Journal of Physics Condensed Matter

A theory of the time-dependent properties of Heisenberg spin chains at infinite temperature

Stephen William Lovesey, Ewald Balcar

Published Feb 7, 1994• 16 citations

Abstract

A coupled-mode theory for the spin response function is used to calculate time-dependent properties of the classical spin chain at infinite temperature. The behaviour of the spin autocorrelation function is consistent with computer simulation data, in that the exponent of the time tail decreases on moving from intermediate to asymptotic time regions. Results for the spin response function and memory function in the two time regions are obtained by a combination of asymptotic analysis and numerical studies. For a large wavevector, near the zone boundary, the spin response function displays a very pronounced oscillatory time dependence. However, there is next to no structure in the corresponding spectral function, in accord with simulation data. The asymptotic analysis shows that, in the infinite-temperature limit, coupled-mode theory does not comply with usual hydrodynamic assumptions, a view that has also emerged from analysis of some of the computer simulation data.

DOI: 10.1088/0953-8984/6/6/027

Related papers

Time-dependent spin correlations in the Heisenberg magnet at infinite temperature

Stephen William Lovesey, E Engdahl, Alessandro Cuccoli +2 more

Journal of Physics Condensed MatterAug 29, 199415 citations

Time-dependent behavior of classical spin chains at infinite temperature

R. W. Gerling, David Paul Landau

An ultrafast, vectorized spin-dynamics method is used to study the time-dependent properties of classical XY and Heisenberg spin chains at infinite temperature. The decay of the energy-energy and spin-spin correlation functions is oscillatory for short times and at long times is consistent with classical diffusion, although the approach to the asymptotic behavior is extremely slow. We have also calculated S(q,\ensuremath{\omega}) and find clear indication of spin-wave peaks in both models.

Physical review. B, Condensed matterNov 1, 199044 citations

Time Dependence of Spin Operators in Finite Heisenberg Linear Chains

Fernando Carboni, Peter M. Richards

Physical ReviewJan 10, 1969104 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

Spin dynamics for the one-dimensional XY model at infinite temperature

Amit Sur, David Jasnow, Irving J. Lowe

Physical review. B, Solid stateNov 1, 1975113 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

Journal of Physics Condensed Matter

A theory of the time-dependent properties of Heisenberg spin chains at infinite temperature

Stephen William Lovesey, Ewald Balcar

Published Feb 7, 1994• 16 citations

Publisher

Abstract

DOI: 10.1088/0953-8984/6/6/027

Related papers

Time-dependent spin correlations in the Heisenberg magnet at infinite temperature

Stephen William Lovesey, E Engdahl, Alessandro Cuccoli +2 more

A coupled-mode theory of spin fluctuations in the d-dimensional Heisenberg magnet at infinite temperature is used to predict the time dependence of various spin correlation functions. The real-space spin autocorrelation function is shown to have a long-time behaviour approximately (1/t) d theta / where theta =(4+d)/2. Properties at intermediate values of the time are extracted from the theory by numerical analysis. In this time window, the reciprocal-lattice spin autocorrelation function, G(q, t), is, to a good approximation, an exponential function of time. The decay rate is proportional to q alpha , where q is the wavevector. Analysis of our numerical data indicates that the exponent alpha depends weakly on d, and it is significantly different from the value 2 which is compatible with a spin diffusion model. In the asymptotic limit, defined by q to 0, t to infinity , and q 2 t to 0, G(q,t) is a function of a single variable=(tq theta ). This result rules against the validity of a diffusion model also in the asymptotic limit.A coupled-mode theory of spin fluctuations in the d-dimensional Heisenberg magnet at infinite temperature is used to predict the time dependence of various spin correlation functions. The real-space spin autocorrelation function is shown to have a long-time behaviour approximately (1/t) d theta / where theta =(4+d)/2. Properties at intermediate values of the time are extracted from the theory by numerical analysis. In this time window, the reciprocal-lattice spin autocorrelation function, G(q, t), is, to a good approximation, an exponential function of time. The decay rate is proportional to q alpha , where q is the wavevector. Analysis of our numerical data indicates that the exponent alpha depends weakly on d, and it is significantly different from the value 2 which is compatible with a spin diffusion model. In the asymptotic limit, defined by q to 0, t to infinity , and q 2 t to 0, G(q,t) is a function of a single variable=(tq theta ). This result rules against the validity of a diffusion model also in the asymptotic limit.

