海森堡七边形本征问题的伽罗瓦作用
Galois actions on the eigenproblem of the Heisenberg heptagon
摘要：
本文从有理数场$\mathbb{Q}$的有限扩张的观点，分析了具有最近邻相互作用的XXX模型中磁七边形（即N=7个自旋为1/2的节点的环）的海森堡哈密顿量的本征问题的精确解。我们指出，作为主要的结果，这些扩展的相关算术结构，自然引入一些伽罗瓦量子位。它们是模型的希尔伯特空间的二维子空间，基于它们相对于相关伽罗瓦群对世俗行列式的不可分解因子的作用的独特性质，它们承认量子信息学解释为（假设的）计算机的基本存储单元。这些伽罗瓦量子比特被嵌套在子场的晶格上，子场涉及几个最小的场，用于确定本征态（复海森堡场），谱（实海森堡场）和磁配置的傅里叶变换（基于简单的7次方根的分圆场）。伽罗瓦群的相应格的结构是根据库默尔理论提出的，其物理解释是根据本征态、能量和密度矩阵的适当排列来表示的。
英文摘要：
We analyse the exact solution of the eigenproblem for the Heisenberg Hamiltonian of magnetic heptagon, i.e. the ring of N=7 nodes, each with spin 1/2, within the XXX model with nearest neighbour interactions, from the point of view of finite extensions of the field $\mathbb{Q}$ of rationals. We point out, as the main result, that the associated arithmetic structure of these extensions makes natural an introduction of some Galois qubits. They are two-dimensional subspaces of the Hilbert space of the model, which admit a quantum informatic interpretation as elementary memory units for a (hypothetical) computer, based on their distinctive properties with respect to the action of related Galois group for indecomposable factors of the secular determinant. These Galois qubits are nested on the lattice of subfields which involves several minimal fields for determination of eigenstates (the complex Heisenberg field), spectrum (the real Heisenberg field), and Fourier transforms of magnetic configurations (the cyclotomic field, based on the simple 7th root of unity). The structure of the corresponding lattice of Galois groups is presented in terms of Kummer theory, and its physical interpretation is indicated in terms of appropriate permutations of eigenstates, energies, and density matrices.
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