Off-diagonal Exchange Research Articles

Erratum: Topological phase transition to Abelian anyon phases due to off-diagonal exchange interaction in the Kitaev spin liquid state [Phys. Rev. B 102 , 174414 (2020)

Received 6 March 2023DOI:https://doi.org/10.1103/PhysRevB.107.139902©2023 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAnyonsQuantum spin liquidTopological materialsPhysical SystemsHoneycomb latticeTechniquesKitaev modelNon-Abelian modelsCondensed Matter, Materials & Applied Physics

Magnetic anisotropy reversal driven by structural symmetry-breaking in monolayer α-RuCl3.

Layered α-RuCl3 is a promising material to potentially realize the long-sought Kitaev quantum spin liquid with fractionalized excitations. While evidence of this state has been reported under a modest in-plane magnetic field, such behaviour is largely inconsistent with theoretical expectations of spin liquid phases emerging only in out-of-plane fields. These predicted field-induced states have been largely out of reach due to the strong easy-plane anisotropy of bulk crystals, however. We use a combination of tunnelling spectroscopy, magnetotransport, electron diffraction and ab initio calculations to study the layer-dependent magnons, magnetic anisotropy, structure and exchange coupling in atomically thin samples. Due to picoscale distortions, the sign of the average off-diagonal exchange changes in monolayer α-RuCl3, leading to a reversal of spin anisotropy to easy-axis anisotropy, while the Kitaev interaction is concomitantly enhanced. Our work opens the door to the possible exploration of Kitaev physics in the true two-dimensional limit.

Dynamics of Visons and Thermal Hall Effect in Perturbed Kitaev Models

A vison is an excitation of the Kitaev spin liquid which carries a $\mathbb Z_2$ gauge flux. While immobile in the pure Kitaev model, it becomes a dynamical degree of freedom in the presence of perturbations. We study an isolated vison in the isotropic Kitaev model perturbed by a small external magnetic field $h$, an offdiagonal exchange interactions $\Gamma$ and a Heisenberg coupling $J$. In the ferromagnetic Kitaev model, the dressed vison obtains a dispersion linear in $\Gamma$ and $h$ and a fully universal low-$T$ mobility, $\mu=6 v_m^2/T^{2}$, where $v_m$ is the velocity of Majorana fermions. In contrast, in the antiferromagnetic Kitaev model interference effects suppress the coherent propagation and an incoherent Majorana-assisted hopping leads to a $T$-independent mobility. The motion of a single vison due to Heisenberg interactions is strongly suppressed for both signs of the Kitaev coupling. Vison bands in the antiferromagnetic Kitaev models can be topological and may lead to a characteristic features in thermal Hall effects in Kitaev materials.

Large-gap quantum anomalous Hall states induced by functionalizing buckled Bi-III monolayer/ Al2O3

Chiral edge modes inherent to the topological quantum anomalous Hall (QAH) effect are a pivotal topic of contemporary condensed matter research aiming at future quantum technology and application in spintronics. A large topological gap is vital to protecting against thermal fluctuations and thus enabling a higher operating temperature. From first-principle calculations, we propose Al$_{2}$O$_{3}$ as an ideal substrate for atomic monolayers consisting of Bi and group-III elements, in which a large-gap quantum spin Hall effect can be realized. Additional half-passivation with nitrogen then suggests a topological phase transition to a large-gap QAH insulator. By effective tight-binding modelling, we demonstrate that Bi-III monolayer/Al$_{2}$O$_{3}$ is dominated by $p_{x}, p_{y}$ orbitals, with subdominant $p_z$ orbital contributions. The topological phase transition into the QAH is induced by Zeeman splitting, where the off-diagonal spin exchange does not play a significant role. The effective model analysis promises utility far beyond Bi-III monolayer/Al$_{2}$O$_{3}$, as it should generically apply to systems dominated by $p_{x}, p_{y}$ orbitals with a band inversion at $\Gamma$.

