Collinear States Research Articles

Size-dependent magnetic transitions in CoFe0.1Cr1.9O4 nanoparticles studied by magnetic and neutron-polarization analysis

Multiferroic, CoCr2O4 bulk material undergoes successive magnetic transitions such as a paramagnetic to collinear and non-collinear ferrimagnetic state at the Curie temperature (TC) and spiral ordering temperature (TS) respectively and finally to a lock-in-transition temperature (Tl). In this paper, the rich sequence of magnetic transitions in CoCr2O4 after mixing the octahedral site with 10% of iron are investigated by varying the size of the particle from 10 to 50 nm. With the increasing size, while the TC increases from 110 to 119 K which is higher than the TC (95 K) of pure CoCr2O4, the TS remains unaffected. In addition, a compensation of magnetization at 34 K and a lock-in transition at 10 K are also monitored in 50 nm particles. Further, we have examined the magnetic-ordering temperatures through neutron scattering using a polarized neutron beam along three orthogonal directions after separating the magnetic scattering from nuclear-coherent and spin-incoherent contributions. While a sharp long-range ferrimagnetic ordering down to 110 K and a short-range spiral ordering down to 50 K are obtained in 50 nm particles, in 10 nm particles, the para to ferrimagnetic transition is found to be continuous and spiral ordering is diffused in nature. Frequency-dependent ac susceptibility (χ) data fitted with different phenomenological models such as the Neel–Arrhenius, Vogel–Fulcher and power law, while ruling out the canonical spin-glass, cluster-glass and interacting superparamagnetism, reveal that both particles show spin-glass behavior with a higher relaxation time in 10 nm particles than in 50 nm. The smaller spin flip time in 50 nm particles confirms that spin dynamics does not slow down on approaching the glass transition temperature (Tg).

Waves Generation by Spiral Magnets at Phase Transitions

The possibility of electromagnetic and acoustic waves generation by magnets with spiral spin structure at phase transition to collinear state has been shown. Intensities and frequencies of generated waves have been calculated. It has been shown that intensities of both electromagnetic and acoustic waves strictly depend on the phase transition kinetics.

Phase transition of anisotropic frustrated Heisenberg model on the square lattice.

We have investigated the J_{1}-J_{2} Heisenberg model with exchange anisotropy on a square lattice and focused on possible AF1-AF2 phase transition below the Néel point and its dependence on the exchange anisotropy, where AF1 and AF2 represent Néel state and collinear state, respectively. We use the double-time Green's-function method and adopt the random-phase approximation. The less the exchange anisotropy, the stronger the quantum fluctuation of the system will be. Both the Néel state and collinear state can exist and have the same Néel temperature for arbitrary anisotropy and spin quantum number S when J_{2}/J_{1}=0.5. Under such parameters, the calculated free energies show that there may occur a first-order phase transition between the Néel state and collinear state for an arbitrary S when anisotropy is not strong.

Magnetic Properties Controlled by Interstitial or Interlayer Cations in Iron Chalcogenides.

By applying density functional theory calculations to iron chalcogenides, we find that magnetic order in Fe1+yTe and magnetic instability at (π, π) in KyFe2Se2 are controlled by interstitial and interlayer cations, respectively. While in Fe1+yTe, magnetic phase transitions occur among collinear, exotic bicollinear and plaquette-ordered antiferronmagnetic states when the height of interstitial irons measured from iron plane or the concentration of interstitial irons is varied, the magnetic instability at (π, π) which is believed to be responsible for the Cooper pairing in iron pnictides is significantly enhanced when y is much smaller than 1 in KyFe2Se2. Our results indicate that, similar to iron pnictides, itinerant electrons play important roles in iron chalcogenides, even though the fluctuating local moments become larger.

Crossing the dividing surface of transition state theory. III. Once and only once. Selecting reactive trajectories.

The purpose of the present work is to determine initial conditions that generate reacting, recrossing-free trajectories that cross the conventional dividing surface of transition state theory (i.e., the plane in configuration space passing through a saddle point of the potential energy surface and perpendicular to the reaction coordinate) without ever returning to it. Local analytical equations of motion valid in the neighborhood of this planar surface have been derived as an expansion in Poisson brackets. We show that the mere presence of a saddle point implies that reactivity criteria can be quite simply formulated in terms of elements of this series, irrespective of the shape of the potential energy function. Some of these elements are demonstrated to be equal to a sum of squares and thus to be necessarily positive, which has a profound impact on the dynamics. The method is then applied to a three-dimensional model describing an atom-diatom interaction. A particular relation between initial conditions is shown to generate a bundle of reactive trajectories that form reactive cylinders (or conduits) in phase space. This relation considerably reduces the phase space volume of initial conditions that generate recrossing-free trajectories. Loci in phase space of reactive initial conditions are presented. Reactivity is influenced by symmetry, as shown by a comparative study of collinear and bent transition states. Finally, it is argued that the rules that have been derived to generate reactive trajectories in classical mechanics are also useful to build up a reactive wave packet.

