薄壁量子化方法进展

Progress in Thin-Layer Quantization

微纳加工技术的快速发展实现了具有非平庸几何结构的低维材料与结构的制备,使得受几何性质影响的有效动力学问题受到关注。作为研究低维弯曲系统中量子力学的有效方法,薄壁量子化方法,它给出的几何势和几何动量也得到实验的证实。本文将回顾薄壁量子化方法,简述其基本计算框架,明确给出几何量子效应是来自于微分同胚变换和局域标架间的旋转变换。这对于理解引力规范场,演生规范场很有帮助,对理解人工规范场所隐含的几何结构也很有帮助。在特定的量子体系中,几何量子效应可表现为共振隧穿、量子阻塞、量子霍尔效应、量子霍尔黏性、量子自旋霍尔效应和单极磁场等,这些结果从不同角度展示了几何与新奇的物理现象之间的联系。

With the rapid development of microtechnology,the low-dimensional materials are fabricated with nontrivial topological structures,and then the action of geometric properties on the effective dynamics receives increasing attention.It is an effective theory that the quantum mechanics of low-dimensional curved systems can be given by using the thin-layer quantization approach,in which the geometric potential and the geometric momentum have been demonstrated experimentally.In the present paper,the thin-layer quantization formalism is first recalled,its fundamental calculation framework is first clarified,and the geometric quantum effects result from the diffeomorphism transformation and the rotation transformation connecting the local frames of different points.The results are helpful to understand the gravitational gauge and emerging gauge,and to image the geometries implied in the artificial gauge.In the particular quantum systems,the novel physical phenomena are briefly recalled that are induced by geometry,such as resonation tunneling,quantum block,quantum Hall effect,quantum Hall viscosity,quantum spin Hall effect,effective monopole magnetic field and so on.The results will shed light from a different angle on the relationships between the geometry and the novel physical phenomenon.