论文标题
超越浆果阶段：量子状态的外部几何形状
Beyond the Berry Phase: Extrinsic Geometry of Quantum States
论文作者
论文摘要
考虑一组量子状态$ | ψ（x）\ rangle $由$ x $参数化，从某些参数空间$ m $中获取。我们演示了状态流形的所有几何属性如何通过标量规数的bargmann不变性$ p^{（3）}（x_1，x_1，x_2，x_3）= \ operatoratorname {tr} [p（x_1） \langleψ（x）| $。从数学上讲，$ p（x）$将一张地图定义为从$ m $到复杂的投影空间$ \ mathbb {c} p^n $，并且该地图由$ p^{（3）}（x_1，x_2，x_3，x_3）$唯一地确定为对称转换。相位$ \ arg p^{（3）}（x_1，x_2，x_3）$可用于计算$ m $中任何闭环的浆果阶段，但是，正如我们证明的那样，它包含其他无法从任何浆果阶段确定的信息。当参数$ x_i $ of $ p^{（3）}（x_1，x_2，x_3）$彼此接近，并降低到领先顺序时，它将减少到熟悉的浆果曲率$ω$和量子$ g $。我们表明，在此扩展中，较高的订单在功能上独立于$ω$和$ g $，并且与$ m $的地图的外部属性有关$ \ mathbb {c} p^n $，从而引起了新的本地量规不变对象，例如完全对称的3 tensor $ t $。最后，我们展示了我们的结果如何立即应用于现代的极化理论，对调制场的电响应的计算以及平面带的物理。
Consider a set of quantum states $| ψ(x) \rangle$ parameterized by $x$ taken from some parameter space $M$. We demonstrate how all geometric properties of this manifold of states are fully described by a scalar gauge-invariant Bargmann invariant $P^{(3)}(x_1, x_2, x_3)=\operatorname{tr}[P(x_1) P(x_2)P(x_3)]$, where $P(x) = |ψ(x)\rangle \langleψ(x)|$. Mathematically, $P(x)$ defines a map from $M$ to the complex projective space $\mathbb{C}P^n$ and this map is uniquely determined by $P^{(3)}(x_1,x_2,x_3)$ up to a symmetry transformation. The phase $\arg P^{(3)}(x_1,x_2,x_3)$ can be used to compute the Berry phase for any closed loop in $M$, however, as we prove, it contains other information that cannot be determined from any Berry phase. When the arguments $x_i$ of $P^{(3)}(x_1,x_2,x_3)$ are taken close to each other, to the leading order, it reduces to the familiar Berry curvature $ω$ and quantum metric $g$. We show that higher orders in this expansion are functionally independent of $ω$ and $g$ and are related to the extrinsic properties of the map of $M$ into $\mathbb{C}P^n$ giving rise to new local gauge-invariant objects, such as the fully symmetric 3-tensor $T$. Finally, we show how our results have immediate applications to the modern theory of polarization, calculation of electrical response to a modulated field and physics of flat bands.
