find that the three different scenarios lead to significantly different magnetization processes in applied magnetic fields. In particular, we show that a sizable off-diagonal interaction $\Gamma_1 > 0$ naturally explains the strongly anisotropic field responses observed in $\alpha$-RuCl$_3$ without the need for a strong anisotropy in the effective $g$ tensor. Moreover, for a generic field direction, it leads to a high-field state with a finite transversal magnetization, which should be observable in $\alpha$-RuCl$_3$.

Unveiling magnetic interactions of ruthenium trichloride via constraining direction of orbital moments: Potential routes to realize a quantum spin liquid

Recent experiments reveal that the honeycomb ruthenium trichloride {\alpha}-RuCl3 is a prime candidate of the Kitaev quantum spin liquid (QSL). However, there is no theoretical model which can properly describe its experimental dynamical response, due to the lack of a full understanding of its magnetic interactions. Here, we propose a general scheme to calculate the magnetic interactions in systems (e.g., {\alpha}-RuCl3) with non-negligible orbital moments by constraining the directions of orbital moments. With this scheme, we put forward a minimal J1-K1-{\Gamma}1-J3-K3 model for {\alpha}-RuCl3 and find that: (I) The third nearest neighbor (NN) antiferromagnetic Heisenberg interaction J3 stabilizes the zigzag antiferromagnetic order; (II) The NN symmetric off-diagonal exchange {\Gamma}1 plays a pivotal role in determining the preferred direction of magnetic moments and generating the spin wave gap. Exact diagonalization study on this model shows that the Kitaev QSL can be realized by suppressing the NN symmetric off-diagonal exchange {\Gamma}1 and the third NN Heisenberg interaction J3. Thus, we not only propose a powerful general scheme for investigating the intriguing magnetism of Jeff=1/2 magnets, but also point out future directions for realizing the Kitaev QSL in the honeycomb ruthenium trichloride {\alpha}-RuCl3.

Classical Spin Liquid Instability Driven By Off-Diagonal Exchange in Strong Spin-Orbit Magnets.

We show that the off-diagonal exchange anisotropy drives Mott insulators with strong spin-orbit coupling to a classical spin liquid regime, characterized by an infinite number of ground states and Ising variables living on closed or open strings. Depending on the sign of the anisotropy, quantum fluctuations either fail to lift the degeneracy down to very low temperatures, or select noncoplanar magnetic states with unconventional spin correlations. The results apply to all 2D and 3D tricoordinated materials with bond-directional anisotropy and provide a consistent interpretation of the suppression of the x-ray magnetic circular dichroism signal reported recently for β-Li_{2}IrO_{3} under pressure.

Spin-Wave Excitations Evidencing the Kitaev Interaction in Single Crystalline α-RuCl_{3}.

Kitaev interactions underlying a quantum spin liquid have long been sought, but experimental data from which their strengths can be determined directly, are still lacking. Here, by carrying out inelastic neutron scattering measurements on high-quality single crystals of α-RuCl_{3}, we observe spin-wave spectra with a gap of ∼2 meV around the M point of the two-dimensional Brillouin zone. We derive an effective-spin model in the strong-coupling limit based on energy bands obtained from first-principles calculations, and find that the anisotropic Kitaev interaction K term and the isotropic antiferromagnetic off-diagonal exchange interaction Γ term are significantly larger than the Heisenberg exchange coupling J term. Our experimental data can be well fit using an effective-spin model with K=-6.8 meV and Γ=9.5 meV. These results demonstrate explicitly that Kitaev physics is realized in real materials.

Dynamically Encircling Exceptional Points: Exact Evolution and Polarization State Conversion.

We show that a two-level non-Hermitian Hamiltonian with constant off-diagonal exchange elements can be analyzed exactly when the underlying exceptional point is perfectly encircled in the complex plane. The state evolution of this system is explicitly obtained in terms of an ensuing transfer matrix, even for large encirclements, regardless of adiabatic conditions. Our results clearly explain the direction-dependent nature of this process and why in the adiabatic limit its outcome is dominated by a specific eigenstate-irrespective of initial conditions. Moreover, numerical simulations suggest that this mechanism can still persist in the presence of nonlinear effects. We further show that this robust process can be harnessed to realize an optical omnipolarizer: a configuration that generates a desired polarization output regardless of the input polarization state, while from the opposite direction it always produces the counterpart eigenstate.

