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Physical review. B, Solid state

Space-Time Correlations in Anisotropic Heisenberg Paramagnets at Elevated Temperatures

Daniel G. McFadden, Raza A. Tahir-Kheli

Published May 1, 1970• 40 citations
Publisher

Abstract

The structure of the space-time-dependent spin-correlation functions for an anisotropic Heisenberg paramagnet with cylindrical symmetry at elevated temperature is analyzed in terms of a two-parameter Gaussian representation representation of the generalized diffusivity. The parameters are chosen so as to exactly preserve the frequency moments, i.e., ${{〈{\ensuremath{\omega}}^{2n}〉}_{\mathrm{K}}}^{\ensuremath{\alpha}\ensuremath{\alpha}}$ ($\ensuremath{\alpha}=z, x, or y$; $n=0, 1, 2$) of the longitudinal and the transverse spectral functions. These moments are calculated for arbitrary spin $S$, arbitrary range of the exchange interactions, and arbitrary dimensionality. The results for the correlation functions are worked out for one, two, and three dimensions. In the extreme transverse limit, the results are compared with the exactly soluble $\mathrm{XY}$ model for a one-dimensional spin-\textonehalf{} system with only nearest-neighbor (transverse) exchange. The agreement of the results is found to be satisfactory.

DOI
10.1103/physrevb.1.3649

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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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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.

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

J.F. Fernández, H. A. Gersch

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.

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Dynamical Spin Correlations in Many-Spin Systems. I. The Ferromagnetic Case

Raza A. Tahir-Kheli

The quantum-statistical mechanics of an anisotropic Heisenberg spin system is studied by a temperature-time-dependent Green's-function formalism. A consistent scheme of higher-order random-phase approximations (RPA) is developed and the second-order (2nd) RPA is examined in detail. It turns out that the 2nd RPA can be defined in two alternative versions, I and II, with equal a priori justification. These two versions of the 2nd RPA have then to be supplemented with either a dynamical or a kinematical sum rule, thus leading to four possible alternative descriptions of the problem. Each of these descriptions can in principle be used to determine the spectral functions of both the longitudinal and the transverse dynamical spin-correlation functions. However, upon detailed examination of the 2nd RPA, it is found that only one of these four possible descriptions satisfies all the various consistency requirements. At low temperatures, this description reproduces the spin-wave results. To facilitate analytical solutions, an approximate version of the 2nd RPA, called the modified (mod) RPA, is introduced which leads to a satisfactory expression for the longitudinal correlation function over the entire range of temperatures. Upon examination it is found that the mod RPA determines the longitudinal correlation to the same accuracy, for the case of the isotropic exchange, as the first RPA determines the transverse correlation function. In addition to this mod RPA version of the consistent 2nd RPA, there also appears to be another relatively satisfactory solution for the longitudinal correlation function which follows from one of the less consistent versions of the 2nd RPA. This is treated as a phenomenological result. The system thermodynamics is analyzed in the region of the transition temperature in terms of both the consistent version of the mod RPA, i.e., the I mod RPA, as well as the phenomenological representation, i.e., the II mod RPA, with results which in addition to being an improvement on those following from the first RPA are also free from the inherent inconsistencies of the first RPA. In conclusion, the related work of other authors is discussed and it is shown that all these works suffer from serious internal inconsistencies which render their results for the longitudinal correlation function completely unacceptable and erroneous.

