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Physical Review

Time Dependence of Spin Operators in Finite Heisenberg Linear Chains

Fernando Carboni, Peter M. Richards

Published Jan 10, 1969• 104 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.177.889

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

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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.

Physical Review•Aug 10, 1968•18 citations
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.
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.
Dynamical spin correlation functions in (q, omega )-space are calculated numerically for linear quantum-mechanical Heisenberg systems containing up to ten spins with s= 1 / 2 and s=1. The authors consider ferro- and antiferromagnets including single-site and exchange anisotropies. They compare their results with various other theoretical treatments and draw conclusions for the dynamics of infinite chains. In the case of the s=1 planar Heisenberg ferromagnet direct comparison is made with inelastic neutron scattering cross sections of CsNiF 3 . The theoretical and experimental results are in good agreement.
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 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.
The two-time spin-pair correlation function σ(R,t) is expressed as a product of a Gaussian distribution function and a power series with respect to time. The width of the Gaussian distribution function is determined by the second derivative with respect to time of the autocorrelation function σ(0,t). The coefficients of the power series are determined up to terms of order t8 for the square and s.c. isotropic Heisenberg and XY magnets of spin ½ at infinite temperature, and up to terms of t10 for the linear magnets. The expressions obtained by truncating the power series to the exactly known terms give a very good fit at short times to the exact expression for the linear XY model, and to the results of the computer simulation due to Windsor for the s.c. Heisenberg ferromagnet. The Fourier-time transforms of the expressions thus obtained are shown to be the Gram-Charlier expansions. Thus we conclude that the Gram-Charlier expansion of the Fourier-time transform of σ(R,t),σ̃(R,ω), and that of its Fourier-space transform I(k,t),S(k,ω), is especially useful in the cases when the short-time behavior of σ(R,t) or I(k,t) is considered important; generally speaking, this is the case at large k or at ω which is not very small. When the width of the Gaussian distribution function is determined by the second derivative of I(k,t), a convergent result is not obtained for small values of k. For larger k, the convergence is as good as for the case when the width is determined by the autocorrelation function.
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.