  The antiferromagnetic quantum Ising chain has a quantum critical point which
belongs to the universality class of the transverse Ising model (TIM). When a
longitudinal field ($h$) is switched on, the phase transition is preserved,
which turns to first-order for $h/\Gamma \to \infty$, $\Gamma$ being the
strength of the transverse field. Here we will re-examine the critical
properties along the phase transition line. During a quantum block
renormalization group calculation, the TIM fixed point for $h/\Gamma>0$ is
found to be unstable. Using DMRG techniques, we calculated the entanglement
entropy and the spin-spin correlation function, both of which signaled a
divergent correlation length at the transition point with the TIM exponents. At
the same time, the bulk correlation function has a jump and the end-to-end
correlation function has a discontinuous derivative at the transition point.
Consequently for finite $h/\Gamma$ the transition is of mixed-order.
