Heisenberg Exchange Research Articles

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The topological textures of a two-dimensional (2D) magnetic monolayer, where ferromagnetic (FM) Heisenberg exchange (HE) and anisotropic Dzyaloshinskii-Moriya (DM) interactions coexist, are investigated with a quantum computational method. We find from our calculated results that when the external magnetic field, exerted perpendicularly to the 2D magnetic monolayer, is gradually increased, FM helical, antiskyrmionic and antivortical lattices (abbreviated as HL, ASL and AVL respectively), spin textures consisting of skyrmions and bimerons, and those of pure bimerons, are induced on the magnetic monolayer successively. The antiskyrmionic lattice states prevail over a broad area in the T-H phase diagram, and the wavelengths of the ASLs and AVLs change discontinuously with the increasing external magnetic field. For comparison, two sets of formulas are used to calculate the topological charge Qav per particle-like spin texture, and the topological charge density, ρi, over the monolayer. Consequently, the calculated ρi contour of every ASL and AVL also forms periodical and symmetric crystals that coincides well with the corresponding spin texture, and divides the ASL or AVL into serval areas with distinct spin structures. For the ASL and spin textures consisting of skyrmions and bimerons, if the formulas of Berg and M. Lüscher (1981) are applied and every bimeron is treated as a skyrmion, the averaged Qav per antiskyrmion and per bimeron are all equal to 1. We also explain why the quantum method is able to work beyond the classical ones.

Valmet invests in filter fabric production in India

The topological textures of a two-dimensional (2D) magnetic monolayer, where ferromagnetic (FM) Heisenberg exchange (HE) and anisotropic Dzyaloshinskii-Moriya (DM) interactions coexist, are investigated with a quantum computational method. We find from our calculated results that when the external magnetic field, exerted perpendicularly to the 2D magnetic monolayer, is gradually increased, FM helical, antiskyrmionic and antivortical lattices (abbreviated as HL, ASL and AVL respectively), spin textures consisting of skyrmions and bimerons, and those of pure bimerons, are induced on the magnetic monolayer successively. The antiskyrmionic lattice states prevail over a broad area in the T-H phase diagram, and the wavelengths of the ASLs and AVLs change discontinuously with the increasing external magnetic field. For comparison, two sets of formulas are used to calculate the topological charge Qav per particle-like spin texture, and the topological charge density, ρi, over the monolayer. Consequently, the calculated ρi contour of every ASL and AVL also forms periodical and symmetric crystals that coincides well with the corresponding spin texture, and divides the ASL or AVL into serval areas with distinct spin structures. For the ASL and spin textures consisting of skyrmions and bimerons, if the formulas of Berg and M. Lüscher (1981) are applied and every bimeron is treated as a skyrmion, the averaged Qav per antiskyrmion and per bimeron are all equal to 1. We also explain why the quantum method is able to work beyond the classical ones.

Spin dynamics from a constrained magnetic tight-binding model

A dynamics of the precession of coupled atomic moments in the tight-binding (TB) approximation is presented. By implementing an angular penalty functional in the energy that captures the magnetic effective fields self-consistently, the motion of the orientation of the local magnetic moments is observed faster than the variation of their magnitudes. This allows the computation of the effective atomic magnetic fields that are found consistent with the Heisenberg's exchange interaction, by comparison with classical atomistic spin dynamics on Fe, Co and Ni magnetic clusters.

