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Topological and magnetic phase transitions in the bilayer Kitaev-Ising model
We investigate the phase diagram of a bilayer Kitaev honeycomb model with Ising interlayer interactions, deriving effective models via perturbation theory and performing Majorana mean-field theory calculations. We show that a diverse array of magnetic and topological phase transitions occur, depending on the direction of the interlayer Ising interaction and the relative sign of Kitaev interactions. When two layers have the same sign of the Kitaev interaction, a first-order transition from a Kitaev spin liquid to a magnetically ordered state takes place. The magnetic order points along the Ising axis and it is (anti)ferromagnetic for (anti)ferromagnetic Kitaev interactions. However, when two layers have opposite signs of the Kitaev interaction, we observe a notable weakening of magnetic ordering tendencies and the Kitaev spin liquid survives up to a remarkably larger interlayer exchange. Our mean-field analysis suggests the emergence of an intermediate gapped ℤ 2 spin-liquid state, which eventually becomes unstable upon vison condensation. The confined phase is described by a highly frustrated 120° compass model. We furthermore use perturbation theory to study the model with the Ising axis pointing along the $\hat{z}$ axis or lying in the 𝑥𝑦 plane. In both cases, our analysis reveals the formation of one-dimensional Ising chains, which remain decoupled in perturbation theory, resulting in a subextensive ground-state degeneracy. Our results highlight the interplay between topological order and magnetic ordering tendencies in bilayer quantum spin liquids.
Theoretical study of the crystal and electronic properties of α - RuI 3
The material α-RuCl 3 , with a two-dimensional Ru honeycomb sublattice, has attracted considerable attention because it may be a realization of the Kitaev quantum spin liquid. Recently, a new honeycomb material, α-RuCl 3 , was prepared under moderately high pressure, and it is stable under ambient conditions. However, different from α-RuCl 3 , α-RuI 3 was reported to be a paramagnetic metal without long-range magnetic order down to 0.35 K. Here, the structural and electronic properties of the quasi-two-dimensional α-RuI 3 are theoretically studied. First, based on first-principles density functional theory calculations, the ABC stacking honeycomb-layer R$\bar{3}$ (No. 148) structure is found to be the most likely stacking order for α-RuI 3 along the c axis. Furthermore, both R$\bar{3}$ and P$\bar{3}$1c are dynamically stable because no imaginary frequency modes were obtained in the phononic dispersion spectrum without Hubbard U. Moreover, the different physical behavior of α-RuI 3 compared to α-RuCl 3 can be understood naturally. The strong hybridization between Ru 4d and I 5p orbitals decreases the “effective” atomic Hubbard repulsion, leading the electrons of RuI3 to be less localized than in RuCl 3 . As a consequence, the effective electronic correlation is reduced from Cl to I, leading to the metallic nature of α-RuI 3 . Based on the DFT+U (Ueff=2 eV) plus spin-orbital coupling, we obtained a spin-orbit Mott insulating behavior for α-RuCl 3 and, with the same procedure, a metallic behavior for α-RuI 3 , in good agreement with experimental results. Furthermore, when introducing large (unrealistic) U eff =6 eV, the spin-orbit Mott gap opens in α-RuI 3 as well, supporting the physical picture we are proposing. Our results provide guidance to experimentalists and theorists working on two-dimensional transition metal tri-iodide layered materials.
First demonstration of tuning between the Kitaev and Ising limits in a honeycomb lattice
Recent observations of novel spin-orbit coupled states have generated interest in 4d/5d transition metal systems. A prime example is the J eff = $\frac{1}{2}$ state in iridate materials and α-RuCl 3 that drives Kitaev interactions. Here, by tuning the competition between spin-orbit interaction (λ SOC ) and trigonal crystal field (Δ T ), we restructure the spin-orbital wave functions into a previously unobserved μ = $\frac{1}{2}$ state that drives Ising interactions. This is done via a topochemical reaction that converts Li 2 RhO 3 to Ag 3 LiRh 2 O 6 . Using perturbation theory, we present an explicit expression for the μ = $\frac{1}{2}$ state in the limit Δ T ≫ λ SOC realized in Ag 3 LiRh 2 O 6 , different from the conventional J eff = $\frac{1}{2}$ state in the limit λ SOC ≫ Δ T realized in Li 2 RhO 3 . The change of ground state is followed by a marked change of magnetism from a 6 K spin-glass in Li 2 RhO 3 to a 94 K antiferromagnet in Ag 3 LiRh 2 O 6 .
