Physical Review

Divergences in the Space-Time Correlation Functions for the Heisenberg Magnet in One Dimension

J.F. Fernández, H. A. Gersch

Published Aug 10, 1968• 18 citations

Abstract

Carboni and Richards have performed exact numerical calculations of the time-dependent two-spin correlation function $〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉$ as $T\ensuremath{\rightarrow}\ensuremath{\infty}$ for a finite one-dimensional Heisenberg system. The non-Gaussian character of their result was characterized by a steep rise near zero frequency for the Fourier transform. We show here that these characteristics result from the inclusion of a Lorentzian form for $S(k, \ensuremath{\omega})$, the paramagnetic scattering function at small wave vectors. We also prove that ${〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉}_{\ensuremath{\omega}}$, the time Fourier transform of $〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉$ in the $T\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, obeys the inequality ${〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉}_{\ensuremath{\omega}}\ensuremath{\ge}\mathrm{const}\ifmmode\times\else\texttimes\fi{}\mathrm{ln}|\frac{1}{\ensuremath{\omega}}|$ as $\ensuremath{\omega}\ensuremath{\rightarrow}0$ for a one-dimensional system. We discuss the probable divergence of the same quantity in two dimensions.

DOI: 10.1103/physrev.172.341

Related papers

Frequency-Dependent Self-Correlation Function for the Heisenberg Spin System in One Dimension

Raza A. Tahir-Kheli, Daniel G. McFadden

Physical ReviewFeb 10, 196925 citations

Dynamic correlation functions for one-dimensional quantum-spin systems: New results based on a rigorous approach

Gerhard H. Muller, Robert E. Shrock

Physical review. B, Condensed matterJan 1, 198465 citations

Space-Time-Dependent Correlation Functions for One-Dimensional Heisenberg Spin Systems in a Lorentzian-Gaussian Approximation

Daniel G. McFadden, Raza A. Tahir-Kheli, G. Bruce Taggart

Physical ReviewSep 10, 19694 citations

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

Raza A. Tahir-Kheli, Daniel G. McFadden

Physical ReviewJun 10, 196963 citations

Two-Time Spin-Pair Correlation Function of the Heisenberg Magnet at Infinite Temperature

Tsuyoshi Horiguchi, Tohru Morita

Physical review. B, Solid stateMar 1, 197310 citations

Dynamical correlation functions for one-dimensional quantum spin systems: New results based on a rigorous approach

Gerhard H. Muller, Robert E. Shrock

Journal of Applied PhysicsMar 15, 198414 citations

Physical Review

Divergences in the Space-Time Correlation Functions for the Heisenberg Magnet in One Dimension

J.F. Fernández, H. A. Gersch

Published Aug 10, 1968• 18 citations

Publisher

Abstract

Carboni and Richards have performed exact numerical calculations of the time-dependent two-spin correlation function

〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉

T\ensuremath{\rightarrow}\ensuremath{\infty}

for a finite one-dimensional Heisenberg system. The non-Gaussian character of their result was characterized by a steep rise near zero frequency for the Fourier transform. We show here that these characteristics result from the inclusion of a Lorentzian form for

S(k, \ensuremath{\omega})

, the paramagnetic scattering function at small wave vectors. We also prove that

{〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉}_{\ensuremath{\omega}}

, the time Fourier transform of

〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉

in the

T\ensuremath{\rightarrow}\ensuremath{\infty}

limit, obeys the inequality

{〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉}_{\ensuremath{\omega}}\ensuremath{\ge}\mathrm{const}\ifmmode\times\else\texttimes\fi{}\mathrm{ln}|\frac{1}{\ensuremath{\omega}}|

\ensuremath{\omega}\ensuremath{\rightarrow}0

for a one-dimensional system. We discuss the probable divergence of the same quantity in two dimensions.