Anisotropic exchange coupling and ground state phase diagram of Kitaev compound YbOCl

Rare-earth chalcohalide REChX (RE = rare earth; Ch = O, S, Se, Te; X = F, Cl, Br, I) is a newly reported family of Kitaev spin liquid candidates. The family offers a platform where a strong spin-orbit coupling meets a van der Waals layered and undistorted honeycomb spin lattice, which outputs highly anisotropic exchange couplings required by the Kitaev model. YbOCl is the first single crystal of the family we grew, with a size up to ~ 15 mm. We have performed magnetization and high magnetic field electron spin resonance measurements from 2 to 300 K. We develop the mean-field scenario for the anisotropic spin system, with which we are able to well describe the experiments and reliably determine the fundamental parameters. The self-consistent simulations give the anisotropic spin-exchange interactions of $J_{\pm}$ (~ -0.3 K) and $J_{zz}$ (~ 1.6 K), and g factors of $g_{ab}$ (~ 3.4) and $g_{c}$ (~ 2.9). Based on the spin-exchange interactions, we employ the exact diagonalization method to work out the ground state phase diagram of YbOCl in terms of the off-diagonal exchange couplings. The phase diagram hosting rich magnetic phases including the spin-disordered one, sheds light on the novel magnetic properties of the family, particularly the Kitaev physics.

Signatures of non-Loudon-Fleury Raman scattering in the Kitaev magnet β−Li2IrO3

We investigate the magnetic excitations of the hyperhoneycomb Kitaev magnet $\beta$-$\text{Li}_2\text{IrO}_3$ by means of inelastic Raman scattering. The spectra exhibits a coexistence of a broad scattering continuum and two sharp low-energy peaks at 2.5 meV and 3 meV, with a distinctive polarization dependence. While the continuum is suggestive of fractional quasi-particles emerging from a proximate quantum spin liquid phase, the sharp peaks provide the first experimental signature of the `non-Loudon-Fleury' one-magnon scattering processes proposed recently [Phys. Rev. B 104, 144412 (2021)]. The corresponding microscopic mechanism is similar to the one leading to the symmetric off-diagonal exchange interaction $\Gamma$ (as it involves a combination of both direct and ligand-mediated exchange paths), but is otherwise completely unexpected within the traditional Loudon-Fleury theory of Raman scattering. The present experimental verification therefore calls for a drastic reevaluation of Raman scattering in similar systems with strong spin orbit coupling and multiple exchange paths.

Thermodynamics and correlations featured quantum criticality of transverse-field Ising model with off-diagonal exchange interactions

We study the quantum phase transitions (QPTs) and quantum criticality of transverse-field Ising model with off-diagonal exchange interactions. In the absence of magnetic field, the clockwise spiral (CWS), counterclockwise spiral (CCWS) and longitudinal ferromagnetic (xFM) phases are revealed by the spin chirality vector κz and longitudinal spin–spin correlation Cxx mutually. The general QPT is diagnosed by the temperature power-law divergence of Grüneisen ratio (GR) Γ∼±T−1. The critical scaling is further done to capture the quantum critical point (QCP) by analyzing κz and specific heat, which offers a direct access to distill the critical exponents (δ, β, α) that fulfill the Essam–Fisher relation α+β(1+δ)=2, providing a new clue to detect the general QPT. Subsequently, some scaling hypothesis equations are proposed to check the scaling analysis. In a magnetic field, an additional field-induced transverse ferromagnetic (zFM) phase is unveiled. Specifically, for the xFM phase transition into zFM one, the transverse-field Ising terms predominates, which approximately keeps the Z2 symmetry at hc=1, giving rise to the self-dual quantum criticality, demonstrated by the constant of GR irrelevant of temperature. Furthermore, the gapped and gapless low-lying excitations of quantum phases are manifested by the thermal Drude weight and specific heat.