Path integral approach to order by disorder selection in partially polarized quantum spin systems

In this paper we study the frustrated J 1−J 2 quantum Heisenberg model on the square lattice for J 2>J 1/2, in a magnetic field. In this regime the classical system is known to have a degenerate manifold of lowest energy configurations, where standard thermal order by disorder occurs. In order to study its quantum version we use a path integral formulation in terms of coherent states. We show that the classical degeneracy in the plane transverse to the magnetic field is lifted by quantum fluctuations. Collinear states are then selected, in a similar pattern to that set by thermal order by disorder, leaving a Z 2 degeneracy. A careful analysis reveals a purely quantum mechanical effect given by the tunneling between the two minima selected by fluctuations. The effective description contains two planar (XY-like) fields conjugate to the total magnetization and the difference of the two sublattice magnetizations. Disorder in either or both of these fields produces the locking of their conjugate observables.

From spiral to ferromagnetic structure in B20 compounds: Role of cubic anisotropy

The cubic noncentrosymmetric structure of the B20 compounds produces the helical (homochiral) structure with the wave vector ${k}_{s}=D/J$ balanced by the competition of two interactions: the large ferromagnetic exchange interaction $J$ and small antisymmetric Dzyaloshinskii-Moriya interaction $D$. The difference in the energies between the ferromagnetic collinear and helical states can be experimentally measured by the critical magnetic field ${H}_{c2}$ needed to transform the helix into the field-induced ferromagnet. We show that the cubic anisotropy, first, can limit the stability of the helix phase in the range of small ${k}_{s}$ and, second, makes its own contribution to the value of the critical field ${H}_{c2}$. We illustrate our findings taking an example of a transformation of the helix structure to the ferromagnet at $x\ensuremath{\rightarrow}{x}_{c}$ in the solid solutions ${\mathrm{Fe}}_{1\ensuremath{-}x}{\mathrm{Co}}_{x}\mathrm{Ge}$. We demonstrate that the mechanism of the transformation is realized via the competition between the cubic anisotropy and the Dzyaloshinskii-Moriya interaction.

Spin reorientation in multiferroic spinel Co0.5Fe0.5Cr2O4 with Mössbauer spectroscopy

We studied the magnetic properties of multiferroic spinel Co0.5Fe0.5Cr2O4, especially focusing on the spin-ordering, spin-reorientation, and charge re-distribution effects. From the Rietveld refinement analysis, the crystal structure was identified to be a normal cubic spinel of Fd-3m. Based on the temperature-dependent measurements of the magnetization and magnetic hyperfine field (Hhf), Curie temperature (TC) of Co0.5Fe0.5Cr2O4 was determined to be around 86 K where Hhf ≅ 0. From the change of slope in the M-T curve, the conical-spiral magnetic ordering temperature was determined to be 20 K, which coincides with Hhf measurement. Also, we have observed the decrease in the slope of the electric quadrupole splitting (ΔEQ) curve above 20 K, suggesting that the change in ΔEQ around TS is originated from charge redistribution due to the spin-relocation associated with the distortion of each tetrahedral site around Fe2+ ion above TS. This indicates that Co0.5Fe0.5Cr2O4 has the noncollinear conical-spiral spin ordering with incommensurate spin structure below TS, while above TS, it has ferrimagnetic spin ordering with commensuration in the collinear state. In addition, the Jahn-teller distortion temperature is measured to be around 155 K, since both ΔEQ and Δ1 of 5T2g band decrease rapidly with increasing temperature and disappear around 155 K.

Electromagnon excitation and longitudinal mode studied on the basis of symmetric spin-dependent electric polarization

Since a symmetric spin-dependent electric polarization makes quantum spin systems couple to an electric field, it is one of a possible origin of electromagnon excitations in noncollinear (spiral) magnetically ordered states. In collinear ordered states, however, it is believed that the electromagnon excitation is difficult to realize. In the present study, we focus on the symmetric spin-dependent electric polarization and study the electromagnon excitation in both collinear and noncollinear states. In one-magnon excitation process, it is found that the symmetric spin-dependent polarization couples to a longitudinal fluctuation of the ordered moment and that the electromagnon measurements probe the longitudinal (Higgs) mode selectively. In noncollinear states, we report that both Higgs and Nambu-Goldstone (transverse) modes are observable.

