超越傳統霍爾效應的非線性霍爾效應 (nonlinear Hall effect) 長期以來一直是凝聚態物理中具有高應用潛力的重要研究領域。與此同時，異向磁阻效應 (Anisotropic magnetoresistance) 亦為該領域的一個重要現象，具有多樣化的應用。盡管這些效應的實際應用層出不窮，理論模型卻仍不完整。本研究致力於彌合一些理論上的現有差距。傳統上，這兩種現象都與材料的內部細節相關，如磁學性質，晶體結構和電子結構：廣泛討論的非線性霍爾效應是由貝里曲率 (Berry curvature) 偶極子引起的，這將現象限制在幾類具有超低晶格對稱性的材料中；而異向磁阻效應通常是由於材料中磁矩排列導致的電子各向異性散射引起的。這項工作中，我們引入了與以往研究有顯著不同的兩個方面：首先，我們聚焦於最簡單的各向同性材料：二維電子氣。因此我們所獲得的結果與材料細節無關，它們純粹是由我們故意創造的幾何不對稱引起。其次，我們利用實空間中的磁場，而非貝里曲率，來產生這些現象。我們將磁場設置為與空間座標 x 成正比，從負值延伸到正值。利用解量子諧振子的算符方法 (operator formalism) 和我們開發的磁化率公式，我們對更高階的霍爾效應進行了解析計算。此外，我們採用對角化方法，計算了更接近真實實驗條件的系統，即外加均勻磁場的彎曲奈米帶。我們的研究發現此系統中存在非線性霍爾效應和異向磁阻效應。

Nonlinear Hall effects extending beyond the conventional ones have long been a significant area of research with high application potential in condensed matter physics. Simultaneously, anisotropic magnetoresistance has also been regarded as an important phenomenon in this field, offering versatile applications. While practical applications of these effects abound, theoretical understanding still lags behind. This work is our dedication to bridge some of the existing gaps in theory. Traditionally, both phenomena are linked to the internal details of materials, such as magnetic properties, crystal structure, and electronic structure. The wildly discussed nonlinear Hall effect is induced by the Berry curvature dipole, which limits the phenomenon to a few classes of materials with ultralow lattice symmetry. On the other hand, anisotropic magnetoresistance is typically caused by the anisotropic scattering of electrons due to the alignment of magnetic moments in the material. In this work, we introduce two significant departures from previous studies. First, we focus on the simplest and isotropic material: two-dimensional electron gases. Hence, the obtained results in our work are independent of material details, as they are solely induced by our intentionally created geometric asymmetry. Second, we utilize the actual magnetic field in real space, rather than the Berry curvature, to generate these phenomena. We orient our magnetic-field dipole linearly along the x-axis, spanning from negative to positive values. Utilizing the operator formalism used for solving the quantum harmonic oscillator and our developed susceptibility formulae, we analytically calculate Hall effects in higher order. Additionally, we employ the diagonalization approach method to numerically compute the case of homogeneous external magnetic fields applied to nanoribbons, aligning more closely with experimental conditions. Our findings reveal both nonlinear Hall effect and anisotropic magnetoresistance in such systems.

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