Revisiting the matrix elements of the position operator in the crystal momentum representation

TL;DR

This paper provides a new, singularity-free method to compute the matrix elements of the position operator in crystal momentum representation, clarifying its structure and broadening its application in physical property calculations.

Contribution

It introduces a direct computation approach for the position operator's matrix elements in CMR, resolving longstanding issues with singular derivatives and degeneracies.

Findings

Position operator is a full matrix in CMR with off-diagonal elements along the position vector

Method avoids singular derivatives and degeneracy problems

Applicable to various physical properties like transitions, polarization, and susceptibilities

Abstract

Fewer operators are more fundamental than the position operator in a crystal. But since it is not translationally invariant in crystal momentum representation (CMR), how to properly represent it is nontrivial. Over half a century, various methods have been proposed, but they often lead to either highly singular derivatives or extremely arcane expressions. Here we propose a resolution to this problem by directly computing their matrix elements between two Bloch states. We show that the position operator is a full matrix in CMR, where the off-diagonal elements in crystal momentum $\bf k$ only appear along the direction of the position vector. Our formalism, free of singular derivative and degeneracy difficulties, can describe an array of physical properties, from intraband transitions, polarization with or without spin-orbit coupling, orbital angular momentum, to susceptibilities.