arXiv subjects

Andreas A. Buchheit

Publications and source records attributed to Andreas A. Buchheit.

12 recordsLinked to original sources

arXiv2026

Exact and fast series expansions for quantum models with long-range interactions

Over the past decade, high-order series expansions based on linked-cluster methods have become an important tool for studying low-energy properties of gapped quantum systems with long-range interactions. We introduce a deterministic framework that removes a central computational bottleneck of this method. Our graph zeta method replaces the costly and statistically noisy Monte Carlo (MC) evaluation of high-dimensional lattice sums by a systematic, high-precision computation that delivers series coefficients within minutes on standard desktop hardware. The full momentum-dependent series is obtained in a single calculation, enabling high-resolution excitation spectra throughout the Brillouin zone. Building on the companion paper [1], the method reformulates graph-embedding sums as graph zeta functions and decomposes them into blocks classified by their treewidth tw. Low-treewidth blocks (tw$\leq2$) admit closed expressions based on Epstein zeta functions, while higher-treewidth blocks (tw$>2$) are evaluated using tensor-network bucket elimination. We benchmark the approach for transverse-field Ising models with power-law interactions in 1d, 2d, and 3d, reproducing previous MC results at a fraction of the computational cost while enabling substantially denser parameter sampling. An open-source implementation makes the method directly applicable to general interactions and large parameter scans. As an application, we compare microscopic interaction models for the stacked quasi-2d transverse-field Ising triangular-lattice antiferromagnet KTmSe$_2$ and find that a model including dipolar interactions best describes existing experimental data. The graph zeta method thus turns high-order linked-cluster expansions into a practical and deterministic tool for fast quantitative momentum-resolved modeling of short- and long-range quantum matter.

cond-mat.str-el↗

arXiv2026

Computation of anisotropic singular sums from high-order derivatives of Epstein zeta functions

The precise and efficient evaluation of large-scale lattice sums involving power-law kernels is a fundamental computational problem in the simulation of classical and quantum systems with long-range interactions. While methods for spatially isotropic kernels, some based on Epstein zeta functions, have advanced considerably in recent years, the anisotropic case has lagged behind, despite its broad relevance to both fundamental and effective interactions such as the dipole interaction in magnetic materials. In this work, we solve this issue by defining and analyzing anisotropic Epstein zeta functions for which we derive stably computable representations obtained from wave vector derivatives of lattice sums over isotropic interaction kernels. These functions find direct application in the analytical and numerical study of anisotropically interacting lattice systems. Going further, they provide the correction term in an exact equivalence between discrete lattices and their continuous analogs in a recent generalization of the classical Euler-Maclaurin summation formula to lattices and summands involving power-law kernels. Their connection to high-order derivatives of zeta functions can be used to improve convergence rates of numerical algorithms, for instance in micromagnetics, or to provide rapidly convergent expansions suitable for precomputations of generalized zeta functions. We derive a stably computable representation of anisotropic Epstein zeta functions, including the possibility for analytically removing Rayleigh--Wood singularities, and we develop a numerical algorithm for their stable evaluation for any lattice, power-law decay exponent and anisotropy order. We benchmark the algorithm against closed-form identities, direct summation and multi-precision results, obtaining machine precision across various lattices, power-law exponents, and anisotropy orders.

math.NA↗

arXiv2026

Non-local edge mode hybridization in the long-range interacting Kitaev chain

In one-dimensional $p$-wave superconductors with short-range interactions, topologically protected Majorana modes emerge, whose mass decays exponentially with system size, as first shown by Kitaev. In this work, we extend this prototypical model by including power law long-range interactions within a self-consistent framework, leading to the self-consistent long-range Kitaev chain (seco-LRKC). In this model, the gap matrix acquires a rich structure where short-range superconducting correlations coexist with long-range correlations that are exponentially localized at both chain edges simultaneously. As a direct consequence, the topological edge modes hybridize even if their wavefunction overlap vanishes, and the edge mode mass inherits the asymptotic scaling of the interaction. In contrast to models with imposed power law pairing, where massive Dirac modes emerge for exponents $ν< d$, we analytically motivate and numerically demonstrate that, in the fully self-consistent model, algebraic edge mode decay with system size persists for all interaction exponents $ν> 0$, despite exponential wave function localization. While the edge mode remains massless in the thermodynamic limit, finite-size corrections can be experimentally relevant in mesoscopic systems with effective long-range interactions that decay sufficiently slowly. Our findings have direct implications for quantum simulations of ultracold microwave-shielded dipolar molecules in one-dimensional optical lattices.

cond-mat.str-el↗

arXiv2026

Zeta expansion for long-range interactions under periodic boundary conditions with applications to micromagnetics

