Momentum space metric, non-local operator, and topological insulators

TL;DR

This paper explores the geometric structure of momentum space in gapped quantum systems using a quantum metric, introduces a non-local operator related to real-space distance, and examines implications for topological phases and interactions.

Contribution

It introduces a non-local operator representing real-space distance squared in momentum space and links it to the Laplacian in curved momentum space, advancing understanding of topological phases.

Findings

Quantum metric measures the second cumulant of position in real space.

The non-local operator corresponds to the Laplacian in curved momentum space.

Analysis of topological invariants and phase transitions using momentum space geometry.

Abstract

Momentum space of a gapped quantum system is a metric space: it admits a notion of distance reflecting properties of its quantum ground state. By using this quantum metric, we investigate geometric properties of momentum space. In particular, we introduce a non-local operator which represents distance square in real space and show that this corresponds to the Laplacian in curved momentum space, and also derive its path integral representation in momentum space. The quantum metric itself measures the second cumulant of the position operator in real space, much like the Berry gauge potential measures the first cumulant or the electric polarization in real space. By using the non-local operator and the metric, we study some aspects of topological phases such as topological invariants, the cumulants and topological phase transitions. The effect of interactions to the momentum space geometry is also discussed.