Classical Heisenberg Antiferromagnet Research Articles

Kosterlitz–Thouless transition in the Heisenberg model with antisymmetric exchange interaction

We calculate the Kosterlitz–Thouless transition temperature for the classical two-dimensional Heisenberg antiferromagnet with Dzyaloshinski–Moriya antisymmetric exchange interaction using a self-consistent harmonic approximation.

Magnetic Susceptibility of Diluted Pyrochlore andSrCr9−9xGa3+9xO19Antiferromagnets

We investigate the magnetic susceptibility of the classical Heisenberg antiferromagnet with nearest-neighbor interactions on the geometrically frustrated pyrochlore lattice, for a pure system and in the presence of dilution with nonmagnetic ions. Using the fact that the correlation length in this system for small dilution is always short, we obtain an approximate but accurate expression for the magnetic susceptibility at all temperatures. We extend this theory to the compound ${\mathrm{SrCr}}_{9\ensuremath{-}9x}{\mathrm{Ga}}_{3+9x}{\mathrm{O}}_{19}$ and provide an explanation of the phenomenological model recently proposed by Schiffer and Daruka [Phys. Rev. B 56, 13 712 (1997)].

Monte Carlo study of the classical Heisenberg antiferromagnet with easy-plane anisotropy on a triangular lattice

In this paper the thermodynamics of the classical Heisenberg antiferromagnet with easy-plane anisotropy on a triangular lattice is investigated by means of Monte Carlo simulations. The topological frustration and the anisotropy of the exchange interaction turn out in a rich phase diagram showing the signatures of two distinct phase transitions, i.e., a Berezinkii–Kosterlitz–Thouless transition, also present in the unfrustrated counterparts of the model and associated with the vortex–antivortex unbinding, and an Ising-like one with a discrete order parameter, the chirality. These effects of the frustration, already observed in the XY model (or planar rotator), were not yet known in the model we studied, which is, being the spins three dimensional vectors, a more realistic description of a spin system. Data are reported in a broad range of temperatures for four values of the easy-plane anisotropy and for lattice sizes ranging from 24×24 to 120×120.

Néel temperature and sublattice magnetiztion of a three-dimensional classical and quantum Heisenberg antiferromagnet

The effective-field theory (EFT) of a two-spin cluster is applied to study the critical behavior of a three-dimensional classical and quantum Heisenberg antiferromagnet (AF). Néel temperature ( T N ) and sublattice magnetization are calculated numerically and compared with the results of these models with ferromagnetic (F) interaction. We find that our results, for the quantum model T N is higher than T c (Curie temperature, F case) but for the classical model T N = T c . The difference between F and AF results for the quantum model obtained here by EFT is in accordance with Monte Carlo simulation and high-temperature expansion.

Phase transitions induced by easy-plane anisotropy in the classical Heisenberg antiferromagnet on a triangular lattice: A Monte Carlo simulation

The antiferromagnetic classical $\mathrm{XXZ}$ model with easy-plane exchange anisotropy on the triangular lattice, which causes frustration of the spin alignment, is studied by means of Monte Carlo simulations. The system shows the signature of a Berezinskii-Kosterlitz-Thouless transition, associated with vortex-antivortex unbinding, and of an Ising-like one due to the chirality, the latter occurring at a slightly higher temperature. Data for internal energy, specific heat, magnetic susceptibility, correlation length, and some properties associated with the chirality are reported in a broad temperature range for lattice sizes ranging from $24\ifmmode\times\else\texttimes\fi{}24$ to $120\ifmmode\times\else\texttimes\fi{}120$; four values of the easy-plane anisotropy are considered. Moving from the strongest towards the weakest anisotropy (1%), as the thermodynamic quantities tend to the isotropic model behavior, the two transition temperatures decrease by about 25% and 22%, respectively.

