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Physical review. B, Solid state

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

Amit Sur, Irving J. Lowe

Published Mar 1, 1975• 28 citations
Publisher

Abstract

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 $t=0$.

DOI
10.1103/physrevb.11.1980

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

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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.

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This paper is concerned with the nonmonotonic frequency dependence of $S(k, \ensuremath{\omega})$, the wave-number-frequency transform of the time-dependent spin-spin correlation functions for a one-dimensional Heisenberg magnet at infinite temperature, as calculated by Carboni and Richards. It is argued that this nonmonotonic feature of $S(k, \ensuremath{\omega})$ is a one-dimensional effect, and supporting evidence is presented via a phenomenological calculation of $S(k, \ensuremath{\omega})$, utilizing the equivalence of the Heisenberg chain with a fermion system.

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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.

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

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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.

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Space-Time Correlations in Exchange-Coupled Paramagnets at Elevated Temperatures

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The frequency-dependent correlations ${\ensuremath{\Phi}}_{n}(\ensuremath{\omega})\ensuremath{\equiv}{〈{{S}_{0}}^{z}(t){{S}_{n}}^{z}({t}^{\ensuremath{'}})〉}_{(\ensuremath{\omega})}$, $n=0,1,2,3, \mathrm{and} 4$, and their inverse-lattice Fourier transforms $\ensuremath{\Phi}(\mathbf{K},\ensuremath{\omega})$, have recently been exactly calculated by Carboni and Richards for finite linear chains consisting of $N$ ($N=6,7,8,9, \mathrm{and} 10$) spins ($S=\frac{1}{2}$) in contact with a heat bath at temperature $T\ensuremath{\rightarrow}\ensuremath{\infty}$ and interacting via a nearest-neighbor isotropic Heisenberg exchange interaction. Carboni and Richards have used plausible extrapolation procedures to predict the corresponding correlations in the thermodynamic limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. As the extension of these exact calculations to two and three dimensions is of prohibitive difficulty, we have constructed an alternative theory, based upon a simple two-parameter Gaussian representation of the generalized diffusivity, for calculating these correlations for general spin and the dimensionality. To test the accuracy of this phenomenological theory (which is, however, free of any arbitrariness in the sense that the diffusivity is exactly specified by the second and the fourth frequency moments of the Fourier transform of the frequency-dependent correlation function, which are known at infinite temperature), we first compare our results for ${\ensuremath{\Phi}}_{n}(\ensuremath{\omega})$ with those given by Carboni and Richards for the one-dimensional spin-\textonehalf{} system and find the agreement to be excellent for $n=0$, good for $n=1,2$ and adequate for $n=3,4$. The comparison of the corresponding results for $\ensuremath{\Phi}(\mathbf{K},\ensuremath{\omega})$ reveals the agreement to be quite satisfactory for $|\mathbf{K}|\ensuremath{\lesssim}\frac{1}{2}\ensuremath{\pi}$ but only adequate for the higher-K range, i.e., $\frac{1}{2}\ensuremath{\pi}\ensuremath{\lesssim}\mathbf{K}\ensuremath{\lesssim}\ensuremath{\pi}$. Further support in favor of our phenomenological theory is obtained from a comparison with another set of available exact "computer experiment" results, whereby ${\ensuremath{\Phi}}_{0}(\ensuremath{\omega})$, ${\ensuremath{\Phi}}_{0}(t)$, and ${\ensuremath{\Phi}}_{1}(t)$ are accurately known for a three-dimensional (simple-cubic) lattice of infinite spins, i.e., $S\ensuremath{\rightarrow}\ensuremath{\infty}$. (This is the limit in which the classical spin system studied by Windsor corresponds to the quantum spin system of interest to us.) The agreement of our result with Windsor's is excellent. The salient features of our results are as follows: (i) In two dimensions, the divergence of ${\ensuremath{\Phi}}_{0}(\ensuremath{\omega})$ for $\ensuremath{\omega}\ensuremath{\rightarrow}0$ becomes less sharp and disappears completely in three dimensions. (ii) The cutoff frequency, beyond which ${\ensuremath{\Phi}}_{0}(\ensuremath{\omega})$ is effectively zero, increases with the dimensionality, being in three dimensions about twice what it is in one dimension. (iii) There exists a system of reduced units, i.e., ${{\ensuremath{\Phi}}_{n}}^{\ensuremath{'}}(\ensuremath{\omega})\ensuremath{\rightarrow}{\ensuremath{\Phi}}_{n}(\ensuremath{\omega}){[S(S+1)]}^{\ensuremath{-}1}$, ${I}^{\ensuremath{'}}\ensuremath{\rightarrow}{[S(S+1)]}^{\frac{1}{2}}I$, and ${\ensuremath{\omega}}^{\ensuremath{'}}\ensuremath{\rightarrow}(\frac{\ensuremath{\omega}}{{I}^{\ensuremath{'}}})$, in which the function ${I}^{\ensuremath{'}}{{\ensuremath{\Phi}}_{n}}^{\ensuremath{'}}({\ensuremath{\omega}}^{\ensuremath{'}})$ is approximately the same for all spins, and the accuracy of this law of corresponding states seems to increase with the increase in the dimensionality.

