Classical Spin Hamiltonian Research Articles

Transverse Magnetization in Spin-Orbit Coupled Antiferromagnets.

Some antiferromagnets under a magnetic field develop magnetization perpendicular to the field as well as more conventional ones parallel to the field. So far, the transverse magnetization (TM) has been attributed to either the spin canting effect or the presence of cluster magnetic multipolar ordering. However, a general theory of TM based on microscopic understanding is still missing. Here, we construct a general microscopic theory of TM in antiferromagnets with cluster magnetic multipolar ordering by considering classical spin Hamiltonians with spin anisotropy that arises from the spin-orbit coupling. First, from general symmetry analysis, we show that TM can appear only when all crystalline symmetries are broken other than the antiunitary mirror, antiunitary twofold rotation, and inversion symmetries. Moreover, by analyzing spin Hamiltonians, we show that TM always appears when the degenerate ground state manifold of the spin Hamiltonian is discrete, as long as it is not prohibited by symmetry. On the other hand, when the degenerate ground state manifold is continuous, TM generally does not appear except when the magnetic field direction and the spin configuration satisfy specific geometric conditions under single-ion anisotropy. Finally, we show that TM can induce the anomalous planar Hall effect, a unique transport phenomenon that can be used to probe multipolar antiferromagnetic structures. We believe that our theory provides a useful guideline for understanding the anomalous magnetic responses of the antiferromagnets with complex magnetic structures.

Classical spin Hamiltonians are context-sensitive languages

Classical spin Hamiltonians are a powerful tool to model complex systems, characterized by a local structure given by the local Hamiltonians. One of the best understood local structures is the grammar of formal languages, which are central in computer science and linguistics, and have a natural complexity measure given by the Chomsky hierarchy. If we see classical spin Hamiltonians as languages, what grammar do the local Hamiltonians correspond to? Here, we cast classical spin Hamiltonians as formal languages, and classify them in the Chomsky hierarchy. We prove that the language of (effectively) zero-dimensional spin Hamiltonians is regular, one-dimensional spin Hamiltonians is deterministic context-free, and higher-dimensional and all-to-all spin Hamiltonians is context-sensitive. This provides a new complexity measure for classical spin Hamiltonians, which captures the hardness of recognizing spin configurations and their energies. We compare it with the computational complexity of the ground state energy problem, and find a different easy-to-hard threshold for the Ising model. We also investigate the dependence on the language of the spin Hamiltonian. Finally, we define the language of the time evolution of a spin Hamiltonian and classify it in the Chomsky hierarchy. Our work suggests that universal spin models are weaker than universal Turing machines.

Statistical approaches to the problem of homogeneous melting of solids in the microcanonical ensemble

Melting is a common phenomenon in our daily life, and although it is understood in thermodynamic (macroscopic) terms, the transition itself has eluded a description from the point of view of microscopic dynamics. While there are studies of metastable states in classical spin Hamiltonians, cellular automata, glassy systems and other models, the statistical mechanical description of the microcanonical superheated solid state is lacking.Our work is oriented to the study of the melting process of superheated solids, which is believed to be caused by thermal vacancies in the crystal or by the occupation of interstitial sites. When the crystal reaches a critical temperature, it becomes unstable and a collective self-diffusion process is triggered. These studies are often observed in a microcanonical environment, revealing long-range correlations due to collective effects, and from theoretical models using random walks over periodic lattices. Our results suggest that the cooperative motion made possible by the presence of vacancy-interstitial pairs (Frenkel pairs) above the melting temperature can induce long-range effective interatomic forces even beyond the neighboring fourth layer. From microcanonical simulations it is also known that an ideal crystal needs a random waiting time until the solid phase collapses. Regarding this, our results also point towards a description of these waiting times using a statistical model in which there is a positive quantity X that accumulates from zero in incremental steps, until it exceeds a threshold value.