Quantum phases of spin-1 system on 3/4 and 3/5 skewed ladders

We study the quantum phase transitions of frustrated antiferromagnetic Heisenberg spin-1 systems on the 3/4 and 3/5 skewed two leg ladder geometries. These systems can be viewed as arising by periodically removing rung bonds from a zigzag ladder. We find that in large systems, the ground state (gs) of the 3/4 ladder switches from a singlet to a magnetic state for J1≥1.82; the gs spin corresponds to the ferromagnetic alignment of effective S=2 objects on each unit cell. The gs of antiferromagnetic exchange Heisenberg spin-1 system on a 3/5 skewed ladder is highly frustrated and has spiral spin arrangements. The amplitude of the spin density wave in the 3/5 ladder is significantly larger compared to that in the magnetic state of the 3/4 ladder. The gs of the system switches between singlet state and low spin magnetic states multiple times on tuning J1 in a finite size system. The switching pattern is nonmonotonic as a function of J1 and depends on the system size. It appears to be the consequence of a higher J1 favoring a higher spin magnetic state and the finite system favoring a standing spin wave. For some specific parameter values, the magnetic gs in the 3/5 system is doubly degenerate in two different mirror symmetry subspaces. This degeneracy leads to spontaneous spin-parity and mirror symmetry breaking, giving rise to spin current in the gs of the system.

Arsenic(III)-Capped 12-Tungsto-2-Arsenates(III) [M2(AsIIIW6O25)2(AsIIIOH)x]n- (M = CrIII, FeIII, ScIII, InIII, TiIV, MnII) and Their Magnetic Properties.

Six arsenic(III)-capped 12-tungsto-2-arsenates(III) of the type [M2(AsIIIW6O25)2(AsIIIOH)x]n- (M = CrIII, 1; FeIII, 2; ScIII, 3; InIII, 4; TiIV, 5; MnII, 6) have been synthesized in aqueous medium by direct reaction of the elements using a one-pot strategy and structurally characterized by FT-IR spectroscopy, single-crystal XRD, and elemental analysis. Polyanions 1-6 are comprised of two octahedrally coordinated guest metal ions M sandwiched between two {AsW6} units, resulting in a structure with C2h point-group symmetry. Polyanions 1-5 contain tri- and tetravalent metal ion guests M (M = CrIII, FeIII, ScIII, InIII, and TiIV, respectively), and they have one {AsIIIOH} group grafted on each {AsW6} unit, whereas the divalent MnII-containing derivative 6 has two such {AsIIIOH} groups grafted on each {AsW6} unit. Magnetic studies on polyanions 3-5 over the temperature range 1.8-295 K and magnetic fields of 0-7 T confirmed that they are diamagnetic. On the other hand, polyanions 1, 2, and 6 are strongly magnetic and follow the Curie-Weiss law above 30 K. The susceptibility plots of 1 and 6 exhibit broad peaks suggesting short-range antiferromagnetic ordering, while the very weak antiferromagnetic ordering of 2 is overshadowed by traces of a paramagnetic impurity. The magnetization data of 1, 2, and 6 at 1.8 K over 0-7 T were analyzed by using the Heisenberg exchange procedure. Small (negative) values of the obtained J values help in understanding the absence of long-range antiferromagnetic ordering.

Transmission-based tomography for spin qubits

We consider a system of static spin qubits embedded in a one-dimensional spin coherent channel and develop a scheme to readout the state of one and two qubits separately. We use unpolarized flying qubits for this purpose that scatter off from the static qubits due to the Heisenberg exchange interaction. Analysing the transmission coefficient as a function of density matrix elements along with additional unitary gates we reconstruct the state of static qubits.

Quantum phases and thermodynamics of a frustrated spin-1/2 ladder with alternate Ising–Heisenberg rung interactions

We study a frustrated two-leg spin ladder with alternate isotropic Heisenberg and Ising rung exchange interactions, whereas, interactions along legs and diagonals are Ising-type. All the interactions in the ladder are anti-ferromagnetic in nature and induce frustration in the system. This model shows four interesting quantum phases: (i) stripe rung ferromagnetic (SRFM), (ii) stripe rung ferromagnetic with edge singlet (SRFM-E), (iii) anisotropic antiferromagnetic (AAFM), and (iv) stripe leg ferromagnetic (SLFM) phase. We construct a quantum phase diagram for this model and show that in stripe rung ferromagnet (SRFM), the same type of sublattice spins (either isotropic S-type or discrete anisotropic σ-type spins) are aligned in the same direction. Whereas, in anisotropic antiferromagnetic phase, both S and σ-type of spins are anti-ferromagnetically aligned with each other, two nearest S spins along the rung form an anisotropic singlet bond whereas two nearest σ spins form an Ising bond. In large Heisenberg rung exchange interaction limit, spins on each leg are ferromagnetically aligned, but spins on different legs are anti-ferromagnetically aligned. The thermodynamic quantities like specific heat C v (T), magnetic susceptibility χ(T) and thermal entropy S(T) are also calculated using the transfer matrix method for various phases. The magnetic gap in the SRFM and the SLFM can be noticed from χ(T) and C v (T) curves.

