Le Journal de Physique Colloques

HIGH-TEMPERATURE SPIN DYNAMICS OF THE CLASSICAL HEISENBERG MAGNET IN ONE, TWO, THREE AND INFINITE DIMENSIONS

Gerhard H. Muller

Published Dec 1, 1988• 1 citation

Abstract

Time-dependent spin-autocorrelation functions at T = ∞ for the classical Heisenberg magnet in dimensions d = 1,2 and 3 are investigated by computer simulations. For the equivalent-neighbor Heisenberg model, the autocorrelation function is calculated analytically.

DOI: 10.1051/jphyscol:19888644

Related papers

SPIN DYNAMICS SIMULATIONS OF CLASSICAL, THREE-DIMENSIONAL HEISENBERG MAGNETS

D. P. Landau, Alex Bunker, KUN CHEN

International Journal of Modern Physics CJun 1, 19963 citations

Time-dependent autocorrelation function for a linear Heisenberg chain at infinite temperature

Amit Sur, Irving J. Lowe

Physical review. B, Solid stateMar 1, 197528 citations

Exact time autocorrelation function of theN-spin classical Heisenberg equivalent neighbor model

Richard A. Klemm, Marco Ameduri

arXiv:cond-mat/0112236Physical review. B, Condensed matterJun 20, 20027 citations

Spin diffusion in classical Heisenberg magnets with uniform, alternating, and random exchange

Niraj Srivastava, Jian-Min Liu, V. S. Viswanath +1 more

We have carried out an extensive simulation study for the spin autocorrelation function at T=∞ of the one-dimensional classical Heisenberg model with four different types of isotropic bilinear nearest-neighbor coupling: uniform exchange, alternating exchange, and two kinds of random exchange. For the long-time tails of all but one case, the simulation data seem incompatible with the simple ∼t−1/2 leading term predicted by spin diffusion phenomenology.

Journal of Applied PhysicsMay 15, 199422 citations

Non-equilibrium critical behavior of the 3D classical Heisenberg model

Evgenii A. Pospelov, Vladimir V. Prudnikov, Pavel Vladimirovich Prudnikov +1 more

Journal of Physics Conference SeriesFeb 1, 20198 citations

Computer studies of spin and energy transport in one-dimensional Heisenberg magnets

N.A. Lurie, David L. Huber, Martin Blume

Physical review. B, Solid stateMar 1, 197499 citations

Le Journal de Physique Colloques

HIGH-TEMPERATURE SPIN DYNAMICS OF THE CLASSICAL HEISENBERG MAGNET IN ONE, TWO, THREE AND INFINITE DIMENSIONS

Gerhard H. Muller

Published Dec 1, 1988• 1 citation

Publisher

Abstract

DOI: 10.1051/jphyscol:19888644

Related papers

SPIN DYNAMICS SIMULATIONS OF CLASSICAL, THREE-DIMENSIONAL HEISENBERG MAGNETS

D. P. Landau, Alex Bunker, KUN CHEN

Spin dynamics simulations have been used to study dynamic critical behavior of classical Heisenberg magnets. The temporal evolutions of the spin configurations were determined numerically from coupled equations of motion by a fourth-order predictor corrector method, with initial spin configurations generated by Monte-Carlo simulations. The neutron scattering function, S (q, ω), was calculated from the space and time displaced spin-spin correlation function and the dynamic critical exponent z was extracted using dynamic finite size scaling theory. For both ferromagnetic and antiferromagnetic models we find good agreement with theoretical predictions and experimental results.Spin dynamics simulations have been used to study dynamic critical behavior of classical Heisenberg magnets. The temporal evolutions of the spin configurations were determined numerically from coupled equations of motion by a fourth-order predictor corrector method, with initial spin configurations generated by Monte-Carlo simulations. The neutron scattering function, S (q, ω), was calculated from the space and time displaced spin-spin correlation function and the dynamic critical exponent z was extracted using dynamic finite size scaling theory. For both ferromagnetic and antiferromagnetic models we find good agreement with theoretical predictions and experimental results.

International Journal of Modern Physics CJun 1, 19963 citations

Time-dependent autocorrelation function for a linear Heisenberg chain at infinite temperature