Magnon condensation in dimerized antiferromagnets with spin-orbit coupling

Bose-Einstein condensation (BEC) of triplet excitations triggered by a magnetic field, sometimes called magnon BEC, in dimerized antiferromagnets gives rise to a long-range antiferromagnetic order in the plane perpendicular to the applied magnetic field. To explore the effects of spin-orbit coupling on magnon condensation, we study a spin model on a distorted honeycomb lattice with dimerized Heisenberg exchange ($J$ terms) and uniform off-diagonal exchange ($\Gamma'$ terms) interactions. Via variational Monte Carlo calculations and spin-wave theory, we find that an out-of-plane magnetic field can induce different types of long-range magnetic orders, no matter if the ground state is a non-magnetic dimerized state or an ordered N\'{e}el state. Furthermore, the critical properties of field-driven phase transitions in the presence of spin-orbit couplings, as illustrated from spin-wave spectrum and interpreted by effective field theory, can be different from the conventional magnon BEC. Our study is helpful to understand the rich phases of spin-orbit coupled antiferromagnets induced by magnetic fields.

Metastable Kitaev Magnets.

Nearly two decades ago, Alexei Kitaev proposed a model for spin- particles with bond-directional interactions on a two-dimensional honeycomb lattice which had the potential to host a quantum spin-liquid ground state. This work initiated numerous investigations to design and synthesize materials that would physically realize the Kitaev Hamiltonian. The first generation of such materials, such as NaIrO, -LiIrO, and -RuCl, revealed the presence of non-Kitaev interactions such as the Heisenberg and off-diagonal exchange. Both physical pressure and chemical doping were used to tune the relative strength of the Kitaev and competing interactions; however, little progress was made towards achieving a purely Kitaev system. Here, we review the recent breakthrough in modifying Kitaev magnets via topochemical methods that has led to the second generation of Kitaev materials. We show how structural modifications due to the topotactic exchange reactions can alter the magnetic interactions in favor of a quantum spin-liquid phase.

Thermodynamics and Correlations Featured Quantum Criticality of Transverse-Field Ising Model with Off-Diagonal Exchange Interactions

Thermodynamics and Correlations Featured Quantum Criticality of Transverse-Field Ising Model with Off-Diagonal Exchange Interactions

Counterdiabatic route for preparation of state with long-range topological order

We propose here a counterdiabatic (CD) strategy for fast preparation of a state with long-range topological order by magnetic field tuning of an initial separable state. For concreteness, we consider the ground state of the honeycomb Kitaev model whose long-range topological order together with the anyonic excitations make it an interesting candidate for fault-tolerant universal quantum computation and storage. The required CD perturbation is found to be local, having the form of the off-diagonal exchange interactions reminiscent of trigonal deformations in Kitaev Hamiltonians. We show that the counterdiabatically produced state can have high fidelity and retain numerous desired entanglement properties.

Quantum criticality and correlations in the Ising-Gamma chain

In this paper, we study quantum phase transitions (QPTs) and critical phenomena of the one-dimensional Ising-Gamma model, which describes the competition between the Ising interactions and the off-diagonal Gamma exchange interactions. The Hamiltonian is analytically solved in virtue of Jordan–Wigner, Fourier and Bogoliubov transformations. The phase diagram consists of three phases, including antiferromagnetic phase, paramagnetic phase and gapless spiral phase. A detailed study of information measures including the Wigner–Yanase skew information (WYSI) and quantum fidelity susceptibility (QFS) is conducted. The second-order QPTs are witnessed both by WYSI and QFS. When the QPT occurs between distinct gapped phases, a logarithmic scaling behavior of local measures and a superextensive behavior of fidelity susceptibility are identified, while they exhibit size-dependent behaviors for the gapped-to-gapless transitions. The correlation-length exponent extracted from various order parameters agrees well with each other.

