一维周期费米子系统中对称性驱动的关联模式：关联函数、纠缠熵和互信息的闭式表达式与精确选择定则
Symmetry-driven correlation patterns in one-dimensional periodic fermionic systems: closed-form expressions and exact selection rules for correlation functions, entanglement entropies and mutual information
- Dipartimento di Scienze Chimiche, Farmaceutiche ed Agrarie（化学、药学与农业科学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文利用循环对称性，将费米子关联函数表示为傅里叶求和，推导出闭式表达式和精确选择定则，揭示一维周期系统中关联的非均匀结构，为理解关联模式提供普适参考。
AI中文摘要：
我们在单Slater行列式层面上对周期费米子系统中关联的空间结构进行了分析研究，重点聚焦于循环情形。利用循环对称性，从局域轨道到分子轨道的变换同时是离散傅里叶变换、单位根上的Vandermonde矩阵、复Hadamard矩阵以及循环群$C_m$的特征标表。在此框架内，关联的空间结构完全由占据的不可约表示（傅里叶模式）决定，而与生成它们的具体哈密顿量无关。特别地，费米子两点关联函数表示为对占据轨道的截断傅里叶求和，从而对对称占据模式的傅里叶模式（$\ell$和$-\ell$对）得到精确解析表达式，这在量子化学中对应于芳香性填充，即满足Hückel的$4n+2$规则的填充。对于半填充情形，我们推导出精确的空间选择定则。特别地，两点关联函数在环上所有偶数距离处恒为零。这一性质传播到约化密度矩阵，并意味着相应轨道之间完全没有互信息。这些结果表明，尽管分子轨道全局离域，循环系统中的关联结构高度非均匀，并由对称性诱导的干涉所支配。虽然以循环晶格系统表述，但本文结果可直接推广到具有周期性条件的一维费米子链，反映了问题的平移对称性。因此，该框架为理解一维周期费米子系统中的关联模式提供了与哈密顿量无关的普适参考。
英文摘要：
We present an analytical study of the spatial structure of correlations in periodic fermionic systems at the single Slater determinant level, focusing on the cyclic case. Exploiting cyclic symmetry, the transformation from localized orbitals to molecular orbitals is simultaneously a discrete Fourier transform, a Vandermonde matrix on the roots of unity, a complex Hadamard matrix, and the character table of the cyclic group $C_m$. Within this framework, the spatial structure of correlations is fully determined by the occupied irreducible representations (Fourier modes), independently of the specific Hamiltonian generating them. In particular, the fermionic two-point correlator is expressed as a truncated Fourier sum over the occupied orbitals, leading to exact analytical expressions for symmetric occupation patterns of Fourier modes ($\ell$ and $-\ell$ pairs), corresponding in quantum chemistry to aromatic fillings, i.e. fillings satisfying Hückel's $4n + 2$ rule. For the half filling case, we derive exact spatial selection rules. In particular, the two-point correlator vanishes identically for all even distances along the ring. This property propagates to reduced density matrices and implies a complete absence of mutual information between the corresponding orbitals. These results reveal that, despite the global delocalization of molecular orbitals, the correlation structure in cyclic systems is highly non-uniform and governed by symmetry-induced interference. Although formulated in terms of cyclic lattice systems, the present results extend straightforwardly to one-dimensional fermionic chains with periodic conditions, reflecting the underlying translational symmetry of the problem. The framework therefore provides a Hamiltonian-independent and universal reference for understanding correlation patterns in one-dimensional periodic fermionic systems.