Amit Sur, Irving J. Lowe

Numerical calculations have been performed to obtain the exact infinite-temperature time-dependent spin autocorrelation function $G(t)$ for a linear chain of $N$ spins ( $S=\frac{1}{2}$ ) interacting by nearest-neighbor Heisenberg exchange for $N=5, 7, 9 \mathrm{and} 11$ by a method different from that of Carboni and Richards. Exact results for the first 20 moments and estimates for ${M}_{22}$ to ${M}_{30}$ of the frequency autocorrelation function for the infinite chain are provided. Excellent agreement is obtained with some recent results obtained by Morita, who, however, gives terms only up to ${M}_{10}$ . $G(t)$ does not show a simple monotonic diffusive ($\ensuremath{\sim}{t}^{\ensuremath{-}\frac{1}{2}}$) behavior within $\frac{4\ensuremath{\hbar}}{J}$, the time domain up to which we believe our result for $N=11$ to be equivalent to that of the infinite chain although it is reduced to about 10% of its value at $t=0$ .Numerical calculations have been performed to obtain the exact infinite-temperature time-dependent spin autocorrelation function $G(t)$ for a linear chain of $N$ spins ($S=\frac{1}{2}$) interacting by nearest-neighbor Heisenberg exchange for $N=5, 7, 9 \mathrm{and} 11$ by a method different from that of Carboni and Richards. Exact results for the first 20 moments and estimates for ${M}_{22}$ to ${M}_{30}$ of the frequency autocorrelation function for the infinite chain are provided. Excellent agreement is obtained with some recent results obtained by Morita, who, however, gives terms only up to ${M}_{10}$. $G(t)$ does not show a simple monotonic diffusive ($\ensuremath{\sim}{t}^{\ensuremath{-}\frac{1}{2}}$) behavior within $\frac{4\ensuremath{\hbar}}{J}$, the time domain up to which we believe our result for $N=11$ to be equivalent to that of the infinite chain although it is reduced to about 10% of its value at $t=0$.

Physical review. B, Solid stateMar 1, 197528 citations

Exact time autocorrelation function of theN-spin classical Heisenberg equivalent neighbor model

Richard A. Klemm, Marco Ameduri

We reduce the autocorrelation function ${\mathcal{C}}_{11}(t)$ of the equivalent neighbor model of N classical spins exhibiting Heisenberg dynamics and exchange coupling J to quadrature. As the temperature $\stackrel{\ensuremath{\rightarrow}}{T}\ensuremath{\infty},{\mathcal{C}}_{11}(t)\ensuremath{\propto}{t}^{\ensuremath{-}N}$ for $\mathrm{Jt}\ensuremath{\gg}1.$ At low T, the antiferromagnetic ${\mathcal{C}}_{11}(t)$ is a simple function of ${(\mathrm{JT})}^{1/2}t,$ exhibiting strong frustration, but the ferromagnetic ${\mathcal{C}}_{11}(t)$ oscillates in a single mode, the frequency of which approaches $\mathrm{NJ}$ as $\stackrel{\ensuremath{\rightarrow}}{T}0.$ We conjecture that as $\stackrel{\ensuremath{\rightarrow}}{T}\ensuremath{\infty},$ the near-neighbor correlation functions of N-spin classical Heisenberg rings are simply obtained from these results.We reduce the autocorrelation function ${\mathcal{C}}_{11}(t)$ of the equivalent neighbor model of N classical spins exhibiting Heisenberg dynamics and exchange coupling J to quadrature. As the temperature $\stackrel{\ensuremath{\rightarrow}}{T}\ensuremath{\infty},{\mathcal{C}}_{11}(t)\ensuremath{\propto}{t}^{\ensuremath{-}N}$ for $\mathrm{Jt}\ensuremath{\gg}1.$ At low T, the antiferromagnetic ${\mathcal{C}}_{11}(t)$ is a simple function of ${(\mathrm{JT})}^{1/2}t,$ exhibiting strong frustration, but the ferromagnetic ${\mathcal{C}}_{11}(t)$ oscillates in a single mode, the frequency of which approaches $\mathrm{NJ}$ as $\stackrel{\ensuremath{\rightarrow}}{T}0.$ We conjecture that as $\stackrel{\ensuremath{\rightarrow}}{T}\ensuremath{\infty},$ the near-neighbor correlation functions of N-spin classical Heisenberg rings are simply obtained from these results.

arXiv:cond-mat/0112236Physical review. B, Condensed matterJun 20, 20027 citations

Spin diffusion in classical Heisenberg magnets with uniform, alternating, and random exchange

Niraj Srivastava, Jian-Min Liu, V. S. Viswanath +1 more

Journal of Applied PhysicsMay 15, 199422 citations

Non-equilibrium critical behavior of the 3D classical Heisenberg model

Evgenii A. Pospelov, Vladimir V. Prudnikov, Pavel Vladimirovich Prudnikov +1 more

We have investigated the non-equilibrium critical behavior of the three-dimensional classical Heisenberg model with evolution from various initial states. Initial states with magnetization m 0 ≠ 0 lead to a new time scale with the exponent k > 0 which produces complex universal scaling forms on the temporal behavior of the magnetization, the autocorrelation function, and the response function. Aging effects have been revealed in two-time behavior of the autocorrelation function with evolution from various initial states. We show that the slowing down of the autocorrelation function during the evolution from high-temperature initial state is considerably more slow than for case of the evolution from low-temperature initial state. Using scaling forms of the magnetization and the autocorrelation function, the values of critical exponents z = 2.035(4), θ ' = 0.490(1), and k = 1/( θ ' + β / νz ) = 1.340(4) were calculated.We have investigated the non-equilibrium critical behavior of the three-dimensional classical Heisenberg model with evolution from various initial states. Initial states with magnetization m 0 ≠ 0 lead to a new time scale with the exponent k > 0 which produces complex universal scaling forms on the temporal behavior of the magnetization, the autocorrelation function, and the response function. Aging effects have been revealed in two-time behavior of the autocorrelation function with evolution from various initial states. We show that the slowing down of the autocorrelation function during the evolution from high-temperature initial state is considerably more slow than for case of the evolution from low-temperature initial state. Using scaling forms of the magnetization and the autocorrelation function, the values of critical exponents z = 2.035(4), θ ' = 0.490(1), and k = 1/( θ ' + β / νz ) = 1.340(4) were calculated.

