BCS Gap Equation on Riemannian Manifolds: A Heat Kernel Analysis
Abstract
In this paper, we investigate the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity formulated on a three dimensional Riemannian manifold. We derive the gap equation in the path integral picture by employing a Hubbard-Stratonovich transformation followed by a saddle point approximation. Under the assumption that the length scale set by the metric is much larger than the healing length, we solve the gap equation to find a slowly varying gap function that exhibits the effects of curvature. To rigorously address the ultraviolet divergences inherent in the gap equation, we utilize heat kernel expansion techniques, which provide a systematic and mathematically transparent renormalization scheme. We explicitly evaluate the renormalized gap equation at both zero and finite temperatures, computing the relevant integrals asymptotically in the weak-coupling limit (Δ≪ μ). Furthermore, we establish a universal, regularization-independent relation connecting the finite-temperature gap to the zero-temperature gap and the temperature itself. Based on this fundamental relation, we analytically prove the monotonic decrease of the energy gap with increasing temperature (∂ Δ/ ∂ T < 0), successfully recovering the standard universal BCS behavior near the critical temperature within a curved space framework.
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BibTeX
@misc{akant2026equation,
title = {{BCS Gap Equation on Riemannian Manifolds: A Heat Kernel Analysis}},
author = {Levent Akant and Emine Ertugrul and O. Teoman Turgut},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14689},
eprint = {2609.14689},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14689},
}
RIS
TY - UNPB
TI - BCS Gap Equation on Riemannian Manifolds: A Heat Kernel Analysis
AU - Levent Akant
AU - Emine Ertugrul
AU - O. Teoman Turgut
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14689
UR - https://arxiv.org/abs/2609.14689
AB - In this paper, we investigate the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity formulated on a three dimensional Riemannian manifold. We derive the gap equation in the path integral picture by employing a Hubbard-Stratonovich transformation followed by a saddle point approximation. Under the assumption that the length scale set by the metric is much larger than the healing length, we solve the gap equation to find a slowly varying gap function that exhibits the effects of curvature. To rigorously address the ultraviolet divergences inherent in the gap equation, we utilize heat kernel expansion techniques, which provide a systematic and mathematically transparent renormalization scheme. We explicitly evaluate the renormalized gap equation at both zero and finite temperatures, computing the relevant integrals asymptotically in the weak-coupling limit (Δ≪ μ). Furthermore, we establish a universal, regularization-independent relation connecting the finite-temperature gap to the zero-temperature gap and the temperature itself. Based on this fundamental relation, we analytically prove the monotonic decrease of the energy gap with increasing temperature (∂ Δ/ ∂ T < 0), successfully recovering the standard universal BCS behavior near the critical temperature within a curved space framework.
ER -