The Lipkin–Meshkov–Glick (LMG) model was devised to test the validity of different approximate formalisms to treat many-particle systems. The model was constructed to be exactly solvable and yet non-trivial, in order to capture some of the main features of real physical systems. In the present contribution, we explicitly review the fact that different many-body approximations commonly used in different fields in physics clearly fail to describe the exact LMG solution. With similar assumptions as those adopted for the LMG model, we propose a new Hamiltonian based on a general two-body interaction. The new model (extended LMG) is not only more general than the original LMG model and, therefore, has a potentially larger spectrum of applicability, but also the physics behind its exact solution can be much better captured by common many-body approximations. At the basis of this improvement lies a new term in the Hamiltonian that depends on the number of constituents and polarizes the system; the associated symmetry breaking is discussed, together with some implications for the study of more realistic systems.

Lipkin-Meshkov-Glick (LMG) 模型旨在测试不同近似形式主义处理多粒子系统的有效性。为了捕捉真实物理系统的一些主要特征，该模型被构建为完全可解且非平凡的。在目前的贡献中，我们明确回顾了一个事实，即物理学中不同领域常用的不同多体近似显然无法描述精确的 LMG 解决方案。使用与 LMG 模型采用的假设类似的假设，我们提出了一个基于一般两体相互作用的新哈密顿量。新模型（扩展 LMG）不仅比原始 LMG 模型更通用，因此具有更大的适用范围，但其精确解背后的物理原理也可以通过常见的多体近似来更好地捕捉。这种改进的基础是哈密顿量中的一个新项，它取决于组成部分的数量并使系统极化；讨论了相关的对称性破坏，以及对更现实系统研究的一些影响。