Spin-wave Hamiltonian Research Articles

Subterahertz and terahertz spin and lattice dynamics of the insulating ferromagnet PbMnBO4

Orthorhombic ${\mathrm{PbMnBO}}_{4}$ is a rare example of an insulating ferromagnet in which the magnetic properties, in particular the strong magnetic anisotropy, are determined by the ${\mathrm{Mn}}^{3+}$ Jahn-Teller ion. Here, we report on the detailed investigation of magnetic excitations in the subterahertz and terahertz frequency ranges, as well as lattice excitations in the terahertz range. Using polarized azimuthally resolved Raman spectroscopy, four branches of magnetic excitations were observed and assigned to one acoustic and three optical one-magnon modes. The coupling between ferromagnetic ordering at ${T}_{C}=31\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ and the Jahn-Teller distortions of $[{\mathrm{MnO}}_{6}]$ octahedra due to the spin-phonon interaction was directly observed for some specific phonon modes involved in these distortions. A spin-wave Hamiltonian was developed that allowed us to determine a unique set of exchange and effective anisotropy parameters. The intensity ratio of the anti-Stokes to Stokes acoustic magnon peaks is found to be anomalous and directly indicates the involvement of the magneto-optical parameters in the magnetic Raman scattering process.

Topological thermal Hall effect driven by spin-chirality fluctuations in frustrated antiferromagnets

By revealing an underlying relation between the Dzyaloshinskii-Moriya interaction (DMI) and the scalar spin chirality, we develop the theory of magnon thermal Hall effects in antiferromagnetic systems. The dynamic fluctuation of the scalar chirality is shown to directly respond to the nontrivial topology of magnon bands. In materials such as the jarosites compounds KFe$_3$(OH)$_6$(SO$_4$)$_2$ and veseignite BaCu$_3$V$_2$O$_8$(OH)$_2$ in the presence of in-plane DMI, the time-reversal symmetry can be broken by the fluctuations of scalar chirality even in the case of coplanar $\mathbf{q}=0$ magnetic configuration. The spin-wave Hamiltonian is influenced by a fictitious magnetic flux determined by the in-plane DMI. Topological magnon bands and corresponding nonzero Chern numbers are presented without the need of a canted non-coplanar magnetic ordering. The canting angle dependence of thermal Hall conductivity is discussed in detail as well. These results offer a clear principle of chirality-driven topological effects in antiferromagnetically coupled systems.

Spin-spin correlation function of the two-dimensionalXYmodel with weak site or bond dilution

The spin-spin correlation function of the 2D XY model decays as a power law at all temperatures below the Berezinskii-Kosterlitz-Thouless transition point with a temperature dependent exponent $\eta=\eta(T/J)$ ($J$ is the ferromagnetic coupling strength). It is known from computer experiments that in the 2D XY model with site or bond dilution this exponent depends on concentration $p$ of removed sites/bonds as well. Knowing the slope $\partial\eta/\partial p$ at point $p=0$, one can predict the value of the exponent for small dilution concentrations: $\eta(p)\simeq\eta(0)+p(\partial\eta/\partial p)|_{p=0}$. As it is shown in this paper, the spin-wave Hamiltonian allows to obtain exact results for this slope: $(\partial\eta/\partial p)|_{p=0} = T/(2J) + O((T/J)^2)$ and $T/(\pi J) + O((T/J)^2)$ for site and for bond dilution, respectively.

Hamiltonian formulation of nonlinear spin-wave dynamics: Theory and applications

A systematic step-by-step development of nonlinear spin-wave theory within the framework of a Hamiltonian formalism is given. The expansion coefficients in the spin-wave Hamiltonian up to the fourth order in the canonical spin-wave amplitudes for a uniformly magnetized ellipsoidal sample are given in general form. This is done for a general orientation of the external field and for a general wave vector. Applications are given for (A) the second-order Suhl spin-wave instability coupling coefficients, (B) the spin-wave wave-vector-dependent nonlinear frequency shift for an in-plane magnetized thin film, and (C) the uniform magnetization mode (ferromagnetic resonance) nonlinear frequency shift for an obliquely magnetized thin film. The analytical results for (C) are compared with results obtained from a time forward numerical analysis of the torque equation of motion and show a near-perfect match.

Magnons in the colossal magnetoresistance pyrochloreTl2Mn2O7

Well defined ferromagnetic spin waves were observed when the spin dynamics of ${\mathrm{Tl}}_{2}{\mathrm{Mn}}_{2}{\mathrm{O}}_{7}$, the first pyrochlore compound found to exhibit colossal magnetoresistance, was measured [J. W. Lynn et al., Phys. Rev. Lett. 80, 4582 (1998)], in stark contrast to the experimental results on the larger family of magnetoresistive manganites with perovskite structure. In this work, we present our calculation for the spin waves in ${\mathrm{Tl}}_{2}{\mathrm{Mn}}_{2}{\mathrm{O}}_{7}$, which we described using the microscopic generic model proposed recently for this compound [C. I. Ventura and M. A. Gusmao, Phys. Rev. B 65, 14422 (2002)]. We have employed a canonical transformation to determine perturbatively the effective spin-wave Hamiltonian, obtaining therefrom the renormalization of the ferromagnetic spin waves related to the localized ${\mathrm{Mn}}^{4+}$ spins, due to their coupling with the conduction electrons present. We have calculated the magnon dispersion relations along different paths in the first Brillouin zone, comparing them with those which are obtained for an ideal isotropic ferromagnet. This comparison evidences an agreement between the ferromagnetic magnons obtained from the generic model and the bare spin waves, providing a simple explanation for the neutron scattering results.

