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Despite of simplicity of the transverse antiferromagnetic Ising model with auniform longitudinal field, its phases and involved quntum phase transitions(QPTs) are nontrivial in comparison to its ferromagnetic counterpart. Forexample, what is the nature of the mixed-order in such a model and does thereexist a disorder phase? Here we use a pattern picture to explore thecompetitions between the antiferromagnetic Ising interaction, the transverseand longitudinal fields and uncover what kind of pattern takes responsibilityof these three competing energy scales, thus determine the possible phases andtheir QPTs or crossovers. Our results not only unveil rich physics of thisparadigmatic model, but also further stimulate quantum simulation by usingcurrent available experimental platforms.

Pattern Description of Quantum Phase Transitions in the Transverse Antiferromagnetic Ising Model with a Longitudinal Field