Efficient Computation of Multidimensional Lattice Sums with Boundaries and Applications to Long-Range Interacting Topological Quantum Systems

Topologically non-trivial quantum states such as in p+ip superconductors exhibit protected edge modes at the material boundary with numerous applications. Recent theoretical results have shown that such states can emerge due to long-range interactions. However, computing the effect of long-range interactions on the edge modes is highly challenging. The complexity of the task arises from the melding of singular long-range interactions with the loss of translational invariance caused by boundaries, rendering standard tools like Ewald's method ineffective. In this work, we lay out a robust framework designed for the efficient computation of multidimensional lattice sums with long-range interactions on bounded lattices. To this end, we show that any lattice sum can be generated from a generalization of the Riemann zeta function to multidimensional non-periodic lattice sums, termed the cut-domain Epstein zeta function. We put forth a new representation of this zeta function together with a numerical algorithm that ensures super-exponential convergence across an extensive range of geometries. Most importantly, we demonstrate that the runtime is solely influenced by the complexity of the structures that the particles form, not by the particle number itself, enabling a rigorous simulation of macroscopic crystals. We showcase the numerical performance of our method by computing interaction energies in three-dimensional crystals with 10²³ particles. Finally, we show how our method can be applied to topological quantum systems.