Southwest Quantum Information and Technology

Extrinsic geometry of Quantum states

Presenting Author: Alexander Avdoshkin, University of California Berkeley
Contributing Author(s): Fedor Popov, NYU

Consider a set of quantum states |ψ(x)⟩ 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⁾(x₁,x₂,x₃) = tr[P(x₁)P(x₂)P(x₃)], where P(x) = |ψ(x)⟩⟨ψ(x)|. Mathematically, P(x) defines a map from M to the complex projective space CPⁿ and this map is uniquely determined by P⁽³⁾(x₁,x₂,x₃) up to a symmetry transformation. The phase arg P⁽³⁾(x₁,x₂,x₃) 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 xᵢ of P⁽³⁾(x₁,x₂,x₃) 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 CPⁿ 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.

Read this article online: https://arxiv.org/abs/2205.15353

(Session 5 : Thursday from 5:00 pm - 7:00 pm)