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Physical review. B, Solid state

Computer studies of spin and energy transport in one-dimensional Heisenberg magnets

N.A. Lurie, David L. Huber, Martin Blume

Published Mar 1, 1974• 99 citations
Publisher

Abstract

We have studied the dynamics of the classical Heisenberg chain at finite and infinite temperatures by computer simulation of an array of 4000 spins. The evaluation of the spin-spin and energy-energy wave-vector- and time-dependent correlation functions out to time $10{J}^{\ensuremath{-}1}$ made it possible to determine the spin and energy diffusion constants. At infinite temperature we find ${D}_{S}=(1.33\ifmmode\pm\else\textpm\fi{}0.10)JS{a}^{2}$ and ${D}_{E}=(3\ifmmode\pm\else\textpm\fi{}1)JS{a}^{2}$. Values of ${D}_{S}$ have also been determined, with less precision, for ferromagnetic and antiferromagnetic interactions at temperatures $\frac{{k}_{B}T}{|J|}=1.0 \mathrm{and} 0.5$. Our results are compared with various theoretical estimates. In addition, we calculate the time-dependent spin and energy self-correlation functions.

DOI
10.1103/physrevb.9.2171

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Dynamical Properties of the Diluted Heisenberg and XY Magnets at Infinite Temperature. I: Spin Diffusion Constant

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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.

