紧束缚哈密顿量的精确矩阵乘积算子的有限状态自动机：分形、准晶体、树和双曲晶格
Finite-state automata for exact matrix product operators of tight-binding Hamiltonians: fractals, quasicrystals, trees and hyperbolic lattices
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中文总结 AI 辅助
本文提出一种基于有限状态自动机的矩阵乘积算子构造方法，用于递归结构晶格上的紧束缚哈密顿量，实现精确解析表示，支持分形、准晶体、树和双曲晶格的大规模谱计算。
中文摘要 AI 辅助
受近期利用张量网络模拟大晶格上紧束缚哈密顿量进展的启发，我们针对递归结构晶格上的单粒子哈密顿量引入了一种系统的矩阵乘积算子（MPO）构造方法。利用这种递归结构，我们将晶格几何编码为有限状态自动机，并借鉴阿贝尔对称张量网络的思想，获得了哈密顿量的精确解析MPO表示，其中张量数量随系统尺寸对数增长，键维数由自动机状态数决定。多个例子证明了该框架的通用性：规则晶格、分形、凯莱树、双曲晶格以及一维和二维斐波那契准晶体。结合核多项式方法，这些MPO表示能够在无需显式构造完整哈密顿量的情况下进行大规模谱性质计算。这为在指数级大系统尺寸下探索广泛类别晶格的电子性质提供了一条统一途径。
英文摘要
Inspired by the recent progress in the simulation of tight-binding Hamiltonians on large lattices using tensor networks, we introduce a systematic matrix product operator (MPO) construction for single-particle Hamiltonians on recursively structured lattices. Taking advantage of this recursive structure, we encode the lattice geometry in a finite-state automaton and, adapting ideas from Abelian-symmetric tensor networks, obtain an exact and analytical MPO representation of the Hamiltonian, where the number of tensors grows logarithmically with the system size and the bond dimension is set by the number of automaton states. Several examples demonstrate the generality of the framework: regular lattices, fractals, Cayley trees, hyperbolic lattices, and one- and two-dimensional Fibonacci quasicrystals. Combined with the kernel polynomial method, these MPO representations enable large-scale calculations of spectral properties without explicitly constructing the full Hamiltonian. This provides a unified route to the exploration of electronic properties across a broad class of lattices at exponentially large system sizes.
发表机构
- Quobly
- Univ. Grenoble Alpes, CNRS, Institut Néel（格勒诺布尔阿尔卑斯大学，法国国家科学研究中心，内耳研究所）
- Univ. Grenoble Alpes, CNRS, LPMMC（格勒诺布尔阿尔卑斯大学，法国国家科学研究中心，低压与微波技术物理实验室）
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