Properties of a Classical Spin Liquid: The Heisenberg Pyrochlore Antiferromagnet

We study the low-temperature behaviour of the classical Heisenberg antiferromagnet with nearest neighbour interactions on the pyrochlore lattice. Because of geometrical frustration, the ground state of this model has an extensive number of degrees of freedom. We show, by analysing the effects of small fluctuations around the ground-state manifold, and from the results of Monte Carlo and molecular dynamics simulations, that the system is disordered at all temperatures, T, and has a finite relaxation time, which varies as 1/T for small T.

Study of the phase diagram and the critical behaviour of the 2D Heisenberg antiferromagnet with small uniaxial anisotropy

We determined a discontinuous character of the field-induced phase transition between the Ising-like and the XY-like phase of the classical two-dimensional Heisenberg antiferromagnet with a small uniaxial anisotropy (UHAF) by Monte Carlo simulations. Additional neutron scattering experiments on Rb 2MnCl 4 yield a description of the critical behaviour of the 2D model system and show a quite good accordance between an experimentally determined 2D phase diagram and a theoretical description of the phase boundaries used so far only for the 3D model system.

Spin dynamics study of the critical dynamics of the anisotropic three-dimensional Heisenberg antiferromagnet (abstract)

Using Monte Carlo and spin-dynamics techniques,1 we studied the dynamic behavior of the body-centered-cubic classical Heisenberg antiferromagnet with single site anisotropy at the critical temperature. In order that we may directly compare our results to experiment, we performed the simulation with the anisotropy term set to match the degree of anisotropy found23 in both FeF2 (strong anisotropy) and MnF2 (weak anisotropy). Through the determination and analysis of the form of the neutron scattering function S(q,ω) at Tc we have studied the critical dynamics. We have compared our results to simulations1 for the isotropic case, and to experimental results23 and theory45 for the anisotropic case. We have found, in agreement with theory, an intense purely diffusive longitudinal component and a suppressed propogative transverse component. We were able to determine the transverse relaxation rate, Γ⊥, at the limit q→0 since at this limit Γ⊥→ωm, the characteristic frequency, which can be measured by the simulation.1 Γ⊥ was found to be noncritical in agreement with theory and experiment. We were able to make a quantitative comparison to experimental results of Γ⊥(0) for both FeF2 and MnF2.

Critical dynamics of the body-centered-cubic classical Heisenberg antiferromagnet.

We have used spin dynamics techniques to perform large-scale simulations of the dynamic behavior of the L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L body-centered-cubic classical Heisenberg antiferromagnet with L\ensuremath{\le}48 at a range of temperatures above and below as well as at the critical point ${\mathit{T}}_{\mathit{c}}$. The temporal evolutions of the spin configurations were determined numerically from coupled equations of motion for individual spins by a fourth-order predictor-corrector method, with initial spin configurations generated by Monte Carlo simulations. The neutron scattering function S(q,\ensuremath{\omega}) was calculated from the space- and time-displaced spin-spin correlation function. We used a previously developed dynamic finite-size scaling theory to extract the dynamic critical exponent z from S(q,\ensuremath{\omega}) at ${\mathit{T}}_{\mathit{c}}$. Our results are in agreement with the theoretical prediction of z=1.5 and with experimental results; however, we find that the asymptotic regime was only entered at L\ensuremath{\approxeq}30. In the analysis of the form of the transverse and longitudinal components of S(q,\ensuremath{\omega}) we found that a central diffusion peak appears below ${\mathit{T}}_{\mathit{c}}$ predominantly in the longitudinal component and remains present through and above ${\mathit{T}}_{\mathit{c}}$. The transverse component of the spin-wave peak is Lorentzian below ${\mathit{T}}_{\mathit{c}}$ but for T\ensuremath{\geqslant}${\mathit{T}}_{\mathit{c}}$ is described best by a more complex functional form. Below ${\mathit{T}}_{\mathit{c}}$ we see evidence of multiple spin-wave peaks in the longitudinal component. \textcopyright{} 1996 The American Physical Society.

Critical dynamics of a dynamical version of the classical Heisenberg model.

