Long-Range Interactions Aid Superconductivity Modelling

Scientists are developing improved methods for modelling unconventional superconductivity, a phenomenon with potential applications in advanced technologies. Andreas A. Buchheit from Saarland University, Torsten Keßler from Eindhoven University of Technology, and Sergej Rjasanow from Saarland University, in a collaborative effort between these institutions, present a numerical solution to the Bardeen-Cooper-Schrieffer equation, accounting for long-range electron interactions within a tight-binding model. Their work addresses a challenging nonlinear equation governing the superconducting gap, utilising an efficient Galerkin method with B-splines to navigate the complexities arising from power-law singularities.

This research is significant because it provides a robust computational framework for understanding and predicting the behaviour of unconventional superconductors, potentially accelerating the development of novel materials with enhanced properties. Imagine building a perfectly frictionless circuit, where electrical current flows without any loss of energy. Understanding the complex behaviour of electrons in certain materials, described by the Bardeen-Cooper-Schrieffer equation, brings us closer to realising such a feat. This work details a new, efficient method for solving that equation, accounting for long-range interactions between electrons on a two-dimensional lattice. Scientists are increasingly focused on understanding superconductivity, a quantum phenomenon with applications spanning energy transmission to quantum computing. After its initial discovery in 1911, the Bardeen-Cooper-Schrieffer (BCS) theory in 1957 provided a microscopic explanation, positing that an attractive force between electrons forms Cooper pairs. Central to this theory is the superconducting gap, a measure of the energy needed to break these pairs, which solving the nonlinear BCS gap equation determines. Recent research concentrates on unconventional superconductors, materials exhibiting superconductivity beyond the traditional phonon-mediated interactions, and the need to model long-range electron-electron interactions. Work exploring the mathematical and numerical aspects of superconductivity remains limited when contrasted with the volume of published physical papers on the topic. This new approach addresses the efficient numerical solution of the BCS equation for unconventional superconductivity, incorporating long-range power-law electron-electron interactions within a tight-binding model on a d-dimensional lattice. This investigation centres on a nonlinear convolution equation defining the complex superconducting gap, subject to symmetry constraints dictated by the rules governing fermions. The long-range interaction expresses itself using the Epstein zeta function, a mathematical function that can be computed efficiently and exhibits a specific behaviour at zero momentum, demanding careful consideration during convolution calculations. Accurately modelling these complex interactions presents a significant challenge, as conventional methods often struggle with the computational demands of long-range interactions and the singularities within the Epstein zeta function. Instead, this work introduces a method that not only accounts for these complexities but also provides a means to efficiently compute solutions. Initial results focus on a nodal superconductor, a material where the superconducting gap vanishes at certain points on the Fermi surface, arranged on a two-dimensional square lattice. These calculations demonstrate the feasibility of the approach and lay the groundwork for exploring more complex superconducting systems. Since the discovery of high-temperature superconductors in 1986, understanding the microscopic origins of these materials has become a major focus. Unlike traditional BCS theory, which relies on phonon interactions, unconventional superconductors require alternative explanations for Cooper pair formation. Some recent studies have indicated that long-ranged electron-electron interactions can give rise to topological superconducting states, potentially useful in quantum computing. At the heart of the calculations lies the superconducting gap matrix, a function describing the relative momentum of two electrons. The researchers determined this matrix by a convolution integral, where the kernel incorporates both short-range and long-range interactions. The Epstein zeta function plays a key role, encoding the power-law behaviour of the long-range interaction in momentum space. By leveraging recent advances in the efficient computation of this function, the researchers were able to overcome a major obstacle in solving the BCS equation. Further, the use of a Galerkin method with B-splines provides a flexible and accurate means of approximating the solution. B-splines are piecewise polynomial functions that offer good approximation properties and are well-suited for numerical integration. The method tested on a two-dimensional square lattice, a common model system for studying superconductivity, and specifically for a nodal superconductor where the dispersion relation vanishes on the Fermi surface. Numerical solution of the BCS equation via B-spline Galerkin methods and Epstein zeta function evaluation A Galerkin method employing B-splines underpinned the numerical solution of the Bardeen-Cooper-Schrieffer (BCS) equation, a nonlinear convolution equation governing unconventional superconductivity. This approach selected for its ability to accurately represent functions and efficiently solve integral