Pressure tuning of Kitaev spin liquid candidate Na 3 Co 2 SbO 6
The search for Kitaev’s quantum spin liquid in real materials has recently expanded with the prediction that honeycomb lattices of divalent, high-spin cobalt ions could host the dominant bond-dependent exchange interactions required to stabilize the elusive entangled quantum state. The layered honeycomb Na 3 Co 2 SbO 6 has been singled out as a leading candidate provided that the trigonal crystal field acting on Co 3d orbitals, which enhances non-Kitaev exchange interactions between $J$ eff = $\frac{1}{2}$ spin-orbital pseudospins, is reduced. Here we show that applied pressure leads to anisotropic compression of the layered structure, significantly reducing the trigonal distortion of CoO 6 octahedra. Ferromagnetic correlations between pseudospins are enhanced in the spin-polarized (3 Tesla) phase up to about 60 GPa. Higher pressures drive a high-spin to low-spin transition destroying the $J$ eff = $\frac{1}{2}$ moments required to map the spin Hamiltonian into Kitaev’s model. The spin transition strongly suppresses the low-temperature magnetic susceptibility and appears to stabilize a paramagnetic phase driven by frustration. The possible emergence of frustrated magnetism of localized $S$ = $\frac{1}{2}$ moments opens the door for exploration of novel magnetic quantum states in compressed honeycomb lattices of divalent cobaltates.
Crystal structure reconstruction in the surface monolayer of the quantum spin liquid candidate α-RuCl3
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Analytic calculation of the vison gap in the Kitaev spin liquid
Although the ground-state energy of the Kitaev spin liquid can be calculated exactly, the associated vison gap energy has to date only been calculated numerically from finite size diagonalization. Here we show that the phase shift for scattering Majorana fermions off a single bond-flip can be calculated analytically, leading to a closed-form expression for the vison gap energy Δ = 0.2633 J . Generalizations of our approach can be applied to Kitaev spin liquids on more complex lattices such as the three dimensional hyper-octagonal lattice.
Disorder-Enriched Magnetic Excitations in a Heisenberg-Kitaev Quantum Magnet Na 2 Co 2 TeO 6
Using optical magnetospectroscopy, we investigate the magnetic excitations of Na 2 Co 2 TeO 6 in a broad magnetic field range (0 T ≤ B ≤ 17.5 T) at low temperature. Our measurements reveal rich spectra of in-plane magnetic excitations with a surprisingly large number of modes, even in the high-field spin-polarized state. Theoretical calculations find that the Na-occupation disorder in Na 2 Co 2 TeO 6 plays a crucial role in generating these modes. Our Letter demonstrates the necessity to consider disorder in the spin environment in the search for Kitaev quantum spin liquid states in practicable materials.
Out-of-plane magnetic phase diagram of the Kitaev quantum magnet Na 2 Co 2 TeO 6
We have investigated the magnetic properties and mapped out the phase diagram of the honeycomb magnet Na 2 Co 2 TeO 6 with Co 3d 7 in out-of-plane magnetic fields. This material has previously been proposed to show nearest-neighbor Kitaev interactions between Co spins and maybe even Kitaev quantum spin liquid behavior in high fields. At 0.01 T, we observe a thermal phase transition at T N = 27K, transitioning from a paramagnetic state to a canonical ferrimagnetic state. Upon increasing the magnetic field, a spin-floplike phase transition occurred before saturation of J = 1/2 between 10 K and T N . Below 10 K, a peak-dip-peak structure emerges between 10 and 17 T in the magnetic susceptibility (dM/dH) before the magnetic saturation, reminiscent of magnetization plateau behavior. The measurement of the magnetocaloric effect also shows dip-peak-dip behavior in this field range. Our data can be explained by an XXZ model with a single ion anisotropy and possibly small Kitaev and Γ exchange interactions. Finally, we also determined the magnetization saturation field that helps constrain the energy scale of the exchange interactions.
