Exact continuum representation of long-range interacting systems

Continuum limits are a powerful tool in the study of many-body systems, yet their validity is often unclear when long-range interactions are present. In this work, we rigorously address this issue and put forth an exact representation of long-range interacting lattices that separates the model into a term describing its continuous analogue, the integral contribution, and a term that fully resolves the microstructure, the lattice contribution. For any system dimension, any lattice, any power-law interaction and for linear, nonlinear, and multi-atomic lattices, we show that the lattice contribution can be described by a differential operator based on the multidimensional generalization of the Riemann zeta function, namely the Epstein zeta function. Our representation provides a broad set of tools for studying the analytical properties of the system and it yields an efficient numerical method for the evaluation of the arising lattice sums. We benchmark its performance by computing forces and energies in relevant physical examples, in which the standard continuum approximation fails: Skyrmions, defects in ion chains, and spin waves in a pyrochlore lattice with dipolar interactions. We demonstrate that our method exhibits the accuracy of exact summation at the numerical cost of an integral approximation. We furthermore extend our method to the study of boundary effects.