We introduce the notion of mixed symmetry quantum phase transition (MSQPT) as singularities in the transformation of the lowest-energy state properties of a system of identical particles inside each permutation symmetry sector $\mu$, when some Hamiltonian control parameters $\lambda$ are varied. We use a three-level Lipkin-Meshkov-Glick model, with $\text{U}\left(3\right)$ dynamical symmetry, to exemplify our construction. After reviewing the construction of $\text{U}\left(3\right)$ unitary irreducible representations using Young tableaux and the Gelfand basis, we first study the case of a finite number $N$ of three-level atoms, showing that some precursors (fidelity susceptibility, level population, etc.) of MSQPTs appear in all permutation symmetry sectors. Using coherent (quasiclassical) states of $\text{U}\left(3\right)$ as variational states, we compute the lowest-energy density for each sector $\mu$ in the thermodynamic $N\to \infty$ limit. Extending the control parameter space by $\mu$, the phase diagram exhibits four distinct quantum phases in the $\lambda \text{−}\mu$ plane that coexist at a quadruple point. The ground state of the whole system belongs to the fully symmetric sector $\mu =1$ and shows a fourfold degeneracy, due to the spontaneous breakdown of the parity symmetry of the Hamiltonian. The restoration of this discrete symmetry leads to the formation of four-component Schrödinger cat states.

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混合置换对称扇区在临界三能级Lipkin-Meshkov-Glick原子模型的热力学极限中的作用

我们引入混合对称量子相变（MSQPT）的概念作为奇异性，以变换每个置换对称扇区内相同粒子系统的最低能态特性 $\mu$，当某些哈密顿控制参数 $\lambda$各种各样。我们使用三级Lipkin-Meshkov-Glick模型，$\text{ü}（3）$动态对称，以举例说明我们的构造。经过审查建设$\text{ü}（3）$ 使用Young tableaux和Gelfand基础进行不可约表示，我们首先研究有限数的情况 $ñ$三能级原子的分布，表明MSQPT的某些前体（保真度，能级总体等）出现在所有排列对称扇区中。使用相干的（准经典）状态$\text{ü}（3）$ 作为变化状态，我们计算每个扇区的最低能量密度 $\mu$ 在热力学中 $ñ\to \infty$限制。通过扩展控制参数空间$\mu$，相图显示出四个不同的量子相 $\lambda \text{-}\mu$在四个点处共存的平面。整个系统的基态属于完全对称的扇区$\mu =\mathrm{1个}$并且由于哈密顿量的奇偶对称性的自发破坏而显示出四倍简并性。这种离散对称性的恢复导致形成了四分量薛定ding猫态。