The emergence of gauge invariance: the stay-at-home gauge versus local-global duality

TL;DR

This paper explores how gauge invariance emerges in condensed matter systems through dualities and local constraints, linking different mechanisms and illustrating their implications in quantum elasticity and gravity.

Contribution

It demonstrates the close relationship between gauge structures arising from dualities and local constraints, providing a unified perspective on their emergence in condensed matter physics.

Findings

Gauge structures are related through dualities and local constraints.

Restoration of symmetry involves superposition of topological defect configurations.

Application to quantum elasticity links nematic phases with linearized gravity.

Abstract

In condensed matter physics gauge symmetries other than the U(1) of electromagnetism are of an emergent nature. Two emergence mechanisms for gauge symmetry are well established: the way these arise in Kramers-Wannier type local-global dualities, and as a way to encode local constraints encountered in (doped) Mott insulators. We demonstrate that these gauge structures are closely related, and appear as counterparts in either the canonical or field-theoretical language. The restoration of symmetry in a disorder phase transition is due to having the original local variables subjected to a coherent superposition of all possible topological defect configurations, with the effect that correlation functions are no longer well-defined. This is completely equivalent to assigning gauge freedom to those variables. Two cases are considered explicitly: the well-known vortex duality in bosonic Mott insulators serves to illustrate the principle. The acquired wisdoms are then applied to the less familiar context of dualities in quantum elasticity, where we elucidate the relation between the quantum nematic and linearized gravity. We reflect on some deeper implications for the emergence of gauge symmetry in general.