Magnetic properties of the quasi two-dimensional centered honeycomb antiferromagnet GdInO3

The crystal structure and magnetic property of the single crystalline hexagonal rare-earth indium oxides ${\mathrm{GdInO}}_{3}$ have been studied by combing experiments and model calculations. The two inequivalent ${\mathrm{Gd}}^{3+}$ ions form the centered honeycomb lattice, which consists of honeycomb and triangular sublattices. The dc magnetic susceptibility and specific heat measurements suggest two antiferromagnetic phase transitions at ${T}_{\text{N1}}=2.3\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ and ${T}_{\text{N2}}=1.02\phantom{\rule{0.28em}{0ex}}\mathrm{K}$. An inflection point is observed in the isothermal magnetization curve, which can be an indication of an up-up-down phase with a 1/3 magnetization plateau, further supported by our theoretical calculation. We also observe a large magnetic entropy change originated from the magnetic frustration in ${\mathrm{GdInO}}_{3}$. By considering a classical spin Hamiltonian, we establish the ground state phase diagram, which suggests that ${\mathrm{GdInO}}_{3}$ has a weak easy-axis anisotropy and is close to the equilateral triangular-lattice system. The theoretical ground-state phase diagram may be used as a reference in NMR, ESR, or $\ensuremath{\mu}\mathrm{SR}$ experiments in future.

Magnetic skyrmions in cylindrical ferromagnetic nanostructures with chiral interactions

We study the geometrical conditions for stabilizing magnetic skyrmions in cylindrical nanostrips and nanotubes of ferromagnetic materials with chiral interactions. We obtain the low-temperature equilibrium state of the system implementing a simulation annealing technique for a classical spin Hamiltonian with competing isotropic exchange and chiral interactions, radial anisotropy and an external field. We address the impact of surface curvature on the formation, the shape and the size of magnetic skyrmions. We demonstrate that the evolution of the skyrmion phase with the curvature is controlled by the competition between two characteristic lengths, namely the curvature radius, $R$ (geometrical length) and the skyrmion radius, $R_{sk}$ (physical length). In narrow nanotubes ($R<R_{sk}$) the skyrmion phase evolves to a stripe phase, while in wide nanotubes ($R>R_{sk}$) a mixed skyrmion-stripe phase emerges. Most interestingly, the mixed phase is characterized by spatially separated skyrmions from stripes owing to the direction of the applied field relative to the surface normal. In the stability regime ($R \gtrsim R_{sk}$) skyrmions remain circular and preserve their size as a consequence of their topological protection. Zero-field skyrmions are shown to be stable on curved nanoelements with free boundaries within the same stability region ($R\gtrsim R_{sk}$). The experimental and technological perspectives from the stability of skyrmions on cylindrical surfaces are discussed.

Rapid fair sampling of the XY spin Hamiltonian with a laser simulator

Coupled oscillators such as lasers, OPO's and BEC polaritons can rapidly and efficiently dissipate into a stable phase locked state that can be mapped onto the minimal energy (ground state) of classical spin Hamiltonians. However, for degenerate or near degenerate ground state manifolds, statistical fair sampling is required to obtain a complete knowledge of the minimal energy state, which needs many repetitions of simulations under identical conditions. We show that with dissipatively coupled lasers such fair sampling can be achieved rapidly and accurately by exploiting the many longitudinal modes of each laser to form an ensemble of identical but independent simulators, acting in parallel. We fairly sampled the ground state manifold of square, triangular and Kagome lattices by measuring their coherence function identifying manifolds composed of a single, doubly degenerate, and highly degenerate ground states, respectively.