Spin-orbit coupledjeff=1/2iridium moments on the geometrically frustrated fcc lattice

Motivated by experiments on the double perovskites ${\mathrm{La}}_{2}{\mathrm{ZnIrO}}_{6}$ and ${\mathrm{La}}_{2}{\mathrm{MgIrO}}_{6}$, we study the magnetism of spin-orbit coupled ${j}_{\mathrm{eff}}=1/2$ iridium moments on the three-dimensional, geometrically frustrated, face-centered cubic lattice. The symmetry-allowed nearest-neighbor interaction includes Heisenberg, Kitaev, and symmetric off-diagonal exchange. A Luttinger-Tisza analysis shows a rich variety of orders, including collinear $A$-type antiferromagnetism, stripe order with moments along the ${111}$ direction, and incommensurate noncoplanar spirals, and we use Monte Carlo simulations to determine their magnetic ordering temperatures. We argue that existing thermodynamic data on these iridates underscores the presence of a dominant Kitaev exchange, and also suggest a resolution to the puzzle of why ${\mathrm{La}}_{2}{\mathrm{ZnIrO}}_{6}$, but not ${\mathrm{La}}_{2}{\mathrm{MgIrO}}_{6}$, exhibits ``weak'' ferromagnetism.

Theory of strain tuning fine structure splitting in self-assembled InAs/GaAs quantum dots

We analytically derive the change in exciton fine structure splitting (FSS) under external stresses in self-assembled InAs/GaAs quantum dots, using the Bir–Pikus model. We find that the FSS change is mainly due to the strain-induced valence band mixing and valence-conduction band coupling. The exciton polarization angle under strain is determined by the argument of the electron-hole off-diagonal exchange integrals. The theory agrees well with the empirical pseudopotential calculations.

Spin-orbit coupling in octamers in the spinel sulfideCuIr2S4: Competition between spin-singlet and quadrupolar states and its relevance to remnant paramagnetism

We theoretically investigate magnetic properties in the low-temperature phase with the formation of eight-site clusters, octamers, in the spinel compound CuIr$_2$S$_4$. The octamer state was considered to be a spin-singlet state induced by a Peieirls instability through the strong anisotropy of $d$ orbitals, the so-called orbital Peierls state. We reexamine this picture by taking into account the spin-orbit coupling which was ignored in the previous study. We derive a low-energy effective model between $j_{\rm eff}=1/2$ quasispins on Ir$^{4+}$ cations in an octamer from the multiorbital Hubbard model with the strong spin-orbit coupling by performing the perturbation expansion from the strong correlation limit. The effective Hamiltonian is in the form of the Kitaev-Heisenberg model but with an additional interaction, a symmetric off-diagonal exchange interaction originating from the perturbation process including both d-d and d-p-d hopping. Analyzing the effective Hamiltonian on two sites and the octamer by the exact diagonalization, we find that there is a competition between a spin-singlet state and a quadrupolar state. The former singlet state is a conventional one, adiabatically connected to the orbital-Peierls state. On the other hand, the latter quadrupolar state is stabilized by the additional interaction, which consists of a linear combination of different total spin momenta along the spin quantization axis. In the competing region, the model exhibits paramagnetic behavior with a renormalized small effective moment at low temperature. This peculiar remnant paramagnetism is not obtained in the Kitaev-Heisenberg model without the additional interaction. Our results renew the picture of the octamer state and provide a scenario for the intrinsic paramagnetic behavior recently observed in a muon spin rotation experiment [K. M. Kojima et al., Phys. Rev. Lett. 112, 087203 (2014)].

