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

Technical Summary: Computation and Properties of the Epstein Zeta Function

Problem Statement
The Epstein zeta function, $Z_{\Lambda, \nu}(x, y)$, generalizes the Riemann zeta function to oscillatory lattice sums in higher dimensions. It is a critical tool in theoretical physics and chemistry for modeling long-range interacting classical and quantum many-body systems, including Madelung constants, Casimir energies, and spin-wave dispersion relations. Despite its importance, a rigorous and comprehensive treatment of its computation and analytic properties has been lacking. Existing methods, such as the Chowla–Selberg formula or Crandall's representation, often face limitations: they may be restricted to special cases of lattice vectors or parameters, suffer from numerical instabilities near singularities (e.g., negative arguments in incomplete gamma functions), or lack a unified library implementation. Furthermore, the joint holomorphic continuation and the precise complex singularity structure of the function in all parameters ($\nu, x, y$) had not been rigorously established, hindering the development of robust numerical algorithms that avoid catastrophic cancellation.

Methodology
The authors address these gaps through a combination of rigorous analytic reformulation and high-performance numerical algorithm design:

1. Analytic Reformulation:

   • The paper introduces a Crandall-like representation based on decomposing the kernel $|z|^{-\nu}$ into "Crandall functions": an upper function $G_\nu$ (superexponentially decaying) and a lower function $g_\nu$ (smooth).
   • By applying the Poisson summation formula to the lower part, the authors derive a representation of the Epstein zeta function as the sum of two superexponentially convergent lattice sums. This formulation serves as the basis for both the numerical algorithm and the analytic proofs.
   • The authors rigorously prove the joint holomorphic continuation of the Epstein zeta function in the exponent $\nu$ and vector arguments $x, y$ within specific complex cones. They characterize the singularity structure, identifying simple poles at $\nu = d$ (when $y \in \Lambda^*$) and removable singularities elsewhere.
   • A regularized Epstein zeta function is defined by subtracting the power-law singularity (derived from the distributional Fourier transform of $| \cdot |^{-\nu}$). This regularization removes singularities without catastrophic cancellation, enabling stable evaluation and efficient precomputation via interpolation.

2. Numerical Algorithm (EpsteinLib):

   • The authors implement a C library, EpsteinLib, with Python, Julia, and Mathematica bindings.
   • Preprocessing: The algorithm rescales lattices to unit volume and projects vectors into elementary lattice cells to exploit translation and scale symmetries.
   • Special Cases: Explicit handling is provided for removable singularities (e.g., $\nu \in -2\mathbb{N}_0$) and poles ($\nu = d$).
   • Truncation and Summation: Rigorous error bounds are derived for truncating the superexponentially decaying sums. To mitigate floating-point roundoff errors in high dimensions, the authors employ Kahan compensated summation.
   • Incomplete Gamma Functions: A custom, high-performance implementation of the incomplete gamma function is developed to handle negative arguments and the full parameter range with full double precision, overcoming limitations of existing libraries.

Key Contributions

• First Comprehensive Analytic Treatment: The paper provides the first proof of the joint holomorphic continuation of the Epstein zeta function in all parameters and offers a complete characterization of its singularity structure.
• Superexponentially Convergent Algorithm: A new algorithm is introduced that computes the function for arbitrary real parameters in any dimension with rigorous error bounds.
• Singularity Removal: A method for decomposing the function into a singular part and a regularized analytic part is presented, allowing for the evaluation of integrals and the treatment of Wood-type anomalies without catastrophic cancellation.
• EpsteinLib Implementation: The first high-performance, open-source library for computing the Epstein zeta function for arbitrary real arguments, featuring full-precision evaluation and extensive benchmarking.

Results

• Accuracy and Performance: The implementation achieves full double-precision accuracy across dimensions $d=1, 2, 3, 4, 6, 8$ and the entire parameter range $\nu \in [-12.5, 12.5]$. Benchmarks against known analytic formulas and arbitrary-precision implementations confirm the algorithm's precision.
• Efficiency: The algorithm demonstrates superexponential convergence. For example, 2D sums are computed in under 20 microseconds, and 8D sums in under 0.5 seconds on a standard laptop. The use of Kahan summation ensures stability even in high dimensions where roundoff errors typically accumulate.
• Applications Demonstrated:
  • Quantum Spin Systems: The method was applied to compute quantum dispersion relations in 3D spin systems with long-range interactions, successfully capturing anomalous scaling behaviors ($\omega(k) \sim |k|^{\nu-d}$) for $d < \nu < d+2$.
  • Casimir Effect: The library was used to compute Casimir energies in 3D geometries, accurately reproducing asymptotic force laws and analyzing energy minimization in deformed boxes.

Significance
The paper claims that this work establishes the Epstein zeta function as a standard special function for computational physics and mathematics. By providing a rigorous analytic foundation and a robust, high-performance implementation, the authors enable the efficient simulation of long-range interacting systems that were previously computationally prohibitive or numerically unstable. The ability to handle general lattices, arbitrary real parameters, and singularities without cancellation errors facilitates new research in theoretical chemistry (stability of matter), condensed matter physics (quantum phase diagrams, devil's staircases), and wave scattering problems. The authors note that the library is already being utilized by the quantum condensed matter community to study exotic phenomena, such as fractional magnetization plateaus and the melting of devil's staircases.