Search

We theoretically study the Ruderman–Kittle–Kasuya–Yosida (RKKY) interaction between two magnetic impurities embedded on the (001) surface of a topological crystalline insulator (TCI), using the Green’s function method. The highly anisotropic band structure of TCI, gives rise to a highly anisotropic magnetic exchange coupling. We show that the interaction is oscillatory; the amplitude and wavelength of oscillations have angular dependence arising from the anisotropy of the surface state band structure. The spatial configurations of the magnetic impurities can also dramatically change the quality and quantity of the RKKY interaction. We find a strong anisotropy of the exchange interaction and the magnetic ground state of two magnetic adatoms can be tuned by changing the rotational configuration of impurities. It is found that the three types of interactions contribute to the magnetic exchange coupling in the (001) TCI surfaces: the Heisenberg, Dzyaloshinsky–Moriya, and Ising types. Our results will open up a new route toward spintronics based on TCIs.

We show theoretically that the magnetic ions, randomly distributed in a two-dimensional (2D) semiconductor system, can generate a ferromagnetic long-range order via the RKKY interaction. The main physical reason is the discrete (rather than continuous) symmetry of the 2D Ising model of the spin-spin interaction mediated by the spin-orbit coupling of 2D free carriers, which precludes the validity of the Mermin-Wagner theorem. Further, the analysis clearly illustrates the crucial role of the molecular field fluctuations as opposed to the mean field. The developed theoretical model describes the desired magnetization and phase-transition temperature ${T}_{c}$ in terms of a single parameter, namely, the chemical potential $\ensuremath{\mu}$. Our results highlight a pathway to reach the highest possible ${T}_{c}$ in a given material as well as an opportunity to control the magnetic properties externally (e.g., via a gate bias). Numerical estimations show that magnetic impurities such as ${\mathrm{Mn}}^{2+}$ with spins $S=5/2$ can realize ferromagnetism with ${T}_{c}$ close to room temperature.

We study the Ruderman-Kittle-Kasuya-Yosida (RKKY) interaction in doped armchair nanotube in the presence of gap parameter. The effects of both next nearest neighbor hopping parameter and electron-Holstein phonon on RKKY interaction have been addressed. RKKY interaction as a function of distance between localized moments have been analyzed. In order to calculate the exchange interaction along arbitrary direction between two magnetic moments, we should obtain the transverse static spin susceptibility of armchair graphene nanoribbon in the presence of electron-phonon coupling and gap parameter. The spin susceptibility components are calculated using Green’s function approach for Holstein model Hamiltonian. The effects of electron doping on dependence of exchange interaction on distance between moments are investigated via calculating correlation function of spin density operators. Our results show the influences of next nearest neighbor hopping parameter on the spatial behavior of RKKY interactions are different in the presence of electron phonon coupling.

A central question in a large class of strongly correlated electron systems, including heavy fermion compounds and iron pnictides, is the identification of different phases and their origins. It has been shown that the antiferromagnetic (AFM) phase in some heavy fermion compounds is induced by Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between localized moments, and that the competition between this interaction and Kondo effect is responsible for quantum criticality. However, conclusive experimental evidence of the RKKY interaction in pnictides is lacking. Here, using high resolution $^{23}\mathrm{Na}$ NMR measurements on lightly Cu-doped metallic single crystals of ${\mathrm{NaFe}}_{1\ensuremath{-}x}{\mathrm{Cu}}_{x}\mathrm{As}$ $(x\ensuremath{\approx}0.01)$ and numerical simulation, we show direct evidence of the RKKY interaction in this pnictide system. Aided by computer simulation, we identify the $^{23}\mathrm{Na}$ NMR satellite resonances with the RKKY oscillations of spin polarization at Fe sites. Our $^{23}\mathrm{Na}$ spin-lattice and spin-spin relaxation data exhibit a signature of an itinerant and inhomogeneous AFM phase in this system, accompanied by a simulation of Cu-induced perturbation to the ordered moments on the Fe sites. Our NMR results indicate coexistence of local and itinerant magnetism in lightly Cu-doped ${\mathrm{NaFe}}_{1\ensuremath{-}x}{\mathrm{Cu}}_{x}\mathrm{As}$.

We theoretically study the indirect interaction between two ferromagnetic contacts located on the surface of a $d$-wave superconductor. When the magnets are connected to a $\{010\}$ edge of the superconductor we find an oscillating RKKY interaction that varies in sign as the distance between the magnetic contacts is varied. However, when coupling the magnets to an $\{110\}$ edge of the superconductor, we find that the presence of midgap states qualitatively changes the results. The ground state of the system is then found to always favor alignment of the magnets as this configuration most strongly suppresses the midgap states, leading to a larger condensation energy which dominates over the intrinsic RKKY interaction., Comment: 10 pages, 8 figures

The emerging and screening of local magnetic moments in solids have been investigated for more than 60 years. Local vacancies as in graphene or in heavy fermions can induce decoupled bound states that lead to the formation of local moments. In this paper, we address the puzzling question how these local moments can be screened and what determines the additionally emerging low-temperature scale. We review the initial problem for half-filled conduction bands from two complementary perspectives: By a single-particle supercell analysis in the uncorrelated limit and by the Lieb-Mathis theorem for systems with a large Coulomb interaction $U$. Applying Wilson's numerical renormalization group approach to a recently developed mapping of the problem onto an effective low-energy description of a Kondo hole with up to ${N}_{f}=7$ correlated impurities as background, we proof that the stable local moments are subject to screening by three different mechanisms. Firstly the local moments are delocalized by a finite $U$ beyond the single-particle bound state. We find a Kosterlitz-Thouless type transition governed by an exponentially suppressed low-energy scale of a counterintuitive Kondo form with ${J}_{\mathrm{eff}}\ensuremath{\propto}{U}^{n}$ for small $U$, where $ng1$ depends on the precise model. Secondly, we show that away from half-filling the local moment phase becomes unstable and is replaced by two types of singlet phases that are adiabatically connected. At a critical value for the band center, the physics is governed by an exponentially suppressed Kondo scale approaching the strong coupling phase that is replaced by a singlet formation via antiferromagnetic RKKY interaction for large deviation from the critical values. Thirdly, we show that the local magnetic moment can be screened by a Kondo hole orbital at finite energy, even though the orbital occupation is negligible: An additional low-energy scale emerges below which the localized moment is quenched. Similarities to the experimental findings in ${\mathrm{Ce}}_{1\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{Pd}}_{3}$ are pointed out.

