A projective symmetry group (PSG) has been regarded as a classification theory of spin liquids. However, it does not include a symmetry-protected topological order of fermionic spinon excitations, and thus the classification of gapped spin liquids is incomplete. We demonstrate the classification beyond PSG by utilizing the Kitaev model on the squareoctagon lattice, where two gapped spin liquids are distinguished by a topological $Z_2$ invariant. This $Z_2$ invariant can be defined solely by the time-reversal and translation symmetries on condition that the time-reversal symmetry is implemented projectively. Thus, it is a hidden class of topological Kitaev spin liquids with helical edge states, which has been ignored for a long time. This suggests that there exists an unknown classification scheme of gapped spin liquids beyond PSG.
We point out that the Kitaev materials may not necessarily support Kitaev spin liquid. It is well-known that having a Kitaev term in the spin interaction is not the sufficient condition for the Kitaev spin liquid ground state. Many other spin liquids may be stabilized by the competing spin interactions of the systems. We thus explore the possibilities of non-Kitaev spin liquids in the honeycomb Kitaev materials. We carry out a systematic classification of gapped $\mathbb{Z}_2$ spin liquids using the Schwinger boson representation for the spin variables. The presence of strong spin-orbit coupli…
We study the gapped phase of Kitaev's honeycomb model (a Z_2 spin liquid) in the presence of lattice defects. We find that some dislocations and bond defects carry unpaired Majorana fermions. Physical excitations associated with these defects are (complex) fermion modes made out of two (real) Majorana fermions connected by a Z_2 gauge string. The quantum state of these modes is robust against local noise and can be changed by winding a Z_2 vortex around a dislocation. The exact solution respects gauge invariance and reveals a crucial role of the gauge field in the physics of Majorana modes.
Mott insulators under sufficiently strong spin-orbit coupling can display quantum spin liquid phases with topological order and fractional excitations. Quantum magnets with pure Kitaev spin exchange interactions can host a gapped quantum spin liquid with a single Majorana edge mode propagating in the counter-clockwise direction when a small positive magnetic field is applied. Here, we show how under a sufficiently strong positive magnetic field a topological transition into a gapped quantum spin liquid with two Majorana edge modes propagating in the clockwise direction occurs. The Dzyaloshinsk…
Thermalisation is a probabilistic process. As such, it is generally expected that when we increase the temperature of a system its classical behaviour dominates its quantum coherences. By employing the Gibbs state of a translationally invariant quantum spin liquid - Kitaev's honeycomb lattice model - we demonstrate that an insulating phase at $T=0$ becomes metallic purely by increasing temperature. In particular, we compute the finite temperature distribution of energies and show that it diverges logarithmically, as we move to small energies. The corresponding wave functions become critical al…
We study quantum phase transitions in the honeycomb Kitaev model under a magnetic field, focusing on the topological nature of Majorana fermion excitations. We find a gapless phase between the low-field gapless quantum spin liquid and the high-field gapped forced-ferromagnetic state for the antiferromagnetic Kitaev model in the [001] field by using the Majorana mean-field theory, in conjunction with the exact diagonalization and the spin-wave theory supporting the validity of this approach. The transition between the two gapless phases is driven by a topological change of the Majorana spectrum…
We introduce a spin-1/2 model in three dimensions which is a generalization of the well-known Kitaev model on a honeycomb lattice. Following Kitaev, we solve the model exactly by mapping it to a theory of non-interacting fermions in the background of a static Z_2 gauge field. The phase diagram consists of a gapped phase and a gapless one, similar to the two-dimensional case. Interestingly, unlike in the two-dimensional model, in the gapless phase the gap vanishes on a contour in the k space. Furthermore, we show that the flux excitations of the gauge field, due to some local constraints, form …
Projective symmetry groups are applied to Raman observations of the Kitaev quantum spin liquids in spherical lattice geometries realized by Platonic and Archimedean polyhedra. Parton single excitations in Kitaev spin polyhedra are characterized by double-valued irreducible representations of their belonging projective symmetry groups, whereas parton geminate excitations relevant to Raman scattering are decomposed into single-valued irreducible representations of the corresponding point symmetry groups. We combine a standard point-symmetry-group analysis of the Loudon-Fleury vertices and an ela…
Low-energy states of quantum spin liquids are thought to involve partons living in a gauge-field background. We study the spectrum of Majorana fermions of Kitaev's honeycomb model on spherical clusters. The gauge field endows the partons with half-integer orbital angular momenta. As a consequence, the multiplicities reflect not the point-group symmetries of the cluster, but rather its projective symmetries, operations combining physical and gauge transformations. The projective symmetry group of the ground state is the double cover of the point group.
