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Journal of Applied Physics

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

Fernando Carboni, Peter M. Richards

Published Feb 1, 1968• 17 citations
Publisher

Abstract

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.

DOI
10.1063/1.1656346

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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.
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.
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.
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.
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.
Various static spin-correlation functions and the weights of the spin excitations, or collective normal modes, in the van Hove response function, observed in neutron scattering experiments, are calculated as a function of temperature for a model quantum-spin chain. The spins interact through an isotropic Heisenberg exchange that extends to nearest-neighbour spins. In our model, spins of magnitude 1/2 form a dimerized chain with alternating exchange interactions. We explore the static and dynamic properties for ferromagnetic and antiferromagnetic exchange interactions. Results for the various properties are shown to be exact in the limit of a high temperature, and we argue that the results are very good at low temperatures. Unlike the case for linear spin-wave theory, the results are also exact in the limit of strong dimerization, i.e. non-interacting coupled pairs of spins.
Quasi-one-dimensional magnetic materials provide important test beds for rigorous models of interacting many-body systems. Nevertheless, there have been few studies of finite temperature effects on spin correlations in low-spin systems, though these are most likely to exhibit quantum effects. Copper benzoate, Cu(C6D5COO)2⋅3D2O, is a good realization of the S=1/2 Heisenberg antiferromagnetic chain (HAFC). We have performed ac susceptibility and inelastic neutron scattering experiments on this material which establish the Hamiltonian and an excitation spectrum which is consistent with the spinon continuum ansatz of Müller et al. We present inelastic neutron scattering measurements of the temperature evolution of the dynamic spin–spin correlation function Szz(π,ω). Comparison will be made to the classical prediction and to a finite temperature field theory of Schulz as applied to this S=1/2 HAFC.