Whenever a quantum system undergoes a cyclic evolution governed by a slow change of parameters, it acquires a phase factor: the geometric phase. Its most common formulations are known as the Aharonov–Bohm phase and the Pancharatnam and Berry phase, but both earlier and later manifestations exist. Although traditionally attributed to the foundations of quantum mechanics, the geometric phase has been generalized and become increasingly influential in many areas from condensed-matter physics and optics to high-energy and particle physics and from fluid mechanics to gravity and cosmology. Interestingly, the geometric phase also offers unique opportunities for quantum information and computation. In this Review, we first introduce the Aharonov–Bohm effect as an important realization of the geometric phase. Then, we discuss in detail the broader meaning, consequences and realizations of the geometric phase, emphasizing the most important mathematical methods and experimental techniques used in the study of the geometric phase, in particular those related to recent works in optics and condensed-matter physics.

每当量子系统经历由缓慢变化的参数控制的循环演化时，它都会获得一个相位因子：几何相位。它最常见的形式被称为Aharonov-Bohm相以及Pancharatnam和Berry相，但既有较早的表现也有较晚的表现。尽管传统上归因于量子力学的基础，但几何相已被概括并在从凝聚态物理和光学到高能和粒子物理学以及从流体力学到重力和宇宙学的许多领域中变得越来越有影响力。有趣的是，几何相位还为量子信息和计算提供了独特的机会。在这篇评论中，我们首先介绍了Aharonov–Bohm效应作为几何相位的重要实现。然后，我们详细讨论更广泛的含义，