长程相互作用量子格点模型的图格点和与图zeta函数
Graph lattice sums and graph zeta functions for long-range interacting quantum lattice models
- Saarland University（萨尔兰大学）
- ETH Zürich（苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究提出一种计算图格点和（图zeta函数）的新方法，将长程相互作用量子格点模型的高阶链接簇展开计算时间从数万核时降至分钟级，并通过FFT和树宽分解实现高效精确求解。
AI中文摘要：
在模拟有能隙的量子格点模型中，抑制希尔伯特空间维度随系统尺寸呈指数增长的问题，对于理解和设计奇异量子材料具有极其重要的意义，其中非局域相互作用尤为引人关注。高阶链接簇展开方法能够求解无限系统的本征值问题，但该方法依赖于计算具有图结构的高维振荡格点和，而此前这类计算只能借助蒙特卡洛方法实现。本研究解决了这一问题，使得所有必需的图格点和（对于涉及幂律的核，称为图zeta函数）均可计算。所提出的方法将最先进的级数展开的计算时间从数万核时缩短至几分钟。在将格点和按块分解后，每个块根据其树宽$\mathrm{tw}$采用最经济的可用策略进行评估。基本块可借助广义zeta函数以解析形式表达。对于$\mathrm{tw}\le 2$的串并联块，可利用基于Epstein zeta函数和快速衰减傅里叶级数的半解析代数，以图节点数和动量网格规模的线性成本进行计算。最后，对于$\mathrm{tw}>2$的情况，该方法与张量网络桶消除相结合，使得数值计算和内存消耗随动量网格规模呈多项式增长，且指数仅随$\mathrm{tw}$增长，而非随顶点数量增长。通过使用快速傅里叶变换（FFT），可以以单次动量评估的成本恢复整个动量网格。我们针对解析和数值基准，对方法的精度和运行时间进行了详细分析。我们还复现了已发表的横场伊辛模型在不同一维、二维和三维格点上的蒙特卡洛数据，结果完全一致。
英文摘要：
Taming the exponential increase of the Hilbert space dimension with system size in the simulation of gapped quantum lattice models is of the highest relevance for understanding and designing exotic quantum materials, where nonlocal interactions are of particular interest. High-order linked-cluster expansions provide access the solution of the eigenvalue problem for the infinite system, yet rely on the computation of high-dimensional oscillatory lattice sums with a graph structure, only approachable with Monte Carlo methods so far. This work resolves this issue, rendering all required graph lattice sums, referred to as graph zeta functions for kernels involving power-laws, computable. The resulting method reduces the evaluation time for state-of-the art series expansions from tenthousands of core-hours to minutes. After factorizing the lattice sum over blocks, each block is evaluated by the cheapest available strategy depending on its treewidth $\mathrm{tw}$. Basic blocks admit analytic forms in terms of generalized zeta functions. Series-parallel blocks with $\mathrm{tw}\le 2$ can be computed at linear cost in the number of graph nodes and in the size of the momentum grid using a semi-analytical algebra based on Epstein zeta functions and rapidly decaying Fourier series. Finally, for $\mathrm{tw}>2$, the method is combined with tensor-network bucket elimination yielding polynomial scaling of numerical work and memory in momentum grid size with exponents only growing with $\mathrm{tw}$ rather than with the number of vertices. Through use of FFT, the full momentum grid is recovered at the cost of a single momentum evaluation. We provide a detailed analysis of the precision and runtime of our method against analytic and numerical benchmarks. We further reproduce published Monte Carlo data for the transverse-field Ising model on different 1D, 2D, and 3D lattices, obtaining full agreement.
