基于几何代数基础与标量投影方法的超导配对对称性
Superconducting Pairing Symmetry on Geometric-Algebra Foundations via the Scalar-Projection Method
- School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文在几何代数 $Cl_{3,0}$ 框架下建立超导配对低能理论，通过两级投影实现代数重构，给出配对通道的代数根源、三个代数定理及多个应用示例。
AI中文摘要：
我们在几何代数 $Cl_{3,0}$ 中构建了超导配对的低能理论。传统的 BCS 平均场理论及其扩展（$s$ 波、$p$ 波、$d$ 波等）在八维实多向量空间中具有代数对应物。该构造是一个两级投影：矩阵块被表示为 Clifford 多向量，其 grade-0 投影几何积提取出用于构建哈密顿量的标量块。结果是一种代数重构，在物理上等价于标准的复矩阵表述。在这种重构中，配对通道的二分性获得了代数根源：它源于费米子恒等式 $B^\dagger(V)=-B^\dagger(V^{\mathsf T})$，而 Fierz 重排通过映射 $\Phi:G\mapsto Ge_{31}$（其中 $e_{31}^2=-1$）实现。这产生了一个核/矩阵字典：在有效核层面，grade $0\oplus3$ 对应自旋单态，grade $1\oplus2$ 对应三重态；在配对矩阵层面，单态平面为 $\mathrm{span}\{e_2,e_{31}\}$。BCS 极限具有双重 grade 恒等式：核 grade 0（标量胶）和顶点 grade 2，一个双向量（$i\sigma_2\leftrightarrow e_{31}$）。我们陈述了三个代数结果：（i）BdG 量子几何的分解定理，具有零纹理判据，应用于 CsV$_3$Sb$_5$ 中的手性态争议；（ii）四维 Minkowski 代数中的 Fierz 符号对偶性，并扩展到双节点 Weyl 半金属模型的手性反转；（iii）在线性响应探针模型下，层析重构的误差界和条件数结构。示例包括胶生成机制、单圈代数重整化流（其馈入常数在数值上与 $\mathfrak{su}(2)$ Casimir 相等）、闭合形式的 $\mu=0$ 节点间谱，以及 UTe$_2$ 两相分析。
英文摘要：
We formulate the low-energy theory of superconducting pairing in the geometric algebra $Cl_{3,0}$. Conventional BCS mean-field theory and its extensions ($s$-, $p$-, $d$-wave, etc.) have algebraic counterparts in an eight-dimensional real multivector space. The construction is a two-level projection: matrix blocks are represented as Clifford multivectors, and their grade-0 projected geometric products extract the scalar blocks from which Hamiltonians are built. The result is an algebraic reconstruction, physically equivalent to the standard complex-matrix formulation. In this reconstruction, the dichotomy of pairing channels acquires an algebraic root: it follows from the fermionic identity $B^\dagger(V)=-B^\dagger(V^{\mathsf T})$, and the Fierz rearrangement is realized by the map $Φ:G\mapsto Ge_{31}$ (with $e_{31}^2=-1$). This yields a kernel/matrix dictionary: at the effective-kernel level grades $0\oplus3$ correspond to spin singlets and grades $1\oplus2$ to triplets; at the pairing-matrix level the singlet plane is $\mathrm{span}\{e_2,e_{31}\}$. The BCS limit has a double grade identity: kernel grade 0 (scalar glue) and vertex grade 2, a bivector ($iσ_2\leftrightarrow e_{31}$). We state three algebraic results: (i) a decomposition theorem for BdG quantum geometry with a null-texture criterion, applied to the chiral-state controversy in CsV$_3$Sb$_5$; (ii) a Fierz sign duality in the four-dimensional Minkowski algebra, with a chirality-reversal extension to two-node Weyl-semimetal models; (iii) an error bound and condition-number structure for tomographic reconstruction under a linear-response probe model. Illustrations include a glue-generation mechanism, a one-loop algebraic renormalization flow whose feeding constant coincides in magnitude with the $\mathfrak{su}(2)$ Casimir, closed-form $μ=0$ inter-node spectra, and a UTe$_2$ two-phase analysis.
