Real space quantum metric of solids

TL;DR

This paper introduces a real space quantum metric for solids, linking local electronic states to geometric properties, measurable quantities, and effects of disorder, with implications for material engineering.

Contribution

It presents a novel real space quantum metric derived from local electronic states, connecting geometry with measurable electronic properties and disorder effects in solids.

Findings

Quantum metric relates to momentum variance in electron systems.

Disorder induces curvature and geometrical structures in real space.

The framework applies to lattice models of metals and topological insulators.

Abstract

By applying the projector to the filled lattice eigenstates on a specific position, or applying the local electron annihilation operator on the many-body ground state, one can construct a quantum state localized around a specific position in a solid. The overlap of two such local states at slightly different positions defines a quantum metric in real space, which manifests even in systems as simple as particles in a box. For continuous systems like electron gas, this metric weighted by the density gives the momentum variance of electrons, which is readily measurable by ARPES. The presence of disorder curves the real space manifold and gives rise to various differential geometrical quantities like Riemann tensor and Ricci scalar, indicating the possibility of engineering differential geometrical properties by disorder, as demonstrated by lattice models of 2D metals and topological insulators.