Finite Temperature and Dynamical Properties of the Random Transverse-Field Ising Spin Chain

We study numerically the paramagnetic phase of the random transverse-field Ising chain, using a mapping to non-interacting fermions. We extend our earlier work, Phys. Rev. 53, 8486 (1996), to finite temperatures and to dynamical properties. Our results show that there are ``Griffiths-McCoy'' singularities in the paramagnetic phase which can be described by a continuously varying exponent $z(\delta)$, where $\delta$ measures the deviation from criticality. The average on-site time dependent correlation function decays with a power law in the paramagnetic phase, namely $\tau^{-1/z(\delta)}$, where $\tau$ is imaginary time. However, the typical value decays with a stretched exponential behavior, $\exp(-c\tau^{1/\mu})$, where $\mu$ ($\approx 2$) does not seem to vary much throughout the paramagnetic phase and may be a constant. We also obtain results for the full probability distribution of time dependent correlation functions at different points in the paramagnetic phase.