We address the efficient computation of power-law-based interaction potentials of homogeneous $d$-dimensional bodies with an infinite $n$-dimensional array of copies, including their higher-order derivatives. This problem forms a serious challenge in micromagnetics with periodic boundary conditions and related fields. Nowadays, it is common practice to truncate the associated infinite lattice sum to a finite number of images, introducing uncontrolled errors. We show that, for general interacting geometries, the exact infinite sum for both dipolar interactions and generalized Riesz power-law potentials can be obtained by complementing a small direct sum by a correction term that involves efficiently computable derivatives of generalized zeta functions. We show that the resulting representation converges exponentially in the derivative order, reaching machine precision at a computational cost no greater than that of truncated summation schemes. In order to compute the generalized zeta functions efficiently, we provide a superexponentially convergent algorithm for their evaluation, as well as for all required special functions, such as incomplete Bessel functions. Magnetic fields and related quantities can thus be evaluated to machine precision in arbitrary cuboidal domains periodically extended along one or two dimensions. We benchmark our method against known formulas for magnetic interactions and against direct summation for Riesz potentials with sufficiently large exponents, consistently achieving full precision. In addition, we identify new corrections to the asymptotic limit of the demagnetization field and tabulate high-precision benchmark values that can be used as a reliable reference for micromagnetic solvers. The techniques developed are broadly applicable, with direct impact in other areas such as molecular dynamics.

math.NA↗

arXiv2026

Computation and properties of the Epstein zeta function with applications to quantum systems

The Epstein zeta function generalises the classical Riemann zeta function to oscillatory lattice sums in higher dimensions and has recently emerged as a key tool in the simulation of long-range interacting classical and quantum many-body systems. Its computation and analytic properties are therefore of significant interest, yet a rigorous and comprehensive treatment has been lacking. We address this gap by introducing a superexponentially convergent algorithm, complete with error bounds, for computing the Epstein zeta function in any dimension with arbitrary real parameters. Our approach is accompanied by a detailed analysis of the analytic properties of the Epstein zeta function. We first present a concise reformulation of its meromorphic continuation, functional equation, and symmetries. We then establish, for the first time, its joint holomorphic continuation in all parameters and offer a complete characterization of the resulting complex singularity structure, which governs convergence rates in numerical algorithms based on the function. Recognizing that the function can be decomposed into power-law singularities and a regularised analytic part, we provide an algorithm for removing singularities without cancellation error. This facilitates the evaluation of integrals and enables fast precomputations through interpolation methods. We present the first high-performance implementation for arbitrary real arguments in EpsteinLib, a C library with Python and Julia bindings, and rigorously benchmark its performance and accuracy, achieving full-precision evaluation against known analytic results in dimensions 1,2,3,4,6, and 8 and against an arbitrary precision implementation across the entire parameter range. Finally, we apply our methods to the computation of quantum dispersion relations in spin systems and Casimir energies in three-dimensional geometries.

math.NA↗

arXiv2026

Epstein zeta method for many-body lattice sums

Many-body interactions arise naturally in the perturbative treatment of classical and quantum many-body systems and play a crucial role in the description of condensed matter systems. In the case of three-body interactions, the Axilrod-Teller-Muto (ATM) potential is highly relevant for the quantitative prediction of material properties. This work solves the long-standing issue of the numerical computation of the resulting energies in $d$-dimensional lattice systems. We present an efficiently computable representation of many-body lattice sums in terms of singular integrals over products of Epstein zeta functions. For three-body interactions in three dimensions, this approach reduces the runtime for computing the ATM lattice sum from weeks to minutes. Our approach further extends to a broad class of $n$-body lattice sums. We demonstrate that the computational cost of our method only increases linearly with $n$, evading the exponential increase in complexity of direct summation. We discuss techniques for numerically computing the arising singular integrals and compare the accuracy of our results against computable special cases and against direct summation in low dimensions, achieving full precision for exponents greater than the system dimension. Finally, we apply our method to study the stability of a three-dimensional lattice system with Lennard-Jones two-body interactions under the inclusion of an ATM three-body term at finite pressure, finding a transition from the face-centered-cubic to the body-centered-cubic lattice structure with increasing ATM coupling strength. This work establishes both the numerical and analytical foundation for an ongoing investigation into the influence of many-body interactions on the stability of matter.

math.NA↗

arXiv2026

Numerical Solution of the Bardeen-Cooper-Schrieffer Equation for Unconventional Superconductors

In this work, we consider the analytical properties and the efficient numerical solution of the Bardeen-Cooper-Schrieffer equation for unconventional superconductivity incorporating long-range power-law electron-electron interactions within a tight-binding model on a $d$-dimensional lattice. It is a nonlinear convolution equation for the complex matrix-valued superconducting gap under symmetry constraints imposed by the fermionic anticommutation rules. The long-range interaction enters in momentum space in the form of the now efficiently computable Epstein zeta function, which exhibits a power-law singularity at zero momentum. This needs to be accounted when evaluating the convolution. After a brief overview of some of the equation's analytical properties, we discuss its efficient numerical solution using a Galerkin method with B-splines. We present numerical results for a nodal superconductor on a two-dimensional square lattice.

math-ph↗

arXiv2025

Exact lattice summations for Lennard-Jones potentials coupled to a three-body Axilrod-Teller-Muto term applied to cuboidal phase transitions