Exact mapping between a laser network loss rate and the classical XY Hamiltonian by laser loss control

AbstractRecently, there has been growing interest in the utilization of physical systems as heuristic optimizers for classical spin Hamiltonians. A prominent approach employs gain-dissipative optical oscillator networks for this purpose. Unfortunately, these systems inherently suffer from an inexact mapping between the oscillator network loss rate and the spin Hamiltonian due to additional degrees of freedom present in the system such as oscillation amplitude. In this work, we theoretically analyze and experimentally demonstrate a scheme for the alleviation of this difficulty. The scheme involves control over the laser oscillator amplitude through modification of individual laser oscillator loss. We demonstrate this approach in a laser network classical XY model simulator based on a digital degenerate cavity laser. We prove that for each XY model energy minimum there corresponds a unique set of laser loss values that leads to a network state with identical oscillation amplitudes and to phase values that coincide with the XY model minimum. We experimentally demonstrate an eight fold improvement in the deviation from the minimal XY energy by employing our proposed solution scheme.

Size and chemical order dependence of magnetic-ordering temperature and spin structure in Fe@Ni and Ni@Fe core-shell nanoparticles.

The effect of particle size and chemical order on the temperature-dependent magnetic properties of Fe@Ni and Ni@Fe core-shell nanoparticles is studied in the framework of a classical spin Hamiltonian and Monte Carlo simulations. We found that the mean temperature-dependent magnetization and magnetic-ordering temperature are strongly affected by both the particle size (in size range of 4 to 16 nm) and core-shell chemical order. As a main result, we report the depression of the magnetic ordering-temperature with decreasing size of the elemental Fe and Ni nanoparticles. More specifically, in the case of Fe and Ni nanoparticles, the magnetic-ordering temperature is lowered by 40 (195 K) to 300 (175 K) compared to the bulk value for nanoparticle diameters ranging from 16 to 4 nm, respectively, consistent with previous theoretical data. We further provide a comprehensive insight into the magnetic properties of Fe@Ni and Ni@Fe nanoparticles, unveiling a rich and distinct magnetic-ordering temperature and spin structure that emphatically depends on the core/shell ratio.

Dynamics of dissipative topological defects in coupled phase oscillators

The dynamics of topological defects in a system of coupled phase oscillators, arranged in one and two-dimensional arrays, was numerically investigated using the Kuramoto model. After a rapid decay of the number of topological defects, a long-time quasi steady state with few topological defects was detected. Two competing time scales governed the dynamics corresponding to the dissipation rate and the coupling quench rate. The density of topological defects scales as a power law function of the coupling quench rate ρ ∼ Cν with ν = 0.25. Reducing the number of topological defects improves the long time coherence and order parameter of the system, enhancing the probability to reach a global minimal loss state that can be mapped to the ground state of a classical XY spin Hamiltonian.

Model of ballistic-diffusive thermal transport in HAMR media

A model is presented that considers the grain size and temperature dependence of the thermal conductivity and specific heat of L10 FePt nanoparticles and is applied to study the temperature distribution within the top recording layer in heat assisted magnetic recording. Ballistic effects on the thermal conductivity are included assuming diffuse scattering of the electron heat carriers at interfaces and grain boundaries. The magnetic contribution to the specific heat during laser heating close to the Curie temperature is determined using Monte Carlo simulations based on the effective classical spin Hamiltonian of chemically ordered L10 FePt. Following the application of a nanosecond laser pulse, the perpendicular heat flow through a FePt/MgO/heat-sink stack is calculated using the finite element method. The temperature drop across the thickness of the FePt layer is shown to be significant for grains of small in-plane size. Size induced grain temperature variations are small due to the temperature dependence of the specific heat of FePt, so the resultant contribution to transition jitter is expected to be limited in extent.

Magnetic properties of nanowires with ferromagnetic core and antiferromagnetic shell

We present a Monte Carlo study of the magnetic properties of thin cylindrical nanowires composed of a ferromagnetic core and an antiferromagnetic shell implementing a classical spin Hamiltonian. We address systematically the impact of interface exchange coupling on the loop characteristics and the magnetization reversal mechanism. We study the effect of shell polycrystallinity on the characteristic fields of the isothermal hysteresis loop (coercivity, exchange-bias). We demonstrate that coupling to a polycrystalline antiferromagnetic shell increases the critical core diameter for transition from transverse to vortex domain walls.