We study the Ruderman–Kittle–Kasuya–Yosida (RKKY) interaction in doped armchair graphene nanoribbon. The effects of both external magnetic field and electron-Holstein phonon on RKKY interaction have been addressed. RKKY interaction as a function of distance between localized moments has been analyzed. It has been shown that a magnetic field along the z-axis mediates an anisotropic interaction which corresponds to a XXZ model interaction between two magnetic moments. In order to calculate the exchange interaction along arbitrary direction between two magnetic moments, we should obtain both transverse and longitudinal static spin susceptibilities of armchair graphene nanoribbon in the presence of electron-phonon coupling and magnetic field. The spin susceptibility components are calculated using the spin dependent Green’s function approach for Holstein model Hamiltonian. The effects of spin polarization on the dependence of exchange interaction on distance between moments are investigated via calculating correlation function of spin density operators. Our results show the influences of magnetic field on the spatial behavior of in-plane and longitudinal RKKY interactions are different in the presence of magnetic field.

We theoretically investigate the current–current correlation of the two-dimensional (2D) parabolic Dirac system in hexogonal lattice. The analytical expressions of the random phase approximation (RPA) susceptibility, Ruderman–Kittel–Kasuya–Yosida (RKKY) Hamiltonian, and the diamagnetic orbital susceptibility in noninteracting case base on the density–density or current–current correlation function are derived and quantitatively analyzed. In noninteracting case, the dynamical polarization within RPA, and spin transverse susceptibility as well as the RKKY interaction (when close to the half-filling) are related to the current–current response in the 2D parabolic Dirac systems. Both the cases of anisotropic dispersion and isotropic dispersion are discussed.

We study theoretically the time-domain dynamics of the spin-dependent Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between driven magnetic impurities localized in a spin-orbit-coupled two-dimensional system. Particular attention is given to the influence of the spin-orbit coupling (SOC) on the system's dynamical response to a time-dependent precessional motion of the localized magnetic moment. We show that, via the RKKY mechanism, a flip of the spin $z$ component of one localized moment affects all $x,\phantom{\rule{4pt}{0ex}}y$, and $z$ spin components of the other localized moment. The Friedel oscillations and the transient spin current triggered by the time-varying localized spin depend strongly on the orientation of the spin with respect to the two-dimensional structure. Our results demonstrate how the dynamic interplay between SOC and the RKKY interaction can be used to manipulate and fine-tune the characteristics of impurity spin reversal and to control the generated spin current and the local magnetization.

We study the RKKY interaction between two magnetic impurities located on the surface of a three-dimensional Dirac semimetal with two Dirac nodes in the band structure. By taking into account both bulk and surface contributions to the exchange interaction between the localized spins, we demonstrate that the surface contribution in general dominates the bulk one at distances larger than the inverse node separation due to a weaker power-law decay. We find a strong anisotropy of the surface term with respect to the spins being aligned along the node separation axis or perpendicular to it. In the many impurity dilute regime, this implies formation of quasi-one-dimensional magnetic stripes orthogonal to the node axis. We also discuss the effects of a surface spin-mixing term coupling electrons from spin-degenerate Fermi arcs., Comment: 7,5 pages, 3 figures (+4 pages of Appendixes)

We determine theoretically the interaction between two magnetic impurities embedded in a spin-split $s$-wave superconductor. The spin-splitting in the superconductor gives rise to two different interaction types between the impurity spins, depending on whether their spins lie in the plane perpendicular to the spin-splitting field (Heisenberg) or not (Ising). For impurity separation distances exceeding $\xi_S$, we find that the magnitude of the spin-splitting can determine whether an antiferromagnetic or ferromagnetic alignment of the impurity spins is preferred by the RKKY interaction. Moreover, the Ising and Heisenberg terms of the RKKY interaction alternate on being the dominant term and their magnitudes oscillate as a function of distance between the impurities., Comment: 9 pages, 6 figures

We theoretically investigate a new stabilization mechanism of a skyrmion crystal (SkX) in centrosymmetric itinerant magnets with magnetic anisotropy. By considering a trigonal crystal system without the horizontal mirror plane, we derive an effective spin model with an anisotropic Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction for a multi-band periodic Anderson model. We find that the anisotropic RKKY interaction gives rise to two distinct SkXs with different skyrmion numbers of one and two depending on a magnetic field. We also clarify that a phase arising from the multiple-Q spin density waves becomes a control parameter for a field-induced topological phase transition between the SkXs. The mechanism will be useful not only for understanding the SkXs, such as that in Gd$$_2$$ 2 PdSi$$_3$$ 3 , but also for exploring further skyrmion-hosting materials in trigonal itinerant magnets.

Spin textures with nontrivial topology, such as vortices and skyrmions, have attracted attention as a source of unconventional magnetic, transport, and optical phenomena. Recently, a new generation of topological spin textures has been extensively studied in itinerant magnets; in contrast to the conventional ones induced, e.g., by the Dzyaloshinskii-Moriya interaction in noncentrosymmetric systems, they are characterized by extremely short magnetic periods and stable even in centrosymmetric systems. Here we review such new types of topological spin textures with particular emphasis on their stabilization mechanism. Focusing on the interplay between charge and spin degrees of freedom in itinerant electron systems, we show that itinerant frustration, which is the competition among electron-mediated interactions, plays a central role in stabilizing a variety of topological spin crystals including a skyrmion crystal with unconventional high skyrmion number, meron crystals, and hedgehog crystals. We also show that the essential ingredients in the itinerant frustration are represented by bilinear and biquadratic spin interactions in momentum space. This perspective not only provides a unified understanding of the unconventional topological spin crystals but also stimulates further exploration of exotic topological phenomena in itinerant magnets., 50 pages, 16 figures, 1 table