Generic Spin Model for the Honeycomb Iridates beyond the Kitaev Limit

Recently, realizations of Kitaev physics have been sought in the A2IrO3 family of honeycomb iridates, originating from oxygen-mediated exchange through edge-shared octahedra. However, for the jeff=1/2 Mott insulator in these materials, exchange from direct d-orbital overlap is relevant, and it was proposed that a Heisenberg term should be added to the Kitaev model. Here, we provide the generic nearest-neighbor spin Hamiltonian when both oxygen-mediated and direct overlap are present, containing a bond-dependent off-diagonal exchange in addition to Heisenberg and Kitaev terms. We analyze this complete model using a combination of classical techniques and exact diagonalization. Near the Kitaev limit, we find new magnetic phases, 120° and incommensurate spiral order, as well as extended regions of zigzag and stripy order. Possible applications to Na2IrO3 and Li2IrO3 are discussed.

Supersolid phase and magnetization plateaus observed in the anisotropic spin-32Heisenberg model on bipartite lattices

We study the spin-3/2 Heisenberg model including easy-plane and exchange anisotropies in one and two dimensions. In the Ising limit, when the off-diagonal exchange interaction J is zero, the phase diagram in magnetic field is characterized by magnetization plateaus that are either translationally invariant or have a two-sublattice order, with phase boundaries that are macroscopically degenerate. Using a site factorized variational wave function and perturbational expansion around the Ising limit, we find that superfluid and supersolid phases emerge between the plateaus for small finite values of J. The variational approach is complemented by a Density Matrix Renormalization Group study of a one-dimensional chain and exact diagonalization calculations on small clusters of a square lattice. The studied model may serve as a minimal model for the layered Ba2CoGe2O7 material compound, and we believe that the vicinity of the uniform 1/3 plateau in the model parameter space can be observed as an anomaly in the measured magnetization curve.

Solute−Solvent Complex Kinetics and Thermodynamics Probed by 2D-IR Vibrational Echo Chemical Exchange Spectroscopy

The formation and dissociation kinetics of a series of triethylsilanol/solvent weakly hydrogen bonding complexes with enthalpies of formation ranging from -1.4 to -3.3 kcal/mol are measured with ultrafast two-dimensional infrared (2D IR) chemical exchange spectroscopy in liquid solutions at room temperature. The correlation between the complex enthalpies of formation and dissociation rate constants can be expressed with an equation similar to the Arrhenius equation. The experimental results are in accord with previous observations on eight phenol/solvent complexes with enthalpies of formation from -0.6 to -2.5 kcal/mol. It was found that the inverse of the solute-solvent complex dissociation rate constant is linearly related to exp(-DeltaH0/RT) where DeltaH0 is the complex enthalpy of formation. It is shown here, that the triethylsilanol-solvent complexes obey the same relationship with the identical proportionality constant, that is, all 13 points, five silanol complexes and eight phenol complexes, fall on the same line. In addition, features of 2D IR chemical exchange spectra at long reaction times (spectral diffusion complete) are explicated using the triethylsilanol systems. It is shown that the off-diagonal chemical exchange peaks have shapes that are a combination (outer product) of the absorption line shapes of the species that give rise to the diagonal peaks.

Effects of domain walls on hole motion in the two-dimensional anisotropict−Jmodel at finite temperature

The t-J model on the square lattice, close to the t-J_z limit, is studied by quantum Monte Carlo techniques at finite temperature and in the underdoped regime. A variant of the Hoshen-Koppelman algorithm was implemented to identify the antiferromagnetic domains on each Trotter slice. The results show that the model presents at high enough temperature finite antiferromagnetic (AF) domains which collapse at lower temperatures into a single ordered AF state. While there are domains, holes would tend to preferentially move along the domain walls. In this case, there are indications of hole pairing starting at a relatively high temperature. At lower temperatures, when the whole system becomes essentially fully AF ordered, at least in finite clusters, holes would likely tend to move within phase separated regions. The crossover between both states moves down in temperature as doping increases and/or as the off-diagonal exchange increases. The possibility of hole motion along AF domain walls at zero temperature in the fully isotropic t-J is discussed.