发表机构
Boğaziçi University（博阿齐希大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在三维黎曼流形上推导并求解BCS能隙方程，利用热核展开处理紫外发散，建立普适关系并解析证明能隙随温度单调递减，恢复弯曲空间中的标准BCS行为。
AI 中文摘要
在本文中，我们研究了在三维黎曼流形上表述的巴丁-库珀-施里弗（BCS）超导理论。我们通过采用Hubbard-Stratonovich变换并随后进行鞍点近似，在路径积分图像中推导出了能隙方程。在假设由度量设定的长度尺度远大于愈合长度的条件下，我们求解了能隙方程，得到了一个缓慢变化的能隙函数，该函数展现了曲率效应。为了严格处理能隙方程中固有的紫外发散，我们利用了热核展开技术，该技术提供了一种系统且数学上透明的重整化方案。我们在零温和有限温度下明确评估了重整化后的能隙方程，在弱耦合极限（$\Delta \ll \mu$）下渐近计算了相关积分。此外，我们建立了一个普适的、与正则化无关的关系，将有限温度能隙与零温度能隙及温度本身联系起来。基于这一基本关系，我们解析地证明了能隙随温度升高而单调递减（$\partial \Delta / \partial T < 0$），在弯曲空间框架内成功恢复了临界温度附近标准的普适BCS行为。
英文摘要
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 ($Δ\ll μ$). 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 ($\partial Δ/ \partial T < 0$), successfully recovering the standard universal BCS behavior near the critical temperature within a curved space framework.
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