Molecular magnetic compounds, which combine the advantages of nanoscale behaviors with the properties of bulk magnetic materials, are particularly attractive in the fields of high-density information storage and quantum computing. Before molecular electronic devices can be fabricated, a crucial task is the measurement and understanding of the transport behaviors. Herein, we consider a magnetic molecular trimer sandwiched between two metal electrodes, and, with the aid of the sophisticated full density matrix numerical renormalization group (FDM-NRG) technique, we study the effect of magnetic anisotropy on the charge transport properties, illustrated by the local density of states (LDOS, which is proportional to the differential conductance), the Kondo effect, and the temperature and inter-monomer hopping robustness. Three kinds of energy peaks are clarified in the LDOS: the Coulomb, the Kondo and the Ruderman-Kittel-Kasuya-Yosida (RKKY) peaks. The local magnetic moment and entropy go through four different regimes as the temperature decreases. The Kondo temperature TK could be described by a generalized Haldane's formula, revealing in detail the process where the local moment is partially screened by the itinerant electrons. A relationship between the width of the Kondo resonant peak WK and TK is built, ensuring the extraction of TK from WK in an efficient way. As the inter-monomer hopping integral varies, the ground state of the trimer changes from a spin quadruplet to a magnetically frustrated phase, then to an orbital spin singlet through two first order quantum phase transitions. In the first two phases, the Kondo peak in the transmission coefficient reaches its unitary limit, while in the orbital spin singlet, it is totally suppressed. We demonstrate that magnetic anisotropy may also induce the Kondo effect, even without Coulomb repulsion, hence it is replaceable in the many-body behaviours at low temperature.

FAPEMIG - Fundação de Amparo a Pesquisa do Estado de Minas Gerais Na presente tese estudamos as propriedades físicas de impurezas magnéticas quânticas em sistemas de matéria condensada. Mais precisamente, estamos interessados em fenômenos de muitos corpos em fios quânticos metálicos com acoplamento spin-órbita (SOC) na presença de impurezas magnéticas. Particularmente, investigamos o efeito Kondo em fios quânticos supercondutores com fases topológicas que apresentem modos de Majorana de energia zero (MZMs) ligados à suas pontas. A primeira parte da tese é dedicada à discussão geral da formação de momentos magnéticos localizados e a emergência do efeito Kondo em sístemas de única impureza. Posteriormente, estudamos um sistema composto por um ponto quântico de um nível acoplado à um reservatório metálico e dois MZMs polarizados. O sistema é descrito por um modelo de Anderson de uma única impureza (SIAM) adaptado para comportar os dois MZMs. Para acessar a física de Kondo utilizamos o método projeção para deduzir um Hamiltoniano do tipo Kondo extraído do SIAM. Nós mostramos a coexistência do singleto de muitos corpos de Kondo e do singleto local de Majorana que resulta da robustez dos pontos fixos de Kondo, mesmo na presença de operadores locais de Majorana. Além disso, mostramos que os MZMs livres do sistema são responsáveis por uma entropia residual da impureza S res = N /2 kB ln(2), onde N é o número de MZMs livres. Esse resultado revela uma entropia residual fracionária para um número impar de MZMs livres associado com um comportamento local de non-Fermi liquid. Ambos os resultados foram confirmados usando o método perturbativo de Anderson, Poor Man’s scaling (PMS), e o método não perturbativo de Wilson, o grupo de renormalização numérico (NRG). A segunda parte da tese é dedicada à física de baixas temperaturas de sistemas de duas impurezas. Nós revisitamos a tradicional interação de Ruderman- Kittel-Kasuya-Yosida (RKKY) entre duas impurezas quânticas em um fio quântico com SOC. Para calcular o acoplamento RKKY usamos a abordagem original de Ruderman e Kittel, baseada na teoria quântica de perturbação. Além da interação RKKY, encontramos também interações adicionais do tipo Dzyaloshinskii-Moriya e Ising entre as impurezas originada do SOC. Nossos resultados mostram que devido ao “spin-lock” do momento, devido ao SOC, o acoplamento RKKY não decai com o inverso da distância entre as impurezas, como no caso usual. Ao invés disso, ele sobrevive a distâncias significativamente grandes e possui um comportamento oscilatório com o vetor de onda característico 2k R, onde k R é o proporcional ao acoplamento spin-órbita de Rashba. In this thesis we study the physical properties of quantum magnetic impurities in con- densed matter systems. More precisely, we are interested in the quantum many-body phenomena emerging in metallic quantum wires with spin-orbit coupling (SOC) in the presence of magnetic impurities. In particular, we investigate Kondo effect in topological superconducting quantum wires sustaining Majorana zero modes (MZMs) bound to their edges. The first part of this work is dedicated to a general digression on the formation of localized magnetic moments in solid state materials and the emergence of the Kondo effect in single-impurity systems. We further study a system composed by a single level quantum dot coupled to a metallic reservoir and to two MZMs with opposite polarizations. The system is described by a version of single impurity Anderson model (SIAM) adapted to account for the MZMs. To access the Kondo physics we used the projection approach to derive an effective Kondo-like Hamiltonian from the SIAM. We show the coexistence of the many-body Kondo singlet and a local Majorana singlet which results from the robustness of the Kondo fixed points even in the presence local Majorana operators. Moreover, we show that the free MZMs of the system are responsible for the impurity residual entropy S res = N/ 2 k B ln(2), where N is the number of free MZMs. This result reveals a fractional residual entropy for an odd number of free MZMs associated with a local non-Fermi liquid behavior. Both results were confirmed by the perturbative Anderson’s Poor Man’s scaling (PMS), and the nonperturbative Wilson’s numerical renormalization group (NRG). The second part of the thesis is dedicated to the low-temperature physics of two-impurity systems. We revisit the well known Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two spatially separated magnetic impurities in a quantum wire with SOC. To compute the RKKY coupling we use the original Ruderman-Kittel approach, based on the traditional second-order perturbation theory. This allows us find also the Dzaloshinskii-Moriya and Ising interactions between the impurities, both originated from the SOC. Our results show that due to momentum-spin-lock, provided by the SOC, the RKKY coupling does not decay with the inverse of the inter-impurities distance, as in the usual case. Instead, the coupling survives even at large distances oscillating with a characteristic wave-vector 2k R, where k R is proportional to the Rashba SOC. Tese (Doutorado)

