  For the spin-glass chain in an external field $h$, a non-zero weight at the
origin of the bond distribution $\rho (J)$ is known to induce a non-analytical
magnetization at zero temperature : for $\rho(J) \sim A | J |^{\mu-1}$ near $J
\to 0$, the magnetization follows the Chen-Ma scaling $M \sim h^{\mu/(2+\mu)}$.
In this paper, we numerically revisit this model to obtain detailed statistical
information on the ground state configuration and on the low-energy two-level
excitations that govern the low temperature properties. The ground state
consists of long unfrustrated intervals separated by weak frustrated bonds: We
accordingly compute the strength distribution of these frustrated bonds, as
well as the length- and magnetization- distributions of the unfrustrated
intervals. We find that the low-energy excitations are of two types (i) one
frustrated bond of the ground state may have two positions that are nearly
degenerate in energy (ii) two neighboring frustrated bonds of the ground state
may be annihilated or created with nearly zero energy cost. For each excitation
type, we compute its probability density as a function of its length. We
moreover show that the contributions of these excitations to various
observables (specific heat, Edwards-Anderson order parameter, susceptibility)
are in full agreement with direct transfer matrix evaluations at low
temperature. Finally, following the special bimodal case $\pm J$, where a
Ma-Dasgupta RG procedure has been previously used to compute explicitly the
above observables, we discuss the possibility of an extended RG procedure : we
find that the ground state can be seen as the result of a hierarchical
`fragmentation' procedure that we describe.