A three-dimensional Kitaev model on a hyperhoneycomb lattice is investigated numerically at finite temperature. The Kitaev model is one of the solvable quantum spin models, where the ground state is given by gapped and gapless spin liquids, depending on the anisotropy of the interactions. This model can be rewritten as a free Majorana fermion system coupled with $Z_2$ variables. The density of states of Majorana fermions shows an excitation gap in the gapped region, while it is semimetallic in the gapless region reflecting the Dirac node. Performing the Monte Carlo simulation, we calculate the…
We study the quantum spin liquid phase in a variant of the Kitaev model where the bonds of the honeycomb lattice are distributed in a Kekulé pattern. The system supports gapped and gapless Z_2 quantum spin liquids with interesting differences from the original Kitaev model, the most notable being a gapped Z_2 spin liquid on a Kagome lattice. Perturbing the exactly solvable model with antiferromagnetic Heisenberg perturbations, we find a magnetically ordered phase stabilized by a quantum `order by disorder' mechanism, as well as an exotic continuous phase transition between the topological spin…
The Kitaev model is a remarkable spin model with gapped and gapless spin liquid phases, which are potentially realized in iridates and $α$-RuCl$_3$. In the recent experiment of $α$-RuCl$_3$, the signature of a nematic transition to the gapped toric code phase, which breaks the $C_3$ symmetry of the system, has been observed through the angle dependence of the heat capacity. We here propose a mechanism by which the nematic transition can be detected electrically. This is seemingly impossible because $J_\textrm{eff}=1/2$ spins do not have an electric quadrupole moment (EQM). However, in the seco…
We study the physical consequences of site dilution in Kitaev's honeycomb model, in both its gapped and gapless phases. We show that a vacancy binds a flux of the emergent $Z_2$ gauge field and induces a local moment. In the gapped phase this moment is free while in the gapless phase the susceptibility has the dependence $χ(h)\sim\ln(1/h)$ on field strength $h$. Vacancy moments have interactions that depend on their separation, their relative sublattice, and the phase of the model. Strikingly, in the gapless phase, two nearby vacancies on the same sublattice have a parametrically larger $χ(h)\…
We study the effects of vacancy disorder on the Kitaev model defined on a hexagonal lattice. We show that the vacancy disorder induces a zero-mode that is localized at the defect site. We derive analytical forms for these localized wave functions in both the gapped and gapless phases of the Kitaev model. We conjecture that the vacancy disorder can be utilized as a probe of the quantum phase transition (from the gapped to gapless phases) in this model. The behavior of the Inverse Participation Ratio (IPR) in the gapless phase and across the transition is also studied numerically. Comments are m…
Gapped Boundaries of Kitaev's Quantum Double Models: A Lattice Realization of Anyon Condensation from Lagrangian Algebras
Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology（南方科技大学量子科学与工程研究院）；International Quantum Academy（国际量子学院）；Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology（南方科技大学广东省量子科学与工程重点实验室）；Hefei National Laboratory, University of Science and Technology of China（中国科学技术大学合肥国家实验室）；Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China（中国科学技术大学微尺度物质科学国家研究中心物理科学学院）
The microscopic modeling of spin-orbit entangled $j=1/2$ Mott insulators such as the layered hexagonal Iridates Na$_2$IrO$_3$ and Li$_2$IrO$_3$ has spurred an interest in the physics of Heisenberg-Kitaev models. Here we explore the effect of lattice distortions on the formation of the collective spin-orbital states which include not only conventionally ordered phases but also gapped and gapless spin-orbital liquids. In particular, we demonstrate that in the presence of spatial anisotropies of the exchange couplings conventionally ordered states are formed through an order-by-disorder selection…