Absence of classical long-range order in anS=12Heisenberg antiferromagnet on a triangular lattice

We study the quantum phase transition of an $S=1/2$ anisotropic $\alpha$ $(\equiv J_z/J_{xy})$ Heisenberg antiferromagnet on a triangular lattice. We calculate the sublattice magnetization and the long-range helical order-parameter and their Binder ratios on finite systems with $N \leq 36$ sites. The $N$ dependence of the Binder ratios reveals that the classical 120$^{\circ}$ N\'{e}el state occurs for $\alpha \lesssim 0.55$, whereas a critical collinear state occurs for $1/\alpha \lesssim 0.6$. This result is at odds with a widely-held belief that the ground state of a Heisenberg antiferromagnet is the 120$^{\circ}$ N\'{e}el state, but it also provides a possible mechanism explaining experimentally observed spin liquids.

Electromagnon as a Probe of Higgs (Longitudinal) Mode in Collinear and Noncollinear Magnetically Ordered States

Electromagnon excitation is investigated on the basis of the symmetric spin-dependent electric polarization in both collinear and noncollinear magnetically ordered states. We focus on quantum spin systems with a large easy-plane single-ion anisotropy, where both massive longitudinal (Higgs) and massless transverse (Nambu–Goldstone) modes exist in the magnetic excitations. In the one-magnon excitation, it is found that the symmetric spin-dependent electric polarization is connected with the longitudinal fluctuation of the ordered moment and that the Higgs mode appears as the electromagnon excitation in collinear states. In noncollinear states, both the Higgs and Nambu–Goldstone modes are observable as the electromagnon excitation.

On the stability of two-layer geostrophic point-vortex multipoles

A point-vortex multipole is an ensemble of m+1 vortices (m = 2, 3, …) possessing a m-fold symmetry, with the “core vortex” being located at the centre and m identical “satellite vortices” located at the vertices of an equilateral m-sided polygon (at m > 2) or at the ends of a straight-line segment (at m = 2). At m = 2, m = 3, and m = 4, the multipole is commonly termed a tripole, a quadrupole, and a pentapole, respectively, and the distance from the core vortex to the satellite vortices, the multipole leg. A multipole is said to be stable if, in response to sufficiently small initial perturbations in the distances between the vortices, the variations in the distances remain small for all times. The main issue of this article is an analytical study of the nonlinear stability of point-vortex tripoles characterized by that their core and satellite vortices reside in different layers of a two-layer f-plane quasigeostrophic model. Also the stability of pentapoles and quadrupoles is discussed. The parameters affecting the stability properties of a multipole are the length of its leg and the intensity of the core vortex relative to the satellite vortices. Among the invariants of the dynamical system that describes the motion of an ensemble of m+1 vortices, there are two ones depending on the distances between the vortices only. To establish the stability/instability of a multipole, we consider the restriction of one of the two invariants to the sheet (in the phase space) constituted by the states at which the second invariant takes the same value as at the multipole equilibrium state. Two versions of the method are presented and employed to analyze the stability of collinear states (m = 2) and non-collinear states (m > 2). For tripoles, complete stability analysis is performed resulting in the determination of the regions of stability/instability in the parameter plane. Depending on the parameters, a multipole can rotate clockwise or counterclockwise, and also can be static. A stable static tripole is shown to minimize the energy of interaction between the vortices. Rigorous stability analysis of pentapoles and quadrupoles is carried out under certain constrains on the permissible perturbations: in pentapoles the perturbations should preserve the central symmetry, and in quadrupoles, the zero linear momentum; only quadrupoles with zero total intensity are considered.