The interaction energy of the indirect-exchange or Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between magnetic spins localized on lattice sites of the $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathcal{T}}_{3}$ model is calculated using linear response theory. In this model, the AB-honeycomb lattice structure is supplemented with C atoms at the centers of the hexagonal lattice. This introduces a parameter $\ensuremath{\alpha}$ for the ratio of the hopping integral from hub to rim and that around the rim of the hexagonal lattice. A valley and $\ensuremath{\alpha}$-dependent retarded Green's function matrix is used to form the susceptibility. Analytic and numerical results are obtained for undoped $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathcal{T}}_{3}$ when the chemical potential is finite and also in the presence of an applied magnetic field. We demonstrate the anisotropy of these results when the magnetic impurities are placed on the A, B, and C sublattice sites. Additionally, a comparison of the behavior of the susceptibility of $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathcal{T}}_{3}$ with graphene shows that there is a phase transition at $\ensuremath{\alpha}=0$.

We investigate the effect of the mass anisotropy on Friedel Oscillations, Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, screening properties, and Boltzmann transport in two dimensional (2D) metallic and doped semiconductor systems. We calculate the static polarizability and the dielectric function within the random phase approximation with the mass anisotropy fully taken into account without making any effective isotropic approximation in the theory. We find that carrier screening exhibits an isotropic behavior for small momenta despite the anisotropy of the system, and becomes strongly anisotropic above a certain threshold momentum. Such an anisotropy of screening leads to anisotropic Friedel oscillations, and an anisotropic RKKY interaction characterized by a periodicity dependent on the direction between the localized magnetic moments. We also explore the disorder limited dc transport properties in the presence of mass anisotropy based on the Boltzmann transport theory. Interestingly, we find that the anisotropy ratio of the short range disorder limited resistivity along the heavy- and light-mass directions is always the same as the mass anisotropy ratio whereas for the long range disorder limited resistivity the anisotropy ratio is the same as the mass ratio only in the low density limit, and saturates to the square root of the mass ratio in the high density limit. Our theoretical work should apply to many existing and to-be-discovered anisotropic 2D systems., 10 pages, 4 figures

Kitaev spin liquids (KSLs) contain both itinerant Majorana fermions and localized $Z_2$ flux excitations. The mobile fermions transport energy, while the fluxes hinder heat flow. By controlling the flux configuration, one can separate a KSL into different domains via $Z_2$ flux walls. We show that at low temperatures, the wall significantly suppresses the Majorana-mediated Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction and blocks the heat transfer between domains., Comment: 6 pages and 4 figures

Pyrochlore rare-earth iridates Ln\(_2\)Ir\(_2\)O\(_7\) (Ln: lanthanides) is a unique frustrated Kondo lattice system composed of localized 4f moments and Ir 5d conduction electrons. Recent active research has revealed that the Kondo coupling between the 4f electron and the Ir 5d bands leads to novel transport properties. First, we will make an overview of the phase diagram of pyrochlore rare-earth iridates Ln\(_2\)Ir\(_2\)O\(_7\). Next, we focus on the phenomena associated with spin ice physics. Ln\(_2\)Ir\(_2\)O\(_7\) (Ln \(=\) Nd, Sm, Eu, \(\ldots \)) exhibits a metal-insulator transition, while Pr\(_2\)Ir\(_2\)O\(_7\) does not show any sign of long range ordering. Both Pr and Nd moments have a local \(\langle 111 \rangle \) Ising anisotropy. In the metallic state, localized 4f moments are coupled through the RKKY interaction. For Pr\(_2\)Ir\(_2\)O\(_7\), a ferromagnetic correlation between Pr moments is developed on cooling. On the other hand, Nd\(_2\)Ir\(_2\)O\(_7\) exhibits a metal-insulator transition at 33 K, and then, all-in all-out magnetic structure of Nd moments emerges below 10 K, as observed in the neutron scattering experiments. For Nd\(_2\)Ir\(_2\)O\(_7\), antiferromagnetic correlation between Nd moments is dominant. Metal insulator transition of Nd\(_2\)Ir\(_2\)O\(_7\) can be suppressed by the application of pressure. The insulating phase disappears above 10 GPa. In the pressure induced metallic state, a new phase transition emerges around 3 K. This phase transition is likely due to ferromagnetic ordering, suggesting an ordered spin ice of Nd moment. In the metallic frustrated magnet Pr\(_2\)Ir\(_2\)O\(_7\), a spontaneous Hall effect is observed at zero field in the absence of uniform magnetization, suggesting an emergence of a chiral spin liquid. The origin of this spontaneous Hall effect is ascribed to chiral spin textures, which are inferred from the magnetic measurements indicating the spin ice-rule formation.

Manipulation of electronic and magnetic properties of topological materials is a topic of much interest in spintronic and valleytronic applications. Perturbation tuning of multiple Dirac cones on the (001) surface of topological crystalline insulators (TCIs) is also a related topic of growing interest. Here we show the numerical evidence for the ferroelectric structural distortion effects on the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two magnetic impurity moments on the SnTe (001) and related alloys. The mirror symmetry breaking between Dirac cones induced by the ferroelectric distortion could be divided into various possible configurations including the isotropically gapped, coexistence of gapless and gapped, and anisotropically gapped phases. Based on the retarded perturbed Green’s functions of the generalized gapped Dirac model, we numerically find the RKKY response for each phase. The distortion-induced symmetry breaking constitutes complex and interesting magnetic responses between magnetic moments compared to the pristine TCIs. In the specific case of coexisted gapless and gapped phases, a nontrivial behavior of the RKKY interaction is observed, which has not been seen in other Dirac materials up until now. For two impurities resided on the same sublattices, depending on the distortion strength, magnetic orders above of a critical impurity separation exhibit irregular ferromagnetic ⇔ antiferromagnetic phase transitions. However, independent of the impurity separation and distortion strength, no phase transition emerges for two impurities resided on different sublattices. This essential study sheds light on magnetic properties of Dirac materials with anisotropic mass terms and also makes TCIs applications relatively easy to understand.