We study exactly both the ground-state fidelity susceptibility and bond-bond correlation function in the Kitaev honeycomb model. Our results show that the fidelity susceptibility can be used to identify the topological phase transition from a gapped A phase with Abelian anyon excitations to a gapless B phase with non-Abelian anyon excitations. We also find that the bond-bond correlation function decays exponentially in the gapped phase, but algebraically in the gapless phase. For the former case, the correlation length is found to be $1/ξ=2\sinh^{-1}[\sqrt{2J_z -1}/(1-J_z)]$, which diverges ar…
One-dimensional gapped phases that avoid any symmetry breaking have drawn enduring attention. In this paper, we study such phases in a bond-alternating spin-1 $K$-$Γ$ chain built of a Kitaev ($K$) interaction and an off-diagonal $Γ$ term. In the case of isotropic bond strength, a Haldane phase, which resembles the ground state of a spin-$1$ Heisenberg chain, is identified in a wide region. A gapped Kitaev phase situated at dominant ferromagnetic and antiferromagnetic Kitaev limits is also found. The Kitaev phase has extremely short-range spin correlations and is characterized by finite $\mathb…
We construct an exactly soluble spin-$\frac{1}2$ model on a honeycomb lattice, which is a generalization of Kitaev model. The topological phases of the system are analyzed by study of the ground state sector of this model, the vortex-free states. Basically, there are two phases, A phase and B phase. The behaviors of both A and B phases may be studied by mapping the ground state sector into a general p-wave paired states of spinless fermions with tunable pairing parameters on a square lattice. In this p-wave paired state theory, the A phase is shown to be the strong paired phase, an insulating …
Topological order, reflected in long range patterns of entanglement, is quantified by the topological entanglement entropy (TEE) $γ$. We show that for gapped quantum spin liquids (QSL) it is possible to extract $γ$ using two-spin local correlators. We demonstrate our method for the gapped $\mathbb{Z}_2$ Kitaev spin liquid on a honeycomb lattice with anisotropic interactions. We show that the $γ= \log 2$ for $\mathbb{Z}_2$ topological order can be simply extracted from local two-spin correlators across two different bonds, with an accuracy comparable or higher than the Kitaev-Preskill construct…
We study the effect of adding disorder to the exactly solvable Kitaev model on the hyper-honeycomb lattice, which hosts both gapped and gapless spin liquid phases with an emergent $\mathbb{Z}_2$ gauge field. The latter has an unusual gapless spectrum of Majorana fermion excitations, with a co-dimension-two Fermi ring. We thus address the question of the interplay of topological physics and disorder by considering the properties of isolated single and pair of vacancies. We show that near the vacancies, the local magnetic response to a field $h_z$ is parametrically enhanced in comparison to the …
We investigate the quantum spin liquid (QSL) ground state of anisotropic Kitaev model with antiferromagnetic (AFM) coupling under the $[001]$ magnetic field with the finite-temperature Lanczos method (FTLM). In this anisotropic AFM Kitaev model with $K_{X}=K_{Y}$, $K_{X}+K_{Y}+K_{Z}=-3K$, and $K_{Z}<-K$, with magnetic field increasing, the gapped QSL experiences a transition to a gapless QSL at $h_{c1}=gμ_{B}H_{z1}/K$, to another gapless QSL with $C_{6}$ rotational symmetry at $h_{c2}$, and to a new $U(1)$ gapless QSL between $h_{c3}$ and $h_{c4}$, respectively. These indicate that magnetic fi…
