Effective Spin Model Research Articles

Finite-temperature magnetic properties of Sm2Fe17Nx using an ab initio effective spin model

In this study, we investigate the finite-temperature magnetic properties of Sm2Fe17Nx (x = 0,3) using an effective spin model constructed based on the information obtained by first-principles calculations. We find that assuming the plausible trivalent Sm3+ configuration results in a model that can satisfactorily describe the magnetization curves of Sm2Fe17N3. By contrast, the model based on the divalent Sm2+ configuration is suitable to reproduce the magnetization curves of Sm2Fe17. These results expand the understanding of how electronic structure affects the magnetic properties of these compounds.

Statistical and analytical approaches to finite-temperature magnetic properties of the compound SmFe12

To investigate the magnetic properties of $\mathrm{Sm}{\mathrm{Fe}}_{12}$, we construct an effective spin model, where magnetic moments, crystal-field (CF) parameters, and exchange fields at 0 K are determined by first-principles calculations. Finite-temperature magnetic properties are investigated by using this model. We further develop an analytical method with strong mixing of states with a different quantum number of angular momentum $J$ ($J$-mixing), which is caused by a strong exchange field acting on the spin component of $4f$ electrons. Comparing our analytical results with those calculated by Boltzmann statistics, we clarify that the previous analytical studies for Sm transition-metal compounds overestimate the $J$-mixing effects. The present method enables us to perform a quantitative analysis of the temperature dependence of magnetic anisotropy (MA) with high reliability. The analytical method with model approximations reveals that the $J$-mixing caused by the exchange field increases the spin angular momentum, which enhances the absolute value of the orbital angular momentum and MA constants via spin-orbit interaction. It is also clarified that these $J$-mixing effects remain even above room temperature. Magnetization of $\mathrm{Sm}{\mathrm{Fe}}_{12}$ shows a peculiar field dependence known as the first-order magnetization process (FOMP), where the magnetization shows an abrupt change at a certain magnetic field. The result of the analysis shows that the origin of FOMP is attributed to competitive MA constants between positive ${K}_{1}$ and negative ${K}_{2}$. The sign of ${K}_{1(2)}$ appears due to an increase in the CF potential denoted by the parameter ${A}_{2}^{0}\ensuremath{\langle}{r}^{2}\ensuremath{\rangle}$ (${A}_{4}^{0}\ensuremath{\langle}{r}^{4}\ensuremath{\rangle}$) caused by hybridization between $3d$-electrons of Fe on the $8i$ ($8j$) site and $5d$ and $6p$ valence electrons on the Sm site. It is verified that the requirement for the appearance of FOMP is given as $\ensuremath{-}{K}_{2}<{K}_{1}<\ensuremath{-}6{K}_{2}$.

Simulating Exotic Phases of Matter with Bond-Directed Interactions with Arrays of Majorana-Cooper Pair Boxes.

It is suggested that networks of Majorana-Cooper pair boxes connected by metallic nanowires can simulate various exotic states of matter. In this simulation Majorana-Cooper boxes play the role of effective spins S=1/2 and the metallic connections generate the Kondo screening and the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. Depending on what prevails-whether it is the Kondo effect or the RKKY exchange, one will have either a Kondo lattice or an effective spin model. The list of exotic states includes the famous hexagonal Kitaev model, a generalization of this model for a Kondo lattice, and various spin models with three-spin interactions. Special emphasis is made on the discussion of the Kondo lattice scenario.

Computational design of f -electron Kitaev magnets: Honeycomb and hyperhoneycomb compounds A2PrO3 ( A= alkali metals)