This work provides a rigorous analysis of Bain-type cuboidal lattice transformations, which connect the face-centered cubic (fcc), mean-centered cubic (mcc), body-centered cubic (bcc) and axially centered cubic (acc) lattices. Our study incorporates a general $(n,m)$ Lennard-Jones two-body potential and a long-range repulsive Axilrod-Teller-Muto (ATM) three-body potential. The two-body lattice sums and their meromorphic continuations are evaluated to full precision using super-exponentially convergent series expansions. Furthermore, we introduce a novel approach to computing three-body lattice sums by converting the multi-dimensional sum into an integral involving products of Epstein zeta functions. This enables us to evaluate three-body lattice sums and their meromorphic continuations to machine precision within minutes on a standard laptop. Using our computational framework, we analyze the stability of cuboidal lattice phases relative to the close-packed fcc structure along a Bain transformation path for varying ATM coupling strengths. We analytically demonstrate that the ATM cohesive energy exhibits an extremum at the bcc phase and show numerically that it corresponds to a minimum for repulsive three-body forces along the Bain path. Our results indicate that strong repulsive three-body interactions can destabilize the fcc phase and render bcc energetically favorable for soft LJ potentials. However, even in this scenario, the bcc phase remains susceptible to further cuboidal distortions. These results suggest that the stability of the bcc phase is, besides vibrational, temperature, and pressure effects, strongly influenced by higher than two-body forces. Because of the wrong short-range behavior of the triple-dipole ATM model the LJ potential is limited to exponents $n>9$ for the repulsive wall, otherwise one observes distortion into a set of linear chains collapsing to the origin.

cond-mat.mtrl-sci↗

arXiv2024

On the computation of lattice sums without translational invariance

This paper introduces a new method for the efficient computation of oscillatory multidimensional lattice sums in geometries with boundaries. Such sums are ubiquitous in both pure and applied mathematics, and have immediate applications in condensed matter physics and topological quantum physics. The challenge in their evaluation results from the combination of singular long-range interactions with the loss of translational invariance caused by the boundaries, rendering standard tools ineffective. Our work shows that these lattice sums can be generated from a generalization of the Riemann zeta function to multidimensional non-periodic lattice sums. We put forth a new representation of this zeta function together with a numerical algorithm that ensures exponential convergence across an extensive range of geometries. Notably, our method's runtime is influenced only by the complexity of the considered geometries and not by the number of particles, providing the foundation for efficient simulations of macroscopic condensed matter systems. We showcase the practical utility of our method by computing interaction energies in a three-dimensional crystal structure with $3\times 10^{23}$ particles. Our method's accuracy is demonstrated through extensive numerical experiments. A reference implementation is provided online along with this article.

math.NA↗

arXiv2023

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

Continuum limits are a powerful tool in the study of many-body systems, yet their validity is often unclear when long-range interactions are present. In this work, we rigorously address this issue and put forth an exact representation of long-range interacting lattices that separates the model into a term describing its continuous analog, the integral contribution, and a term that fully resolves the microstructure, the lattice contribution. For any system dimension, any lattice, any power-law interaction, and for linear, nonlinear, and multi-atomic lattices, we show that the lattice contribution can be described by a differential operator based on the multidimensional generalization of the Riemann zeta function, namely the Epstein zeta function. We employ our representation in Fourier space to solve the important problem of long-range interacting unconventional superconductors. We derive a generalized Bardeen--Cooper--Schrieffer gap equation and find emerging exotic phases in two-dimensional superconductors with topological phase transitions. Finally, we utilize non-equilibrium Higgs spectroscopy to analyze the impact of long-range interactions on the collective excitations of the condensate. We show that the interactions can be used to fine-tune the Higgs mode's stability, ranging from exponential decay of the oscillation amplitude up to complete stabilization.

cond-mat.str-el↗

arXiv2021

Singular Euler-Maclaurin expansion

We present the singular Euler--Maclaurin expansion, a new method for the efficient computation of large singular sums that appear in long-range interacting systems in condensed matter and quantum physics. In contrast to the traditional Euler--Maclaurin summation formula, the new method is applicable also to the product of a differentiable function and a singularity. For suitable non-singular functions, we show that the approximation error decays exponentially in the expansion order and polynomially in the characteristic length scale of the non-singular function, where precise error estimates are provided. The sum is approximated by an integral plus a differential operator acting on the non-singular function factor only. The singularity furthermore is included in a generalisation of the Bernoulli polynomials that form the coefficients of the differential operator. We demonstrate the numerical performance of the singular Euler--Maclaurin expansion by applying it to the computation of the full non-linear long-range forces inside a macroscopic one-dimensional crystal with $2\times 10^{10}$ particles. A reference implementation in Mathematica is provided online.

math.NA↗

arXiv2021

Singular Euler-Maclaurin expansion on multidimensional lattices

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 two-dimensional lattices, which are of high relevance in solid state and quantum physics. An implementation in Mathematica is provided online along with this article.

math.NA↗