We are pleased to announce the FAU LMQ Talk on Thursday, January 29th, at 14:15, at the Department of Physics, Lecture Hall E, Staudtstraße 5, Erlangen. The talk, titled “Graph zeta methods for efficient simulations of long-range interacting quantum lattices” will be presented by Andreas Buchheit from Saarland University.
This event is open to all and is organized and hosted by Kai Phillip Schmidt.
Abstract:
I first present the Singular Euler—Maclaurin expansion, an extension of the 300-year-old classical Euler-Maclaurin summation formula to long-range interactions on high-dimensional lattices with applications in spin systems [1,2]. This method allows for the exact representation of a discrete lattice in terms of its continuous analog, with corrections given in terms of a generalization of the Riemann zeta function, the so-called Epstein zeta function. With the properties and efficient computation of this function analyzed in [4] and a high-performance implementation available in our library EpsteinLib [5], a new toolset is provided for studying arbitrary long-range interacting lattices. I briefly discuss recent extensions to systems with boundaries [6] as well as to micromagnetics systems [7]. Building on this framework, I subsequently study 2D unconventional BCS superconductors with long-range interactions, finding a rich phase diagram with topologically non-trivial and mixed-parity phases as well as stabilization of Higgs modes in the non-equilibrium dynamics [3].
In the second part of the talk, I present how generalized zeta functions, built from the Epstein zeta function, can allow for the precise evaluation of n-body interaction energies in chemistry given by (n-1) d-dimensional lattice sums, reducing the scaling from exponential to linear in the number of interaction partners n [8]. For cuboidal lattices with a two-body Lennard–Jones potential coupled to a three-body Axilrod–Teller–Muto potential, we demonstrate that increasing the three-body coupling can drive a structural transition from fcc to bcc [8,9].
In long-range-interacting quantum lattices, current graph decomposition methods, such as pCUT, transform the problem of an exponentially growing Hilbert space dimension into the computation of high-dimensional lattice sums associated with graphs, which are usually computed with low-precision Monte Carlo methods. In the final part of the talk, I present ongoing work on computing the arising graph zeta functions efficiently and precisely, laying the foundation for exploring quantum systems in regimes inaccessible to other methods.
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/epsteinlib, pip install epsteinlib
[6] On the computation of lattice sums without translational invariance, Andreas A. Buchheit, Torsten Keßler, and Kirill Serkh, Math. Comp. 94 (2025), 2533-2574
[7] Zeta expansion for long-range interactions under periodic boundary conditions with applications to micromagnetics, arXiv:2509.26274 (2025)
[8] Exact lattice summations for Lennard-Jones potentials coupled to a three-body Axilrod–Teller–Muto term applied to cuboidal phase transitions, Andres Robles-Navarro, Andreas A. Buchheit, et. al., J. Chem. Phys. 163, 094104 (2025)
[9] Epstein zeta method for many-body lattice sums, Andreas A. Buchheit, Jonathan K. Busse, arXiv:2504.11989 (2025)
We are pleased to announce the FAU LMQ Talk on Thursday, January 29th, at 14:15, at the Department of Physics, Lecture Hall E, Staudtstraße 5, Erlangen. The talk, titled “Graph zeta methods for efficient simulations of long-range interacting quantum lattices” will be presented by Andreas Buchheit from Saarland University.
This event is open to all and is organized and hosted by Kai Phillip Schmidt.
Abstract:
I first present the Singular Euler—Maclaurin expansion, an extension of the 300-year-old classical Euler-Maclaurin summation formula to long-range interactions on high-dimensional lattices with applications in spin systems [1,2]. This method allows for the exact representation of a discrete lattice in terms of its continuous analog, with corrections given in terms of a generalization of the Riemann zeta function, the so-called Epstein zeta function. With the properties and efficient computation of this function analyzed in [4] and a high-performance implementation available in our library EpsteinLib [5], a new toolset is provided for studying arbitrary long-range interacting lattices. I briefly discuss recent extensions to systems with boundaries [6] as well as to micromagnetics systems [7]. Building on this framework, I subsequently study 2D unconventional BCS superconductors with long-range interactions, finding a rich phase diagram with topologically non-trivial and mixed-parity phases as well as stabilization of Higgs modes in the non-equilibrium dynamics [3].
In the second part of the talk, I present how generalized zeta functions, built from the Epstein zeta function, can allow for the precise evaluation of n-body interaction energies in chemistry given by (n-1) d-dimensional lattice sums, reducing the scaling from exponential to linear in the number of interaction partners n [8]. For cuboidal lattices with a two-body Lennard–Jones potential coupled to a three-body Axilrod–Teller–Muto potential, we demonstrate that increasing the three-body coupling can drive a structural transition from fcc to bcc [8,9].
In long-range-interacting quantum lattices, current graph decomposition methods, such as pCUT, transform the problem of an exponentially growing Hilbert space dimension into the computation of high-dimensional lattice sums associated with graphs, which are usually computed with low-precision Monte Carlo methods. In the final part of the talk, I present ongoing work on computing the arising graph zeta functions efficiently and precisely, laying the foundation for exploring quantum systems in regimes inaccessible to other methods.
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/epsteinlib, pip install epsteinlib
[6] On the computation of lattice sums without translational invariance, Andreas A. Buchheit, Torsten Keßler, and Kirill Serkh, Math. Comp. 94 (2025), 2533-2574
[7] Zeta expansion for long-range interactions under periodic boundary conditions with applications to micromagnetics, arXiv:2509.26274 (2025)
[8] Exact lattice summations for Lennard-Jones potentials coupled to a three-body Axilrod–Teller–Muto term applied to cuboidal phase transitions, Andres Robles-Navarro, Andreas A. Buchheit, et. al., J. Chem. Phys. 163, 094104 (2025)
[9] Epstein zeta method for many-body lattice sums, Andreas A. Buchheit, Jonathan K. Busse, arXiv:2504.11989 (2025)