The Kitaev spin model offers an exact quantum spin liquid in the ground state, which has stimulated exploration of its material realization over the last decade. Thus far, most of the candidates are found in $4d$- and $5d$-electron compounds, in which the low-spin $d^5$ electron configuration subject to strong spin-orbit coupling comprises a Kramers doublet with the effective angular momentum $j_{\rm eff}=1/2$ and gives rise to the bond-dependent anisotropic interactions in the Kitaev model. Here we theoretically investigate other candidates in $4f$-electron compounds with the $f^1$ electron configuration on both quasi-two-dimensional honeycomb and three-dimensional hyperhoneycomb structures, $A_2$PrO$_3$ with $A$=Li, Na, K, Rb, and Cs. Based on {\it ab initio} calculations, we show that the electronic structures of these compounds host a spin-orbital entangled Kramers doublet with $j_{\rm eff}=1/2$ in the $\Gamma_7$ state. By constructing the tight-binding Hamiltonian and performing a perturbation expansion, we find that the low-energy magnetic properties of $A_2$PrO$_3$ are well described by an effective spin model with the $J$-$K$-$\Gamma'$ model. The most remarkable feature is that the Kitaev interaction $K$ is antiferromagnetic, in contrast to the ferromagnetic one in the $d^5$ candidates at hand. The exchange interactions are systematically modulated by changing the $A$-site cations. As a consequence, the compounds with $A$=Li and Na may have a dominant antiferromagnetic $K$, but $J$ dominates $K$ and $\Gamma'$ in the cases with $A$=Rb and Cs. Also, by computing the ground states of the $J$-$K$-$\Gamma'$ model by using the exact diagonalization, we map out the systematic evolution of the model parameters in the phase diagram. Our results will stimulate material exploration of the antiferromagnetic Kitaev interaction in $f$-electron compounds.

Local force method for the ab initio tight-binding model: Effect of spin-dependent hopping on exchange interactions

To estimate the Curie temperature of metallic magnets from first principles, we develop a local force method for the tight-binding model having spin-dependent hopping derived from spin-density-functional theory. While spin-dependent hopping is crucial for the self-consistent mapping to the effective spin model, the numerical cost to treat such nonlocal terms in the conventional Green's function scheme is formidably expensive. Here, we propose a formalism based on the kernel polynomial method (KPM), which makes the calculation dramatically efficient. We perform a benchmark calculation for bcc-Fe, fcc-Co, and fcc-Ni and find that the effect of the magnetic nonlocal terms is particularly prominent for bcc-Fe. We also present several local approximations to the magnetic nonlocal terms for which we can apply the Green's function method and reduce the numerical cost further by exploiting the intermediate representation of the Green's function. By comparing the results of the KPM and local methods, we discuss which local method works most successfully. Our approach provides an efficient way to estimate the Curie temperature of metallic magnets with a complex spin configuration.

Skyrmionic order and magnetically induced polarization change in lacunar spinel compounds GaV4S8 and GaMo4S8 : Comparative theoretical study

We show how low-energy electronic models derived from the first-principles electronic structure calculations can help to rationalize the magnetic properties of two lacunar spinel compounds GaM4S8 with light (M=V) and heavy (M=Mo) transition-metal elements, which are responsible for different spin-orbit interaction strength. In the model, each magnetic lattice point was associated with the M4S4 molecule, and the model itself was formulated in the basis of molecular Wannier functions constructed for three magnetic t2 bands. The effects of rhombohedral distortion, spin-orbit interaction, band filling, and the screening of Coulomb interactions in the t2 bands are discussed in details. The electronic model is further treated in the superexchange approximation, which allows us to derive an effective spin model for the energy and electric polarization ($P$) depending on the relative orientation of spins in the bonds, and study the properties of this model by means of classical Monte Carlo simulations with the emphasis on the possible formation of the skyrmionic phase. While isotropic exchange interactions clearly dominate in GaV4S8, all types of interactions -- isotropic, antisymmetric, and symmetric anisotropic -- are comparable in the case of GaMo4S8. Particularly, large uniaxial exchange anisotropy has a profound effect on the properties of GaMo4S8. On the one hand, it raises the Curie temperature by opening a gap in the spectrum of magnon excitations. On the other hand, it strongly affects the skyrmionic phase by playing the role of a molecular field, which facilitates the formation of skyrmions, but makes them relatively insensitive to the external magnetic field in the large part of the phase diagram. We predict reversal of the magnetic dependence of $P$ in the case of GaMo4S8 caused by the reversal of direction of the rhombohedral distortion.