Journal of Physics Conference SeriesFeb 1, 20198 citations

Computer studies of spin and energy transport in one-dimensional Heisenberg magnets

N.A. Lurie, David L. Huber, Martin Blume

We have studied the dynamics of the classical Heisenberg chain at finite and infinite temperatures by computer simulation of an array of 4000 spins. The evaluation of the spin-spin and energy-energy wave-vector- and time-dependent correlation functions out to time $10{J}^{\ensuremath{-}1}$ made it possible to determine the spin and energy diffusion constants. At infinite temperature we find ${D}_{S}=(1.33\ifmmode\pm\else\textpm\fi{}0.10)JS{a}^{2}$ and ${D}_{E}=(3\ifmmode\pm\else\textpm\fi{}1)JS{a}^{2}$. Values of ${D}_{S}$ have also been determined, with less precision, for ferromagnetic and antiferromagnetic interactions at temperatures $\frac{{k}_{B}T}{|J|}=1.0 \mathrm{and} 0.5$ . Our results are compared with various theoretical estimates. In addition, we calculate the time-dependent spin and energy self-correlation functions.We have studied the dynamics of the classical Heisenberg chain at finite and infinite temperatures by computer simulation of an array of 4000 spins. The evaluation of the spin-spin and energy-energy wave-vector- and time-dependent correlation functions out to time $10{J}^{\ensuremath{-}1}$ made it possible to determine the spin and energy diffusion constants. At infinite temperature we find ${D}_{S}=(1.33\ifmmode\pm\else\textpm\fi{}0.10)JS{a}^{2}$ and ${D}_{E}=(3\ifmmode\pm\else\textpm\fi{}1)JS{a}^{2}$. Values of ${D}_{S}$ have also been determined, with less precision, for ferromagnetic and antiferromagnetic interactions at temperatures $\frac{{k}_{B}T}{|J|}=1.0 \mathrm{and} 0.5$. Our results are compared with various theoretical estimates. In addition, we calculate the time-dependent spin and energy self-correlation functions.

Physical review. B, Solid stateMar 1, 197499 citations

Numerical calculations have been performed to obtain the exact infinite-temperature time-dependent spin autocorrelation function

G(t)

for a linear chain of

N

spins (

S=\frac{1}{2}

) interacting by nearest-neighbor Heisenberg exchange for

N=5, 7, 9 \mathrm{and} 11

by a method different from that of Carboni and Richards. Exact results for the first 20 moments and estimates for

{M}_{22}

{M}_{30}

of the frequency autocorrelation function for the infinite chain are provided. Excellent agreement is obtained with some recent results obtained by Morita, who, however, gives terms only up to

{M}_{10}

G(t)

does not show a simple monotonic diffusive ($\ensuremath{\sim}{t}^{\ensuremath{-}\frac{1}{2}}$) behavior within $\frac{4\ensuremath{\hbar}}{J}$, the time domain up to which we believe our result for

N=11

to be equivalent to that of the infinite chain although it is reduced to about 10% of its value at

t=0

We reduce the autocorrelation function

{\mathcal{C}}_{11}(t)

of the equivalent neighbor model of N classical spins exhibiting Heisenberg dynamics and exchange coupling J to quadrature. As the temperature $\stackrel{\ensuremath{\rightarrow}}{T}\ensuremath{\infty},{\mathcal{C}}_{11}(t)\ensuremath{\propto}{t}^{\ensuremath{-}N}$ for $\mathrm{Jt}\ensuremath{\gg}1.$ At low T, the antiferromagnetic

{\mathcal{C}}_{11}(t)

is a simple function of

{(\mathrm{JT})}^{1/2}t,

exhibiting strong frustration, but the ferromagnetic

{\mathcal{C}}_{11}(t)

oscillates in a single mode, the frequency of which approaches

\mathrm{NJ}

as $\stackrel{\ensuremath{\rightarrow}}{T}0.$ We conjecture that as $\stackrel{\ensuremath{\rightarrow}}{T}\ensuremath{\infty},$ the near-neighbor correlation functions of N-spin classical Heisenberg rings are simply obtained from these results.

{D}_{S}

have also been determined, with less precision, for ferromagnetic and antiferromagnetic interactions at temperatures

\frac{{k}_{B}T}{|J|}=1.0 \mathrm{and} 0.5

. Our results are compared with various theoretical estimates. In addition, we calculate the time-dependent spin and energy self-correlation functions.