We study two-dimensional (2D) droplets of noninteracting electrons in a strong magnetic field, placed in a confining potential with arbitrary shape. Using semiclassical methods adapted to the lowest Landau level, we obtain near-Gaussian energy eigenstates that are localized on level curves of the potential and have a position-dependent height. This one-particle insight allows us to deduce explicit formulas for expectation values of local many-body observables, such as density and current, in the thermodynamic limit. In particular, correlations along the edge are long-ranged and inhomogeneous. As we show, this is consistent with the system’s universal low-energy description as a free 1D chiral conformal field theory of edge modes, known from earlier works in simple geometries. A delicate interplay between radial and angular dependencies of eigenfunctions ultimately ensures that the theory is homogeneous in terms of the canonical angle variable of the potential, despite its apparent inhomogeneity in terms of more naïve angular coordinates. Finally, we propose a scheme to measure the anisotropy by subjecting the droplet to microwave radiation; we compute the corresponding absorption rate and show that it depends on the droplet’s shape and the waves’ polarization. These results, both local and global, are likely to be observable in solid-state systems or quantum simulators of 2D electron gases with a high degree of control on the confining potential.

中文翻译：

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各向异性量子霍尔液滴

我们研究强磁场中非相互作用电子的二维 (2D) 液滴，并将其置于任意形状的限制电势中。使用适应最低朗道能级的半经典方法，我们获得了近高斯能量本征态，它们位于势能的能级曲线上，并且具有与位置相关的高度。这种单粒子洞察力使我们能够推导出局部多体可观测量的期望值的明确公式，例如热力学极限内的密度和电流。特别是，沿边缘的相关性是长范围的且不均匀的。正如我们所展示的，这与系统作为边缘模式的自由一维手性共形场论的普遍低能量描述是一致的，从早期的简单几何工作中可知。本征函数的径向和角度依赖性之间的微妙相互作用最终确保了该理论在势的正则角度变量方面是同质的，尽管它在更朴素的角坐标方面具有明显的不均匀性。最后，我们提出了一种通过使液滴受到微波辐射来测量各向异性的方案；我们计算了相应的吸收率，并表明它取决于液滴的形状和波的偏振。这些局部和全局的结果很可能在固态系统或二维电子气的量子模拟器中观察到，并对限制电势进行高度控制。