DOI: 10.1103/physrev.172.341

Related papers

Frequency-Dependent Self-Correlation Function for the Heisenberg Spin System in One Dimension

Raza A. Tahir-Kheli, Daniel G. McFadden

The exact numerical calculations by Carboni and Richards for ${〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉}_{(\ensuremath{\omega})}$ in a one-dimensional Heisenberg system of $S=\frac{1}{2}$ spins at elevated temperatures are compared with the predictions of a simple, two-parameter Gaussian representation of the generalized diffusivity. The parameters of the diffusivity are completely determined by the known second and fourth frequency moments, and the procedure is free of arbitrary parameters. The results of the calculation are found to be in good agreement with Carboni and Richards. The calculation is also carried out for $S>\frac{1}{2}$ , and it is found that with the increase in the magnitude of spin, ${〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉}_{(\ensuremath{\omega})}$ gradually loses its characteristic hump and approaches a shape roughly similar to the one which would result from the use of an appropriate Lorentzian for the function ${〈{{S}_{f}}^{z}(t){{S}_{g}}^{z}(0)〉}_{(k,\ensuremath{\omega})}$.The exact numerical calculations by Carboni and Richards for ${〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉}_{(\ensuremath{\omega})}$ in a one-dimensional Heisenberg system of $S=\frac{1}{2}$ spins at elevated temperatures are compared with the predictions of a simple, two-parameter Gaussian representation of the generalized diffusivity. The parameters of the diffusivity are completely determined by the known second and fourth frequency moments, and the procedure is free of arbitrary parameters. The results of the calculation are found to be in good agreement with Carboni and Richards. The calculation is also carried out for $S>\frac{1}{2}$, and it is found that with the increase in the magnitude of spin, ${〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉}_{(\ensuremath{\omega})}$ gradually loses its characteristic hump and approaches a shape roughly similar to the one which would result from the use of an appropriate Lorentzian for the function ${〈{{S}_{f}}^{z}(t){{S}_{g}}^{z}(0)〉}_{(k,\ensuremath{\omega})}$.

Physical ReviewFeb 10, 196925 citations

Dynamic correlation functions for one-dimensional quantum-spin systems: New results based on a rigorous approach

