一维准晶的张量网络有限状态自动机
One-dimensional quasicrystals with tensor-network finite-state automata
- Aalto University（阿尔托大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文利用替换规则构造有限状态自动机，将转移矩阵映射为矩阵乘积态张量，实现一维准晶哈密顿量的精确张量网络表示，并高效计算超长链的谱密度。
AI中文摘要：
准晶在晶体的平移有序和无序的非晶态物质之间占据独特的位置。模拟这种物理一直具有挑战性，因为准周期结构缺乏晶体所利用的平移对称性，通常需要对大型有限近似体进行昂贵的对角化。准晶序允许两种等价的描述：从高维周期晶体进行切割投影的方案，以及作用于有限字母表的一组离散替换规则。我们表明，后者以适应该替换的记数系统书写，定义了一个带输出的确定性有限自动机：输入一个位点索引的数字，自动机返回占据该位点的字母。我们进一步利用与另一种构造的等价性，转移矩阵恰好是矩阵乘积态的张量，其键维是自动机状态的数量，并且与系统大小无关。这使得在张量列语言中能够高效表示极大的紧束缚哈密顿量，从而在任何系统大小下为准晶哈密顿量生成精确的矩阵乘积算符。我们展示了该框架如何适用于两个一维准晶族，即金属均值和$k$-波那契族，并明确构造了斐波那契、银均值和三波那契准晶。通过利用高效的张量网络压缩和核多项式方法，我们计算了超过$10^9$个位点链的谱密度，并直接分辨出谱的层次结构。
英文摘要：
Quasicrystals occupy a distinctive position between the translational order of crystals and the disordered amorphous matter. Simulating this physics has remained challenging, since quasiperiodic structures lack the translational symmetry exploited for crystals and generally require costly diagonalization of large finite approximants. Quasicrystalline order admits two equivalent descriptions, a cut-and-project scheme from a higher-dimensional periodic crystal, and a discrete set of substitution rules acting on a finite alphabet. We show that the latter, written in a numeration system adapted to the substitution, defines a deterministic finite automaton with output, the digits of a site index are fed, and the automaton returns the letter occupying that site. We further exploit another equivalence to a different construction, the transition matrices are exactly the tensors of a matrix product state, whose bond dimension is the number of automaton states and is independent of system size. This allows efficient representation of extremely large tight-binding Hamiltonians in the tensor-train language, thereby yielding an exact matrix product operator for the quasicrystal Hamiltonian at any system size. We show how this framework works for two families of one-dimensional quasicrystals, the metallic-mean and $k$-bonacci families and we explicitly construct the Fibonacci, silver-mean, and Tribonacci quasicrystals. By leveraging efficient tensor-network compression and the kernel polynomial method, we compute spectral densities for chains with more than $10^9$ sites and directly resolve the hierarchical structure of the spectrum.