We have performed large-scale molecular dynamics simulations of a dynamical variant of the classical Heisenberg model in which the spins are replaced by interacting "linear molecules," each of which has two rotational degrees of freedom. The Hamiltonian consists of rotational kinetic-energy terms as well as nearest-neighbor Heisenberg-like interactions $\ensuremath{-}J{\mathrm{s}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{s}}_{j}$, where ${\mathrm{s}}_{i}$ is a unit vector specifying the orientation of the molecule at lattice site $i$. Systems of size up to ${64}^{3}$ spins on the simple cubic lattice have been studied, and thermostatting is used to maintain strictly constant temperature. We determined the dynamic structure function from the time- and space-displaced correlation functions, and found good agreement with the predictions of the dynamic scaling theory with a dynamic critical exponent of $z=1.5$. Results are compared and contrasted with data from spin-dynamics calculations on classical Heisenberg ferromagnets and antiferromagnets.

Two-dimensional solitons in the classical Heisenberg antiferromagnet with nonmagnetic impurities

Experimental evidence is presented for the existence of nonlinear localized spin excitations (solitons) in the classical two-dimensional Heisenberg magnet doped with nonmagnetic impurities. These solitons are shown to provide the dominant mechanism for spin relaxation in electron paramagnetic resonance linewidths immediately above the ordering temperature for layered Mn compounds with a small impurity concentration. The new gapless excitations are pinned by the nonmagnetic impurities thereby becoming topologically unstable and lower in energy than the nonlinear excitations (skyrmions) that may exist in the pure material. Concentration-dependent results preclude a spin-wave mechanism for this effect.

Analytical study of the thermodynamic behavior of an antiferromagnetic Heisenberg model.

By introducing a trial action ${\mathit{S}}_{0}$, the thermodynamic properties of the two-dimensional (2D) and 3D quantum and classical antiferromagnetic Heisenberg model are studied and compared analytically with the variational cumulant expansion method. The free energy of the two models are expanded up to the fourth order and the critical temperature ${\mathit{T}}_{\mathit{N}}$ is given for each order and for different lattice structures. On the simple cubic lattice and square lattice, the sublattice magnetization ${\mathit{M}}_{\mathit{s}}$ and the staggered susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathit{s}}$ are calculated.

Monte Carlo simulation of the Heisenberg antiferromagnet on a triangular lattice: Topological excitations.

We have simulated the classical Heisenberg antiferromagnet on a triangular lattice using a local Monte Carlo algorithm. The behavior of the correlation length $\xi$, the susceptibility at the ordering wavevector $\chi(\bf Q)$, and the spin stiffness $\rho$ clearly reflects the existence of two temperature regimes -- a high temperature regime $T > T_{th}$, in which the disordering effect of vortices is dominant, and a low temperature regime $T < T_{th}$, where correlations are controlled by small amplitude spin fluctuations. As has previously been shown, in the last regime, the behavior of the above quantities agrees well with the predictions of a renormalization group treatment of the appropriate nonlinear sigma model. For $T > T_{th}$, a satisfactory fit of the data is achieved, if the temperature dependence of $\xi$ and $\chi(\bf Q)$ is assumed to be of the form predicted by the Kosterlitz--Thouless theory. Surprisingly, the crossover between the two regimes appears to happen in a very narrow temperature interval around $T_{th} \simeq 0.28$.

Monte Carlo study of the Heisenberg antiferromagnet on the triangular lattice.

We report a Monte Carlo study of the classical antiferromagnetic Heisenberg model on the triangular lattice. The free-energy cost for the formation of free vortices is obtained from a vorticity modulus. Evidence of a Kosterlitz-Thouless type of defect-mediated phase transition at a finite temperature is found.

Magnetic properties of the dipolar Heisenberg antiferromagnet Gd in GdBa2Cu3O6+x.

