arXiv:cond-mat/9805337 · Physical review. B, Condensed matter

Quantum-classical crossover in the spin-1/2 XXZ chain

K. Fabricius, Barry M. McCoy

Published Jan 1, 1999• 15 citations

Abstract

We compute, by means of exact diagonalization of systems of $N=16$ and $18$ spins, the correlation function $〈{\ensuremath{\sigma}}_{0}^{z}{\ensuremath{\sigma}}_{n}^{z}〉$ at nonzero temperature for the $\mathrm{XXZ}$ model with anisotropy $\ensuremath{\Delta}.$ In the gapless attractive region $\ensuremath{-}1<\ensuremath{\Delta}<0$ for fixed separation the temperature can always be made sufficiently low so that the correlation is always negative for $n\ensuremath{\ne}0.$ However we find that for sufficiently large temperatures and fixed separation or for fixed temperature greater than some ${T}_{0}(\ensuremath{\Delta})$ and sufficiently large separations the correlations are always positive. This sign changing effect has not been previously seen and we interpret it as a crossover from quantum to classical behavior.

DOI: 10.1103/physrevb.59.381

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arXiv:cond-mat/9805337 · Physical review. B, Condensed matter

Quantum-classical crossover in the spin-1/2 XXZ chain

K. Fabricius, Barry M. McCoy

Published Jan 1, 1999• 15 citations

arXiv HTML PDF

Abstract

We compute, by means of exact diagonalization of systems of

N=16

and

18

spins, the correlation function

〈{\ensuremath{\sigma}}_{0}^{z}{\ensuremath{\sigma}}_{n}^{z}〉

at nonzero temperature for the

\mathrm{XXZ}

model with anisotropy

\ensuremath{\Delta}.

In the gapless attractive region

\ensuremath{-}1<\ensuremath{\Delta}<0

for fixed separation the temperature can always be made sufficiently low so that the correlation is always negative for

n\ensuremath{\ne}0.

However we find that for sufficiently large temperatures and fixed separation or for fixed temperature greater than some

{T}_{0}(\ensuremath{\Delta})

and sufficiently large separations the correlations are always positive. This sign changing effect has not been previously seen and we interpret it as a crossover from quantum to classical behavior.

DOI: 10.1103/physrevb.59.381

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We compute, by means of exact diagonalization of systems of

N=16

and

18

spins, the correlation function

〈{\ensuremath{\sigma}}_{0}^{z}{\ensuremath{\sigma}}_{n}^{z}〉

at nonzero temperature for the

\mathrm{XXZ}

model with anisotropy

\ensuremath{\Delta}.

In the gapless attractive region

\ensuremath{-}1<\ensuremath{\Delta}<0

for fixed separation the temperature can always be made sufficiently low so that the correlation is always negative for

n\ensuremath{\ne}0.

However we find that for sufficiently large temperatures and fixed separation or for fixed temperature greater than some

{T}_{0}(\ensuremath{\Delta})

and sufficiently large separations the correlations are always positive. This sign changing effect has not been previously seen and we interpret it as a crossover from quantum to classical behavior.

We study the correlation $〈{\ensuremath{\sigma}}_{0}^{z}{\ensuremath{\sigma}}_{n}^{z}〉$ for the

\mathrm{XXZ}

chain in the massless attractive (ferromagnetic) region at positive temperatures by means of a numerical study of the quantum transfer matrix. We find that there is a range of temperature where the behavior of the correlation for large separations is oscillatory with an incommensurate period which depends on temperature.

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.

The random antiferromagnetic spin-

1∕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

\mathcal{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.