Kitaev interactions in the Co honeycomb antiferromagnets Na3Co2SbO6 and Na2Co2TeO6

${\mathrm{Co}}^{2+}$ ions in an octahedral crystal field stabilize a ${j}_{\mathrm{eff}}=1/2$ ground state with an orbital degree of freedom and have been recently put forward for realizing Kitaev interactions, a prediction we have tested by investigating spin dynamics in two cobalt honeycomb lattice compounds, ${\mathrm{Na}}_{2}{\mathrm{Co}}_{2}{\mathrm{TeO}}_{6}$ and ${\mathrm{Na}}_{3}{\mathrm{Co}}_{2}{\mathrm{SbO}}_{6}$, using inelastic neutron scattering. We used linear spin wave theory to show that the magnetic spectra can be reproduced with a spin Hamiltonian including a dominant Kitaev nearest-neighbor interaction, weaker Heisenberg interactions up to the third neighbor, and bond-dependent off-diagonal exchange interactions. Beyond the Kitaev interaction that alone would induce a quantum spin liquid state, the presence of these additional couplings is responsible for the zigzag-type long-range magnetic ordering observed at low temperature in both compounds. These results provide evidence for the realization of Kitaev-type coupling in cobalt-based materials, despite hosting a weaker spin-orbit coupling than their $4d$ and $5d$ counterparts.

Topological phase transition to Abelian anyon phases due to off-diagonal exchange interaction in the Kitaev spin liquid state

We investigate how the Kitaev spin liquid state described by Majorana fermions coupled with $Z_2$ gauge fields is affected by non-Kitaev interactions which exist in real candidate materials of the Kitaev magnet. It is found that the off-diagonal exchange interaction referred to as the $\Gamma'$ term dramatically changes the Majorana band structure in the case of the antiferromagnetic Kitaev interaction, and gives rise to a topological phase transition from a non-Abelian topological phase with the Chern number equal to $\pm 1$ to an Abelian phase with the Chern number equal to $\pm 2$, in which the $Z_{2}$ vortices behave as Abelian anyons. On the other hand, other non-Kitaev interactions such as the Heisenberg interaction and the $\Gamma$ term, only affect the bandwidth of the Majorana band as long as the spin liquid state is not destabilized.

Lifshitz phase transitions in a one-dimensional Gamma model.

In this paper, we study quantum phase transitions and magnetic properties of a one-dimensional spin-1/2 Gamma model, which describes the off-diagonal exchange interactions between edge-shared octahedra with strong spin-orbit couplings along the sawtooth chain. The competing exchange interactions between the nearest neighbors and the second neighbors stabilize the semimetallic ground state in terms of spinless fermions, and give rise to a rich phase diagram, which consists of three gapless phases. We find distinct phases are characterized by the number of Weyl nodes in the momentum space, and such changes in the topology of the Fermi surface without symmetry breaking produce a variety of Lifshitz transitions, in which the Weyl nodes situating at k=π change from type I to type II. A coexistence of type-I and type-II Weyl nodes is found in phase II. The information measures including concurrence, entanglement entropy, and relative entropy can effectively signal the second-order transitions. The results indicate that the Gamma model can act as an exactly solvable model to describe Lifshitz phase transitions in correlated electron systems.

High-field quantum disordered state in α−RuCl3 : Spin flips, bound states, and multiparticle continuum

Layered $\alpha$-RuCl3 has been discussed as a proximate Kitaev spin liquid compound. Raman and THz spectroscopy of magnetic excitations confirm that the low-temperature antiferromagnetic ordered phase features a broad Raman continuum, together with two magnon-like excitations at 2.7 and 3.6 meV, respectively. The continuum strength is maximized as long-range order is suppressed by an external magnetic field. The state above the field-induced quantum phase transition around 7.5 T is characterized by a gapped multi-particle continuum out of which a two-particle bound state emerges, together with a well-defined single-particle excitation at lower energy. Exact diagonalization calculations demonstrate that Kitaev and off-diagonal exchange terms in the Fleury-Loudon operator are crucial for the occurrence of these features in the Raman spectra. Our study firmly establishes the partially-polarized quantum disordered character of the high-field phase.

