Mixed-order transition in the antiferromagnetic quantum Ising chain in a field

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 $h\mathrm{/}\mathrm{\Gamma }\to \mathrm{\infty }$, $\mathrm{\Gamma }$ 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 $h\mathrm{/}\mathrm{\Gamma }>0$ 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 $h\mathrm{/}\mathrm{\Gamma }$ the transition is of mixed-order.