Gerhard H. Muller, Robert E. Shrock

We present new results on the time-dependent correlation functions ${\ensuremath{\Xi}}_{n}(t)=4〈{S}_{0}^{\ensuremath{\xi}}(t){S}_{n}^{\ensuremath{\xi}}〉$, $\ensuremath{\xi}=x,y$, at zero temperature of the one-dimensional $S=\frac{1}{2}$ isotropic $\mathrm{XY}$ model ($h=\ensuremath{\gamma}=0$) and of the transverse Ising (TI) model at the critical magnetic field ($h=\ensuremath{\gamma}=1$). Both models are characterized by special cases of the Hamiltonian $H=\ensuremath{-}J\ensuremath{\Sigma}{l}^{}[(1+\ensuremath{\gamma}){S}_{l}^{x}{S}_{l+1}^{x}+(1\ensuremath{-}\ensuremath{\gamma}){S}_{l}^{y}{S}_{l+1}^{y}+h{S}_{1}^{z}]$. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions ${\ensuremath{\Xi}}_{0}(0)$ and on the singularities of their frequency-dependent Fourier transforms ${\ensuremath{\Phi}}_{0}^{\ensuremath{\xi}\ensuremath{\xi}}(\ensuremath{\omega})$. We have also determined the latter functions by high-precision numerical calculations. The functions ${\ensuremath{\Phi}}_{0}^{\ensuremath{\xi}\ensuremath{\xi}}(\ensuremath{\omega})$, $\ensuremath{\xi}=x,y$, have singularities at the infinite sequence of frequencies $\ensuremath{\omega}=m{\ensuremath{\omega}}_{0}$, $m=0,1,2,3,\dots{}$ , where ${\ensuremath{\omega}}_{0}=J$ for the $\mathrm{XY}$ model and ${\ensuremath{\omega}}_{0}=2J$ for the TI model. In the TI case, the leading singularities in ${\ensuremath{\varphi}}_{0}^{\mathrm{xx}}(\ensuremath{\omega})$ are alternately one-sided and two-sided power-law singularities, the first two of which (at $\ensuremath{\omega}=0,2J$) are divergent. The dominant singularities in the $\mathrm{XY}$ case are alternate one-sided power laws and two-sided power laws with logarithmic corrections, the first two of which (at $\ensuremath{\omega}=0,J$) are divergent. The singularities at higher frequencies in both models are finite and become increasingly weaker. We point out that the nonanalyticities at $\ensuremath{\omega}\ensuremath{\ne}0$ are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis (Luttinger model). At least the most prominent features of our new results should be observable in low-temperature dynamical experiments on quasi-one-dimensional compounds such as the $\mathrm{XY}$ -like substances ${\mathrm{Cs}}_{2}$ Co ${\mathrm{Cl}}_{4}$ and Pr ${\mathrm{Cl}}_{3}$ and the $S=\frac{1}{2}$ Ising-like substance CsCo ${\mathrm{Cl}}_{3}$ ${\mathrm{\ifmmode\cdot\else\textperiodcentered\fi{}}2\mathrm{H}}_{2}$O.We present new results on the time-dependent correlation functions ${\ensuremath{\Xi}}_{n}(t)=4〈{S}_{0}^{\ensuremath{\xi}}(t){S}_{n}^{\ensuremath{\xi}}〉$, $\ensuremath{\xi}=x,y$, at zero temperature of the one-dimensional $S=\frac{1}{2}$ isotropic $\mathrm{XY}$ model ($h=\ensuremath{\gamma}=0$) and of the transverse Ising (TI) model at the critical magnetic field ($h=\ensuremath{\gamma}=1$). Both models are characterized by special cases of the Hamiltonian $H=\ensuremath{-}J\ensuremath{\Sigma}{l}^{}[(1+\ensuremath{\gamma}){S}_{l}^{x}{S}_{l+1}^{x}+(1\ensuremath{-}\ensuremath{\gamma}){S}_{l}^{y}{S}_{l+1}^{y}+h{S}_{1}^{z}]$. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions ${\ensuremath{\Xi}}_{0}(0)$ and on the singularities of their frequency-dependent Fourier transforms ${\ensuremath{\Phi}}_{0}^{\ensuremath{\xi}\ensuremath{\xi}}(\ensuremath{\omega})$. We have also determined the latter functions by high-precision numerical calculations. The functions ${\ensuremath{\Phi}}_{0}^{\ensuremath{\xi}\ensuremath{\xi}}(\ensuremath{\omega})$, $\ensuremath{\xi}=x,y$, have singularities at the infinite sequence of frequencies $\ensuremath{\omega}=m{\ensuremath{\omega}}_{0}$, $m=0,1,2,3,\dots{}$, where ${\ensuremath{\omega}}_{0}=J$ for the $\mathrm{XY}$ model and ${\ensuremath{\omega}}_{0}=2J$ for the TI model. In the TI case, the leading singularities in ${\ensuremath{\varphi}}_{0}^{\mathrm{xx}}(\ensuremath{\omega})$ are alternately one-sided and two-sided power-law singularities, the first two of which (at $\ensuremath{\omega}=0,2J$) are divergent. The dominant singularities in the $\mathrm{XY}$ case are alternate one-sided power laws and two-sided power laws with logarithmic corrections, the first two of which (at $\ensuremath{\omega}=0,J$) are divergent. The singularities at higher frequencies in both models are finite and become increasingly weaker. We point out that the nonanalyticities at $\ensuremath{\omega}\ensuremath{\ne}0$ are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis (Luttinger model). At least the most prominent features of our new results should be observable in low-temperature dynamical experiments on quasi-one-dimensional compounds such as the $\mathrm{XY}$-like substances ${\mathrm{Cs}}_{2}$Co${\mathrm{Cl}}_{4}$ and Pr${\mathrm{Cl}}_{3}$ and the $S=\frac{1}{2}$ Ising-like substance CsCo${\mathrm{Cl}}_{3}$${\mathrm{\ifmmode\cdot\else\textperiodcentered\fi{}}2\mathrm{H}}_{2}$O.