Journal of Physics Condensed MatterAug 29, 199415 citations

Time-dependent behavior of classical spin chains at infinite temperature

R. W. Gerling, David Paul Landau

Physical review. B, Condensed matterNov 1, 199044 citations

Time Dependence of Spin Operators in Finite Heisenberg Linear Chains

Fernando Carboni, Peter M. Richards

Numerical calculations of quantum-mechanical time correlation functions are reported for finite Heisenberg linear chains containing up to 10 spin-\textonehalf{} particles. The results, obtained by manipulation of eigenvalues and eigenfunctions, give Fourier transforms in histogram form. Two-spin correlation functions show a sharp rise near zero frequency and highly non-Gaussian behavior at infinite temperature. The short-time behavior is estimated and compared with classical calculations and other theories. Four-spin functions are calculated and compared with results of a simple decoupling approximation. Spatial Fourier transforms of two-spin functions as needed for neutron scattering cross sections are computed. These are limited to fairly short wavelengths by the finiteness of the system. The temperature dependence of two-spin functions shows a decrease in the near-zero frequency component with decreasing temperature. Computed correlation functions are used to predict the frequency dependence of the ESR linewidth in the linear-chain salt ${\mathrm{C}\mathrm{u}(\mathrm{N}{\mathrm{H}}_{3})}_{4}$ S ${\mathrm{O}}_{4}$ \ifmmode\cdot\else\textperiodcentered\fi{} ${\mathrm{H}}_{2}$ O, and comparison is made with experiment. Very good agreement is found using the same exchange constant $J$ as inferred from specific-heat and magnetic-susceptibility data. The Gaussian approximation, on the other hand, is in extremely poor agreement.Numerical calculations of quantum-mechanical time correlation functions are reported for finite Heisenberg linear chains containing up to 10 spin-\textonehalf{} particles. The results, obtained by manipulation of eigenvalues and eigenfunctions, give Fourier transforms in histogram form. Two-spin correlation functions show a sharp rise near zero frequency and highly non-Gaussian behavior at infinite temperature. The short-time behavior is estimated and compared with classical calculations and other theories. Four-spin functions are calculated and compared with results of a simple decoupling approximation. Spatial Fourier transforms of two-spin functions as needed for neutron scattering cross sections are computed. These are limited to fairly short wavelengths by the finiteness of the system. The temperature dependence of two-spin functions shows a decrease in the near-zero frequency component with decreasing temperature. Computed correlation functions are used to predict the frequency dependence of the ESR linewidth in the linear-chain salt ${\mathrm{C}\mathrm{u}(\mathrm{N}{\mathrm{H}}_{3})}_{4}$S${\mathrm{O}}_{4}$\ifmmode\cdot\else\textperiodcentered\fi{}${\mathrm{H}}_{2}$O, and comparison is made with experiment. Very good agreement is found using the same exchange constant $J$ as inferred from specific-heat and magnetic-susceptibility data. The Gaussian approximation, on the other hand, is in extremely poor agreement.

Physical ReviewJan 10, 1969104 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

Spin dynamics for the one-dimensional XY model at infinite temperature

Amit Sur, David Jasnow, Irving J. Lowe

The infinite-temperature space- and time-dependent spin-correlation functions ${g}_{r}^{x}(t)$ are studied for the one-dimensional $\mathrm{XY}$ model. Numerical calculations are performed to obtain the exact autocorrelation function ${g}_{0}^{x}(t)$ for chains containing 5, 7, and 9 spins ( $S=\frac{1}{2}$ ). This yields exact results for the first 16 moments of the frequency autocorrelation function of the infinite chain, and estimates for a few of the higher moments as well. The analysis suggests that ${g}_{0}^{x}(t)$ for the infinite chain is identical to $\frac{\mathrm{exp}(\ensuremath{-}{J}^{2}{t}^{2})}{4}$. We show that ${g}_{r}^{x}(t)$ for $r\ensuremath{\ne}0$ vanishes identically for all values of time, implying a wave-vector-independent relaxation shape function. Our result for ${g}_{0}^{x}(t)$ is compared with that obtained by Huber for the classical ($S=\ensuremath{\infty}$) chain.The infinite-temperature space- and time-dependent spin-correlation functions ${g}_{r}^{x}(t)$ are studied for the one-dimensional $\mathrm{XY}$ model. Numerical calculations are performed to obtain the exact autocorrelation function ${g}_{0}^{x}(t)$ for chains containing 5, 7, and 9 spins ($S=\frac{1}{2}$). This yields exact results for the first 16 moments of the frequency autocorrelation function of the infinite chain, and estimates for a few of the higher moments as well. The analysis suggests that ${g}_{0}^{x}(t)$ for the infinite chain is identical to $\frac{\mathrm{exp}(\ensuremath{-}{J}^{2}{t}^{2})}{4}$. We show that ${g}_{r}^{x}(t)$ for $r\ensuremath{\ne}0$ vanishes identically for all values of time, implying a wave-vector-independent relaxation shape function. Our result for ${g}_{0}^{x}(t)$ is compared with that obtained by Huber for the classical ($S=\ensuremath{\infty}$) chain.

Physical review. B, Solid stateNov 1, 1975113 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

{\mathrm{C}\mathrm{u}(\mathrm{N}{\mathrm{H}}_{3})}_{4}

{\mathrm{O}}_{4}

\ifmmode\cdot\else\textperiodcentered\fi{}

{\mathrm{H}}_{2}

O, and comparison is made with experiment. Very good agreement is found using the same exchange constant

J

as inferred from specific-heat and magnetic-susceptibility data. The Gaussian approximation, on the other hand, is in extremely poor agreement.

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

The infinite-temperature space- and time-dependent spin-correlation functions

{g}_{r}^{x}(t)

are studied for the one-dimensional

\mathrm{XY}

model. Numerical calculations are performed to obtain the exact autocorrelation function

{g}_{0}^{x}(t)

for chains containing 5, 7, and 9 spins (

S=\frac{1}{2}

). This yields exact results for the first 16 moments of the frequency autocorrelation function of the infinite chain, and estimates for a few of the higher moments as well. The analysis suggests that

{g}_{0}^{x}(t)

for the infinite chain is identical to $\frac{\mathrm{exp}(\ensuremath{-}{J}^{2}{t}^{2})}{4}$. We show that

{g}_{r}^{x}(t)

for $r\ensuremath{\ne}0$ vanishes identically for all values of time, implying a wave-vector-independent relaxation shape function. Our result for

{g}_{0}^{x}(t)

is compared with that obtained by Huber for the classical ($S=\ensuremath{\infty}$) chain.

{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.