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

RESUMO

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.