Nodal-line semimetals (NLSMs) are typically characterized by a nodal ring, a closed line of Dirac points in the Brillouin zone, which features a torus-shaped Fermi surface, as well as a quantized Berry phase $\ensuremath{\pi}$. By investigating the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in NLSMs, a close relationship between the magnetic interaction and the radius ${k}_{0}$ of the nodal ring is revealed. For impurities deposited in the direction normal to the nodal-ring plane, when the long-range interaction at half-band filling follows the decaying law ${R}^{\ensuremath{-}3}$ with impurity distance, which is a result of anisotropic dispersion of semi-Dirac semimetal, the decaying rate for the relatively short-range impurity distance would be prolonged by nodal-ring radius ${k}_{0}$, which is prominent for the intermediate radius. This phenomenon originates from the competition between the contribution from the electron states inside and outside the nodal ring. In contrast, for impurities distributed in the nodal-ring plane, the RKKY interaction $J\ensuremath{\propto}{k}_{0}^{2}\mathrm{cos}(2{k}_{0}R){R}^{\ensuremath{-}4}$ shows an oscillation, from whose period one can easily determine the parameter ${k}_{0}$ of the nodal ring. Furthermore, at finite Fermi energy, we find a distinct signature of the RKKY interaction to characterize the topological trivial and topological nontrivial phases in NLSMs.

We consider a spin-fermion model consisting of free electrons coupled to classical spins where the latter are embedded in a quasi-one-dimensional superlattice structure consisting of spin blocks separated by spinless buffers. Using a spiral ansatz for the spins, we study the effect of the electron mediated Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction on the $T=0$ ground state of the system. We find that the RKKY interaction can lead to ferromagnetic, antiferromagnetic, or intermediate spiral phases for different system parameters. When the width is much larger than the length of the individual blocks, the spiral phases are suppressed, and the ground state oscillates between ferromagnetic and antiferromagnetic orders as the size of the buffer regions is varied. This is accompanied by a corresponding oscillation in the Drude weight reflecting an increased conductivity in the ferromagnetic state compared to the antiferromagnetic one. These results are reminiscent of classic giant magnetoresistance phenomena observed in a similar geometry of thin sandwiched magnetic and nonmagnetic layers. Our analysis provides a robust framework for understanding the role of the RKKY interaction on the ground-state order and corresponding transport properties of such systems, extending beyond the conventional perturbative regime.

We study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two magnetic impurities resided on the (001) plane of topological crystalline insulator (TCI) SnTe associated with four anisotropic Dirac cones protected by the mirror symmetry. The lattice Green's functions governing the correlation of both the sublattices and spins are deduced by developing a two-band effective model. We explore the RKKY Heisenberg interaction of the system established by the interference between four Dirac cones. Our key finding is that, independent of the position of magnetic impurities on the same sublattices or different ones, multiple wave vectors with different orders describe the RKKY coupling physics in TCIs. Further, both the undoped and doped spin susceptibility give rise to the same decay rates like graphene. Magnetic impurities on the same and different sublattices propose the ferromagnetic and antiferromagnetic base phase of the undoped system, respectively. Here we demonstrate that RKKY exchange in TCIs, undergoes a ferromagnetic-to-antiferromagnetic transition and vice versa in response to switching on/off the electron and hole doping. Analytic expressions of doped TCIs are obtained to draw the magnetic phase diagram of the distance between two magnetic impurities and the doping concentration, clarifying the contributions of short and long distances as well as low and high concentrations to the RKKY coupling. Our results may tailor the magnetic features of TCIs for further research and application.

The standard Metropolis algorithm and the parallel tempering method are used to study the influence of the long-range RKKY interaction on the formation of magnetization plateaus in the 3D Ising model with magnetically coupled Shastry-Sutherland layers. It is shown that depending on the Fermi momentum kF the model exhibits the rich spectrum of magnetic solutions which is demonstrated by the appearance of the large number of magnetization plateaus on the magnetization curves. In particular, the following set of individual fractional magnetization plateaus with macroscopic stability regions is observed: m∕ms = 1/8, 1/7, 1/6, 1/5, 2/9, 1/4, 1/3, 3/8, 2/5, 5/12, 3/7, 1/2, 5/9 and 2/3. In the regime of kF ~ 2π∕1.24, corresponding to the real situation in TmB4, we have confirmed the existence of the magnetization plateau with m∕ms = 1∕8, the nature of which was intensively discussed in recent experimental works. Since the change of kF can be induced by doping, our results provide predictions of the complete sequences of magnetization plateaus that could appear in tetraboride solid solutions. And finally, the current results are confronted with our previous ones obtained for the two-dimensional Ising model with the RKKY interaction in order to reveal the role of system dimension on the formation of magnetization plateaus.

The Green's function~(GF) of two localized magnetic moments embedded in the electron gas is calculated exactly. The electrons are treated in the effective mass approximation and the magnetic moments are coupled with electrons by a delta-like~$s-d$ interaction. The resulting GF is obtained by the exact summation of the Born series using a generalization of the method developed by Slater-Koster and Ziman to non-commuting spin operators with the use of the Woodbury identities. The exact GF crucially depends on the value of the one-electron GF at the origin, denoted as~$g_0$. The Born series is convergent only if~$g_0$ is finite, which holds for electrons in parabolic energy bands in~$1D$, but not in~$2D$ and~$3D$. In the general case, the exact GF includes nonlinear combination of localized spins operators. A method of calculating matrix elements of these operators is given. For spins~$S_a=S_b = 1/2$ the exact GF is expressed as a linear combination of components of~$\hS_a, \hS_b$, and the exact range function~${\cal J}(r)$ is obtained as a double integral over analytical expression. For electron energy~$E=0$ and~$J g_0/2 = 2$ or~$J g_0/2=-2/3$ the range function and GF are singular. Poles of GF occur in the vicinities of singularity points and the resulting energies of bound states are calculated. There are three regimes of~$J$. For small $J$ the range function resembles RKKY one: it has the same period~$\pi/k_F$, the same decay character and a slightly different amplitude, usually within a few percent. This regime occurs for most frequently in the nature. For~$|J|$ comparable to~$|g_0|^{-1}$ the exact range function differs qualitatively from RKKY one,. For large $|J|$ the exact range function oscillates with the same period and power-like decay as the usual RKKY function but it has much lower amplitude decaying with growing~$|J|$., Comment: 22 pages, 3 figures and 8 pages of Supplemental material