Multiple-Q magnetism by anisotropic bilinear-biquadratic interactions in momentum space

The noncoplanar multiple-Q magnetic orderings are investigated on the basis of an effective spin model with bilinear-biquadratic interactions in momentum space. By performing the simulated annealing for the model on a triangular lattice, we find various types of the multiple-Q magnetic orderings including the skyrmion crystals and magnetic chiral states at zero magnetic field. We show that the effective positive biquadratic interaction leads to a multiple-Q instability depending on the amplitude of the anisotropic parameter. The nearly isotropic biquadratic interaction induces two types of skyrmion crystals with the topological number of two and triple-Q chiral state, while the easy-axis one favors the triple-Q collinear bubble crystal and the easy-plane one becomes a source of the double-Q and triple-Q coplanar magnetic states without a net scalar chirality. We also find that the multiple-Q states in the region for large easy-axis or easy-plane anisotropy can turn into the multiple-Q chiral states by introducing the external magnetic field.

Rethinking α−RuCl3

We argue that several empirical constraints strongly restrict parameters of the effective microscopic spin model describing $\alpha$-RuCl$_3$. In particular, such constraints dictate a substantial \emph{positive} off-diagonal anisotropic coupling, $\Gamma^\prime\!>\!0$, not anticipated previously. The renormalization by quantum fluctuations allows to reconcile larger values of the advocated bare parameters with their earlier assessments and provides a consistent description of the field evolution of spin excitations in the paramagnetic phase. We assert that large anisotropic terms inevitably result in strong anharmonic coupling of magnons, necessarily leading to broad features in their spectra due to decays, in accord with the observations in $\alpha$-RuCl$_3$. Using duality transformations, we explain the origin of the pseudo-Goldstone mode that is ubiquitous to the studied parameter space and is present in $\alpha$-RuCl$_3$. Our analysis offers a description of $\alpha$-RuCl$_3$ as an easy-plane ferromagnet with antiferromagnetic further-neighbor and strong off-diagonal couplings, which is in a fluctuating zigzag ground state proximate to an incommensurate phase that is continuously connected to a ferromagnetic one.

Effective low-energy spin model for narrow zigzag graphene nanoribbons

The magnetic properties of narrow zigzag graphene nanoribbons with periodically embedded atoms of transition metals have been studied in the framework of Heisenberg spin Hamiltonian. We have proposed the simple effective model to give a semi-qualitative description of the peculiarities of magnetization profiles of the systems under consideration. This model can be used for an arbitrary value of spin of the embedded atoms of transition metals. Our analytical and numerical calculations confirm the correctness of the proposed model.

Confinement, reduced entanglement, and spin-glass order in a random quantum spin-ice model

We study an effective spin model derived perturbatively from random transverse-field Ising model on the pyrochlore lattice. The model consists of spin-configurations on the pyrochlore lattice, restricted to the spin-ice subspace, with spins interacting with random Ising exchange couplings as well as ring exchanges along the hexagons of the lattice. This model is studied by exact diagonalization upto N=64 site systems. We calculate spin-glass correlation functions and local entanglement entropy $S_T$ between spins in a single tetrahedron and the rest of the system. We find that the model undergoes two phase transitions. At weak randomness the model is in a quantum spin-ice phase where $S_T=\ln{6}$. Increasing randomness first leads to a frozen phase, with long-range spin-glass order and $S_T=\ln{2}$ corresponding to the Cat states associated with Ising order. Further increase in randomness leads to a random resonating-hexagon phase with a frozen backbone of spins and a broad distribution of entanglement entropies. The implications of these studies for non-Kramers rare-earth pyrochlores are discussed.

Time-reversal symmetry breaking Abelian chiral spin liquid in Mott phases of three-component fermions on the triangular lattice

