Computation of generalized zeta functions with applications to long-range interacting classical and quantum lattices

The Epstein zeta function generalizes the Riemann zeta function to oscillatory lattice sums in higher dimensions, with broad applications in condensed matter physics, theoretical chemistry, and quantum field theory, particularly for long-range interacting lattices. In this talk, we present the computation of this function, including its higher-order derivatives, to full precision, make the resulting advancement publicly available through the high-performance library EpsteinLib, and present key physical applications. For the ferromagnetic Heisenberg model on the honeycomb lattice, we show that generalized zeta functions provide direct access to the phase diagram, where a new phase emerges upon inclusion of power-law long-range interactions. Derivatives of the Epstein zeta function can further be used to compute winding numbers in long-range topological superconductors, where we observe a fractional winding number in three dimensions. Extending our approach, we demonstrate that derivatives of generalized zeta functions associated with n-dimensional lattices in d-dimensional space enable the numerically exact computation of magnetic fields and related quantities in long-range interacting micromagnetic textures with periodic boundary conditions, such as Skyrmion and Hopfion lattices. In this context, we identify new asymptotic corrections to the demagnetization field. The method is general and readily extends to ferroelectrics, spin dynamics in thin films, and molecular dynamics with power-law interactions.