Magnetoentropic mapping and computational modeling of cycloids and skyrmions in the lacunar spinels GaV4S8 and GaV4Se8

We report the feasibility of using magnetoentropic mapping for the rapid identification of magnetic cycloid and skyrmion phases in uniaxial systems, based on the GaV4S8 and GaV4Se8 model skyrmion hosts with easy-axis and easy-plane anisotropies respectively. We show that these measurements can be interpreted with the help of a simple numerical model for the spin Hamiltonian to yield unambiguous assignments for both single phase regions and phase boundaries. In the two lacunar spinel chemistries, we obtain excellent agreement between the measured magnetoentropic features and a minimal spin Hamiltonian built on Heisenberg exchange, single-ion anisotropy, and anisotropic Dzyaloshinskii-Moriya interactions. In particular, we identify characteristic high-entropy behavior in the cycloid phase that serves as a precursor to the formation of skyrmions at elevated temperatures and is a readily-measurable signature of this phase transition. Our results demonstrate that rapid magnetoentropic mapping guided by numerical modeling is an effective means of understanding the complex magnetic phase diagrams innate to skyrmion hosts. One notable exception is the observation of an anomalous, low-temperature high-entropy state in the easy-plane system GaV$_4$Se$_8$, which is not captured in the numerical model. Possible origins of this state are discussed.

Heisenberg and anisotropic exchange interactions in magnetic materials with correlated electronic structure and significant spin-orbit coupling

The Dzyaloshinskii-Moriya (DM) interaction, as well as symmetric anisotropic exchange, are important ingredients for stabilizing topologically non-trivial magnetic textures, such as, e.g., skyrmions, merons and hopfions. These types of textures are currently in focus from a fundamental science perspective and they are also discussed in the context of future spintronics information technology. While the theoretical understanding of the Heisenberg exchange interactions is well developed, it is still a challenge to access, from first principles theory, the DM interaction as well as the symmetric anisotropic exchange, which both require a fully-relativistic treatment of the electronic structure, in magnetic systems where substantial electron-electron correlations are present. Here, we present results of a theoretical framework which allows to compute these interactions in any given system and demonstrate its performance for several selected cases, for both bulk and low-dimensional systems. We address several representative cases, including the bulk systems CoPt and FePt, the B20 compounds MnSi and FeGe as well as the low-dimensional transition metal bilayers Co/Pt(111) and Mn/W(001). The effect of electron-electron correlations is analyzed using dynamical mean-field theory on the level of the spin-polarized $T$-matrix + fluctuating exchange (SPTF) approximation, as regards the strength and character of the isotropic (Heisenberg) and anisotropic (DM) interactions in relation to the underlying electronic structure. Our method can be combined with more advanced techniques for treating correlations, e.g., quantum Monte Carlo and exact diagonalization methods for the impurity solver of dynamical mean-field theory. We find that correlation-induced changes of the DM interaction can be rather significant, with up to five-fold modifications in the most distinctive case.