Spin-wave Hamiltonian in double-exchange systems

A simple derivation of the effective spin-wave Hamiltonian for a double-exchange system with infinitely large Hund's-rule coupling is demonstrated. The formalism can be applied to models with arbitrary range of hopping as well as those with randomness. The result shows that, within the leading order of the 1/ S expansion, one magnon excitation spectrum can be described by the Heisenberg model.

Spin Waves in Layered Structures of Antiferromagnetic F.C.C. MnTe

Spin waves in single thin films and exchange-coupled bilayers of f.c.c. MnTe are analyzed theoretically in the low temperature limit. Long-range antiferromagnetic order of AF-III type is assumed. The spin-wave Hamiltonian includes exchange interactions between nearest and next nearest neighbours, magnetic anisotropy and Zeeman term.

Spin waves in several Heisenberg systems:Three-sublattice with different exchange constants (Jab=Jbc≠Jca) anda superlattice with the elementary unit of four or three different layers

The quadratic Holstein-Primakoff spin-wave Hamiltonian for three-sublattice Heisenberg systems with different exchange constants (J(ab)=J(bc)not equal J(ca)) was diagonalized by a three-step procedure. The classical ground state was examined and the effects of quantum fluctuation on the ground state were discussed. The 0 K spin reorientation angles due to quantum fluctuations were found to originate from the asymmetry of the system. As an extended application of the spin-wave study of the multisublattice systems, the spin-wave spectra of a Heisenberg superlattice, which has four or three magnetic atoms or spins, in each magnetic unit cell in the x direction, were solved analytically in terms of creation and annihilation operators. It has been found that the spin-wave spectra of the present superlattice systems depend on the exchange constants J and J(1), and that the degeneracy of the spin-wave spectra remains for the superlattice of four atomic layers and is partially removed for the superlattice of three atomic layers mainly due to the asymmetry of the system. Another type of splitting of the energy level occurs which differs for those levels with different degeneracies. The mixing of the interface and the bulk modes were found for the spin-wave dispersion of the superlattices. It was predicted that even for different superlattices having the same exchange constants (either J or J'), i.e., without the oscillation of the exchange coupling, the asymmetry can lead to the oscillation of magnetic properties, such as magnetization and giant magnetoresistance.

Dipolar antiferromagnetism in the spin-wave approximation

The spin-wave Hamiltonian for a two-sublattice antiferromagnet with purely dipolar interactions is diagonalized. The simple-cubic (sc) lattice is solved as an example, and is found to exhibit zero-point-motion corrections to the Ne\ifmmode\acute\else\textasciiacute\fi{}el state that are much larger than those of the nearest-neighbor sc Heisenberg antiferromagnet. The two-dimensional square dipolar lattice is found not to exhibit long-range order at finite temperature in this approximation.

Renormalization of the quantum antiferromagnet in two dimensions.

We discuss the renormalization of the quantum antiferromagnet in two dimensions. With an analysis of the spin-wave Hamiltonian of the quantum Heisenberg model or the quantum nonlinear \ensuremath{\sigma} model (NL\ensuremath{\sigma}M), we consider O(n) invariant observable 〈${\mathbf{S}}_{\mathit{l}}$\ensuremath{\cdot}${\mathbf{S}}_{\mathit{j}}$〉 and evaluate \ensuremath{\beta} functions in the dimensional regularization method up to two-loop order. We show that quantum and classical NL \ensuremath{\sigma} M are connected by a simple crossover function. It follows from this crossover phenomenon that the critical temperature ${\mathit{T}}_{\mathit{c}}$ for dg2 is reduced due to the quantum effects and that the thermal fluctuations reduce the critical coupling constant ${\mathit{g}}_{\mathit{c}}$. The disappearance of ${\mathit{g}}_{\mathit{c}}$ indicates thermal restoration of the asymptotic freedom. Finally we present an exact expression of the correlation length.

Spin-wave results for the triangular Heisenberg antiferromagnet.

We perform the standard spin-wave analysis of the triangluar Heisenberg quantum antiferromagnet. Contrary to the variational calculation of Anderson, we diagonalize exactly the quadratic part of the spin-wave Hamiltonian and obtain results in a 1/S expansion. We compute the ground-state staggered magnetization per spin and the ground-state energy. For spin (1/2, the agreement with the variational values of Huse and Elser suggests that the trangular lattice has long-range magnetic order.