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Time Correlations and Magnetic-Resonance Linewidth in Finite Heisenberg Linear Chains

Fernando Carboni, Peter M. Richards

We have performed exact calculations of two-spin and four-spin time-correlation functions, such as 〈Siz(t)Sjz〉 and 〈Siz(t)Sj+(t)SkzSl−〉, for finite linear chains with periodic boundary conditions containing as many as ten spin-½ particles coupled by a nearest-neighbor Heisenberg exchange interaction. Previous workers have computed thermodynamic properties of finite linear chains, but this to our knowledge is the first calculation of time-dependent properties, which requires the eigenfunctions as well as eigenvalues. Determination of the correlation functions enables us to compute the variation of magnetic-resonance linewidth with frequency in a linear-chain salt such as Cu(NH3)4SO4·H2O by applying the methods of Kubo and Tomita. Values of linewidth computed with our exact correlation functions give very much better agreement with Rogers' experimental values for Cu(NH3)4SO4·H2O than the usual Gaussian approximation. We also show results for the simple two-spin correlation function, such as measured by inelastic neutron scattering. Its frequency transform is non-Gaussian, showing a steep rise near zero frequency.

Journal of Applied Physics•Feb 1, 1968•17 citations
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∼t\ensuremath−12\ensuremath{\sim}{t}^{\ensuremath{-}\frac{1}{2}}\ensuremath∼t\ensuremath−21​) behavior within 4\ensuremathℏJ\frac{4\ensuremath{\hbar}}{J}J4\ensuremathℏ​, 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 infinite-temperature space- and time-dependent spin-correlation functions grx(t){g}_{r}^{x}(t)grx​(t) are studied for the one-dimensional XY\mathrm{XY}XY model. Numerical calculations are performed to obtain the exact autocorrelation function g0x(t){g}_{0}^{x}(t)g0x​(t) for chains containing 5, 7, and 9 spins (S=12S=\frac{1}{2}S=21​). 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 g0x(t){g}_{0}^{x}(t)g0x​(t) for the infinite chain is identical to $\frac{\mathrm{exp}(\ensuremath{-}{J}^{2}{t}^{2})}{4}$. We show that grx(t){g}_{r}^{x}(t)grx​(t) for $r\ensuremath{\ne}0$ vanishes identically for all values of time, implying a wave-vector-independent relaxation shape function. Our result for g0x(t){g}_{0}^{x}(t)g0x​(t) is compared with that obtained by Huber for the classical ($S=\ensuremath{\infty}$) chain.
This paper is concerned with the nonmonotonic frequency dependence of $S(k, \ensuremath{\omega})$, the wave-number-frequency transform of the time-dependent spin-spin correlation functions for a one-dimensional Heisenberg magnet at infinite temperature, as calculated by Carboni and Richards. It is argued that this nonmonotonic feature of $S(k, \ensuremath{\omega})$ is a one-dimensional effect, and supporting evidence is presented via a phenomenological calculation of $S(k, \ensuremath{\omega})$, utilizing the equivalence of the Heisenberg chain with a fermion system.
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 Cu(NH3)4{\mathrm{C}\mathrm{u}(\mathrm{N}{\mathrm{H}}_{3})}_{4}Cu(NH3​)4​SO4{\mathrm{O}}_{4}O4​\ifmmode\cdot\else\textperiodcentered\fi{}H2{\mathrm{H}}_{2}H2​O, and comparison is made with experiment. Very good agreement is found using the same exchange constant JJJ as inferred from specific-heat and magnetic-susceptibility data. The Gaussian approximation, on the other hand, is in extremely poor agreement.
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.
The frequency-dependent correlations ${\ensuremath{\Phi}}_{n}(\ensuremath{\omega})\ensuremath{\equiv}{〈{{S}_{0}}^{z}(t){{S}_{n}}^{z}({t}^{\ensuremath{'}})〉}_{(\ensuremath{\omega})}$, n=0,1,2,3,and4n=0,1,2,3, \mathrm{and} 4n=0,1,2,3,and4, and their inverse-lattice Fourier transforms $\ensuremath{\Phi}(\mathbf{K},\ensuremath{\omega})$, have recently been exactly calculated by Carboni and Richards for finite linear chains consisting of NNN (N=6,7,8,9,and10N=6,7,8,9, \mathrm{and} 10N=6,7,8,9,and10) spins (S=12S=\frac{1}{2}S=21​) in contact with a heat bath at temperature $T\ensuremath{\rightarrow}\ensuremath{\infty}$ and interacting via a nearest-neighbor isotropic Heisenberg exchange interaction. Carboni and Richards have used plausible