贝里相位输运的霍奇结构：拓扑、几何与噪声
The Hodge structure of Berry-phase transport: topology, geometry, and noise
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中文总结 AI 辅助
该研究通过霍奇-德拉姆分解将贝里曲率扇区与输运、涨落关联，证明电流噪声谱可双向分离全局带拓扑与局域带几何。
中文摘要 AI 辅助
我们证明，贝里曲率的霍奇-德拉姆分解能将布洛赫带的输运及其涨落组织在单一几何结构中。将曲率分解为L²正交的调和、正合与余正合扇区，为电流的各阶矩建立了对应关系。在平均响应中，调和扇区携带拓扑反常霍尔电导率，正合扇区携带费米面几何与极性金属的反对称贝里曲率偶极，三维情况下余正合扇区携带手征反常，其由外尔节点电荷量子化。在由涨落-耗散定理（FDT）约束的粒子守恒随机玻尔兹曼方程描述的涨落中，调和扇区无贡献，因此拓扑输运无噪声，而场致噪声由几何扇区提供（二维中仅由正合扇区提供）；因此电流噪声谱可将全局带拓扑与局域带几何分离。我们证明调和零空间保护与维度无关，并明确了剩余扇区：余正合（单极）扇区无守恒律，且在热采样测度下与正合扇区以O(1)量级混合，故无法提供清晰的噪声可观测量。因此噪声提供的分离是双向的，在二维和三维中均可区分拓扑与几何。
英文摘要
We show that the Hodge-de Rham decomposition of the Berry curvature organises the transport of a Bloch band and its fluctuations within a single geometric structure. Splitting the curvature into $L^2$-orthogonal harmonic, exact, and co-exact sectors yields a dictionary for both moments of the current. In the mean response the harmonic sector carries the topological anomalous-Hall conductivity, the exact sector the Fermi-surface geometry and the antisymmetric Berry-curvature dipole of polar metals, and, in three dimensions, the co-exact sector the chiral anomaly, quantised by the Weyl-node charges. In the fluctuations, described by a particle-conserving stochastic Boltzmann equation constrained by the fluctuation-dissipation theorem (FDT), the harmonic sector is silent, so topological transport is noiseless, while the field-driven noise is sourced by the geometric sectors (solely the exact sector in two dimensions); current-noise spectroscopy therefore separates global band topology from local band geometry. We prove that the harmonic null-space protection is dimension-independent, and we settle the remaining sector: the co-exact (monopole) sector carries no conservation law and, under the thermal sampling measure, mixes with the exact sector at $\mathcal{O}(1)$, so it furnishes no clean noise observable. The separation the noise provides is therefore two-way, topology versus geometry, in both two and three dimensions.