The nuclear spin-lattice and spin-spin relaxation times ({ital T}{sub 1},{ital T}{sub 2}) and the rf-enhancement factor {eta} of {sup 63,65}Cu in GdBa{sub 2}Cu{sub 3}O{sub 6+{ital x}} were investigated at low temperatures (1.3--4.2 K) and in magnetic fields up to 7 T. {ital T}{sub 1} and {ital T}{sub 2} of the Cu(1) and Cu(2) sites were measured in the whole temperature range for various oxygen concentrations. The phase transition of the Gd sublattice is well resolved with {ital T}{sub {ital N}}=2.19 K for {ital x}=0, decreasing slightly to {ital T}{sub {ital N}}=2.09 K for {ital x}=1. The temperature dependence of the Gd fluctuations is in good agreement with the two-dimensional (2D) Ising model, indicating a breaking of the expected Heisenberg symmetry for the {sup 8}{ital S}{sub 7/2} Gd spins by a uniaxial anisotropy. We propose that this is due to the dipolar interaction, similar to the behavior observed in the classical 2D dipolar Heisenberg antiferromagnets of the K{sub 2}MnF{sub 4} family. The critical field was determined for the semiconducting samples as a function of temperature by a maximum in {ital T}{sub 2}{sup {minus}1}. Extrapolation of the critical field to {ital T}=0 yields {mu}{sub 0}{ital H}{sub {ital c}}({ital T}=0)=3.1 T. Enhancement factorsmore » up to {eta}=15 were observed at the spin-flop field {mu}{sub 0}{ital H}{sub sf}=0.6 T in the nonsuperconducting samples. From the spin-flop and the critical field we find an in-plane exchange {ital J}/{ital k}{sub {ital B}}{approx}75 mK and an anisotropy field {mu}{sub 0}{ital H}{sub {ital a}}=0.1 T, which is in accord with the dipolar anisotropy calculated for the Gd sublattice.« less

Dynamics of vortices in a two-dimensional easy-plane antiferromagnet.

Dynamical behavior of vortices is investigated for a two-dimensional classical Heisenberg antiferromagnet with easy-plane anisotropy. Equations of motion for the vortices are derived with the use of the Lagrangian formulation. The effect of a magnetic field applied in the XY plane is considered. The implications of the theory developed here to the calculations of dynamical correlation functions are presented. However, since pair interactions are neglected, these calculations are relevant only for temperatures below the Kosterlitz-Thouless transition temperature.

The static and dynamic critical behaviour of the two-dimensional Heisenberg antiferromagnet KFeF4

KFeF4 is a well characterized planar antiferromagnet with S=5/2 and is an excellent example of a two-dimensional Heisenberg antiferromagnet. Using neutron-scattering techniques, a study of both the static and dynamic critical behaviour above the Neel temperature, TN=136.75+or-0.35 K, has been undertaken. It is found that in the region above 1.04TN, the static correlation length, the structure-factor peak intensity and the width of the dynamic scattering are all qualitatively described by the theories developed for Heisenberg antiferromagnets. The data have been compared with both the theory for a quantum Heisenberg antiferromagnet. As developed for La2CuO4, and also the model for a classical Heisenberg antiferromagnet. Both these theories describe the data equally well. Below 1.04TN the system is expected to behave like a two-dimensional Ising system.

Numerical method for large-scale non-Hermitian matrices and its application to percolating Heisenberg antiferromagnets.

A numerical method is developed for calculating the spectral density of states, eigenvalues, and eigenvectors of very large non-Hermitian matrices with real eigenvalues. We also present an efficient method to calculate the dynamic correlation function of the system described by non-Hermitian matrices. The effectiveness of the method is demonstrated by applying it to percolating classical Heisenberg antiferromagnets.

MONTE CARLO SIMULATION OF A 3-D FRUSTRATED HEISENBERG MODEL

I present simulations of the classical antiferromagnetic Heisenberg model on a 3-D stacked triangular lattice. The program simulates three independent systems, one third of each system being updated in parallel. It performs between 600 and 800 MegaFlops on a 32 nodes CM-5.

Spin stiffness in frustrated antiferromagnets.

The classical antiferromagnetic Heisenberg model on the triangular lattice is studied using Monte Carlo techniques. The behavior of the antiferromagnetic structure factor, antiferromagnetic correlation length, and spin stiffness at low temperatures is consistent with recent low-temperature renormalization-group predictions based on a nonlinear sigma model (NLσ) description of the system. Our results support the validity of the NLσ model approach to the study of frustrated systems