Spiral antiferromagnets beyond the spin-wave approximation: FrustratedXYand Heisenberg models on the honeycomb lattice

We examine the stability of classical states with a generic incommensurate spiral order against quantum fluctuations. Specifically, we focus on the frustrated spin-1/2 $XY$ and Heisenberg models on the honeycomb lattice with nearest-neighbor $J_1$ and next-nearest-neighbor $J_2$ antiferromagnetic couplings. Our variational approach is based on the Jastrow wave functions, which include quantum correlations on top of classical spin waves. We perform a systematic optimization of wave vectors and Jastrow pseudo-potentials within this class of variational states and find that quantum fluctuations favor collinear states over generic coplanar spirals. The N\'eel state with ${\bf Q}=(0,0)$ extends its stability well beyond the classical value $J_2/J_1=1/6$. Most importantly, the collinear states with ${\bf Q}=(0,2\pi/\sqrt{3})$ (and the two symmetry-related states) are found to be stable in a large regime with intermediate frustration, while at the classical level they are limited to the point $J_2/J_1=0.5$. For large frustration, the $120^\circ$ state is stabilized for finite values of $J_2/J_1$ in both models.

Frustrated spin-12Heisenberg antiferromagnet on a chevron-square lattice

The coupled cluster method (CCM) is used to study the zero-temperature properties of a frustrated spin-half ($s={1}{2}$) $J_{1}$--$J_{2}$ Heisenberg antiferromagnet (HAF) on a 2D chevron-square lattice. Each site on an underlying square lattice has 4 nearest-neighbor exchange bonds of strength $J_{1}>0$ and 2 next-nearest-neighbor (diagonal) bonds of strength $J_{2} \equiv x J_{1}>0$, with each square plaquette having only one diagonal bond. The diagonal bonds form a chevron pattern, and the model thus interpolates smoothly between 2D HAFs on the square ($x=0$) and triangular ($x=1$) lattices, and also extrapolates to disconnected 1D HAF chains ($x \to \infty$). The classical ($s \to \infty$) version of the model has N\'{e}el order for $0 < x < x_{{\rm cl}}$ and a form of spiral order for $x_{{\rm cl}} < x < \infty$, where $x_{{\rm cl}} = {1}{2}$. For the $s={1}{2}$ model we use both these classical states, as well as other collinear states not realized as classical ground-state (GS) phases, as CCM reference states, on top of which the multispin-flip configurations resulting from quantum fluctuations are incorporated in a systematic truncation scheme, which we carry out to high orders and extrapolate to the physical limit. We calculate the GS energy, GS magnetic order parameter, and the susceptibilities of the states to various forms of valence-bond crystalline (VBC) order, including plaquette and two different dimer forms. We find that the $s={1}{2}$ model has two quantum critical points, at $x_{c_{1}} \approx 0.72(1)$ and $x_{c_{2}} \approx 1.5(1)$, with N\'{e}el order for $0 < x < x_{c_{1}}$, a form of spiral order for $x_{c_{1}} < x < x_{c_{2}}$ that includes the correct three-sublattice $120^{\circ}$ spin ordering for the triangular-lattice HAF at $x=1$, and parallel-dimer VBC order for $x_{c_{2}} < x < \infty$.

Zero-temperature phase diagram of the classical Kane-Mele-Heisenberg model

The classical phase diagram of the Kane-Mele-Heisenberg model is obtained using three complementary methods: Luttinger-Tisza, variational minimization, and the iterative minimization method. Six distinct phases were obtained in the space of the couplings. Three phases are commensurate with long-range ordering: planar N\'eel states in horizontal plane (phase I), planar states in the plane vertical to the horizontal plane (phase VI), and collinear states normal to the horizontal plane (phase II). However, the other three are infinitely degenerate due to the frustrating competition between the couplings, and they are characterized by a manifold of incommensurate wave vectors. These phases are planar helical states in a horizontal plane (phase III), planar helical states in a vertical plane (phase IV), and non-coplanar states (phase V). Employing the linear spin-wave analysis, it is found that the quantum fluctuations select a set of symmetrically equivalent states in phase III through the quantum order-by-disorder mechanism. Based on some heuristic arguments, it is argued that the same scenario may also occur in the other two frustrated phases VI and V.

Collinear and noncollinear spin ground state of wurtzite CoO

Collinear and noncollinear spin structures of wurtzite phase CoO often appearing in nanosized samples are investigated using first-principles density functional theory calculations. We examined the total energy of several different spin configurations, electronic structure, and the effective magnetic coupling strengths. It is shown that the AF3-type antiferromagnetic ordering is energetically most stable among possible collinear configurations. Further, we found that a spiral spin order can be stabilized by including the relativistic spin-orbit coupling and the noncollinearity of spin direction. Our result suggests that a noncollinear spin ground state can be observed in the transition-metal-oxide nanostructures, which adds an interesting aspect to the nanomagnetism study.