Prioritization of the influence of different exchange interactions and uniaxial anisotropy on the generation and morphology of skyrmions

Different magnetic interactions and anisotropy in the multi-layer structure have different effects on the formation and morphology of skyrmions under zero field. In this paper, we show that the Ruderman–Kittel–Kasuya–Yosida (RKKY) coupling plays a dominant role in the generation of skyrmions but has no significant effect on the size of skyrmions. When the RKKY coupling is weak, if the Dzyaloshinskii–Moriya interaction (DMI) is strong enough, some irregular cruciform domain states can be formed. With the increase of the DMI, the shape of skyrmions is altered from Néel skyrmions to skyrmioniums, and when the Heisenberg exchange is relatively low, complex domain states, such as the labyrinth are formed. However, when the Heisenberg coupling is comparatively high, the spin states of the Néel-skyrmions are reversed. Both the Heisenberg exchange interaction and uniaxial anisotropy have a positive influence on reducing the diameters of skyrmions. However, skyrmions are difficult to generate when the Heisenberg exchange and anisotropy are strong enough. We find the prioritization of the effects of different magnetic interactions and uniaxial anisotropy on the formation and morphology of skyrmions by analyzing the energy diagrams, combined with the magnetization configuration in different pairs of the exchange interactions and anisotropy, in which the RKKY coupling has the greatest influence, followed by the DMI, then the Heisenberg exchange and finally the uniaxial anisotropy.

Magnetic phase diagram of single-layer CrBr3 * *Project supported by the National Natural Science Foundation of China (Grant No. 11625415). Z.-G. Fu acknowledges the Innovation Development Foundation of China Academy of Engineering Physics (Grant No. ZYCX1921-02).

We theoretically provide a magnetic phase diagram for the single-layer (SL) CrBr3, which could be effectively tuned by both strain engineering and charge doping in SL-CrBr3. Through systematical first-principles calculations and Heisenberg model Hamiltonian simulations, three different magnetic phases in SL-CrBr3, which are off-plane ferromagnetic, in-plane ferromagnetic and in-plane Néel-antiferromagnetic phases, are found in the strain and charge doping regimes we studied. Furthermore, our results show that higher order Heisenberg exchange parameters and anisotropy exchange parameters should be taken into account for accurately illustrating the magnetic phase transition in SL-CrBr3. As a result, we find from the SpinW simulation that the Curie temperature is about T c = 38.4 K, which is well consistent with the experimental result 34 K [Nano Lett. 19 3138 (2019)]. The findings here may be confirmed in future experiments, and may be useful for the potential applications of SL-CrBr3 in spintronics field.

Heisenberg Exchange and Dzyaloshinskii–Moriya Interaction in Ultrathin Pt(W)/CoFeB Single and Multilayers

We present results of the analysis of Brillouin light-scattering (BLS) measurements of spin waves performed on ultrathin single and multirepeat CoFeB layers with adjacent heavy metal layers. From a detailed study of the spin-wave dispersion relation, we independently extract the Heisenberg exchange interaction (also referred to as symmetric exchange interaction), the Dzyaloshinskii–Moriya interaction (DMI, also referred to as antisymmetric exchange interaction), and the anisotropy field. We find a large DMI in CoFeB thin films adjacent to a Pt layer and nearly vanishing DMI for CoFeB films adjacent to a W layer. Furthermore, the influence of the dipolar interaction on the dispersion relation and on the evaluation of the Heisenberg exchange parameter is demonstrated. Eventually, an experimental analysis of the impact of the DMI on the spin-wave lifetime is presented.

Molecular-field-theory fits to magnetic susceptibilities of antiferromagnetic GdCu2Si2, CuO, LiCrO2, and [formula omitted]-CaCr2O4 single crystals below their Néel temperatures

A recently-developed molecular field theory (MFT) has been used to fit single-crystal magnetic susceptibility χ versus temperature T data below the respective antiferromagnetic ordering temperature TN for a variety of collinear and coplanar noncollinear Heisenberg antiferromagnets. The spins in the system are assumed to interact by Heisenberg exchange and to be identical and crystallographically equivalent. The fitting parameters for χ(T) of collinear antiferromagnets are measurable quantities: the Weiss temperature θp in the Curie–Weiss law, TN,χ(TN), and the spin S. For coplanar noncollinear helix and cycloid structures, an additional fitting parameter is the turn angle between layers of ferromagnetically-aligned spins. Here MFT fits to anisotropic χ(T) data from the literature for single crystals of the collinear antiferromagnets GdCu2Si2 and CuO and the noncollinear antiferromagnets LiCrO2 with a 120° cycloidal structure and α-CaCr2O4 with a 120° helical structure below their respective Néel temperatures are presented. The MFT fit to the anisotropic χ(T⩽TN) data for CuO is poor, whereas the fits to the data for GdCu2Si2, LiCrO2, and α-CaCr2O4 are quite good. The poor fit for CuO is attributed to the influence of strong quantum fluctuations associated with the small Cu spin and the quasi-one-dimensional magnetism that are not taken into account by the MFT. The magnetic contribution to the zero-field heat capacity of the collinear antiferromagnet GdNiGe3 at T⩽TN is also fitted by the MFT.

