一种在大局部Hilbert空间模拟低维量子晶格模型的高效灵活方法
Efficient and Flexible Approach to Simulate Low-Dimensional Quantum Lattice Models with Large Local Hilbert Spaces
摘要：
具有大局部希尔伯特空间的量子晶格模型出现在量子多体物理的各个领域。诸如费米子和声子之间的相互作用、相互作用玻色子的BCS-BEC交叉、量子模拟器中的退相干等问题已经在理论和实验上得到了广泛的研究。近年来，张量网络方法已经成为数值处理格点系统的最成功的工具之一。然而，具有大局部希尔伯特空间的系统仍然具有挑战性。在这里，我们介绍了一个映射，允许构建人工$U(1)$对称的任何类型的晶格模型。利用生成的对称性，与局部自由度相关的数值费用显着减少。这允许有效地处理具有大的局部尺寸的系统。进一步探索这种映射，我们揭示了一个密切的联系之间的施密特值的相应的矩阵产品状态表示和单位点约化密度矩阵。我们的研究结果激发了一个直观的物理图片的截断发生在典型的算法，我们给出了界限的数值复杂性相比，标准的方法，不利用这种人为的对称性。我们展示了这种新的映射，提供了一个现有的代码的实现配方，并在半填充的荷斯坦模型进行示例计算。我们研究了系统具有非常大的数量的晶格站点高达$L=501$，而占$N_{\rm ph}=63$声子每个站点具有高精度的CDW阶段。
英文摘要：
Quantum lattice models with large local Hilbert spaces emerge across various fields in quantum many-body physics. Problems such as the interplay between fermions and phonons, the BCS-BEC crossover of interacting bosons, or decoherence in quantum simulators have been extensively studied both theoretically and experimentally. In recent years, tensor network methods have become one of the most successful tools to treat lattice systems numerically. Nevertheless, systems with large local Hilbert spaces remain challenging. Here, we introduce a mapping that allows to construct artificial $U(1)$ symmetries for any type of lattice model. Exploiting the generated symmetries, numerical expenses that are related to the local degrees of freedom decrease significantly. This allows for an efficient treatment of systems with large local dimensions. Further exploring this mapping, we reveal an intimate connection between the Schmidt values of the corresponding matrix-product-state representation and the single-site reduced density matrix. Our findings motivate an intuitive physical picture of the truncations occurring in typical algorithms and we give bounds on the numerical complexity in comparison to standard methods that do not exploit such artificial symmetries. We demonstrate this new mapping, provide an implementation recipe for an existing code, and perform example calculations for the Holstein model at half filling. We studied systems with a very large number of lattice sites up to $L=501$ while accounting for $N_{\rm ph}=63$ phonons per site with high precision in the CDW phase.
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