Spin-Current Order in Anisotropic Triangular Antiferromagnets

We analyze instabilities of the collinear up-up-down state of a two-dimensional quantum spin-S spatially anisotropic triangular lattice antiferromagnet in a magnetic field. We find, within the large-S approximation, that near the end point of the plateau, the collinear state becomes unstable due to the condensation of two-magnon bound pairs rather than single magnons. The two-magnon instability leads to a novel two-dimensional vector chiral phase with alternating spin currents but no magnetic order in the direction transverse to the field. This phase breaks a discrete Z(2) symmetry but preserves a continuous U(1) one of rotations about the field axis. It possesses orbital antiferromagnetism and displays a magnetoelectric effect.

Ground-state phase diagram of the asymmetric Hubbard model with geometrical frustration

The variational cluster approximation is used to study the ground state of the spin-asymmetric Hubbard model at half filling with geometrical frustration defined on the square lattice. We show that the three phases with N\'eel, collinear, and disorder states appear in the parameter space and we determine the orders of the phase transitions between the three phases. We corroborate quantitatively that the asymmetry due to the spin-dependent hopping parameters, which gives rise to the Ising anisotropy in spin space, suppresses the quantum fluctuations of the system and destabilizes its disordered spin state induced by the geometrical frustration.

The competing spin orders and fractional magnetization plateaus of the classical Heisenberg model on Shastry-Sutherland lattice: Consequence of long-range interactions

The competing spin orders and fractional magnetization plateaus of the classical Heisenberg model with long-range interactions on a Shastry-Sutherland lattice are investigated using Monte Carlo simulations, in order to understand the fascinating spin ordering sequence observed in TmB4 and other rare-earth tetraborides. The simulation reproduces the experimental 1/2 magnetization plateau at low temperature by considering multifold long range interactions. It is found that more local long range interactions can be satisfied in the 1/2 plateau state than those in the 1/3 plateau state, leading to the stabilization of the extended 1/2 plateau. The phase boundaries in the magnetic field at zero temperature are determined, demonstrating the simulation results. When the energies of the Neel state and the collinear state are degenerated, the former state is more likely to be stabilized due to the competitions among the local collinear spin orders. The present work provides a comprehensive proof of the phase transitions to the Neel state at nonzero temperature, in complimentary to the earlier predictions for the Fe-based superconductors.

Environment-dependent noncollinear magnetic orders and spin-wave spectra of Fe chains and stripes

The ground-state magnetic properties of free-standing Fe wires having linear, zigzag, and rectangular geometries are investigated in the framework of density-functional theory. The stability of various wires geometries and types of magnetic orders is analyzed as a function of the spin-density-wave (SDW) number $q$. In particular, we consider ferromagnetic (FM) and antiferromagnetic (AF) collinear states as well as periodic and alternating noncollinear (NC) states. The relaxed interatomic bond lengths $d$, frozen-magnon dispersion relations $\ensuremath{\Delta}E(q)$, local magnetic moments ${\ensuremath{\mu}}_{i}$, spin-polarized electronic densities of states ${\ensuremath{\rho}}_{d\ensuremath{\sigma}}(\ensuremath{\varepsilon})$, and effective exchange couplings ${J}_{ij}$ are determined. Iron zigzag chains show FM order ($q=0$) at the equilibrium lattice parameter $a$ with $\ensuremath{\Delta}E(q)$ increasing monotonously with $q$. In contrast, biatomic rectangular ladders (RLs) develop spontaneously a spiral SDWs along the wire, keeping FM dimer couplings in the perpendicular direction. This NC order is further stabilized by compressing the wire. However, stretched RLs show FM order. In all cases, strong ${\ensuremath{\mu}}_{i}$ are found within the Wigner-Seitz spheres, which decrease moderately with increasing $q$. The electronic structures of collinear and NC states are analyzed by comparing the density of states ${\ensuremath{\rho}}_{d\ensuremath{\sigma}}(\ensuremath{\varepsilon})$ for different representative values of $q$. Effective exchange-interaction parameters ${J}_{ij}$ between the local moments ${\ensuremath{\mu}}_{i}$ and ${\ensuremath{\mu}}_{j}$ are derived by fitting the ab initio magnon dispersion relation $\ensuremath{\Delta}E(q)$ to a classical Heisenberg model. The interplay between the various competing ${J}_{ij}$ is shown to depend decisively on the lattice parameter and wire geometry. Comparing the ab initio results within a model phase diagram provides useful insights on the magnetic order of Fe wires from a local perspective.