Itinerant Topological Magnons in SU(2) Symmetric Topological Hubbard Models with Nearly Flat Electronic BandsSupported by the National Natural Science Foundation of China (Grant No. 11774152), and National Key R&D Program of China (Grant No. 2016YFA0300401).

We show that a suitable combination of flat-band ferromagnetism, geometry and nontrivial electronic band topology can give rise to itinerant topological magnons. An SU(2) symmetric topological Hubbard model with nearly flat electronic bands, on a Kagome lattice, is considered as the prototype. This model exhibits ferromagnetic order when the lowest electronic band is half-filled. Using the numerical exact diagonalization method with a projection onto this nearly flat band, we can obtain the magnonic spectra. In the flat-band limit, the spectra exhibit distinct dispersions with Dirac points, similar to those of free electrons with isotropic hoppings, or a local spin magnet with pure ferromagnetic Heisenberg exchanges on the same geometry. Significantly, the non-flatness of the electronic band may induce a topological gap at the Dirac points, leading to a magnonic band with a nonzero Chern number. More intriguingly, this magnonic Chern number changes its sign when the topological index of the electronic band is reversed, suggesting that the nontrivial topology of the magnonic band is related to its underlying electronic band. Our work suggests interesting directions for the further exploration of, and searches for, itinerant topological magnons.

Spin diffusion and spin conductivity in the two-dimensional Hubbard model

We study the spin diffusion and spin conductivity in the square lattice Hubbard model by using the finite-temperature Lanczos method. We show that the spin diffusion behaves differently from the charge diffusion and has a nonmonotonic $T$ dependence. This is due to a progressive liberation of charges that contribute to spin transport and enhance it beyond that active at low temperature due to the Heisenberg exchange. We further show that going away from half-filling and zero magnetization increases the spin diffusion, but that the increase is insufficient to reconcile the difference between the model calculations and the recent measurements on cold-atoms.

A DMI Guide to Magnets Micro-World

Dzyaloshinskii-Moriya interaction, DMI in short, represents an antisymmetric type of magnetic interactions that favour orthogonal orientation of spins and competes with Heisenberg exchange. Being introduced to explain weak ferromagnetism in antiferromagnets without an inversion center between magnetic atoms such an anisotropic interaction can be used to analyze other non-trivial magnetic structures of technological importance including spin spirals and skyrmions. Despite the fact that the corresponding DMI contribution to the magnetic energy of the system has a very compact form of the vector product of spins, the determination of DMI from first-principles electronic structure is a very challenging methodological and technical problem whose solution opens a door into the fascinating microscopic world of complex magnetic materials. In this paper we review a few such methods developed by us for calculating DMI and their applications to study the properties of real materials.

New Syntheses, Analytic Spin Hamiltonians, Structural and Computational Characterization for a Series of Tri-, Hexa- and Hepta-Nuclear Copper (II) Complexes with Prototypic Patterns