Two separate gate electrodes with opposite gate voltages can drive an AB-stacked bilayer graphene (BLG) under them into different quantum valley Hall states. And between the two topological domains there appears a one-dimensional boundary, dubbed the topological channel (TC). By means of the Lanczos method, we theoretically study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two magnetic impurities in such a TC of BLG. We find that the RKKY interaction is long-range with an ${R}^{\ensuremath{-}1}$ decay rate as the distance $R$ between two magnetic impurities increases, whereas the slowest decay rate in any graphene structure is ${R}^{\ensuremath{-}2}$ according to previous literature. Moreover, we also find that the strength and sign of the RKKY interaction in the TC can be readily controlled by altering the gate voltage or carrier doping. Such a tunable and ${R}^{\ensuremath{-}1}$ decaying RKKY interaction implies the possibility of establishing and controlling a long-range magnetic ordering state in a dilute magnetic impurity-doped BLG.

We theoretically explore the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction mediated by spin-3/2 quasiparticles in half-Heusler topological semimetals in quasi-two-dimensional geometries. We find that while the Kohn-Luttinger part of the effective Hamiltonian gives rise to generalized Heisenberg coupling of the form ${\mathcal{H}}_{\mathrm{RKKY}}\ensuremath{\propto}{\ensuremath{\sigma}}_{1,i}{\mathcal{I}}_{ij}{\ensuremath{\sigma}}_{2,j}$ with a symmetric matrix ${\mathcal{I}}_{ij}$, the addition of a small antisymmetric linear spin-orbit coupling term leads to a Dzyaloshinskii-Moriya type of coupling with an antisymmetric matrix ${\mathcal{I}}_{ij}^{\ensuremath{'}}$. As a general aspect, we demonstrate that besides revealing the usual oscillatory dependence on the distance, all coupling strengths depend strongly on the relative orientation of the two impurities with respect to the lattice. This yields a strongly anisotropic behavior for ${\mathcal{I}}_{ij}$ such that by only rotating one impurity around another at a constant distance, we can see further oscillations of the RKKY couplings. As we will explain, this unprecedented effect is very unique to electronic systems such as half-Heusler topological semimetals, which combine spin-orbit coupling with strongly anisotropic Fermi surfaces. We further find that all of the RKKY terms have two common features: (i) a tetragonal warping in their map of spatial variations in a similar way to the two-dimensional Fermi lines of the system, and (ii) a complex beating pattern consisting of different periodicities. Intriguingly, all these features survive in all dopings, and we see them in both electron- and hole-doped cases. In addition, due to the lower dimensionality combined with the effects of different spin-orbit couplings, we see that only one symmetric off-diagonal term, ${\mathcal{I}}_{xy}$, and two antisymmetric (Dzyaloshinskii-Moriya) components, ${\mathcal{I}}_{xz}^{\ensuremath{'}}$ and ${\mathcal{I}}_{yz}^{\ensuremath{'}}$, are nonvanishing, while the remaining three off-diagonal components are identically zero. This manifests in another drastic difference of RKKY interaction in half-Heusler topological semimetals compared to the electronic systems with a spin-1/2 effective description.

We present a theory for the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction mediated by a two-dimensional (2D) electron system subjected to periodic driving. This is demonstrated for a 2D metal with two ferromagnetic chains deposited in parallel. Our calculations reveal new non-analytic features in the time-averaged spin susceptibility. For weak light–matter coupling, the RKKY interaction shows oscillations with a period tunable by the light amplitude and frequency. For stronger light–matter coupling, the interaction becomes non-oscillatory and remains purely ferromagnetic. Our findings open a path forward for realizing dynamic control of the indirect exchange interaction in 2D magnetic structures.

Using the Hubbard model in the framework of the tight-binding formulation, we studied the effects of the electron–electron (e–e) interaction on the indirect magnetic exchange coupling between the magnetic impurities embedded in triangular graphene nanoflakes. The results show that the magnitude of the coupling enhances in the presence of the e–e interaction and Rashba spin–orbit interaction (RSOI). The RKKY coupling magnitude depends on the impurity positions in nanoflake and the size of the system, as well.

We theoretically study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in two-dimensional black phosphorus, phosphorene. The RKKY interaction enhances significantly for the low levels of hole doping owing to the nearly valence flat band. Remarkably, for the hole-doped phosphorene, the highest RKKY interaction occurs when two impurities are located along the zigzag direction and it tends to a minimum value with changing the direction from the zigzag to the armchair direction. We show that the interaction is highly anisotropic and the magnetic ground-state of two magnetic adatoms can be tuned by changing the rotational configuration of impurities. Owing to the anisotropic band dispersion, the oscillatory behavior with respect to the angle of the rotation is well-described by $\sin(2k_{\rm F}R)$, where the Fermi wavelength $k_{\rm F}$ changes in different directions. We also find that the tail of the RKKY oscillations falls off as $1/R^2$ at large distances., Comment: 11 pages, 9 figures

Motivated by the growing interest in solving the obstacles of spintronics applications, we study the Ruderman-Kittel-Kasuya-Yosida (RKKY) effective pairwise interaction between magnetic impurities interacting through the π -electrons embedded in both electronically doped-semiconducting and metallic armchair graphene nanoribbons. In terms of the Green’s function formalism, treated in a tight-binding approximation with hopping beyond Dirac cone approximation, the RKKY coupling is an attraction or a repulsion depending on the magnetic impurities distances. Our results show that the RKKY coupling in semiconducting nanoribbons is much more affected by doping than metallic ones. Furthermore, we found that the RKKY coupling increases with ribbon width, while there exist some critical electronic concentrations in RKKY interaction oscillations. On the other hand, we find an unusual incoming wave-vector direction for electrons which describes more clearly the ferro- and antiferromagnetic spin configurations in such system. Also, the RKKY coupling at low and high-temperature regions has been addressed for both ferro- and antiferromagnetic spin arrangements.

