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arXiv:cond-mat/9711146 · Physical review. B, Condensed matter

Spin diffusion and the spin-1/2 XXZ chain at T=∞ from exact diagonalization

K. Fabricius, Barry M. McCoy

Published Apr 1, 1998• 77 citations
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Abstract

We study the long time behavior of the $\mathrm{zz}$ and $\mathrm{xx}$ time-dependent autocorrelation function of the spin-1/2 $\mathrm{XXZ}$ chain at $T=\ensuremath{\infty}$ by exact diagonalizations on a chain of 16 sites. We find that the numerical results for the $\mathrm{zz}$ correlation are very well fit by the formula ${t}^{\ensuremath{-}d}[{A+Be}^{\ensuremath{-}\ensuremath{\gamma}(t\ensuremath{-}{t}_{0})}\mathrm{cos}\ensuremath{\Omega}(t\ensuremath{-}{t}_{0})].$ From this we estimate $d$ as a function of the anisotropy of the chain and study the crossover from ballistic to diffusive behavior.

DOI
10.1103/physrevb.57.8340

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We study the long time behavior of the zz\mathrm{zz}zz and xx\mathrm{xx}xx time-dependent autocorrelation function of the spin-1/2 XXZ\mathrm{XXZ}XXZ chain at T=\ensuremath∞T=\ensuremath{\infty}T=\ensuremath∞ by exact diagonalizations on a chain of 16 sites. We find that the numerical results for the zz\mathrm{zz}zz correlation are very well fit by the formula t\ensuremath−d[A+Be\ensuremath−\ensuremathγ(t\ensuremath−t0)cos\ensuremathΩ(t\ensuremath−t0)].{t}^{\ensuremath{-}d}[{A+Be}^{\ensuremath{-}\ensuremath{\gamma}(t\ensuremath{-}{t}_{0})}\mathrm{cos}\ensuremath{\Omega}(t\ensuremath{-}{t}_{0})].t\ensuremath−d[A+Be\ensuremath−\ensuremathγ(t\ensuremath−t0​)cos\ensuremathΩ(t\ensuremath−t0​)]. From this we estimate ddd as a function of the anisotropy of the chain and study the crossover from ballistic to diffusive behavior.
No spin diffusion in the spin 1/2 XXZ chain at T=∞T=\inftyT=∞: Numerical asymptotics
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The random antiferromagnetic spin-1∕21∕21∕2 XX and XXZ chain is studied numerically for varying strength of the disorder, using exact diagonalization and stochastic series expansion methods. The spin-spin correlation function as well as the stiffness display a clear crossover from the pure behavior (no disorder) to the infinite randomness fixed point or random singlet behavior predicted by the real space renormalization group. The crossover length scale is shown to diverge as $\ensuremath{\xi}\ensuremath{\sim}{\mathcal{D}}^{\ensuremath{-}\ensuremath{\gamma}}$, where D\mathcal{D}D is the variance of the random bonds. Our estimates for the exponent $\ensuremath{\gamma}$ agrees well within the error bars with the one for the localization length exponent emerging within an analytical bosonization calculation. Exact diagonalization and stochastic series expansion results for the string correlation function are also presented.
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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.