Quantum spin liquid is a disordered magnetic state with fractional spin excitations. Its clearest example is found in an exactly solved Kitaev honeycomb model where a spin flip fractionalizes into two types of anyons, quasiparticles that are neither fermions nor bosons: a pair of gauge fluxes and a Majorana fermion. Here we demonstrate this kind of fractionalization in the Kitaev paramagnetic state of the honeycomb magnet $α$-RuCl$_3$. The spin-excitation gap measured by nuclear magnetic resonance consists of the predicted Majorana fermion contribution following the cube of the applied magneti…
The ruby lattice is a four-valent lattice interpolating between honeycomb and triangular lattices. In this work we investigate the topological spin-liquid phases of a spin Hamiltonian with Kitaev interactions on the ruby lattice using exact diagonalization and perturbative methods. The latter interactions combined with the structure of the lattice yield a model with $\mathbb{Z}_2 \times \mathbb{Z}_2$ gauge symmetry. We mapped out the phase digram of the model and found gapped and gapless spin-liquid phases. While the low energy sector of the gapped phase corresponds to the well-known topologic…
We introduce valence bond fluctuations, or bipartite fluctuations associated to bond-bond correlation functions, to characterize quantum spin liquids and their entanglement properties. Using analytical and numerical approaches, we find an identical scaling law between valence bond fluctuations and entanglement entropy in the two-dimensional Kitaev spin model and in one-dimensional chain analogues. We also show how these valence bond fluctuations can locate quantum phase transitions between the three gapped and the gapless Majorana semi-metal phases in the honeycomb model. We then study the eff…
We analyse in detail the effect of non-trivial band topology on the area law behaviour of the entanglement entropy in Kitaev's honeycomb model. By mapping the translationally invariant 2D spin model into 1D fermionic subsystems, we identify those subsystems responsible for universal entanglement contributions in the gapped phases and those responsible for critical entanglement scaling in the gapless phases. For the gapped phases we analytically show how the topological edge states contribute to the entanglement entropy and provide a universal lower bound for it. For the gapless semi-metallic p…
Frustrated magnetic systems exhibit many fascinating phases. Prime among them are quantum spin liquids, where the magnetic moments do not order even at zero temperature. A subclass of quantum spin liquids called Kitaev spin liquids are particularly interesting, because they are exactly solvable, can be realized in certain materials, and show a large variety of gapless and gapped phases. Here, we show that non-symmorphic symmetries can enrich spin liquid phases, such that the low-energy spinon degrees of freedom form three-dimensional Dirac cones or nodal chains. In addition, we suggest how suc…
We study the Kitaev-Heisenberg-$Γ$ model with antiferromagnetic Kitaev exchanges in the strong anisotropic (toric code) limit to understand the phases and the intervening phase transitions between the gapped $Z_2$ quantum spin liquid and the spin-ordered (in the Heisenberg limit) as well as paramagnetic phases (in the pseudo-dipolar, $Γ$, limit). We find that the paramagnetic phase obtained in the large $Γ$ limit has no topological entanglement entropy and is proximate to a gapless critical point of a system described by an equal superposition of differently oriented stacked one-dimensional $Z…
We study the excitations in a three dimensional version of Kitaev's spin-1/2 model on the honeycomb lattice introduced by the present authors recently. The gapped phase of the system is analyzed using a low energy effective Hamiltonian which is defined on the diamond lattice and consists of plaquette operators. The excitations of the effective Hamiltonian form loops in an embedded lattice. The elementary excitations, which are the shortest loops, are fermions. Moreover, the excitations obey nontrivial braiding rules: when a fermion winds through a loop, the wave function acquires a phase $π$.
