Singular Euler-Maclaurin expansion on multidimensional lattices

Abstract

We extend the classical Euler-Maclaurin expansion to sums over multidimensional lattices that involve functions with algebraic singularities. This offers a tool for the precise quantification of the effect of microscopic discreteness on macroscopic properties of a system. First, the Euler-Maclaurin summation formula is generalised to lattices in higher dimensions, assuming a sufficiently regular summand function. We then develop this new expansion further and construct the singular Euler-Maclaurin (SEM) expansion in higher dimensions, an extension of our previous work in one dimension, which remains applicable and useful even if the summand function includes a singular function factor. We connect our method to analytical number theory and show that all operator coefficients can be efficiently computed from derivatives of the Epstein zeta function. Finally we demonstrate the numerical performance of the expansion and efficiently compute singular lattice sums in infinite twodimensional lattices, which are of high relevance in solid state and quantum physics. An implementation in Mathematica is provided online along with this article.

Exact Continuum Representation of Long-range Interacting Systems and Emerging Exotic Phases in Unconventional Superconductors

“…Therefore, it focuses on infinite lattices in order to avoid additional boundary terms, leading to an exact description of infinite systems. For finite systems, geometry-dependent terms arise, which can be described within our method as well [14]. These terms can be of relevance, e.g., in mesoscopic systems or in quantum-Hall type topological materials that can exhibit soliton-like edge states [93].…”

“…Indeed, for g = P a polynomial of arbitrary degree, a key result of Ref. [14] shows that With this result and for g sufficiently differentiable, we can now expand g in a Taylor series around x of order 2 + 1. The cutoff function allows us to exchange the sum due to the Taylor series with the sum-integral and the β-limit [22], resulting in a representation of the lattice contribution in terms of derivatives of g…”

“…The representation of the lattice contribution in Eq. ( 4) is called the Singular Euler-Maclaurin (SEM) expansion, a full derivation of which is provided in [13][14][15]. For Ω = R d the SEM operator D ( ) takes the particularly simple form…”

“…with f x (y) = g(y)/|y − x| ν . As such differences will reappear often in our considerations, it is useful to introduce the corresponding operator on the right hand side, which is called the sum-integral [14]. In this work, we focus on the case of an infinite lattice Ω = R d to avoid additional geometry-dependend contributions due to boundaries.…”

“…We show that the discrete lattice problem can be separated into a term that describes its continuous analogue, the continuum contribution, and a term that includes all information about the microstructure, the lattice contribution, hence demonstrating an equivalence between lattice and continuum. To this end, we apply the recently developed Singular Euler-Maclaurin (SEM) expansion [13][14][15], which generalizes the 300-year old Euler-Maclaurin summation formula, and extend it to nonlinear and multiatomic systems. The singular lattice sum is expressed in terms of an integral and a lattice contribution described by a differential operator, both of which are efficiently computable.…”