Physical review. B, Condensed matterJan 1, 198465 citations

Space-Time-Dependent Correlation Functions for One-Dimensional Heisenberg Spin Systems in a Lorentzian-Gaussian Approximation

Daniel G. McFadden, Raza A. Tahir-Kheli, G. Bruce Taggart

We comment on the recent work of Fernandez and Gersch on the frequency Fourier transform of the selfcorrelation function for a one-dimensional, $S=\frac{1}{2}$ , Heisenberg spin system with nearest-neighbor exchange. Using a more embellished version of their phenomenological theory, we have computed the frequency Fourier transform of the self-correlation, as well as those for several spatially separated correlations. The results are in good agreement with those calculated by Carboni and Richards with an exact ab initio procedure.We comment on the recent work of Fernandez and Gersch on the frequency Fourier transform of the selfcorrelation function for a one-dimensional, $S=\frac{1}{2}$, Heisenberg spin system with nearest-neighbor exchange. Using a more embellished version of their phenomenological theory, we have computed the frequency Fourier transform of the self-correlation, as well as those for several spatially separated correlations. The results are in good agreement with those calculated by Carboni and Richards with an exact ab initio procedure.

Physical ReviewSep 10, 19694 citations

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

Raza A. Tahir-Kheli, Daniel G. McFadden

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.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.

Physical ReviewJun 10, 196963 citations

Two-Time Spin-Pair Correlation Function of the Heisenberg Magnet at Infinite Temperature

Tsuyoshi Horiguchi, Tohru Morita

The two-time spin-pair autocorrelation function $\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{0},t)$ and the Fourier-space transform $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},t)$ of the two-time spin-pair correlation function $\ensuremath{\sigma}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}_{\mathrm{if}},t)$, are expressed in terms of the "friction function" occurring in the generalized Langevin equation for the spin operator ${s}_{i}^{\ensuremath{\sim}}$ and its Fourier-space transform ${S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{z}$, respectively. The friction function is determined in the form of a product of a Gaussian distribution function and a power series as a function of time $t$ for the isotropic Heisenberg magnet and the $\mathrm{XY}$ magnet of spin \textonehalf{} at infinite temperature. By truncating the power series to the exactly known term, satisfactory results are obtained for $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},t)$ of the isotropic Heisenberg magnet of the linear chain and sc lattice. Satisfactory results for $\ensuremath{\sigma}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}_{\mathrm{if}},t)$ are achieved by taking an inverse Fourier-space transform of the thus determined $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},t)$.The two-time spin-pair autocorrelation function $\ensuremath{\sigma}(\stackrel{\ensuremath{\rightarrow}}{0},t)$ and the Fourier-space transform $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},t)$ of the two-time spin-pair correlation function $\ensuremath{\sigma}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}_{\mathrm{if}},t)$, are expressed in terms of the "friction function" occurring in the generalized Langevin equation for the spin operator ${s}_{i}^{\ensuremath{\sim}}$ and its Fourier-space transform ${S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{z}$, respectively. The friction function is determined in the form of a product of a Gaussian distribution function and a power series as a function of time $t$ for the isotropic Heisenberg magnet and the $\mathrm{XY}$ magnet of spin \textonehalf{} at infinite temperature. By truncating the power series to the exactly known term, satisfactory results are obtained for $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},t)$ of the isotropic Heisenberg magnet of the linear chain and sc lattice. Satisfactory results for $\ensuremath{\sigma}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}_{\mathrm{if}},t)$ are achieved by taking an inverse Fourier-space transform of the thus determined $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},t)$.

