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

J. Kisker and A.P. Young
Phys. Rev. B

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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