We analytically derived the effective two-body interaction for a finite thickness quantum Hall system with a harmonic perpendicular confinement and an in-plane magnetic field. The anisotropic effective interaction in the lowest Landau level (LLL) and first Landau level (1LL) are expanded in the basis of the generalized pseudopotentials (PPs), and we analyze how the coefficients of some prominent isotropic and anisotropic PPs depend on the thickness of the sample and the strength of the in-plane magnetic field. We also investigate the stability of the topological quantum Hall states, especially the Laughlin state and its emergent guiding center metric, which we can now compute analytically. An interesting reorientation of the anisotropy direction of the Laughlin state in the 1LL is revealed, and we also discuss various possible experimental ramifications for this quantum Hall system with broken rotational symmetry.

中文翻译：

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倾斜磁场下量子霍尔系统的各向异性pseudo势表征

我们通过分析得出了具有谐波垂直限制和面内磁场的有限厚度量子霍尔系统的有效两体相互作用。在广义伪势（PPs）的基础上扩展了最低Landau能级（LLL）和第一Landau能级（1LL）上的各向异性有效相互作用，并且我们分析了一些主要的各向同性和各向异性PP的系数如何取决于厚度。样品和面内磁场的强度。我们还研究了拓扑量子霍尔状态的稳定性，尤其是拉夫林状态及其出现的引导中心度量，我们现在可以对其进行分析计算。揭示了1LL中Laughlin态的各向异性方向的有趣重新定向，