Pattern description of quantum phase transitions in the transverse antiferromagnetic Ising model with a longitudinal field

A uniform longitudinal field applied to the transverse Ising model (TIM) distinguishes the antiferromagnetic Ising interaction from its ferromagnetic counterpart. While the ground state of the latter shows no quantum phase transition (QPT), the ground state of the former exhibits rich phases: paramagnetic, antiferromagnetic, and possibly disordered phases. Although the first two are clearly identified, the existence of the disordered phase remains controversial. Here, we use the pattern picture to explore the competition among the antiferromagnetic Ising interaction J, the transverse field h_x and the longitudinal field h_z, and uncover which patterns are responsible for these three competing energy scales, thereby determining the possible phases and the QPTs among them. The system size ranges from L = 8 to 128 and the transverse field h_x is fixed at 1. Under these parameters, our results show the existence of the disordered phase. For a small h_z, the system transitions from a disordered phase to an antiferromagnetic phase as J increases. For a large h_z, the system undergoes two phase transitions: from paramagnetic to disordered, and then to antiferromagnetic phase. These results not only unveil the rich physics of this paradigmatic model but also stimulate quantum simulation by using currently available experimental platforms.