Visons in Kitaev Spin Liquids with Majorana Fermi Surfaces
Universidade Federal do Rio Grande do Norte（北里奥格兰德联邦大学）；International Institute of Physics, Universidade Federal do Rio Grande do Norte（北里奥格兰德联邦大学国际物理研究所）
We theoretically study laser driven nonequilibrium states in the Kitaev honeycomb model with a magnetoelectric cross coupling. We show that a topological spin liquid with a gapless chiral edge mode emerges when we apply an elliptically or circularly polarized laser. This is a strongly correlated quantum spin version of the Floquet topological insulator. In the topological phase, the edge mode is made from Majorana fermions and the bulk has gapped non-Abelian anyon excitations.
We present a solution of Kitaev's spin model on the honeycomb lattice and of related topologically ordered spin models. We employ a Jordan-Wigner type fermionization and find that the Hamiltonian takes a BCS type form, allowing the system to be solved by Bogoliubov transformation. Our fermionization does not employ non-physical auxiliary degrees of freedom and the eigenstates we obtain are completely explicit in terms of the spin variables. The ground-state is obtained as a BCS condensate of fermion pairs over a vacuum state which corresponds to the toric code state with the same vorticity. We…
We determine the phase diagram of a bilayer, Yao-Lee spin-orbital model with inter-layer interactions (J), for several stackings and moiré superlattices. For AA stacking, a gapped Z2 quantum spin liquid phase emerges at a finite Jc. We show that this phase survives in the well-controlled large-J limit, where an isotropic honeycomb toric code emerges. For moiré superlattices, a finite-q inter-layer hybridization is stabilized. This connects inequivalent Dirac points, effectively `untwisting' the system. Our study thus provides insight into the spin-liquid phases of bilayer spin-orbital Kitaev m…
Kitaev-Heisenberg model on the square-hexagon-dodecagon lattice
Institute for Advanced Studies in Basic Sciences (IASBS)（基础科学高等研究所）；Indian Institute of Technology Madras（马德拉斯印度理工学院）；The Institute of Mathematical Sciences（数学科学研究所）；Perimeter Institute for Theoretical Physics（理论物理珀米特研究所）；Donostia International Physics Center（多诺西亚国际物理中心）；Université Paris-Saclay（巴黎萨克雷大学）；Sorbonne Université（索邦大学）
Higher-order topological insulators are distinguished by the existence of topologically protected modes with codimension two or higher. Here, we report the manifestation of a second-order topological insulator in a two dimensional frustrated quantum magnet, which exhibits topological corner modes. Our exactly-solvable model is a generalization of the Kitaev honeycomb model to the Shastry-Sutherland lattice that, besides a chiral spin liquid phase, exhibits a gapped spin liquid with Majorana corner modes, which are protected by two mirror symmetries. This second-order Kitaev spin liquid remains…
We present a spin model, namely, the Kitaev model augmented by a loop term and perturbed by an Ising Hamiltonian and show that it exhibits both confinement-deconfinement transitions from spin liquid to antiferromagnetic/spin-chain/ferromagnetic phases and topological quantum phase transitions between gapped and gapless spin liquid phases. We develop a Fermionic mean-field theory to chart out the phase diagram of the model and estimate the stability of its spin liquid phases which might be relevant for attempts to realize the model in optical lattices. We also conjecture that some of the confin…
We study a quantum spin Kitaev model with zigzag edges to clarify the effects of anisotropy in the exchange couplings on the spin propagation. We simulate the spin and Majorana dynamics triggered by a magnetic pulse, using the real-space time-dependent Majorana mean-field theory. When the anisotropy is small, the dispersion of the itinerant Majorana fermions remains gapless, where the velocity of the spin propagation matches the group velocity of the itinerant Majorana fermions at the nodal points. On the other hand, in the gapped system with a large anisotropy, the spin propagation is strongl…
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 …
We present a general classification of the perturbations to the Kitaev model on the basis of their effect on it's spin correlation functions. We derive a necessary and sufficient condition for the spin correlators to exhibit a long ranged power-law behavior in the presence of such perturbations. We substantiate our result by a study of the phase diagram of the Kitaev model augmented by a loop term and perturbed by an Ising term, within a RVB mean-field theory. We estimate the stability of the spin-liquid phase against such perturbations and show that this model exhibits both confinement-deconf…
We explore the nature of the quantum phase transition between a magnetically ordered state with collinear spin pattern and a gapless $Z_2$ spin liquid in the Heisenberg-Kitaev model. We construct a slave particle mean field theory for the Heisenberg-Kitaev model in terms of complex fermionic spinons. It is shown that this theory, formulated in the appropriate basis, is capable of describing the Kitaev spin liquid as well as the transition between the gapless $Z_2$ spin liquid and the so-called stripy antiferromagnet. In particular, within a mean field theory, we have a discontinuous transition…
We present an exactly solvable lattice Hamiltonian to realize gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also present the parallel algebraic/categorical structure of gapped boundaries.