Progress of Theoretical Physics•Oct 1, 1980•2 citations
We have studied the dynamics of the classical Heisenberg chain at finite and infinite temperatures by computer simulation of an array of 4000 spins. The evaluation of the spin-spin and energy-energy wave-vector- and time-dependent correlation functions out to time 10J\ensuremath−110{J}^{\ensuremath{-}1}10J\ensuremath−1 made it possible to determine the spin and energy diffusion constants. At infinite temperature we find DS=(1.33\ifmmode±\else\textpm\fi0.10)JSa2{D}_{S}=(1.33\ifmmode\pm\else\textpm\fi{}0.10)JS{a}^{2}DS​=(1.33\ifmmode±\else\textpm\fi0.10)JSa2 and DE=(3\ifmmode±\else\textpm\fi1)JSa2{D}_{E}=(3\ifmmode\pm\else\textpm\fi{}1)JS{a}^{2}DE​=(3\ifmmode±\else\textpm\fi1)JSa2. Values of DS{D}_{S}DS​ have also been determined, with less precision, for ferromagnetic and antiferromagnetic interactions at temperatures kBT∣J∣=1.0and0.5\frac{{k}_{B}T}{|J|}=1.0 \mathrm{and} 0.5∣J∣kB​T​=1.0and0.5. Our results are compared with various theoretical estimates. In addition, we calculate the time-dependent spin and energy self-correlation functions.
A theoretical criterion for the occurrence of the spin diffusion is given in terms of the "friction function" or "memory function" which occurs in the kinetic equation of the Fourierspace transform $I(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, t)$ of the two-time spin-pair correlation function. The friction function in the limit of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ensuremath{\rightarrow}0$ is determined with the aid of the short-time expansion of the two-time spin-pair correlation function for the square, sc, and bcc isotropic Heisenberg magnets at infinite temperature, and the criterion is checked. The spin diffusion constants for spin \textonehalf{} are 0.860Ja20.860J{a}^{2}0.860Ja2, 0.619Ja20.619J{a}^{2}0.619Ja2, and 0.509Ja20.509J{a}^{2}0.509Ja2, respectively, for these lattices, where JJJ is the exchange integral and aaa is the lattice constant. The spin diffusion constants for larger spins are also provided. It should be noted that 0.509Ja20.509J{a}^{2}0.509Ja2 is in complete agreement with the experimental value $(0.525\ifmmode\pm\else\textpm\fi{}0.06)J{a}^{2}$ for the bcc solid He3{\mathrm{He}}^{3}He3. For the one-dimensional isotropic Heisenberg magnet and for the square, sc, and bcc XY\mathrm{XY}XY magnets, the friction function could not be determined in the present calculation. Suggested values of the spin diffusion constant are given for these lattices. Comparison is made with the results of Windsor's computer-simulation calculation. There exists no spin diffusion for the one-dimensional XY\mathrm{XY}XY model.
The energy and spin diffusion behaviors in the one-dimensional classical Heisenberg spin chain have been systematically investigated using the equilibrium diffusion method. The spatiotemporal autocorrelation functions for energy and spin are calculated at finite and infinite temperatures. As conserved quantities, the spreading of excess energy and spin can be used to determine their actual diffusion behaviors. At low temperatures, the energy diffusion shows almost ballistic behavior, and spin shows superdiffusion behavior for finite chain size. For energy diffusion, normal diffusion behavior can be obtained when the temperature is higher than 0.75. For spin diffusion, normal diffusion behavior is observed at infinite temperature.
We use the functional renormalization approach for quantum spin systems developed by Krieg and Kopietz [Phys. Rev. B 99, 060403(R) (2019)] to calculate the spin-spin correlation function $G(\mathbit{k},\ensuremath{\omega})$ of quantum Heisenberg magnets at infinite temperature. For small wave vectors $\mathbit{k}$ and frequencies $\ensuremath{\omega}$ we find that $G(\mathbit{k},\ensuremath{\omega})$ assumes in dimensions d>2d>2d>2 the diffusive form predicted by hydrodynamics. In three dimensions our result for the spin-diffusion coefficient D\mathcal{D}D is somewhat smaller than previous theoretical predictions based on the extrapolation of the short-time expansion, but is still about 3030%30 larger than the measured high-temperature value of D\mathcal{D}D in the Heisenberg ferromagnet ${\mathrm{Rb}}_{2}{\mathrm{CuBr}}_{4}\ifmmode\cdot\else\textperiodcentered\fi{}2{\mathrm{H}}_{2}\mathrm{O}$. In reduced dimensions $d\ensuremath{\le}2$ we find superdiffusion characterized by a frequency-dependent complex spin-diffusion coefficient $\mathcal{D}(\ensuremath{\omega})$ which diverges logarithmically in d=2d=2d=2, and as a power-law $\mathcal{D}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{-}1/3}$ in d=1d=1d=1. Our result in one dimension implies scaling with dynamical exponent z=3/2z=3/2z=3/2, in agreement with recent calculations for integrable spin chains. Our approach is not restricted to the hydrodynamic regime and allows us to calculate the dynamic structure factor $S(\mathbit{k},\ensuremath{\omega})$ for all wave vectors. We show how the short-wavelength behavior of $S(\mathbit{k},\ensuremath{\omega})$ at high temperatures reflects the relative sign and strength of competing exchange interactions.
Computer calculations of the time-dependent spin and energy correlation functions of the classical one-dimensional XY\mathrm{XY}XY model at infinite temperature are reported. Plots of 〈SziSzi(t)〉〈{S}_{z}^{i}{S}_{z}^{i}(t)〉〈Szi​Szi​(t)〉, 〈SziSzi+1(t)〉〈{S}_{z}^{i}{S}_{z}^{i+1}(t)〉〈Szi​Szi+1​(t)〉, $〈{\ensuremath{\epsilon}}^{i}{\ensuremath{\epsilon}}^{i}(t)〉$, and (12)[〈SxiSxi(t)〉+〈SyiSyi(t)〉](\frac{1}{2})[〈{S}_{x}^{i}{S}_{x}^{i}(t)〉+〈{S}_{y}^{i}{S}_{y}^{i}(t)〉](21​)[〈Sxi​Sxi​(t)〉+〈Syi​Syi​(t)〉] out to times Jt=9Jt=9Jt=9 are given (${\ensuremath{\epsilon}}^{i}$ is the energy-density operator and $\ensuremath{\hbar}J$ is the exchange integral). Comparison is made with the exact results for the spin-1/2 XY\mathrm{XY}XY model. After an appropriate scaling of the exchange integral and normalization to the classical value at t=0t=0t=0 the values of the spin-1/2 functions are close to the values of their classical counterparts for times up to $\mathrm{Jt}\ensuremath{\approx}4$.
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.