Exchange bias effect in cylindrical nanowires with ferromagnetic core and polycrystalline antiferromagnetic shell

We study numerically the exchange bias effect in thin cylindrical nanowires composed of a ferromagnetic core and an antiferromagnetic shell implementing a classical spin Hamiltonian and Monte Carlo simulations. We address systematically the effect of shell polycrystallinity on the characteristic fields of the isothermal hysteresis loop (coercivity, exchange-bias) and their angular dependence upon the direction of the reversing and cooling fields. We relate the observed trends to modifications of the underlying magnetization reversal mechanism. We fit our simulation results to an extended Stoner-Wohlfarth model with effective off-axis unidirectional anisotropy and demonstrate that shell polycrystallinity leads to maximum exchange bias effect in an off-axis direction. Our results are in qualitative agreement with recent experimental studies of Co/CoO nanowires.

Bayesian Approach to Effective Model of NiGa2S4 Triangular Lattice with Boltzmann Factor

We propose a method for inducting the Boltzmann factor to extract effective classical spin Hamiltonians from mean-field-type electronic structural calculations by means of the Bayesian inference. This method enables us to compare electronic structural calculations with experiments according to the classical model at a finite temperature. Application of this method to the unrestricted Hartree–Fock calculations for NiGa2S4 led to the estimation that the superexchange interaction between the nearest neighbor sites is ferromagnetic at low temperature, which is consistent with magnetic experiment results. This supports the theory that competition between the antiferromagnetic third neighbor interaction and ferromagnetic nearest neighbor interaction may lead to the quantum spin liquid in NiGa2S4.

Up-up-down-down magnetic chain structure of the spin- 12 tetragonally distorted spinel GeCu2O4

GeCu2O4 exhibits a tetragonal spinel structure due to the strong Jahn-Teller distortion associated with Cu2+ ions. We show that its magnetic structure can be described as slabs composed of a pair of layers with orthogonally oriented spin 1/2 Cu chains in the basal ab plane. The spins between the two layers within a slab are collinearly aligned while the spin directions of neighboring slabs are perpendicular to each other. Interestingly, we find that spins along each chain form an unusual up-up-down-down (UUDD) pattern, suggesting a non-negligible nearest-neighbor biquadratic exchange interaction in the effective classical spin Hamiltonian. We hypothesize that spin-orbit coupling and orbital mixing of Cu2+ ions in this system is non-negligible, which calls for future calculations using perturbation theory with extended Hilbert (spin and orbital) space and calculations based on density functional theory including spin-orbit coupling and looking at the global stability of the UUDD state.

On Complexity of the Quantum Ising Model

We study complexity of several problems related to the Transverse field Ising Model (TIM). First, we consider the problem of estimating the ground state energy known as the Local Hamiltonian Problem (LHP). It is shown that the LHP for TIM on degree-3 graphs is equivalent modulo polynomial reductions to the LHP for general k-local ‘stoquastic’ Hamiltonians with any constant \({k \ge 2}\). This result implies that estimating the ground state energy of TIM on degree-3 graphs is a complete problem for the complexity class \({\mathsf{StoqMA}}\) —an extension of the classical class \({\mathsf{MA}}\). As a corollary, we complete the complexity classification of 2-local Hamiltonians with a fixed set of interactions proposed recently by Cubitt and Montanaro. Secondly, we study quantum annealing algorithms for finding ground states of classical spin Hamiltonians associated with hard optimization problems. We prove that the quantum annealing with TIM Hamiltonians is equivalent modulo polynomial reductions to the quantum annealing with a certain subclass of k-local stoquastic Hamiltonians. This subclass includes all Hamiltonians representable as a sum of a k-local diagonal Hamiltonian and a 2-local stoquastic Hamiltonian.

