## Dynamical Critical Properties of the Random Transverse-Field Ising
Spin Chain

J. Kisker
and
A.P. Young

Phys. Rev. B
**Abstract**

We study the dynamical properties of the random transverse-field
Ising chain at criticality
using a mapping to free fermions, with which we can obtain
numerically exact results for system sizes, $L$, as large as 256.
The probability distribution of the local imaginary time correlation
function $S(\tau)$ is investigated and found to be simply a function of
$\alpha \equiv -\log S(\tau) / \log \tau$.
This scaling behavior implies that the {\em typical}
correlation function decays
algebraically, $S_{{\rm typ}}(\tau) \sim \tau^{-\alpha_{\rm typ}}$,
where the exponent $\alpha_{\rm typ}$
is determined from
$P(\alpha)$, the distribution of $\alpha$.
The precise value for $\alpha_{\rm typ}$ depends on exactly how the
``typical'' correlation function is defined. The form of $P(\alpha)$ for
small $\alpha$ gives a contribution to the {\em average} correlation function,
$S_{{\rm av}}(\tau)$, namely
$S_{{\rm av}}(\tau) \sim(\log \tau)^{-2x_m}$, where
$x_m$ is the bulk magnetization exponent, which was obtained recently in
Europhys. Lett. {\bf 39}, 135 (1997). These results represent a type of
``multiscaling'' different from the well-known
``multifractal'' behavior.

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