Time-dependent spin autocorrelation functions and their spectral densities for the semi-infinite one-dimensional 1 XY and XXZ models at T= oo are determined in part by rigorous calculations in the fermion representation and in part by the recursion method in the spin representation. Boundary effects yield valuable new insight into the different dynamical processes which govern the transport of spin fluctuations in the two models. The results obtained for the X X X model bear the unmistakable signature of spin diffusion in the form of a squareroot infrared divergence in the spectral density.
Time-dependent spin autocorrelation functions and their spectral densities for the semi-infinite one-dimensional 1 XY and XXZ models at T= oo are determined in part by rigorous calculations in the fermion representation and in part by the recursion method in the spin representation. Boundary effects yield valuable new insight into the different dynamical processes which govern the transport of spin fluctuations in the two models. The results obtained for the X X X model bear the unmistakable signature of spin diffusion in the form of a squareroot infrared divergence in the spectral density.
Time-dependent spin-autocorrelation functions at T=\ensuremath{\infty} and (in particular) their spectral densities for the bulk spin and the boundary spin of the semi-infinite spin-1/2 XXZ model (with exchange parameters Jx=Jy\ensuremath{\equiv}J, Jz) are investigated on the basis of (i) rigorous bounds in the time domain and (ii) a continued-fraction analysis in the frequency domain. We have found strong numerical evidence for spin diffusion in quantum spin models. For Jz/J increasing from zero, the results of the short-time expansion indicate a change of the bulk-spin xx-autocorrelation function from Gaussian decay to exponential decay. The continued-fraction analysis of the same dynamic quantity signals a change from exponential decay to power-law decay as Jz/J approaches unity and back to a more rapid decay upon further increase of that parameter. By contrast, the change in symmetry at Jz/J=1 has virtually no impact on the bulk-spin zz-autocorrelation function (as expected). Similar contrasting properties are observable in the boundary-spin autocorrelation functions.
The spectral densities of dynamical spin autocorrelation functions at infinite temperature are studied for the S=1/2XXZ model (with exchange couplings Jx=Jy\ensuremath{\equiv}J,Jz) on the linear chain, the square lattice, and the simple cubic lattice. The low-frequency behavior of a given spectral density is inferred from certain characteristic properties of its continued-fraction coefficients as determined from computed frequency moments. The analysis yields estimates for the Jz/J dependence of the infrared-singularity exponent. In the d=1 case, the exponent for spin fluctuations perpendicular to the O(2) symmetry axis responds sensitively as the anisotropy parameter sweeps across the O(3) symmetry point Jz/J=1, while the exponent for the parallel fluctuations shows little variation. In the cases d=2 and d=3 the same observations are made for autocorrelation functions of aggregate spins in chains and lattice planes, respectively.
The recursion method is applied to the T=0 dynamics of the S=1/2 XXZ model on a linear chain and a square lattice. By means of new calculational techniques for the analysis of the continued-fraction coefficients pertaining to specific dynamical quantities, we obtain reliable information on the type of ordering in the ground state, on the size of gaps in the dynamically relevant excitation spectrum, on the bandwidths of dominant structures in spectral densities, on the exponents of infrared singularities, and on the detailed shape of spectral-weight distributions. We investigate some characteristic properties of the dynamic structure factors ${\mathit{S}}_{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(q,\ensuremath{\omega}) and the spin autocorrelation functions ${\mathit{S}}^{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(\ensuremath{\omega})=${\mathit{N}}^{\mathrm{\ensuremath{-}}1}$ tsumq${\mathit{S}}_{\mathrm{\ensuremath{\mu}}\mathrm{\ensuremath{\mu}}}$(q,\ensuremath{\omega}), specifically their dependence on the uniaxial anisotropy, i.e., on the parameter which controls the type of ordering and the amount of quantum fluctuations in the ground state. We find, for example, that the different degrees of ordering in the planar regime of the one-dimensional and two-dimensional systems (criticality versus antiferromagnetic long-range order) have characteristic signatures in the dynamical properties which are conspicuously displayed in our results.
The infinite-temperature space- and time-dependent spin-correlation functions grx(t) are studied for the one-dimensional XY model. Numerical calculations are performed to obtain the exact autocorrelation function g0x(t) for chains containing 5, 7, and 9 spins (S=21). This yields exact results for the first 16 moments of the frequency autocorrelation function of the infinite chain, and estimates for a few of the higher moments as well. The analysis suggests that g0x(t) for the infinite chain is identical to $\frac{\mathrm{exp}(\ensuremath{-}{J}^{2}{t}^{2})}{4}$. We show that grx(t) for $r\ensuremath{\ne}0$ vanishes identically for all values of time, implying a wave-vector-independent relaxation shape function. Our result for g0x(t) is compared with that obtained by Huber for the classical ($S=\ensuremath{\infty}$) chain.
We present new results on the time-dependent correlation functions Ξn(t) =4〈Sξ0(t)Sξn〉, ξ=x,y at zero temperature of the one-dimensional S=1/2 isotropic XY model (h=γ=0) and of the transverse Ising model (TI) at the critical magnetic field (h=γ=1). Both models are characterized by special cases of the Hamiltonian H=−J∑l[(1+γ)SxlSxl+1 +(1−γ)SylSyl+1 +hSzl]. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions (ACF’s) Ξ0(t) and on the singularities of their frequency-dependent Fourier transforms Φξξ0(ω). We have also determined the latter functions by high-precision numerical calculations. The functions Φξξ0(ω), ξ=x,y have singularities at the infinite sequence of frequencies ω=mω0, m=0, 1, 2, 3, ... where ω0=J for the XY model and ω0=2J for the TI model. In both models the singularities in Φxx0 (ω) for m=0, 1 are divergent, whereas the nonanalyticities at higher frequencies become increasingly weaker. We point out that the nonanalyticities at ω≠0 are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis.
We address the nature of spin transport in the integrable XXZ spin chain, focusing on the isotropic Heisenberg limit. We calculate the diffusion constant using a kinetic picture based on generalized hydrodynamics combined with Gaussian fluctuations: we find that it diverges, and show that a self-consistent treatment of this divergence gives superdiffusion, with an effective time-dependent diffusion constant that scales as D(t)∼t^{1/3}. This exponent had previously been observed in large-scale numerical simulations, but had not been theoretically explained. We briefly discuss XXZ models with easy-axis anisotropy Δ>1. Our method gives closed-form expressions for the diffusion constant D in the infinite-temperature limit for all Δ>1. We find that D saturates at large anisotropy, and diverges as the Heisenberg limit is approached, as D∼(Δ-1)^{-1/2}.