Cluster Monte Carlo methods for the FePt Hamiltonian

Cluster Monte Carlo methods for the classical spin Hamiltonian of FePt with long range exchange interactions are presented. We use a combination of the Swendsen–Wang (or Wolff) and Metropolis algorithms that satisfies the detailed balance condition and ergodicity. The algorithms are tested by calculating the temperature dependence of the magnetization, susceptibility and heat capacity of L10-FePt nanoparticles in a range including the critical region. The cluster models yield numerical results in good agreement within statistical error with the standard single-spin flipping Monte Carlo method. The variation of the spin autocorrelation time with grain size is used to deduce the dynamic exponent of the algorithms. Our cluster models do not provide a more accurate estimate of the magnetic properties at equilibrium.

Magnetic correlations beyond the Heisenberg model in an Fe monolayer on Rh(0 0 1)

Motivated by a recent experimental observation of a complex magnetic structure (Takada et al 2013 J. Magn. Magn. Mater. 329 95) we present a theoretical study of the magnetic structure of an Fe monolayer deposited on Rh(0 0 1). We use a classical spin Hamiltonian with parameters obtained from ab initio calculations and go beyond the usual anisotropic Heisenberg model by including isotropic biquadratic interactions. Zero-temperature Landau–Lifshitz–Gilbert spin dynamics simulations lead to a complex collinear spin configuration that, however, contradicts experimental findings. We thus conclude that higher order multi-spin interactions are likely needed to account for the magnetic ordering of the system.

Model Selection of NiGa2S4 Triangular Lattice by Bayesian Inference

We propose a novel method for extracting effective classical spin Hamiltonians from mean-field type electronic structural calculations by means of the Bayesian inference. We apply the method for a NiS2 triangular lattice in NiGa2S4, where the ground state is a spin disordered state. Starting from unrestricted Hartree–Fock calculations for the spin configurations of 16 Ni sites, we estimated that not only the strongest superexchange interaction between the third nearest neighbor-sites but also those between the nearest and the second nearest neighbor-sites should be considered for the effective classical spin Hamiltonian for NiGa2S4.

Finite size effects in L1o-FePt nanoparticles

Finite size effects on the temperature dependence of the uniaxial magnetic anisotropy, longitudinal and transverse susceptibilities and specific heat are examined for L1o-ordered FePt nanoparticles using an atomistic model based on an effective classical spin Hamiltonian. At low temperatures below criticality, we study the intrinsic uniaxial magnetic anisotropy energy (MAE) K1 and its scaling with magnetization K1(T)∼Ms(T)δ and using Langevin dynamics simulations we show that the dependence of the exponent δ on the size L and aspect ratio of the grain arises from decomposition of the MAE into bulk and surface dependent terms. Monte Carlo simulations in the critical regime near the Curie temperature Tc, show that the temperature variation of the specific heat and longitudinal susceptibility is given by finite size scaling relations c=Lα/νc̃(L1/νϵ) and χ=Lγ/νχ̃(L1/νϵ), respectively, where ϵ=(T−Tc)/Tc is the reduced temperature, and the susceptibility scaling function χ̃ can be approximated by a Lorentzian. Our estimates of the critical exponents α,γ, and ν appear to be in agreement with the universality class of the 3D Ising model.

Theory of half-metallic double perovskites. II. Effective spin Hamiltonian and disorder effects

Double perovskites like Sr$_2$FeMoO$_6$ are materials with half-metallic ground states and ferrimagnetic T$_{\rm{c}}$'s well above room temperature. This paper is the second of our comprehensive theory for half metallic double perovskites. Here we derive an effective Hamiltonian for the Fe core spins by "integrating out" the itinerant Mo electrons and obtain an unusual double square-root form of the spin-spin interaction. We validate the classical spin Hamiltonian by comparing its results with those of the full quantum treatment presented in the companion paper "Theory of Half-Metallic Double Perovskites I: Double Exchange Mechanism". We then use the effective Hamiltonian to compute magnetic properties as a function of temperature and disorder and discuss the effect of excess Mo, excess Fe, and anti-site disorder on the magnetization and T$_{\rm{c}}$. We conclude with a proposal to increase T$_{\rm{c}}$ without sacrificing carrier polarization.