In addition to the Kitaev ($K$) interaction, candidate Kitaev materials also possess Heisenberg ($J$) and off-diagonal symmetric ($Γ$) couplings. We investigate the quantum ($S = 1/2$) $K$-$J$-$Γ$ model on the honeycomb lattice by a variational Monte Carlo (VMC) method. In addition to the "generic" Kitaev spin liquid (KSL), we find that there is just one proximate KSL (PKSL) phase, while the rest of the phase diagram contains different magnetically ordered states. The PKSL is a gapless Z$_2$ state with 14 Majorana cones, which in contrast to the KSL has a gapless spin response. In a magnetic f…
Recent numerical studies indicate that the antiferromagnetic Kitaev honeycomb lattice model undergoes a magnetic-field-induced quantum phase transition into a new spin-liquid phase. This intermediate-field phase has been previously characterized as a gapless spin liquid. By implementing a recently developed variational approach based on the exact fractionalized excitations of the zero-field model, we demonstrate that the field-induced spin liquid is gapped and belongs to Kitaev's 16-fold way. Specifically, the low-field non-Abelian liquid with Chern number $C=\pm 1$ transitions into an Abelian…
The Kitaev model with an applied magnetic field in the $H||[111]$ direction shows two transitions: from a non-abelian gapped quantum spin liquid (QSL) to a gapless QSL at $H_{c1} \simeq 0.2K$ and a second transition at a higher field $H_{c2} \simeq 0.35K$ to a gapped partially polarized phase, where $K$ is the strength of the Kitaev exchange interaction. We identify the intermediate phase to be a gapless U(1) QSL and determine the spin structure function $S({\bf k}) $ and the Fermi surface $ε_F^S({\bf k})$ of the gapless spinons using the density matrix renormalization group (DMRG) method for …
Quantum spin liquid is an exotic quantum state of matter in magnets. This state is a spin analogue of the liquid helium which does not solidify down to the lowest temperature due to strong quantum fluctuations. In conventional fluids, liquid and gas possess the same symmetry and adiabatically connect to each other by bypassing the critical end point. We find that the situation is qualitatively different in quantum spin liquids realizing in a three-dimensional Kitaev model; both gapless and gapped quantum spin liquid phases at low temperatures are always distinguished from the high-temperature …
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 $Δ_p$ becomes lower than…
Motivated by the recent experimental observation of the half-quantized thermal Hall conductivity for the candidate material of the Kitaev spin liquid, $α$-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 conduct…
This paper studies fault-tolerant quantum computation with gapped boundaries. We first introduce gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories using their Hamiltonian realizations. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also provide a commuting Hamiltonian to realize defects between boundaries in any quantum double model. Next, we present the algebraic/categorical structure of gapped boundaries and boundary defects, which will be used to describe topologically protec…
We present a study of longitudinal thermal transport in the Kitaev spin model on the honeycomb lattice, focusing on the role of anisotropic exchange to cover both, gapless and gapped phases. Employing a complementary combination of exact diagonalization on small systems and an average gauge configuration approach for up to $\sim O(10^4)$ spinful sites, we report results for the dynamical energy current auto-correlation function as well as the dc thermal conductivity over a wide range of temperatures and exchange anisotropies. Despite a pseudogap in the current correlation spectra, induced by e…
