Beyond the Berry Phase: Extrinsic Geometry of Quantum States

Abstract

Consider a set of quantum states $|\psi (x)⟩$ 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 $P^{(3)}(x_1,x_2,x_3)=\mathrm{tr}[P(x_1)P(x_2)P(x_3)]$, where $P(x)=|\psi (x)⟩⟨\psi (x)|$. 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 $\omega$ and quantum metric $g$. We show that higher orders in this expansion are functionally independent of $\omega$ 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.

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• Quantum Geometric Origin of the Intrinsic Nonlinear Hall Effect
• Symmetry-enforced double Weyl points, multiband quantum geometry, and singular flat bands of doping-induced states at the Fermi level
• Geometric semimetals and their simulation in synthetic matter
• Quantum geometric tensors from sub-bundle geometry
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• Quantum Geometric Helical Superconductivity
• Estimation of multivariate traces of states given partial classical information
• Exploring Many-Body Quantum Geometry Beyond the Quantum Metric with Correlation Functions: A Time-Dependent Perspective