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The investigation of the first-order quantum phase transition (QPT) is farfrom clarity in comparison to that of the second-order or continuous QPT, inwhich the order parameter and associated broken symmetry can be clearlyidentified and at the same time the concepts of universality class and criticalscaling can be characterized by critical exponents. Here we present a comparedstudy of these two kinds of QPT in the transverse Ising model. In the absenceof a longitudinal field, the ground state of the model exhibits a second-orderQPT from paramagnetic phase to ferromagnetic one, which is smeared out once thelongitudinal field is applied. Surprisingly, the first excited state involves afirst-order QPT as the longitudinal field increases, which has not beenreported in literature. Within the framework of a pattern picture we clearlyidentify the difference between these two kinds of QPT: for the continuous QPTonly the pattern flavoring ferromagnetic phase is always dominant over theothers, and on the contrary, there exist at least two competitive patterns inthe first-order QPT, which is further indicated by patterns' occupanciescalculated by pattern projections on the ground and first excited stateswavefunctions. Our result has not only a fundamental significance in theunderstandings of the nature of QPTs, but also a practical interest in quantumsimulations used to test the present finding.

First-Order Excited-State Quantum Phase Transition in the Transverse Ising Model with a Longitudinal Field