Lattice models consisting of high-dimensional local degrees of freedom without global particle-number conservation constitute an important problem class in the field of strongly correlated quantum many-body systems. For instance, they are realized in electron-phonon models, cavities, atom-molecule resonance models, or superconductors. In general, these systems elude a complete analytical treatment and need to be studied using numerical methods where matrix-product states (MPSs) provide a flexible and generic ansatz class. Typically, MPS algorithms scale at least quadratic in the dimension of the local Hilbert spaces. Hence, tailored methods, which truncate this dimension, are required to allow for efficient simulations. Here, we describe and compare three state-of-the-art MPS methods each of which exploits a different approach to tackle the computational complexity. We analyze the properties of these methods for the example of the Holstein model, performing high-precision calculations as well as a finite-size-scaling analysis of relevant ground-state observables. The calculations are performed at different points in the phase diagram yielding a comprehensive picture of the different approaches.

由没有全局粒子数守恒的高维局部自由度组成的晶格模型构成了强关联量子多体系统领域的一个重要问题类别。例如，它们在电子-声子模型、腔、原子-分子共振模型或超导体中实现。通常，这些系统无法进行完整的分析处理，需要使用数值方法进行研究，其中矩阵乘积状态 (MPS) 提供了灵活且通用的 ansatz 类。通常，MPS 算法在局部Hilbert 空间的维度上至少进行二次缩放. 因此，需要截断该维度的定制方法来进行有效的模拟。在这里，我们描述并比较了三种最先进的 MPS 方法，每种方法都利用不同的方法来解决计算复杂性。我们以荷斯坦模型为例分析了这些方法的特性，执行高精度计算以及相关基态可观测值的有限尺寸缩放分析。计算是在相图中的不同点进行的，产生了不同方法的综合图。