Ferromagnetic Kitaev interaction and the origin of large magnetic anisotropy in α-RuCl3

$\alpha$-RuCl$_3$ is drawing much attention as a promising candidate Kitaev quantum spin liquid. However, despite intensive research efforts, controversy remains about the form of the basic interactions governing the physics of this material. Even the sign of the Kitaev interaction (the bond-dependent anisotropic interaction responsible for Kitaev physics) is still under debate, with conflicting results from theoretical and experimental studies. The significance of the symmetric off-diagonal exchange interaction (referred to as the $\Gamma$ term) is another contentious question. Here, we present resonant elastic x-ray scattering data that provides unambiguous experimental constraints to the two leading terms in the magnetic interaction Hamiltonian. We show that the Kitaev interaction ($K$) is ferromagnetic, and that the $\Gamma$ term is antiferromagnetic and comparable in size to the Kitaev interaction. Our findings also provide a natural explanation for the large anisotropy of the magnetic susceptibility in $\alpha$-RuCl$_3$ as arising from the large $\Gamma$ term. We therefore provide a crucial foundation for understanding the interactions underpinning the exotic magnetic behaviours observed in $\alpha$-RuCl$_3$.

Competition between spin-singlet and -triplet superconducting states in the doped extended Kitaev-Heisenberg model

Recently, the extended Kitaev-Heisenberg model has been proposed to describe spin-orbital Mott insulators, such as iridate oxides and ruthenium chloride with honeycomb lattice. Using mean-field theory, we obtain the linear gap equations to find all possible superconducting phases in terms of different exchanges and doping levels. Our calculation based on the hole-doped model, in the presence of the off-diagonal exchange , shows the spin-triplet states can be stable in a larger area related to the doped Kitaev-Heisenberg model with K0. However, the finite ferromagnetic off-diagonal exchange solely cannot generate the triplet pairing instabilities in competition with the antiferromagnetic-Heisenberg and -Kitaev exchanges.

Magnon damping in the zigzag phase of the Kitaev-Heisenberg- Γ model on a honeycomb lattice

We calculate magnon dispersions and damping in the Kitaev-Heisenberg model with an off-diagonal exchange $\Gamma$ and isotropic third-nearest-neighbor interaction $J_3$ on a honeycomb lattice. This model is relevant to a description of the magnetic properties of iridium oxides $\alpha$-Li$_2$IrO$_3$ and Na$_2$IrO$_3$, and Ru-based materials such as $\alpha$-RuCl$_3$. We use an unconventional parametrization of the spin-wave expansion, in which each Holstein-Primakoff boson is represented by two conjugate hermitian operators. This approach gives us an advantage over the conventional one in identifying parameter regimes where calculations can be performed analytically. Focusing on the parameter regime with the zigzag spin pattern in the ground state that is consistent with experiments, we demonstrate that one such region is $\Gamma = K>0$, where $K$ is the Kitaev coupling. Within our approach we are able to obtain explicit analytical expressions for magnon energies and eigenstates and go beyond the standard linear spin-wave theory approximation by calculating magnon damping and demonstrating its role in the dynamical structure factor. We show that the magnon damping effects in both Born and self-consistent approximations are very significant, underscoring the importance of non-linear magnon coupling in interpreting broad features in the neutron-scattering spectra.

Large off-diagonal exchange couplings and spin liquid states in C3 -symmetric iridates

Iridate oxides on a honeycomb lattice are considered promising candidates for realization of quantum spin liquid states. We investigate the magnetic couplings in a structural model for a honeycomb iridate K$_2$IrO$_3$, with $C_3$ point group symmetry at the Ir sites, which is an end member of the recently synthesized iridate family K$_x$Ir$_y$O$_2$. Using \textit{ab-initio} quantum chemical methods, we elucidate the subtle relationship between the real space symmetry and magnetic anisotropy and show that the higher point group symmetry leads to high frustration with strong magnetic anisotropy driven by the unusually large off-diagonal exchange couplings ($\Gamma$'s) as opposed to other spin-liquid candidates considered so far. Consequently, large quantum fluctuations imply lack of magnetic ordering consistent with the experiments. Exact diagonalization calculations for the fully anisotropic $K$-$J$-$\Gamma$ Hamiltonian reveal the importance of the off-diagonal anisotropic exchange couplings in stabilizing a spin liquid state and highlight an alternative route to stabilize spin liquid states for ferromagnetic $K$.