Silicene offers an ideal platform for exploring the phase transition due to strong spin-orbit interaction and its unique structure with strong tunability. With applied electric field and circularly polarized light, silicone is predicted to exhibit rich phases. We propose that these intricate phase transitions can be detected by measuring the bulk Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. We have in detail analyzed the dependence of RKKY interaction on phase parameters for different impurity configurations along zigzag direction. Importantly, we present an interesting comparison between different terms of RKKY interaction with phase diagram. It is found that the in-plane and out-of-plane terms can exhibit the local extreme value or change of sign at the phase critical point and remarkable difference in magnitude for different phase regions. Consequently, the magnetic measurement provides unambiguous signatures to identify various types of phase transition simultaneously, which can be carried out with present technique.

The competition between the indirect exchange interaction (IEC) of magnetic impurities in metals and the Kondo effect gives rise to a rich quantum phase diagram, the Doniach Diagram (Doniach, 1977). A Kondo screened phase is separated from a spin ordered phase when the local exchange coupling J and the concentration of magnetic moments n M are varied. In disordered metals, both the Kondo temperature and the IEC are widely distributed due to the scattering of the conduction electrons from the impurity potential. Therefore, it is a question of fundamental importance, how this Doniach diagram is modified by the disorder, and if one can still identify separate phases. Recently, Nejati et al. (2017) investigated the effect of Ruderman–Kittel–Kasuya–Yosida (RKKY) correlations on the Kondo effect of two magnetic impurities, renormalizing the Kondo interaction based on the Bethe–Salpeter equation and performing the poor men’s renormalization group (RG) analysis with the RKKY-renormalized Kondo coupling. In the present study, we extend this theoretical framework, allowing for different Kondo temperatures of two RKKY-coupled magnetic impurities due to different local exchange couplings and density of states. As a result, we find that the smaller one of the two Kondo temperatures is suppressed more strongly by the RKKY interaction, thereby enhancing their initial inequality. In order to find out if this relevance of inequalities between Kondo temperatures modifies the distribution of the Kondo temperature in a system of a finite density of randomly distributed magnetic impurities, we present an extension of the RKKY coupled Kondo RG equations. We discuss the implication of these results for the interplay between Kondo coupling and RKKY interaction in disordered electron systems and the Doniach diagram in disordered electron systems.

In this study, we investigate the spin–spin correlation of two magnetic impurities in zigzag edge MoS2 nanoflakes using a three-orbital tight-binding model. In particular, we consider the dependence of the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction on the local electric field strength for providing a degree of tunability in triangular and rectangular nanoflakes. Our results indicate that manipulating the local electric field at the zigzag edges could allow the control of all the different components of the RKKY interaction in MoS2 nanoflakes. Interestingly, the Dzyaloshinskii–Moriya JDM oscillates but does not decay due to strong spin–orbit coupling. Moreover, the electric field induces a sizable JDM interaction as a function of the magnetic separation along zigzag edges for various Fermi energies. Our findings provide a method for enhancing the RKKY exchange interactions with magnetic arrays in these materials.

A study of the physical properties of spin glass systems (SG) is carried out within the Monte Carlo technique. The studied systems are the “classic” CuMn and AuFe metallic alloys, where the RKKY interaction is used as a model for the interaction among the magnetic (Mn, Fe) atoms. A careful discussion about the approach to the equilibrium is addressed. By introducing a cutoff radius in the range on the spin-spin interaction, the results show the influence of this range on the spin-glass behavior of the alloys, where the SG behavior is only observed when the full (long range) RKKY is taken into account, specially for low concentrations x

We derive a theory of carrier induced local moment magnetization of p-type Sn1−xMnxTe based on the Hubbard model, k → · π → electronic structure method ( k → is the electronic wave vector and π → is the relativistic momentum operator) and the statistical paramagnetic approach for the localized moments. The Hubbard model is used to derive an internal exchange magnetic field. The difference in exchange self-energy is expressed in terms of an internal exchange field that is proportional to the parameter U, the onsite Coulomb repulsion, and the spin-density of carriers. In the present theory, the k → · π → + U model is integrated with the statistical paramagnetic theory for localized spins, which is then solved in a self-consistent manner by adding the exchange field to the applied field. The technique is applied to study the magnetic properties of p-type Sn1−xMnxTe, an important material for spintronics devices. The local moment magnetization calculated using the total magnetic field self-consistently agrees with the experimental observations. Magnetization and the exchange field studied as functions of the applied field, temperature and carrier concentration yield results on expected lines. Ours is a mechanism that is different from the RKKY interaction, normally invoked for carrier induced ferromagnetism and is thus a novelty.

