Beyond the Berry Phase: Extrinsic Geometry of Quantum States


Consider a set of quantum states ∣ψ(x)⟩| \psi(x) \rangle parameterized by xx taken from some parameter space MM. We demonstrate how all geometric properties of this manifold of states are fully described by a scalar gauge-invariant Bargmann invariant P(3)(x1,x2,x3)=tr⁑[P(x1)P(x2)P(x3)]P^{(3)}(x_1, x_2, x_3)=\operatorname{tr}[P(x_1) P(x_2)P(x_3)], where P(x)=∣ψ(x)⟩⟨ψ(x)∣P(x) = |\psi(x)\rangle \langle\psi(x)|. Mathematically, P(x)P(x) defines a map from MM to the complex projective space CPn\mathbb{C}P^n and this map is uniquely determined by P(3)(x1,x2,x3)P^{(3)}(x_1,x_2,x_3) up to a symmetry transformation. The phase arg⁑P(3)(x1,x2,x3)\arg P^{(3)}(x_1,x_2,x_3) can be used to compute the Berry phase for any closed loop in MM, however, as we prove, it contains other information that cannot be determined from any Berry phase. When the arguments xix_i of P(3)(x1,x2,x3)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 Ο‰\omega and quantum metric gg. We show that higher orders in this expansion are functionally independent of Ο‰\omega and gg and are related to the extrinsic properties of the map of MM into CPn\mathbb{C}P^n giving rise to new local gauge-invariant objects, such as the fully symmetric 3-tensor TT. 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

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