Physical Review•Jul 10, 1967•26 citations
The structure of the space-time-dependent spin-correlation functions for an anisotropic Heisenberg paramagnet with cylindrical symmetry at elevated temperature is analyzed in terms of a two-parameter Gaussian representation representation of the generalized diffusivity. The parameters are chosen so as to exactly preserve the frequency moments, i.e., 〈\ensuremathω2n〉K\ensuremathα\ensuremathα{{〈{\ensuremath{\omega}}^{2n}〉}_{\mathrm{K}}}^{\ensuremath{\alpha}\ensuremath{\alpha}}〈\ensuremathω2n〉K​\ensuremathα\ensuremathα (\ensuremathα=z,x,ory\ensuremath{\alpha}=z, x, or y\ensuremathα=z,x,ory; n=0,1,2n=0, 1, 2n=0,1,2) of the longitudinal and the transverse spectral functions. These moments are calculated for arbitrary spin SSS, arbitrary range of the exchange interactions, and arbitrary dimensionality. The results for the correlation functions are worked out for one, two, and three dimensions. In the extreme transverse limit, the results are compared with the exactly soluble XY\mathrm{XY}XY model for a one-dimensional spin-\textonehalf{} system with only nearest-neighbor (transverse) exchange. The agreement of the results is found to be satisfactory.
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.
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=12S=\frac{1}{2}S=21​ 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>12S>\frac{1}{2}S>21​, 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 longitudinal and transverse space and time dependent spin correlation functions for the one-dimensional Heisenberg-Ising model are derived asymptotically in the weak-coupling-low-frequency-long-wavelength limit within the framework of ordinary many-body theory (field theory). By means of the Jordan-Wigner transformation and using methods developed previously for the Tomonaga model (1950), the zero-temperature correlation functions is expressed as functional integrals. The associated one-body problem is explicitly soluble in the asymptotic regions and the correlation functions assume the form of Gaussian integrals which are evaluated analytically in the weak-coupling-low-frequency-long-wavelength limit. Apart from a linear correction to the transverse exchange constant agreement with Luther and Peschel (1975) is obtained.
We comment on the recent work of Fernandez and Gersch on the frequency Fourier transform of the selfcorrelation function for a one-dimensional, S=12S=\frac{1}{2}S=21​, 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.
Carboni and Richards have performed exact numerical calculations of the time-dependent two-spin correlation function 〈S1z(t)S1z(0)〉〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉〈S1​z(t)S1​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 〈S1z(t)S1z(0)〉〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉〈S1​z(t)S1​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.
The quantum-statistical mechanics of an anisotropic Heisenberg spin system is studied by a temperature-time-dependent Green's-function formalism. A consistent scheme of higher-order random-phase approximations (RPA) is developed and the second-order (2nd) RPA is examined in detail. It turns out that the 2nd RPA can be defined in two alternative versions, I and II, with equal a priori justification. These two versions of the 2nd RPA have then to be supplemented with either a dynamical or a kinematical sum rule, thus leading to four possible alternative descriptions of the problem. Each of these descriptions can in principle be used to determine the spectral functions of both the longitudinal and the transverse dynamical spin-correlation functions. However, upon detailed examination of the 2nd RPA, it is found that only one of these four possible descriptions satisfies all the various consistency requirements. At low temperatures, this description reproduces the spin-wave results. To facilitate analytical solutions, an approximate version of the 2nd RPA, called the modified (mod) RPA, is introduced which leads to a satisfactory expression for the longitudinal correlation function over the entire range of temperatures. Upon examination it is found that the mod RPA determines the longitudinal correlation to the same accuracy, for the case of the isotropic exchange, as the first RPA determines the transverse correlation function. In addition to this mod RPA version of the consistent 2nd RPA, there also appears to be another relatively satisfactory solution for the longitudinal correlation function which follows from one of the less consistent versions of the 2nd RPA. This is treated as a phenomenological result. The system thermodynamics is analyzed in the region of the transition temperature in terms of both the consistent version of the mod RPA, i.e., the I mod RPA, as well as the phenomenological representation, i.e., the II mod RPA, with results which in addition to being an improvement on those following from the first RPA are also free from the inherent inconsistencies of the first RPA. In conclusion, the related work of other authors is discussed and it is shown that all these works suffer from serious internal inconsistencies which render their results for the longitudinal correlation function completely unacceptable and erroneous.