Correlation functions of harmonic lattices in d-dimensional space

TL;DR

The paper addresses the computation of vacuum correlation matrices $Q$ and $P$ for dynamical variables and their conjugate momenta in a $d$-dimensional harmonic lattice. It demonstrates that in the thermodynamic limit these correlators can be expressed analytically through Lauricella's C-type hypergeometric series $F_C^{(d)}$, independent of boundary conditions, enabling efficient evaluation of entanglement-related quantities. It provides explicit finite-lattice formulas for periodic and Dirichlet boundaries and proves that near the lattice center the Dirichlet results converge to the periodic ones, reinforcing a boundary-condition independence in the bulk. The findings have practical implications for fast, precise computation of information quantities such as Renyi entropies and motivate future work on concise hypergeometric representations and the calculation of corner and edge contributions in higher dimensions.

Abstract

We study the correlation functions between the dynamical variables and between their conjugate momenta at sites of a harmonic lattice in the $d$-dimensional Euclidean space. We show that at the thermodynamic limit, they can be expressed in terms of Lauricella's C-type hypergeometric series. Furthermore, using these expressions, we explicitly demonstrate that the correlators near the center of the lattice satisfying Diriclet boundary conditions coincide with those for the lattice with the periodic boundary conditions. By utilizing these expressions, we expect to make it easier to create programs that compute fast and highly precise for the quantum information quantities of subsystems within lattices.