The problem of spin diffusion is studied numerically in the one-dimensional classical Heisenberg model using a deterministic odd-even spin precession dynamics. We demonstrate that spin diffusion in this model is normal in the infinite-temperature limit and one obtains a long-time diffusive tail in the decay of the autocorrelation function. Some variations of the model with different coupling schemes and anisotropy are also studied and we find normal diffusion in all of them. A systematic finite-size analysis also suggests normal diffusive spreading of spin fluctuations, contrary to previous claims of anomalous diffusion. We also briefly discuss spin diffusion in this model at finite temperatures.
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
0.860Ja2,
0.619Ja2, and
0.509Ja2, respectively, for these lattices, where
J is the exchange integral and
a is the lattice constant. The spin diffusion constants for larger spins are also provided. It should be noted that
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. For the one-dimensional isotropic Heisenberg magnet and for the square, sc, and bcc
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 model.
We report a systematic study of finite-temperature spin transport in quantum and classical one-dimensional magnets with isotropic spin interactions, including both integrable and nonintegrable models. Employing a phenomenological framework based on a generalized Burgers' equation in a time-dependent stochastic environment, we identify four different universality classes of spin fluctuations. These comprise, aside from normal spin diffusion, three types of superdiffusive transport: the Kardar-Parisi-Zhang universality class and two distinct types of anomalous diffusion with multiplicative logarithmic corrections. Our predictions are supported by extensive numerical simulations on various examples of quantum and classical chains. Contrary to common belief, we demonstrate that even nonintegrable spin chains can display a diverging spin diffusion constant at finite temperatures.
We study the one-dimensional Heisenberg spin-
1∕2 alternating bond chain in which the nearest neighbor exchange couplings are ferromagnetic (FM) and antiferromagnetic (AF) alternatively. By using exact diagonalization and the density matrix renormalization group method, we discuss how the system approaches the AF uniform spin-1 chain under certain conditions. When the ratio of AF to FM coupling strengths, $\ensuremath{\alpha}$ $(\ensuremath{\alpha}={J}_{AF}∕{J}_{F})$, is very small, physical properties of the alternating bond chain such as spin-spin correlation, the string correlation function, and the spin density coincide with those of the AF uniform spin-1 chain. The edge state problem is discussed in the present model with the small $\ensuremath{\alpha}$ limit. In addition, the Haldane gap of the AF uniform spin-1 chain is four times the gap of the system considered.
A coupled-mode theory of spin fluctuations in the d-dimensional Heisenberg magnet at infinite temperature is used to predict the time dependence of various spin correlation functions. The real-space spin autocorrelation function is shown to have a long-time behaviour approximately (1/t) d theta / where theta =(4+d)/2. Properties at intermediate values of the time are extracted from the theory by numerical analysis. In this time window, the reciprocal-lattice spin autocorrelation function, G(q, t), is, to a good approximation, an exponential function of time. The decay rate is proportional to q alpha , where q is the wavevector. Analysis of our numerical data indicates that the exponent alpha depends weakly on d, and it is significantly different from the value 2 which is compatible with a spin diffusion model. In the asymptotic limit, defined by q to 0, t to infinity , and q 2 t to 0, G(q,t) is a function of a single variable=(tq theta ). This result rules against the validity of a diffusion model also in the asymptotic limit.