Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics Condensed Matter

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

Stephen William Lovesey, E Engdahl, Alessandro Cuccoli, V. Tognetti, Ewald Balcar

Published Aug 29, 1994• 15 citations
Publisher

Abstract

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.

DOI
10.1088/0953-8984/6/35/001

Related papers

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

Stephen William Lovesey, Ewald Balcar

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.

Journal of Physics Condensed Matter•Feb 7, 1994•16 citations

Spin diffusion in the one-dimensional classical Heisenberg model

Debarshee Bagchi

The problem of spin diffusion is studied numerically in the one-dimensional classical Heisenberg model using a deterministic odd-even spin precession dynamics. We demonstrate that spin diffusion in this model is normal in the infinite-temperature limit and one obtains a long-time diffusive tail in the decay of the autocorrelation function. Some variations of the model with different coupling schemes and anisotropy are also studied and we find normal diffusion in all of them. A systematic finite-size analysis also suggests normal diffusive spreading of spin fluctuations, contrary to previous claims of anomalous diffusion. We also briefly discuss spin diffusion in this model at finite temperatures.

arXiv:1212.2829•Physical Review B•Feb 21, 2013•27 citations

Dynamical Properties of the Diluted Heisenberg and XY Magnets at Infinite Temperature. I: Spin Diffusion Constant

T. Horiguchi, T. Morita

The exact short-time expansion coefficients of the two-time spin-pair correlation function σ(Rif,t)=≪siz(t)sfz(0)≫c at infinite temperature are used in order to estimate the spin diffusion constants of the diluted Heisenberg and XY magnets of spin 1/2. The used coefficients are those up to the term of t10 for the Heisenberg magnet on the linear chain, and those up to the term of t8 for the Heisenberg and XY magnets on the square, sc and bcc lattices. Our calculations are based on the “linear extrapolation approximation”, in which unknown coefficients in Mori's continued fraction representation for ~Is(k), the Laplace-time Fourier-space transform of σ(Rif,t), are linearly extrapolated from the last two of the known coefficients. The obtained values of the diffusion constant and their concentration dependences show a reasonable trend of convergence for successive approximations in a range of concentration not close to the percolation limit. Discussions are also given on the diffusion constants obtained by the “two-parameter Gaussian approximation in time” applied to the memory function.

Progress of Theoretical Physics•Oct 1, 1980•2 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 $t=0$.

Physical review. B, Solid state•Mar 1, 1975•28 citations

LONG-TIME RELAXATION ON SPIN LATTICE AS A MANIFESTATION OF CHAOTIC DYNAMICS

Boris V. Fine

The long-time behavior of the infinite temperature spin correlation functions describing the free induction decay in nuclear magnetic resonance and intermediate structure factors in inelastic neutron scattering is considered. These correlation functions are defined for one-, two- and three-dimensional infinite lattices of interacting spins, both classical and quantum. It is shown that, even though the characteristic time-scale of the long-time decay of the correlation functions considered is non-Markovian, the generic functional form of this decay is either simple exponential or exponential multiplied by cosine. This work contains (i) the summary of the existing experimental and numerical evidence of the above asymptotic behavior; (ii) theoretical explanation of this behavior; and (iii) semi-empirical analysis of various factors discriminating between the monotonic and the oscillatory long-time decays. The theory is based on a fairly strong conjecture that, as a result of chaos generated by spin dynamics, a Brownian-like Markovian description can be applied to the long-time properties of ensemble average quantities on a non-Markovian time-scale. The formalism resulting from that conjecture can be described as "correlated diffusion in finite volumes."

arXiv:cond-mat/9911230•International Journal of Modern Physics B•Mar 30, 2004•25 citations

Spin Diffusion in the Heisenberg Magnets at Infinite Temperature

Tohru Morita

A theoretical criterion for the occurrence of the spin diffusion is given in terms of the "friction function" or "memory function" which occurs in the kinetic equation of the Fourierspace transform $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, t)$ of the two-time spin-pair correlation function. The friction function in the limit of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ensuremath{\rightarrow}0$ is determined with the aid of the short-time expansion of the two-time spin-pair correlation function for the square, sc, and bcc isotropic Heisenberg magnets at infinite temperature, and the criterion is checked. The spin diffusion constants for spin \textonehalf{} are $0.860J{a}^{2}$, $0.619J{a}^{2}$, and $0.509J{a}^{2}$, respectively, for these lattices, where $J$ is the exchange integral and $a$ is the lattice constant. The spin diffusion constants for larger spins are also provided. It should be noted that $0.509J{a}^{2}$ is in complete agreement with the experimental value $(0.525\ifmmode\pm\else\textpm\fi{}0.06)J{a}^{2}$ for the bcc solid ${\mathrm{He}}^{3}$. For the one-dimensional isotropic Heisenberg magnet and for the square, sc, and bcc $\mathrm{XY}$ magnets, the friction function could not be determined in the present calculation. Suggested values of the spin diffusion constant are given for these lattices. Comparison is made with the results of Windsor's computer-simulation calculation. There exists no spin diffusion for the one-dimensional $\mathrm{XY}$ model.

