久保公式為線性響應理論中計算電導率的標準框架，然其速度規範與長度規範的表達式存在差異。速度規範的表達式同時包含抗磁項與順磁項，並帶有 1/ω 的規範係數，其中 ω 對應外加微擾場的震盪 (以角頻率表示)；相較之下，長度規範僅包含順磁項，且規範係數為 1/∆ωn,n′ ，其中 ∆ωn,n′ 為躍遷頻率 (以角頻率表示)，對應於兩條能帶的能階差。在準靜態極限下，速度規範的結果會發散，進一步增加了解釋上的困難。
本研究透過含時微擾理論與古典電磁學的架構，來研究二維系統的觀察量期望值，並在結論中討論一階響應中的電流與自旋流。在兩種規範中，本研究得到了一致的表達式－與張量虛部相關，消除了不同規範中係數不一致的情形，從而在準靜態極限下保持一致性。此外，我們發現了一個帶有相位差與規範係數的額外貢獻－與張量實部相關，經典久保公式的框架中並未提及此貢獻。該相位差的存在，使得本研究結果與線性響應理論的結果分歧，此分歧需要進一步地在理論上進行調和，或透過實驗觀測此相位差。此外，本研究指出不同規範的適用範圍，在直流極限下速度規範失效，而長度規範則適用在低頻範圍內。

The Kubo formula provides the standard framework for evaluating conductivity in linear response theory, yet discrepancies persist between the velocity and length gauge formulations. In the velocity gauge, the expression includes both diamagnetic and paramagnetic terms with a 1/ω factor, where ω is the angular frequency of the external perturbation. By contrast, the length gauge contains only the paramagnetic term with a 1/∆ωn,n′ factor, where ∆ωn,n′ denotes the transition frequency between energy bands. In the quasi-static limit, the velocity gauge diverges, complicating physical interpretation.
Using time-dependent perturbation theory in combination with classical electrodynamics, we examine expectation values of observables in two-dimensional systems, focusing on first-order charge and spin current responses. We eliminate gauge-dependent factors and reproduce an equivalent expression for the term — associated with the imaginary part of the tensor — in both gauges, ensuring consistency in the quasi-static limit. In addition, we identify an extra gauge-dependent term linked to the real part of the tensor and characterized
by a phase shift—absent in the conventional Kubo framework. The presence of this phase shift deviates from the standard results of linear response theory, underscoring the need for
further theoretical reconciliation or experimental verification. In addition, we delineate the regime of validity: while the velocity gauge breaks down in the DC limit, the length gauge provides a more reliable description at low frequencies.

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