The range of non-Kitaev terms and fractional particles in α-RuCl 3
Significant efforts have focused on the magnetic excitations of relativistic Mott insulators, predicted to realize the Kitaev quantum spin liquid (QSL). This exactly solvable model involves a highly entangled state resulting from bond-dependent Ising interactions that produce excitations which are non-local in terms of spin flips. A key challenge in real materials is identifying the relative size of the non-Kitaev terms and their role in the emergence or suppression of fractional excitations. Here, we identify the energy and temperature boundaries of non-Kitaev interactions by direct comparison of the Raman susceptibility of α-RuCl 3 with quantum Monte Carlo (QMC) results for the Kitaev QSL. Moreover, we further confirm the fractional nature of the magnetic excitations, which is given by creating a pair of fermionic quasiparticles. Interestingly, this fermionic response remains valid in the non-Kitaev range. Our results and focus on the use of the Raman susceptibility provide a stringent new test for future theoretical and experimental studies of QSLs.
Pressure tuning of Kitaev spin liquid candidate Na$_3$Co$_2$SbO$_6$goo
The search for Kitaev's quantum spin liquid (KQSL) state in real materials has recently expanded with the prediction that honeycomb lattices of divalent, high-spin cobalt ions could host the dominant bond-dependent exchange interactions required to stabilize the elusive entangled quantum state. The layered honeycomb Na$_3$Co$_2$SbO$_6$ has been singled out as a leading candidate provided that the trigonal crystal field acting on Co $3d$ orbitals, which enhances non-Kitaev exchange interactions between $J_{\rm eff}=\frac{1}{2}$ spin-orbital pseudospins, is reduced. We find that applied pressure leads to anisotropic compression of the layered structure, significantly reducing the trigonal distortion of CoO$_6$ octahedra. A strong enhancement of ferromagnetic correlations between pseudospins is observed in the spin-polarized (3 Tesla) phase up to about 60 GPa. Higher pressures drive a spin transition into a low-spin state destroying the $J_{\rm eff}=\frac{1}{2}$ local moments required to map the spin Hamiltonian into Kitaev's model. The spin transition strongly suppresses the low-temperature magnetic susceptibility and appears to stabilize a paramagnetic phase driven by frustration. Although applied pressure fails to realize a KQSL state, the possible emergence of frustrated magnetism of localized, low-spin $S=\frac{1}{2}$ moments opens the door for exploration of novel magnetic quantum states in compressed honeycomb lattices of divalent cobaltates.
Introducing the monoclinic polymorph of the honeycomb magnet Na 2 Co 2 TeO 6
Recent theoretical studies have suggested that the low-energy Hamiltonian of honeycomb cobaltate systems could be dominated by anisotropic Kitaev interactions. Motivated by the theory, a honeycomb layered material Na 2 Co 2 TeO 6 with a hexagonal unit cell has been studied and found to exhibit antiferromagnetic (AFM) ordering at 27 K with two spin reorientation transitions at 15 and 5 K. Here we report a monoclinic polymorph of Na 2 Co 2 TeO 6 , also with honeycomb layered structure but with a single AFM transition at 9.6 K and without spin reorientation transitions at lower temperatures. Here, using neutron diffraction, we identify an in-plane zigzag AFM order in the ground state with the spins canted out of the honeycomb planes and ferromagnetically coupled between them. The zigzag order is suppressed by a magnetic field of 6T. The lower critical temperature and field, positive Curie-Weiss temperature, and out-of-plane canting of spins in the monoclinic Na 2 Co 2 TeO 6 suggest enhanced frustration in this polymorph compared to the hexagonal one.
Impact of disorder on dynamics and ordering in the honeycomb-lattice iridate Na 2 IrO 3
Kitaev's honeycomb spin-liquid model and its proposed realization in materials such as α-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. Finally, 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.
