A Gauge Invariant Formulation of Interband and Intraband Currents in Solids

TL;DR

This paper develops gauge-invariant definitions for interband and intraband currents in solids, clarifying their physical significance and ensuring consistent calculations regardless of the gauge choice.

Contribution

It introduces gauge-invariant formulations for interband and intraband currents and instantaneous band populations in solid-state physics.

Findings

Gauge-invariant definitions enable consistent calculations.

Clarifies physical meaning of interband and intraband currents.

Addresses gauge dependence issues in high harmonic generation simulations.

Abstract

Experiments and simulations in solid-state high harmonic generation often make use of the distinction between interband and intraband currents. These two contributions to the total current have been associated with qualitatively different processes, as well as physically measurable signatures, for example in the spectral phase of harmonic emission. However, it was recently argued [P. F\"oldi, Phys. Rev. B 96, 035112 (2017)] that these quantities can depend on the gauge employed in calculations. Since physical quantities are expected to have gauge-independent values, this raises the question of whether the decomposition of the total current into interband and intraband contributions is physically meaningful, or merely a feature of a particular mathematical representation of nature. In this article, we explore this apparent ambiguity. We show that a closely related issue arises when calculating instantaneous band populations. In both the case of inter/intraband currents and in the case of instantaneous band populations, we propose definitions which are gauge-invariant, and thus allow these quantities to be calculated consistently in any gauge.