We propose an RKKY-type interaction that is mediated by a spin liquid. If a spin liquid ground state exists such an interaction could leave a fingerprint by ordering underlying localized moments such as nuclear spins. This interaction has a unique phenomenology that is distinct from the RKKY interaction found in fermionic systems; most notably the lack of a Fermi surface and absence of the requirement for itinerant electrons, since most spin liquids are insulators. As a working example we investigate the two-dimensional spin-1/2 kagome antiferromagnet (KAFM), although the treatment remains general and can be extended to other spin liquids and dimensions. We find that several different nuclear spin orderings minimize the RKKY-type energy induced by the KAFM but are unstable due to a zero-energy flat magnon band. Despite this we show that a small magnetic field is able to gap out this magnon spectrum for some of the orderings resulting in an intricate nuclear magnetism., Comment: Published in Scientific Reports. 13 pages, 6 figures. Supplementary material available (see journal reference)

We examine the RKKY interactions of CeB$_6$ between multipole moments based on the effective Wannier model obtained from the bandstructure calculation including 14 Ce-$f$ orbitals and 60 conduction orbitals of Ce-$d,s$ and B-$p,s$. By using the $f$-$c$ mixing matrix elements of the Wannier model together with the conduction band dispersion, the multipole couplings with the RKKY oscillation are obtained for the active moments in $\Gamma_{8}$ subspace. Both of the $\Gamma_{5g}$ quadrupole $O_{xy}$ and the $\Gamma_{2u}$ octupole $T_{xyz}$ couplings are largely enhanced with $\bm{q}=(\pi,\pi,\pi)$ which naturally explains the antiferro-quadrupolar phase of the phase II, and are also enhanced with $\bm{q}=(0,0,0)$ corresponding to the elastic softening of $C_{44}$. Also the couplings of the $\Gamma_{5u}$ octupole $T_{z}^{\beta}$ is quite large for $\bm{q}=(0,0,\pi)$ which is related to the antiferro-octupolar ordering of a possible candidate for the phase IV of Ce$_{x}$La$_{1-x}$B$_6$., Comment: 6 pages, 2 figures. arXiv admin note: text overlap with arXiv:1904.06469. Accepted for publication in JPS Conference Proceedings(SCES2019)

We study the Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling between localized magnetic impurities in the ${\text{MoS}}_{2}$ nanoribbons. Our calculations are based on a three-orbital tight-binding model for a ${\text{MoS}}_{2}$ finite nanoribbon system with periodic boundary conditions. We consider impurities hybridized to different d orbitals of the host transition metal. The large intrinsic spin-orbit coupling leads to the spatial anisotropy in the RKKY interaction amplitude and appearance of collinear and noncollinear terms. We find that the RKKY interaction is sensitive to the Fermi energy values and changes dramatically in doped systems. We analyze the exponent decay behavior of the RKKY interaction envelope for doping levels inside the band gap of the infinite layer.

By means of the Lanczos method, a numerically efficient theoretical approach, we study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in a graphene bubble. We find that the RKKY interaction in a graphene bubble can be larger or smaller than the corresponding result in pristine graphene by a few orders of magnitude, depending on the sublattice attribution and pseudomagnetic field strength where two magnetic impurities are positioned, which is due to the sublattice polarization of the low-energy electronic states in a strong pseudomagnetic field. If two magnetic impurities are both in the bubble region, the ${R}^{\ensuremath{-}3}$ decay rate of the RKKY interaction found in pristine graphene breaks down. But it recovers when one magnetic impurity is far away from the bubble center no matter where another impurity is located. When the Fermi level deviates from the Dirac point by carrier doping, the antiferromagnetic RKKY interaction between two magnetic impurities located at the opposite sublattices can be inverted to be ferromagnetic by altering the bubble height properly.

We propose a physical mechanism to generate and selectively amplify anisotropic Rudermann-Kittel-Kasuya-Yosida (RKKY) interactions between two local spins. The idea is to combine the deflection of the carrier velocity by a P-N interface and the locking of this velocity to the carrier spin orientation via spin-orbit coupling. We provide analytical and numerical results to demonstrate this mechanism on the surface of a topological insulator P-N junction. This work identifies the P-N interface as a second knob which, together with the carrier density, enables independent control of the strength and anisotropy of the RKKY interaction over a wide range. These findings may be relevant to scalable quantum computation and two-impurity quantum criticality., 10 pages, 6 figures

We theoretically investigate the indirect signatures of the tilted anisotropic Dirac cones on Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction in a two dimensional polymorph of boron atoms. Unlike the case of isotropic non-tilted Dirac material-graphene, here we observe that the tilting of the Dirac cones exhibits a significant impact on the RKKY exchange interaction in terms of the suppression of oscillation frequency. The reason can be attributed to the behavior of the Fermi level and the corresponding density of states with respect to the tilting parameter. The direct measurement of the period of the RKKY interaction can thus be a possible probe of the tilt parameter associated with the tilted Dirac cones. We also obtain the direction dependent analytical expressions of the RKKY exchange interaction, in terms of Meijer G-function. However, the effects due to tilting of the Dirac cones on the RKKY interaction depend on the spatial alignments of the two magnetic impurities with respect to the direction of tilting., This is the published version

In our publication from 8 years ago (Phys. Rev. B {\bf 84}, 115119 (2011)) we calculated RKKY interaction between two magnetic impurities adsorbed on graphene at zero temperature. We show in this short paper that the approach based on Matsubara formalism and perturbation theory for the thermodynamic potential in the imaginary time and coordinate representation which was used then, can be easily generalized, and calculate RKKY interaction between the magnetic impurities at finite temperature., LaTex, 4 pages. arXiv admin note: text overlap with arXiv:1108.1306

The relationship between magnetism and superconductivity has been one of the most discussed topics in iron-based superconductors. Using first-principles calculations, we have studied the electronic structure of 1144-type iron-based superconductor EuRbFe4As4. We find the crystal field splitting of EuRbFe4As4 is unique, such that the $$d_{z^2}$$ d z 2 orbitals are closer to the Fermi level $$\epsilon _{\mathrm F}$$ ϵ F than the dxy orbitals. The Ruderman−Kittel−Kasuya−Yosida (RKKY) interaction strength is approximately 0.12 meV in pristine EuRbFe4As4. Upon Ni-doping on the Fe site, the RKKY interaction strength is barely changed upon Ni-doping due to the highly anisotropic Fermi surfaces and multiband effect, despite the drastically reduced dzx(y) density of state at $$\epsilon _{\mathrm F}$$ ϵ F . Finally, in both pristine and doped compounds, the RKKY interaction is primarily mediated through bands due to Fe-$$d_{z^2}$$ d z 2 orbitals. Our calculations suggest the RKKY interaction mediated by $$d_{z^2}$$ d z 2 orbital is probably responsible for the magnetism in EuRbFe4As4 and doesn’t change upon Ni-doping.