Correlation functions of harmonic lattices in d-dimensional space

TL;DR

This paper derives explicit formulas for correlation functions in d-dimensional harmonic lattices using hypergeometric series, enabling efficient computation of quantum information measures.

Contribution

It introduces a new hypergeometric series representation for correlation functions in harmonic lattices, facilitating faster and more precise calculations.

Findings

Correlation functions expressed via Lauricella's hypergeometric series.

Boundary condition effects on correlators are explicitly demonstrated.

Formulas enable efficient quantum information computations in lattice subsystems.

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.