Physical review. B, Solid state•Nov 1, 1972•36 citations
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 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.
The problem of spin diffusion is studied numerically in the one-dimensional classical Heisenberg model using a deterministic odd-even spin precession dynamics. We demonstrate that spin diffusion in this model is normal in the infinite-temperature limit and one obtains a long-time diffusive tail in the decay of the autocorrelation function. Some variations of the model with different coupling schemes and anisotropy are also studied and we find normal diffusion in all of them. A systematic finite-size analysis also suggests normal diffusive spreading of spin fluctuations, contrary to previous claims of anomalous diffusion. We also briefly discuss spin diffusion in this model at finite temperatures.
The exact short-time expansion coefficients of the two-time spin-pair correlation function σ(Rif,t)=≪siz(t)sfz(0)≫c at infinite temperature are used in order to estimate the spin diffusion constants of the diluted Heisenberg and XY magnets of spin 1/2. The used coefficients are those up to the term of t10 for the Heisenberg magnet on the linear chain, and those up to the term of t8 for the Heisenberg and XY magnets on the square, sc and bcc lattices. Our calculations are based on the “linear extrapolation approximation”, in which unknown coefficients in Mori's continued fraction representation for ~Is(k), the Laplace-time Fourier-space transform of σ(Rif,t), are linearly extrapolated from the last two of the known coefficients. The obtained values of the diffusion constant and their concentration dependences show a reasonable trend of convergence for successive approximations in a range of concentration not close to the percolation limit. Discussions are also given on the diffusion constants obtained by the “two-parameter Gaussian approximation in time” applied to the memory function.
Numerical calculations have been performed to obtain the exact infinite-temperature time-dependent spin autocorrelation function G(t)G(t)G(t) for a linear chain of NNN spins (S=12S=\frac{1}{2}S=21​) interacting by nearest-neighbor Heisenberg exchange for N=5,7,9and11N=5, 7, 9 \mathrm{and} 11N=5,7,9and11 by a method different from that of Carboni and Richards. Exact results for the first 20 moments and estimates for M22{M}_{22}M22​ to M30{M}_{30}M30​ 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 M10{M}_{10}M10​. G(t)G(t)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=11N=11N=11 to be equivalent to that of the infinite chain although it is reduced to about 10% of its value at t=0t=0t=0.
The long-time behavior of the infinite temperature spin correlation functions describing the free induction decay in nuclear magnetic resonance and intermediate structure factors in inelastic neutron scattering is considered. These correlation functions are defined for one-, two- and three-dimensional infinite lattices of interacting spins, both classical and quantum. It is shown that, even though the characteristic time-scale of the long-time decay of the correlation functions considered is non-Markovian, the generic functional form of this decay is either simple exponential or exponential multiplied by cosine. This work contains (i) the summary of the existing experimental and numerical evidence of the above asymptotic behavior; (ii) theoretical explanation of this behavior; and (iii) semi-empirical analysis of various factors discriminating between the monotonic and the oscillatory long-time decays. The theory is based on a fairly strong conjecture that, as a result of chaos generated by spin dynamics, a Brownian-like Markovian description can be applied to the long-time properties of ensemble average quantities on a non-Markovian time-scale. The formalism resulting from that conjecture can be described as "correlated diffusion in finite volumes."
A theoretical criterion for the occurrence of the spin diffusion is given in terms of the "friction function" or "memory function" which occurs in the kinetic equation of the Fourierspace transform $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, t)$ of the two-time spin-pair correlation function. The friction function in the limit of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ensuremath{\rightarrow}0$ is determined with the aid of the short-time expansion of the two-time spin-pair correlation function for the square, sc, and bcc isotropic Heisenberg magnets at infinite temperature, and the criterion is checked. The spin diffusion constants for spin \textonehalf{} are 0.860Ja20.860J{a}^{2}0.860Ja2, 0.619Ja20.619J{a}^{2}0.619Ja2, and 0.509Ja20.509J{a}^{2}0.509Ja2, respectively, for these lattices, where JJJ is the exchange integral and aaa is the lattice constant. The spin diffusion constants for larger spins are also provided. It should be noted that 0.509Ja20.509J{a}^{2}0.509Ja2 is in complete agreement with the experimental value $(0.525\ifmmode\pm\else\textpm\fi{}0.06)J{a}^{2}$ for the bcc solid He3{\mathrm{He}}^{3}He3. For the one-dimensional isotropic Heisenberg magnet and for the square, sc, and bcc XY\mathrm{XY}XY magnets, the friction function could not be determined in the present calculation. Suggested values of the spin diffusion constant are given for these lattices. Comparison is made with the results of Windsor's computer-simulation calculation. There exists no spin diffusion for the one-dimensional XY\mathrm{XY}XY model.