We provide numerical evidence in favor of spontaneous chiral symmetry breaking and the concomitant appearance of an Abelian chiral spin liquid for three-component fermions on the triangular lattice described by an SU(3) symmetric Hubbard model with hopping amplitude $-t$ ($t>0$) and on-site interaction $U$. This chiral phase is stabilized in the Mott phase with one particle per site in the presence of a uniform $\pi$-flux per plaquette, and in the Mott phase with two particles per site without any flux. Our approach relies on effective spin models derived in the strong-coupling limit in powers of $t/U$ for general SU$(N)$ and arbitrary uniform charge flux per plaquette, which are subsequently studied using exact diagonalizations and variational Monte Carlo simulations for $N=3$, as well as exact diagonalizations of the SU($3$) Hubbard model on small clusters. Up to third order in $t/U$, and for the time-reversal symmetric cases (flux $0$ or $\pi$), the low-energy description is given by the $J$-$K$ model with Heisenberg coupling $J$ and real ring exchange $K$. The phase diagram in the full $J$-$K$ parameter range contains, apart from three already known, magnetically long-range ordered phases, two previously unreported phases: i) a lattice nematic phase breaking the lattice rotation symmetry and ii) a spontaneous time-reversal and parity symmetry breaking Abelian chiral spin liquid. For the Hubbard model, an investigation that includes higher-order itinerancy effects supports the presence of a phase transition inside the insulating region, occurring at $(t/U)_{\rm c}\approx 0.07$ [$(U/t)_{\rm c} \approx 13$] between the three-sublattice magnetically ordered phase at small $t/U$ and this Abelian chiral spin liquid.

Magnetic hedgehog lattices in noncentrosymmetric metals

The magnetic hedgehog lattice (HL) is a noncoplanar magnetic texture with a periodic array of magnetic monopoles and anti-monopoles. Despite phenomenological and numerical studies thus far, there remain open issues on the microscopic origin, especially with respect to the recent experimental findings of two different types of HLs even at zero magnetic field. Here, we study the stability of the HLs for an effective spin model with long-range interactions arising from itinerant nature of electrons. By variational calculations and simulated annealing, we find that the HLs are stabilized in the ground state at zero magnetic field by the synergetic effect of the anti-symmetric exchange interactions generated by the spin-orbit coupling and the multiple-spin interactions generated by the spin-charge coupling. We also clarify the phase diagram in the magnetic fields, which includes topological phase transitions with pair annihilation of the monopoles and anti-monopoles depending on the field directions.

Calculating Curie temperatures for rare-earth permanent magnets: Ab initio inspection of localized magnetic moments in d -electron ferromagnetism

We present a data set of calculated Curie temperatures for the main-phase compounds of rare-earth permanent magnets. We employ ab initio electronic structure calculations for the itinerant ferromagnetism and an effective spin model for the finite-temperature magnetism. Curie temperatures are derived on the basis of a classical Heisenberg model mapped via Liechtenstein's formula for atomic-pair-wise exchange couplings. Relative trends with respect to the species of rare-earth elements in calculated Curie temperatures for R2Fe14B are in agreement with experimental trends. Quantitative comparison between calculation and experimental data found in the literature point to an effective range of the exchange couplings imposing a limit on the validity range of the effective spin model.

Classical phase diagram of the Rashba-Hubbard model in the strongly correlated limit on square lattice

In this work, we investigate the interacting electrons on a square lattice in the presence of Rashba spin-orbit coupling. We first obtain the effective spin model from the Rashba-Hubbard model in the strongly correlated limit using the second perturbation theory. The effective spin model includes isotropic Heisenberg terms, nearest- and next-nearest-neighbor interactions, as well as the anisotropic ones as Kane-Mele and Dzyaloshinski-Moriya interactions. We proceed to study the influence of Rashba spin-orbit coupling on the stability of the magnetic phases of isotropic Heisenberg using Luttinger-Tisza and variational minimization classical methods. Our classical calculations show that the anisotropic terms leads to the instability of the Neel, classical degenerate and collinear phases of the isotropic Heisenberg model on the square lattice into an incommensurate planar phase. The spiral magnetic order in the two-dimensional frustrated magnets can be disordered by considering the quantum fluctuations. In a heterostructure including a noncollinear magnet and a singlet superconductor, singlet Cooper pairs can be converted to triplet pairings due to the broken spin rotational symmetry. Therefore, we can engineer a topological superconductor using noncollinear magnet in a heterostructure system.

