We are pleased to announce the FAU LMQ Talk on Tuesday, April 29th, at 14:15, at the Department of Physics (HF), Staudtstraße 5, Erlangen. The talk, titled “Numerical methods for quantum matter with long-range interactions” will be presented by Andreas Buchheit from Universität des Saarlandes.
This event is open to all and is organized and hosted by Kai Phillip Schmidt.
Abstract:
In the first part of this talk, I present the Singular Euler-Maclaurin expansion, a generalization of the 300-year-old Euler-Maclaurin summation formula to physically relevant long-range interactions [1,2], and discuss applications in magnetic systems [3]. A key ingredient of this method is the Epstein zeta function, a generalization of the Riemann zeta function to multidimensional oscillatory lattice sums, now available in a high-performance C library with a Python wrapper [4,5]. I further discuss extensions of our method to systems with boundaries [6].
Building on this analytical foundation, I explore the impact of long-range effective electron-electron interactions in unconventional Bardeen-Cooper-Schrieffer (BCS) superconductors in two dimensions [3]. The resulting phase diagrams exhibit a rich structure, including mixed-parity and topologically nontrivial phases. Out of equilibrium, long-range interactions can give rise to novel dynamical phenomena: they can stabilize Higgs oscillations and synchronize mixed-parity gap components.
Next, I examine the real-space structure of the superconducting gap in three-dimensional lattices with long-range interactions. For s-wave and other non-nodal states, the gap function exhibits the expected exponential decay. In contrast, for nodal states, we numerically demonstrate and analytically prove that the gap exhibits algebraic decay, with scaling determined by the interaction exponent and system dimension.
I conclude with an efficient numerical method for computing lattice sums arising in systems with many-body interactions, such as the Axilrod–Teller three-body potential. By reducing the required runtime from weeks to minutes, this method enables rigorous studies of the role of three-body forces in the stability of matter. In particular, I show that three-body interactions can significantly influence the relative stability of the fcc compared to the bcc phase in noble gas crystals.
References:
[1] Singular Euler-Maclaurin expansion on multidimensional lattices, Andreas A. Buchheit and Torsten Keßler, Nonlinearity 35 3706 (2022)
[2] On the Efficient Computation of Large Scale Singular Sums with Applications to Long-Range Forces in Crystal Lattices, Andreas A. Buchheit and Torsten Keßler, J. Sci. Comput. 90, 53 (2022)
[3] Exact Continuum Representation of Long-range Interacting Systems and Emerging Exotic Phases in Unconventional Superconductors, Andreas A. Buchheit, Torsten Keßler, Peter K. Schuhmacher, and Benedikt Fauseweh, Phys. Rev. Research 5, 043065 (2023)
[4] Computation and properties of the Epstein zeta function with high-performance implementation in EpsteinLib, Andreas A. Buchheit, Jonathan Busse, and Ruben Gutendorf, arXiv preprint 2412.16317 (2025)
[5] Github Repository: github.com/epsteinlib, pip install epsteinlib
[6] On the computation of lattice sums without translational invariance, Andreas A. Buchheit, Torsten Keßler, and Kirill Serkh, Mathematics of Computation (October 2024 issue)
We are pleased to announce the FAU LMQ Talk on Tuesday, April 29th, at 14:15, at the Department of Physics (HF), Staudtstraße 5, Erlangen. The talk, titled “Numerical methods for quantum matter with long-range interactions” will be presented by Andreas Buchheit from Universität des Saarlandes.
This event is open to all and is organized and hosted by Kai Phillip Schmidt.
Abstract:
In the first part of this talk, I present the Singular Euler-Maclaurin expansion, a generalization of the 300-year-old Euler-Maclaurin summation formula to physically relevant long-range interactions [1,2], and discuss applications in magnetic systems [3]. A key ingredient of this method is the Epstein zeta function, a generalization of the Riemann zeta function to multidimensional oscillatory lattice sums, now available in a high-performance C library with a Python wrapper [4,5]. I further discuss extensions of our method to systems with boundaries [6].
Building on this analytical foundation, I explore the impact of long-range effective electron-electron interactions in unconventional Bardeen-Cooper-Schrieffer (BCS) superconductors in two dimensions [3]. The resulting phase diagrams exhibit a rich structure, including mixed-parity and topologically nontrivial phases. Out of equilibrium, long-range interactions can give rise to novel dynamical phenomena: they can stabilize Higgs oscillations and synchronize mixed-parity gap components.
Next, I examine the real-space structure of the superconducting gap in three-dimensional lattices with long-range interactions. For s-wave and other non-nodal states, the gap function exhibits the expected exponential decay. In contrast, for nodal states, we numerically demonstrate and analytically prove that the gap exhibits algebraic decay, with scaling determined by the interaction exponent and system dimension.
I conclude with an efficient numerical method for computing lattice sums arising in systems with many-body interactions, such as the Axilrod–Teller three-body potential. By reducing the required runtime from weeks to minutes, this method enables rigorous studies of the role of three-body forces in the stability of matter. In particular, I show that three-body interactions can significantly influence the relative stability of the fcc compared to the bcc phase in noble gas crystals.
References:
[1] Singular Euler-Maclaurin expansion on multidimensional lattices, Andreas A. Buchheit and Torsten Keßler, Nonlinearity 35 3706 (2022)
[2] On the Efficient Computation of Large Scale Singular Sums with Applications to Long-Range Forces in Crystal Lattices, Andreas A. Buchheit and Torsten Keßler, J. Sci. Comput. 90, 53 (2022)
[3] Exact Continuum Representation of Long-range Interacting Systems and Emerging Exotic Phases in Unconventional Superconductors, Andreas A. Buchheit, Torsten Keßler, Peter K. Schuhmacher, and Benedikt Fauseweh, Phys. Rev. Research 5, 043065 (2023)
[4] Computation and properties of the Epstein zeta function with high-performance implementation in EpsteinLib, Andreas A. Buchheit, Jonathan Busse, and Ruben Gutendorf, arXiv preprint 2412.16317 (2025)
[5] Github Repository: github.com/epsteinlib, pip install epsteinlib
[6] On the computation of lattice sums without translational invariance, Andreas A. Buchheit, Torsten Keßler, and Kirill Serkh, Mathematics of Computation (October 2024 issue)