Susceptibility of a Heisenberg antiferromagnet

The spin-wave Hamiltonian for the two-sublattice Heisenberg antiferromagnet is obtained with the use of a Maleev transformation. The dipole-dipole interaction is restricted to nearest neighbors. The double-time Green-function formulation is used to obtain the expression, up to first order, of the 16 elementary two-operator Green functions. These are used to obtain the frequency-dependent susceptibility.

The Heisenberg model: Dynamical interactions including odd-boson terms

A systematic treatment is given of a rather general spin-wave Hamiltonian containing up to four-boson terms. The linear boson terms of leading order in the 1/ S expansion disappear, applying an appropriate boson shift or rotation in spin space prior to the Holstein-Primakoff transformation. The remaining one- and all three-boson terms are eliminated using a Fröhlich transformation which yields an effective Hamiltonian consisting of two- and four-boson terms apart from a constant. The free energy corresponding to this Hamiltonian is obtained using a variational type of approach.

The spin-wave theory of solid3He. Pseudo-six-dimensional spin method

A systematic group-theoretic approach is used to treat the spin-wave theory for the uudd phase of solid3He based on a four-spin exchange Heisenberg model. The method is based on the fact that the bilinear products of four kinds of magnon creation and annihilation operators form a Lie algebra ofU(1)×O(4, 2). The thermodynamic properties at low temperature are studied in detail in terms of the lowest order spin-wave Hamiltonian. An analytic formula for the dipolar torque is obtained for the first time from the spin-wave theory under the local equilibrium assumption. The temperature dependence of the spin contraction, the dipolar torque, and the parallel susceptibility is shown to start fromT 2. For comparison with the data of Osheroff, Cross, and Fisher, a numerical analysis is made of such physical quantities as spin contractionΔS atT=0 and the antiferromagnetic zero-field resonance frequenceΩ(T)/2π, within a two-parameter (J t ,K p ) model.

Two-magnon Raman scattering in quadratic double-layer antiferromagnets

Two-magnon Raman spectra of the quadratic double-layer antiferromagnet ${\mathrm{K}}_{3}$${\mathrm{Mn}}_{2}$${\mathrm{F}}_{7}$ (${T}_{N}=58.3$ K) have been investigated from 2 K up to beyond the ordered regime. Following an analysis of the Raman-scattering Hamiltonian on the basis of symmetry, the spectra are interpreted in terms of a two-dimensional four-sublattice spin-wave theory appropriate to the double layer. The method used is the equation of motion of Green functions. It appears essential to include those quartic terms in the spin-wave Hamiltonian that couple the lower and upper magnon branches, but in the random-phase approximation do not contribute to renormalization of the magnon dispersion. Spectra calculated with first-order decoupling agree well with the spectra observed at 2 K. In second order, the damping of magnons near the zone boundary is expressed in an approximate analytical form. The computed second-order spectra account for the observed temperature dependence of both the peak position and the width of the scattering up to 1.1 ${T}_{N}$. Fits to spectra of ${\mathrm{K}}_{3}$${\mathrm{Mn}}_{2}$${\mathrm{F}}_{7}$, with the anisotropy set at $\ensuremath{\Delta}=g{\ensuremath{\mu}}_{\mathrm{B}}\frac{{H}_{A}}{5}|J|S=0.002$, yield a nearest-neighbor exchange $J=\ensuremath{-}7.68\ifmmode\pm\else\textpm\fi{}0.07$ K, in agreement with findings from the sublattice magnetization and susceptibility. From the low-temperature spectrum of the isomorphic ${\mathrm{K}}_{3}$${\mathrm{Ni}}_{2}$${\mathrm{F}}_{7}$ taken from the literature, we derive $J(1+1.04\ensuremath{\Delta})=\ensuremath{-}95.4\ifmmode\pm\else\textpm\fi{}0.5$ K.

Symmetry of a magnet and structure of spin-wave Hamiltonian

Symmetry of a magnet and structure of spin-wave Hamiltonian

Spin-Wave Contribution to the Deviation of the Magnetization from Saturation

A first-order macroscopic spin-wave Hamiltonian is used to describe and calculate the deviation of the magnetization from saturation for materials with a uniaxial magnetic anisotropy perpendicular to an applied magnetic field. The calculations predict a spin-wave deviation from saturation prior to the onset of any actual rotation of the magnetization. When the magnetic anisotropy creates an easy axis of magnetization, the spin-wave deviations from saturation occur, for a decreasing applied field, at a larger field than the nucleation field given by the variational and/or micromagnetic approaches. When the anisotropy creates a hard axis of magnetization, spin-wave deviations from saturation occur; whereas, the variational and micromagnetic approaches predict no deviation from saturation for finite fields in this case. It is concluded that spin-wave contributions to the deviation from saturation are important.

Some Exact Solutions to the Two‐Dimensional Spin Wave Hamiltonian

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