The recent realization of twisted, two-dimensional, bilayers exhibiting strongly correlated states has created a platform in which the relation between the properties of the electronic bands and the nature of the correlated states can be studied in unprecedented ways. The reason is that these systems allow extraordinary control of the electronic bands’ properties, for example by varying the relative twist angle between the layers forming the system. In particular, in twisted bilayers the low energy bands can be tuned to be very flat and with a nontrivial quantum metric. This allows the quantitative and experimental exploration of the relation between the metric of Bloch quantum states and the properties of correlated states. In this work we first review the general connection between quantum metric and the properties of correlated states that break a continuous symmetry. We then discuss the specific case when the correlated state is a superfluid and show how the quantum metric is related to its superfluid stiffness. To exemplify such relation we show results for the case of superconductivity in magic angle twisted bilayer graphene. We conclude by discussing possible research directions to further elucidate the connection between quantum metric and correlated states’ properties.

最近实现的扭曲、二维、双层表现出强相关状态创造了一个平台，在该平台中可以以前所未有的方式研究电子带的特性与相关状态的性质之间的关系。原因是这些系统允许对电子带的特性进行非凡的控制，例如通过改变形成系统的层之间的相对扭曲角。特别是，在扭曲的双层中，低能带可以调整为非常平坦并且具有非平凡的量子度量。这允许对布洛赫量子态的度量与相关态的性质之间的关系进行定量和实验探索。在这项工作中，我们首先回顾了量子度量与打破连续对称性的相关态的性质之间的一般联系。然后我们讨论相关状态是超流体的具体情况，并展示量子度量如何与其超流体刚度相关。为了举例说明这种关系，我们展示了魔角扭曲双层石墨烯中超导情况的结果。最后，我们讨论了可能的研究方向，以进一步阐明量子度量与相关态属性之间的联系。