Structural Distortions and Short‐Range Magnetism in a Honeycomb Iridate Cu 3 ZnIr 2 O 6
Layered honeycomb iridates receive significant attention in the materials chemistry and physics fields due to the relevance of their crystal structures to the Kitaev model of a quantum spin liquid (QSL). In quest of liquid‐like magnetic ground state signatures, first‐generation alkali metal iridates A 2 IrO 3 ≡ A 3 [AIr 2 ]O 6 (A = Li, Na) and second‐generation iridates T 3 [AIr 2 ]O 6 ( T = Cu, Ag, H) are developed. T 3 [AIr 2 ]O 6 is synthesized from A 3 [AIr 2 ]O 6 via metathesis reactions replacing alkali ions located between honeycomb layers. Herein, the next level of chemical and structural complexity is introduced by synthesizing the honeycomb iridate, Cu 3 ZnIr 2 O 6 , in which alkali ions between and within the honeycomb layers are both selectively exchanged with two different transition metals. Analysis of powder X‐Ray diffraction data reveals corrugation of the honeycomb layers in Cu 3 ZnIr 2 O 6 that hinders complete magnetic frustration and results in a spin glass behavior observed from magnetization and specific heat data. Thus, Cu 3 ZnIr 2 O 6 represents yet another model, which broadens understanding of intricate relationships between intralayer distortions and magnetism of prospective Kitaev QSL compounds.
Superconductivity from Luttinger surfaces: Emergent Sachdev-Ye-Kitaev physics with infinite-body interactions
The pairing of two electrons on a Fermi surface due to an infinitesimal attraction between them always results in a superconducting instability at zero temperature (T = 0). The equivalent question of pairing instability on a Luttinger surface (LS)—190a contour of zeros of the propagator—instead leads to a quantum critical point (QCP) that separates a non-Fermi liquid and superconductor. A surprising and little understood aspect of pair fluctuations at this QCP is that their thermodynamics maps to that of the Sachdev-Ye-Kitaev (SYK) model in the strong coupling limit. Here, we offer a simple justification for this mapping by demonstrating that (i) LS models share the reparametrization symmetry of the q → ∞ SYK model with q-body interactions close to the LS, and (ii) the enforcement of gauge invariance results in a $\frac{1}{√τ}$ (τ ~ T -1 ) behavior of the fluctuation propagator near the QCP, as is the feature of SYK conformal Green functions, but leaves the overall form of the free energy map robust.
Impurity-induced moment freezing in NaFe x Ru 1–x O 2
We report the impact of magnetic impurity substitution on the quantum disordered magnetic ground state of NaRuO 2 . Local S = 5/2 moments are introduced into the frustrated triangular lattice of J eff = 1/2 Ru moments via Fe substitution in NaFe x Ru 1–x O 2 , and the evolution of the magnetic ground state is reported. Local spin freezing associated with conventional spin-glass behavior is observed upon Fe substitution, marking an impurity-induced freezing of the primarily dynamic magnetic ground state in NaRuO 2 . Furthermore, local Fe moments induce a Curie-Weiss magnetic behavior in the uniform magnetic susceptibility, and the local moment magnitude is best described by dynamic Ru moments polarized about impurity sites. Furthermore, our results establish an impurity-doping phenomenology consistent with inherently dynamic moments in NaRuO 2 that are pinned by local magnetic impurities, similar to “swiss cheese” models of impurity-substituted copper oxides.
Quantum Spin Dynamics Due to Strong Kitaev Interactions in the Triangular-Lattice Antiferromagnet CsCeSe 2
The extraordinary properties of the Kitaev model have motivated an intense search for new physics in materials that combine geometrical and bond frustration. In this Letter, we employ inelastic neutron scattering, spin wave theory, and exact diagonalization to study the spin dynamics in the perfect triangular-lattice antiferromagnet (TLAF) CsCeSe 2 . This material orders into a stripe phase, which is demonstrated to arise as a consequence of the off-diagonal bond-dependent terms in the spin Hamiltonian. By studying the spin dynamics at intermediate fields, we identify an interaction between the single-magnon state and the two-magnon continuum that causes decay of coherent magnon excitations, level repulsion, and transfer of spectral weight to the continuum that are controlled by the strength of the magnetic field. Our results provide a microscopic mechanism for the stabilization of the stripe phase in TLAF and show how complex many-body physics can be present in the spin dynamics in a magnet with strong Kitaev coupling even in an ordered ground state. Published by the American Physical Society 2024