Physical review. B, Solid stateMar 1, 197310 citations

Dynamical correlation functions for one-dimensional quantum spin systems: New results based on a rigorous approach

Gerhard H. Muller, Robert E. Shrock

We present new results on the time-dependent correlation functions Ξn(t) =4〈Sξ0(t)Sξn〉, ξ=x,y at zero temperature of the one-dimensional S=1/2 isotropic XY model (h=γ=0) and of the transverse Ising model (TI) at the critical magnetic field (h=γ=1). Both models are characterized by special cases of the Hamiltonian H=−J∑l[(1+γ)SxlSxl+1 +(1−γ)SylSyl+1 +hSzl]. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions (ACF’s) Ξ0(t) and on the singularities of their frequency-dependent Fourier transforms Φξξ0(ω). We have also determined the latter functions by high-precision numerical calculations. The functions Φξξ0(ω), ξ=x,y have singularities at the infinite sequence of frequencies ω=mω0, m=0, 1, 2, 3, ... where ω0=J for the XY model and ω0=2J for the TI model. In both models the singularities in Φxx0 (ω) for m=0, 1 are divergent, whereas the nonanalyticities at higher frequencies become increasingly weaker. We point out that the nonanalyticities at ω≠0 are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis.We present new results on the time-dependent correlation functions Ξn(t) =4〈Sξ0(t)Sξn〉, ξ=x,y at zero temperature of the one-dimensional S=1/2 isotropic XY model (h=γ=0) and of the transverse Ising model (TI) at the critical magnetic field (h=γ=1). Both models are characterized by special cases of the Hamiltonian H=−J∑l[(1+γ)SxlSxl+1 +(1−γ)SylSyl+1 +hSzl]. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions (ACF’s) Ξ0(t) and on the singularities of their frequency-dependent Fourier transforms Φξξ0(ω). We have also determined the latter functions by high-precision numerical calculations. The functions Φξξ0(ω), ξ=x,y have singularities at the infinite sequence of frequencies ω=mω0, m=0, 1, 2, 3, ... where ω0=J for the XY model and ω0=2J for the TI model. In both models the singularities in Φxx0 (ω) for m=0, 1 are divergent, whereas the nonanalyticities at higher frequencies become increasingly weaker. We point out that the nonanalyticities at ω≠0 are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis.

Journal of Applied PhysicsMar 15, 198414 citations

Carboni and Richards have performed exact numerical calculations of the time-dependent two-spin correlation function

〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉

T\ensuremath{\rightarrow}\ensuremath{\infty}

S(k, \ensuremath{\omega})

, the paramagnetic scattering function at small wave vectors. We also prove that

{〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉}_{\ensuremath{\omega}}

, the time Fourier transform of

〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉

in the

T\ensuremath{\rightarrow}\ensuremath{\infty}

limit, obeys the inequality

{〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉}_{\ensuremath{\omega}}\ensuremath{\ge}\mathrm{const}\ifmmode\times\else\texttimes\fi{}\mathrm{ln}|\frac{1}{\ensuremath{\omega}}|

\ensuremath{\omega}\ensuremath{\rightarrow}0

for a one-dimensional system. We discuss the probable divergence of the same quantity in two dimensions.

The exact numerical calculations by Carboni and Richards for ${〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉}_{(\ensuremath{\omega})}$ in a one-dimensional Heisenberg system of

S=\frac{1}{2}

spins at elevated temperatures are compared with the predictions of a simple, two-parameter Gaussian representation of the generalized diffusivity. The parameters of the diffusivity are completely determined by the known second and fourth frequency moments, and the procedure is free of arbitrary parameters. The results of the calculation are found to be in good agreement with Carboni and Richards. The calculation is also carried out for

S>\frac{1}{2}

, and it is found that with the increase in the magnitude of spin, ${〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉}_{(\ensuremath{\omega})}$ gradually loses its characteristic hump and approaches a shape roughly similar to the one which would result from the use of an appropriate Lorentzian for the function ${〈{{S}_{f}}^{z}(t){{S}_{g}}^{z}(0)〉}_{(k,\ensuremath{\omega})}$.