Optimizing configurations for determining the magnetic model based on ab initio calculations

In this paper, it is presented a novel strategy to optimize the determination of magnetic couplings by using ab initio calculations of the energy. This approach allows determining efficiently, in terms of a proposed effective magnetic spin model, an optimal set of magnetic configurations to be simulated by DFT methods. Moreover, a procedure to estimate the values of the coupling constants and their error bounds from the estimated energies is proposed. This method, based on Monte Carlo sampling, takes into account the accuracy of the ab-initio simulations. A strategy to refine models reusing previously computed configuration energies is also presented. We apply the method to determine a magnetic model for the recently synthesized material Bi3Mn4O12(NO3). Finally, an open source software that implements and automatizes the whole process is presented.

Entanglement structure of the two-component Bose-Hubbard model as a quantum simulator of a Heisenberg chain

We consider a quantum simulator of the Heisenberg chain with ferromagnetic interactions based on the two-component 1D Bose-Hubbard model at filling equal to two in the strong coupling regime. The entanglement properties of the ground state of the two-component Bose-Hubbard model are compared to those of the effective spin model as the interspecies interaction approaches the intraspecies one. A numerical study of the entanglement properties of the two-component Bose-Hubbard model is supplemented with analytical expressions derived from the effective spin Hamiltonian. When the pure ferromagnetic Heisenberg chain is considered, the entanglement properties of the effective Hamiltonian are not properly predicted by the quantum simulator.

Ferromagnetic Mott state in Twisted Graphene Bilayers at the Magic Angle.

We address the effective tight-binding Hamiltonian that describes the insulating Mott state of twisted graphene bilayers at a magic angle. In that configuration, twisted bilayers form a honeycomb superlattice of localized states, characterized by the appearance of flat bands with fourfold degeneracy. After calculating the maximally localized superlattice Wannier wave functions, we derive the effective spin model that describes the Mott state. We suggest that the system is an exotic ferromagnetic Mott insulator, with well-defined experimental signatures.

Field-induced incommensurate ordering in Heisenberg chains coupled by Ising interaction: Model for ytterbium aluminum perovskite YbAlO3

We study isotropic antiferromagnetic Heisenberg chains coupled by antiferromagnetic Ising interaction as an effective spin model for the ytterbium aluminum perovskite YbAlO$_3$. Using the density-matrix renormalization group (DMRG) method we calculate the magnetization curve, local spin, central charge, and dynamical spin structure factors in the presence of magnetic field. From the fitting of the experimental magnetization curve, the effective intrachain and interchain couplings are estimated as $J=2.3$K and $J_{\rm ic}=0.8$K, respectively. We can quantitatively explain the experimental observations: (i) phase transition from antiferromagnetic to incommensurate order at field 0.35T, and (ii) quantum critical behaviors at the saturation field of 1.21T. Furthermore, the low-energy excitations in the experimental inelastic neutron scattering spectra can be well described by our DMRG results of the dynamical structure factors.

Numerical investigation of spin excitations in a doped spin chain

We study the doping evolution of spin excitations in a 1D Hubbard model and its downfolded spin Hamiltonians, by using exact diagonalization combined with cluster perturbation theory. In all models, we observe hardening (softening) of spin excitations upon electron (hole) doping, which are reminiscent of recent experiments on 2D cuprate materials. We also find that the 3-site and even higher-order terms are crucial for the low-energy effective spin models to reproduce the magnetic spectra of doped Hubbard systems at a quantitative level. To interpret the numerical results, we further employ a strong coupling slave-boson mean-field theory. The mean-field theory provides an intuitive understanding of the overall compact support of dynamic spin structure factors, including the shift of zero-energy modes and change of spin excitation bandwidth with doping. Our results can serve as predictive benchmarks for future inelastic x-ray or neutron scattering experiments on doped 1D antiferromagnetic Mott insulators.

Spin-imbalance-induced transverse magnetization in the Hofstadter-Hubbard model

The fermionic, time-reversal invariant Hofstadter-Hubbard model with a population difference between the two spin states is investigated. In the strongly interacting regime, where the system can be described by an effective spin model, we find an exotic spin structure by means of classical Monte-Carlo calculations. Remarkably, this spin structure exhibits a transverse net magnetization perpendicular to the magnetization induced by the population imbalance. It is thus inherently different from canted antiferromagnetism. We further investigate effects of quantum fluctuations within the dynamical mean-field approximation and obtain a rich phase diagram including ferromagnetic, anti-ferromagnetic, ferrimagnetic, and transverse magnetization phases.