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

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