Finite-temperature and dynamical properties of the random transverse-field Ising spin chain

We study numerically the paramagnetic phase of the spin-$1/2$ random transverse-field Ising chain, using a mapping to noninteracting fermions. We extend our earlier work, Phys. Rev. B 53, 8486 (1996), to finite temperatures and to dynamical properties. Our results are consistent with the idea that there are Griffiths-McCoy singularities in the paramagnetic phase described by a continuously varying exponent $z\left(\delta \right),$ where δ measures the deviation from criticality. There are some discrepancies between the values of $z\left(\delta \right)$ obtained from different quantities, but this may be due to corrections to scaling. The average on-site time dependent correlation function decays with a power law in the paramagnetic phase, namely, ${\tau }^{-1/z\left(\delta \right)},$ where τ is imaginary time. However, the typical value decays with a stretched exponential behavior, $\exp \left(-c{\tau }^{1/\mu }\right),$ where μ may be related to $z\left(\delta \right).\mathrm{}$ We also obtain results for the full probability distribution of time dependent correlation functions at different points in the paramagnetic phase.