S=\frac{1}{2}

isotropic

\mathrm{XY}

model ($h=\ensuremath{\gamma}=0$) and of the transverse Ising (TI) model at the critical magnetic field ($h=\ensuremath{\gamma}=1$). Both models are characterized by special cases of the Hamiltonian $H=\ensuremath{-}J\ensuremath{\Sigma}{l}^{}[(1+\ensuremath{\gamma}){S}_{l}^{x}{S}_{l+1}^{x}+(1\ensuremath{-}\ensuremath{\gamma}){S}_{l}^{y}{S}_{l+1}^{y}+h{S}_{1}^{z}]$. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions ${\ensuremath{\Xi}}_{0}(0)$ and on the singularities of their frequency-dependent Fourier transforms ${\ensuremath{\Phi}}_{0}^{\ensuremath{\xi}\ensuremath{\xi}}(\ensuremath{\omega})$. We have also determined the latter functions by high-precision numerical calculations. The functions ${\ensuremath{\Phi}}_{0}^{\ensuremath{\xi}\ensuremath{\xi}}(\ensuremath{\omega})$, $\ensuremath{\xi}=x,y$, have singularities at the infinite sequence of frequencies $\ensuremath{\omega}=m{\ensuremath{\omega}}_{0}$,

m=0,1,2,3,\dots{}

, where ${\ensuremath{\omega}}_{0}=J$ for the

\mathrm{XY}

model and ${\ensuremath{\omega}}_{0}=2J$ for the TI model. In the TI case, the leading singularities in ${\ensuremath{\varphi}}_{0}^{\mathrm{xx}}(\ensuremath{\omega})$ are alternately one-sided and two-sided power-law singularities, the first two of which (at $\ensuremath{\omega}=0,2J$) are divergent. The dominant singularities in the

\mathrm{XY}

case are alternate one-sided power laws and two-sided power laws with logarithmic corrections, the first two of which (at $\ensuremath{\omega}=0,J$) are divergent. The singularities at higher frequencies in both models are finite and become increasingly weaker. We point out that the nonanalyticities at $\ensuremath{\omega}\ensuremath{\ne}0$ are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis (Luttinger model). At least the most prominent features of our new results should be observable in low-temperature dynamical experiments on quasi-one-dimensional compounds such as the

\mathrm{XY}

-like substances

{\mathrm{Cs}}_{2}

{\mathrm{Cl}}_{4}

and Pr

{\mathrm{Cl}}_{3}

and the

S=\frac{1}{2}

Ising-like substance CsCo

{\mathrm{Cl}}_{3}

${\mathrm{\ifmmode\cdot\else\textperiodcentered\fi{}}2\mathrm{H}}_{2}$O.

We comment on the recent work of Fernandez and Gersch on the frequency Fourier transform of the selfcorrelation function for a one-dimensional,

S=\frac{1}{2}

, Heisenberg spin system with nearest-neighbor exchange. Using a more embellished version of their phenomenological theory, we have computed the frequency Fourier transform of the self-correlation, as well as those for several spatially separated correlations. The results are in good agreement with those calculated by Carboni and Richards with an exact ab initio procedure.

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.

t

for the isotropic Heisenberg magnet and the

\mathrm{XY}

magnet of spin \textonehalf{} at infinite temperature. By truncating the power series to the exactly known term, satisfactory results are obtained for $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},t)$ of the isotropic Heisenberg magnet of the linear chain and sc lattice. Satisfactory results for $\ensuremath{\sigma}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}_{\mathrm{if}},t)$ are achieved by taking an inverse Fourier-space transform of the thus determined $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},t)$.