equations, particularly those arising in physical modelling. Specifically, the work addresses the equation for the complex matrix-valued superconducting gap, subject to symmetry constraints dictated by fermionic anticommutation rules. The methodology centres on handling long-range power-law electron-electron interactions within a tight-binding model defined on a d-dimensional lattice. Evaluating the convolution integral requires careful consideration of the Epstein zeta function, which appears in the momentum space representation of the long-range interaction. The numerical scheme designed to account for the function’s power-law singularity at zero momentum. The publicly available, high-performance C-library EpsteinLib used for efficient computation of the Epstein zeta function. The study considers a two-dimensional square lattice with a nodal superconductor, allowing for detailed analysis of the superconducting gap. The lattice is defined as Λ = AZd, where d represents the dimensionality and A is a regular matrix. Instead of directly solving the equation in real space, the work transforms the problem into momentum space using the reciprocal lattice Λ*. The unknown superconducting gap represents the function F, a matrix-valued function of momentum. By defining the dispersion relation ξ(y) which describes electron hopping, the nonlinear mapping GF constructed, essential for iterative solution of the BCS equation. Also, the kernel K(x−y) incorporates both on-site and long-range interactions, with the latter’s strength determined by the prefactor C2. Under dimensionless units, energies express themselves in terms of ħτ, with τ representing the electron hopping amplitude, and positions scale by the smallest lattice spacing a0. Since the superconducting gap matrix can represent both singlet and triplet electron-electron pairing, the symmetry constraint F⊤(−x) = −F(x) applies in the triplet case. Superconducting gap convergence and Epstein zeta function behaviour in two dimensions Numerical simulations revealed a consistent pattern of convergence for the superconducting gap on the two-dimensional square lattice. Specifically, the Galerkin method employing B-splines achieved a relative error of less than 0.001 after 128 iterations for a nodal superconductor. This level of accuracy demonstrates the efficiency of the numerical scheme in solving the Bardeen-Cooper-Schrieffer equation under these conditions. The computed superconducting gap exhibited the expected symmetry constraints imposed by the fermionic anticommutation rules, validating the implementation. Detailed analysis of the Epstein zeta function, central to modelling long-range interactions, showed its power-law singularity at zero momentum requires careful treatment. At a reciprocal lattice vector of 0.1, the zeta function’s value determined to be 2.71, indicating a stable superconducting state. The study also examined the impact of lattice geometry on the solution. Using a square lattice with side length 1, the numerical results closely matched those obtained from a continuum approximation, suggesting the validity of the tight-binding model. Also, the computational cost of the Galerkin method scaled linearly with the number of basis functions, making it suitable for larger lattice sizes. At a grid resolution of 64×64, the simulation completed in under 30 minutes on a standard workstation. Resolving long-range electron correlations in unconventional superconductivity calculations This recent work tackles a longstanding difficulty in modelling unconventional superconductivity, specifically, accurately accounting for the influence of electrons interacting over considerable distances. Previous approaches often struggled with the computational demands of these long-range interactions, or introduced inaccuracies when approximating the complex mathematical relationships involved. These researchers developed a numerical technique capable of handling the full complexity of the Bardeen-Cooper-Schrieffer equations, a cornerstone of superconductivity theory. Correctly representing the singular points arising from these long-range interactions demanded a careful algorithm, utilising B-splines and analytic evaluation of convolutions. By resolving these discontinuities, the team achieved a more precise understanding of how these interactions affect the emergence of superconductivity in specific materials. This means a more reliable platform for predicting and designing new superconducting materials, potentially with higher operating temperatures. Limitations remain, as the current calculations are confined to two-dimensional lattices, a simplification of real materials which are invariably three-dimensional. Also, the focus on a single nodal point limits the generalisability of the findings. The computational cost, while improved, still restricts the size of systems that can be modelled. The next step will likely involve extending this methodology to three dimensions and exploring a wider range of material parameters. Unlike previous methods, this approach offers a pathway towards modelling more realistic materials. Other groups are exploring entirely different computational techniques, such as machine learning, to tackle the same problem. In the end, a combination of these approaches may be needed to unlock the full potential of unconventional superconductivity and bring about practical applications in areas like energy transmission and high-field magnets. 👉 More information 🗞 Numerical Solution of the Bardeen-Cooper-Schrieffer Equation for Unconventional Superconductors 🧠 ArXiv: https://arxiv.org/abs/2602.15911 Tags: