We study a system involving a single quantum degree of freedom per site of the lattice interacting with a few neighbors (up to second neighbors), with the interactions chosen so as to produce frustration. At zero temperature, this system undergoes several quantum phase transitions from both gapped to gapless and gapless to gapless phases, providing a very rich phase structure with disordered, homogeneous, and modulated ordered phases meeting in a quantum Lifshitz point. The gapless phases spontaneously break spatial lattice translations as well as internal symmetries of the form $\text{U}{\left(1\right)}^{{\mathsf{N}}_c}$, where ${\mathsf{N}}_c$ is the number of independent pitch vectors that arise in the homogeneous and modulated ordered phases. We carry out a detailed analysis of the quantum critical behavior, discussing the mechanism leading to the phase transitions. We also discuss a proper characterization of all the gapless phases as well as the nature of the Goldstone excitations. We study the behavior of the correlation functions and identify regions in the phase diagram where the system exhibits generalized symmetries such as polynomial shift symmetry. This type of symmetry plays an important role in the so-called fractonic phase, which is an exotic form of matter recently discovered.

中文翻译：

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自发对称破坏和受挫阶段

我们研究了一个系统，涉及每个晶格点的单个量子自由度与几个邻居（最多第二个邻居）相互作用，选择的相互作用会产生挫折。在零温度下，该系统经历了从有间隙到无间隙和从无间隙到无间隙相的几次量子相变，提供了非常丰富的相结构，具有无序、均匀和调制有序的相在量子 Lifshitz 点相遇。无间隙相自发地打破空间晶格平移以及形式的内部对称性$\text{你}{\left(1\right)}^{{\mathsf{N}}_C}$， 在哪里 ${\mathsf{N}}_C$是在均匀和调制有序相位中出现的独立音调矢量的数量。我们对量子临界行为进行了详细分析，讨论了导致相变的机制。我们还讨论了所有无间隙相的正确表征以及戈德斯通激发的性质。我们研究相关函数的行为并识别相图中系统表现出广义对称性（例如多项式平移对称性）的区域。这种对称性在所谓的分形相中起着重要作用，这是最近发现的一种奇特的物质形式。