extrapolation procedures to predict the corresponding correlations in the thermodynamic limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. As the extension of these exact calculations to two and three dimensions is of prohibitive difficulty, we have constructed an alternative theory, based upon a simple two-parameter Gaussian representation of the generalized diffusivity, for calculating these correlations for general spin and the dimensionality. To test the accuracy of this phenomenological theory (which is, however, free of any arbitrariness in the sense that the diffusivity is exactly specified by the second and the fourth frequency moments of the Fourier transform of the frequency-dependent correlation function, which are known at infinite temperature), we first compare our results for ${\ensuremath{\Phi}}_{n}(\ensuremath{\omega})$ with those given by Carboni and Richards for the one-dimensional spin-\textonehalf{} system and find the agreement to be excellent for n=0n=0n=0, good for n=1,2n=1,2n=1,2 and adequate for n=3,4n=3,4n=3,4. The comparison of the corresponding results for $\ensuremath{\Phi}(\mathbf{K},\ensuremath{\omega})$ reveals the agreement to be quite satisfactory for $|\mathbf{K}|\ensuremath{\lesssim}\frac{1}{2}\ensuremath{\pi}$ but only adequate for the higher-K range, i.e., $\frac{1}{2}\ensuremath{\pi}\ensuremath{\lesssim}\mathbf{K}\ensuremath{\lesssim}\ensuremath{\pi}$. Further support in favor of our phenomenological theory is obtained from a comparison with another set of available exact "computer experiment" results, whereby ${\ensuremath{\Phi}}_{0}(\ensuremath{\omega})$, ${\ensuremath{\Phi}}_{0}(t)$, and ${\ensuremath{\Phi}}_{1}(t)$ are accurately known for a three-dimensional (simple-cubic) lattice of infinite spins, i.e., $S\ensuremath{\rightarrow}\ensuremath{\infty}$. (This is the limit in which the classical spin system studied by Windsor corresponds to the quantum spin system of interest to us.) The agreement of our result with Windsor's is excellent. The salient features of our results are as follows: (i) In two dimensions, the divergence of ${\ensuremath{\Phi}}_{0}(\ensuremath{\omega})$ for $\ensuremath{\omega}\ensuremath{\rightarrow}0$ becomes less sharp and disappears completely in three dimensions. (ii) The cutoff frequency, beyond which ${\ensuremath{\Phi}}_{0}(\ensuremath{\omega})$ is effectively zero, increases with the dimensionality, being in three dimensions about twice what it is in one dimension. (iii) There exists a system of reduced units, i.e., ${{\ensuremath{\Phi}}_{n}}^{\ensuremath{'}}(\ensuremath{\omega})\ensuremath{\rightarrow}{\ensuremath{\Phi}}_{n}(\ensuremath{\omega}){[S(S+1)]}^{\ensuremath{-}1}$, ${I}^{\ensuremath{'}}\ensuremath{\rightarrow}{[S(S+1)]}^{\frac{1}{2}}I$, and ${\ensuremath{\omega}}^{\ensuremath{'}}\ensuremath{\rightarrow}(\frac{\ensuremath{\omega}}{{I}^{\ensuremath{'}}})$, in which the function ${I}^{\ensuremath{'}}{{\ensuremath{\Phi}}_{n}}^{\ensuremath{'}}({\ensuremath{\omega}}^{\ensuremath{'}})$ is approximately the same for all spins, and the accuracy of this law of corresponding states seems to increase with the increase in the dimensionality.
We have performed exact calculations of two-spin and four-spin time-correlation functions, such as 〈Siz(t)Sjz〉 and 〈Siz(t)Sj+(t)SkzSl−〉, for finite linear chains with periodic boundary conditions containing as many as ten spin-½ particles coupled by a nearest-neighbor Heisenberg exchange interaction. Previous workers have computed thermodynamic properties of finite linear chains, but this to our knowledge is the first calculation of time-dependent properties, which requires the eigenfunctions as well as eigenvalues. Determination of the correlation functions enables us to compute the variation of magnetic-resonance linewidth with frequency in a linear-chain salt such as Cu(NH3)4SO4·H2O by applying the methods of Kubo and Tomita. Values of linewidth computed with our exact correlation functions give very much better agreement with Rogers' experimental values for Cu(NH3)4SO4·H2O than the usual Gaussian approximation. We also show results for the simple two-spin correlation function, such as measured by inelastic neutron scattering. Its frequency transform is non-Gaussian, showing a steep rise near zero frequency.