Austrich-Olivares Joan A, Vergara Jose David
Departamento de Física de Altas Energías, Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, Ciudad de México 04510, Mexico.
Entropy (Basel). 2022 Sep 2;24(9):1236. doi: 10.3390/e24091236.
We introduce a quantum geometric tensor in a curved space with a parameter-dependent metric, which contains the quantum metric tensor as the symmetric part and the Berry curvature corresponding to the antisymmetric part. This parameter-dependent metric modifies the usual inner product, which induces modifications in the quantum metric tensor and Berry curvature by adding terms proportional to the derivatives with respect to the parameters of the determinant of the metric. The quantum metric tensor is obtained in two ways: By using the definition of the infinitesimal distance between two states in the parameter-dependent curved space and via the fidelity susceptibility approach. The usual Berry connection acquires an additional term with which the curved inner product converts the Berry connection into an object that transforms as a connection and density of weight one. Finally, we provide three examples in one dimension with a nontrivial metric: an anharmonic oscillator, a Morse-like potential, and a generalized anharmonic oscillator; and one in two dimensions: the coupled anharmonic oscillator in a curved space.
我们在具有参数依赖度规的弯曲空间中引入一个量子几何张量，它包含作为对称部分的量子度规张量以及对应反对称部分的贝里曲率。这种参数依赖度规修改了通常的内积，通过添加与度规行列式关于参数的导数成比例的项，从而在量子度规张量和贝里曲率中引发修改。量子度规张量通过两种方式获得：利用参数依赖弯曲空间中两个态之间无穷小距离的定义以及通过保真度敏感性方法。通常的贝里联络获得一个附加项，借助弯曲内积，该附加项将贝里联络转变为一个作为权重为一的联络和密度进行变换的对象。最后，我们给出一维中具有非平凡度规的三个例子：一个非简谐振荡器、一个类莫尔斯势和一个广义非简谐振荡器；以及二维中的一个例子：弯曲空间中的耦合非简谐振荡器。