We present a series of pyrazolato-bridged copper complexes with interesting structures that can be considered prototypic patterns for tri-, hexa- and hepta- nuclear systems. The trinuclear shows an almost regular triangle with a μ3-OH central group. The hexanuclear has identical monomer units, the Cu6 system forming a regular hexagon. The heptanuclear can be described as two trinuclear moieties sandwiching a central copper ion via carboxylate bridges. In the heptanuclear system, the pyrazolate bridges are consolidating the triangular faces, which are sketching an elongated trigonal antiprism. The magnetic properties of these systems, dominated by the strong antiferromagnetism along the pyrazolate bridges, were described transparently, outlining the energy levels formulas in terms of Heisenberg exchange parameters J, within the specific topologies. We succeeded in finding a simple Kambe-type resolution of the Heisenberg spin Hamiltonian for the rather complex case of the heptanuclear. In a similar manner, the weak intermolecular coupling of two trimer units (aside from the strong exchange inside triangles) was resolved by closed energy formulas. The hexanuclear can be legitimately proposed as a case of coordination-based aromaticity, since the phenomenology of the six-spins problem resembles the bonding in benzene. The Broken-Symmetry Density Functional Theory (BS-DFT) calculations are non-trivial results, being intrinsically difficult at high nuclearities.

Collective excitations in the tetravalent lanthanide honeycomb antiferromagnetNa2PrO3

Thermomagnetic and inelastic neutron scattering measurements on Na2PrO3 are reported. This material is an antiferromagnetic honeycomb magnet based on the tetravalent lanthanide Pr4+ and has been proposed to host dominant antiferromagnetic Kitaev interactions. These measurements reveal magnetic fluctuations in Na2PrO3 below an energy of 2 meV as well as crystal-field excitations around 230 meV. The latter energy is comparable to the scale of the spin-orbit interaction and explains both the very small effective moment of around 1.0 {\mu}B per Pr4+ and the difficulty to uncover any static magnetic scattering below the ordering transition at TN = 4.6 K. By comparing the low-energy magnetic excitations in Na2PrO3 to that of the isostructural spin-only compound, Na2TbO3, a microscopic model of exchange interactions is developed that implicates dominant and surprisingly large Heisenberg exchange interactions J approximately equal to 1.1(1) meV. Although antiferromagnetic Kitaev interactions with K less than or equal to 0.2J cannot be excluded, the inelastic neutron scattering data of Na2PrO3 is best explained with a delta = 1.24(2) easy-axis XXZ exchange anisotropy.

Interplay between multipolar spin interactions, Jahn-Teller effect, and electronic correlation in a Jeff=32 insulator

In this work we study the complex entanglement between spin interactions, electron correlation and Janh-Teller structural instabilities in the 5d$^1$ $J_{eff}=\frac{3}{2}$ spin-orbit coupled double perovskite $\rm Ba_2NaOsO_6$ using first principles approaches. By combining non-collinear magnetic calculations with multipolar pseudospin Hamiltonian analysis and many-body techniques we elucidate the origin of the observed quadrupolar canted antifferomagnetic. We show that the non-collinear magnetic order originates from Jahn-Teller distortions due to the cooperation of Heisenberg exchange, quadrupolar spin-spin terms and both dipolar and multipolar Dzyaloshinskii-Moriya interactions. We find a strong competition between ferromagnetic and antiferromagnetic canted and collinear quadrupolar magnetic phases: the transition from one magnetic order to another can be controlled by the strength of the electronic correlation ($U$) and by the degree of Jahn-Teller distortions.

A potential all-electronic route to the charge-density-wave phase in monolayer vanadium diselenide

The transition metal dichalcogenides offer significant promise for the tunable realisation and application of correlated electronic phases. However, tuning their properties requires an understanding of the physical mechanisms underlying their experimentally observed ordered phases, and in particular the extent to which lattice vibrations are a necessary ingredient. Here we present a potential mechanism for charge-density-wave formation in monolayers of vanadium diselenide in which the key role at low energies is played by a combination of electron–electron interactions and nesting. There is a competition between superconducting and density-wave fluctuations as sections of the Fermi surface are tuned to perfect nesting. This competition leads to charge-density-wave order when the effective Heisenberg exchange interaction is comparable to the effective Coulomb repulsion. When all effective interactions are purely repulsive, it results instead in d-wave superconductivity. We discuss the possible role of lattice vibrations in enhancing the effective Heisenberg exchange during the earlier stages of the renormalisation group flow.