Random-bond Ising chain in a magnetic field at low temperatures

The low-temperature behavior ($T\ll h, J$) of a random-bond Ising chain in a magnetic field is considered [$H=-\left(\frac{1}{2}\right)J\Sigma T_i{\sigma }_i{\sigma }_{i+1\mathrm{}}-h\Sigma {\sigma }_i, {\sigma }_i=\pm 1$, $\left\{T_i\right\}$ is a fixed random sequence of numbers +1 and -1 with concentrations $c_1$ and $c_2=1-c_1$, respectively]. The ground-state energy $E_0$, magnetization ${\mu }_0$ and zero-point entropy $S_0$ are calculated exactly. It is shown that ${\mu }_0$ and $S_0$ are discontinuous functions of magnetic field having jumps at $h=\frac{J}{n}$, $n=1, 2, \dots$.